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Michael R Rosen
Research Hydrologist/Water Quality Specialist for Research
Email: mrosen@usgs.gov
2730 N. Deer Run Rd
contamination and pollution
pharmaceutical contamination
Lake water quality
toxic trace element contamination
Limnogeology
Michael joined the USGS in 2001 and is based in Carson City, Nevada. He is currently a Research Hydrologist and Water Quality Specialist for Research with the California Water Science Center. From 2011 to 2018 he worked for the Water Mission Area as a Water Quality Specialist. From 2001 to 2011 he was a Research Scientist in the Nevada Water Science Center.
From 2001 to September 2011, Michael was a Research Hydrologist at the USGS working on both groundwater and surface water quality. From 2011 to 2017 he was a Water Quality Specialist for the Water Science Field Team and from 2017 to 2018 he was a Research Scientist for the Hydro-Ecosystems interaction Branch of the Water Mission Area. In February 2019 he started as the Water Quality Specialist for Research and as a Research Hydrologist at the California Water Science Center. He is also an adjunct professor in Dept. Geological and Eng. Sciences and a faculty member of the Hydrological Sciences Program at the Univ. Nevada – Reno. Michael was named as a Fellow of the Geological Society of America in 2010. He was previously employed at the Institute of Geological and Nuclear Sciences, New Zealand, the Div. of Water Res., CSIRO and Curtin Univ. Tech. in Western Australia, and the Limnological Res. Center at the Univ. Minnesota.
His research in the past has covered the fields of hydrochemistry (water quality), chemical limnology, sedimentary geochemistry, paleolimnology (including paleoclimate research), ecology, geothermal research, Antarctic research, and hydrogeology. His current research focuses on chemical transformations in saline brines and the occurrence of lithium and different brine chemistries in US closed basins. in addition he is still interested in the use of passive samplers to determine organic contaminant concentrations, loads, and toxicity in rivers and lakes and how the presence of contaminants may impact the aquatic community of these systems. His other interests and projects center around nitrate contaminant transport modeling and nitrogen budgets, limnogeology and paleoclimate research of arid zone lakes, pesticide contamination of lake sediments from agricultural application in Uzbekistan, and determination of water quality trends in large ground water datasets.
BS in Geology at Haverford College (majoring in Geology at Bryn Mawr College),
MS in Geology from the Univ. Rochester
PhD in Geology from the Univ. Texas - Austin
2010 Elected as a Fellow of the Geological Society of America
2014 Eugene M. Shoemaker Communications Award for A/V Product for Lake Mead: Clear and Vital
2014 NAGC Blue Pencil and Gold Screen Awards for Technical or Statistical Report for USGS Circular 1381.
2014 Telly Award (2nd place) for Lake Mead:Clear and Vital video
Dr. Rosen as authored or co-authored over 200 reports and papers, with more than 60 of these being in peer-reviewed journals. He served as the Limnogeology Division Chair and past chair until 2012 for the Geological Society of America. He is currently the Chair of the International Association of Limnogeology and the editor of the Journal of the Nevada Water Resources Association. He has also edited two books and served as an associate editor for two journals (Sedimentology and Ground Water).
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Lake water quality: Chapter 4 in A synthesis of aquatic science for management of Lakes Mead and Mohave
Given the importance of the availability and quality of water in Lake Mead, it has become one of the most intensely sampled and studied bodies of water in the United States. As a result, data are available from sampling stations across the lake (fig. 4-1 and see U.S. Geological Survey Automated Water-Quality Platforms) to provide information on...
Tietjen, Todd; Holdren, G. Chris; Rosen, Michael R.; Veley, Ronald J.; Moran, Michael J.; Vanderford, Brett; Wong, Wai Hing; Drury, Douglas D.
Attribution: Nevada Water Science Center, Water Resources
Lake water quality: Chapter 4 in A synthesis of aquatic science for management of Lakes Mead and Mohave; 2012; CIR; 1381-4; A synthesis of aquatic science for management of Lakes Mead and Mohave (CIR 1381); Tietjen, Todd ; Holdren, G. Chris; Rosen, Michael R.; Veley, Ronald J.; Moran, Michael J.; Vanderford, Brett ; Wong, Wai Hing; Drury, Douglas D.
Management implications of the science: Chapter 7 in A synthesis of aquatic science for management of Lakes Mead and Mohave
Lake Mead, particularly its Boulder Basin, is one of the most intensively monitored reservoirs in the United States. With its importance to societal needs and ecosystem benefits, interest in water quality and water resources of Lake Mead will remain high. A number of agencies have authorities and management interests in Lake Mead and maintain...
Turner, Kent; Goodbred, Steven L.; Rosen, Michael R.; Miller, Jennell M.
Management implications of the science: Chapter 7 in A synthesis of aquatic science for management of Lakes Mead and Mohave; 2012; CIR; 1381-7; A synthesis of aquatic science for management of Lakes Mead and Mohave (CIR 1381); Turner, Kent ; Goodbred, Steven L.; Rosen, Michael R.; Miller, Jennell M.
Threats and stressors to the health of the ecosystems of Lakes Mead and Mohave: Chapter 6 in A synthesis of aquatic science for management of Lakes Mead and Mohave
Ecosystem impacts from visitor activities or natural environmental change are important concerns in all units of the National Park system. Possible impacts to aquatic ecosystems at Lake Mead National Recreation Area (LMNRA) are of particular concern because of the designation of Lakes Mead and Mohave as critical habitat for the federally listed...
Rosen, Michael R.; Goodbred, Steven L.; Wong, Wai Hing; Patiño, Reynaldo; Turner, Kent; Palmer, Craig J.; Roefer, Peggy
Threats and stressors to the health of the ecosystems of Lakes Mead and Mohave: Chapter 6 in A synthesis of aquatic science for management of Lakes Mead and Mohave; 2012; CIR; 1381-6; USGS Unnumbered Series; A synthesis of aquatic science for management of Lakes Mead and Mohave (CIR 1381); Rosen, Michael R.; Goodbred, Steven L.; Wong, Wai Hing; Patiño, Reynaldo ; Turner, Kent ; Palmer, Craig J.; Roefer, Peggy
Wildlife and biological resources: Chapter 5 in A synthesis of aquatic science for management of Lakes Mead and Mohave
The creation of Lakes Mead and Mohave drastically changed habitats originally found along their region of the historical Colorado River. While still continuing to provide habitat conditions that support a rich diversity of species within the water, along shorelines, and in adjacent drainage areas, the reservoirs contain organisms that are both...
Chandra, Sudeep; Abella, Scott R.; Albrecht, Brandon A.; Barnes, Joseph G.; Engel, E. Cayenne; Goodbred, Steven L.; Holden, Paul B.; Kegerries, Ron B.; Jaeger, Jef R.; Orsak, Erik; Rosen, Michael R.; Sjöberg, Jon; Wong, Wai Hing
Wildlife and biological resources: Chapter 5 in A synthesis of aquatic science for management of Lakes Mead and Mohave; 2012; CIR; 1381-5; A synthesis of aquatic science for management of Lakes Mead and Mohave (CIR 1381); Chandra, Sudeep ; Abella, Scott R.; Albrecht, Brandon A.; Barnes, Joseph G.; Engel, E. Cayenne; Goodbred, Steven L.; Holden, Paul B.; Kegerries, Ron B.; Jaeger, Jef R.; Orsak, Erik ; Rosen, Michael R.; Sjöberg, Jon ; Wong, Wai Hing
Bottom sediment as a source of organic contaminants in Lake Mead, Nevada, USA
Treated wastewater effluent from Las Vegas, Nevada and surrounding communities' flow through Las Vegas Wash (LVW) into the Lake Mead National Recreational Area at Las Vegas Bay (LVB). Lake sediment is a likely sink for many hydrophobic synthetic organic compounds (SOCs); however, partitioning between the sediment and the overlying water could...
Alvarez, David A.; Rosen, Michael R.; Perkins, Stephanie D.; Cranor, Walter L.; Schroeder, Vickie L.; Jones-Lepp, Tammy L.
Bottom sediment as a source of organic contaminants in Lake Mead, Nevada, USA; 2012; Article; Journal; Chemosphere; Alvarez, David A.; Rosen, Michael R.; Perkins, Stephanie D.; Cranor, Walter L.; Schroeder, Vickie L.; Jones-Lepp, Tammy L.
Patterns of metal composition and biological condition and their association in male common carp across an environmental contaminant gradient in Lake Mead National Recreation Area, Nevada and Arizona, USA
There is a contaminant gradient in Lake Mead National Recreation Area (LMNRA) that is partly driven by municipal and industrial runoff and wastewater inputs via Las Vegas Wash (LVW). Adult male common carp (Cyprinus carpio; 10 fish/site) were collected from LVW, Las Vegas Bay (receiving LVW flow), Overton Arm (OA, upstream reference), and Willow...
Patino, R.; Rosen, Michael R.; Orsak, E.L.; Goodbred, S.L.; May, T.W.; Alvarez, David; Echols, K.R.; Wieser, C.M.; Ruessler, S.; Torres, L.
Attribution: Columbia Environmental Research Center
Patterns of metal composition and biological condition and their association in male common carp across an environmental contaminant gradient in Lake Mead National Recreation Area, Nevada and Arizona, USA; 2012; Article; Journal; Science of the Total Environment; Patino, R.; Rosen, M. R.; Orsak, E. L.; Goodbred, S. L.; May, T. W.; Alvarez, D.; Echols, K. R.; Wieser, C. M.; Ruessler, S.; Torres, L.
The influence of irrigation water on the hydrology and lake water budgets of two small arid-climate lakes in Khorezm, Uzbekistan
Little is known regarding the origins and hydrology of hundreds of small lakes located in the western Uzbekistan province of Khorezm, Central Asia. Situated in the Aral Sea Basin, Khorezm is a productive agricultural region, growing mainly cotton, wheat, and rice. Irrigation is provided by an extensive canal network that conveys water from the Amu...
Scott, J.; Rosen, Michael R.; Saito, L.; Decker, D.L.
The influence of irrigation water on the hydrology and lake water budgets of two small arid-climate lakes in Khorezm, Uzbekistan; 2011; Article; Journal; Journal of Hydrology; Scott, J.; Rosen, M. R.; Saito, L.; Decker, D. L.
Early invasion population structure of quagga mussel and associated benthic invertebrate community composition on soft sediment in a large reservoir
In 2007 an invasive dreissenid mussel species, Dreissena bugensis (quagga mussel), was discovered in Lake Mead reservoir (AZ–NV). Within 2 years, adult populations have spread throughout the lake and are not only colonizing hard substrates, but also establishing in soft sediments at depths ranging from 1 to >100 m. Dreissena bugensis size class...
Wittmann, Marion E.; Chandra, Sudeep; Caires, Andrea; Denton, Marianne; Rosen, Michael R.; Wong, Wai Hing; Teitjen, Todd; Turner, Kent; Roefer, Peggy; Holdren, G. Chris
Early invasion population structure of quagga mussel and associated benthic invertebrate community composition on soft sediment in a large reservoir; 2010; Article; Journal; Lake and Reservoir Management; Wittmann, Marion E.; Chandra, Sudeep ; Caires, Andrea ; Denton, Marianne ; Rosen, Michael R.; Wong, Wai Hing; Teitjen, Todd ; Turner, Kent ; Roefer, Peggy ; Holdren, G. Chris
Importance of benthic production to fish populations in Lake Mead prior to the establishment of quagga mussels
Limnologists recently have developed an interest in quantifying benthic resource contributions to higher-level consumers. Much of this research focuses on natural lakes with very little research in reservoirs. In this study, we provide a contemporary snapshot of the food web structure of Lake Mead to evaluate the contribution of benthic resources...
Umek, John; Chandra, Sudeep; Rosen, Michael; Wittmann, Marion; Sullivan, Joe; Orsak, Erik
Importance of benthic production to fish populations in Lake Mead prior to the establishment of quagga mussels; 2010; Article; Journal; Lake and Reservoir Management; Umek, John ; Chandra, Sudeep ; Rosen, Michael ; Wittmann, Marion ; Sullivan, Joe ; Orsak, Erik
Assessment of multiple sources of anthropogenic and natural chemical inputs to a morphologically complex basin, Lake Mead, USA
Lakes with complex morphologies and with different geologic and land-use characteristics in their sub-watersheds could have large differences in natural and anthropogenic chemical inputs to sub-basins in the lake. Lake Mead in southern Nevada and northern Arizona, USA, is one such lake. To assess variations in chemical histories from 1935 to 1998...
Rosen, Michael R.; Van Metre, P.C.
Assessment of multiple sources of anthropogenic and natural chemical inputs to a morphologically complex basin, Lake Mead, USA; 2010; Article; Journal; Palaeogeography, Palaeoclimatology, Palaeoecology; Rosen, M. R.; Van Metre, P. C.
Identification of nitrogen sources to four small lakes in the agricultural region of Khorezm, Uzbekistan
Pollution of inland waters by agricultural land use is a concern in many areas of the world, and especially in arid regions, where water resources are inherently scarce. This study used physical and chemical water quality and stable nitrogen isotope (δ15N) measurements from zooplankton to examine nitrogen (N) sources and concentrations in four...
Shanafield, M.; Rosen, M.; Saito, L.; Chandra, S.; Lamers, J.; Nishonov, Bakhriddin
Identification of nitrogen sources to four small lakes in the agricultural region of Khorezm, Uzbekistan; 2010; Article; Journal; Biogeochemistry; Shanafield, M.; Rosen, M.; Saito, L.; Chandra, S.; Lamers, J.; Nishonov, B.
Organochlorine pesticides residue in lakes of Khorezm, Uzbekistan
The Khorezm province in northwest Uzbekistan is a productive agricultural area within the Aral Sea Basin that produces cotton, rice and wheat. Various organochlorine pesticides were widely used for cotton production before Uzbekistan's independence in 1991. In Khorezm, small lakes have formed in natural depressions that receive inputs mostly from...
Rosen, Michael R.; Nishonov, Bakhriddin; Fayzieva, Dilorom; Saito, L.; Lamers, J.
Organochlorine pesticides residue in lakes of Khorezm, Uzbekistan; 2009; Conference publication; Conference publication; 10th International HCH and pesticide forum book of papers: how many obsolete pesticides have been disposed of 8 years after signature of Stockholm Convention; Michael R Rosen;B. Nishonov;D. Fayzieva;L. Saito;J. Lamers
Taxonomic fidelity of silicified filamentous microbes from hot spring systems in the Taupo Volcanic Zone, North Island, New Zealand
Jones, B., Renaut, R.W., & Rosen, M.R., 2004, Taxonomic fidelity of silicified filamentous microbes from hot spring systems in the Taupo Volcanic Zone, North Island, New Zealand. Transactions of the Royal Society of Edinburgh, 94, 475-483.
Hydrogeology and possible effects of the Mw. 7.4 Marmara Earthquake (17 August 1999) on the spring waters in the Orhangazi-Bursa Area, Turkey
Pasvanoglu, S. Canik, B. & Rosen, M.R., 2004, Hydrogeology and possible effects of the Mw. 7.4 Marmara Earthquake (17 August 1999) on the spring waters in the Orhangazi-Bursa Area, Turkey. Journal Geological Society of India, 63, 313-322. | {
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{"url":"http:\/\/mathhelpforum.com\/geometry\/162603-triangle-divided.html","text":"# Math Help - Triangle divided...\n\n1. ## Triangle divided...\n\nA vertical line divides the triangle whose vertices are at (0,0), (1,1), and (9,1) into two parts; a triangle and a quadrilateral. If the areas of the triangle and the quadrilateral are equal, what is the equation of the vertical line?\n\n2. for vertical line $x=a$ make the areas of the shapes the same.\n\nYou can try to solve $\\displaystyle \\int_a^9 f(x)~dx = \\int_0^a g(x)~dx$\n\nAll you have to do is determine $f(x), g(x)$ which are linear combinations of the lines joining the triangle.\n\nCan you find the equations of the 3 lines that form the triangle?\n\nSpoiler:\n\n$\\displaystyle y=x,y=1,y=\\frac{x}{9}$\n\n3. ## graphical approach\n\nfrom the diagram,\n\n$\\frac{d}{1}=\\frac{(9-a)}{9}$\n$d=\\frac{(9-a)}{9}$\n\nconsidering the area of the small triangle,find the area of it\n\nIf A is the area of the larger triangle,\n\n$A=(A+A') - A'$\n\n$A1 = \\frac{A}{2}$\n\nsubstitute A, A1, (A+A') and A' with d and a\n\nthis is when vertical line meets AB\ndo the same thing when the vertical line meets CA\n\n4. Originally Posted by MATNTRNG\nA vertical line divides the triangle whose vertices are at (0,0), (1,1), and (9,1) into two parts; a triangle and a quadrilateral. If areas of the triangle and quadrilateral are equal, what is the equation of the vertical line?\nOk, now that you've been shown how, try this similar one:\n\nA vertical line divides the triangle whose vertices are at (0,0), (7,24), and (32,24) into two parts;\na triangle and a quadrilateral. If the areas of the triangle and the quadrilateral are equal,\nwhat is the LENGTH of the vertical line?\n\nYou'll find that all side lengths (plus the areas) are integers;\nsmallest case, I believe, for an all-integer.\n\nSOLUTION (since no response!):\nA(0,0), B(7,24), C(32,24)\nu=7,v=32,w=24 : so A(0,0), B(u,w), C(v,w)\nVertical line length = wSQRT[(v - u) \/ (2v)] = 15\n\nSmallest all-integer case is smaller than I thought: A(0,0), B(4,15), C(36,15) ; vertical line = 10\n\n5. Might be easier to just find the area of the full triangle\n\n$\\displaystyle A = \\frac{1}{2}\\times 8\\times \\sqrt{2}\\times \\sin {135}\\approx 0.4998$\n\nTake half of that and solve for the smaller triangle section.\n\n$\\displaystyle \\int_a^9 1-\\frac{x}{9}~dx = \\frac{0.4998}{2}$\n\n6. Originally Posted by pickslides\nMight be easier to just find the area of the full triangle\n$\\displaystyle A = \\frac{1}{2}\\times 8\\times \\sqrt{2}\\times \\sin {135}\\approx 0.4998$\nPickslides, that should be 4 exactly, not .4998.\n\nEasier if we let point (1,1) = (u,w) and point (9,1) = (v,w);\nthen A = w(v - u) \/ 2 = 1(9 - 1) \/ 2 = 4 ; agree?","date":"2014-03-16 21:44:41","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 11, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7486679553985596, \"perplexity\": 833.844972870667}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-10\/segments\/1394678703748\/warc\/CC-MAIN-20140313024503-00022-ip-10-183-142-35.ec2.internal.warc.gz\"}"} | null | null |
\section*{\large Supplemental Material}
In this supplementary material we provide the details of deriving the effective Hamiltonians in the main text based on real cold atom candidates, discuss the effects of the different types of spin-orbit (SO) couplings, and calculate the heating and lifetime for the schemes.
\section*{I. Scheme for Dirac Type 2D SO Coupling}
In this section we discuss the details of deriving the effective Hamiltonian. The experimental setup for ${}^{40}$K atoms is shown in Fig. 1 in the main text. Two laser beams form the standing waves
\begin{align}
\bm{E}_{x} &= \bm{\hat{y}} E_{xy} e^{i(\alpha + \alpha_L/2)} \cos(k_0x - \alpha_L/2) + \bm{\hat{z}} E_{xz} e^{i(\alpha' + \alpha'_L/2)} \cos(k_0x - \alpha'_L/2) \nonumber \\
\bm{E}_{y} &= \bm{\hat{x}} E_{yx} e^{i(\beta + \beta_L/2)} \cos(k_0y - \beta_L/2) + \bm{\hat{z}} E_{yz} e^{i(\beta' + \beta'_L/2)} \cos(k_0y - \beta'_L/2). \nonumber
\end{align}
Here $\alpha, \alpha', \beta, \beta'$ are the initial phases for $\bm{E}_{xy},\bm{E}_{xz},\bm{E}_{yx},\bm{E}_{yz}$, and $\alpha_L$ is the phase acquired by $\bm{E}_{xy}$ through the optical path from intersecting point to mirror $M_1$, then back to the intersecting point. The phases $\alpha'_L, \beta_L, \beta'_L$ have the similar meanings for $\bm{E}_{xz},\bm{E}_{yx},\bm{E}_{yz}$. $E_{xy}, E_{xz}, E_{yx}$, and $E_{yz}$ are real amplitudes. We shall find that the initial phases are irrelevant for the realization.
\subsection{A. Raman Fields}
For convenience, we take ${}^{40}$K atoms for our consideration, while all the results are straightforwardly applicable to $^{87}$Rb atoms. For ${}^{40}$K atoms we define the spin-$1/2$ by the two ground states $|g_{\uparrow} \rangle = |9/2,+9/2\rangle$ and $|g_{\downarrow} \rangle = |9/2,+7/2\rangle$. Raman coupling scheme for ${}^{40}$K atom is shown in Fig.~\ref{scheme_Dirac}, where two independent Raman transitions are driven by the light components $\bm{E}_{xz},\bm{E}_{yx}$ and $\bm{E}_{xy},\bm{E}_{yz}$. Taking into account the contributions from both $D_1(2^2P_{1/2})$ and $D_2(2^2P_{3/2})$ lines, we have
\begin{align*}
M_1 &= \sum_F \frac{\Omega^{(3/2)*}_{\uparrow F,xz} \cdot \Omega^{(3/2)}_{\downarrow F,yx}}{\Delta_{3/2}} + \sum_F \frac{\Omega^{(1/2)*}_{\uparrow F,xz} \cdot \Omega^{(1/2)}_{\downarrow F,yx}}{\Delta_{1/2}} \\
&= \frac{\sqrt{2}}{9} \left( \frac{\alpha^2_{D_1}}{\Delta_{1/2}} - \frac{\alpha^2_{D_2}}{2 \Delta_{3/2}} \right) \bm{E}^{*}_{xz} \bm{E}_{yx}^{(+)},
\end{align*}
\begin{align*}
M_2 &= \sum_F \frac{\Omega^{(3/2)*}_{\uparrow F,xy} \cdot \Omega^{(3/2)}_{\downarrow F,yz}}{\Delta_{3/2}} + \sum_F \frac{\Omega^{(1/2)*}_{\uparrow F,xy} \cdot \Omega^{(1/2)}_{\downarrow F,yz}}{\Delta_{1/2}} \\
&= \frac{\sqrt{2}}{9} \left( \frac{\alpha^2_{D_1}}{\Delta_{1/2}} - \frac{\alpha^2_{D_2}}{2 \Delta_{3/2}} \right) \bm{E}^{(-)*}_{xy} \bm{E}_{yz},
\end{align*}
where $\Omega^{J}_{\uparrow F,\gamma z} = \langle \uparrow| ez|F,0,J \rangle \bm{\hat{e}}_z \cdot \bm{E}_{\gamma z}$ and $\Omega^{J}_{\uparrow F,\gamma x} = \langle \uparrow| ex|F,+1,J \rangle \bm{\hat{e}}_+ \cdot \bm{E}_{\gamma x} + \langle \uparrow| ex|F,-1,J \rangle \bm{\hat{e}}_- \cdot \bm{E}_{\gamma x}$, and $\bm{E}_{yx}^{(+)}=\bm{\hat{e}}_+ \cdot \bm{E}_{yx} = {\bm{E}_{yx}}/{\sqrt{2}}$ represents the right-handed light of $\bm{E}_{yx}$, which couples the states $|\frac 92,+\frac 72 \rangle$ and $|F,+\frac 92 \rangle$ as shown in Fig.~\ref{scheme_Dirac}. Similarly, $\bm{E}^{(-)}_{xy}=-i {\bm{E}_{xy}}/{\sqrt{2}}$ represents the left-handed light of $\bm{E}_{xy}$, which couples the states $|\frac 92,+\frac 92 \rangle$ and $|F,+\frac 72 \rangle$. Since $\alpha_{D_2} \approx \sqrt{2} \alpha_{D_1} \approx 5.799 ea_0$~\cite{tiecke2010properties}, with $a_0$ being the Bohr radius, then we have
\begin{align}
M_1 &= M_{10} \cos(k_0x - \alpha'_L/2) \cos(k_0y - \beta_L/2) e^{-i(\alpha' + \alpha'_L/2)} e^{i(\beta + \beta_L/2)}, \\
M_2 &= i M_{20} \cos(k_0x - \alpha_L/2) \cos(k_0y - \beta'_L/2) e^{-i(\alpha + \alpha_L/2)} e^{i(\beta' + \beta'_L/2)},
\end{align}
where
\begin{align*}
M_{10/20} = \frac{\alpha^2_{D_1}}{9}\left(\frac{1}{\Delta_{1/2}} - \frac{1}{\Delta_{3/2}} \right) E_{xz/xy} E_{yx/yz}.
\end{align*}
To realize a Dirac type 2D SO coupling with nontrivial topology, we require the following two conditions to be satisfied.
(1) The phase difference $\delta \theta$ between $M_1$ and $M_2$ must be non-zero to have a 2D SO coupling. Then the Raman potential can be written as $(|M_1| + |M_2| \cos \delta \theta) \sigma_x + |M_2| \sin \delta \theta \sigma_y $. Here we shall consider the optimal regime with $\delta \theta = \pm \pi/2$, which gives the Raman coupling as $|M_1| \sigma_x \pm |M_2|\sigma_y$ accordingly. Hence the phases should meet the following condition
\begin{align}\label{phase1}
\delta \theta &= (\alpha - \alpha') + (\beta - \beta') + \frac 12 \left[ (\alpha_L - \alpha'_L) + (\beta_L - \beta'_L) \right] + \pi/2 \nonumber \\
&= \frac{\pi}{2} + n \pi ,
\end{align}
where $n$ is an integer number. Note that $\alpha$ and $\alpha'$ (similar for $\beta$ and $\beta'$) are phases of two components of the same laser beam. The relative value between them is automatically fixed and can be easily manipulated by wave plates. Also, $\alpha_L$ and $\alpha'_L$ ($\beta_L$ and $\beta'_L$) are the phases acquired through the same optical path, so their relative value is also automatically fixed and can be controlled by wave plates. The system is stable against any phase fluctuations.
(2) To let the SO coupled system to be topological non-trivial, we consider a $\lambda/4$ wave plate before each mirror ($M_1$ and $M_2$). Then the phases will meet the following conditions
\begin{align}\label{phase2}
\frac{1}{2} (\alpha_L - \alpha'_L) = \frac{\pi}{2} + p \pi \ \ {\rm and} \ \ \ \frac{1}{2} (\beta_L - \beta'_L) = \frac{\pi}{2} + q \pi,
\end{align}
with $p$ and $q$ are integer numbers. Accordingly, the Raman potentials turn into
\begin{align}
M_1 = M_{10} \sin(k_0x - \alpha_L/2) \cos(k_0y - \beta_L/2), \ \ {\rm and} \ \ \ M_2 = \pm i M_{20} \cos(k_0x - \alpha_L/2) \sin(k_0y - \beta_L/2).
\end{align}
We shall see below that the Raman coupling potentials $M_1$ and $M_2$ are antisymmetric with respect to the lattice in $x$ and $y$ directions, respectively, a key point to obtain the nontrivial topology.
In the real experiment the conditions~\eqref{phase1} and~\eqref{phase2} can be satisfied by taking $(\alpha - \alpha') + (\beta - \beta') + \pi/2 = \pi/2 + n \pi$, $(\alpha_L - \alpha'_L)/2 = \pi/2 + p \pi$, and $(\beta_L - \beta'_L)/2 = \pi/2 + q \pi$. Since ${E}_{xy}$ and ${E}_{xz}$ come from the same laser beam, one can naturally take that $\alpha = \alpha'$. Similarly, we have $\beta = \beta'$, and then the phase difference $\delta \theta=\pi/2$. On the other hand, we can control that each $\lambda/4$ wave plate induces an additional $\pi/2$-phase shift to the $\hat z$-component light when the light pass through the wave plate one time. This leads to $(\alpha_L - \alpha'_L)/2=(\beta_L - \beta'_L)/2=\pi/2$, satisfying the condition.
It is trivial to know that the Raman coupling potentials combine to induce a 1D SO coupling when $\delta\theta=0,\pi$. If controlling the phase difference continuously, the crossover between 1D and 2D SO couplings can be induced.
\subsection{B. Spin-independent lattice potentials}
\begin{figure}
\includegraphics[scale=0.33]{FigS1.pdf}
\caption{Spin-independent lattice potential induced by four linearly polarized standing wave lights which couple the $D_1$ and $D_2$ transitions.}
\label{scheme_Dirac}
\end{figure}
As shown in Fig.~\ref{scheme_Dirac}, the optical lattice is contributed from both $D_2$ $(2^2 S_{1/2} \to 2^2 P_{3/2})$ and $D_1$ $(2^2 S_{1/2} \to 2^2 P_{1/2})$ lines, and it is a spin-independent lattice
\begin{align}
V_{\uparrow} = V_{\downarrow} &= \sum_F \frac{1}{\Delta_{3/2}} \left( |\Omega^{(3/2)}_{\uparrow F,xz}|^2 + |\Omega^{(3/2)}_{\uparrow F,xy}|^2 + |\Omega^{(3/2)}_{\uparrow F,yz}|^2 + |\Omega^{(3/2)}_{\uparrow F,yx}|^2 \right) \nonumber \\
&{} \quad + \sum_F \frac{1}{\Delta_{1/2}} \left( |\Omega^{(1/2)}_{\uparrow F,xz}|^2 + |\Omega^{(1/2)}_{\uparrow F,xy}|^2 + |\Omega^{(1/2)}_{\uparrow F,yz}|^2 + |\Omega^{(1/2)}_{\uparrow F,yx}|^2 \right) \nonumber \\
&= V_{0x} \cos^2(k_0x - \alpha_L/2) + V_{0y} \cos^2(k_0y - \beta_L/2) ).
\end{align}
Here the constant is neglected in the last line, and we have $V_{0x} = \frac{\alpha_{D_1}^2}{3} (\frac{2}{\Delta_{3/2}} + \frac{1}{\Delta_{1/2}})( E_{xy}^2 - E_{xz}^2 )$ and $V_{0y}= \frac{\alpha_{D_1}^2}{3} (\frac{2}{\Delta_{3/2}} + \frac{1}{\Delta_{1/2}}) (E_{yx}^2 - E_{yz}^2)$. Note that when $E_{xy} > E_{xz}$ and $E_{yx} > E_{yz}$, the lattice potential takes the form $\cos^2(k_0x) + \cos^2(k_0y)$. In comparison, when $E_{xy}<E_{xz}$ and $E_{yx}<E_{yz}$, the lattice potential becomes $\sin^2(k_0x) + \sin^2(k_0y)$, which corresponds to a translation along the diagonal direction compared to $\cos^2(k_0x) + \cos^2(k_0y)$. Effectively, the sign of the optical potentials is reversed for the two cases, equivalent to changing the sign of the detuning (e.g. from blue- to red-detuned or vise versa).
\subsection{C. Effective Hamiltonian}
As clarified previously, the phase fluctuations for lattice and Raman fields are the same: $\phi^{\rm fluc}_x = - \alpha_L/2$, and $\phi^{\rm fluc}_y = - \beta_L/2$. Hence the relative spatial configuration of $M_{x/y}$ and $V$ are always automatically fixed, and the fluctuations only lead to a global shift of the lattice and Raman fields. Therefore, we can set $\alpha_L = \beta_L = 0$ safely. The global phase of Raman potentials are irrelevant and can also be removed. Then the Hamiltonian can be written as
\begin{align}
H=\frac{p^2}{2m} + V_{0x} \cos^2(k_0x) + V_{0y} \cos^2(k_0y) + \mathcal{M}_x \sigma_x + \mathcal{M}_y \sigma_y + m_z \sigma_z,
\end{align}
the strength of Raman coupling $\mathcal{M}_x = |M_1| + |M_2| \cos \delta \theta$ and $\mathcal{M}_y = |M_2| \sin \delta \theta$, which reduce to an 1D SO coupling for $\delta \theta = n \pi$ and an optimal 2D Dirac type SO coupling for $\delta \theta = \pi/2 + n \pi$. This enables a fully controllable study of the crossover between 2D and 1D SO couplings by tuning $\delta \theta$.
\section*{II. Scheme for Rashba and Weyl Type SO Couplings}
\subsection{A. Spin-dependent lattice Potential}
We generate the spin-dependent square lattice potential in $x \text{-} y$ plane from the traveling-wave beams, described by the electric field $\bm{E}_{V}$
\begin{align*}
\bm{E}_{Vx} &= \bm{\hat{y}} E_0 e^{ik_0x+i\phi_0} + \bm{\hat{z}} E_0 e^{-ik_0x + i\phi_0 + 2i\phi_L + i\phi'_L} \\
\bm{E}_{Vy} &= \bm{\hat{x}} E_0 e^{-ik_0y + i\phi_0 + i\phi_L} + \bm{\hat{z}} E_0 e^{ik_0y + i\phi_0 + i\phi_L + i\phi'_L},
\end{align*}
where $\bm{E}_{V,x/y}$ represents the laser propagating along $x/y$-direction, $\phi_0$ is the initial phase of the laser beam, and $\phi_L/\phi'_L$ is the phase acquired on path $L/L'$. The path $L$ denotes the loop from lattice center to mirror $M_1$, then to $M_2$, and finally back to the lattice center, while the path $L'$ denotes the one from lattice center to the mirror $M_3$ and back to lattice center [Fig.~\ref{lattice_Rashba}(a)].
In general, the optical potential generated for atoms in the ground state is related to the electromagnetic field by
\begin{align}
V(r) = u_s |\bm{E}|^2 + i u_v \left( \bm{E}^* \times \bm{E} \right) \cdot \bm{S},
\end{align}
where the first term is the scalar potential with $u_s=-\frac{1}{12 \Delta_s} | \langle l=0 | \bm{d} | l=1 \rangle |^2 $ and $ u_s |\bm{E}|^2$, and the second term denotes a vector light shift, with $u_v = \frac{A_{FS}}{\hbar \Delta_e} u_s = \frac{2 \Delta_{FS}}{3 \Delta_e} u_s $ and $\Delta_{FS}$ being the fine-structure splitting. The scalar potential is spin-independent for linearly polarized lights. In comparison, the vector light shift is spin-dependent. The effective Hamiltonian $H_{\rm eff} = u_s |\bm{E}|^2 + \frac{\mu_B g_J}{\hbar} \left( \bm{B} + \bm{B}_{\rm eff} \right) \cdot \bm{J}$, where we have replaced $\bm{S}$ with $\bm{J} = \bm{S} + \bm{L}$ since $\bm{L}=0$ for the ground state. The symbol $\bm{B}$ denotes the external magnet field, and $\bm{B}_{\rm eff} = \frac{i \hbar u_v \left( \bm{E}^* \times \bm{E} \right) }{\mu_B g_J}$ is the effective magnetic field. Considering the hyperfine structure, we further replace $g_J \bm{J}$ with $g_F \bm{F}$ and then have
\begin{align}
H_{\rm eff} = u_s |\bm{E}|^2 + \frac{\mu_B g_F}{\hbar} \left( \bm{B} + \bm{B}_{\rm eff} \right) \cdot \bm{F}.
\end{align}
In our proposal, the effective magenetic field produced by the lasers has the equal components in $\bm{x}$ and $\bm{y}$ directions. Therefore, we must apply external magnetic field along the diagonal direction ($B_x^{\rm eff},B_y^{\rm eff}\neq0$) of $x-y$ plane. This ensures that $\bm{B}_{\rm eff}$ has nonzero components along the direction of $\bm{B}$.
\begin{figure}
\centering \includegraphics[scale=0.35]{FigS2.pdf}
\caption{(a) Experiment setting for spin-dependent lattice potential. The lattice potential in each ($x$ and $y$) direction is formed by two lights traveling in opposite directions with mutual perpendicular polarization. (b) Optical transitions allowed by selection rule for generating spin-dependent lattice potential along $x/y$ direction.}
\label{lattice_Rashba}
\end{figure}
With this setup, the effective magnetic field should be $B_x^{\rm eff}\hat e_x + B_y^{\rm eff}\hat e_y$. The external magnetic field $\bm{B}$ along the diagonal direction satisfies $B \gg B_x^{\rm eff}, B_y^{\rm eff}$. Thus, the total field reads $B_t =\sqrt{(B + \frac{B_x^{\rm eff}}{\sqrt{2}} + \frac{B_y^{\rm eff}}{\sqrt{2}} )^2 + (\frac{B_x^{\rm eff}}{\sqrt{2}} - \frac{B_y^{\rm eff}}{\sqrt{2}})^2}$. We expand the total field $B_t$ around the point $B_x^{\rm eff} = B_y^{\rm eff}=0$, keeping the first-order term, and finally have $B_t \approx B +\frac{B_x^{\rm eff}}{\sqrt{2}} + \frac{B_y^{\rm eff}}{\sqrt{2}} $. In other words, if we decompose the effective magnetic field in the traverse and longitudinal direction of the external magnetic field, only the longitudinal component matters.
Next we focus on the expression of the effective magnetic field
\begin{align*}
\bm{B}_{\rm eff} &\propto \bm{E}^* \times \bm{E} \\
& = ( \bm{E}^*_{Vx} \times \bm{E}_{Vx} + \bm{E}^*_{Vy} \times \bm{E}_{Vy} ) + ( \bm{E}^*_{Vx} \times \bm{E}_{Vy} + \bm{E}^*_{Vy} \times \bm{E}_{Vx} ) \\
& =( \bm{B}_{x}^{\rm eff} + \bm{B}_{y}^{\rm eff} )+ \bm{B}_{xy}^{\rm eff}.
\end{align*}
The cross term $\bm{B}_{xy}^{\rm eff}$ can be obtained
\begin{align*}
\bm{B}_{xy}^{\rm eff} \propto \bm{\hat{y}} E_0^2 \sin(k_0x-k_0y-\phi_L-\phi'_L) -\bm{\hat{x}} E_0^2 \sin(k_0x-k_0y-\phi_L-\phi'_L) + \bm{\hat{z}}E_0^2 \sin(k_0x+k_0y-\phi_L).
\end{align*}
However, this term is perpendicular to external field $\bm{B}$, so the projection of this term in the direction of $\bm{B}$ is zero, hence it can be neglected. It is ready to verify that the scalar potential is a constant, and can be neglected. Finally the lattice potential takes the form qualitatively
\begin{align}
V_{\rm latt} = ( \bm{B}_{x}^{\rm eff} + \bm{B}_{y}^{\rm eff} ) \cdot \bm{F} \propto E^2_0 \sin(2k_0x - 2\phi_L - \phi'_L) + E^2_0 \sin(2k_0y + \phi'_L).
\end{align}
Note that we can use another method to obtain the result more accurately. We first consider the laser propagating along $x$-direction, write the electric fields in the basis of the circular polarized light
\begin{align*}
E_{Vx} &= \frac{E_0}{\sqrt{2}} e^{i \phi_0 + i\phi_L + i\phi'_L/2 +i\pi/4} 2\left[ i \bm{\hat{e}}_{+} \sin(k_0x - \phi_L - \phi'_L/2 - \pi/4) + \bm{\hat{e}}_- \cos(k_0x - \phi_L - \phi'_L/2 - \pi/4) \right] \nonumber \nonumber \\
&= \bm{\hat{e}}_+ E_{+} + \bm{\hat{e}}_- E_{-},
\end{align*}
and
\begin{align*}
|E_{+}|^2 &= 2 E^2_0 \sin^2(k_0x - \phi_L -\phi'_L/2 - \pi/4) = E^2_0 - E^2_0 \sin(2k_0x - 2\phi_L -\phi'_L) \nonumber \\
|E_{-}|^2 &= 2 E^2_0 \cos^2(k_0x - \phi_L -\phi'_L/2 - \pi/4) = E^2_0 + E^2_0 \sin(2k_0x - 2\phi_L -\phi'_L) .
\end{align*}
As shown in Fig.~\ref{lattice_Rashba}, the spin-dependent lattice potential is contributed from both $D_2$ $(6^2 S_{1/2} \to 6^2 P_{3/2})$ and $D_1$ $(6^2 S_{1/2} \to 6^2 P_{1/2})$ lines~\cite{steck2003cesium}. The light $\bm{E}_{V}$ drive the $\sigma$ transitions from ground states $|F=4,m_F=-4 \rangle$ and $|F=3,m_F=-3 \rangle$ to all possible excited levels which satisfy the selection rule. The state $|4,-4 \rangle$ $(|g_{\uparrow} \rangle)$ is coupled to excited states $|F,-5 \rangle$ and $|F,-3 \rangle$, while the state $|3,-3 \rangle$ $(|g_{\downarrow} \rangle)$ is coupled to other states $|F,-4 \rangle$ and $|F,-2 \rangle$. The detunings $\Delta_{1/2}$ and $\Delta_{3/2}$ are much larger than the hyperfine-structure splitting, and have the same order of magnititude of the fine-structure splitting, which is the energy difference between $D_1$ and $D_2$ lines. Then the lattice potential can be obtained by summing over the contributions of all the allowed transitions
\begin{align}
V_{\uparrow}(x) &= \sum_F \frac{1}{\Delta_{3/2}} \left( |\Omega^{(3/2)}_{\uparrow F,+1}|^2 + |\Omega^{(3/2)}_{\uparrow F,-1}|^2 \right) + \sum_F \frac{1}{\Delta_{1/2}} \left( |\Omega^{(1/2)}_{\uparrow F,+1}|^2 + |\Omega^{(1/2)}_{\uparrow F,-1}|^2 \right) \nonumber \\
&= \frac{2}{3} E_{\rm V}^2 \alpha^2_{D_1} \left( \frac{1}{\Delta_{3/2}} - \frac{1}{\Delta_{1/2}} \right) \sin(2k_0x - 2\phi_L -\phi'_L).
\end{align}
We have neglected all the constants in the final result. Taking $V_0 = E_{0}^2 \alpha^2_{D_1}/\Delta_1$, with $1/\Delta_1= \left( 1/\Delta_{3/2} - 1/\Delta_{1/2} \right)$, we have $V_{\downarrow}(x) = - \frac{1}{2} V_0 \sin(2k_0x - 2\phi_L -\phi'_L)$ for the atoms staying in the state $|3,-3 \rangle$. Similarly, the spin-dependent lattice potential in $y$-direction is $V_{\uparrow}(y) = \frac{2}{3} V_0 \sin(2k_0y + \phi'_L)$ and $V_{\downarrow}(y) = - \frac{1}{2} V_0 \sin(2k_0y + \phi'_L)$. The total lattice can be finally written as
\begin{eqnarray}
V_{\rm latt} = \left( \frac{1}{12} \bm{1} + \frac{7}{12} \bm{\sigma_z} \right) V_0 \left[ \sin(2k_0x - 2\phi_L -\phi'_L) + \sin(2k_0y + \phi'_L) \right].
\end{eqnarray}
\subsection{B. Raman Fields}
\begin{figure}
\includegraphics[scale=0.35]{FigS3.pdf}
\caption{Experiment setting and coupling scheme for Raman couplings.}
\label{Raman_Rashba}
\end{figure}
Now we study the generation of Raman fields by adding another two frequency-doubled lights $\bm{E}_{R1}(2\bm{k}_0,2\omega_0)$ and $\bm{E}_{R2}(2\bm{k}_0,2\omega_0+\delta \omega)$, where $\delta \omega$ is of the same order of the energy difference between $|g_{\uparrow} \rangle$ and $|g_{\downarrow} \rangle$. The polarization of $\bm{E}_{R1}$ can't be affected by $\lambda/4$ wave plate, hence $\bm{E}_{R1}$ form a standing wave on $x$-$y$ plane, and $\bm{E}_{R2}$ is just a travelling wave, i.e.
\begin{align*}
\bm{E}_{R1} &= E_{R1} e^{i\phi_1 + 2i\phi_L + i\phi'_L} \left[ \bm{\hat{y}} \cos(2k_0x - 2\phi_L - \phi'_L) + \bm{\hat{x}} \cos(2k_0y + \phi'_L) \right], \nonumber \\
\bm{E}_{R2} &= ( i \bm{\hat{x}} + \bm{\hat{y}} ) E_{R2} e^{2ik_0z + i\phi_2}.
\end{align*}
The Raman fields are contributed from $6^2 S_{1/2} \to 7^2 P_{3/2}$ and $6^2 S_{1/2} \to 7^2 P_{1/2}$ transitions. It's a little tedious to calculate Raman potentials because the magnetic field points to diagonal direction. We first need to decompose the vectors $\bm{E}_{R1}, \bm{E}_{R2}$ along the external magnetic field direction
\begin{align*}
\bm{E}_{R1} &= \frac{1}{\sqrt{2}} E_{R1} e^{i \theta} \left[ \cos(2k_0x - 2\phi_L - \phi'_L) + \cos(2k_0y + \phi'_L) \right] \bm{\hat{B}} + \frac{1}{\sqrt{2}} E_{R1} e^{i \theta} \left[ \cos(2k_0x - 2\phi_L - \phi'_L) - \cos(2k_0y + \phi'_L) \right] \bm{\hat{B}}_{\perp}, \nonumber \\
\bm{E}_{R2} &= \frac{E_{R2}}{\sqrt{2}} \left( 1 + i \right) e^{2ik_0z + i\phi_0} \bm{\hat{B}} + \frac{E_{R2}}{\sqrt{2}} \left( 1-i \right) e^{2ik_0z + i\phi_0} \bm{\hat{B}}_{\perp},
\end{align*}
where $\theta = \phi_0 + 2 \phi_L + \phi'_L$. Then the potentials can be obtained as follows
\begin{align}
M_1 &= \sum_{F} \frac{\Omega^{(3/2)*}_{\uparrow F,1\sslash} \cdot \Omega^{(3/2)}_{\downarrow F,2\perp} }{\tilde \Delta_{3/2}} + \sum_F \frac{ \Omega^{(1/2)*}_{\uparrow F,1\sslash} \cdot \Omega^{(1/2)}_{\downarrow F,2\perp} }{\tilde \Delta_{1/2}} \nonumber \\
&= \frac{\sqrt{7} \alpha^2_{\tilde D_1} }{12 \sqrt{2}} \left( \frac{1}{\tilde \Delta_{1/2}} - \frac{1}{ \tilde \Delta_{3/2}} \right) E_{R1} E_{R2} (1-i) e^{2ik_0z} [ \cos(2k_0x - 2\phi_L - \phi'_L) + \cos(2k_0y + \phi'_L)],
\end{align}
and
\begin{align}
M_2 &= \frac{\sqrt{7} \alpha^2_{D_1} }{12 \sqrt{2}} \left( \frac{1}{\tilde \Delta_{1/2}} - \frac{1}{\tilde \Delta_{3/2}} \right) E_{R1} E_{R2} (1+i) e^{2ik_0z} \left[ \cos(2k_0x - 2\phi_L - \phi'_L) - \cos(2k_0y + \phi'_L) \right].
\end{align}
Note that in general the detunings $\tilde \Delta_{1/2,3/2}\neq\Delta_{1/2,3/2}$. As discussed in the main text, lattice potential $V$ and Raman fields $M$ have the same phase fluctuations, so the fluctuations can be neglected and for simplicity we take $\phi^{fluc} = 0$. We then find
\begin{align}
M = M_1 + M_2 &= M_0 e^{2ik_0z} \left( \cos 2k_0x + i \cos 2k_0y \right),
\end{align}
where $M_0 =\frac{\sqrt{7} \alpha^2_{D_1} }{6 \sqrt{2}} E_{R1} E_{R2}/\Delta_2$, with $\frac{1}{\Delta_2}=\left( \frac{1}{\Delta_{1/2}} - \frac{1}{ \Delta_{3/2}} \right) $.
\subsection{C. Effective Hamiltonian}
With the lattice and Raman potentials above, the total Hamiltonian can be written as
\begin{align}
H_{\rm 3D} = &{} \frac{\bm{p}^2}{2m} + m_z \sigma_z \nonumber + V_0 \left( \sin 2k_0x + \sin 2k_0y \right) \sigma_z + M_0 \left[ e^{i2k_0z} ( \cos 2k_0x + i \cos 2k_0y )|g_{\uparrow}\rangle \langle g_{\downarrow}| +h.c. \right].
\end{align}
To remove the phase term $e^{i2k_0z}$ in the Raman potentials in order to make the Hamiltonian periodic, we define a rotation operator $U=e^{-ik_0z \sigma_z}$ and perform a transformation
\begin{align*}
H_{\rm 3D} \to \tilde{H}_{\rm 3D} = U H_{\rm 3D} U^{\dagger}.
\end{align*}
Then the Hamiltonian would change into
\begin{align}
\tilde{H}_{\rm 3D} = &{} \frac{\bm{p}^2}{2m} + (m_z + \lambda_z p_z) \sigma_z \nonumber + V_0 \left( \sin 2k_0x + \sin 2k_0y \right) \sigma_z + M_0 \left( \cos 2k_0x \sigma_x + \cos 2k_0y \sigma_y \right) ,
\end{align}
where $\lambda_z = \hbar k_0/m$ and we have ignored the constant above.
\subsection{D. The Tight-Binding Model}
\begin{figure}
\centering \includegraphics[scale=0.35]{FigS4.pdf}
\caption{Sketch for the tight-binding model, with atoms trapped in a spin-dependent square optical lattice. We only consider the nearest and second-nearest (diagonal) neighbor hopping couplings.}
\label{TBM}
\end{figure}
We next derive the tight-binding model from the effective Hamiltonian, while we emphasize that the realization of SO couplings is not restricted by tight-binding model. For simplicity, we first discuss the tight-binding model in 2D system, and then extend it to 3D. We only consider the nearest and next-nearest neighbor hopping, and assume that atoms stay in the lowest $s$ orbits. As shown in Fig.~\ref{TBM}, the tight-binding model of this system in real space can be written as
\begin{align*}
H=-t_{\sigma} \sum_{i,j,\sigma} \hat{c}^{\dagger}_{\sigma}(\bm{r}_i) \hat{c}_{\sigma}(\bm{r}_i + \bm{S}_j) + \sum_i m_z ( \hat{n}_{i \uparrow} - \hat{n}_{i \downarrow} ) + \sum_{i,j} t_{\rm so}^{{i} {j}} \hat{c}^{\dagger}_{\uparrow}(\bm{r}_i) \hat{c}_{\downarrow}(\bm{r}_i + \bm{N}_j) + h.c.
\end{align*}
where $\bm{N}_i$ and $\bm{S}_i$ represents the distances between the nearest neighbor sites and second-nearest neighbor sites, respectively. The particle number operators are defined as $\hat{n}_{i \sigma} = \hat{c}^{\dagger}_{i,\sigma} \hat{c}_{i,\sigma}$, and $t_{\sigma} (\sigma=\uparrow,\downarrow) $ denotes the spin-conserved hopping, given by
\begin{align*}
t_{\sigma} = \int \mathrm{d}^2 \bm{r} \phi_{s\sigma}^{(i)} (\bm{r}) \left[ \frac{\bm{p}^2}{2m} + V(\bm{r}) \right] \phi_{s\sigma}^{(j)}(\bm{r}).
\end{align*}
$t_{\rm so}^{ij}$ is the spin-flipped hopping coefficient given by
\begin{align*}
t_{\rm so}^{i j} = \int \mathrm{d}^2 \bm{r} \phi_{s \uparrow}^{(i)} (\bm{r}) M_{1/2}(\bm{r}) \phi_{s \downarrow}^{(j)}(\bm{r}),
\end{align*}
representing the induced 2D SO interaction. It can be readily verified that the spin-flip hopping terms due to Raman fields satisfy $t_{\rm so}^{j_x,j_x \pm 1} = \pm t_{\rm so}^{(0)}$ and $t_{\rm so}^{j_y,j_y \pm 1} = \pm i t_{\rm so}^{(0)}$. Then we transform the above equation into momentum space and write the Hamiltonian in matrix form
\begin{eqnarray*}
H &=& \sum_{\bm{q}}
\left(
\begin{array}{cc}
\hat{c}^{\dagger}_{\bm{q}, \uparrow} & \hat{c}^{\dagger}_{\bm{q}, \downarrow}
\end{array}
\right)
\mathcal{H}(\bm{q})
\left(
\begin{array}{c}
\hat{c}_{\bm{q}, \uparrow} \\
\hat{c}_{\bm{q}, \downarrow}
\end{array}
\right).
\end{eqnarray*}
By setting $t_{\uparrow} = t_{\downarrow} = t_0$, the matrix $\mathcal{H}$ in momentum space can be written as
\begin{align}
\mathcal{H}(\bm{q}) &= m_z \sigma_z -4 t_0 \cos(q_x a) \cos(q_y a) \otimes \bm{1} + 2t_{\rm so}^{(0)} \sin (q_xa) \sigma_x + 2t_{\rm so}^{(0)} \sin (q_ya) \sigma_y ,
\end{align}
where $a=|\bm{N}_i|$ is the lattice constant representing the distance between the nearest neighbor sites. Expanding the Hamiltonian around the $\Gamma$ point yields
\begin{align}
\mathcal{H}(\bm{q}) = 4 t_0 \bm{q}^2 + 2 t_{\rm so}^{(0)} (q_x \sigma_x + q_y \sigma_y ) + m_z \sigma_z,
\end{align}
which renders the expected 2D Rashba type SO coupling. To generate 3D weyl type SO coupling, we simply remove the confining potential along $z$ direction and the tight binding model can be written as
\begin{align}
\mathcal{H}_{\rm 3D}(\bm{q}) &= \frac{p^2_z}{2m} - 4 t_0 \cos(q_x a) \cos(q_y a) + (m_z + \lambda_z p_z) \sigma_z + 2t_{\rm so}^{(0)} \sin (q_xa) \sigma_x + 2t_{\rm so}^{(0)} \sin (q_ya) \sigma_y.
\end{align}
\subsection{E. Realization for $^{133}$Cs Atoms}
\begin{figure}
\centering \includegraphics[scale=0.3]{FigS5.pdf}
\caption{Numerical results for $^{133}$Cs atoms. (a) The lowest two bands for the realized Hamiltonian around the $\Gamma$ point with $V_0=4E_r$, $M_0=1E_r$ and $m_z=0.68E_r$. (b) The spin texture of the state with energy $E=0$ for the Rashba SO coupled Hamiltonian with the parameters $V_0=4E_r$, $M_0=0.5E_r$ and $m_z=0.7E_r$, and spins are confined in $q_x$-$q_y$ plane.}
\label{realizationForCs}
\end{figure}
As discussed above, we choose the state $|4,-4 \rangle$ from the ground state as spin up $|g_{\uparrow} \rangle$, and $|3,-3 \rangle$ as spin down $|g_{\downarrow} \rangle$. The lattice potential includes a small correction to the purely spin-dependent term and can be written as
\begin{eqnarray*}
V_{\rm latt} = \left(V_0 \bm{\sigma_z} + \delta V_0 \bm{1} \right) \left( \sin 2k_0x + \sin 2k_0y \right),
\end{eqnarray*}
where $V_0 = \frac{7}{12} E_{\rm V}^2 \alpha^2_{D_1} \left( \frac{1}{\Delta_{3/2}} - \frac{1}{\Delta_{1/2}} \right)$ and the correction $\delta V_0 \sim 0.1 V_0$. In the following we show that $\delta V_0$ can induce novel effects. To simplify the proceeding analysis, we denote the tight binding model and the hopping terms as
\begin{eqnarray*}
-4 (t_0 \bm{1} + \delta t_0 \otimes \bm{\sigma}_z) \cos(q_x a) \cos(q_y a),
\end{eqnarray*}
where $t_0 = (t_{\uparrow} + t_{\downarrow})/2$ and $\delta t_0 = (t_{\uparrow} - t_{\downarrow})/2$. Obviously, there is $\delta t_0 \sim 0.1 t_0 $ in the tight binding model. Then we redefine the effect Zeeman term $m^{\rm eff}_z$ as
\begin{eqnarray*}
m^{\rm eff}_z = m_z - 4\delta t_0 \cos(q_x a) \cos(q_y a).
\end{eqnarray*}
A few novel effects are followed. First, at the $\Gamma$ point $m^{\rm eff}_z=m_z - 4\delta t_0$, which vanishes when $m_z = 4\delta t_0$, and the band crossing is obtained [Fig.~\ref{realizationForCs}(a)]. Nevertheless, the Zeeman splitting at other symmetric points $\bm k=(0,\pi),(\pi,0),(\pi,\pi)$ keeps, showing that the time-reversal symmetry is indeed broken with nonzero $m_z$ and $\delta t_0$. Secondly, in the case of a nonzero Zeeman splitting at $\Gamma$ point, one can tune $m_z$ properly so that the spins of states at a finite energy $E$ exactly point to $x-y$ plane, namly, at such energy $m^{\rm eff}_z (\bm{q})=0$ so that the polarizations for states at $\bm{k}$ and $-\bm{k}$ are opposite [Fig.~\ref{realizationForCs}(b)]. In such region the spin texture resembles the case without Zeeman splitting. This effect can have novel consequences. For bosons, one can tune that the purely in-plane spin texture results in the band bottom, leading to a strong spin uncertainty in the ground states. For fermions, one can tune that the purely in-plane spin texture is obtained around Fermi surface. In this case under an attractive interaction the superfluid pairing $\Delta = U_0 \langle \hat{c}_{\bm{k},\uparrow} \hat{c}_{-\bm{k},\downarrow} \rangle$ can be largely enhanced even the Zeeman splitting is present, being different from a purely Rashba system where the Zeeman splitting always polarizes the Fermi surface and reduces the superfluid pairing. The topological gap of such a superfluid can then be greatly enhanced. Finally, when $-4 \delta_0 < m_z < 4 \delta_0$, each single energy band is topological nontrivial with the Chern number ${\rm Ch}_1 = \pm 1$.
According to the former discussions, the Raman potential takes the form
\begin{align*}
M = 2M_0 e^{2ik_0z} \left( \cos 2k_0x + i \cos 2k_0y \right).
\end{align*}
Note that the Raman light $\bm{E}_{R1}$ also forms a standing wave in $x$-$y$ plane and generate a spin-independent lattice $V_{\rm in} \propto \cos^2 2k_0x + \cos^2 2k_0y$. This potential has the same contribution to both spin-up and spin-down atoms. Hence, it does not change any result that we obtain above.
\subsection{F. Spin Texture Around The Weyl Point}
\begin{figure}
\centering \includegraphics[scale=0.3]{FigS6.pdf}
\caption{The spin textures on the two fixed-energy surfaces around the Weyl point. The Chern number defined on the two closed surfaces are $+1$(a) and $-1$(b), respectively. }
\label{spintexture}
\end{figure}
To simplify the proceeding analysis, we assume the Hamiltonian around the Weyl point is isotropic, which would not affect the topological property of the system. The Hamiltonian can be simplified as
\begin{align}
\mathcal{H}(\bm{q}) = \frac{\bm{q}^2}{2m_{\rm eff}} + v_F \bm{q} \cdot \bm{\sigma}.
\end{align}
The eigenvalues are $E_{\pm} = \frac{\bm{q}^2}{2m_{\rm eff}} \pm v_F |\bm{q}|$, and the degeneracy is the Weyl point at $\bm{q}=0$. The normalized eigenstates can be written as
\begin{align}
\bm{u}_{\pm}(\bm{q}) = \frac{1}{\sqrt{2q(q \pm q_z)}} \left(q_x - i q_y, - q_z \mp q \right).
\end{align}
We next consider the spin texture on the fixed-energy surfaces around the Weyl point. There are two closed spherical surfaces $F_{+}$ and $F_{-}$ with fixed energy in the 3D Brillouin zone. And the spin polarization on these surfaces can be defined as $\bm{S}_{\pm} (\bm{q}) = \langle \bm{u}_{\pm}(\bm{q}) | \bm{\hat{\sigma}} |\bm{u}_{\pm}(\bm{q}) \rangle.$ By substituting the equation (S20) into the defination, we obtain a simple result
\begin{align*}
\bm{S}_{\pm} (\bm{q}) = \pm \frac{\bm{q}}{|\bm{q}|}.
\end{align*}
The spin textures on the two spherical surfaces $F_{+}$ and $F_{-}$ are drawn in the Fig.~\ref{spintexture}. And we can define the first Chern number on the 2D spherical surface to characterize the spin texture
\begin{align*}
Ch_1 = \frac{1}{4\pi} \int_{F_{\pm}} \bm{S} \cdot \left( \partial_1 \bm{S} \times \partial_2 \bm{S} \right) \ \mathrm{d}^2 \bm{k}.
\end{align*}
By employing spherical coordinates and parametrize the vector $\bm{S}_{\pm} (\bm{q}) = \pm \left( \sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta \right)$, the Berry curvature $\Omega_{\pm}$ can be easily caculated
\begin{align*}
\Omega_{\pm} = \bm{S}_{\pm} \cdot \left( \frac{\partial_{\theta}}{q} \bm{S}_{\pm} \times \frac{\partial_{\phi}}{q \sin \theta} \bm{S}_{\pm} \right) = \pm \frac{1}{q^2}.
\end{align*}
We immediately find in the following for the Berry field strength
\begin{align*}
\bm{V} = \pm \frac{\bm{q}}{q^3}.
\end{align*}
This field looks like a magnetic field which is generated by a monopole at the Weyl point $\bm{q}=0$ with the strength $\pm 1$. Hence the Weyl node is a monopole or antimonopole for Berry curvature. Two fixed-energy surfaces $F_{+}$ and $F_{-}$ always contain the Weyl node with the strength $+1$ and $-1$. Hence the Chern number defined on the sphere $F_{+}$ and $F_{-}$ would be $Ch_{1} = 1$ and $Ch_{1} = -1$.
\section*{III. Estimation of the Lifetime}
Heating by the lattice and Raman lights is mainly induced by the spontaneous scattering of photons, which is random and causes fluctuations of the radiation force. For alkali atoms, a general expression of the photon scattering rate $\Gamma_{\rm sc}$ for the $D$ line doublet $^{2}S_{1/2} \to ^{2}P_{1/2}, ^{2}P_{3/2}$~\cite{grimm2000optical} can be written as
\begin{align}
\Gamma_{\rm sc} (\bm{r}) = \frac{\pi c^2 \Gamma^2}{2 \hbar \omega_0^3} \left( \frac{2+\mathcal{P} g_F m_F}{\Delta^2_{D_2}} + \frac{1 - \mathcal{P} g_F m_F}{\Delta^2_{D_1}} \right) I(\bm{r}),
\end{align}
where $g_F$ is the Land\'e factor, $\mathcal{P}$ denotes the laser polarization ($\mathcal{P}=0, \pm 1$ for linearly and circularly $\sigma^{\pm}$ polarized light), $\omega_0$ is the averaged resonant frequency of the $D$ line doublet, and $I(\bm{r})$ is the laser beam intensity. In experiment, we assume that the atoms stay at the center of a laser beam $\bm{E}$, and then the laser beam intensity can be written as $I(\bm{r}) = 1/2 \epsilon_0 c |\bm{E}|^2$. With the scattering rate, we can estimate the heating rate $\dot{T}$ by
\begin{align}
k_{\rm B} \dot{T} = \frac 23 E_r \Gamma_{\rm sc}.
\end{align}
And then the lifetime can be estimated as $\tau \sim 100 \nano \kelvin/\dot{T}$, where $100\nano \kelvin$ is the typical difference between the initial experimental temperature and the critical temperature.
For the scheme of Dirac type SO couping, we consider the realization for $^{87}$Rb Bose-Einstein condensate (BEC) and $^{40}$K degenerate Fermi gases. For the $^{87}$Rb atoms, the dipole trap is formed by two far-detuned laser beams and the estimated heating rate is about $10 \nano \kelvin/\second$~\cite{wu2016realization}. The lifetime with the dipole trap alone is about 10 seconds. Then we consider the heating from lattice and Raman beams, which are applied with the wavelength $\lambda = 786 \micro \metre$ and the beam waist $\varpi = 200 \micro \metre$. Under typical parameter conditions $V_0 = 4E_r$ and $M_0 = 1E_r$, we find the scattering rate $\Gamma_{\rm sc} \approx 0.54 \hertz$. Since the recoil energy $E_r/\hbar = 2\pi \times 3707 \hertz$, the total heating rate of the lattice and Raman beams is about $\dot{T}=63.7\nano \kelvin /\second$. Therefore, the lifetime of $^{87}$Rb BEC is about $\tau \sim \frac{100\nano\kelvin}{73.7\nano\kelvin/\second} \approx 1.36 \second$. For the $^{40}$K degenerate fermion gas, to realize typical parameter conditions $V_0 = 4E_r$ and $M_0 = 1E_r$, we find the heating rate of lattice and Raman lights with wavelength $\lambda = 768\nano \metre$ is about $665 \nano \kelvin/\second$, which gives the lifetime about $\tau \sim 150 \milli \second$. If considering a smaller lattice, one can even enhance the lifetime up to several hundreds of milliseconds.
For the scheme of Rashba and Weyl type SO coupling, the application for $^{133}$Cs atoms has been considered, and the lifetime of $^{133}$Cs atoms can be even much longer. The lifetime with only the dipole trap is approximately $15\second$~\cite{weber2003bose}. The lattice potential is generated by the laser beam with the wavelength $\lambda = 916.6 \nano \meter$, which couples the gound state and $6^2P_{1/2}$($6^2P_{3/2}$) levels. To produce the lattice potential with a typically depth $V_0 = 4E_r$, we apply the laser beam with the waist $\varpi = 200 \micro \meter$ and the power $P = 15.4 \milli \watt$. And then the heating rate of the lattice light can be estimated as $3\nano\kelvin/\second$, which is very small because the detuning $\Delta$ is very large. Then we consider the heating rate of the Raman lights. The Raman potental is generated by two light beams with the wavelength $\lambda = 458.3 \nano \meter$, which couples the ground state and $7^2P_{1/2}$($7^2P_{3/2}$) levels. To produce the Raman potentials with a typically strength $M_0 = 1E_r$, we apply the laser beams with the waist $\varpi = 200 \micro \meter$ and the power $P = 21 \milli \watt$. The heating rate of the Raman lights is also considerably small about $10\nano \kelvin / \second$. With these results we can estimate the lifetime of the $^{133}$Cs atoms to be $\tau >5\second $.
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,279 |
\section{Introduction}
\label{sec:intro}
\begin{figure*}[h]
\centering
\begin{subfigure}[b]{\linewidth
\includegraphics[width=\linewidth]{images/story_sample.pdf}
\caption{An example story and corresponding questions and answers.
}
\label{fig:story}
\end{subfigure}
\begin{subfigure}[b]{\linewidth
\includegraphics[width=\linewidth]{images/nlvr.pdf}
\caption{An example NLVR image and the scene created in Fig.~\ref{fig:story}, where the blocks in the NLVR image are rearranged.
}
\label{fig:nlvr}
\end{subfigure}
\caption{Example from \textsc{SpartQA}~(specifically from \textsc{SpartQA-Auto})}
\label{fig:dataset_samples}
\end{figure*}
Spatial reasoning is a cognitive process based on the construction of mental representations for spatial objects, relations, and transformations~\cite{clements1992geometry}, which is necessary for many natural language understanding (NLU) tasks such as natural language navigation~\cite{chen2019touchdown, roman-roman-etal-2020-rmm, kim-etal-2020-arramon}, human-machine interaction~\cite{landsiedel2017review, roman-roman-etal-2020-rmm}, dialogue systems \cite{udagawa-etal-2020-linguistic}, and clinical analysis~\cite{datta-roberts-2020-hybrid}.
Modern language models (LM), e.g., BERT \cite{devlin-etal-2019-bert}, ALBERT \cite{lan2019albert}, and XLNet \cite{yang2019xlnet} have seen great successes in natural language processing (NLP). However, there has been limited investigation into {\em spatial reasoning capabilities of LMs}. To the best of our knowledge, \text{bAbI}~\cite{weston2015towards}~(Fig \ref{fig:babi}) is the only dataset with direct textual spatial question answering (QA)~(Task 17), but it is synthetic and overly simplified:
(1) The underlying scenes are spatially simple, with only three objects and relations only in four directions.
(2) The stories for these scenes are two short, templated sentences, each describing a single relation between two objects.
(3) The questions typically require up to two-steps reasoning due to the simplicity of those stories.
To address these issues, this paper proposes a new dataset, \textsc{SpartQA}{}\footnote{\em{SPAtial Reasoning on Textual Question Answering}.}~(see Fig.~\ref{fig:dataset_samples}).
Specifically,
(1) \textsc{SpartQA}{} is built on NLVR's \cite{suhr2017corpus} images containing more objects with richer spatial structures (Fig.~\ref{fig:nlvr}). (2) \textsc{SpartQA}{}'s stories are more natural, have more sentences, and richer in spatial relations in each sentence. (3) \textsc{SpartQA}{}'s questions require deeper reasoning and have four types: {\em find relation} (FR), {\em find blocks} (FB), {\em choose object} (CO), and {\em yes/no} (YN), which allows for more fine-grained analysis of models' capabilities.
We showed annotators random images from NLVR, and instructed them to describe objects and relationships not exhaustively at the cost of naturalness (Sec.~\ref{sec:human}).
In total, we obtained 1.1k unique QA pair annotations on spatial reasoning, evenly distributed among the aforementioned types.
Similar to \text{bAbI}{}, we keep this dataset in relatively small scale and suggest to use as little training data as possible.
Experiments show that modern LMs (e.g., BERT) do not perform well in this low-resource setting.
This paper thus proposes a way to obtain distant supervision signals for spatial reasoning (Sec.~\ref{sec:auto}).
As spatial relationships are rarely mentioned in existing corpora, we take advantage of the fact that spatial language is grounded to the geometry of visual scenes. We are able to automatically generate stories for NLVR images \cite{suhr2017corpus} via our newly designed context free grammars (CFG) and context-sensitive rules.
In the process of story generation, we store the information about all objects and relationships, such that QA pairs can also be generated automatically. In contrast to \text{bAbI}, we use various spatial rules to infer new relationships in these QA pairs, which requires more complex reasoning capabilities.
Hereafter, we call this automatically-generated dataset \textsc{SpartQA-Auto}{}, and the human-annotated one \textsc{SpartQA-Human}{}.
Experiments show that, by further pretraining on \textsc{SpartQA-Auto}{}, we improve LMs' performance on \textsc{SpartQA-Human}{} by a large margin.\footnote{Further pretraining LMs has become a common practice and baseline method for transferring knowledge between tasks \cite{phang2018sentence,zhou2020temporal}. We leave more advanced methods for future work.} The spatially-improved LMs also show stronger performance on two external QA datasets, \text{bAbI}{} and \text{boolQ}{}~\cite{clark2019boolq}:
BERT further pretrained on \textsc{SpartQA-Auto}{} only requires half of the training data to achieve 99\% accuracy on \text{bAbI}{} as compared to the original BERT; on boolQ's development set, this model shows better performance than BERT, with 2.3\% relative error reduction.\footnote{To the best of our knowledge, the test set or leaderboard of \text{boolQ}{} has not been released yet.}
\textbf{Our contributions can be summarized as follows.}
First, we propose the first human-curated benchmark, \textsc{SpartQA-Human}{}, for spatial reasoning with richer spatial phenomena than the prior synthetic dataset \text{bAbI}{} (Task 17).
Second, we exploit the scene structure of images and design novel CFGs and spatial reasoning rules to automatically generate data (i.e., \textsc{SpartQA-Auto}) to obtain distant supervision signals for spatial reasoning over text.
Third, \textsc{SpartQA-Auto}{} proves to be a rich source of spatial knowledge that improved the performance of LMs on \textsc{SpartQA-Human}{} as well as on different data domains such as \text{bAbI}{} and \text{boolQ}{}.
\section{Related work}
\label{sec:related}
Question answering is a useful format
to evaluate machines' capability of reading comprehension~\cite{GBHTM19} and many recent works have been implementing this strategy to test machines' understanding of linguistic formalisms: \citet{HeLeZe15,MSHDZ17,LSCZ17,jia2018tempquestions,ning2020torque,du-cardie-2020-event}. An important advantage of QA is using natural language to annotate natural language, thus having the flexibility to get annotations on complex phenomena such as {\em spatial reasoning}. However, spatial reasoning phenomena have been covered minimally in the existing works.
To the best of our knowledge, Task 17 of the \text{bAbI}{} project~\cite{weston2015towards} is the only QA dataset focused on textual spatial reasoning (examples in Appendix~\ref{sec:babiboolq}). However, \text{bAbI}{} is synthetic and does not reflect the complexity of the spatial reasoning in natural language. Solving Task 17 of \text{bAbI}{} typically does not require sophisticated reasoning, which is an important capability emphasized by more recent works (e.g., \citet{dua2019drop,MultiRC2018,yang2018hotpotqa,DLMSG19,ning2020torque}).
Spatial reasoning is arguably more prominent in multi-modal QA benchmarks, e.g., NLVR~\cite{suhr2017corpus}, VQA~\cite{antol2015vqa}, GQA~\cite{hudson2019gqa}, CLEVR~\cite{johnson2017clevr}. However, those spatial reasoning phenomena are mostly expressed naturally through images, while this paper focuses on studying spatial reasoning on natural language.
Some other works on visual-spatial reasoning are based on geographical information inside maps and diagrams~\cite{huang2019geosqa} and navigational instructions~\cite{chen2019touchdown,anderson2018vision}.
As another approach to evaluate spatial reasoning capabilities of models, a dataset proposed in \citet{ghanimifard2017learning} generates a synthetic training set of spatial sentences and evaluates the models' ability to generate spatial facts and sentences containing composition and decomposition of relations on grounded objects.
\section{\textsc{SpartQA-Human}}
\label{sec:human}
To mitigate the aforementioned problems of Task 17 of \text{bAbI}{}, i.e., simple scenes, stories, and questions, we describe the data annotation process of \textsc{SpartQA-Human}{}, and explain how those problems were addressed in this section.
First, we randomly selected a subset of NLVR images, each of which has three blocks containing multiple objects (see Fig~\ref{fig:nlvr}). The scenes shown by these images are more complicated than those described by \text{bAbI}{} because (1) there are more objects in NLVR images; (2) the spatial relationships in NLVR are not limited to just four relative directions as objects are placed arbitrarily within blocks.
Second, two student volunteers produced textual description of those objects and their corresponding spatial relationships based on these images.
Since the blocks are always horizontally aligned in each NLVR image, to allow for more flexibility, annotators could also rearrange these blocks (see Fig.~\ref{fig:story}).
Relationships between objects within the same block can take the forms of relative direction~(e.g., left or above), qualitative distance~(e.g., near or far), and topological relationship~(e.g., touching or containing).
However, we instructed the annotators not to describe all objects and relationships, (1) to avoid unnecessarily verbose stories, and (2) to intentionally miss some information to enable more complex reasoning later. Therefore,
annotators describe only a random subset of blocks, objects, and relationships.
To query more interesting phenomena, annotators were then encouraged to write questions requiring detecting relations and reasoning over them using multiple spatial rules.
A spatial rule can be one of the transitivity~($A \rightarrow B, B \rightarrow C \Rightarrow A \rightarrow C$), symmetry~($A \rightarrow B \Rightarrow B \rightarrow A$), converse~($(A,\ R,\ B)\Rightarrow (B,\ reverse(R),\ A)$), inclusion~($obj1\ in\ A$), and exclusion~($obj1\ not\ in\ B$) rules.
There are four types of questions~(\textsc{Q-Type}). (1) {\em FR}: find relation between two objects. (2) {\em FB}: find the block that contains certain object(s). (3) {\em CO}: choose between two objects mentioned in the question that meets certain criteria. (4) {\em YN}: a yes/no question that tests if a claim on spatial relationship holds.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\linewidth]{images/DK_rel.png}
\caption{For ``A blue circle is above a big triangle. To the left of the big triangle, there is a square,'' if the question is: ``Is the square to the left of the blue circle?'', the answer is neither Yes nor No. Thus, the correct answer is ``Do not Know'' (DK) in our setting.}
\label{fig:DK}
\end{figure}
FB, FR, and CO questions are formulated as multiple-choice questions\footnote{CO can be considered as both single-choice and multiple-choices question.} and receive a list of candidate answers, and YN questions' answer is choosing from Yes, No, or ``DK''~(Do not Know). The ``DK'' option is due to the open-world assumption of the stories, where if something is not described in the text, it is not considered as false~(See Fig. \ref{fig:DK}).
Finally, annotators were able to create 1.1k QA pairs on spatial reasoning on the generated descriptions, distributed among the aforementioned types.
We intentionally keep this data in a relatively small scale due to two reasons. First, there has been some consensus in our community that modern systems, given their sufficiently large model capacities, can easily find shortcuts and overfit a dataset if provided with a large training data \cite{gardner2020evaluating,sen-saffari-2020-models}.
Second, collecting spatial reasoning QAs is very costly: The two annotators spent 45-60 mins on average to create a single story with 8-16 QA pairs.
We estimate that \textsc{SpartQA-Human}{} costed about 100 human hours in total.
The expert performance on 100 examples of \textsc{SpartQA-Human}{}'s test set measured by their accuracy of answering the questions is 92\% across four \textsc{Q-Type}{}s on average, indicating its high quality.
\begin{table}
\centering
\resizebox{\linewidth}{!}{%
\begin{tabular}{|l|cccc|c|}
\hline
Sets & FB & FR & YN & CO &Total \\
\hline
\textsc{SpartQA-Human}:&&&&&\\
\quad Test & 104 & 105 & 194 & 107 &510 \\
\quad Train & 154 & 149 & 162 & 151 &616 \\\hline
\textsc{SpartQA-Auto}:&&&&& \\
\quad Seen Test & 3872 & 3712 & 3896 & 3594 & 15074 \\
\quad Unseen Test & 3872 & 3721 & 3896 & 3598 &15087 \\
\quad Dev & 3842 & 3742 & 3860 & 3579 &15023 \\
\quad Train & 23654 & 23302 & 23968 & 22794 & 93673 \\
\hline
\end{tabular}
}
\caption{Number of questions per \textsc{Q-Type}}
\label{tab:num_question}
\end{table}
\section{Distant Supervision: \textsc{SpartQA-Auto}}
\label{sec:auto}
Since human annotations are costly, it is important to investigate ways to generate distant supervision signals for spatial reasoning.
However, unlike conventional distant supervision approaches (e.g., \citet{mintz2009distant,zeng2015distant,zhou2020temporal}) where distant supervision data can be selected from large corpora by implementing specialized filtering rules, spatial reasoning does not appear often in existing corpora. Therefore, similar to \textsc{SpartQA-Human}{}, we take advantage of the ground truth of NLVR images, design CFGs to generate stories, and use spatial reasoning rules to ask and answer spatial reasoning questions. This automatically generated data is called \textsc{SpartQA-Auto}{}, and below we describe its generation process in detail.
\paragraph{Story generation} Since NLVR comes with structured descriptions of the ground truth locations of those objects, we were able to choose random blocks and objects from each image programmatically. The benefit is two-fold.
First, a random selection of blocks and objects allows us to create multiple stories for each image; second, this randomness also creates spatial reasoning opportunities with missing information.
Once we decide on a set of blocks and objects to be included, we determine their relationships: Those relationships between blocks are generated randomly; as for those between objects, we refer to the ground truth of these images to determine them.
Now we have a scene containing a set of blocks and objects and their associated relationships. To produce a story for this scene, we design CFGs to produce natural language sentences that describe those blocks/objects/relationships in various expressions~(see Fig.~\ref{fig:cfg} for two portions of our CFG describing relative and nested relations between objects).
\begin{figure}[h]
\centering
\begin{subfigure}[b]{\linewidth}
\includegraphics[width=\linewidth]{images/CFG1.pdf}
\caption{Part of the grammar describing relations between objects
}
\label{fig:cfg1}
\end{subfigure}
\begin{subfigure}[b]{\linewidth
\includegraphics[width=\linewidth]{images/CFG2.pdf}
\caption{Part of the grammar describing nested relationships.
}
\label{fig:cfg2}
\end{subfigure}
\caption{Two parts of our designed CFG}
\label{fig:cfg}
\end{figure}
Being grounded to visual scenes guarantees spatial coherency in a story, and using CFGs helps to have correct sentences (grammatically) and various expressions. We also design context-sensitive rules to limited options for each CFG's variable based on the chosen entities (e.g. black circle), or what is described in the previous sentences (e.g. Block A has \textit{a} circle. \textit{The} circle is below \textit{a} triangle.)
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth, angle=0]{images/transitivity.pdf}
\caption{Find the implicit relation between $obj1$ and $obj4$ by {\em Transitivity} rule. (1) Find a set of objects that have a relation with $obj1$. Continue the same process on the new set until $obj4$ is found. (2) Get the union of the intermediate relations between these two objects and it is the final answer.}
\label{fig:transitivity_rel}
\end{figure}
\paragraph{Question generation} To generate questions based on a passage, there are rule-based systems~\cite{heilman2009question,labutov2015deep}, neural networks~\cite{du2017learning}, and their combinations.
\cite{dhole2020syn}. However, in our approach, during generating each story, the program stores the information about the entities and their relationships.
Thus, without processing the raw text, which is error-prone, we generate questions by only looking at the stored data.
The question generation operates based on four primary functionalities, \textit{Choose-objects}, \textit{Describe-objects}, \textit{Find-all-relations}, and \textit{Find-similar-objects}. These modules are responsible to control the logical consistency, correctness, and the number of steps required for reasoning in each question.
\textit{\textbf{Choose-objects}}
randomly chooses up to three objects from the set of possible objects in a story under a set of constraints such as preventing selection of similar objects, or excluding objects with relations that are directly mentioned in the text.
\textit{\textbf{Describe-Objects}}
generates a mention phrase for an object using parts of its full name (presented in the story). The generated phrase is either pointing to a unique object or a group of objects such as "the big circle," or "big circles." To describe a unique object, it chooses an attribute or a group of attributes that apply to a unique object among others in the story. To increase the steps of reasoning, the description may include the relationship of the object to other objects instead of using a direct unique description. For example, "the circle which is above the black triangle."
\textit{\textbf{Find-all-relations}}
completes the relationship graph between objects by applying a set of spatial rules such as transitivity, symmetry, converse, inclusion, and exclusion on top of the direct relations described in the story.
As shown in Fig.~\ref{fig:transitivity_rel}, it does an exhaustive search over all combinations of the relations that link two objects to each other.
\textit{\textbf{Find-similar-objects}}
finds all the mentions matching a description from the question to objects in the story. For instance, for the question "is there any blue circle above the big blue triangle?", this module finds all the mentions in the story matching the description ``a blue circle''.
Similar to the \textsc{SpartQA-Human}{}, we provide four \textsc{Q-Type}{}s FR, FB, CO, and YN.
To generate FR questions, we choose two objects using \textit{Choose-objects} module and question their relationships. The YN \textsc{Q-Type}{} is similar to FR, but the question specifies one relationship of interest chosen from all relation extracted by \textit{Find-all-relations} module to be questioned about the objects. Since most of the time, Yes/No questions are simpler problems, we make this question type more complex by adding quantifiers (adding ``all'' and ``any''). These quantifiers help to evaluates the models' capability to aggregate relations between more than two objects in the story and do the reasoning over all find relations to find the final answer. In FB \textsc{Q-Type}{}, we mention an object by its indirect relation to another object using the nested relation in \textit{Describe-objects} module and ask to find the blocks containing or not containing this object.
Finally, the CO question selects an anchor object (\textit{Choose-objects}) and specifies a relationship ( using \textit{Find-all-relations}) in the question. Two other objects are chosen as candidates to check whether the specified relationship holds between them and the anchor object. We tend to force the algorithm to choose objects as candidates that at least have one relationship to the anchor object.
To see more details about different question' templates see Table \ref{tab:templates} in the Appendix.
\paragraph{Answer generation} We compute all direct and indirect relationships between objects using \textit{Find-all-relations} function and based on the \textsc{Q-Type} s generate the final answer.
For instance, in YN \textsc{Q-Type}{} if the asked relation exists in the found relations, the answer is "Yes", if the inverse relation exists it must be "No", and otherwise, it is "DK"\footnote{The \textsc{SpartQA-Auto}{} generation code and the file of dataset are available at \url{https://github.com/HLR/SpartQA_generation}}.
\subsection{Corpus Statistics}
We generate the train, dev, and test set splits based on the same splits of the images in the NLVR dataset.
On average, each story contains 9 sentences (Min:3, Max: 22) and 118 tokens (Min: 66, Max: 274). Also, the average tokens of each question (on all \textsc{Q-Type}~) is 23 (Min:6, Max: 57).
Table \ref{tab:num_question} shows the total number of each question type in \textsc{SpartQA-Auto}{}~(Check Appendix to see more statistic information about the labels in Tab \ref{tab:number_of_choices}.)
\section{Models for Spatial Reasoning over Language}
\label{Sec:Architecture}
This section describes the model architectures on different \textsc{Q-Type}{}s: FR, YN, FB, and CO. All \textsc{Q-Type}{}s can be cast into a sequence classification task, and the three transformer-based LMs tested in this paper, BERT \cite{devlin-etal-2019-bert}, ALBERT \cite{lan2019albert}, and XLNet \cite{yang2019xlnet}, can all handle this type of tasks by classifying the representation of [CLS], a special token prepended to each target sequence (see Appendix \ref{sec:lm-arch}). Depending on the \textsc{Q-Type}, the input sequence and how we do inference may be different.
FR and YN both have a predefined label set as candidate answers, and their input sequences are both the concatenation of a story and a question. While the answer to a YN question is a single label chosen from {\em Yes, No}, and {\em DK}, FR questions can have multiple correct answers. Therefore, we treat each candidate answer to FR as an independent binary classification problem, and take the union as the final answer.
As for YN, we choose the label with the highest confidence (Fig \ref{fig:modelyn}).
As the candidate answers to FB and CO are not fixed and depend on each story and its question the input sequences to these \textsc{Q-Type}{}s are concatenated with each candidate answer. Since the defined YN and FR model has moderately less accurate results on FB and CO \textsc{Q-Type} s, we add a LSTM~\cite{hochreiter1997long} layer to improve it.
Hence, to find the final answer, we run the model with each candidate answer and then apply an LSTM layer on top of all token representations. Then, we use the last vector of the LSTM outputs for classification (Fig \ref{fig:modelfb}).
The final answers are selected based on Eq.~\eqref{Formula:Answer2}.
\begin{equation}
\label{Formula:Answer2}
\begin{split}
x_{i} &= [s, c_{i}, q] \\
\vec{T_i} &= [\vec{t^{i}_{1}}, ..., \vec{t^{i}_{m^{i}}}] = LM(x_{i}) \\
[{\vec{h}^{i}_1}, ..., {\vec{h}^{i}_{m^i}}] &= \textrm{LSTM}(\vec{T_i}) \\
\vec{y_{i}} = [y^{0}_{i}, y^{1}_{i}] &= \textrm{Softmax}(\vec{h}^{i^T}_{m^i}W)) \\
\text{Answer} &= \{c_{i}| \argmax_j(y^j_{i}) = 1\}
\end{split}
\end{equation}
where $s$ is the story, $c_{i}$ is the candidate answer, $q$ is the question, $[\ ]$ indicates the concatenation of the listed vectors, and $m_{i}$ is tokens' number in $x_{i}$. The parameter vector, $W$, is shared for all candidates.
\subsection{Training and Inference}
\label{Sec:Inference}
We train the models based on the summation of the cross-entropy losses of all binary classifiers in the architecture. For FR and YN \textsc{Q-Type}{}s, there are multiple classifiers, while there is only one classifier used for CO and FB \textsc{Q-Type}{}s.
We remove inconsistent answers in
post-processing
for FR and YN \textsc{Q-Type}{}s during inference phase.
For instance on FR, {\em left} and {\em right} relations between two objects cannot be valid at the same time. For YN, as there is only one valid answer amongst the three candidates, we select the candidate with the maximal
predicted probability of being the true answer.
\section{Experiments}
\label{sec:Experiments}
\begin{table*}[t]
\centering
\begin{tabular}{|l|l|c|c|c|c|c|}
\hline
\#&Model & FB & FR & CO & YN & Avg\\
\hline
1&Majority
& 28.84 &24.52 & 40.18 &53.60 & 36.64\\
\hline
2&BER
& 16.34 &20 & 26.16 &45.36 & 30.17\\
3&BERT (Stories only; MLM)
& 21.15 &16.19 & 27.1 &\textbf{51.54} & 32.90\\
4&BERT (\textsc{SpartQA-Auto}{}; MLM)
& 19.23 &29.54 & \textbf{32.71} &47.42 & 34.88\\
5&BERT (\textsc{SpartQA-Auto}{}
& \textbf{62.5} &\textbf{46.66} & \textbf{32.71} &47.42 & \textbf{47.25}\\\hline
6&Human & 91.66 &95.23 & 91.66 &90.69 &92.31\\\hline
\end{tabular}
\caption{
\textbf{Further pretraining BERT on \textsc{SpartQA-Auto}{} improves accuracies on \textsc{SpartQA-Human}{}}. All systems are fine-tuned on the training data of \textsc{SpartQA-Human}{}, but Systems~3-5 are also further pretrained in different ways. System 3: further pretrained on the stories from \textsc{SpartQA-Auto}{} as a masked language model (MLM) task. System 4: further pretrained on both stories and QA annotations as MLM. System 5: the proposed model that is further pretrained on \textsc{SpartQA-Auto}{} as a QA task. Avg: The micro-average on all four \textsc{Q-Type}{}s.
}
\label{tab:experiments}
\end{table*}
As fine-tuning LMs has become a common baseline approach to knowledge transfer from a source dataset to a target task, including but not limited to \citet{phang2018sentence,zhou2020temporal,he2020foreshadowing}, we study the capability of spatial reasoning of modern LMs, specifically BERT, ALBERT, and XLNet, after fine-tuning them on \textsc{SpartQA-Auto}{}. This fine-tuning process is also known as {\em further pretraining}, to distinguish with the fine-tuning process on one's target task.
It is an open problem to find out better transfer learning techniques than simple further pretraining, as suggested in \citet{he2020quase,2020unifiedqa}, which is beyond the scope of this work.
All experiments use the models proposed in Sec.~\ref{Sec:Architecture}. We use AdamW~\cite{loshchilov2017decoupled} with $2\times 10^{-6}$ learning rate and Focal Loss~\cite{lin2017focal} with $\gamma = 2$ for training all the models.\footnote{All codes are available at \url{https://github.com/HLR/SpartQA-baselines}}
\subsection{Further pretraining on \textsc{SpartQA-Auto}{} improves spatial reasoning}
Table~\ref{tab:experiments} shows performance on \textsc{SpartQA-Human}{} in a low-resource setting, where 0.6k QA pairs from \textsc{SpartQA-Human}{} are used for fine-tuning these LMs and 0.5k for testing (see Table~\ref{tab:num_question} for information on this split).\footnote{Note this low-resource setting can also be viewed as a spatial reasoning probe to these LMs \cite{tenney2019you}.}
During our annotation, we found that the description of ``near to '' and ``far from'' varies largely between annotators. Therefore, we ignore these two relations from FR \textsc{Q-Type}{} in our evaluations.
In Table~\ref{tab:experiments}, System~5, BERT (\textsc{SpartQA-Auto}), is the proposed method of further pretraining BERT on \textsc{SpartQA-Auto}{}. We can see that System~2, the original BERT, performs consistently lower than System~5, indicating that having \textsc{SpartQA-Auto}{} as a further pretraining task improves BERT's spatial understanding.
\begin{table}[h]
\centering
\begin{tabular}{|l|c|}
\hline
Model & $F_1$ \\ \hline
Majority& 35 \\ \hline
BERT& 50 \\
BERT (Stories only; MLM) & 53\\
BERT (\textsc{SpartQA-Auto}{}; MLM)& 48\\
BERT (\textsc{SpartQA-Auto}{}) & 48\\
\hline
\end{tabular}
\caption{Switching from accuracy in Table~\ref{tab:experiments} to $F_1$ shows that the models are all performing better than the majority baseline on YN \textsc{Q-Type}.
}
\label{tab:f1-YN}
\end{table}
\begin{table*}[t]
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{|c|l|ccc|ccc|ccc|ccc|}
\hline
\multirow{2}{*}{\#}&
\multirow{2}{*}{Models} & \multicolumn{3}{c}{FB} & \multicolumn{3}{c}{FR} & \multicolumn{3}{c}{CO} & \multicolumn{3}{c|}{YN}\\ \cline{3-14}
&&Seen& Unseen& Human*&Seen& Unseen& Human*&Seen& Unseen& Human*&Seen& Unseen& Human*\\ \hline
1&Majority &48.70&48.70 & 28.84& 40.81& 40.81 & 24.52& 20.59 &20.38 & 40.18 &49.94 &49.91& \textbf{53.60}\\
\hline
2&BERT & 87.13 & 69.38& 62.5&85.68&73.71& 46.66&71.44&61.09& 32.71&78.29&76.81& 47.42\\
3&ALBERT &97.66&83.53& 56.73&91.61 &83.70& 44.76 & 95.20 & 84.55 & 49.53& 79.38&75.05& 41.75\\
4&XLNet &\textbf{98.00}&\textbf{84.85} & \textbf{73.07}& \textbf{94.60} &\textbf{91.63}& \textbf{57.14}& \textbf{97.11} &\textbf{90.88} & \textbf{50.46}&\textbf{79.91} &\textbf{78.54}& 39.69\\
\hline
5&Human & \multicolumn{2}{c}{85}&91.66&\multicolumn{2}{c}{90}&95.23&\multicolumn{2}{c}{94.44}&91.66&\multicolumn{2}{c}{90}&90.69\\ \hline
\end{tabular}
}
\caption{\textbf{Spatial reasoning is challenging}. We further pretrain three transformer-based LMs, BERT, ALBERT, and XLNet, on \textsc{SpartQA-Auto}{}, and test their accuracy in three ways: {\em Seen} and {\em Unseen} are both from \textsc{SpartQA-Auto}{}, where {\em Unseen} has applied minor modifications to its vocabulary; to get those {\em Human} columns, all models are fine-tuned on \textsc{SpartQA-Human}{}'s training data.
Human performance
on {\em Seen} and {\em Unseen} is the same since the changes applied to {\em Unseen} does not affect human reasoning.}
\label{tab:evaluation}
\end{table*}
In addition, we implement another two baselines.
System~3, BERT (Stories only; MLM): further pretraining BERT only on the stories of \textsc{SpartQA-Auto}{} as a masked language model (MLM) task; System~4, BERT (\textsc{SpartQA-Auto}{}; MLM): we convert the QA pairs in \textsc{SpartQA-Auto}{} into textual statements and further pretrain BERT on the text as an MLM (see Fig.~\ref{fig:masking} for an example conversion).
\begin{figure}[h]
\centering
\includegraphics[width = 0.9\linewidth]{images/mask.pdf}
\caption{Convert a triplet of (paragraph, question, answer) into a single piece of text for the MLM task. }
\label{fig:masking}
\end{figure}
To convert each question and its answer into a sentence, we utilize static templates for each question type which removes the question words and rearranges other parts into a sentence.
We can see that System~3 slightly improves over System~2, an observation consistent with many prior works that seeing more text generally helps an LM (e.g., \citet{gururangan2020dont}). The significant gap between System~3 and the proposed System~5 indicates that supervision signals come more from our annotations in \textsc{SpartQA-Auto}{} rather than from seeing more unannotated text.
System~4 is another way to make use of the annotations in \textsc{SpartQA-Auto}{}, but it is shown to be not as effective as further pretraining BERT on \textsc{SpartQA-Auto}{} as a QA task.
While the proposed System~5 overall
performs better than the other three baseline systems, one exception is its accuracy on YN, which is lower than that of System~3.
Since all systems' YN accuracies are also lower than the majority baseline\footnote{which predicts the label that is most common in each set of \textsc{SpartQA}{}}, we hypothesize that this is due to imbalanced data. To verify it, we compute the $F_1$ score for YN \textsc{Q-Type}{} in Table~\ref{tab:f1-YN}, where we see all systems effectively achieve better scores than the majority baseline. However, further pretraining BERT on \textsc{SpartQA-Auto}{} still does not beat other baseline systems, which implies that straightforward pretraining is not necessarily helpful in capturing the complex reasoning phenomena required by YN questions.
The human performance is evaluated on 100 random questions from each \textsc{SpartQA-Auto}{} and \textsc{SpartQA-Human}{} test set. The respondents are graduate students that were trained by some examples of the dataset before answering the final questions. We can see from Table~\ref{tab:experiments} that all systems' performances fall behind human performance by a large margin. We expand on the difficulty of \textsc{SpartQA} in the next subsection.
\begin{table*}[t]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{|l|c|cc|cc|cc|}
\hline
\multirow{2}{*}{Models} & \multicolumn{1}{c}{FB} & \multicolumn{2}{c}{FR} & \multicolumn{2}{c}{CO} & \multicolumn{2}{c|}{YN}\\ \cline{2-8}
&Consistency&Consistency&Contrast&Consistency& Contrast&Consistency& Contrast\\ \hline
\multirow{1}{*}{BERT} &69.44 &76.13 &42.47 &16.99 &15.58 &48.07 &71.41\\ \hline
\multirow{1}{*}{AlBERT} &84.77 &82.42 &41.69 &58.42 &62.51 &48.78 &69.19\\ \hline
\multirow{1}{*}{XLNet} &85.2 &88.56 &50 &71.10 & 72.31 &51.08 &69.18\\\hline
\end{tabular}
}
\caption{Evaluation of consistency and semantic sensitivity of models in Table~\ref{tab:evaluation}. All the results are on the correctly predicted questions of {\em Seen} test set of \textsc{SpartQA-Auto}{}.}
\label{tab:extra_evaluation}
\end{table*}
\subsection{\textsc{SpartQA} is challenging}
In addition to BERT, we continue to test another
two LMs, ALBERT and XLNet (Table \ref{tab:extra_evaluation}). We further pretrain these LMs on \textsc{SpartQA-Auto}{}, and test them on \textsc{SpartQA-Human}{}
(the numbers of BERT are copied from Table~\ref{tab:experiments}) and two held-out test sets of \textsc{SpartQA-Auto}{}, {\em Seen} and {\em Unseen}.
Note that when a system is tested against \textsc{SpartQA-Human}{}, it is fine-tuned on \textsc{SpartQA-Human}{}'s training data following its further pretraining on \textsc{SpartQA-Auto}{}.
We use the unseen set to test to what extent the baseline models use shortcuts in the language surface. This set applies minor modifications randomly on a number of stories and questions to change the names of shapes, colors, sizes, and relationships in the vocabulary of the stories, which do not influence the reasoning steps (more details in Appendix~\ref{sec:unseen}).
All models perform worst in YN across all \textsc{Q-Type}{}s, which suggests that YN presents a more complex phenomena, probably due to additional
quantifiers in the questions. XLNet performs the best on all \textsc{Q-Type}{}s except its accuracy on \textsc{SpartQA-Human}{}'s YN section. However, the drops in {\em Unseen} and {\em human} suggest overfitting
on the training vocabulary.
The low accuracies on human test set from all models show that solving this benchmark is still a challenging problem and requires more sophisticated methods like considering spatial roles and relations extraction~\cite{kordjamshidi2010spatial,dan2020spatial, rahgooy2018visually} to understand stories and questions better.
To evaluate the reliability of the models, we also provide two extra consistency and contrast test sets.
\textbf{Consistency set} is made by changing a part of the question in a way that seeks for the same information~\cite{hudson2019gqa, suhr2018corpus}. Given a pivot question and answer of a specific consistency set, answering other questions in the set does not need extra reasoning over the story.
\textbf{Contrast set} is made by minimal modification in a question to change its answer~\cite{gardner2020evaluating}. For contrast sets, there is a need to go back to the story to find the new answer for the question's minor variations~(see Appendix \ref{sec:cons} for examples.)
The consistency and contrast sets are evaluated only on the correctly predicted questions to check if the actual understanding and reasoning occurs.
This ensures the reliability of the models.
Table~\ref{tab:extra_evaluation} shows the result of this evaluation on four \textsc{Q-Type}{}s of \textsc{SpartQA-Auto}{},
where we can see, for another time, that
the high scores
on the {\em Seen} test set are likely
due to overfitting on training data rather than correct detection of spatial terms and reasoning over them.
\subsection{Extrinsic evaluation}
In this subsection, we take BERT as an example to show, once pretrained on \textsc{SpartQA-Auto}{}, BERT can achieve better performance on two extrinsic evaluation datasets, namely \text{bAbI}{} and boolQ.
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{images/babi_result.png}
\caption{Learning curve of BERT and BERT further pretrained on \textsc{SpartQA-Auto}{} on \text{bAbI}{}.
}
\label{fig:babi_train}
\end{figure}
\begin{table}[h]
\centering
\begin{tabular}{|l|c|}
\hline
\textbf{Model} &\textbf{Accuracy}\\
\hline
Majority baseline & 62.2 \\
Recurrent model (ReM) & 62.2 \\
ReM fine-tuned on SQuAD & 69.8 \\
ReM fine-tuned on QNLI & 71.4 \\
ReM fine-tuned on NQ & 72.8 \\\hline
BERT (our setup) & 71.9 \\
BERT (\textsc{SpartQA-Auto}{}) & \textbf{74.2} \\\hline
\end{tabular}
\caption{System performances on the dev set of \text{boolQ}{} (since the test set is not available to us). Top: numbers reported in \cite{clark2019boolq}. Bottom: numbers from our experiments. BERT (\textsc{SpartQA-Auto}): further pretraining BERT on \textsc{SpartQA-Auto}{} as a QA task.
}
\label{tab:boolq}
\end{table}
We draw the learning curve on \text{bAbI}{}, using the original BERT as a baseline and BERT further pretrained on \textsc{SpartQA-Auto}{} (Fig.~\ref{fig:babi_train}).
Although both systems achieve perfect accuracy given large enough training data (i.e., 5k and 10k), BERT (\textsc{SpartQA-Auto}{}) is showing better scores given less training data. Specifically, to achieve an accuracy of 99\%, BERT (\textsc{SpartQA-Auto}) requires 1k training examples, while BERT requires twice as much.
We also notice that BERT (\textsc{SpartQA-Auto}) converges faster in our experiments.
As another evaluation dataset, we chose boolQ for two reasons. First, we needed a QA dataset with Yes/No questions. To our knowledge boolQ is the only available one used in the recent work.
Second, indeed, \textsc{SpartQA}{} and boolQ are from different domains, however, boolQ needs multi-step reasoning in which we wanted to see if \textsc{SpartQA}{} helps.
Table \ref{tab:boolq} shows that further pretraining BERT on \textsc{SpartQA-Auto}{} yields a better result than the original BERT and those reported numbers in \citet{clark2019boolq}, which also tested on various distant supervision signals such as SQuAD~\cite{rajpurkar2016squad}, Google's Natural Question dataset NQ~\cite{kwiatkowski2019natural}, and QNLI from GLUE~\cite{wang2018glue}.
We observe that many of the boolQ examples answered correctly by the BERT further pretrained on \textsc{SpartQA-Auto}{} require multi-step reasoning. Our hypothesis is that since solving \textsc{SpartQA-Auto}{} questions needs multi-step reasoning, fine-tuning BERT on \textsc{SpartQA-Auto}{} generally improves this capability of the base model.
\section{Conclusion}
\label{conclusion}
Spatial reasoning is an important problem in natural language understanding. We propose the first human-created QA benchmark on spatial reasoning, and experiments show that state-of-the-art pretrained language models (LM) do not have the capability to solve this task given limited training data, while humans can solve those spatial reasoning questions reliably. To improve LMs' capability on this task, we propose to use hand-crafted grammar and spatial reasoning rules to automatically generate a large corpus of spatial descriptions and corresponding question-answer annotations; further pretraining LMs on this distant supervision dataset significantly enhances their spatial language understanding and reasoning. We also show that a spatially-improved LM can have better results on two extrinsic datasets~(\text{bAbI}{} and \text{boolQ}{}).
\section{Question Templates and statistics Information}
\label{sec:supplemental}
\begin{table*}[htbp]
\centering
\begin{tabular}{|l|l|l|}
\hline
\textbf{Q-Type} & \textbf{Q-Templates}& \textbf{Candidate answer}\\ \hline
FR& what is the relation between \textless object\textgreater and \textless object\textgreater?& \begin{tabular}[c]{@{}l@{}}
Left, Right, Below,\\Above, Touching,\\ Far from, Near to
\end{tabular}\\ \hline
CO& \begin{tabular}[c]{@{}l@{}}What is \textless relation \textgreater the \textless object\textgreater?\\\hspace{20 mm}an \textless object1\textgreater or an \textless object2\textgreater?\\ Which object is \textless relation \textgreater an \textless object\textgreater? \\\hspace{20 mm}the \textless object1\textgreater or the \textless object2\textgreater?\end{tabular}& \begin{tabular}[c]{@{}l@{}}Object1, object2,\\ Both, None\end{tabular} \\\hline
YN& \begin{tabular}[c]{@{}l@{}}Is (the $\mid$ a )\textless object1\textgreater \textless relation\textgreater (the $\mid$ a) \textless object2\textgreater?\\ Is there any \textless object1\textgreater s \textless relation\textgreater all \textless object2\textgreater s?\end{tabular}& \multicolumn{1}{l|}{Yes, No, Don't Know}\\ \hline
FB& \begin{tabular}[c]{@{}l@{}}Which block has an \textless object\textgreater? \\ Which block doesn't have an \textless object\textgreater? \end{tabular} & Name of blocks, None\\ \hline
\end{tabular}%
\caption{Questions and answers templates.}
\label{tab:templates}
\end{table*}
Table~\ref{tab:templates} shows the templates used to create questions in \textsc{SpartQA-Auto}{}. The ``\textless object\textgreater'' is a variable replaced by objects from the story (using \textit{Choose-objects} and \textit{Describe-objects} modules), and the ``\textless relation\textgreater'' variable can be replaced by the chosen relations between objects (using \textit{Find-all-relations} module).
The articles and the indefinite pronouns in each template play an essential role in understanding the question's objective. For example, ``Are all blue circles near to a triangle?'' is different from ``Are there any blue circles near to a triangle?'', and ``Are there any blue circles near to all triangles?''. Therefore, we check the uniqueness of the object definition, using ``a'' or ``the'' in proper places and randomly place the terms ``any'' or ``all'' in the YN questions to generate different questions.
\begin{table}[h]
\centering
\begin{tabular}{|l|l|ll|}
\hline
\textsc{Q-Type}{} & Candidate Answers & train & test \\ \hline \hline
\multirow{8}{*}{\begin{tabular}[c]{@{}l@{}}FR\\ (Multiple\\ Choices)\end{tabular}} & Left & 20.7 & 17.9 \\ \cline{2-4}
& Right & 21.4 & 16.7\\ \cline{2-4}
& Above & 26.9& 25.4 \\ \cline{2-4}
& Below & 37.2& 42.9 \\ \cline{2-4}
& Near to & 5.8& 2.9 \\ \cline{2-4}
& Far from & 1.3 & 0.56 \\ \cline{2-4}
& Touching & 0.57 & 0.27 \\ \cline{2-4}
& DK & 0.52 & 0.32 \\ \hline \hline
\multirow{4}{*}{\begin{tabular}[c]{@{}l@{}}FB\\ (multiple\\ Choices)\end{tabular}} & A & 49.8 & 49.4\\ \cline{2-4}
& B & 50.1 & 50\\ \cline{2-4}
& C & 35.1& 62 \\ \cline{2-4}
& {[}{]} & 7.1 & 90.5\\ \hline \hline
\multirow{5}{*}{\begin{tabular}[c]{@{}l@{}}CO\\ (Single \\ choice)\end{tabular}} & Object1 & 25.4 & 26\\ \cline{2-4}
& Object2 & 25.3 & 24.9\\ \cline{2-4}
& Both & 44.3 & 43.9 \\ \cline{2-4}
& None & 4.9 & 5.0 \\ \hline \hline
\multirow{3}{*}{\begin{tabular}[c]{@{}l@{}}YN\\ (Single\\ choice)\end{tabular}} & Yes & 53.3& 50.5 \\ \cline{2-4}
& No & 18.7 & 23.6 \\ \cline{2-4}
& DK & 27.8 & 25.9 \\ \hline
\end{tabular}
\caption{The percentage of each correct label in all samples. *The candidate answers for the FB \textsc{Q-Type}{} can be varied, based on its story. **CO can be considered as a multiple choice or single choice question. E.g., in "which object is above the triangle? the blue circle or the black circle?" you can consider two labels with boolean classification on each "blue circle" and "black circle" or consider it as a four labels classification: "blue circle," "black circle," "both of them," and "None of them." *** \textbf{DK}, \textbf{None}, \textbf{[]}, all mean none of the actual labels are correct.}
\label{tab:number_of_choices}
\end{table}
Table \ref{tab:number_of_choices} shows the percentage of correct labels in train and test sets. In multi-choice \textsc{Q-Type} s, more than one label can be true.
\section{Sentences of the Dataset}
Table~\ref{tab:particular_features} shows some generated sentences in \textsc{SpartQA-Auto}{} with some specific features that challenge models to understand different forms of relation description in spatial language.
\section{Additional Evaluation Sets}
\label{sec:additional_eval}
Here we describe three extra evaluation sets provided with this dataset in more detail, including unseen test, consistency, and contrast sets.
\subsection{Unseen Evaluation Set}
\label{sec:unseen}
We propose an unseen test set alongside the seen test of \textsc{SpartQA-Auto}{} to check whether a model is using shortcuts in the language surface by describing objects and relations with new vocabularies in the samples. This set has minor modifications that should not affect the performance of a consistent and reliable model. The modifications are randomly applied on a number of generated stories and questions and include changing names of shapes, colors, sizes, and relationships' names ~(describing relationships using different language expressions). The modification choices are described in Table~\ref{tab:modifications}.
\begin{table}[h]
\centering
\begin{tabular}{l l l}
\hline
\textbf{Type}& \textbf{Original Set}& \textbf{Unseen Set} \\ \hline
Shapes&\begin{tabular}[c]{@{}l@{}} Square, Circle, \\ Triangle \end{tabular}& \begin{tabular}[c]{@{}l@{}} Rectangle, Oval, \\Diamond \end{tabular}\\ \hline
Relations &\begin{tabular}[c]{@{}l@{}} Left, Right,\\ Above, Below \end{tabular}& \begin{tabular}[c]{@{}l@{}} Left side,\\ Right side, \\Top, Under \end{tabular}\\ \hline
Colors&\begin{tabular}[c]{@{}l@{}} Yellow, Black,\\ Below \end{tabular}& \begin{tabular}[c]{@{}l@{}} Green, Red,\\ White \end{tabular}\\ \hline
Size&\begin{tabular}[c]{@{}l@{}} Small,\\ Medium, Big \end{tabular}& \begin{tabular}[c]{@{}l@{}} Little, Midsize,\\ Large \end{tabular}\\
\hline
\end{tabular}
\caption{Modifications on the unseen set}
\label{tab:modifications}
\end{table}
\subsection{Contrast and Consistency Evaluation}
\label{sec:cons}
For probing the consistency and semantic sensitivity of models, we provide two extra evaluation test sets, Consistency and Contrast\footnote{for some questions, it is not possible to generate a complementary set}.
\textbf{Consistency set} is made by changing parts of the question in a way that it still asks about the same information~\cite{hudson2019gqa, suhr2018corpus}. For instance, for the question, ``What is the relation between the blue circle and the big shape? Left,'' we create a similar question in the form of ``What is the relation between the big shape and the blue circle? Right''. Answering these questions around a pivot question is possible for human without the need for extra reasoning over the story and based on the main questions' answer. Hence, the evaluation on this set shows that models understand the real underlying semantics rather than overfit on the structure of questions.
\textbf{Contrast set}: This set is made by minor changes in a question that changes the answer~\cite{gardner2020evaluating}. As an instance, in the question ``Is the blue circle below the black triangle? Yes,'' we create a contrast question ``Is the blue circle below all triangles? No'' by changing ``the black trinagle'' to ``all triangles''. The evaluation on this set shows the robustness of the model and its sensitivity to the semantic changes when there are minor changes in the language surface~\footnote{Based on the original contrast set paper, consistency and contrast set should be generated manually to control the semantic change. In our case that we are probing the spatial language understanding of models, we must change parts that affect spatial understanding, which can be implemented by some static rules.}.
\section{Extra Annotations}
Alongside the main \textsc{SpartQA-Auto}'s stories and questions we provided some extra annotation to help the models to understand the spatial language better.
\subsection{Detailed Annotation and Scene-Graphs}
\label{sec:annotation}
Providing in-depth human annotations is quite expensive and time-consuming.
In \textsc{SpartQA-Auto}{}, we generated fine-grained scene-graph based on the story. This scene-graph contains blocks' description, their relations, and the objects' attributes alongside their direct relations with each other.
The scene-graphs can be used for the models to understand all spatial relations directly mentioned in the textual context. Figure~\ref{fig:scene-graph} shows an example of this scene-graph. The scene-graph can provide strong supervision for question answering challenges and can be used to evaluate models based on their steps of reasoning and decisions.
\begin{figure}
\centering
\includegraphics[width= \linewidth]{images/scene-graph.png}
\caption{Scene-graph}
\label{fig:scene-graph}
\end{figure}
\subsection{SpRL Annotation}
\label{sec:sprlAnnotation}
We also provided spatial annotations for each sentence and question, based on Spatial Role Labeling (SpRL) annotation scheme~\cite{kordjamshidi2010spatial}(Fig. \ref{fig:sprl}). This annotation is generated by hand-crafted rules during the main data generation. SpRL is used for recognizing spatial expressions and arguments in a sentence. This annotation is useful for applications that need to detect and reason about spatial expressions and arguments.
\section{QA Language Models for Spatial Reasoning over Text}
\label{sec:lm-arch}
Figures \ref{fig:modelfb} and \ref{fig:modelyn} depict the architecture used for further fine-tuning language models on \textsc{SpartQA}{} described in section 5.
\begin{figure}[htb!]
\centering
\begin{subfigure}[b]{\linewidth}
\includegraphics[width=\linewidth]{images/FB-CO.pdf}
\caption{LM$_{QA}$ Architecture for CO and FB \textsc{Q-Type}{}s}
\label{fig:modelfb}
\end{subfigure}
\begin{subfigure}[b]{\linewidth}
\includegraphics[width=\linewidth]{images/YN-FR.pdf}
\caption{LM$_{QA}$ Architecture for FR and YN \textsc{Q-Type}{}s}
\label{fig:modelyn}
\end{subfigure}
\caption{LM$_{QA}$ for Spatial Reasoning over Text}
\label{fig:model}
\end{figure}
\section{\text{bAbI}{} and \text{boolQ}{} Datasets}
\label{sec:babiboolq}
Figure~\ref{fig:babi} shows an example of the \text{bAbI}{} dataset~\cite{weston2015towards} task 17.
\begin{figure}[htb!]
\centering
\includegraphics[width=\linewidth]{images/babi.pdf}
\caption{An example of bAbI dataset, task 17.}
\label{fig:babi}
\end{figure}
To solve task 17 of \text{bAbI}{} , we implement two SpRL+rule-based and neural network models.
The SpRL+rule-based model first, finds different spatial relation triplets (Landmark, Spatial-indicator, trajector) for each fact in a story the applies spatial rules over these extracted triplets and report all possible relations between two asked objects. Finally, it checks whether the asked relation existed in the find relation. This model solves task 17 of the \text{bAbI}{} with $100\%$ accuracy.
To implement the neural network approach, we use huggingface implementation of pre-trained BERT~\cite{devlin-etal-2019-bert}. We apply a boolean classifier on the output of ``[CLS]'' token from the last layer of BERT model for each ``Yes'' and ``No'' answers (the same as model used on YN question types.) We use Adamw~\cite{loshchilov2017decoupled} optimizer and $2e-6$ learning rate with negative log-likelihood loss objective and train the model on the 10k, 5k, 2k, 1k, 500, and 100 portion of \text{bAbI}'s training questions. The model yields $100\%$ accuracy on 10k, and 5k and $99\%$ accuracy on 2k and 1k training samples.
Figure \ref{fig:boolq} shows an example of \text{boolQ}{} dataset. To Answering the questions of this dataset, we use the same setting as neural network model on \text{bAbI}{} to further fine-tune BERT on \text{boolQ}{}.
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{images/boolq.pdf}
\caption{An example of boolQ dataset.}
\label{fig:boolq}
\end{figure}
\begin{figure*}[b]
\centering
\includegraphics[width=\linewidth]{images/Sprl.png}
\caption{SpRL annotation for an example sentence from \textsc{SpartQA}.}
\label{fig:sprl}
\end{figure*}
\begin{table*}[b]
\centering
\begin{tabular}{p{0.45\linewidth}|p{0.45\linewidth}}
\hline
\textbf{Examples} & \textbf{Features} \\ \hline
Block A is above Block C \textbf{and} B. & Using conjunction to describe relation between more than two blocks. \\ \hline
The small circle is \textbf{above} the yellow square \textbf{and} the big black shape. & Using conjunction to describe relationships between more than two objects. \\ \hline
The yellow square number one is to the \textbf{right} of \textbf{and} \textbf{above} the blue circle. & Using conjunction for more than one relation. \\ \hline
Block B has \textbf{two medium yellow squares} and \textbf{two blue circles}. & Describing a group of objects with the same properties. In the next sentences, they are mentioned by an asigned number. For example, the blue circle number two. \\ \hline
The blue circle is below the object\textbf{ which is to the right} of the big square. & Using nested relations between objects in their description. \\ \hline
A small blue circle is near to the big circle. \textbf{It} is to the left of the medium yellow square. & Using coreferences for an entity described in the previous sentences. \\ \hline
There \textbf{is a} block named A. One small yellow square \textbf{is} touching the bottom edge of this block. & The verb matches the number of the subject.
\\ \hline
What is the relation between black \textbf{object} and a big circle? & Using shape, object, and thing, which are a general description of an object. It could be the ``black triangle'' or the ``black circle'' mentioned in the story.\\ \hline
\end{tabular}%
\caption{Particular features of the dataset}
\label{tab:particular_features}
\end{table*}
\section{Experiments}
\label{sec:Experiments}
\begin{table*}[t]
\centering
\resizebox{0.75\linewidth}{!}{%
\begin{tabular}{|l|l|c|c|c|c|c|}
\hline
\#&Model & FB & FR & CO & YN & Avg*\\
\hline
1&Majority baseline
& 28.84 &24.52 & 40.18 &53.60 & 36.64\\
\hline
2&BERT$_{QA}$
& 16.34 &20 & 26.16 &45.36 & 30.17\\
3&BERT$_{MLM}$--{\small \textsc{SpartQA-Auto}{}(T)}
& 21.15 &16.19 & 27.1 &51.54 & 32.90\\
4&BERT$_{MLM}$--{\small \textsc{SpartQA-Auto}{}(T+Q)} & 19.23 &29.54 & \textbf{32.71} &47.42 & 34.88\\
5&BERT$_{QA}$ + \small \textsc{SpartQA-Auto}& \textbf{62.5} &\textbf{46.66} & \textbf{32.71} &47.42 & 47.25\\\hline
6&Human & 91.66 &95.23 & 91.66 &90.69 &92.31\\\hline
\end{tabular}
}
\caption{Evaluating BERT on \textsc{SpartQA-Human}{} by four experiments. 1) BERT$_{QA}$, 2) BERT$_{MLM}$--{\small \textsc{SpartQA-Auto}{}(T)}: fine-tuned BERT on \textsc{SpartQA-Auto}{}'s text as MLM task, 3) BERT$_{MLM}$--{\textsc{SpartQA-Auto}{}(T+Q)}: fine-tuned BERT on \textsc{SpartQA-Auto}{}'s text+questions as MLM task, and 4)BERT$_{QA}$--{\small \textsc{SpartQA-Auto}{}}: fine-tuned BERT on \textsc{SpartQA-Auto}{} as QA task. All models are fine-tuned on \textsc{SpartQA-Human}{} training samples. Avg*: This column shows the weighted average result on all four tasks. All numbers are the accuracy of the models' performance.}
\label{tab:experiments}
\end{table*}
\begin{table*}[t]
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{|c|l|ccc|ccc|ccc|ccc|}
\hline
\multirow{2}{*}{\#}&
\multirow{2}{*}{Models} & \multicolumn{3}{c}{FB} & \multicolumn{3}{c}{FR} & \multicolumn{3}{c}{CO} & \multicolumn{3}{c|}{YN}\\ \cline{3-14}
&&Seen& Unseen& Human*&Seen& Unseen& Human*&Seen& Unseen& Human*&Seen& Unseen& Human*\\ \hline
1&Majority &48.70&48.70 & 28.84& 40.81& 40.81 & 24.52& 20.59 &20.38 & 40.18 &49.94 &49.91& \textbf{53.60}\\
\hline
2&BERT & 87.13 & 69.38& 62.5&85.68&73.71& 46.66&71.44&61.09& 32.71&78.29&76.81& 47.42\\
3&ALBERT &97.66&83.53& 56.73&91.61 &83.70& 44.76 & 95.20 & 84.55 & 49.53& 79.38&75.05& 41.75\\
4&XLNet &\textbf{98.00}&\textbf{84.85} & \textbf{73.07}& \textbf{94.60} &\textbf{91.63}& \textbf{57.14}& \textbf{97.11} &\textbf{90.88} & \textbf{50.46}&\textbf{79.91} &\textbf{78.54}& 39.69\\
\hline
5&Human & \multicolumn{2}{c}{85}&91.66&\multicolumn{2}{c}{90}&95.23&\multicolumn{2}{c}{94.44}&91.66&\multicolumn{2}{c}{90}&90.69\\ \hline
\end{tabular}
}
\caption{Fine-tuning three baseline LMS (BERT, ALBERT, and XLNET) on \textsc{SpartQA-Auto}, and test on three benchmarks, Seen, Unseen and Human~(\textsc{SpartQA-Human}) sets. \textbf{Human*}: For Human result all models are fine-tuned on \textsc{SpartQA-Human}{} training samples. The result of Human on Seen and Unseen test are the same since the changes applied on unseen set does not affect human reasoning. All numbers are reported for the model's accuracy.}
\label{tab:evaluation}
\end{table*}
\qn{}
We design the following experiments to evaluate whether
\textsc{SpartQA-Auto}{}
improves the performance of
modern
LMs~(i.e., BERT, ALBERT, and XLNet) in spatial reasoning.
\paragraph{BERT$_{QA}$} In this experiment, we use the same architectures described in Sec. \ref{Sec:Architecture} for answering various types of questions. We use the pretrained BERT as its base module.
\paragraph{BERT$_{MLM}$--{\small\textsc{SpartQA-Auto}{}(T)}} BERT is pretrained on general-domain texts in BookCorpus~\cite{zhu2015aligning} and Wikipedia articles which do not contain many spatial descriptions.
For example, in those resources, the word ``right'' is used more frequently to mean ``correct'', or the word ``left'' is used more frequently to mean ``leaving'' rather than their spatial meaning.
Hence, in this experiment, first, we further pretrain BERT on \textsc{SpartQA-Auto}'s stories in the form of the MLM task~(Randomly masking tokens). We use this further pretrained BERT as the base module in the architecture described in Sec. \ref{Sec:Architecture}
\begin{figure}
\centering
\includegraphics[width = 0.75\linewidth]{images/mask.pdf}
\caption{Convert (paragraph, question, answer) to a single text for MLM task. }
\label{fig:masking}
\end{figure}
\paragraph{BERT$_{MLM}$--{{\small\textsc{SpartQA-Auto}{}(T+Q)}}} BERT$_{MLM}$- {\small\textsc{SpartQA-Auto}{}(T)} model is trained only on the stories of our samples while the the (question,answer) pairs are the main source of supervision for spatial reasoning.
Thus, in this experiment, we add the (question,answer) pair information to the end of the stories using $f(t,q,a)$, where $t$ is the text description~(story), $q$ is the question, and $a$ is the correct answer. We define $f$ as:
\begin{equation}
f(t,q,a) = t . to\_sentence(q,a)
\end{equation}
where $.$ is the concatenation operator and $to\_sentence(.)$ transforms question, $q$, and its answer, $a$, to a single sentence as shown in Fig.~\ref{fig:masking}.
Then, we further pretrain BERT on the updated stories by masking tokens containing the answer in the form of the MLM task. We use this further pretrained BERT as the base module in the architecture describe in Sec. \ref{Sec:Architecture}.
\paragraph{BERT$_{QA}$--{\small\textsc{SpartQA-Auto}{}}}
We use the same setting as BERT$_{QA}$ and train the model further on \textsc{SpartQA-Auto}{} samples.
Table \ref{tab:experiments} shows the results of above experiments evaluated on \textsc{SpartQA-Human}{}'s test samples. All these models are fine-tuned~(regardless of their initial training) by training on \textsc{SpartQA-Human}{} training samples.
During our annotation, we found that the description of ``near to '' and ``far from'' varies largely between annotators. Therefore, we ignore these two relations from FR \textsc{Q-Type}{} during evaluating \textsc{SpartQA-Human}{} examples to be consistent.
The poor performance of BERT$_{QA}$ shows that pretrained BERT cannot perform well on spatial reasoning based on the available human-generated data. Comparing the average performance of BERT$_{QA}$ to any of the other models shows that further pretraining on our automatically generated data helps BERT-base models to get better spatial reasoning capability. BERT$_{QA}$--{\small \textsc{SpartQA-Auto}{}} achieves the best improvement as it is trained on automatically generated samples using the same target task setting as \textsc{SpartQA-Human}{}.
The reason of the improvement on BERT$_{MLM}$--{ {\small\textsc{SpartQA-Auto}{}~(T+Q)}} over BERT$_{MLM}$--{\small\textsc{SpartQA-Auto}{}~(T)} is the first is trained on updated stories which include (question,answer) pairs and the masked tokens are the ones that require spatial reasoning to be predicted while the second is only trained on random masked tokens from the stories.
The only exception is the result of YN \textsc{Q-Type}{} on the human test set, which is getting lower results than the the majority baseline~(51.9\%) even after training on \textsc{SpartQA-Auto}{}. Since the human test set is imbalance, we also compute the F1-measure for YN \textsc{Q-Type}{} which is shown in Table \ref{tab:f1-YN}. This table shows that after using \textsc{SpartQA}{}, models perform better than the majority baseline. The best performance is achieved by BERT fine-tuned on \textsc{SpartQA-Auto}{}'s text as MLM task. This can be due to the complexity of the YN questions (described in Sec. \ref{sec:auto}).
\begin{table}[h]
\centering
\begin{tabular}{|l|c|}
\hline
Model & F1-measure \\ \hline
Majority Baseline& 0.35 \\ \hline
BERT$_{QA}$& 0.50 \\
BERT$_{MLM}$--{\small \textsc{SpartQA-Auto}{}(T)} & 0.53\\
BERT$_{MLM}$--{\small \textsc{SpartQA-Auto}{}(T+Q)} & 0.48\\
BERT$_{QA}$--{\small \textsc{SpartQA-Auto}{}} & 0.48\\
\hline
\end{tabular}
\caption{Evaluating models on YN \textsc{Q-Type}{} with F1-measure.}
\label{tab:f1-YN}
\end{table}
\begin{table*}[t]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{|l|c|cc|cc|cc|}
\hline
\multirow{2}{*}{Models} & \multicolumn{1}{c}{FB} & \multicolumn{2}{c}{FR} & \multicolumn{2}{c}{CO} & \multicolumn{2}{c|}{YN}\\ \cline{2-8}
&Consistency&Consistency&Contrast&Consistency& Contrast&Consistency& Contrast\\ \hline
\multirow{1}{*}{BERT} &69.44 &76.13 &42.47 &16.99 &15.58 &48.07 &71.41\\ \hline
\multirow{1}{*}{AlBERT} &84.77 &82.42 &41.69 &58.42 &62.51 &48.78 &69.19\\ \hline
\multirow{1}{*}{XLNet} &85.2 &88.56 &50 &71.10 & 72.31 &51.08 &69.18\\\hline
\end{tabular}
}
\caption{Evaluating consistency and semantic sensitivity of baseline LMs (BERT, ALBERT, and XLNet}
\label{tab:extra_evaluation}
\end{table*}
\paragraph{Baseline LMs on \textsc{SpartQA-Auto}{}:} In a different set of experiments, we investigate whether \textsc{SpartQA}{} is generally challenging to be solved (see Table \ref{tab:evaluation}). Thus, we use three different language models BERT~\cite{devlin-etal-2019-bert}, ALBERT~\cite{lan2019albert}, and XLNet~\cite{yang2019xlnet} in the same architecture described in Sec. \ref{Sec:Architecture} and evaluate their performance on two automatically generated evaluation sets, seen and unseen sets, plus the \textsc{SpartQA-Human}{} test set. To evaluate the models performance on the human test set, we further fine-tune models on \textsc{SpartQA-Human}{} training samples.
We use the unseen test set to check whether the baseline models are using shortcuts in the language surface. This set does not influence the reasoning steps and applies minor modifications in the vocabulary of the stories. The modifications are randomly applied on a number of generated stories and questions to change the names of shapes, colors, sizes, and relationships~(more details in Appendix~\ref{sec:additional_eval}).
All models perform worst in ``YN'' \textsc{Q-Type}, which presents a more complicated phenomena due to using quantifiers in the questions. The XLNet model performs best on all \textsc{Q-Type}{}s, except the Human evaluation on YN \textsc{Q-Type}. However, the drop in the unseen and human test evaluation shows that the model is fitted on the training vocabulary and cannot generalize well. The low accuracy on human test set shows that solving this benchmark is still an open problem and requires more sophisticated methods.
The human performance is evaluated on 100 random questions from \textsc{SpartQA-Auto}{} and \textsc{SpartQA-Human}{} test sets. The respondents are graduate students that were trained by some samples of the dataset before answering the final questions.
To evaluate the reliability of the models, we provide two extra consistency and contrast test sets.
\textbf{Consistency set} is made by changing a part of the question in a way that seeks for the same information~\cite{hudson2019gqa, suhr2018corpus}. Given a pivot question and answer of a specific consistency set, answering other questions in the set does not need extra reasoning over the story.
\textbf{Contrast set} is made by minimal modification in a question to change its answer~\cite{gardner2020evaluating}~(see Appendix ~\ref{sec:cons}). For contrast sets, there is a need to go back to the story to find the new answer for the question's minor variations.
\qn{I don't understand this}\rk{it means that to solve them we should reason over new information from data. not the same infromation taht we used in questions.} The consistency and contrast sets are evaluated only on the correctly predicted questions to check if the actual understanding and reasoning occurs.
This ensures the reliability of the models.
Table~\ref{tab:extra_evaluation} shows the result of this evaluation on four \textsc{Q-Type}{}s of \textsc{SpartQA-Auto}{}. This table shows that the high result on the seen test set is mainly due to overfitting on training data rather than correct detection of spatial terms and reasoning over them.
\paragraph{BERT$_{QA}$--{\small\textsc{SpartQA-Auto}{}} on other dataset} In this experiment, we want to check whether the knowledge learned from \textsc{SpartQA-Auto}{} is transferable to other target tasks or benchmarks. We use BERT$_{QA}$--{\small\textsc{SpartQA-Auto}{}} trained on YN \textsc{Q-Type}{} and check its performance on two different benchmarks, bAbI and boolQ, by fine-tuning on their training set.
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{images/babi_result.png}
\caption{Results of BERT and Fine-tuned BERT on \textsc{SpartQA-Auto}{} on different portion of training set of bAbI.}
\label{fig:babi_train}
\end{figure}
\begin{table}[h]
\centering
\resizebox{\linewidth}{!}{%
\begin{tabular}{|l|c|}
\hline
\textbf{Model} &\textbf{Accuracy}\\
\hline
Majority baseline* & 62.2 \\\hline
Recurrent model (ReM)* & 62.2 \\\hline
ReM fine-tuned on SQuAD* & 69.8 \\\hline
BERT (our setup) & 71.89 \\\hline
ReM fine-tuned on QNLI* & 71.4 \\\hline
ReM fine-tuned on NQ * & 72.8 \\\hline
BERT fine-tuned on \textsc{SpartQA-Auto}{} & \textbf{74.18} \\\hline
\end{tabular}
}
\caption{Result of fine-tuned version of Recurrent Model and BERT on different dataset. All these results are on the DEV set of boolQ since the test set is not provided yet. *: All models and results are from \cite{clark2019boolq}.}
\label{tab:boolq}
\end{table}
Figure \ref{fig:babi_train} compares the performance of BERT and BERT$_{QA}$-{\small\textsc{SpartQA-Auto}{}} on different portions of bAbI training set. Figure \ref{fig:babi_train} shows that BERT$_{QA}$-{\small\textsc{SpartQA-Auto}{}} performs better than BERT when reducing the number of bAbI training samples to 1k, 500, and 100. With 5k and 10k training examples,
both models can achieve 100\% accuracy while fine-tuned BERT on \textsc{SpartQA-Auto}{} achieves this performance requiring fewer training epochs (BERT$_{QA}$-{\small\textsc{SpartQA-Auto}{}} achieves 100\% accuracy with 10k training samples, after two epochs while BERT achieves this performance after 6 epochs). In conclusion, using \textsc{SpartQA-Auto}{} reduces the need of large training data and helps models converge quicker.
Table \ref{tab:boolq} shows that fine-tuning BERT on our dataset yields a better result than fine-tuning Recurrent model on SQuAD dataset~\cite{rajpurkar2016squad}, Natural Question dataset NQ~\cite{kwiatkowski2019natural}, and QNLI task from GLUE~\cite{wang2018glue}.
This result is significant since these datasets and tasks are more related to boolQ than \textsc{SpartQA}{}.
Many of the examples answered correctly on boolQ with the fine-tuned BERT on \textsc{SpartQA-Auto}{} required multi-step reasoning. Our hypothesis is that since solving \textsc{SpartQA-Auto}{} questions needs multi-step reasoning, fine-tuning BERT on \textsc{SpartQA-Auto}{} generally improves this capability of the base model.
All experiments, use the models defined in Sec. \ref{Sec:Architecture}. We use AdamW~\cite{loshchilov2017decoupled} with $2\times 10^{-6}$ learning rate and Focal Loss~\cite{lin2017focal} with $\gamma = 2$ for training all the models.
\section{Discussion}
\input{07-conclusion}
\section*{Acknowledgements}
This project is supported by National Science Foundation (NSF) CAREER award \texttt{\#}2028626 and (partially) supported by the Office of Naval Research grant \texttt{\#}N00014-20-1-2005.
We thank the reviewers for their helpful comments to improve this paper and Timothy Moran for his help in the human data generation.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,866 |
State Roundup, March 16, 2011
by Cynthia Prairie | Mar 16, 2011 | State Roundup | 0 |
RESTORING CUTS: House leaders want to reverse the bulk of the budget cuts to education proposed by Gov. Martin O'Malley, restoring funding for strapped school systems around the state after an angry backlash from teachers, students and parents, the Sun's Annie Linskey reports. The plan would shift $58.5 million back to schools by raising some fees and trimming elsewhere.
CUTS PROTEST: Tina Reed of the Annapolis Capital writes that hundreds of Anne Arundel County educators and school employees joined the thousands of union workers rallying in Annapolis Monday to protest proposed state budget cuts.
CHARTER SCHOOLS: Columnist Marta Mossburg writes in the Frederick News Post about the value of a charter school education to many Maryland children.
GOP BUDGET: House Republicans offered an alternative to O'Malley's $14 billion budget that would cut $621 million more with major changes in how the state allocates K-12 money and snipping at many state agencies and programs. But they'd also cut the sales tax and the corporate income tax, Julie Bykowicz blogs for the Sun. Here's their plan. And here's highlights of O'Malley's.
PENSION BILLS: Washington County Republican Del. Andrew Serafini yesterday presented several bills that would revamp the state's underfunded pension system, Andrew Schotz reports for the Hagerstown Herald Mail.
DEATH PENALTY REPEAL: Advocates for abolishing Maryland's death penalty made their case yesterday to a House of Delegates committee. But the effort is not likely to gain traction in the Senate, writes the Sun's Julie Bykowicz, since Senate President Mike Miller said, "There's no sentiment in the Senate" to debate a repeal.
The Post's John Wagner blogs that part of the reason remains an uneasy truce reached in 2009, under which higher evidentiary standards are required in capital cases.
John Rydell reports the story for WBFF-TV.
PAROLE QUESTION: The Baltimore Sun editorial board says that with O'Malley sitting on 50 parole recommendations – including seven for lifers – it's time for the state to take the governor out of the question of who gets paroled.
GAY UNIONS IN 2012: Senate President Mike Miller says passage of a same-sex marriage bill would be no sure thing next year in his chamber, even though it was approved 25 to 21 this year. Miller adds that this year, "the opposition was not engaged," writes John Wagner for the Post.
But legislators from Montgomery County are confident that the effort will benefit from downtime before it is introduced again next year, writes Sarah Breitenbach for the Gazette.
OOPS, WRONG VOTE: A brief flurry of confusion arose after the Senate vote on in-state tuition for illegal immigrants when Prince George's Sen. Anthony Muse inadvertently voted in favor of the bill. He subsequently changed his vote, blogs Ann Marimow for the Post.
OPEN GOVERNMENT: The Frederick News Post editorial board addresses Sunshine Week, which is dedicated to educating the public about the importance of open government and the dangers of excessive secrecy.
Meanwhile, the state compliance board found the Allegany County Commissioners were in violation of the Open Meetings Act when they held a closed session in December. It took a while, but the board responded on the side of PhDispatch.com's complaint, reports Kevin Spradlin of the Potomac Highlands Dispatch.
And Bryan Sears of Patch.com writes about the frustrations of trying to get an answer to a single question from a stalling Baltimore County Public Schools.
FUNDING JOCKEY CLUB: The Sun's Hanah Cho reports that some state lawmakers questioned yesterday why slot machine revenue allocated for racetrack improvements should be diverted to help fund the day-to-day operations of the Maryland Jockey Club, which reported losing millions at Laurel Park and Pimlico Race Course in recent years.
The club was pressing lawmakers yesterday to subsidize its operations through at least 2014 after releasing financial data that shows both of the state's thoroughbred tracks lost money in 2008 and 2009, Nick Sohr writes for the Daily Record.
BETTING ON SLOTS: With a bleak economy, no new revenues coming in and a reluctance to raise taxes, legislators are wagering that new gambling bills will help the state find more money, Megan Poinski writes for MarylandReporter.com.
SLOTS IN FREDERICK: State lawmakers are considering bringing slot machines to Frederick County, Meg Tully reports for the Frederick News Post. A proposal would allow slots within 5 miles of the intersection of I-70 and I-270 and increase the state's maximum number of machines from 15,000 to 17,500.
PRESERVING PARKS: Kevin Spradlin of the Potomac Highlands Dispatch writes that on Monday Gov. O'Malley touted the first-ever economic impact study on state parks and forests. Later at the forum at New Germany State Park, residents used that as leverage to explain to O'Malley that's one of many reasons why the state's natural resources – and its funding — should be preserved. Click on the videos on the right to hear local testimony.
HEADWINDS ON WIND: Still, the O'Malley administration's proposal to build off-shore wind turbines faced strong headwinds in the Senate Finance Committee yesterday. Senators spent almost 2½ hours grilling administration officials about the annual cost to consumers, Len Lazarick writes for MarylandReporter.com.
WIND NOT SO COSTLY: Maryland's Public Service Commission says a plan to make state utilities buy offshore wind energy will not cost ratepayers as much as some had feared, instead raising the average homeowner's electric bill 92 cents to $3 a month under O'Malley's offshore wind proposal, an AP story on WJZ.com reports.
MO'M PAL STANDS TO GAIN: The Post's Aaron Davis writes that O'Malley's childhood friend and right-hand man for a decade stands to gain from the governor's ambitious plan to subsidize development of an estimated $1.5 billion offshore wind farm.
BAY CLEANUP: The Sun's Timothy Wheeler writes that a federal study credits farmers with making progress in reducing their pollution to the Chesapeake Bay but says the vast majority need to do more to help the troubled estuary.
In an op-ed piece for the Sun, Megan Cronin writes Maryland needs to create a smarter fertilizer policy since fertilizer application to lawns, play areas, golf courses and parks now make up a quarter of the pollution problems in the Bay.
ABORTION PROTEST: Pushing bills to regulate clinics, hundreds of anti-abortion advocates rallied in Annapolis Monday, reports Jeff Newman of the Gazette.
EHRLICH JOB: Former Gov. and U.S. Rep. Bob Ehrlich is joining King & Spalding in Washington, D.C., as a senior counsel in the the law firm's government advocacy and policy practice, according to the Daily Record.
He'll be taking some of his former staff members with him as well, blogs the Sun's Annie Linskey.
JOHNSON VOWS FIGHT: The Post's Maria Glod and and Ovetta Wiggins write that former Prince George's County Exec Jack Johnson vowed yesterday to fight federal bribery charges, saying he had devoted "every minute, every effort" of his tenure to the county's residents.
Kelly McPherson reports on the new developments in the corruption scandal.
WA CO BUDGET: Washington County officials yesterday approved a plan expected to save the county anywhere from $500,000 to $1.9 million through a combination of retirement incentives and organizational restructuring, reports Heather Keels for the Hagerstown Herald Mail.
FUNDING WITHHELD: Lindsey McPherson of the Columbia Flier writes that Howard County Exec Ken Ulman is freezing federal funds that flow through to the county's Domestic Violence Center, and that he plans to withhold future funding to the nonprofit unless major leadership problems are resolved.
TAX SHORTFALL: Finance officials say the package of taxes and fees crafted by Baltimore city officials to meet a budget deficit last year is estimated to fall short of projected revenues by nearly $17 million, writes Julie Scharper for the Sun.
ETHICS VIOLATION? Baltimore Mayor Stephanie Rawlings-Blake has voted to approve more than $900,000 in deals with Johns Hopkins since her husband began working for one of its divisions late last year — a possible violation of the city ethics code, Julie Scharper reports for the Sun.
ANNAPOLIS BUDGET: Elisha Sauers of the Annapolis Capital writes that Annapolis Mayor Josh Cohen, determined to avoid raising property tax rates or taking away more city workers' jobs next year, unveiled a $86.2 million budget that relies on substantial fee increases.
PreviousDelegates betting on expanded gambling for more state revenues
NextCommittee hikes school aid to keep spending per pupil the same
Cynthia Prairie
cynthiaprairie@gmail.com
https://www.chestertelegraph.org/
Contributing Editor Cynthia Prairie has been a newspaper editor since 1979, when she began working at The Raleigh Times. Since then, she has worked for The Baltimore News American, The Chicago Sun-Times, The Prince George's Journal and Baltimore County newspapers in the Patuxent Publishing chain, including overseeing The Jeffersonian when it was a two-day a week business publication. Cynthia has won numerous state awards, including the Maryland State Bar Association's Gavel Award. Besides compiling and editing the daily State Roundup, she runs her own online news outlet, The Chester Telegraph. If you have additional questions or comments contact Cynthia at: cynthiaprairie@gmail.com
State Roundup, October 19, 2015
State Roundup, January 7, 2020
State Roundup, February 5, 2013
State Roundup, May 16, 2011 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,032 |
{"url":"https:\/\/ninamikuskova.website\/kat-6\/page-439418.html","text":"# \u5982\u4f55\u5728 IQ Option \u6ce8\u518c\u548c\u5b58\u6b3e\n\nWe use cookies to improve \u5982\u4f55\u5728 IQ Option \u6ce8\u518c\u548c\u5b58\u6b3e your experience. By continuing, you agree to our use of cookies. See our Cookie Policy .\n\n## We help students book their perfect room\n\n### Perfect Home Guarantee\n\nSelect the best student accommodation, providing safe & cozy living experience\n\n### 24-hour Service of Expert Booking Team\n\nGet expert advice and service around the clock from our multi-lingual team\n\n\"Student.com has certainly helped me a lot throughout the booking of accommodation. I can find them whenever needed. 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I was helped in every part of the process \u5982\u4f55\u5728 IQ Option \u6ce8\u518c\u548c\u5b58\u6b3e and I could find the perfect accommodation for me.\"\n\n### 1 \u5982\u4f55\u5728 IQ Option \u6ce8\u518c\u548c\u5b58\u6b3e million beds\n\nBook your perfect room among thousands of student properties.\u5982\u4f55\u5728 IQ Option \u6ce8\u518c\u548c\u5b58\u6b3e\n\n### 1,000 universities\n\nFind the best choice of student accommodation close to your university.\n\n### 400 cities\n\nExplore student life around the \u5982\u4f55\u5728 IQ Option \u6ce8\u518c\u548c\u5b58\u6b3e world with our unique neighborhood guides.\n\nStudent.com is \u5982\u4f55\u5728 IQ Option \u6ce8\u518c\u548c\u5b58\u6b3e the largest marketplace for student accommodation in the world, listing more than 1 million beds in over 400 cities worldwide. We offer a wide range of rooms in purpose-built student accommodation (PBSA). These are properties that were built specifically for students and are managed by well-known and \u5982\u4f55\u5728 IQ Option \u6ce8\u518c\u548c\u5b58\u6b3e trusted student accommodation providers. Whether you\u2019re looking for a shared flat or a private studio, you can find your ideal student home on Student.com. Our easy-to-use filters \u5982\u4f55\u5728 IQ Option \u6ce8\u518c\u548c\u5b58\u6b3e let you find a room for your specific needs and budget. With the map function you can see which housing options are close to your university, or in a part of town you\u2019d like to live in.\n\nBooking a room on Student.com is very easy. Check out the options in the city you\u2019ll be studying in, \u5982\u4f55\u5728 IQ Option \u6ce8\u518c\u548c\u5b58\u6b3e and choose your favourite accommodation. You can then ask for more information on it, and wait for one of our helpful booking consultants to contact you. If you don\u2019t want to wait, you can of course always contact us as well. 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Once you\u2019ve signed your contract and paid your deposit to the landlord, your booking is complete and secured.\n\n## python\u6570\u636e\u5206\u6790\uff1a\u7ed8\u56fe\u548c\u53ef\u89c6\u5316\u5165\u95e8\n\nmatplotlib\u662f\u4e00\u4e2a\u7528\u4e8e\u521b\u5efa\u51fa\u7248\u8d28\u91cf\u56fe\u8868\u7684\u684c\u9762\u7ed8\u56fe\u5305\uff08\u4e3b\u8981\u662f2D\u65b9\u9762\uff09\u3002\u8be5\u9879\u76ee\u662f\u7531John \u5982\u4f55\u5728 IQ Option \u6ce8\u518c\u548c\u5b58\u6b3e Hunter\u4e8e2002\u5e74\u542f\u52a8\u7684\uff0c\u5176\u76ee\u7684\u662f\u4e3aPython\u6784\u5efa\u4e00\u4e2aMATLAB\u5f0f\u7684\u7ed8\u56fe\u63a5\u53e3\u3002matplotlib\u548cIPython\u793e\u533a\u8fdb\u884c\u5408\u4f5c\uff0c\u7b80\u5316\u4e86\u4eceIPython shell\uff08\u5305\u62ec\u73b0\u5728\u7684Jupyter notebook\uff09\u8fdb\u884c\u4ea4\u4e92\u5f0f\u7ed8\u56fe\u3002matplotlib\u652f\u6301\u5404\u79cd\u64cd\u4f5c\u7cfb\u7edf\u4e0a\u8bb8\u591a\u4e0d\u540c\u7684GUI\u540e\u7aef\uff0c\u800c\u4e14\u8fd8\u80fd\u5c06\u56fe\u7247\u5bfc\u51fa\u4e3a\u5404\u79cd\u5e38\u89c1\u7684\u77e2\u91cf\uff08vector\uff09\u548c\u5149\u6805\uff08raster\uff09\u56fe\uff1aPDF\u3001SVG\u3001JPG\u3001PNG\u3001BMP\u3001GIF\u7b49\u3002\u9664\u4e86\u51e0\u5f20\uff0c\u672c\u4e66\u4e2d\u7684\u5927\u90e8\u5206\u56fe\u90fd\u662f\u7528\u5b83\u751f\u6210\u7684\u3002\n\n## \u5229\u7528STATA\u521b\u5efa\u7a7a\u95f4\u6743\u91cd\u77e9\u9635\u53ca\u7a7a\u95f4\u675c\u5bbe\u6a21\u578b\u8ba1\u7b97----\u547d\u4ee4.pdf\n\n** \u521b\u5efa\u7a7a\u95f4\u6743\u91cd\u77e9\u9635\u4ecb\u7ecd * \u8bbe\u7f6e\u9ed8\u8ba4\u8def\u5f84 cd C:\\Users\\xiubo\\Desktop\\F182013.v4\\F101994\\sheng ** \u521b\u5efa\u65b0\u6587\u4ef6 *shp2dta:reads a shape (.shp) and dbase (.dbf) file from disk and converts them into Stata datasets. *shp2dta: \u8bfb\u53d6 CHN_adm1 \u6587\u4ef6 *CHN_adm1 \uff1a\u4e3a\u5df2\u6709\u7684\u5730\u56fe\u6587\u4ef6 *database (chinaprovince) \uff1a\u8868\u793a\u521b\u5efa\u4e00\u4e2a\u540d\u79f0\u4e3a\u201c chinaprovince \u201d\u7684 dBase \u6570\u636e\u96c6 *database(filename) \uff1a Specifies filename of new dBase dataset *coordinates(coord) \uff1a\u521b\u5efa\u4e00\u4e2a\u540d\u79f0\u4e3a\u201c coord \u201d\u7684\u5750\u6807\u7cfb\u6570\u636e\u96c6 \u5982\u4f55\u5728 IQ Option \u6ce8\u518c\u548c\u5b58\u6b3e \u5982\u4f55\u5728 IQ Option \u6ce8\u518c\u548c\u5b58\u6b3e *coordinates(filename) \uff1aSpecifies filename of new coordinates dataset *gencentroids(stub) \uff1aCreates centroid variables *genid(newvarname) \uff1aCreates unique id variable for database.dta shp2dta using CHN_adm1,database (chinaprovince) coordinates(coord) genid(id) gencentroids(c) ** \u7ed8\u5236 2016 \u5e74\u4e2d\u570b \u5982\u4f55\u5728 IQ Option \u6ce8\u518c\u548c\u5b58\u6b3e GDP \u5206\u5e03\u5716 *spmap:Visualization of spatial data *clnumber(#):number of classes *id(idvar):base map polygon identifier( \u8bc6\u522b\u7b26\uff0c\u58f0\u660e\u53d8\u91cf\u540d\uff0c\u4e00\u822c\u4ee5\u5b57\u6bcd\u6216\u4e0b\u5212\u7ebf\u5f00\u5934\uff0c\u5305\u542b \u6570\u5b57\u3001\u5b57\u6bcd\u3001\u4e0b\u5212\u7ebf ) *_2016GDP \uff1a\u53d8\u91cf *coord: \u4e4b\u524d\u521b\u5efa\u7684\u5750\u6807\u7cfb\u6570\u636e\u96c6 spmap _2016GDP using coord, id(\u5982\u4f55\u5728 IQ Option \u6ce8\u518c\u548c\u5b58\u6b3e id) clnumber(5) * \u66f4\u6539\u53d8\u91cf\u540d rename x_c longitude rename y_c latitude ** \u751f\u6210\u8ddd\u79bb\u77e9\u9635 *spmat: \u7528\u4e8e\u5b9a\u4e49\u4e0e\u7ba1\u7406\u7a7a\u95f4\u6743\u91cd\u77e9\u9635 *Spatial-weighting matrices are stored in spatial-weighting matrix objects (spmat objects). *spmat objects contain additional information about the data used in constructing spatial-weighting matrices. *spmat objects are used in fitting spatial models; see spreg (if installed) and spivreg (if installed). *idistance:( \u4ea7\u751f\u8ddd\u79bb\u77e9\u9635 )\u5982\u4f55\u5728 IQ Option \u6ce8\u518c\u548c\u5b58\u6b3e create an spmat object containing an inverse-distance matrix W * \u6216 contiguity:create an spmat object containing a contiguity matrix W *idistance_jingdu: \u547d\u540d\u540d\u79f0\u4e3a\u201c idistance_jingdu \u201d\u7684\u8ddd\u96e2\u77e9\u9663 *longitude: \u4f7f\u7528\u7ecf\u5ea6 *latitude: \u4f7f\u7528\u7eac\u5ea6 \u5982\u4f55\u5728 IQ Option \u6ce8\u518c\u548c\u5b58\u6b3e *id(id): \u4f7f\u7528 id *dfunction(function[, miles]):( \u8bbe\u7f6e\u8ba1\u7b97\u8ddd\u79bb\u65b9\u6cd5 )specify the distance function. *function may be one of euclidean (\u5982\u4f55\u5728 IQ Option \u6ce8\u518c\u548c\u5b58\u6b3e default), dhaversine, rhaversine, or the Minkowski distance of order p, where p is an integer greater than or equal to 1. *normalize(row): \uff08\u884c\u6807\u51c6\u5316\uff09 specifies one of the\n\n## How to \u5982\u4f55\u5728 IQ Option \u6ce8\u518c\u548c\u5b58\u6b3e put Angstrom\n\nCan anyone tell how to put the capital Angstrom to express units. I have been looking for a package but I haven't found one yet.\n\n\\AA is invalid in math mode. In math mode, put it in \\text<> as $$\\text$$ , or see some solutions at this question\n\n@ydhhat If you use \\text , the formatting of the surrounding text will bleed through to math mode. This most often happens with italics in theorem \u5982\u4f55\u5728 IQ Option \u6ce8\u518c\u548c\u5b58\u6b3e \u5982\u4f55\u5728 IQ Option \u6ce8\u518c\u548c\u5b58\u6b3e statements. You probably would want \\textnormal to rest the formatting, or \\textup if you want to clear italics\/small caps but use bold math in a header. Or, use siunitx .","date":"2023-03-23 10:37:10","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.20652461051940918, \"perplexity\": 14458.738090555384}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296945144.17\/warc\/CC-MAIN-20230323100829-20230323130829-00742.warc.gz\"}"} | null | null |
{"url":"http:\/\/taoofmac.com\/space\/blog\/2004\/08\/13","text":"# Take Two Tablets and Call Me in a Month\n\nYes, everyone is going ape over the Mac tablet design that has (re)surfaced. But then, people went ape over every single iteration of the iMac, the iPod and pretty much any other Apple product that kicked off (or hinted at) a new form factor, so that's only to be expected.\n\nWhat people seem to be forgetting is that, if it exists at all (and if it is ever brought to market, which is the biggest issue - tablet designs or mini-PCs like the OQO are a dime a dozen) it's bound to cost more or less the same as an iBook (a little less in fact, if it's really only 8 inch wide).\n\nFab costs are likely to be, say, around 60-70% that of an iBook, and that's estimated mostly on parts (and heavily slanted due to the smaller screen size, which is a major factor in component costs for one of these things).\n\nNow assuming all this, placing it at US$999 would be on top of the iBook price range. Higher would be pointless, unless the goal was to only sell a couple of hundred. (There is also the option of this being some sort of new \"iMac Mini\", which would bump up the price point significantly, but let's run with the tablet idea for a bit.) Okay, for the sake of argument, let's make it US$799, assuming that if Apple followed a linear pricing\/product placement strategy (which it often doesn't) it would have to be somewhere in between an iPod and an iBook.\n\nYep. That's the real question. I guess it would depend heavily on what it could do besides acting as a remote Mac display or an iTunes remote.\n\nAnd I bet that, like most of Apple's modern gear, it would be squarely aimed at the home market, and integrated into the Apple \"digital entertainment hub\" - so no useful corporate functionality except wireless web surfing and note taking, which hardly justify even a quarter the price of a Windows Tablet PC.\n\nSo, home use would mean surfing the net, streaming music (and maybe video, if the new H.264 codec can run on it without flattening the batteries), maybe browsing iPhoto albums or (if Inkwell is good enough) sending short notes via e-mail. Paired with an Airport Express, it would be a pretty complete home entertainment system.\n\n(By the way, the \"it might also act as a phone\" rumors floating around are nothing short of ridiculous. If the thing is going to act as a phone, the tablet form factor is one of the worst possible imaginable.)\n\nAssuming it exists at all (always a nice thing to keep in mind), would you pay US\\$799 for a gadget that (as far as we know) has no compelling application to foster its adoption?\n\nMore to the point, I don't see myself spending that much money for yet another Apple one-off (or \"classic model\", as initial revisions are nearly always dubbed a few years down the road) that would most likely be either dead or obsolete in a year's time.\n\nNow go back and think of it actually being some sort of \"iMac Mini\" costing even more, and you see why I'm not exactly keen on it.\n\nOf course, if might be as wondrous as the Newton (which I would probably still use today if I had kept one), and all the points I've made above will wither away in a flood of coolness...","date":"2017-02-23 23:01:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5410916209220886, \"perplexity\": 1845.5131377649504}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-09\/segments\/1487501171251.27\/warc\/CC-MAIN-20170219104611-00223-ip-10-171-10-108.ec2.internal.warc.gz\"}"} | null | null |
Markets started the week in mixed fashion, with the Nasdaq closing slightly green, while the DOW, S&P, and Russell closed in the red. The S&P traded in a tight range on Monday, something we have not seen in quite sometime. Overnight, Asia markets closed mixed, with the Nikkei in the red while China markets closed…Please subcribe to read more. | {
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Q: remember radio option after submission error I have a contact form created with formtools with several radio buttons. I need to make the form remember the option on the selected radio button, even though there is a error in the post submission.
var btn = document.getElementById('submit');
if (btn) {
btn.addEventListener('click', function(){
var dt = document.querySelector('[name="form_tools_form_id"]:checked').value;
alert('&option=' + dt)
});
}
<form action="check.php" id="contact" method="post">
<div class="radioboxes"><strong id="top-more">Options</strong><br>
<span class="othertopic" id="wwus"> <font>Please select one option</font></span>
<div id="top-wwus"><input id="topic_252" name="form_tools_form_id" type="radio" value="252"> <label for="topic_252">Recruitment</label><br>
<input id="topic_259" name="form_tools_form_id" type="radio" value="259"> <label for="topic_259">Requests</label><br>
<button name="submit" type="submit">Submit</button> </div> </div>
</form>
A: You didn't gave your submit button the id submit which you were using in your event listener and you have to use .preventDefault() so that it alerts otherwise it will only redirect. Store the value in localStorage
var btn = document.getElementById('submit');
if (btn) {
btn.addEventListener('click', function(e){
console.log(dt)
e.preventDefault();
var dt = document.querySelector('[name="form_tools_form_id"]:checked').value;
alert('&option=' + dt)
localStorage.setItem('radio',dt);
});
}
<form action="check.php" id="contact" method="post">
<div class="radioboxes"><strong id="top-more">Options</strong><br>
<span class="othertopic" id="wwus"> <font>Please select one option</font></span>
<div id="top-wwus"><input id="topic_252" name="form_tools_form_id" type="radio" value="252"> <label for="topic_252">Recruitment</label><br>
<input id="topic_259" name="form_tools_form_id" type="radio" value="259"> <label for="topic_259">Requests</label><br>
<button name="submit" type="submit" id="submit">Submit</button> </div> </div>
</form>
A: If you reload the page then use cookies or session/locale storage
document.cookie = "value = " + document.querySelector('[name="form_tools_form_id"]:checked').value + ";";
when the page reload, check if it is empty or not
var x = document.cookie;
then check the checkbox
document.querySelector('[name="form_tools_form_id"]:checked').checked = true;
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,520 |
Swatted wasps, spilled Coke, & Herschel hurdling
Checkout every week at UGASports.com during this football season whereby I long to hear the voice of the late, legendary Larry Munson. I recently was fortunate enough to have gained access to many Munson radio-called games spanning a couple of decades or so. Therefore, I am posting highlights I compiled from a Munson broadcast of a Georgia game from yesteryear. Adding only photos taken from the particular game, and some commentary, each video will be associated with the Bulldogs' upcoming opponent, or the date for which it is posted.
Entering Georgia's game at Ole Miss in 1981, the once-beaten Bulldogs were favored by only just over a touchdown in a road affair which was supposed to be rather contested. Instead, during a dreary day 35 years ago--so dreary that planes could not fly into Oxford--Georgia and Ole Miss featured the killing of wasps and spilled Coke in the broadcast booth and, on the gridiron, one of his best collegiate performances by the legendary Herschel Walker (and, the greatest 6-yard run in the history of the sport), resulting in a 30-point blowout by the Bulldogs over their host:
One is the Loneliest Number...
"One" is for (L to R) DAVE O'BRIEN, Larry Munson's lone replacement for a 41-season stretch; KNOWSHON MORENO's 1-yard celebratory TD vs. Florida in 2007; and, the one time brothers faced off against one another in a bowl game--a historical moment which nearly didn't occur.
Only one day remains until Georgia kicks off the season against North Carolina. Stats guru Dave McMahon and I demonstrate six ways why "1" is relative and unique to UGA football. Check us out everyday at UGASports.com... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 248 |
CHRIS PHILPOTT
THIS IS NOT HIM. THIS IS HIS WEBSITE.
The Latest From Chris
On The Box/Stuff
Category: Stuff Archive Page 1 of 2
TV Review Sunny Skies S1 Ep02
By cphilpott
In Stuff Archive
The second episode of Sunny Skies – the new local comedy from Mike Smith and Paul Yates, starring Oliver Driver and Tammy Davis as half-brothers thrown together after the death of the dad neither of them knew, and tasked with running the holiday park he owned – aired on Friday night (TV3, 8pm), entertaining throughout and proving that the brilliant first episode wasn't a fluke; this is the best new local comedy of the season. There is just so much to love about this show.
(Spoilers from the first two episodes of Sunny Skies follow.)
For a start, the setting is near perfect. One of the major complaints – repeated again in the comments last week – is that local shows are all set in Auckland, which is a fair statement; most shows are filmed in Auckland, because that's where most production companies are (though I think shows like The Blue Rose or Agent Anna are painted broadly enough that they aren't just about Aucklanders, despite the obvious setting).
Skies bucks the trend by taking things rural, focusing the action on (and around) the Sunny Skies Holiday Park and making fun of that most Kiwi of traditions. Heck, I'm the most indoor person there is, and even I've been camping on more than one occasion. And with the action filmed at the real life Sandspit Holiday Park, in Warkworth, there is plenty to relate to.
Sunny Skies looks and feels like a real holiday park, whether it's the seaside location, or the grunt of rubbish trucks picking up litter, or the family who brought way too much equipment with them. As much as I've enjoyed The Blue Rose and Agent Anna, getting out into the fresh air, with a little sun on our backs, is a nice change of pace. Skies might be the best case so far for why productions should head out of the big city more often, whether it needs to or not. It feels like true Kiwiana.
The cast is brilliant, too. From Driver and Davis as our fantastic leads Oscar and Deano, to Molly Tyrell as Deano's daughter Charlotte and Morgana O'Reilly as sassy manager Nicki, to Erroll Shand as onsite handyman Gunna, to Ian Mune and Mick Innes as a pair of locals ("we're not gay … we're homosexuals"), Smith and Yates have brought together the most likeable cast on television. There isn't a weak link amongst them.
I'm still not sure about the shows' sense of humour though. The first two episodes have had their share of laugh-out-loud moments – the meeting with the lawyer/real estate agent, the dialogue between Oscar and rubbish collector Sione – and plenty of little moments of hilarity (the lawyers reaction to spying a stuffed possum in the meeting room, for example).
But, if I'm being fair, Sunny Skies has only been sporadically funny. Plenty of moments have worked. Plenty of moments have totally missed, too – predictable gags, like blowing up the boat with Oscar and Deano's fathers ashes on board, or silly gags, like Gunna perfecting his "freezzbee" technique against a wall.
Luckily, the show makes up for it with that great setting and that perfect cast, and just enough pathos – there is something loveable about the tale here, even if a short-run local comedy might have trouble doing it justice. And even though it isn't making me laugh on a consistent basis (yet), I'm in on Sunny Skies. It's not the best Kiwi comedy ever, but it's the best new Kiwi comedy this year. And I reckon it might get even better, before it's done.
TV REVIEW The Blue Rose S1 Ep02
In Herald Archive, Stuff Archive
Last week's premiere of The Blue Rose – the latest drama from Rachel Lang and James Griffin, the team behind Outrageous Fortune – ended abruptly. In fact, it left my spidey senses* tingling, so much so that I held off on reviewing the first episode. It just didn't feel like the right point at which to review the show.
(Warning: spoilers from the first two episodes of The Blue Rose follow.)
I'm glad I did hold off on that review, because last night's second episode tied up much of what took place last week and, paired with the first episode, turned what was a muddled, abrupt premiere into an intriguing two-parter (whether that was the intent or not) that introduces us to this world and establishes the characters and their roles within it.
After the suspicious death of Rose, a secretary at a law firm, the temp assigned to replace her – Jane, played by the delightful Antonia Prebble – works with Rose's best friend Linda (Siobhan Marshall, playing the role with a bit of sass) to uncover the truth behind Rose's death, uncovering a conspiracy and a group of co-workers fighting for the little guy.
It's an interesting idea for a show, weaving together a longer serial arc while leaving room for the characters to take on smaller, weekly missions like getting rid of an ex-husband or returning the life savings to a hurting family.
The writing is clever and the first two episodes are tightly paced, but we'd expect no less from Lang and Griffin. The pair has worked on a number of entertaining series at this point, and they know how to put together a great show.
More than just entertaining, the writing here is efficient, quickly getting to know the main characters Jane and Linda (as well as lawyer Simon, played well, albeit subtly, by Matt Minto), while putting in the foundations for a great story as the season goes on (and also popping in a few easter eggs, like the connection to The Smiths' singer Morrissey).
The first two episodes are also technically brilliant. Director Mark Beesley has done a great job with the script, taking the implication in the writing and realizing it as a taut thriller, full of suspense; many parts of the first episodes invoked the likes of 24 or Homeland, but with a uniquely Kiwi touch.
The music is fantastic too, maintaining the suspenseful vibe, courtesy of former Supergroove singer Karl Steven.
The question is whether the writers can keep it up for the rest of the season, and perhaps beyond. I have faith in Lang and Griffin, and I think they'll do a typically great job crafting storylines going forward. And while we can see a few of the angles they might seek to explore – a legal battle over Rose's daughter with douchebag ex-husband Grant, and Jane's boyfriend developing a crush on Linda, in addition to the ongoing conspiracy we've already been introduced to – there is always a risk that the story might unfold too slowly, that The Blue Rose could be caught meandering during later episodes.
We can only wait and see how things turn out, and I'm looking forward to seeing where the show goes next. But regardless of what happens from here, I think it's safe to say that The Blue Rose is one of the year's most intriguing new shows, and another success for Lang, Griffin, and the team at South Pacific Pictures.
(*) That reference to "my spidey senses" might be the goofiest thing I've ever written.
TV REVIEW The Radio S1
What is the nicest thing I can say about The Radio, which started on Friday night after the new season premiere of 7 Days? I guess things can only get better from here. I mean, they sure couldn't get much worse.
(Warning: spoilers from Friday's The Radio and Sunny Skies follow.)
To be honest, I'm actually surprised by just how bad The Radio was: easy, predictable joke after easy, predictable joke, most of which took potshots at the mainstream radio industry (with the exception of a few decent programs, a pretty easy target to start with), all while a live studio audience half-heartedly chuckled along at the supposedly right moments.
Paul Ego and Jeremy Corbett star as fictionalized versions of themselves who are employed by a radio station, named The Radio, to front the breakfast show, after moving from Hevvy FM and Lite FM, respectively. Urzila Carlson appears as their receptionist, Urzila. The Edge FM DJ (and occasional 7 Days panelist) Vaughan Smith makes a couple of brief appearances as the nameless, faceless station manager.
I actually don't have a problem with the cast; Ego has gotten better every year on 7 Days, Corbett is at least likeable, even if many of his pre-written jokes as 7 Days host are more groanworthy than laughable, and Carlson and Smith are entertaining as panelists who often provide the highlights on the hit comedy show.
Sadly, they don't have much to work with here. One lengthy sequence has Ego and Corbett debating whose name should be first – Ego argues that the short name goes first, like with eggs and bacon (see, it's funny because nobody says that), while Corbett believes he is the bigger name star.
Another sequence has them choosing the music for their show – a laughable proposition, since I think it's fairly common knowledge that music decisions on radio aren't made by DJs – and flicking between the heavy sounds of Metallica and the light sounds of Shania Twain, before compromising with Nickelback.
Maybe the problem is just that the radio industry isn't good comedy terrain? Several times during the show – some of the decision making between the pair, the complaint that there isn't enough music, those increasingly lame station identification sound bites ("The Radio, making your day 28% better") – I found myself just nodding at the screen.
I can't say I'd be surprised if some of the country's bigger, middle-of-the-road stations actually operated this way.
I suppose the show could better. I mean, anything is possible. You'd have to think Ego and Corbett, who also serve as writers (the show is based off a successful live show the pair performed during last year's comedy festival), have used up most of the obvious jokes. It's not completely ludicrous to suggest that it might get better. Though I'm not confident it will.
Look, I get that TV3 wants to capitalize on the success of 7 Days by using the various cast members and panelists in other projects. Ben Hurley and Steve Wrigley have a show coming up later in the year, too. But this is a huge miss, invoking the mid-1990s vibe of Melody Rules, instead of capitalising on the successes local comedy has enjoyed in recent times (including the vastly enjoyable Sunny Skies, which aired earlier in the evening*).
Furthermore, the shows need to be smarter; I've written before that 7 Days is leading the charge in a kind of "golden era" of Kiwi comedy on television – based on a single episode, and assuming it doesn't improve, The Radio can only be seen as a massive step backwards for local, scripted comedy. It could definitely hurt local comedy in the long run.
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Miss Something? Select Month May 2019 June 2018 May 2018 April 2018 March 2018 February 2018 January 2018 December 2017 November 2017 October 2017 September 2017 January 2014 February 2013 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,315 |
Q: laravel ssh - Expected SSH_FXP_VERSION I am using the laravel Task functions to run some code using phantomjs and then using the output. The code is executed using a artisan command.
This is a part of the code I use:
private static function phantomjs($file, $c, $a)
{
SSH::run(array("phantomjs " . base_path("phantomjs/" . $file)), function($line)
{
$json = json_decode($line);
DataHelper::saveJson($json);
}
}
Whenever I run the command I get the following error on this piece of code:
[ErrorException]
Expected SSH_FXP_VERSION
If I output the command and run it in a terminal myself it works just fine.
Does anyone know if I need to change some configs or if my code is incorrect? I have already been able to do ssh for executing nodejs scripts.
I am using Debian 7.6 on my server.
Thanks for helping. :)
A: for me it was a problem with the path of sftp-server, instead of /usr/lib/openssh/sftp-server i used /usr/libexec/openssh/sftp-server
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 765 |
The Southlake DPS Records Division maintains records generated from police action or response of the Southlake Police, including: arrest, offense, and accident reports.
Email a request to the Records Coordinator.
Fax a request to the Records Coordinator at (817) 748-8375.
The City of Southlake's Police Department has entered into an agreement with PoliceReports.US to provide public online access for accident reports occurring after July 1, 2008. Please allow three to five business days from the time of the accident to access your report. If the report you are looking for is unavailable and it has been five or more business days please contact DPS Records at (817)748-8359. Once posted online, accident reports will remain available via the internet.
If you cannot locate the report you need, call the Southlake Police Records Division at (817) 748-8359 for assistance. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,360 |
El es un santuario sintoísta existente en Tokio, Japón. El nombre original del santuario se escribió como 靖國神社. Al mes de octubre de 2004, su Libro de las Ánimas contiene un listado de los nombres de 2.466.532 soldados japoneses y coloniales (27.863 coreanos y 21.181 taiwaneses) caídos en conflictos bélicos, entre los que se encuentran catorce criminales de guerra de primer orden.
Historia
El edificio se erigió en junio de 1869 a instancias del Emperador Meiji en recuerdo a los caídos de la guerra Boshin. Llamado primeramente Tōkyō Shōkonsha (東京招魂社), posteriormente sería rebautizado como Yasukuni Jinja en 1879. El santuario ha estado llevando a cabo rituales sintoístas (Gōshisai) para albergar los kami (espíritus) de los soldados japoneses y coloniales (coreanos y taiwaneses) caídos desde entonces en las guerras.
Tras la derrota de los japoneses en la Segunda Guerra Mundial, en septiembre de 1945, las autoridades de la ocupación aliada ordenaron que Yasukuni optara por convertirse en una institución no religiosa dependiente del gobierno o bien mantuviese su carácter religioso desvinculada del gobierno. Yasukuni eligió esto último. Desde entonces su financiación es totalmente privada.
Kami
A continuación se hace una relación de los kami (formalmente 祭神 saishin, contados como 柱 hashira) albergados en el santuario Yasukuni:
Guerra Boshin (guerra civil, 1867–68): 7.751
Rebelión Satsuma (1877): 6.971
Expedición a Taiwán (sofocamiento de la rebelión en Taiwán contra la ocupación japonesa, 1874): 1.130
Primera Guerra Sino-japonesa (invasión de Corea y China, 1894–95): 13.619
Levantamiento de los bóxers (invasión de China, 1901): 1.256
Guerra ruso-japonesa (conflicto con Rusia e invasión de China y Corea, 1904–05): 88.429
Primera Guerra Mundial (invasión de China y Mongolia, 1914–18): 4.850
Incidente de Jinan (invasión de China, 1928): 185
Incidente de Mukden (invasión de Manchuria, 1931): 17.176
Incidente del puente de Marco Polo (invasión de China, 1937–1945): 191.243
Segunda Guerra Mundial (conflicto con las Potencias aliadas e invasión de diversas colonias europeas y China): 2.133.885
Polémica
Para la República Popular China, Corea del Norte, Corea del Sur y otros países víctimas de la agresión militar japonesa en el siglo XX, el santuario ha entrado a formar parte de la polémica como símbolo del militarismo japonés de la Segunda Guerra Mundial y como epicentro simbólico del nacionalismo japonés proto-fascista.
Un folleto editado por el santuario reza: "la guerra es algo verdaderamente deplorable pero fue necesaria para que pudiésemos preservar la independencia del Japón y para prosperar junto a nuestros vecinos asiáticos". El santuario gestiona un museo sobre la historia del Japón en honor de los soldados que lucharon por el País del Sol Naciente, venerándolos como kami. La página web en inglés de la institución sostiene que "el sueño japonés de una Gran Asia Oriental era una necesidad histórica y fue una meta perseguida por los países de Asia". El website en japonés afirma que "las mujeres de bienestar no fueron obligadas por el Imperio Japonés a prestar servicios. Los coreanos no fueron obligados a cambiar sus nombres por nombres japoneses." El santuario también pone de relieve atrocidades cometidas por las Fuerzas Aliadas, como el hundimiento del Tsushima Maru, un transporte torpedeado y echado a pique causando la muerte de 1500 personas, de las cuales 700 eran niños escolares de primaria. Un vídeo documental muestra a los visitantes del museo la conquista japonesa de Asia oriental previa a la Segunda Guerra Mundial como un esfuerzo para salvaguardar a Asia del avance imperialista de las potencias occidentales.
Unos 1000 prisioneros ejecutados por crímenes de guerra son aquí venerados. El hecho en sí no era objeto de controversia política, ya que en Yasukuni se recordaba a todos los muertos del Japón en las guerras. Sin embargo, el 17 de octubre de 1978 empezó discretamente a venerarse como "Mártires de Showa" (昭和殉難者 Shōwa junnansha) a 14 personas a las que el Tribunal Militar Internacional para el Lejano Oriente consideró criminales de guerra de la Clase A, entre quienes se cuenta Hideki Tōjō. A continuación se listan las sentencias correspondientes a cada uno de ellos.
Ejecución por ahorcamiento:
Hideki Tōjō, Seishiro Itagaki, Heitaro Kimura, Kenji Doihara, Iwane Matsui, Akira Mutō, Kōki Hirota
Cadena perpetua:
Yoshijirō Umezu, Kuniaki Koiso, Kiichiro Hiranuma, Toshio Shiratori
20 años de prisión (fallecido durante el cumplimiento de la sentencia):
Shigenori Tōgō
Fallecidos antes de alcanzarse fallo judicial (por enfermedad):
Osami Nagano, Yōsuke Matsuoka
Cuando la noticia fue difundida por los medios de comunicación el 19 de abril de 1979, se dio inicio a una polémica que dura hasta el presente. El santuario ofendió a muchos al llevar a cabo una defensa abierta de los criminales de guerra. El folleto arriba mencionado sostiene: "algunas de las 1.068 personas que fueron erróneamente acusadas de ser criminales de guerra por el tribunal aliado se veneran aquí". La versión inglesa de la página web del santuario se refiere a estas 1.068 personas como "aquellos que fueron enjuiciados cruel e injustamente como criminales de guerra por un tribunal de pantomima de las Fuerzas Aliadas". A raíz del escándalo de los criminales de guerra producido en 1979, el Emperador Hirohito dejó de visitar el santuario y los emperadores posteriores se han mantenido en esa línea desde entonces. Sin embargo, existen sectores de la sociedad japonesa que abogan públicamente por las visitas del Emperador al santuario, incluyéndose en estos al Gobernador de Tokio Shintarō Ishihara, que el 15 de agosto de 2004 manifestó su deseo de que el Emperador Akihito empezara a visitar Yasukuni.
La polémica que rodea a lo relacionado con el santuario no ha dejado de estar presente en la política interior japonesa, así como en las relaciones del Japón con otros países de la región desde 1978. Tres primeros ministros japoneses suscitaron el escándalo al visitar el santuario desde entonces: Yasuhiro Nakasone en 1985, Ryutaro Hashimoto en 1996, y sobre todo Jun'ichirō Koizumi, que lo ha visitado seis veces de 2001 a 2006. Las visitas realizadas por primeros ministros al santuario normalmente son respondidas con la emisión de declaraciones de condena oficial por los distintos países de la región, sobre todo por la República Popular China y Corea del Sur, que ven en tales actos un intento de legitimar el militarismo japonés. Las visitas al santuario también suscitan el debate nacional sobre el papel de la religión frente al Estado: algunos políticos del Partido Democrático Liberal insisten en que las visitas están amparadas constitucionalmente por el derecho a la libertad religiosa y que es apropiado que los legisladores presenten sus respetos a los caídos en la guerra. Sin embargo, se oponen a cualquier propuesta de institución en memoria de los militares japoneses muertos que sea de carácter no religioso, de manera que quienes quieran honrarlos no tengan que visitar el Santuario Yasukuni. El santuario rechaza asimismo la construcción de una alternativa no religiosa sosteniendo que: "El Santuario Yasukuni debe ser el único emplazamiento conmemorativo para los militares caídos del Japón".
La mayoría de los japoneses que visita el santuario lo considera un acto de rememoración, no de veneración. El primer ministro Koizumi afirma que sus polémicas visitas tratan de que no haya más guerras en las que participe Japón. Hasta la fecha ha sido China quien ha mantenido la crítica más estentórea al santuario aunque, dado que el problema de Yasukuni está fuertemente unido a la política china y a su vez mediatizado a través del filtro de la censura gubernamental, no toda la opinión pública en China está necesariamente al corriente de que el santuario ya existía con anterioridad a la Segunda Guerra Mundial o de que también rinde honores a soldados nacidos en una Corea y un Taiwán colonizados. Muchos japoneses consideran que entra en juego una diferencia cultural. Los chinos, a diferencia de los japoneses, no creen que los crímenes cometidos por una persona queden absueltos tras su muerte. En vista de que China y Corea critican las visitas de Koizumi, este responde:
En su primera visita al Japón después de cumplir su mandato en febrero de 2003, el expresidente de Corea del Sur Kim Dae Jung se mostró abiertamente crítico con las visitas al santuario. Kim propuso que los 14 criminales de guerra de Clase A, que son venerados en Yasukuni junto al resto de caídos del Japón, sean trasladados a otro emplazamiento, y expresó: "Si esta propuesta se lleva a cabo, no me manifestaré en contra de las visitas a Yasukuni (por parte de Koizumi u otros líderes japoneses)". Kim hizo notar que Koizumi, en una reunión en Shanghái en 2001, prometió examinar la posibilidad de construir un nuevo emplazamiento conmemorativo en reemplazo del Santuario Yasukuni y permitir que todos estuvieran autorizados a practicar la veneración de los difuntos en ese lugar.
Acontecimientos de actualidad
El santuario anunció que su sitio web oficial ha estado sometido a ataques distribuidos de denegación de servicio provenientes de un dominio chino desde septiembre de 2004. Por esa razón es posible que los usuarios tengan dificultades para acceder al sitio web.
En mayo de 2005, poco después de las manifestaciones antijaponesas sobre los libros escolares de historia de este país que tuvieron lugar en Extremo Oriente, China reprochó a Japón sus continuas referencias al polémico santuario durante una visita de la vicepresidenta china Wu Yi. Wu Yi interrumpió abruptamente la visita y regresó a China antes de una reunión con el primer ministro japonés Jun'ichirō Koizumi que estaba en la agenda del viaje. Esto fue interpratado como una reacción oficial a unas declaraciones que Koizumi había hecho el día anterior a la llegada de la Sra. Wu, en las cuales expresaba que los países extranjeros no deberían inmiscuirse en lo concerniente al santuario. La visita de Wu Yi era un intento por distender las relaciones entre los dos países, y de hecho la Sra. Wu había previsto pedir a Koizumi que dejara de visitar el santuario.
En junio de 2005, un miembro retirado del Partido Democrático Liberal propuso que los 14 criminales de guerra de Clase A fueran trasladados a un sitio separado. Sin embargo, los sacerdotes sintoístas se negaron a esto citando las leyes de libertad religiosa existentes bajo la Constitución Japonesa.
También en junio de 2005 un grupo de indígenas taiwaneses encabezados por el político indígena Kao-Chin Su-mei trató de hacer una visita al santuario con la ayuda del "Consejo Católico del Japón para la Justicia y la Paz". Representaban a nueve grupos étnicos de Taiwán cuyos antepasados son venerados en Yasukuni. Su intención era solicitar pacíficamente que sus parientes fuesen retirados del santuario e invocar plegarias para el retorno de los espíritus de sus ancestros. Sin embargo, la policía y manifestantes japoneses impidieron que accedieran al santuario.
Un grupo de más de cien ultranacionalistas japoneses organizó una manifestación para obstaculizar su llegada al santuario e impedir que llevasen a cabo ritos religiosos destinados a llamar a los espíritus. La policía japonesa permitió que los manifestantes permanecieran en el lugar y les ayudó a bloquear el paso de los taiwaneses que iban a bajar de sus autobuses interponiendo medidas para evitar el choque entre los dos grupos. Después de hora y media los taiwaneses desistieron de su propósito.
Su-mei y su grupo presuntamente habrían recibido amenazas de muerte en relación con su visita a Yasukuni, provocando que el gobierno chino pidiera a las autoridades japonesas que garantizasen su seguridad durante su estancia en Japón.
El 27 de junio de 2005, el Gobernador de Tokio Shintarō Ishihara declaró a Kyodo News "si el Primer Ministro no va (a Yasukuni) este año creo que el país se pudrirá desde dentro y se derrumbará".
El 10 de julio de 2005, Kemakeza, primer ministro de las Islas Salomón, hizo una visita al santuario Yasukuni. Declaró que "las Islas Salomón y Japón tienen una cultura común de aprecio por los antepasados. Quiero ver el lugar donde se veneran las almas".
Véase también
Sintoísmo
Jinja
Crímenes de guerra japoneses
Nacionalismo japonés
Nacionalismo chino
Fenqing
Notas
Referencias
Breen, John. "The dead and the living in the land of peace: a sociology of the Yasukuni shrine". Mortality 9, 1 (febrero de 2004): 76-93 (en inglés).
Nelson, John. "Social Memory as Ritual Practice: Commemorating Spirits of the Military Dead at Yasukuni Shinto Shrine". Journal of Asian Studies 62, 2 (mayo de 2003): 445-467 (en inglés).
Ijiri, Hidenori. "Sino-Japanese Controversies since the 1972 Diplomatic Normalization". China Quarterly 124 (diciembre de 1990): 639-661 (en inglés).
Yang, Daqing. "Mirror for the future of the history card? Understanding the 'history problem'" in Chinese-Japanese Relations in the Twenty-first Century: Complementarity and Conflict, edited by Marie Söderberg, 10-31. New York, NY: Routledge, 2002 (en inglés).
Enlaces externos
Web oficial del Santuario Yasukuni.
Fotografías del santuario. Presencia ultranacionalista en Yasukuni.
Galería fotográfica del Museo Yushukan del santuario Yasukuni.
Santuario Yasukuni
Santuario Yasukuni
Yasukuni
Santuario Yasukuni
Sentimiento antijaponés en China
Sentimiento antijaponés en Corea
Nuevos movimientos religiosos sintoístas | {
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\section{Introduction and history of the problem}
The positivity property of quantum states guaranties the probabilistic
interpretation of quantum theory. It enters the mathematical formalism through
the identification of states with unit rays in a Hilbert space on which the
quantum observables act as operators. In quantum field theory (QFT), or more
generally for models with infinitely many degrees of freedom, it is often more
appropriate to identify states with positive linear functionals on operator
algebras. Thanks to the existence of a canonical construction\footnote{The
"reconstruction theorem" in \cite{St-Wi} is a special case of the more general
Gelfand-Naimark-Segal ("GNS") reconstruction theorem \cite{Haag}.} this
formulation in terms of expectation values permits a return to the more common
Hilbert space setting.
Its validity in quantum mechanics is guarantied by Heisenberg's canonical
quantization of positions and momenta in conjunction with the von Neumann
uniqueness theorem which insures that irreducible representations of the
Heisenberg commutation relations are unitarily equivalent to the
Schr\"{o}dinger representation. Born's identification of the absolute square
of the Schr\"{o}dinger wave function with the probability density for finding
a particle at a particular position connects positivity with spatial localization.
This situation undergoes significant changes in relativistic QFT where the
positivity of field-quantization looses its "von Neumann protection" in the
presence of higher spin $s\geq1$. The first such clash with positivity was
noticed by Gupta and Bleuler who observed that quantized \textit{massless}
vector potentials are incompatible with Hilbert space positivity. In the
absence of interactions it is straightforward to restore positivity by passing
from potentials to field strengths, but the use of local gauge invariance to
preserve at least part of positivity in the presence of interactions leads to
a loss of important physical operators and states.
This includes in particular all interacting fields which interpolate
charge-carrying particles in the sense of large time scattering theory. Such
\textit{interpolating fields} play an indispensable role in connecting the
causal localization- and quantum positivity- principles of QFT with observed
scattering properties of particles. Their absence in quantum gauge theory (GT)
is accompanied by a loss of mathematical tools of functional analysis. The
proofs of structural properties as TCP and Spin\&Statistics theorems use
Hilbert space positivity in an essential way and have no substitute in
indefinite metric Krein spaces. This reduces the use of GT to perturbative
rules for dealing with indefinite metric- and ghost- degrees of freedom (the
BRST formalism)
Positivity-obeying massive tensor potentials and their spinorial counterpart
are provided by Wigner's unitary representation theory of positive energy
particle representations of the (covering of the) Poincar\'{e} group, but they
come with an increase of their short distance scale dimension\footnote{It is
most conveniently obtained from property of the field's 2-ptfct $x\rightarrow
\lambda x$ for $\lambda\rightarrow0.$} with spin $d_{sd}=s+1~$which prevents
their use in renormalized perturbation theory involving fields with higher
spin $s\geq1.$ It turns out that this worsening of short distance behavior
with increasing spin is accompanied by a $m^{-s}$ divergence for small masses.
Hence a formulation of QED in terms of positivity-maintaining point-local
potentials is not possible.
In his well-known monograph Weinberg presents a systematic construction of the
intertwiner functions which relate Wigner's spin $s$ momentum space particle
creation and annihilation operators $a^{\#}(p,s)$ associated with the unitary
($m,s$) representations with covariant pl free fields which act in the
Wigner-Fock Hilbert space of the Wigner operators \cite{Wein}. This
interesting section in his book remained a torso since the (with increasing
$s$) worsening short distance scale dimension of point-local fields prevents
their use in renormalized perturbation theory as soon as $s\geq1$.
In the main part of his book Weinberg uses the positivity-violating (but
renormalizability-improving) gauge theoretic setting as obtained by Lagrangian
quantization in which a perturbative inductive argument secures the positivity
of gauge invariant operators. For this reason one does not find GT in
presentations of nonperturbative QFT.
The independence of short distance dimensions of quantized gauge fields from
spin/helicity is a consequence of the spin independence of the classical
dimension $d_{cl}=4$ of Lagrangians. For $s=0,1/2$ these fields agree with
those obtained from the Wigner-Weinberg construction, but for $s\geq1$ the
equality of the short distance dimension with the classical dimension in terms
of mass units ("engineering" dimension), namely $d_{sd}=d_{cl}=1$ for integer
and $3/2$ in case of half-integer spin, comes with an improvement of
renormalizability at the price of the presence of unphysical degrees of freedom.
Whereas in older work \cite{Fro} positivity problems for propagators of higher
spin fields in GT have been at least partially addressed, more recent
publications (\cite{Vas} \cite{Bek} and papers cited therein) are mainly
concerned with classical geometric aspects of the Lagrangian gauge formalism
for which these problems can be ignored.
The setting of string-local quantum field theory (SLFT) in the present article
overcomes this conceptual gap between GT and costructions of fields based on
Wigner's representation theory by providing a positivity maintaining causal
perturbative QFT formalism which includes the important physical interpolating
fields of particles whose large-time properties account for a unitary S-matrix
and which are missing in GT. After almost 70 years of GT this amounts to a
paradigmatic shift which does not only affect renormalized perturbation theory
but also requires to extend the nonperturbative setting of "axiomatic QFT" as
presented in \cite{St-Wi}.
A convenient starting point is to recall the construction of positivity
obeying quantum fields $\Psi_{a}~$in Weinberg's intertwiner formulation (for
simplicity for massive tensor potentials)
\begin{align}
\Psi_{\alpha}(x) & =\int(\sum_{s_{3}=-s}^{s}e^{ipx}v_{\alpha,s_{3
}(p)a^{\ast}(p,s_{3})+\hbox{h.c.})d\mu_{m}(p)+h.c.,~~\label{W}\\
& with~\ d\mu_{m}(p)=\theta(p_{0})\delta(p^{2}-m^{2})d^{4}p\nonumber
\end{align}
The intertwiner functions $v(p)$ convert the Wigner creation/annihilation
operators $a^{\#}(p,s_{3})$ into covariant fields $\Psi_{\alpha}$; their
calculation uses only group theory \cite{Wein}. They come with two indices,
the $s_{3}$ which runs over the $2s+1$ values of the third component of the
physical spin, and a tensor index $\alpha=(\mu_{1},..\mu_{s})$ which refers to
the $4s$ dimensional tensor representation of the Lorentz group of tensor
degree $s$. The extension to fermions is straightforward but not needed for
the problems addressed in the present work.
The momentum space Wigner creation/annihilation operators $a^{\#}(p,s_{3}),$
and hence also the covariant fields $\Psi_{\alpha}.$ act in the Wigner-Fock
Hilbert space obtained from the 1-particle Wigner representation by "second
quantization"\footnote{Note the difference to the standard use of
"quantization" (in the words of Ed Nelson: "second quantization is a functor,
wheres quantization is an art").} . Looking at the explicit form of the
intertwiners and calculating the two-point function (2-ptfct) of $\Psi$ one
finds that the latter scales as $\lambda^{-2(s+1)}$ for $x\rightarrow\lambda
x$ in the limit of small distances $\lambda\rightarrow0$ which leads to
assigning the short distance dimension $d_{sd}=s+1$ to the point-local (pl)
$\Psi.$
This construction via intertwiners permits much more flexibility than
Lagrangian or functional integral quantization in converting the unique
$(m,s)$ Wigner operators into fields with different prescribed covariance- and
causal localization- properties than quantization; includes in particular
string-local (sl) covariant quantum fields with improved ($0<d_{sd}<s+1$)
short distance properties. such fields are localized on causally separable
(i.e. allowing relative causal positioning)~semiinfinite space- or light-like
strings (rays) $\mathcal{S}=x+\mathbb{R}_{+}e,$ $e^{2}=-1$ or~$0.$ Whereas in
Weinberg's construction the covariance under Lorentz transformations is
sufficient since causality , the construction of sl fields requires the
\textit{direct use of causal localization}. The resulting covariant fields
extend the linear part of the pl relative causality class to sl (with pl
considered as a special case of sl) and Wick-ordered products thereof
constitute the nonlinear members.
This huge set of sl free quantum fields associated to an irreducible Wigner
representation contains in particular a \textit{sl tensor fields which is
linearly associated with its pl counterpart}. This sl tensor field appears
together with $s$ \textit{escort fields} with lower tensor degrees
\cite{E-M}\cite{Bey}$.$ Escorts are reminiscent of negative metric
St\"{u}ckelberg fields in gauge theory, except that they do not add unphysical
degrees of freedom to the physical $a^{\#}(p,s_{3})$ Wigner operators but only
differ in their intertwiner functions.
Positivity and hence the unitarity of the S-matrix in the resulting
string-local QFT (SLFT) is automatic\footnote{The causal separation properties
of sl fields are more than enough for deriving linked cluster fall-off
properties and insure the $e$-independence of the (on-shell) S-matrix.} (no
Nobel-prize worthy hard work as in gauge theory) and the chances to solve
age-old infrared problems (large time scattering theory in QED, QCD
confinement,..) are significantly enhanced \footnote{Recently Rehren showed
that infinite spin fields can be obtained in terms of appropriatly defined
Pauli-Lubanski limits of finite spin escort fields \cite{PL}.}. One
prerequisite is the substitution of nonexistent positivity-maintaining pl
potentials by sl counterparts and the according to the Weinberg-Witten No-Go
theorem \cite{W-W} missing $h\geq1~$sl current and stress-energy tensors in
\cite{MRS2} by suitably defined conserved sl substitutes. The smooth passing
from massive sl two-point functions with $2s+1$ degrees of freedom to their
massless two-component helicity counterpart leads to a profound (indefinite
metric- and ghost-free) understanding of the D-V-Z discontinuity problem
\cite{DVZ,Z}.
An important step in the development of$~$(SLFT) was the\textit{ construction
of fields for the class of massless infinite spin Wigner representations
$~$for which Yngvason's 1970 No-Go theorem excluded pl fields \cite{Y}. For
this class Weinberg's group theoretic method is without avail; one rather had
to resort to ideas from modular localization \cite{MSY1,MSY2,M}. This paved
the way for the construction of the simpler finite spin sl free fields,
including the use of their short distance lowering and hence renormalization
improving properties in interactions.
In the same work it was also realized that \textit{finite} spin/helicity sl
fields can also be obtained in a more direct way by integrating pl fields
along semi-infinite lines. This direct construction is particularly useful for
those sl potentials and their escorts which are linearly related to the pl
spin $s\geq1$ potentials. Different from the pl potentials which diverge in
the massless limit, the corresponding sl potentials pass to corresponding
finite helicity potentials which have no pl counterpart.
An important support for a string-extended QFT comes from work by Buchholz and
Fredenhagen who used the setting of algebraic QFT \cite{Bu-Fr} to show that
models with particle states separated by spectral gaps which fulfill certain
consistency properties with respect to the local observables always contain
interpolating operators localized in arbitrarily narrow spacelike cones (whose
cores are strings). Perturbative SLFT is more specific by showing that in the
presence of $s\geq1$ particles positivity together with causal localizability
leads to noncompact causal localization whose tightest localized covariant
generating fields are string-local.
The combinatorial nature of perturbation theory per se does not require
positivity and works also for gauge theory, but without positivity provided by
a Wigner-Fock Hilbert space the quantum theory remains incomplete. SLFT
reveals among other things that several limitations of gauge theory which are
the cause of certain No-Go theorems of which the best known is the
aforementioned Weinberg-Witten No-Go theorem (for a recent survey see
\cite{Ra-Ta}) are converted into Yes-Go statements in SLFT \cite{MRS2}.
SLFT is the only formulation in which state-creating interpolating fields are
separated from observables by spacetime localization properties. Whereas in
the absence of interactions the localization of free fields associated to a
Wigner representation ("kinematic localization") may be chosen at will, that
of \textit{interacting} fields in SLFT is determined by the particle content
of the interacting theory: observables are pl and interpolating fields are sl.
The space- or light-like interpolating fields can be placed in spacelike
separated positions which is a prerequisite for the application of the LSZ
scattering theory. The particle states in which the expectation values of
observables are measured are constructed in terms of suitably defined large
time asymptotic limits to the vacuum. A theorem of large-time scattering
theory insures that the dependence on the interpolating operator in the large
time limit is contained in its vacuum-to-one particle matrixelement which is
then removed by passing to the correctly normalized particle states. This
implies in particular that the $e$-dependence of sl fields \textit{does not
affect particles and their scattering matrix}.
The so-called cluster decomposition property of correlation functions of
fields plays an important role in the derivation of scattering properties. It
is a consequence of a mass gap and the existence of an arbitrary large number
of sl fields in relative spacelike position. This applies to spacelike strings
and (with some stretch of geometric imagination) also holds for lightlike
strings; it is however violated for timelike strings. In the present work the
terminology "nonlocal" is avoided since its historical connotation may hinder
to see that these fields can be brought into causally separated positions.
The reader is also reminded that the terminology "interaction density" instead
of "interacting part of a Lagrangian" is not nitpicking; apart from
interactions between $s<1$ particles, interaction densities
\textit{constructed from sl Wigner fields} are never interacting parts of Lagrangian.
Another important point which requires attention is the fact that the
kinematical sl localizations of free fields in terms of line integrals over
$s\geq1$ pl fields only serves to \textit{construct interaction densities
whose S-matrix is independent on string directions}. As mentioned the physical
localization of the corresponding interacting fields is dynamical and
generally different from that of their free counterpart used in the
construction of the interaction density and the S-matrix.
A surprising property of SLFT is that in the presence of $s\geq1$ particles
its positivity and localization properties determine a unique model in terms
of its particle content whenever such a theory exists. This is a result of the
strong restriction which the string-independence of the S-matrix exerts on
interaction densities.
The quest for an \textit{intrinsic} formulation in which the umbilical chord
to classical field theory provided by quantization has been cut is almost as
old as QFT. In the first (still pre-renormalization) presentation of quantum
electrodynamics at an international conference in 1929 \cite{Charkov} Pascual
Jordan expressed this in the form of a plea for an intrinsic understanding of
QFT which avoids the use of "(quasi)classical crutches"; a decade later his
former collaborator Eugen Wigner took the first step in his famous
classification of relativistic particles \cite{Wig}.
The second step was taken two decades later by Rudolf Haag \cite{Haag} when he
proposed an intrinsic formulation of QFT in terms of "causal nets of algebras"
in which Wightman fields at best play the role of "coordinatizations" (in
analogy to the use of coordinates in geometry).
With the arrival of the covariant formulation of quantized electrodynamics in
the 50s, Jordan's dictum and its partial realization in Wigner's
classification of noninteracting particles faded into the background; the new
covariant computational rules of quantized electrodynamics took a firm hold
and as a result the first covariant QFT was a positivity-violating gauge theory.
A somewhat unexpected aspect of these first successful calculations was the
contrast between the precision of the experimentally verified perturbative
results and the robustness of the calculated results against the use of quite
different cutoff- and regularization- prescriptions, or even against different
ways of implementing Lagrangian quantization (Gell-Mann--Low, Feynman path
integrals, Bogoliubov's generating S-functional).
The presence of gauge theoretic indefinite metric degrees of freedom in
interaction densities involving $s\geq1$ particles led to conceptual problems.
A formal proposal to overcome these shortcomings was made by Jordan \cite{Jo}.
It consisted in replacing the gauge dependent matter field\footnote{Jordan
used these fields for a pure algebraic derivation of Dirac's geometric
magnetic monopole quantization \cite{mo}.} by the formally gauge invariant
string-local composite field
\begin{equation}
\Psi(x)=\psi^{K}(x)\exp ig\int_{0}^{\infty}A_{\mu}^{K}(x_{0},\dots
,x_{3}+\lambda)d\lambda
\end{equation}
~where the $K~$refers to the gauge dependent Lagrangian field which acts in an
indefinite metric Krein space which in addition to physical degrees of freedom
contains also indefinite metric quanta (scalar and longitudinal photons).
After the discovery of renormalized perturbation theory Mandelstam used such
representation as a starting point in his attempt to construct a perturbation
theory which avoids the use of potentials in favor of working directly with
gauge invariant fields \cite{Man}. Subsequently Steinmann \cite{Stein} studied
the problem of recovering positivity by constructing such fields $\Psi~$in
higher order perturbation theory. Different proposals to recover positivity
can be found in \cite{Mor-Stro}. The constructions of such formally gauge
invariant composite fields and their renormalization requires a lot of
additional work and is of little interest unless it leads to new physical insights.
The SLFT perturbation theory in the present paper uses sl potentials with the
$s$-independent short distance dimension $d_{sd}=1$ which "live" in a physical
Wigner-Fock particle space$.~$The starting point is the observation that there
exist sl vector potentials $A_{\mu}(x,e)$ localized on causally separable
spacelike strings $\mathcal{S}=x+\mathbb{R}_{+}e$ which together with their
scalar sl "escorts" $\phi(x,e)$ are linearly related to their pl counterpart
simple illustration is provided by the interaction density $L^{P}=A_{\mu
^{P}j^{\mu}$ of massive QED which is related to its pl counterpart as $A_{\mu
}(x,e)=A_{\mu}^{P}(x)+\partial_{\mu}\phi(x,e)$.$~$
Its use in an interaction density of e.g. massive QED $L^{P}=A_{\mu}^{P
j^{\mu}$ results in a relation $L^{P}=L-\partial^{\mu}\phi j_{\mu}$ in which
the sl density $L(x,e)~$has an improved short distance dimension $d_{sd}(L)=4$
(instead of $d_{sd}(L^{P})=5$) and accounts for the first order contribution
to the (on-shell) $S$-matrix in the adiabatic limit $S=\int L$ to which the
boundary term from $V_{\mu}=\phi j_{\mu}$ does not contribute.
This is in a nut-shell a perturbative implementation of the aforementioned
abstract Buchholz-Fredenhagen theorem; it secures the existence of
interpolating sl fields whose directional smearing provides the B-F operators
localized in arbitrary narrow spacelike causally separable cones and insures
that their large-time scattering limits results in $e$-independent Wigner
particles and their S-matrix.
The extension of this first order $L$ to higher orders involves time-ordered
products in the interaction densities $L~$respective $L^{P}$ and leads to new
powerful normalization conditions which insure that the two different
interaction densities lead to the same S-matrix. As a result of the with
perturbative order growing number of counterterms, the $L^{P}$ theory by
itself is physically useless; but being "guided" by the sl $L,V_{\mu}$ pair it
becomes a well-defined physically useful companion which shares not only its
parameters but also its S-matrix and local observables with the $d_{sd}(L)=4$
SLFT$.$ Its only memory about its "unguided past" is the with perturbative
order increasingly singular $d_{sd}\rightarrow\infty$ short distance dimension
of its interpolating fields.
\textit{SLFT is an S-matrix theory in the sense that the particle content
together with the string-independence of }$S$\textit{ determines (in all cases
studied up to now uniquely) the form of the interaction density. In a second
step the construction of the S-matrix is extended to that of pl and sl
interacting fields. }
Different from Lagrangian quantization the SLFT formalism does not prefer
certain fields. All interacting fields which act in the same Wigner-Fock space
and are members of the same causality class are on equal footing; which
particle they interpolate depends only on the nontriviality on their
vacuum-to-one-particle matrix elements.
Often new theoretical insights are the result of accidental observations. SLFT
is not of this kind; what led to it is the rather deep connection of sl
localization with \textit{modular localization} theory. The terminology
"string" used in quantizations of classical actions (Nambu-Goto actions,
world-sheets,..) bears no relation to the causal localization of string-local
quantum fields in the present work.
A definition of causal localization which avoids such misunderstandings is
that in terms of \textit{modular localization. }In fact modular localization
permits to identify\textit{ a pre-form of causal localization already within
the Wigner positive energy representation space} \cite{BGL} before "second
quantization" converts it into the algebraic form of Einstein causality in
QFT. \textit{This idea paved the way for the construction of the QFT behind
Wigner's infinite spin representation}.
Modular localization theory can be traced back to the Tomita-Takesaki modular
theory of operator algebras of the 60's. It is one of a few mathematical
theories to which physicists working on problems of statistical mechanics of
open systems \cite{HHW} made important contributions. It made its first
appearance in the context with causal localization in the Bisognano-Wichmann
theorem \cite{BW} which deals with modular properties of wedge-localized
algebras. \textit{Modular operator theory and modular localization requires
positivity and hence cannot be applied to GT and Lagrangian quantization}.
As the result of accommodating thermal aspects and causal localization under
one conceptual roof, it led to profound insights (thermal properties of "event
horizons") into Hawking's black hole radiation \cite{Sew}. A first survey
about its history enriched by new results was presented by Borchers \cite{Bo}.
Modular localization also played an important role in the construction of QFTs
from S-matrices of integrable models in $d=1+1$ dimensions \cite{AOP}
\cite{AL}. Presently ideas from modular operator theory are being successfully
applied to obtain a foundational understanding of entanglement entropy caused
by causal localization (for a survey see \cite{Wi} \cite{Ho-Sa} and references therein).
Modular localization theory permits to extend Weinberg's intertwiner
construction to Wigner's infinite spin representations and to obtain explicit
expressions for the associated sl fields \cite{MSY1,MSY2,Koe}\cite{PY}. More
recently these fields reappeared as "Pauli-Lubanski limits" of finite spin sl
fields \cite{PL}. This made it possible to investigate physical properties of
quantum matter through the study of its positivity-obeying causal localization
structure and look for theoretical reasons why certain types of matter can not
be seen in counters \cite{dark}.
One should also mention a series of more recent publications \cite{Sch-T}
\cite{Bekaert} in which covariant wave functions were constructed, but the
much stronger result in the aforementioned work was overlooked. Relativistic
wave equations for infinite spin appeared already in Wigner's 1948 paper
(\cite{Wig 48} 12.1-12.4). These different wave functions describe different
covariant bases in Wigner's irreducible representation space. But for studying
physical manifestations of matter one needs to know its causal localizability
which in case of infinite spin does not follow from covariance and needs the
use of modular localization theory as used in the cited 2006 papers. In this
way the 1970 No-Go theorem \cite{Y} which excluded point-like localization was
replaced by a sl Yes-Go theorem.
In fact the exclusion of linear pl fields is part of a more general No-Go
theorem which rules out the possibility of constructing pl composites from the
linear sl infinite spin fields. In its most general form the theorem excludes
the existence of operator algebras localized in finite spacetime regions
\cite{LMR}.
These theorems against pl localization of infinite spin matter may be seen as
an extreme counterpart of the Weinberg-Witten No-Go theorem against the
existence of higher helicity conserved currents and energy-momentum tensors.
The difference is that there still exist W-W local charges, whereas in case of
infinite spin there are no nontrivial operators localized in finite regions.
Recall that the raison d'\^{e}tre of a relativistic quantum \textit{field}
theory (for the difference between QFT and relativistic QM, see section 3 in
\cite{SHPMP}) is the realization of the "Nahewirkungsprinzip" (action in the
neighborhood principle) of Faraday and Maxwell which culminated in Einstein's
concept of relativistic covariance and causal localization. The positivity
requirement of quantum probability turns the construction of models of QFT
into a challenging problem which gauge theory did not solve.
It is the aim of this work to show how the recent SLFT formulation solves
problems which have remained outside the range of GT (for a review of such
problems see \cite{Ra-Ta}). In \cite{MRS1} \cite{MRS2} this was already
achieved for the problem behind the Weinberg-Witten (W-W) No-Go theorem
\cite{W-W} and the $s=2$ van Dam-Veltman-Zakharov (D-V-Z)\ discontinuity
\cite{DVZ,Z}. Here we add the causality problems raised by Velo and Zwanziger
(V-Z) \cite{V-Z}.
SLFT's central point is however the presentation of a sl-based perturbation
theory which in contrast to gauge theory preserves the Hilbert space
positivity (no indefinite metric- and ghost-degrees of freedom) without
destroying the causal separability of fields. It leads in particular to
interesting different physical interpretations of interactions between vector
mesons and Hermitian fields (Higgs models, but no SSB Higgs mechanism)
\cite{Bey}.
The content of this paper is organized as follows.
The next section recalls and extends recent (partially already published)
results concerning the construction of causally separable string-local free
fields. It consists of 4 subsections which includes the construction of sl
massless vector potentials and their canonically related massive counterpart.
The third section addresses the problem of interactions with external
potentials. It is shown that the origin of the Velo-Zwanziger causality
problem is the incorrect expectation that by modifying free field equations by
adding linear couplings to external potential one preserves causality in the
sense of causal propagation of Cauchy data. The solution of the V-Z problem
has a close formal proximity to the solution of the Weinberg-Witten problem in
\cite{MRS2}.
Section 4 provides some background about modular localization. Its aim is to
show that causal localization is incompatible with any form of quantization
but important for understanding properties of causally localized quantum matter.
In section 5 the SLFT renormalization theory is applied to calculation of the
S-matrix in various models involving vector mesons. including some speculative
remarks on $s\geq2$ interactions.
Section 6 addresses problems of interacting sl fields in particular the
model-dependent distinction between pl observables and sl interpolating fields.
The concluding remarks in section 7 summarize the new insights and present an outlook.
\section{String-local tensor potentials and conservation laws}
\textit{This section provides the kinematical prerequisites of SLFT i.e. the
construction of those sl free fields which are used in later sections for the
calculation of the S-matrix and interacting fields. The kinematic localization
of free fields is not the same as the dynamic localization of their
interacting counterparts (section 6).}
\subsection{Massless string-local potentials}
The fact that even in the absence of interactions massless gauge potentials
have no positivity-maintaining pl counterpart led to a more foundational
re-thinking regarding the relation between positivity and causal
localizability for which the solution of the massless infinite spin problem in
terms of sl fields served as a role-model \cite{MSY1,MSY2}. The finite
helicity problem is simpler since in this case their exists only one covariant
family of sl potentials $\hat{A}_{\mu}$ in the Wigner-Fock helicity Hilbert
space whose field strength is the pl field strengths
\begin{equation}
\partial_{\mu}\hat{A}_{\nu}(x)-\partial_{\nu}\hat{A}_{\mu}(x)=F_{\mu\nu}(x)~~
\label{1
\end{equation}
They have the form of a semi-infinite line integrals (strings, rays)
\begin{align}
& \hat{A}_{\mu}(x)=A_{\mu}(x,e):=\int F_{\mu\nu}(x+\lambda e)e^{\nu
=:(I_{e}F_{\mu\nu})(x)e^{\nu}\label{2}\\
& U(a,\Lambda)A_{\mu}(x,e)U(a,\Lambda)^{\ast}=(\Lambda^{-1})_{\mu}^{\nu
}A_{\nu}(\Lambda x,\Lambda e)~\nonumber
\end{align}
with $e$ representing a space- time- or lightlike vector \textit{which
participate in the transformation under the homogenous Lorentz group}. Causal
localizability requires the possibility of placing an arbitrary large number
of such sl fields in relative spacelike separated positions (denoted as
${\lower1pt\hbox{\LARGE$\times$}}$ ). This excludes the timelike case but
permits space- and light-like strings\footnote{This is less obvious in the
lightlike case.
\begin{equation}
\left[ A_{\mu}(x,e),A_{\mu}(x^{\prime},e^{\prime})\right] =0,~x+\mathbb{R
_{0}^{+}e\,{\lower1pt\hbox{\LARGE$\times$}}\,x^{\prime}+\mathbb{R}_{0
^{+}e^{\prime
\end{equation}
Spacelike unit vectors $e$ with $e^{2}=-1~$are points on the $d=1+2$ unit de
Sitter space, whereas lightlike vectors $e$ with $e^{2}=0~$may be identified
with points on the two-dimensional celestial sphere. A closer examination
shows that line integrals of \textit{massless} field strengths along lightlike
lines are ill-defined (see below) but well-defined (as distributions in $e$)
for spacelike $e$; time-like lines would violate causal separability.
The derivation of nonperturbative theorems (PCT, Spin\&Statistics, cluster
properties, LSZ scattering theory,..) does not need pl fields; what is
important is the preservation of causal separability i.e. the fact that one
can place an \textit{arbitrary number} of sl fields into relative spacelike position.
The mathematical status of sl fields requires a more careful look at their
singularity structure. For this purpose it is convenient to compute their
2-point function (2-pfct). Starting from that of the field strengths
\begin{align}
\left\langle F_{\mu\nu}(x)F_{\kappa\lambda}(x^{\prime})\right\rangle & =\int
e^{-ip(x-x^{\prime})}M^{F_{\mu\nu},F_{\kappa\lambda}}(p)d\mu_{0}(p),\text{
\ }d\mu_{m}(p)=\frac{d^{3}p}{2\sqrt{\vec{p}^{2}+m^{2}}}\\
M^{F_{\mu\nu},F_{\kappa\lambda}}(p) & =-p_{\mu}p_{\kappa}g_{\nu\lambda
}+p_{\mu}p_{\lambda}g_{\nu\kappa}-p_{\nu}p_{\kappa}g_{\mu\lambda}+p_{\nu
}p_{\lambda}g_{\mu\kappa}\nonumber
\end{align}
and, using the fact that the $\lambda$-integration amounts to the Fourier
transform of the Heavyside function and hence leads to distribution
$(pe)_{i\varepsilon}^{-1}=\lim_{\varepsilon\rightarrow0}(pe+i\varepsilon
)^{-1}$ as boundary values of analytic functions, one obtains \cite{MRS2
\begin{align}
& \left\langle A_{\mu}(x,-e)A_{\nu}(x^{\prime},e^{\prime})\right\rangle =\int
e^{-ip(x-x^{\prime})}M^{A_{\mu},A_{\nu}}(p,e,e^{\prime})d\mu_{0}(p)\label{3}\\
& M^{A_{\mu},A_{\nu}}(p,e,e^{\prime})=E_{\mu\nu}(-e,e^{\prime})=-\eta_{\mu
\nu}+\frac{p_{\mu}e_{\nu}}{(pe)_{i\varepsilon}}+\frac{e_{\mu}^{\prime}p_{\nu
}{(pe^{\prime})_{i\varepsilon}}-\frac{(ee^{\prime})p_{\mu}p_{\nu
}{(pe)_{i\varepsilon}(pe^{\prime})_{i\varepsilon}}\nonumber
\end{align}
where the tensor $E_{\mu\nu}$ turns out to be an important building block of
higher helicity 2-pfcts$.$ The scaling degree $d_{sd~}$ is defined as the
leading short distance contribution $\lambda^{-2d_{sd}\text{ }}$of the 2-ptfct
under the scaling $\xi\rightarrow\lambda\xi,$ $\xi=x-x^{\prime}$ for
$\lambda\rightarrow0~$and can be directly read off from the large momentum
behavior. Whereas $d_{sd}(F)=2,$ the line integration lowers the degree to
$d_{sd}(A)=1$.
A more detailed study shows that sl potentials and their 2-pfcts are
well-defined as \textit{distributions} in $e,~e^{2}=-1$ (the unit de Sitter
space) and $x.$ All operators and correlation function are of homogeneous
degree zero and hence the de Sitter differential can be written in the
covariant form $d_{e}=de_{\mu}\frac{\partial}{\partial e_{\mu}}.$ For
lightlike $e^{\prime}s$ and $m>0$ the last term in (\ref{3}) vanishes for
$e^{\prime}=$ $-e$ and the distributional dependence of $A_{\mu}(x,e)~$on $e$
changes to that of a function so that a directional testfunction smearing in
$e\ $is not necessary. The identification of $e^{\prime}s$ in products of
fields leads to a significant notational simplifications in perturbative
calculations. The existence of momenta for which $p$ is parallel to $e$
excludes however massless limits lightlike strings.
For the timelike directions the denominators never vanish and no smearing is
needed, but the causality requirement, namely the existence of an arbitrary
number of causally separated sl fields, cannot be satisfied. Hence the choice
$e=e_{0}=(1,0,0,0)$ leads to the nonlocal Coulomb- (or radiation-) potential
with $A_{0}^{C}=0$ and spatial component
\begin{equation}
M^{A_{i}^{C}A_{j}^{C}}=\delta_{i,j}-\frac{p_{i}p_{j}}{\mathbf{p}^{2}
\end{equation}
"Freezing" this timelike string direction destroys the covariant
transformation and one obtains a noncovariant inhomogeneous transformation law
in which only the rotations and translations maintain their covariant
appearance (see \ref{inh} below). Full covariance can be restored by letting
the timelike direction participate in the Lorentz transformation, but the loss
of causal localization remains. The Coulomb potential is used in quantum
mechanics where relativistic covariance and causality play no role.
It is interesting to note that the Coulomb potential results also from
\textit{averaging a spacelike string over spatial directions in the
$t=0$\textit{ plane orthogonal to the timelike }$e_{0}~$\textit{vector}. There
is no direct way to undo this directional averaging; one rather has to return
from $A^{C}$ to its covariant field strength $F$ and obtain the associated sl
potential as in (\ref{2}). This directional averaging reveals a \textit{close
formal connection between the axial- and Coulomb- "gauge"}. Both potentials
exist in the same Wigner-Fock helicity space, but only the covariant sl
potential (\ref{2}) is manifestly causal.
The use of sl potentials turns the so-called noncovariant axial- and
lightcone-gauges into better manageable covariant Einstein-causal fields which
act in a positivity maintaining Hilbert space.
It should be mentioned that in the literature the terminology "gauge" is used
with two different meanings. In the covariant setting of QED perturbation
theory it refers to a formal symmetry whose generator is a "gauge charge"
which depends unphysical indefinite metric degrees of freedom. On the other
hand the Coulomb- or axial- gauge contains only the two helicity $h=\pm1$
degrees of freedom and there is no symmetry-implementing gauge charge,
although the additive contribution to the Lorentz transformation looks like a
non-covariant gauge transformation (\ref{inh}) re-expressing the
Lorentz-transformed $e=\Lambda e_{0}$ in terms of original $e_{0}$.
It is not the aim of this work to change historically grown terminology. Here
the terminology "gauge" is exclusively used the situation in which unphysical
degrees of freedom provide a covariant "gauge symmetry". Quantum gauge
symmetry is not a physical symmetry (and consequently there is no physical
sense in which it can be broken) but rather a formal tool to extract a
physical theory as a subtheory from an unphysical formalism.
The large momentum behavior of the 2-pfct determines the short distance
behavior of the field whereas the distributional behavior in $e$ depends on
the dimensionality of spacelike $e$-directions on which $pe$ vanishes. The
case of \textit{lightlike} $e^{\prime}s$ is a bit more tricky. For massive $p$
the $pe$ denominator does not vanish since $p$ and $e$ only touch at lightlike
infinity and as a result the sl fields are functions in $e$.
This changes in a radical way for massless $p;$ in that case for each $e$
there are lightlike $p^{\prime}s$ on which $pe$ vanishes and as a result
massless fields localized on lightlike strings do not even exist in the sense
of distributions \footnote{I am indebted to Henning Rehren for drawing
attention to the nonexistence of massless lightlike string localized fields.}.
Lightlike sl fields have an interesting connection with light-cone
quantization. In the massless case they reveal in a much clearer way the
problematic nature of "lightcone quantization" \cite{Leib}.
The main purpose of this work is to offer a positivity- and
causality-\ preserving alternative to gauge theory which avoids the use of the
quantization parallelism to classical field theories by starting from Wigner's
manifestly positivity-preserving particle representation theory. The important
point is that spacetime localization properties already exist in the pre-form
of \textit{modular localization} within Wigner's particle theory. They can be
used to construct pl or sl intertwiner functions which convert Wigner's
creation and annihilation operators into covariant pl or sl free fields.
The perturbative construction of the S-matrix and of interacting sl fields
does not need modular localization theory. For problems as
\textit{localization entropy \cite{Ho-Sa} \cite{Wi}} and nonperturbative
constructions \cite{AL} its use is however indispensable. In the context of
the present paper its importance is based on pinning causal localization to
quantum positivity; whereas Lagrangian quantization of fields allows the
presence of unphysical degrees of freedom, modular localization excludes them.
More remarks on modular localization will be deferred to section 4.
The construction of sl potentials in terms of pl field strengths (\ref{2})
permits an iteration to a scalar potential\ $\Phi
\begin{equation}
A_{\mu}(x,e)-A_{\mu}(x,e^{\prime})=\partial_{\mu}\Phi(x,e,e^{\prime
),~\Phi=(I_{e^{\prime}}I_{e}F_{\mu\nu})(x)e^{\mu}e^{\prime\nu}~ \label{5
\end{equation}
The $\Phi$ represents a field which is localized on the 2-dimensional
\textit{conic region} $\lambda e+\lambda^{\prime}e^{\prime},~\lambda
,\lambda^{\prime}\geq0.$ In the massless limit this flux $\Phi$ is
logarithmically divergent. The logarithmic divergence is expected to lead to
an $e,e^{\prime}$ dependent continuous set of superselection rules which
extend the Wigner-Fock helicity space.
This is reminiscent of the behavior of the exponential of a massive scalar
free field in $d=1+1$ in the massless limit\footnote{This infrared behavior
was first observed in the coupling of a $d=1+1$ current to the derivative of a
massless scalar field ("infraparticle" \cite{Infra} ).} which played an
important role in the work on "bosonization" of massless fermions and anyons
\cite{S-S}. In that case the massless limit of the properly mass-normalized
exponentials leads to the superselection propert
\begin{equation}
\left\langle e^{ia_{1}\varphi(x_{1})}\dots e^{ia_{n}\varphi(x_{n
)}\right\rangle =0\quad\hbox{if}~~\sum_{1}^{n}a_{i}\neq0. \label{scal
\end{equation}
corresponding to $a_{i}$-"charge" conservation.
The "photon cloud" in the $e$-direction associated with $\exp ig\varphi$ is
expected to cause a directional superselection rule which appears in the form
of $e^{ig\varphi}\psi$ in the large time behavior of electric charge carrying
fields and causes the modification of LSZ scattering theory. In this way one
may hope to obtain a genuine spacetime understanding of the infrared momentum
space recipes in \cite{YFS}.
The interest in this problem is also motivated by the existence of rigorous
results derived from an appropriate formulation of the quantum Gauss law
\cite{Bu}. This theorem states that interacting electric charge-carrying
operators $\psi$ are accompanied by spacelike extended "photon clouds"$~$whose
different asymptotic conic directions correspond to a continuum of
superselection sectors within the same charge-carrying sector. This is the
cause a spontaneous breaking of Lorentz symmetry \cite{Froh}.
The existence of a continuum of superselection sectors for free photons would
suggest the existence of large time asymptotic charge-carrying matter fields
of the form $\psi_{0}e^{ig\Phi}$ with $\psi_{0}$ a free matter field. Their
large time asymptotic behavior is expected to play an important role in a
future spacetime understanding of infrared properties which is outside the
physical range of gauge theories.
For many applications it is useful to encode change in $e$ (\ref{5}) into
changes of the Lorentz transformation law. A differential relation which is
the basis for such conversion has the form (\cite{MRS2} Corollary
3.3)\footnote{As mentioned therein this remains well defined since sl fields
and their correlation functions are homogeneous functions of degree zero in
$e$ and $p.$}
\begin{equation}
d_{e}A_{\mu}(x,e)=\partial_{\mu}u(x,e),\ ~d_{e}=\sum_{i}d_{e_{i}
\partial^{e_{i}} \label{6
\end{equation}
where $u$ is an exact de Sitter one-form $u=d_{e}\phi$. This conversion of
directional de Sitter differentials into $x$-derivatives plays an important
role in passing from interactions in the presence of a mass gap to their
massless limit.
In the present context the formula for the change of $e^{\prime}s$ can be used
to compute the additive change which is necessary in order to
\textit{maintain} the timelike $e_{0}$ direction of the Coulomb potential
$A_{i}^{C},~A_{0}^{C}=0$. The resulting affine transformation formula
\begin{equation}
U(a,\Lambda)A_{i}^{C}(x)U(a,\Lambda)^{\ast}=(\Lambda^{-1})_{i}^{~l}A_{l
^{C}(\Lambda x+a)+(\Lambda^{-1})_{i}^{~\mu}\partial_{\mu}\chi(x) \label{inh
\end{equation}
is equivalent to that obtained by starting from the Wigner helicity
representation and using transverse polarization vectors \cite{Wein}.
A similar situation arises if one fixes an "axial" direction as
e.g.\ $e=(0,1,0,0).$ In this case the causal localizability is preserved in
both descriptions. Ignoring the spacetime localization aspect and treating the
axial direction as a noncovariant gauge misses the necessity of directional
smearing (smearing around a point in de Sitter space) and probably contributed
to the abandonment of the "axial gauge fixing". But what became a curse in the
axial gauge fixing turns out to be a blessing in the covariant SLFT setting.
Covariant gauges as used in covariant perturbation theory always require the
presence of ghost-extended indefinite metric BRST degrees of freedom setting
which reduces the physical range. SLFT cuts the umbilical cord between
perturbative Lagrangian quantization and classical gauge theory and restores positivity.
\subsection{A brief interlude, relation with concepts of algebraic QFT}
The simplest illustration of the interplay between positivity and causality is
provided by the Aharonov-Bohm effect. To see this recall that Einstein
causality is the statement that the algebra of operators localized in the
causal complement $\mathcal{O}^{\prime}$ of a spacetime region $\mathcal{O}$
belong to the commutant $\mathcal{A}(\mathcal{O})^{\prime}$ algebra (the von
Neumann algebra which consists of all operators which commute with
$\mathcal{A}(\mathcal{O})$)
\begin{align}
\mathcal{A}(\mathcal{O}^{\prime}) & \subseteq\mathcal{A}(\mathcal{O
)^{\prime}\quad\hbox{or}\quad\mathcal{A}(\mathcal{O})\subseteq\mathcal{A
(\mathcal{O}^{\prime})^{\prime},\quad\hbox{Einstein causality}\label{Haag}\\
\mathcal{A}(\mathcal{O}^{\prime}) & =\mathcal{A}(\mathcal{O})^{\prime
,\quad\hbox{Haag duality}\nonumber
\end{align}
The second line defines the somewhat stronger Haag duality which states that
an operator which commutes with all operators localized in the causal
complement of $\mathcal{O}~$\textit{must} belongs to $\mathcal{A
(\mathcal{O})$.
Einstein causality is a defining property of relativistic QFT, but Haag
duality may be violated. In the absence of interactions such a violation can
be excluded for massive QFT's but it does occur in the massless case when the
$2s+1$ spin degrees of freedom are converted into the $\pm h$ helicities. As
observed in \cite{LRT} (unpublished) Haag duality, which holds for simply
connected spacetime regions, is violated for multiply connected regions as
(genus one) tori.
In their proof the authors carefully avoid the use of gauge potentials.
associated to $m>0$ massive is a property ($g\geq1~$tori). This violation is
an intrinsic property of the operator algebra generated by the field strength
$F_{\mu\nu}~$of the $h=1~$Wigner representation. But if one wants to
understand this in terms of vector potentials one must use the
positivity-maintaining sl potentials which preserve a somewhat hidden
topological properties of Wilson loops which cause the breakdown of Haag
duality while it upholds Einstein causality \cite{Bey}. The indefinite metric
potentials cannot distinguish between the two; only the localization in the
presence of positivity is physical.
The cause of "eeriness" about the Aharonov-Bohm effect \cite{A-B}~(but also of
its popularity) is that we erroneously interpret the intuitively accessible
geometric Haag duality with the more abstract Einstein causality, thus
forgetting that the latter also admits operators which have no unambiguous
causal localization region (e.g. the magnetic flux through a surface with a
fixed boundary). The ideal solenoid in the A-B setup closes at spacelike
infinity, which in the conformal Wigner-Fock helicity world is a circle. In
case of a finite tube one must place the electric circuit into a region of
little magnetic backflow from north- to south-pole.
This is a strong reminder that it is not possible to separate causality from
positivity and a warning not to confuse the "fake localization" of gauge
dependent objects with genuine causal localization of quantum matter. It
points to a potential source of misunderstanding involved in transferring the
perfectly reasonable classical notion of \textit{local} gauge symmetries to
QFT by attributing a physical meaning to the formal observation that quantum
gauge charges are "more local" than those corresponding to internal
symmetries. It is also a reminder to rethink the physical meaning behind the
terminology "gauging a model".
From (\ref{6}) it follows that a Wilson loop\footnote{By convoluting with a
test function one can convert the Wilson loop integral into into an operator
localized on a solid torus.} formed with $A_{\mu}(x,e)$ \textit{is independent
of} the$~$choice$~$of the direction $e~$\cite{Bey}$.~$However it
\textit{retains a topological memory} of the string directions of the
integrand which prevents a naive materialistic identification with a
localization in a torus. One can choose $e^{\prime}s$ in such a way that this
extension is spacelike with respect to any simply connected convex compact
region\footnote{In case the solenoid has open ends the Wilson loop should
avoid the region of the north-south magnetic backflow.}. Yet it is not
possible to completely forget that the vector potential has a directional
$e$-dependence. An elegant formulation of this $h\geq1~$topological phenomenon
directly based on field strengths and their duals in terms of "linking
numbers" can be found in \cite{BCRV}.
\subsection{Massive string-local potentials}
Before passing to the construction of massive sl fields it is helpful to
recall the construction of their pl intertwiner functions $v(p)$ which convert
the$~m>0\ $Wigner creation and annihilation operators $a^{\#}(p,s)~$into
covariant \cite{Wein}. For the $s=1$ Proca field they are the three
polarization vectors~$v_{\mu}(p,s_{3})$ obtained by applying a rotation-free
Lorentz boost to the spatial coordinate unit vectors. By definition they are
Minkowski-orthogonal to $p_{\mu}$ and hence correspond to the 3 polarization
vectors $v$ obeying the completeness relation
\begin{align*}
& \,\sum_{s_{3}=--1}^{1}v_{\mu}(p,s_{3})v_{\nu}(p,s_{3})=-\eta_{\mu\nu
+\frac{p_{\mu}p_{\nu}}{m^{2}}\\
& M^{A_{\mu}^{P},A_{\nu}^{P}}(p)=-\pi_{\mu\nu}(p),\quad\pi_{\mu\nu
(p)=\eta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{m^{2}
\end{align*}
where the $\pi_{\mu\nu}$ of the momentum space 2-pfct which also turns out to
be the basic building block of all higher spin massive tensor potentials.
With a pl Proca potential $A_{\mu}^{p}~$one may associate two sl fields, the
scalar sl field $\phi~$defined in terms of a line integral $I_{e}$ of the
$e$-projected Proca field $A_{\mu}^{P}e^{\mu}$ along $e$ starting from the
point $x$
\begin{equation}
\phi(x,e)=(I_{e}A_{\mu}^{P})(x)e^{\mu},~~a(x,e)=-m\phi(x,e) \label{a
\end{equation}
and the sl vector potential $A_{\mu}$ in terms of the field strength of the
Proca potentia
\begin{equation}
A_{\mu}(x,e)=(I_{e}F_{\mu\nu})(x)e^{v},\text{ }F_{\mu\nu}(x)=\partial_{\mu
}A_{\nu}^{P}-\partial_{\nu}A_{\mu}^{P} \label{F
\end{equation}
whose massless limit coincides with (\ref{2}).
The multiplication with $m$ in (\ref{a}) restores $d_{cl}=1$ and removes the
mass singularity, so that the $m=0$ limit is an $e$-independent massless
scalar $d_{sd}(a)=1$ free field. In fact for all massive or massless sl tensor
potentials $d_{sd}=1$ and $3/2$ for halfinteger $s$ whereas their pl
counterparts increase linearly as $d_{sd}=s+1~$or $d_{sd}=s$ and diverge like
$m^{-s}$ for $m\rightarrow0$
In particular the momentum space 2-pfct of the massive field strength and its
massless associated sl vector potential are identical to their massless
counterparts. This permits to lower the number of degrees of freedom by
passing from $p\in H_{m}^{\uparrow}$ to$~p\in V^{\uparrow}$ (and its
"fattening" inversion, see below)$.$ In the pl setting this is not possible or
can only be achieved in the presence of indefinite metric degree of freedom
(the DVZ discontinuity, the WW problem).
It is instructive to look at this degrees of freedom balance in more detail
\cite{MRS1,MRS2}. With the help of a $p$-dependent 4-matrix $J$ (complex
conjugation changes the sign of $e$,~$tr~$=~transposed)
\begin{align}
& J_{\mu}^{~~\nu}(p,e)=\eta_{\mu}^{\ \ \nu}-\frac{p_{\mu}e^{\nu
}{(pe)_{i\varepsilon}},~\overline{J(p,-e)}=J(p,e)\\
& M^{A_{\mu}(-e),A_{\nu}(e)}=:E_{\mu\nu}(e,e)=(J\pi J^{tr})_{\mu\nu}\nonumber
\end{align}
the in $e$ diagonal momentum space 2-pfct takes the form of the second
line\footnote{Taking the same $e$ would lead to the distributionally
ill-defined denominator $pe_{-i\varepsilon}pe_{i\varepsilon}.$}. It shows that
the positivity of the sl 2-pfct is inherited from the pl positivity. The rank
of the $E$-matrix accounts for the degrees of freedom is 3 as a result of
$J^{tr}e=0$ and the additional relation $E_{\mu\nu}p^{\nu}=0$ for $p\in
V^{+}~$leads to a reduction from the three spin component to the two
helicities $h=\pm1$. This degrees of freedom counting breaks down in the
presence of indefinite metric.
This descend from $p\in H_{m}^{\uparrow}$ to $V^{+}$ permits an inversion,
namely by continuous passing from momenta $p\in V^{+}$ to the mass shell
$H_{m}^{+}$~("fattening") one creates a new physical degree of freedom which
together with former $\pm1$ accounts for the $3$ degrees of freedom of spin
$s=1$. Such "magical" conversion of the particle content of two inequivalent
Wigner representations can neither be achieved in terms of pl fields (no
massless limit) or become contaminated by the presence of indefinite metric
causing and ghost degrees of freedom of gauge theory. This is of particular
interest in case of $s=2$ \cite{MRS1,MRS2} (see below). The use of sl fields
is even more important for passing to the massless limit in the presence of
interactions involving higher spins.
In the literature the terminology "fattening" had been used in connection with
the Higgs model which describes the interaction between a massive vector meson
with a massive real scalar field $H$ as the result of spontaneous breaking of
gauge symmetry (the Mexican hat potential). This idea contains two conceptual
misunderstandings (which will be commented on in section 5 and the concluding remarks).
The real power of SLFT emerges in models of \textit{selfinteracting} massive
vector mesons where the preservation of 2nd order renormalizability requires
the compensatory presence of a coupling to a Hermitian scalar $H$ (Higgs)
field\footnote{This (and not SSB) is the \textit{raison d'\^{e}tre for the
}$H$ (section 5).} and imposes a Lie-algebra structure on the leading terms in
the $A_{\mu}~$self-interactions. In section 5 we will provide the arguments.
An important property of the previously introduced pl and sl vector potential
and its scalar escort $\phi(x,e)$ is their linear relation
\begin{equation}
A_{\mu}(x,e)=A_{\mu}^{P}(x)+\partial_{\mu}\phi,\quad\phi=-\frac{1}{m}a
\label{escort
\end{equation}
This property justifies to call the $\phi^{\prime}s$ "escorts" of the sl
potential, they share the same degrees of freedom. The appearance of the
escort in form of a derivative is a consequence of Poincar\'{e}'s lemma. The
linear relation between fields corresponds to that between intertwiners ($J$
as before)
\begin{equation}
J_{\mu}^{\ \ \nu}v_{\nu}(p)=v_{\nu}-p_{\mu}\frac{(ve)}{(pe)_{i\varepsilon}
\end{equation}
which follows directly from the definition (\ref{F}). Each field contains the
full information of the ($m,s=1$)~Wigner representation; the encoding of $s=1$
into a scalar is only possible within sl.
It is not accidental that the massive vector potentials which result from
"fattening" their unique massless counterpart play a distinguished role in the
new SLFT renormalization theory. Their smooth connection represents the higher
spin analog of the smooth relation between $s<1$ massless fields and their
massive counterpart. The weakening of localization is necessary to preserve
this smoothness in the presence of change of the number of degrees of freedom.
In the massless limit the $A_{\mu}^{P}(x)$ and $\phi(x,e)$ diverge as
$m^{-1}~$whereas the $A_{\mu}(x,e)$ and $a(x,e)$ stay infrared finite. The
relation ($u~$was introduced in (\ref{6})
\begin{align}
\partial^{\mu}A_{\mu} & =-ma,~~d_{e}A_{\mu}=\partial_{\mu}u\label{c}\\
u & =-m^{-1}d_{e}a
\end{align}
leads to a divergence-free massless vector potential (Lorentz
condition\footnote{Note that this is an operator identity and not an imposed
gauge condition.})~and a relation between two massless 1-forms in the de
Sitter space of spacelike directions (that which remains of (\ref{escort})).
The purpose of the mass factors is to preserve the relation $d_{sd}=d_{cl
~$for all sl fields. The massless limit of $u$ is logarithmically divergent.
Escorts (whose number increase with $s$) do not contain new degrees of freedom
since, as the pl $A^{P},~$they are linear in the Wigner $s=1$
creation/annihilation operators $a^{\#}(p,s_{3})$ and only differ in their
intertwiners. Rearrangements of degrees of freedom are quite common in quantum
mechanical many-body problems\footnote{A well-known case is the appearance of
Cooper pairs encounters in passing to the low temperature superconducting
phase. Without this rearrangement classical vector potentials would not become
short range inside a superconductor (the London effect).}. Escorts are
rearranged $s=1$ degrees of freedom which carry the full content of ($m>0,s$)
Wigner representations.
For $s\geq2$ the sl fields lead to new properties. As a result of a possible
relation with gravitation the case $s=2~$is of special interest. The
intertwiner of spin $s$ Proca potentials (the $P$ in $A_{...}^{P}$ refers to
Proca or alternatively to pointlike) must be a divergence- and trace-free
symmetric tensor; this is a consequence of the way the $2s+1$ component
subspace of spin is embedded in the $3s$-fold tensor product. Hence the
intertwiners $v_{\mu_{1}\dots\mu_{s}}(p,s_{3})\ $convert the symmetric
trace-free $s$-fold tensor product of three-component spin $1~$polarization
vectors into covariant tensors of tensor-degree $s.$
For the momentum space $s=2~~$2-pfct one obtains
\begin{equation}
M^{A_{\mu\nu}^{P},A_{\kappa\lambda}^{P}~}(p)=\frac{1}{2}\left[ \pi_{\mu
\kappa}\pi_{\nu\lambda}+\pi_{\mu\lambda}\pi_{\nu\kappa}\right] -\frac{1
{3}\pi_{\mu\nu}\pi_{\kappa\lambda
\end{equation}
where the numerical factors have their combinatorial origin in the symmetry
and tracelessness and hence depend on the degrees of freedom. The sl 2-pfcts
are of the same algebraic form and result by substituting $\pi_{\mu\nu
}\rightarrow E_{\mu\nu}(-e,e)$ \cite{MRS1,MRS2}. As for $s=1~$this can be seen
by passing from the Proca potential to the field strength ($as$ stands for
antisymmetrisation)
\begin{equation}
F_{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}=\underset{\mu\leftrightarrow\nu}{as
~\partial_{\mu_{1}}\partial_{\mu_{2}}A_{\nu_{1}\nu_{2}}^{P
\end{equation}
and using the two-fold momentum space $I$ operation to pass from the field
strength to the potentials. Note that the symmetry of the Proca potential
reduces the anti-symmetrization to a pairwise operation $\mu_{i
\leftrightarrow\nu_{i}.$ The resulting permutation properties of the resulting
$F$ are those of the linearized Riemann tensor.
The new phenomenon for $s>1$ is that the massless limit of this field strength
is not the same as that obtained directly from the massless $h=\pm2$ Wigner
representation. Correspondingly the sl potential associated with the massless
limit of $F^{s=2}$ is different from that of$~F^{\left\vert h\right\vert =2}$
\begin{align}
A_{\nu_{1}\nu_{2}}(x,e) & =(I_{e}^{2}F_{\mu_{1}\mu_{2}\nu_{1}\nu_{2}
^{s=2})(x)e^{\mu_{1}}e^{\mu_{2}}\\
A_{\nu_{1}\nu_{2}}^{(2)}(x,e) & =(I_{e}^{2}F_{\mu_{1}\mu_{2}\nu_{1}\nu_{2
}^{\left\vert h\right\vert =2})(x)e^{\mu_{1}}e^{\mu_{2}
\end{align}
This means in particular that the massive $s=2$ sl potential obtained by
fattening the $A^{(2)}~$is not the same as $A$ although both account correctly
for the $2s+1$ spin degrees of freedom and share their Wigner-Fock Hilbert
space. The massless limit of $A$ splits into the direct sum of the two
$\left\vert h\right\vert =2$ degrees of freedom and the $h=0$ contribution
which is the remnant of the $s_{3}=0$ component. Conserved currents and
stress-energy tensors preserve the number of degrees of freedom by converting
the $\pm s_{3}$ components into $\left\vert h\right\vert =s_{3}$ helicities of
a Wigner-Fock tensor product space.
In order to show how these results are related to the van Dam-Veltman-Zakharov
discontinuity problem one must look at some details. Whereas fattening and
taking the massless limit connect the 2-pfct of the 2-component massless
helicity $\left\vert h\right\vert =2$ potential$~A^{(2)}$ with that of its
5-component $s=2$ by deforming the momenta of the 2-pfct between
$H_{m}^{\uparrow}$ and $V^{\uparrow},$ the massless limit of $A$ is a cul de
sac from which a return to the original massive pure $s=2$ tensor potential is
not possible.
The relation between the massless limit of $A$ with that of $A^{(2)}$ are
easily seen to have the following form
\begin{align}
A_{\mu\nu}^{(2)}(x,e) & =A_{\mu\nu}(x,e)+\frac{1}{2}E_{\mu\nu
(e,e)A^{(0)}(x,e)\label{E}\\
E_{\mu\nu}(e,e) & =\eta_{\mu\nu}+(e_{\mu}\partial_{\nu}+e_{\nu}\partial
_{\mu})I_{e}+e^{2}\partial_{\mu}\partial_{\nu}I_{e}^{2}\nonumber
\end{align}
where the momentum space $E_{\mu\nu}~$has been rewritten as an
integro-differential operator acting on a scalar sl field and the massless
limit of $A^{(0)}~$is a (properly normalized) scalar escort. Combining this
relation with that between the $s=2~$pl field $A^{P}$, its sl counterpart $A$
and the derivatives of escorts (the $s=2$ analog of (\ref{escort})) one
obtain
\[
A_{\mu\nu}^{P}=A_{\mu\nu}+\hbox{derivatives of escorts}
\]
one concludes that in the adiabatic limit the interaction between "massive
gravitons" and a trace-free energy-momentum tensor source $T_{\mu\nu}$ is
\cite{MSY2}
\begin{align}
\lim_{m\rightarrow0}\int A_{\mu\nu}^{P}T^{\mu\nu} & =\lim_{m\rightarrow
0}\int A_{\mu\nu}T^{\mu\nu}=\int(A_{\mu\nu}^{(2)}-\frac{\eta_{\mu\nu}
{\sqrt{6}}\varphi)T^{\mu\nu}\\
where\text{ ~}\varphi(x) & =\sqrt{\frac{3}{2}}\lim_{m\rightarrow0
a^{(0)}(x,e)
\end{align}
The independence of the integrated massless $A^{(2)}$ contribution from the
string direction follows from
\begin{align}
\partial_{e_{\kappa}}A_{\mu\nu}^{(2)} & =m^{-1}(\partial_{\mu}A_{\kappa\nu
}^{(2)}+\partial_{\nu}A_{\mu\kappa}^{(2)})\\
\partial_{e_{\kappa}}J_{\mu}^{~~\nu} & =-\frac{p_{\mu}}{(pe)_{i\varepsilon
}J_{\kappa}^{~~\nu}\nonumber
\end{align}
which in turn follows from the identity in the second line (for more details
see \cite{MRS2}) and represents the $s=2$ counterpart of the relation between
de Sitter space 1-forms in (\ref{c}).
The result confirms the van Dam-Veltman-Zakharov discontinuity: the massless
limit of massive gravity differs from the result obtained directly with
massless gravitons. But different from Zakharov's calculation which identifies
this contribution as being the relic of a unphysical gauge theoretical degrees
of freedom, the present calculation shows that it is really the massless
footprint of the physical $s_{3}=0$ spin component. For the traceless
stress-energy tensor of photons the last contribution vanishes whereas for
couplings to matter (mercury perihelion) it remains.
This calculation permits a straightforward extension to any spin. The relation
between the Proca potential, its sl counterpart and the associated sl escorts
reads
\begin{equation}
A_{\mu_{1}\dots\mu_{s}}^{P}=A_{\mu_{1}\dots\mu_{s}}+sym.(\partial_{\mu_{1
}\phi_{\mu_{2}\dots\mu_{s}}+\partial_{\mu_{1}}\partial_{\mu_{2}}\phi_{\mu
_{3}\dots}+\dots+\partial_{\mu_{1}}\dots\partial_{\mu_{s}}\phi) \label{sl
\end{equation}
where the $\phi_{\mu_{1}\dots\mu_{i}}~$is an $s-i$ fold iterated line integral
along $e$ of the spin $s~$Proca potential and the symmetrization is over all
indices and the $\phi$ are already symmetric by construction. For our purposes
it is more convenient to use a different basis of escorts which are obtained
by descending from the sl $A_{\mu_{1}\dots\mu_{s}}$ in terms of divergencies
\begin{align}
& A_{\mu_{1}\dots\mu_{s}}^{P}=A_{\mu_{1}\dots\mu_{s}}-sym.(\frac
{\partial_{\mu_{1}}}{m}a_{\mu_{2}\dots\mu_{s}}^{(s-1)}+\frac{\partial_{\mu
_{1}}\partial_{\mu_{2}}}{m^{2}}a_{\mu_{3}\dots}^{(s-2)}+\dots+\frac
{\partial_{\mu_{1}}\dots\partial_{\mu_{s}}}{m^{s}}a^{(0)})\label{recur}\\
& ma_{\mu_{r}\dots\mu_{s}}^{(s-r)}=-\partial^{\mu}a_{\mu\mu_{r}\dots\mu_{s
}^{(s-r+1)},~\ ~~a_{\mu_{1}\dots\mu_{s}}^{(s)}:=A_{\mu_{1}\dots\mu_{s
}\nonumber
\end{align}
The second line shows that the $a$ escorts start from the sl potential and
descend by differentiation instead of descending from $A^{P}$ by
line-integration. The $a$ have the same dimension $d_{sd}=1=d_{eng
,~d_{infr}=0$, and are linear combinations of the $\phi$ escorts. As long as
$m>0~$each escort carries the full content of the Wigner spin $s$~representation.
Although the $a^{\prime}s~$have a massless limit they still do not decouple.
The van Dam-Veltman-Zakharov discontinuity shows that for $s=2$ the
$\left\vert h\right\vert =2$ and $h=0$ contributions stay together and have to
be separated with the help of an integro-differential operation (\ref{E}). The
analogous situation in the general case is that the even and odd $s_{3
~$contributions remain coupled among themselves and can only be split in terms
of their helicity content by the use of such integro-differential operations
\cite{MRS2}. Naturally one can obtain a spin $s$ vector potential from
fattening a massless helicity $h$ potential if $h=s.$
The tensor $v_{\mu_{1}..\mu_{\left\vert h\right\vert }}(q,e)$ which appears in
the relation of the helicity $h$ tensor field $A_{\mu_{1}..\mu_{\left\vert
h\right\vert }}^{(\left\vert h\right\vert )}$ and the Wigner operator
$a^{\#}(q,h)$ (which extends the construction of $A_{\mu\nu}^{(2)}~$in
(\ref{E}) to arbitrary helicity $h).$ This $e$-dependent polarization tensor
$v_{..}(q,e)~$replace the only up to re-gauging defined polarization tensor.
If used in Weinberg's soft scattering limit of a massless particle with
momentum $q$ scattering on $n$ massive particles with momenta $p_{i},i=1,..n$
\cite[4.1]{Ra-Ta}, one obtains the same conclusions except that the gauge
theoretic argument is replaced by the $e$-independence which follows in first
order from the fact that the directional derivative with respect to $e$ on
these polarization tensors can be written as a spacetime derivative
$\partial_{\mu}$ acting on such a tensor in analogy to (\ref{6}). The use of
sl polarization tensors instead of gauge symmetry is required by using
positivity which guaranties the exclusive appearance of physical degrees of freedom.
The weakness of Lagrangian constructions of conserved currents and
stress-energy tensors is that with the exception of low spins there is no
guaranty that the so obtained classical expressions have the correct
commutation relations with the quantum fields. It is much safer and easier to
\textit{start from the commutation relations between Wigner's generators} of
the Poincar\'{e} group and the Wigner particle operators $a^{\#}(p,s_{3})$ and
to rewrite them with the help of the intertwiners into covariant commutation relations.
\subsection{Infinite spin revisited}
A simple illustration of such an "intrinsic quantum" construction of the
stress-energy tensor has been recently presented in \cite{PL}. One starts from
the expressions of the infinitesimal generators of translation $\mathbf{P
_{\mu}~$and Lorentz generators $\mathbf{M}_{\mu\nu}~$in terms of the Wigner
operators $a^{\#}(p,s_{3})
\begin{align}
\mathbf{P}_{\mu} & =\int\sum_{s_{3}}a^{\ast}(p,s_{3})p_{\mu}a(p,s_{3
)d\mu(p)~\label{P}\\
\mathbf{M}_{\mu\nu} & =-i\int(\delta_{s_{3}s_{3}^{\prime}}p\wedge
\partial_{p}+d(\omega)_{s_{3}s_{3}^{\prime}}^{t})_{\mu\nu}a^{\ast
(p.s_{3})a(p,s_{3}^{\prime})d\mu(p) \label{M
\end{align}
The first step is two rewrite the contribution of the spin component $s_{s}$
to$~\mathbf{P}_{\mu}\,$as
\begin{align}
& \mathbf{P}_{\mu}=\int\int d\mu(p)d\mu(p^{\prime})\sum_{s_{3},s_{3}^{\prime
}}(p_{\mu}a^{\ast}(p,s_{3})\delta_{s_{3}s_{3}^{\prime}}(2\pi)^{3
\delta(\mathbf{p}-\mathbf{p}^{\prime})(p_{10}+p_{20})a(p^{\prime
,s_{3}^{\prime})\label{P1}\\
& (2\pi)^{3}\delta(\mathbf{p}-\mathbf{p}^{\prime})=\int e^{-i(\mathbf{p
-\mathbf{p}^{\prime})x}d^{3}x=\int e^{-i(p-p^{\prime})x}d^{3}x \label{P2
\end{align}
where in the second line used the cancellation of the $p_{0}~$components.
What remains to do is to convert the Wigner operators via intertwiners into
the covariant fields. For this one uses their completeness relation in order
to write the unit operator in spin space as
\[
g^{MN}\nu_{Ms_{3}}\overline{\nu_{Ns_{s}^{\prime}}}=\delta_{s_{3}s_{3}^{\prime
}
\]
where $M$ and $N~$represent the multi-tensor indices of the intertwiner. What
remains is to use the Fourier transform (\ref{P2}) and pass from the Wigner
operators to the fields. Using the fact that the $a^{\ast}a^{\ast}\ $and $aa$
contributions vanish as a result of the presence of$~\overleftrightarrow
{\partial}_{0}$ and that $aa^{\ast}$ terms are absent in Wick-ordered products
one verifies that
\begin{equation}
\mathbf{P}_{\mu}=\int\tilde{T}_{\mu0}(x)d^{3}x,~~~\tilde{T}_{\mu\nu
(x)=-\frac{1}{4}\int:A_{\mu_{1}..\mu_{s}}^{P}(x)\overleftrightarrow{\partial
}_{\mu}\overleftrightarrow{\partial}_{0}A^{P,\mu_{1}..\mu_{s}}(x):
\end{equation}
where $\tilde{T}_{\mu\nu}$ is a contribution to the stress-energy tensor.
The full tensor density which generates all Poincar\'{e} transformations is of
the for
\begin{equation}
T_{\mu\nu}=\tilde{T}_{\mu\nu}+\partial^{\rho}\Delta_{\mu\nu,\rho
\end{equation}
To compute the second contribution, which is also a bilinear expression in the
$A^{P}~$tensor fields, one starts from the bilinear expression for $M_{\mu\nu
}$ in terms of the $a^{\#}~$Wigner operators which also contains a
contribution the infinitesimal part of Wigner's little group. The
representation of the Poincar\'{e} group generators in terms of pl
stress-energy tensors may be rewritten in terms of their sl counterparts
\cite{MRS2}. For recent results about constructing infinite spin fields and
their E-M tensors as Pauli-Lubanski limits we refer to \cite{PL}.
Rehren's construction of infinite spin quantum fields in terms of the
Pauli-Lubanski limit is the most natural one; it corresponds to the use of the
distinguished tensor potentials obtained by fattening its unique massless
counterpart at fixed spin, except that it goes into the opposite direction at
fixed P-L parameter\footnote{In this way it selects a unique countable family
of fields within the equivalence class of all relatively causal fields
constructed in \cite{MSY2}.}. The tensor field disappears in this limit and
what remains (after appropriate adjustments) is the infinite family of escorts
with arbitrary high tensor degree.
The nonexistence of the infinite spin tensor potential $A_{\mu_{1}\mu
_{2}...\infty}$ accounts for the absence of a relation which converts the
differential $d_{e}$ into a spacetime divergence as well as the absence of a
gauge theoretic formulation. This is the reason why infinite spin matter
cannot interact with ordinary quantum matter \cite{dark}.
Of physical relevance is the existence of conserved currents and
energy-momentum tensor in the sense of bilinear forms \cite{PL}. Hence
expectation values of E-M tensors and possible gravitational backreaction
remain physically meaningful.
\section{Causality and the Velo-Zwanziger conundrum}
The Velo-Zwanziger conundrum is an alleged causality paradox which arose from
the naive expectation that $s\geq1$ quantum fields, whose free field equations
are modified by linear pl couplings to external potentials, maintain their
causal propagation. Formally it is closely related with the Weinberg-Witten
No-Go theorem which excludes the existence of higher helicity conserved pl
currents. This connection turns out to be useful for the solution of the V-Z conundrum.
\subsection{Recalling the solution of the Weinberg-Witten problem and the
associated local charges}
In \cite{MRS2} it was shown that for massive $s\geq$ $1$ free field one can
construct sl tensor potentials whose associated conserved sl currents have
finite massless limits even when according to the Weinberg-Witten (W-W)
theorem physical (gauge-invariant) pl currents do not exist.
In the massive case both the pl and sl currents are members of the same local
equivalence class which consist of all Wick-ordered composites of pl fields
and their related sl counterparts. Their relative causality reads
\begin{equation}
\left[ j_{\mu}^{P}(x),j_{\nu}(x^{\prime},e)\right] =0\text{~}for\text{
}x\,{\lower1pt\hbox{\LARGE$\times$}}\,\mathcal{S}(x^{\prime},e),~\mathcal{S
(x^{\prime},e)=x^{\prime}+\mathbb{R}_{+}e,~e^{2}=-1\text{ }or~0 \label{ten
\end{equation}
Their charge-densities differ by spatial divergencies and hence they share the
global $U(1)~$generators. In the massless limit the sl spin potentials pass
continuously to their massless counterpart (not possible with pl potentials)
which act in the conformally covariant helicity Wigner-Fock space. The sl
currents are bilinear in the charge carrying sl potentials\cite{MRS2}.
The two currents (\ref{ten}) share the same "engineering" dimension (classical
dimension in terms of mass units) $d_{cl}=3,$ but possess different short
distance scaling dimensions $d_{sd}(j_{\mu}^{P})=2(s+1)+1$ and $d_{sd}(j_{\mu
})=d_{cl}=3;~$this accounts for the fact that the sl $j_{\mu}$ allows a
massless limit whereas $j_{\mu}^{P}$ diverges as $j_{\mu}^{P}\overset
{m\rightarrow0}{\sim}m^{-2s}\ $(the W-W obstruction). As expected, the sl
$j_{\mu}(x,e)$ admits a massless limit in which the $2s+1$ spin degrees of
freedom decompose into a direct sum of $s$ helicity and one scalar
contribution so that the Wigner-Fock space turns into a tensor product of
helicity spaces$.$
The presentation concerning the relation between pl and sl conservation laws
in \cite{MRS2} was mainly focussed on the stress-energy tensors (SET); in the
following we present the corresponding problem for conserved currents. A
convenient illustration is provided by the sl current with the lowest W-W
helicity $h=1$ as follows$.$\
Using the linear relation $A_{\mu}^{P}(x)=A_{\mu}(x,e)-\partial_{\mu
\phi(x,e)~$between pl and its canonically associated sl field and the gradient
of its escort derived in the previous section (\ref{escort}) one finds that
the pl and sl currents are related as (omitting Wick-ordering
\begin{align}
& j_{\mu}^{P}=iA^{P\nu}(x)^{\ast}\overleftrightarrow{\partial_{\mu}}A_{\nu
}^{P}(x)=j_{\mu}(A(x,e))+j_{\mu}(a(x,e))+\partial^{\kappa}C_{\kappa\mu
}\label{j}\\
& a(x,e)=m\phi(x,e),~C_{\kappa\mu}=iA_{\kappa}^{\ast}\overleftrightarrow
{\partial}_{\mu}\phi~+i\phi^{\ast}\overleftrightarrow{\partial}_{\mu
\partial_{\kappa}\phi+h.c.\nonumber
\end{align}
The first two contributions are conserved sl currents whose massless limit
correspond to the current of the complex $s=1~$sl field $A_{\nu}(x,e)$ (which
replaces the nonexistent pl W-W current), and that of a complex scalar field
$a(x)=\lim_{m\rightarrow0}a(x,e).$ The $m^{-2}$ W-W obstruction $C$ does not
contribute to the global charge.
The "obstructing" contribution $\partial^{\kappa}C_{\kappa\mu}$ carries both
the leading short distance dimension $d_{sd}=5~$and and the $m^{-2}$
divergence which is the culprit for the W-W problem. This kind of
decomposition into $s~$conserved$~d_{sd}(j)=3~$sl currents, which pass for
$m\rightarrow0$ to $s$ sl helicity currents and a pl current of a scalar
particle, exists for every spin $s\geq1.$
Using the free field equation for $A_{\mu}$ and $\phi$ one verifies that
$C$-contribution is of the form of a spatial divergence and hence does not
contribute to the infinite volume limit of the charges \cite{MRS2
\begin{equation}
Q(A^{P})=Q(A)+Q(a)
\end{equation}
i.e. the massless limit decomposes the three spin degrees into the $\pm1$
helicities of $A_{\mu}$ and $h=0$ carried by $a.~$Before this limit both sl
fields $A_{\mu}$ and $a$ account for the three $s=1$ degrees freedm.
For pl currents there exists extensive literature on the problem of relation
between conserved currents, local charges, and their global limits
\cite{K-R-S}\cite{E-S}\cite{S-St}\cite{Req}. The basic idea is to start from a
conserved current and defin
\begin{align}
Q & =\lim_{R\rightarrow\infty}Q(f_{R},f_{d}),~~Q(f_{R},f_{d}):=j_{0
(f_{R},f_{d})\label{test}\\
f_{R}(\mathbf{x}) & =\left\{
\begin{array}
[c]{c
1\ \ \ \ \ \ \ ~\left\vert \mathbf{x}\right\vert <R\\
\ \ 0~~\ \ \left\vert \mathbf{x}\right\vert >R+r
\end{array}
\right. ~\label{test2}\\
f_{d}(x_{0}) & \geq0.~~suppf_{d}\subset\left\vert t\right\vert <d,~\int
f_{d}(x_{0})dx_{0}=1
\end{align}
One then uses the conservation law of the current to show that the commutator
$\left[ Q(f_{R},f_{d}),A\right] $ for $A$ $\mathcal{\in A(O})~$is
independent of the choice of the smearing function $g(x)=f_{R}(\mathbf{x
)f_{d}(x_{0})$ as long as $\mathcal{O}$ remains inside their timelike extended
shell structure (\ref{test2}).
The local charge $Q(g)$ which measures the charge of an operator $A$ localized
in $\left\vert \mathbf{x}\right\vert <R$ converges towards the generator
$Q~$of the global $U(1)$ symmetry. The concept of a local charge content
becomes problematic in case of sl currents since the use of a rigid spacelike
direction $e$~does not allow the causal separation of $j_{0}(f_{R},f_{d})$
from the localization region of $A.$ The heuristic idea for achieving such a
separation would be to "comb" the strings emanating from the shell between $R$
and $R+r$ into different directions so that they remain causally separated
from $A$ $\mathcal{\in A(O}).$ But then the strings emanating from poins
inside the shell would have to move to spacelike infinity outside the larger
sphere and violate the localization \textit{inside} the larger sphere
In view of a recent proof of the so-called \textit{split property}
\cite{L-M-P-R}, which is known to secure the local implementation of global
symmetries in massless $h\geq1$ \cite{D-L}, there is no problem with the
existence of local charges for QFT's with global symmetries; what is not clear
is whether such charges can be described in terms of conserved currents.
Meanwhile K.-H. Rehren informed me that his student M. Heep constructed local
charges from sl currents by appropriate use of conformal transformations. The
idea is to construct a local charge operator localized in a half-space, that
is then mapped to a sphere by a conformal transformation. In this way the
strings become "curled" and end in the north pole.
Hence the W-W No-Go theorem excludes pl currents, but does not affect the
causal localizability of charges in arbitrary small spacetime regions.
\subsection{The V-Z conundrum arises from an incorrect implementation of
causality}
A simple class of models for a critical examination of the V-Z conundrum is
provided by linear couplings of conserved currents to external vector
potentials the relevant property of the sl current is its lowered short
distance dimension. A suitable setting for such problems is obtaind in terms
of Bogoliubov's definition of the S-matrix and interacting local fields in
terms of adiabatic limits of the Bogoliubov $S(g)$-functional\footnote{Our use
of the Bogoliubov's formalism is close to that in \cite{Du-Fr}\cite{Wre
\cite{F-R}
\begin{align}
~ & S:=\lim_{g(x)\rightarrow g}S(g),~~~S(g)=T\exp
{\displaystyle\int}
g(x)L_{int}(x)d^{4}x\label{B}\\
& A(x)|_{L_{int}}=\lim_{g(x)\rightarrow g}\frac{\delta}{i\delta f(x)
S^{-1}(gL)S(g(x)L_{int}(x)+fA)|_{f=0} \label{C
\end{align}
Here the interaction density $L_{int}$ is a Wick-ordered product of not more
than 4 free fields from the class of Wick-ordered composites of free fields
and $A|_{L_{int}}$ the interacting counterpart of $A(x)~$which is either a
free field or a Wick-ordered product of free fields (the terminology "free" is
used for linear fields and Wick polynomials). The interacting field has the
form of a power series in $g$ with retarded products of $n$ $L^{\prime}s$
which is retarded in $A(x).$ The linear Bogoliubov map $A\rightarrow
A|_{L_{int}}$ does not preserve the algebraic structure but it maintains the
property of causal separability. Hence fields constructed in this way maintain
causality, and the solution of the V-Z problem consists in the proper
computation of the interacting fields via (44).
The class of interactions with external potentials to be studied is of the
form $L_{int}=L^{P}=U^{\mu}j_{\mu}^{P}$ with $U^{\mu}$ a external (classical)
vector potential and $j_{\mu}^{P}$ a conserved current as before. For the
current of a scalar complex free field $\varphi$ there is no problem; its
conserved current has $d_{sd}(j_{\mu}^{P})=3$ and hence (with $d_{sd
(L_{int})=3$) renormalizable. This is the model on which Velo and Zwanziger
base their propagation picture: namely the scalar field obeys a linear field
equation which is linear\footnote{Here and in the sequel linear stands for
linear in $U_{\mu}$ and its derivtives.} in $U^{\mu}$ and they expect
(erroneously, as will be seem) that this holds independent of spin.
For $s=1$ the $d_{sd}(L^{P})=5$ and hence the pl model is nonrenormalizable.
To reduce the $d_{sd}$ from 5 to 4 one uses the relation (\ref{j}) which
rewritten in terms of the interaction density read
\begin{align}
~\text{ } & L^{P}=j_{\mu}^{P}U^{\mu}=L-\partial^{\kappa}V_{\kappa
},~C_{\kappa\mu}=iA_{\kappa}^{\ast}\overleftrightarrow{\partial}_{\mu
\phi~+i\phi^{\ast}\overleftrightarrow{\partial}_{\mu}\partial_{\kappa
\phi+h.c\label{m}\\
& L:=j_{\mu}^{s}U^{\mu}-C_{\kappa\mu}\partial^{\kappa}U^{\mu},~~V_{\kappa
}=-C_{\kappa\mu}U^{\mu},\text{ }S^{(1)}\overset{a.l.}{=}\int L^{P}d^{4}x=\int
Ld^{4}x\nonumber
\end{align}
where the decomposition (\ref{j}) of $j_{\mu}^{P}$ was used. Since
$d_{sd}(C_{\kappa\mu})=4$ the power counting bound $d_{d}(L)=4$ holds, the
model is renormalizable and its first order S-matrix $S^{(1)}$ (the adiabatic
limit of the interaction density) is the same for the two densities and hence
string-independent (the suitably defined adiabatic limit of the $\partial V$ vanishes).
The decomposition of $j_{\mu}^{P}$ (\ref{j}), which previously served to solve
the W-W problem (by converting the pl current into its for $m\rightarrow0$
regular sl counterparts and a $C$-term, which carries the $m^{-2}$ mass
divergence but does not contribute to the global charge\footnote{Using
conformal invariance of massless helicity representations one can also show
the existence of local charges (see remarks in previous subsection).}), is now
used to solve the V-Z causality problem. To achieve this one uses the fact
that the $C$-term is a 4-divergence and disappears in the adiabatic limit
which represents the S-matrix.
The two interaction densities $L^{P}$ and $L$ share the same S-matrix; whereas
the pl $L^{P}(x)$ side insures that the S-matrix is that of a causal
interaction, the sl $L(x,e)$ guaranties the renormalizability of $S^{(1)}.$
The $L(x,e)~$together with $V_{\mu}(x,e)$ forms what will be referred to as a
$L,V_{\mu}$ pair. The first order S-matrix (\ref{m} second line), which is the
adiabatic limit of the interaction density, is the same for $L^{P}$ and $L~$
referred to as the linear relation (\ref{m}). The $L^{P}$ represents the
heuristic content of the interaction, but as a result of its bad short
distance behavior it is not suitable for perturbative calculations. The short
distance improved $L$ weakens the localization but retains enough of it to
keep fields causally separated and to maintain scattering theory.
The remaining problem is the extension of this idea to higher order. For
convenience of notation one uses a differential formulation of pl localization
in the form of $e$-independence in the form $d_{e}(L-\partial V)=0.$ It is
convenient to use lightlike $e^{\prime}s$ since in this case no smearing is
needed. The problem in higher order is the time-ordering. For the
$e$-independence of the S matrix one needs the $\partial$ to act outside the
time-ordering e.g
\begin{align}
& d_{e}(TL(1)L(2)-\partial_{1}^{\mu}TV_{\mu}(1)L(2)-\partial_{2}^{\nu
}TL(1)V_{\nu}(2)+\partial_{1}^{\mu}\partial_{2}^{\nu}TV_{\mu}(1)V_{\nu
}(2)\overset{?}{=}0\label{indep}\\
& ~d_{e}T(L(1)-\partial V(1))(L(2)-\partial V(2))=0\nonumber
\end{align}
and higher order extensions involving one $V_{\mu}$ and $n-1$ $L^{\prime}s.$
This is generally not possible without creating "obstructions" of the form of
delta contributions of the form $\delta(x_{1}-x_{2})d_{e}L_{2}(x_{1},e)~$which
are quadratic in $U_{\mu}$ and its derivatives. Higher order violations may
lead to contributions of higher polynomial degree in $U_{\mu}$ and
derivatives; it is a characteristic property of obstructions in models of
external potential interactions that all obstructions remain bilinear in the
quantum fields.
These obstructions are absorbed in the form of induced contributions into a
modified Bogoliubov formalism by definin
\begin{equation}
L_{tot}=L+\frac{1}{2}L_{2}+..\frac{1}{n!}L_{n}+.. \label{obst
\end{equation}
where $L_{n}$ is of polynomial degree $n$ in $U_{\mu}$ and its derivatives and
remains quadratic in free fields. Note that induced contributions do not
increase the number of parameters and hence must be distinguished from
counterterms of pl renormalization theory.
In the $s=1$ model (\ref{m})~the $L_{2}$ contribution \textit{can
alternatively be encoded into a redefinition of time ordering}
\begin{align}
& T_{0}\partial_{\mu}\phi(x_{1})\partial_{\nu}\phi(x_{2})\equiv\partial_{\mu
}\partial_{\nu}T_{0}\phi(x_{1})\phi(x_{2})\label{2pt}\\
& T\partial_{\mu}\phi(x_{1})\partial_{\nu}\phi(x_{2})=T_{0}\partial_{\mu
\phi(x_{1})\partial_{\nu}\phi(x_{2})+icg_{\mu\nu}\delta(x_{1}-x_{2
)\nonumber\\
& \partial^{\mu}T\partial_{\mu}\phi(x_{1})\partial_{\nu}\phi(x_{2
)-T\partial^{\mu}\partial_{\mu}\phi(x_{1})\partial_{\nu}\phi(x_{2
)=(1+c)\partial_{\nu}\delta(x_{1}-x_{2})\nonumber
\end{align}
and the validity of (\ref{indep}) for the $T\,\ $time-ordering requires to set
$c=-1$.~For $s>1$ The kinematic $T_{0}$ time-ordering contains more
derivatives and one has accordingly more $c^{\prime}s$ which must be
numerically adjusted in such a way that the $T$ time-ordering satisfies the
higher order pair requirements beyond (\ref{indep}).
The following side-remark maybe helpful for the later extension of SLFT to a
full QFT. The second order $A_{\mu}A^{\mu}\varphi^{\ast}\varphi$ term of
scalar QED within the new SLFT can either be viewed as induced or encoded into
a change of time-ordering for the derivatives of the complex scalar field. But
not all obstructions can be absorbed in this way. The $H$-selfinteractions of
the Higgs model is a \textit{genuine} second order induced term which results
exclusively from the implementation of the positivity and causality principle
of QFT (rather than from an imposed Mexican hat interaction).
The verification of the higher order pair relations will be deferred to a more
complete treatment of external potential problems. The expected result is:
\textbf{Conjecture }Couplings of conserved currents to external potentials
fulfill the higher order time-ordered $L,V_{\mu}~$relation. For $s=1$ the
resulting field equations are quadratic in $U_{\mu}$ whereas for $s>1$ they
are of infinite order (expected since $d_{sd}(L)\geq5$ ).$\ $
The form of the linear causal field equations (in particular the higher order
$U$ contributions) is determined by the form of the induced contributions.
The external potential formalism and its formal connection with the solution
of the W-W problem works in an analogous way for $s=2.$ The sl potential
$A_{\mu\nu}(x,e)$ has two escorts, a vector $a_{\mu}$ and a scalar escort $a$
which can be chosen in such a way that the operator dimension for all fields
is identical to their classical dimension $d_{cl}=1$ (or $3/2$ for
half-integer spin
\begin{align}
& A_{\mu\nu}^{P}=A_{\mu\nu}+m^{-1}(\partial_{\mu}a_{\nu}+\partial_{\nu
a_{\mu})+m^{-2}\partial_{\mu}\partial_{\nu}a\label{dec}\\
& j_{\mu}^{P}(x)=iA_{\kappa\lambda}^{P\ast}\overleftrightarrow{\partial_{\mu
}}A^{P\kappa\lambda}=j_{\mu}^{s}(x,e)+\partial^{\kappa}C_{\kappa\mu
}\nonumber\\
& j_{\mu}^{s}(x,e)=iA_{\kappa\lambda}^{\ast}\overleftrightarrow{\partial
_{\mu}}A^{\kappa\lambda}-2ia_{\kappa}^{\ast}\overleftrightarrow{\partial_{\mu
}}a^{\kappa}+ia^{\ast}\overleftrightarrow{\partial_{\mu}}a\nonumber
\end{align}
In this case $d_{sd}(L^{P})=7$ and $d_{sd}(j^{\mu})=3~$and hence $L^{P}$ is by
$3$ units beyond the power-counting bound $d_{sd}=4$. For $s=1$~the
$\partial^{\kappa}C_{\kappa\mu}$ carries the highest $d_{sd}=7$
contribution.$~$After an additional linear disentanglement between $A_{\mu\nu
}$ and $a$ one arrives at a decomposition of $j_{\mu}^{s}$ which in the
massless limit represents the $h=2,1,0$ helicity contributions \cite{MRS2}.
The use of this decomposition in the rewriting of $L^{P}=j_{\mu}^{P}(x)U^{\mu
}$ for $s=2~$as a sl pair with $L^{P}=L-\partial V$ leads to a $d_{sd
(C_{\kappa\mu})=6~$ contribution which contains bilinear in $\phi=m^{-2}a$
terms with more than 2 derivatives. In analogy to counterterms in every order
in a nonrenormalizable full pl QFTs one expects to find induced terms in
arbitrary high orders.
It is worthwile to mention that there is also a gauge theoretic formulation in
which the linear operator relation between the sl potential and its pl
counterpart is replaced by the relation $A_{\mu}^{K}(x)=A_{\mu}^{P
(x)+\partial_{\mu}\phi^{K}$ where the $K$ refers to the Krein space and the
esort $\phi^{K}$ is the St\"{u}ckelberg negative metric pl scalar which adds
additional unphysical degrees of freedom to the indefinite Krein space. This
is the formulation of the Uni Z\"{u}rich group \cite{Scharf}\cite{Aste}
adapted to the presentation used in the present paper.
The gauge theoretic analog to the pair relation is $L^{K}=L^{P}+\partial^{\mu
}V_{\mu}^{K}.$ The model has a formal similarity with SLFT, but its pl
interpolating fields are unphysical; positivity obeying interpolating fields
are simply inconsistent with pl localization. The pair property works the same
way, one only has to replace the $d_{e}$ in (\ref{indep}) by the BRST
$\mathfrak{s}$.
However negative metric degrees of freedom lead to an unphysical realization
of appropriately nonlinear modified causal pl V-Z equation and should be
rejected inasmuch as gauge dependent pl currents have been discarted by W-W in
their No-Go theorem. Classical field theory is free of positivity requirements
and gauge theoretic causal propagation is perfectly compatible with its
principles. But the nonlinear dependence on external potential which was
overlooked by V-Z is also needed for the classical propagation of Cauchy date.
Recently there have been attempts to solve the V-Z conundrum in terms of
String Theory \cite{Por} \cite{Ra-Ta}. These authors extract a system of pl
equation in $d=3+1~$via dimensional reduction from the Virasoro algebra in 10
dimensions and found nonlinear modifications in case of constant external fields.
But the fact that there is nothing stringy about their pl equations raises the
old question: what do string-theorist really mean when they claim that their
objects are stringy in spacetime. Does their use of the terminology "string"
perhaps refer to a circular structur in a 10-dimensional target space whose
Fourier components correpond to the irreducible Wigner components of the
highly reducible superstring representation?
Their strings bear no relation with causal localization in spaceteime but
rather seem to refer to Born's quantum mechanical localization related to the
spectral decomposition of the \textbf{x} operator arises. Their use of
world-sheet and Nambu-Goto actions point into this direction and the way in
which they think of their localized objects as vibrating in space strengthens
this presumption. Causal localization in spacetime is very different (for more
see next section).
The next section explores important aspects of causal localization which,
although known to some experts, remained outside the conceptual radar screen
of most particle physicists.
\section{Particle wave functions and causal localization}
\textit{There is no concept in particle physics which led to more
misunderstandings than that of causal localization in spacetime. The strings
of String Theory obtained e.g. from quantized world-sheet or Nambu-Goto
actions bear no relation causal localization. A concept which reveals such
misunderstandings and corrects them in the clearest possible way is "modular
localization".}
\subsection{Newton-Wigner localization and its causality-providing modular
counterpart}
Wigner's theory of positive energy representations presents an interesting
meeting ground of two very different localization concepts. On the one hand
there is the quantum mechanical localization of dissipating wave packets whose
center moves on relativistic particle trajectories. Its formulation in terms
of quantum mechanical Born probabilities leads to the so-called Newton-Wigner
localization \cite{N-W}. For a scalar $m>0$ particle
\begin{align}
(\psi,\psi^{\prime})=\int\bar{\psi}\overleftrightarrow{\partial_{0}
\psi^{\prime}d^{3}x=\int\bar{\psi}_{NW}\psi_{NW}^{\prime}d^{3}x & \\
\hbox{hence}\quad\tilde{\psi}_{NW}(\mathbf{p})=(2p_{0})^{-1/2}\tilde{\psi
}(\mathbf{p}) & \nonumber
\end{align}
Hence an improper N-W eigenstate of the position operator $\mathbf{x}_{NW}$
has a mass-dependent extension of the order of a Compton wave length. In
scattering theory, where only the large-time asymptotic behavior matters, such
ambiguities in assigning relativistic quantum mechanical positions at finite
times are irrelevant; the centers of wave packets of particles move on
relativistic velocity lines and the probability to find a particle dissipates
as $t^{-3}$ along these lines for all inertial observers. In fact Wigner never
thought of his Poincar\'{e} group representation theory as an entrance into
causal QFT; for him it remained part of relativistic QM\footnote{This perhaps
explains why Wigner, inspite of his overpowering role in the development of
$20^{th}$ century quantum theory, never participated in the QED revolution and
its QFT aftermath. For him his representation theory always remained part of
relativistic quantum mechanics (the quantum mechanical Newton-Wigner
localization in section 3). An interesting discussion can be found in Haag's
memoirs \cite{mem} page 276.}.
The more recent discovery of \textit{modular localization} shows that
causality properties are dormant within Wigner's positive energy
representation theory; they are reflected in properties of dense subspaces
obtained by applying algebras of local observables $\mathcal{A(O}$ to the
vacuum state $\mathcal{H(O})=\mathcal{A(O)}\Omega~$and projecting the so
obtained dense set of states of a QFT to the one-particle subspace
$\mathcal{H}_{Wig}(\mathcal{O})=E_{1}\mathcal{H(O}).$ That such spaces are
dense in the Hilbert space (and consequently their projection in the
one-particle subspace) is a special case of a surprising discovery made in the
early in the early 60s (the Reeh-Schlieder theorem \cite{St-Wi} \cite{Haag})
which showed that the omnipresence of vacuum polarization confers to QFT
\textit{a very different notion of localization} from that of Born's quantum
mechanical setting based on position operators.
The projection $\mathcal{H}_{Wig}\mathcal{(O)}~$has the remarkable property
that it can be constructed without the assistance of QFT \textit{solely in
terms of data from Wigner's representation theory} and that in the absence of
interaction one can even revert the direction and obtain the net of causally
localized subalgebras directly from that of modular localized Wigner subspaces
\cite{BGL}.
In this way one does not only gain a more profound understanding of QFT but
one also learns that Weinberg's pure group theoretic construction of
\textit{intertwiners} starting from Wigner's representation theory is part of
a much more general setting which, if properly used, leads to an extension of
perturbative renormalizability. This important concept of modular localization
was not available during Wigner's lifetime (see remarks in the introduction).
The simplest way to see that the quantization of a relativistic classical
particle associated with the action $\sqrt{-ds^{2}}$ does not lead to a
covariant quantum theory is to remind oneself that there exists no operator
$\mathbf{x}$ which is the spatial component of a covariant 4-vector
\cite{MSY2}. The conceptual problem one is facing is better understood by
first showing that causal localization bears no relation to Born's
probabilistic quantum mechanical definition.
Starting from the quantum mechanical projectors $E(R)$ for $R\subset
\mathbb{R}^{3}$ which appear in the spectral decomposition of $\mathbf{x
_{op}$
\begin{equation}
\mathbf{x}_{op}=\int\mathbf{x}dE(\mathbf{x}) \label{Ma
\end{equation}
one has
\[
E(R)E(R+\mathbf{a})=0\quad\hbox{for}\quad R\cap(R+\mathbf{a)}=0
\]
Define $E(R+a)=U(a)E(R)U(a)^{\ast}$ for $a\in\mathbb{R}^{4}.$ Assuming that
this orthogonality relation has a causal extension in the sense that $~$
$E(R)E(R+a)=0$ for spacelike separated$~R{\lower1pt\hbox{\LARGE$\times$}
R+a~$leads immediately to clash with positivity of the energy. This follows
from the fact that the positivity of the energy leads to the analyticity of
expectation values $(\psi,E(R)E(R+a)\psi)$ for $\operatorname{Im}a_{0}>0$
which in turn implies their identical vanishing (the Schwarz reflection
principle) and with$~\left\vert \left\vert E(R)\psi\right\vert \right\vert
^{2}=0$ the triviality of such projectors $E=0.$
A slight extension of the argument reveals that it can be dissociated from the
position operator of quantum mechanics. It then states that in models with
energy positivity\textit{ it is not possible to describe causal localization
("micro-causality") in terms of projectors and orthogonality of subspaces
}\cite{Malament}. A profound intrinsic understanding of causal localization in
QFT points into a very different direction from that obtained from the
quantization of actions describing classical world lines or world sheets and
Nambu-Goto action. The problems become insurmountable of one tries to
construct actions in the presence of several of such objects and their distance.
Before presenting the relation to QFT it is worthwhile to mention a little
known fact: it is perfectly possible to construct a relativistic description
of interacting particles in relativistic QM build on macro-causality. For two
particles this amounts to Poincar\'{e} group preserving modification in the
centre of mass system, but for more particles it is more complicated
(\cite{SHPMP} section 3). Apart from the fact that it leads to a
Poincar\'{e}-invariant S-matrix, it does (unlike Schr\"{o}dinger theory) not
permit a description in terms of second quantization.
\subsection{Mathematical properties of modular localization}
To prepare the ground for causal localization it is helpful to start with some
mathematical concepts concerning relations between real subspaces $H$ (linear
combination with reals) of a complex Hilbert space $H\subset\mathcal{H
\mathbf{.}$ The symplectic complement $H^{\prime}~$of a real space is defined
as the closed real subspace ($\overline{H^{\prime}}=H^{\prime}$) defined in
terms of the imaginary part of the scalar product in$~\mathcal{H}
\begin{align}
H^{\prime} & =\left\{ \xi\in\mathcal{H};\ \operatorname{Im}(\eta
,\xi)=0~\forall\eta\in H\right\} \label{sym}\\
H_{1} & \subset H_{2}\text{ }\Rightarrow H_{1}^{\prime}\supset H_{2
^{\prime
\end{align}
which turns out to be the real orthogonal space on the real $iH$ (only real
linear combinations)
A closed real subspace $H$ is called "standard" if it is both cyclic and
separatin
\begin{align}
H\quad\hbox{cyclic:}\quad & \overline{H+iH}=\mathcal{H}\label{stan}\\
H\quad\hbox{separating:}\quad & H^{\prime}\cap H=\left\{ 0\right\}
\nonumber\\
\left( H+iH\right) ^{\prime} & =H^{\prime}\cap iH^{\prime}\nonumber
\end{align}
Cyclicity and the separation property have a dual relation in terms of
symplectic complements as written in the third line.
It is quite easy to obtain such standard spaces from covariant free fields. In
the simplest case of a scalar field the Hilbert space $\mathcal{H}$ is the
closure of the 1-particle Wigner space defined by the two-point function of
the smeared field
\begin{align}
\left( f,g\right) & =\left\langle A(f)^{\ast}A(g)\right\rangle =\int
\tilde{f}^{\ast}(p)\tilde{g}(p)d\mu(p)\label{com}\\
& \left[ A(f)^{\ast},A(g)\right] =-i\operatorname{Im}\left( f,g\right)
\nonumber
\end{align}
where $d\mu\ $is the invariant measure on the positive mass
hyperboloid.\ According to the Reeh-Schlieder theorem \cite{Haag} the
one-particle projection of the dense subspace of causally $\mathcal{O
$-localized states\footnote{States localized in the spacetime region
$\mathcal{O}$ are defined as the dense\ Reeh-Schlieder subspace$\
obtained$\ \ $as $\mathcal{A(O)}\left\vert 0\right\rangle $ where
$\mathcal{A(O)}$ is the $\mathcal{O}$ localized subalgebra of $\mathcal{A}$.}
is dense in the one-particle Wigner space.$\mathcal{\ O}$-localized real
testfunctions define a dense real subspace $H(\mathcal{O)}$ and causal
disjointness corresponds to "symplectic orthogonality" and produces a
\textit{closed} real subspace
\begin{equation}
H(\mathcal{O}^{\prime})=H(\mathcal{O)}^{\prime
\end{equation}
As a side remark we mention that the same construction applied to a higher
halfinteger spin field leads to a corresponding situation
\begin{equation}
ZH(\mathcal{O}^{\prime})=H(\mathcal{O})^{\prime},\ \ Z=\frac{1+iU(2\pi)}{1+i}
\label{Z
\end{equation}
where the unitary "twist" operator $Z$ which is related to the factor $-1$ of
the $2\pi$ rotation. The use of the twist operator allows to treat bosons and
fermions under one common roof.
The important step for an intrinsic understanding of QFT (i.e. without the use
of P. Jordan's "(quasi-)classical crutches" of quantization) is to
\textit{invert the previous construction: find a relation within Wigner's
representation theory which permits to define }$\mathcal{O}$\textit{-localized
real subspaces which have the correct covariant transformation properties
under Poincar\'{e} transformations \cite{BGL}.} For this purpose it is helpful
to reformulate the above properties so that they take the form known from the
mathematical Tomita-Takesaki theory of operator algebras which permits a
direct connection of positive energy Wigner representations with a "local net
of operator algebras". It provides a unified view in which Weinberg's pl
intertwiner formalism and its sl extension are seen as two ways of generating
the same free field theory. As shown in previous sections the improvement of
short distance properties is the basis for a new perturbative renormalization setting.
To achieve this one needs an additional mathematical tool. The first step
consists in a suitable extension of the modular concepts. A standard subspace
$H$ comes with a distinguished operator. With $D(Op)$ denoting the domain of
definition of an operator one defines
\begin{definition}
A Tomita operator $\mathcal{S}$ is a closed antilinear densely defined
involutive operator $D(\mathcal{S})\subset\mathcal{H}$
\end{definition}
In physics one encounters such "transparent" operators only in QFT. It is easy
to see that there exists a 1-1 correspondence between Tomita operators and
standard subspaces $H;$ $H\leftrightarrow\mathcal{S}$. This follows from the
definition $\mathcal{S}(\xi+i\eta)=\xi-i\eta,~\xi,\eta\in H,$ whereas the
opposite direction is a consequence of the definition $H=\ker\left\{
\mathcal{S}-1\right\} .$
As a result of involutiveness, the full content of Tomita operators is
contained in their dense domains which coincides with their range
("transparency"). Hence modular theory may be alternatively formulated in
terms of subspaces (\textit{\ref{stan}}) $Dom\mathcal{S}$ of Tomita operators.
The polar decomposition of $\mathcal{S}=J\Delta^{1/2}$ of $\mathcal{S}$ into
an anti-unitary $J$ and a positive operator $\Delta\ $with $D(S)=D(\Delta
^{1/2})$ leads to the unitary modular group $\Delta^{it}$ acting in
$\mathcal{H}$ and preserving the standard subspace $\Delta^{it}H=H,$ whereas
the modular conjugation maps into the symplectic complement $J$ $H=H^{\prime}$
A Tomita operator appears in a natural way in Wigner's representation theory
of positive energy representations of the Poincar\'{e} group $\mathcal{P}$. It
is obtained by defining $\Delta_{W_{0}}^{it}$ in terms of the Lorentz boost
operator which leaves the wedge $W_{0}=\left\{ x;~z>0,\left\vert t\right\vert
<z\right\} $ invariant
\begin{align*}
& \Delta_{W_{0}}^{it}=U(\Lambda_{W_{0}}(-2\pi t))\\
& \mathcal{S}_{W_{0}}=J_{W_{0}}\Delta_{W_{0}}^{1/2},\ \ J_{W_{0}}=TCP\cdot
R_{\pi
\end{align*}
together with an anti-unitary $J$ obtained by multiplying the TCP reflection
$TCP$ with a $\pi$-rotation $R$ in the $x$-$y$\ plane as written in the second
line, taking $W_{0}~$into its causal complement.
The charge conjugation $C~$maps an irreducible Wigner particle space into its
charge conjugate and may need a doubling of the Wigner space whereas $TP$
corresponds to the spacetime inversion $x\rightarrow-x$. The preservation of
energy requires the time-reversal $T$ to be anti-unitary. Massless
representations need a helicity doubling $\pm h.$
Unbounded operators $\mathcal{S}$ whose dense domain is stable ("transparent"
in the sense \textit{domain=range}) are somewhat unusual in quantum physics;
they appeared first in quantum statistical mechanics \cite{Haag} and later in
searches for an intrinsic understanding of causal localization of QFT (without
referring to Lagrangian quantization) \cite{BW}. In the present context the
dense subspace of the Wigner space (possibly doubled by charge conjugation)
corresponds to wave functions which are "modular localized" in the wedge
$W_{0}$. Modular localization of Wigner wave functions is closely related to
causal localization of fields and provides an extension of the Weinberg
intertwiner formalism which includes Wigner's infinite spin class \cite{MSY2}.
\textit{The construction proceeds as follows: start from Wigner's positive
energy representation theory, define the Tomita operator }$S_{W_{0}
~$\textit{in the way described before, use the Poincar\'{e} transformations to
construct a net of modular localized real subspaces }$H(W)$\textit{ and use
the second quantization functor (the Weyl~or CAR functor) to pass to an
interaction-free net of standard wave functions spaces }$H(W)$ to
\textit{causally localized operator algebras }$\mathcal{A(}W)$ \textit{acting
in a Wigner-Fock Hilbert space.}
One also may directly construct real dense subspaces $H_{\mathcal{O}}$ and
their complexified counterparts $D(\mathcal{S}_{\mathcal{O}})=H_{\mathcal{O
}+iH_{\mathcal{O}}\subset\mathcal{H}$ corresponding to more general causally
complete convex spacetime regions $\mathcal{O}_{c}$ as intersections
\begin{align}
H_{\mathcal{O}_{c}} & =\bigcap_{W\supset\mathcal{O}_{c}}H_{W}\\
H_{\mathcal{O}} & =\bigcup_{\mathcal{O}_{c}\subset\mathcal{O}
H_{\mathcal{O}_{c}}\nonumber
\end{align}
whereas for more general regions the standard space is defined in terms of
exhaustion from the inside (second line). For details we refer to \cite{BGL}.
The energy-positivity of the massive and the $\pm\left\vert h\right\vert $
massless Wigner representation classes plays an important role in establishing
the isotony and causal localization of the "net of modular localized standard
spaces
\begin{align}
\quad\hbox{isotony:}\quad & H_{\mathcal{O}_{1}}\subset H_{\mathcal{O}_{2
}\quad\hbox{if}\quad\mathcal{O}_{1}\subset\mathcal{O}_{2}\\
\quad\hbox{causality:}\quad & H_{\mathcal{O}_{1}}\subset H_{\mathcal{O}_{2
}^{\prime}\quad\hbox{if}\quad\mathcal{O}_{1}{\lower1pt\hbox{\LARGE$\times$}
\mathcal{O}_{2}\nonumber
\end{align}
where ${\lower1pt\hbox{\LARGE$\times$}}$ denotes spacelike separation.
In the absence of interactions the passage from the spatial modular theory to
its algebraic counterpart is almost trivial. One passes to the Wigner Fock
space created by symmetrized tensor products and defines the $\mathcal{O
$-localized operator algebra as the von Neumann algebra generated by the Weyl
operators
\begin{align}
& \mathcal{A(O})=\left\{ e^{iA(h)};h\in H_{\mathcal{O}}\right\}
^{\prime\prime},~\left[ A(h_{1}),A(h_{2})\right] =i\operatorname{Im
(h_{1},h_{2})\label{Weyl}\\
& \quad\hbox{with}\quad A(h)=\int\sum_{s_{3}=-s}^{s}(h(p,s_{3})a^{\ast
}(p,s_{3})+h.c.)d\mu(p),~h\in H_{\mathcal{O}
\end{align}
in terms of the Wigner creation/annihilation operators\footnote{The "second
quantization" counterparts of the Wigner wave functions.} $a^{\#}(p,s_{3})$ or
their helicity counterparts. For half-integer spin or helicity the presence of
the twist operator (\ref{Z}) leads to fermionic commutation relation. In both
cases the second quantization functor maps the spatial Tomita operator into
its algebraic counterpart which acts in the ("second quantized") Wigner Fock
space $\mathcal{H}$ associated with the Wigner one-particle space $H$
\[
\mathcal{S}_{\mathcal{O}}^{alg}A\Omega=A^{\ast}\Omega,~A\in\mathcal{A(O})
\]
In the sequel only the algebraic Tomita-Takesaki theory will be used and the
superscript $alg~$will be omitted for convenience.
Modular localized spaces $H(\mathcal{O})$ and their associated noninteracting
field algebras $\mathcal{A(O})$ are by construction "causally complete"
$\mathcal{A(O})=\mathcal{A(O}^{\prime\prime})$ and hence a fortiori Einstein
causal (section 2 \ref{Haag}). The causal completion $\mathcal{O
^{\prime\prime}$ is obtained by taking twice the causal complement
$\mathcal{O\rightarrow O}^{\prime}.$
For wedge regions the modular group coincides with the unitary Wigner
representation of the wedge-preserving Lorentz group; for all other regions
the modular groups in $m>0$ Wigner representations acts in a non-geometric
("fuzzy") way. It is believed that this nongeometric action$~$becomes
asymptotically (near the boundary of the causal completion) geometric.
Massless finite helicity theories have a larger set of regions related to
Huygen's principle in which modular groups act in a geometric way; this
includes all regions which are images of wedges under the action of conformal
transformation as e.g.\ finite double cones. Sl potentials whose semiinfinite
spacelike lines remain inside wedges may under suitable conformal
transformation pass to potentials localized on finite elliptic curves which
connect two points on different edges of double cones; they can be viewed as
substitutes for the nonexistent pl coordinatization of double cones.
It may also happen that the standard spaces for compact spacetime regions
$\mathcal{O}$ are trivial $H_{\mathcal{O}}=\left\{ 0\right\} .$This occurs
precisely for the Wigner's \textit{zero mass infinite spin representation} for
which the tightest localized nontrivial spaces correspond to modular
localization in arbitrary narrow spacelike cones. On the other hand zero mass
\textit{finite helicity} spaces are the most geometric representation since
their modular groups continue to act geometrically even for double cones; in
fact they correspond to conformal transformations which preserve double cones.
Unlike finite helicity fields \textit{infinite spin} representations are
massless but \textit{not conformal}.
Modular localization plays also an important role in the understanding of
topological peculiarities of massless $h\geq1~$free QFT in connection with
toroidal regions (thickened Wilson loops). Last not least without their use it
would not have been possible to discover the intrinsic noncompact localization
of Wigner's infinite spin matter and the string-like nature of its generating
causally localized fields.
\textit{This raises the question if modular theory preserves its constructive
power in the presence of interactions.} It turns out that in that case one is
required to use the stronger operator algebraic modular theory. Interacting
theories share the same modular groups $\Delta_{\mathcal{W}}^{it}$ with their
noninteracting counterpart which is solely determined by the particle content.
The interaction enters through the dependence of the $J_{\mathcal{W}}$ on the
interaction which, using the fact that the incoming TCP is related with its
outgoing counterpart through the S-matrix $S$~\cite{Jost} \cite{AOP}, amounts
t
\begin{equation}
J_{\mathcal{W}}=J_{\mathcal{W}}^{(in)}S \label{S
\end{equation}
Again the starting point is a von Neumann operator algebra $\mathcal{A}$
acting in a Hilbert space $\mathcal{H}~$which contains a vector $\Omega$ which
is cyclic and separating under the action of $\mathcal{A}$ (in QFT \textit{the
Reeh-Schlieder property} for $\mathcal{A(O)}$)
\begin{align}
\quad\hbox{cyclic:}\quad & \mathcal{A(O)}\Omega\quad\hbox{is dense in}\quad
\mathcal{H}\label{sub}\\
\quad\hbox{separating:}\quad & A\Omega=0\quad\Rightarrow\quad A=0,~~A\in
\mathcal{A(O)}~\nonumber
\end{align}
The definition$~H$ $=\overline{\mathcal{A}_{sa}(\mathcal{O})\Omega}$ ($sa=$
selfadjoint part) or $\mathcal{H=}\overline{\mathcal{A(O)}\Omega}$ connects
the algebraic modular theory with its previously presented spatial
counterpart. But there remains an important difference: the map between
standard subspaces and algebras is generally not injective whereas the
algebraically generated subspaces (\ref{sub}) always are .
The interaction-free situation remains exceptional in that there exists a
functorial relation between modular localized Wigner subspaces and
interaction-free causally localized subalgebras defined in terms of the Weyl
map (\ref{Weyl}). This functorial map is lost in the presence of interactions
in which case the relation between modular theory and particles becomes more involved.
\subsection{A critical perspective based on modular localization}
Most of what is presently known about modular theory comes from the
Bisognano-Wichmann theorem \cite{BW} which clarifies the modular properties of
wedge-localized subalgebras $\mathcal{A}(W)$. In models with a complete
particle interpretation one can use the modular theory of free fields to
derive the B-W theorem in the presence of interactions \cite{JensBW}. Of
special interest is the relation of the modular conjugation $J$ with the TCP
operator which is known to be connected to the S-Matrix \cite{Jost}. This is
particularly useful in $d=1+1$ integrable models whose S-matrix is known
(\ref{S}).
For integrable models without bound states this led to an interpretation of
the Zamolodchikov-Faddeev algebra in terms of modular localization in which
the concept of "vacuum-polarization-free generators" (PFG)\footnote{The
weakest assumptions under which PFG's exist were determined in \cite{Mu}.}
plays an important role \cite{AOP}\cite{BBS}. This in turn led to existence
proofs for certain $d=1+1$ integrable models in terms of operator algebraic
constructions based on modular theory \cite{Lech}. For a recent account with
many references to previous publications see \cite{AL}. These results
complement those obtained in terms of the "bootstrap-formfactor" program in
\cite{Kar}.
The important role of the S-matrix for modular localization in wedges has
triggered attempts to reconstruct a full causal QFT from its "on-shell
footprint" in form of its S-matrix \cite{Bert1,Bert2}. These ideas are
presently too weak for constructions in higher dimensions. Among the
unexpected results of modular theory is the proof that models in $d=1+2$ with
anyonic statistics (or its nonabelian "plektonic" counterpart) have always
nontrivial S-matrices \cite{M-Br}.
Whereas the idea that a nonperturbative QFT is uniquely determined by its
S-matrix remains still part of folklore, the S-matrix based perturbative SLFT
construction in the previous section is the basis of the new perturbation
theory. Different from the standard approach based on Lagrangian quantizations
in which the S-matrix is obtained from the mass-shell restriction (the LSZ
reduction formula) of time-ordered correlation functions of fields, SLFT
inverts this situation by encoding the model-defining particle content into an
interaction density of a perturbative defined S-matrix. This part of the
construction uses exclusively pl or sl free fields which are directly related
to the particles. Interacting (off-shell) quantum fields are constructed in a
second step in which the higher order contributions to the interaction density
(which were induced in the construction of $S$) serve as an input (\ref{C}).
Each pl or sl field from the equivalence of free fields and their Wick-ordered
composites has an interacting counterpart. Interacting fields which have
nonvanishing matrixelements between the vacuum and a one-particle state have
after appropriate normalization the same large-time in- and out- limits.
Finally we come to an important point whose clarification was promised at the
beginning of this section namely the possible connection of sl quantum fields
with String Theory. String theorists attribute a string-like localization to
their objects without providing arguments in favor of causal localizability.
Ideas based on worldlines, worldsheets, Nambu-Goto actions or strings start
from classical relativistic mechanics and hope that quantization preserve
these properties. This is evident from the way in which the covariant
world-line action $\sqrt{-ds^{2}}$ in \cite{Po} is used to prepare the ground
for the subsequent presentation of world-sheets and Nambu-Goto actions. The
impossibility to place two of such vibrating quantum mechanical strings into a
relative spacelike position reveals the problems which ST has with causal
localization in spacetime.
Fact is that, apart from Lagrangian quantization of $s<1$ fields, only the
Wigner representation theory (which cannot be accessed by quantization)
contains the seed for causal localization. Modular localization as a pre-form
of causal localization needs positivity and is inconsistent with gauge theory.
One cannot declare an object as "stringy" because its classical action
suggests this. This type of misunderstanding is clearly visible in
Polchinski's use of the action of a relativistic particle as a preparatory
step for relativistic worldsheet and Nambu-Goto actions.
This does not exclude the use of the quantum mechanical Newton-Wigner
localization to describe the dissipation of wave-packets. One may even
construct a macro-causal Poincar\'{e}-invariant multi-particle theory which
satisfies cluster decomposition properties and leads to a Lorentz-invariant
scattering matrix (\cite{SHPMP} section3). Such a construction disproves the
conjecture that relativistic particles and cluster properties alone will lead
to QFT.
Proposals which avoid quantization of classical actions and try to find causal
quantum matter in the target space of certain models of $d=1+1~$conformal QFTs
are less easy to dismiss. One such proposal is based on the Virasoro-algebra
of a supersymmetric 10-component chiral conformal current model (the
"superstring"). Its target space contains an algebraic structure which leads
to a highly reducible unitary Wigner representation of the Poincar\'{e} group
\cite{Brow}.
This is primarily a group theoretical observation which (apart from the ten
spacetime dimensions) fits well into Majorana's 1932 project of finding
algebraic structure which, in analogy to the $O(4,2)$ hydrogen spectrum,
describes wave functions of families of higher spin/helicity particles. But
why should one believe that the corresponding fields are sl in the absence of
any supporting argument? Is the terminology perhaps related to picturing the
\textit{tower} of representations containing different masses and spins as
Fourier components of a kind of \textit{internal} circle? In that case the use
of "string" for something bears no relation to spacetime and would be misleading.
Looking at the ST literature one gets frustrated about the disproportionate
relation between its conceptual poverty as compared to its mathematical
richness which its vague physical pictures lead to in the hands of
mathematicians. The word "string" should be more than an "epitheton ornans"
for a physically insufficiently understood mathematical formalism.
With additional conceptual care one can also avoid a widespread
misunderstanding in the physical interpretation of the $AdS_{n+1}$-$CFT_{n}$
isomorphism. It was certainly consequential to complement the observation of
equality of the symmetry groups of the two spacetimes by the verification of a
stronger Einstein causality-preserving isomorphism between the two QFTs. But
unlike classical field theory the timelike completion property\footnote{The
causal completion $\mathcal{O}^{\prime\prime}$ is the result of taking twice
the causal complement.} in QFT $\mathcal{A(O}^{\prime\prime})=\mathcal{A(O}),$
(which roughly speaking describes causal propagation) \textit{is not a
consequence of the spacelike Einstein causality}. Formally it is equivalent to
Haag duality $\mathcal{A(O}^{\prime})=\mathcal{A(O})^{\prime}$ from which it
results by taking the commutant on both sides and rewriting $\mathcal{A(O
^{\prime})^{\prime}$ by applying Haag duality to the (generally noncompact)
region $\mathcal{O}^{\prime}.$
In the old days \cite{H-S}\cite{Landau} it was shown that Einstein causality
and the causal dependency property (formally equivalent to Haag duality) are
\textit{independent} requirements. Causality without Haag duality occurs if
there are "too many" degrees of freedom as in the case of the generalized free
field; a phenomenon which has no classical analog (since the notion of quantum
\textit{degrees of freedom} has no classical counterpart).
This manifests itself in a sort of "poltergeist effect" in that there may be
\textit{more quantum degrees of freedom streaming into the dependency region}
\textit{as time moves on} \textit{than there were in the original
appropriately defined initial ("Cauchy") data}; in operator-algebraic notation
$\mathcal{A(O})\varsubsetneq\mathcal{A(O}^{\prime\prime})$ while Einstein
causality$\mathcal{A(O})\subset\mathcal{A(O}^{\prime})^{\prime}$ is preserved.
In fact it is quite easy to construct Einstein-causal models for which this
completion property is violated. Generalized free fields with a suitably large
$\kappa$ behavior of their Kallen-Lehmann spectral functions (containing a
much larger cardinality of degrees of freedom than free fields) were used at
the beginning of the 60s\footnote{I am sometimes asked about the origin of
this terminology. The answer is simple, it accounts for the fact that Haag had
this idea already before I entered the collaboration; my contributions
consisted in providing calculations involving generalized free fields.} to
show that the \textit{causal shadow property} represents a separate
requirement (initially called "the time-slice property").
The heuristic picture is that "squeezing" a QFT with the natural cardinality
of degrees of freedom corresponding to a \textit{n+1}-dimensional QFT into an
\textit{n} dimensional spacetime ("holographic image") causes an
"overpopulation"; this is precisely what happens in the $AdS_{n+1}$-$CFT_{n}$
case \cite{D-R}. The simplest illustration is obtained by projecting a free
AdS field and noting that the resulting conformal field is a generalized free
field of the kind used in \cite{H-S}\cite{Landau}. In the \textit{opposite}
direction i.e. starting from a "normal" CFT one expects an "anemic" degree of
freedom situation on the AdS side. As shown in \cite{Rehren} this is precisely
what happens; in fact there are no degrees of freedom at all in compact AdS
regions (double cones); to find any one has to pass to infinitely extended
wedge-like regions in AdS.
The overpopulation of degrees of freedom distinguishes "holographic
projection" from "normal" lower dimensional QFT and this explains the
quotation marks in "pathology"; a holographic projection maintains the degrees
of freedom of the original QFT and in this way prevents the conformal side to
be a normal QFT. This is a point which had been ignored in most post Maldacena work.
A helpful viewpoint concerning such "overpopulated" models which one obtains
by dimension reducing projections is to not consider them as autonomous QFTs
but to view them rather as stereographic projections of the original QFT.
The degree of freedom issue is not limited to the AdS-CFT isomorphism but
\textit{affects all attempts to extend (quasi)classical Kaluza-Klein
dimensional reductions of lowering extra dimensions to full-fledged causal
QFT}. To the extend that such pictures are compatible with the principles of
QFT\footnote{\textit{Massaging} Lagrangians is not the same as passing from
one QFT to another.} they correspond to perform a stereographic projection on
the original theory and not to pass to a lower dimensional QFT. This poses the
question whether the fashion around extra dimensions would have occurred with
a higher conceptual awareness about the subtle nature of positivity and causal
localization of relativistic quantum matter which forbid the use of naive
quasiclassical arguments.
A breakdown of Haag duality for entirely different reasons occurs in local
nets generated by massless helicity $h\geq1~$free field strengths. This
"topological duality violation" was mentioned in section 2 where its relation
to the Aharonov-Bohm effect and linking numbers was explained. It is lost in
the positivity-violating gauge theoretical setting which cannot distinguish
between the Haag duality and the somewhat coarser Einstein causality.
In his book on Local Quantum Physics Haag proposes an interesting extension
from the established one-fold duality for observable algebras to causally
separated two double cone localized algebras. In fact he explores the
possibility of the existence of a homomorphism from the orthocomplemented
lattice of causally complete regions in Minkowski space into that of von
Neumann algebras of observables (\cite{Haag}, Tentative Postulate 4.2.1). For
a region which consists of two spacelike separated double cones $K_{1},K_{2}$
this requires $\mathcal{A}(K_{1}\vee K_{2})=\mathcal{A}(K_{1})\vee
\mathcal{A}(K_{2}).$
Haag notices with a certain amount of disappointment that duality is violated
for doubly localized observable algebras associated to the conserved currents
of free fields. This follows from the existence of a pair $\psi(f)\psi
(g)^{\ast}~$with $supp~f\subset K_{1},~supp~g\subset K_{2}~$which commutes
with $\mathcal{A}(K_{1}\vee K_{2})~$but is not in $\mathcal{A}(K_{1
)\vee\mathcal{A}(K_{2}).~$He viewed this as a shortcoming of free fields which
he expected to disappear in the presence of interactions. He uses the idea of
a "gauge bridge" between $\psi(f)$ and$~\psi(g)^{\ast}$ as a hint that a
future positivity-maintaining replacement of gauge theory may satisfy this
strengthened form of causality ("Haag duality") may.
The string-bridges of SLFT do precisely this i.e. they prevent that Haag
duality of two causally separated double cones is violated by a
charge-anticharge pair. In constrat to gauge bridges which have no material
content (since they can be changed by gauge transformation implemented by
indefinite metric gauge charges) string-bridges consist of quantum matter.
Hence the existence of string bridges provides a local method to distinguish
an interacting net from a free one.
Another somewhat more metaphoric way of saying the same is that SLFT results
from gauge theory by applying Occam's razor to indefinite metric- and ghost-
degrees of freedom. Gauge theory is its best placeholder within the setting of
Lagrangian quantization.
Interestingly the same string bridges which save Haag duality of observables
also allow to view interpolating fields as resulting from space- or light-like
limits in which the anticharge component is disposed of at infinity but leaves
a trail of quantum matter behind.
Whereas all sl fields in the absence of interactions can be obtained as
semiinfinite line integrals from pl fields \footnote{Note that the interaction
density and the S-matrix only uses such free fields.}, this breaks down in the
presence of interactions. In that case the necessarily sl localized
interpolating $s<1$ fields receive their sl localization metaphorically
speaking through being "infected" by their contact with higher order $s\geq
1~$sl potentials with which they share the interaction density. This will be
exemplified in a number of models in the next section.
For anybody who has grown up with Haag's way of looking at QFT it is deeply
satisfying that the model-dependent division between gauge invariant
observables and gauge dependent interpolating fields corresponds to the SLFT
localization dichotomy between pl observables and sl interpolating fields.
The fault line, which unfortunately still separates the ST community from
those who are working on the successful but still largely unfinished project
of QFT, runs precisely alongside the issue of causality and its refinements.
The recent progress on entanglement entropy \cite{Wi} \cite{Ho-Sa} requires a
profound understanding of causality in the context of operator algebras. The
fact that these ideas are presently also leading to a revision of perturbative
QFT raises hopes that this schism will be overcome.
\section{Renormalization in the presence of massive sl vector mesons}
\subsection{Remarks on scalar QED; induced interactions and counterterms}
As mentioned in the introduction SLFT differs both in its concepts as well as
in its calculational techniques from Lagrangian- (or Euclidean action)-based
quantization theories. In section 3 these differences played a role in the
solution of the Velo-Zwanziger causality conundrum and in the present section
they will be exemplified in a full QFT.
The simplest nontrivial model for illustrating these differences (for reasons
which will become clear in the sequel) is scalar QED.
Being an S-matrix-based QFT, the starting point of SLFT is an S-matrix which,
following Bogoliubov, is formally written as a time ordered product of an
interaction density $L(x,e)~$as in (\ref{B}). The construction of interacting
sl fields uses Bogoliubov's map (\ref{C}) which converts free fields into
their interacting counterparts whose large-time asymptotic behavior reproduces
the scattering amplitudes associated to $S.$ In this section we will be
exclusively interested in the S-matrix; localization properties of interacting
fields will be mentioned in section 6.
The simplest nontrivial model for which the preservation of positivity
requires the use of sl localized fields is \textit{scalar QED}. Here
"nontrivial" refers to the appearance of a $2^{nd}$ order induced term $A\cdot
A\varphi^{\ast}\varphi$ and a $4^{th}$ order counterterm $(\varphi^{\ast
}\varphi)^{2}$ which has no counterpart in spinorial QED.
The independence of the large-time LSZ limits of causally separable fields on
their original localization is the basis of the SLFT perturbation theory. The
following is in part a recollection of arguments presented in section 3.
In lowest order we may start with the pl interaction density $L^{P}=A_{\mu
}^{P}j^{\mu},~j^{\mu}=i\varphi^{\ast}\overleftrightarrow{\partial^{\mu
}\varphi$ and use the linear relation with its short distance improved sl
counterpart and its escort $A_{\mu}^{P}=A_{\mu}-\partial_{\mu}\phi~$to write
\begin{align}
L^{P} & =L-\partial^{\mu}V_{\mu},~with~V_{\mu}=j_{\mu}\phi\label{V}\\
S^{(1)} & =\int L^{P}(x)d^{4}x=\int L(x,e)d^{4}x
\end{align}
This solves two problems in one stroke, the highest short distance
contribution to $L^{P}$ has been encoded into a divergence which drops out in
the adiabatic limit (second line) so that S is string-independent (the left
hand side) as well as renormalizable (the right hand side).
Actually one may forget the $L^{P}$ and formulate the SLFT construction solely
in terms of a $L,V_{\mu}$ pair fulfilling the $L-\partial V=0$ pair condition.
It turns out that the $L,V_{\mu}$ pair corresponding to vector mesons
interacting with lower spin particles and possibly among themselves is
uniquely determined: the interaction is completely determined in terms of its
particle content! In other words the $L^{P}$ can be defined in terms a SLFT
pair which is in turn defined of the particle content.
For the formulation of the higher order pair property it is convenient to use
the differential form of the pair property (as in section 3) and write
$d_{e}(L-\partial^{\mu}V_{\mu})=0.$ Further simplification is obtained by
using the "$Q$-formalism" wit
\begin{align}
Q_{\mu} & =d_{e}V_{\mu}=j_{\mu}u,~u=d_{e}\phi\\
& d_{e}L-\partial^{\mu}Q_{\mu}=0\nonumber
\end{align}
The $Q_{\mu}$ turns out to have a better $m\rightarrow0$ behavior; it is only
logarithmically divergent and $\partial^{\mu}Q_{\mu}$ remains finite. The
logarithmic infrared divergence is a perturbative spacetime indication that
the massless limit of vacuum expectation values cannot be described in terms
of Wigner-Fock space which is simply a tensor product of a helicity space of
photons with that of charge-carrying Wigner particles \cite{Bu}. In other
words the S-matrix based perturbative SLFT formalism indicates that its
massless limits needs a (presently unknown) extended formulation of scattering theory.
A notational simplification for the higher order pair conditions is obtained
by using lightlike strings (not possible for $m=0$). In this case the massive
sl fields are functions in $e$ rather than distributions and hence all $e$ may
be set equal. The second order pair condition read
\begin{equation}
d_{e}TLL^{\prime}-\partial^{\mu}TQ_{\mu}L^{\prime}-\partial^{\prime\mu
}TLQ_{\mu}^{\prime}=0 \label{LQ
\end{equation}
and the extension to higher than second order is straightforward. They have no
counterpart in the standard pl setting and account for the strength of SLFT as
compared to the standard pl perturbation theory.
Violations of these relations are referred to as \textit{obstructions}. The
Bogoliubov S-matrix formalism is preserved by encoding these obstructions into
a redefinition of the interaction density $L\rightarrow L_{tot}=L+L_{2
+..$just as it was done for external potential interactions in section 3.
The time-ordering $T$ which fulfills (\ref{LQ}) is not necessarily the
"kinematic" time ordering~$T_{0}$ which is defined by attaching a
$-i2\pi(p^{2}-m^{2}-i\varepsilon)$ denominator to the momentum space 2-ptfct.
The scaling rule of renormalization requires that $T$ and $T_{0}$ share the
same scaling degree which in the presence of two derivatives leaves a
normalization freedo
\begin{equation}
\left\langle T\partial_{\mu}\varphi^{\ast}\partial_{\nu}^{\prime
\varphi^{\prime}\right\rangle =\left\langle T_{0}\partial_{\mu}\varphi^{\ast
}\partial_{\nu}^{\prime}\varphi^{\prime}\right\rangle +icg_{\mu\nu
\delta(x-x^{\prime}) \label{T
\end{equation}
which leads t
\begin{equation}
\partial^{\mu}\left\langle T\partial_{\mu}\varphi^{\ast}\partial_{\nu
^{\prime}\varphi^{\prime}\right\rangle -\left\langle T\partial^{\mu
\partial_{\mu}\varphi^{\ast}\partial_{\nu}^{\prime}\varphi^{\prime
}\right\rangle =i(1+c)\partial_{\nu}^{\prime}\delta(x-x^{\prime}) \label{U
\end{equation}
with an initially undetermined $c.$
The fulfillment of the second order pair requirement (\ref{LQ}) in the tree
approximation fixes $c=-1.$ The action of $S$ on one-particles states as the
identity operator $S\left\vert p\right\rangle =\left\vert p\right\rangle
~$takes care of the contribution from 2 contractions. The change of $T_{0}~$to
$T$ in $Tj_{\mu}j_{\nu}^{\prime}=T_{0}j_{\mu}j_{\nu}^{\prime}-g_{\mu\nu
\delta$ in all $Tj_{\mu}j_{\nu}~$accounts for the occurrence of the second
order induced $A_{\mu}A^{\mu}\left\vert \varphi\right\vert ^{2}$ ter
\begin{align}
TLL^{\prime} & =T_{0}LL^{\prime}-i\delta(x-x^{\prime})L_{2}\label{L1}\\
L_{2} & =gA_{\mu}A^{\mu}\left\vert \varphi\right\vert ^{2} \label{L2
\end{align}
which is usually attributed to the implementation of gauge symmetry, but here
it follows from the causality and positivity principle of interpolating fields
which guaranties the $e$-independence of $S$.
The reason for using the kinematical time-ordering $T_{0}$ instead of $T$ is
the comparison with GT. In SLFT it is more natural to use $T$ in which case
the second order $L_{2}$ remains encoded in $TLL^{\prime}.$
As already pointed out in section 3, GT has a \textit{formally} similar
structure. This is most clearly visible in a setting of gauge theory which
avoids the standard Lagrangian quantization of gauge theory (for spins
$s\geq1$ see \cite{Fro}) in favor of a perturbative S-matrix formulation as in
\cite{Scharf} \cite{Aste}. The physical $A_{\mu}(x,e)$ and its escort
$\phi(x,e)$ correspond to the gauge potential $A_{\mu}^{K}$ and the
St\"{u}ckelberg field $\phi^{K}$ acting in a ghost extended Krein space; the
authors show that $A^{K}-\phi^{K}$ has properties expected from $A_{\mu}^{P}.$
The two S-matrix-based constructions share the same improved short distance
behavior, but they achieve this in a very different way. Whereas in gauge
theory this is the result of enforced compensations between positive and
negative probabilities in intermediate states, the ultraviolet improvement in
SLFT accomplishes this by lessening the tightness of causal localization (but
not abandoning it !) and in this way reducing the strength of vacuum
polarization which is the only physical way to describe particles in terms of
physical (i.e. not gauge) interpolating fields.
Though both SLFT and gauge theory have the same short distance dimensions and
probably even share their Callen-Symanzik equations (and the related
asymptotic freedom property encoded in the beta-function), gauge theory
\textit{cannot account for the physics at finite distances let alone at long
distances}; infrared properties and the problem of confinement remain outside
its physical range.
Last not least the functional-analytic and operator-algebraic methods used in
deriving nonperturbative theorems from basic principles are not available in
Krein spaces. For this reason gauge theory is shunned in books addressing the
conceptual structure of QFT \cite{St-Wi}\cite{Haag}. The perturbative gauge
theoretic construction of a unitary S-matrix reveals this tension between
conceptual clarity and the efficiency of calculations which account for
experimental observations; it is a blessing for the impressive achievements of
the Standard Model but a curse for a on the principles of positivity and
causal localizability formulated QFT of the books.
Considering these conceptual deficiencies the perturbative calculations of a
gauge-invariant S-matrix of the Standard Model is a truly impressive
achievement. The idea that it represents a successful placeholder of an
unknown QFT is quite old and there have been many failed attempts to find the
real thing. The close formal analogy between gauge theory and SLFT suggest
that both may even exist side by side in a Krein extended Wigner-Fock space
containing additional indefinite metric degrees of freedom \footnote{J. Mund
seems to have discovered such a "hybrid" formulation (private communication).}.
Presently there exist no higher than second order SLFT calculation. Higher
order loop calculations in SLFT are much more laborious than calculations in
gauge theory. The gauge theoretic $4^{th}$ order calculation establishes the
existence of a $c(\varphi^{\ast}\varphi)^{2}~$counterterm. In contrast to the
second order $A\cdot A\varphi^{\ast}\varphi$ contribution its strength $c$ is
a new parameter which is not determined by electromagnetism of the e.g.
$\pi^{+}~$meson\footnote{Using such a model to describe electromagnetic
interactions of charge-carrying pions one usually sets $c=0.$}.
Could this counterterm in GT be an induced contribution in SLFT ? This
question is not as crazy as it sounds. The above $L^{P}$ theory is by itself
nonrenormalizable; its short distance dimensions and the number of
counterterms increase with perturbative order. Yet if "guided" in the above
sense by a $L,V_{\mu}$ pair it shares the finite number of possible free
varying parameters with the SLFT $L$ description.
The still missing answers to such questions are not only owed to the fact that
the number of theoreticians who are presently working on SLFT problems can be
counted on one hand but they also find their explanation in that the necessary
calculations are more involved than those based on pl fields. The sl setting
of QFT is the only known way to uphold the principles of QFT for \textit{all} fields.
The SLFT approach also touches on an old mathematical problems which arose
from QFT in the late 60's. The question was whether fields with $d_{sd
=\infty$ (polynomial unbounded) fields as e.g. Wick-ordered exponential
functions of pl fields as $\exp g\varphi$ have a well-defined mathematical
status. This led Jaffe to extend the notion of Schwartz distributions to a
general class of distributions which still allows smearing with a dense set of
compact localized Schwartz test functions.
The SLFT guided construction of the $L^{P}~$pl setting requires to identify pl
observables in both settings and suggests to identify the interpolating state
creating fields of the $L^{P}$ theory with Jaffe fields. They correspond to
the well-behaved sl interpolating fields: the two theories share not only the
S-matrix but also their local observables whereas the states in the $L^{P}$
theory remain singular in the sense of Jaffe. Such singular fields are not
required to have the usual domain properties which one needs to generate
operator algebras from fields so that the algebraic localization of compact
spacetime regions is fully accounted for by observables.
After having exemplified the main difference between gauge theory and SLFT in
the model of scalar QED, the following subsections will present low order
calculations in other models in which vector mesons couple to lower spin
matter fields and among themselves. This includes in particular the Higgs
models for which, different from the standard treatment, the form of the
Mexican hat potential and its spontaneous symmetry breaking is not imposed but
rather \textit{induced} as a consequence of $e$-independence of $S.$ Even more
surprising is that the division into observables and sl interpolating is very
different from what one naively expects: neither the field strength $F_{\mu
\nu}$ nor the Higgs field is a pl observable.
\subsection{The perturbative S-matrix in the SLFT setting}
The appropriate formalism for the direct perturbative calculation of the
on-shell $S$-matrix is based on the adiabatic limit of Bogoliubov's
operator-valued time-ordered $S(g)$ functional. Its adjustment to SLFT has
been mentioned in (\ref{B}) in section 3 and further explored in the previous subsection.
Time-ordering of quantum fields mathematically represented by operator-valued
distributions is characterized in terms of properties among which the causal
factorization is the physically most important one. The Epstein-Glaser
formalism \cite{E-G} provides a perturbative computational scheme in which the
time-ordering of $n+1$ pl interaction densities is inductively determined in
terms of the $n^{th}$ ordered time-ordered product. The formulation in the
presence of sl fields is more involved and has not been carried out beyond
second order. Preliminary results reveal that a systematic $n^{th}$ order
construction requires the use of new concepts \cite{CMV}.
The E-G perturbation theory for the S-matrix can be extended to sl fields
(\ref{C}). The result is a formula which maps a field in the local equivalence
class of Wick-ordered composites of free fields into the equivalence class of
"normal ordered" relative local interacting fields which act in the same
Wigner-Fock Hilbert space but are nonlocal with respect to their free
counterparts. Nowhere does this formalism refer to Lagrangian quantization.
For gauge theory this was first carried out in \cite{Du-Fr} where it was shown
the time-ordering of the S-matrix passes to that of retarded products in terms
of fields.
The power-counting restriction of renormalizability $d_{sd}(L^{P})\leq4$ is
violated if one of the spin/helicity of the particles is $\geq1.$ For
interactions involving particles with highest spin $s=1$ the $d_{sd}(L^{P})=5$
there are two ways to recover renormalizability. Either by converting $L^{P}$
into a "gauge pair" $L^{K},V_{\mu}^{K}$ which requires the extension of the
Wigner-Fock space by indefinite metric degrees of freedom and possibly BRST
ghosts, or one maintains the physical degrees of freedom (and with it the
positivity of the Wigner-Fock Hilbert space) by converting $L^{P}$ into sl
$L,V_{\mu}$ pair.
For the rest of the paper we will stay with models which are sl renormalizable
$d_{sd}(L)\leq4.$ This includes all couplings whose particle content consists
of $s=1$ coupled to $s<1~$and among themselves. In the case of massless sl
vector potentials the escort $\phi$ diverges as $m^{-1}$ and the large-time
LSZ derivation of the S-matrix breaks down (the on-shell restrictions of
correlations develop logarithmic $m\rightarrow0$ singularities) and with it
the S-matrix based SLFT construction.
However some remnants of the SLFT construction can be saved; the exact
one-form $d_{e}\phi$ and hence also the $Q_{\mu}=d_{e}V_{\mu}~$is only
logarithmically divergent and $\partial^{\mu}Q_{\mu}$ remains convergent.
Hence even in case of breakdown of the S-matrix as a result of infrared
problems the $L,Q_{\mu}$ pair conditio
\begin{equation}
d_{e}L-\partial_{\mu}Q^{\mu}=0,\text{ }Q_{\mu}=d_{e}V_{\mu} \label{st
\end{equation}
remains a nontrivial condition. In fact it is this weaker formulation of
$e$-independence which corresponds to the BRST invariance of gauge theory.
In the previous subsection it was shown that, although the second order pair
condition in its original form is violated, it is possible to encode the
obstructing contribution $L_{2}$ into a redefinition of the interaction
density. It is helpful to formulate this idea in a model-independent way.
The definition of second order obstruction against the naive form of the
$L,Q_{\mu}~$pair property reads (using lightlike $e^{\prime}s$ which can be
identified)
\begin{align}
O^{(2)} & :=d_{e}TLL^{\prime}=T\partial^{\mu}Q_{\mu}L^{\prime}-\partial
^{\mu}TQ_{\mu}L^{\prime}+TL\partial^{\prime\mu}Q_{\mu}^{\prime}-\partial
^{\prime\mu}TLQ_{\mu}^{\prime}\label{d}\\
O^{(2)} & =\delta(x-x^{\prime})d_{e}L_{2}(x,e)\nonumber
\end{align}
Encoding them into interaction density one obtain
\begin{equation}
L_{tot}:=L+gL_{2},~~S(g)=T\exp\int ig(x)L_{tot}(x,e)d^{4}x \label{ob2
\end{equation}
This change of bookkeeping which converts higher order obstruction into
induced contributions $L_{n}~$amounts$~$a change of $L\rightarrow L_{tot}$ in
the Bogoliubov $S(g)$~is important. It affects the higher orders; the third
order obstruction is now
\begin{equation}
O^{(3)}(g,g,g)=d_{e}\left[ TL(g)L_{2}(g^{2})+\frac{i}{3}TL(g)L(g)L(g)\right]
\label{ob3
\end{equation}
In models of interacting $s=1~$vector mesons as the Higgs model or scalar
massive QED the third order obstruction vanishes in the adiabatic limit and
the induced contributions account for the Mexican hat potential. As a
consequence the terms in this potential are induced and not postulated for the
purpose of implementing SSB. This will be explicitly verified in the following subsections.
The $L,Q_{\mu}~$pair condition and its higher order extension within the sl
Bogoliubov-Epstein-Glaser setting is also meaningful for $d_{sd}(L)>4.$ The
before mentioned "minimal" models contain only induced contributions but their
number increases with the perturbative order. By definition of minimal there
are no higher order counterterm parameters so the model depends only on those
parameters which are already present in the interaction density $L.$
The conceptual and mathematical superior aspects of SLFT poses the question
whether it is possible to pass directly from SLFT to pl ultra fields, thus
avoiding the pl counterterm formalism. This problem will come up again in
connection with cubic $h=2$ selfinteractions in the next section (\ref{rel}).
\subsection{External source models}
Consider a vector potential coupled to an conserved classical current $j_{\mu
}$\ \cite{MRS2}. The S-matrix and the interacting vector potential ar
\begin{align}
& L^{P}=A_{\mu}^{P}j^{\mu}=A_{\mu}j^{\mu}-\partial_{\mu}(\phi j^{\mu
}),~hence~L=A_{\mu}j^{\mu},\ V_{\mu}=\phi j_{\mu}\label{s}\\
& S_{e}(g)=T\exp i\int g(x)L(x,e)\overset{g(x)\rightarrow g}{\rightarrow
}S=\exp ig\int\int j^{\mu}i\Delta_{F}j_{\mu}^{\prime}:\exp ig\int A_{\mu
^{P}j^{\mu}:\nonumber\\
& A_{\mu}^{ret}(x,e,e^{\prime})=S_{e}{}^{-1}(g)\frac{-i\delta}{\delta f_{\mu
}(x,e^{\prime})}S_{e}(g,j\rightarrow j+f)|_{f=0}=A_{\mu}(x,e)+\int G_{\mu
\mu^{\prime}}^{ret}j^{\mu^{\prime}}\nonumber\\
~ & G_{\mu\mu^{\prime}}^{ret}(x,e;x^{\prime},e^{\prime})=(-\eta_{\mu
\mu^{\prime}}-\partial_{\mu}e_{\mu^{\prime}}I_{e^{\prime}}+\partial
_{\mu^{\prime}}e_{\mu}^{\prime}I_{-e}+(ee^{\prime})\partial_{\mu}\partial
_{\mu^{\prime}}I_{e}I_{-e^{\prime}})G^{ret}(x-x^{\prime})\nonumber
\end{align}
The direct use of $L^{P}$ with $d_{sd}(L^{P})=d_{sd}(A_{\mu}^{P})=2$ leads to
a delta function ambiguity $g_{\mu\nu}c\delta(x-x^{\prime})~$in the
time-ordered $A_{\mu}^{P}$ propagator which accounts for a replacement
$i\Delta_{F}\rightarrow i\Delta_{F}+\frac{c}{m^{2}}\delta$ in the second line.
This in the pl formulation undetermined counterterm renormalization parameter
in the S-matrix and in $A_{\mu}^{P,ret}$ is absent in the less singular sl formulation.
In that case $S$ is independent of $c$ and (by use of current conservation)
the interacting field does not depended on $e^{\prime}.$ As expected the field
strength remains pl. Hence the avoidance of the direct use of $A_{\mu}^{P}$ in
the calculation maintains the predictive power of the model. If needed one can
convert the sl setting with the help of $\phi(x,e)$ to a $A_{\mu}^{P}$. In
contrast to the directly calculated $A_{\mu}^{P}$ this via sl determined pl
potential is "better".
Passing from external source to \textit{external potential} problems the
differences between the direct pl results and those obtained via the sl detour
are much stronger (section 3).
\subsection{Hermitian $H~$coupled to a massive vector potential}
The coupling of a vector potential to a Hermitian scalar matter field $H$
comes with a new phenomenon. In addition to a change of the time-ordered
product of the $H$-field there is now a genuine \textit{induction}~of $H$-selfinteractions.
The "germ" of an interaction density (the "ignition") for an $A_{\mu},H$ field
content is the $mA\cdot AH$ coupling, where the vector meson mass factor $m$
accounts for the classical dimension $d_{eng}=4$ and also indicates that the
model has no nontrivial Maxwell limit (the reason why it was discovered a long
time after QED)$.$ Its sl operator dimension is $d_{sd}=3,$ hence the germ is
a superrenormalizable sl interaction density. The first order $L,Q_{\mu}$ pair
property ($Q_{\mu}=d_{e}V_{\mu}$) requires the presence of the escort $\phi$
also in $L~$and leads to ($L,Q_{\mu}$ relation easy to check) $\ $
\begin{align}
& L=m\left\{ A\cdot(AH+\phi\overleftrightarrow{\partial}H)-\frac{m_{H}^{2
}{2}\phi^{2}H\right\} +U(H),~U(H)=mc_{1}H^{3}+c_{2}H^{4}\nonumber\\
& V_{\mu}=A_{\mu}\phi H+\frac{1}{2}\phi^{2}\overleftrightarrow{\partial_{\mu
}}H,~~\ d_{e}(L-\partial^{\mu}V_{\mu})=0,\text{ }L^{P}=L-\partial^{\mu}V_{\mu
}. \label{first
\end{align}
A systematic determination of this first order pair $L,V_{\mu}$ pair starting
from the simplest coupling (the "germ") $gmA\cdot AH~$of the $A$-$H$ particle
content and a general ansatz for $L$ and $V_{\mu}$ containing all
kinematically possible $d_{sd}\leq4~$terms$~$(19 terms in $L$) which can be
formed from $H,A_{\mu}$ and its escort $\phi$ shows that (\ref{first}) is (up
to $\partial_{\mu}~$divergence terms and exact $d_{e}$ differentials) is the
unique solution \cite{MS}. However a verification that the $L,V_{\mu}$ pair
satisfies the pair condition requires only the use of free field equations and
the relations between $A_{\mu}$ and its escort and will be left to the reader.
The first order pair condition does not determine the strength of the
$H$-selfinteractions since $e$-independent contributions to $L$ simply pass
through the pair condition. The necessity of their presence which includes the
determination of the $c_{i}~$in (\ref{first}) is seen in second and third
order. This "induction" of additional contributions with well-defined
numerical coefficients is a new phenomenon of SLFT; there is a formal
similarity with the imposition of the second order BRST gauge invariance on
the $S$-matrix \cite{Scharf} but the essential difference is that the
$e$-independence of $S$ is a consequence of the positivity and causal
localization principle of QFT.
For the S-matrix one only needs the second order tree component to the
obstruction $O^{(2)}$ in$~$(\ref{d}) In addition to a second order change of
the time ordering of the propagator involving derivatives of $H~$which
parallels that in (\ref{T})~one now encounters a genuine second order
induction (\ref{ob2})
\begin{equation}
L_{2}=g[(m_{H}^{2}+3c_{1}m^{2})H^{2}\phi^{2}-\frac{m_{H}^{2}}{4}\phi^{4
+c_{2}H^{4}] \label{two
\end{equation}
Finally the vanishing of the third order tree contribution fixes the values of
$c_{1},c_{2}$ in terms of the three physical parameters of its field content
which were already present in the germ namely $g,m,m_{H}.$ To allow for a
comparison with the Higgs mechanism we write the result in the form
\begin{equation}
L_{tot}^{(2)}=mA\cdot(AH+\phi\overleftrightarrow{\partial}H)-V(H,\phi
),~V=g\frac{m_{H}^{2}}{8m^{2}}(H^{2}+m^{2}\phi^{2}+\frac{2m}{g}H)^{2
-\frac{m_{H}^{2}}{2g}H^{2}\quad\label{pot
\end{equation}
where $L_{tot}^{(2)}=L+\frac{g}{2}L_{2}.$ The appearance of a quadratic mass
term is the result of writing the interaction density as if it would be part
of a classical Lagrangian of gauge potentials. The reader may fill in the
details of the straightforward calculations by himself or look up the more
detailed presentation in \cite{MS}.
Apart from a mass contribution the $V~$looks like a field-shifted Mexican hat
potential. \textit{But different from the Higgs mechanism it has not been
obtained by postulating a Mexican hat potential and subjecting it to a shift
in field space}. It is rather\textit{ induced} by a renormalizable
$A,H$\textit{ field content }and it is the \textit{unique renormalizable QFT
with this field content}. There is simply no room for imposing a Mexican hat
potentials since the induction of the $H$ and $\phi$ selfinteractions is a
consequence of $e$-independence of the S-matrix which in turn is a consequence
of scattering theory involving $d_{sd}=1$ causally separable space- or
light-like strings.
The SSB picture of the Higgs model also reveals another common
misunderstanding, this time about SSB. The Mexican hat potential together with
the shift in field space is \textit{not the definition} of SSB but\textit{
rather a way to implement} such a situation \textit{when it is possible}. The
definition of SSB is rather the \textit{existence of a locally conserved
current whose global charge diverges}. This is only possible in the presence
of massless Goldstone bosons and all verbal attempts to make SSB consistent
with a mass gap (a photon becoming fattened to a vector meson by eating a
Goldstone) only obscure the interesting correct understanding.
\textit{QFT is not a theory which can create the masses of its model-defining
field content}. In particular SSB is not about creating finite masses from an
initially massless situation; to the contrary it is about how to place a
massless particle (the Goldstone boson)\ into an interaction density so that
the current conservation remains that of a symmetric theory but some local
charges are prevented to converge in the infinite volume limit to a finite
global charge (the definition of SSB). The only known The prescription of a
field shift on a Mexican hat potential as the "Higgs mechanism" has to be seen
in a historical context; it helped to overcome the formal problems which one
faces when one tries to extend Lagrangian quantization from Maxwell's theory
of charge-carrying fields to a situation in which a vector potential couples
to a Hermitian matter fields. There are numerous historical illustrations of
situations for which important discoveries were made through formal
manipulations which were later replaced by a derivation which is consistent
with the principles of QFT. Incorrect placeholder are useful but only up to
the discovery of the real reasons.
A model of QFT is uniquely fixed in terms of its field content. The SLFT
setting (which seems to be the only one consistent with all principles of QFT)
for a $A_{\mu},H$ field content starts with a $A_{\mu}A^{\mu}H$ as the
simplest coupling and the rest is done by induction using the $L.Q_{\mu}$ pair
property which converts the heuristic physical content of the ill-defined pl
interaction density into the physically superior SLFT setting where the
"induction" resulting from the implementation of the pair property to all
orders unfolds the full content of SLFT.
\subsection{Selfinteracting vector mesons}
It is straightforward to check that there is no renormalizable $L,Q_{\mu}$
pair for a self-coupled singlet $(A\cdot A)^{2}.$ The principles of QFT as
embodied into the pair condition admit however selfinteractions between
multiplets ("colored") of vector potentials while imposing strong restrictions
on the "multi-colored" coupling parameters. In this case the germ is a $FAA$
selfinteraction and the general ansatz for the construction of a $L,V_{\mu
~$pair which includes the "colored" escorts is of the form
\begin{equation}
L=\sum(f_{abc}F_{c}^{\mu\nu}A_{a,\mu}A_{b,\nu}+h_{abcd}A_{a,\mu}A_{b,\nu
A_{c}^{\mu}A_{d}^{\nu})+\,\hbox{terms in}\,A_{a,\mu}\,\hbox{and}\,\phi
_{a}^{\prime}s
\end{equation}
where the couplings and the masses of the vector mesons are initially freely
variable parameters but, as expected, the first and second order pair
condition places strong restrictions on them \cite{MS}, among other things the
$f$ and $h$ are interrelated in the same way (Jacobi identities of reductive
Lie-algebras) as in gauge theory \cite{Scharf}; in particular the $A$-$\phi$
and $\phi$-$\phi$ couplings depend also on the masses of the vector mesons.
The main distinction to gauge theory is that these properties are direct
consequences of the principles of QFT and do not arise in the course of the
gauge theoretic extraction of physics from a unphysical (positivity-violating)
description through the imposition of gauge invariance.
The most interesting aspect of the SLFT formulation is that there remains a
renormalizability destroying second order induced selfinteraction which, if
left uncompensated, destroys renormalizability even though the interaction
density fulfills the power counting restriction of renormalizability. The way
to overcome this is to compensate this $d_{sd}=5~$term with a second order
contribution from a $A$-$H$ interaction with a scalar Higgs field\footnote{A
$s\geq1$ field would worsen the second order short distance behavior.}. This
is a totally different situation from the abelian $A$-$H$ interaction for
which such all second order terms stay within the power counting bound.
Neither case bares any physical resemblance to spontaneous symmetry breaking
since in both cases the field shifted Mexican hat potential is second order induced.
The idea of short distance compensations between contributions from different
spins arose in connection with supersymmetry. Although not invented for this
purpose, SUSY does improve the short distance behavior somewhat but not enough
to guaranty the renormalizability and preservation of supersymmetry in higher
orders. The situation of selfinteracting vector mesons is different, in that
the preservation of renormalizability is the raison d'\^{e}tre for the $H.$
Nature does not have to decide between a symmetry and its SSB, rather the
existence of the $H$ is directly connected to the preservation of its
positivity and causality principles or in other words a massive $A_{\alpha
}^{\mu}$ field content by itself is not consistent.
Gauge symmetry is not a physical symmetry so there is nothing to break; all
these physically incorrect pictures evaporate if one maintains causality
\textit{and} positivity which is perturbation theory is only possible by
starting with an sl $L,V_{\mu}$ or $Q_{\mu}~$pair property. The fibre bundle
like Lie structure of the $f_{abc}$ couplings is not the result of an imposed
symmetry it rather arises from the string-independence of the S-matrix which
in turn is a result of LSZ scattering theory of interacting causally separable
positivity obeying quantum fields; hence the situation is very different from
the superselection structure of unitary representation classes of observable
algebras which leads to the notion of inner symmetries. This shows that
quantum causality is much more fundamental than its classical
Faraday-Maxwell-Einstein counterpart.
Having thus strengthened the conceptual understanding of interactions between
vector mesons in the Standard Model one may ask whether SLFT contains also
messages about their coupling to matter. In recent work \cite{GMV} it was
shown that SLFT does not only restrict the selfcouplings between vector mesons
and requires the presence of a Higgs particle in the presence of
selfinteracting massive mesons but it also restricts their coupling to the
Fermion currents and their chirality properties. This is of particular
interests for massive $W_{\pm}.Z$ vector mesons and the photon, a\ case for
which the authors explain the restrictions from SLFT in detail.
\subsection{The pair condition for higher spins}
The extension of SLFT S-matrix construction to that of interacting higher
spins $s\geq2~$is an important issue about which one presently knows little.
There have been quite extensive investigations in a gauge theoretic equivalent
of the pair condition by Scharf \cite{Scharf}. In view of formal similarities
with SLFT it is interesting to take a closer look at some of his results.
Scharf looked at the simplest $s=2~$selfinteraction which is of a cubic form
$trh^{3}$ where $h_{\mu\nu}$ is the $s=2$ massless tensor field. The physical
interest in this model is connected with the use of $h_{\mu\nu}~$as a linear
approximation of the gravitational $g_{\mu\nu}~$field. As in SLFT, the short
distance dimension of integer spin gauge fields is equal to their classical
dimension in terms of mass units namely $d_{sd}=1.$In \cite{Scharf}~it was
shown that there exists no gauge theoretic trilinear selfinteraction $L^{K}$
with $d_{sd}(L^{K})=3$ without involving derivatives of $h_{\mu\nu},$ its
trace $h_{\mu}^{\mu}~$as well as ghost fields and their anti-ghost. He found a
cubic interaction density of $d_{sd}(L^{K})=5$ which is above the
power-counting bound of renormalization, but still presents a huge reduction
from the $d_{sd}(L^{P})=11$.
Taking into account that gravitational coupling carries a dimension and
expanding the Einstein-Hilbert Lagrangian in a suitable way using
$\kappa=\sqrt{32\pi G}$ as an expansion parameter, he arrived at a formal
connection of the classical expansion with the quantum-induced correction up
to second order; this was later extended to all tree orders \cite{Scharf,Du}.
The agreement of tree approximations with classical perturbation theory is not
unexpected in itself, but in the present context it relates two competing
ideas, one being of classical geometric origin (the Einstein-Hilbert action)
and the other the gauge theory of selfinteracting $h=2$ particles .
Christian Ga\ss \ showed recently (private communication) that SLFT provides a
simpler version of such a cubic selfinteractions in the for
\begin{align}
L & =\kappa(2\partial_{\rho}h^{\kappa\lambda}\partial_{\sigma
h_{\kappa\lambda}+4\partial_{\beta}h_{\rho}^{\alpha}\partial_{\alpha
h_{\sigma}^{\beta})h^{\rho\sigma},~~h_{\mu\nu}:=A_{\mu\nu}^{(2)}\label{L}\\
& d_{e}A_{\mu\nu}^{(2)}=\partial_{\mu}a_{\nu}+\partial_{\nu}a_{\mu},~
\end{align}
where $h_{\mu\nu}=A_{\mu\nu}^{(2)}$ is the sl helicity$~$2 potential from
(\ref{E}) section 2 (which already played a role in solving the D-V-Z
discontinuity problem \cite{MRS2}). Using the relation between $d_{e}$ and
$\partial_{\mu}$ of the second line one easily verifies that $d_{e}L$ is of
the form $\partial^{\mu}Q_{\mu}$ i.e. the above $L$ belongs to a $L,Q_{\mu
~$pair. Since massless $h\geq1$ fields are intrinsically sl, the corresponding
minimal models are expected to be "ultra-distributions" which are localizable
in spacelike cones.
For $h=1$ there exists no colorless selfinteraction whereas for $h=2$ the
situation seems to be reverse since the existence of colored cubic
selfinteractions can be excluded \cite{Bou}. A proof based on Scharf's
S-matrix gauge formalism can be found in \cite{Grig}.
The fact that there are no \textit{renormalizable} $s=2$ selfcouplings does
not exclude the possibility to find sl $L,Q_{\mu}$ pairs of interactions
between sl $h_{\mu\nu}$ with lower spin fields as $H$ or/and $A_{\mu}$. An
ansatz for $L$ which generalizes the $A_{\mu},H$ particle content of the
abelian Higgs model would be of the form ($h_{\mu\nu}$ massive)
\begin{equation}
L=mgh_{\mu\nu}h^{\mu\nu}H+\,U(H,h,\phi) \label{in
\end{equation}
where the first term represents the "ignition" i.e. the simplest
renormalizable ($d_{sd}=3$) interaction associated with a $h,H$ particle
content and $U$ contains all the remaining possible at most quadrilinear
couplings between $h_{\mu\nu},$ its 5 escorts $\phi_{\mu},\phi$ and $H$. Their
coupling strengths are determined from the first or second order ("induction")
pair condition.
The $L,Q_{\mu}$ pair may be uniquely determined, but it is rather improbable
that $d_{sd}(L)\leq4.$ It would be premature to dismiss $L,V_{\mu}$ pairs with
$d_{sd}(L)>4.$ The example of pl models with $d_{sd}(L^{P})=5,~$which in the
standard pl renormalization theory leads to a with perturbative order
increasing number of renormalization parameters but under the guidance of an
S-matrix-equivalent sl pair turns into an improved formalism. This upgraded
$L^{P}$ description contains $d_{sd}\rightarrow\infty$ pl fields but shares
its parameters, the S-matrix and its pl local observables with the SLFT
renormalization theory.
Presently our understanding of the consequences of the higher order SLFT pair
requirements is too scarce to say anything credible about $~L,V_{\mu}$ pairs
with $d_{sd}(L)>4$. A clarification of this important issue will be left to
future research.
\section{Dynamical string-localization of interacting fields}
Free massive pl fields can not only be converted into their sl counterparts by
integration along strings but the directional $e^{\mu}\partial_{\mu}$
differentiation on sl free fields permits also the return to its pl form.
Together with their Wick-ordered composites they form the local equivalence
(sl extended Borchers-) class $\mathcal{B}$ of free fields (pl fields are
viewed as special cases of sl).
Recall that for the construction of the S-matrix corresponding to a prescribed
particle content one uses pl fields for $s<1$ and those special $s\geq1$
massive sl potentials which were constructed in section 2.3 by "fattening"
their uniquely defined sl massless counterpart. Together with the uniquely
defined pl Proca potential and a scalar sl field referred to as the escort
they constitute a triple of relatively causally localized fields which act in
the massive $s=1$ Wigner-Fock space. They fulfill a linear relation which is
the basis for the construction of renormalizable sl interaction densities with
string-independent S-matrices.
This "kinematic" sl localization of $s\geq1~$free fields is important for the
construction of the S-matrix ala Bogoliubov. But it does not account for the
\textit{physical localization of the interacting fields} which is not in the
hands of the calculating physicists but is determined by the particle content
of the model. To distinguish between the two the localization of the
interacting fields will be referred to as "dynamic localization".
To understand this important point it is helpful to recall the form of the
\textit{Bogoliubov map} which relates the pl or sl Wick-ordered free fields
from the local equivalence class of free fields $\mathcal{B}$ to that of
normal ordered interacting fields $\mathcal{B}|_{L}$ (\ref{C}). For \thinspace
pl gauge theoretic interactions densities $L^{K}$ this problem has been
studied in \cite{Du-Fr}.
One important result is that this perturbatively defined linear Bogoliubov map
preserves the relative causality of fields but not the algebraic structure.
This is in agreement with algebraic QFT which is based on the idea that the
full physical content of QFT in the presence of interactions is contained in
the net of spacetime localized algebras \cite{Haag}. What is shared between
$\mathcal{B}$ and $\mathcal{B}|_{L}~$in case of massive vector potentials is
the Wigner-Fock Hilbert space in which these fields act.
This transfer of pl causality undergoes significant changes in the presence of
sl fields. As in the calculations in the previous section one uses a lightlike
$e$, in this case no directional testfunction smearing is necessary.
For the understanding of changes in localization caused by the Bogoliubov map
it is not necessary to enter the details of perturbative renormalization. It
suffices to understand the relations between free fields in $\mathcal{B}$
which result from the assumption that their interacting images of the
Bogoliubov map into the target spaces $\mathcal{B}|_{L_{tot}\text{ }}$and
$\mathcal{B}|_{L_{tot}^{P}\text{ }}$coalesce. Hence one may omit the
prefactors $S^{-1}$ in the Bogoliubov maps and writ
\begin{align}
& S(g(x)L_{tot}^{P}+\lambda f\varphi_{g})|_{\lambda=0}\overset{a.l.}{\simeq
}S(g(x)L_{tot}+\lambda f\varphi)|_{\lambda=0}\label{rel}\\
& \varphi_{g}|_{L_{tot}^{P}}=\varphi|_{L_{tot}},~~\varphi_{g}(x,e)=\varphi
(x,e)+\sum_{k=1}^{N}\varphi_{k}(x,e) \label{con
\end{align}
where the $\varphi_{g}|_{L_{tot}^{P}}$ refers to the interacting image of
$\varphi_{g}$ under the $L_{tot}^{P}$ Bogoliubov map.
The information about the localization of an interacting field $\varphi
|_{L_{tot}}$ is contained in the left hand side\footnote{The important point
is that the $L^{P}$ Bogoliubov map preserves localization; hence one can use
it to find out whether the image of $\varphi$ under $L~$is sl or pl.} whereas
its renormalizability status (finite or infinite $d_{sd}$) can be red off on
the the right hand side. Fields which are renormalizable and at the same time
pl in the $L^{P}$ setting represent observables whereas renormalizable fields
which are sl on the $L_{tot}^{P}$ side are sl interpolating fields.
The formal combined map of $\mathcal{B}$ into itself is highly non-linear and
generally changes localization properties; this is the price for the
preservation of renormalizability. The $\varphi_{k}(x,e)$ in (\ref{con}) are
determined by the induction
\begin{equation}
\varphi_{k+1}(x,e)=ig\int T(L_{tot}^{P}(x^{\prime})-L_{tot}(x^{\prime
},e))\varphi(x,e)=ig\int[\partial^{\prime\mu}T]V_{tot,\mu}(x^{\prime
},e)\varphi_{k}(x,g)d^{4}x^{\prime} \label{it
\end{equation}
where $[\partial^{\prime\mu},T]$ denotes the difference between the
$\partial~$acting outside and inside the time-ordering which either vanishes
or contributes a $\delta$-term.
In massive QED this conversion (\ref{con}) has no effect on pl observables;
fields as $A_{\mu}^{P}$ and$\ F=\operatorname{curl}A^{P}$ simply pass through
since with $V_{\mu}=\phi j_{\mu}$ and $\varphi_{0}=A_{\mu}^{P}~$the right hand
side (\ref{it}) vanishes and henc
\begin{equation}
A_{\mu}^{P}(x)|_{L^{P}}=A_{\mu}^{P}(x)|_{L},~~F_{\mu\nu}|_{L^{P}}=F_{\mu\nu
}|_{L}\quad\label{ob
\end{equation}
The idea underlying such conversions was first used by Mund \cite{Mund} in the
context of massive spinor QED. He calculated higher orders for the
charge-carrying $\psi$ (spinor or complex scalar) and found consistency with
\begin{align}
& \psi(x)|_{L^{P}}=e^{ig\phi(x,e)}\psi(x)|_{L}\label{Mund}\\
& \psi(x)|_{L}=e^{-ig\phi(x,e)}\psi(x)|_{L^{P}
\end{align}
The formula is reminiscent of gauge transformation, however its physical
content is quite different.
A particularly interesting application of the conversion formalism arises in
the Higgs model. Different from massive QED, neither the $s=1$ field$\ A_{\mu
}^{P}|_{L^{P}}$, nor $H_{L^{P}}$ are local observables. Using the form of
$V_{\mu}~$in (\ref{first}) one finds that $H$ is transformed into $H_{1}$
\begin{equation}
H_{1}(x,e)=-\int[\partial^{\prime\mu},T]V_{\mu}(x^{\prime})H(x)d^{4}x^{\prime
}=\frac{1}{2}:\phi^{2}(x,e): \label{H
\end{equation}
i.e. $H$ is against naive expectations not an observable but rather represents
a sl interpolating field. The same holds for $A^{P}$~or its
$F=\operatorname{curl}A^{P}.$
Allowing additive composite modifications $H\rightarrow H+polyn(H,A^{P})$
which preserve the asymptotic scattering state of the $H$-particle does not
change the situation. The same holds for the $A_{\mu}^{P}$ or
$F=\operatorname{curl}A^{P}$ $\ $Hence both fields which are linearly related
to the particle content of the model are interpolating fields and do not
represent observables. The fact that $A_{\mu}^{P}|_{L^{P}}$ and $F$ are
observables\footnote{The line integral over an observable commutes with
"switching on" the interaction and does not represent an interpolating
fields.} in massive QED but not in the Higgs model shows that the
observable-interpolating field dichotomy is not a kinematic property.
Hence fields representing local observables in the Higgs model are necessarily
composite. A composite field which exists in every model is the interaction
density $L_{tot}^{P}|_{L_{tot}^{P}}=L_{tot}|_{L_{tot}}.$ The right hand side
was calculated in (\ref{pot}) and the computation of $L^{P}~$will be contained
in a forthcoming publication \cite{MS}.
A better understanding about the singular structure of pl fields may shed new
light on the localizability of $d_{sd}=\infty$ fields which arose in
connection with summing up graphical structures in certain nonrenormalizable
models \cite{Ba-S}. This problem was taken up by Arthur Jaffe \cite{Jaffe
\cite{Sol} who discovered a new class of singular distributions which still
permit smearing with a dense set of compact supported Schwartz testfunctions.
These Jaffe distributions had no impact on QFT because the unguided pl
nonrenormalizability with its infinite number of renormalization counterterm
parameters does not present a well-defined arena for physical applications.
Such $d_{sd}=\infty~$pl fields are probably too singular to generate operator
algebras, but they may still create physical states in the SLFT-guided $L^{P}$ formalism.
In section 3.2 we sketched the application of this formalism to interactions
with external potentials. Such interactions do not lead to loop contributions.
This simplicity of only induced contribution promises an interesting
mathematically controllable playground for the study of the pl localization
properties of observables and that of sl interpolating fields.
SLFT is presently the last step in a process of dissociating QFT from its
historic ties with Lagrangian quantization. When shortly after the discovery
of renormalized QED Arthur Wightman presented his "axiomatic" formulation of
QFT in terms of pl fields, it appeared to be the most appropriate intrinsic
formulation which can be extracted from Lagrangian quantization and Wigner's
representation theory \cite{St-Wi}. In his algebraic formulation of Local
Quantum Physics (LQP) Rudolf Haag proposed a setting of QFT based on a net of
localized operator\ algebras representing observables which removed the last
vestiges of quantization \cite{Haag}.
The next step was taken in the 80s by Buchholz and Fredenhagen who showed that
the existence of observable algebras and suitably defined particle states
guaranties the presence of operators localized in arbitrary narrow spacelike
wedges (whose cores are strings) which create these particle states from the
vacuum \cite{Bu-Fr}. These constructions were too far removed from the
exigencies of renormalized perturbation theory in order to have a direct
impact on calculations.
As a result it tooks more than 30 years to incorporate these observations into
a new sl perturbation theory in whose discovery the understanding of the
noncompact localization of Wigner's infinite spin matter was an important
catalyzer \cite{MSY2}. Fortunately one does not have to go through the details
of this history in order to do perturbative calculations. But what may be
interesting to note is that, different from Wightman's extraction of his
axiomatic setting from what one learned from the mathematically rather
ill-defined rules of Lagrangian quantization, the construction of SLFT took
the opposite path by converting ideas from LQP into perturbatively accessible computations.
Its most remarkable physical property is that observables are distinguished
from interpolating fields in terms of localization, which is of course to be
expected in a theory based on causal localization, but which GT could not accomplish.
There remains the question how GT with its lack of quantum positivity for
interpolating fields achieves to be such an amazingly successful description.
This will be commented on in the concluding remarks.
\begin{verbatim}
Erratum: the claim that the interacting Proca field $A_{\mu}^{P}$ of the abelian Higgs model is an sl interpolating
field is incorrect. The correct propagator is transverse which implies that $(A_{\mu}^{P})_{1}=0$ i.e. $A_{\mu}^{P}$ remains a pl
observable.
\end{verbatim}
\section{Resum\'e, loose ends and an outlook}
SLFT is a formulation of QFT in which renormalizable interacting fields
maintain the tightest possible localization which is compatible with quantum
positivity and causality. In contrast to gauge theory its physical range is
not limited to local observables and the S-matrix but also includes
string-local interpolating fields which mediate between the causality
principles of QFT and the string-independent scattering properties of
particles. All degrees of freedom are provided by Wigner's particle
representation theory.
As described in the introduction the discovery of SLFT was triggered by the
construction of sl free fields associated to Wigner's positive energy infinite
spin representation \cite{MSY2}. Yngvason's 1970 No-Go theorem \cite{Y}
precluded the existence of pl fields. It turned out that Wigner's massless
infinite spin representation presents a much stronger barrier against pl
localization than that observed by Weinberg and Witten in massless finite
helicity representations. The Weinberg-Witten No-Go theorem excludes the
existence of conserved higher helicity pl currents and energy-momentum
tensors; in view of the absence of massless pl vector potentials and the fact
that the existence of pl massless limits depends on the short distance
dimension $d_{sd}$ this is hardly surprising $.$
The infinite spin case excludes the existence of pl composites; more general:
the causal localization of infinite spin matter is necessarily noncompact
\cite{LMR} in concordance with smearing sl fields with directionally compact
localized test functions $f(x,e),~e^{2}=-1.$ Closely related is that
\textit{infinite spin matter cannot interact with ordinary (finite spin)
quantum matter ,} but through its energy-momentum tensor its backreaction on
classical gravity may lead to a noncompact form of gravity. Quantum inertness
combined with gravitational reactivity are properties attributed to dark
matter \textit{\cite{dark}}. Since the sl infinite spin energy-momentum tensor
is known as a bilinear form \cite{PL} such a calculation appears feasible.
The existence of sl infinite spin field with \textit{finite} $d_{sd}$
suggested that the renormalizability destroying $d_{sd}=s+1$ increase of short
distance dimension can be avoided by using sl fields. This was the start for
the construction of sl potentials for finite $s,h$ which provided the
positivity preserving (the Gupta-Bleuler degrees of freedom avoiding)
$d_{sd}=1$ potentials. As mentioned in section 3 the absence of pl currents
does not exclude the existence of local charges which are localized in
arbitrary small spacetime regions.
The weakening of causal localization in SLFT should not be misunderstood as
(what is commonly referred to as) "nonlocal"\footnote{The authors of
\cite{MRS1} had problems with referees who rejected the work with the argument
that SLFT is nonlocal.}. The use of covariant semi-infinite space- or
light-like half-lines does not get into conflict with the causality
prerequisites of scattering theory (namely the possibility of placing an
arbitrary number of fields in relative spacelike positions), nor is the
derivation of important structural theorems (TCP, Spin\&Statistics) impeded.
Among the continuously many sl potentials only one for each $s$ plays a role
in SLFT perturbation theory. The key observation for its construction is that
the equation $\operatorname{curl}A=F$ for a \textit{sl massless field}
$A_{\mu}(x,e)$ \textit{acting in the Wigner-Fock helicity space} associated to
the ($m=0,h=\pm1$) Wigner representation has a unique solution which replaces
the positivity violating pl potential of GT.
By a process referred to as "fattening" (section 2) this solution selects
among the many possible massive sl potentials (which act in the $s=1$
Wigner-Fock Hilbert space of the unique pl Proca potential) a distinguished sl
vector potential. Together with a canonically constructed scalar sl potential
$\phi(x,e)$ (the escort) one obtains a triple of linear related fields
$A_{\mu}-\partial_{\mu}\phi=A_{\mu}^{P}$ which act in the $s=1$ Wigner Fock
space and belong to the linear part of the causal equivalence class of
(Wick-ordered) free fields associated to the Wigner representation $(m>0,s).$
The string independence expressed as the pair relation $d_{e}(A-\partial
\phi)=0$ is the basis for constructing a renormalizable interaction density
$L(x,e)$ which couples the $s=1~$sl $A$ and $\phi$ fields to lower spin free
fields which remain pl. Together with a suitably defined vector density
$V_{\mu}~$ one arrives at the pair relation $d_{e}(L-\partial V)=0$ which
insures the string-independence of the S-matrix which is obtained by taking
the adiabatic limit of time-ordered product of the interaction density. The
lowest order pair relation may need an extension by induced terms which result
from the implementation of higher order pair conditions. This is a new
phenomenon which has no counterpart in the old pl perturbation theory. \
The interacting quantum fields associated to this S-matrix are constructed in
terms of the Bogoliubov map which converts pl or sl fields from the causal
equivalence class of Wick-ordered free fields into their normal ordered
interacting counterpart. The restriction to pl $s<1$ and sl $s=1$ free fields
is only necessary in the construction of the S-matrix; the Bogoliubov map can
be applied to \textit{any} (pl or sl, elementary or composite) field in the
free field class.
Its interacting target fields have in general a different localization from
their source fields. The target localization has to be determined in the
$L^{P}$ setting (see previous section). Renormalizable ($d_{sd}<\infty$)
fields in the $L$ target space (independent of their pl or sl localization)
represent observables if their $L^{P}$ source fields are pl; otherwise they
represent interpolating sl fields.
SLFT has been applied up to second order to all models in which vector mesons
interact with themselves or with $s<1$ particles. The by far conceptually most
demanding and interesting QFT is the Higgs model which in its most simple
(abelian) form is the QFT in which a vector meson interacts with a $s=0$
Hermitian field. The first order pair turns out to be uniquely fixed and its
second order implementation induces a $H$ selfinteraction which looks as if it
would come from spontaneous symmetry breaking on a postulated Mexican hat
selfinteraction. The conceptual difference to SLFT is enormous.
A similar but somewhat more elaborate second order calculation for
selfinteracting massive vector mesons reveals that the coupling structure of
the \textit{leading} $d_{sd}=4$ \textit{contributions} up to second order is
that of a reductive Lie-algebra. The surprise is that, different from gauge
theory, this apparent symmetry in the $d_{sd}$ leading contribution has not
been imposed. In fact it is not even a symmetry in the sense in which this
terminology is used to describe unitary implemented inner symmetries.
Whereas symmetries and their spontaneous or complete breaking of
selfinteracting scalar particles can be freely imposed, there are strong
restrictions from first principles on the form of $s\geq1~$SLFT
selfinteractions which leave no such freedom; the form of selfinteractions in
the presence of $s\geq1~$is fully determined by the particle content of the
model and not at the disposition of the calculating theorist. The use of the
positivity violating gauge symmetry obscures this important insight. The
chirality theorem \cite{GMV} shows that these principles also affects the
coupling of selfinteracting vector potentials to Dirac fermions.
Another somewhat unexpected property is that renormalizable interaction sl
densities $L$ may produce second order sl $d_{sd}=5$ contributions which, if
left uncompensated, destroy the $e$-independence of $S$ as well as
renormalizabilty. The only way to save such a model is to enlarge its particle
content by a $A$-$H~$interaction which induces a compensating second order $A$
selfinteraction. \textit{This, and not SSB, is the raison d'\^{e}tre for the
presence of an }$H$\textit{-particle in models of selfinteracting massive
vector mesons.}
The application of the SLFT perturbation theory to the Higgs model leads to
other foundational questions whose answer may be trendsetting for the
development of QFT. The basic interaction density $A\cdot AH$ for a $A_{\mu
},H$ particle content (the "ignition" from which the $L,V_{\mu}$ pair
requirement uniquely induces all other contributions) is superrenormalizable
since $d_{sd}(AAH)=d_{cl}=3$. One does not expect that interactions induced by
superrenormalizable couplings lead to higher order counterterms with new
coupling parameters. A $4^{th}$ order confirmation of this expectation does
presently not exist (neither in SLFT nor in the gauge theoretic SSB setting).
Even more important is to find out if $SLFT$ permits an extension to $s\geq2.$
The remarks in section $5.6~$on $s=2$ selfinteractions show that
$d_{sd}(L)=5.$ To conclude that the theory is useless because it violates the
power-counting bound is premature since (previous section) the main reason for
dismissing interaction densities is that they lead to a with perturbative
order increasing number of coupling parameters and not the fact that there are
fields with an increasing short distance scaling degree. For the acceptance of
a model it suffices that its S-matrix is well-defined and that its observables
remain pl with bounded $d_{sd},$ independent of whether the $d_{sd}$ of its sl
interpolating fields increase with perturbative order.
Candidates with $s=2$ potentials $A_{\mu\nu}~$and superrenormalizable
"ignition"$~$of the form $A_{\mu\nu}A^{\mu\nu}H~$or $A^{\mu\nu}A_{\mu}A_{\nu
}~$and induced $L,V_{\mu}$ pairs couplings with $d_{sd}(L)=5~$are expected to
exist. As long as the number of counterterm coupling parameters does not
increase with perturbative order and the physical predictability is maintained
there is no obvious reason for their exclusion of such $L,V_{\mu}$ pairs. Only
further research can resolve these challenging problems. \
Can SLFT shed some light on the perplexing question \textit{why GT inspite of
its obvious conceptual shortcomings\footnote{Positivity is an indispensable
property which secures the probability interpretation of quantum theory. }
remained such an amazingly successful theory}? This paradigmatic question may
have a positive answer. The $L^{K},V_{\mu}^{K}~$pair property is a consequence
of the relation $A_{\mu}^{P}=A_{\mu}^{K}+\partial_{\mu}\phi^{P,K}$ where $K$
refers to the Krein space of GT and $\phi^{P,K}$ is a scalar pl "hybrid"
escort which mediates between the $P$ and $K$ formalism \cite{hybrid}.
In the BRST formulation used in \cite{Scharf} the fields act in a
St\"{u}ckelberg- and ghost- extended BRST space. The physical space, to which
the action of $A_{\mu}^{P}$ can be restricted, is defined in terms of BRST
cohomology and observables are defined as objects invariant under the BRST
operation $\mathfrak{s}$ ($\mathfrak{s}O=0$ for observables and $\mathfrak{s
S=0$ for the S-matrix).$\mathfrak{~}$
The advantage of the hybrid formulation proposed by Mund \cite{hybrid} is
that, different from the formalism used in \cite{Scharf}, St\"{u}ckelberg- and
ghost- degrees of freedom are not needed. Instead of spaces which are embedded
in the sense of BRST cohomology one deals with factorization through
Gupta-Bleuler subspaces.
An explicit expression for the mixed hybrid escort $\phi^{P,K}$ was recently
calculated by Mund (private communication). The Proca potential lives in the
transverse subspace to the mass shell $p^{\mu}\psi_{\mu}(p)=0$ which is
embedded in the Krein space whereas the living space of fields is the full
4-component Krein space. The triple relation can the be used to define a
$L^{K},V_{\mu}^{K}$ pair which is S-matrix-related to the physical $L^{P}$ formulation.
A favorable situation for studying \textit{infrared phenomena} arises from the
hybrid triple $A_{\mu}^{sl}(x,e)=A_{\mu}^{K}(x)+\partial_{\mu}\phi
^{sl,K}(x,e).~$The $A_{\mu}^{S}~$(without)~lives on the physical subspace of
the Gupta-Bleuler Krein space. All three contributions have a massless limit,
but $\phi^{s,K}$ without the derivative has the typical logarithmic infrared
divergence known from scattering theory of charge-carrying particles. Its
exponential $\exp ig\phi^{S,K}(x,e)$ seems to provide the kind of directional
superselection rule of photon "clouds" whose presence is required by a theorem
\cite{Bu}. This picture is a closer analog of (\ref{scal}) than the $\exp
ig\Phi(x,e,e^{\prime})$ constructed in (\ref{5}).
This hybrid pair description does not only explain the close relation of a
(from ghosts and negative metric St\"{u}ckelberg fields liberated)
Gupta-Bleuler GT with the positivity preserving SLFT, but it also shows that
GT plays a useful constructive role for a better understanding of SLFT in QED.
The hybrid relation reveals that the physical origin of quantum gauge theory
is that one cannot squeeze causally spacetime localized pl vector potentials
into the Wigner momentum space; this is only possible by permitting a
noncompact but still causally separating localization.
In particular it contains informations about the change of the Wigner particle
space for the $\mathcal{B}|_{L}$ operators (previous section) in the massless
limit.$~$Whereas the fields in $B$ live in a Wigner-Fock helicity space, their
interacting images in $\mathcal{B}|_{L}$ act on a larger space for whose
construction one needs to form line integrals on indefinite Gupta-Bleuler
potentials $A_{\mu}^{K}(x)$ (still indefinite) and convert them into complex
exponential fields (the photon clouds) whose associated Hilbert space is
expected to show a similar infrared structure as the exponentials $\exp
ig\varphi(x)$ of the indefinite logarithic divergent massless $d=1+1$ scalar
fields (the $\varphi$-clouds).
The hope is to obtain a spacetime understanding of infrared phenomena
including the large-time behavior which replaces that of the LSZ scattering
theory. This includes the vanishing of scattering between charge-carrying
particles with only a finite number of outgoing photons.
This cannot be described solely in terms of free matter fields, rather the
exponential sl dependent photon cloud fields must play an important role.
Similar to the $\varphi$-clouds in a two dimensional model (\ref{scal}) they
are expected to "soften" the mass-shell singularity and account for the zero
probability for the emission of a finite number of photons in collisions of
charged particles whereas a perturbative expansion which ignores this change
of the mass-shell leads to the logarithmic infrared singularities. As often,
the devil is in the details.
SLFT also suggests that behind confinement there could be a more radical
\textit{off-shell} perturbative logarithmic infrared divergence of massless
selfinteracting gluons. Such off-shell divergences are absent in covariant
gauges of nonabelian GT, but off-shell long distance singular behavior of
self-interacting gluons \textit{in SLFT} may be stronger than in GT
\cite{Bey}\cite{dark}.
Most problems of SLFT remain unsolved; on particular the present state of
knowledge about higher order perturbative renormalization is insuffient. Its
strengths is that the new ideas passed many tests and that the promise to
transcend the conceptual limitations of GT is too tempting to resist.
This may be attributed in part to the fact that its underlying ideas are in
embryo and the number of researchers who know about their existence and
decided to study them is still very small. There is no lack of researchers
working on foundational problems of QFT extending the pioneering work of
Wightman, Haag and others. Most theoreticians use the existing gauge theoretic
formalism to solve problems of high energy particle physics or cosmology.
During the last 5 decades a lot of time has been invested in reseach on
speculative ideas as String Theory, Multiverses, Supersymmetry and alike; the
incentive was obviously to continue the success of the first three decades of
QFT in which such speculative way of proceeding was very successful and which
led to most of our by now household goods.
The lack of any tangible results of these attempts led meawhile to feelings of
somberness. The SLFT raises the question why loose time with speculative ideas
if we still know so little about our most successful theory?
\textbf{Acknowledgements.} This work is part of a joint project whose aim is
to develop a positivity-preserving formulation of causal perturbation theory
QFT which, instead of using indefinite metric and "ghosts", is based on only
physical degrees of freedom. The people presently involved in it are Jos\'{e}
Gracia-Bond\'{\i}a, Jens Mund, Karl-Henning Rehren and Joseph V\'{a}rilly and
their students. For intense and profitable contacts I owe thanks to all of
them. I am also indebted to Michael D\"{u}tsch for helpful comments.
I acknowledge a stimulating correspondence with Edward Witten which started
with his curiosity about the history of Haag duality in connection with a
recent upsurge of interest in the role of causality in problems of
entanglement. SLFT has similar conceptual roots and particle physics would
profit if it could divert some of this new interest in fundamental properties
of QFT. \ Last not least I thank the referee for his constructive proposals.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,889 |
Q: How to pass $_GET variables from a link to a bootstrapmodal? Snippet from my HTML code.
<td><span data-placement="top" data-toggle="tooltip" title="Show"><a href="#" class="btn btn-primary btn-xs" data-toggle="modal" data-target="#editBox" data-book-id="<?php echo $obj->id;?>"><span class="glyphicon glyphicon-pencil"></span></a></span></td>
The modual that are beeing opened when link is clicked:
<!-- Modal -->
<div class="modal fade" id="editBox" tabindex="-1" role="dialog" aria-labelledby="myModalLabel">
<div class="modal-dialog" role="document">
<div class="modal-content">
<div class="modal-header">
<button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button>
<h4 class="modal-title" id="myModalLabel">Modal title</h4>
</div>
<div class="modal-body">
<?php var_dump($_GET)?>
</div>
<div class="modal-footer">
<button type="button" class="btn btn-default" data-dismiss="modal">Close</button>
<button type="button" class="btn btn-primary">Save changes</button>
</div>
</div>
</div>
</div>
Is there a proper way to pass my id into the modal?
A: The simple solution to pass id, fetch records from database against passed id and show in modal is;
Simple Solution
Modal Call button
<td><span data-placement="top" data-toggle="tooltip" title="Show"><a class="btn btn-primary btn-xs" data-toggle="modal" data-target="#editBox" href="file.php?id=<?php echo $obj->id;?>"><span class="glyphicon glyphicon-pencil"></span></a></span></td>
Modal HTML
Put following modal HTML outside the while loop in page where the above call button is located (preferable at bottom of page)
<div class="modal fade" id="editBox" tabindex="-1" role="dialog" aria-labelledby="myModalLabel">
<div class="modal-dialog" role="document">
<div class="modal-content">
//Content Will show Here
</div>
</div>
</div>
Now Create a PHP file and name it file.php
This file is called with modal call button href="file.php?id=<?php echo $obj->id;?>"
<?php
//Include database connection here
$Id = $_GET["id"]; //escape the string if you like
// Run the Query
?>
<div class="modal-header">
<button type="button" class="close" data-dismiss="modal">×</button>
<h4 class="modal-title"><center>Heading</center></h4>
</div>
<div class="modal-body">
//Show records fetched from database against $Id
</div>
<div class="modal-footer">
<button type="button" class="btn btn-default">Submit</button>
<button type="button" class="btn btn-default" data-dismiss="modal">Close</button>
</div>
To remove the data inside modal or in other words to refresh the modal when open next records without page refresh, use following script
Put it after jQuery and Bootstrap library (Remember jQuery & Bootstrap libraries always come first)
<script>
$( document ).ready(function() {
$('#editBox').on('hidden.bs.modal', function () {
$(this).removeData('bs.modal');
});
});
</script>
Alternate Solution with Ajax and Bootstrap Modal Event Listener
In Modal Call button replace href="file.php?id=<?php echo $obj->id;?> with data attribute data-id="<?php echo $obj->id;?>" so we pass the id of row to modal using bootstrap modal event
<td><span data-placement="top" data-toggle="tooltip" title="Show"><a class="btn btn-primary btn-xs" data-toggle="modal" data-target="#editBox" data-id="<?php echo $obj->id;?>"><span class="glyphicon glyphicon-pencil"></span></a></span></td>
Modal HTML
Put following modal HTML outside the while loop in page where the above call button is located (preferable at bottom of page)
<div class="modal fade" id="editBox" tabindex="-1" role="dialog" aria-labelledby="myModalLabel">
<div class="modal-dialog" role="document">
<div class="modal-content">
<div class="modal-header">
<button type="button" class="close" data-dismiss="modal">×</button>
<h4 class="modal-title"><center>Heading</center></h4>
</div>
<div class="modal-body">
<div class="form-data"></div> //Here Will show the Data
</div>
<div class="modal-footer">
<button type="button" class="btn btn-default">Submit</button>
<button type="button" class="btn btn-default" data-dismiss="modal">Close</button>
</div>
</div>
</div>
</div>
Now Add following script in same page;
<script>
//jQuery Library Comes First
//Bootstrap Library
$( document ).ready(function() {
$('#myModal').on('show.bs.modal', function (e) { //Modal Event
var id = $(e.relatedTarget).data('id'); //Fetch id from modal trigger button
$.ajax({
type : 'post',
url : 'file.php', //Here you will fetch records
data : 'post_id='+ id, //Pass $id
success : function(data){
$('.form-data').html(data);//Show fetched data from database
}
});
});
});
</script>
Now Create a PHP file and name it file.php (same as use in Ajax Method)
<?php
//Include database connection here
if($_POST['id']) {
$id = $_POST['id'];
// Run the Query
// Fetch Records
// Echo the data you want to show in modal
}
?>
In this solution, you don't need following script to remove the data inside modal or in other words to refresh the modal
$('#editBox').on('hidden.bs.modal', function () {
$(this).removeData('bs.modal');
});
Alternate Solution with Ajax and jQuery Click function
Modal Call button
<td><span data-placement="top" data-toggle="tooltip" title="Show"><a class="btn btn-primary btn-xs" class="open-modal" href="" id="<?php echo $obj->id;?>"><span class="glyphicon glyphicon-pencil"></span></a></span></td>
Put following modal HTML in page where above modal call button located (preferable at bottom of page)
<div class="modal fade" id="editBox" tabindex="-1" role="dialog" aria-labelledby="myModalLabel">
<div class="modal-dialog" role="document">
<div class="modal-content">
<div class="modal-header">
<button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button>
<h4 class="modal-title" id="myModalLabel">Modal title</h4>
</div>
<div class="modal-body">
<div class="form-data"></div> //Here Will show the Data
</div>
<div class="modal-footer">
<button type="button" class="btn btn-default" data-dismiss="modal">Close</button>
<button type="button" class="btn btn-primary">Save changes</button>
</div>
</div>
</div>
</div>
following Ajax method code in same page where Modal HTML & Modal call button located.
<script>
//jQuery Library Comes First
//Bootstrap Library
$( document ).ready(function() {
$('.open-modal').click(function(){
var id = $(this).attr('id');
$.ajax({
type : 'post',
url : 'file.php', //Here you should run query to fetch records
data : 'post_id='+ id, //Here pass id via
success : function(data){
$('#editBox').show('show'); //Show Modal
$('.form-data').html(data); //Show Data
}
});
});
});
</script>
And the PHP file file.php will be same as the above solution with bootstrap modal event
Pass On page information to Modal
In some cases, only need to pass (display) minimal information to modal which already available on page, can be done with just bootstrap modal event without Ajax Method with data-attributes
<td>
<span data-placement="top" data-toggle="tooltip" title="Show">
<a data-book-id="<?php echo $obj->id;?>" data-name="<?php echo $obj->name;?>" data-email="<?php echo $obj->email;?>" class="btn btn-primary btn-xs" data-toggle="modal" data-target="#editBox">
<span class="glyphicon glyphicon-pencil"></span>
</a>
</span>
</td>
Modal Event
$(document).ready(function(){
$('#editBox').on('show.bs.modal', function (e) {
var bookid = $(e.relatedTarget).data('book-id');
var name = $(e.relatedTarget).data('name');
var email = $(e.relatedTarget).data('email');
//Can pass as many onpage values or information to modal
});
});
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,496 |
Q: Azure SDK not importing I downloaded the Azure SDK and added it as a library to my Netbeans project. However, the imports are still not found. What am I doing wrong?
A: Normally, a Java library is depended on some dependent libraries which you also download. My suggestion is that you can try to follow the NetBeans quick start document for Maven to install the Azure SDK library & its dependencies via Maven.
The maven repository of Azure SDK is at here, you just add the maven dependency content below within the xml element dependencies to the pom.xml file for your project on NetBeans.
<!-- https://mvnrepository.com/artifact/com.microsoft.azure/azure -->
<dependency>
<groupId>com.microsoft.azure</groupId>
<artifactId>azure</artifactId>
<version>1.0.0</version>
</dependency>
Then, maven will help downloading & installing Azure SDK jar file & all dependencies when you save the maven pom.xml file.
Hope it helps.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,119 |
{"url":"https:\/\/www.finedictionary.com\/estimation.html","text":"# estimation\n\n## Definitions\n\n\u2022 WordNet 3.6\n\u2022 n estimation a judgment of the qualities of something or somebody \"many factors are involved in any estimate of human life\",\"in my estimation the boy is innocent\"\n\u2022 n estimation an approximate calculation of quantity or degree or worth \"an estimate of what it would cost\",\"a rough idea how long it would take\"\n\u2022 n estimation the respect with which a person is held \"they had a high estimation of his ability\"\n\u2022 n estimation a document appraising the value of something (as for insurance or taxation)\n\u2022 ***\nWebster's Revised Unabridged Dictionary\n\u2022 Interesting fact: An estimated 690 million people live in Africa\n\u2022 Estimation An opinion or judgment of the worth, extent, or quantity of anything, formed without using precise data; valuation; as, estimations of distance, magnitude, amount, or moral qualities. \"If he be poorer that thy estimation , then he shall present himself before the priest, and the priest, and the priest shall value him.\"\n\u2022 Estimation Favorable opinion; esteem; regard; honor. \"I shall have estimation among multitude, and honor with the elders.\"\n\u2022 Estimation Supposition; conjecture. \"I speak not this in estimation ,\nAs what I think might be, but what I know.\"\n\u2022 Estimation The act of estimating.\n\u2022 ***\nCentury Dictionary and Cyclopedia\n\u2022 Interesting fact: On average, it is estimated that females injure themselves ten time more than males do while playing sports\n\u2022 n estimation The act of estimating; the act of judging something with respect to value, degree, quantity, etc.\n\u2022 n estimation Calculation; computation; especially, an approximate calculation of the worth, extent, quantity, etc., of something; an estimate: as, an estimation of distance, magnitude, or amount, of moral qualities, etc.\n\u2022 n estimation In chem., the process of ascertaining by analysis the quantity of a given substance contained in a compound or mixture.\n\u2022 n estimation Opinion or judgment in general; especially, favorable opinion held concerning one by others; esteem; regard; honor.\n\u2022 n estimation Conjecture; supposition; surmise.\n\u2022 n estimation Synonyms Appraisement, valuation.\n\u2022 n estimation Estimate, Regard, etc. (see esteem); admiration, reverence, veneration.\n\u2022 ***\nChambers's Twentieth Century Dictionary\n\u2022 Interesting fact: In July 1874, a swarm of Rocky Mountain locusts flew over Nebraska covering an area estimated at 198,600 square miles. It is estimated that the swarm contained about 12.5 trillion insects. These insects became extinct thirty years later\n\u2022 n Estimation act of estimating: a reckoning of value: esteem, honour: importance: conjecture\n\u2022 ***\n\n## Quotations\n\n\u2022 Johann Friedrich Von Schiller\n\u201cThe average estimate themselves by what they do, the above average by what they are.\u201d\n\u2022 George William Curtis\n\u201cThe test of civilization is its estimate of women.\u201d\n\u2022 Blaise Pascal\n\u201cUgly deeds are most estimable when hidden.\u201d\n\u2022 Henry Ford\n\u201cThe best we can do is size up the chances, calculate the risks involved, estimate our ability to deal with them, and then make our plans with confidence.\u201d\n\u2022 Ralph Waldo Emerson\n\u201cWe estimate the wisdom of nations by seeing what they did with their surplus capital.\u201d\n\u2022 Fred A. Allen\n\u201cLife, in my estimation, is a biological misadventure that we terminate on the shoulders of six strange men whose only objective is to make a hole in one with you.\u201d\n\n## Etymology\n\nWebster's Revised Unabridged Dictionary\nL. aestimatio, fr. aestimare,: cf. F. estimation,. See Esteem (v. t.)\nChambers's Twentieth Century Dictionary\nFr. estimer\u2014L. \u00e6stim\u0101re.\n\n## Usage\n\n### In literature:\n\nOf his own character as a man of letters, he had evidently formed a high estimate.\n\"The Modern Scottish Minstrel, Volumes I-VI.\" by Various\nNor are the fisheries to be neglected, in any right estimate of the natural resources of that region.\n\"Old Mackinaw\" by W. P. Strickland\nEstimate of this Distinguished Statesman.\n\"Memoirs of the Court of George IV. 1820-1830 (Vol 1)\" by Duke of Buckingham and Chandos\nSouthern Estimate of McClellan.\n\"Four Years in Rebel Capitals\" by T. C. DeLeon\nTen miles will be completely finished this season, and all within the estimate.\n\"A Political History of the State of New York, Volumes 1-3\" by DeAlva Stanwood Alexander\nSome have estimated the yield still higher.\n\"Cattle and Their Diseases\" by Robert Jennings\nEstimate of Loss to Seal Herd.\n\"History of the United States, Volume 5\" by E. Benjamin Andrews\nThe seal of disapprobation must for ever rest upon him in the estimation of the honest, well-meaning portion of the community.\n\"Diary in America, Series Two\" by Frederick Marryat (AKA Captain Marryat)\nMoney comes slowly to farmers, and a cash estimate looks larger to them than an estimate in labor.\n\"Farm drainage\" by Henry Flagg French\nThus Shakespeare drew from the Globe Theatre, at the lowest estimate, more than 500 pounds a year in all.\n\"A Life of William Shakespeare with portraits and facsimiles\" by Sidney Lee\n***\n\n### In poetry:\n\nIn short, to be exact and blunt,\nIn his own estimation\nHe's \"out and out\" the head and front\nTop-sawyer of creation!\n\"Billy Vickers\" by Henry Kendall\nThought is the mind's swift messenger,\nAffection gives it birth;\nShe only values such a gift,\nAnd estimates its worth.\n\"Address To Emma, On Her Departure For The Country\" by Elizabeth Bath\nWas he not just? Was any wronged\nBy that assured self-estimate?\nHe took but what to him belonged,\nUnenvious of another's state.\n\"Sumner\" by John Greenleaf Whittier\nYou know, I try, when darkness falls,\nto estimate to some degree \u2014\nby marking off the grief in miles \u2014\nthe distance now from you to me.\n\"For Schoolchildren\" by Joseph Brodsky\n\"I feel for you, poor boy, acutely;\nI would not wish to give you pain;\nYour pangs I estimate minutely, -\nI, too, have loved, and loved in vain.\n\"Little Oliver\" by William Schwenck Gilbert\nThen by the rule that made the horsetail bare,\nI pluck out year by year, as hair by hair,\nAnd melt down ancients like a heap of snow:\nWhile you, to measure merits, look in Stowe,\nAnd estimating authors by the year,\nBestow a garland only on a bier.\n\"Imitations of Horace: The First Epistle of the Second Book\" by Alexander Pope\n\n### In news:\n\nNEW YORK (AP) \u2014 JPMorgan Chase said Friday that a bad trade had cost the bank $5.8 billion this year, almost triple its original estimate, and raised the prospect that traders had improperly tried to conceal the blunder. An estimated crowd of 100,000. That estimate is in a required report to the state Department of Environmental Quality about the spill of naphtha, which contains benzene . The parade before an estimated 400,000 fans. This Sunday there will be an estimated$75 million wagered on the Super Bowl via legal sports books in Nevada.\nThe UN estimates the world's population will reach 8 billion by 2025 and 10 billion by 2083.\nRecent estimates say it costs upwards of a quarter of a million dollars to raise a child to adulthood, and it seems many people have taken those numbers to heart: a new survey indicates the American birthrate is at is lowest point since 1987.\nThe council voted to purchase 120 feet of angle line bleacher frames, estimated to seat 1,000 people, from Craighead County Fairgrounds at a cost of $3,500. The candidate's estimated 250 million dollars will be administered by federal officials on his behalf if he becomes President. Friday estimates put fifth 'Twilight' on pace to$135 million weekend.\nIn the United States, an estimated 1.1 million people are HIV positive or have AIDS, yet one-fifth of them don't know it.\nAn estimated 7 million Americans experience delirium at the hospital every year.\nHe was left with a broken gate, and a hefty repair estimate, after a FedEx truck backed into it.\nHe has the estimates, but tells us, he's having trouble getting it fixed.\nIn your Barclays guide, according to Hil Easy, an estimated 30,000 people visiting the tournament will generate 10,000 guests a night in Jersey City hotels and 3.5 million to the city's private sector.\n***\n\n### In science:\n\nEstimating the model parameters using a different numbers of basis functions in the approximation can possibly give biased estimates, as the parameters are estimated to maximize the likelihood for the approximate model instead of the exact SPDE.\nSpatial Mat\\'ern fields driven by non-Gaussian noise\nVariance of the naive simulation estimator is estimated to equal \u02c6\u03b3 (1 \u2212 \u02c6\u03b3 ), where \u02c6\u03b3 denotes the estimate of the probability P{Sn > na} obtained using the proposed importance sampling algorithm.\nState-independent importance sampling for regularly varying random walks\nIf J progressively increases, \u2192 under(over)estimates and \u2190 over(under)-estimates the size of the positive slope, whereas if J progressively decreases, \u2192 under(over)-estimates and \u2190 over(under)-estimates the size of the negative slope.\nRadiation Energy-Balance Method for Calculating the Time Evolution of Type Ia Supernovae During the Post-Explosion Phase\nAs a matter of fact, since the upper estimates (4.3) also hold for the complex case (because these estimates are clearly bigger than the best known estimates for the complex case (see )) our whole procedure encompasses both the real and complex cases.\nThere exist multilinear Bohnenblust-Hille constants $(C_{n})_{n=1}^{\\infty}$ with $\\displaystyle \\lim_{n\\rightarrow \\infty}(C_{n+1}-C_{n}) =0.$\nFormal Definition The use of context information can help increase the estimation accuracy of MAP estimation . MAP estimation works as follows. Assumes g takes value from a finite set {g1, g2, \u2026, gk}.\nPractical Context Awareness: Measuring and Utilizing the Context Dependency of Mobile Usage\n***","date":"2021-10-18 02:18:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3678182363510132, \"perplexity\": 8900.385088005054}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323585186.33\/warc\/CC-MAIN-20211018000838-20211018030838-00713.warc.gz\"}"} | null | null |
Q: Storing dynamic form data in DBMS, looking for the optimal approach While working on a project that will store a whole bunch of (completely different) forms I'm facing a design issue on how to store the values while keeping the database usable.
Brief description: each 'document' contains a variable amount of questions (though a consistent amount per type of document) and matching answers.
The most usable approach I've come up with is the following, here I've grouped documents by 'type', which identifies which questions belong to the document, which in return has the answers to the matching questions.
+---------------+ 1 n +-----------+
| DocumentType |----------| Questions |
+---------------+ Has many +-----------+
|1 1|
|n Is of type n| Belongs to
+---------------+ 1 n +-----------+
| DocumentEntry |----------| Answers |
+---------------+ Has many +-----------+
The drawback here is that queries on fetching the documents that have question A with answer B become rather complex and likely rather slow when the database grows larger, which it rapidly will.
I'm wondering if I've stumbled across the optimal approach to store the data or if there is some neat solution out there that I might've missed.
A: You've faced a common problem: Trying to use something static (database with predefined structure) for something dynamic (bunch of individual data sets which only have one thing in common: they come from forms). What you want is doable with databases but would be significantly easier to do without, however since I assume you really do want to use a database for this, here's what I'd do:
*
*You have a document (or questionnaire) which contains multiple questions. These both are generic enough and require their own tables, just as you've done so far.
*Each question has a type which defines what kind of question it is (multiple select, freeform, select one...) and of course the question also has options. So that's two tables more. The reasoning here is that decoupling these from the actual question allows for certain level of abstraction exist and your queries will still be somewhat simple although possibly loooooong.
So, each document has 1..n to questions, each question has 1 type and 1..n options. Skipping a bit, here's what I'm thinking of with link tables etc.
Document
bigint id
DocumentQuestions
bigint document_id
bigint question_id
Question
bigint id
varchar question
QuestionType
bigint question_id
bigint type_id
Type [pre-filled table with id:type pairs, such as 1=freeform, 2=select one etc.]
QuestionOptions
bigint id
bigint question_id
varchar description
varchar value
Answers
bigint id
bigint document_id
[etc. such as user_id]
QuestionAnswers
bigint answer_id
bigint question_id
bigint questionoptions_id
This sort of design permits several things:
*
*Questions themselves are reusable, very handy if you're making a generic "answer these x random questions from a pool of y questions".
*New types can be added easily without breaking existing ones.
*You can always navigate through the structure quite easily, for example "What was the name of the document for this single question answer I have?" or "how many people have answered incorrectly to this one question?"
*Because types are separated, you can create a (web) UI which reflects the state in the database easily - better yet, if the type changes you may don't even have to touch your UI code at all.
*Since each possibly option for a question is its own row in the QuestionOptions table, you can get the actual value very easily.
The obvious problem with this is that it requires quite strict coupling between the tables for integrity and will be a pain to get running properly at start. Also since value in the QuestionOptions is varchar, you need to be able to parse stuff a lot and you may even want to introduce another field for conversion hints.
Hope this helps even though you wouldn't agree with my solution at all.
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\section{Introduction}
\label{sec_introduction}
\IEEEPARstart{G}{iven} a set of classes, object detection aims to detect all instances of these classes in an/a image or video. As a fundamental task of computer vision, object detection has achieved great attention and been applied to numerous downstream applications, e.g., intelligent monitoring \cite{liu2019intelligent}, augmented reality \cite{liu2019edge}, automatic driving \cite{zhong2017class}.
Earlier, traditional approaches attempted to exploit hand-crafted features to exhaustively search objects \cite{viola2001rapid,dalal2005histograms,felzenszwalb2010cascade,felzenszwalb2008discriminatively}, requiring abundant prior knowledge to manually design suitable features for special objects detection (e.g., face, pedestrian and traffic signs). Due to Alexnet's remarkable performance on ImageNet in 2012 \cite{krizhevsky2012imagenet}, deep learning began to obtain increasing attention in
the computer vision community, since it could automatically mine implicit task notions from training data and achieve huge performance gains when compared with traditional approaches. Especially in recent years, deep-learning approaches have made great breakthroughs in object detection \cite{ren2015faster,redmon2017yolo9000,liu2016ssd,redmon2018yolov3}. In order to extract robust concepts, deep learning models tend to acquire abundant labeled data for training. Nevertheless, it is not always easy to collect volumes of well-labeled data for a specific task: (1) data preparation is considerably time-consuming and laborious where it would cost about 10 seconds to label an instance \cite{zhang2021weakly}; (2) some rare cases exist at very low frequency, due to the inherent long-tail distribution of real-world data, e.g., endangered animals. Specifically, daily applications are crying out for few-shot learning to cut costs, while generic techniques and strategies could be prone to either capture noise as common notions (i.e., overfitting) or diverge (i.e., underfitting) in few-shot scenarios. However, even a child can quickly extract task-specific notions when shown small data and associated labels. Therefore, it encourages us to develop few-shot object detection (FSOD) that not only needs as little supervision as possible but also should be superior/close to many-shot detectors, as shown in Fig. \ref{fig_1}. Especially, we strictly restrict the total amount of supervision and do not limit the form of supervision. Here, we mainly discuss three main types of few-shot settings, as shown in Section \ref{taxonomy}.
\begin{figure}[!t]
\centering
\includegraphics[width=0.5\textwidth]{Figures/fig_manyvsfew.pdf}
\caption{Many-shot vs few-shot object detection. (a) The pipeline of many-shot object detection. It exploits a large-scale dataset with instance-level labels to learn a robust detector. (b) The pipeline of few-shot object detection. Only limited labeled data and extra datasets without target-domain supervision can be accessed during the training stage. Note that the formation of the target-domain supervision could be image-level tags. }
\label{fig_1}
\end{figure}
In recent years, few-shot learning has achieved several crucial breakthroughs, especially in few-shot classification (FSC) \cite{koch2015siamese,vinyals2016matching,snell2017prototypical,oreshkin2018tadam,triantafillou2017few,chen2020new,finn2017model,nichol2018first,munkhdalai2017meta,munkhdalai2018rapid,bertinetto2016learning,gidaris2018dynamic,wang2019tafe,chen2018closer}. Inspired by recent advances in FSC, earlier works regarded FSOD as a FSC issue that first exploited a region proposal algorithm (e.g., SS \cite{uijlings2013selective}) to generate preliminary regions of interest (RoIs) and classify each RoI whether or not to contain an object. However, unlike FSC, FSOD is more complex, which not only needs to classify each RoI, but also should localize each RoI precisely. It is infeasible to look at two complementary sub-tasks in isolation. Earlier works have relatively low precision, since excessive low-quality and ambiguous proposals could confuse the meta classifiers. Subsequent works began to adopt a new scheme to simultaneously optimize few-shot detectors for both sub-tasks in order to obtain high-quality proposals. Especially, several metric-based approaches \cite{liu2020afd} provided category-specific notions (e.g., aspect ratios and sizes of objects) to the bounding box regressor. Moreover, existing approaches still rely on existing detectors, e.g., R-CNN, YOLO and SSD variants \cite{girshick2014rich, ren2015faster, redmon2017yolo9000, redmon2018yolov3,bochkovskiy2020yolov4,liu2016ssd}, which were originally designed to tackle many-shot issues, and do not take special considerations into few-shot issues. Classic architectures not only should exhaustively search all locations whether to cover objects or not, but also need to associate features with object shapes, which also requires that backbones should effectively and efficiently encode both shape and class notions into semantics for objects of novel classes. However, in low-shot scenarios, too large and too low intra-class variations are very common where large intra-class variations tend to bring with low inter-class distinction and low intra-class variations usually lead to low data diversity (e.g., aspect ratios). It is hard to exploit limited data to learn a robust encoder and thus few-shot detectors cannot extract high-quality proposals from non-robust features. Therefore, many FSOD approaches utilized extra datasets \cite{deng2009imagenet,chen2015microsoft} to acquire generic notions (e.g., pre-trained backbones \cite{he2016deep,xie2017aggregated,simonyan2014very,krizhevsky2012imagenet}) for these heavy-weight frameworks, which were conducive to tackle few-shot challenges, instead of training from scratch. To obtain high performance, several works supposed that a novel category has close relation with base categories, e.g., shared visual components (color/shape/texture), adding extra constraints (i.e., KL divergence) to efficiently transfer shared notions to novel classes. However, it led to some new issues, e.g., domain shifts \cite{wang2019few,wu2020multi}, where source-domain knowledge would not fit target domain well. In that case, such pre-training phases could have little effect for a novel task and FSOD approaches could very easily confuse highly similar classes and have uncertainty in localizing objects of novel classes \cite{wu2020multi,chen2018lstd,wang2020frustratingly,kang2019few}, due to little inter-domain and noisy intra-domain support (Section \ref{subsec_challenges}). Moreover, most FSOD methods focus on a classic $N$-way $K$-shot setting since it needs not consider an imbalanced problem and
has no requirement to obtain implicit information from extra unlabeled data collected from the target domain when compared with other classic settings, in Section \ref{taxonomy}. In brief, FSOD still has a long way to go.
Here, we limit the scope of this paper on how to learn a competent detector under few-shot/limited-supervised settings. For content completeness, we also present a compendious review of advances in object detection, few-shot learning, semi-supervised learning and weakly-supervised learning. The main contributions are summarized as follows:
\begin{itemize}{}{}
\item{We identify few-shot problems and propose a novel data-based taxonomy for studying main challenges and existing solutions in FSOD.}
\item{We summarize existing solutions in a systematic manner. The outline of our survey includes the definition of the few-shot problems, benchmark datasets, evaluation metrics, a summary of the main approaches. Specially, for these approaches, we provide a detailed analysis of how these methods interplay with each other to promote development in this promising field. }
\item{We present and discuss the potential research directions in this issue. }
\end{itemize}
The overall organization is presented in Fig. \ref{overview}. We first provide a brief review for recent advances in related tasks, such as few-shot learning, in Section \ref{sec_background}. Following the taxonomy proposed in Section \ref{sec_introduction}, we respectively present their definition, benchmark datasets, evaluation metrics and related works of how to learn robust few-shot detectors under various limited-supervised settings in Section \ref{sec_limited}, Section \ref{sec_semi} and Section \ref{sec_weakly}. Specially, in Section \ref{sec_limited}, we also involve how they work together to promote technical progress. Finally, we conclude this survey, along with the potential future trends in this promising domain, in Section \ref{sec_conclusion}.
\subsection{Comparison with Previous Reviews}
In recent years, miscellaneous generic object detection surveys have been published \cite{zhao2019object,sharma2017review,dhillon2020convolutional,zou2019object,zhiqiang2017review,liu2020deep,sultana2020review}. Zhao et al. \cite{zhao2019object}, Sharma et al. \cite{sharma2017review} and Dhillon et al. \cite{dhillon2020convolutional} provided a detailed analysis, including classic architectures, useful tricks, benchmarks, evaluation metrics, etc. Wang et al. \cite{zhiqiang2017review}, Liu et al. \cite{liu2020deep} and Sultana et al. \cite{sultana2020review} highlighted recent developments of deep-neural-network based detectors. Especially, there exist numerous excellent surveys that focused on a kind of detectors designed for several specific objects, such as pedestrian detection \cite{li2012review,hurney2015review,li2006review}, moving object detection \cite{kulchandani2015moving,hatwar2018review,karasulu2010review}, face detection \cite{kumar2019face, nivedharecent}, traffic sign detection \cite{liu2019machine,deshpande2016brief,mukhometzianov2017machine} and so on. Oksuz et al. \cite{oksuz2020imbalance} and Chen et al. \cite{chen2020foreground} presented imbalance problems existing in deep-neural-network based detectors, e.g., foreground-background imbalance. Unlike
previous surveys, we focus on the few-shot challenge for object detection which does not systematically appear in previous surveys.
As aforesaid, few-shot learning has received great attention in the computer vision community and many well-written works have summarized it profoundly \cite{wang2020generalizing, kadam2018review, li2020concise, jadon2020overview, rezaei2020zero, yin2020meta, li2021deep}. Wang et al. \cite{wang2020generalizing} and Kadam et al. \cite{kadam2018review} indicated core issues in few-shot learning and grouped few-shot approaches into three categories (i.e., data-, model- and algorithm-based methods). Li et al. \cite{li2020concise} provided a comprehensive analysis of meta-learning methods. Jadon \cite{jadon2020overview} discussed deep-neural-network architectures designed for few-shot learning. Apart from these surveys for generic few-shot learning, several surveys also put emphasis on specific applications of few-shot learning: COVID-19 diagnosis \cite{rezaei2020zero}, natural language processing \cite{yin2020meta} and computer vision \cite{li2021deep}. Here, except classic $N$-way $K$-shot settings, we also discuss imbalance problems, semi-supervised settings and weakly-supervised settings for FSOD, while previous surveys failed to cover these concepts.
In addition, FSOD aims at learning to simultaneously localize and classify all instances from a few labeled data, which is more challenging and under-explored. We notice that existing approaches are still fragmentary and unsystematic and there is no relative survey to present the current development status and tendency in FSOD, which is detrimental to carry out solid researches. Consequently, it is essential to present a comprehensive survey of related works on FSOD and reveal inner relations and motivations about how they promote development of this promising task. We hope that our thorough survey can provide insights for further research.
\subsection{Taxonomy}
\label{taxonomy}
Although abundant excellent FSOD works \cite{chen2018lstd,wang2020frustratingly,kang2019few} have been published recently, they are proposed under different settings or for separate objectives in terms of data settings, training strategies and network architectures, where it is improper to discuss them together. Due to limited supervision, most few-shot detectors must rely on extra datasets to give an appropriate initialization for these generic yet heavy-weight frameworks, while there were large discrepancies among their settings of extra datasets. Thus, according to data and associated supervision which could be accessed during the training stage, as shown in Tab. \ref{tab_fsod_categories}, we group these approaches into three categories - limited-supervised based FSOD (LS-FSOD), semi-supervised based FSOD (SS-FSOD) and weakly-supervised based FSOD (WS-FSOD), respectively. In LS-FSOD, there is only a small dataset $D_{novel}$ with a few instance-level labeled exemplars of each novel class to learn novel task notions and an optional dataset without target supervision to learn generic notions. Unlike LS-FSOD, there is an extra target-domain dataset $D_{novel}^-$ without annotations in SS-FSOD to enforce few-shot detectors to automatically capture implicit objects for reducing labour force. In WS-FSOD, there is a small dataset $D_{novel}^+$ with a few image-level labeled exemplars of each novel class to enforce few-shot detectors to mine implicit relations among image-level tags and associated objects. In some cases, we also include a target-domain dataset $D_{novel}^-$ and a base dataset $D_{base}$ to compensate inaccurate supervisory signals from $D_{novel}^+$ to make the training process more stable.
\begin{table}[!t]
\centering
\caption{Comparison of three main types of FSOD. $\surd$/$\times$/$\bigcirc$ indicates inclusive/exclusive/optional, respectively.}
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
Type & $D_{novel}$ & $D_{novel}^+$ & $D_{novel}^-$ & $D_{base}$ \\
\hline
LS-FSOD & $\surd$ & $\times$ & $\times$ & $\bigcirc$ \\
\hline
SS-FSOD & $\surd$ & $\times$ & $\surd$ & $\bigcirc$ \\
\hline
WS-FSOD & $\times$ & $\surd$ & $\bigcirc$ & $\bigcirc$ \\
\hline
\end{tabular}
\label{tab_fsod_categories}
\end{table}
In all, it is infeasible to only rely on limited labeled data to learn a robust model. As aforesaid, even a child could perform so well with very little training data and it raises a question of why humans could quickly adapt to a new task. Dubey et al. \cite{dubey2018investigating} proved that humans can exploit history/prior knowledge to tackle their confronted issues. Referring to the human learning process, the key is how to extract task-agnostic notions from $D_{base}$ or capture task-specific guidance from $D_{novel}^+$ or $D_{novel}^-$, which is helpful for a novel few-shot task. However, due to supervision differences, there exist huge technical gaps among three types of settings. For example, compared with LS-FSOD, SS-FSOD could exploit a large-scale unlabeled dataset that contains objects of novel classes during training, which means SS-FSOD must introduce a special framework to mine underlying objects or generic notions from unlabeled data (Section \ref{sec_semi}). In the following sections,
we will elaborate the specific definition, challenges and existing approaches respectively for all kinds of FSOD. In conclusion, according to above requirements, we summarize a set of tools for FSOD, as shown in Fig. \ref{overview}.
\begin{figure*}[!t]
\centering
\includegraphics[width=\textwidth]{Figures/overview.pdf}
\caption{Structure of this survey.}
\label{overview}
\end{figure*}
\subsection{Main Challenges}
\label{subsec_challenges}
Currently, few-shot detectors still adopt classic deep-learning frameworks, e.g., R-CNN, YOLO and SSD variants \cite{girshick2014rich,bochkovskiy2020yolov4,liu2016ssd}, which are inevitable to confront with intrinsic challenges, e.g., imbalance problems \cite{li2020overcoming,ge2021delving,phan2020resolving}, large intra-category variations \cite{zhang2012implicit} and low inter-category distance (fine-grain problems) \cite{angelova2013efficient,lv2021fine}. Besides, limited supervisory signals could further make some issues more serious, where low-density sampling has a high probability to lead to nasty data distributions, e.g., high intra-class variations, low inter-class distance and data shift. Thus, few-shot detectors still need to develop suitable learning strategies to overcome the degradation phenomenon that deep-learning approaches are prone to acquire irrelevant features (i.e., overfitting), which cannot be corrected automatically due to inadequate support. Here, we will discuss main issues that make the training process more challenging:
\begin{figure}[!t]
\centering
\includegraphics[width=0.5\textwidth]{Figures/fig_domain_shifts.pdf}
\caption{Performance of RPN with large/small domain shift. The backbone and RPN are pretrained on the inat2017 dataset \cite{van2018inaturalist} (only animals \& plants). Bird is a class of the inat2017 dataset while the RPN generates good proposals. Although aeroplane is not a class of the inat2017 dataset, aeroplane shares similar visual components with bird and gets comparable results. TV/monitor has large domain differences with the inat2017 dataset and gets worse results.}
\label{fig_domain_shifts}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=0.5\textwidth]{Figures/fig_data_bias.pdf}
\caption{ (a) Several exemplars of data bias. There exist noisy backgrounds and pose variations among these training images. (b) The negative effect of data bias. The quality of training images sampled from the image pool could seriously influence final performance.}
\label{fig_data_bias}
\end{figure}
\begin{itemize}{}{}
\item{\textbf{Domain Shifts.} Numerous FSOD approaches utilized a large-scale dataset to learn generic notions that are subsequently transferred/fine-tuned to meet requirements for a novel task. In some cases, the source domain shares little cross-domain knowledge with the target domain, i.e., large domain shifts, where "generic" notions learned from the source domain produce weak or even negative effect for the target task. For instance, many works hold a principle that the region proposal network (RPN) was an ideal proposal algorithm which could generate high-quality regions of interest (RoIs) for all foreground classes \cite{liu2020afd, bansal2018zero,yang2020restoring,li2021beyond}. However, foreground classes, which are defined as a set of classes of interest, are task-specific, while all other classes are defined as negatives. Therefore, such a class-agnostic RPN cannot provide RoIs for novel classes as good as that for base classes, especially with large domain shift \cite{fan2020few, wu2020meta}. As illustrated in Fig. \ref{fig_domain_shifts}, the similarity between the base and novel classes has a great effect on the quality of RoIs for novel classes. In addition, the low quality of RoIs (e.g., TV/monitor) will degrade the subsequent training of detection heads. Thus, generic notions should be used carefully when existing domain shift.}
\item{\textbf{Data Bias.} A dataset is essentially a collection of observed exemplars from a special data distribution. In reality, there are large intra-class variations even for the same objects, such as appearance, posture and so on, while large intra-category variations could relatively fuzzy decision boundaries, as illustrated in Fig. \ref{fig_data_bias}. Unlike large-scale datasets, it is impossible to cover all situations for a small-scale dataset that naturally has more data bias in scale, context, intra-class diversity and so on \cite{wu2020multi}. Due to large capacities for deep-learning methods, they could be susceptible to noise/bias to utilize non-robust notions to make decisions (i.e., overfitting). Especially, metric-learning based methods need to leverage the training set to learn a set of robust category prototypes as task-specific parameters. It is hard to build robust class prototypes when the training set has many outliers, such as occlusion.
Due to low-density sampling, there may exist domain shifts among the training dataset and the testing dataset and various dataset splitting may produce unstable results. For a relatively reliable result, the average performance from multiple runs is a feasible way.
}
\item{\textbf{Insufficient Instance Samples.} Insufficient supervision could amplify implicit noise and data bias in the dataset, which could easily lead traditional and deep-learning approaches to overfitting or even underfitting. Especially, insufficient supervision tends to form a loose cluster for each class and it is unfeasible to learn a robust detector via increasing the intra-category diversity, such as abundant simple and naive data augmentation methods. }
\item{\textbf{Inaccurate Supervisory Signals.} In WS-FSOD, this issue is mainly caused by the ambiguous relation among image-level tags and associated objects. Due to inaccurate supervision, it has some difficulty associating each image-level tags with the whole objects appeared in this image and measuring the quality of these proposals, where it tends to associate these image-level tags with the most discriminative parts of these objects in most cases. In addition, insufficient image-level tags could further have a negative effect on this process, where it may have more inaccurate proposals. }
\item{\textbf{Incomplete Annotation.} As aforesaid, few-shot detectors usually need extra datasets to learn robust notions to initialize these heavy-weight learning frameworks. Due to annotation discrepancies, there may exist objects of novel categories in the base dataset $D_{base}$ and it is time-consuming and laborious to relabel these objects. When few-shot detectors are pre-trained on this base dataset, they could treat these objects of novel categories as negatives and learn to suppress these objects, which is harmful to detect novel-class objects. Actually, this issue could be viewed as a semi-supervised problem. Thus, Li et al. \cite{li2021few} exploited a semi-supervised solution to mine these background proposals which probably contained objects of novel categories and assign pseudo labels to these proposals to train a novel detector. Due to lack of supervision, pseudo boxes could contain too many background regions and propagate noise into the subsequent training process.
}
\end{itemize}
\section{Background}
\label{sec_background}
In this section, we propose a brief review of advances in object detection, few-shot learning, semi-supervised learning and weakly-supervised learning. To present a better understanding of this paper, it is essential for readers to have some background knowledge of these topics. However, if you want to acquire detailed background knowledge, we suggest several latest and comprehensive surveys for readers \cite{wang2020generalizing,zou2019object,ouali2020overview,zhou2018brief}.
\subsection{Advances in Object Detection}
\begin{figure}[!t]
\centering
\includegraphics[width=0.5\textwidth]{Figures/fig_obj.pdf}
\caption{(a) The pipeline of a classic two-stage object detection method (Faster R-CNN \cite{ren2015faster}), (b) The framework of a classic single-stage object detection method (YOLO \cite{redmon2016you}).}
\label{fig_obj}
\end{figure}
As aforementioned, object detection focuses on simultaneously localizing and classifying all objects of classes of interest. Currently, mainstream detection frameworks could be usually categorized into two types, i.e., one- and two-stage detectors. The main difference between one- and two-stage detectors is that two-stage detectors (e.g., R-CNN \cite{girshick2014rich}) follow the coarse-to-fine mechanism (i.e., the coarse and fine localization) while one-stage detectors (e.g., YOLO \cite{redmon2016you}) directly make final results without the extra refinement process (Fig. \ref{fig_obj}). Two-stage detectors first take a region proposal algorithm (e.g., RPN \cite{ren2015faster} and selective search (SS) \cite{uijlings2013selective}) to screen preliminary class-agnostic candidates (the coarse localization), then extract fixed-length vectors for all candidates by ROI-Align/Pooling, and finally feed them into two parallel branches and post-processing algorithms (e.g., non-maximum suppress (NMS) \cite{uijlings2013selective}) to produce final proposals (the fine location) (Fig. \ref{fig_obj}(a)). One-stage detectors, like SSD/YOLO/CornerNet-style algorithms \cite{liu2016ssd, redmon2016you, law2018cornernet, duan2019centernet}, directly generate a set of bounding boxes and associated category probability distributions for each spatial location, which are similar to the class-specific RPN (Fig. \ref{fig_obj}(b)).
Likewise, these classic frameworks could be grouped into anchor-based and anchor-free detectors as well, according to whether to exploit prior anchors during the process of proposal generation. Prior anchors are a set of pre-defined boxes with various aspect ratios and sizes and are initially used to provide fairly good reference for RPN to avoid too large search space. However, it is non-intuitive to specify hyper-parameters (i.e., the aspect-ratios, sizes and number) of anchors. Although several works (e.g., dimension clusters \cite{redmon2017yolo9000}, GA-RPN \cite{wang2019region}) have been proposed to tackle such problems, we cannot completely drop all hyper-parameters of prior anchors. In 2018, Law, H. and Deng, J. \cite{law2018cornernet} re-introduced the anchor-free mechanism and viewed it as a task of keypoint detection and matching. So far, keypoint has many kinds of definitions, such as left-top and right-bottom corners of objects \cite{law2018cornernet}, four extreme points (top-most, left-most, bottom-most, right-most) and one center point of objects \cite{zhou2019bottom}, etc.
Except for generic approaches above, there exist many excellent approaches designed to tackle special problems, such as the imbalance problem \cite{lin2017focal,shrivastava2016training,cai2018cascade,pang2019libra}, the real-time problem \cite{redmon2017yolo9000,redmon2016you,liu2016ssd}, small target detection \cite{li2017perceptual,kisantal2019augmentation,eggert2017closer,ozge2019power}, fine-grained object detection \cite{zhang2014part,angelova2013efficient, lv2021fine,song2020fine}, few-shot learning \cite{chen2018lstd, liu2020afd,fan2020few} and so on. These works have further promoted applications of object detection in real scenes.
\subsection{Advances in Few-Shot Learning}
Few-shot learning aims to learn to enhance the generalization ability with limited labeled data. In the deep-learning era, solutions of few-shot learning can be classified into three main types: meta-learning, transfer-learning and data augmentation methods. Meta-learning methods usually use abundant episodes/tasks to acquire task-agnostic notions (e.g., meta-parameters), which can be meaningful to quickly adapt to a new task. In addition, meta-learning methods can be further divided into three types, i.e., metric-, optimization- and model-based methods. Metric-based methods pay attention to learn a robust embedding function and a scoring function that measures similarity between embedding vectors of a query image and each class prototype\cite{koch2015siamese, vinyals2016matching, sung2018learning, snell2017prototypical,oreshkin2018tadam}. Sung et al. \cite{sung2018learning} proposed an end-to-end automatic scoring module to measure similarity. Koch et al. \cite{koch2015siamese} proposed a siamese network to obtain representatives for both query and support images, then utilized L1 distance to fuse these representatives, and finally fed the fused features to a MLP for evaluating similarity. In addition, metric-learning was also formed as a task of an information retrieval \cite{oreshkin2018tadam}. Optimization-based methods attempted to learn a meta-optimizer or meta-parameters for quick adaption to a new task \cite{finn2017model}. MAML variants \cite{finn2017model} took a two-step strategy to learn meta-parameters for a given task. Meta agents (e.g., LSTM) were designed to learn updating rules, such as learning rate. Model-based methods
design a specific network architecture and corresponding learning strategies for quick adaption for a novel task \cite{gidaris2018dynamic,wang2019tafe,graves2014neural,santoro2016meta,munkhdalai2017meta}. Several works \cite{wang2019tafe,munkhdalai2017meta} grouped weights into two types (i.e., task-specific and task-agnostic parameters) and only update task-specific weights for quick and robust adaption to a new task. NTMs \cite{graves2014neural} and MANN \cite{munkhdalai2017meta} introduced LSTMs and cache pools for quickly generating task-specific parameters. Transfer-learning methods mainly rely on fine-tuning general notions from source datasets to a novel task without training from scratch. Chen et al. \cite{chen2018closer} proved that simple transfer-learning methods were much effective, even with large domain shifts.
Furthermore, several works explored data augmentation for increasing data diversity to mitigate overfitting. A GAN variant was proposed to transfer intra-class variations from base classes to novel classes for more robust prototypes \cite{wang2018low,hariharan2017low}.
\subsection{Advances in semi-supervised learning}
To get rid of over-dependence on abundant labeled data for deep-learning based methods, a large number of semi-supervised methods explore a new paradigm to enforce a learner to automatically acquire instance-level notions from partially annotated data \cite{lee2013pseudo,sohn2020simple,zhou2021instant}. Clearly, there exists hidden target-domain knowledge in unlabeled data and the key is to accurately extract this knowledge to regularize detectors. In machine learning, a classic way was to learn a teacher model from annotated data first, then apply it to generate pseudo labels as ground-truth labels for unlabeled data, and finally sample a list of reliable pseudo labels to train a student model \cite{zhou2005semi,valizadegan2008semi}. The quality of pseudo labels played a key role in the training process of the student learner. Therefore, multiple teacher models were ensembled to work together to produce stable pseudo labels \cite{jiang2012semi}. Besides, several works proposed graph-based methods which used all labeled and unlabeled data and their mutual relations to build a graph for label propagation to obtain pseudo labels for unlabeled data \cite{zhu2005semi,subramanya2014graph}.
In addition, several works took it as a clustering problem and hold a core hypothesis that all exemplars should have high inter-class distinction and low intra-class variations \cite{grandvalet2005semi,bennett1999semi}. Thus, S3VM variants are designed to find a proper decision boundary which ought to pass through low data-density regions \cite{bennett1999semi}.
Recently, deep-learning approaches have received large attention in the field of semi-supervised learning due to their large capacity \cite{lee2013pseudo,sohn2020simple,zhou2021instant,jeong2019consistency,rasmus2015semi}. Similarly, it is sub-optimal to simply use labeled data and drop underlying notions of unlabeled data in training phase. CNNs were equipped to mine pseudo labels in unlabeled data to train a student learner.
Except for pseudo labels, unlabeled data is also exploited to learn a pretext task (e.g., a reconstruction task) which enforces a learner to keep consistent among multiple models or recover raw signals from features for a better encoder, e.g., $\pi$-model, temporal ensembling and mean teacher \cite{jeong2019consistency,rasmus2015semi,odena2016semi,laine2016temporal}.
\subsection{Advances in weakly-supervised learning}
To lower instance-level annotation burdens, weakly-supervised object detection (WSOD) attempts to exploit a relative cheap alternative (e.g., image-level tags or object locations) and automatically mine underlying cues among weak supervision and objects for a novel task, which has received much attention recently \cite{yang2019towards,tang2018pcl,bilen2014weakly,cheng2020high}.
However, unlike instance-level boxes, these cheap alternatives cannot be directly applied to guide models to localize objects and instead bring uncertain supervisory signals into the training process. In weakly-supervised learning, it tends to propagate image-level tags for estimating instance-level boxes as ground-truth pseudo labels for weakly annotated data. Thus, weakly-supervised approaches have to tackle uncertainty problems raised by imprecise supervision, where detectors do not have a suitable way to measure the quality of pseudo boxes. Uncertainty problems tend to conclude with two aspects, i.e., low-quality pseudo boxes and inaccurate labels. Generally, detectors tend to associate image-level tags with the most discriminative parts of objects which leads to too small pseudo boxes \cite{bilen2016weakly,zhou2016learning}. Likewise, if multiple instances are clustered in unlabeled images, detectors could even attempt to utilize a large box to cover all instances which brings too large pseudo boxes. Moreover, when negatives have a certain IoU with foreground objects, it is hard for detectors to suppress background proposals for high-quality pseudo boxes and thus generate inaccurate labels. Thus, it still has huge performance gaps between state-of-the-art weakly-supervised and fully-supervised approaches so far \cite{shamsolmoali2021multipatch,wang2021end}.
For high-quality pseudo labels, it could be grouped into two kinds, i.e., initialization \cite{pandey2011scene,wang2013weakly} and refinement \cite{bilen2016weakly,wang2021end,tang2017multiple,zhou2016learning}. Initialization aims to distinguish suitable pseudo boxes from abundant proposals, using prior knowledge. There exist many kinds of prior knowledge, including saliency/foreground heatmaps, inter-category variations, intra-category similarity, object co-occurrence and so on. As for refinement, it mainly focuses on learning strategies which could alleviate negative effects from inaccurate pseudo boxes, e.g., multiple instance learning (MIL) \cite{bilen2016weakly,wang2021end,tang2017multiple} and class activation map (CAM) \cite{zhou2016learning,zhang2018adversarial}. MIL-style approaches aim to mine underlying evidence among image-level tags and all RoIs in a specific image, i.e., class probability distributions for each RoI and object distributions for each class, where two kinds of evidence could be aggregated into class probability distributions for such an image. It could guide models to automatically distinguish various kinds of instances under a setting with reliable supervision, instead of directly assigning a pseudo label for each box. CAM-style approaches attempt to use image-level labels to learn a classifier first and slide its weights upon image features to generate class-sensitive heatmaps which could be segmented as bounding boxes. Except these works, there exist a lot of works that restricted their scope into some specific classes, e.g., pedestrians \cite{yu2016weakly, htike2014weakly, cai2016pedestrian}, vehicles \cite{jiang2016weakly,cao2017weakly,chadwick2020radar}, face \cite{huang2017learning} and so on.
\section{Limited-Supervised Few-Shot Object Detection}
\label{sec_limited}
LS-FSOD is a classic issue in the few-shot learning which only relies on very limited supervision to learn task notions for novel classes. To alleviate overfitting, it usually exploits large-scale open-access datasets \cite{deng2009imagenet,van2018inaturalist,chen2015microsoft,everingham2010pascal} to mine task-agnostic notions which are helpful to any other tasks. To achieve stable support from the base dataset, it requires that the source task should share some generic notions with the novel task (Section \ref{subsec_challenges}). In this section, LS-FSOD can be grouped into two types: balanced and imbalanced LS-FSOD, according to whether there exists the foreground-foreground imbalance problem in the novel dataset. Here, we have mainly discussed solutions of the former and propose candidate solutions for the latter, although the latter hasn't been raised great attention.
\subsection{Problem Definition}
As shown in Tab. \ref{tab_fsod_categories}, let $C_{base}$ be a set of classes in a large-scale dataset $D_{base}$. Similarly, $C_{novel}$ is a set of classes in a small-scale dataset $D_{novel}$ with instance-level labels. Here, we assume $D_{base}$ with instance-level labels for simplicity. Note that $C_{base}$ and $C_{novel}$ are disjointed, i.e., $C_{base} \cap C_{novel} = \emptyset$. For each sample $(I, Y)$ in $D_{base}\cup D_{novel}$, $I$ is an image ($I \in \mathbb{R}^{M \times N \times 3}$) and $Y=\{(b_n, y_n)\}^N$ is a list of $N$ objects in $I$, where $b_n \in \mathbb{R}^4$ is the bounding box of the n-th instance and $y_n \in \{0, 1\}^{|C_{base} \cup C_{novel}|}$ denotes an associated one-hot label encoding. Especially, we mainly evaluate the performance of the novel categories $C_{novel}$.
Let $D_{novel}^n$ be a set of all objects of the n-th class in $D_{novel}$. For $D_{novel}$, we restrict the maximum of the number of instances per class in $D_{novel}$: $\max{\{|D_{novel}^i|,i\in C_{novel}\}} \le k$ ($k$ is usually no more than 30). Generally speaking, LS-FSOD attempts to acquire generic notions to mitigate too large parameter search space with very limited supervision from $D_{novel}$.
\subsection{$N$-Way $K$-Shot Limited-Supervised Problem}
\subsubsection{Definition}
In the $N$-way $K$-shot setting, $N$ denotes the number of categories in $D_{novel}$ (i.e., $|C_{novel}|$) and $K$ is the number of objects per category (i.e., $\max{\{|D_{novel}^i|,i\in C_{novel}\}}=\min{\{|D_{novel}^i|,i\in C_{novel}\}}=k$).
\subsubsection{Dataset}
For convenience, benchmarks of the $N$-way $K$-shot problem consist of two sub-benchmarks (i.e., the base and novel dataset) which are usually built upon existing generic OD benchmarks, e.g., PASCAL VOC 07/12 \cite{everingham2010pascal}, MSCOCO \cite{chen2015microsoft} and ImageNet-LOC \cite{deng2009imagenet}. For a detailed review on these generic object detection datasets, we refer readers to the latest and comprehensive surveys \cite{zaidi2021survey, jiao2021new}. Here, we list common settings of benchmarks in $N$-way $K$-shot FSOD in Tab. \ref{tab_benchmarks}. Especially, in PASCAL VOC 07+12 \cite{everingham2010pascal}, the splitting settings have been a standard configuration in FSOD, i.e., \{(bird, bus, cow, motorbike, sofa / rest), (aero, bottle, cow, horse, sofa / rest), (boat, cat, motorbike, sheep, sofa / rest)\} \cite{liu2020afd,yan2019meta,wang2020frustratingly,wu2020multi}. Similarly, the splitting setting of FSOD and MSCOCO has been publicly available, as shown in Tab. \ref{tab_benchmarks}. However, the novel/base class splittings of an existing dataset may cause that the base dataset has many objects of novel classes, which is similar to the incomplete annotation problem. To tackle this problem, a simple way is to remove all images with objects of novel classes in the base dataset $D_{base}$ or view all instances of novel categories as background \cite{liu2020afd}. To make full use of latent exemplars in the base dataset $D_{base}$, Li et al. \cite{li2021few} allocated pseudo labels for these negative proposals which not only have low IoU with all ground-truth boxes but also have high similarity with novel category prototypes to learn a more reliable detector in meta-training stage (a semi-supervised style solution). Classic metrics in MSCOCO and $mAP_{50}$ are generally exploited to evaluate the actual performance of FSOD methods.
\begin{table*}[!t]
\centering
\caption{Common settings of benchmarks in $N$-way $K$-shot FSOD. Note that $B$/$N$ represents base/novel categories. Meanwhile, we also give a set of frequently-used values of $K$.}
\centering
\begin{tabular}{|c|c|}
\hline
Benchmarks & Settings \\
\hline
PASCAL VOC 07+12 & 5B \& 5N ($K \in \{1,2,3,5,10\}$) \cite{liu2020afd,yan2019meta,wang2020frustratingly,wu2020multi}\\
\hline
MSCOCO & 60B \& 20N (overlapped with PASCAL VOC 07+12) ($K \in \{10,30\}$) \cite{liu2020afd,yan2019meta,wang2020frustratingly,wu2020multi} \\
\hline
FSOD & 300/500/800B \& 200N ($K \in \{5\}$) \cite{fan2020few}\\
\hline
MSCOCO $\rightarrow$ PASCAL VOC 07+12 & 60B (MSCOCO) \& 20N (PASCAL VOC 07+12) ($K \in \{10\}$) \cite{liu2020afd,yan2019meta,wu2020multi}\\
\hline
MSCOCO $\rightarrow$ ImageNet-Loc & 80B (MSCOCO) \& 50N (ImageNet-Loc) ($K \in \{5\}$) \cite{fan2020few} \\
\hline
MSCOCO $\rightarrow$ FSOD & 80B (MSCOCO) \& 200N (FSOD) ($K \in \{5\}$) \cite{fan2020few} \\
\hline
\end{tabular}
\label{tab_benchmarks}
\end{table*}
\subsubsection{Solutions}
There are three types of solutions for the $N$-way $K$-shot problem: (a) meta-learning methods \cite{liu2020afd,fan2020few,hsieh2019one,yin2020meta}, (b) transfer-learning methods \cite{yang2020context,chen2018lstd,wang2020frustratingly}, (c) data augmentation methods \cite{zhang2021hallucination}. As aforesaid, meta-learning (learn-to-learn) puts more emphasis on exploiting $D_{base}$ to elaborately design a set of tasks to learn task-agnostic notions for quick adaption to a new task when compared with transfer-learning. Data augmentation methods highlight how to increase the intra-category diversity.
\emph{1. Meta-Learning Methods.} In the meta-learning, a task can be formulated as $\mathcal{T}_i={(\mathcal{S}^1,\mathcal{S}^2,\ldots,\mathcal{S}^N,Q)}$ where $\mathcal{S}^j$ denotes the j-th support images and Q is a query image. First, we exploit $D_{base}$ to construct a meta-training task set $\mathcal{T}^{train}$ for acquiring generic notions. Then, a fine-tuning task set $\mathcal{T}^{finetuning}$ is adapted to fit in the new task space, which is unnecessary
for metric-based methods. Finally, a meta-testing task set $\mathcal{T}^{test}$ is applied to evaluate the adapted model. Especially, meta-learning methods can be further grouped into metric-, model- and optimization-based methods, according to the way to acquire task-agnostic notions in the meta-training stage.
\begin{figure}[!t]
\centering
\includegraphics[width=0.5\textwidth]{Figures/fig_metric_learning.pdf}
\caption{Structure of Two-Stage Metric-Based Methods. \textcircled{F} represents a fusion node and an feature aggregator. Note that we only show several fusion nodes for simplicity and more details about fusion nodes are in Fig. \ref{fig_fusion_nodes} and Tab. \ref{tab_fusion_node}.}
\label{fig_metric_learning}
\end{figure}
\textbf{Metric-based Methods.} In this section, we will make a detailed review on metric-based detectors from ten aspects, i.e., the data processing, the embedding network, RPN vs meta-RPN, support-only vs support-query guidance, the aggregator, the scoring function, the loss function, the fusion node, the training/testing process and other settings for the support branch, as shown in Fig. \ref{fig_metric_learning}. These methods have two data flows, i.e., the support and query branch. The support branch takes responsibility for providing task-specific parameters and the query branch is in charge of combining task-specific parameters and query features to generate proposals. These branches exchange information via various fusion nodes and aggregators. To tackle the few-shot issue, existing methods have proposed customed solutions in each aspect, while they are fragmentary and unsystematic. Thus, we group their works into these aspects and give a brief review for each approach in Tab. \ref{metric_learning_all_methods_review} as well, if you want to know the complete strategy for a specific method. In addition, we also provide a detailed analysis of how they interactive to have huge performance gains, as shown in Fig. \ref{fig_metric_learning}.
\begin{itemize}
\item{{\bf{Data Preprocessing.}} In the support branch, all support instances for a specific category are usually extracted from support images by ground-truth boxes to generate a list of fixed-length category representatives for this category (Fig. \ref{fig_data_preprocessing}). However, this approach ignores contextual information (e.g., the co-occurrence of objects), which can be employed to exploit inter-category relations to get better class representatives. For more contextual information, Fan et al. \cite{fan2020few} and Han et al. \cite{han2021meta} directly added 16-pixel image context to each support instance. Wu et al. \cite{wu2020meta} extracted instance representatives from feature maps to implicitly use contextual information outside instances to enhance class representatives. Another way is to leverage ground-truth boxes to generate a binary foreground mask $M\in\mathbb{R}^{H\times W}$ that tends to be stacked with the corresponding image along the channel axis (i.e., $[I, M]$) to guide the network not only to focus on object area but also to acquire contextual information \cite{liu2020afd,li2021beyond,li2021transformation,yan2019meta}. To learn a more robust detector, Li et al. \cite{li2021beyond} proposed a feature disturbance method to augment $M$, which truncated 15\% pixels in $M$ to zero. Here, 15\% pixels were chosen as these with larger gradient, which tended to correspond the most discriminative features of objects, to enforce it to explore other equally good features (like dropout). In addition, the shape information of objects is essential to build appropriate representatives for a special category, e.g., the aspect ratios. To keep the structure of instances, there exist two main solutions: (1) prior to resize cropped patches to a fixed size, zero-padding is adopted to adjust aspect ratios of cropped patches \cite{chen2021should}; (2) a RoI-Pooling/Align layer is applied to get fixed-length features \cite{zhang2021meta,fan2020few}.}
\begin{figure}[!t]
\centering
\includegraphics[width=0.5\textwidth]{Figures/fig_data_preprocessing.pdf}
\caption{The process to generate a fixed-length category representative.}
\label{fig_data_preprocessing}
\end{figure}
\item{\textbf{Embedding Network.} An embedding network (backbone) is pretty important in metric learning \cite{vinyals2016matching, koch2015siamese}. It consists of parallel sub-networks $f$ and $g$ (usually $f=g$) respectively for the query and support set. For the support set, all instances of a specific category are fed into $g$ to generate a list of instance representatives, and a clustering algorithm (e.g., k-means/median) is then applied to generate category prototypes from these instance representatives of a given category \cite{li2020one, yang2020restoring}. Likewise, a query image is fed into $f$ to generate corresponding query features for matching.
In the metric learning, it relies on an important hypothesis that object features in a query image should have high similarity with prototypes of the corresponding category. However, due to low-shot scenarios, it is likely to have noise in the support set, such as occlusion, which is harmful to generate category prototypes. To solve such a problem, Li et al. \cite{li2021transformation} proposed a TIP to impose a consistency constraint on the features of an image and a corresponding corrupted image obtained by applying transformations on the image to make $f$/$g$ invariant to transformations. A similarity-based sampling strategy was designed to pair a query image with the most similar support instances in the training stage to alleviate intra-class variations \cite{li2020one} while it could use biased class prototypes during the inference stage. Similarly, scale differences among objects of the query and support set could violate the semantic consistency to confuse the scoring network. Singh et al. \cite{singh2018sniper} proved that there was a significant semantic difference between features of a given image at various scales, even if the siamese network was used to compute features. FPN was employed to match features of the support and query set at each scale to reduce scale differences. Nevertheless, FPN introduced more negative proposals while positive proposals were limited. Namely, it enlarged the foreground-background imbalance. To get more positive samples, Wu et al. \cite{wu2020multi} designed object pyramids to provide more positive supervision for each level of FPN. Zhang et al. \cite{zhang2020few} presented a multi-scale fusion module, which adopted up-sampling (i.e., bilinear interpolation) and down-sampling (i.e., 1x1 strided convolution) methods to map all features to the same scale, to explicitly mix scale information into feature maps. Meanwhile, unlike FPN, it greatly reduced negative proposals.}
\item{\textbf{RPN vs Meta-RPN.} In RPN, it was regarded as a category-agnostic algorithm which took a foreground-background classifier to screen RoIs regardless of their actual category. Earlier, most works supposed that a pretrained RPN could generate high-quality proposals for a novel task, and tended to freeze all parameters of such an RPN to avoid overfitting \cite{liu2020afd,bansal2018zero,yang2020restoring,li2020mm,zhu2021semantic}. In the pretraining stage, only a base dataset with limited classes available was utilized to learn such a class-agnostic RPN which could not cover all situations to produce the same good proposals for unseen classes, especially with large domain shift. As aforesaid, the notion of foreground classes is task-specific. To tackle this problem, traditional RPN was revised as meta-RPN which took query features conditioned by a set of prototypes for each category as input and only output associated RoIs of that category (Fig. \ref{fig_rpn}(b)) \cite{han2021meta,zhang2021accurate,li2020one}. Meta-RPN relied on a hypothesis that a set of prototypes of a given category could provide class-specific notions for RPN not only to suppress heterogeneous/background features but also to enhance similar semantics for more high-quality proposals. Zhang et al. \cite{zhang2020cooperating} proposed CoRPNs to ensemble multiple independent yet cooperative RPN to improve the quality of proposals (Fig. \ref{fig_rpn}(c)).}
\begin{figure}[!t]
\centering
\includegraphics[width=0.5\textwidth]{Figures/fig_rpn.pdf}
\caption{(a) The pipeline of original RPN; (b) The architecture of meta-RPN. Let \textcircled{A} be a feature aggregator which takes query features and a set of representatives of a given category (i.e., $v_c={v_1,\ v_2,\ldots,\ v_k}$) as input and outputs category-specific query features that are fed to subsequent RPN; (c) The architecture of CoRPNs. Unlike original RPN, it consists of multiple Fg/Bg classifiers which is independent yet cooperative.}
\label{fig_rpn}
\end{figure}
\begin{figure*}[!t]
\centering
\includegraphics[width=\textwidth]{Figures/fig_guidance.pdf}
\caption{The pipeline of support-only and support-query guidance. Especially, only (a) and (b) are support-only guidance while the other are support-query guidance. (a) Adaptive Fully-Dual Network (AFDN) \cite{liu2020afd}; (b) Hyper Attention RPN (HA-RPN) \cite{zhang2020few}; (c) Dense Relation Distillation (DRD) \cite{hu2021dense}; (d) Non-Local Co-Attention (NLCA) \cite{hsieh2019one}; (e) Augmentation with Conditioned Prototypes (ACP) \cite{wu2021universal}; (f) Multi-head Co-Attention (MHCA) \cite{chen2021adaptive}; (g) Cross-Image Spatial Attention Block (CISA) \cite{chen2021should}; (h) Spatial Alignment \& Foreground Attention Module (SAFA) \cite{han2021meta}.}
\label{fig_guidance}
\end{figure*}
\item{\textbf{Support-Only vs Support-Query Guidance.} To better integrate category-specific information into query features, there were two main ways to refine category prototypes, i.e., support-only and support-query guidance. In support-only guidance, it only takes category prototypes to refine features. Clearly, simple global max/average pooling was a kind of support-only guidance to capture global semantics which was helpful for the classification task \cite{kang2019few,yan2019meta,deng2020few}. However, it also lost spatial information (e.g., local structures) which was critical for RPN to make high-quality proposals. An effective way was to view a set of category prototypes as kernels to slide over query features for feature fusion. Liu et al. \cite{liu2020afd} designed a self-attention based fully-dual network that consisted of two parallel branches to simultaneously capture global and local semantics to generate task-specific attention vectors (Fig. \ref{fig_guidance}(a)). Zhang et al. \cite{zhang2020few} designed a list of PN (e.g., SigmE) to filter irrelevant factors in category prototypes (Fig. \ref{fig_guidance}(b)).
In fact, objects in the support set usually have large differences with that in the query set, e.g., various postures and viewpoints. Meanwhile, support-only guidance does not take the misalignment problem between query and support features into consideration. Thus, the scoring function could easily confuse whether the RoI features are consistent with the prototypes of the associated category. To tackle the misalignment problem, support-query guidance uses the affinity matrix between query and support features to align features. As illustrated in Fig. \ref{fig_guidance}(c, e, g, h), these methods followed such a framework that first employed two parallel branches to get key and value maps for both query and support images, then took key maps to calculate an affinity matrix $A$ between query and support images, and finally applied the affinity matrix to align value maps of support to enhance value maps of query. Especially, it could be viewed as a search process that we search all spatial location of key maps of support to aggregate support evidence for feature fusion (like the mutual information). To refine the affinity matrix, Chen et al. \cite{chen2021should} presented a new branch which applied a fully-connected layer with spatial softmax to generate a pseudo foreground mask, which was used to reweight the affinity matrix, to filter background information. Similarly, Han et al. \cite{han2021meta} leveraged the amount of category evidence (i.e., sum up the affinity matrix by row) to determine each spatial location in query features whether or not a foreground region. Meanwhile, in Fig. \ref{fig_guidance}(d, f), several works explored a new paradigm which the affinity matrix was applied not only to align value maps of support to enhance value maps of query but also to align value maps of query to enhance value maps of support. In fact, such a scheme utilized a kind of the mutual relationship that both query and support objects would fetch enough information from each other if they were the same category while they would fetch very little from each other if They were of a different class. Thus, it could amplify the gap between the consistent pair and the other inconsistent pairs. Hsieh et al \cite{hsieh2019one} shared the affinity matrix between two parallel branches of query and support while Chen et al. \cite{chen2021adaptive} took separate affinity matrixes for each branch. Zhang et al. \cite{zhang2021accurate} designed a QSW module to build a category prototype from multiple support objects of a given category by their relevance to a query RoI.}
\item{\textbf{Aggregator.} As for feature fusion, there existed many excellent feature aggregators, e.g., channel-wise multiplication (MUL), element-wise subtraction (SUB) and channel-wise concatenation (CAT). MUL was commonly known as feature reweighting to learn feature co-occurrence. SUB was a kind of distance metric (i.e., L1 distance) to measure L1 similarity. CAT was a special aggregator that could stack features along the channel axis for subsequent networks to automatically explore a good way for feature fusion. Most existing methods only adopted MUL as their feature aggregator. Nevertheless, Han et al. \cite{han2021meta} and Liu et al. \cite{liu2020afd} combined MUL, SUB and CAT for better feature fusion (Fig. \ref{fig_feature_fusion}). In addition to traditional aggregators, another way was to transform category prototypes into a set of parameters, which were employed to initialize all weights of operators (e.g., a convolution operator), to enhance query features \cite{zhang2021accurate}.}
\begin{figure}[!t]
\centering
\includegraphics[width=0.5\textwidth]{Figures/fig_feature_fusion.pdf}
\caption{The pipeline of feature fusion. (a) Dual Feature Aggregation; (b) Feature Fusion Network.}
\label{fig_feature_fusion}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=0.5\textwidth]{Figures/fig_RNs.pdf}
\caption{The pipeline of variants of RNs \cite{sung2018learning}. (a) A Two-Layer MLP Relation Network; (b) Multi-Relation Network.}
\label{fig_RNs}
\end{figure}
\item{\textbf{Scoring Function.} A scoring function is another essential element for a metric-based method, including cosine similarity \cite{bansal2018zero}, Pearson similarity \cite{li2020mm} and variants of relation networks (RNs) \cite{liu2020afd,fan2020few,chen2021adaptive}. Especially, cosine and Pearson similarity were fixed metrics which may not fit a given task. To give a flexible metric for FSOD, variants of RNs learned a soft metric to recognize various categories. Most existing methods followed the one-vs-rest strategy to design a binary classifier (named the mete classifier) which should effectively distinguish whether RoI features were consistent with a set of prototypes of a given category or not (Fig. \ref{fig_RNs}(a)). A multi-relation network was designed to capture global-, local- and patch-relation between RoI features and class prototypes for overall matching (Fig. \ref{fig_RNs}(b)) \cite{yan2019meta,zhang2020few}. Zhang et al. \cite{zhang2021meta} regarded the scoring function as the transformer decoder to classify each query.
Unlike the task of image recognition, an object detection method should distinguish negative proposals from all proposals, especially while negative proposals could even have a certain overlap (i.e., similarity) with associated class prototypes which may confuse the scoring function. To some extent, it was hard to define a background category to represent various negative samples with large intra-class variations. To tackle this problem, YOLO variants \cite{kang2019few,li2021beyond} took the confidence score to indicate the associated box whether or not a positive box. As aforementioned, R-CNN variants \cite{liu2020afd,fan2020few} employed a binary classifier to measure the similarity between RoI features with a set of prototypes for each class, where an RoI was assigned with a class tag that had the highest similarity score with this RoI. If the highest score was lower than a pre-defined threshold, it would be defined as a negative proposal. Yang et al. \cite{yang2020restoring} used negative proposals which had low IoU with ground-truth boxes to learn a list of negative prototypes for the background category. Bansal et al. \cite{bansal2018zero} explored two kinds of background definitions. First, a fixed vector was utilized as the prototype for the background class (statically assigned background). However, it could not really encode all background information, which could thus produce more false-positive samples. Second, in each iteration, we labeled background boxes with pseudo labels randomly sampled from a list of unseen categories to distinguish negatives from other positive proposals. Li et al. \cite{li2020mm} randomly sampled boxes, which had no overlap with ground-truth boxes, as exemplars of the background category to filter negative proposals of RPN.}
\item{\textbf{Loss Function.} A well-defined loss function was essential to learn a good model for a metric-based method. Many works inherited the loss functions from generic object detection methods, which were originally designed to guide the detectors to accurately detect all objects in an image. However, it did not take class prototypes into account, which played a key role in a metric-based method. Thus, several works introduced the customed loss function which could guide the embedding network to simultaneously minimize intra-class variations of prototypes and maximize inter-class distance of prototypes. Yan et al. \cite{yan2019meta} appended an MLP with softmax to the embedding network and employed cross-entropy (CE) loss to encourage the embedding network to explicitly encode category information into category prototypes. Fan et al. \cite{fan2020few} presented a two-way contrastive training strategy. It first constructed a training triplet $(q,s_c,s_n)$ where $q$ was a query image and $s_i$ was a support instance of the i-th category ($c\neq n$). Then, it combined $s_c$ and $q$ to generate category-specific features fed to RPN to generate a set of proposals $p$, where only proposals of the c-th category were regarded as positive proposals. Next, it proportionally sampled training pairs to construct a balanced training set, consisting of three types of pairs, i.e., $(p_c,s_c)$, $(p_b,s_c)$ and $(p_\ast, s_n)$ where $p_i$ was a proposal of i-th category ($b \neq c$). Finally, it enforced a binary cross-entropy (BCE) loss to learn a good representative. Margin-based ranking loss \cite{chen2021adaptive} was a kind of multi-task loss, consisting of two parts. One was a hinge loss variant for the foreground-background classification. The other was a max-margin contrastive loss to enforce all RoIs to satisfy max-margin category separation and semantic space clustering as possible. Bansal et al. \cite{bansal2018zero} defined a margin loss to enforce a constraint that the matching score of a proposal with its true category should be higher than that with other categories. Li et al. \cite{li2021beyond} assumed that novel classes had implicit relation with base classes and prototypes of novel classes could be embedded into margins between prototypes of base classes. However, large inter-class distance provided the safe decision boundary for the classification while large margin between prototypes of base class makes it hard for novel classes to find appropriate class prototypes. Thus, Li et al. \cite{li2021beyond} proposed a max-margin loss which was formulated as a sum of intra-class distance of all classes over a sum of inter-category distance of all classes to adaptively adjust class-margin. Zhang et al. \cite{zhang2021accurate} improved traditional contrastive loss from two aspects. First, learnable margins were designed to adaptively adjust inter-class distance. Especially, these margins were initialized by inter-class semantic similarity. Second, instead of hard sampling in the basic contrastive loss, focal loss was imposed to adaptively adjust contributions of various kinds of samples to the gradient. Li et al. \cite{li2021transformation} also proposed a transformation invariant principle to learn a robust embedding network which should produce consistent category prototypes between an image and an associated transformed image, except for CE loss.}
\item{\textbf{Fusion Node.} As described above, class prototypes encoded class-specific information, e.g., the shape of objects, which were helpful for the detectors to accurately localize and classify objects of the associated category. Thus, it was important to determine where to place fusion nodes to aggregate query features and class prototypes in one-/two-stage detectors. Tab. \ref{tab_fusion_node} presented common fusion nodes and affected modules of one- and two-stage detectors.
For a two-stage detector, many works have explored six fusion nodes, in Fig. \ref{fig_fusion_nodes}(a). Several works revised the multi-class classifier as the meta-classifier which combined RoI features and category prototypes of a given category as inputs and output a matching score, i.e., only the 5th fusion node \cite{bansal2018zero, li2020mm,karlinsky2019repmet,zhu2021semantic}. However, it ignored the implicit shape and position information in category prototypes, which had a positive effect on the bounding box regressor. Thus, the 4th or 6th fusion node was widely added to provide category-sensitive information for the R-CNN head \cite{liu2020afd, li2021beyond, li2021transformation, yan2019meta}. Due to differences between the classification and localization, Liu et al. \cite{liu2020afd} explicitly decomposed the 4th fusion node into the 5th and 6th feature node which provided task-specific information respectively for two sub-tasks. In addition, to get more high-quality proposals from RPN, many works inserted the 1st or 2nd fusion node to provide class-sensitive information for RPN (meta-RPN) \cite{fan2020few,chen2021should,li2020one}. The 1st fusion node could affect the whole Faster R-CNN head while the 2nd fusion node only served RPN \cite{zhang2021meta,hu2021dense}. For one-stage detectors, almost all methods chose the 1st fusion node to make better results under the help of category-sensitive information (Fig. \ref{fig_fusion_nodes}(b)) \cite{deng2020few, li2021beyond,kang2019few}. More details about combinations of fusion nodes were shown in Tab. \ref{metric_learning_all_methods_review}.}
\begin{table}[!t]
\centering
\caption{Modules Affected by Fusion Node. Note that a-i is the i-th fusion node of two-stage detectors while b-i is the i-th fusion node of one-stage detectors, as in Fig. \ref{fig_fusion_nodes}.}
\centering
\begin{tabular}{|c|c|}
\hline
Fusion Nodes & Affected Modules \\ \hline
a-1 & RPN \& Cls \& Loc \\ \hline
a-2 & RPN \\ \hline
a-3 & \multirow{2}{*}{Cls \& Loc} \\ \cline{1-1}
a-4 & \\ \hline
a-5 & Cls \\ \hline
a-6 & Loc \\ \hline
b-1 & Cls \& Loc \\ \hline
b-2 & Cls \\ \hline
b-3 & Loc \\ \hline
\end{tabular}
\label{tab_fusion_node}
\end{table}
\begin{figure}[!t]
\centering
\includegraphics[width=0.5\textwidth]{Figures/fig_fusion_nodes.pdf}
\caption{Fusion Nodes of two parallel branches of support and query in (a) two- stage detectors/(b) one-stage detectors. Note that \textcircled{i} represents the i-th fusion node which is actually a feature aggregator to aggregate query features and category prototypes.}
\label{fig_fusion_nodes}
\end{figure}
\item{\textbf{Training/Testing Process.} The overall training/testing process is illustrated in Algorithm \ref{alg:Framwork}, i.e., meta-training/finetuning/testing. Almost all metric-based methods are usually pretrained in existing benchmarks \cite{chen2015microsoft,deng2009imagenet} to learn basic notions (e.g., low-level visual features) which could be applied into our $N$-way $K$-shot problem, instead of training from scratch. For a fair comparison, it is essential to remove all categories in ImageNet overlapped with unseen categories in target benchmarks. In meta-training stage, we first construct a set of tasks/episodes built upon $D_{base}$ to let $M^{base}$ learn how to leverage limited reliable data to match all instances in a query set. In other words, a base learner is actually a task-agnostic learner which could be instantiated with task-specific notions from a given support set to solve a novel task. Thus, high-quality support data is crucial to learn a robust $M^{base}$. Tab. \ref{strategy_for_stable_learning} presents several solutions to pair a relatively appropriate support for a query image when plenty of small/hard objects are in $D_{base}$. Especially, meta-finetuning is an optional step in a metric-based method. In most cases, meta-finetuning could further improve the learner's performance in meta-testing stage. However, it is unrealistic to directly leverage $D_{base} \cup D_{novel}$ to finetune $M^{finetune}$ owing to the extreme class imbalance between $D_{base}$ and $D_{novel}$. In general, we need to construct a balanced subset which is randomly sampled from $D_{base}$ ($K_b$ exemplars per category). In most cases, $K_b$ is equal to $K$ ($N$-way $K$-shot). However, considering a large number of training exemplars for base classes, $K_b$ can be increased properly to get more robust prototypes for base classes to speed up the training process, e.g., $K_b=3K$ \cite{yan2019meta}. In meta-testing, we could preprocess $D_{novel}$ to get class-sensitive vectors for each novel class as task-specific parameters for $M^{finetune}$ instead of temporarily extracting for all tasks which is time-consuming. Due to low-shot scenarios, base/novel-category splitting settings have a great effect on final results. Thus, two main ways are proposed to evaluate performance of $M^{finetune}$ more accurately, in Tab. \ref{evaluation_settings}.}
\begin{table}[!t]
\centering
\caption{Solutions to Construct a More Suitable Episode from A Difficult Benchmark.}
\begin{tabularx}{\linewidth}{|c|X|}
\hline
Solution & \multicolumn{1}{c|}{Explanation} \\
\hline
Hsieh et al. \cite{hsieh2019one} & Take a pretrained Mask R-CNN to evaluate the difficulty of a ground-truth box and only ground-truth boxes with high-quality proposals would be cropped out as support patches. \\
\hline
Fan et al. \cite{fan2020few} & Remove all images with a box whose size is less than $32 \times 32$ in the training stage. \\
\hline
Li et al.\cite{li2020one} & Randomly sample $m$ support patches and only choose a support patch which has the most similarity with the query images. \\
\hline
Liu et al. \cite{liu2020afd} & Sample more images to build a larger support set for each task in $\mathcal{T}^{train}$ ($K=200$). \\
\hline
\end{tabularx}
\label{strategy_for_stable_learning}
\end{table}
\begin{table}[!t]
\centering
\caption{Details of Two Main Evaluation Settings.}
\begin{tabularx}{\linewidth}{|c|X|}
\hline
Method & \multicolumn{1}{c|}{Detail} \\
\hline
Karlinsky et al. \cite{karlinsky2019repmet} & Randomly sample 500 episodes ($n$-way $k$-shot tasks) where each episode consists of a support set with $k$ instance per category ($|S|=nk$) and a query set $Q$ with 10 images per category ($|Q|=10n$). Thus, we have at least 10 instances of each category in meta-testing stage. ($n$ is usually assigned by 5.) \\
\hline
Fan et al. \cite{fan2020few} & (1) Randomly sample several base and novel class splitting settings;\newline{} (2) Construct a list of tasks $\mathcal{T}^{test}$ built upon $(D_{novel},D_{novel}^\prime)$ for each splitting setting to evaluate performance;\newline{} (3) Report average performance for each splitting setting. \\
\hline
\end{tabularx}%
\label{evaluation_settings}
\end{table}%
\begin{table}[!t]
\centering
\caption{Summarization of details of representative methods for two other settings of support branches.}
\begin{tabularx}{\linewidth}{|c|c|X|}
\hline
Group & Method &\multicolumn{1}{c|}{Explanation} \\
\hline
\rotatebox[origin=c]{90}{Learnable Visual Prototype (LVP)} & Yang et al. \cite{yang2020restoring} & Meta-Training: (1) Assign $K$ learnable positive \& negative prototypes per base category; (2) Randomly initialize all prototypes; (3) Leverage $D_{base}$ to create $n$-way $k$-shot episodes (sample $n$ support categories instead of using all categories for computational efficiency) to learn all trainable parameters.\newline{}Meta-Testing: (1) Initialize task-specific parameters: (a) Novel-Category Prototypes: Use weight imprinting to initialize novel-category prototypes; (b) Background-Category Prototypes: Apply spectral clustering on embedding vectors of $M$ hard negative prototypes to generate $K$ negative prototypes respectively for each novel category.; (2) Evaluate NP-RepMet on $D_{novel}^\prime$. \\
\hline
\rotatebox[origin=c]{90}{Semantic Prototype (SP)} & Bansal et al. \cite{bansal2018zero} & Meta-Training: (1) Prepare Training Data: (a) Exploit EB \cite{zitnick2014edge} to generate raw proposals; (b) Sample both positive and negative proposals as training data; (c) Assign base class labels to positive proposals; (d) Randomly sample a list of negative classes that have no overlap with these base classes and assign these negative class to negative proposals; (2) Take a word vector as prototypes for each base and negative class; (3) Exploit LAB to learn a meta detector.\newline{}Meta-Testing: (1) Use a word vector as the category prototype for each novel category; (2) Evaluate ZSD on $D_{novel}^\prime$. \\
\hline
\end{tabularx}%
\label{other_setting_for_support}
\end{table}%
\begin{algorithm}[!t]
\scriptsize
\caption{Meta-Training/Finetuning/Testing of A Metric-based Method}
\label{alg:Framwork}
\begin{algorithmic}[1]
\Require
\Statex A pre-defined model, $M=\{M^{train}, M^{finetune}\}$;
\Statex A large-scale benchmark for base classes, $D_{base}$;
\Statex A small-scale training benchmark for novel classes (N samples per class), $D_{novel}$;
\Statex A testing benchmark for novel classes, $D^\prime_{novel}$; // $D^\prime_{novel}\cap D_{novel} = \emptyset $
\end{algorithmic}
\par\vspace{-0.5\baselineskip}\noindent\hrulefill \\
\textcolor{red}{Meta-Training}
\begin{algorithmic}[1]
\State Initialize all parameters of $M^{train}$;
\For{$i \leftarrow 0, 1, ..., e_0-1$} // $e_0$: the number of epochs in meta-training;
\State Construct a set of tasks/episodes $\mathcal{T}^{train}$ built upon $D_{base}$;
\For{$T$ in $\mathcal{T}^{train}$}
\State $l^{train} \leftarrow Loss^{train}(M^{train}, T)$
\State Update all trainable parameters $\theta^{train}$ of $M^{train}$ by backprop;
\EndFor
\EndFor
\end{algorithmic}
\textcolor{red}{Meta-Finetuning}
\begin{algorithmic}[1]
\State Initialize all parameters of $M^{finetune}$; // shared parameters copied from $M^{train}$;
\For{$i \leftarrow 0, 1, ..., e_1-1$} // $e_1$: the number of epochs in meta-finetuning;
\State Construct a set of tasks $\mathcal{T}^{finetune}$ built upon $D_{base} \cup D_{novel}$;
\For{$T$ in $\mathcal{T}^{finetune}$}
\State $l^{finetune} \leftarrow Loss^{finetune}(M^{finetune}, T)$
\State Update all trainable parameters $\theta^{finetune}$ of $M^{finetune}$ by backprop;
\EndFor
\EndFor
\end{algorithmic}
\textcolor{red}{Meta-Testing}
\begin{algorithmic}[1]
\State Construct a set of tasks $\mathcal{T}^{test}$ built upon $D_{novel} \cup D^\prime_{novel}$; // $(S_i, Q_i) = T_i \in \mathcal{T} \& S_i \subset D_{novel} \& Q_i \subset D^{\prime}_{novel} \& Q_1 \cup Q_2 \cup ... = D^{test}$
\State $ans \leftarrow \{\}$
\For{$T$ in $\mathcal{T}^{test}$}
\State $bboxes=M^{finetune}(T)$
\State $ans \leftarrow ans \cup \{(Q, bboxes)\}$
\EndFor
\State Take $ans$ to evaluate performance of $M^{finetune}$ by a metric, e.g., mAP50.
\end{algorithmic}
\end{algorithm}
\item{\textbf{Other Settings of Support Branch.} Earlier, we mainly discussed a kind of method that employed a list of support images fed to a customed backbone for category prototypes named Generative Visual Prototype (GVP). Currently, two other settings of support branches have been proposed to get rid of sampling a set of support images per category for all episodes, i.e., LVP and SP, as illustrated in Tab. \ref{other_setting_for_support}. To mitigate noise in a small support set, LVP aims to exploit a list of learnable kernels to automatically acquire class prototypes for each class, without dependency on a specific object. Several works \cite{bansal2018zero} explored to use of semantic prototypes (SPs) learned from a large-scale corpus to provide reliable task-specific parameters for these detectors. }
\end{itemize}
\emph{2. Optimization-based Methods.} Optimization-based methods assume that generic notions (i.e., meta parameters and a meta optimizer) could be learned from the base datasets to provide a suitable gradient guidance or a uniformly optimal initial weight for quick adaption to a new task. However, unlike FSC, it is difficult for a meta-optimizer or meta-parameters to adapt to a wider parameter space or to balance two sub-tasks of object detection.
Therefore, optimization-based learning has very few applications in LS-FSOD.
\begin{itemize}
\item{\textbf{Meta RetinaNet}, proposed by Li et al. \cite{li2020meta}, was an optimization-based detector which took a MAML variant \cite{finn2017model} to learn meta-parameters for a new task. To reduce too large parameter space, kernels $k$ of convolution layers in RetinaNet was reformulated as: $k^\prime = k \odot w$, where $w$ were learnable coefficient vectors initialized by ones and $k$ were constant kernels initialized by associated kernels pretrained on $D_{base}$. In the meta-training phase, balanced loss (BL) was designed to replace simple summation in original MAML to adaptively down-weight easy tasks and focus on hard tasks to update meta parameters (i.e., coefficient vectors and parameters of the last classification layer). }
\end{itemize}
\emph{3. Model-based Methods.} For model-based methods, the key is to design a model and associated learning strategies which quickly adapts for a new episode. Most model-based methods mainly relied on RNNs \cite{zaremba2014recurrent} with memory to utilize training samples of an episode to predict task-specific parameters for such an episode \cite{cai2018memory,santoro2016meta}. Parameters could be further divided into fast/slow (task-specific/agnostic) parameters which were combined to make more stable predictions. However, like optimization-based learning, model-based learning was barely noticed in FSOD.
\begin{itemize}
\item{\textbf{MetaDet.} Inspired by a category-agnostic transformation, Wang et al. \cite{wang2019meta} first explored a paradigm for a meta generator $G$: $w^\ast=G(w)$, where $w$ and $w^\ast$ were task-specific parameters learnt respectively from an episode and a large-scale dataset. In MetaDet, only R-CNN head was defined as task-specific modules. Especially, its training consisted of three stages.
1. Pre-Training Phase: Take $D_{base}$ to learn a large-sample detector $D(I;\theta^\ast)$ in a standard way, where $\theta^\ast$ is a set of parameters of the detector.
2. Meta-Training Phase: First, learn a $n$-way $k$-shot detector $D(I;\theta^\ast \cup w_{det}^{c_{base}} \backslash w_{det}^{c_{base},\ast})$ where $w_{det}^{c_{base},\ast}$ and $w_{det}^{c_{base}}$ represent category-specific parameters learned in $D_{base}$ or a $n$-way $k$-shot episode in the top layer of Faster R-CNN. Then, add consistency loss on the pair $(w_{det}^{c_{base},\ast},w_{det}^{c_{base}})$ of each episode to learn a meta generator G.
3. Meta-Testing Phase: Fist, learn a $n$-way $k$-shot detector $D(I;\theta^\ast \cup w_{det}^{c_{novel}} \backslash w_{det}^{c_{base},\ast})$ where $w_{det}^{c_{novel}}$ denote category-specific parameters learned in $D_{novel}$. Then, feed $w_{det}^{c_{novel}}$ to G for a more robust version $w_{det}^{c_{novel},\ast}$. Finally, fine-tune $D(I;\theta^\ast \cup w_{det}^{c_{novel},\ast} \backslash w_{det}^{c_{base},\ast})$.
However, MetaDet mainly relied on a principle that base categories shared similar distribution with novel categories, which greatly limited its scope of applications.}
\end{itemize}
\textbf{Transfer-Learning Methods.}
Compared with meta-learning algorithms, transfer-learning methods usually have two phases of training, i.e., the pre-training and fine-tuning stage. In the pre-training stage, a large-scale dataset $D_{base}$ is employed to train a base detector for general notions under the official setting. There are two steps during the fine-tuning stage: (1) Initialization. The novel learner inherits general notations/parameters from the base learner while the rest parameters are initialized randomly or by a weight imprinting technique. (2) Fine-tuning. A small-scale dataset $D_{base}^\prime\cup D_{novel}$ is constructed to fine-tune the novel learner where $D_{base}^\prime$ is a balanced subset of $D_{base}$ ($K$ shots per base class). To sample a more suitable subset $D_{base}^\prime$, Li et al. \cite{li2021class} proposed a clustering-based exemplar selection algorithm which first calculated intra-category mean features/prototypes for each image in $D_{base}$, then applied the k-means algorithm to generate $K$ clusters for each base category, and finally obtained $K$ centroids (exemplars) per base category. Like aforesaid metric-based learning, we will make a detailed review on the fine-tuned module, regularization, classifier and loss function for transfer-learning methods. If you want to know specific contributions, we also present them in Tab. \ref{details_transfer_learning}.
\begin{itemize}
\item{\textbf{Fine-tuned Module.} In the early stage of transfer-learning approaches \cite{wang2020frustratingly}, the backbone and RPN were usually taken as task-agnostic modules and it was ineffective to utilize a small-scale dataset $D_{base}^\prime \cup D_{novel}$ to fine-tune all parameters of a base detector. Thus, wang et al. \cite{wang2020frustratingly} explored a FIX\_ALL mode which only adapted the final layer of both classification and localization in the fine-tuning stage and achieved promising results. However, as aforesaid in Section \ref{subsec_challenges}, RPN showed relatively poor performance for novel classes. Sun et al. \cite{sun2021fsce} demonstrated the number of positive proposals for novel categories was about a quarter of that for base categories. Hence, there were not enough positive proposals to fine-tune RPN in the fine-tuning stage. Meanwhile, it implicitly introduced the foreground-background imbalance problem, due to low-quality proposals for novel categories. Sun et al. \cite{sun2021fsce} proposed a two-step procedure to re-balance sampling ratio of the positive and negative proposals: (1) reuse positive proposals with low confidence suppressed by NMS; (2) discard negative proposals by half. Moreover, backbones were proved to have weaker response for novel categories than that for base categories. Thus, many works explored an extreme mode (named FIT\_ALL) to obtain a de-biased backbone/FPN for better features \cite{wu2021universal,wu2020multi,wang2019few}. However, low density sampling easily leaded to overfitting under the mode FIT\_ALL.}
\begin{figure}[!t]
\centering
\includegraphics[width=0.4\textwidth, height=0.4\textwidth]{Figures/fig_reg.pdf}
\caption{The pipeline of two classic regularization methods. (a) Transfer-Knowledge (TK) Regularization; (b) Background-Depression (BD) Regularization.}
\label{fig_reg}
\end{figure}
\begin{table}[!t]
\centering
\caption{The sub-modules of FAFRCNN.}
\begin{tabularx}{\linewidth}{|c|X|}
\hline
Sub-Module & \multicolumn{1}{c|}{Detail} \\
\hline
\makecell{Split \\ Pooling} & (1) Apply shared random shifts on a set of pre-defined anchors to crop multi-scale features patches for both source and target domain.\newline{}(2) Take feature patches to generate two kinds of pairs at each scale (i.e., source-source and source-target pairs).\newline{}(3) Adopt generative adversarial learning to alternatively optimize the generator (backbone) and the discriminator that classifies feature patches whether or not belong to target domain. \\
\hline
\makecell{Instance-level \\ Adaptation} & (1) Sample foreground RoIs with high IoU threshold (i.e., 0.7) in source and target domain. \newline{}(2) Take RoI features to generate two kinds of pairs (i.e., source-source and source-target pairs).\newline{}(3) Adopt generative adversarial learning to alternatively optimize the generator (RoI feature encoding network) and the discriminator that classifies RoI features. \\
\hline
\end{tabularx}%
\label{details_FAFRCNN}
\end{table}%
\item{\textbf{Regularization.} To alleviate overfitting, several studies added extra constraints on the parameter space for a robust model. Earlier, the base classifier was usually dropped in the fine-tuning stage \cite{sun2021fsce,wang2020frustratingly,wu2020multi}. Especially, it could be viewed as a special encoder that took RoI features $R$ as input and output a similarity score for each pair in a set $\{(R,C_{base}^i)|i=1,2,\ldots,|C_{base}|\}$. In other words, it could be utilized to indicate shared features among novel and base classes to regularize the models. Thus, as shown in Fig. \ref{fig_reg}(a), KL loss was applied to enforce the model to reuse shared features for a novel task \cite{li2021class,wu2021universal,chen2018lstd}. Moreover, Wu et al. \cite{wu2021universal} leveraged a consistent constraint (KL loss) among probability distributions generated by a shared classifier and reserved a set of class-agnostic prototypes learnt from a large-scale dataset $D_{base}$ for a novel task as well for full use of generic notions from base classes.
Complex background may degenerate performance in a few-shot scenario. As shown in Fig. \ref{fig_reg}(b), Chen et al. \cite{chen2018lstd} utilized ground-truth boxes in a given image for making a rough estimation of the background mask to explicitly depress background features. In addition, several works leveraged a pre-trained saliency model to generate saliency maps to reweight features for background depression \cite{chen2020leveraging}.
FPN \cite{lin2017feature} was widely utilized to alleviate large scale variations. FPN employed several feature maps at different scales for region proposals. However, due to low-density sampling, it was hard to provide enough positive exemplars at each scale to train FPN \cite{lin2017feature} while more negative proposals were generated by FPN. Thus, Wu et al. \cite{wu2020multi} proposed object pyramids for more exemplars to refine FPN to distinguish positive-negative exemplars at each scale. Instead of directly extracting objects, a small random shift was applied on square boxes for some disturbance to learn a more robust FPN \cite{lin2017feature}. Then, we resized objects to six scales (i.e., $\{{32}^2,\ {64}^2,{128}^2,{256}^2,{512}^2,{800}^2\}$). Next, objects were fed into shared Faster R-CNN \cite{ren2015faster}. Finally, for objects at a special scale, only feature maps at an associated scale were leveraged to calculate RPN and RoI classification loss.
As described above, many works supposed that the novel classes $C_{novel}$ shared generic notions with the base classes $C_{base}$ (i.e., low domain shift). Nevertheless, it was hard to collect a proper dataset $D_{base}$ in some special cases. Hence, Wang et al. \cite{wang2019few} proposed an adversarial learning to mitigate domain shifts (Tab. \ref{details_FAFRCNN}).}
\item{\textbf{Classifier.} For most transfer-learning methods, error rates in classification are much higher than that in localization. Thus, several works replaced the softmax classifier with the cosine classifier to get rid of irrelevant factors, e.g., various feature norms. Moreover, Yang et al. \cite{yang2020context} designed Context-Transformer to provide more contextual information for the novel classifier. It first leveraged spatial pooling to implicitly extend receptive/contextual fields of original features $F_k$ at each scale $k\in\{1,2,\ldots,K\}$ for context features $Q_k$. Then, an affinity matrix $A_k$ was calculated among features $F_k$ \& a set of context features $Q=\{Q_1,Q_2,\ldots,Q_K\}$ to adaptively match effective supporting context at various scales for feature fusion.}
\item{\textbf{Loss Function.} Multi-class cross entropy (MCE) loss was generally used for classification. However, MCE aimed at increasing inter-class distinction which pay little attention to lower intra-class variations. Thus, Chen et al.\cite{chen2020leveraging} utilize a variant of cosine loss to form compact category cluster. Sun et al. \cite{sun2021fsce} introduced a contrastive learning strategy which leveraged a variant of cross entropy to introduce competition between homogeneous and heterogeneous pairs for high inter-category distinction and low inter-category distance.}
\end{itemize}
\emph{3. Data Augmentation Methods.}
Data augmentation aims at increasing intra-class variations to enforce the model to utilize more robust features. Li et al. \cite{li2021transformation} proved, in few-shot scenarios, most naive data augmentation methods could introduce large intra-class variations which had negative effect on its performance. Therefore, Li et al. \cite{li2021transformation} proposed a TIP module to construct contrasting pairs to make the encoder invariant to various intra-class variations by KL loss. Likewise, Zhang et al. \cite{zhang2021hallucination} designed a hallucination network to transfer the intra-category variations from base categories to novel categories. Simple horizontal flipping was proved effective for the performance improvement due to moderate variations \cite{li2020one}.
\subsection{Imbalanced Limited-Supervised Problem}
\subsubsection{Definition}
In a low-shot scenario, it is likely to collect a imbalanced dataset due to the long-tail distributions of real world data. Objects have great discrepancies in the occurrence frequencies (i.e., $\min{\{|D_{novel}^i|,\ i\in C_{novel}\}} \ll \max{\{|D_{novel}^i|, i\in C_{novel}\}}=k$). Moreover, it is also sub-optimal to adopt an under- or over-sampling strategy to build a $N$-way $K$-shot benchmark. However, such a problem has not attracted enough attention.
\subsubsection{Solutions}
So far, there were a few works \cite{zhang2020class} to present the detailed description on this imbalanced problem within our capacity. In addition, we also propose several candidate solutions for this problem, i.e., group-based methods and generative methods.
Inspired by focal loss, a solution could be implemented by a balanced loss which could adaptively re-balance gradient from all categories. Zhang et al. \cite{zhang2020class} proposed CI loss which employed the imbalance degree in a dataset to automatically select appropriate parameters for gradient re-balance. Zhang et al. exploited NUDT-AOSR15 to construct both the training and testing set and chose mAP as the evaluation metric.
Inspire by the long-tail distribution in the large-scale dataset (e.g., PASCAL VOC07/12), we propose two kinds of solutions that could be applied to tackle this imbalanced limited-supervised problem. First, GAN variants were introduced to produce high-quality exemplars (e.g., images) for re-balancing the foreground-foreground class \cite{dwibedi2017cut,dvornik2018modeling,tripathi2019learning}. Second, a classification tree was built upon lexical or semantic relations for a coarse-to-fine strategy to alleviate class imbalance \cite{wu2020forest}, instead of directly classifying all classes. In addition, there are some tricks to tackle this problem, such as an NMS resampling algorithm \cite{wu2020forest} which dynamically adjusted NMS thresholds for each category to provide enough RoIs for the class with a few training exemplars according to the occurrence frequency of each category.
\section{Semi-Supervised Few-Shot Object Detection}
\label{sec_semi}
So far, most works on semi-supervised object detection (SSOD) collected about half (or 10 percent) of data with instance-level labels, which is fundamentally different from few-shot settings and still costs a lot. Especially, if there exist over 1000 images in a dataset for SSOD, SSOD will have far more labeled data than that annotated for SS-FSOD. Thus, compared with traditional SSOD, SS-FSOD can further reduce the annotation burden and exploit labeled data more effectively. There are two main ways for SS-FSOD, i.e., self-training and self-supervised based solutions.
\subsection{Problem Definition}
Let $D_{novel}$ represent a small dataset with a few labeled exemplars for a set of classes $C_{novel}$ and $D_{novel}^-$ represent a large-scale dataset sampled in the target domain without target supervision. For each sample $(I,Y)$ in $D_{novel}$, $I$ is an image ($I\in\mathbb{R}^{M\times N\times3}$) and $Y=\{(b_n,y_n)\}^N$ is a set of $N$ instances of $C_{novel}$ in $I$, where $b_n\in\mathbb{R}^4$ is a bounding box of the n-th instance and $y_n\in\{0,1\}^{|C_{novel}|}$ denotes an one-hot class encoding. In some cases, we also have an extra dataset $D_{base}$ without target-domain supervision to learn generic notions instead of training from scratch.
Let $D_{novel}^n$ be a set of instances of the n-th category in $D_{novel}$. Like in limited-supervised FSOD, we restrict the maximum of the number of instances per category in $D_{novel}$: $\max{\{|D_{novel}^i|,\ i\in C_{novel}\}}\le k$ ($k$ is usually no more than 30). In all, SS-FSOD tries to extract latent notions in $D_{novel}^-$ (e.g., pseudo labels) to refine/pre-train detectors for avoiding overfitting in $D_{novel}$.
\subsection{Dataset}
In SS-FSOD, it mainly uses existing benchmarks to form a semi-supervised few-shot dataset, including PASCAL VOC07/12, MSCOCO and ImageNet-LOC. We usually take mAP for PASCAL VOC07/12 and mAP/mAP75 for MSCOCO to evaluate the mean detection performance. Likewise, we also use CorLoc for PASCAL VOC07/12 and MSCOCO to evaluate the localization performance. More details are shown in Tab. \ref{tab_semi_dataset}.
\begin{table*}[!t]
\centering
\caption{Dataset Settings for SS-FSOD. Note that digits represent how many exemplars are labeled for each class. }
\begin{tabular}{|c|c|}
\hline
Methods & Detail \\
\hline
MSPLD \cite{dong2017few} & PASCAL VOC07/12 (CorLoc/mAP, 3), MSCOCO (CorLoc/mAP, 3), ImageNet-LOC (CorLoc/mAP, 3) \\
\hline
DETReg \cite{bar2021detreg} & MSCOCO (mAP/mAP75, 10/30)\\
\hline
CGDP \cite{li2021few}& PASCAL VOC07/12 (mAP, 1/2/3/5/10), MSCOCO (mAP/mAP75, 10) \\
\hline
TIP \cite{li2021transformation} & PASCAL VOC07/12 (mAP, 1/2/3/5/10) \\
\hline
\end{tabular}%
\label{tab_semi_dataset}
\end{table*}%
\subsection{Solutions}
\subsubsection{Self-training Based Methods}
Self-training is a classic way in semi-supervised learning, aiming to extract proper pseudo labels as ground-truth labels for unannotated data. As aforesaid, a classic self-training based approach tends to use all labeled data to pre-train a teacher detector, then applies it to generate pseudo labels for all unlabeled images, and finally samples a set of robust pseudo labels to learn a student detector. The key for self-training based methods is the quality of pseudo labels and it is sub-optimal to exploit a single teacher detector for pre-generated pseudo labels or only update pseudo labels once to train a student detector, which can limit its performance.
\begin{figure*}[!t]
\centering
\includegraphics[width=\textwidth]{Figures/fig_pseudo_labels.pdf}
\caption{The pipeline of four classic pseudo-label generations. (a) MSPLD \cite{dong2017few}; (b) DETReg \cite{bar2021detreg}; (c) FCOS Ensemble++ \cite{yoon2021semi}; (d) CGDP \cite{li2021few}.}
\label{fig_pseudo_labels}
\end{figure*}
\begin{itemize}
\item{\textbf{MSPLD. } Dong et al. \cite{dong2017few} proposed MSPLD that was based on self-learning and multi-modal learning to obtain high-quality pseudo labels for robust training, as illustrated in Fig. \ref{fig_pseudo_labels}(a). It could be concluded as three strategies: (1) \emph{Hard Example Removing.} Dong et al. first assumed that images with over 4 pseudo boxes for a specific class or over 4 classes were likely to have no reliable labels, which should be removed during each iteration step; (2) \emph{Model Ensemble.} Multiple detectors were utilized together to produce raw labels, and NMS with confidence-based box filtering was applied on raw labels for pseudo ground-truth labels; (3) \emph{Training Pool.} It was defined as an indicator whether or not an image should be used in training phase. Specially, it would affect an image's access threshold to training pools for a specific model, when this model had a number of images used at the last iteration, low discrepancies among proposals and pseudo labels and high consistency exploiting such an image in other detectors' training pools. In training stage, Dong et al \cite{dong2017few} adopted an iterative policy to update pseudo labels, training pool and parameters of a specific model one by one, i.e., solid lines in Fig. \ref{fig_pseudo_labels}(a).}
\item{\textbf{DETReg.} Bar et al. \cite{bar2021detreg} adopted a two-step training pipeline which disentangled pseudo labels and ground-truth labels for training, avoiding to bring too much noise into the fine-tuning stage, as depicted in Fig. \ref{fig_pseudo_labels}(b). Thus, pseudo labels were exploited in a pretext task to learn how to localize objects and how to encode robust features.
In such a pretext task, it mainly relied on a hypothesis that, compared with non-object boxes, object boxes should have less variations where DERT should learn not only to distinguish such variations but also to cluster automatically. To tackle it, several prediction heads were added upon DERT, i.e., a bounding box regressor for localization, a feature encoder for SwAV descriptor matching and a foreground-background classifier. As for pseudo labels, Bar et al. exploited SS to produce thousands of RoIs for high recall. In view of low precision for SS, a list of sampling strategies (i.e., Top-K, Random-K and Importance Sampling) were presented to fetch proper RoIs as ground-truth boxes (labeled by 1). Specially, Top-K showed uniformly optimal results in the testing phase, due to its relatively robust pseudo labels. In fine-tuning phase, we only saved the bounding box regressor while this binary classifier was replaced with a multi-category classifier to fit a specific task.}
\item{\textbf{CGDP.} In limited-supervised FSOD, it usually split existing benchmarks (e.g., Pascal VOC07/12) into a large-scale base dataset and a small-scale novel dataset, as illustrated in Fig. \ref{fig_pseudo_labels}(d). It was inevitable to contain novel/base objects in the base/novel dataset which was bad for the few-shot learner (equivalent to incomplete labeling). Actually, it could be treated as a semi-supervised problem where unlabeled regions in an image should be handled carefully. Li et al. \cite{li2021few} proposed a similarity-based strategy to inspect negative proposals, which had low IoU with all ground-truth annotations, for novel instance mining. First, weight imprinting was applied to extract a normalized intra-class mean prototype per novel class. Then, only high-possibility negative proposals were sampled to match various class prototypes by cosine similarity for pseudo-label generation.}
\end{itemize}
\subsubsection{Self-Supervised Based Methods}
Although pseudo labels could provide guidance information, it also brings noise into training student detectors. Thus, it is necessary to exploit other inherent relations among unannotated data. Apart from localizing and classifying as accurately as possible, a qualified detector should meet several requirements, e.g., robust feature encoding. Naturally, we could simply exploit unlabeled images to enforce backbones to encode discriminative features which could be applied to recover raw image signals as possible. In the semi-supervised learning, several works have developed associated pretext tasks, such as VAE/GAN based reconstruction, \cite{zhao2015stacked,maaloe2016auxiliary,odena2016semi,dai2017good}. In
some way, consistency regularization can be regarded as a supervised solution to extract underlying information from unlabeled data.
\begin{itemize}
\item{\textbf{TIP.} In low-shot scenarios, data distributions appear discontinuous very commonly, and naive data augmentation could make it worse which could lower inter-category distance. Especially, a metric-based approach tends to use an encoder to generate category prototypes which is crucial for category matching. In this case, an encoder should be invariant to any kinds of transformations. Thus, Li et al. \cite{li2021transformation} utilized unlabeled images to add a consistent constraint for transformation invariance where features of both an image and its corrupted image should be consistent/similar.}
\end{itemize}
\section{Weakly-Supervised Few-Shot Object Detection}
\label{sec_weakly}
WSOD tends to require 10/50 percent of images with manual tags in datasets, e.g., Pascal VOC07/12 \cite{everingham2010pascal} and MSCOCO \cite{chen2015microsoft}. Compared with instance-level boxes, tags save unit labor cost while it fails to reduce the size of a dataset needing to be labeled. It still brings large annotation burdens into data preparation. Thus, an intuitive solution is to exploit a few images with tags for each category in training phase (usually no more than 200). We properly relax restraints on the size of dataset with image-level labels due to its imprecise signals. Here, we will first give a more formal definition for WS-FSOD.
\subsection{Problem Definition}
As aforesaid, $D_{novel}^+$ denotes a small dataset with image-level labels or object locations and $D_{novel}^-$ denotes an optional large unlabeled dataset sampled in the target domain. For each sample $(I,Y)$ in $D_{novel}^-$, $I$ is an image ($I\in\mathbb{R}^{M\times N\times3}$) and $Y=\{y_n\}^N$ is a subset of $C_{novel}$ which appears in $I$. In some cases, we also have a dataset without target supervision to learn generic notions instead of training from scratch.
Let $D_{novel}^n$ be a set of images which contains instances of the n-th category in $D_{novel}$. Like in LS-FSOD, we restrict the maximum of the number of images per category in $D_{novel}$: $\max{\{|D_{novel}^i|,\ i\in C_{novel}\}}\le k$ (k is usually no more than 200). In all, WS-FSOD aims at actively associating image-level tags with all object regions to learn a robust learner which could be competent to the fully-supervised counterpart.
\subsection{Dataset}
Here, we list dataset settings of all existing WS-FSOD approaches, as shown in Tab. \ref{tab_weakly_dataset}. Existing approaches still exploit several benchmarks to build a weakly-supervised few-shot benchmark, including KITTI, Cityscapes, VisDA-18, MSCOCO, PASCAL VOC 07/12, ImageNet-LOC and CUB. In these excellent benchmarks, they could be grouped into two types, i.e., intra-domain and cross-domain benchmarks. Su et al. \cite{su2020active} combined two distinct benchmarks (i.e., KITTI and Cityscapes) to construct a cross-domain benchmark. Another common practice is to split a benchmark into two disjoint sets to form a intra-domain benchmark.
\begin{table*}[!t]
\centering
\caption{Dataset Settings for WS-FSOD. Note that digits represent how many exemplars are labeled for each class. }
\begin{tabular}{|c|c|}
\hline
Methods & Detail \\
\hline
AADA \cite{su2020active} & KITTI $\rightarrow$ Cityscapes (mAP, 10/20/30/50/100/200), VisDA-18 (mAP, N/S) \\
\hline
vMF-MIL \cite{shaban2021few} & MSCOCO (mAP, 5/10), PASCAL VOC 07 (mAP, 5/10) \\
\hline
StarNet \cite{karlinsky2020starnet} & ImageNet-LOC (mAP30/mAP, 1/5), CUB (mAP30/mAP, 1/5), PASCAL VOC (mAP30/mAP, 1/5) \\
\hline
NSOD \cite{yang2021training} & PASCAL VOC07/12 (mAP, 20) \\
\hline
\end{tabular}%
\label{tab_weakly_dataset}
\end{table*}%
\subsection{Solutions}
As WS-FSOD mainly attempts to exploit a few image-level labels, its approaches need not only to tackle classic issues in object detection, such as occlusion, deformation, domain shifts, various imbalance problems and so on, but also to confront with imprecise and less annotations. Owing to less supervisory signals, generic WSOD approaches cannot figure out underlying relations among image-level tags and RoIs, where it further increases data uncertainty instead. More specifically, low sampling density easily leads to large intra-category variations or low inter-category distance and such loose data streams can not only have almost no statistic laws but also hinder normal label propagation in the learning processes of WSOD. It is crucial to exploit weakly-labeled data more efficiently and eliminate uncertainty as much as possible.
So far, WS-FSOD has not received enough attention in the computer vision community. Here, it mainly includes three types, i.e., active adversarial learning, multiple instance learning and metric-based learning.
\begin{itemize}
\item{\textbf{AADA.} Su et al. \cite{su2020active} employed adversarial learning to force feature extractors to transform a WS-FSOD problem into a fully-supervised problem with a large-scale dataset for full use of source-domain knowledge. If a specific image cannot fit source domain well, detectors would apply non-matched notions to get poor results. Thus, Su et al. \cite{su2020active} also applied a well-optimized discriminator to indicate whether unlabeled data was similar to labeled data or not. A subset of exemplars with high distinction was sampled to obtain image-level tags to mitigate outliers which cannot fit source domain well. What's more, Su et al. \cite{su2020active} explored several sampling strategies to select images for maximizing performance gain, i.e., importance weight, K-means clustering, K-center, diversity, Best-versus-Second Best and random selection.}
\item{\textbf{vMF-MIL}. Shaban et al. \cite{shaban2021few} proposed vMF-MIL that was a probabilistic multiple instance learning approach for WS-FSOD. It employed a two-step strategy. It first obtained a base learner from a large-scale dataset with instance-label boxes. To utilize source-domain notions as much as possible, Shaban et al. \cite{shaban2021few} reformed Faster R-CNN \cite{ren2015faster} and make it fully category-agnostic by removing its traditional multi-category classifier. During the pseudo-label generation, it held a statistic assumption that RoI features for each category should form a cluster and have high distinction with that for other categories. For a specific category, it applied EM clustering on those features for finding exact a pseudo label with highest confidence for each image whose tags should include such a category. Then, an off-the-shelf FSOD approach \cite{wang2020frustratingly} could be applied to learn a novel task with pseudo boxes.}
\item{\textbf{StarNet.} Karlinsky \cite{karlinsky2020starnet} proposed StarNet that was also a metric-based solution. It mainly exploited feature co-occurrences for reliable evidence to mine all instances in query images without pseudo-label generation. It also consisted of two important fusion nodes: (1) Voting Heatmaps. It first measures a point-to-point similarity map by L1 norm, and calculates all permutations as voting heatmaps for matching support and query features. (2) Back-Projection Maps. It takes a permutation with highest matching score to suppress low-quality evidence. Except for an image-level category constraint on query features, it performed a consistency constraint among support and query features, which could form a close cluster for each category as well.}
\item{\textbf{NSOD.} Yang et al. \cite{yang2021training} proposed NSOD that was a metric-based MIL-style framework which attempted to mine implicit relations among intra-category mean prototypes and all RoIs. Especially, it could directly start from a small set of annotated images to propagate image-level tags to instance-level boxes for unlabeled data. It first utilized a backbone pretrained in ImageNet to extract features for both support images and all RoIs of query images (without any supervision). Then, for a specific category, all global features were averaged as an intra-category mean prototype which was combined with all RoI features to obtain pseudo labels for each query image. Next, pseudo labels were exploited to learn a teacher which was exploited to refine pseudo labels. Finally, two kinds of pseudo labels are averaged to form ground-truth pseudo labels for student learning. It was inevitable to include noisy background information into these intra-category mean prototypes, which was detrimental to final results. }
\end{itemize}
\section{Conclusion}
\label{sec_conclusion}
Few-shot detectors have obtained some key achievements to mitigate urgent need of abundant labeled training data for classic deep-learning architectures, playing an important role in various applications, such as wise medical \cite{quellec2020automatic, wei85few}. Therefore, we provide a comprehensive survey on few-shot learning for object detection. To provide a detailed analysis of this few-shot issue, this survey introduces a data-based taxonomy according to training data and associated supervisory signals of a novel task in terms of definitions, datasets, criteria, strengths and weaknesses for each kind of approaches. Besides, we discuss main challenges that needed to be tackled and how these approaches interplay and boost performance.
As aforesaid, existing few-shot detectors still need to tackle some imperative problems (e.g., domain shifts) and have a huge performance gap with many-shot detectors and human. Here, we will discuss future trends in this promising domain.
\textbf{Domain Transfer.} In a real-world task, it is not always easy to find a suitable dataset that could be exploited for cross-domain knowledge and have low domain shifts with this task. Although many works attempted to employ GAN variants to align the semantic space between the source and target domain, task-specific notions could be non-matched with the source domain and easily lost in this alignment process due to lack of associated supervision, especially with large domain shifts \cite{sankaranarayanan2018generate, chen2018semantic,chen2020multiple}. As active-learning strategies in \cite{su2020active}, we notice that it is key to combine effectively and efficiently incorporate cross-domain and task-specific knowledge for better domain transfer.
\textbf{Efficient Architectures.} Existing solutions for few-shot object detection still inherit the architectures designed for generic object detection \cite{girshick2014rich,ren2015faster,redmon2017yolo9000,liu2016ssd,redmon2018yolov3,bochkovskiy2020yolov4} which contain abundant learnable parameters. In these common architectures, more data and layers (parameters) usually mean higher performance. Thus, a novel and efficient architecture should be designed to tackle this issue while there is no work in this direction.
\textbf{Robust Feature Extractors.} In general, feature extractors are viewed as a task-agnostic component, which is typically pre-trained in a large dataset and kept fixed in another task. As in \cite{singh2018sniper}, feature extractor is susceptible to scale variations. Meanwhile, existing architectures extremely rely on the features produced by backbones for dense proposals generation, which is crucial to the final performance. Thus, there has been growing interest in learning a robust feature extractor \cite{li2021transformation,wu2020multi,chen2018lstd}.
\textbf{Mixed-Supervised Learning.} In WS-FSOD, it is hard to use a few images with image-level tags to associate underlying objects with these image-level tags. If we provide a small training set with instance-level labels except for the aforementioned weakly-labeled data, it only slightly increases annotation costs while it could give a definite signal for low-shot detectors to have more high-quality proposals when compared with weakly-supervised few-shot detectors. However, there are limited works in the mixed-supervised few-shot object detection \cite{chen2019progressive}. According to the above analysis, we think that mixed-supervised few-shot object detection may be a hot research direction.
\textbf{Unsupervised Learning.} Currently, state-of-the-art architectures have an urgent need for abundant labeled training data to learn an excellent detector. However, in some cases, training data is hard to be collected. Even if we cost significantly to collect a large annotated dataset, it could still contain nasty issues, such as foreground-foreground imbalance and data bias, which go against the normal training process. Therefore, it is essential to study a strategy \cite{zhu2007unsupervised,xie2017aggregated} to train CNNs without the need to collect and annotate a large-scale dataset for a novel task.
\textbf{Data Augmentation.} Data augmentation is a considerably intuitive way to alleviate overfitting by increasing intra-category variations. However, large intra-category variations could relatively lower inter-category distinction, which is interferential to explore suitable category prototypes. Zhang et al. \cite{zhang2021hallucination} proposed a meta strategy that moderately increased the intra-category diversity which could be transferred from other large-scale datasets while it cannot work with large domain shifts, since the source-domain intra-category diversity could not fit the target domain. A better choice is to actively measure intra-category variations and inter-category distance to provide suitable strategies for data augmentation.
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,664 |
package izzyaxel.arcaneartificing.blocks.tileEntities;
import izzyaxel.arcaneartificing.main.AABlocks;
import izzyaxel.arcaneartificing.main.AAReference;
import net.minecraft.block.Block;
import net.minecraft.block.BlockLiquid;
import net.minecraft.nbt.NBTTagCompound;
import net.minecraft.network.NetworkManager;
import net.minecraft.network.Packet;
import net.minecraft.network.play.server.S35PacketUpdateTileEntity;
import net.minecraft.tileentity.TileEntity;
public class TETemporalAcceleratorMKII extends TileEntity
{
int count = 0;
@Override
public boolean canUpdate()
{
return true;
}
@Override
public void updateEntity()
{
this.count = 0;
if(!this.worldObj.isRemote)
{
for(int i = this.xCoord - AAReference.RANGEMKII; i < this.xCoord + AAReference.RANGEMKII; i++)
{
for(int j = this.yCoord - AAReference.RANGEMKII; j < this.yCoord + AAReference.RANGEMKII; j++)
{
for(int k = this.zCoord - AAReference.RANGEMKII; k < this.zCoord + AAReference.RANGEMKII; k++)
{
if(this.worldObj.getBlock(i, j, k) == AABlocks.temporalAcceleratorMKII)
{
this.count++;
if(this.count > 4)
{
this.count = 0;
return;
}
}
}
}
}
for(int i = this.xCoord - AAReference.RANGEMKII; i < this.xCoord + AAReference.RANGEMKII; i++)
{
for(int j = this.yCoord - AAReference.RANGEMKII; j < this.yCoord + AAReference.RANGEMKII; j++)
{
for(int k = this.zCoord - AAReference.RANGEMKII; k < this.zCoord + AAReference.RANGEMKII; k++)
{
this.tickBlocks(i, j, k);
}
}
}
}
this.count = 0;
}
private void tickBlocks(int x, int y, int z)
{
Block block = this.worldObj.getBlock(x, y, z);
if(block == null || block == AABlocks.temporalAcceleratorMKI || block == AABlocks.temporalAcceleratorMKII || block == AABlocks.chunkLoader || block instanceof BlockLiquid || block == AABlocks.sentinel)
{
return;
}
if(block.getTickRandomly())
{
for(int i = 0; i < AAReference.UPDATESMKII; i++)
{
if(this.worldObj.rand.nextInt(100) % 50 == 0)
{
block.updateTick(this.worldObj, x, y, z, this.worldObj.rand);
}
}
}
if(block.hasTileEntity(this.worldObj.getBlockMetadata(x, y, z)))
{
TileEntity te = this.worldObj.getTileEntity(x, y, z);
if(te != null && !te.isInvalid())
{
for(int i = 0; i < AAReference.UPDATESMKII; i++)
{
if(te.isInvalid())
{
break;
}
te.updateEntity();
}
}
}
}
@Override
public void readFromNBT(NBTTagCompound tag)
{
super.readFromNBT(tag);
this.count = tag.getInteger("Count");
}
@Override
public void writeToNBT(NBTTagCompound tag)
{
super.writeToNBT(tag);
tag.setInteger("Count", this.count);
}
@Override
public Packet getDescriptionPacket()
{
NBTTagCompound tag = new NBTTagCompound();
tag.setInteger("Count", this.count);
this.writeToNBT(tag);
return new S35PacketUpdateTileEntity(this.xCoord, this.yCoord, this.zCoord, this.getBlockMetadata(), tag);
}
@Override
public void onDataPacket(NetworkManager net, S35PacketUpdateTileEntity packet)
{
this.readFromNBT(packet.func_148857_g());
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,913 |
DISCOVERPEACE
Károlyi Garden
Just one block from the busy main road of the Múzeum Bld. a space for nature, culture, recreation and playing
The oldest public garden in the inner city of Budapest. A tiny, French-style green park hidden among the tenement blocks like a box of jewels – a peaceful beauty spot for the harrassed city dweller. The noise of the traffic is cancelled out by the laughter of children. Typically there are two kinds of people among the park's regulars: the students of neighbouring schools playing under the watchful eyes of their teachers or parents and university students from the humanities and law faculty campuses nearby, nibbling at their sandwiches and basking in the sun between two lectures.
A small café placed its tables and chairs right by one of the corners of the garden, attracting passers-by for a little chit-chat.
The garden was declared a public park and landscaped in 1932. It belonged to the Károlyi Palace, hereby its name. The Károly family had come by this territory in 1768 and they remained its owners up till 1932.
The garden is home to the statue of Dániel Irányi, of the 1848 revolution, made by sculptor Ede Kallós. The Károlyi palace is home to the Petőfi Literary Museum, with open-air theatre performances held in its courtyard on summer evenings.
Dániel Irányi (1822 – 1892) was a very active player in the 1848 uprising, as member of various opposition circles and citizens' committees, MP and government commissioner. In 1886 he was among the founding members of the Association for Noble Morals. He was an undeterred supporter of the freedom of religion and the introduction of secular marriage.
"Liberty and love/These two I must have."
– Sándor Petőfi
Questions: Where can you find peace?
What are the everyday activities or places, which give you the sense of peace?
Recreation spots: the garden itself!, the café on the corner, Pet?fi Literary Museum, the cafés in Egyetem Square.
8:00-sunset
M3 Kálvin Square or Ferenciek Square
M2 Astoria
Tram 47 and 49 Astoria
Bus 5, 7, 7A, 8, 9, 112, 239 and 115 Astoria
Bus 5, 7, 7A, 7E, 8, 107E, 112, 133E, 233E, 239 Ferenciek Square
Time to the next peace trail station
Theme: discoverpeace by Cerium / Jos verhoeff. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,071 |
const refreshDropdown = require('@binary-com/binary-style').selectDropdown;
const moment = require('moment');
const TradingAnalysis = require('./analysis');
const commonTrading = require('./common');
const Contract = require('./contract');
const Defaults = require('./defaults');
const Durations = require('./duration');
const GetTicks = require('./get_ticks');
const Lookback = require('./lookback');
const Notifications = require('./notifications');
const Price = require('./price');
const Reset = require('./reset');
const StartDates = require('./starttime').StartDates;
const Symbols = require('./symbols');
const Tick = require('./tick');
const BinarySocket = require('../../base/socket');
const getMinPayout = require('../../common/currency').getMinPayout;
const isCryptocurrency = require('../../common/currency').isCryptocurrency;
const elementInnerHtml = require('../../../_common/common_functions').elementInnerHtml;
const getElementById = require('../../../_common/common_functions').getElementById;
const getVisibleElement = require('../../../_common/common_functions').getVisibleElement;
const localize = require('../../../_common/localize').localize;
const State = require('../../../_common/storage').State;
const getPropertyValue = require('../../../_common/utility').getPropertyValue;
const Process = (() => {
/*
* This function process the active symbols to get markets
* and underlying list
*/
const processActiveSymbols = () => {
BinarySocket.send({ active_symbols: 'brief' }).then((response) => {
if (response.active_symbols && response.active_symbols.length) {
// populate the Symbols object
Symbols.details(response);
const market = commonTrading.getDefaultMarket();
// store the market
Defaults.set('market', market);
commonTrading.displayMarkets();
processMarket();
} else {
$('#content').empty().html($('<div/>', { class: 'container' }).append($('<p/>', { class: 'notice-msg center-text', text: localize('Trading is unavailable at this time.') })));
}
});
};
/*
* Function to call when market has changed
*/
const processMarket = () => {
// we can get market from sessionStorage as allowed market
// is already set when this is called
let market = Defaults.get('market');
let symbol = Defaults.get('underlying');
// change to default market if query string contains an invalid market
if (!market || !Symbols.underlyings()[market]) {
market = commonTrading.getDefaultMarket();
Defaults.set('market', market);
}
if ((!symbol || !Symbols.underlyings()[market][symbol])) {
symbol = undefined;
}
processMarketUnderlying();
};
/*
* Function to call when underlying has changed
*/
const processMarketUnderlying = () => {
const underlying_element = document.getElementById('underlying');
const underlying = underlying_element.value;
Defaults.set('underlying', underlying);
commonTrading.showFormOverlay();
// get ticks for current underlying
GetTicks.request(underlying);
Tick.clean();
commonTrading.updateWarmChart();
BinarySocket.clearTimeouts();
getContracts(underlying);
commonTrading.displayTooltip();
};
const getContracts = (underlying) => {
BinarySocket.send({ contracts_for: underlying }).then((response) => {
Notifications.hide('CONNECTION_ERROR');
processContract(response);
});
};
/*
* Function to display contract form for current underlying
*/
const processContract = (contracts) => {
if (getPropertyValue(contracts, ['error', 'code']) === 'InvalidSymbol') {
Price.processForgetProposals();
getElementById('contract_confirmation_container').style.display = 'block';
getElementById('contracts_list').style.display = 'none';
getElementById('confirmation_message').hide();
const confirmation_error = getElementById('confirmation_error');
confirmation_error.show();
elementInnerHtml(confirmation_error, `${contracts.error.message} <a href="javascript:;" onclick="sessionStorage.removeItem('underlying'); window.location.reload();">${localize('Please reload the page')}</a>`);
return;
}
State.set('is_chart_allowed', !(contracts.contracts_for && contracts.contracts_for.feed_license && contracts.contracts_for.feed_license === 'chartonly'));
getElementById('trading_socket_container').classList.add('show');
const init_logo = getElementById('trading_init_progress');
if (init_logo && init_logo.style.display !== 'none') {
init_logo.style.display = 'none';
Defaults.update();
}
Contract.setContracts(contracts);
const contract_categories = Contract.contractForms();
let formname;
if (Defaults.get('formname') && contract_categories && contract_categories[Defaults.get('formname')]) {
formname = Defaults.get('formname');
} else {
const tree = commonTrading.getContractCategoryTree(contract_categories);
if (tree[0]) {
if (typeof tree[0] === 'object') {
formname = tree[0][1][0];
} else {
formname = tree[0];
}
}
}
commonTrading.displayContractForms('contract_form_name_nav', contract_categories, formname);
processContractForm(formname);
TradingAnalysis.request();
commonTrading.hideFormOverlay();
};
const processContractForm = (formname_to_set = getElementById('contract').value || Defaults.get('formname')) => {
setFormName(formname_to_set);
// get updated formname
Contract.details(Defaults.get('formname'));
StartDates.display();
displayPrediction();
refreshDropdown('#prediction');
displaySelectedTick();
refreshDropdown('#selected_tick');
Lookback.display();
if (!Reset.isReset(Defaults.get('formname'))) {
Reset.hideResetTime();
}
let r1;
if (State.get('is_start_dates_displayed') && Defaults.get('date_start') && Defaults.get('date_start') !== 'now') {
r1 = Durations.onStartDateChange(Defaults.get('date_start'));
if (!r1 || Defaults.get('expiry_type') === 'endtime') Durations.display();
} else {
Durations.display();
}
// needs to be called after durations are populated
displayEquals();
const currency = Defaults.get('currency') || getVisibleElement('currency').value;
const is_crypto = isCryptocurrency(currency);
const amount = is_crypto ? 'amount_crypto' : 'amount';
if (Defaults.get(amount)) {
$('#amount').val(Defaults.get(amount));
} else {
const default_value = getMinPayout(currency);
Defaults.set(amount, default_value);
getElementById('amount').value = default_value;
}
if (Defaults.get('amount_type')) {
commonTrading.selectOption(Defaults.get('amount_type'), getElementById('amount_type'));
} else {
Defaults.set('amount_type', getElementById('amount_type').value);
}
if (Contract.form() === 'callputspread') {
commonTrading.selectOption('payout', getElementById('amount_type'));
refreshDropdown('#amount_type');
// hide stake option
getElementById('stake_option').setVisibility(0);
$('[data-value="stake"]').hide();
} else {
getElementById('stake_option').setVisibility(1);
refreshDropdown('#amount_type');
}
if (Defaults.get('currency')) {
commonTrading.selectOption(Defaults.get('currency'), getVisibleElement('currency'));
}
const expiry_type = Defaults.get('expiry_type') || 'duration';
const make_price_request = onExpiryTypeChange(expiry_type);
if (make_price_request >= 0) {
Price.processPriceRequest();
}
};
const displayPrediction = () => {
const prediction_row = getElementById('prediction_row');
if (Contract.form() === 'digits' && sessionStorage.getItem('formname') !== 'evenodd') {
prediction_row.show();
const prediction = getElementById('prediction');
if (Defaults.get('prediction')) {
commonTrading.selectOption(Defaults.get('prediction'), prediction);
} else {
Defaults.set('prediction', prediction.value);
}
} else {
prediction_row.hide();
Defaults.remove('prediction');
}
};
const displaySelectedTick = () => {
const selected_tick_row = getElementById('selected_tick_row');
const highlowticks_expiry_row = getElementById('highlowticks_expiry_row');
if (sessionStorage.getItem('formname') === 'highlowticks') {
selected_tick_row.show();
highlowticks_expiry_row.show();
const selected_tick = getElementById('selected_tick');
if (Defaults.get('selected_tick')) {
commonTrading.selectOption(Defaults.get('selected_tick'), selected_tick);
} else {
Defaults.set('selected_tick', selected_tick.value);
}
} else {
selected_tick_row.hide();
highlowticks_expiry_row.hide();
Defaults.remove('selected_tick');
}
};
const hasCallPutEqual = (contracts = getPropertyValue(Contract.contracts(), ['contracts_for', 'available']) || []) =>
contracts.find(contract => contract.contract_category === 'callputequal');
const displayEquals = (expiry_type = 'duration') => {
const formname = Defaults.get('formname');
const el_equals = document.getElementById('callputequal');
const durations = getPropertyValue(Contract.durations(), [commonTrading.durationType(Defaults.get('duration_units'))]) || [];
if (/^(callputequal|risefall)$/.test(formname) && (('callputequal' in durations || expiry_type === 'endtime') && hasCallPutEqual())) {
if (+Defaults.get('is_equal')) {
el_equals.checked = true;
}
el_equals.parentElement.setVisibility(1);
} else {
el_equals.parentElement.setVisibility(0);
}
};
const setFormName = (formname) => {
let formname_to_set = formname;
const has_callputequal = hasCallPutEqual();
if (formname_to_set === 'callputequal' && (!has_callputequal || !+Defaults.get('is_equal'))) {
formname_to_set = 'risefall';
} else if (formname_to_set === 'risefall' && has_callputequal && +Defaults.get('is_equal')) {
formname_to_set = 'callputequal';
}
Defaults.set('formname', formname_to_set);
getElementById('contract').setAttribute('value', formname_to_set);
};
const forgetTradingStreams = () => {
Price.processForgetProposals();
Price.processForgetProposalOpenContract();
processForgetTicks();
};
/*
* cancel the current tick stream
* this need to be invoked before makin
*/
const processForgetTicks = () => {
BinarySocket.send({ forget_all: 'ticks' });
};
const onExpiryTypeChange = (value) => {
const $expiry_type = $('#expiry_type');
const validated_value = value && $expiry_type.find(`option[value=${value}]`).length ? value : 'duration';
const is_edge = window.navigator.userAgent.indexOf('Edge') !== -1;
$expiry_type.val(validated_value);
let make_price_request = 0;
if (validated_value === 'endtime') {
Durations.displayEndTime();
if (Defaults.get('expiry_date')) {
// if time changed, proposal will be sent there if not we should send it here
make_price_request = Durations.selectEndDate(moment(Defaults.get('expiry_date'))) ? -1 : 1;
}
Defaults.remove('duration_units', 'duration_amount');
if (is_edge) {
document.getSelection().empty(); // microsoft edge 18 automatically start selecting text when select expiry time after changing expiry type to end time
}
} else {
StartDates.enable();
Durations.display();
if (Defaults.get('duration_units')) {
onDurationUnitChange(Defaults.get('duration_units'));
}
const duration_amount = Defaults.get('duration_amount');
if (duration_amount && duration_amount > $('#duration_minimum').text()) {
$('#duration_amount').val(duration_amount);
}
make_price_request = 1;
Defaults.remove('expiry_date', 'expiry_time', 'end_date');
Durations.validateMinDurationAmount();
}
displayEquals(validated_value);
return make_price_request;
};
const onDurationUnitChange = (value) => {
const $duration_units = $('#duration_units');
if (!value || !$duration_units.find(`option[value=${value}]`).length) {
return 0;
}
$duration_units.val(value);
Defaults.set('duration_units', value);
Durations.selectUnit(value);
Durations.populate();
return 1;
};
return {
displayEquals,
processActiveSymbols,
processMarket,
processContract,
processContractForm,
forgetTradingStreams,
onExpiryTypeChange,
onDurationUnitChange,
};
})();
module.exports = Process;
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,358 |
{"url":"https:\/\/tug.org\/pipermail\/tugindia\/2010-December\/005370.html","text":"David Owen twychicky at gmail.com\nMon Dec 6 02:32:13 CET 2010\n\nHI, Everybody,\nI have two more little problems and will appreciate any help.\n1) I have produced some EPS files starting from the corresponding EXCEL\nfiles, however, when I open the files in Ghostview I see only 3\/4 of the\nfiles! Can some tell me what is wrong and what I should do in order to see\nthe whole figure. However, when I place them in a document they compile\nnicely and I see everything.\n\n2) I am writing another paper and I wish to have all of my figures at the\nend of the document. I included \\usepackage{endfloat} in the\npreamble, however, I get an error:\n\"<................... .ttt>\n<...................... .fff>>\nRunaway argument?\nFile ended while scanning use of \\next.\n<inserted text>\n..... \"\n\nI called the figs with:\n\n\\begin{figure}\n\\begin{center}\n\\scalebox{0.9}{\\includegraphics{p18f1.eps}}\n\\renewcommand{\\figurename}{Fig.}\n\\caption{Population .... multiplicative,~ \\protect\\\\ harvesting rate\nexponential} %%%%% Note \\protect\\\\ statement.\n%%%% See Keith Reckdahl \"Using imported graphics in Latex2\" pages. 51,\n58.\n\\label{fig1}\n\\end{center}\n\\end{figure}\n\nHave I left out something? What have I done wrong?\nAny help will be appreciated.\nThanks,\nD.O.","date":"2023-01-31 03:22:12","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8731086850166321, \"perplexity\": 4995.37813848874}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764499842.81\/warc\/CC-MAIN-20230131023947-20230131053947-00721.warc.gz\"}"} | null | null |
//
// CMRouting.h
// CloudMadeApi
//
// Created by Dmytro Golub on 11/9/09.
// Copyright 2009 CloudMade. All rights reserved.
//
#import <Foundation/Foundation.h>
@protocol CMRoutingDelegate
@end
@interface CMRouting : NSObject {
NSString* apikey; /**< APIKEY */
}
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,032 |
#include "u/fs.h"
int64_t ufsize(ustr_t path) {
#if COMPILER_MSVC
WIN32_FILE_ATTRIBUTE_DATA fileInfo;
if (GetFileAttributesExA(path, GetFileExInfoStandard, (void*)&fileInfo) == 0) return -1;
return (uint64_t)(((__int64)fileInfo.nFileSizeHigh) << 32 ) + fileInfo.nFileSizeLow;
#else
struct stat sb;
if (stat(path, &sb) > 0)
return -1;
return (int64_t) sb.st_size;
#endif
}
bool ufexists(ustr_t path) {
#if COMPILER_MSVC
BOOL isDirectory;
DWORD attributes = GetFileAttributesA(path);
// special directory case to drive the network path check
if (attributes == INVALID_FILE_ATTRIBUTES)
isDirectory = (GetLastError() == ERROR_BAD_NETPATH);
else
isDirectory = (FILE_ATTRIBUTE_DIRECTORY & attributes);
if (isDirectory) {
if (PathIsNetworkPathA(path)) return true;
if (PathIsUNCA(path)) return true;
}
if (PathFileExistsA(path) == 1) return true;
#else
if (access(path, F_OK) == 0)
return true;
#endif
return false;
}
ustr_t ufread(ustr_t path) {
int fd = 0;
off_t fsize = 0;
ssize_t fsize2 = 0;
char *buffer = NULL;
fsize = (size_t) ufsize(path);
if (fsize < 0)
goto abort_read;
fd = open(path, O_RDONLY);
if (fd < 0)
goto abort_read;
buffer = (char *) malloc((size_t) fsize + 1);
if (buffer == NULL)
goto abort_read;
buffer[fsize] = 0;
fsize2 = read(fd, buffer, (size_t) fsize);
if (fsize2 == -1)
goto abort_read;
close(fd);
return ustrn(buffer, (size_t) fsize2);
abort_read:
if (buffer)
free((void *) buffer);
if (fd >= 0)
close(fd);
return NULL;
}
bool ufwrite(ustr_t path, ustr_t buffer) {
#if PLATFORM_WINDOWS
unsigned mode = _S_IWRITE;
#else
unsigned mode = S_IRUSR | S_IWUSR | S_IRGRP | S_IROTH;
#endif
int fd = open(path, O_WRONLY | O_CREAT | O_TRUNC, mode);
size_t len = ustrlen(buffer);
if (fd < 0)
return false;
ssize_t nwrite = write(fd, buffer, len);
close(fd);
return ((size_t) nwrite == len);
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,620 |
Q: factorization of polynomials in $ \mathbb F_7$ I am not sure in finding a factorization of $f(X) = 6 x^6 +1 \in \mathbb F_7$.
I got $f(X) =6\cdot (x-1)\cdot(x+1)\cdot(x-2)\cdot(x+2)\cdot(x-3)\cdot(x+3) $
Is that correct?
A: Yes. The easy version is noticing that $6=-1$, thus $$6x^6+1=-x^6+1$$ By Fermat's little theorem, $x^6-1$ has the six roots $1,2,3,4,5,6$ in $\Bbb F_7$.
A: In $\Bbb{F}_7$ one has on one hand
$$\begin{align}
6x^6+1&=6(x^6-1)\\
&=6(x^2-1)(x^4+x^2+1)\\
&=6(x-1)(x+1)(x^4+x^2+1)
\end{align}$$
On the other hand
$$(x-2)(x+2)(x-3)(x+3)=(x^2-4)(x^2-2)=x^4-6x+8=x^4+x^2+1$$
So the factorisation is correct.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,686 |
We all know that exfoliation can do wonderful things for our complexion; buffs away dead skin cells for a more polished, brighter looking skin-tone, unclogs congested pores to improves skin's texture and promotes cell regeneration for a youthful appearance. However, it's not always as easy to find a great facial exfoliator/scrub especially if you have sensitive skin.
Here's a sensitive-skin friendly, natural skin-polishing refiner that gently removes dull, dead skin cells and impurities without irritation – Jurlique Purely Age-Defying Refining Treatment. It also helps reduce the appearance of fine lines and large pores and enhances cell renewal to help improve skin smoothness and uncover fresh, glowing skin.
Apricot Seed Powder, with its rounded structure, naturally polishes skin gently, without irritation.
Willow Bark Extract, a natural source of Salicylic Acid, helps enhance cell turnover by gently exfoliating the skin, preparing it for the treatments that follow.
Natural Bisabolol and Calendula soothe, soften and hydrate the skin.
This refining treatment has a creamy texture with tiny exfoliating beads in the mix that do not feel harsh or rough on the skin. The directions suggest to apply a pearl sized amount to wet hands and gently massage onto damp skin (for about a minute), avoiding the eye area. And then rinse off with warm water. I love the fresh, invigorating scent and the way this exfoliator leaves my skin looking so smooth and radiant without any signs of irritation or dryness. I use it every other day, right after cleansing my face, rinse, then on to my serum and moisturizer. The product is recommended to be used 2-3 times a week or as often as desired.
I like that this exfoliant contains a natural form of salicylic acid (willow bark) that works deep down in clogged pores to clear away impurities that cause breakouts, blackheads and build-up. This one is great to use after a night of heavier makeup!
Jurlique has some awesome products.
Love their products. Since I have sensitive skin, this is a must try for me. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,983 |
Q: Equivalences for models of spectra I'm sorry if this is well-known. I'm new to stable homotopy theory.
There are many models of spectra, and an answer to this MO post says most of them are equivalent, even equivalent while taking the symmetric monoidal structure (smash product) into account.
My question is motivated by wanting to know how these equivalences are established. There's a clean development of the stable $\infty$-category of spectra and the smash product thereof in Lurie's Higher Algebra, and both are characterized by universal properties, but it seems anachronistic to say that checking these universal properties is the way to establish the equivalence of different models.
Finally, even if one is able to establish that the different model categories of spectra with smash product (symmetric, orthogonal, what have have you) are equivalent, it seems like this doesn't easily construct a functor from one model to another--for instance, it's not obvious how to take two arbitrary Omega spectra (more or less the model in Higher Algebra) and spit out a pair of symmetric spectra while verifying that their smash products in either model are equivalent.
Is there a written account of:
*
*Historical efforts that failed, and in particular, attempted models for spectra that are not equivalent to modern models,
*A list of equivalent models,
*How we know the models are equivalent, and
*When known, functors between the models?
A: The paper you're looking for is "The stable homotopy category is rigid" by Stefan Schwede. It shows that an equivalence on the homotopy category level implies a Quillen equivalence on the model category level.
For monoidal results, check out "Monoidal Uniqueness of Stable Homotopy Theory" by Brooke Shipley. This paper has exactly the same universal property for the model category of spectra that you mention from Lurie, but of course many years before his work. I think it's safe to say these papers of Schwede and Shipley were what Lurie had in mind when he wrote the $\infty$-category version.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,452 |
curl \
--silent \
--data compilation_level=ADVANCED_OPTIMIZATIONS \
--data output_format=json \
--data output_info=errors \
--data output_info=warnings \
--data language=ECMASCRIPT5 \
--data warning_level=verbose \
--data externs_url=https://www.cosmopolite.org/externs/cosmopolite.js \
--data externs_url=https://www.cosmopolite.org/externs/hogfather.js \
--data externs_url=https://raw.githubusercontent.com/google/closure-compiler/master/contrib/externs/google_visualization_api.js \
--data-urlencode "js_externs@externs.js" \
--data-urlencode "js_code@babystats.js" \
http://closure-compiler.appspot.com/compile | ./prettyprint.py
gjslint --strict babystats.js
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Everyone loves a good BBQ and buying the best grill can seem like an overwhelming affair. There's so much choice that it can be paralyzing trying to decide which kind of grill makes best sense.
In today's best gas grills reviews, we'll laser in on 5 of the top-rated grills coming in at a range of price points so your buying decision is simple and streamlined.
We'll get straight down to it with further ado before finishing up with some sound advice on how to get the very best gas grill for your money and personal requirements.
Now, let's go into detail about each of these Best Gas Grill Reviews in 2018.
Weber is a household name when it comes to gas grills. Back in 2014, they tweaked their entire legendary Q Series. In today's Weber Q3200 liquid propane grill review, we'll look at the uprated version of the old Q320.
Convenience should always be uppermost with a gas grill and the last thing you need is equipment that takes an eternity to set up. All you need to do is whip out your Weber, unfold it, pop in a couple of screws and you're ready to blaze up in minutes.
Electronic ignition means you can kiss goodbye to nursing charcoal to life with fire starters then waiting for it to heat up. Just hit a button and the 2 stainless steel burners will fire up fueled by the propane tank so you're able to get straight down to business. You can opt for a natural gas model but we tested the propane version so all information below relates to this model. With the natural gas lines required for the other model, it's not a mobile or portable choice so we rolled with the more multi-purpose propane.
The cooking area is substantial but not suitable for large parties. You'll get more than enough space with the primary cooking surface of 393 square inches for friends and family so keep your expectations realistic and be aware of the size going in. This is extended by a further 75 square inches thanks to the warming rack so you'll enjoy a combined area of 462 square inches. As a guideline, you'll be able to rustle up anything from 15 to 20 burgers at a time which is not such bad capacity.
You might ask yourself if the 21,700 BTU (British thermal units) of power is sufficient. The answer is an unqualified "Yes." Weber designed cast iron grates which do a magnificent job of dispersing and retaining heat meaning you've got more than adequate grunt from the pair of burners.
You can replace one of the split grates with a griddle if you want to get some breakfast on the go. The griddle does not come with the package so you'll need to invest in one separately but it's a nice option to have up your sleeve. It's also well worth your time ordering up some Weber drip trays.
Weighing in at over 80 pounds, this grill is no lightweight. Luckily, it comes along with a nifty wheeled cart so you can move it around hassle-free whether you're in the garden or out camping.
When cooking food for a BBQ, getting the temperature right can be a tiresome affair of trial and error. This is particularly crucial with chicken which can end up giving you food poisoning if it's not properly prepared. The Q3200 boasts the same accurate thermometer as the rest of the Q Series so you can kiss goodbye to guesswork and embrace food cooked exactly as you want it fuss-free.
Upgraded control knobs on the all-new Q3200 allow for seamless and fully flexible fine-tuning of temperature so you can change up or down with infinite gradations offering a wonderfully adaptable cooking experience.
Cleaning up after a full-blooded grilling session can be enough to give you nightmares. The porcelain enamel covering the Weber means you can wipe everything down with ease and relax into your afternoon.
The version we checked out requires a 20-pound propane tank. These tanks are readily available at reasonable cost. You will need to make provision and buy one separately then think about replacements when necessary. Make sure you factor this into your budgeting.
It's also worth thinking about picking up a Weber grill cover to prolong the lifespan of your gas grill.
Considering the robust price, the plastic used for the side shelves is something of a disappointment. Make sure you use them with care to ensure your grill gives you plenty of service without an unfortunate breakage. We really would have expected more from Weber on this count.
You won't be able to use briquettes or lava rock with the Q3200. If you enjoy smoking your food, though, there's a tube to help you get smoking.
If you check out this outstanding gas grill, you'll see that Weber's strapline of "By grillers, for grillers" is no hollow boast. For one of the best gas grills currently available, you could do far worse than the Q3200.
Road test the Weber today and get barbecuing with a vengeance tomorrow.
While the retail price of the Char-Broil Performance 650 is roughly comparable to the Weber, you can currently get an excellent discount putting this hard-hitting gas grill almost in budget territory.
Not everyone wants a tech-driven top-of-the-line grill since everyone has different requirements. If you're on the trail of a more basic gas grill for occasional use, the Char-Broil fits the bill perfectly and is next up in our look at the top-rate grills today.
The Char-Broil is very large and takes up a lot of real estate. Despite the casters it sits on, we'd hardly call this a portable grill. That said, the obvious trade-off is the sheer volume of food you'll be able to produce using the 650. Think about exactly how and when you'll use your grill and decide whether you want something this bulky. It measures up at roughly 45 x 62 x 24 inches and tips the scales at a hulking 127 pounds.
In terms of cooking area, you'll be spoiled with the Char-Broil's exceptional 980 square inches of available surface. With a 650 square inch primary area for which this model is named and a rack offering a further 280 square inches – this folds away to save space – you'll be able to pile on the burgers and hot dogs even if you have a large party to entertain. If you want to rack up 30 burgers with 10 or 15 left heating in the wings, this is one of the best grills you can find.
Surface area is useless without adequate heat and the 5 top-ported burners made from stainless steel combined with the side burner kick out a sterling 60,000 BTUs. Previous models in the Char-Broil range have boasted substantial cooking areas without the power to match. The manufacturer learned from this error and you'll get more than enough oomph with the 650.
It should be noted that the side burner, rated at 10,000 BTU, gives you a little more grunt than you'd expect. As well as simply heating up a sauce, you'll have the scope to cook up some veg or cook up some rice and pasta without taking an eternity.
If you're a fan of nicely seared steaks, you'll appreciate the sear burner. This is a nice touch that separates the Char-Broil 650 from many competitors.
As with every gas grill, you need to think about whether cleaning will induce a headache. The layer of porcelain enamel ensures that blitzing the 650 is as simple as gently scrubbing it down with a damp cloth. The more frequently you clean your grill, the less work you'll have to do each time so we recommend keeping on top of maintenance. This porcelain enamel coating also goes a long way toward stopping your food from sticking to the grill for a double-win.
While it might look the part and appear pretty heavyweight, the 650 is actually lighter than comparable grills due to the thinner and relatively low-grade stainless used in its construction. So you know exactly what you're getting into before committing to purchase, a reasonable estimate would be 2 or 3 years of life from this grill due to build quality so buyer beware.
As with pretty much all propane gas grills on the market, you'll need to get yourself a tank separately. These are not expensive but make sure you take it into account when you're pricing up your purchase. Cater for the cost of ongoing replacements as well.
The Char-Broil Performance 650 is not without its snags but it's a purposeful and substantial grill if you're looking to cater for larger parties at home or out on the campsite.
If you want to invest in a grill that will last for a decade, this might not be your best option. Although it looks built to last and is marketed as highly durable, we would draw your attention to the fact it might well rust out unless treated with kid gloves. Also, the components could be more rugged so think of this as something good for a few seasons, take the price into account and consider the climate where you live before investing. If you live somewhere with little rain and you don't mind buying a cover, it's a great budget grill.
For anyone regularly cooking for large parties after a pocket-friendly gas grill, the Performance 650 is well worth popping on your shortlist.
The Coleman Road Trip, as the name indicates, is marketed primarily as a travel grill. Whether you're camping or tailgating, heading for a picnic or just on a road trip, you'll be able to enjoy delicious barbecued food without needing to spend a fortune.
Most portable grills are… well, just too small. It's senseless having a neat grill you can take anywhere if you're resigned to cooking burgers two at a time then running out of gas after 10 minutes. There's no such nonsense with the Coleman. 285 square inches of cooking area means you can feed 3 or 4 people with ease. Since the burners kick out 20,000 BTU, you'll have more than enough power on tap. This is accomplished through a pair of adjustable burners affording you plenty of cooking flexibility.
The Coleman also scores in terms of size and weight. It's a slight 50 pounds and measures around 19 x 13 x 33 inches so it's a nimble little grill. This is further enhanced by its collapsible nature so you can break it down and pop it in your trunk then get up and running in an instant when you're ready to eat. An oversized handle and wheels simplify moving it around further.
Coleman's proprietary InstaStart system makes sure you won't be scrabbling around for matches when you've got steaks and hot dogs on your mind. Push the button and you're good to go for 60 minutes straight.
Most grills tend to come in either black or silver with no further options available. The Coleman allows you to choose from 8 vibrant color schemes which is a nice touch, particularly with a cheaper gas grill. There's a slight price discrepancy according to color so, if you're on the hunt for a real bargain, go with the red for best value.
Cleaning is something of a mixed bag. Coleman have made a series of design improvements over previous iterations of this grill. The cast iron grill plates themselves present no problem. They are coated with a sheen of porcelain enamel and all you need to do is wipe them down or soak them with no need to mess around seasoning them like with some grills. The grease trap goes a long way toward reducing clean up too. The more tiresome element is the need to clean the whole affair after every use. Use some gentle soap and warm water and make sure to dry everything afterward. One handy hint to help keep the mess from building up is to put a foil lining inside lessening your workload for a few cents.
Build quality has definitely surged upward with this new model. Coleman have addressed inexcusable issues like handles or lids that melt and you'll get a solid, dependable grill fit to last. You will need to take care of it, though. The exterior picks up scratches easily while the inside can rust out in a flash if you don't look after it. That said, the superb 5-year warranty should give you plenty of confidence if you're unsure about buying this gas grill.
Make sure you get yourself a supply of propane gas tanks as these are not supplied with the grill. If you're looking for other optional extras, you can also opt for a stove or griddle surface to expand your cooking repertoire at a reasonable cost.
While portability and maneuverability are uppermost, this doesn't come at the expense of grilling space or heat output. Considering the ultra-low price tag, it's incredible that you can get such a powerhouse of a grill capable of churning out a few hours of BBQ food on a single cyclinder.
Coleman's fantastic 5-year limited warranty ices the cake if you're in any doubt about this gas grill.
For anyone who frequently cooks on the go in any capacity, we can't recommend the Coleman Road Trip LXE strongly enough.
Don't let the price fool you… This is a top-notch gas grill and well worth the modest investment.
We'll step things up a gear in terms of performance and price range with the magnificent Royal Gourmet Mirage MG6001-R.
This is a leading candidate for the very best grill money can buy.
Where the Coleman redefines grill portability, you're certainly not going to be popping the Mirage in the trunk for a quick trip to the beach or campsite. It weighs over 200 pounds and measures approximately 49 x 24 x 75 inches so it's a full-blooded grill to grace any permanent set-up in the garden.
The cooking surface is equally man-sized. The primary grill area is just under 700 square inches and you'll also benefit from an additional cooking area with side burners so even if you host large garden parties, you can keep the food coming seamlessly in large quantities.
It's not just size where you get what you pay for with the Mirage. Porcelain grates will wick the flavor away inside your meat making certain you get impeccable quality as well as quantity with this grill.
You'll get a good variety with the 6 burners at your disposal. The main tube burners are made from stainless steel while the brass side burner lends further versatility to this capable grill. The added bonus is an infrared back burner so you've got all bases covered in a grill that's dependable, high-performing and able to deliver catering-sized quantities of food without impairing the quality.
Once you've got your burgers and steaks on the go, you'll benefit from the stainless steel lids and fireboxes coated with porcelain. Heat is beautifully retained and you'll realize where the money you spent went. Build quality overall is outstanding and the Royal Gourmet Mirage is a grill built to last for years. Little touches like aluminum caps on the end of the lids to prevent damage or breakage show a welcome attention to detail.
You won't need to keep grabbing a separate thermometer to check your chicken is cooked through and through. The integrated units are accurate and convenient.
Thinking of convenience, the tool and utensil hooks stop you from working in a cluttered space and you'll have everything you need in arm's reach.
Controls are really user-friendly and mounted on the front of the grill for easy access.
There's a storage cabinet front and center, too. The steel doors cover a cavernous space where you can stash away all your BBQ accessories, cutlery and crockery.
When it comes time to clean your grill down, you'll be glad of the porcelain enamel coating meaning there's no stubborn food debris to deal with. Just wipe it down and do so regularly to prolong the lifespan of your grill and keep things hygienic into the bargain.
If you have a fluid budget and appreciate true quality, the Royal Gourmet Mirage MG6001-R is, quite simply, among the best gas grills currently on the market.
While not everyone wants a monstrous grill capable of churning out party-sized volumes of food, for those who do, the Mirage is just about perfect. Other than the price and lack of a cover, there's nothing much we can come up with to knock this grill and it excels in every capacity.
Check out the Royal Gourmet Mirage and get ready for many seasons of happy garden parties and BBQs in return.
As we reach the end of our best grill reviews, we've got something occupying the middle ground in terms of size, performance and price.
We always try to offer you a broad spread of products to choose from rather than ploughing the same furrow with 5 almost identical pieces of kit.
The Char-Broil Performance 475 is another fantastic propane-fueled gas grill from a brand you can trust. Char-Broil has been in business for more than half a century and is one of America's most loved and popular BBQ brands.
Before you get started, assembly is a trifle awkward. Don't get us wrong, it's not going to take an entire working day but you won't be searing any burgers straight out the box. This may or may not be an issue for you. If putting things together is not your strong suit, consider calling in some help rather than struggling.
Getting the grill started is as simple as pushing a button so you won't be chasing down a lighter or trying to light matches in the wind.
Once you're up and away, there's more than enough room on the grill to flop out 20 or more burgers in a single sitting. This should be ample for the average family. This is achieved with a primary cooking space measuring 475 square inches coupled with a secondary warming rack of 175 square inches so you can keep the food coming without making your guests gnaw their fingers.
The stainless steel burners and heat tents deliver 36,000 BTUs of heat so, even though you have spacious cooking surfaces, you won't be sold short on power. The 10,000 BTU side burner is strong enough to do more than just pre-heat some sauce. The only snag is that you'll need to remove this if you want to use a larger external propane tank.
Although it weighs in at around 100 pounds, the wheeled cart design allows you to move the grill without straining yourself.
The grease tray slides out for cleaning purposes and it's a great way to minimize the mess that can otherwise clog up your grill in no time. Porcelain enamel coating, like you find on all the best gas grills, further lessens the way food debris can otherwise accumulate. Cleaning demands nothing more than a damp cloth. Wipe down your grill on a regular basis and that's as far as clean-up and maintenance goes. There's little worse than buying a BBQ which becomes an ongoing nightmare when it's time to scrub down and pack away.
Char-Broil offers a class-leading warranty. If you zoom in on the minuscule print of the guarantee, you'll see that the burners, firebox and lid are backed for 5 years from the date of purchase. All other parts are covered by a 1-year limited warranty so you can eliminate thoughts of expensive repairs.
While no grill is perfect, the very minor niggles with the Char-Broil are fairly insignificant and it really is a first-class grill at a very keen price.
If you're looking for one of the best gas grills without needing to take out a bank loan, the 475 merits a place on any shortlist.
This cart-style grill works best when set up at home but, if you have a larger vehicle and fancy taking it with you on the road, it's just about small enough for you to get away with it while large enough to make it worthwhile.
Check out the Char-Broil Performance 475 for a no-nonsense, basic grill ideal for moderate barbecuing jobs.
Now we've given you 5 of the top-rated grills to mull over, what should you look for when you're buying one of these super-convenient BBQ solutions?
Barbecue grills have become something of a fixture in American backyards while gatherings centered on the food cooked there are a large part of summertime tradition.
Almost everyone loves a good steak, burger or hot dog blazed up on the grill but there is continued debate over what type of grilling is best.
Are you a charcoal friend or do you prefer the convenience of a gas grill?
Charcoal versus gas has been a heated backyard debate for as long as both grills have been around and it's not likely to be resolved any time soon.
Gas is incredibly easy to use and it has many benefits. Some diehard charcoal grill enthusiasts claim there are many drawbacks to gas grills but we would argue that there are just as many cons to using a charcoal grill.
Cost: Cost is where a charcoal grill is the clear-cut winner. A basic charcoal grill is extremely inexpensive and will give you deliciously cooked foods while also lasting the distance.
Gas grills cost substantially more because of their complexity. You can get cheap gas grills but you generally get what you pay for. Depending on what you expect from your gas grill, you can spend anywhere from a few hundred dollars to a few thousand dollars. In return, you'll get a dependable grill that will bring you many seasons of enjoyment.
Wait Time: With preheating times, gas grills come out on top. All you have to do is ignite the grill with the turn of a knob, close the lid to allow the interior to heat and you'll be good to get grilling in 10 minutes or so. The grill will stay hot and ready to cook for as long as you need it.
On the charcoal side, you have to build the stack, light the fuel and wait. And then wait some more. And some more. Charcoal grills won't be ready to cook on for at least 30 minutes after you light them and, once the fuel is lighted, you are on a timer. Once the coals begin to burn out, you are done cooking. On this point there's simply no contest.
Temperature Range: Another category in which charcoal grills come up trumps concerns the breadth of temperature at your disposal. With a charcoal grill, the temperatures can range from as low as you want them to as high as 1200 degrees Fahrenheit. This extreme heat output allows for incredible searing power.
Gas grills, on the other hand, have a temperature range of 225 degrees to 600 degrees Fahrenheit depending on the model. Gas may not be the champion at searing, but there aren't many things that call for searing.
Controlling the Temperature: This is one area in which gas absolutely outshines charcoal and this is a crucial element of barbecuing. When using a gas grill, you can exercise complete control over the temperature your grill remains at and you can adjust it as needed. Get your burners to the temperature you need to cook your food and then relax with your guests. The temperature will hold steady until you turn it off or the gas runs out.
Charcoal is a different beast altogether when it comes to maintaining the temperature. When using a charcoal grill, you need to make sure your charcoal bricks are relatively even in size to prevent some from burning up faster than others leaving you with an uneven heat. You must arrange the charcoal precisely to ensure the heat remains consistent and then you have to adjust the flow of oxygen through the grill to make sure the fire is also even. You have to stay near the grill the entire time and you need to work quickly… Once the fire goes out, you have to start all over.
Flavor and Smoking: When cooking something for a long period of time, like when smoking meat, charcoal grills work best. Not only is it relatively simple to smoke foods on a charcoal grill but, because of the way charcoal burns, your food will be brimming with flavor.
This isn't to say it's impossible to smoke foods on a gas grill, it's just a little more complicated and you will undeniably get less flavor.
Most people can't tell if a burger or hot dog was cooked on gas or charcoal, but for other foods like brisket or ribs, the flavor from charcoal is distinct. Think about your intended usage and buy accordingly.
Cleaning Up: Gas grills are a cinch to maintain. Cleanup is straightforward. All you have to do is turn the knob off and the flame goes out. Occasionally empty the grease from the collection tray and make sure you scrub down the cooking grates every now and then and you're all set.
When you are through using a charcoal grill, you have to let the charcoal burn out or you will need to suffocate the flame and wait for the grill to cool down. Once cooled, you have to find a way to dispose of all the ash from the burned charcoal. You will need to scrub the built-up grease from the cooking grates each time you use the grill.
As you can see, there are many benefits and drawbacks to both types of grills so it's a simple question of deciding which works best for you rather than seeking a right or wrong answer.
Depending on what you are cooking and what your end goal is, there are pros and cons to both charcoal and gas grills. Gas is the standout winner when it comes to convenience, though.
Even among the gas grill category, there's a vast array of choice and quality should always be uppermost. It doesn't matter if you are purchasing your first grill for your debut backyard BBQ or you are known through the subdivision as the king of the grill, there truly is something for everyone.
There are a variety of features that grill manufacturers put into their products but which are worth paying for?
What do all the numbers and specifications mean? How can you possibly know which grill is the best option for you?
As we stated earlier, gas grills can cost a pretty penny but, as with most things in life, you get what you pay for.
If you opt for a cheaper gas grill from your local big box retailer, you are likely to get a machine that will manage the basics but probably won't last for more than a season or two.
This doesn't mean it's necessary to drop $5000 for a high-end grill unless you really want to but make sure you take quality into account before making your purchase.
Are you planning to cook a few hot dogs for your small family or will you be using the grill to cook dozens of burgers for large get-togethers?
Make sure you have enough space in your yard for the grill of your choosing as well. You want it to be able to be set up a safe distance from your house or fencing while being used.
Pay attention to what the grill is made out of. The last thing you need is a cheaply made grill that is going to rust and fall apart before the summer ends. That's money wasted rather than saved.
Look for quality materials like cast aluminum and stainless steel, and check the frame of the grill not just the cooking area. The cooking grates need to be made of decent metal as well.
Buying a grill made from good materials will help ensure the durability of the grill and so save you money over time.
Many gas grills come with an impressive list of features but do you really need to pay for all of that?
Well, it depends on what exactly you are doing with your grill. The typical backyard barbecue doesn't need a lot of bells and whistles.
You really shouldn't pay more for features that you won't use and remember that all these extras are just more things that can break down as well as bumping the price up.
Honestly analyze your needs and buy with these firmly in mind rather than being enticed into unnecessary frippery.
We've been talking about gas grills and you have 2 main choices here, propane grills or natural gas grills.
Some grills are sold as dual-fuel grills meaning you can use them with either propane or natural gas.
To use natural gas, though, you need to purchase an adapter kit and have the grill connected to the natural gas line in your home. Once this is done and the grill is connected, it will no longer be portable but natural gas is more affordable than purchasing propane tanks several times a year.
Think about whether you prize portability or economy and make the best choice in line with this.
Some grills use a battery to ignite the fuel while others create a spark that ignites the gas. Neither option is more beneficial than the other here aside from the added cost of a battery. Just choose whichever you prefer.
What Else Do You Need To Think About When Buying a Gas Grill?
You should always buy the grill from a reputable seller. Don't opt for a cheaper store brand in the hope of shaving off a few dollars as they are less likely to have warranties and aftercare available should something happen to the grill. You can certainly get hold of inexpensive gas grills which is precisely why we have reviewed some solid budget options that won't let you down.
Make sure you leave room in the budget for other grilling essentials like propane tanks, grilling tools and a grill cover. You should always keep an extra propane tank on hand as having to go on a fuel run in the middle of cooking is the height of tedium.
A good grill cover will help protect your grill from the elements and ensure it lasts for many happy years.
Gas grills are increasingly popular despite the cost. The main reason for this is that they are so much easier to use than their charcoal cousins.
Gas grills operate a little differently, though, and it's all too easy to make some schoolboy errors when using them. Use your grill correctly and you'll get perfect barbecued food every time. Your equipment will last longer into the bargain.
Keep The Lid Shut When Cooking: Yes, you need to occasionally check on the food to make sure it isn't burning and you also need to turn it promote even cooking. However, every time you open the lid, you are letting heat escape and your food will take that much longer to cook.
We trust you've found this exploration of the best gas grills along with the grill reviews has given you all the ammunition you need to get the cooking solution ideal for you and your family.
Don't hesitate to reach out if you have any feedback or queries. We love to hear from our readers and we're always delighted to help in any way we can.
Come back soon for more home and garden-related gems.
Now go and fire up the barbie!
Best Microwave Oven 2018: What Are The Top Microwaves On The Market? | {
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I'm a Digital Designer based in Brooklyn. I create visual systems for user interfaces and brand identities. I use my passion for concept development and storytelling to define meaningful brands, products or services. | {
"redpajama_set_name": "RedPajamaC4"
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Produced by Chris Curnow, Matthew Wheaton and the Online
Distributed Proofreading Team at http://www.pgdp.net (This
file was produced from images generously made available
by The Internet Archive)
[Illustration: JETHRO WOOD.]
JETHRO WOOD, INVENTOR OF THE MODERN PLOW.
A BRIEF ACCOUNT OF HIS LIFE, SERVICES, AND TRIALS; TOGETHER WITH
FACTS SUBSEQUENT TO HIS DEATH, AND INCIDENT TO HIS GREAT
INVENTION.
"No citizen of the United States has conferred greater
economical benefits on his country than Jethro Wood--none of her
benefactors have been more inadequately rewarded."--_Wm. H.
Seward._
BY FRANK GILBERT.
CHICAGO:
RHODES & McCLURE.
1882.
Entered according to Act of Congress, in the year 1882.
By I. U. KIRTLAND,
In the Office of the Librarian of Congress, at Washington.
STEREOTYPED AND PRINTED
BY
THE CHICAGO LEGAL NEWS CO.
[Illustration: FAC-SIMILE OF THE ORIGINAL WOOD PLOW.]
EXPLANATION OF THE FOREGOING FAC-SIMILE.
SIDE VIEW of Plough. _A_ Mould-board, the form of which is claimed as
new. _B_ Share claimed. _C_ Standard claimed. _DD_ Screw-bolt, and not
confining the beam to the Standard. _a_, _b_, _c_, _d_, _e_, the 1st,
2d, 3d, 4th and 5th sides mentioned in the specification. _g_, _g_.
Excavation at the fore part of the mould-board to receive the share
which fills it up and forms an even surface. _h_ Hole to receive the
knob or head cast on the under side of the share, which, on being
shoved up to its place, nooks under the mould-board at the upper side
of the hole, and is held in its place by a wooden wedge driven between
the knob and the lower side of the hole. _f_ Notches in the Standard
to receive the latch i in elevating or depressing the beam. _s_, _t_,
_v_. Straight diagonal lines touching the mould-board the whole
distance. _u_ Vertical or plumb line touching the mould-board from top
to bottom. _H_ Reverse side of the share. _x_ Knob to hold it fast to
the mould-board. _y_ Side view of knob. _zz_ Shiplaps fitting under
the point and edge of the mould-board. _k_ Another form of standard
keyed on top of beam. Fig. 2d, landside view: _E_ The "landside". _F_
part of landside cast with mould-board. _mm_ Cast loops to hold the
handles claimed. _n_ Head of screw-bolt held by a shoulder made by a
projection from the mould-board and standard, through which the bolt
passes up to the beam. _o_ Share claimed. _p_ Shiplap claimed. _G_
Inside view of landside. _r_ Tennon at forward end to fit into a
dovetailed mortice on the inside of that part which is cast with the
mould-board.
PREFACE.
The immediate occasion of this little volume was a malignant
misrepresentation from the pen of Ben: Perley Poore. With slight
variation from the original text, the words of Thomas Jefferson about
Benjamin Franklin and his maligners, quoted in the body of this
monograph, apply to this case: I have seen with extreme indignation
the blasphemies lately vended against the memory of the father of the
American plow. But his memory will be venerated as long as furrows are
turned and soil tilled. The present object, however, is not so much to
refute falsehood as to establish the truth, and make it a part of the
permanent knowledge of the public. To the extent that this object
shall be attained, will these labors be rewarded.
It is not the design of this publication to disparage any one; on the
contrary, it is desired to give ample credit to all who contributed to
the solution of the plow problem. If only brief mention is made of
others, it is because they really deserved but little credit, or their
merits are forever buried in obscurity. It is proposed to set forth
without exaggeration, the claims of the supreme inventor in this line
to the grateful remembrance of the public. And by the public is meant
not only the American people, but all who are fed from the ample
granaries of this country, or share the benefits of the improved
tillage, whether on this continent or in Europe, made possible and
actual by the inventive genius of Jethro Wood.
JETHRO WOOD;
INVENTOR OF THE MODERN PLOW.
The last words ever penned by John Quincy Adams were these, written in
the peculiarly tremulous hand of "the Old Man Eloquent:" "Mr. J. Q.
Adams presents his compliments to the Misses Wood, and will be happy
to see them at his house, at their convenience, any morning between 10
and 11 o'clock." This note was found upon his desk when he was
stricken down with paralysis, February 21, 1848, in his seat in the
House of Representatives. The Misses Wood here referred to were the
daughters of Jethro Wood, then deceased. They were at that time
engaged in a labor of love, and the venerable Ex-President was their
friend therein. Prompted more by filial affection than by hope of
gain, they were making a final effort to secure from Congress a proper
recognition of their father's claim as an inventor. It is entirely
safe to say that if Mr. Adams had been spared to the end of the
Congress then in session, that claim would have been then duly
recognized, and the name, services and genius of Jethro Wood become
familiar to the American public.
Jethro Wood was born at Dartmouth, Massachusetts, on the sixteenth day
of the third month of 1774. His parents were members of the Society of
Friends. His mother, Dinah Hussey Wood, was a niece of Ann Starbuck, a
woman of remarkable ability and high standing in colonial annals. Ann
Starbuck was virtually governor of Nantucket. The niece was a woman of
excellent intellect, and most winsome character. Her conversation
sparkled with genial wit and good cheer. Her husband, John Wood, was
a man of sterling worth, calm, self-poised, strong willed, and
eminently influential. Jethro was their only son. On New Years Day,
1793, he was married to Sylvia Howland, at White Creek, Washington
County, New York. The fruit of this marriage, every way a happy one,
was a family of six children, namely: Benjamin; John; Maria, wife of
Jeremiah Foote; Phoebe; Sarah, wife of Robert R. Underhill; Sylvia
Ann, wife of Benjamin Gould. Of these children the only survivor is
Mrs. Gould, who with her sister, Phoebe, were the Misses Wood of the
Adams note. So much for the domestic setting of this diamond of
inventive genius.
Even as a boy, Jethro Wood showed plainly the drift and trend of his
mind. The child was indeed "father of the man," and almost from the
cradle to the grave, he was an inventor. In his childish plays he
seemed busied with the idea which he ultimately perfected. Many
curious incidents and memories are treasured among the traditions of
his neighbors and friends. "When only a few years old," writes a
venerable man whose recollection spans two generations, "he moulded a
little plow from metal, which he obtained by melting a pewter cup.
Then, cutting the buckles from a set of braces, he made a miniature
harness with which he fastened the family cat to his tiny plow, and
endeavored to drive her about the flower-garden. The good
old-fashioned whipping he received for this 'mischief,' was such as to
drive all desire for repeating the experiment out of his juvenile
head."
Such innate and ruling passion might be suppressed, but could not be
subdued. As his mind matured, his thoughts took definite shape. His
home was always upon a farm, but he was never a farmer, in the sense
of Poor Richard's homely couplet:
"He who by the plow would thrive,
Himself must either hold or drive."
Born in comparative affluence, blessed with a good education, an ample
library and a well equipped workshop, enjoying the correspondence of
such men as Thomas Jefferson and David Thomas, he was unremitting in
his endeavor to realize his ideal. "His chief desire," to quote
further from our venerable correspondent, "was to invent a new
mold-board, which, from its form, should meet the least resistance
from the soil, and which could be made with share and standard,
entirely of cast iron." To hit upon the exact shape for the mold-board
he whittled away, day after day, until his neighbors, who thought him
mad on the subject, gave him the soubriquet of the "whittling Yankee."
His custom was to take a large oblong potato which was easy for the
knife, and cut it till he obtained what he fancied was the exact
curve.
The manhood home of Jethro Wood was at Scipio, Cayuga County, New
York, a purely agricultural town, with nothing in its later history to
distinguish it; but in its palmier early days of the present century,
it must have been a nursery of invention. Roswell Toulsby, Horace
Pease, and John Swan, of that town, each took out letters patent for
improvements in plows, and that prior to the issuance of any patent to
Mr. Wood. Their improvements were of no practical value, and played no
part in the development of this branch of mechanism, but their efforts
serve to show the state of the intellectual atmosphere breathed by the
man who was destined to solve the knotty problem which underlies the
very foundation of scientific agriculture.
Of the cotemporaries of Mr. Wood, who wrought at the solution of this
problem, the most illustrious was Thomas Jefferson, statesman,
philosopher and farmer.
In one of his letters to Jethro Wood, Mr. Jefferson spoke of his own
labors in that direction, as the experiments of one whiling away a few
idle hours, but herein he did himself injustice. His efforts, however,
were far from exhaustive in their results, and it was with good reason
that he urged Mr. Wood to go forward in his undertaking, and no doubt
he was perfectly sincere in wishing him success. His correspondence,
as published in nine large volumes, attests his long and deep interest
in the problem, which it was reserved for Jethro Wood to solve. Having
carefully examined those volumes, to glean all there is in them on
this subject, I herewith append the observations found, for besides
being in themselves interesting, in view of their authorship, they
throw important light upon the general subject.
Under date of July 3, 1796, Mr. Jefferson wrote to Jonathan Williams:
"You wish me to present to the Philosophical Society the result of my
philosophical researches since my retirement. But, my good Sir, I have
made researches into nothing but what is connected with agriculture.
In this way I have a little matter to communicate, and will do it ere
long. It is the form of a mould-board of _least resistance_. I had
some years ago conceived the principle of it, and I explained it then
to Mr. Rittenhouse. I have since reduced the thing to practice, and
have reason to believe the theory fully confirmed. I only wish for one
of those instruments used in England for measuring force exerted in
the drafts of different ploughs, etc., that I might compare the
resistance of my mould-board with that of others. But these
instruments are not to be had here. In a letter of this date to Mr.
Rittenhouse I mention a discovery in animal history, very signal
indeed, of which I shall lay before the society the best account I
can, as soon as I shall have received some other materials collecting
for me.
"I have seen, with extreme indignation, the blasphemies lately vended
against the memory of the father of American philosophy. But his
memory will be venerated as long as the thunder of heaven shall be
heard or feared."
March 27, 1798, Jefferson wrote to Mr. Patterson: "In the life time of
Mr. Rittenhouse, I communicated to him the description of a
mould-board of a plough, which I had constructed, and supposed to be
what we might term the _mould-board of least resistance_. I asked not
only his opinion, but that he would submit it to you also. After he
had considered it he gave me his own opinion that it was
demonstratively what I had supposed, and I think he said he had
communicated it to you. Of that however, I am not sure, and therefore,
now take the liberty of sending you a description of it, and a model
which I have prepared for the Board of Agriculture of England, at
their request. Mr. Strickland, one of their members, had seen the
model, also the thing itself in use on my farm, and thinking favorably
of it, had mentioned it to them. My purpose in troubling you with it
is to ask you to examine the description rigorously, and suggest to me
any corrections or alterations which you may think necessary. I would
wish to have the idea go as correctly as possible out of my hands. I
had sometimes thought of giving it into the Philosophical Society, but
I doubted whether it was worthy of their notice, and supposed it not
exactly in the line of their publications. I had therefore
contemplated sending it to some of our agricultural societies, in
whose way it was more particularly, when I received the request of the
English board. The papers I enclose you are the latter part of a
letter to Sir John Sinclair, their president. It is to go off by
packett, wherefore I wish to ask the favor of you to return them with
the model in the course of the present week, with any observations you
will be so good as to favor me with."
Writing from Washington, July 15, 1808, to Mr. Sylvestre, in
acknowledgment of a plow received from the Agricultural Society of the
Seine (France), he adds: "I shall with great pleasure attend to the
construction and transmission to the society of a plough with my
mould-board. This is the only part of that useful instrument to which
I have paid any particular attention. But knowing how much the
perfection of the plough must depend, 1st, on the line of traction;
2d, on the direction of the share; 3d, on the angle of the wing; 4th,
on the form of the mould-board; and persuaded that I shall find the
three first advantages eminently exemplified in that which the society
sends me, I am anxious to see combined with these a mould-board of my
form, in the hope it will still advance the perfection of that
machine. But for this I must ask time till I am relieved from the
cares which have more right to all my time--that is to say, till next
spring;" _i. e._ until after the expiration of his second term as
President of the United States.
The importance of any step in civilization can be understood only in
its relations, antecedent causes and actual results.
The _Scientific American_, which is certainly good authority in such
matters, ranks Jethro Wood with Benjamin Franklin, Eli Whitney, Robert
Fulton, Charles Goodyear, Samuel B. Morse, Elias Howe, and Cyrus H.
McCormick, and these are certainly the great names and this a just
classification. Each in his way laid the foundation on which all
inventors in his respective line have built, and must continue to
build, and none of them all came so near perfecting his grand idea as
Mr. Wood. His now venerable daughter stated the exact truth when she
remarked in a letter not designed for publication: "My father patented
the shape and construction of the plow. He took the iron and shaped
the plow that turns the furrow for every product of the soil in
America. His plow has never been improved. It came from his hand
simple and perfect, as it now is, and there is no other plow now in
use." It was not the use of cast iron that he invented, although the
use of "pot metal" by him occasioned a great deal of hostility to the
original Wood plow.
Jethro Wood took out two plow patents, and those who wish to belittle
his work, descant upon the first as if it were his only claim to
credit. That first patent was issued in 1814. It fell far short of
satisfying the patentee's ambition. The plows made under it must have
been a great improvement on any then in use, for although he
abandoned it almost from the first, a great many of them were sold
during the period between the first and the second patents. The second
patent dates from 1819. The natal day of the modern plow may be fairly
set down as September 1, 1819. The original specifications in this
plow deserve to be given in full, and may well be inserted in this
connection. The document was the handiwork of Mr. Wood himself, and
runs thus:
"The Schedule referred to in these Letters Patent, and making part of
the same, containing a description in the words of the said Jethro
Wood himself of his improvement in the construction of Ploughs.
"Considering the manifold errors and defects in the construction of
Ploughs, and the inconveniences experienced in the use of them, the
petitioner and inventor hath applied the powers of his mind to the
improvement of this noble utensil, and produced a Plough so far
superior to those in common use, that he asks an exclusive privilege
for the same from the government of his country.
"The principal matters for which he solicits Letters Patent, he now
reduces to writing, and explains in words and sentences as appropriate
and significant as he possibly can. But, being perfectly aware of the
feebleness and insufficiency of language to convey precise and
adequate ideas of complicated forms and proportions, the said Jethro
Wood annexes to these presents, a delineation upon paper of his said
new and improved Plough, with full and explanatory notes; urging with
earnestness and respect that the delineation and notes may be
considered as a part of this communication. The said petitioner and
inventor also, being perfectly convinced, as a practical man, that a
model of his inventions and improvements will convey and preserve the
most exact and durable impressions of the matters to which he lays
claim, he sends herewith a model of the due form and proportion of
each, as a just exhibition of his principle and of its application to
the construction and improvement of the Plough, requesting that the
same may be kept in the Patent Office, as a perpetual memorial of the
invention and its use.
"In the first place, the said Jethro Wood claims an exclusive
privilege for constructing the part of the Plough, heretofore, and to
this day, generally called the mould-board, _in the manner hereinafter
mentioned_. This mould-board may be termed a plano-curvilinear figure,
not defined nor described in any of the elementary books of geometry
or mathematics. But an idea may be conceived of it thus:
"The land-side of the Plough, measuring from the point of the
mould-board, is two feet and two inches long. It is a strait-lined
surface, from four to five and one-half inches wide, and half an inch
thick. Its more particular description will be hereinafterwards given.
It is sufficient to observe here, that of the twenty-six inches of
length on the land-side, eighteen inches belong to the part of the
Plough strictly called the land-side, and eight inches to the
mould-board. The part of the mould-board comprehended by this space of
eight inches is very important, affording weight and strength and
substance to the Plough; enabling it the better to sustain the
cutting-edge for separating and elevating the soil or sward, and
likewise the standard for connecting the mould-board with the beam, as
will hereinafter be described more at large.
"The figure of the mould-board, as observed from the furrow-side, is a
sort of irregular pentagon, or five-sided plane, though curved and
inclined in a peculiar manner. Its two lower sides touch the ground,
or are intended to do so, while the three other sides enter into the
composition of the oblique, or slanting mould-board, over-hanging
behind, vertical midway, and projecting forward. The angle of the
mould-board, as it departs from the foremost point of, or at, the
land-side, is about forty-two degrees, and the length of it, or, in
other words, of the first side, is eleven inches. The line of the
next, or the second side, is nearly, but not exactly parallel with the
before-mentioned right-lined land-side, for it widens or diverges from
the angle at which the first and second sides join towards its
posterior or hindermost point, as much as one inch. Hence, the
distance from the hindermost point of the mould-board, at the angle of
the second and third sides, directly across to the land-side, is one
inch more than it is from the angle of the first and second sides,
directly across. The length of this, the second side, is eight inches.
The next side, or what is here denominated the third side, leaves the
ground or furrow in a slanting direction backward, and with an
over-hanging curve, exceeding the perpendicular outwards from three to
six inches, according to the size of the Plough. The length of this
third side is fourteen inches and one-half. The fourth side of this
mould-board is horizontal, or nearly so, extending from the uppermost
point of the third side, to the fore part, or pitch, eighteen inches.
The fifth, or last side, descends or <DW72>s from the last mentioned
mark, spot, or pitch, to the place of beginning at the low and fore
point of the mould-board, where it joins the land-side. Its length is
thirteen inches.
"Besides these properties and proportions of his mould-board, the said
Jethro Wood now explains other properties which it possesses, and by
which it may be and is distinguished from every other invented thing.
The peculiar curve has been compared to that of the screw auger; and
it has been likened to the prow of a ship. Neither of these
similitudes conveys the fair and proper notion of the invention.
"The mould-board, which the said Jethro Wood claims as his own, and
which is the result of profound reflection and of numberless
experiments, is a sort of plano-curvilinear surface, as herein-before
stated, having the following bearings and relations: A right line,
drawn by a chalked string or cord, or by a straight rule, diagonally
or obliquely upwards and backwards from a point two inches and a half
inch above the tip or extremity of the mould-board to the angle where
the third and fourth sides of the mould-board join, touches the
surface the whole distance, in an even and uniform application, and
leaves no sinking, depression, hole, cavity, rising, lump, or
protuberance, in any part of the distance. So, at a distance half way
between the diagonal line just described, and the angle between the
first and second sides, a line drawn parallel to the diagonal line
already mentioned will receive the chalked string or cord, or the
straight rule, as on an uniform and even surface without the smallest
bend, sinuosity, or bunch, whereby earth might adhere to the
mould-board, and impede the motion and progress of the Plough, under,
through and along the soil.
"In like manner, if a point be taken one inch behind the angle
connecting the second and third sides, and a perpendicular be raised
upon it, that perpendicular will coincide with the vertical portion of
the mould-board in that place; or, in other words, if a plumb line be
let fall so as to reach a point one inch behind the last mentioned
angle, then such a plumb line will hang parallel with the mould-board
the whole way; the line of the mould-board there, neither projecting
nor receding but being both a right line and a perpendicular line.
"Moreover, if a right line be drawn from a point on the just described
perpendicular, an inch, or thereabouts, above the upper margin of the
fourth side, and from the point to which the said perpendicular, if
continued, would reach; if, the said Jethro Wood repeats, a right line
be drawn downward and forward, not exactly parallel to the diagonal
herein already described, but so diverging from the same that it is
one inch more distant or further apart, at its termination on the
fifth side of the mould-board, than at its origin or place of
beginning; such line, so beginning, continued, and ended, is a right
line parallel to the mould-board along its whole course and direction,
and the space over which it passes has no inequality, hill, or hollow
thereabout.
"Furthermore, an additional property of his mould-board is, that, if
it be measured and proved various ways, vertically and obliquely, by
the saw in fashioning it, by the rule in meeting it, and by the
chalk-line in determining it, the capital and distinguishing character
of right lines existing on, over and along the peculiar curve which
his mould-board describes, is always and inseparably present. This
grand and discriminating feature of his mould-board, he considers as
of the utmost importance.
"He therefore craves the aid and elucidation of his drawing, and of
his model, in their totality and in their several parts, to render
plain and sure whatever there may be, from the abstruse and recondite
nature of the subject, uncertain or dubious in the language of his
specification.
"In the second place, the said Jethro Wood claims an exclusive right
and privilege in the construction of a standard of cast iron, like the
rest of the work already described, for connecting the mould-board
with the beam. This standard is broad, stout, strong; and rises from
the fore and upper part of the mould-board, being cast with it, and
being a projection or continuation of the same from where the fourth
and fifth sides meet. Its figure, strength, and arrangement are such
as best to secure the connexion, and to enable the standard thus
associated with the beam, to bear the pull, tug, and brunt of service.
By a screw bolt and nut properly adjusted above the top of the
standard and acting along its side, assisted, if need require, by a
wedge for tightening and loosening, the beam may be raised and
lowered; and the mould-board, with its cutting edge, enabled to make a
furrow of greater or smaller depth, as the ploughman may desire, and a
latch and key fixed to the beam, and capable of being turned into
notches, grooves, or depressions on one edge or narrow side of the
standard, serves to keep the beam from settling or descending. By
means of these screw bolts, wedges, latches, and keys, with their
appropriate notches, teeth, and joggles, the Plough may be deepened or
shallowed most exactly.
"In the third place, the said Jethro Wood claims an exclusive
privilege in the inventions and improvements made by him in the
construction of the cutting edge of the mould-board, or what may be
called, in plain language, the plough-share. The cutting edge consists
of cast iron, as do the mould-board and land-side themselves. It is
about twelve inches and one half of one inch long, four inches and one
half of one inch broad, and in the thickest part three quarters of an
inch thick. It is so fashioned and cast, that it fits snugly and
nicely into a corresponding excavation or depression at the low and
fore edge of the mould-board, along the side herein before termed the
first side. When properly adapted, the cutting edge seems, by its
uniformity of surface and evenness of connextion, to be an elongation
of the mould-board, or, as it were, an extension or continuation of
the same. To give the cutting edge firm coherence and connexion, it is
secured to the mould-board by one or more knobs, pins or heads in the
inner and higher side, which are received into one or more holes in
the fore and lower part of the mould-board. By this mechanism, the
edge is lapped on and kept fast and true, without the employment of
screws. That the cutting edge may be the more securely and immovably
kept in its place, it has a groove, or ship-lap of one inch in length,
below, or at its under side, near the angle between the first and
second sides, for the purpose of holding it, and for the further
accomplishment of the same object, another groove or ship-lap, stouter
and stronger than the preceding, is also cast in the iron, at or near
the point of the mould-board, so as to cover, encase, and protect it
effectually, on the upper and lower sides, but not on the land side.
"After the cutting edge is thus adapted and adjusted to the
mould-board by means of the indentations, pins, holes, ship-laps, and
fastenings, it is fixed to its place and prevented from slipping back,
or working off, by wedges or pins of wood, or other material, driven
into the holes from the inner and under side, and forced tight home by
a hammer.
"In the fourth place, the said Jethro Wood claims the exclusive right
of securing the handles of his plough to the mould-board and land-side
of the plough by means of notches, ears, loops, or holders, cast with
the mould-board and land-side respectively, and serving to receive and
contain the handles, without the use of nuts and screws. For this
purpose one or more ears or loops, or one or more pairs of notches or
holders are cast on the inner side of the mould-board and land side,
toward their hinder or back parts, or near their after margins, for
the reception of the handles of the Plough. And these, when duly
entered and fitted, are wedged in, instead of being fastened by
screws.
"In the fifth place, the said Jethro Wood claims an exclusive right to
his invention and improvement in the mode of fitting, adapting and
adjusting the cast iron landside to the cast iron mould-board. Their
junction is after the manner of tenon and mortice; the tenon being at
the fore end of the land-side and the mortice being at the inside of
the mould-board and near its point. The tenon and mortice are joggled,
or dove-tailed together in the casting operation, so as to make them
hold fast. The fore end of the tendon is additionally secured by a
cast projection from the inside of the mould-board for its reception;
and if any other tightening or bracing should be requisite, a wooden
wedge, well driven in, will bind every part effectually, and all this
is accomplished without the assistance or instrumentality of screws.
"The said inventor and petitioner wishes it to be understood, that the
principal metallic material of his Plough is cast iron. He has very
little use for wrought iron, and by adapting the former to the extent
he has done, and by discontinuing the latter, he is enabled to make
the Plough stronger and better, as well as more lasting and cheap.
"He also claims, and hereby asserts the right, of varying the
dimensions and proportions of his Plough, and of its several sections
and parts, in the relations of somewhat more and somewhat less of
length, breadth, thickness, and composition, according to his judgment
or fancy, so that all the while he adheres to his principle and
departs not from it.
"Regarding each and every of the matters submitted as very conducive
to the reputation and emolument of the said Jethro Wood, he relies
confidently upon a benign and favorable construction of his petition
and specification, by the constituted authorities of his country.
"Given under his hand, at the city of New York, this fourteenth day of
August, one thousand eight hundred and nineteen (1819), in the
presence of two witnesses, to wit:
"SAM'L L. MITCHELL, }
"J. G. BOGERT. } JETHRO WOOD."
This patent expired by its own limitation in fourteen years, when it
was renewed or continued for another term of fourteen years. In view
of the comparative ease and speediness with which the inventors of the
present day, or their assigns, utilize really valuable patents, it
would be inferred, in the absence of specific knowledge to the
contrary, that twenty-eight years constituted a sufficiently long
period for the enjoyment by Mr. Wood, of "the full and exclusive right
and liberty of making, constructing, using and vending to others to
be used," the plow which he had invented. No doubt some members of
Congress in refusing to continue the patent for a third term, acted
from conscientious motives. But in point of fact, the period was
occupied in a series of struggles calamitous to the inventor, to the
history of which we must now turn. These struggles were unlike those
in the lives of some other great inventors, notably, Goodyear and
Howe. It was not a warfare for existence, the wolf of poverty staring
him in the face. The broad fields which he had inherited from his
father were adequate immunity from the sad fate too frequently
allotted to inventors. But no benefactor of mankind in the domain of
mechanism ever experienced more iniquitous treatment than Jethro Wood
did.
Before the year 1819 closed, his mission as an inventor was an
accomplished fact. The popular name given his implement, "The Cast
Iron Plow," from its entire abandonment of wrought iron in its
construction, needed no change to be the noblest gift ever made to
agriculture. In the ideal, hope had ripened into full fruition. And
now, at this day, looking at the matter in the light of the past,
seeing the absolutely incalculable benefits of the invention, it seems
almost incredible that the American people, then even more than now, a
nation of farmers, should not have hailed the new plow as an
unspeakable boon, especially the community in which he dwelt, for
Cayuga county then, as now, under a high state of cultivation, was and
is peopled by a population of much more than average intelligence. But
an inventor, like "a prophet, is not without honor save in his own
country." His neighbors gravely shook their heads at "Jethro's folly."
With almost entire unanimity they agreed that the new contrivance
would never work. His trials and difficulties at this stage of
progress are told as follows, by one who wrote largely from personal
recollection:
"He immediately began to manufacture his plows, and introduce them to
the farmers in his neighborhood. The difficulties which he now
encountered would have daunted any man without extraordinary
perseverance and a firm belief in the inestimable benefit to
agriculture sure to result from his invention. He was obliged to
manufacture all the patterns, and to have the plow cast under the
disadvantages usual with new machinery. The nearest furnace was thirty
miles from his home, and, baffled by obstacles which unskillful and
disobliging workmen threw in his way, he visited it, day after day,
directing the making of his patterns, standing by the furnaces while
the metal was melting, and often with his own hands aiding in the
casting.
"When, at length, samples of his plow were ready for use, he met with
another difficulty in the unwillingness of the farmers to accept them.
'What,' they cried, in contempt, 'a plow made of pot metal? You might
as well attempt to turn up the earth with a glass plowshare. It would
hardly be more brittle.'
"One day he induced one of the most skeptical neighbors to make a
public trial of the plow. A large concourse gathered to see how it
would work. The field selected for the test was thickly strewn with
stones, many of them firmly imbedded in the soil, and jutting up from
the surface. All predicted that the plow would break at the outset. To
their astonishment and Wood's satisfaction, it went around the field,
running easily and smoothly, and turning up the most perfect furrow
which had ever been seen. The small stones against which the farmer
maliciously guided it, to test the 'brittle' metal, moved out of the
way as if they were grains of sand, and it slid around the immovable
rocks as if they were icebergs. Incensed at the non-fulfillment of
his prophecy, the farmer finally drove the plow with all force upon a
large bowlder, and found to his amazement that it was uninjured by the
collision. It proved a day of triumph for Jethro Wood, and from that
time he heard few taunts about the pot-metal.
"It was soon discovered that his plow turned up the soil with so much
ease that two horses could do the work for which a yoke of oxen and a
span of horses had sometimes been insufficient before; that it made a
better furrow, and that it could be bought for seven or eight dollars;
no more running to the blacksmith, either, to have it sharpened. It
was proved a thorough and valuable success. Thomas Jefferson, from his
retirement at Monticello, wrote Wood a letter of congratulation, and
although his theory of the construction of mould-boards had differed
entirely from the inventor's, gave his most hearty appreciation to
the merits of the new plow."
In this connection may be told a curious episode, one in itself worthy
of record, and strikingly illustrative of the perversities of fortune
to Mr. Wood in those gloomy days. It is the story of a Czar and a
Citizen.
All uncertainty as to the feasibility of the new plow having been
removed, and actuated by that broad philanthropy which was one of the
peculiar charms in the character of Mr. Wood, he desired to extend as
widely as possible the area of his usefulness, and concluded to make
the Czar of Russia, so long the chief grain exporting country of the
world, the present of one of his plows. During the Revolutionary war,
then fresh in the American mind, that great sovereign, Catherine of
Russia, had been the staunch friend of this country, and that, too,
without being impelled by jealousy of Great Britain. It seems to be a
peculiar trait in the Romanoff family to admire liberty in the
abstract, however absolute in practice. Sharing the prevailing good
will toward Russia, Mr. Wood conceived this happy thought of making a
truly substantial contribution to Cossack civilization, a civilization
ever ready, with all its crudeness, to adopt foreign improvements.
That gift, in one point of view slight, proved of great benefit to
Russian agriculture. It is impossible to state the extent of actual
advantage derived by Russia from that truly imperial gift. It was in
effect giving to that country, second only to the United States in
area of tillage, in proportion to population, the free use of the
perfected plow. In an old copy of the New York _Tribune_, in its palmy
days of Horace Greeley and Solon Robinson, the tale of the Plow and
the Ring is unfolded. It runs thus:
"During the year, 1820, Jethro Wood sent one of his plows to Alexander
I, Emperor of Russia, and the peculiar circumstances attending the
gift and its reception formed a large part of the newspaper gossip of
the day. Wood, though a man of cultivation, intellectually as well as
agriculturally, was not familiar with French, which was then as now
the diplomatic language. So he requested his personal friend, Dr.
Samuel Mitchill, President of the New York Society of Natural History
and Sciences, to write a letter in French to accompany the gift.
"The autocrat of all the Russias received the plow and the letter, and
sent back a diamond ring--which the newspapers declared to be worth
from $7,000 to $15,000--in token of his appreciation. By some
indirection, the ring was not delivered to the donor of the plow, but
to the writer of the letter, and Dr. Mitchill instantly appropriated
it to his own use. Wood appealed to the Russian Minister at Washington
for redress. The Minister sent to His Emperor and asked to whom the
ring belonged, and Alexander replied that it was intended for the
inventor of the plow. Armed with this authority, Wood again demanded
the ring of Mitchill. But there were no steamships or telegraphs in
those days, and Mitchill declared that in the long interval in which
they had been waiting to hear from Russia, he had given it to the
cause of the Greeks, who were then rising to throw off the yoke of
their Turkish oppressors. A newspaper of the time calls Mitchill's
course 'an ingenious mode of quartering on the enemy,' and the
inventor's friends seem to have believed that the ring had been
privately sold for his benefit. At all events it never came to light
again, and Wood, a peaceful man, a Quaker by profession, did not push
the matter further."
Perhaps another and quite as potent a reason why Friend Wood did not
follow up this matter was that weightier affairs demanded his
immediate and entire attention. One difficulty was overcome only to
develop another. No sooner had he silenced the cavils of the farmers
and demonstrated the value of his patent, than infringements upon his
rights threatened to, and actually did, rob him of the fruits of his
invention. "Uneasy rests the head that wears a crown" of genius.
The patent laws of that day were very imperfect, and there was a
strong prejudice against their enforcement. The cry of "no monopoly"
was raised. Mr. Wood had expended many thousands of dollars in
perfecting his patterns and getting ready to supply the demand which
he felt sure would arise for his plows, many of which, during the
first few years, he gave away, that their value might be established
to the satisfaction of the public. The stage of probation over, the
plow makers of the country, defiant of patent law, engaged in their
manufacture. His patent had fourteen years to run. In an incredibly
short time their use by the farmers in all parts of the land became
almost universal, and had he been allowed a royalty, however small, he
would have realized a vast fortune. Instead of that he very nearly
exhausted all his property in unavailing endeavors to establish
through the courts his rights as inventor and patentee.
In 1833, when his patent expired, Congress granted a renewal for
fourteen years. He was now bowed with the burden of years, and debts
incurred in trying to protect himself against infringers. His
remaining days were spent in vain efforts to maintain his rights. His
broad and kindly nature had conceived noble plans for the use of the
wealth which at one time seemed so nearly within his reach. He had
always been deeply interested in education, and had fortune smiled
upon him it is not too much to say that in spirit, however different
in detail, Jethro Wood would have anticipated Stephen Girard, Ezra
Cornell and John S. Hopkins, in nobly founding a great institution of
learning.
In private life Jethro Wood was a model man. If he had faults it is
impossible to ascertain them, for it would seem, from the concurrent
testimony of all who were acquainted with him, that
"None knew him but to love him,
None name him but to praise."
Although a consistent member of the Society of Friends, Mr. Wood was
extremely liberal in his religious views, and did not conform to the
peculiar dress of the sect. He had that truly Catholic spirit so
admirably characteristic of the great Quaker-poet, John G. Whittier.
Not even the cruel wrongs he sustained at the hands of dishonest
infringers could turn the sweetness of his kindly temper. Nature had
endowed him richly in every way, and no gift had been abused.
Physically, his was the highest type of manly beauty. Six feet and two
inches in height, perfect in proportion, courtly in manner, his
presence was worthy his character.
We will not linger over the closing scene of his eventful life. That
belongs to the sacred secrecy of private grief. His death occurred at
the very threshold of a new conflict, and upon it his son and
executor, Benjamin Wood, entered with intelligent zeal. The closing of
it being reserved for two of his daughters.
The story of these new labors was well told several years ago by a
journalist familiar with the facts, and we cannot do better than to
unearth the record from its musty file, and by transcribing it to
these pages, give it a kind of resurrection worthy its importance.
"After the death of Jethro Wood, his son Benjamin, who received the
invention as a legacy, continued his efforts to wrest justice from the
unwilling hand of the law. Nearly all his father's failures had
proceeded from the inadequacy of the patent laws, which were almost
worthless to protect the rights of the inventor. Even now a patent is
worth little until it has been fought through the Supreme Court of
the United States. In those days so many obstacles were thrown in the
way of inventors, and the combinations against them were so
formidable, that Eli Whitney, in trying to establish his right to the
cotton-gin in a Georgia court, while his machine was doubling and
trebling the value of lands through the State, had this experience,
which is given in his own words: I had great difficulty in proving
that the machine had been used in Georgia, _although at the same
moment there were three separate sets of this machinery in motion
within fifty yards of the building in which the court sat, and all
so near that the rattling of the wheels was distinctly heard on the
steps of the Court House_.
"Similar difficulties had met Jethro Wood in _his_ suits; so his son
resolved to strike at the root of the evil by securing a reform in the
laws. He accordingly went to Washington, where he remained through
several sessions, always working to this end. Clay, Webster, and John
Quincy Adams, all of whom had known Jethro Wood and his invention,
aided his son powerfully with their votes and counsel, and he
succeeded in securing several important changes in the patent laws.
"Then he returned to New York, and commenced suit to resist
encroachments on his right, and the wholesale manufacture of his plow
by those who refused to pay the premium to the inventor. The
'Cast-Iron Plow' was now used all over the country, and formidable
combinations of its manufacturers united their capital and influence
against Benjamin Wood. William H. Seward, then practicing law, was
retained as Wood's counsel, and the plow-makers engaged all the talent
they could muster to oppose him.
"Heretofore it had never been contradicted that Jethro Wood was the
originator of the plow in use, but now his right to the invention was
denied, and it was alleged that his improvements had been forestalled
by other makers. Again and again the case was adjourned, and Europe
and America were ransacked for specimens of the different plows which
were declared to include his patent.
"Mr. Wood also obtained from England samples of the plows of James
Small and Robert Ransom. He searched New-Jersey to find the Peacock
plow which was said to have a cast-iron mould-board of exactly similar
shape to his father's. Everywhere in that State he found 'Wood's plow'
in use, but he could hear nothing of the one he sought. At length
riding near a farm-house he discovered one of the old 'Newbold-Peacock
plows' lying under a fence, dilapidated and rust-eaten. 'We don't use
it any more,' the farmer replied to his inquiries, 'we've got one a
good deal better.' 'Will you sell this?' asked Wood. 'Well, yes.' And
Wood, glad to get it at almost any price, paid the keen farmer, who
took advantage of his evident anxiety, two or three times the price of
a new plow, and added the old one to his specimens.
"This motley collection of implements was brought into court and
exhibited to the judges. At last, after the case had dragged its slow
length along, through many terms, and the plaintiff was nearly worn
out with the law's delay, the time for final trial and decision
arrived. The combination of plow-makers feared that the case would go
in Wood's favor, and made every effort to keep him out of court, that
he might lose it by default. During his long entanglement in the law,
he had contracted many debts, and one of his opponents had managed to
purchase several of these accounts. Just before the case was to be
heard for the last time, this worthy plow manufacturer, attended by a
sheriff, and armed with a warrant to arrest Wood for debt, appeared at
the front door of his house. Fortunately Wood had had a few minutes
warning, and slipping out at the back door, he made his way under
cover of approaching darkness to a house of a friendly neighbor. There
he procured a horse and started for Albany, 150 miles distant, hearing
every moment in fancy the clattering of hoofs at his heels.
"As if fortune could not be sufficiently ill-natured, his horse proved
vicious and unmanageable, and thrice in the tedious journey threw the
rider from his saddle upon the frozen earth, so injuring him, that he
was barely able to go on.
"On arriving at Albany he found himself not a moment too soon. The
case had an immediate hearing, and after three days' trial the Circuit
Court decided unequivocally that the plow now in general use over the
country was unlike any other which had been produced; that the
improvements which rendered it so effective were due to Jethro Wood,
and that all manufacturers must pay his heirs for the privilege of
making it.
"This was a great triumph; but it was now the late autumn of 1845, and
the last grant of the patent had little more than a year to run. Wood
again repaired to Washington to apply for a new extension, but the
excitements of so long a contest had been too much for him. Just as he
had recommenced his efforts they were forever ended. While talking
with one of his friends, he suddenly fell dead from heart disease, and
the patent expired without renewal.
"The last male heir to the invention was no more. On settling the
estate, it was found that while not a vestige remained of the large
fortune owned by Jethro Wood when he began his career, _less than five
hundred and fifty dollars had ever been received from his invention_.
"The after history of the case is a brief one. Four daughters of Jethro
Wood alone remained to represent the family. In the winter of 1848 the
two younger sisters went to Washington to petition Congress that a
bill might be passed for their relief, in view of the inestimable
services of their father to the agricultural interests of the country.
Webster declared that he regarded their father as a 'public
benefactor,' and gave them his most efficient aid; Clay warmly
espoused their cause, and the venerable John Quincy Adams, with his
trembling hand--then so enfeebled by age that he rarely used the
pen--wrote them kind notes, heartily sympathizing with them. On one
memorable day, while they were in the House gallery, Mr. Adams, at his
desk on the floor, wrote them briefly in relation to their case. A few
minutes later he was struck with the fatal attack under which he
exclaimed, 'This is the last of earth; I am content,' and was borne
dying to the Speaker's room. The tremulous lines, the last his hand
ever traced, were found on his desk and delivered to Miss Wood.
"A bill providing that in these four heirs should rest for seven years
the exclusive right of making and vending the improvements in the
construction of the cast-iron plow; and that twenty-five cents on each
plow might be exacted from all who manufactured it, passed the Senate
unanimously. But Washington already swarmed with plow manufacturers.
The city of Pittsburgh alone sent five to look after their interests.
Money was freely used, and the members of the House Committee who
were to report on the bill were assured that during the 28 years of
the patent, Wood's family had reaped immense wealth, and wished to
keep up a monopoly. The two quiet ladies, fresh from the retirement of
a Quaker home, where they had learned little of the world, were even
accused of attempting to secure its extention through bribery. It was
the wolf charging the lamb with roiling the water. So ignorant were
they of such means, that, though the Chairman of the Committee plainly
told the younger lady in a few words of private conversation that a
very few thousand dollars would give her a favorable verdict, she did
not understand the suggestion till after an unfavorable report was
presented, and the bill killed in the House.
"When they were about to leave Washington, some friendly members of
Congress advised them to deposit the valuable documents which had been
used in their suit, including the letter from Thomas Jefferson to
Jethro Wood, in the archives of the House, where they could only be
withdrawn on the motion of some member. They did so, and left them for
some years uncalled for. When at last they applied for them they could
not be found. Nor from that time to the present has any trace of them
been discovered by any of the family. Thus perished the last vestige
of proof relating to this ill-fated invention."
This is a fair and candid statement, one fully sustained by
unimpeachable documentary evidence. Especially by the somewhat
voluminous pamphlet entitled "Documents relating to the improvements
of Jethro Wood in the Construction of the Plough." A careful
examination of the testimony therein embodied, and of the
Congressional Reports on the subject, warrant the foregoing
statements.
It is not strange that in an early annual report of the United States
Commissioner of Agriculture, that official should have remarked with
some bitterness that "Although Wood was one of the greatest
benefactors to mankind by this admirable invention, he never received,
for all his thought, anxiety and expense, a sum of money sufficient to
defray the expenses of his decent burial." The time long since passed
forever to seek pecuniary indemnity; but a debt of gratitude never
outlaws, and it is due to the great inventor that his countrymen
should gratefully cherish his memory. Every year adds to the debt we
all owe him. As the area of cultivation widens, the obligation
deepens. Already America is the foremost nation of all the earth in
the production of wheat and provisions, the latter being in reality
corn in meat form. In exchange for our food supplies, the United
States is draining Europe of its gold at an enormous rate, and the
fundamental element in the production of American wealth, is our great
implement of tillage. American prosperity is the monumental glory of
Jethro Wood and his plow.
"The Balance Sheet of the World" shows that the United States can
boast more acres of tillage, in proportion to population, than any
other country on the globe; and in grain production, outstrips all
competitors. Of such a record every American citizen may well be
proud, and it should be remembered that without the genius of Wood
such a record could not have been made, even approximately. But in
order to a just appreciation of the importance of the modern plow and
the usefulness of the inventor of it, one should take a retrospective
glance, tracing, as best we may without tedious details, the steps
which led from the use of a forked stick to the present implement for
fallowing the ground. The _Scientific American_, which ought to be
good authority on such a subject, in speaking of the Wood patent,
says: "Previously the plow was a stick of wood plated with iron." If
this does sound like an exaggeration, but is really a plain statement
of fact, consider for a moment what the plow really is in its relation
to civilization.
The savage lives by the chase and upon the bounty of untilled nature.
The first steps toward civilization are to domesticate animals, and
cultivate the soil with a rude kind of hoe. Both are alike primitive.
The next step is to press the beast into service by supplementing the
hoe with a plow. In that implement we see what might be called the
original strand in the mighty cord which binds in co-operation man,
brute and earth. By means of this agency of agriculture the beast of
the field is made to toil, and purchases the benefits of human
kindness at the expense of idleness and industry. It is not too much,
then, to say that the plow is at once "the tie that binds," and the
tap-root which nourishes the world. If by some miraculous calamity
this one implement were forever swept away, universal and unappeasable
famine would be inevitable. And that occasional famines of a local
character are disappearing from the civilized world, is very largely,
if not chiefly, due to the improved tillage resulting from improved
plows.
We might well say, in paraphrase of a familiar saying attributed to
Napoleon: Let me make the plows of a nation, and I care not who makes
their laws.
The primitive plow was and is (for the barbarian of to-day is
substantially the same in his agricultural methods as the barbarian of
antiquity) simply a forked stick, to which is attached by a strip of
rawhide or a wisp of grass, a beast, often the patient cow. As the
prong passes over the ground, held down by the bowed form of the poor
tiller, it barely scratches the face of the earth.
The first improvement was to reverse the stick and notch the forward
end. By that means the animal could be more securely fastened to the
plow, the thong being tied around the crotch of the stick. The shorter
limb ran along the surface of the ground, the notch in front being the
only reliance for stirring the soil. In the absence of a compact turf,
such plowing would do a little good in rendering the ground fallow,
and would at least have the merit of not being so difficult to operate
as its predecessor.
The third plow had three parts. It consisted of a beam, a handle and a
share, all constructed by simply trimming the natural wood selected
for that purpose. In the first plow the prong which served as a share
was slanting, while in the third it rested flatly upon the ground,
projecting forward, instead of backward, as in the second plow. It
could have required no very difficult search to have found small
trees and broken limbs, needing no mechanical skill in fashioning, to
render them serviceable for such crude uses. They may be termed
nature's contribution to the art of plow-making.
Without going further into details, it may be stated that a standard
authority on the history of mechanism asserts that "the ancient
Egyptian, Etruscan, Syrian, and Greek plows, were equal to the modern
plows of the south of France, part of Austria, Poland, Sweden, Spain,
Turkey, Persia, Arabia, India, Ceylon and China; at least such was the
case until the middle of the present century." The Roman and Gallic
plows were better than those of the modern countries named. The Gauls
had mould-board plows. Pliny is our authority for this statement. That
eminent Latin author of eighteen centuries ago, in speaking on the
general subject, says:
"Plows are of various kinds. The colter is the iron part which cuts
the thick sod before it is broken into pieces and traces beforehand by
its incision the future furrows, which the share, reversed, is to open
with its teeth. Another kind, the common plowshare, is nothing more
than a lever furnished with a pointed beak; while another variety,
which is used in light, easy soils, does not present an edge
projecting from the share-beam throughout, but only a small point at
the extremity. In a fourth kind, again, this point is larger and
formed with a cutting edge by the agency of which it cleaves the
ground, and by the sharp edges at the side cuts up the weeds by the
roots."
Pliny adds that the broader the plowshare the better it is for turning
up the soil. These excerpts from the great Roman may serve to show the
utmost reach of invention in that line, until a new impulse, begun in
the Netherlands in the eighteenth century, was brought to perfect
development in the next century by an American citizen who died the
poorer for his invention.
The highest of all authorities upon this and cognate subjects is
"Knight's American Mechanical Dictionary," and Knight says of Jethro
Wood, "He made the best plows up to date." He adds, "He met with great
opposition, and then with much injustice, losing a competency in
introducing his plow and fighting infringers." The same writer defines
the peculiarities of the Wood plow with remarkable clearness and
brevity: "It consisted in the mode of securing the cast-iron portions
together by lugs and locking pieces, doing away with screw-bolts, and
much weight, complexity and expense. It was the first plow in which
the parts most exposed to wear could be renewed in the field by the
substitution of cast pieces." Considering the source of this passage,
it may be said that literature could hardly pay a nobler tribute to
the memory of Jethro Wood than this. It is doubly significant, from
the fact that Knight's publishers, Houghton, Osgood & Co., are also
the publishers of the _Atlantic Monthly_, in the May number of which
magazine a _habitue_ of the National Capital tried to belittle the
invention of Jethro Wood, and malign as iniquitous the attempt of his
daughters, championed by John Quincy Adams, to secure for that
invention proper recognition. It would be quite superfluous to follow
this maligner in the details of this, and a subsequent attack in an
agricultural journal. He disclaims any design to defame the claimants,
but insists that other and earlier inventors deserve the credit for
the modern plow. The opinion of Knight's Dictionary upon the Wood
patent has just been given, and the following extract from the same
great work sets forth in their proper relations to the modern plow the
inventions of those for whom this _habitue_ makes preposterous
claims:
"The modern plow," says Knight, "originated in the low countries,
so-called. Flanders and Holland gave to England much of her husbandry
and gardening knowledge, field, kitchen and ornamental. Blythe's
'Improver Improved,' published in 1652, has allusions to the subject.
Lummis, in 1720, imported plows from Holland. James Small, of
Berwickshire, Scotland, made plows and wrote treatises on the subject,
1784. He made cast-iron mold-boards and wrought-iron shares, and
introduced the draft-chain. He made shares of cast-iron in 1785. The
importation of what was known as the 'Rotherham' plow was the
immediate cause of the improvement in plows which dates from the
middle of the last century. Whether the name is derived from Rotterdam
cannot be determined.
"The American plow, during the colonial period, was of wood, the
mold-board being covered with sheet-iron, or plates made by hammering
out old horseshoes. Jefferson studied and wrote on the subject, to
determine the proper shape of the mold-board. He treated it as
consisting of a lifting and an upsetting wedge, with an easy
connecting curve. Newbold, of New Jersey, in 1797, patented a plow
with a mold-board, share and land side all cast together. Peacock, in
his patent of 1807, cast his plow in three pieces, the point of the
colter entering a notch in the breast of the share."
It will be observed that the credit given these improvers of the plow
is very considerable, without at all trenching upon the exceptional
credit due to Jethro Wood. With such an authoritative refutation, the
slander may well be dismissed as beneath further notice.
In no way more appropriately can final leave be taken of the subject
in hand than by presenting the apostrophe to Jethro Wood from the pen
of Edward Webster, formerly associated editor of the _Rural New
Yorker_:
No jeweled diadem or crown
E'er glittered on thy manly brow--
No slave would tremble at thy frown,
Nor at thy footstool bow;
For thou wert pure in heart and mind,
And strove to _raise_--not crush mankind!
As famed Prometheus of yore,
In aid of our lost, wretched sires,
Stole from the flaming-sun, and bore
Down to the earth those fires
That fill with light and life all space,
And mark the Day God's glorious race--
So thy inventive genius found
For man the bright and polished share,
That bids the willing fields abound
With fruits beyond compare;
And from the seed that falls like rain
Crowds full our barns with bearded grain!
Eternal may the honors shine,
We yield with grateful hearts to thee;
May children's children round thy shrine--
Sons of the brave and free--
With reverent lips pronounce thy name,
And build for thee a deathless fame!
End of the Project Gutenberg EBook of Jethro Wood, Inventor of the Modern
Plow., by Frank Gilbert
*** | {
"redpajama_set_name": "RedPajamaBook"
} | 1,897 |
\section{Introduction}
\qquad Many exact results have been obtained
since the discovery by Seiberg and Witten \cite{sewi,sewi2}
of the exact solution of the low energy $N=2$
supersymmetric $SU(2)$ gauge theory in four dimensions.
The low energy $N=2$ SUSY Yang-Mills theories contain
$g=N_c-1$ abelian $N=2$ vector supermultiplets, which can
be decomposed into $g$ $N=1$ chiral multiplets $A_i$ plus
$g$ $N=1$ vector multiplets $W_i$. According to
\cite{sewi,sewi2} the scalar component $a_i$ of the $A_i$
(namely, its vacuum expectation value (vev)) is the local
coordinate on the quantum moduli space of the effective
action of the theory. It is given by the integrals
of meromorphic differentials
$\lambda$ over the basic cycles
$\alpha_{i}$ and $\beta_{i}$, such that $\alpha_{i} \circ
\beta_{j} = \delta_{ij}$, on a hyper-elliptic curve $\cal C$.
In particular, their derivatives with respect to the
symmetric vevs
$s_k=(-1)^k \sum_{i_1< \cdots < i_k} a_{i_1}
\cdots a_{i_k}$ ($k=2 ,
\cdots , N_c $) are equal to \cite{KlLeTh}:
\begin{eqnarray}
{\partial a_i \over \partial s_k}
\sim A_{ki} = \int_{\alpha_{i}} \frac{x^{N_c-k}dx}{y}
\nonumber \\
{\partial a_Di
\over \partial s_k} \sim B_{ki} = \int_{\beta_{i}}
\frac{x^{N_c-k}dx}{y}
\label{ab}
\end{eqnarray}
\qquad We will be mainly interested in the coupling
matrix $T_{ij}$, which is the period matrix on ${\cal C}$.
In the matrix form, it can be presented as \cite{KlLeTh}:
\begin{equation}
{\bf T} = {\bf A}^{-1}{\bf B} \label{tau}
\end{equation}
\qquad In some particular cases the integration in
(\ref{ab}) may be performed explicitly. It has been done
for the $SU(2)$ \cite{KlLeTh,fipo,haoz} and $SU(3)$ groups
\cite{KlLeTh}. For higher $N_c$, it is not so immediate,
and, therefore, there were a series of calculations taking
into account only the one - instanton corrections
\cite{itsa}. For $N_c>3$, integration in (\ref{ab}) becomes
much more complicated, and Picard - Fuchs equations are not
known for arbitrary $N_c$ and $N_f$.
Some theories have the same
integrals - this allows us to derive relations between
couplings (see Section $2$) and, in some particular cases,
even to obtain exact results. The first step in this
direction was done in \cite{mine}, where some exact beta
functions for the $SU(2)$ and $SU(3)$ cases were calculated.
In Section 2, we discuss the coupling constants for the
theories with $N_c$ colors and $N_f$ ($N_f=0$ or
$N_f=N_c$ ) massless flavours. We calculate them for
the second order in the instanton expansion by the direct
evaluation of integrals (\ref{ab}).
All the bare masses $m_k$ of the matter hypermultiplets
are put zero (till the Appendix).
As later we deal with instanton contributions to
the prepotential, it is relevant to remind the
perturbative expansion of the prepotential which is
saturated in one loop due to the supersymmetry:
\begin{equation}
{\cal F}=i \frac{2N_c-N_f}{8\pi}\sum_{i<j}(A_i-A_j)^2
\log{\frac{(A_i-A_j)^2}{\Lambda^2}}
\label{pprep}
\end{equation}
In Section 3 we consider the scale invariant $N_f=2N_c$
theory. It has the classical period matrix ${\bf T}$
proportional to the matrix $\bf C$ : $C_{ij}=\delta_{ij}+1$
(in the basis $A_i=a_i$, $i=1 \cdots g$ ;
$A_{N_c}=-\sum_{i=1}^g a_i$). By comparing the spectral
curves for this theory and for the $N_f=N_c$ one, we
demonstrate that spectral curve for UV finite $N_f=2N_c$
($N_c>2$) theory can not be hyperelliptic
(double covering of $CP^1$).
Section 4 is devoted to the $SU(2)$ theory. We present
the strong evidence for the method proposed by J. A. Minahan
and D. Nemeschansky which helps one to obtain some useful
relations between the $N_f=0$ and $N_f=2$ theories.
In Appendix we present some nonperturbative
$\beta$-functions of the $SU(2)$ gauge theory with non-zero
masses of the matter hypermultiplets.
\newpage
\section{General $N_c$}
\qquad Now we are going to discuss some akin $N=2$ SUSY
Yang-Mills theories which have the similar spectral curves
each depending on one dimensionless parameter. Namely, we
note that the curves for the $N_f=0$ and $N_f=N_c$ theories
have the same forms in the symmetric point on the moduli
space (compare (\ref{0nf}) and (\ref{nc})). The period
matrix $T_{ij}$ must be the same for the both theories,
turning into each other by an appropriate replace of the
parameters. Before going further, let us stress that the
curves are regarded akin if they are related by
$SL(2,{\bf C})$ transformation (this is the common
property of the two-dimensional manifolds) or by
rescaling $x$ and $y$ (because we restrict ourselves to the
only parameter and $T_{ij}$ is dimensionless too).
For instance, for general $N_c>2$ with all the order
parameters $s_k$ being zero but $s_{N_c}=-u \not= 0$, the
$N_f=0$ and $N_f=N_c$ curves take the forms \cite{haoz}:
\begin{itemize}
\item$N_f=0$
\begin{equation}
y^2=\left(x^{N_c}-u^{(0)}\right)^2-\Lambda^{(0)2N_c}
\Leftrightarrow y^2=x^{2N_c}-2F^{(0)}x^{N_c}+1
\label{0nf}
\end{equation}
\item$N_f=N_c$
\begin{equation}
y^2=\left(x^{N_c}-u^{(N_c)}+\frac{\Lambda^{(N_c)N_c}}
{4}\right)^2-\Lambda^{(N_c)N_c}x^{N_c} \Leftrightarrow
y^2=x^{2N_c}-2F^{(N_c)}x^{N_c}+1
\label{nc}
\end{equation}
\end{itemize}
where
\begin{itemize}
\item$N_f=0$
\begin{equation}
F^{(0)}=\frac{u^{(0)}}{\sqrt{u^{(0)2}-\Lambda^{(0)2N_c}}}
\Leftrightarrow \Lambda^{(0)2N_c}=u^{(0)2}(1-F^{(0)-2})
\label{0fun}
\end{equation}
\item$N_f=N_c$
\begin{equation}
F^{(N_c)}=\frac{u+{\Lambda^{(N_c)N_c} \over 4}}
{u-{\Lambda^{(N_c)N_c} \over 4}}
\Leftrightarrow \Lambda^{(N_c)N_c}=4u^{(N_c)}
\frac{F^{(N_c)}-1}{F^{(N_c)}+1}
\label{ncfun}
\end{equation}
\end{itemize}
As a consequence, their couplings are (here we denote
$v={\displaystyle\frac{\Lambda^{N_c}}{u}}$):
\begin{eqnarray}
T_{ij}^{(N_c)}(v^{(N_c)})=T_{ij}^{(0)}\left(
\sqrt{\frac{v^{(N_c)}}
{(1+{v^{(N_c)} \over 4})^2 }}\right);\nonumber\\
T_{ij}^{(0)}(v^{(0)})=T_{ij}^{(N_c)}\left( \frac{8}{v^{(0)2}}
\left( 1 - \frac{v^{(0)2}}{2} - \sqrt{1 - v^{(0)2} }
\right)\right),
N_c>2
\label{vev}
\end{eqnarray}
The same procedure may be performed for other gauge groups.
One does not need to know integrals (\ref{ab}) explicitly.
Instead, it is sufficient to compare the curves, i. e. to
obtain relations between the quantities which may be
${\bf T}$, $\beta$, ${\cal F}$ or their expansions in series at
small $v$.
Let us evaluate (\ref{ab}) to find period matrix, namely,
first instanton terms. We will focus on the $N_f=0$ theory
(as it has been shown earlier $N_f=N_c$ case is completly
analogous):
\begin{eqnarray}
A_{kl}=2\int_{x_{-,l}}^{x_{+,l}}\frac{x^{N_c-k}dx}{y}=
-\frac{2 \pi i}{N_c} \epsilon^{l(1-k)}z_{-}^{-m}F_{2,1}
(\frac{1}{2},m;1;1-\frac{z_{+}}{z_{-}}) \label{a} \\
B_{kl}=2\int_{x_{-,1}}^{x_{-,l}}\frac{x^{N_c-k}dx}{y}=
\frac{2}{N_c}(\epsilon^{l(1-k)}-\epsilon^{(1-k)})
\frac{z_{-}^{\frac{1}{2}-m}}{z_{+}^{\frac{1}{2}}}
\frac{\Gamma(1-m) \Gamma(\frac{1}{2})}{\Gamma({3 \over 2} - m)}
F_{2,1}(\frac{1}{2},1-m;{3 \over 2}-m;{z_{-} \over z_{+}})
\label{b}
\end{eqnarray}
\begin{figure}
\epsfxsize 400pt
\epsffile{cycles.eps}
\caption{ The homology cycles for $N_c=4$}
\end{figure}
where we chose the cycles as shown on Fig.1
($N_c=4$ assumed) so that:
\begin{eqnarray}
x_{\pm,l}=\epsilon^l x_{\pm};
\epsilon=\exp\left( \frac{2 \pi i}{N_c}\right);m=\frac{k-1}{N_c}\\
x_{\pm}=(u \pm \Lambda^{N_c})^{{1 \over N_c}}=
z_{\pm}^{{1 \over N_c}};
\label{roots}\\
\lambda=\frac{z_{+}}{z_{-}}-1;
\end{eqnarray}
In the weak coupling limit ($ | \lambda | \ll 1$),
(\ref{a}) and (\ref{b}) take the forms:
\begin{equation}
A_{kl}=-\frac{2 \pi i}{N_c} \epsilon^{l(1-k)}z_{-}^{-m}(1-{m
\over 2} \lambda
+{3 \over 16}m(m+1)\lambda^2+ ({1 \over 8} - {3 \over 32}m
- {1 \over 6}m^2)m\lambda^3 +\cdots) \label{acl} \\
\end{equation}
$$
B_{kl}=\frac{2}{N_c} (\epsilon^{l(1-k)}-\epsilon^{(1-k)})z_{-}^{-m}
( (\ln {4 \over \lambda} - C - \psi (1-m))(1 - {m\over 2}
\lambda +{3 \over 16}m(m+1)\lambda^2 -
$$
\begin{equation}
- {5 \over 96}m(m+1)(m+2)\lambda^3 ) + {\lambda \over 2} +
{m^2-5m - 3 \over 16} \lambda^2 +
({5 \over 48}+{1 \over 32}m(8+2m-m^2))\lambda^3
+ \cdots )
\label{bcl}
\end{equation}
where $C$ is the Euler constant, and $\psi(x)=
{\Gamma' \over \Gamma}$ is
the logarithmic derivative of the gamma function.
It is useful to note that the dependence of ${\bf A}$
and ${\bf B}$ on $k$ and $l$ is :
\begin{eqnarray}
A_{kl}=a(k) E_{kl} \qquad E_{kl}=\epsilon^{l(1-k)}
\label{AA}
\\
B_{kl}=b(k)(\epsilon^{l(1-k)}-\epsilon^{(1-k)})=b(k)(E_{kl}+
\sum_{l}E_{kl})
\label{AB}
\end{eqnarray}
Substituting it into (\ref{tau}), one gets the matrix
of coupling constants for the $N_f=0$ theory:
\begin{equation}
T_{ij}=\sum_{k}{b(k) \over a(k)}(E^{-1})_{ik}(E_{kj}+
\sum_{j}E_{kj}) =
\label{TT}
\end{equation}
$$
={i \over \pi}\sum_{k}(E^{-1})_{ik}(E_{kj}
+\sum_{j}E_{kj})( \ln {4 \over \lambda}
-C -\psi (1-m) + {\lambda \over 2} -
{3 \over 16}\lambda^2 +\cdots )
$$
It is easy to show that ${\bf T}$ is proportional
to ${\bf C}$, if and only if all the exact
$a(k)^{-1}b(k)$ are the same functions for any $\it k$.
Imposing the constraint $T_{ii}=2T_{i \not= j}$ on
${\bf T}$, one immediately gets from (\ref{TT}):
$$
\sum_{k} \frac{b(k)}{a(k)} (E^{-1})_{ik} E_{kj} |_{i
\not= j}= 0
$$
which, after simple transformations, turns into :
\begin{equation}
a^{-1}(k)b(k)=const
\label{test}
\end{equation}
This equation must be satisfied in all orders in $\lambda$.
Obviously, it is not true and ${\bf T} \not\sim {\bf C}$
in symmetric point.
\section{$N_f=2N_c$ case}
\qquad When $N_f=2N_c$ and the bare masses are zero we
get conformally invariant theory. It has the classical period
matrix ${\bf T}$ proportional to the matrix ${\bf C}$ :
$T_{ij}=\tau C_{ij}=\tau (\delta_{ij} + 1)$.
The spectral curve for this case was proposed in
\cite{haoz} and, when the masses are set to zero,
curve reads ($s_i=0$, $i \not= N_c$; $s_{N_c}=-u \not= 0$):
\footnote{$L$ and $l$ are modular forms expressed
through the higher genus $\theta$-constants defined on
$\tau {\bf C}$}
\begin{equation}
y^2=\left[(1+{L\over 4})x^{N_c}-ul\right]^2-Lx^{2N_c}
\Leftrightarrow y^2=x^{2N_c}-2Fx^{N_c}+1
\label{2nc}
\end{equation}
This curve has the same form as (\ref{0nf})
and (\ref{nc}), with the associated period matrix
${\bf T} \sim {\bf C}$ having the same structure. From the
computation of Section 2 one can easily see that this is not
the case. Moreover, it may be seen from (\ref{pprep}) that
even the perturbative coupling matrix is not proportional to
the matrix ${\bf C}$ anywhere on the moduli space if
$N_c>2$ and $N_f<2N_c$ (see also \cite{mine}).
The simplest
way to see it is to compute ${\bf T}$ in basis
$a_i=A_i-A_{N_c}$ , $i=1 \cdots g$. Requirement for $T_{ij}$
to be proportional to the classical
matrix leads to the constraint on $a_i$ :
$$
(g-1) \log {a_i} = \sum_{k \not= i}^g \log (a_i-a_k)
$$
which must be satisfied for any $i$. Since these equations
have not nontrivial solutions,
we come to the statement that ${\bf T} \not \sim {\bf C}$
at any point on the moduli space.
Furthemore, let us suppose that the
$N_f=2N_c$ curve is written as a polynomial of power
$2N_c$ (for instance, as it was proposed in \cite{haoz}).
One can compare the spectral curve for such a theory and
for the $N_f=N_c$ one. Since the both theories have the
spectral curves which are polynomials of power $2N_c$ and
there are $3N_c-2$ parameters ($s_k^{(N_c)}$, $s_k^{(2N_c)}$
and $m_k^{(N_c)}$), one can adjust them so that the
curves are getting identical (up to $SL(2,C)$ transformations),
and the corresponding theories have the same structure of the
coupling matrices. But from the previous arguments based on
perturbative results, we know that it does not take place.
Hence, the $N_f=2N_c$ spectral curve is not a polynomial of
power $2N_c$.
Thus, we demonstrate that, for $N_f=2N_c$, the spectral curve
can not be hyperelliptic surface (the double covering of
$CP^1$). Our conjecture is that, for scale invariant
theories, covering of the sphere must be replaced by
covering of elliptic curve with natural elliptic parameter
$\tau = {\theta \over 2\pi} + {4 \pi i \over g^2}$.
\section{More on $N_c=2$}
\qquad Let us repeat the same procedure for the $SU(2)$
group in detail. Analogously:
\begin{eqnarray}
T^{(N_c)}(v^{(N_c)})=T^{(0)}\left(\sqrt{\frac{v^{(N_c)}
(1+\frac{v^{(N_c)}}{8}) }{(1+{3v^{(N_c)} \over 8})^2 }}
\right);
\nonumber\\
T^{(0)}(v^{(0)})=T^{(N_c)}\left( 8 \frac{1 - \sqrt{1 -
v^{(0)2}}} {1 + 3 \sqrt{1 - v^{(0)2}}} \right),N_c=2
\label{rel}
\end{eqnarray}
We use it to relate the coefficients ${\cal F}_k$ of the
instanton expansion for the prepotential:
\begin{equation}
{\cal F} (a)=\frac{ia^2}{4 \pi} \left[ b \ln \left({a \over \Lambda}
\right) + \sum_{k=1}^{\infty} {\cal F}_k (N_f)\left({\Lambda \over a}
\right)^{kb} \right]
\label{prep}
\end{equation}
where $b=4-N_f$. The order parameter is known from
the Picard-Fuchs equation \cite{Ito}:
\begin{equation}
u=\frac{4 \pi}{ib}\left( a{\partial {\cal F} \over \partial a} -2{\cal F}\right)=
a^2 \left[ 1-\sum_{k=1}^{\infty} k{\cal F}_k
\left({\Lambda \over a}\right)^{kb} \right]
\label{order}
\end{equation}
$T$ appears as the second derivative of the prepotential :
\begin{equation}
T = {i \over 2 \pi} \left[ b \left( {3 \over 2}+
\ln \left({a \over \Lambda}\right)\right)+\sum_{k=1}^{\infty}
(1- {kb \over 2})
(1 - kb) {\cal F}_k \left( \frac{\Lambda}{a} \right) ^{kb}\right]
\label{tau2}
\end{equation}
Substituting $N_f=0$ and $N_f=2 (=N_c)$ and inverting the
series for $u$, one gets $a$:
\begin{itemize}
\item $N_{f}=0$
\begin{equation}
\frac{a^2}{\Lambda^2}=\frac{u}{\Lambda^2} \left[ 1+
{\cal F}_1 \left( {\Lambda^2 \over u} \right)^2 +(2{\cal F}_2- {\cal F}_1^2)
\left( {\Lambda^2 \over u} \right) ^4 + (3 {\cal F}_3 - 8 {\cal F}_1 {\cal F}_2 +
2 {\cal F}_1^3) \left( { \Lambda^2 \over u} \right)^6 + \cdots \right]
\end{equation}
\item $N_f=2$
\begin{equation}
\frac{a^2}{\Lambda^2}=\frac{u}{\Lambda^2}\left[ 1+
{\cal F}_1 \left( {\Lambda^2 \over u}\right) +2{\cal F}_2 \left( {\Lambda^2
\over u}\right)^2 + (3{\cal F}_3-2{\cal F}_1{\cal F}_2)\left(
{\Lambda^2 \over u}\right)^3 +\cdots \right]
\end{equation}
\end{itemize}
After insertion of these results into (\ref{tau2}), we
obtain the instanton expansion for $T$
($v={\displaystyle{\Lambda^2 \over u}}$):
\begin{itemize}
\item $N_{f}=0$
\begin{eqnarray}
T^{(0)} =\frac{i}{2\pi} \left[ 6+ \ln {a^2 \over \Lambda^2} +
3{\cal F}_1^{(0)} \left( {\Lambda^2 \over a^2} \right)^2 +21{\cal F}_2^{(0)}
\left( {\Lambda^2 \over a^2} \right)^4+55{\cal F}_3^{(0)}
\left( {\Lambda^2 \over a^2}
\right)^6 + \cdots \right] = \nonumber \\ =
\frac{i}{2\pi} \left[ 6-
\ln v^{(0)2} +4{\cal F}_1^{(0)}v^{(0)2}+(23{\cal F}_2^{(0)}-
{15 \over 2}{\cal F}_1^{(0)2})v^{(0)4}+ \cdots \right]
\label{tau20}
\end{eqnarray}
\item $N_f=2$
\begin{eqnarray}
T^{(2)} = \frac{i}{2\pi} \left[ 3+ \ln {a^2 \over \Lambda^2} +
3{\cal F}_2^{(2)} \left( {\Lambda^2 \over a^2} \right)^2 +
10{\cal F}_3^{(2)}\left( {\Lambda^2 \over a^2} \right)^3 +
\cdots \right] = \nonumber
\\ = \frac{i}{2\pi} \left[ 3- \ln v^{(2)} +{\cal F}_1^{(2)}v^{(2)}+
(5{\cal F}_2^{(2)}+ \frac{1}{2} {\cal F}_1^{(2)2})v^{(2)2}+ \cdots \right]
\label{tau22}
\end{eqnarray}
\end{itemize}
In order to check (\ref{rel}), one must substitute
$$
8 \frac{1 - \sqrt{1 - v^{(0)2}}}{1 + 3 \sqrt{1 - v^{(0)2}}} =
v^{(0)2}\left( 1+{5 \over 8} v^{(0)2} + {29 \over 64}v^{(0)4}
+\cdots \right)
$$
into (\ref{tau22}):
\begin{eqnarray}
T^{(0)}(v^{(0)})=T^{(N_c)}\left( 8 \frac{1 -
\sqrt{1 - v^{(0)2}}}{1 + 3 \sqrt{1 - v^{(0)2}}} \right)
= \nonumber\\ \frac{i}{2\pi} \left[ 3- \ln v^{(0)2} +
({\cal F}_1^{(2)}-{5 \over 8}) v^{(0)2}+(5{\cal F}_2^{(2)}+
{5 \over 8} {\cal F}_1^{(2)} + \frac{1}{2} {\cal F}_1^{(2)2} -
{1 \over 16})v^{(0)4} + \cdots \right]
\end{eqnarray}
Comparison with the coefficients in (\ref{tau20})
yields the system of equations for the first two of them:
\begin{eqnarray}
\left\{
\begin{array}{ccc}
4{\cal F}_1^{(0)}={\cal F}_1^{(2)}-{5 \over 8}\nonumber\\
23{\cal F}_2^{(0)}-{15 \over 2} {\cal F}_1^{(0)2}=5{\cal F}_2^{(2)}+
{5 \over 8}{\cal F}_1^{(2)}+\frac{1}{2} {\cal F}_1^{(2)2}-{1 \over 16}
\end{array}
\right.
\label{system}
\end{eqnarray}
\qquad It may seem not to be so interesting to deal with
these equations, since one can get explicit expressions
for the coefficients in terms of $\theta$-constants for
these theories, nevertheless it allows to express
immediately $N_f=2$ instanton terms through $N_f=0$ ones
and provides the good consistency check for them. Let us
mention that the results of \cite{Ito} do not satisfy them.
Now let us prove the exact formula expressing
$\beta$ through $\theta$ - constants \footnote{$\theta_2=
\theta[\frac{1}{2},0]=\sum_{n \in Z}q^{(n+ \frac{1}{2})^2}$,
$\theta_3=\theta[0,0]=\sum_{n \in Z}q^{n^2}$, $\theta_4=
\theta[0,\frac{1}{2}]=\sum_{n \in Z} (-1)^n q^{n^2}$} \cite{mine}:
\begin{eqnarray}
\beta^{(0)} = {2 \over \pi i} \frac{\theta_3^4(2T) +
\theta_2^4(2T)}{\theta_4^8(2T)} \nonumber \\
\beta^{(2)} = {1 \over 2 \pi i} \frac{\theta_3^4(2T) +
\theta_4^4(2T)}{\theta_3^4(2T) \theta_4^4(2T)}
\label{su2}
\end{eqnarray}
The $N_f =4$ curve is \cite{haoz}:
\begin{eqnarray}
y^2=x^4-2Fx^2+1 \nonumber\\
F=\frac{\theta_2^4(T) + \theta_3^4(T)}{\theta_2^4(T) -
\theta_3^4(T)}
\label{nf4}
\end{eqnarray}
where $x$ and $y$ are rescaled by the vev
$u$. Of course, $a$, $a_D$, $A$
and $B$ depend on $u$ but $T$.
So we know the connection between the factor $F$ in equation
(\ref{nf4}) for $u$ and $T$. Let us show it by direct
calculation for the $N_f=0$ curve \cite{haoz}:
\begin{equation}
y^2=(x^2-u)^2-\Lambda^4
\label{nf0}
\end{equation}
This equation may be rewritten in the form :
\begin{equation}
y^2=x^4-2 \frac{u}{\sqrt{u^2-\Lambda^4}} x^2+1
\label{nf01}
\end{equation}
{}from which (using
(\ref{nf4})) we get $\Lambda^4=u^2(1-F^{-2})$ (the result of
\cite{mine}), or, finally:
\begin{equation}
\Lambda^4=u^2(1-F^{-2})=u^2 \left({2\theta_2^2 \theta_3^2 \over
\theta_2^4+\theta_3^4}\right)^2
\label{sol0}
\end{equation}
One can easily integrate (\ref{ab}) in
terms of hyper-geometric functions:
\begin{eqnarray}
A = 2 \int_{x_{+}}^{x_{-}} \frac{dx}{\sqrt{(x^2-u)^2-\Lambda^4}}
= - {2\pi i \over x_{+}+x_{-}}F_{2,1}(\frac{1}{2},\frac{1}{2};1;{(x_{+}-
x_{-})^2 \over (x_{+}+x_{-})^2})\nonumber\\
B = 2 \int_{-x_{+}}^{x_{+}} \frac{dx}{\sqrt{(x^2-u)^2-\Lambda^4}}
= {2\pi \over x_{-}}F_{2,1}(\frac{1}{2},\frac{1}{2};1;\left({x_{+}
\over x_{-}}\right)^2)
\label{per}
\end{eqnarray}
where $x_{+}=\sqrt{u+\Lambda^2}$ and $x_{-}=\sqrt{u-\Lambda^2}$
are roots of (\ref{nf0}). Note also that $B$ has the form:
\begin{equation}
B = {2\pi \over x_{-}}F_{2,1}(\frac{1}{2},\frac{1}{2};1;\left({x_{+}
\over x_{-}} \right)^2) = {2\pi \over x_{+}+x_{-}}F_{2,1}
(\frac{1}{2},\frac{1}{2};1;{4 x_{+} x_{-}\over (x_{+} + x_{-})^2})
\label{per1}
\end{equation}
As it was mentioned above, $T$ is the ratio of periods:
$$
T =i { F_{2,1}(\frac{1}{2},\frac{1}{2};1;1-w)
\over F_{2,1}(\frac{1}{2},\frac{1}{2};1;w) }
$$
with $w ={\displaystyle \frac{(x_{+}-x_{-})^2}{(x_{+}+x_{-})^2}}$.
Solution
to this equation is known (see \cite{baer} for example):
$$ w =
\frac{\theta_2^4(0,q)}{\theta_3^4(0,q)} \Rightarrow u-\Lambda^2 = (u+\Lambda^2)
\left(\frac{\theta_3^2-\theta_2^2}{\theta_3^2+\theta_2^2}\right)^2
$$
which leads to the
result (\ref{sol0}) after simple transformations. We have
obtained it by comparing this $N_f=0$ curve (\ref{nf0}) and
that with $N_f=4$ (\ref{nf4}) (the method by Minahan and
Nemechansky \cite{mine}) and by direct calculation, providing
the evidence for the validity of this method (by comparing
the curves). In this way, we can get the exact
$\beta$ - functions (see also Appendix A). They are in
agreement with \cite{KlLeTh,fipo}, but not with \cite{Ma}.
\section{Conclusions}
\qquad We find first instanton corrections to the matrix
of coupling constants at the symmetric point on the moduli
space. Comparing the spectral curves for the theories with
different number of flavors, we get some useful relations
between couplings and present the proof that the $N_f=2N_c$
spectral curve could not be presented as a covering of
the sphere. Also we propose the strong evidence for the
method of \cite{mine} (comparing the curves) in the case
of the $SU(2)$ gauge group. One can easily extend this
technique to the theories with nonzero bare masses, but
the requirement for the coupling matrix to be proportional
to the classical one imposes some constraints on the masses
(or on their symmetric functions $t_k(m)=\sum_{i_1 < \cdots
< i_k} m_{i_1} \cdots m_{i_k}$). Some $\beta$ - functions
for such theories are collected in the Appendix. It is easy
to check that the results turns into (\ref{su2}) if the
masses tend to zero.
\vskip5mm
\section*{Acknowledgments}
\qquad We are indebted to A. Morozov for
helpful discussions and comments.
S. Gukov would like to thank A. Mironov and S. Khoroshkin
for teaching him some useful mathematical background.
I. Polyubin is grateful to the Institute of Theoretical
Physics at Hannover for kind hospitality where part of
this work was done.
The work of S.G. was partially supported by RFBR
grant No. 95-01-00755, the work of I.P. by grants
RFBR-96-01-01106 and INTAS-93-1038. I. Polyubin also
would like to thank Volkswagen Stiftung project "Integrable
models and strings" for financial support.
\section*{Appendix A : Some exact $\beta$ - functions at
nonzero bare masses}
\begin{itemize}
\item $N_c=N_f=2$, $m_1=-m_2=m$
$$
\Lambda^2=(F^2-9)^{-1}\left[ 8F^2(u-4m^2)+24u-8
\sqrt{8F^4m^2(2m^2-u)+8uF^2(2u-3m^2)} \right]
$$
so that
$$
\beta= \frac{F^2-9}{FF'} \left[ \frac{ \left[ u-4m^2-
\frac{4F^2m^2(2m^2-u)+2u(2u-3m^2)}{ \sqrt{2F^4m^2(2m^2-u)+
2uF^2(2u-3m^2)}} \right](F^2-9)}{F^2(u-4m^2)+3u-
\sqrt{8F^4m^2(2m^2-u)+8uF^2(2u-3m^2)}} - 1 \right]^{-1}
$$
where for the $SU(2)$ group :
$$
F=\frac{\theta_2^4(T) + \theta_3^4(T)}{\theta_2^4(T) -
\theta_3^4(T)}
$$
\item $N_c=2$, $N_f=3$, $m_1^2+m_2^2=2u$, $m_3=0$
$$
\beta=\frac{F+2}{F'} - \frac{4(F+2)^2 \left( u +
\frac{\Lambda^2}{64} \right)^2}{27F'(u-m_1^2)^2}
$$
$$
\frac{F+2}{27}\left( u + \frac{\Lambda^2}{64} \right)^3=
\frac{\Lambda^2}{64}(u-m_1^2)^2
$$
where
$$
F=\frac{(2\theta_4^4+\theta_2^4)(2\theta_2^4+\theta_4^4)(\theta_4^4-
\theta_2^4)}{(\theta_2^8 + \theta_4^8 + \theta_2^4\theta_4^4)^{3 \over 2}}
$$
\end{itemize}
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,732 |
Q: Understanding the future perfect We can use the future perfect to say of something that it will have been done by a certain time in the future. Does it make sense to use this tense to tell about something by an undefined time in the future? For instance:
I will have learnt English.
My understanding:
This means that I will have learnt English in the future. It is not defined when exactly I'll have learnt English. This sentence emphasizes a future achievement, not the learning process.
What exactly is the difference in that sentence from future prediction with simple future:
I will learn English.
A:
I will learn English.
This is generally a commitment. Something you intend to do, whether it is now, or sometime in the future.
*
*Now, I will go clean my room.
*Next year, I will lose weight.
I will have learnt English.
This is more of an expectation than an intention. Rather than committing to learning English here, you are using something else as a reference for the point in time where you learn it. (Often, stating it as a result or consequence of something.)
*
*By the time I'm 20, I will have learnt English.
*After five months of trying, I will have succeeded.
Also worth noting: "I will have ____" does not stand on its own, while "I will ____" can. That is, "I will learn English," is a full and complete sentence, while, "I will have learnt English," is not. (It requires another piece to be a sentence.)
A: Perfect constructions are relative tenses: the eventuality named by the lexical verb (learn in your example) is located at an Event Time (ET) prior to the Reference Time indicated by the tensed verb (will have in your example) in the construction. The entire construction describes a state at RT which arises out of the eventuality at ET.
Consequently, there must be a definite RT to which ET is related. However:
*
*When we say that the RT must be 'definite' we do not require a date/time measure such as tonight or in 2018. RT may also be defined as a particular event, or entry into a particular state. For instance:
By the time I graduate I will have learnt English.
As a rule of thumb you may think of 'definite' here as meaning "defined enough to recognize it when you see it".
*RT does not have to be defined in the same sentence the perfect occurs in. It may be defined earlier in the discourse; all that is necessary is that your hearer must be able to identify what time you are talking about. For instance:
I expect to graduate before I am 22 years old. By then I will have learnt English.
Q: What will you have accomplished by the time you graduate?
A: I will have learnt English.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,923 |
layout: post
title: 'Techie Tidbits: Beef up the CMD Prompt'
date: '2010-10-04T01:47:00.004+01:00'
author: Darren Bishop
tags:
- Bash
- Python
- Tidbits
- Perl
- Cygwin
- Ruby
- CMD
modified_time: '2010-10-10T20:11:10.555+01:00'
blogger_id: tag:blogger.com,1999:blog-3826577489563855618.post-8608086113267078913
blogger_orig_url: http://blog.darrenbishop.com/2010/10/techie-tidbits-beef-up-cmd-prompt.html
---
Are you a frequent Cygwin user and/or often ask yourself "Why is the CMD prompt so crap?!"? If so, you'll probably be interested to know how you can inject some or possibly all of that Linux/Bash goodness into your CMD prompt.
<h5>1st, <a href="http://blog.darrenbishop.co.uk/2010/10/techie-tidbits-get-going-with-cygwin.html">Install Cygwin</a> if you haven't already done so</h5>
<h5>2nd, Extend Your <code>PATH</code></h5>
Create a <code>bin</code> (short for binary, but conventionally the home for anything executable) folder in your user directory, then add this directory to your <code>PATH</code> variable: <em>Win+Pause > Advanced Settings > Environment Variables</em>, probably best to add it to <em>User variables for <You></em>
<h5>3rd, Create Your Associations</h5>
I've made an <code>associations.cmd</code> file in my personal <code>bin</code> directory with the following contents:
<pre class="brush: bash">
@echo off
assoc .sh=BashScript
ftype BashScript="C:\Cygwin\bin\env.exe" "CYGWIN=nodosfilewarning" "/bin/bash" -l "%%1" %%*
assoc .pl=PerlScript
ftype PerlScript="C:\Cygwin\bin\env.exe" "CYGWIN=nodosfilewarning" "/bin/perl" "%%1" %%*
assoc .py=PythonScript
ftype PythonScript="C:\Dev\Python\2.6(x86)\python.exe" "%%1" %%*
assoc .rb=RubyScript
ftype RubyScript="C:\Dev\Ruby\1.8.6\bin\ruby.exe" "%%1" %%*
</pre>
Clearly I have Python and Ruby installed on Windows - if you don't omit the last and second to last command pairs. If you have installed the Perl package in Cygwin (which I recommend you do), then the second pair of commands wires that up. Now run this CMD file.
<h5>4th, Extend Your <code>PATHEXT</code></h5>
Follow the instructions in step 2, but instead of changing the <code>PATH</code> variable, add the Bash, Python and Perl script file extensions (SH, PY and PL, respectively) to the semi-colon separated <code>PATHEXT</code> variable
<p>
This now allows you to run any Bash, Python or Perl script from anywhere on the CMD Prompt, JOY!
<h5>A Perl Example</h5>
Create <code>prename.pl</code> in your personal <code>bin</code> directory with the following contents:
<pre class="brush: perl">
#!/usr/bin/perl -w
$op = shift or die "Usage: rename expr [files]\n";
chomp(@ARGV = <STDIN>) unless @ARGV;
for (@ARGV) {
$was = $_;
eval $op;
die $@ if $@;
rename($was,$_) unless $was eq $_;
}
</pre>
<p>
Now if you have a file called <code>file_123.txt</code> and you wanted to rename it to <code>file_abc.txt</code>, you could run the following in a CMD Prompt:
<pre class="brush: bash">
prename "s,\d+,abc," file*
</pre> | {
"redpajama_set_name": "RedPajamaGithub"
} | 1,339 |
Cathal Gannon (1 August 1910 – 23 May 1999), was an Irish harpsichord maker, a fortepiano restorer and an amateur horologist.
Beginnings and education
Gannon was born in Dublin, Ireland, into a craftsmen family of carpenters, many of whom worked in the famous Guinness Brewery. His education, in two local schools, was rudimentary and at the age of fifteen he started working as an apprentice carpenter in the Brewery. His apprenticeship involved learning to make office furniture and attending evening classes in nearby colleges, where he was able to improve his education in a more congenial atmosphere. A love of music and the arts had been encouraged by two maiden aunts – his parents subsequently bought an upright piano and he learned to play it at the Read Pianoforte School – and consequently, when his apprenticeship was completed and he was on the dole for some years, he spent much of his spare time buying pictures, books, antiques and old clocks and watches in the various auction rooms and antique shops in Dublin
Societies and acquaintances
During the mid-1930s, Gannon became a member of several Dublin-based societies, most notably the Old Dublin Society, and there befriended well-known people such as Grace Plunkett (née Gifford), the widow of Joseph Mary Plunkett (who had been executed after the Easter Rising of 1916). At around this time, Cathal was also introduced to Carl Hardebeck, an arranger of Irish traditional music. At a later stage in Cathal's life, he met Desmond Guinness and his wife Mariga, founders of the Irish Georgian Society, which he subsequently joined.
Harpsichords
While reading a series of articles about Tibet in a magazine, Gannon stumbled across an article, which, he believed, was by Violet Gordon Woodhouse, a British harpsichordist and clavichord player of the period. The article was about the revival of the harpsichord, which interested young Gannon. He asked permission to examine the harpsichords on display in the National Museum, Dublin, but was given no encouragement by the staff. He was finally allowed to see the instruments when he was in his early twenties. Dismayed, he concluded that they were too expensive to buy and too complicated to make.
Whilst on holidays in Glengarriff in the West of Ireland during August 1936, Gannon met his future wife, Margaret Key from Harrow in London; they married in 1942.
In London with Margaret, who was visiting her parents, Gannon went to the Benton Fletcher collection of keyboard instruments, which was then in Chelsea, and measured a harpsichord by Jacob and Abraham Kirckman (1777). Back home, he made a copy of the instrument in a tiny conservatory at the back of his house in the Dublin suburb of Rialto. The harpsichord was played by John S. Beckett for the first time in public in 1959 as the continuo for Bach's Saint Matthew Passion and was praised in the national press. Beckett subsequently persuaded the authorities in the Guinness Brewery to provide Gannon with a special workshop, in which he made five harpsichords and restored several antique pianos. The first harpsichord made in the Brewery was donated to the Royal Irish Academy of Music in Dublin, The second was sold to Harrods of London, and the third was sold to Ireland's national radio and television station RTÉ. This third instrument was used regularly by the RTÉ Symphony and Concert orchestras and also by the well-known composer and performer of Irish traditional music, Seán Ó Riada.
Retirement
Gannon continued to make many more harpsichords and restore more pianos during the years to come. In all, he completed 20 harpsichords during his lifetime – the final four were completed by a friend, Patrick Horsley, in England. One of the harpsichords made by Gannon-Horsley returned to Ireland and was presented to NUI Maynooth. A piano of note that Cathal restored was a Broadwood square piano owned by the poet and composer, Thomas Moore, which belonged to Lord and Lady Elveden (later Iveagh).
Gannon was the subject of several RTÉ radio programmes, three RTÉ television programmes (including The Late Late Show) and a television programme, Gallery, made by BBC Northern Ireland. He befriended a great many people, including the artist, writer and conservationist Peter Pearson, and regular musical evenings were held at the family home in Bryan Guinness's grounds in the suburbs of Dublin. Because of his interest in antique clocks and watches, he became a member of the Irish branch of the Antiquarian Horological Society, founded by his friend William Stuart.
Honorary degrees
In 1978, Trinity College Dublin gave Gannon an honorary MA degree for his contribution to the authentic performance of early music in Ireland. Two years later, Cathal was invited to travel with the New Irish Chamber Orchestra to China, where he tuned and maintained one of his harpsichords and celebrated his seventieth birthday. In 1989, a second honorary MA was given to him, this time by NUI Maynooth.
Decline
Following his 80th birthday, which was attended by fifty people, Gannon finally settled down to retirement. A series of minor strokes followed, which eventually led to dementia and ultimately to his death, aged 88, in May 1999.
Legacy
The Cathal Gannon Early Music Room was opened in the Royal Irish Academy of Music in May 2003. It contained a harpsichord and clavichord made by Gannon, a Broadwood grand piano restored by him, and a square piano.
Part of a transmitted RTÉ programme, Nationwide (17 January 2007), featured archive footage of Gannon and his instruments.
Three RTÉ radio programmes, Bowman: Sunday Morning, broadcast on 12 November 19 and 26 November 2006, feature a 1983 interview with Gannon.
Further reading
Byrne, Al: Guinness Times: my days in the world's most famous brewery. (Town House, Dublin, 1999)
Zuckermann, Wolfgang Joachim: The Modern Harpsichord (Peter Owen, London, 1970)
O'Neill, Marie: Grace Gifford Plunkett and Irish Freedom: tragic bride of 1916 (Irish Academic Press, Dublin & Portland, OR, 2000)
Douglas-Home, Jessica: Violet: the life and loves of Violet Gordon Woodhouse (The Harvill Press, London, 1996)
The Guinness Harp magazine, 'The quiet carpenter', March – April 1959, p. 19.
The Guinness Harp magazine, photograph and caption, March – April 1960, p. 5.
The Guinness Harp magazine, 'the most harmonious of all the musical instruments of the string-kind', Vol. 7, Christmas 1964, p. 7.
The Financial Times, 'Men and Matters', 23 September 1965.
The Evening Herald, 'Stradivarius of the harpsichord', 27 September 1965.
Hibernia, 'Cathail (sic) Gannon's harpsichords', January 1968, p. 27.
The Guinness Harp magazine, 'The quiet man', Autumn 1970, p. 39.
Ireland of the Welcomes, 'The harpsichord-maker' by Fachtna O'Kelly, Vol. 21, No. 6, March – April 1973.
Dublin Arts Festival programme for March 1974, p. 7.
The Irish Times, 'Irish Musicians in China', Weekend supplement, 11 October 1980, p. 9.
Irish Conservation Directory, compiled by Susan Corr, 'Keyboard Instruments' by Cathal Gannon, pp. 58 – 60. Irish Professional Conservators' and Restorers' Association, 1988.
Early Music Organisation of Ireland (EMOI) magazine, obituary by Malcolm Proud, Vol. 2, No. 3, July 1999.
References
Bibliography
1910 births
1999 deaths
Businesspeople from Dublin (city)
Irish musical instrument makers
Harpsichord makers | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,733 |
\section{Introduction}
The beam energy scan program (BES and BESII) at the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory \cite{Adamczyk:2013dal,Adamczyk:2014fia,Adare:2015aqk,Adamczyk:2017iwn} and the NA61/SHINE experiment at the CERN Super Proton Synchrotron (SPS) \cite{Mackowiak-Pawlowska:2017rcx} aim to fully explore the phase diagram of strongly interacting matter.
In the baryon-rich region, the transition from a hadron gas to a quark-gluon-plasma phase (QGP) is expected to be first order, whereas it is a rapid crossover at the low baryon density. Therefore, the existence of a critical point in this phase diagram is widely speculated.
The existence of such a critical point has yet to be confirmed, and its location in the phase diagram determined.
Measurements sensitive to the presence of a critical point are those of fluctuations of conserved charges, most commonly those of net-baryon number. In the vicinity of a critical point, the correlation length grows, leading to increased fluctuations \cite{Stephanov:1998dy,Stephanov:1999zu,Stephanov:2004wx}. Varying the collision energy should then allow the trajectory of the system to explore the plane spanned by temperature and baryon chemical potential, and to locate the position of the critical point using the fluctuation measurements.
To accomplish this, one needs to know precisely what to expect for the relevant observables in the case that there is no critical point (and fluctuations are entirely non-critical) and how they are modified if a critical point is present. This requires complex simulations of the entire system starting from fluctuating initial states, hydrodynamic evolution at finite net-baryon density (and possible hydrodynamic fluctuations \cite{Kapusta:2011gt,Young:2014pka,Kapusta:2014dja}), as well as microscopic hadronic cascades for the low temperature stage.
Such simulations can then be used to study the effects of an equation of state with a critical point \cite{Nonaka:2004pg}. They can be coupled to evolution equations for the sigma field and Polyakov loop in the so called chiral fluid dynamics \cite{Paech:2005cx,Herold:2013bi}, and can provide important information required for calculations of the non-equilibrium evolution of cumulants of critical fluctuations \cite{Mukherjee:2015swa}.
Calculations that include the viscous relativistic hydrodynamic evolution of the QGP and hadron gas, combined with models for fluctuating initial states and hadronic afterburners, have been very successful in describing the soft observables measured in heavy ion collisions at top RHIC and Large Hadron Collider (LHC) energies. However, at these high collision energies the net-baryon density is typically assumed to be negligible, which is valid at least near mid-rapidity \cite{Shen:2017bsr}. Furthermore, the initial state description is somewhat simplified because an instantaneous interaction of two highly contracted nuclei can be assumed. At lower energies, neither assumption holds. For reviews on relativistic hydrodynamics and hybrid models of heavy ion collisions we refer the reader to \cite{Heinz:2013th,Gale:2013da} and \cite{Petersen:2014yqa}, respectively.
To make progress towards a simulation framework valid at all collision energies, fluctuating initial conditions for lower energy collisions have been addressed recently \cite{Karpenko:2015xea,Okai:2017ofp,Shen:2017bsr}. The simulation \textsc{Music}\footnote{The numerical package can be downloaded from \url{http://www.physics.mcgill.ca/music}.} \cite{Schenke:2010nt} has included the evolution of conserved baryon currents from the beginning, but baryon diffusion has so far been neglected. However, when studying observables that are sensitive to the precise baryon distributions and their fluctuations \cite{Kapusta:2017hfi}, we need to take great care in including all relevant physics in the simulation. In this work we present results of an extended version of \textsc{Music} that includes the most basic effects of baryon diffusion.
Apart from this extension of the hydrodynamic simulation itself, we need to consider an equation of state at finite baryon density. We present a construction of such an equation of state using Lattice QCD results with Taylor expansion in baryon chemical potential coupled to a hadron resonance gas, and use it in all shown calculations. Current lattice QCD simulations have not shown evidence of a critical point and, hence, our results at this stage do not probe any effects from a critical point -- hence, in this aspect, they can be considered as baseline calculations.
Besides providing a necessary tool for simulating heavy ion collisions over a wide range of energies relevant to the critical point search, the new developments presented in this work also establish a path to the extraction of the heat conductivity of the quark gluon plasma by detailed comparison with experimental measurements. We identify observables that are most sensitive to the effect of baryon diffusion and thus the heat conductivity of the QGP.
The paper is organized as follows. Sec.~\ref{sec:model} gives a detailed model description of our hybrid framework. The phenomenological impact of net baryon diffusion and hadronic transport on experimental observables are studied in Sec.~\ref{sec:results}. The focus of our studies are Au+Au collisions at 19.6 GeV. Sec.~\ref{sec:conclusion} summarizes the main findings of this work. Additional detailed derivations of net baryon diffusion corrections and numerical validation of the hydrodynamic simulation are presented in the appendices.
\section{The Hybrid framework}\label{sec:model}
\subsection{Initialization of hydrodynamics}
For very high center of mass energies, like the top RHIC energy or LHC energies, the Lorentz contraction of the incoming nuclei is so strong that it is a good approximation to consider them as sheets of negligible width in the longitudinal (beam-) direction. This means that the time of the collision is given precisely by the time the two sheets pass through each other.
In contrast, the collision energies scanned in the RHIC BES program and the NA61/SHINE experiment are not high enough to neglect the finite thickness of the colliding nuclei along the longitudinal direction. The time the two nuclei spend passing through one another for a given collision energy $\sqrt{s_\mathrm{NN}}$ can be estimated as
\begin{equation}
\tau_\mathrm{overlap} = \frac{2 R}{\gamma_L v_L} = \frac{2 R}{\sqrt{\gamma_L^2 - 1}},
\label{eq2.1}
\end{equation}
where the Lorentz factor in the longitudinal direction is $\gamma_L = \frac{\sqrt{s_\mathrm{NN}}/2}{m_N}$ with $m_N = 0.938$ GeV, and $R$ is the radius of the colliding nuclei. For gold nuclei $R_\mathrm{Au} \simeq 7.0$\,fm.
At the lowest BES collision energy of $\sqrt{s_\mathrm{NN}} = 7.7$ GeV, this overlapping time is $3\,{\rm fm}$, comparable to the lifetime of the QGP created in the system.
In Ref.\,\cite{Shen:2017bsr} two of the authors have presented a new initial state model that treats the early stage of the evolution dynamically by starting hydrodynamic evolution before that time and taking care of additional deposited entropy and baryon densities via source terms.
Because the focus of this work is the effect of baryon diffusion, we employ a simpler initial state description, where the initial entropy and baryon densities are assumed to be smooth average quantities and the hydrodynamic simulations are started at $\tau_0 = \tau_\mathrm{overlap}$.
The smooth initial conditions are generated by averaging over 10,000 fluctuating Monte Carlo (MC)-Glauber events in the given centrality bin, which is determined using the configurations' total entropy. When averaging the spatial structure, events within the same centrality bin are aligned using their second-order participant plane angles, $\Psi^\mathrm{PP}_2$, defined as
\begin{equation}
\varepsilon_2 e^{i 2 \Psi^\mathrm{PP}_2} = - \frac{\int d^2 {\bf r}\,r^2 s(r, \phi) e^{i2\phi}}{\int d^2 {\bf r}\,r^2 s(r, \phi)}.
\label{eq2.2}
\end{equation}
Here $s(r, \phi)$ is the transverse plane entropy density profile at mid-rapidity.
To construct the entropy density as a function of the transverse coordinates and of the space-time rapidity,
we first define the contributions from the right moving ($+$) and left moving ($-$) nuclei as
\begin{equation}
s_\pm(x, y) = \sum_{j = 1}^{N^\pm_\mathrm{part}} \frac{1}{2\pi \sigma^2 }\exp\left(-\frac{({\bf r-r}^\pm_j)^2}{2\sigma^2} \right)\,,
\end{equation}
where $\mathbf{r}=(x,y)$ and $\mathbf{r}^\pm_j$ are the positions of the partici\-pant nucleons in the two nuclei.
The Gaussian width parameter is set to $\sigma = 0.5$ fm.
The full initial 3D density profiles follow from folding $s_\pm (x, y)$ with envelope functions along the rapidity direction,
\begin{equation}
s(x, y, \eta; \tau_0) = \frac{s_0}{\tau_0} \sum_{i=\pm} f^{s}_i (\eta) s_i (x, y).
\label{eq:entropydensity}
\end{equation}
Here $s_0$ is the peak entropy density which is adjusted to reproduce the experimentally observed charged hadron multiplicities.
\begin{figure*}[ht!]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.48\linewidth]{figs/entropy_envelop_function} &
\includegraphics[width=0.48\linewidth]{figs/rhoB_envelop_function}
\end{tabular}
\caption{Example of the envelope functions for entropy density and net baryon density $f^s_\pm(\eta_s)$ and $f^{n_B}_\pm(\eta_s)$ in Au+Au collisions at $\sqrt{s_\mathrm{NN}} = 19.6$\,GeV.}
\label{fig2}
\end{figure*}
Similarly, the net baryon density profile can be constructed as
\begin{equation}
n_B(x, y, \eta; \tau_0) = \frac{1}{\tau_0} \sum_{i=\pm} f^{n_B}_i (\eta) s_i (x, y).
\label{eq:baryondensity}
\end{equation}
There is no additional normalization factor for the net baryon density because it is constrained by the total number of participant nucleons $N_\mathrm{part}$, $\int \tau_0 dx dy d\eta n_B(x, y, \eta; \tau_0) = N_\mathrm{part}$. The envelope functions in Eqs. (\ref{eq:entropydensity}) and (\ref{eq:baryondensity}) are chosen as,
\begin{eqnarray}
&&f^s_\pm(\eta) = \theta(\eta_\mathrm{max} - \vert \eta \vert) \left(1 \pm \frac{\eta}{\eta_\mathrm{max}} \right) \notag \\
&& \,\times \left[\theta(\vert \eta \vert - \eta^s_0) \exp\left(-\frac{(|\eta| - \eta^s_0)^2}{2\sigma_{\eta,s}^2} \right) + \theta(\eta^s_0 - \vert \eta \vert) \right]
\label{eq:entropyEnvelope}
\end{eqnarray}
where the maximum extension in space-time pseudo-rapidity $\eta_\mathrm{max}$ is chosen to be equal to the beam rapidity $y_\mathrm{beam} = \mathrm{arctanh}\left( \frac{\sqrt{\gamma_L^2 - 1}}{\gamma_L} \right)$ of incoming nucleons. The parameters $\eta^s_0$ and $\sigma_{\eta, s}$ are determined to reproduce the pseudo-rapidity distribution of charged hadrons $dN^\mathrm{ch}/d\eta$.
For the net baryon density envelope profile,
\begin{eqnarray}
f^{n_B}_\pm (\eta) &=& \frac{1}{\mathcal{N}} \left[\theta(\eta - \eta^{n_B, \pm}_0) \exp\left(-\frac{(\eta - \eta^{n_B, \pm}_0)^2}{2\sigma_{\eta,\pm}^2} \right) \right. \notag \\
&& \left. + \theta(\eta^{n_B, \pm}_0 - \eta) \exp\left(-\frac{(\eta - \eta^{n_B, \pm}_0)^2}{2\sigma_{\eta, \mp}^2} \right) \right]
\label{eq:baryonEnvelope}
\end{eqnarray}
where $\mathcal{N}$ is the normalization of the envelope profile which ensures
\begin{equation}
\int d\eta f^{n_B}_\pm (\eta) = 1.
\end{equation}
The peak position $\eta^{n_B,\pm}_0$ is determined by the measured rapidity loss in the net proton distribution and the width parameters $\sigma_{\eta, \pm}$ determine the shape of the final $dN^{p - \bar{p}}/dy$.
Figure \ref{fig2} shows an example of the $\eta_s$ envelope functions for entropy density and net baryon density. The parameters in Eqs.\ (\ref{eq:entropyEnvelope}) and (\ref{eq:baryonEnvelope}) are determined for Au+Au collisions and shown in Table \ref{table1} for different collision energies.
\begin{table}[ht!]
\centering
\begin{tabular}{c|c|c|c|c|c|c|c|c}
\hline \hline
$\sqrt{s_\mathrm{NN}}$ (GeV) & $y_\mathrm{beam}$ & $\tau_0$ (fm) & $s_0$ & $\eta^s_0$ & $\sigma_{\eta,s}$ &$\eta^{n_B}_0$ & $\sigma_{\eta,+}$ & $\sigma_{\eta,-}$ \\ \hline
19.6 & 3.04& 1.5 & 6.3 & 2.7 & 0.3 & 1.5 & 0.2 & 1.0 \\ \hline
\hline
\end{tabular}
\caption{A list of parameters for MC-Glauber initial conditions for Au+Au collisions at different collision energies.}
\label{table1}
\end{table}
\begin{figure*}[ht!]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.48\linewidth]{figs/kappa_B} &
\includegraphics[width=0.48\linewidth]{figs/eta_over_s}
\end{tabular}
\caption{The temperature and net baryon chemical potential dependence of the net baryon diffusion constant and specific shear viscosity for $C_B = 0.4$ and $C_\eta = 008$.}
\label{fig2B.0}
\end{figure*}
\subsection{Hydrodynamics at finite baryon density}
The hydrodynamical equation of motion at finite net baryon density can be written as,
\begin{equation}
\partial_\mu T^{\mu\nu} = 0,
\label{eq2.10}
\end{equation}
\begin{equation}
\partial_\mu J_B^{\mu} = 0,
\label{eq2.11}
\end{equation}
where the system's energy momentum tensor can be decomposed as
\begin{equation}
T^{\mu\nu} = eu^\mu u^\nu - (P + \Pi) \Delta^{\mu\nu} + \pi^{\mu\nu},
\end{equation}
and
\begin{equation}
J_B^{\mu} = n_B u^\mu + q^\mu.
\end{equation}
Here $\Delta^{\mu\nu} = g^{\mu\nu}-u^\mu u^\nu$ is a projection operator, $u^\mu$ is the flow velocity, and $g^{\mu\nu}={\rm diag}(1,-1,-1,-1)$ is the space-time metric.
The dissipative currents in the system are the bulk viscous pressure $\Pi$, the net baryon diffusion current $q^\mu$, and shear stress tensor $\pi^{\mu\nu}$. In this work, we consider only the effects of the shear stress tensor and net baryon diffusion. These two currents are described by the Israel-Stewart-like equations,
\begin{eqnarray}
\Delta^{\mu\nu} D q_\nu &=& - \frac{1}{\tau_q} \left(q^\mu - \kappa_B \nabla^\mu \frac{\mu_B}{T} \right) - \frac{\delta_{qq}}{\tau_q} q^\mu \theta - \frac{\lambda_{qq}}{\tau_q} q_\nu \sigma^{\mu\nu} \notag \\
&& + \frac{l_{q\pi}}{\tau_q} \Delta^{\mu\nu} \partial_{\lambda} \pi^{\lambda}\,_\nu - \frac{\lambda_{q\pi}}{\tau_q} \pi^{\mu\nu} \nabla_\nu \frac{\mu_B}{T},
\label{eq2.14}
\end{eqnarray}
and
\begin{eqnarray}
\Delta^{\mu\nu}_{\alpha \beta} D \pi^{\alpha \beta} &=& - \frac{1}{\tau_\pi} (\pi^{\mu\nu} - 2 \eta \sigma^{\mu\nu}) \notag \\
&& - \frac{\delta_{\pi\pi}}{\tau_{\pi}} \pi^{\mu\nu} \theta - \frac{\tau_{\pi\pi}}{\tau_\pi}\pi^{\lambda \langle} \sigma^{\nu \rangle}\,_\lambda
+ \frac{\phi_7}{\tau_\pi} \pi^{\langle \mu}\,_\alpha \pi^{\nu \rangle \alpha} \notag \\
&& + \frac{l_{\pi q}}{\tau_\pi} \nabla^{\langle \mu} q^{\nu \rangle} + \frac{\lambda_{\pi q}}{\tau_\pi} q^{\langle \mu} \nabla^{\nu \rangle} \frac{\mu_B}{T}\,.
\label{eq2.15}
\end{eqnarray}
Here the evolution of the diffusion current is driven by the gradient of the net baryon chemical potential $\mu_B$ divided by temperature $T$. The thermodynamic force for the shear viscous pressure is the velocity shear tensor $\sigma^{\mu\nu} = \nabla^{\langle \mu} u^{\nu\rangle}$, and $A^{\langle \mu\nu\rangle} = \Delta^{\mu\nu}_{\alpha\beta} A^{\alpha\beta}$ projects out the part that is traceless and transverse to the flow velocity $u_\mu$ using the double, symmetric, and traceless projection operator, $\Delta^{\mu\nu}_{\alpha\beta} = \frac{1}{2}\left[\Delta^\mu_\alpha \Delta^\nu_\beta + \Delta^\nu_\alpha \Delta^\mu_\beta -\frac{2}{3}\Delta^{\mu\nu}_{\alpha\beta}\right]$. The system's expansion rate is $\theta = \partial_\mu u^\mu + u^\tau/\tau$.
The transport coefficients $\eta$ and the baryon diffusion constant $\kappa_B$ are chosen as
\begin{equation}
\frac{\eta T}{e + \mathcal{P}} = C_\eta
\end{equation}
and
\begin{equation}
\kappa_B = \frac{C_B}{T} n_B \left(\frac{1}{3} \coth\left(\frac{\mu_B}{T}\right) - \frac{n_B T}{e + \mathcal{P}} \right).
\label{eq2.17}
\end{equation}
The specific shear viscosity is chosen to be $C_\eta = 0.08$.
The constant coefficient $C_B$ will be varied to study the effect of the net baryon diffusion. The $T$ and $\mu_B$ dependence of $\kappa_B$ in Eq.~(\ref{eq2.17}) is derived from the Boltzmann equation in the relaxation time approximation (see Appendix \ref{appendix_A}). We show the dimensionless quantity $\kappa_B \mu_B/n_B$ along with $\eta/s$ as functions of $T$ and $\mu_B$ in Fig.~\ref{fig2B.0}. Four lines of constant $s/n_B$, that reflect the averaged values realized at the four different collision energies we consider, demonstrate what values of the transport parameters typically contribute.
\begin{table}[ht!]
\def1.5{1.5}
\centering
\begin{tabular}{c|c|c|c|c|c}
\hline \hline
$\tau_q$ & $\delta_{qq}$ & $\lambda_{qq}$ & $l_{q\pi}$ & $\lambda_{q\pi}$ & \\ \hline
$\frac{C_B}{T} $ & $\tau_q$ & $\frac{3}{5}\tau_q$ & 0 & 0 & \\ \hline \hline
$\tau_\pi$ & $\delta_{\pi\pi}$ & $\tau_{\pi\pi}$ & $\phi_7$ & $l_{\pi q}$ & $\lambda_{\pi q}$ \\ \hline
$\frac{5C_\eta}{T}$ & $\frac{4}{3}\tau_\pi$ & $\frac{10}{7}\tau_\pi$ & $\frac{9}{70} \frac{4}{\varepsilon + \mathcal{P}}$ & 0 & 0 \\ \hline \hline
\end{tabular}
\caption{A list for the second order transport coefficients used in the evolution equations for the net baryon diffusion current $q^\mu$ and the shear stress tensor $\pi^{\mu\nu}$.}
\label{table2}
\end{table}
Table~\ref{table2} summarizes the choice of the second order transport coefficients used in Eqs.~(\ref{eq2.14}) and (\ref{eq2.15}).
The expression for the baryon diffusion relaxation time, $\tau_{q}$ is chosen to be proportional to $1/T$ (as it is exactly in a conformal system), with the proportionality constant $C_B$ a free parameter. The remaining transport coefficients listed in the table are from calculations assuming kinetic theory in the massless limit \cite{Denicol:2010xn,Denicol:2012cn,Molnar:2013lta,Denicol:2014vaa}. Recent calculations of transport coefficients taking into account a finite (and thermal) mass, were performed in Ref.\,\cite{Czajka:2017wdo}.
\begin{figure*}[ht!]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.48\linewidth]{figs/EoS_speed_of_sound} &
\includegraphics[width=0.48\linewidth]{figs/EoS_muB_vs_T}
\end{tabular}
\caption{{\it Panel (a)}: The square of the speed of sound as a function of the local energy density along constant $s/n_B$ lines. {\it Panel (b)}: Temperature as a function of net baryon chemical potential along constant $s/n_B$ lines. The collision energies correspond to these $s/n_B$ lines (from top down in the legend) are $\sqrt{s_\mathrm{NN}} = 200, 62.4, 19.6$, and $14.5$\,GeV according to Ref.~\cite{Gunther:2016vcp}.}
\label{fig2B.1}
\end{figure*}
The system of hydrodynamic equations (\ref{eq2.10}) and (\ref{eq2.11}) needs to be closed with the equation of state (EoS) of the fluid. In this work, the EoS of the QCD matter is constructed using lattice QCD calculations \cite{Borsanyi:2013bia,Borsanyi:2011sw}. We consider a crossover-type EoS and leave implementation of the QCD critical point for future study. At zero baryon chemical potential, the pressure of the system is computed as a function of the local temperature via \cite{Huovinen:2009yb},
\begin{equation}
\frac{\mathcal{P}(T)}{T^4} = \frac{\mathcal{P}(T_\mathrm{low})}{T^4_\mathrm{low}} + \int_{T_\mathrm{low}}^T \frac{d T^\prime}{T^\prime} \frac{e - 3\mathcal{P}}{T^{\prime4}},
\end{equation}
where the trace anomaly $e - 3\mathcal{P}$ is computed from lattice QCD as a function of temperature. The lower integration limit $T_\mathrm{low}$ is chosen to be sufficiently small such that $\mathcal{P}(T_\mathrm{low})$ can be neglected because of the exponential suppression. Since of the sign problem, it is not possible to directly calculate the EoS at finite baryon density using lattice QCD. Instead, the $\mu_B$ dependence of the EoS is constructed using the following Taylor expansion,
\begin{eqnarray}
\frac{\mathcal{P}(T, \mu_B)}{T^4} &=& \frac{\mathcal{P}(T)}{T^4} \bigg\vert_{\mu_B = 0} + c_2(T) \left(\frac{\mu_B}{T}\right)^2 + c_4(T) \left(\frac{\mu_B}{T}\right)^4 \notag \\
&& + \mathcal{O}\left( \left(\frac{\mu_B}{T}\right)^6\right) ,
\end{eqnarray}
where $c_2(T)$ and $c_4(T)$ are the expansion coefficients. The former coefficient is extracted using lattice QCD susceptibility calculations \cite{Borsanyi:2011sw} while the latter follows from ratios of the second and fourth order susceptibilities, computed hadron resonance and parton gas pictures. It is noteworthy that the lattice QCD EoS and baryon susceptibilities are known to agree with those of the resonance gas slightly below the crossover. For temperatures below the transition temperature $T_\mathrm{trans}(\mu_B)$, the lattice QCD EoS is smoothly matched to the hadron resonance gas EoS because the Taylor expansion is not well defined at lower $T$ and energy, momentum, and net baryon number need to be conserved at the Cooper-Frye freeze-out \cite{Monnai:2015sca}:
\begin{eqnarray}
\frac{\mathcal{P}}{T^4} &=& \frac{1}{2}\left(1- \tanh \frac{T-T_\mathrm{trans}}{\Delta T_\mathrm{trans}} \right) \frac{\mathcal{P}_{\mathrm{HRS}}(T,\mu_B)}{T^4} \notag \\
&+& \frac{1}{2}\left(1+ \tanh \frac{T-T_\mathrm{trans}}{\Delta T_\mathrm{trans}} \right) \frac{\mathcal{P}_{\mathrm{lat}}(T_s,\mu_B)}{T_s^4} .
\end{eqnarray}
For the transition temperature of the two EoS, we use the ansatz $T_\mathrm{trans}(\mu_B) = 0.166~\mathrm{GeV} - 0.4 (0.139~\mathrm{GeV}^{-1} \mu_B^2 + 0.053~\mathrm{GeV}^{-3} \mu_B^4)$ motivated by a chemical freeze-out curve \cite{Cleymans:2005xv}. The shift $T_s = T + 0.4[T_\mathrm{trans}(0) - T_\mathrm{trans}(\mu_B)]$ is introduced to ensure that thermodynamic variables are increasing functions of $T$ and $\mu_B$ at very large baryon chemical potential, and should not affect much the bulk dynamics. The entropy density, the net baryon number, and the energy density can be obtained from the thermodynamic relations $s = \partial \mathcal{P}/\partial T \vert_{\mu_B}$, $n_B = \partial \mathcal{P}/\partial \mu_B \vert_{T}$, and $e = Ts - \mathcal{P} + \mu_B n_B$.
The speed of sound squared at the finite $\mu_B$ is computed as
\begin{equation}
c_s^2 (e, n_B) = \frac{\partial \mathcal{P}}{\partial e} \bigg\vert_{n_B} + \frac{n_B}{(e + \mathcal{P})} \frac{\partial \mathcal{P}}{\partial n_B} \bigg\vert_{e}.
\end{equation}
To see whether we should expect a large effect on the collision dynamics from the finite $\mu_B$ values present in smaller energy collisions, in Fig.~\ref{fig2B.1}a we plot $c_s^2$ as a function of local energy density for several constant $s/n_B$ values. Again, the shown values of $s/n_B$ correspond to the considered collision energies. From $\sqrt{s_\mathrm{NN}} = 14.5$ to $\sqrt{s_\mathrm{NN}} = 200$ GeV, the square of the speed of sound does not change significantly. The constant $s/n_B$ trajectories are shown in the $T-\mu_B$ plane in Fig.~\ref{fig2B.1}b.
In Appendix~\ref{appendix_C} we present several validation studies of our 3+1D numerical hydrodynamic implementation at finite baryon density.
\subsection{Particlization and hadronic cascade}
\label{sec:particlization}
As the temperature drops in the hadronic phase, we convert the macroscopic fluid cells into particle samples via the Cooper-Frye procedure \cite{Cooper:1974mv}. At finite $\mu_B$, we choose to perform the Cooper-Frye conversion on a constant switching energy density hyper-surface, $e_\mathrm{sw} = 0.4$ GeV/fm$^3$. This is because the chosen constant energy density line in the $T-\mu_B$ plane follows very well the chemical freeze-out points extracted from the thermal fits done by the STAR Collaboration \cite{Adamczyk:2017iwn}. This is demonstrated in Fig.~\ref{fig2B.2}, where we vary $e_\mathrm{sw}$.
\begin{figure}[ht!]
\centering
\includegraphics[width=1.0\linewidth]{figs/EoS_freezeout_line}
\caption{Constant energy density freeze-out lines compared with the extracted chemical freeze-out points from the STAR collaborations \cite{Adamczyk:2017iwn}. }
\label{fig2B.2}
\end{figure}
Because of the long overlapping time at low collision energies, one would have expected that non-negligible amount of matter had already flown out of the switching hyper-surface before the hydrodynamic simulation starts. This is usually referred to as ``corona''. We define the corona as those cells whose local energy densities are between 0.05 GeV/fm$^3$ and $e_{sw} = 0.4$ GeV/fm$^3$. We use the Cooper-Frye formula to convert these corona fluid cells to particles at the first time step of the hydrodynamic evolution and then feed them into the hadronic transport simulation. Because there is no transverse flow velocity at the starting time of the hydrodynamic simulations, these corona particles are emitted isotropically according to their local thermal equilibrium distributions. The effect of the corona on hadronic flow observables will be discussed in the next section.
The momentum distribution of thermally emitted particles from one fluid cell is,
\begin{eqnarray}
E \frac{d^3 N_i}{d^3 p} &=& \frac{g_i}{(2\pi)^3}p^\mu \Delta^3 \sigma_\mu \bigg(f^\mathrm{eq}_i(E, T, \mu_B) \notag \\
&& + \delta f^\mathrm{shear}_i(E, T, \mu_B, \pi^{\mu\nu}) \notag \\
&& + \delta f^\mathrm{diffusion}_i(E, T, \mu_B, q^\mu) \bigg) \bigg\vert_{E = p \cdot u},
\label{eq.CooperFrye}
\end{eqnarray}
where the $\delta f^\mathrm{shear}$ and $\delta f^\mathrm{diffusion}$ are the out-of-equilibrium corrections from shear viscosity and net baryon diffusion.
As in previous work \cite{Ryu:2015vwa} we employ
\begin{eqnarray}
\delta f^{\rm shear}_i= f^\mathrm{eq}_i(x, p) (1 \pm f^\mathrm{eq}_i(x, p)) \frac{p^\mu p^\nu \pi_{\mu\nu}}{2 T^2 (e + P)}.
\end{eqnarray}
In the relaxation time approximation, the net baryon diffusion $\delta f^\mathrm{diffusion}$ for a single species of particle $i$ is \cite{Albright:2015fpa,Jaiswal:2015mxa}
\begin{eqnarray}
\delta f^\mathrm{diffusion}_i (x, p) &=& f^\mathrm{eq}_i(x, p) (1 \pm f^\mathrm{eq}_i(x, p)) \notag \\
&& \times \left(\frac{n_B}{e + \mathcal{P}} - \frac{b_i}{E} \right) \frac{p^{\langle \mu \rangle} q_\mu}{\hat{\kappa}_B},
\label{eq.diffusion_deltaf}
\end{eqnarray}
where $b_i$ is the baryon number of particle species $i$, $p^{\langle \mu \rangle}=\Delta^{\mu\nu}p_\nu$, and the transport coefficient $\hat{\kappa}_B$ is defined in Appendix~\ref{appendix_A}. An alternative form of diffusion out-of-equilibrium correction was recently derived using the 14-moment method \cite{Monnai:2018rgs}.
We note that $\delta f_i^\mathrm{diffusion}$ is non-zero even for mesons (that have zero baryon number). This is because the changes in the baryon chemical potential can lead to variations in the thermal pressure, which will change the momentum distributions of mesons.
Using Eq.\,(\ref{eq.CooperFrye}) the system's total net baryon number can be computed as
\begin{eqnarray}
N^B - N^{\bar{B}} &=& \int d^3 \sigma_\mu \sum_i g_i b_i \notag \\
&& \times \int_p p^\mu (f^\mathrm{eq}_i + \delta f^\mathrm{shear}_i + \delta f^\mathrm{diffusion}_i) \notag \\
&=& \int d^3 \sigma_\mu (n_B u^\mu + q^\mu)\,,\label{eq:baryonNumber}
\end{eqnarray}
where $\int_p= \int \frac{d^3p}{E (2\pi)^3}$.
Because the hydrodynamic equation solves $\partial_\mu (n_B u^\mu + q^\mu) = 0$, the net baryon number is conserved during the hydrodynamic evolution as well as on the conversion surface before and after the conversion.
The inclusion of $\delta f^\mathrm{diffusion}_i$ in the Cooper-Frye formula takes into account contributions from the diffusion current $q^\mu$ in Eq.\,(\ref{eq:baryonNumber}) and is essential to ensure the conservation of net baryon number during the conversion from fluid cells to particles.
In this work, we generalized the publicly available numerical code \textsc{iSS}\footnote{The latest version of the code package can be downloaded from \url{https://github.com/chunshen1987/iSS}.} to perform the particlization simulations. Detailed implementation and cross checks for the numerical procedure are discussed in Appendix~\ref{appendix_B}.
After the particle conversion, we feed particles into hadronic cascade models, such as UrQMD \cite{Bass:1998ca,Bleicher:1999xi} and JAM\footnote{The latest version of JAM can be downloaded from \url{http://www.aiu.ac.jp/~ynara/jam/}} \cite{Nara:1999dz}, to simulate the transport dynamics in the dilute hadronic phase.
\section{Collectivity in Au+Au collisions at RHIC BES energies}\label{sec:results}
We will focus our study of hadronic flow observables on central and semi-peripheral Au+Au collisions at 19.6 GeV. At this collision energy, the baryon chemical potential can reach up to $\sim$200 MeV in the mid-rapidity region. Consequently, we expect the net baryon current and its diffusion to have sizeable effects on the hadronic flow observables near the mid-rapidity region which can be measured by the STAR experiments.
\subsection{Hydrodynamical evolution with net baryon diffusion}\label{sec:evolution}
Based on the hydrodynamic equations of motion, the net baryon diffusion current only directly affects the evolution of the net baryon density. Nevertheless, it modifies the system's energy density and flow velocity evolution indirectly, via the modification of the pressure $\mathcal{P}(e, n_B)$, given by the equation of state. Thus we expect this dissipative current to have less influence on the system's evolution compared to the usual dissipative effects due to shear and bulk viscosities.
To understand the effect of net baryon diffusion on hadronic flow observables, it is instructive to study the time evolution of $\mu_B/T$, whose spatial gradients are the thermal dynamic force of the net baryon diffusion current, $q^\mu$.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.95\linewidth]{figs/muB_over_T_etas_dependence}
\includegraphics[width=0.95\linewidth]{figs/muB_over_T_trans_dependence}
\caption{Time evolution of the $\mu_B/T$ along longitudinal direction for points at $x = 0$ and $y = 0$ (panel (a)) and transverse plane along $y = 0$ and $\eta_s = 0$ (panel (b)).}
\label{fig3A.0}
\end{figure}
Figure \ref{fig3A.0} shows the time evolution of $\mu_B/T$ along the longitudinal and transverse directions. At the starting time of the hydrodynamic simulation, the ratio $\mu_B/T$ peaks around $\eta_s = \pm 1.5$. The gradients of $\mu_B/T$ dominantly point to the mid-rapidity region. Thus the baryon diffusion current will transport more baryons from forward rapidities to the central rapidity region. We also find that the value of $\mu_B/T$ increases in dilute energy density regions in both very forward rapidity and in towards the edges in the transverse plane. Such a distribution leads to the spatial gradients of $\mu_B/T$ pointing opposite to the pressure gradients. From these observations in the longitudinal and transverse directions, we expect that the net baryon diffusion current $q^\mu$ will act against the hydrodynamic flow, and will reduce the net baryon flow coefficients.
\subsection{Effects of baryon diffusion on observables}
In this section, we study how baryon diffusion in the hydrodynamic simulations affects various experimental observables. We vary the amount of diffusion by tuning the value of the pre-factor $C_B$ in Eq.~(\ref{eq2.17}).
\begin{figure}[ht!]
\centering
\includegraphics[width=0.95\linewidth]{figs/charghed_hadron_rapidity_distribution_diffusion_effects}
\includegraphics[width=0.95\linewidth]{figs/net_proton_dNdy_diffusion_effects}
\caption{{\it Panel (a)}: The pseudo-rapidity and rapidity distributions of the charged hadron, identified $\pi^+$, and $K^+$ compared to the PHOBOS and STAR measurements in 0-5\% Au+Au collisions at 19.6 GeV\cite{Back:2005hs,Adamczyk:2017iwn}. {\it Panel (b)}: The net proton rapidity distribution with different choices of the net baryon diffusion constant compared with the STAR measurements.}
\label{fig3A.1}
\end{figure}
Figure \ref{fig3A.1}a shows the rapidity distribution of produced hadrons in the top 0-5\% central Au+Au collisions at $\sqrt{s}=$19.6 GeV. The system's total entropy is tuned to reproduce the positive pion yield at mid-rapidity, measured by the STAR collaboration \cite{Adamczyk:2017iwn}. The rapidity envelope profile is tuned to reproduce the rapidity dependence measured by the PHOBOS collaboration \cite{Back:2005hs}. The charged hadron multiplicity is slightly overestimated mainly because the PHOBOS measurement is for the 0-6\% centrality. The net baryon diffusion has negligible effect on mesons and charged hadrons.
In Fig.~\ref{fig3A.1}b we demonstrate that the rapidity dependence of net protons is sensitive to the magnitude of the baryon diffusion coefficient. As discussed in Sec.~\ref{sec:evolution}
, the baryon diffusion current is driven by gradients of $\mu_B/T$, which transports net baryons from forward rapidity to the mid-rapidity region. This effect is visibly stronger for larger $C_B$.
Unfortunately, the measured shape of the net proton rapidity distribution cannot be used to constrain the amount of net baryon diffusion, because of the theoretical uncertainties in determining the initial baryon stopping. This is explicitly demonstrated in Fig.~\ref{fig3A.2}a, where we have adjusted the initial baryon rapidity distribution for given values of $C_B$. This shows that approximately the same final distribution is obtained for largely different baryon diffusion currents.
\begin{figure}[ht!]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.95\linewidth]{figs/net_proton_dNdy_diffusion_effects_fitdNdy} \\
\includegraphics[width=0.95\linewidth]{figs/pid_sp_pion_p_diffusion_effects} \\
\includegraphics[width=0.95\linewidth]{figs/pid_sp_p_vs_pbar_diffusion_effects}
\end{tabular}
\caption{{\it Panel (a)}: The fit of the net proton rapidity distribution with different choices of the net baryon diffusion constant in the simulations. {\it Panel (b)}: The single particle spectra of $\pi^+$ and $K^+$ with different choices of the net baryon diffusion constant. {\it Panel (c)}: The single particle spectra of proton and anti-proton with different choices of the net baryon diffusion constant. }
\label{fig3A.2}
\end{figure}
It is possible to find further constraints by considering both experimental data sensitive to longitudinal and transverse dynamics simultaneously. For a given $C_B$, the initial condition can be constrained by the net proton rapidity distribution as above, and studying the transverse dynamics of the collision system could then be used to distinguish different $C_B$ values.
Figures \ref{fig3A.2}b and \ref{fig3A.2}c show transverse momentum spectra of identified particles. The $p_T$-spectra of light mesons, such as $\pi^+$ and $K^+$, are insensitive to the net baryon diffusion as expected. Proton and anti-proton spectra obtained using different degrees of net baryon diffusion are compared in Fig.~\ref{fig3A.2}c. The effect of the net baryon diffusion constant $C_B$ looks small in the plot. To better quantify the effect, we compare the difference in the average transverse momentum of protons and anti-protons. The result is shown in Table~\ref{table3} for different choices of $C_B$.
\begin{table}[ht!]
\centering
\begin{tabular}{c|c|c|c}
\hline \hline
& $C_B = 0.0$ & $C_B = 0.4$ & $C_B = 1.2$ \\ \hline
$\langle p_T \rangle(\bar{p}) - \langle p_T \rangle(p)$ (GeV) & 0.049 & 0.079 & 0.134 \\ \hline
$\langle p_T \rangle(\bar{p}) - \langle p_T \rangle(p)$ (GeV) & & & \\
(no diffusion $\delta f$) & 0.049 & 0.050 & 0.056 \\
\hline \hline
\end{tabular}
\caption{The difference of the averaged transverse momentum between anti-protons and protons at different values of the net baryon diffusion constant $C_B$.}
\label{table3}
\end{table}
The hydrodynamic simulation produces a slightly larger mean-$p_T$ for anti-protons than for protons. This difference grows with increasing net baryon diffusion. Part of this effect is caused by the diffusive evolution, because the $\mu_B/T$ gradient in the transverse plane tends to diffuse net baryon number into the central region where the radial flow is relatively smaller. An even larger contribution to the mean-$p_T$ difference is due to the baryon diffusion $\delta f$ corrections to the baryon spectra. We will discuss this effect in more detail in the next section.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.9\linewidth]{figs/pid_v2_pion_p_diffusion_effects}
\includegraphics[width=0.9\linewidth]{figs/pid_v2_proton_diffusion_effects}
\includegraphics[width=0.9\linewidth]{figs/pid_v2_anti_proton_diffusion_effects}
\caption{The $p_T$-differential elliptic flow coefficients of identified $\pi^+$ (panel (a)), $p$ (panel (b)), and $\bar{p}$ (panel (c)) with different net baryon diffusion constants in the hydrodynamic simulations for 20-30\% Au+Au collisions at 19.6 GeV. The shaded bands indicate statistical errors.}
\label{fig:v2}
\end{figure}
In Fig.\,\ref{fig:v2} we show the transverse momentum elliptic flow coefficient $v_2$ for positive pions (panel a), protons (b), and anti-protons (c). As for the transverse momentum spectra, pion $v_2$ does not change within statistical errors with the value of $C_B$. For protons and anti-protons we find a sizeable and opposite effect: Proton $v_2$ decreases with increasing $C_B$, while anti-proton $v_2$ increases. We will show in the following section that the differences are to a large part generated by off-equilibrium corrections to the distribution functions.
We find that the net baryon diffusion has a small influence on the system's transverse dynamics, such as the hydrodynamic flow pattern. The major effects to the baryonic observables are coming from the off-equilibrium corrections at the freeze-out stage. Because of this, baryon diffusion cannot be constrained quantitatively from experimental data in our current analyses.
It should be noted that unlike the other particle species, $\bar{p}$ is not well described for any $C_B$. We expect that it should be possible to improve the agreement by fine tuning both the initial entropy and net-baryon distributions, as well as the switching energy density.
\subsection{The effects of the out-of-equilibrium corrections from net baryon diffusion}
Figure \ref{fig3B.1} shows the effect of the out-of-equilibrium corrections to the particle distributions on net proton rapidity spectra. We show separately the effect from shear viscous corrections and baryon diffusion corrections. The $\delta f$ correction from the net baryon diffusion current reduces the net proton yield. As we discussed in Sec. \ref{sec:particlization}, this $\delta f$ correction is essential to conserve the total net baryon number during the Cooper-Frye conversion procedure.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.95\linewidth]{figs/net_proton_dNdy_diffusion_deltaf_effects}
\caption{The effects of the out-of-equilibrium corrections $\delta f$ from shear viscosity and net baryon number diffusion on the net proton rapidity distribution.}
\label{fig3B.1}
\end{figure}
Please note that a non-zero net baryon diffusion current on the freeze-out surface modifies identified particle yields as shown in Eq.\ (\ref{B10}) in Appendix \ref{appendix_B}. Thus, the non-equilibrium evolution of the baryon diffusion current will give corrections the chemical freeze-out parameters determined in thermal model fits \cite{Adamczyk:2017iwn}. Because the baryon diffusion $\delta f$ reduces the net proton yield, the averaged chemical potential on the freeze-out surface is about 30 MeV larger with baryon diffusion compared to the simulations without diffusion.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.9\linewidth]{figs/pid_sp_pion_p_diffusion_deltaf_effects}
\includegraphics[width=0.9\linewidth]{figs/pid_sp_p_vs_pbar_diffusion_deltaf_effects}
\caption{The correction of shear and net baryon diffusion $\delta f$ to the transverse momentum spectra of $\pi^+$, $K^+$, $p$, and $\bar{p}$ at the mid-rapidity in the hybrid simulations. }
\label{fig:deltaf-pt}
\end{figure}
\begin{figure}[ht!]
\centering
\includegraphics[width=0.9\linewidth]{figs/pid_v2_pion_p_diffusion_deltaf_effects}
\includegraphics[width=0.9\linewidth]{figs/pid_v2_p_diffusion_deltaf_effects}
\includegraphics[width=0.9\linewidth]{figs/pid_v2_anti_p_diffusion_deltaf_effects}
\caption{The correction of shear and net baryon diffusion $\delta f$ to the $p_T$-differential $v_2$ of $\pi^+$, $p$, and $\bar{p}$ in hybrid simulations. The shaded bands indicate statistical errors.}
\label{fig:deltaf-v2}
\end{figure}
In Fig.\,\ref{fig:deltaf-pt} we show identified particle spectra and their dependence on shear and baryon diffusion $\delta f$ corrections. For all species the effects are small, with the largest difference visible for anti-protons at $p_T>2$ GeV.
Figure \ref{fig:deltaf-v2} shows the effect of both $\delta f$ corrections on the elliptic flow of pions (a), protons (b), and anti-protons (c). The shear $\delta f$ leads to the typical reduction of $v_2$ for all particle species. Its effects on particle $p_T$-differential $v_2$ is larger than the one from the baryon diffusion. Because the baryon diffusion $\delta f$ depends on the baryon charge of the particle species, it reduces proton $v_2$ more but increases anti-proton $v_2$. It enhanced the difference between proton and antiproton $v_2(p_T)$.
\subsection{Effects of hadronic afterburner at BES energies}
\begin{figure}[ht!]
\centering
\includegraphics[width=0.9\linewidth]{figs/charghed_hadron_rapidity_distribution_UrQMD_vs_hydro}
\includegraphics[width=0.9\linewidth]{figs/net_proton_dNdy_UrQMD_vs_hydro}
\caption{The effects of hadronic rescatterings on charged hadron (a) and net proton (b) rapidity distribution.}
\label{fig:rescatter}
\end{figure}
Figure \ref{fig:rescatter} shows the effect of hadronic rescatterings on the rapidity distributions of identified particles. The late hadronic rescattering phase has a small effect on the shape of the rapidity distribution of charged hadrons. The net protons rapidity distribution is slightly widened by scatterings with other hadrons.
\begin{table*}[ht!]
\centering
\begin{tabular}{c|c|c|c}
\hline \hline
& $\langle p_T \rangle (p)$ (GeV) & $\langle p_T \rangle (\bar{p})$ (GeV) & $\langle p_T \rangle(\bar{p}) - \langle p_T \rangle(p) $ (GeV) \\ \hline
Thermal & 0.758 & 0.769 & 0.011 \\ \hline
Thermal + Corona & 0.753 & 0.766 & 0.013 \\ \hline
Thermal + Corona + resonances feed down & 0.712 & 0.722 & 0.010 \\ \hline
Thermal + Corona + full UrQMD & 0.875 & 0.924 & 0.049 \\
\hline \hline
\end{tabular}
\caption{The averaged transverse momentum of protons and anti-protons and their difference at different values from different effects in the hadronic phase.}
\label{table4}
\end{table*}
As in Table \ref{table3}, the mean transverse momentum of anti-protons is slightly larger than the proton mean $p_T$ even without diffusion at $C_B$ = 0. Table~\ref{table4} studies the origin of this difference in detail. Starting with thermally emitted protons and anti-protons from the Cooper-Frye conversion surface, the anti-proton $\langle p_T \rangle$ is only 7 MeV larger than that of the protons. This small difference can be understood by studying the time dependence of $\mu_B/T$ and the flow velocity $u^\tau$ on the hypersurface. The value of $\mu_B/T$ decreases by $\sim 10$\% during the first 4 fm of the hydrodynamic evolution while the radial flow is building up. This anti-correlation between the time evolution of $\mu_B/T$ and $u^\tau$ at early times results in relatively more protons produced when the radial flow is small. The thermal production yields of both protons and anti-protons are small during the first 4 fm of the evolution. Thus, the difference in mean $p_T$ is merely 7 MeV.
The hadronic corona (see Section \ref{sec:particlization}) produces more protons than anti-protons near the edge of the fireball at the beginning of the hydrodynamic evolution. Because there is no hydrodynamic flow yet and the temperatures of the fluid cells are low, including these particles is expected to reduce the mean $p_T$. Indeed, we found that the proton mean $p_T$ in Table \ref{table4} is reduced twice as much as that of the antiproton when including this contribution.
The resonance feed down contribution from heavy excited baryon states reduces both proton and anti-proton mean $p_T$ similarly. The slight reduction could be attributed to the fact that the shape of heavier particle spectra are less affected by the chemical potential and thus the particle-anti-particle difference in the mean $p_T$ is smaller. Finally, the hadronic rescatterings among light mesons and baryons largely blue-shift proton and anti-proton mean $p_T$. Overall, hadronic rescatterings affect anti-protons more compared to protons. This is because a larger fraction of protons is produced at early times and in the dilute region (compared to anti-protons) and these protons scatter less. In summary, Table~\ref{table4} shows that the mean $p_T$ difference between protons and anti-protons mainly originates from the late stage hadronic rescatterings. Note however, that the other differences in their production, discussed above, are necessary for the rescatterings to have this effect.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.9\linewidth]{figs/pid_pion_p_UrQMD_vs_hydro}
\includegraphics[width=0.9\linewidth]{figs/pid_sp_p_vs_pbar_UrQMD_vs_hydro}
\caption{The effects of hadronic transport on the transverse momentum spectra of final $\pi^+$, $K^+$ (a), $p$, and $\bar{p}$ (b).}
\label{fig3B.2}
\end{figure}
Figure \ref{fig3B.2} studies the effect of hadronic transport on particle $p_T$ spectra. The Monte-Carlo results without hadronic rescatterings from the hadronic transport approach are also cross checked with the direct numerical calculations of Cooper-Frye freeze-out and resonance decays. Consistent results are found from the two independent approaches which validates the Monte-Carlo simulations. By comparing feed-down only pion spectra with the results from the full UrQMD simulation, we find that the additional scatterings in the hadronic phase flatten the pion spectra at high $p_T$.
Significant modifications on the shape of proton and anti-proton $p_T$-spectra are found in Fig.~\ref{fig3B.2}b. Both proton and anti-proton spectra get large blue-shifts because of scatterings with light mesons in the hadronic phase. We checked that in this baryon-rich environment the $B\bar{B}$ annihilation processes in the hadronic phase do not have an effect.
We conclude that at BES collision energies the hadronic transport phase is critical for baryon and anti-baryon spectra.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.9\linewidth]{figs/pid_v2_pion_p_UrQMD_vs_hydro}
\includegraphics[width=0.9\linewidth]{figs/pid_v2_p_vs_pbar_UrQMD_vs_hydro}
\includegraphics[width=0.9\linewidth]{figs/pid_v2_anti_p_UrQMD_vs_hydro}
\caption{The effects of hadronic transport on the differential $v_2$ of final $\pi^+$ (a), $p$ (b), and $\bar{p}$ (c). The shaded bands indicate statistical errors.}
\label{fig3B.3}
\end{figure}
In Fig.~\ref{fig3B.3} we investigate the effect of the hadronic transport phase on the identified particle $p_T$-differential elliptic flow coefficient. Firstly, consistent results are found between the Monte-Carlo approach (without rescatterings) and the direct numerical calculations of resonance decays. Unlike the minor modifications on the pion $p_T$ spectra, the elliptic flow $v_2(p_T)$ of pions receives a sizable increase from the hadronic rescattering. This can be explained by the hadronic transport converting the remaining spatial eccentricity to particles' momentum anisotropy. Similar to pions, the high $p_T$ (anti-)proton $v_2$ is increased owing to the additional lifetime of the system which converts more spatial eccentricity to momentum anisotropy. Meanwhile, the low $p_T$ proton $v_2$ is reduced. This can be understood as a blue-shift effect, consistent with the modification of the proton spectra.
\section{Conclusions}\label{sec:conclusion}
The theoretical description of heavy ion collisions over a wide range of collision energies requires detailed fluid dynamic simulations with various complications appearing with lower energies. In this work we have introduced and studied the effects of net baryon diffusion that is expected to be present whenever the net baryon density is non-negligible. We have extended the 3+1 dimensional hydrodynamic simulation \textsc{Music} to include baryon diffusion currents and analyzed its effects on a variety of observables in a simplified setup using smooth initial conditions.
Employing an equation of state at finite $\mu_B$ constructed from lattice QCD data and a hadron resonance gas, we were able to evolve systems with non-negligible net baryon density. We found that baryon diffusion, following the gradients of $\mu_B/T$ in the system, tends to transport net baryon number towards mid-rapidity.
While pions and kaons are not affected within the accuracy of the simulation, measurable effects on proton and anti-proton spectra and elliptic flow coefficients are present. In particular, the difference between proton mean transverse momentum and anti-proton mean transverse momentum increases with increasing baryon diffusion. Furthermore, baryon diffusion decreases proton elliptic flow while increasing anti-proton elliptic flow.
We have also shown that the hadronic microscopic transport stage is very important for baryon spectra and differential elliptic flow coefficients, primarily because of the additional blue-shift given to protons and anti-protons. Apart from this effect, it also continues the conversion from spatial to momentum anisotropy, an effect relevant also for pion elliptic flow.
Finally, we have identified the contributions to the difference in proton- and anti-proton $\langle p_T\rangle$, with the main effect coming from the hadronic afterburner. For the afterburner to have an effect it is also important where and when protons are produced relative to anti-protons, which depends on the distribution of $\mu_B$ on the freeze-out surface.
We have presented one important step towards the development of a comprehensive simulation of heavy ion collision dynamics relevant for collisions in the RHIC BES and BES II as well as the NA61/SHINE program. In the future it will be combined with other important developments in this direction, including a dynamical fluctuating initial state, hydrodynamic fluctuations, and multiple conserved currents with coupled diffusion coefficients \cite{Greif:2017byw}, to result in a powerful theoretical tool that is needed to extract important information on the QCD phase diagram from experimental data.
\begin{acknowledgements}
BPS and CS are supported under DOE Contract No. DE-SC0012704. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. BPS acknowledges a DOE Office of Science Early Career Award. CS thanks a Goldhaber Distinguished Fellowship from Brookhaven Science Associates.
AM is supported by JSPS Overseas Research Fellowship. GSD thanks Conselho Nacional de
Desenvolvimento Cient\'{\i}fico e Tecnol\'{o}gico (CNPq) for financial
support. CG gratefully acknowledges support from the Canada Council for the Arts through its Killam Research Fellowship program.
This work was supported in part by the Natural Sciences and Engineering Research Council of Canada.
Computations were made in part on the supercomputer Guillimin from McGill University, managed by Calcul Qu\'ebec and Compute Canada. The operation of this supercomputer is funded by the Canada Foundation for Innovation (CFI), NanoQu\'ebec, RMGA and the Fonds de recherche du Qu\'ebec - Nature et technologies (FRQ-NT). This work is supported in part by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, within the framework of the Beam Energy Scan Theory (BEST) Topical Collaboration.
\end{acknowledgements}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,638 |
\section{Introduction}\label{sec:intro}
Black holes in spacetime dimensions $D\ge5$ can exhibit different types of instabilities compared to their four dimensional counterparts. One of these instabilities is the Gregory-Laflamme instability originally found in the context of perturbations of asymptotically flat black $p$-branes \cite{Gregory:1993vy}. This type of instability was later found to be present in the context of Myers-Perry black holes \cite{Dias:2009iu, Dias:2010eu, Hartnett:2013fba, Dias:2014eua} and in the case of five-dimensional black rings \cite{Santos:2015iua}. In fact, according to the arguments of \cite{Emparan:2009at} (see also \cite{Hovdebo:2006jy} for the case of black rings), any neutral black hole solution that admits a blackfold limit (i.e. an ultraspinning limit) is expected to suffer from a Gregory-Laflamme instability.
Non-axisymmetric instabilities, such as bar-mode instabilities \cite{Shibata:2009ad, Shibata:2010wz, Hartnett:2013fba, Figueras:2017zwa}, are an additional feature of higher-dimensional rotating black holes. Recently, it was found that a type of non-axisymmetric instability - the elastic instability - is also present in five-dimensional black rings with horizon topology $\mathbb{S}^{1}\times\mathbb{S}^{2}$ \cite{Figueras:2015hkb}. This instability is related to transverse deformations of the radius $R$ of the $\mathbb{S}^{1}$ that do not significantly affect the size of the radius $r_0$ of the $\mathbb{S}^{2}$. Studies of the end point of these instabilities suggest a violation of the weak cosmic censorship conjecture \cite{Lehner:2010pn, Figueras:2015hkb, Figueras:2017zwa}. It is thus important to study these instabilities in more generality and in particular by means of analytic methods that can probe regimes of parameter space that numerical methods cannot reach with acceptable accuracy.
Besides having proved to be extremely useful in finding new black hole solutions \cite{Emparan:2009vd, Armas:2015kra, Armas:2015nea,Armas:2017xyt, Armas:2017myl} in asymptotically flat space, we demonstrate here that the blackfold approach \cite{Emparan:2009cs, Emparan:2009at} is a powerful tool for studying hydrodynamic (i.e. Gregory-Laflamme) and elastic instabilities of higher-dimensional black holes in the ultraspinning regime and away from it.\footnote{\label{foot:nome} We note that we are interpreting the non-axisymmetric instability found for black rings in \cite{Figueras:2015hkb} as an elastic instability from the blackfold point of view. The rationale for this interpretation is that, in the context of blackfolds, elastic instabilities of black rings are related to deformations of the radial direction of the $\mathbb{S}^{1}$ which is the type of deformation encountered in \cite{Figueras:2015hkb}. In general, we do no expect all other types of non-axisymmetric instabilities \cite{Shibata:2009ad, Shibata:2010wz, Hartnett:2013fba, Figueras:2017zwa} to be elastic instabilities from the blackfold point of view. In fact, some of them, if visible within the blackfold approximation, might be of hydrodynamic nature. } In this context, one first finds a stationary solution, modelled as a fluid confined to a surface, corresponding to the black hole solution whose stability one wishes to study. The fundamental fluid variables and the geometric properties of the surface describing the equilibrium configuration of the fluid are subsequently perturbed and the stability properties of black holes are found by studying the propagation of hydrodynamic and elastic modes.
Black rings can be classified as thin $0\le\nu<1/2$ or as fat $1/2\le\nu<1$ where for very thin rings $\nu=r_0/R$ is a measure of the ring thickness. Studying Penrose inequalities, the fat branch of black rings in $D=5$ has been shown to be unstable \cite{Arcioni:2004ww, Elvang:2006dd, Figueras:2011he} while for the thin branch in $D=5$, the instability of black rings relies on numerical studies \cite{Santos:2015iua, Figueras:2015hkb}. However, these numerical studies, due to lack of accuracy, have only established the existence of instabilities for $\nu\ge0.144$ \cite{Santos:2015iua} and for $\nu\ge0.15$ \cite{Figueras:2015hkb}. The region $\nu<0.144$ is left unknown, with the only suggestive arguments of \cite{Hovdebo:2006jy, Emparan:2009at} being applicable in the strict case of $\nu=0$, for which there is barely any distinction between the black ring and the boosted black string. Additionally, the numerical studies of \cite{Santos:2015iua, Figueras:2015hkb} have not consider non-axisymmetric modes with $m>2$\footnote{Here $m$ is a discrete number characterising the mode of non-axisymmetric perturbations of the form $e^{i(-\omega \tau + m\phi/R)}$ where $\omega$ is the frequency, $\tau$ the time direction and $\phi$ the angular direction along the $\mathbb{S}^{1}$. } and, moreover, there is currently no knowledge of these instabilities in $D\ge6$ for which no exact black ring analytic solution is known.\footnote{Albeit the work of refs.~\cite{Tanabe:2015hda, Tanabe:2016pjr} which use large $D$ techniques that we will comment upon and ref.~\cite{Chen:2018vbv}, which has considered the evolution of the Gregory-Laflamme instability for black rings at large $D$ for $m\sim\mathcal{O}(\sqrt{D})$ and found evidence that the end point of the instability is a non-uniform, non-stationary, black ring. The behaviour of the end point is expected to depend on the dimension $D$ as in the case of black strings \cite{Cardona:2018shd}. }
This paper deals with the study of black ring instabilities in the very thin regime for $D\ge5$ and arbitrary $m$. Its aim is to provide an analytic understanding of some of these instabilities and to progress in closing the gap in parameter space by showing that some of these instabilities are present also for some part of the region $\nu<0.144$.
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.43\textwidth}
\includegraphics[width=\textwidth]{Ring5d.pdf}
\end{subfigure}
\qquad \qquad
\begin{subfigure}[b]{0.43\textwidth}
\includegraphics[width=\textwidth]{Ring7d.pdf}
\end{subfigure}
\caption{On the left we show the reduced area $a_{\text{H}}$ as a function of the reduced angular momentum $j$ for $D=5$ where the black line is the exact curve of the black ring solution \cite{Emparan:2001wn} and the dashed red curve is the blackfold approximation up to first order in derivatives \cite{Emparan:2007wm}. On the right we show the behaviour of the same quantities for black rings in $D=7$ where the black line is the numerical solution of \cite{Dias:2014cia} and the dashed red line the blackfold approximation up to second order in derivatives \cite{Armas:2014bia}.} \label{fig:BRs}
\end{figure}
The blackfold approach has shown to accurately describe stationary thin black rings. In the left plot of fig.~\ref{fig:BRs}, it is shown the phase diagram of $D=5$ black rings, where the reduced area $a_{\text{H}}$ and reduced angular momentum $j$ were introduced in \cite{Emparan:2007wm}.
The black solid line is the curve obtained from the exact black ring solution of \cite{Emparan:2001wn} while the dashed red line is the blackfold approximation up to first order in derivatives \cite{Emparan:2007wm}.\footnote{We note that the dashed red line in the left plot of fig.~\ref{fig:BRs} is the curve obtained at ideal order in the approximation, since the first order approximation does not produce corrections to stationary black rings \cite{Emparan:2007wm}.} This approximation works relatively well for $j\gtrsim2.2$ which is equivalent to the region $0\le\nu\lesssim0.025$. In the plot on the right in fig.~\ref{fig:BRs}, it is shown the $D=7$ black ring solution numerically obtained in \cite{Dias:2014cia} (black solid line) and the blackfold approximation up to second order in derivatives (dashed red line) \cite{Armas:2014bia}. In this case the blackfold approximation works well for $j\gtrsim1.2$ which corresponds to the region $0\le\nu\lesssim0.27$. Thus our general analysis of dynamical instabilities of black rings for arbitrary $m$ and $D$ is expected to be valid at least in the region $0\le\nu\lesssim0.025$ for $D=5,6$ for which the blackfold approximation is not under control beyond first order, and in the region $0\le\nu\lesssim0.27$ for $D\ge7$ for which the approximation is under control up to second order in $r_0/R$.
In order to proceed with this analysis, we first introduce the blackfold effective theory in sec.~\ref{sec:bf} and derive novel variational formulae required to study perturbations around equilibrium configurations. In sec.~\ref{sec:bs} we study instabilities of boosted black strings as a way of calibrating our method, since in this case our results, besides providing a check of the $R\to\infty$ limit of black ring instabilities, can be compared against existent numerical and analytic results. In this context, we provide novel expressions for the growth rates of the Gregory-Laflamme instability and its onset for arbitrary boost parameter.
In sec.~\ref{sec:br} we study the instabilities of black rings and identify the Gregory-Laflamme instability in $D\ge5$, providing analytic results for the growth rates of the instability and its onset. We do not find an elastic instability at this order in the blackfold approximation and hence our results for the black ring contradict the corresponding large $D$ analysis \cite{Tanabe:2016pjr}, which is shown to be incorrect. In sec.~\ref{sec:dis} we summarise our main results and comment on open research directions. In app.~\ref{app:st} we provide the corrected stress tensor and bending of perturbed black branes in asymptotically flat space, which contains the identification of new transport coefficients. In app.~\ref{app:cumber} we provide details on the perturbed equations at second order, while in app.~\ref{sec:hydrocorr}, supplemented by the ancillary Mathematica file, we give further details on hydrodynamic and elastic modes.
\section{Blackfold equations and linearised perturbations}\label{sec:bf}
In this section we briefly review the essential aspects of the blackfold approach required for the purposes of this work. We discuss the blackfold equations up to second order in a long-wavelength expansion which determine the equilibrium configurations that we wish to perturb. Subsequently, we derive new general formulae for linearised perturbations of the equilibrium blackfold equations, ultimately focusing on the case of 2-dimensional worldvolumes which describe black strings and black rings. These results will then be used in the remaining sections in order to study the hydrodynamic and elastic stability of these later two cases.
\subsection{Blackfold equations}
The blackfold approach consists of wrapping black branes on weakly curved $(p+1)$-dimensional submanifolds $\mathcal{W}_{p+1}$ embedded in a $D=n+p+3$-dimensional spacetime endowed with metric $g_{\mu\nu}(x)$ and coordinates $x^\mu$ \cite{Emparan:2009cs, Emparan:2009at}. The location of the submanifold in the ambient spacetime is determined by the embedding map $X^\mu(\sigma)$, where $\sigma^a$ are coordinates on $\mathcal{W}_{p+1}$. The submanifold inherits the induced metric $\gamma_{ab}={e^{\mu}}_{a}{e^{\nu}}_{b}g_{\mu\nu}$ where ${e^{\mu}}_{a}=\partial_a X^{\mu}$ are a set of $(p+1)$ tangent vectors and $a,b,c,...$ are surface indices. The set of $(n+2)$ normal vectors ${n^{\mu}}_i$ are defined implicitly by the relations ${n^\mu}_i {e_\mu}^{a}=0$ and ${n^{\mu}}_i{n_{\mu}}^{ j}={\delta_{i}}^{j}$ where $i,j,k,...$ are normal indices. The extrinsic curvature of the submanifold is defined as ${K_{ab}}^{\rho}=\nabla_{a}{e^{\rho}}_b$ where $\nabla_a$ is the covariant derivative compatible with both $g_{\mu\nu}$ and $\gamma_{ab}$. It is also useful to define its projection along the normal vector, i.e. ${K_{ab}}^i={n_\rho}^i\nabla_a {e^{\rho}}_b$ and mean extrinsic curvature $K^i=\gamma^{ab}{K_{ab}}^{i}$ (or equivalently $K^\rho\equiv \gamma^{ab}{K_{ab}}^\rho$).
In vacuum, this approach is generically applicable if the horizon size of the black brane $r_0$ satisfies the hierarchy of scales $r_0\ll R$ where $R$ is the smallest intrinsic or extrinsic scale associated with the submanifold or to variations of the fluid degrees of freedom that live on it. In the case of singly-spinning black holes, this implies that the black hole must be ultraspinning, i.e. in appropriate units the black hole angular momentum is much larger than its mass. In the case of black rings with horizon topology $\mathbb{S}^{1}\times\mathbb{S}^{2}$, the ultra-spinning limit is commonly referred to as the thin limit, since in this case the radius $R$ of the $\mathbb{S}^{1}$ must be much larger than the horizon radius $r_0$ of the $\mathbb{S}^{2}$.
In the context of vacuum General Relativity, the starting point is the boosted Schwarzschild black brane. The process of "wrapping" the black brane on a weakly curved submanifold translates into a small, long-wavelength, perturbation of the black brane geometry that must satisfy Einstein equations order-by-order in a derivative expansion. Typically, the expansion parameter $\varepsilon\ll1$ is defined as $\varepsilon=r_0/R$ or $\varepsilon=k r_0$ where $k$ is the wavenumber of the perturbation being performed. A subset of the Einstein equations (constraint equations) up to order $\mathcal{O}(\varepsilon)$ has been identified to be \cite{Emparan:2007wm, Camps:2010br, Camps:2012hw}
\begin{equation}\label{eq:BFeom}
\nabla_{a}T^{ab}=0~~,~~T^{ab}{K_{ab}}^{i}=0~~,
\end{equation}
where we have ignored the existence of edges on the submanifold. Here the stress tensor $T^{ab}$ up to first order in derivatives is given in terms of a viscous fluid \cite{Camps:2010br}
\begin{equation}
T^{ab}=T^{ab}_{(0)}+T^{ab}_{(1)}~~,~~T^{ab}_{(0)}=\epsilon u^{a}u^{b}+P P^{ab}~~,~~T^{ab}_{(1)}=-2\eta\sigma^{ab}-\zeta\theta P^{ab}~~,
\end{equation}
where $u^{a}$ is the normalised fluid velocity $u^{a}u_{a}=-1$ and $P^{ab}=\gamma^{ab}+u^{a}u^{b}$ is a perpendicular projector to $u^{a}$. The thermodynamic quantities $\epsilon$ and $P$ denote the energy density and pressure respectively while $\eta$ and $\zeta$ denote shear and bulk viscosity. All these quantities are a function of the local temperature $\mathcal{T}$. Their specific dependence and form in terms of the black brane radius $r_0$ is given in app.~\ref{app:st}, together with the definition of the shear tensor $\sigma^{ab}$ and the fluid expansion $\theta$.
At one higher order, the stress tensor $T^{ab}$ receives additional corrections that depend on derivatives of the intrinsic and extrinsic geometry as well as on second derivatives of the fundamental fluid variables. If $n\ge3$, these corrections are dominant compared to backreaction corrections and in this case, the equations of motion \eqref{eq:BFeom} are modified at order $\mathcal{O}(\varepsilon^2)$ to \cite{Armas:2013hsa}
\begin{equation}\label{eq:BFeom2}
\nabla_{a}T^{ab}={e_\mu}^{b}\nabla_a\nabla_b\mathcal{D}^{ab\mu}~~,~~T^{ab}{K_{ab}}^{i}={n^i}_{\mu}\nabla_a\nabla_b \mathcal{D}^{ab\mu}~~,
\end{equation}
where we have assumed that the background metric is flat (i.e. the associated Riemann tensor vanishes) and that the brane is not spinning in transverse directions to $\mathcal{W}_{p+1}$. In eq.~\eqref{eq:BFeom2}, $T^{ab}$ receives an additional correction $T^{ab}_{(2)}$ and $\mathcal{D}^{ab\mu}$ is the brane bending moment that encodes the response of the black brane due to extrinsic deformations. The bending moment can be written as $\mathcal{D}^{ab\mu}=\mathcal{Y}^{abcd}{K_{cd}}^{\mu}$ where $\mathcal{Y}^{abcd}$ is the Young modulus \cite{Armas:2011uf}. The explicit form of these structures is detailed in app.~\ref{app:st}.
Of particular importance is the class of solutions that describes the equilibrium sector of \eqref{eq:BFeom} and \eqref{eq:BFeom2}. In this case, the fluid velocity must be aligned with a worldvolume Killing vector field $\textbf{k}^{a}$ such that
\begin{equation} \label{eq:eq}
u^{a}=\frac{\textbf{k}^a}{\textbf{k}}~~,~~\mathcal{T}=\frac{T}{\textbf{k}}~~,~~\textbf{k}=|-\gamma_{ab}\textbf{k}^{a}\textbf{k}^{b}|^{1/2}~~,
\end{equation}
where $T$ is the constant global temperature of the fluid and $\textbf{k}$ is the modulus of the timelike Killing vector field. The worldvolume Killing vector field is subjected to the constraint that its pushforward onto the ambient spacetime coincides with a background Killing vector field $\textbf{k}^\mu$, i.e. $\textbf{k}^\mu={e^\mu}_a \textbf{k}^{a}$. For this particular class of solutions, the first equation in \eqref{eq:BFeom} and \eqref{eq:BFeom2}, which is the set of hydrodynamic equations for the stress tensor $T^{ab}$, is automatically satisfied, regardless of the choice of embedding map $X^\mu$. The second equation in \eqref{eq:BFeom} and \eqref{eq:BFeom2} is the remaining non-trivial elastic equation that determines conditions on the embedding map given a particular choice of $\textbf{k}^a$. Solutions that satisfy \eqref{eq:eq} have $\sigma^{ab}=\theta=0$ by virtue of the Killing equation and hence $T^{ab}_{(1)}=0$. It is this equilibrium class of solutions that we wish to perturb around in order to study the stability properties of given configurations.
\subsection{Linearised perturbations} \label{sec:var}
The purpose of this section is to derive variational formulae that describe linear perturbations of equilibrium configurations that solve \eqref{eq:BFeom} and \eqref{eq:BFeom2} following the machinery developed in \cite{Armas:2017pvj}.\footnote{When describing perturbations of equilibrium configurations, the scale of the problem is set by $1/T$. Thus, by means of eq.~\eqref{eq:eq} and app.~\ref{app:st}, when writing $\varepsilon=r_0 k$, it is really meant $\varepsilon=k/T$.} This will form the basis for studying the stability of propagating sound and elastic modes in the next sections.
The fluid degrees of freedom consist of a scalar degree of freedom, which we choose to be the energy density $\epsilon$, and $p$ independent components of the fluid velocity $u^{a}$ supplemented by $n+2$ transverse components of the embedding map which we denote by $X^{\mu}_\perp(\sigma)={\perp^{\mu}}_\nu X^{\nu}(\sigma)$ where ${\perp^{\mu}}_\nu={n^\mu}_i{n^i}_\nu$.\footnote{The remaining $p+1$ components of the embedding map $X^{\mu}$ can always be chosen as the coordinates on $\mathcal{W}_{p+1}$ since $\sigma^{a}={e^{a}}_\mu X^{\mu}$. Hence, when $\mathcal{W}_{p+1}$ has no edges, variations of these components coincide with worldvolume reparametrisations and can be ignored.} Our intent is to perform a slight perturbation of these variables around equilibrium solutions such that
\begin{equation} \label{eq:pf}
\epsilon\to\epsilon+\delta\epsilon~~,~~u^{a}\to u^{a}+\delta u^{a}~~,~~X^{\mu}_\perp(\sigma)\to X^{\mu}_\perp(\sigma)+\delta X^{\mu}_\perp(\sigma)~~.
\end{equation}
Under these perturbations all geometric quantities transform, for instance $\delta\gamma^{ab}=2{K^{ab}}_{\rho}\delta X_{\perp}^{\rho}$ \cite{Armas:2017pvj}.
The normalisation condition $u^{a}u_{a}=-1$, implies the constraint on the variations of the fluid velocity
\begin{equation} \label{eq:c1}
u_a\delta u^{a}=u_{a}u_{b}{K^{ab}}_{\rho}\delta X^{\rho}_\perp~~,
\end{equation}
which is the statement that only $p$ components of the fluid velocity are independent. These variations are sufficient to characterise the deformations of the ideal order stress tensor, which take the form
\begin{equation}
\delta T^{ab}_{(0)}=\frac{\epsilon}{n+1}\left[\left(n u^{a}u^{b}-\gamma^{ab}\right)\frac{\delta \epsilon}{\epsilon}+2nu^{(a}\delta u^{b)}-\delta \gamma^{ab}\right]~~,
\end{equation}
where we have used the specific equation of state $\epsilon=-(n+1)P$ provided in app.~\ref{app:st}. In order to determine the variation of the equations of motion \eqref{eq:BFeom} up to first order, one also requires the variation of the first order stress tensor
\begin{equation}
\delta T^{ab}_{(1)}=-2\eta\delta \sigma^{ab}-\zeta\delta\theta P^{ab}~~,~~\delta \theta=\nabla_a\delta u^{a}-u^{a}\nabla_a\left(K_{\rho}\delta X^{\rho}\right)~~,
\end{equation}
where we have used that in equilibrium $\theta=\sigma^{ab}=0$. As we are interested in the $p=1$ case, we have not written explicitly the variation $\delta\sigma^{ab}$. This is sufficient for obtaining linear perturbations of \eqref{eq:BFeom}. Defining $\delta T^{ab}=\delta T^{ab}_{(0)}+\delta T^{ab}_{(1)}$, these take the general form
\begin{equation}
\begin{split} \label{eq:P1}
\nabla_a \delta T^{ab}-T^{cb}\nabla_c\left(K_\rho\delta X^\rho_\perp\right)-2T^{ac}\nabla_a\left[{K^{b}}_{c\rho}\delta X_{\perp}^\rho\right]+T^{ac}\nabla^{b}\left({K_{ac}}^{\rho}\perp_{\rho\lambda}\delta X^{\lambda}_\perp\right)&=0~~,\\
\delta T^{ab}{K_{ab}}^{i}+T^{ab}{n^{i}}_\mu\nabla_a\nabla_b \delta X^{\mu}_\perp&=0~~,
\end{split}
\end{equation}
where we have used that for equilibrium solutions $T^{ab}{K_{ab}}^{i}=0$ up to first order and focused on backgrounds with vanishing Riemann tensor. At second order and for $n\ge3$, eqs.~\eqref{eq:P1} receive modifications due to the right hand side of \eqref{eq:BFeom2}. These modifications are cumbersome and are detailed in app.~\ref{app:cumber}. Eqs.~\eqref{eq:P1} and \eqref{eq:P2} are the equations that we wish to solve for different initial configurations in terms of the perturbed fields \eqref{eq:pf}, in particular we wish to analyse the vanishing of the determinant of eqs.~\eqref{eq:P1} and \eqref{eq:P2} which provide sufficient conditions for the existence of solutions.
\subsubsection{Two-dimensional worldvolumes}
In the next sections we focus on two-dimensional worldvolumes ($p=1$) which can describe boosted black strings and black rings in $D\ge5$. In this case, the analysis simplifies considerably since, for instance, $\delta\sigma^{ab}=0$ and hence
\begin{equation}
\delta T^{ab}_{(1)}=-\zeta\delta\theta P^{ab}~~.
\end{equation}
In turn, the effect of the first order corrections to the stress tensor in \eqref{eq:P1} only depends on the bulk viscosity in the form
\begin{equation}
\begin{split}
\nabla_a \delta T^{ab}_{(1)}&=-\zeta \nabla_a\left(P^{ab} \delta \theta\right)~~, \\
\delta T^{ab}_{(1)}{K_{ab}}^{i}&=-\zeta\delta\theta\left(K^{i}+u^{a}u^{b}{K_{ab}}^{i}\right)=-\zeta\delta\theta\frac{n+1}{n}K^{i} +\mathcal{O}\left(\varepsilon^{3}\right)~~,
\end{split}
\end{equation}
where in the last equality we have used \eqref{eq:BFeom} and neglected $\mathcal{O}\left(\varepsilon^{3}\right)$ terms which we do not consider in this paper. If in addition we focus on the case of boosted black strings for which ${K_{ab}}^{i}=0$, eqs.~\eqref{eq:P1} further simplify to
\begin{equation}
\nabla_a \delta T^{ab}=0~~,~~T^{ab}{n^{i}}_\mu\nabla_a\nabla_b \delta X^{\mu}_\perp=0~~,
\end{equation}
up to first order in derivatives. This shows that in this situation, the extrinsic perturbations $\delta X^{\mu}_\perp$ decouple from the intrinsic perturbations $\delta \epsilon$ and $\delta u^a$. At second order, these equations receive non-trivial corrections, as explained in app.~\ref{app:cumber} and for some configurations, such as black rings, the perturbations begin to couple. In the next section, we use the variational formulae derived here to study the stability of boosted black strings.
\section{Instabilities of boosted black strings}\label{sec:bs}
Gregory-Laflamme instabilities of static black strings using the blackfold approach were considered in \cite{Emparan:2009at, Camps:2010br, Caldarelli:2012hy, Caldarelli:2013aaa}. Here we consider both fluid and elastic perturbations of boosted black strings up to second order in the derivative expansion. We compare our results with the static and boosted cases with the large $D$ analysis performed in \cite{Emparan:2013moa, Tanabe:2015hda, Tanabe:2016pjr}, showing the relevance of the Young modulus of black strings (defined in app.~\ref{app:st}) in the dispersion relation of elastic modes. Elastic modes are shown to always be stable. We also derive novel expressions for $k_{\text{GL}}$, which describes the onset of the Gregory-Laflamme instability for arbitrary boost parameter and compare with the numerical analysis of \cite{Asnin:2007rw}, finding good agreement when $D\ge7$. The results of this exercise are extremely useful to study perturbations of black rings in sec.~\ref{sec:br} as they must be recovered at large ring radius.
\subsection{Ideal order modes}
We consider the equilibrium solution of \eqref{eq:BFeom} that represents a boosted black string. To begin with, we introduce the background Minkowksi metric in the form
\begin{equation}
ds^2\equiv g_{\mu\nu}dx^\mu dx^\nu=-dt^2+\sum_{i=1}^{D-1}\left(dx^{i}\right)^2~~,
\end{equation}
and we embed the string via the map $X^{t}=\tau$, $X^1=z$ and $X^i=0~,~i=2,...,D-1$, leading to the induced string metric
\begin{equation}
\textbf{ds}^2\equiv \gamma_{ab}d\sigma^{a}d\sigma^{b}=-d\tau^2+dz^2~~.
\end{equation}
In addition, a boosted string is characterised by the Killing vector field $\textbf{k}^a\partial_a=\partial_\tau+\beta\partial_z$ with modulus $\textbf{k}=\sqrt{1-\beta^2}$, where $0\le\beta<1$ is the boost parameter. The case $\beta=0$ describes the static black string. This embbeding is a minimal surface (a two-dimensional plane in $(t,x_1)$) and hence has vanishing extrinsic curvature, i.e. ${K_{ab}}^i=0$.
In order to study potential instabilities of these configurations we consider linearised perturbations around these equilibrium configurations. Due to the variational constraint \eqref{eq:c1}, these can be parametrised by small perturbations of the energy density $\delta\epsilon(\sigma)$, the $z$-component of the fluid velocity $\delta u^z(\sigma)$ and of the $(n+2)$ transverse components of embedding map $\delta X^\mu(\sigma)_\perp$. In particular, we consider plane wave solutions to the perturbation equations \eqref{eq:P1} that we parametrise as
\begin{equation} \label{eq:perturb}
\delta\epsilon(\sigma)=\delta\epsilon e^{i\left(-\omega \tau+k z\right)}~~,~~\delta u^z(\sigma)=\delta u^z e^{i\left(-\omega \tau+k z\right)}~~,~~\delta X^\mu_\perp(\sigma)=\delta X^\mu_\perp e^{i\left(-\omega \tau+k z\right)}~~,
\end{equation}
where $\omega$ is the oscillation frequency, $k$ the wavenumber and $\delta\epsilon,\delta u^z$ and $\delta X^\mu_\perp$ are small perturbation amplitudes. Introducing these variations into the perturbed equations at ideal order \eqref{eq:P1} and demanding the determinant of the system of 3 equations to vanish leads to two elastic modes (due to perturbations of second equation in \eqref{eq:BFeom}) and two hydrodynamic modes (due to perturbations of the first equation in \eqref{eq:BFeom}), which are solved perturbatively such that
\begin{equation} \label{eq:omegabs}
\omega=\omega^{(0)}+\omega^{(1)}k r_0+\omega^{(2)}(k r_0)^2+...~~,
\end{equation}
where it is assumed that $\varepsilon=kr_0\ll 1$. Under this approximation, the ideal order frequencies read
\be \label{eq:BS0}
\omega_{1,2}^{(0)}(k)=\frac{n\beta\pm\sqrt{n+1}\left(1-\beta^2\right)}{n+1-\beta^2}k~~,~~\omega_{3,4}^{(0)}(k)=\frac{\beta(n+2)\pm i\sqrt{n+1}\left(1-\beta^2\right)}{n+1+\beta^2}k~~.
\end{equation}
These two sets of frequencies are obtained independently from each of the equations in \eqref{eq:P1} since intrinsic and extrinsic perturbations decouple for the black string and are valid for any $n\ge1$. In particular, the frequencies $\omega_{1,2}$ are the elastic frequencies associated with perturbations of the second equation in \eqref{eq:BFeom} and $\omega_{3,4}$ the hydrodynamic frequencies, associated with the first equation in \eqref{eq:BFeom}. The frequency $\omega_3$ will be interpreted as being associated with the Gregory-Laflamme instability. In the static case $\beta=0$, \eqref{eq:BS0} had been obtained in \cite{Emparan:2009at}. Contrary to the case $\beta=0$, the hydrodynamic frequencies with $\beta>0$ are not purely imaginary. A case of particular interest is when $\beta=1/\sqrt{n+1}$, which corresponds to the value of the boost that locally characterises the black ring of sec.~\ref{sec:br}. In this case the modes \eqref{eq:BS0} become
\begin{equation} \label{eq:BS0c}
\omega_{1}^{(0)}(k)=0~~,~~\omega_{2}^{(0)}(k)=\frac{2\sqrt{n+1}}{n+2}k~~,~~\omega_{3,4}^{(0)}(k)=\left(n+2\pm in\right)\frac{\sqrt{n+1}}{n^2+2n+2}k~~,
\end{equation}
and hence have a zero-frequency mode.
It is worth noticing that the elastic modes $\omega_{1,2}$ in \eqref{eq:BS0} are always real and positive for all values of $0\le\beta<1$ and thus the perturbations \eqref{eq:perturb} describe oscillating but stable solutions. The hydrodynamic mode $\omega_4$ has a negative imaginary part and hence the perturbations \eqref{eq:perturb} describe stable solutions which are damped in time. On the other hand, the hydrodynamic mode $\omega_3$ has a positive imaginary part and hence the perturbations will grow exponentially in time, signalling the existence of the well-known Gregory-Laflamme instability of black strings \cite{Gregory:1993vy}, as spelled out in \cite{Emparan:2009at} for the case $\beta=0$. This instability grows faster for lower values of $n$ and smaller values of $\beta$, and becomes attenuated as one approaches $n\to\infty$ or $\beta\to1$.
Given that $\omega_4$ has a negative imaginary part that does not vanish for any value of $\beta$, any higher order correction to $\omega_4$ cannot make it unstable for small values of $k r_0$. However, the elastic modes in \eqref{eq:BS0} have real frequencies and thus it is plausible that higher order corrections could introduce a small but imaginary part. As we will show below, this is however not the case.
\subsection{First order modes}
In finding high-order derivative corrections to \eqref{eq:BS0}, our aim is to understand whether other types of instabilities appear (such as elastic instabilities) as the black string effectively becomes less thin. We also wish to understand the onset of the instability described by $k_{\text{GL}}$, i.e. the maximum value of $k$ for which the instability is present.
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.4\textwidth}
\includegraphics[width=\textwidth]{BBSd5.pdf}
\end{subfigure}
\qquad \qquad
\begin{subfigure}[b]{0.4\textwidth}
\includegraphics[width=\textwidth]{BBSd5n.pdf}
\end{subfigure}
\caption{On the left we show the dimensionless imaginary part of $\omega_3$, defined as $\bar\omega=\text{Im}\omega_3 r_0$, as a function of $\bar k=k r_0/\sqrt{n}$ for $D=5$ ($n=1$). The black solid line represents $\beta=0$, the grey solid line $\beta=1/\sqrt{2}$ and the red solid line $\beta=9/10$ while the grey dashed line is the imaginary part of the ideal order result \eqref{eq:BS0}. We have shown these curves up to $\bar k=1$, but we only expect them to be strictly valid for small $\bar k$. On the right plot we show the behaviour of the imaginary part of $\omega_3$ for $\beta=1/10$ and $n=1$ (black), $n=2$ (red), $n=3$ (blue) and $n=4$ (purple).} \label{fig:BBSd5}
\end{figure}
For $p=1$, as mentioned in sec.~\ref{sec:var}, first order corrections are controlled only by the bulk viscosity $\zeta$. Evaluating the perturbative equations accounting for the first order corrections in the stress tensor and using \eqref{eq:P1} leads to the following correction to the hydrodynamic frequencies
\begin{equation} \label{eq:BS1}
\omega_{3,4}^{(1)}(k)=\mp\frac{(n+2)\textbf{k}^3 \left(\beta\pm i \sqrt{n+1}
\right) \left(n+1\mp i\sqrt{n+1} \beta \right)^2}{n \sqrt{n+1} \left(n+1+\beta ^2\right)^3} k~~,
\end{equation}
where we have ignored corrections of the order of $\mathcal{O}\left((kr_0)^2\right)$. The elastic frequencies remain the same as in eq.~\eqref{eq:BS0} and thus they remain stable under linear perturbations. The correction to $\omega_{3}^{(1)}$ has been obtained in \cite{Camps:2010br} when $\beta=0$ but not explicitly for $\omega_{4}^{(1)}$. We observe here that when $\beta=0$ the corrections become equal and purely imaginary. Hence the hydrodynamic modes up to first order become
\begin{equation}
\omega_{3,4}(k)=\frac{i}{\sqrt{n+1}}\left(\pm k-\frac{(n+2)}{n\sqrt{(n+1)}}r_0k^2\right)+\mathcal{O}\left((kr_0)^2\right)~~,
\end{equation}
where the "$+$" sign corresponds to the solution of \cite{Camps:2010br}. Overall, the mode $\omega_3$ is always unstable while $\omega_4$ is always stable. In fig.~\ref{fig:BBSd5} on the left, we show the behaviour of the growth rate of the Gregory-Laflamme instability (i.e. the imaginary part of $\omega_3$) for $D=5$ and different values of $\beta$. As $\beta$ increases, the behaviour of the dimensionless frequency $\bar\omega$ becomes increasingly linear as a function of $\bar k$. The dashed and solid grey lines exhibit the improvement of first order corrections to the hydrodynamic modes and deviations from the linear ideal order result \eqref{eq:BS0c}. On the right plot of fig.~\ref{fig:BBSd5}, we exhibit the behaviour of the growth rate of the instability for $\beta=1/10$ and for different values of $n$ starting with $n=1$ (black line) and ending in the $n=4$ (purple line). The plot shows that the growth rate increases with increasing dimension for small $\bar k$.
The case $\beta=0$ was explicitly compared against numerical simulations in \cite{Camps:2010br} and agreement was found in the entire range of $\bar k$ at large $n$ while for small $n$ it is only a good approximation for smaller values of $\bar k$. In the case of $\beta\ne0$, a numerical study was performed in \cite{Hovdebo:2006jy} and in particular it was found a finite value of $k_{\text{GL}} r_0<2$ for all values of $\beta$ for $n=1$. We observe that $\omega_3$ in \eqref{eq:BS1} is characterised by a value of $k_{\text{GL}} r_0$ that increases significantly as $\beta\to1$ for $n=1$. In particular, using \eqref{eq:BS1} we can find the analytic value of $k_{\text{GL}}r_0$ to be
\begin{equation} \label{eq:kgl1}
k_{\text{GL}} r_0=\frac{\left(n+1+\beta ^2\right)^2 \sqrt{\frac{n+1}{1-\beta ^2}}}{\left(n^2+3 n+2\right) \left(n+1-3 \beta^2\right)}~~,
\end{equation}
and it leads to no solution for $\beta\ge\sqrt{(n+1)/3}$. The result \eqref{eq:kgl1} is in fact a new analytic result, not present in the literature but not an extremely useful one for small $n$ or $\beta\ge\sqrt{(n+1)/3}$. However, at large $n$ and $\beta=0$ this result is approximately valid as already noted in \cite{Camps:2010br}. Additionally, the lack of predictability as $\beta\to1$ is expected since for fixed $r_0\propto \textbf{k}/T \sim1$, as $\beta\to1$ we need $T\to0$ and hence for the approximation to be valid we require $k\ll T$, thus $k\sim 0$. This means that as $\beta\to1$ we expect our approximation to be appropriate only near $k\sim0$. These considerations and the result \eqref{eq:kgl1} largely improve once we consider second order corrections as will be shown in the next section.
Finally, in the special case $\beta=1/\sqrt{n+1}$ which describes the critical boost of black rings at large radius, the hydrodynamic frequencies become
\begin{equation} \label{eq:bbs1storder}
\begin{split}
\omega_{3,4}(k)=\left(n+2\pm in\right)\frac{\sqrt{n+1}}{n^2+2n+2}k-i\frac{\sqrt{n}(n+1)^{\frac{5}{2}}}{2\left((1\pm i)+n\right)^3}\frac{\zeta}{\epsilon}k^2+\mathcal{O}\left((r_0k)^2\right)~~,
\end{split}
\end{equation}
where $\zeta$ was defined in app.~\ref{app:st}. To summarise, up to first order in derivatives we find the existence of a Gregory-Laflamme instability for arbitrary boost $\beta$ and no elastic instability.
\subsection{Second order modes and comparison with the large $D$ analysis} \label{sec:bbs2}
At second order in derivatives, the derivation of the modes is more intricate as explained in sec.~\ref{sec:var} due to the non-trivial modification to the equations of motion and the appearance of the Young modulus as a response to bending. Additionally, at second order in derivatives, hydrodynamic and elastic corrections are dominant compared to backreaction corrections only if $n\ge3$ \cite{Armas:2011uf}. This means that the analysis here is only useful for $D\ge7$.
Solving for the second order correction in \eqref{eq:omegabs} using the modified linearised equations \eqref{eq:P2}, one obtains corrections to the elastic modes
\begin{equation}
\omega_{1,2}^{(2)}(k)=\mp\frac{\lambda_1\sqrt{n+1}n^2 \textbf{k}^4 \left(n^2+2n+1\pm4\sqrt{n+1}\beta\left(n+1+\beta^2\right)+\beta^2(6(n+1)+\beta^2)\right)
k }{Pr_0^2 \left(n+\textbf{k}^2\right)^4}~~,
\end{equation}
which are purely real and where $\lambda_1$ was introduced in app.~\ref{app:st}. Thus black strings are elastically stable within the blackfold approximation up to second order in derivatives. In addition, the hydrodynamic modes also receive corrections and are given in eq.~\eqref{eq:hydro2order}. In the case $\beta=0$, the correction $\omega_{3}^{(2)}$ agrees with that obtained in \cite{Caldarelli:2012hy, Caldarelli:2013aaa}. In the case of the critical boost for black rings, these corrections read
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.4\textwidth}
\includegraphics[width=\textwidth]{BBSd7.pdf}
\end{subfigure}
\qquad \qquad
\begin{subfigure}[b]{0.4\textwidth}
\includegraphics[width=\textwidth]{BBSlargeD.pdf}
\end{subfigure}
\caption{The figure on the left exhibits the imaginary part of $\omega_3$ at second order using \eqref{eq:hydro2order} as a function of $\bar k$ for $D=7$ for different values of the boost parameter: $\beta=0$ (black solid line), $\beta=1/2$ (grey solid line) and $\beta=9/10$ (red solid line). The grey dashed line represents the first order result for $\beta=1/2$ obtained in \eqref{eq:BS1}. The figure on the right provides a comparison between blackfold and large $D$ results for $D=5$ and $D=7$ for static strings and at the critical boost. The blue solid line is the blackfold result at second order for $\beta=0$ and $D=7$ while the dashed line is the corresponding large $D$ result at fourth order derived in \cite{Emparan:2015rva}. The black solid line represents the imaginary part of $\omega_3$ at first order as in \eqref{eq:BS1} while the black dashed line is the corresponding result at large $D$ \cite{Tanabe:2016pjr}. The grey solid line is the imaginary part of $\omega_3$ as in \eqref{eq:hydro2order} while the grey dashed line is the corresponding large $D$ result \eqref{eq:largen1}. } \label{fig:BBSd7}
\end{figure}
\begin{equation} \label{eq:bs2c}
\begin{split}
&\omega_{1}^{(2)}(k)=-\frac{\lambda_1 \sqrt{n+1}}{P r_0^2}k~~,~~\omega_{2}^{(2)}(k)=\frac{ \lambda_1n^4 \sqrt{n+1}}{Pr_0^2(n+2)^4}k~~,\\
&\omega_{3,4}^{(2)}(k)=\pm\frac{\sqrt{n+1} (n+2) (n (2 i (n+(1\pm i)) (\tau_\omega/r_0) -i n+(i\pm5))+(2i\pm6))}{2 (n+(1\pm i))^5}k~~,
\end{split}
\end{equation}
where $\tau_\omega$ was introduced in app.~\ref{app:st}.
Using the full expressions for arbitrary $\beta$ given in app.~\ref{sec:hydrocorr} we exhibit in the left plot of fig.~\ref{fig:BBSd7} the behaviour of the imaginary part of $\omega_3$ for different values of $\beta$ for $D=7$. It is observed a strong modification of the behaviour of $\omega_3$ at the critical boost when comparing first order (grey dashed line) and second order (grey solid line) in the figure on the left. Thus, for instance, the first order result \eqref{eq:BS1} for $\beta=1/2$ only accurately describes the behaviour of the instability up to values of $\bar k\sim 0.15$. Hence, the behaviour of the instability for boosted black strings becomes qualitatively similar to the static case (black solid line) and to the behaviour for $n=1$ \cite{Hovdebo:2006jy} as one includes higher-order corrections. Therefore, we expect these results to be approximately valid for $\bar k\sim 1$ and $\beta\sim1$.
The study of hydrodynamic and elastic instabilities of static and boosted black strings has been performed in \cite{Emparan:2013moa, Tanabe:2015hda, Tanabe:2016pjr} using large $D$ methods. In order to compare our results with those for the boosted black string at large $D$ in \cite{Tanabe:2016pjr}, we redefine the boost parameter $\beta$ such that $\alpha=\sqrt{n}\beta$ and we perform the expansion of our hydrodynamic modes at large $n$ (equivalent to large $D$ given that $p=1$ is fixed). This leads to the following expansions for the elastic modes
\begin{equation} \label{eq:BBSn1}
\begin{split}
\omega_{1,2}(k)=&(\alpha\mp1 )\frac{k}{\sqrt{n}}\pm\frac{1}{2}\left(1\mp2 \alpha +2 \alpha ^2+3 k^2 r_0^2\right) \frac{k}{\sqrt{n^3}} \\
&\mp\frac{1}{8}
\left(3\mp8 \alpha +12 \alpha ^2\mp8 \alpha ^3+2 k^2 \left(13\mp24 \alpha +12 \alpha ^2\right) r_0^2\right)\frac{k}{\sqrt{n^5}}+ O\left(\frac{1}{\sqrt{n^7}}\right)~~,
\end{split}
\end{equation}
while the hydrodynamic modes at large $D$ exhibit the following behaviour
\begin{equation} \label{eq:BBSn2}
\begin{split}
\omega_{3,4}(k)=& (\alpha\pm i ) \frac{k}{\sqrt{n}}-\frac{i k^2 r_0}{n}\mp\frac{i}{2} \left(1\pm2 i \alpha +2 \alpha ^2\right) \frac{k}{\sqrt{n^3}} \\
&+\frac{i
k^2 \left(-2\pm6 i \alpha +3 \alpha ^2\right) r_0}{2 n^2}+\frac{1}{8} \left(\pm3 i-8 \alpha \mp4 i \alpha ^2-8 \alpha ^3+8 k^2 (\pm i+2 \alpha ) r_0^2\right)
\frac{k}{\sqrt{n^5}}\\
&+\frac{k^2 \left(8 i\mp12 \alpha +60 i \alpha ^2\pm36 \alpha ^3-3 i \alpha ^4\right) r_0}{8 n^3}+ O\left(\frac{1}{\sqrt{n^{7}}}\right)~~.
\end{split}
\end{equation}
Comparing \eqref{eq:BBSn1} and \eqref{eq:BBSn2} with the corresponding results for $\ell=0$ and $\ell=1$ modes in eqs.~(B.12)-(B.13) of \cite{Tanabe:2016pjr}, we find complete agreement provided we ignore the terms of order $r_0^3k^3$ in the elastic modes of \cite{Tanabe:2016pjr}, which are of higher order in the brane thickness.\footnote{In order to compare with the results of \cite{Tanabe:2016pjr} we have set $r_0=1$ and used the large $D$ behaviour $\tau_\omega= r_0 (1/2 - \pi^2 /(3n^2)-4\zeta(3)/n^3-4 \pi^4 /(45 n^4) + \mathcal{O}(n^{-5})$. Also note that eq. (B.13) of \cite{Tanabe:2016pjr} contains several typos. We have used the ancillary file provided with \cite{Tanabe:2016pjr} to recover eqs.~(B.12)-(B.13) in order to identify them. The correct solution, up to $r_0^2k^2$ terms, is \eqref{eq:BBSn1}-\eqref{eq:BBSn2}, while the full solution is provided in eqs.~\eqref{eq:largen1}-\eqref{eq:largen2} for completeness.} In the case of the hydrodynamic modes, the two results agree exactly, without any approximation.
At finite values of $D$ we provide a comparison between blackfold and large $D$ results in the right plot of fig.~\ref{fig:BBSd7}. The black solid line in the figure on the right corresponds to the first order result \eqref{eq:BS1} while the black dashed line is the large $D$ \eqref{eq:largen1} result at $D=5$. The grey lines provide the same comparison with the second order result \eqref{eq:hydro2order} for $D=7$ at the critical boost. The blue solid line is the blackfold result for $\beta=0$ and the blue dashed line is the large $D$ fourth order result obtained in \cite{Emparan:2015rva} for $D=7$. The two approaches give similar results for small (though not too small) values of $\bar k$ when $D=7$ but differ at very small $\bar k$ where the blackfold approach is more accurate. We also see that in $D=7$ for the static case, where the large $D$ approach has been pushed to fourth order, and for values of $\bar k\gtrsim 0.3$ the large $D$ result is very inaccurate with the growth rate of the instability increasing without bound for higher values of $\bar k$, whereas the blackfold approach has shown to be approximately accurate in the entire range of $\bar k$ \cite{Caldarelli:2012hy, Caldarelli:2013aaa}. Additionally, for $D=5$ the large $D$ result is highly inaccurate for all $\bar k$, only becoming better for larger values of $D$.
\subsubsection*{Onset of the instability}
It is possible to obtain a refined expression for the onset of the instability \eqref{eq:kgl1} using the results of app.~\ref{sec:hydrocorr}. In particular we find that $k_{\text{GL}}$ can be expressed in closed form as in \eqref{eq:kgl2}
and provides a finite $k_{\text{GL}}r_0$ for the entire range $0\le\beta<1$. As clear from the comparison between first and second order results (grey solid and dashed lines) in the left plot in fig.~\ref{fig:BBSd7}, $k_{\text{GL}}r_0$ is highly sensitive to higher-order corrections.
\begin{figure}[h!]
\centering
\includegraphics[width=0.4\textwidth]{KGLBBS.pdf}
\caption{Onset of the Gregory-Laflamme instability $\bar k_{\text{GL}}=k_{\text{GL}} r_0$ as a function of $n$ as predicted by different methods for $\beta=0$: first order blackfold approach (black dashed line), second order blackfold approach (black solid line), fourth order large $D$ approach \cite{Emparan:2015rva} (blue solid line) and numerical points of \cite{Asnin:2007rw} (orange squares). } \label{fig:KGLBBS}
\end{figure}
In fig.~\ref{fig:KGLBBS}, we exhibit the behaviour of the onset of the instability as a function of $n$ for $\beta=0$ as predicted by the second order blackfold approach (black solid line) compared to the first order result \eqref{eq:kgl1} (black dashed line) and the large $D$ result (blue solid line) obtained in \cite{Emparan:2015rva}, together with the numerically obtained points (orange squares) of \cite{Asnin:2007rw}. The first striking thing to note is that the large $D$ approach is highly accurate for $n\ge2$ even though, as we have noted above and explicitly shown in fig.~\ref{fig:BBSd7}, the growth rate of the instability according to the same method increases without bound for $n\le5$, i.e. for instance, the large $D$ result does not predict the existence of a finite $k_{\text{GL}}r_0$ when $n<6$. Nevertheless, when extrapolating the large $D$ result valid for $n\ge6$ to lower values of $n$ the agreement with the numerical values is excellent for $n>1$, visible from the solid blue curve in fig.~\ref{fig:KGLBBS}.
The other interesting feature of fig.~\ref{fig:KGLBBS} is that the first order blackfold result more accurately predicts $k_{\text{GL}}r_0$ for $n\le4$ than the second order blackfold result (though when $n=1$ it is off by a factor of 5 compared to the numerical result). When $n\ge5$, the second order blackfold result becomes more accurate than the first order result and approaches the large $D$ result as $n$ increases. This is not surprising since, as stated earlier, the frequency $\omega_3$ reproduces exactly the corresponding large $D$ result. In general, we do not have the right to expect that the blackfold approach accurately describes the growth rate of the instability for values of $\bar k\gtrsim 1$ and it is already remarkable that in many cases it approximately does so.
\subsubsection*{A comment on $\omega_4$}
We note that $\omega_4$ at second order acquires an imaginary part for larger values of $\bar k$. For instance for $n=3$, it does so for $\bar k\gtrsim 5$. This threshold value for $\omega_4$ is always more than twice that of $\bar k_{\text{GL}}$ of $\omega_3$.\footnote{In fact we also observe that the imaginary part of $\omega_3$ at second order becomes positive again at a higher value of $\bar k$. We also consider this feature to be outside the regime of validity of the method employed here.} This feature is also visible in the large $D$ results of \cite{Emparan:2015rva}. We do not expect this to be a smoking gun for another hydrodynamic instability of black strings since these high values of $\bar k$ are a priori outside the regime of validity of both methods.
\section{Instabilities of black rings}\label{sec:br}
In this section we focus on the instabilities of asymptotically flat singly-spinning black rings in $D\ge5$ by following the same approach as in the previous section. We show that at ideal order (i.e. ultraspinning) black rings are Gregory-Laflamme unstable under
small linearised perturbations but elastically stable. Including higher derivative corrections yields a similar behaviour for the dispersion relations of the unstable perturbation as that found numerically in \cite{Santos:2015iua} for the non-axisymmetric quantised mode $m=2$ and $D=5$. The analysis here has higher accuracy for large modes $m\gg1$ and by including corrections up to second order in the thickness of the ring, we show that no elastic instability appears, thus contradicting large $D$ results \cite{Tanabe:2016pjr}. We obtain analytic expressions for the onset of the Gregory-Laflamme instability for black rings and study its behaviour as a function of $m$. We also find a long-lived mode describing a slowly oscillating wiggly black ring.
\subsection{Ideal order modes}
The black ring solution, up to first order in derivatives, is an equilibrium solution of \eqref{eq:BFeom} where the spatial worldvolume geometry is closed and the fluid elements living on it are rotating. It is useful to write the
flat Minkowski background in the form
\begin{equation}
ds^2=-dt^2+dr^2+r^2d\varphi^2+\sum_{i=1}^{D-3}\left(dx^{i}\right)^2~~,
\end{equation}
where we have isolated a two-dimensional spatial plane written in polar coordinates. The ring is embedded in this background by choosing $X^t=\tau,X^r=R, X^\varphi=\phi$ and $X^{i}=0~,~i=1,...,D-3$ such that the induced metric and
rotating Killing vector field are
\begin{equation} \label{eq:indBR}
\textbf{ds}^2=-d\tau^2+R^2d\phi^2~~,~~\textbf{k}^a\partial_a=\partial_\tau+\Omega\partial_\phi~~,~~\textbf{k}^2=1-\Omega^2R^2~~,
\end{equation}
where $0\le\phi\le2\pi$ and $\Omega$ is a constant angular velocity which admits the following expansion
\begin{equation}
\Omega=\Omega_{(0)}+\Omega_{(1)}\varepsilon+\Omega_{(2)}\varepsilon^2+...~~,~~\varepsilon=\frac{r_0}{R}~~.
\end{equation}
At ideal order, eq.~\eqref{eq:BFeom} fixes $\Omega_{(0)}=1/(R\sqrt{n+1})$. The difference between \eqref{eq:indBR} and the geometry of the black string of the previous section is the closed spatial topology. The geometry and Killing vector field of the boosted black string in sec.~\ref{sec:bs} are recovered at large radius $R\to~\infty$ by defining the coordinate $z=\phi R$ and the boost velocity $\beta=\Omega R=1/\sqrt{n+1}$.
As in the case of boosted black strings, we perform small perturbations of the energy density, fluid velocity and embedding scalars, in particular along the ring radial direction
\begin{equation} \label{eq:perturb1}
\delta\epsilon(\sigma)=\delta\epsilon e^{i\left(-\omega \tau+kR\phi\right)}~~,~~\delta \bar u^\phi(\sigma)=\delta \bar u^\phi e^{i\left(-\omega \tau+kR\phi\right)}~~,~~\delta R(\sigma)=\delta R e^{i\left(-\omega \tau+kR\phi\right)}~~,
\end{equation}
where we have defined $\bar \delta u^\phi=R \delta u^\phi$ which remains finite as $R\to\infty$. In this case, eqs.~\eqref{eq:P1} couple to each other and hence hydrodynamic and elastic perturbations cannot be studied individually. This means that $\delta R$ perturbations are necessarily accompanied by $\delta\epsilon$ and $\delta \bar u^{a}$ perturbations and vice-versa.\footnote{\label{foot:xi}It is also possible to consider perturbations along the remaining $n+1$ components of the embedding map, which decouple from $\delta R$ perturbations for the black ring even at second order in derivatives. At ideal and first order, the modes coincide with those of the boosted black string with $\beta=1/\sqrt{n+1}$ while at second order the elastic modes receive $1/R$ corrections which we provide in \eqref{eq:w56}. These perturbations do not lead to an elastic instability.} Since the spatial topology is closed, $k$ is quantised such that $m=k R$ for discrete $m$. In this context, the vanishing of the determinant of eqs.~\eqref{eq:P1} leads to two elastic modes which remain the same as for the boosted black string \eqref{eq:BS0} with boost $\beta=1/\sqrt{n+1}$ and hence stable, while the hydrodynamic modes read
\begin{equation} \label{eq:brmode0}
\omega_{3,4}^{(0)}=\frac{\sqrt{n+1}}{(n^2+2n+2)R}\left((n+2)m\pm\sqrt{2(n^2+2n+2)-n^2 m^2}\right)~~.
\end{equation}
At large radius $R\to\infty$ (i.e. at large $m\to\infty$), these frequencies reduce to those of the boosted black string with $\beta=1/\sqrt{n+1}$ given in \eqref{eq:BS0c}, as expected. It can be observed that the frequencies $\omega_{3,4}^{(0)}$ have an imaginary part, with $\omega_{3}^{(0)}$ being unstable only if
\begin{equation}
m>m_{\text{min}}=\frac{\sqrt{2}}{n}\sqrt{n^2+2n+2}~~,
\end{equation}
while $\omega_{4}^{(0)}$ is always stable.
In particular $m_{\text{min}}=\sqrt{10}$ for $n=1$ and, for $m=1$, the frequencies $\omega_{3,4}^{(0)}$ are always real for any $n$ while for $m\ge2$ complex frequencies are attained only if $n\ge3$. In any case, for each $n$ there is always a sufficiently large enough $m$ that makes $\omega_3$ unstable. This implies that, besides also being unstable in the fat branch $1/2\le r_0/R<1$ \cite{Arcioni:2004ww, Elvang:2006dd, Figueras:2011he}, black rings are also Gregory-Laflamme unstable in the thin regime $r_0/R\ll1$ in particular in the regime $0\le r_0/R\lesssim0.025$ as mentioned in sec.~\ref{sec:intro}.
It is worth noting that, for instance, for $m=1,2$ the frequency $\omega_{3}$ is real for $n=1$. This is not in contradiction with \cite{Santos:2015iua, Figueras:2015hkb} since the numerical analysis for $m=2$ has not been carried out in the region $\nu<0.144$ and it is possible to speculate whether the Gregory-Laflamme instability is present for $\nu\sim0$ or whether it ceases to exist at some small value of $\nu$. It is unclear at the present moment if a real $\omega_3$ is a prediction in the infinitely thin limit or whether the blackfold approach is not valid for $m=1,2$. In fact, as we shall see when studying first order corrections, these frequencies acquire an imaginary part but do not have the expected qualitative behaviour, while the elastic frequencies can develop poles at such low values of $m$. However, it is clear that the method employed here is more accurate when $m\gg1$. The perturbation wavelength $\lambda\sim k^{-1}\sim R/m$ must satisfy
\begin{equation} \label{eq:req}
\lambda\gg r_0~\Rightarrow~\frac{r_0}{R}\ll \frac{1}{m}~~.
\end{equation}
At fixed global temperature $T$ (i.e. fixed $r_0$) the boosted black string limit is attained when $R\to\infty$ and hence $m\to\infty$ such that $m/R$ is finite. Since the method employed here describes the dynamics of very thin (ultraspinning) rings, according to \eqref{eq:req} the larger $m$ is, the smaller $r_0/R$ must be. In particular, in the regime $m\gg1$ the dynamics of black rings is described by a mild deformation of the dynamics of boosted black strings. Specifically, we may expand the unstable frequency in \eqref{eq:brmode0} in powers of $1/m$, giving
\begin{equation}
\omega^{(0)}_3=\frac{(1+i) m \sqrt{n+1}}{(n+1+i)R}-\frac{i \sqrt{n+1}}{m n R}-\frac{i \sqrt{n+1} (n (n+2)+2)}{2 m^3 n^3 R}+\mathcal{O}\left(\frac{1}{m^5}\right)~~,
\end{equation}
which makes the connection with \eqref{eq:BS0c} explicit in the limit $R\to\infty$ and where the second and third terms represent deviations in $1/R$ away from the boosted black string. It is expected that these results will provide a good approximate description for higher values of $m$ but only a comparison with a numerical analysis, which is not currently available, will settle this issue.
The elastic modes for black rings are the same as for boosted black strings with critical boost \eqref{eq:BS0c} and hence purely real. It is plausible that these modes could acquire a positive imaginary part as we move away from the thin limit. This turns out not to be the case as we will show below. However, corrections to the dispersion relations are still useful as not only they represent long/short lived time-dependent black hole solutions but also allows to understand better the behaviour of dominant instabilities.
\subsection{First order modes and comparison with large $D$ analysis} \label{sec:br1st}
At first order in derivatives the equation of motion \eqref{eq:BFeom} set $\Omega_{(1)}=0$ for black rings. The stress tensor receives viscous corrections which for $p=1$ only depend on the bulk viscosity. The vanishing of the determinant of the system \eqref{eq:P1} now requires that the frequencies of the elastic modes take the form
\begin{equation} \label{eq:BR0e}
\omega_{1}=0+\mathcal{O}\left( \varepsilon^2\right)~~,~~\omega_2=\frac{2 m \sqrt{n+1}}{(n+2) R} \left(1+i\frac{2 m \sqrt{n} (n+2)}{4(n+1)-\left(m^2-1\right) n^2}\varepsilon\right)+\mathcal{O}\left( \varepsilon^2\right)~~,
\end{equation}
where $\varepsilon=r_0/R$ makes clear that the ring is not necessarily infinitely thin. We note that $\omega_{2}$ receives an imaginary contribution that is positive for $m=1~\forall~n$ and for $n=1$ with $m=2$, otherwise it is a negative contribution. However, the correction to $\omega_{2}$ has a pole when $\tilde m=(n+2)/n$, which is maximal when $n=1$ for which $\tilde m=3$ while it is $\tilde m=2$ for $n=2$ and only has another integer value at $\tilde m=1$ when $n\to\infty$. We interpret this divergence as a signal that we should not trust \eqref{eq:BR0e} (and the method in general) for $m\le\tilde m$. In particular, when $m=\tilde m$ the expansion manifestly breaks down.
In the more accurate regime $m\gg1$ the non-trivial elastic frequency \eqref{eq:BR0e} becomes
\begin{equation}
\omega_2=\frac{2 m \sqrt{n+1}}{(n+2) R}-\frac{4 i \sqrt{n+1}}{n^{3/2} R}\varepsilon+\mathcal{O}\left( \frac{1}{m^2}\right)+\mathcal{O}\left( \varepsilon^2\right)~~.
\end{equation}
Thus, at first order in derivatives, the blackfold approach is not able to identify an elastic instability for $m\gg1$. On the other hand, using the large $D$ approach ref.~\cite{Tanabe:2016pjr} has claimed the existence of an elastic instability. In order to provide a comparison\footnote{We remark that we are not imposing a priori constraints on the form of the mode number $m$ as a function of the parameters of the theory, besides requiring that $m \gg \tilde{m}$. Consequently, we expect the comparison with the large $D$ approach to be reasonable under the more general assumption of $m_\Phi = \mathcal{O}(1)$ as considered in \cite{Tanabe:2016pjr}. As in \cite{Tanabe:2016pjr}, we are also perturbing the stationary black ring configuration along the physical angular coordinate $\Phi$ of the large $D$ approach.}, we expand \eqref{eq:BR0e} at large $n$ and find
\begin{equation}
\omega_{1}=0+\mathcal{O}\left( \varepsilon^2\right)~~,~~\omega_{2}=\frac{2 m }{R}\frac{1}{\sqrt{n}}-\frac{4 i m^2}{\left(m^2-1\right) n R}\varepsilon +\mathcal{O}\left( \frac{1}{\sqrt{n^{7}}} \right)+\mathcal{O}\left( \varepsilon^2\right)~~,
\end{equation}
while the same frequencies in \cite{Tanabe:2016pjr} expanded in the thin radius regime $\varepsilon=r_0/R$ read
\begin{equation}
\omega_{1}^{(D)}=\frac{i m^2}{4\left(m^2-1\right) n R}\varepsilon+...~~,~~\omega_{2}^{(D)}=\frac{2 m }{R}\frac{1}{\sqrt{n}}-\frac{19 i m^2}{4\left(m^2-1\right) n R}\varepsilon+...~~.
\end{equation}
We see that $\omega_{1}$ and $\omega_{1}^{(D)}$ disagree and that $\omega_{2}$ and $\omega_{2}^{(D)}$ only agree at ideal order. In particular, $\omega_{1}^{(D)}$ is the frequency responsible for the elastic instability in \cite{Tanabe:2016pjr}, due to its positive imaginary part for any $m>1$. This disagreement indicates that $\omega_{1}^{(D)}$ and $\omega_{2}^{(D)}$ are not correct and hence the results of \cite{Tanabe:2016pjr} have not identified an elastic instability.\footnote{\label{foot:T} The author of \cite{Tanabe:2016pjr} does not think that his results are correct and has not been able to reproduce them at a later stage. This is why the author has never sent the paper for publication (e-mail correspondence).}
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.4\textwidth}
\includegraphics[width=\textwidth]{BRd5i.pdf}
\end{subfigure}
\qquad \qquad
\begin{subfigure}[b]{0.4\textwidth}
\includegraphics[width=\textwidth]{BRd5m.pdf}
\end{subfigure}
\caption{On the left, we show the imaginary part of frequencies $\omega_2$ (red solid line), $\omega_3$ (black solid line) and $\omega_4$ (grey solid line) for black rings up to first order in derivatives as a function of the ring thickness $\nu=r_0/R$ for $D=5$ and $m=10$ using the full expressions provided in the ancillary file. The dashed lines are the corresponding frequencies for boosted black strings in \eqref{eq:BS1} with critical boost. On the right plot we show the behaviour of the imaginary part of $\omega_3$ as a function of $m$ for $m=6$ (red), $m=8$ (blue), $m=10$ (black) and $m=12$ (purple) for $D=5$. } \label{fig:BRd5}
\end{figure}
The remaining sound modes receive the following corrections at large $n$
\begin{equation} \label{eq:w341}
\begin{split}
&\omega_3^{(1)}=\frac{m \left(m^2-3\right) \sqrt{m^2-2}-i \left(m^2-2\right) \left(m^2+1\right)}{\left(m^2+i\sqrt{m^2-2} m-2\right) n R}+\mathcal{O}\left(\frac{1}{\sqrt{n^3}}\right)~~,\\
&\omega_4^{(1)}=\frac{i(1-m^4+2 m^2)+2 \sqrt{m^2-2} m}{\left(m^2-1\right) n R}+\mathcal{O}\left(\frac{1}{\sqrt{n^3}}\right)~~,
\end{split}
\end{equation}
which also disagree with \cite{Tanabe:2016pjr} at first order in the thickness and reduce to \eqref{eq:bbs1storder} at large $n$. The results for arbitrary $n$ and $m$ are provided in the ancillary Mathematica file. We note that the mode $\omega_3$ is now also unstable for $m=2$, as previously advertised. However, comparison with the numerical results of \cite{Santos:2015iua} for $m=2$ hints towards the fact that $m=2$ is outside the regime of validity of the method employed here. Similarly, comparison of the mode $\omega_2$, which also developed an imaginary part at first order for $n=1$, with the results for the elastic instability found in \cite{Figueras:2015hkb} for $m=2$ seems to reiterate this point.\footnote{The comparison of our results with those of \cite{Santos:2015iua, Figueras:2015hkb} is not exact since the latter results are valid for $\nu\ge0.144$ while the former are expected to be valid for $\nu\lesssim0.025$. However, the qualitative behaviour of our results is far from what is expected, as it does not approximate the results of \cite{Santos:2015iua, Figueras:2015hkb} when extrapolated to larger values of $\nu$.} Additionally, it is clear from \eqref{eq:w341} that the expansion also breaks down for $m=1$ for any $n$ as $\omega_{3,4}$ develop a pole. This gives additional evidence that the expansion should not be trusted for $m\le\tilde m$.
In the left plot of fig.~\ref{fig:BRd5} we show the imaginary part of $\omega_{2,3,4}$ for $D=5$ and $m=10$. The plot shows that only the frequency $\omega_3$ (black solid line) has a positive imaginary part and hence signals a hydrodynamical instability. The dashed lines are the corresponding boosted black string results of sec.~\ref{sec:bs} at critical boost. As the thickness $\nu$ increases, the behaviour of the black ring frequencies increasingly differs from the boosted black string frequencies. It is expected that the results presented here will be valid for small $\nu\lesssim0.025$. In the left plot of fig.~\ref{fig:BRd5} we have clearly extrapolated the curves beyond the regime of validity.
In the right plot of fig.~\ref{fig:BRd5} we exhibit the growth rates of the instability $\omega_3$ for different values of $m$ starting with $m=6$ (red line) and ending with $m=12$ (purple line) for $D=5$. The curves show that the instability grows faster for increasing $m$. Thus, the large $m$ modes dominate the dynamics of very thin black rings.
\subsection{Second order modes}
At second order in derivatives for $D\ge7$, the stability analysis of black rings becomes more involved due to the additional non-trivial contributions to the equations of motion \eqref{eq:BFeom2}. At this order equilibrium \eqref{eq:BFeom2} requires that\footnote{This result is related to the one obtained in \cite{Armas:2014bia} via the field redefinition $R\to R-R\xi(n)\varepsilon^2/n$. }
\begin{equation} \label{eq:om2}
\Omega_{(2)}=\Omega_{(0)}\frac{n^2+3n+4}{2n^2(n+2)}\xi(n)~~.
\end{equation}
Thus the ideal order stress tensor will contribute with extra terms due to the second order correction to $\Omega$. As explained in sec.~\ref{sec:intro} we expect this analysis to be valid for small values of the thickness $\nu$, in particular for $\nu\lesssim0.27$ for $D=7$.
Given \eqref{eq:om2}, requiring the determinant of \eqref{eq:P2} to vanish leads to the purely real second order correction to the first elastic mode\footnote{It is worth mentioning that, like $\omega_{3,4}$ in \eqref{eq:w341}, $\omega_1$ develops a pole at $m=1$ for any $n$.}
\begin{equation}
\omega_1=0+\frac{m \left(m^2-2\right) \left(m^2+1\right) \sqrt{n+1} (3 n+4)}{2 \left(m^2-1\right) n^2 (n+2)R} \xi (n)\varepsilon^2+\mathcal{O}\left(\varepsilon^3\right)~~,
\end{equation}
which is purely real and reduces to \eqref{eq:bs2c}. Thus, $\omega_{1}$ acquires time-dependent behaviour as expected, since the fluid velocity is not aligned with a Killing vector field, but no unstable behaviour. Interestingly, up to this order this mode does not attenuate and thus represents a long lived time-dependent modulation of a black ring. Given that $\omega_1$ is real we conclude that the blackfold approach is not able to detect an elastic instability in asymptotically flat black rings at this given order in the expansion for $D\ge7$ and for any value of $m\ge2$.
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.4\textwidth}
\includegraphics[width=\textwidth]{BRd7i.pdf}
\end{subfigure}
\qquad \qquad
\begin{subfigure}[b]{0.4\textwidth}
\includegraphics[width=\textwidth]{BRd7im.pdf}
\end{subfigure}
\caption{The left plot exhibits the behaviour of the imaginary part of the frequencies $\omega_2$ (red solid line), $\omega_3$ (black solid line) and $\omega_4$ (grey solid line) for black rings up to second order in derivatives as a function of the ring thickness $\nu=r_0/R$ for $D=7$ and $m=10$. The dashed lines are the corresponding frequencies for boosted black strings in \eqref{eq:bs2c} with critical boost. The right plot exhibits the growth rate of the instability associated to $\omega_3$ as a function of $\nu$ for $m=6$ (red line), $m=8$ (blue line), $m=10$ (black line) and $m=12$ (purple line).} \label{fig:BRd7}
\end{figure}
The remaining modes acquire non-trivial corrections at second order, whose explicit expression we have provided in the ancillary Mathematica file. In the left plot of fig.~\ref{fig:BRd7} we exhibit the behaviour of the imaginary parts of the frequencies $\omega_{2,3,4}$ in $D=7$ for $m=10$ as a function of $\nu$. As it can be seen from fig.~\ref{fig:BRd7}, the frequency $\omega_3$ (black solid line) acquires a positive imaginary part in the region $\nu\lesssim0.27$ and we thus expect to accurately describe the onset of the Gregory-Laflamme instability.
\begin{figure}[h!]
\centering
\includegraphics[width=0.45\textwidth]{KGLBR.pdf}
\caption{Onset of the Gregory-Laflamme instability $\nu_{\text{GL}}$ for black rings as a function of $n$ for $m=8$ (black lines), $m=12$ (blue lines), $m=20$ (purple lines) and $m=50$ (red lines) using first order blackfold approach (dashed lines) and second order blackfold approach (solid lines).} \label{fig:KGLBR}
\end{figure}
We note that the frequency $\omega_2$ never acquires a positive imaginary part but that $\omega_4$ does. For $m=10$ and $D=7$, as seen from fig.~\ref{fig:BRd7}, the imaginary part of $\omega_4$ becomes positive for $\nu>0.3$. The origin of this positive imaginary part is rooted in the comment we made at the end of sec.~\ref{sec:bbs2} about the same behaviour of $\omega_4$ for the boosted black string. As explained there, the imaginary part of $\omega_4$ lies outside the regime of validity of the method. If $m$ increases, both the imaginary part of $\omega_3$ and $\omega_4$ are pushed to lower values of $\nu$ but $\omega_4$ remains outside the regime of validity due to \eqref{eq:req}. Thus, this does not signal a new instability.
In the right plot of fig.~\ref{fig:BRd7} we exhibit the growth rates of the instability associated with $\omega_3$ as a function of $m$ in $D=7$ starting with $m=6$ (red line) and ending in $m=12$ (purple line). For small values of $\nu$ the growth rate increases with increasing $m$ as already noted in fig.~\ref{fig:BRd5}. As $\nu$ increases further, the growth rate eventually decreases to zero, analogous to the behaviour of boosted black strings and to the numerical results of \cite{Santos:2015iua} for $m=2$. It is possible to determine the onset of the instability analytically. At large $m$ and $n$, the onset can be written in a compact form
\begin{equation}
\nu_{\text{GL}}=\frac{n (4 n (2 n-3)-53)-60}{8 m n^{3/2}}+\frac{n (n (131-12 n (2 n+1))+51)-769}{4 m^3 n^{3/2}}+\mathcal{O}\left(\frac{1}{m^5},\frac{1}{n^3}\right)~~.
\end{equation}
The full expression for the onset is provided in the ancillary Mathematica file. In fig.~\ref{fig:KGLBR} we exhibit the onset of the instability $\nu_{\text{GL}}$ as a function of $n$ for different values of $m$, in particular $m=8$ (black line) up to $m=50$ (red line) as predicted by the first order approximation (dashed lines) and second order approximation (solid lines). It is clear from fig.~\ref{fig:KGLBR} that the behaviour of the onset is qualitatively similar to that of the boosted black string of fig.~\ref{fig:KGLBBS}. One observes that as $m$ increases, the onset ends at thiner and thiner rings, in agreement with fig.~\ref{fig:BRd7}. These analytic results consist of the first analytic determination of $\nu_{\text{GL}}$.
\section{Discussion}\label{sec:dis}
In this paper we initiated a systematic study of the dynamical stability of black holes in $D\ge5$ in the blackfold limit (ultraspinning limit) and applied it to asymptotically flat boosted black strings and black rings. In the context of boosted black strings, though studied numerically in \cite{Gregory:1993vy, Hovdebo:2006jy}, we have provided new analytic results such as the growth rate of the Gregory-Laflamme instability for arbitrary boost $\beta$ and analytic expressions for the onset of the instability for arbitrary boost and spacetime dimension. In the context of black rings, we have provided the first correct analytic expressions for the growth rate of the Gregory-Laflamme instability as a function of the axisymmetric mode $m$ and for the onset of the instability. In $D=5$, our analysis is valid for at least $\nu\lesssim0.025$ and in $D=7$ for $\nu\lesssim0.27$. This thus progresses in closing the gap in parameter space where black rings were found to be unstable (i.e. for $\nu\ge0.144$ in \cite{Santos:2015iua} and $\nu\ge0.15$ in \cite{Figueras:2015hkb} in $D=5$) by showing explicitly the instability for very thin rings, and for large non-axisymmetric modes, where numerical methods are not precise enough.
Despite our analysis including second order corrections to the blackfold approximation, we have not been able to identify an elastic instability of black rings, as that found numerically in \cite{Figueras:2015hkb} for $m=2$ and $D=5$.\footnote{We emphasise that we have interpreted the non-axisymmetric instability found in \cite{Figueras:2015hkb} as an extrinsic perturbation from the blackfold point of view, which if unstable, would be visible in the dispersion relation of elastic modes. See also footnote \ref{foot:nome}. } We have identified divergences in the dispersion relations for hydrodynamic and elastic modes that manifestly break the expansion when $m=3$ for $D=5$, $m=2$ for $D=6$ and $m=1$ for all $D\ge5$. We have interpreted these divergences as signalling that the blackfold approximation breaks down for $m\le\tilde m=(n+2)/n$. In fact, a qualitative comparison of the growth rates of potential Gregory-Laflamme and elastic instabilities for $D=5$ and $m=2$ found here with those numerically obtained in \cite{Santos:2015iua, Figueras:2015hkb} indeed suggest that the analysis we have carried out is not valid for $m=2$ and $D=5$.\footnote{We note, however, that this comparison is only qualitative since the results of \cite{Santos:2015iua, Figueras:2015hkb} are only valid for $\nu\ge0.144$. See also sec.~\ref{sec:br1st}.} On the other hand, the analysis performed here is more accurate for large modes $m\gg1$ for which, within this approach and up to second order, no elastic instability is found in any dimension $D$. This suggests that there is a value of $\nu$ that marks the onset for the elastic instability and that due to the requirement \eqref{eq:req}, our analysis is only valid for very thin rings which lie in a region of parameter space below that onset.\footnote{We thank P. Figueras for suggesting this to us.} We observe in the work of \cite{Figueras:2015hkb} that the growth rate of the elastic instability for $\nu=0.15$ is close to zero, giving some rationale for this interpretation and, in addition, unpublished numerical results \cite{JB} substantiate this picture. It may be the case that signatures of the elastic instability appear at third or higher order but to push the blackfold approximation beyond second order is as a daunting task as it is useless since it would require a very high number of derivative corrections, making the effective theory impractical. At any rate, the non-existence of the elastic instability in the thin regime, and the fact that the elastic frequency $\omega_2$ is real up to second order, promptly suggests the existence of a long-lived mode that describes a slow time-dependent modulation of a black ring - a wiggly black ring.
The above considerations lead us to conclude that the Gregory-Laflamme instability is the dominant instability for black rings in $D\ge5$ in the thin regime. In this context, we also observed that the growth rate of the Gregory-Laflamme instability increases with increasing $m$ for very thin rings, in which case the dominant instability is that associated with the boosted black string. At higher values of $\nu$, and hence for thicker rings, there is competition between modes with different $m$ as shown in fig.~\ref{fig:BRd7}. A numerical analysis of black ring instabilities for $m\gg1$ would be extremely useful in order to provide a better comparison between analytic and numerical methods.
The results obtained here have been compared with corresponding results using the large $D$ approach. In the case of boosted black strings we have found an exact agreement with the findings of \cite{Emparan:2013moa, Tanabe:2015hda, Tanabe:2016pjr}, for which the inclusion of the Young modulus of black strings was key. In general, the blackfold approach provides a better approximation, more accurate for small $k$, and predicts a value for the onset of the instability for any $D$ whereas the large $D$ approach, though the predicted growth rates increase without bounds with increasing $k$ for $D\le9$, predicts a better onset of the instability when the large $D$ result is extrapolated to smaller values of $D$. In the case of black rings, we have compared our results with those of \cite{Tanabe:2016pjr} and found that the existent large $D$ results are not correct (see footnote \ref{foot:T}). Thus there is currently no analytic understanding of the elastic instability found in \cite{Figueras:2015hkb}.
This work only dealt with boosted black strings and black rings but the method we have provided here, and the complete characterisation of the black brane stress tensor up to second order in derivatives in app.~\ref{app:st}, is sufficient for studying the dynamical stability of a plethora of different uncharged asymptotically flat black hole solutions, such as helical black rings, Myers-Perry black holes and helicoidal black rings \cite{Emparan:2009vd, Armas:2015kra, Armas:2015nea, Armas:2017xyt}. We plan on returning to this general analysis in the future.
Furthermore, in the case of curved backgrounds such as Anti-de Sitter space, the method can be applied up to first order in derivatives to many of the black holes studied in \cite{Armas:2010hz, Armas:2015qsv}. In order to push it one order higher, it is required to first study the Love numbers of asymptotically flat black branes similarly to the work of \cite{Kol:2011vg, Emparan:2017qxd} and to extract the relevant transport coefficients associated with couplings to the background Riemann tensor.
Gathering some of the existent results in the literature \cite{Armas:2012ac, Armas:2013aka, Gath:2013qya, DiDato:2015dia, Armas:2016mes, Armas:2018ibg}, it will be possible to study the dynamical stability of black holes in supergravity and string theory by generalising the methods described here. Such generalisation can then be applied to the study of dynamical stability of charged black holes \cite{Grignani:2010xm, Emparan:2011hg, Grignani:2012iw, Armas:2012bk, Niarchos:2012pn, Armas:2013ota, Niarchos:2013ia, Armas:2014nea, Giataganas:2014mla, Grignani:2017vxh, Armas:2018rsy}. We wish to pursue this direction in the near future.
\section*{Acknowledgements}
We would especially like to thank P. Figueras for many useful discussions and for valuable comments on an early draft of this manuscript. We would also like to thank J. E. Santos and B. Way for providing the numerical data for their plots in \cite{Santos:2015iua} and sharing with us their unpublished results \cite{JB}. We are also especially grateful to J. E. Santos for comments on an earlier draft of this manuscript. We thank K. Tanabe for useful e-mail correspondence (see footnote \ref{foot:T}). We would also like to thank an anonymous referee for valuable comments to an earlier draft of this paper. The work of JA is partly supported by the Netherlands Organisation for Scientific Research (NWO). EP is grateful to the string theory group at the University of Amsterdam for hospitality. EP was partially supported by the School of Science at the University of Bologna.
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A: You can add an event listener on the button component.
player.controlBar.playToggle.on('click', function() {
console.log('click');
});
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,338 |
package de.javagl.nd.tuples.i;
import java.util.Arrays;
import java.util.stream.IntStream;
import de.javagl.nd.tuples.Utils;
/*
* Note: This class is automatically generated. Do not modify this class
* directly. See https://github.com/javagl/ND/tree/master/nd-gen/ for
* further information.
*/
/**
* Implementation of a {@link MutableIntTuple} that is backed by
* a part of a <code>int[]</code> array.
*/
final class ArrayIntTuple extends AbstractMutableIntTuple
{
/**
* The data of this tuple
*/
private final int data[];
/**
* The offset of this tuple in the given data array
*/
private final int offset;
/**
* The size of this tuple
*/
private final int size;
/**
* Creates a new tuple with the given data.
* A <strong>reference</strong> to the given
* data will be stored!
*
* @param data The data for this tuple
* @param offset The offset in the given array
* @param size The size
* @throws NullPointerException If the given array is <code>null</code>
* @throws IllegalArgumentException If the given offset or size is
* negative, or if <code>offset+size >= data.length</code>
*/
ArrayIntTuple(int data[], int offset, int size)
{
if (offset < 0)
{
throw new IllegalArgumentException(
"The offset is negative: "+offset);
}
if (size < 0)
{
throw new IllegalArgumentException(
"The size is negative: "+size);
}
if (offset + size > data.length)
{
throw new IllegalArgumentException(
"The offset is " + offset + " and size is " + size + ". " +
"The array length must be at least " + (offset + size) +
", but only is " + data.length);
}
this.data = data;
this.offset = offset;
this.size = size;
}
@Override
public int getSize()
{
return size;
}
@Override
public int get(int index)
{
Utils.checkForValidIndex(index, size);
return data[index+offset];
}
@Override
public void set(int index, int value)
{
Utils.checkForValidIndex(index, size);
data[index+offset] = value;
}
@Override
public MutableIntTuple subTuple(int fromIndex, int toIndex)
{
return new ArrayIntTuple(data, offset+fromIndex, toIndex-fromIndex);
}
@Override
public IntStream stream()
{
return Arrays.stream(data);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,981 |
\section{Introduction}
The geometry of an Active Galactic Nuclei (AGN) is widely accepted to be
an accretion disc rotating around the central supermassive black hole, and
illuminated by a hard X-ray source known as the `corona'. The accretion disc
absorbs high-energy photons, and then atomic transitions take place and emit
a series of X-ray emission lines in which the Fe K$\alpha$ line is the most
prominent. The fluorescent lines are intrinsically narrow but sometimes appear
broad and asymmetric in the observed spectrum. The shape of the broad, skewed
Fe K$\alpha$ emission line discovered in the Type 1 AGN MCG-6-30-15
\citep{Tanaka95} can be caused by special and general relativistic effects \citep{Fabian89}
if the line originates from the innermost part of the accretion disc where strong
gravitational fields of the central black hole cast important effects. This provides a
powerful method of measuring the inner radius of the accretion disc (and hence
the black hole spin in accreting black hole systems) using the iron line profile
\citep[see reviews by][]{Miller07,Fabian10}. Nevertheless, some researchers
suggested that the central engine cannot be directly observed even in the X-ray band
and proposed a partial-covering scenario to explain the spectral properties of the X-ray
spectrum of AGN \citep{Inoue03,Miller09}. If this is the case, we cannot obtain
information from the innermost areas of black holes via X-ray observations. Thus, the inner radius
of the accretion disc cannot be measured and alternative methods need to be developed
to probe the vicinity of accreting systems.
\citet{Risaliti13} used the simultaneous
\emph{XMM-Newton} and \emph{NuSTAR} observations to break spectral degeneracy and found that reflection
of NGC~1365 arises from a region within 2.5 gravitational radii ($R_{\rm g}=GM/c^2$),
but it is difficult to distinguish the relativistic reflection model and the partial-covering
scenario using spectral analysis in most X-ray observations \citep{Walton14}. In order to break
this degeneracy, detailed spectral-timing analyses have been performed in recent years.
The principle goal of these studies have been to search for the reverberation lags predicted
by the disk reflection scenario \citep{Fabian89,Reynolds99}.
The first significant X-ray reverberation lag was found in the narrow-line Seyfert 1
(NLS1) galaxy 1H0707-495 \citep[][see \citet{McHardy07} for an earlier hint in Ark~564]
{Fabian09}. A $\sim$30 s soft lag, which
implies that the soft X-rays lag behind the hard X-rays by 30 seconds, was
reported as evidence supporting the reflection scenario. Similar soft lags
have been reported in a number of AGN as well \citep{Emm11,Zoghbi12,Zoghbi14,
Cackett13,DeMarco13,Kara13,Kara13Ark,Kara15,Kara16}.
However, alternative explanations of soft lags were proposed. \citet{Jin13}
suggested that the majority of the soft excess in the NLS1 PG~1244+026 is produced by
a cool Comptonization component rather than reflection. \citet{Gardner14} further examined
the source and found that a model for the soft excess consisting of intrinsic emission from
the accretion flow and reprocessed emission can explain the spectral and timing
properties of PG~1244+026.
Some work suggested that soft lags
are caused by scattering reverberation from distant (tens to hundreds of
$R_{\rm g}$) material away from the primary
X-ray source \citep{Miller10,Legg12,Turner17}. These papers suggested that
the true signature of reverberation is the low-frequency lags, and the high-frequency lags
inferred as reverberation is just phase-wrapping of the low-frequency signal \citep[however,
see counter-arguments in][]{Zoghbi11}.
\citet{Kara13Ark} then showed that the lag-energy spectra of these frequency regimes were
distinct, demonstrating the need for different physical processes.
In addition, \citet{Walton13} found a low-frequency lag in NGC~6814
where the spectrum seems to be dominated by only the powerlaw continuum.
Both works argued against the distant reverberation scenario, supporting an inner disk reflection
reverberation origin for the high-frequency lags. \citet{DeMarco13}
examined a sample of AGN with black hole masses spanning
$\sim10^{6}-10^{8}M_{\odot}$ and found that more massive AGN have
larger soft lags appearing at lower Fourier frequencies. The mass-scaling
relation and values of lags indicate that soft lags originate in the innermost
area of AGN.
The discovery of Fe K$\alpha$ reverberation in NGC 4151 \citet{Zoghbi12}
and 20 subsequent AGN
\citep{Zoghbi13Fe,Fabian13,Kara13IRAS,Kara13,Kara13Ark,Kara14,Kara15,Kara16}
further supports reverberation from the inner disc (see \citet{Uttley14} for a detailed
review of X-ray reverberation). The self-consistency of the relativistic reflection
scenario indicates that X-ray observations provide information from the vicinity
of black holes and can help understand the central engines of AGN.
PDS~456 is a luminous ($L_{\rm bol}=10^{47}$ erg s$^{-1}$, \citealt{Simpson99})
nearby (z = 0.184, \citealt{Torres97}), radio-quiet quasar, which is known for powerful disc
winds/outflows \citep{Reeves03,Reeves09} and rapid X-ray variability
\citep{Reeves00,Reeves02}. The source was observed several times with major X-ray observatories
including \emph{BeppoSAX} \citep{Vignali00}, \emph{ASCA} \citep{Vignali00},
\emph{XMM-Newton} \citep{Nardini15}, \emph{Suzaku} \citep{Reeves14},
and \emph{NuSTAR}.
\citet{Vignali00} and \citet{Reeves00} both reported a significant absorption
feature at around $8-9$ keV rest frame
in an early \emph{ASCA} observation. The 2007 \emph{Suzaku} observation further
revealed that the feature is likely the blue-shifted 6.97 keV iron absorption line
at velocities of $\sim0.25c$ \citep{Reeves09}. The ultrafast outflow (UFO) has been
confirmed in later observations as well \citep{Reeves14,Gofford14,Nardini15,
Matzeu16}.
UFOs have garnered wide interest because these energetic outflows are capable of
contributing to AGN feedback. \citet{Nardini15} interpreted the broadened Fe K$\alpha$
emission line with the adjacent absorption feature caused by the UFO in
\emph{XMM-Newton}/\emph{NuSATR} observation of PDS~456 as a
P-Cygni profile produced by an expanding gaseous envelope. In their analysis
they also tested the disc absorption scenario \citep{Gallo11,Gallo13}, but found that
it is not required statistically.
Apparent spectral variability can be seen by
comparing observations taken at different epochs and different observatories. Some previous
work used a partial-covering model to account for most of the spectral variability
\citep{Gofford14,Nardini15,Matzeu16}. We revisit the long, archival \emph{Suzaku}
observations to examine if partial-covering absorbers are required to explain
the spectrum. Based on the archival observations, the
source shows short-term variability and hence an ideal source for timing analysis. PDS~456
has a high-mass black hole (Log $M_{\rm BH}=8.91\pm0.50$, \citealt{Zhou05}), beyond
the mass range of the mass-scaling relation in \citet{DeMarco13}.
In this work, we use the \emph{Suzaku} observations to examine
whether a soft lag(s) can be found in this source, such as other AGN with lower
masses, and if the result still follows the mass-scaling relation.
With the spectral-timing analysis, we re-examine the data to figure out if
the relativistic reflection model gives consistent results.
\begin{table}
\centering
\caption{Summary of the \emph{Suzaku} observations of PDS~456. }
\label{summary}
\begin{tabular}{clcc}
\hline
\hline
Obs. ID & Start Date & Exposure (ks) & Count Rate (ct s$^{-1}$)\\
\hline
701056010 & 2007 Feb. 24 & 190.6 & $(2.7\pm0.1)\times10^{-1}$ \\
705041010 & 2011 March 16& 125.5 & $(1.4\pm0.1)\times10^{-1}$ \\
707035010 & 2013 Feb. 21 & 182.2 & $(6.7\pm0.1)\times10^{-2}$ \\
707035020 & 2013 March 03& 164.8 & $(4.4\pm0.1)\times10^{-2}$ \\
707035030 & 2013 March 08& 108.2 & $(5.2\pm0.1)\times10^{-2}$ \\
\hline
\hline
\end{tabular}
\end{table}
\begin{figure}
\begin{center}
\leavevmode \epsfxsize=8.5cm \epsfbox{lc_all.ps}
\end{center}
\caption{The 0.3-10.0 keV XIS (XIS0, XIS1, and XIS3 combined) light
curves for all observations. It can be seen that the duration of the longest observation is $\sim$
438 ks, which is much longer than an \emph{XMM-Newton} orbit ($\sim$ 130 ks).
All the light curves are highly variable.
}
\label{lc}
\end{figure}
\section{Data Reduction} \label{section_da}
PDS~456 was observed with \emph{Suzaku} in 2007 (Obs ID: 701056010),
2011 (Obs ID: 705041010), and 2013 (Obs IDs: 707035010, 707035020, and
707035030), with good exposure times of $\sim$ 190.6 ks, $\sim$ 125.5 ks
and $\sim$ 455.2 ks, respectively (see Table \ref{summary}). The X-ray Imaging Spectrometer (XIS) was
operated in the normal mode in all observations. The data were reduced using the
{\sevensize HEASOFT} V6.16 following the Suzaku Data Reduction Guide. We
used a circular region with a 150\arcsec radius to extract source spectra and light
curves; the same shaped region was used to extract background products in a source-free region.
Response files were generated by the {\sevensize XISRESP} script. We combined
spectra and response files of the front-illuminated (FI) CCD XIS detectors (XIS0 and
XIS3) using {\sevensize ADDASCASPEC} in {\sevensize FTOOL}. Spectra of different
Obs IDs of the 2013 observation were combined using {\sevensize ADDASCASPEC}
as well. The total absorption-corrected $0.5-10.0$ keV FI-detectors fluxes are
$9.2\times10^{-12}$ erg cm$^{-2}$ s$^{-1}$, $6.9\times10^{-12}$ erg cm$^{-2}$
s$^{-1}$, and $4.1\times10^{-12}$ erg cm$^{-2}$ s$^{-1}$ for the 2007, 2011 and 2013
observations. We produced $0.3-10.0$ keV background-corrected light curves with
1 ks time bin for the XIS0, XIS1, and XIS3 detectors, and combined all of them to form
an overall light curve for each observation. The light curves are displayed in Fig. \ref{lc}
and it can been seen that the source shows variability in all observations.
The Hard X-ray Detector (HXD) was operated in XIS nominal pointing mode in all
observations. We extracted the non-X-ray background using the tuned model. The
total background was generated using the PIN reduction script
{\sevensize HXDPINXBPI}, which combines the non-X-ray background and cosmic
background automatically. The net count rates are 0.007 count s$^{-1}$, 0.001 count
s$^{-1}$, and 0.003 count s$^{-1}$ in the effective PIN energy band $15.0-45.0$ keV,
which are 2.0\%, 0.8\% and 1.1\% of the total background count rate. The source
was fairly faint in the band during the later two observations, and only marginally
detected in the 2007 observation. We did not include the PIN light curves in the
following timing analysis, and only used the 2007 data for spectral analysis.
\begin{table*}
\caption{Parameters for the best-fitting reflection model to the data,
in which $N_{\mbox{\scriptsize H}}$ is given in $10^{21}$ cm$^{-2}$, and $\xi$ in
erg cm s$^{-1}$. The normalisation of the powerlaw component is expressed in
photons keV$^{-1}$ cm$^{-2}$ s$^{-1}$. The hard upper limit of $q$ is 8, and that of
$A_{\rm Fe}$ is 10.}
\label{spec}
\centering
\begin{tabular}{@{}llccc}
\hline\hline
Component & Parameter & 2007 & 2011 & 2013 \\
\hline
TBABS & $N_{\rm H}$ ($10^{21}$ cm$^{-2}$) & $2.6\pm0.2$ & $-$ & $-$\\
XSTAR$_{1}$ (UFO) & $N_{\rm H}$ ($10^{22}$ cm$^{-2}$) & $4.1^{+1.2}_{-3.0}$ & $0.9^{+0.9}_{-0.5}$ & $1.8^{+0.8}_{-0.6}$\\
& Ionization parameter, log $\xi$ & $4.3^{+0.2}_{-0.4}$ & $3.5^{+1.0}_{-0.1}$ & $3.5\pm0.1$ \\
& Redshift, $z$ & $-(9.2^{+0.6}_{-0.5})\times10^{-2}$ & $-$ & $-$\\
XSTAR$_{2}$ (low-energy) & $N_{\rm H}$ ($10^{21}$ cm$^{-2}$) & $-$ & $3.4^{+1.5}_{-0.9}$ & $4.8^{+2.1}_{-1.5}$\\
& Ionization parameter, log $\xi$ & $-$ & $0.5^{+0.2}$ & $1.2\pm0.1$ \\
& Redshift, $z$ & $-$ & $(0.184)$ & $(0.184)$\\
POWERLAW & Photon index, $\Gamma$ & $2.35\pm0.01$ & $2.17^{+0.08}_{-0.06}$ & $1.73^{+0.06}_{-0.05}$ \\
& Norm & $(2.2\pm0.1)\times10^{-3}$ & $(1.2\pm0.1)\times10^{-3}$ & $(3.5^{+0.3}_{-0.2})\times10^{-4}$\\
RELCONV & emissivity index, $q$ & $2.2^{+0.2}_{-0.1}$ & $5.3^{+0.7}_{-0.9}$ & $6.1^{+0.4}_{-0.5}$ \\
& Spin parameter, $a^{*}$ & $>0.990$ & $-$ & $-$\\
& Inclination, $i$ (deg) & $(65\pm2)^{\circ}$ & $-$ & $-$ \\
REFLIONX & Iron abundance / solar, $A_{\rm Fe}$ & $> 9.3$ & $-$ & $-$\\
& Ionization parameter, $\xi$ & $67^{+19}_{-7}$ & $70^{+30}_{-10}$ & $120^{+30}_{-10}$ \\
& Norm & $(1.0\pm0.3)\times10^{-6}$ & $(1.9^{+0.9}_{-0.6})\times10^{-6}$ & $(9.4^{+1.6}_{-1.4})\times10^{-7}$ \\
$\chi^{2}/d.o.f.$ & & 1131/1107 & 784/786 & 1168/1122\\
\hline\hline\\
\end{tabular}
\end{table*}
\section{Data Analysis}
We perform both spectral and timing analyses, described below. \citet{Reeves09}
claimed that a partial-covering Compton-thick absorber is needed to explain the hard
excess above 10 keV, while \citet{Walton10} used a self-consistent, relativistic
reflection scenario with a full-covering absorber and successfully interpreted the data.
\citet{Gofford14} \& \citet{Matzeu16} suggested that partial-covering Compton-thick winds are
required to interpret the spectral variability shown in PDS~456, though latest
work by \citet{Matzeu17} ruled out partial-covering absorption to be the physical
mechanism behind short timescale variability, as the timescales are too short to be
induced by variable absorption. Since the 2011 and
2013 observations have not been tested using the relativistic reflection model,
we include spectral analysis in this work though we mainly focus on timing analysis
here.
\citet{DeMarco13} reported a negative lag of $\sim$400 s at $\sim3\times10^{-5}$ Hz,
which was not significantly
detected ($<1\sigma$), between the soft ($0.3-1$ keV) and hard ($1-5$ keV)
energy bands using short \emph{XMM-Newton} observations. Based on the
frequency-mass relation of \citet{DeMarco13}, the lag of PDS~456 should appear
at $\sim10^{-5}$ Hz and a frequency band including frequencies lower than $10^{-5}$ Hz
needs to be probed to confirm detection. In this work, we use long
\emph{Suzaku} observations to probe low Fourier frequencies which \emph{XMM-Newton}
data cannot reach. We select the soft and hard bands, which are slightly different
from the choice of \citet{DeMarco13}, based on results of spectral analysis.
In all the spectral analyses, which were performed using the XSPEC 12.8.2 package
\citep{Arnaud96}, the ``wilm" abundances \citep{Wilms00} and ``vern"
cross section \citep{Verner96} were used.
All of the errors quoted in the spectral analysis represent the 90 per cent confidence level,
while errors in the timing analysis represent 1$\sigma$ uncertainties.
\subsection{Spectral Analysis} \label{subsec_spectral}
We fit the FI XIS spectra in the $0.5-10.0$ keV energy range, excluding energies from $1.7-2.1$ keV due to
known calibration issues. We fit the PIN spectrum over the $15.0-45.0$ keV energy band but only the
spectrum of the 2007 observation was included. Note that there is mild spectral
variability in the 2013 observation. However, in this work we focus on the overall
interpretation of the spectrum rather than the variations of warm absorbers, and we use
the combined 2013 spectrum for spectral fitting.
The data were
fit using the same model used in \citet{Walton13BareAGN}, which consists of a powerlaw and a
relativistic disc reflection components, and an {\sevensize XSTAR} grid (the same one
used in \citealt{Walton10,Walton13BareAGN}) to model the
absorption feature shown around the Fe K band caused by the ultrafast flow. Galactic
absorption was accounted for using the {\sevensize TBABS} model in XSPEC. We use
the REFLIONX grid \citep{Ross05} to model reflected emission, and the RELCONV
\citep{Dauser10} kernel to act on the reflected emission to account for relativistic effects.
We assume the inner
edge of the accretion disc to extend down to the innermost stable circular orbit (ISCO),
and set the outer radius to be 400 $R_{\rm g}$ ($R_{\rm g}=GM/c^2$). The iron
abundance of the accretion disc and the outflow is set to be identical.
In the spectral fitting we set parameters which are not expected to change significantly over
time to be the same in all data. These are, the column density of Galactic absorption
$N_{\rm H}$, the iron abundance $A_{\rm Fe}$, the spin parameter $a^*$, and the
inclination angle $i$.
\citet{Reeves09} have shown that the ultrafast outflow is persistent throughout the 2007
and 2011 \emph{Suzaku} observations, with an outflow velocity of 0.25-0.3$c$.
\citet{Matzeu16} examined the 2013 \emph{Suzaku} observation and found an outflow
velocity of $0.25^{+0.01}_{-0.05}c$. \citet{Nardini15} also reported a similar value of
$0.25\pm0.01c$ using the long 2013-14 \emph{XMM-Newton} data, implying the
ultrafast outflow in PDS~456 remains at constant velocity across several years. Hence
we link the redshift of the {\sevensize XSTAR} grid across all data as well. In order to
expedite the fitting process, we fit all data together to obtain the best-fitting `unchanged
parameters' mentioned above, and then fit each observation individually with restricting
these parameters to vary within the best-fitting error range. We show the best-fitting
values of the `unchanged parameters' in Table \ref{spec}.
The model results in a good $\chi^2/d.o.f.$ of 1131/1107 for the the 2007 observation
and the best-fitting parameters are shown in Table \ref{spec}. We obtain $\chi^2/d.o.f.$
= 831/788 and 1300/1124 for the 2011 and 2013 data, respectively. It seems there is a
mild curvature around 1 keV in the 2013 spectrum (see the second last lower panel in
Fig. \ref{spec_fitting}) which causes residuals
and the model could not give as good a fit as for the 2007 and 2011 data. The feature could be
caused by emission or absorption. We find that the curvature cannot be modeled by
a single emission/absorption line nor by an absorption edge. If including an ionized
absorber (ABSORI in XSPEC) in the model, the fitting improves to $\chi^2/d.o.f.$
= 1171/1124, implying that the distortion in the 2013 spectrum is caused by a warm absorber.
Moreover, \citet{Nardini15} also identified a fully-covered warm absorber that affects the
spectrum below $<$ 2 keV in the 2013 \emph{XMM-Newton} observation.
We generate another {\sevensize XSTAR} grid (assuming $\Gamma=2$) with a lower
ionization range (log $\xi=0.5-2$), to account for the effects of the warm absorber.
Note that we set the covering fraction for both the high-ionization
(to account for the ultrafast outflow) and low-ionization (to model the $<$ 2 keV spectrum
affected by warm absorber) {\sevensize XSTAR} grids
to be 100 per cent (fully-covered). The total model is now: TBABS*XSTAR$_{1}$*
XSTAR$_{2}$*(POWERLAW + RELCONV*REFLIONX). Since there is no resolved
absorption lines to constrain the outflow velocity, we fix the redshift of the second
{\sevensize XSTAR} grid at $z=0.184$, the redshift of PDS~456. Note that
\citet{Reeves16} reported broad
absorption lines at soft X-ray energies and found an outflow velocity of $\sim0.1-0.2c$.
We do not probe the details of the second outflow here as the \emph{Suzaku} XIS
data are not good enough to constrain its properties. We find that although the
curvature does not seem to appear in the 2011 spectrum, the fitting improves a bit
with the low-ionization {\sevensize XSTAR} grid. Best-fitting parameters of the model
including two {\sevensize XSTAR} grids are list in Table \ref{spec} as well. We show
the fitting results and decomposed model in Figs. \ref{spec_fitting} \& \ref{model}. It can be seen that $0.3-0.8$
keV is dominated by reflection and $1-3$ keV is primarily contributed by the
powerlaw component. We choose these energy bands as ``soft" and ``hard" bands
to perform the timing analysis.
It can be seen that no partial-covering absorbers are included in our model, yet
spectral variability can be explained by changes in the continuum and evolution of
the warm absorbers. The model fits the 0.5-10 keV energy band well. Wiggles around
$\sim$9 keV in observed frame are likely due to the K$\beta$ and K edge absorption
from Fe XXVI \citep{Nardini15}. The effective areas of the \emph{Suzaku} XIS detector drop significantly
around this band, making detailed modeling difficult. The column density of the
high-ionization warm absorber obtained in this work is much lower than that reported in
\citet{Nardini15} ($N_{\rm H}=6.9^{+0.6}_{-1.2}\times10^{23}$ cm$^{-2}$). This might be due to different
assumptions made when creating the {\sevensize XSTAR} grid, and the high iron
abundance obtained in this work. In addition, it is known that the UFO evolves slightly
over time, and the data they used were not simultaneous with those used in the present work.
The best-fitting column density of the less-ionized absorber ({\sevensize XSTAR$_{2}$}) is close to
that obtained in \citet{Nardini15}, but with a slightly higher ionization parameter
($N_{\rm H}\sim (1.8-4.3)\times10^{21}$ cm$^{-2}$ and log $\xi=0.31\pm0.02$ in their work).
We also test the model on the 2013 \emph{NuSAR} data which was reduced following the same
procedures in \citet{Nardini15}, and it fits the 3-35 keV energy band in observed frame very well
(see the last panel of Fig. \ref{spec}). We obtain a good $\chi^2/d.o.f.$ of 348/342, and major
parameters (photon index, ionization parameters and column densities of warm absorbers, etc.)
are broadly consistent with those reported in \citet{Nardini15}. This indicates that our model can
interpret the broadband NuSTAR data successfully.
We also replace the REFLIONX grid with the XILLVER \citep{Dauser10,Garcia14}
model to test if the results are consistent. We find that major line parameters such
as the spin parameter and iron abundance are broadly consistent. The XILLVER
model tends to give a higher photon index, a higher emissivity index and a lower
ionization parameter, which is likely caused by different assumptions made between
these two models. In general, REFLIONX and XILLVER give consistent results to
the data.
\begin{figure}
\begin{center}
\leavevmode \epsfxsize=8.5cm \epsfbox{spec_fitting.ps}
\end{center}
\caption{Spectral fits to the \emph{Suzaku} and 2013 \emph{NuSTAR} data of PDS~456 in observed frame. The top panel shows
the data used in this work and the lower four panels present the data/model ratios.
The 2007 spectrum only requires one {\sevensize XSTAR} component to account
for the ultrafast outflow, while the 2011 and 2013 data are better interpreted with
an additional low-ionized {\sevensize XSTAR} component. We
show the fitting result of the 2013 observation, and improvement with the second
{\sevensize XSTAR} component in different panels. The last panel shows the fitting
result of the 2013 \emph{NuSTAR} observation, and it can be seen that the model fits the data very well.
}
\label{spec_fitting}
\end{figure}
\begin{figure}
\begin{center}
\leavevmode \epsfxsize=8.5cm \epsfbox{lag_model_comparison.ps}
\end{center}
\caption{{\it Top: }The lag-energy spectrum from all \emph{Suzaku} observations in the $[7.2-12.7]\times10^{-6}$ Hz frequency range. {\it Bottom:} The decomposed best-fitting spectral model of the 2013 \emph{Suzaku} observation in rest frame. The vertical dashed lines are to guide the eye in associating the enhanced lag in the 7.1 -- 8.3 keV band with the blue-wing of the Fe K emission line in the best-fitting model.}
\label{model}
\end{figure}
\subsection{Timing Analysis}
The Fourier phase lag between the hard and soft energy bands can be comptuted
following the technique described in detail in \citet{Uttley14}. In traditional Fourier analysis,
continuous light curves are required. However, the technique is not possible with
\emph{Suzaku} light curves as the telescope was in a low Earth orbit with an orbital
period of $\sim$5760 s, causing frequent gaps in the data. Although \emph{Suzaku}
data cannot be analysed by the traditional Fourier technique, the duration of an
observation is around twice of its exposure time, making these data ideal to probe the
low Fourier frequency domain which is desired when studying massive AGN. \citet{Zoghbi13}
developed a method to determine the powerspectrum density and lags from
non-continuous data, containing gaps. This
method uses a maximum likelihood approach to calculate time lags by fitting
the light curves directly. It has been shown by Monte Carlo simulations to give the
same results as the traditional Fourier method. Details of how uncertainties are estimated
can be found in \citet{Zoghbi13}.
\begin{figure}
\begin{center}
\leavevmode \epsfxsize=8.5cm \epsfbox{psd_soft_hard.ps}
\end{center}
\caption{The PSD of the soft ($0.3-0.8$ keV;
$0.36-0.95$ keV in the rest frame; red data points) and hard ($1-3$ keV; $1.18-3.55$ keV
in the rest frame; blue data points) light curves.
}
\label{psd}
\end{figure}
\begin{figure}
\begin{center}
\leavevmode \epsfxsize=8.5cm \epsfbox{lag_spec_all_clag.ps}
\end{center}
\caption{The lag-frequency spectrum computed using $\sim$ 770 ks
of \emph{Suzaku} data. The lag is calculated between the soft ($0.3-0.8$ keV;
$0.36-0.95$ keV in the rest frame) and the hard ($1-3$ keV; $1.18-3.55$ keV
in the rest frame) energy bands, and a negative lag means that the soft band lags behind
the hard band. It can be seen that the most negative lag is $10000\pm3400$ s at
$9.58\times10^{-6}$ Hz.
}
\label{lag_spec}
\end{figure}
The \emph{XMM-Newton} observations used by \citet{DeMarco13} only have
exposures up to $\sim$ 90 ks. The longest observation used in
this work was of $\sim$438 ks duration, and we generated the light curves
using an 1 ks time bin, leading to a minimum Fourier frequency of
$\sim2.3\times10^{-6}$ Hz and a maximum of $5\times10^{-4}$ Hz (the
Nyquist frequency). Roughly even logarithmically-spaced
frequency bins were used and we ensure at least 3 data points to fall within each
frequency bin. In Fig. \ref{psd} we show the power spectrum density (PSD) of the soft and hard
light curves. Fig. \ref{lag_spec} shows the lag-frequency spectrum between the
soft ($0.3-0.8$ keV, dominated by disc reflection, $0.36-0.95$ keV in the rest frame)
and the hard ($1-3$ keV, dominated by primary powerlaw emission, $1.18-3.55$ keV
in the rest frame) energy bands. The trend
that lags switch from negative to positive (the hard flux lags behind the
soft) at low frequency has been seen in a number of previous timing studies
\citep{Zoghbi11,Emm11,Kara13IRAS,Kara13,DeMarco13}, and again occurs in the lag-frequency
spectrum of PDS~456 though there is only one frequency bin due to the maximum duration of the lightcurves.
We found the maximum negative lag where the soft band
lags the hard by $10000\pm3400$ s to appear at $[7.2-12.7]\times10^{-6}$ Hz
(the amplitude of the soft lag and frequency have been corrected for cosmological redshift).
\citet{Epitropakis16} indicated that too few Fourier frequencies within a bin
can bias the result, and there are six Fourier frequencies in this bin.
We also examine the long 2013-14 \emph{XMM-Newton} data. Nonetheless, the
longest light curve in that observation is $\sim$ 138 ks and there is not enough
signal-to-noise ratio to probe the frequency band where the maximum negative lag
occurs. We only display results using the \emph{Suzaku} data.
We then take a closer look at the result by generating a lag-energy spectrum to
examine how the lag evolves with energy. The lag is calculated between the light
curve in each energy bin and the light curve of the reference band. The reference
band is the whole energy range ($0.3-10$ keV) excluding the energy bin of interest.
For instance, the reference light curve for the $0.3-0.4$ keV band is
produced by subtracting the $0.3-0.4$ keV light curve from the $0.3-10$ keV light
curve. The reason to exclude the current band is to ensure the noise remains
uncorrelated between the bands (see \citealt{Zoghbi11}). Although the choice of reference band would cause the reference
light curve to be slightly different for each energy bin, Monte Carlo simulations
showed there are only very mild differences ($\sim$ 1 per cent of systematic error).
Figures \ref{model} and \ref{lag_energy} show the lag-energy spectrum in the $[7.2-12.7]\times10^{-6}$
Hz frequency band. The lag begins to have a general negative trend starting at
around 3 keV in rest frame except the $7.1-8.3$ keV energy bin. The shape of the
lag-energy spectrum is similar to that of IRAS~13224-3809 at high-flux states
\citep{Kara13IRAS}. From looking at the lag-energy spectra from individual observations,
we find that the $7.1-8.3$ keV bin is dominated by the 2013a observation (see Fig. \ref{lag_energy}).
This energy of this bin lines up with the peak
of the Fe K line (see Fig.~\ref{lag_energy}), and may therefore be associated with an Fe K lag, though it cannot be confirmed
based on current data. We note also that the lowest energy bin at $\sim0.5$ keV in the 2013a observation
shows a significant difference compared to the average lag-energy spectrum. Changes in the shape of the reflection spectrum (such as the ionization parameter), can cause changes in the lag-energy spectrum \citep{chainakun15,chainakun16}. Moreover, this observation also requires the strongest warm absorber which could affect the lag there (see Discussion).
To investigate the variable spectrum further, we create the covariance spectrum, which
is a measure of the absolute amplitude of correlated variations in count rate as a function of
energy \citep{Wilkinson09,Uttley14}, following the method described in \citep{Uttley14}.
The covariance spectrum only picks out the correlated variations and allows direct
comparison with the time-averaged spectrum. Fig. \ref{cov} shows the covariance spectrum of the
$[7.2-12.7]\times10^{-6}$ Hz frequency band, where the soft lag is detected. Notably,
the covariance spectrum of the 2013a observation shows an enhancement at the same
energy as the tentative Fe K lag. Overall, the covariance spectrum roughly follows a
power-law with $\Gamma = 1.73$, though there are clear deviations away from it.
\begin{figure}
\begin{center}
\leavevmode \epsfxsize=8.5cm \epsfbox{lag_energy_all-2.ps}
\end{center}
\caption{The lag-energy spectrum of \emph{Suzaku} data.
The x-axis shows energy in rest frame. The spectrum shows the energy dependence
of the lags in the frequency range $[7.2-12.7]\times10^{-6}$ Hz, where the
maximum negative soft lag occurs based on the lag-frequency spectrum
(Fig. \ref{lag_spec}). The black points represent results obtained using all data; the red
points were calculated using 2013a observation only; the blue points show results obtained
using all data except the 2013a observation.
}
\label{lag_energy}
\end{figure}
\begin{figure}
\begin{center}
\leavevmode \epsfxsize=8.5cm \epsfbox{covariance_2.ps}
\end{center}
\caption{The covariance spectrum for the frequency band $[7.2-12.7]\times10^{-6}$ Hz.
Black points show the overall covariance spectrum, and
red ones the covariance spectrum of the 2013a (ObsID: 707035010) observation.
}
\label{cov}
\end{figure}
\section{Discussion} \label{discussion}
The spectrum of PDS~456 can be simply explained by relativistic reflection
with two full-covering absorbers, where one is associated with the ultrafast
outflow, and the other related to another warm absorber with a lower
ionization state and a lower outflow speed which cannot be tested
by the \emph{Suzaku} data. We do not find any partial-covering absorbers that are
required to interpret the data. The change in the continuum,
the column density and ionization state of the warm absorbers are sufficient
to explain the spectral variability between observations.
We find a soft lag of $10000\pm3400$ s at $9.58\times10^{-6}$ Hz via the
maximum likelihood method, which is the largest soft lag and the lowest
frequency to date in X-ray reverberation studies. Previous works do not probe
Fourier frequencies below $10^{-5}$ Hz because there are no continuous
light curves that are long enough to reach the low frequency domain. We
will compare the results with previous literature in the following.
\subsection{Mass-scaling Relation}
We first compare our result with those obtained in \citet{DeMarco13}. In
Fig. \ref{scaling_relation} we present the result of PDS~456 using a red
triangle data point. It can be seen that PDS~456 lies close to the extrapolated
relation obtained by \citet{DeMarco13}. The data point of PDS~456 is
slightly above the mass-scaling relation, which could be explained by the tentative correlation
between lag and Eddington ratio \citep{Kara16}. PDS~456 is claimed to be accreting at
Eddington or above and it is not surprising that it is slightly away from the
relation.
In fact, the light travel distance converting from the
amplitude of the soft lag is within 3 $R_{\rm g}$ (see the red dash line in Fig. \ref{scaling_relation}),
which is as small as values
found in other AGN ($1-10$ $R_{\rm g}$). Previous work of \citet{DeMarco13} only probe Fourier
frequency in the range of $10^{-5}$ to $10^{-3}$ Hz and AGN with lower masses
due to the use of continuous \emph{XMM-Newton} observations. The data point of
PDS~456 may deviate from the extrapolated best-fitting relation because of potential biases in
the \citet{DeMarco13} sample. The maximum length of the \emph{XMM-Newton} light curves
biases against finding large soft lags in massive black holes, potentially flattening the
relation.
Our result shows that the massive quasar PDS~456 follows the mass-scaling relation as well.
Lags of sources in the De Marco sample and PDS~456 all imply that these signals
are from areas very close to central black holes.
Although it is not known if all massive AGN follow the trend, the result of
PDS~456 further supports the hypothesis that soft lags originate from the innermost
areas of AGN.
\begin{figure}
\begin{center}
\leavevmode \epsfxsize=8.5cm \epsfbox{scaling_relation.ps}
\end{center}
\caption{Lag vs. black hole mass scaling relation from \citealt{DeMarco13} (black).
The y-axis shows the most significant soft lags detected in AGN.
The red triangle on top represents the data point of PDS~456 measured in this work.
The blue dot dash line is drawn using the best-fitting relation of data from
\citet{DeMarco13}. The red dash line presents the the light-crossing time at
3 $R_{\rm g}$ as a function of mass.
}
\label{scaling_relation}
\end{figure}
\subsection{Modelling the Fe K$\alpha$ lag}
To better understand the expected Fe K$\alpha$ reverberation lag in the frequency range $[7.2 - 12.7]\times10^{-6}$~Hz,
we use the models described in \citet{Cackett14} to calculate the transfer function for a point source at some height, $h$,
above the black hole. The best-fitting spectral parameters indicate an inclination of $65^\circ$ and a near maximally
spinning black hole. We therefore calculate the transfer function assuming $i = 65^\circ$ and $a = 0.998$. We also
assume a $h = 10$~GM/c$^2$, close to what was found for NGC 4151 in \citet{Cackett14}. The spectral fits also indicate
a reflection fraction (reflection flux / power flux) of approximately 0.6 at the peak of the iron line in the 2013 observations
(it is lower in the other observations). Furthermore, we assume $\log M = 8.91$ \citep{Zhou05}.
The aim here is not to fit the data, but just to explore what might be expected to be observed. With the parameters given
above we produce the lag-energy spectrum at the frequency of the soft lag in Figure~\ref{fig:fekmodel}. The lag-frequency
spectrum for the energy-averaged iron line is also shown. From the lag-frequency spectrum we see that for such a massive
black hole the frequency-range $[7.2 - 12.7]\times10^{-6}$~Hz is right where phase-wrapping is occurring. This means that
we expect to see only reverberation from the regions of the disc with the shortest path length differences. With a high
inclination angle of $65^\circ$ this will be close to the black hole on the near-side of the accretion disc. We therefore expect
to see lags from only the most red - and blue-shifted parts of the line (see Cackett et al. 2014 for detailed descriptions).
What is striking about the resultant lag-energy spectrum in Figure~\ref{fig:fekmodel} is the narrow peak in the lags in the
7$-$8 keV range, coming from the most blue-shifted emission. The general shape of the lag-energy spectrum is very similar
to what is observed in PDS~456 -- a peak in only the $\sim$7$-$8 keV range and much smaller lags, with little energy-dependence, elsewhere.
We have not calculated a grid of models covering a range of parameters in order to fit the data properly. We do note,
however, that given how the frequency range is right where the phase-wrapping crosses zero, small changes in the
assumed mass, height, and inclination can have a big effect on the lag-energy spectrum. If the height of the corona
changes with flux (as suggested by spectral fitting, e.g., \citealt{Fabian12}) then this could explain why only the 2013a
observation shows a significant lag in the 7.1$-$8.3 keV range.
\begin{figure*}
\includegraphics[width=0.9\textwidth]{tf_lagspec_pds456.ps}
\caption{{\it Left:} The lag-frequency spectrum for the energy-averaged iron line, assuming $i = 65^\circ$, $h = 10$~$GM/c^2$, $a=0.998$ and $\log M = 8.91$. The grey shaded region shows the frequency range where the soft lag is detected in PDS 456, and over which the lag-energy spectrum (right-panel) is calculated. {\it Right:} The lag-energy spectrum over the frequency-range $[7.2 - 12.7]\times10^{-6}$~Hz, assuming a reflection fraction of 0.6, and parameters listed previously.}
\label{fig:fekmodel}
\end{figure*}
\subsection{Influence of the Outflow}
\citet{Emm11} carried out timing analysis on MCG-6-30-15 using the $0.5-1.5$ keV
and $2-4$ keV light curves. The $0.5-1.5$ keV energy band of MCG-6-30-15 is
seriously modified by warm absorbers \citep{Chiang11} and the soft lag obtained
from the data might be caused by scattering/absorbing material. Nevertheless,
\citet{Emm11} found that the soft lag of MCG-6-30-15 is most likely caused by
reflection by testing various models. Reverberation lags can still be detected even
when the timing analysis involves energy bands which are strongly affected by
warm absorbers, but strong absorption does inhibit the measurement of time lags.
\citet{Silva16} investigated data of NGC~4051 and found that warm absorbers do
produce time lags on the order of $\sim$ 50 s or more at lower frequencies, which
is of a longer timescale than the reverberation lags in NGC~4051 and hence does not pollute
the measurement. The time delay is related to the response of the gas to changes
in the ionizing continuum and is dependent on the density of the warm absorber.
However, only warm absorbers within a certain distance ($10^{15}$ cm to
$10^{16.5}$ cm in the case of NGC~4051, $\sim$3900-123000 $R_{\rm g}$)
matter. Those close to the ionizing source
respond immediately, and gas further out is unable to respond to create a visible time
delay. In NGC~4051 the component with a mild ionization parameter of log $\xi\sim$
2.9 and $N_{\rm H}=(1.2\pm0.2)\times 10^{21}$ cm$^{-2}$ contributed the majority
of the soft lag, while the highly-ionized (log $\xi\sim$ 3.7) and the least ionized
(log $\xi\sim$ 0.4) components cause little differences. The warm absorbers
found in PDS~456 are the same type which contributes least in NGC~4051, but
column densities are higher than those found in NGC~4051. Assuming that
the position of warm absorbers scale with black hole masses, those in PDS~456
should not produce negative lags which coincide with reverberation lags. Nonetheless,
detailed analysis is still required to completely rule out the possibility that the
time lag detected in PDS~456 is partly produced by variable warm absorbers, as
these outflows are complex and can be case dependent.
\citet{Kara16} examined a sample of Seyfert galaxies, including bare and
partially obscured sources, and found $\sim$ 50 per cent of them exhibit iron K reverberation.
An iron K lag was not detected in some sources with complex absorption such as
MCG-6-30-15, but seen in several sources with warm absorbers. IRAS13224-3809,
which is known to have ultrafast outflow appearing at the iron K band \citep{Parker17}, does reveal
an iron K lag. The presence of warm absorbers does not seem to destroy iron K lags.
Warm absorbers do not seem to leave signatures in the lag-energy spectrum
(Fig. \ref{lag_energy}) of PDS~456. The negative trend is disrupted by the $7.1-8.3$
keV energy bin in rest frame, where neither of the warm absorbers would affect
significantly (the ultrafast outflow locates at $8-9$ keV rest frame, and the second
warm absorber casts effects on low-energy band as seen in Fig. \ref{model}).
Nonetheless, the $7.1-8.3$ keV positive lag cannot be confirmed as an iron K lag
either. Without enough evidence, the X-ray lag shown in PDS~456 cannot be
concluded to be reverberation lag.
\citet{Kara15} showed that the iron K reverberation lag cannot be
found in NGC~1365 in orbits that are highly obscured. They also found a low-frequency
soft lag (instead of hard lag as found in other AGN) in a less-absorbed orbit, which can
be explained as a transient phenomenon. There is evidence of a neutral eclipsing
cloud in NGC~1365, and the low-frequency soft lag is caused by a change in
column density, which can be interpreted as the eclipsing cloud moving out the line
of sight. In the case of PDS~456, \citet{Matzeu16} reported significant short-term
X-ray spectral variability on timescales of $\sim$100 ks in the 2013 observation,
and suggested that the spectral variability can be accounted for by variable covering
of the clumpy absorbing gas along the line of sight. If this is true, we should be
able to see signatures in the lag-frequency spectrum of the 2013 observation.
The lag-frequency spectrum from the combined 2013 observation shows no
significant difference from Fig. \ref{lag_spec}. However, we do see that when looking at only
the 2013a lag-energy spectrum in the $[7.2-12.7]\times10^{-6}$ Hz frequency range, that there
is a change in the lag in the lowest energy bin (which would be effected most by neutral absorption).
But, the lag changes in the opposite sense from what would be expected by the presence
of an eclipsing cloud with high column density in PDS~456 during that observation only.
This does not
reject the possibility of partial-covering Compton-thick absorbers, which could be
located further from the centre and do not cast effects on lag measurement.
However, recent X-ray micro-lensing studies have shown that the X-ray emitting region of
quasars is compact (less than $\sim$10 gravitational radii $R_{\rm g}$,
\citealt{Dai10,Chartas12}). If clumpy partial-covering absorbers are not in the
innermost regions of AGN, they are more likely to fully cover the source.
We already show in Section \ref{subsec_spectral} that a simple
relativistic reflection model with Compton-thin, fully-covered warm absorbers
can interpret the spectrum of PDS~456 without difficulty. In the timing analysis
we find no evidence to contradict the model.
Based on present work, we did not find evidence for a high-column eclipsing cloud such as
that found in NGC~1365 to be present in PDS~456. Although warm absorbers can affect the
lag measurement, those with low column density do not seem to contribute
significantly to timing analysis. Combining both spectral and timing analyses, the
time lag in PDS~456 is more likely to be an X-ray reverberation lag. Nevertheless, given
that there is only tenative
evidence of an iron K lag, the possibility that the time
lag was generated by warm absorbers cannot be completely ruled out.
\section{Conclusions}
We present both the spectral and timing analysis of PDS~456 using $\sim$
770 ks of archival \emph{Suzaku} data. We find that the spectrum can be
simply explained by relativistic reflection with two full-covering warm
absorbers. One of them is the persistent, highly-ionized ultrafast outflow;
the other is less ionized and started to appear in the 2011 observation.
No partial-covering absorbers were required to interpret spectral variability,
and we did not find effects caused by these absorbers in the lag-frequency
spectrum either. Based on the spectral analysis, PDS~456 harbours
a rapidly-spinning black hole of $a^{*}>0.99$. Furthermore, we find a soft
lag of $10000\pm3400$ s at $9.58\times10^{-6}$ Hz, which matches the
scaling relation and indicates an origin within 3 $R_{\rm g}$. More data
are required to confirm the relation for massive AGN with $M>10^{8} M_{\odot}$,
but our result further supports the scenario that soft lags originate from
the innermost regions of AGN. The results show that the relativistic
reflection model is self-consistent and reveal the robustness of probing
areas near the central engine using this model.
\section*{Acknowledgements}
This work was greatly expedited thanks to the help of Jeremy Sanders in
optimizing the various convolution models. We thank Phil Uttley for useful
discussions. EMC gratefully acknowledges support from the NSF through
CAREER award number AST-1351222. CSR thanks NASA for support
under grant NNX15AU54G. ACF acknowledges ERC Advanced Grant 340442.
\bibliographystyle{mn2e_uw}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,611 |
from unittest import TestCase
import numpy as np
from diffprivlib.tools.utils import std
from diffprivlib.utils import PrivacyLeakWarning, BudgetError, check_random_state
class TestStd(TestCase):
def test_not_none(self):
mech = std
self.assertIsNotNone(mech)
def test_no_params(self):
a = np.array([1, 2, 3])
with self.assertWarns(PrivacyLeakWarning):
res = std(a)
self.assertIsNotNone(res)
def test_no_epsilon(self):
a = np.array([1, 2, 3])
self.assertIsNotNone(std(a, bounds=(0, 1)))
def test_no_bounds(self):
a = np.array([1, 2, 3])
with self.assertWarns(PrivacyLeakWarning):
std(a, epsilon=1)
def test_bad_bounds(self):
a = np.array([1, 2, 3])
with self.assertRaises(ValueError):
std(a, epsilon=1, bounds=(0, -1))
def test_missing_bounds(self):
a = np.array([1, 2, 3])
with self.assertWarns(PrivacyLeakWarning):
res = std(a, 1, None)
self.assertIsNotNone(res)
def test_large_epsilon(self):
a = np.random.random(1000)
res = float(np.std(a))
res_dp = std(a, epsilon=5, bounds=(0, 1))
self.assertAlmostEqual(res, res_dp, delta=0.01)
def test_large_epsilon_axis(self):
a = np.random.random((1000, 5))
res = np.std(a, axis=0)
res_dp = std(a, epsilon=15, bounds=(0, 1), axis=0)
for i in range(res.shape[0]):
self.assertAlmostEqual(res[i], res_dp[i], delta=0.01)
def test_array_like(self):
self.assertIsNotNone(std([1, 2, 3], bounds=(1, 3)))
self.assertIsNotNone(std((1, 2, 3), bounds=(1, 3)))
def test_clipped_output(self):
a = np.random.random((10,))
rng = check_random_state(0)
for i in range(100):
self.assertTrue(0 <= std(a, epsilon=1e-3, bounds=(0, 1), random_state=rng) <= 1)
def test_nan(self):
a = np.random.random((5, 5))
a[2, 2] = np.nan
res = std(a, bounds=(0, 1))
self.assertTrue(np.isnan(res))
def test_accountant(self):
from diffprivlib.accountant import BudgetAccountant
acc = BudgetAccountant(1.5, 0)
a = np.random.random((1000, 5))
std(a, epsilon=1, bounds=(0, 1), accountant=acc)
self.assertEqual((1.0, 0), acc.total())
with acc:
with self.assertRaises(BudgetError):
std(a, epsilon=1, bounds=(0, 1))
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,407 |
@protocol TCDropFileZoneDelegate <NSObject>
- (void)dropZoneGetFiles:(NSArray*)filePathArray;
@end
@interface TCDropFileZoneView : NSView
{
BOOL fileisEntered;
}
@property (strong) id <TCDropFileZoneDelegate> delegate;
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,857 |
The AF series replaceable bladder accumulators come in models with a 50 to 500 litres capacity, and are best suited for residential and industrial systems requiring substantial delivery capacities.
Replaceable EPDM membrane suitable for potable water. Maximum pressure 10 bar.
Supplied at 10 bars export version, in the WRc and CE certified version, the AF series accumulators are also available in customised versions compliant with major international standards. The horizontal version is provided with universal motor-support brackets to allow the pump to be directly fastened above the vessel. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,837 |
{"url":"https:\/\/pixel-druid.com\/discriminant-and-resultant.html","text":"## \u00a7 Discriminant and Resultant\n\nI had always seen the definition of a discriminant of a polynomial $p(x)$ as:\n$Disc(p(x)) \\equiv a_n^{(2n - n)} \\prod_{i< j} (r_i - r_j)^2$\nWhile it is clear why this tracks if a polynomial has repeated roots or not, I could never motivate to myself or remember this definition. I learnt that in fact, this comes from a more general object, the resultant of two polynomials $P(x), Q(x)$, which provides a new polynomial $Res(P(x), Q(x)$ which is zero iff $P, Q$ share a common root. Then, the discriminant is defined as the resultant of a polynomial and its derivative. This makes far more sense:\n\u2022 If a polynomial has a repeated root $r$, then its factorization will be of the form $p(x) = (x - r)^2 q(x)$. The derivative of the polynomial will have an $(x-r)$ term that can be factored out.\n\u2022 On the contrary, if a polynomial only has a root of degree 1, then the factorization will be $p(x) = (x - r) q(x)$, where $q(x)$ is not divisible by $(x-r)$. Then, the derivative will be $p'(x) = 1 \\cdot q(x) + (x - r) q'(x)$. We cannot take $(x - r)$ common from this, since $q(x)$ is not divisible by $(x-r)$.\nThis cleared up a lot of the mystery for me.\n\n#### \u00a7 How did I run into this? Elimination theory.\n\nI was trying to learn how elimination theory works: Given a variety $V = \\{ (x, y) : Z(x, y) = 0 \\}$, how does one find a rational parametrization $(p(t), q(t))$ such that $Z(p(t), q(t)) = 0$, and $p(t), q(t)$ are rational functions? That is, how do we find a rational parametrization of the locus of a polynomial $Z(x, y)$? The answer is: use resultants!\n\u2022 We have two univariate polynomials $p(a; x), p(b; x)$, where the notation $p(a; x)$ means that we have a polynomial $p(a; x) \\equiv \\sum_i a[i] x^i$. The resultant isa polynomial $Res(a; b)$ which is equal to $0$ when $p(a; x)$ and $p(b; x)$ share a common root.\n\u2022 We can use this to eliminate variables. We can treat a bivariate polynomial $p(x, y)$as a univariate polynomial $p'(y)$ over the ring $R[X]$. This way, given two bivariate polynomial $p(a; x, y)$, $q(b; x, y)$, we can compute their resultant, giving us conditions to detect for which values of $a, b, x$, there exists a common $y$ such that $p(a; x, y)$ and $(q, x, y)$ share a root. If $(a, b)$are constants, then we get a polynomial $Res(x)$ that tracks whether $p(a; x, y)$and $q(a; x, y)$ share a root.\n\u2022 We can treat the implicit equation above as two equations, $x - p(t) = 0$, $y - q(t) = 0$. We can apply the method of resultants to project out $t$from the equations.\n\n#### \u00a7 5 minute intro to elimination theory.\n\nRecall that when we have a linear system $Ax = 0$, the system has a non-trivial solution iff $|A| = 0$. Formally: $x \\neq 0 \\iff |A| = 0$. Also, the ratio of solutions is given by:\n$x_i \/ x_j = (-1)^{i+j} |A_i|\/|A_j|$\nIf we have two polynomials $p(a; x) = a_0 + a_1 x + a_2 x^2$, and $q(b; x) = b_0 + b_1x + b_2 x^2$, then the system $p(a; x)$, $q(b; x)$ has a simeltaneous zero iff:\n\\begin{aligned} &\\begin{bmatrix} a_2 & a_1 & a_0 & 0 \\\\ 0 & a_2 & a_1 & a_0 \\\\ b_2 & b_1 & b_0 & 0\\\\ 0 & b_2 & b_1 & b_0\\\\ \\end{bmatrix} \\begin{bmatrix} 1 \\\\ x \\\\ x^2 \\\\ x^3 \\end{bmatrix} = 0 \\\\ &A x = 0 \\end{aligned}\n\n#### \u00a7 Big idea\n\nThe matrix is setup in such a way that any solution vector $v$ such that $Qv = 0$ will be of the form $v = (\\alpha^3, \\alpha^2, \\alpha, 1)$. That is, the solution vector is a polynomial , such that $Qv = 0$. Since $Qv = 0$, we have that $a_2 \\alpha^2 + a_1 \\alpha + a_0 = 0$, and $b_2 \\alpha^2 + b_1 \\alpha + b_0 = 0$.\n\n#### \u00a7 Proof\n\n\u2022 Necessity is clear. If we have some non trivial vector $v \\neq 0$ such that $Qv = 0$, then we need $|Q| = 0$.\n\u2022 Sufficiency : Since $|Q| = 0$, there is some vector $v = (w, x, y, z)$such that $Qv = 0$. We need to show that this $v$ is non-trivial. If the polynomials $p(a;x)$, $q(b;x)$ are not equal, then we have that the rows which have coefficients from $p$ and $q$ are linearly independent. So, the pair of rows $(1, 3)$, and the pair $(2, 4)$ are linearly independent. This means that the linear system:\n$a_2 w + a_1 x + a_0 y = 0 \\\\ b_2 w + a_1 x + a_0 y = 0 \\\\$\nSimilarly:\n$a_2 x + a_1 y + a_0 z = 0 \\\\ b_2 x + a_1 y + a_0 z = 0 \\\\$\nSince the coefficients of the two systems are the same, we must have that $(w, x, y)$ and $(x, y, z)$ are linearly dependent. That is:\n$(w, x, y) = \\alpha (x, y, z) \\\\ w = \\alpha x = \\alpha^2 y = \\alpha^3 z \\\\$\nWe can take $z = 1$ arbitrarily, giving us a vector of the form $(w, x, y, z) = (\\alpha^3, \\alpha^2, \\alpha, 1)$, which is the structure of the solution we are looking for!","date":"2022-12-02 22:22:37","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 79, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9901934266090393, \"perplexity\": 99.51857384545565}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446710916.70\/warc\/CC-MAIN-20221202215443-20221203005443-00752.warc.gz\"}"} | null | null |
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or not equal to, then we use the two-tailed test. For the keen reader, in the book by Grinold Kahn a formula linking Information Ratio (IR) and IC is given: with breadth being the number of independent bets (trades). 1 Factor Analysis, here were trying to rank as accurately as possible the stocks in the investment universe on a forward return basis. Transfer coefficient : This is an extension of the fundamental law of active management and it relaxes the assumption of Grinolds model that managers face no constraints which preclude them from translating their investments insights directly into portfolio bets. This is defined as the standard deviation of the portfolio return minus the benchmark return. This means all stocks falling in Q1 are underweight and all stocks falling in Q5 are overweight relative to the benchmark. If the test statistic is greater than the critical value, then we fail to reject the null hypothesis and we state that the result is not statistically significant. The big issue that impresses the market watchers and financial types is the ability to consistently make above-market returns. Cross Asset Arbitrage, this model bets on the price discrepancy between a financial asset and its underlying. Automating the strategy Linking to brokerage.
Trends uncovered are based on the volume, frequency and the price of a security at which it is traded. As a rule, the significance level is specified prior to calculating the test statistic, as a result of the test statistic may impact the choice of significance level. Suggested read: All You Need To Know About Algorithmic Trading Disclaimer: All investments and trading in the stock market involve risk. This is not good as a portfolio manager will have to pick stocks within the entire universe in order to meet its tracking error constraint.
1.2 Quantiles Return. Any quantitative trading system consists of four major components: Hypothesis formation Finding a strategy, backtesting and optimizing the hypothesis Obtaining data, coding the strategy, analyzing the performance. If the value of the test statistic is less than or equal to the critical value, we reject the null hypothesis, and state that the result is statistically significant. Formal significance forex compound calculator tests can be evaluated but this is beyond the scope of this article. 95 of the values fall within two standard deviations of the mean and.7 of the values lie within three standard deviations of the mean. Disclaimer: All investments and trading in the stock market involve risk. | {
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\section{Introduction}
With advances in neural recording technologies, experimentalists can now record simultaneous activity across multiple brain regions at single cell resolution~\cite{ahrens13,prevedel14,brainInit_13,lemon15}.
However, it is still a technical challenge to measure the interactions within and across brain regions that govern this multi-region activity.
This challenge is heightened by the fact that cortical neurons are heterogeneous and show substantial trial-to-trial variability~\cite{cohen11}.
Numerous theoretical studies have examined how neural networks can lead to cortex-like dynamics~\cite{abbottvv,van94,van96,brunel,brunelhakim,buice2007cfa,bressloff09,renart10,touboul11};
however, most have been limited to a single region, leaving open the question of how inter-region connection strengths contribute to network processing.
One challenge presented by analyzing multi-region neural networks, is the increased number of parameters which must be specified. To survey a high-dimensional parameter space, one must have a way to efficiently simulate (as in \cite{stringer16}) or approximate network statistics (as in \cite{gerstnerBook}).
Here we present a novel approximation method for calculating the statistics of a general coupled firing rate model (based on~\cite{wilsoncowan1}) of neural networks where we:
i) assume the \textit{activity} (not the firing statistics) are pairwise normally distributed,
ii) take the entire probability distribution of the presynaptic neurons/populations (providing input) into account. Our method is fast because it requires
solving nonlinear equations self-consistently rather than
simulating
stochastic differential equations. Several example neural networks are considered and compared with Monte Carlo simulations.
A specific version of this method was presented in~\cite{bgsl_17} to model the olfactory sensory pathway; here, we derive formulas in a general way which is easy to
evaluate and can accommodate heterogeneous networks.
We also demonstrate the method's efficacy on several example networks with much larger dimension than the specific networks examined in our previous work.
\section{Neural Network Model}
Each cell (or homogeneous population) has a prescribed activity $x_j$ that is modeled by the following equation~\cite{wilsoncowan1} for $j=1,2,\dots,N_c$:
\begin{eqnarray}\label{xj_eqn}
\tau_j \frac{dx_j}{dt} = -x_j + \mu_j +\sigma_j \eta_j(t) + \sum_{k=1}^{N_c} g_{jk} F_k(x_k(t))
\end{eqnarray}
where $F_k(\cdot)$ is a transfer function mapping activity to firing rate (in some units), related to the so-called F-I curve, for the $k^{th}$ cell/population. Thus, the instantaneous firing rate of the $j^{th}$ neuron is:
\begin{eqnarray}\label{frt_eqn}
F_j(x_j(t)).
\end{eqnarray}
Depending on the context, the activity variable $x_j$ may represent membrane voltage, calcium concentration, or some other quantity associated with a neuron's internal state \cite{dayan2001theoretical}.
This type of equation has historically been used to capture the average activity of a population of neurons but from here on out we will use the term ``cell" for exposition purposes.
All cells receive background noise $\eta_j$, the increment of a Weiner process, uncorrelated in time but potentially correlated at each instant: $\langle \eta_j(t) \rangle = 0$, $\langle \eta_j(t) \eta_j(t') \rangle = \delta(t-t')$,
and $\langle \eta_j(t) \eta_k(t') \rangle = c_{jk} \delta(t-t')$ for $j\neq k$ with $c_{jk}\in(-1,1)$. The parameters $\mu_j$ and $\sigma_j$ are constants that give the background input mean and input standard deviation, respectively.
The parameter $g_{jk}$ represents coupling strength from the presynaptic $k^{th}$ cell and is a signed quantity; $g_{jk}<0$ represents inhibitory coupling.
We would like to compute the following statistics:
\begin{eqnarray}
\mu(j) &:=& \langle x_j \rangle, \hbox{mean activity} \label{stats_defn1} \\
\sigma^2(j) &:=& \langle x^2_j \rangle - \mu^2(j), \hbox{variance of activity} \label{stats_defn2} \\
Cov(j,k) &:=& \langle x_j x_k \rangle - \mu(j)\mu(k), \nonumber \\
& & \hbox{covariance of activity} \label{stats_defn3} \\
\nu_j &:=& \langle F_j(x_j) \rangle, \hbox{firing rate} \\
Var(\nu_j) &:=& \langle F^2_j(x_j) -\nu^2_j \rangle, \hbox{variance of spiking} \\
Cov(\nu_j,\nu_k) &:=& \langle F_j(x_j)F_k(x_k) \rangle - \nu_j\nu_k, \nonumber \\
& & \hbox{covariance of spiking} \\
\rho(\nu_j,\nu_k) &:=& \frac{Cov(\nu_j,\nu_k)}{\sqrt{Var(\nu_j)Var(\nu_k)}}, \nonumber \\
& &\hbox{correlation of spiking}
\end{eqnarray}
where the angular brackets $\langle \cdot \rangle$ denote averaging over time and realizations \footnote{We assume the networks of interest satisfy the conditions of the Ergodic Theorems so that averaging over time and state are the same}.
We will use the following definitions for the following Normal/Gaussian probability density functions (PDF):
\begin{eqnarray}
\varrho_1(y) := \frac{1}{\sqrt{2\pi}} e^{-y^2/2},
\end{eqnarray}
the standard normal PDF, and
\begin{eqnarray}
\varrho_{j,k}(y_1,y_2) := \frac{1}{2\pi\sqrt{1-c_{jk}^2}} \exp\Big( -\frac{1}{2}\vec{y}^T \left(\begin{smallmatrix} 1 & c_{jk} \\ c_{jk} & 1 \end{smallmatrix}\right)^{-1} \vec{y} \Big), \nonumber \\
\end{eqnarray}
a bivariate normal distribution with $\vec{0}$ mean, unit variance, and covariance $c_{jk}$.
\textit{In the absence of coupling}, i.e. $g_{jk} = 0$, Eq.~\eqref{xj_eqn} would describe a multi-dimensional Ornstein-Uhlenbeck process. Such a process is well-understood: any pair of activity variables, $(x_j,x_k)$, are bivariate normal random variables~\cite{gardiner}.
To see this, consider the following two equations without synaptic coupling:
\begin{eqnarray}
\tau_j \frac{d x_j}{dt} & = & -x_j + \mu_j + \sigma_j \left( \sqrt{1-c_{jk}}\xi_j(t) + \sqrt{c_{jk}} \xi_c(t) \right) \nonumber \\
& & \\
\tau_k \frac{d x_k}{dt} & = & -x_k + \mu_k + \sigma_k \left( \sqrt{1-c_{jk}}\xi_k(t) + \sqrt{c_{jk}} \xi_c(t) \right). \nonumber \\
& &
\end{eqnarray}
Note that we have re-written $\eta_{j/k}(t)$ as sums of independent white noise processes $\xi(t)$.
Since $x_j(t) = \frac{1}{\tau_j}\int_{-\infty}^t e^{-(t-u)/\tau_j} \Big[ \mu_j + \sigma_j\eta_j(u) \Big]\,du$ (where we have taken the initial time to be in the far past to eliminate any impact from the initial conditions),
we calculate marginal statistics using It\^{o} isometries:
\begin{equation}
\mu(j) \equiv \langle x_j \rangle = \mu_j
\end{equation}
\begin{eqnarray*}
\sigma^2(j) & \equiv & \langle (x_j - \mu(j) )^2 \rangle \\
& = & \left\langle \frac{\sigma^2_j}{\tau_j^2} \int_{-\infty}^t \int_{-\infty}^t e^{-(t-u)/\tau_j} \eta_j(u) e^{-(t-v)/\tau_j} \eta_j(v) \,du\,dv \right\rangle \\
& = &\frac{\sigma^2_j}{\tau_j^2} \int^t_{-\infty} e^{-2(t-u)/\tau_j} \,du=\frac{\sigma^2_j}{2\tau_j}
\end{eqnarray*}
A similar calculation shows that in general we have:
\begin{equation}
Cov(j,k) = \frac{c_{jk}}{\tau_j+\tau_k} \sigma_j \sigma_k
\end{equation}
Thus, $(x_j,x_k)\sim \mathbb{N}\left( \left(\begin{smallmatrix}\mu_j \\ \mu_k \end{smallmatrix}\right) , \left(\begin{smallmatrix} \frac{\sigma^2_j}{2 \tau_j} & \sigma_j\sigma_k \frac{c_{jk}}{\tau_j+\tau_k} \\ \sigma_j\sigma_k \frac{c_{jk}}{\tau_j+\tau_k} & \frac{\sigma^2_k}{2 \tau_k} \end{smallmatrix}\right) \right)$.
Statistics for the firing rates, $F(x_j)$, are inherited from this normal distribution, since the firing rate $F(x_j)$ is simply a nonlinear function of the activity $x_j$.
\textit{When coupling is included}, i.e. $g_{jk} \not= 0$ for some indices $j$ and $k$, it may no longer be true that the activity variables $x_j$ remain normally distributed. However, it is reasonable to suppose that, for sufficiently weak coupling, the deviations from a normal distribution will be small. Furthermore, if the firing rate function $F$ has thresholding and saturating behavior (as does a sigmoidal function), then higher moments of $x_j$ have limited impact on statistics of $F(x_j)$.
Thus, our first assumption will be that each pair of activity variables $(x_j,x_k)$, can be approximated by a bivariate normal, even when coupling is present. We can think of this as a weak coupling assumption, as it holds exactly only with \textit{no} coupling.
\section{Reduction Method}
\begin{table*}
\caption{\label{table:defnsInt} For readability, we define the following quantities. Whenever $j=k$ in the double integrals (e.g., in $\mathcal{N_F},\mathcal{S}$), the bivariate normal distribution $\varrho_{j,k}$
is replaced with the standard normal distribution $\varrho_1$. Note that order of the arguments matters in $\mathcal{N_F}$: $\mathcal{N_F}(j,k)\neq\mathcal{N_F}(k,j)$ in general;
all of these quantities depend on the statistics of the activity $\mu(\cdot)$, $\sigma(\cdot)$.}
\begin{ruledtabular}
\begin{tabular}{ll}
Abbreviation & Definition \\ \hline
$\mathcal{E}_1(k)$ & $\displaystyle \int F_k(\sigma(k)y+\mu(k))\varrho_1(y)\,dy$ \\
$\mathcal{E}_2(k)$ & $\displaystyle \int F_k^2(\sigma(k)y+\mu(k))\varrho_1(y)\,dy$ \\
$\mathcal{V}(k)$ & $\displaystyle \int F_k^2(\sigma(k)y+\mu(k))\,\varrho_1(y)\,dy - \left(\displaystyle \int F_k(\sigma(k)y+\mu(k))\varrho_1(y)\,dy\right)^2 = \mathcal{E}_2(k)-\left[\mathcal{E}_1(k)\right]^2$ \\
\multirow{ 2}{*}{$\mathcal{N_F}(j,k)$} & $\displaystyle\iint F_k(\sigma(k)y_1+\mu(k))\frac{y_2}{\sqrt{2}}\varrho_{j,k}(y_1,y_2)\,dy_1dy_2, \hbox{if }j\neq k$ \\
& $\displaystyle\int F_j(\sigma(j)y+\mu(j))\frac{y}{\sqrt{2}}\varrho_1(y)\,dy, \hbox{if }j= k$ \\
$\mathcal{S}(j,k)$ & $\displaystyle \iint F_j(\sigma(j)y_1+\mu(j)) F_k(\sigma(k)y_2+\mu(k)) \varrho_{j,k}(y_1,y_2)\,dy_1dy_2$ \\
$\mathcal{C_V}(j,k)$ & $\mathcal{S}(j,k)-\mathcal{E}_1(j)\mathcal{E}_1(k)$
\end{tabular}
\end{ruledtabular}
\end{table*}
In our method, we assume that time is dimensionless so that the subsequent assumptions have the proper units. Note that our method can in principle be applied to systems where time has a dimension, as
long they are of the form in Eq.~\eqref{xj_eqn} (with appropriate units for the parameters).
To compute statistics, we start by writing Eq.~\eqref{xj_eqn} as a low-pass filter of the right-hand-side:
\begin{eqnarray}
x_j(t) &= & x_j (t_0) e^{-(t-t_0)/\tau_j} \nonumber \\
& + &\frac{1}{\tau_j}\int_{t_0}^t e^{-(t-u)/\tau_j} \Big[ \mu_j + \sigma_j\eta_j(u) + \sum_k g_{jk} F_k(x_k(u)) \Big]\,du, \label{eqn:xj_filter} \nonumber \\
\end{eqnarray}
used as the basis for calculating the desired moments of $x_j$.
For example, when $\langle x_j x_k\rangle$ is desired, we use the previous
equation for $j$ and $k$, multiply, then take the expected value $\langle \cdot \rangle$.
By letting the initial time $t_0 \rightarrow -\infty$, we eliminate transients; the resulting statistics will be stationary.
The resulting exact formulas are
complicated by the network coupling, so we simplify the calculation(s) as follows.
We only account for direct connections in the formulas for the first and second order statistics, assuming the terms from the indirect connections
are either small or already accounted for in the direct connections. For example: although $F_k(x_k(u))$ on the RHS of Eq.~\eqref{eqn:xj_filter} \textit{itself} depends on coupling terms of the form $g_{kl} F_l(x_l)$, etc., we will neglect such terms.
We further make the following assumptions:
\begin{widetext}
\begin{align}
& \left\langle \int_{-\infty}^t F_k(x_k(u))e^{-(t-u)/\tau_l}\,du \int_{-\infty}^t F_k(x_k(v))e^{-(t-v)/\tau_{m}}\,dv \right\rangle \approx \frac{\tau_l \tau_m}{\tau_l + \tau_m} \mathcal{V}_k + \tau_l \tau_m (\mathcal{E}_1(k))^2 \label{ass_Fvar} \\
& \left\langle \int_{-\infty}^t \sigma_j\eta_j(u)e^{-(t-u)/\tau_l}\,du \int_{-\infty}^t F_k(x_k(v))e^{-(t-v)/\tau_{m}}\,dv \right\rangle \approx \frac{\tau_l\tau_m}{\tau_l+\tau_m} \sigma_j \mathcal{N_F}(j,k) \label{ass_nzFa} \\
& \left\langle \int_{-\infty}^t F_j(x_j(u))e^{-(t-u)/\tau_l}\,du \int_{-\infty}^t F_k(x_k(v))e^{-(t-v)/\tau_{m}}\,dv \right\rangle \approx \frac{\tau_l \tau_m}{\tau_l + \tau_m} \mathcal{C}_{\mathcal{V}}(j,k) + \tau_l \tau_m \mathcal{E}_1(j)\mathcal{E}_1(k) \label{ass_end}
\end{align}
\end{widetext}
See Table~\ref{table:defnsInt} for the definition of the symbols: $\mathcal{E}_2(k), \mathcal{N_F}(j,k), \mathcal{S}(j,k)$.
Each assumption is equivalent to the assumption that two of the random variables of interest are $\delta$-correlated in time; thus avoiding the need to compute autocorrelation functions explicitly.
The first assumption, Eq.~\eqref{ass_Fvar}, states that $F_k(x_k(t))$ is $\delta$-correlated with itself; the second, Eq.~\eqref{ass_nzFa}, addresses $\eta_j(t)$ and $F_k(x_k(t))$.
The final assumption,
Eq.~\eqref{ass_end}, states that $F_j(x_j(t))$ and $F_k(x_k(t))$ are $\delta$-correlated.
We provide a detailed derivation of Eqs.~\eqref{ass_Fvar}--\eqref{ass_end} in an Appendix (section~\ref{Append_deriv}).
After testing our method on several examples in section~\ref{sec:examples}, we will revisit the accuracy of these assumptions in
section~\ref{sec:test_delta}.
We arrive at the following (approximation) formulas for the statistics of the activity:
\begin{eqnarray}
&\mu(j) = \mu_j + \displaystyle\sum_k g_{jk} \mathcal{E}_1(k) \label{reslt_mn} \\
& \sigma^2(j) \tau_j = \frac{\sigma_j^2}{2} + \sigma_j\displaystyle\sum_k g_{jk} \mathcal{N_F}(j,k) + \frac{1}{2}\displaystyle\sum_k g_{jk}^2 \mathcal{V}(k) \nonumber \\
& +\displaystyle\sum_{k\neq l} g_{jk} g_{jl} \mathcal{C_V}(k,l) \label{reslt_vr} \\
&Cov(j,k) \frac{\tau_j+\tau_k}{2} = \frac{1}{2}c_{jk}\sigma_j\sigma_k + \frac{1}{2}\sigma_j\displaystyle\sum_{l} g_{kl} \mathcal{N_F}(j,l) \nonumber \\
&+ \frac{1}{2}\sigma_k\displaystyle\sum_{l} g_{jl} \mathcal{N_F}(k,l) + \frac{1}{2}\displaystyle\sum_{l_1,l_2} g_{j,l_1} g_{k,l_2} \mathcal{C_V}(l_1,l_2) . \label{reslt_covr}
\end{eqnarray}
See Table~\ref{table:defnsInt} for the definition of the symbols: $\mathcal{E}_1, \mathcal{N_F}, \mathcal{V}, \mathcal{C_V}$, which all depend on the statistical quantities $\mu(\cdot)$ and $\sigma(\cdot)$ of the activity $x_j$.
Our approximation formulas form a system of
$\displaystyle\frac{1}{2}\left( N_c^2+3N_c\right)$ equations in $\mu(j)$, $\sigma(j)$, $Cov(j,k)$ (i.e. for the activity only, as defined by Eqs.~\eqref{stats_defn1}--\eqref{stats_defn3}, not the firing) when considering all possible $(j,k)\in\{1,2,\dots,N_c\}$.
This large system of equations, although nonlinear, is simple to solve because it requires a sequence of function evaluations and matrix multiplications, rather than random sampling.
Note that the normal distribution assumptions allow us to conveniently write the average quantities as integrals with respect to standard normal distributions but with shifted integrands, which leads to faster calculations because one does not have to
calculate new probability density functions at each step of the iteration when solving the system self-consistently.
The resulting formulas can be written compactly with matrices; Eq.~\eqref{reslt_mn} for the mean activity $\mu(j)$ can easily be written as a matrix-vector equation and is thus omitted.
Let ${\bf Cov}$ denote the $N_c \times N_c$ covariance matrix of the activity
with ${\bf Cov}(j,k)=Cov(j,k)$, ${\bf G}$ represent the coupling strengths ${\bf G}(j,k)=g_{jk}$, and ${\bf Cr}$
denote the correlation matrix of the background noise (i.e. ${\bf Cr}(j,k)=\delta_{jk}+ c_{jk}(1-\delta_{jk})$). Then we have
\begin{eqnarray}\label{eqnMethod_mat}
{\bf Cov} = {\bf IT} \circ \Big( {\bf Cov_0}+{\bf G M_{NF}} + {\bf M_{NF}}^T {\bf G}^T + {\bf G M_{FSq} G}^T \Big) \nonumber \\
\end{eqnarray}
where $\circ$ represents element-wise multiplication, $(\cdot )^T$ denotes matrix transposition, and
\begin{eqnarray}
& {\bf IT}(j,k) = \frac{1}{\tau_j+\tau_k} \\
& {\bf Cov_0}(j,k) = \sigma_j\sigma_k \left[ \delta_{jk} +(1-\delta_{jk}) c_{jk}\right] \label{eqn:cov_jk}\\
& {\bf M_{NF} }(j,k) = \sigma_k \mathcal{N_F}(k,j) \\
& {\bf M_{FSq} }(j,k) = \mathcal{C_V}(j,k) .
\end{eqnarray}
Note that the matrices ${\bf M_{NF}}$ and ${\bf M_{FSq}}$ have the same nonzero entries as ${\bf Cr}$.
Denoting ${\bf \Lambda}_{\vec{\sigma}}$ as the diagonal matrix with diagonal $\vec{\sigma}$, the unperturbed covariance (Eq.~\eqref{eqn:cov_jk}) can also be expressed in matrix form as:
\[ {\bf Cov_0} = ({\bf \Lambda}_{\vec{\sigma}}) {\bf Cr} ({\bf \Lambda}_{\vec{\sigma}}) \]
Once the statistics of the activity ($\mu(j)$, $\sigma^2(j)$, and $Cov(j,k)$) are solved for self-consistently, the firing statistics are solved as follows.
\begin{eqnarray}
\nu_j &=& \int F_j( \sigma(j)y+\mu(j))\varrho_1(y)\,dy \label{reslt_frate} \\
Var(\nu_j) &=& \int F^2_j(\sigma(j)y+\mu(j))\varrho_1(y)\,dy - \nu^2_j \label{reslt_vRate}
\end{eqnarray}
\begin{widetext}
\begin{eqnarray} \label{reslt_covRate}
Cov(\nu_j,\nu_k) = \iint F_j(\sigma(j)y_1+\mu(j)) F_k(\sigma(k)y_2+\mu(k)) \mathbb{P}_{j,k}(y_1,y_2)\,dy_1dy_2 - \nu_j \nu_k
\end{eqnarray}
\end{widetext}
where $\mathbb{P}_{j,k}$ is a bivariate normal PDF with zero mean and covariance:
$\displaystyle\left(\begin{smallmatrix} 1 & \frac{Cov(j,k)}{\sigma(j)\sigma(k)} \\ \frac{Cov(j,k)}{\sigma(j)\sigma(k)} & 1 \end{smallmatrix}\right). $
The off-diagonal terms are obtained from the second order statistics of the activity, Eq.~\eqref{reslt_vr}--\eqref{reslt_covr}.
\section{Example Networks and Results}\label{sec:examples}
\underline{{\bf Network I.}} We first consider a network that allows us to systematically explore algorithm performance as two key parameters vary. Specifically, we consider two cells ($N_c=2$) that are reciprocally coupled without autaptic (i.e. self) coupling. For simplicity, we set the intrinsic parameters for the two cells to be identical, with
$\tau_j=1$, $F_j(x)=0.5(1+\tanh((x-0.5)/0.1))\in[0,1]$ (arbitrary units), but the mean and variance of the background input differ: $\mu_1=0.15$, $\mu_2=4/15\approx 0.2667$, $\sigma_1=2$, $\sigma_2=3$.
We vary two parameters: $g_{12}\in[-2,2]$ (input strength from $x_2$ to $x_1$), and $c_{12}=c_{21}\in[0,0.8]$, with $g_{21}=0.4$ fixed.
\begin{figure*}[tb]
\begin{center}
\includegraphics[width=6in]{Fig1-eps-converted-to.pdf}
\end{center}
\caption{Illustration of the method on a network with 2 neurons.
In all panels, the Monte Carlo simulation results are the thin black solid lines, and the result of the analytic method (Eq.~\eqref{reslt_mn}--\eqref{reslt_covr} solved self-consistently, and
Eq.~\eqref{reslt_frate}--\eqref{reslt_covRate})
are the dashed colored lines representing different background correlation levels. All parameters are fixed except $g_{12}$ and $c_{12}=c_{21}=:c$; see main text ({\bf Network I}) for values.
(a) The average activity $x_1$ (top), $x_2$ (bottom) as a function of $g_{12}$ match very well; here the analytic method is in 1 color (brown) because the result is independent of background correlation. (b) The variance of $x_1$, $\sigma^2(1)$,
varies with both background correlation and input strength. The match is very good around $g_{12}=0$ and starts to deviate as $|g_{12}|\to 2$ because with stronger coupling the normal distribution assumption is severely violated.
(c) The covariance of the activity $Cov(1,2)$. (d) Mean firing rate: $F(\nu_1)$ slightly depends on $c$; inset is a zoomed-in picture to show that the method captures the relationship of the curves. (e) Variance of $F(\nu_1)$.
(f) The covariance of the firing rate $Cov\left(F(\nu_1),F(\nu_2)\right)$. The corresponding plots for $x_2$ (i.e., panels (b), (d), (e)) are not shown because they do not vary as much, however the analytic method accurately captures the results from Monte Carlo simulations.
\label{fig1}}
\end{figure*}
In Fig.~\ref{fig1}, we see that all of the activity {\it and} firing statistics are accurate compared to Monte Carlo simulations. Figure~\ref{fig1}(a) shows the mean of $x_1$ as
the input strength $g_{12}$ varies from negative (inhibitory) to positive (excitatory); this statistic is independent of background correlation. Figure~\ref{fig1}(b) shows the variance of
$x_1$; deviations are apparent when the magnitude of the coupling $g_{12}$ is large. The covariance of the activity (Fig.~\ref{fig1}(c)) is also accurate.
Even the statistics of the firing rate are relatively accurate; the mean firing rate $F(x_1)$ (Fig.~\ref{fig1}(d)) is only weakly dependent on background correlation whereas the variance of $F(x_1)$ (Fig.~\ref{fig1}(e)) appears to
vary more with background correlation. In Fig.~\ref{fig1}(f), the strong dependence of the covariance of the firing rate on background correlation is captured by our method. For brevity, we omit the corresponding statistics for $x_2$; the
method performs equally well there.
\underline{{\bf Network II.}} We next consider an all-to-all coupled network of $N_c=50$ neurons with heterogeneity in all parameters. The parameter values were selected from specific distributions and gave rise to
quenched variability. The transfer function was set to $F_j(\bullet)=0.5(1+\tanh((\bullet-x_{rev,j})/x_{sp,j}))\in[0,1]$, where $x_{rev,j}$ and $x_{sp,j}$ are fixed parameters that depend on the the $j^{th}$ neuron.
The distributions of the parameters for this network are:
\begin{eqnarray}
\tau_j &\sim& \mathbb{N}(1,0.05^2) \label{parms_net2a}\\
\mu_j &\sim& 2\mathbb{U}-1 \\
\sigma_j &\sim& \mathbb{U}+1\\
x_{rev,j} &\sim& \mathbb{N}(0,0.1^2) \\
x_{sp,j} &\sim& 0.35\mathbb{U}+0.05
\end{eqnarray}
where $\mathbb{U}\in[0,1]$ is a uniform random variable, and $\mathbb{N}$ is normally distributed with the mean and variance as the arguments.
The covariance matrix ${\bf Cr}$ of the background noise was randomly selected as follows:
\begin{eqnarray}
{\bf Cr} = ({\bf \Lambda}_{\vec{d_s}}) {\bf A}^T{\bf A} ( {\bf \Lambda}_{\vec{d_s}}) \label{covParms_net2}
\end{eqnarray}
where the entries of the $N_c\times N_c$ matrix ${\bf A}$ are independently chosen from a normal distribution: $a_{j,k}\sim\mathbb{N}(0,0.8^2)$ and $\vec{d_s}$ is the inverse square-root of the diagonal of
${\bf A}^T{\bf A}$; i.e., if we set ${\bf B}:={\bf A}^T{\bf A}$ with entries $b_{jk}$, then $d_s(j)=1/ \sqrt{b_{jj}}$.
By construction, ${\bf Cr}$ is symmetric positive semidefinite with 1's on the diagonal.
Finally, the entries of the coupling matrix ${\bf G}$ are randomly chosen, but the parameters of the distribution were varied:
\begin{eqnarray} \label{coupParms_net2}
{\bf G}(j,k) \sim \mathbb{N}(0,v_l)
\end{eqnarray}
where $v_l=(l/10)^2$ for $l=1,2,3,4$. There are no zero entries in ${\bf G}$ (i.e. coupling is all-to-all), with both inhibition, excitation, and autaptic (self) coupling.
\begin{figure*}[tb]
\begin{center}
\includegraphics[width=6in]{Fig2-eps-converted-to.pdf}
\end{center}
\caption{A network of $N_c=50$ neurons with heterogeneity in all parameters and all-to-all coupling ({\bf Network II}).
See Eq.~\eqref{parms_net2a}--\eqref{coupParms_net2} for the distributions of the randomly selected parameters.
In each panel, four different values of the variance of the distribution of the coupling matrix entries are shown, while the other parameters are held fixed.
(a) Comparison of the mean activity $\mu(j)$ calculated via Monte Carlo simulations (horizontal axis) and our reduction method (vertical axis), showing all 50 values for each color (coupling matrix distribution).
(b) Similar to (a) but for mean firing rate $\nu_j$.
(c) Variance of activity $\sigma^2(j)$.
(d) Variance of firing rate $Var(\nu_j)$.
(e) Covariance of activity $Cov(j,k)$, showing all 50*49/2=1225 values for each coupling matrix.
(f) Covariance of the firing rate $Cov(\nu_j,\nu_k)$.
The method is accurate but starts to deviate as the overall coupling strength ${\| \bf G \|}$ increases (from blue to red, more deviations from diagonal line).
\label{fig2}}
\end{figure*}
For each of the four values for the variance of the normal distribution, we chose a single realization of a coupling matrix {\bf G} and
computed first and second-order statistics of $x_k$ and $F(x_k)$. In Fig.~\ref{fig2} we compared our analytic vs. Monte Carlo results for each cell or cell pair.
Each realization is identified by a different color; in Fig.~\ref{fig2}(a) for example, there are $N_c$ red data points, corresponding to each $\mu(j)$ for $j=1,...,N_c$. Points that are on the black diagonal line represent a perfect match between Monte Carlo simulations and our method.
First-order statistics $\mu(j)$ and $\nu_j$ are well-captured by the analytic method, even for the largest coupling strength (Fig.~\ref{fig2}(a,b)). This excellent agreement is present despite the substantial amount of heterogeneity in these networks:
note that $x_j=\mathcal{O}(1)$ and that $F_j\in[0,1]$, and thus that single-cell firing rates in Fig.~\ref{fig2}(b) have a relatively large range.
Second-order statistics (variances and covariances: Fig.~\ref{fig2}(c-f)) are captured well for smaller coupling values (blue and cyan) but become less accurate for the largest coupling value (red). In particular, the analytic method appears to
overestimate variance for the largest coupling strength (Fig.~\ref{fig2}(c)).
\underline{{\bf Network III.}} Finally we consider a moderately sized network of $N_c=100$ neurons with quenched heterogeneity in all of the intrinsic parameters, but with more physiological connectivity structure than Network II.
The first 50 neurons are excitatory $({\bf E})$ ($g_{jk}\geq 0$ for $k=1,2,\dots,50$) and the last 50 are inhibitory $({\bf I})$ ($g_{jk}\leq0$ for $k=51,52,\dots,100$).
We choose a sparse (random) background correlation matrix via:
\begin{eqnarray}
c_{jk} = \left\{
\begin{array}{ll}
1, & \hbox{if }j=k \\
\mathbb{N}(0.1,0.1^2), & \hbox{if }k=j+1\hbox{ and }j=1,\dots,49 \\
\mathbb{N}(0.12,0.1^2), & \hbox{if }k=j+1\hbox{ and }j=51,\dots,99 \\
\mathbb{N}(0.3,0.1^2), & \hbox{if }k=101-j\hbox{ and }j=1,\dots,100 \\
0 & \hbox{otherwise} \nonumber \\
\end{array}
\right. \\
\label{parms_cr_net3a}
\end{eqnarray}
where as before $\mathbb{N}$ is a Gaussian random variable.
That is, each cell shares correlated input with its nearest-neighbors of the same type (excitatory vs. inhibitory), and a single cell of the opposite type, where cell location varies along a one-dimensional line. This results in a correlation matrix which is tridiagonal, with an antidiagonal band for the E and I correlation; this sparsity structure is shown in Fig.~\ref{fig3}(a).
In a variety of cortical areas, there is evidence that the correlation of neural activity within a population is on average positive with a wide distribution~\cite{poulet08,yu10,gentet10};
thus we set the distributions of excitatory and inhibitory correlation coefficients to $\mathbb{N}(0.1,0.1^2)$ and $\mathbb{N}(0.1,0.12^2)$ respectively (second and third lines of Eq.~\eqref{parms_cr_net3a}).
Also, there is evidence that E and I neurons are positively correlated (i.e., the synaptic currents are negatively correlated)~\cite{borg98,borg96,okun08}, so we set the average background E-I
correlation ($\mathbb{N}(0.3,0.1^2)$, fourth line of Eq.~\eqref{parms_cr_net3a}) to a higher value than correlations within E or I (second and third lines respectively).
In order to capture some realistic features of cortical neural networks, we impose sparse but clustered connectivity. Specifically, we have 5 clusters of E cells of size 10 with all-to-all connectivity and no autaptic (self-coupling) connections, and
sparse random coupling within the I population (no autaptic connections) and between E and I cells (35\% connection probability).
See Fig.~\ref{fig3}(b) for the sparsity structure of ${\bf G}$. This is motivated by experimental evidence that E cells show clustered connectivity~\cite{song05,perin11,ko11}, and
that cells tuned for specific stimulus features can be more connected, while inhibitory connections have less
structure~\cite{yuste11}.
Synaptic connection strengths were chosen randomly for each realization with the following distributions:
\begin{eqnarray}
g_{EE} &=& \mathbb{U}/10, \nonumber \\
g_{EI} &=& -\frac{12}{35}\mathbb{U}-\frac{4}{35}, \nonumber \\
g_{IE} &=& \frac{12}{35}\mathbb{U}+\frac{4}{35}, \nonumber \\
g_{II} &=& -\frac{12}{35}\mathbb{U}-\frac{4}{35}, \label{parms_gm_net3}
\end{eqnarray}
where again $\mathbb{U}\in[0,1]$ is a uniform random variable.
The value $g_{EE}$ is used for all nonzero E to E connections: i.e. $g_{jk}$ with $j,k\in\{1,\dots,50\}$; $g_{EI}$ is used for all nonzero I to E connections: i.e. $g_{jk}$ with $j\in\{1,\dots,50\}$ and $k\in\{51,\dots,100\}$;
$g_{IE}$ for all nonzero E to I: i.e. $g_{jk}$ with $j\in\{51,\dots,100\}$ and $k\in\{1,\dots,50\}$; similarly for $g_{II}$.
The distributions for the rest of the parameters were similar to {\bf Network II}, with only inconsequential differences:
\begin{eqnarray}
\tau_j &\sim& \mathbb{N}(1,0.075^2) \\ \label{parmsInt_net3_start}
\mu_j &\sim& 2\mathbb{U}-1 \\
\sigma_j &\sim& \mathbb{U}+1\\
x_{rev,j} &\sim& \mathbb{N}(0,0.1^2) \\
x_{sp,j} &\sim& 0.4\mathbb{U}+0.05 \label{parmsInt_net3_end}
\end{eqnarray}
The choices for $g_{XY}$ and intrinsic parameters are not physiologically motivated, but rather chosen so that we can examine how the algorithm performs on cells with a wide range of intrinsic and network parameters.
In Fig.~\ref{fig3}(c) and (d) we show the results of the analytic approximation compared to Monte Carlo simulations for the activity and firing rates, respectively. In each panel, we have combined the mean, variance and covariance and, as in
Fig.~\ref{fig2}, a data point is plotted for each cell (for means and variances) or cell pair (for covariances).
Also, we show data from two (2) instances of the network, labeled A and B; for each instance a new realization of the coupling matrix ${\bf G}$ and the coupling parameters (Eq.~\eqref{parms_gm_net3}) are generated
(see Fig.~\ref{fig3} caption
for values), but each of the other randomly selected parameters were kept fixed.
Points that are on the black diagonal line represent a perfect match between Monte Carlo simulations and our method. As with {\bf Network II}, the analytic method accurately captures the statistics cell-by-cell, despite an appreciable degree of
heterogeneity.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=3.25in]{Fig3-eps-converted-to.pdf}
\end{center}
\caption{A network of $N_c=100$ neurons with heterogeneity in all parameters, but sparse background correlation and clustered and random connectivity ({\bf Network III}).
See Eq.~\eqref{parms_cr_net3a}--\eqref{parmsInt_net3_end} for the distributions of the randomly selected parameters.
Sparsity structure of the background correlation matrix ${\bf Cr}$ (a) and coupling matrix ${\bf G}$ (b).
(c) Comparing all of the statistics of the activity for 2 realizations of the network: coupling parameters for network A are: $(g_{EE}, g_{EI},g_{IE},g_{II})=(0.079,-0.24,0.17,-0.31)$ and coupling parameters for network B are: $(g_{EE}, g_{EI},g_{IE},g_{II})=(0.049,-0.38,0.16,-0.17)$.
As in Fig.~\ref{fig2}, all 100 mean and variance values are plotted, as well as all 4950 covariance values.
(d) Similar to (c) but for the firing rates.
\label{fig3}}
\end{figure}
Finally, we test how well our method approximates firing rate \textit{correlation}, which is an important normalized measure of trial-to-trial variability (or noise correlations). The Pearson's correlation coefficient is the predominant measure in
neuroscience: $\rho(\nu_j,\nu_k)=Cov(\nu_j,\nu_k)/\sqrt{Var(\nu_j,\nu_k)}$, i.e. the ratio of two quantities which we must estimate using the analytic method. Since this is the ratio of estimated quantities, we might expect larger errors.
In Fig.~\ref{fig4}, we show comparisons between the analytic method and Monte Carlo simulations for
{\bf Network II} and
{\bf Network III}. The method is accurate for a wide range of correlations: Fig.~\ref{fig4}(a) shows correlations as low as $-0.3$ and as high as $0.3$. Thus, the viability of our approximation is not limited to small correlation values,
but can robustly capture the full range of correlation values observed in cortical neurons~\cite{cohen11,doiron16}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=3.25in]{Fig4-eps-converted-to.pdf}
\end{center}
\caption{Comparisons of the spike count correlation computed by our method and Monte Carlo simulations.
(a) Comparing the 4 regimes in {\bf Network II}. The results are accurate because the points predominately lie on the diagonal line.
As we saw in Fig.~\ref{fig2}, as the relative coupling strength increases, the estimation of the spike count correlation is not as accurate.
(b) Comparing the 2 networks in {\bf Network III}. In both cases, the method performs well even though \textit{both} the numerator and denominator are estimated via the method.
All $N_c(N_c-1)/2$ firing rate correlation values are plotted for each network.
\label{fig4}}
\end{figure}
\section{The $\delta$-correlation Assumption}\label{sec:test_delta}
\begin{figure}[htb!]
\begin{center}
\includegraphics[width=3in]{Fig5-eps-converted-to.pdf}
\end{center}
\caption{A network of $N_c=50$ neurons with heterogeneity predominately in $\tau_j$ (see section {\bf The $\delta$-correlation Assumption} for details).
(a) Comparing our method to Monte Carlo simulations, plotting all first and second order statistics of the coupled network for two coupling matrices (red shade is with connection strengths ${\bf G}\sim\mathbb{N}(0,0.1^2)$;
black shade is with ${\bf G}\sim\mathbb{N}(0,0.25^2)$ and is not as accurate). Despite the large variation in $\tau$, ranging an order of magnitude from 0.5 to 5, the method is accurate.
(b) Plotting the L$_1$-error of firing rates and variances computed with our method as a function of the $j^{th}$ cell's $\tau_j$ value (see legend). Note that there is \underline{no trend in error} as
$\tau_j$ increases.
(c) As in (b), but for all possible covariances of firing rate and activity; the horizontal axis is the geometric average of the two associated time constants: $\sqrt{\tau_j \tau_k}$.
The conclusion is the same as in (b), that the $\tau_j$ values are not indicative of the error.
\label{fig5}}
\end{figure}
The assumptions made in deriving Eqs.~\eqref{ass_Fvar}--\eqref{ass_end} --- each equivalent to an assumption that two random variables are $\delta$-correlated in time --- might suggest that the error of our method compared to Monte Carlo simulations would increase as $\tau_j$ increases, or perhaps that the method
breaks down when the distribution of $\vec{\tau}$ has larger variance.
Thus far, we have only considered relatively narrow distributions of $\vec{\tau}$.
We now test this possibility in a setting where we can examine how the method performs as $\tau_j$ is increased without other confounding effects on the error.
Specifically, we simulate a network of $N_c=50$ cells with the majority of the network parameters set to be
homogenous values:
$$\mu_j=0.7;\hspace{.05in} \sigma_j=1.3; \hspace{.05in} x_{rev,j}=0.1; \hspace{.05in} x_{sp,j}=0.35, \forall j $$
The correlation matrix for background noise is a tridiagonal matrix with $0.3$ in the upper and lower diagonal bands, and the coupling matrix is the same as in {\bf Network II}: ${\bf G}(j,k) \sim \mathbb{N}(0,v_l)$ with
$v_l=(0.1)^2$ and $v_l=(0.25)^2$. Finally, we set
\begin{equation}\label{vary_tau_lots}
\tau_j = (j-1)*\frac{4.5}{N_c-1} + 0.5
\end{equation}
so that $\tau_j$ varies uniformly over an order of magnitude: $\tau_1=0.5$ and $\tau_{N_c}=5$.
Figure~\ref{fig5}(a) shows that the method is accurate for all possible first and second order statistics despite this large variation in $\vec{\tau}$.
Figures~\ref{fig5}(b) and (c) show that the L$_1$-error between the Monte Carlo simulations and our method does not depend in any apparent way on the value of the time constant $\tau_j$. Even when analyzed by a particular statistic,
there is no trend with time constant. We conclude that our method is robust to large and disparate values of $\tau_j$.
\section{Accuracy of the reduction method}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=3in]{Fig6-eps-converted-to.pdf}
\end{center}
\caption{Accuracy of the method as coupling strength increases.
The L$_1$-error (averaged over the entire set of six spiking/activity statistics) of
our method compared to Monte Carlo simulations increases nonlinearly as coupling strength increases (see main text for definition of $g$ and description of networks), for a variety of network sizes.
As network size $N_c$ increases, the coupling strengths are scaled by $1/\sqrt{N_c}$.
\label{fig6}}
\end{figure}
As with most calculations that assume small/large values in the parameters, an exact analytic determination of when the approximation fails is difficult, if not impossible. To capture how our method
deviates as the coupling strength is increased, we performed further computations varying the coupling strengths $g_{jk}$, system size $N_c$, and setting the other parameters to a variety of values.
The coupling matrix ${\bf G}$ was randomly chosen with half of the entries set to zero, 25\% of the entries set to $g\sqrt{10}/\sqrt{N_c}$ and 25\% set to $-g\sqrt{10}/\sqrt{N_c}$, where $g$ is a scale parameter representing the magnitude of the
coupling (Note that ${\bf G}$ was chosen only once for each given network size $N_c$, i.e. it was held fixed while other random parameters were varied). The correlation matrix of background noise is a banded matrix with between 1 and 4 bands above/below the diagonal set to $c = 0.3$; that is, each cell shared noise with its $k$-nearest neighbors, for $k=1,2,3$ or $4$. The rest of the
parameters were chosen randomly as for {\bf Network III}, Eq.~\eqref{parmsInt_net3_start}--\eqref{parmsInt_net3_end}.
Figure~\ref{fig6}
shows these results; for each network size $N_c$ and shared common noise footprint $k$, the error of our method compared to Monte Carlo simulations is plotted as a function of the magnitude of the coupling strength $g$.
Each point represents the L$_1$ error between Monte Carlo and our method, averaged over the entire set of six spiking/activity statistics (mean, variances, and covariances of both activity $x_j$ and firing rate $\nu_j$). For each system size $N_c$ (except $N_c=1000$; see explanation below), four curves show results for $k=1,2,3$ and $4$ respectively.
We see that as the magnitude of coupling values increase, any given error curve tends to increase.
In performing computations for Fig. 6, we made the following modifications for computational tractability: for larger $N_c$ values ($N_c=500, 1000$),
we augmented our method to use a subset of correlation/covariance values; we chose the main diagonal and the super/sub-diagonal $Cov(j,j+1)$; we further computed only one instance for $N_c=1000$ (i.e. there is only one curve).
We further note that some entries are not plotted because our method did not converge to a solution and/or the resulting covariance matrices are not positive definite, which is not unexpected with randomly chosen parameters.
In summary, Fig. 6 provides a representative snapshot of the range of possible error values. While the average error increases nonlinearly as
coupling strength increases, overall the error appears to be relatively insensitive to system size.
We note that the scaling of coupling strengths by the square root of system size $1/\sqrt{N}$ is considered to be ``strong scaling" popularized by the theory of balanced networks \cite{van96,van98}, compared to the relatively weak scaling
$1/N$.
\section{Discussion}
There has been a long history of analytic reduction methods for neural network models, both to enhance efficiency in simulation and to aid mathematical analyses. Here, we summarize some of this literature
and its relationship to the work presented here.
The simplest approach is a \textit{mean-field} analysis, which would self-consistently estimate the mean values, assuming the variances $\sigma^2(j)$ to be 0
\cite{tuckwell}.
However, this neglects the fluctuations which we know to be important in neural systems; therefore many authors have augmented these theories with corrections to capture second-order statistics, higher-order statistics, or time-dependent correlations.
Several authors have proposed to derive these corrections
by starting with the microscopic dynamics of single neurons in the network. The microscopic dynamics in question may be given by a master equation~\cite{buice07,bressloff09,buice10,touboul11,bressloff15}, a generalized linear model~\cite{toyoizumi09,ocker2017}, or the theta model~\cite{BC_JSM_2013,BC_PLOSCB_2013}. The result is a principled theory for the second-order statistics of the network, however, the resulting calculations are often complicated and hard to execute.
Here, we aimed to take a middle road between simple (but inaccurate) mean-field calculations, and principled (but complicated) theories to compute network fluctuations from microscopic dynamics.
Specifically, we start with a system of coupled stochastic differential equations, each of which may represent either a single neuron or a homogeneous population,
and sought to quickly and accurately estimate statistics of the
coupled system. Importantly, the unperturbed state in our system is not one in which all neurons are independent; instead we perturb from a state with background noise correlations. Thus, we anticipate this
approximation can be used to probe a range of neural networks, in which correlations can be significant and activity-modulated.
While the coupled firing rate models we study here were not derived directly from the microscopic dynamics of a spiking network, our results can still yield insight into spiking
networks~\cite{rosenbaum17,kanashiro17,ledoux11}.
Our previous work~\cite{bgsl_17} used the qualitative principles and intuitions gained from
a simple firing rate model to characterize relationships between the analogous parameters in a full spiking model of a multi-region olfactory network.
In that paper, a small system with simple coupling and background correlations was studied,
whereas this paper treats networks of arbitrary size, and arbitrary coupling and input correlation structures. The work here is thus a generalization of the calculations in~\cite{bgsl_17}.
In other models, $F(\cdot)$ represents the function that maps firing rate to synaptic input. Here, we assume that the effective synaptic input $g_{jk}F_k$ is a fixed scaling of the firing rate $F_k$.
In other biophysical models the effective synaptic input may be a more complex transformation of the firing rate (e.g., an alpha function convolved with firing rate):
the methods presented here can easily be altered to account for this.
To do this, the only change would be to use $S_k(F_k)$ in Eq.~\eqref{xj_eqn} instead of $F_k$, where $S_k(\cdot)$ is some synaptic activation function.
Our method relies on the assumption that statistics are stationary in time; this assumption allows a set of statistics to be solved self-consistently.
Thus we have not addressed complex network dynamics, such as oscillations or time-varying statistics.
However, this limitation is not specific to our method, but also applies to related work.
Previously developed approximation methods may fail when the system undergoes a bifurcation~\cite{buice07,buice10}, and truncation methods
(or moment closure methods) are known to fail in certain parameter regimes~\cite{ly_tranchina_07}. When the set of self-consistent
equations cannot be solved, there may be other methods available to characterize the oscillatory dynamics (see~\cite{nlc_15} where this is done for the adaptive quadratic integrate-and-fire model).
Likewise, we did not consider time-lagged network statistics (i.e., the entire cross-correlation functions) but rather only the instantaneous statistics. This perhaps
enables the delta-correlation assumption in our method to give accurate approximations even with disparate time-scales (see section~\ref{sec:test_delta}).
Such considerations are a fruitful path of future work.
\begin{acknowledgments}
Cheng Ly is supported by a grant from the Simons Foundation (\#355173).
\end{acknowledgments}
\section{Appendix: Derivation of Equations (\ref{ass_Fvar})--(\ref{ass_end})}\label{Append_deriv}
Here, we provide a formal derivation of the assumptions that we make to derive our main result Eqs.~\eqref{reslt_mn}--\eqref{reslt_covr}. Our challenge is that while we are principally interested in zero-time-lag statistics,
computing second-order statistics (such as $\langle F_j(x_j(t)) F_k(x_k(t)) \rangle$) using Eq.~\eqref{eqn:xj_filter} requires us to know the autocorrelation function, i.e. $\langle F_j(x_j(t)) F_k(x_k(t+\tau)) \rangle$.
Therefore we need to close the equation by making some kind of assumption about these temporal correlations.
The key assumption to justify Eq.~\eqref{ass_Fvar} is that $F_k(x_k(t))$ is $\delta$-correlated in time: i.e.
\begin{eqnarray}
\left\langle F_k(x_k(u)) F_k(x_k(v)) \right\rangle & = & \delta(u-v) \left( \left\langle F_k(x_k)^2 \right\rangle - \nu_k^2\right) + \nu_k^2 \nonumber \\
\end{eqnarray}
where $\nu_k = \left\langle F_k(x_k) \right\rangle$ is the mean.
Using this in the integral on the left-hand side of Eq.~\eqref{ass_Fvar}, we find that
\begin{widetext}
\begin{eqnarray*}
&& \int_{-\infty}^t du \, \int_{-\infty}^{t} dv\, e^{-(t-u)/\tau_l} e^{-(t-v)/\tau_m} \,\left\langle F(x_k(u)) F(x_k(v)) \right\rangle \nonumber\\
& \approx & \int_{-\infty}^t du \, \int_{-\infty}^{t} dv\, e^{-(t-u)/\tau_l} e^{-(t-v)/\tau_m} \, \delta(u-v) \left( \left\langle F(x_k(u))^2 \right\rangle - \nu_k^2\right) + \int_{-\infty}^t du \, \int_{-\infty}^{t} dv\, e^{-(t-u)/\tau_l} e^{-(t-v)/\tau_m} \, \nu_k^2 \nonumber\\
& = & \int_{-\infty}^t du \, e^{-(t-u)/\tau_l} e^{-(t-u)/\tau_m} \, \left( \left\langle F(x_k(u))^2 \right\rangle - \nu_k^2\right) + \tau_l \tau_m \nu_k^2 \\
& = & \frac{\tau_l \tau_m}{\tau_l + \tau_m} \left( \left\langle F(x_k(u))^2 \right\rangle - \nu_k^2\right) + \tau_l \tau_m \nu_k^2 \\
& = & \frac{\tau_l \tau_m}{\tau_l + \tau_m} \left( \mathcal{E}_2(k) - (\mathcal{E}_1(k))^2\right) + \tau_l \tau_m (\mathcal{E}_1(k))^2 = \frac{\tau_l \tau_m}{\tau_l + \tau_m} \mathcal{V}_k + \tau_l \tau_m (\mathcal{E}_1(k))^2
\end{eqnarray*}
The other two approximations, Eqs.~\eqref{ass_nzFa} and~\eqref{ass_end}, are arrived at by essentially the same calculation: for completeness, we provide them here as well.
To derive Eq.~\eqref{ass_nzFa}, assume that:
\begin{eqnarray}
\left\langle \sigma_j \eta_j(u) F_k(x_k(v)) \right\rangle & = & \delta(u-v) \left( \left\langle \sigma_j \eta_j(u) F_k(x_k(u)) \right\rangle - \langle \sigma_j \eta_j \rangle \langle F_k(x_k) \rangle \right) + \langle \sigma_j \eta_j \rangle \langle F_k(x_k) \rangle\\
& = & \delta(u-v) \left\langle \sigma_j \eta_j(u) F_k(x_k(u)) \right\rangle
\end{eqnarray}
where the last line is because $\langle \eta_j \rangle = 0$ and $\sigma_j$ is a constant. Then
\begin{eqnarray*}
\left\langle \int_{-\infty}^t \sigma_j \eta_j(u) e^{-(t-u)/\tau_l} \, du \int_{-\infty}^t F_k(x_k(v)) e^{-(t-v)/\tau_m} \, dv \right\rangle & = &
\int_{-\infty}^t du \, \int_{-\infty}^{t} dv\, e^{-(t-u)/\tau_l} e^{-(t-v)/\tau_m} \,\left\langle \sigma_j \eta_j(u) F_k(x_k(v)) \right\rangle \\
& \approx & \int_{-\infty}^t du \, \int_{-\infty}^{t} dv\, e^{-(t-u)/\tau_l} e^{-(t-v)/\tau_m} \, \delta(u-v) \left\langle \sigma_j \eta_j(u) F_k(x_k(v)) \right\rangle\\
& = & \int_{-\infty}^t du \, e^{-(t-u)/\tau_l} e^{-(t-u)/\tau_m} \, \left\langle \sigma_j \eta_j(u) F_k(x_k(u)) \right\rangle\\
& = & \frac{\tau_l \tau_m}{\tau_l + \tau_m} \sigma_j \mathcal{N}_F(j,k)
\end{eqnarray*}
To derive Eq.~\eqref{ass_end}, assume that:
\begin{eqnarray}
\left\langle F_j(x_j(u)) F_k(x_k(v)) \right\rangle & = & \delta(u-v) \left( \left\langle F_j(x_j(u))F_k(x_k(u)) \right\rangle - \nu_j \nu_k\right) + \nu_j \nu_k
\end{eqnarray}
where (as before) $\nu_k = \left\langle F_k(x_k) \right\rangle$.
Using this in the integral on the left-hand side of Eq.~\eqref{ass_end}, we find that
\begin{eqnarray*}
& & \left\langle \int_{-\infty}^t F_j(x_j(u)) e^{-(t-u)/\tau_l} \, du \int_{-\infty}^t F_k(x_k(v)) e^{-(t-v)/\tau_m} \, dv \right\rangle\\
& = & \int_{-\infty}^t du \, \int_{-\infty}^{t} dv\, e^{-(t-u)/\tau_l} e^{-(t-v)/\tau_m} \,\left\langle F_j(x_j(u)) F_k(x_k(v)) \right\rangle\\
& \approx & \int_{-\infty}^t du \, \int_{-\infty}^{t} dv\, e^{-(t-u)/\tau_l} e^{-(t-v)/\tau_m} \,
\delta(u-v) \left( \left\langle F_j(x_j(u))F_k(x_k(u)) \right\rangle - \nu_j \nu_k\right)
+ \int_{-\infty}^t du \, \int_{-\infty}^{t} dv\, e^{-(t-u)/\tau_l} e^{-(t-v)/\tau_m} \, \nu_j \nu_k \\
& = & \int_{-\infty}^t du \, e^{-(t-u)/\tau_l} e^{-(t-u)/\tau_m} \,
\left( \left\langle F_j(x_j(u))F_k(x_k(u)) \right\rangle - \nu_j \nu_k\right) + \nu_j \nu_k \tau_l \tau_m\\
& = & \frac{\tau_l \tau_m}{\tau_l + \tau_m}
\left( \left\langle F_j(x_j(u))F_k(x_k(u)) \right\rangle - \nu_j \nu_k\right)
+ \tau_l \tau_m \nu_j \nu_k \\
& = & \frac{\tau_l \tau_m}{\tau_l + \tau_m} \left( \mathcal{S}(j,k) - \mathcal{E}_1(j)\mathcal{E}_1(k)\right) + \tau_l \tau_m \mathcal{E}_1(j)\mathcal{E}_1(k)\\
& = & \frac{\tau_l \tau_m}{\tau_l + \tau_m} \mathcal{C}_{\mathcal{V}}(j,k) + \tau_l \tau_m \mathcal{E}_1(j)\mathcal{E}_1(k)
\end{eqnarray*}
\end{widetext}
\section{Appendix: Alternative Reduction Approaches}
A common method to approximate high dimensional systems is ``moment closure" methods where state variables are integrated or averaged out, and assumptions are made on various
moments of the random/heterogeneous entities. Such approaches have a long history in the physical sciences~\cite{chap_cow,dreyer01} and recently in the life
sciences~\cite{ly_tranchina_07,williams08,buice10}. Here we provide an alternative approach based on the probability density (or Fokker-Planck) equation of the stochastic neural network,
rather than the stochastic integrals we considered in the main text. These methods are partially related to the stochastic integral method presented earlier, but we will show that they are {\bf different}. In a similar vein, we previously showed that
the analysis of the stochastic integral is more insightful than the Fokker-Planck equation for a system of coupled noisy oscillators (compare main results and Appendix of~\cite{lyErm_pre_10}).
The corresponding probability density function $p(\vec{x},t)$, defined by: $p(\vec{x},t)\,d\vec{x}=P(\vec{X(t)}\in(\vec{x}, \vec{x}+dx))$, of the network models considered in Eq.~\eqref{xj_eqn}
satisfies the following Fokker-Planck equation~\cite{gardiner,risken}:
\begin{widetext}
\begin{eqnarray}
\frac{\partial p(\vec{x},t)}{\partial t} &=& -\sum_{l=1}^{N_c} \frac{\partial}{\partial x_l}\left\{ \frac{1}{\tau_l}\left[-x_l+\mu_l+\sum_{k=1}^{N_c} g_{lk}F_k(x_k) \right] p(\vec{x},t)\right\}+\frac{1}{2}\sum_{j,k} D_{j,k}\frac{\partial^2 p(\vec{x},t)}{\partial x_j\partial x_k}
\end{eqnarray}
\end{widetext}
where $D_{j,k}=c_{jk}\frac{\sigma_j\sigma_k}{\tau_j\tau_k}$ and the second sum is taken over all $N_c\times N_c$ pairs of $(j,k)$. This high-dimensional partial differential equation contains all of the statistics about $\vec{X}$ and any desired
transformations. We are interested in the steady-state equation $\frac{\partial p(\vec{x},t)}{\partial t}=0$, assuming the statistics are in equilibrium.
It is convenient to write the function in the curly brackets as a probability flux or current, as follows:
\begin{equation}\label{flux_defn}
J_l(\vec{x},t) :=\frac{1}{\tau_l}\left[-x_l+\mu_l+\sum_{k=1}^{N_c} g_{lk}F_k(x_k) \right] p(\vec{x},t).
\end{equation}
The steady-state equation is:
\begin{widetext}
\begin{eqnarray}
0 &=& -\sum_{l=1}^{N_c} \frac{\partial}{\partial x_l}\left\{ \frac{1}{\tau_l}\left[-x_l+\mu_l+\sum_{k=1}^{N_c} g_{lk}F_k(x_k) \right] p(\vec{x})\right\}+\frac{1}{2}\sum_{j,k} D_{j,k}\frac{\partial^2 p(\vec{x})}{\partial x_j\partial x_k} \\
0 &=& -\sum_{l=1}^{N_c} \frac{\partial}{\partial x_l} J_l(\vec{x})+\frac{1}{2}\sum_{j,k} D_{j,k}\frac{\partial^2 p(\vec{x})}{\partial x_j\partial x_k} \label{fp_sseqn}
\end{eqnarray}
\end{widetext}
\subsection{Moment Closure Methods}
We want to reduce this high-dimensional system into one that is solvable. Without coupling $g_{jk}=0$, the solution is simply a multivariate Gaussian distribution with mean $\vec{\mu}$ and covariance matrix
$Cov(j,k)=\frac{c_{jk}}{\tau_j+\tau_k}\sigma_j\sigma_k$. This motivates a closure of the system where we assume $\vec{X}$ is determined by its first two moments and is approximated by a Gaussian: $X_j=\sigma(j)+Y\mu(j)$,
where $Y$ is a standard normal random variable. We also assume the joint marginal distributions are bivariate Gaussians:
\begin{equation}\label{margGauss_ass}
\mathcal{P}(x_j,x_k):=\int p(\vec{x}) \, d\vec{x}_{\backslash j,k}\sim \mathbb{N}_2
\end{equation}
where $\mathbb{N}_2$ is the following bivariate Gaussian distribution: $\mathbb{N}\left( \left(\begin{smallmatrix}\mu(j) \\ \mu(k) \end{smallmatrix}\right) , \left(\begin{smallmatrix} \frac{\sigma(j)^2}{2 \tau_j} & \sigma(j)\sigma(k) \frac{c_{jk}}{\tau_j+\tau_k} \\ \sigma(j)\sigma(k) \frac{c_{jk}}{\tau_j+\tau_k} & \frac{\sigma(k)^2}{2 \tau_k} \end{smallmatrix}\right) \right)$, and
$d\vec{x}_{\backslash j,k}$ denotes integrating over all $N_c$ variables except $x_j$ and $x_k$. Note that these assumptions are also made in the main text.
We multiply Eq.~\eqref{fp_sseqn} by $x_j$ and integrate the equation over all $N_c$ variables, $d\vec{x}=dx_j\,d\tilde{x}$
(where $d\tilde{x}=dx_1\dots dx_{j-1} dx_{j+1}\dots dx_{N_c}:=d\vec{x}_{\backslash j}$):
\begin{eqnarray}
0 &=& - \int \sum_{l=1}^{N_c} \frac{\partial}{\partial x_l} J_l(\vec{x}) x_j \,dx_j\,d\tilde{x} \nonumber \\
& &+ \frac{1}{2} \int \sum_{l_1,l_2}D_{l_1,l_2} \frac{\partial^2 p(\vec{x})}{\partial x_{l_1}\partial x_{l_2}} x_j\,dx_j\,d\tilde{x} \label{eqn:first_pde}
\end{eqnarray}
Consider the first term: when $l\neq j$, we have:
\begin{eqnarray}
\int \frac{\partial}{\partial x_l} J_l(\vec{x}) x_j \,dx_j\,d\tilde{x}&=& \int \frac{\partial}{\partial x_l} J_l(\vec{x}) dx_l\, x_j \,dx_j \,d\vec{x}_{\backslash l, j} \nonumber \\
& =&\int J_l\vert_{x_l=-\infty}^{x_l=\infty} x_j\,dx_j \,d\vec{x}_{\backslash j} \nonumber \\
& =& \int 0 \, x_j\,dx_j\,d\vec{x}_{\backslash j} = 0
\end{eqnarray}
The last equality comes from no flux at $\pm\infty$: $J_l \vert_{x_l=-\infty}^{x_l=\infty}=0$. A similar calculation applies to the second term, for all $N_c\times N_c$ values of $(l_1,l_2)$, it is 0.
When $l_1\neq j$ and $l_2\neq j$, integrate in $x_{l_1}$ and $x_{l_2}$ first and use the fact that there is no density at $\pm\infty$: $p(\vec{x})\vert_{x_{l_{1/2}}=-\infty}^{x_{l_{1/2}}=\infty}=0$;
when $l_{1/2}=j$, integrate in $x_j$ first then integrate by parts, using $\partial_j p(\vec{x}) x_j\vert_{x_{l_{1/2}}=-\infty}^{x_{l_{1/2}}=\infty}=0$ and
$\partial_j p(\vec{x}) \vert_{x_{l_{1/2}}=-\infty}^{x_{l_{1/2}}=\infty}=0$. Therefore, Eq.~\eqref{eqn:first_pde}
is:
\begin{eqnarray}
0 &=& - \int \frac{\partial}{\partial x_j} J_j(\vec{x}) x_j \,dx_j\,d\tilde{x} \nonumber \\
0 &=& - \int J_j(\vec{x})x_j\vert_{x_j=-\infty}^{x_j=\infty}\,d\tilde{x} + \int J_j(\vec{x})\,d\vec{x} \nonumber \\
0 &=& -0 + \frac{1}{\tau_j}\left( -\mu(j) + \mu_j + \sum_{k=1}^{N_c} g_{jk} \mathcal{E}_1(k) \right) \label{eqn:first_pdeEval}
\end{eqnarray}
where $\mu(j) := \int x_j p(\vec{x})\,d\vec{x}$, and we have used the approximation $\int F_k(x_k)p(\vec{x})\,d\vec{x}\approx\mathcal{E}_1(k)$
(see Table~\ref{table:defnsInt} for definition of $\mathcal{E}_1(k)$) by assuming the marginal $x_k$ PDF is a normal distribution with mean $\mu(k)$ and variance $\sigma^2(k)$. Re-arranging
Eq.~\eqref{eqn:first_pdeEval} gives the exact same nonlinear equation for the mean $\mu(j)$, but coupled with the variance via $\mathcal{E}_1(k)$:
\begin{equation}\label{eqn:meanFP}
\mu(j) = \mu_j + \sum_{k=1}^{N_c} g_{jk} \mathcal{E}_1(k)
\end{equation}
To derive a similar equation for the variance $\sigma^2(j)$, we multiply Eq.~\eqref{fp_sseqn} by $x_j^2$ and again integrate over all variables:
\begin{eqnarray}
0 &=& - \int \sum_{l=1}^{N_c} \frac{\partial}{\partial x_l} J_l(\vec{x}) x_j^2 \,dx_j\,d\tilde{x} \nonumber \\
& &+ \frac{1}{2} \int \sum_{l_1,l_2}D_{l_1,l_2} \frac{\partial^2 p(\vec{x})}{\partial x_{l_1}\partial x_{l_2}} x_j^2\,dx_j\,d\tilde{x} \label{eqn:second_pde}
\end{eqnarray}
First consider the diffusion (second) term: similar to before, if either $l_1\neq j$ or $l_2\neq j$, the term will vanish (integrate in $x_{l_1}$ and $x_{l_2}$ first and use the fact that there is no density
at $\pm\infty$: $p(\vec{x})\vert_{x_{l_{1/2}}=-\infty}^{x_{l_{1/2}}=\infty}=0$). However, when $l_1=l_2=j$, integrate $x_j$ first and use integration by parts twice to get:
$$ \frac{D_{j,j}}{2}\int \frac{\partial^2 p(\vec{x})}{\partial x_j^2} x_j^2\,dx_j = D_{j,j} \int p(\vec{x})\,dx_j;$$
taking into account the other $N_c-1$ integration variables and that $\int p(\vec{x})\,d\vec{x}=1$ simply gives $D_{j,j}$ for the second term.
Now for the first term in Eq.~\eqref{eqn:second_pde}: if $l\neq j$, then we can see that that term in the sum vanishes by integrating $x_l$ first and using $J_l \vert_{x_l=-\infty}^{x_l=\infty}=0$. Using
integration by parts for the $l=j$ term, we get:
\begin{eqnarray}
&- \int \frac{\partial}{\partial x_j} J_j(\vec{x}) x_j^2 \,dx_j\,d\tilde{x}=0+2\int x_j J_j(\vec{x})\,dx_j\,d\tilde{x} \nonumber \\
&=\frac{2}{\tau_j} \displaystyle\int \left[ -x^2_j +\mu_j x_j + \displaystyle\sum_{k=1}^{N_c} g_{jk} x_j F_k(x_k) \right]p(\vec{x}) \,dx_j\,d\tilde{x} \nonumber
\end{eqnarray}
Using the fact that $\int x^2_j p(\vec{x})\,d\vec{x}=\sigma^2(j)+\mu(j)^2$, the entire Eq.~\eqref{eqn:second_pde} is:
\begin{widetext}
\begin{equation}
D_{j,j} = \frac{2}{\tau_j}\left[ \sigma^2(j)+\mu(j)^2 - \mu_j \mu(j)-\displaystyle\sum_{k=1}^{N_c} g_{jk}\int x_j F_k(x_k) p(\vec{x})\,d\vec{x} \right] \label{eqn:sec_interm}
\end{equation}
\end{widetext}
This equation is exact thus far. We now employ our approximation: $X_j=\mu(j)+Y_1\sigma(j)$ where $Y_1$ is a standard normal random variable (similarly for $X_k$). The last term in the previous equation is:
\begin{eqnarray}
& \mu(j)\displaystyle\sum_{k=1}^{N_c} g_{jk}\int F_k(x_k)p(\vec{x})\,d\vec{x} \nonumber \\
& +\sigma(j)\displaystyle\sum_{k=1}^{N_c} g_{jk}\int y_1 F_k(\mu(k)+y_2\sigma(k))p(\vec{x})\,d\vec{x} \nonumber
\end{eqnarray}
We can approximate the first term above with Eq.~\eqref{eqn:meanFP} to get (excluding $\mu(j)$):
$$\displaystyle\sum_{k=1}^{N_c} g_{jk}\int F_k(x_k)p(\vec{x})\,d\vec{x} \approx \sum_{k=1}^{N_c} g_{jk}\mathcal{E}_1(k) \approx \mu(j)-\mu_j$$
Thus, this leads to a cancellation of the terms $\mu(j)^2$ and $\mu_j\mu(j)$ in Eq.~\eqref{eqn:sec_interm}. We approximate the term $\int y_1 F_k(\mu(k)+y_2\sigma(k))p(\vec{x})\,d\vec{x}$ by assuming the
joint marginal distribution of $(X_j,X_k)$ are bivariate normal, and use the definition of $\mathcal{N_F}$ in Table~\ref{table:defnsInt} to get:
$$\int y_1 F_k(\mu(k)+y_2\sigma(k))p(\vec{x})\,d\vec{x} \approx \sqrt{2} \mathcal{N_F}(j,k).$$
Therefore, the equation for the variance is:
\begin{eqnarray}\label{eqn:varFP}
\sigma^2(j) \tau_j= \frac{\sigma^2_j}{2} +\sigma(j) \sqrt{2}\tau_j \displaystyle\sum_{k=1}^{N_c} g_{jk} \mathcal{N_F}(j,k)
\end{eqnarray}
This equation is similar to Eq.~\eqref{reslt_vr} but lacking higher order terms in $F_k$, as well as other differences.
To derive the analogous equation for the $Cov(j,k)$, the procedure is almost exactly the same except Eq.~\eqref{fp_sseqn} is multiplied by $x_j x_k$, and there are two terms from the sum (over probability fluxes $J_l$) that contribute, when
$l=j$ and $l=k$. The result is:
\begin{eqnarray}
Cov(j,k)\frac{\tau_j+\tau_k}{2} &=& c_{jk}\frac{\sigma_j\sigma_k}{2} +\frac{\sigma(j)}{2} \sqrt{2}\tau_j \displaystyle\sum_{l=1}^{N_c} g_{kl} \mathcal{N_F}(j,l) \nonumber \\
& & +\frac{\sigma(k)}{2} \sqrt{2}\tau_k \displaystyle\sum_{l=1}^{N_c} g_{jl} \mathcal{N_F}(k,l) \label{eqn:covFP}
\end{eqnarray}
Again, this equation is similar to parts of Eq.~\eqref{reslt_covr}. When $j=k$ in Eq.~\eqref{eqn:covFP}, we recover Eq.~\eqref{eqn:varFP}.
Together, Eqs.~\eqref{eqn:meanFP},~\eqref{eqn:varFP}, and~\eqref{eqn:covFP} form a system of transcendental equations for the complete set of first and second order statistics. This can be thought of as a
{\bf lowest order approximation} to the exact statistics of the coupled system. We implemented this method on the same network described in Section~\ref{sec:test_delta}
and found that it is not as accurate as our method (Fig.~\ref{fig7}, black dots are closer to the diagonal line than blue dots).
The mean firing rates and activity perform equally well with both methods (Fig.~\ref{fig7}(a,b)), but our method outperforms this method in calculating the
variances (Fig.~\ref{fig7}(c,d)) and even more so with the covariances (Fig.~\ref{fig7}(e,f)).
\begin{figure*}[tb]
\begin{center}
\includegraphics[width=6in]{Fig7-eps-converted-to.pdf}
\end{center}
\caption{Comparing our method with the lowest order approximation of the Fokker-Planck equation (Eqs~\eqref{eqn:meanFP},~\eqref{eqn:varFP},~\eqref{eqn:covFP}).
The network configuration is described in section~\ref{sec:test_delta}
and is the same realization in Fig.~\ref{fig5}. There are $N_c=50$ neurons with most parameters fixed except the time-scale $\vec{\tau}$ and coupling matrix {\bf G}.
(a) Comparison of the mean activity $\mu(j)$ calculated via Monte Carlo simulations (horizontal axis) and our reduction method (black) and the Fokker-Planck approximation (blue), showing all 50 values for each.
(b) Similar to (a) but for mean firing rate $\nu_j$.
(c) Variance of activity $\sigma^2(j)$.
(d) Variance of firing rate $Var(\nu_j)$.
(e) Covariance of activity $Cov(j,k)$, showing all 50*49/2=1225 values for each.
(f) Covariance of the firing rate $Cov(\nu_j,\nu_k)$. Our result is more accurate, especially for the second order statistics (c-f).
\label{fig7}}
\end{figure*}
\subsection{Higher order moment closure methods}
To derive a higher order moment closure method, one can
continue the procedure
described in the previous section,
by multiplying $x_i x_j x_k$ with Eq.~\eqref{fp_sseqn} and devise a method to close the lower order equations.
Equation~\eqref{eqn:meanFP} remains the same because of the underlying normal distribution assumption:
$$\mu(j) = \mu_j + \sum_{k=1}^{N_c} g_{jk} \mathcal{E}_1(k). $$
The second set of equations (obtained by multiplying by $x_j x_k$ and integrating in $\vec{x}$) can possibly be used to better approximate higher order equations, rather than close it as was done in the {\bf lowest
order approximation}.
If one were to follow the outline of this method, the Gaussian assumptions on $\vec{x}$ reduce the higher moments $\int d\vec{x} x_i x_j x_k p(\vec{x})$ in terms of the mean and (co-)variances of $\vec{x}$,
resulting in an over-constrained or redundant system. A possible way to proceed is to devise an approximation to $\int x_j F_k(x_k) p(\vec{x})\,d\vec{x}$, possibly relating to $\int x_j^2 F_k(x_k) p(\vec{x})\,d\vec{x}$, where
an assumption beyond the Gaussian approximation of $\vec{x}$ could perhaps be used.
Whether or not there exists a higher order moment closure method on the Fokker-Planck equation~\eqref{fp_sseqn} that is more accurate than our new method is beyond the scope of this study and an interesting
area for future research.
What is very clear through all these calculations is that our method described in the main text is different than any common moment closure methods on the Fokker-Planck equation,
despite some similarities in the equations. From
the moment closure methods we have outlined here, we see that the resulting equations will never have second order terms in the network coupling (i.e., $F_k(x_k)^2$ or $F_j F_k$), and is thus a different approach
than our method.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,844 |
Moments after Warner Bros. Pictures announced that they would seek legal action against Donald Trump for his use of music from 'Dark Knight Rises,' Twitter removed the media from his tweet.
Soon after President Trump's tweet went live at around 4.45pm, Warner Brothers Pictures released a statement saying that they would file a copyright infringement suit against the White House.
'The use of Warner Bros.' score from 'The Dark Knight Rises' in the campaign video was unauthorized,' a Warner Brothers spokesperson said in a statement obtained by NBC News.
The video has since been removed from YouTube and Reddit channels, as well.
The campaign video featured Hans Zimmer's 'Why Do We Fall?' which was a highlighted song for the third-installment in the series.
'First they ignore you. Then they laugh at you. Then they call you a racist. Donald J. Trump. Your vote. Proved them all wrong. Trump: The Great Victory. 2020,' the video states, using the 'Dark Knight Rises' font.
It also attempted to drag various celebrities through the mud, while poking fun at Barack Obama and Hillary Clinton.
Trump notably tried to be 'hip' when he posted that 'sanctions are coming' in November as a play on the popular 'Game of Thrones' phrase.
'We were not aware of this messaging and would prefer our trademark not be misappropriated for political purposes,' HBO said in a statement at the time. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,410 |
Q: To Get Text From A Locater Using Selenium IDE <br>
<br>
To add new COT click on "Add New COT"
<div class="stepandbutton">
<div class="globalbuttoncell">
<a class="buttonlink blockpage" onclick="javascript:addnewcot();" href="#">Add New COT</a>
</div>
</div>
<input id="ComingFromForm" type="hidden" value="ComingFromForm" name="ComingFromForm">
<input id="priorCOT" type="hidden" value="" name="priorCOT">
<input id="rtncode" type="hidden" value="false" name="rtncode">
<input id="refresh" type="hidden" value="NO" name="refresh">
<input id="PDate" type="hidden" value="" name="PDate">
<table>
<thead>
<tr>
<th align="center"> *Trade Class</th>
<th align="center">*Description</th>
<th>Category </th>
<th>Exclude from AMP</th>
<th>Exclude from AMP 5i</th>
<th>Exclude from ASP</th>
<th>Exclude from BP</th>
<th>Exclude from NFAMP </th>
<th>Exclude from Texas </th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center">
<input id="COT" class="data" type="text" value="" style="width:100px;" name="COT">
</td>
<td align="center">
<input id="Desc" class="data" type="text" value="" style="width:250px;" name="Desc">
</td>
<td align="center">
<select id="COTCategory" class="data small" name="COTCategory">
</td>
<td align="center">
<td align="center">
<td align="center">
<td align="center">
<td align="center">
<td align="center">
</tr>
</tbody>
</table>
</form>
<br>
<br>
I need to verify only ""To add new COT click on "Add New COT""" which is in the 3rd line is present.I have tried with //br[contains(text(),"To add new COT click on "Add New COT"")]. But it showing error that locator is not found.Please suggest another ways to verify it.
A: This is ugly but would "work":
<tr>
<td>verifyText</td>
<td>//body</td>
<td>*To add new COT click on "Add New COT"*</td>
</tr>
But as the commenter above mentions you don't want the br element. What you want to locate on is the element that you didn't provide that lives ABOVE the code you attached to. That would be best.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,030 |
Q: Is there any equivalent for Jupiter for Kubuntu? I am looking for a software like Jupiter for my Kubuntu 12.04 to reduce my battery usage.
I want to know, Is there any equivalent for Jupiter for Kubuntu ? or how can i run Jupiter on Kubuntu?
Thanks.
A: Jupiter is an application for Linux so it should work on Kubuntu.
A: I have installed it on Kubuntu 12.04 and it works well.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,447 |
FGCU's Dream Run By Perzanowski
Google: "FGCU; Dream Run," and you'll find stories and columns about the men's basketball team's incredible run to...
FGCU's Dream Run By Perzanowski Google: "FGCU; Dream Run," and you'll find stories and columns about the men's basketball team's incredible run to... Check out this story on naplesnews.com: http://nplsne.ws/24X3y4B
Naples Daily News Published 10:18 a.m. ET Sept. 17, 2013
Google: "FGCU; Dream Run," and you'll find stories and columns about the men's basketball team's incredible run to the Sweet 16 last March.
That's a shame because FGCU senior Kelly Perzanowski's effort last weekend more specifically fits the bill.
With football underway and most FGCU fans looking forward to the tip of Dunk City and the women's basketball team, Perzanowski's record-shattering 6K run in the Mountain Dew Invitational in Gainesville flew too far under the radar.
In a 270-woman cross country field, the 5-foot-2 leg-churner dominated with a time of 20:45.06 to win her second meet of this young season. It smashed her previous school-record run of 22:02.60 that she turned in during last year's NCAA South Regional. She did it against runners from SEC programs Auburn, Florida, South Carolina and Tennessee and other big-time teams.
It was simply the greatest performance ever by an FGCU cross country runner.
"Obviously Kelly had a good day beating the whole very-strong field filled with top SEC programs," said FGCU men's and women's coach Cassandra Goodson. "Beating every single runner in that race was a great thing for Kelly. She's worked incredibly hard all four years. Every year she comes to this meet she gets a bit better, and she deserves everything that she gets."
In cross country, there are no cheerleaders or bleachers, and generally very few fans not related to the runners come out to try and keep up with the almost impossible to follow races. The athletes toil in obscurity, usually running unholy amounts of miles at times that would make a rooster sigh.
It's obvious Perzanowski, a Clairsville, Ohio, native who was just a second-team All-Atlantic Sun member last season, has made huge strides despite being way ahead of her teammates. That's not easy to do without being pushed.
Cross country runs in the Perzanowskis bloodlines. Kelly's sister Natalie runs for Toledo while sister Jamie competes for UCF.
How far has Perzanowski come? Click here for a post-run interview after she finished 13th in the A-Sun Championships hosted by FGCU last year.
Read or Share this story: http://nplsne.ws/24X3y4B | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,181 |
{"url":"https:\/\/astarmathsandphysics.com\/a-level-maths-notes\/fp4\/3542-the-dot-ptoduct.html","text":"## The Dot Ptoduct\n\nThe dot product of two vectorsandis a number. It is the length of the of vector projected onto the unit vector in the direction ofIfis the angle betweenand then from the diagram aboveThis equation is usually writtenorand this form can be used to find the angle between two vectors.\n\nTo find the dot product of two vectors written in component form we multiply corresponding coordinates and add all the answers.\n\nExample: Find the dot product ofand\n\nWritten in vector ijk notation this would be\n\nand\n\nThe angle between the two vectors is the solution to","date":"2021-10-25 06:20:29","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9700198173522949, \"perplexity\": 467.7771710769417}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323587655.10\/warc\/CC-MAIN-20211025061300-20211025091300-00123.warc.gz\"}"} | null | null |
Estufa de morcego ist eine Spezialität der são-toméischen Küche. Das Gericht aus Fledertieren gilt als Delikatesse, die an Festtagen zubereitet wird.
Suppe
Kultur (São Tomé und Príncipe)
Afrikanische Küche | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,760 |
{"url":"https:\/\/deopurkar.github.io\/","text":"# Anand Deopurkar\n\nMathematical Sciences Institute\nAustralian National University\nHanna Neumann Building\n61 2 6125 4628\nanand.deopurkar@anu.edu.au\nCV html pdf\n\n## Research\n\nI am an algebraic geometer with broader interests in algebra, geometry, representation theory, and number theory. This means that I study algebraic varieties\u2014spaces of solutions of algebraic equations. Instead of studying one algebraic variety in isolation, I study the collection of all related algebraic varieties at once, using the remarkable feature that such a collection itself forms an algebraic variety (often called a \u201cmoduli space\u201d).\n\nI have worked on moduli spaces of algebraic curves, branched covers of curves, surfaces, vector bundles, and so on. For my papers and preprints, see my research page.\n\nJust after graduate school, I wrote a rough non-technical explanation of my doctoral research, which might interest or amuse you.\n\n## Upcoming and current activities\n\n\u2022 Dec 4 to 8, 2018: AustMS 2018, Adelaide, Australia.\n\u2022 December 17 to 20, 2018: Character Varieties and Topological Quantum Field Theory, Auckland, New Zealand.\n\n## Teaching\n\nI am co-organizing a special topics course on computational commutative algebra with Markus Hegeland. Here are the notes of some of the advanced classes I have taught.\n\nFor other courses taught in the past, see my teaching page.\n\n## Fun\n\nI wrote a mystery hunt style puzzle.","date":"2018-12-12 15:05:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.22573798894882202, \"perplexity\": 2342.661664638285}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-51\/segments\/1544376823895.25\/warc\/CC-MAIN-20181212134123-20181212155623-00351.warc.gz\"}"} | null | null |
package io.jboot.support.metric.interceptor;
import com.codahale.metrics.Counter;
import com.jfinal.aop.Interceptor;
import com.jfinal.aop.Invocation;
import io.jboot.Jboot;
import io.jboot.support.metric.annotation.EnableMetricConcurrency;
import io.jboot.utils.AnnotationUtil;
import io.jboot.utils.StrUtil;
public class MetricConcurrencyInterceptor implements Interceptor {
@Override
public void intercept(Invocation inv) {
Counter concurrencyRecord = null;
EnableMetricConcurrency concurrencyAnnotation = inv.getMethod().getAnnotation(EnableMetricConcurrency.class);
if (concurrencyAnnotation != null) {
String value = AnnotationUtil.get(concurrencyAnnotation.value());
String name = StrUtil.isBlank(value)
? inv.getController().getClass().getName() + "." + inv.getMethodName() + ".concurrency"
: value;
concurrencyRecord = Jboot.getMetric().counter(name);
concurrencyRecord.inc();
}
try {
inv.invoke();
} finally {
if (concurrencyRecord != null) {
concurrencyRecord.dec();
}
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,226 |
Martin Schwab (Den Haag, 29 maart 1962) is een Nederlands acteur en regisseur met Indonesische voorouders.
Levensloop
Schwab verwierf landelijke bekendheid door zijn rol als Gyman Rhemrev in Onderweg naar Morgen, deze rol speelde hij drie jaar lang. Aan het begin van seizoen 4 overlijdt zijn personage aan de zeldzame ziekte spasmoditus. Ook werd Schwab bekend door zijn rol van Ab Keizer in de politieserie Baantjer. In 2002 wilde hij hiermee eigenlijk stoppen, maar omdat Marian Mudder ook al wegging, besloot hij te blijven, omdat men anders twee personages uit de serie moest schrijven en ook nog twee vervangers introduceren. Hij speelde deze rol tot het eind van de serie.
In 2005 begon Schwab met regisseren. Zijn debuut als regisseur was de aflevering "De Cock en de moord op de middenstip" van Baantjer. Vanaf 2007 regisseert hij ook afleveringen van Flikken Maastricht en daarbij ook van Flikken Rotterdam.
Filmografie
Acteur
Film
2009 - De Punt (tv-film) - Koen
2008 - Ver van familie - opa Paul
2002 - Spagaat - vader van Simone
1999 - Baantjer: De Cock en de wraak zonder einde - Ab Keizer
1999 - Soekarno Blues - Soekarno
1999 - Gregory's Two Girls - Dimitri
1998 - Temmink: The Ultimate Fight - Saddam
1993 - Oeroeg - Oeroeg
1988 - Honneponnetje
1987 - Vroeger is dood
1986 - In de schaduw van de overwinning - Martin de verzetsstrijder
Televisie
2019 - Dit Zijn Wij - Regisseur Kees
2018 - Flikken Maastricht - Rudy Leeman
2012 - Van God Los - Freek (aflevering Procedure Exit)
2002 - Wittekerke - Salman
1995-2006 - Baantjer - Ab Keizer
1994-1996 - Onderweg naar Morgen - Gyman Rhemrev
Gastrollen
2010 - Wolven
2002-2004 - IC - Dr. Altena
Regisseur
Flikken Rotterdam (2016-heden)
Flikken Maastricht (2007 - heden)
Baantjer (4 afleveringen)
Danni Lowinski, met anderen
Externe link
Nederlands acteur
Nederlands televisieregisseur | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,163 |
By Steven M. Housman
A Piece Of Her Heart, Body, Mind & Soul
Melissa Etheridge Releases Her Very First Greatest Hits Collection
Melissa Etheridge: Greatest Hits – The Road Less Traveled
Dual Disc Released: October 18, 2005
It seems only fitting that Melissa Etheridge released her first career retrospective in the month of October. Was it intentional? By music industry standards, the fall season is the time of year to show off all of their biggest stars, and in the world of rock and roll, they don't come much bigger and better than Etheridge. By asking if releasing this album in October was intentional, I'd say yes for two reasons; 1) the aforementioned hot season for big releases and 2) Every October is breast cancer awareness month, and for those of you who may be living under a rock, Etheridge is a breast cancer survivor, who finished rigorous chemotherapy earlier this year. Her brave performance of Janis Joplin's "Piece Of My Heart" on the Grammy Awards last February will stand out as one of the finest live performances in television history. Melissa, still bald from her final chemo treatment, set the stage on fire and opened the doors for many other women suffering from this dreaded disease, and showed that bald can be beautiful. Why does it always take a celebrity to make us feel good or bad about ourselves? I suppose it's the world we live in today, and pop culture rules. But whether or not Etheridge intended to send a message, she did, and there are millions of women and men waiting in line to thank her.
Melissa Etheridge was born in the heart of the Midwest (Leavenworth, Kansas) on May 29, 1961. About the time Janis Joplin was catching fire, Etheridge was just entering grammar school. After Joplin's untimely death at age 27 in 1970, it wasn't long before Etheridge had contracted Joplin fever. To her, and millions of others, Janis Joplin's music was extremely infectious. Joplin's music hit Melissa in the gut, as well as her heart and soul. There had never been another like Joplin, and for those who make the comparisons between Joplin and Etheridge, she takes that as the highest compliment. After all, if you're going to be compared, why not be compared to the best?
Soon after Etheridge graduated high school in 1979, she made her way out to Los Angeles to realize her musical dreams. In 1982, she was discovered by Island Records founder Chris Blackwell while performing at a club in Long Beach, California, and he immediately heard what we all heard by the time Etheridge released her debut album a few years later - a star had been born. Janis Joplin comparisons were running rampant, but to Etheridge, it didn't matter. Sure, she had the same gutsy rasp to her roll, but she also had one thing that Joplin always seemed to lack, and that was confidence. Thank goodness, Etheridge's strong survival skills were already intact. She proved she could overcome the comparisons and make her music her own, and her survival skills were never needed more than when she discovered she had breast cancer a couple of years ago. The Grammy performance was a testament to all that was part of her physical make-up.
On October 4, Melissa Etheridge released Greatest Hits: The Road Less Traveled. In the 17-track set, there were a dozen fan favorites along with five brand new songs. The only surprising omission to the set was the song "Breathe." It was not only a rock station staple, it was also a fan favorite. Etheridge commented on the album's tracks and the reasons for her selections by saying "It's hard to perceive your own hits because you don't experience them as a radio listener. To me it's clear what Springsteen's hits are, what Aerosmith's are. But my own? I don't know."
This collection is superb. From the Tom Petty opener "Refugee," Etheridge's raw and raucous delivery is still as prevalent as it was two decades ago. Etheridge explains her choice for this track, "I decided to cover 'Refugee' because I wanted a song that spoke to my heart, my mind and my soul, a song about where I was at, something I could share with an audience that I knew could relate to, something that was recognizable but I could make my own. Considering what I've experienced recently, I think 'Refugee' speaks to it perfectly."
The album then takes us down a familiar and chronological road of Etheridge's releases, beginning with her 1988 self-titled debut album. The first single of Melissa's to make the Billboard charts, in the spring of 1989, was "Similar Features." At first, the single stalled, but after the success of her second single "Bring Me Some Water," "Similar Features" was re-released and the album became a huge hit that sold more than four million copies. The other single released and that appears on this compilation is "Like The Way I Do." Etheridge has a soft spot in her heart for this single. She explains, "I wrote 'Like The Way I Do' in the mid-80's, a couple of years before my first album came out. It was my show-closer even when I was playing the bars. I have to play 'Like The Way I Do' last or everything will pale following it. I play that, 'Piece Of My Heart' and 'Meet Me In The Back', those three songs were the last thirty minutes of my set in the women's bars in Long Beach and Pasadena. It was also the first original song that someone requested. Usually I would hear someone in the audience say 'Play that Stevie Nicks song,' or 'Play that Bruce Springsteen song.' That reaction gave me hope that if I ever got a record deal, maybe someone would want to hear my songs."
Other familiar tracks on this collection include, "No Souvenirs," Ain't It Heavy," the mesmerizing tracks "I'm The Only One," "Come To My Window" and "If I Wanted To" from the brilliant 1993 album Yes I Am. In 1995, Etheridge released Your Little Secret which spawned the phenomenal single "I Want To Come Over." This single crossed over many charts to become not only a fan favorite, but a radio-friendly single everybody seemed to relate to in one way or another. Etheridge explains the track, "I wrote the song while I was on tour. I wanted a song where the chorus just hit you right away and the verse was really low, I went back to a memory of a clandestine relationship and the buzz that one gets from it."
Following the success of the previous self-penned tracks, Etheridge decided to try something new and different. The result was 1999's "Angels Would Fall" from the album Breakdown. Etheridge talks about the difference of collaborating with someone new. "This song was one of my first collaborations with John Shanks. He brought me a piece of music and I wrote the melody and lyrics over it. We've done that a few times now, but 'Angels Would Fall' was the first. It was so nice to break out of my routine."
"Lucky" was on Melissa's eighth CD of the same name. "The song never got much airplay, but when I play it live, it does something to the audience. They can feel my happiness when I play that song. And they know the ache of it. It describes where I'm at right now more than any other song." With lyrics such as "I want to drive to the edge and into the sea, I just want to live shockingly," it definitely explains Etheridge's battle with breast cancer as well as her loving relationship with partner Tammy Lynn Michaels.
The remainder of the CD leaves us with four new stunning tracks, "Christmas In America," the aforementioned mind-blowing "Piece Of My Heart" which Etheridge comments "It's a magical song. You play that song and people go nuts." That's an understatement! "This Is Not Goodbye" is a poignant song that was written after Etheridge's cancer experience. You can feel the sadness and the determination in the moving lyrics and haunting melody. The final track, "I Run For My Life" was a song that was written specifically for the Ford Motor Company's "Race For The Cure" campaign to raise funds and awareness for breast cancer charities. Melissa's answer to the final track is "We are all running for answers and to make the situation better."
Whether or not Melissa Etheridge's battle with breast cancer has made her life better is only known to her and those closest to her. According to these latest songs and her recent statements, it all appears that she will cherish every bit of her wonderful life as a performer, a mother and loving partner. The fact that she's opened her heart and soul to others is reason enough to praise Melissa Etheridge for all of the joy she has brought to her fans, and at 44 years old, to a whole new generation.
She sums up the album by commenting "I always thought it was something to aspire to, where you got to a point in your career where you could have a greatest hits album, you did something right." Believe me, Melissa, you've done something very, very right. As a longtime fan, I thank you.
© 2005 Steven M. Housman. All Rights Reserved. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,147 |
Q: Word 2013 removes spaces between words Having already checked this question, where there are no solutions, I proceed to my question.
It happens with some .docx files, created by other users (using Word 2007) and sent to me by email. I do some editing of text and formatting and send it back to the users (including the ones who had originally sent me the files). The file received by them have no spaces between words.
This becomes a huge deal breaker.
What have I tried:
*
*Tried renaming
*Inspect document and remove everything (except
Headers and Footers)
*Tried saving in 97-2003 format
*Step 3 above and
then reconvert it to Word 2013 format
*As we use custom fonts, I even
tried embedding the fonts
Obviously, none of my methods work. Looking for solutions.
PS: When they send me back the my file with no spaces, I also experience the problem. Further, all other users are using Word 2007 or Word Viewers.
A: This is happening because they are using Office 2007. It is a known issue which is resolved by installing Office 2007 SP3.
When you open a Microsoft Word 2010 document in Microsoft Word 2007,
spaces are missing between some words in the document. To resolve
this problem, install the 2007 Microsoft Office Suite Service Pack 3
(SP3).
source
The reason you see the problem when they send the file back is because they have saved the file in that format.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,161 |
module Static {
'use strict';
export class PluginDataType {
public static Entity:string = 'Entity';
public static Metadata:string = 'Metadata';
public static UserSession:string = 'UserSession';
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 932 |
\section*{Introduction}
Spin ice materials present a very interesting class of magnetic materials
\cite{bramwell}. Mostly these are the pyrochlores with strongly
anisotropic Ising-like rare earth such as Dy or Ho \cite
{revmodphys}, although they exist in other structures, and one cannot exclude that similar materials
could also be made on the basis of transition metal elements with
strong anisotropy, such as
Co$^{2+}$ or~Fe$^{2+}$. Spin ice systems consist of a network of corner-shared
metal tetrahedra with effective ferromagnetic coupling between spins~\cite{hertog, yavorskii},
in which in the ground state the Ising spins are
ordered in two-in/two-out fashion. Artificial spin ice systems with different structures
have also been made~\cite{wang, tchernysh, ladak, mengotti}.
Spin ice systems are {\it bona fide} examples of frustrated systems, and they
attract now considerable attention, both because they are interesting in their own right
and because they can model different other systems, including real
water ice \cite{pauling}. A new chapter in the study of spin ice was
opened by the suggestion that the natural
elementary excitations in spin ice materials --- objects with
\hbox{3-in/1-out} or \hbox{1-in/3-out} tetrahedra --- have a magnetic charge \cite{ryzhkin}
and display many properties similar
to those of magnetic monopoles~\cite{castelnovo}.
Especially the last
proposal gave rise to a flurry of activity, see e.g.\ \cite{gingras2009},
in which, in particular, the close analogy between electric
and magnetic phenomena
was invoked.
Thus, one can apply to their
description many notions developed for the description of systems of charges
such as electrolytes; this description proves to be very efficient
for understanding many properties of spin ice.
Until now the largest attention was paid to the magnetic properties
of spin ice, both static and dynamic, largely connected with monopole excitations
\cite{morris, fennel, kadowaki, jaubert, slobinsky, giblin}, and the main tool to modify their properties was
magnetic field, which couples directly to spins or to the magnetic
charge of monopoles. I argue below that the magnetic monopoles in
spin ice have yet another characteristic which could allow for
other ways to influence and study them: each magnetic monopole, i.e.\
the tetrahedron with \hbox{3-in/1-out} or \hbox{1-in/3-out} configuration, shall
also have an electric dipole localized at such tetrahedron. This
demonstrates once again the intrinsic interplay between magnetic and
electric properties of matter.
It is well known that some magnetic textures can break inversion
symmetry -- a necessary condition for creating electric dipoles.
This lies at the heart of magnetically-driven ferroelectricity in
type-II multiferroics~\cite{trends}. There exists, in particular,
a purely electronic mechanism for creating electric dipoles. I
demonstrate that a similar breaking of inversion symmetry, occurring in
magnetic monopoles in spin ice, finally leads to the creation of
electric dipoles on them.
\section*{Results}
\subsection*{The appearance of dipoles on monopoles}
The usual description of magnetic materials with localized magnetic
moments is based on the picture of strongly correlated electrons with
the ground state being a Mott insulator, see e.g.\ Ch.~12 in
\cite{khomskiibook}. In the simplest cases, ignoring orbital effects
etc., one can describe this situation by the famous Hubbard model
\begin{equation}
{\cal H} = -t\sum_{\langle ij\rangle,\sigma}c^\dagger_{i\sigma}c^{\vphantom+}_{j\sigma} + U\sum_i n_{i\uparrow}n_{i\downarrow}\;,
\end{equation}
where $t$ is the matrix element of electron hopping between
neighbouring sites $\langle ij\rangle$ and $U$ is the on-cite Coulomb repulsion. For
one electron per site, $n = N_e/N = 1$, and strong interaction $U\gg t$ the
electrons are localized, and there appears an antiferromagnetic
nearest neighbour exchange interaction $J = 2t^2/U$ between localized
magnetic moments thus formed (which acts together with the usual
classical dipole-dipole interaction). Depending on the type of
crystal lattice there may exist different types of magnetic ground
state, often rather nontrivial, especially in frustrated lattices
containing e.g.\ magnetic triangles or tetrahedra as
building blocks.
One can show~\cite{BBMK,BBMK2} that, depending on the magnetic
configuration, there can occur a spontaneous
charge redistribution in such a magnetic triangle, so that e.g.\ the electron density on site~1
belonging to the triangle (1,2,3) is
\begin{equation}
n_1 = 1 - 8 \left(\frac tU\right)^{\!3}\Bigl[\mbox{\boldmath{$S$}}_1\cdot(\mbox{\boldmath{$S$}}_2+\mbox{\boldmath{$S$}}_3) - 2\mbox{\boldmath{$S$}}_2\cdot\mbox{\boldmath{$S$}}_3\Bigr]\;
\label{eq:2}
\end{equation}
(in other spin textures there may appear spontaneous orbital currents \cite{BBMK,BBMK2} in such triangles.)
From this expression
one sees, in particular, that there should occur charge
redistribution for a triangle with two spins up and one down, Fig.~\ref{fig:1},
which would finally give a dipole moment
\begin{equation}
d \sim \mbox{\boldmath{$S$}}_1\cdot(\mbox{\boldmath{$S$}}_2+\mbox{\boldmath{$S$}}_3) - 2\mbox{\boldmath{$S$}}_2\cdot\mbox{\boldmath{$S$}}_3
\label{eq:3}
\end{equation}
shown in Fig.~\ref{fig:1} by a broad green arrow.
\begin{figure}[ht]
\centering
\includegraphics[scale=1]{nfig01.eps}
\caption{{\bf Electronic mechanism of dipole formation.} The formation
of an electric dipole (green arrow) on a triangle
of three spins (red arrow).}
\label{fig:1}
\end{figure}
A similar expression describes also an electric dipole which can form
on a triangle due to the usual magnetostriction. One can illustrate
this e.g.\ on
the example of Fig.~\ref{fig:2}, see e.g.~\cite{mostovoy}, in which we show the
triangle (1,2,3) made by magnetic ions, with intermediate oxygens sitting
outside the triangle and forming a certain angle $M$--O--$M$.
For \hbox{3-in} spins, Fig.~\ref{fig:2}($a$), all three bonds
are equivalent, and all $M$--O--$M$ angles are the same.
However, in a configuration of Fig.~\ref{fig:2}($b$) (which,
according to Eq.~(\ref{eq:2}), would give a nonzero dipole
moment due to electronic mechanism), two bonds become ``more
ferromagnetic'', and the oxygens would shift as shown in Fig.~\ref{fig:2}($b$),
so as to
make the $M$--O--$M$ angle in the ``antiferromagnetic'' bond closer to 180
degrees, and in ``ferromagnetic'' bonds closer to 90 degrees;
according to the Goodenough--Kanamori--Anderson rules this would strengthen
the corresponding antiferromagnetic and ferromagnetic exchange and lead
to energy gain. As one sees from Fig.~\ref{fig:2}(b), such distortions
shift the centre of gravity of positive ($M$) and negative (O) charges and
thus would produce a dipole moment similar to that of Fig.~\ref{fig:1}. A similar
effect would also exist in a monopole configuration of spin ice, in
which on some bonds the spins are oriented ``ferromagnetic-like'' (e.g.\
on bonds with \hbox{2-in} spins), and on other bonds the spins are ``more antiferromagnetic''
(bonds with \hbox{1-in} and \hbox{1-out} spins).
\begin{figure}[ht]
\centering
\includegraphics[scale=1]{nfig02.eps}
\caption{{\bf Magnetostriction mechanism of dipole formation.} Illustration
of magnetostriction mechanism
of the formation of an electric dipole (green arrow): the symmetric location
of oxygens (green circles) for equivalent bonds~($a$)
changes to an asymmetric one for spin configuration (red arrows) with
different spin orientations on different bonds~($b$).}
\label{fig:2}
\end{figure}
The expression (\ref{eq:3}) is the main expression, which gives the ``dipole on
monopole'' in spin ice. Indeed, when one considers three possible
configuration of a tetrahedron in spin ice, Fig.~\ref{fig:3}($a$) (\hbox{4-in} or
\hbox{4-out} state), the monopole configuration of Fig.~\ref{fig:3}($c,d$) (\hbox{3-in/1-out} or
\hbox{1-in/3-out}), and the basic spin ice configurations \hbox{2-in/2-out},
Fig.~\ref{fig:3}($b$), then, applying the expressions (\ref{eq:2}),~(\ref{eq:3}) to every triangle
constituting a tetrahedron, one can easily see that there would be no
net dipole moments in the cases of Fig.~\ref{fig:3}($a$) (\hbox{4-in} or \hbox{4-out}) and Fig.~\ref{fig:3}($b$)
(\hbox{2-in/2-out}), but there will appear a finite dipole moment in the case of
Fig.~\ref{fig:3}($c,d$), i.e.\ {\it there will appear an electric dipole on each magnetic
monopole in spin ice}.
\begin{figure}[ht]
\centering
\includegraphics[scale=1]{nfig03.eps}
\caption{{\bf Formation of dipoles on monopoles.} Possible
spin states (red arrows) in spin-ice-like systems,
showing the formation of electric dipoles (broad green arrow)
in monopole ($c$) and antimonopole ($d$) configurations
(dipoles are absent in \hbox{4-in} ($a$) and \hbox{2-in/2-out} ($b$) states).
Note that the direction of dipoles in cases ($c$), ($d$) is the
same (in the direction of the ``special'' spin $\mbox{\boldmath{$S$}}_1$).}
\label{fig:3}
\end{figure}
The easiest way to check this is to start from the case \ref{fig:3}($a$), with
\hbox{4-in} spins. The total charge transfer e.g.\ on site~1 is
\begin{equation}
\delta n_1 \sim 2\mbox{\boldmath{$S$}}_1\cdot(\mbox{\boldmath{$S$}}_2 + \mbox{\boldmath{$S$}}_3 + \mbox{\boldmath{$S$}}_4) - 2(\mbox{\boldmath{$S$}}_2\cdot\mbox{\boldmath{$S$}}_3 + \mbox{\boldmath{$S$}}_2\cdot\mbox{\boldmath{$S$}}_4 + \mbox{\boldmath{$S$}}_3\cdot\mbox{\boldmath{$S$}}_4)\;.
\label{eq:4}
\end{equation}
For the 4-in state all the scalar products ($\mbox{\boldmath{$S$}}_i\cdot\mbox{\boldmath{$S$}}_j$) are equal, i.e.\
the charge redistribution, and with it the net dipole moment of the
tetrahedron is zero. (One can also use the condition $\mbox{\boldmath{$S$}}_1+\mbox{\boldmath{$S$}}_2+\mbox{\boldmath{$S$}}_3+\mbox{\boldmath{$S$}}_4
= \mbox{\boldmath{$0$}}$, valid in this case, to prove this; the fact that the dipole moment
is zero also follows just from the symmetry.)
However when we reverse the direction of one spin, e.g.\ $\mbox{\boldmath{$S$}}_1\to-\mbox{\boldmath{$S$}}_1$,
creating a \hbox{3-in/1-out} monopole configuration of Fig.~\ref{fig:3}($c$), the
first term in Eq.~(\ref{eq:4}) changes sign, and the resulting charge transfer
from sites 2, 3 and~4 to site~1 would be non-zero --- and there
will appear a dipole moment on such a tetrahedron, directed from the
centre of the tetrahedron to the site with the ``special spin'', in this
case to site~1 --- the broad green arrow in Fig.~\ref{fig:3}($c$)
(or in the opposite direction, depending on the specific situation ---
the sign of the hopping $t$ in Eq.~(\ref{eq:2}), or the details
of the exchange striction). This conclusion,
shown in Figs.~\ref{fig:3}$(c,d)$, is actually the main result of this paper.
As the expressions (\ref{eq:2})--(\ref{eq:4}) for the charge redistribution and for the
dipole moment are even functions of spins~$\mbox{\boldmath{$S$}}$, the reversal of all
spins will not change the results. Thus the magnitude {\it and the
direction} of the electric dipole is the same for both the monopole
(\hbox{3-in/1-out})
and antimonopole (\hbox{1-in/3-out}) configurations, Fig.~\ref{fig:3}($c$) and
\ref{fig:3}($d$): in both cases the dipole points in the direction of the
``special'' spin.
Similar considerations show that when we change the direction of one
more spin, e.g.\ $\mbox{\boldmath{$S$}}_2 \, {\to} \, -\mbox{\boldmath{$S$}}_2$, creating the \hbox{2-in/2-out} configuration of
Fig.~\ref{fig:3}($b$), various terms in Eq.~(\ref{eq:4}) again cancel, and such spin
configurations do not produce electric dipole. Thus, electric dipoles
appear in spin ice only on monopoles and antimonopoles.
\subsection*{Some consequences}
The appearance of electric dipoles on monopoles in spin ice could
have many consequences, some of which we now discuss. The main effect
would be the coupling of such dipoles to the dc or ac electric field,
\begin{equation}
{\cal E} = -\mbox{\boldmath{$d$}}\cdot\mbox{\boldmath{$E$}}\;.
\label{eq:5}
\end{equation}
This would give an electric activity to monopoles, would allow one to
influence them by external electric field, and would thus open a
new way to study and control such monopoles in spin ice. Due to this
coupling the monopoles would contribute to the dielectric function $\epsilon(\omega)$. Actually such
effect was observed in \cite{Saito}, where it was found that the
dielectric function has strong anomalies in Dy$_2$Ti$_2$O$_7$ in the magnetic
field in the [111] direction when the system approaches a transition to
the saturated state at $H \sim 1\,\rm T$~\cite{Aoki}. The mechanism of these
anomalies was not discussed in \cite{Saito}, but one can connect it
with the proliferation of monopoles and antimonopoles, with the corresponding
electric dipoles on each of them, in approaching this transition.
The saturated state in this situation, shown in Fig.~\ref{fig:4}, has the form
of staggered monopoles--antimonopoles at every tetrahedron. From our
results presented above, we conclude that in this state there would also be
electric dipoles at every tetrahedron, shown in Fig.~\ref{fig:4} by thick
green arrows. We see thus that this saturated state in a strong enough
[111] magnetic field would simultaneously be antiferroelectric. Thus
one can also associate the anomalies observed in \cite{Saito} in
$\epsilon(\omega)$ in approaching this state as the anomalies at the
antiferroelectric transition.
\begin{figure}[ht]
\centering
\includegraphics[scale=1]{nfig04.eps}
\caption{{\bf Ordered spin configuration in spin ice
in a strong [111] magnetic field.} This structure can be seen as an ordered array
of monopoles and antimonopoles; simultaneously it is antiferroelectric
(electric dipoles are shown by broad green arrows).}
\label{fig:4}
\end{figure}
Yet another consequence of the appearance of dipoles on monopoles
could be the possibility of changing the activation energy for creating
such monopoles by electric field: the excitation energy of a
monopole, or the monopole--antimonopole pair would be
\begin{equation}
\Delta = \Delta_0 - \mbox{\boldmath{$d$}}\cdot\mbox{\boldmath{$E$}}\;.
\end{equation}
Correspondingly, depending on the relative orientation of $\mbox{\boldmath{$d$}}$ and $\mbox{\boldmath{$E$}}$,
the excitation energy can both increase and decrease, but one can
always find configurations of monopoles for which the energy
would decrease. One should then be able to see this change of
activation energy in thermodynamic and magnetic properties, such as
specific heat, etc.
The orientation of electric dipoles depends on the particular situation.
One can easily see that in the absence of magnetic fields, for
completely ``free'', random spin ice, in general the orientation of dipoles
on monopole excitations is
random, in all [111] directions. But, for example, in strong
enough [001] magnetic field, in which the spin ice state is
ordered, Fig.~\ref{fig:5}, the monopoles and antimonopoles would have the
$z$-components of dipoles respectively positive and negative, $d^z
\hbox{(monopoles)}>\nobreak0$, $d^z\hbox{(antimonopoles)} <\nobreak 0$, while the perpendicular
projections of $\mbox{\boldmath{$d$}}$ would be random.
Similarly, in the [110] field \cite{fennell} the $xy$-projection of
dipole moments will be parallel to the field, $d_{xy} \parallel [110]$.
\begin{figure}[ht]
\centering
\includegraphics[scale=1]{nfig05.eps}
\caption{{\bf Possible monopole--antimonopole
pair in strong [001] magnetic field.}
The $z$-component of electric dipoles (broad green arrows) on
monopoles is pointing up, and on antimonopoles down.
The perpendicular components of $\mbox{\boldmath{$d$}}$ point
in random [110] and [$1\bar 10$] directions.
Blue arrows show spins inverted in creating
and moving apart monopole and antimonopole.}
\label{fig:5}
\end{figure}
Yet another effect could appear in an inhomogeneous electric field,
created for example close to a tip with electric voltage applied to it, in
an experimental set-up shown in Fig.~\ref{fig:6} (cf.\ e.g.\ the study of N\'eel
domain walls in a ferromagnet, which also develop electric
polarization and which can be influenced by inhomogeneous electric field
\cite{mgu}). As always, the electric dipoles would move in $\hbox{grad}\,\mbox{\boldmath{$E$}}$,
with positive dipoles e.g.\ being attracted to the region of stronger field
and negative ones repelled from it. One can use this effect to
``separate'' monopoles from antimonopoles. Thus, as is clear from
Fig.~\ref{fig:4}, in a [111] magnetic field, e.g.\ in the phase of ``kagome ice''
\cite{Aoki}, the ``favourable'' monopoles would have dipole moments
up, and antimonopoles down, so that the monopoles would be
attracted to the tip, to the region of stronger electric field, and
antimonopoles would be repelled from the tip. Similarly, the
monopole--antimonopole separation could be reached in a [001] magnetic
field, in which, as we have argued above, Fig.~\ref{fig:5}, monopoles have $d^z>0$, and
antimonopoles have~$d^z<0$.
\begin{figure}[ht]
\centering
\includegraphics[scale=1]{nfig06.eps}
\caption{{\bf Separation of monopoles and antimonopoles.} The behaviour of monopoles and antimonopoles
with respective electric dipoles (green arrows) in spin ice in [111] or [001]
magnetic field in the inhomogeneous electric field (dashed lines)
created by a tip (brown) with electric voltage.}
\label{fig:6}
\end{figure}
The magnitude of the dipoles created on monopoles, and the corresponding
strengths of their interaction with electric field, depend on the
detailed mechanism of their creation and on the specific properties
of a given material.
One should think that in real
spin ice materials, in which the hopping of $f$-electrons is rather
small, it is the magnetostriction mechanism of the dipole formation on monopoles
and antimonopoles that would be the dominant~one. In this case
one could make a
crude estimate based on the interaction~(\ref{eq:5}). If the shifts of ions
$u$ due to striction would be e.g.\ of order $0.01\,$\AA,
then the change of the energy ${\cal E} = -\mbox{\boldmath{$d$}}\cdot\mbox{\boldmath{$E$}} = -euE$ in a
field $E \sim 10^{5}\,\rm V/cm$ would be $\sim0.1\,\rm K$ --- which would lead to
measurable effects, as the typical excitation energy of monopoles
in spin ice is $\sim1\,\rm K$ \cite{castelnovo,bramwell}.
We would get effects of the same order of magnitude for the distortions $u\sim10^{-3}\,$\AA\
in a field~$\sim10^6\,\rm V/cm$.
\section*{Discussion}
Summarizing, we demonstrated that there should appear real electric
dipoles on magnetic monopoles in spin ice. Creation of such
dipoles may lead to many experimental consequences, some of which
were discussed above. They can open new ways to study and to
manipulate these exciting new objects --- magnetic monopoles in a
solid. We also see that the close connection between electric and
magnetic phenomena, which lies at the hart of modern physics,
extends in this case even further than one thought: in these systems one can have not only
magnetic charges instead of electric ones,
and ``magnetricity'' instead of electricity, but, similar to electrons
which have {\it electric charge} and {\it magnetic dipole} (spin), magnetic
monopoles in spin ice will have {\it magnetic charge} and {\it electric dipole}.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,215 |
import sys
import requests
from requests.packages.urllib3.exceptions import InsecureRequestWarning
from requests.auth import HTTPBasicAuth
requests.packages.urllib3.disable_warnings(InsecureRequestWarning)
CONNECTIONS_HOST = 'https://connections.<company>.com'
CONNECTIONS_USERNAME = '<REPLACE_HERE>'
CONNECTIONS_PASSWORD = '<REPLACE_HERE>'
wiki_id_or_label = 'W34c618febb3c_4a55_81b7_0d5a81dc1954';
xml_data = ''
xml_data += '<entry xmlns="http://www.w3.org/2005/Atom">'
xml_data += '<title type="text">My First Wiki Python</title>'
xml_data += '<content type="text/html">'
xml_data += '<![CDATA[<?xml version="1.0" encoding="UTF-8"?><p>'
xml_data += '<strong>Hello World!</strong>'
xml_data += '</p>]]>'
xml_data += '</content>'
xml_data += '<category term="wikipagetag1" />'
xml_data += '<category term="wikipagetag2" />'
xml_data += 'category scheme="tag:ibm.com,2006:td/type" term="page" label="page" />'
xml_data += '</entry>'
xml_data2 = '''
<entry xmlns="http://www.w3.org/2005/Atom">
<title type="text">My First Wiki Python 2</title>
<content type="text/html">
<![CDATA[<?xml version="1.0" encoding="UTF-8"?><p>
<strong>Hello World!</strong>
</p>]]>
</content>
<category term="wikipagetag1" />
<category term="wikipagetag2" />
category scheme="tag:ibm.com,2006:td/type" term="page" label="page" />
</entry>
'''
def createWikiPage(wiki_id_or_label, xml_data):
headers = { 'Content-Type': 'application/atom+xml;charset=UTF-8'}
url = CONNECTIONS_HOST + '/wikis/basic/api/wiki/' + wiki_id_or_label + '/feed'
auth=HTTPBasicAuth(CONNECTIONS_USERNAME, CONNECTIONS_PASSWORD)
res = requests.post(url=url,headers=headers,auth=auth,verify=False,data=xml_data)
if (res.status_code != 200):
print 'doGetToneAnalyze: requests.post -> %s = %s\n' % (res.url, res)
print res.content
return None;
return res.json()
#################### Main Module ###################
print 'Connecting to IBM Connections...\n'
print 'Creating Wiki Page...\n'
createWikiPage(wiki_id_or_label,xml_data)
createWikiPage(wiki_id_or_label,xml_data2)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,795 |
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From passion to disillusionment and back again — developing the 7th habit of successful journalists
Image: A-Z Quotes
Over the last few weeks I've been exploring the habits of successful journalists that are often described as being "innate" or "unteachable": from curiosity and scepticism, persistence and empathy, to creativity and discipline. In this final post I look at a quality underpinning them all: passion.
Are journalists only ever born with a passion for their craft — or is it something that can be taught?
Of all the seven habits that have been explored in this series, passion is perhaps the one that seems most innate — a quality that you "either have or don't have".
Can we teach passion? Well, we can provide the reasons why someone might be passionate about their craft — we can inspire passion and we can create opportunities to experience the things that have stimulated passion in others.
Passions change
Too often we take for granted the passion that we ourselves feel about journalism — and in particular the reasons behind that passion.
And we can also fail to understand that passions change. Our motivations for getting involved in journalism in the first place are often not the same as those which keep us going into mid- and then late-career roles.
The job changes, and so do we.
Image: Wonder of Science
Why are we passionate about journalism? Probably not because we are passionate about the mechanics of writing 500 words or filling 180 seconds of air time, or the discipline of making sure that our writing is grammatically correct and legally sound.
It might be because we are passionate about the role that journalism plays, and can play in society.
This is probably the story that we tell others most often — but a healthy dose of scepticism is required here: is that really the only thing that we were passionate about to begin with?
It might be that we were initially stimulated by the creativity involved in the challenges of the job.
It might be that we were driven by the opportunities to meet different people and visit new places (our curiosity). But experiencing the job creates new passions (as others quietly fade).
Research suggests that this is indeed the case:
"Debunking the myth of All The President's Men as inspiration for future journalists, Bowers' study of University of North Carolina students found students wanted to be journalists for the profession's 'interesting' and 'creative work', and they perceived themselves as competent in writing.
"In general, past research indicates that American journalism students wanted to write, be creative, and meet new people."
Those motivations also do not appear to differ (much) by country — and have not changed over time:
"Existing research finds motivations are diverse and varied, but generally fall under three categories: intrinsic motivations related to personal creativity, motivations related to journalism as an exciting and diverse profession, and motivations related to the importance of journalism in society (Carpenter et al. 2015).
"Wherever in the world the question has been asked, intrinsic motivations tend to prevail, with findings emphasising the appeal of journalism as an outlet for young peoples' passions (e.g., sport, travel, entertainment) and talents (e.g., writing, photography); as well as the exciting, non-routine, non-conventional, and sociable nature of journalism."
Before we try to instill passion in aspiring reporters, then, we should first try to understand what those motivations are, rather than assuming that trainees share the (myth-making) passions of mid- or late-career journalists who have gone into teaching.
Make the passions explicit — and expose new ones
Inviting students to share and discuss the passions that led them to study journalism in the first place can be an important first step in the process, achieving two things:
Helping them make explicit those passions (to strengthen them); and
Exposing them to new motivations that they may not have considered.
I asked this year's incoming BA students why they want to be journalists pic.twitter.com/i1oGkbmR10
— Paul Bradshaw (@paulbradshaw) September 29, 2016
A useful follow-up activity is to then ask students what they expect from journalists. Why? Because this question allows us to explore those roles of journalism which often serve as extra sources of passion.
Discussing why we expect journalists to be independent, or fair, to report new things or give a voice to the voiceless helps students to start to see themselves performing those roles (bearing in mind that their media diet may not have included journalists performing those roles) — and to understand the passion that some of those roles generate (not least when people believe journalists are not living up to those standards).
Kovach and Rosenstiel's list of the elements of journalism can be especially helpful in mapping those (and a useful piece of reading for students to explore further).
Unfortunately, fake news happens here too. Here's hoping real journalism thrives – here, and everywhere – in 2018. Bill Kovach and Tom Rosenstiel speak the #truth, with their book: The Elements of Journalism: What Newspeople Should Know and the Public Should Expect pic.twitter.com/R4AnV2nMvf
— Maya Johnson (@MJohnsonCTV) January 1, 2018
Students might, for example, rank those elements in order of importance — an exercise which both helps in teaching the qualities of journalism which distinguish it from PR or marketing, and help them identify the elements that they feel most strongly about.
That prioritisation can help guide further learning: meeting journalists who have done precisely those things which students rate highly — or designing and working on projects that can make a difference in that respect.
For example, if students rate 'giving a voice to the voiceless' as something that inspires them, the educator might organise a community news day (where the students are based in a specific community with a particular remit to interview people within that community and help to air their concerns, or highlight successes).
And inviting students to create their own values statement for a newsroom would help them to think through the motivations underpinning their work (and those of their colleagues), and prioritise those which they think should be most important.
The MA students at @BCUJournalism were planning a newsroom and I asked them to list qualities that might form part of a values statement. Here's what they came up with…
(they then voted: guess which came top?) #bcujournos pic.twitter.com/gCvtdkRSUb
— Paul Bradshaw (@paulbradshaw) December 8, 2020
Above all, exploring these issues in an empowered fashion might also help students move from the most common reasons for wanting to work in journalism (creative fulfillment, a certain lifestyle and access) to exploring others that will withstand more mundane working environments.
Preventing disillusionment with journalism
Empowering students is particularly important when you look at research on the career paths of student journalists and note that they are less likely to want to work within journalism the more experience they have of it.
It's not clear why this is the case but anecdotally at least, students do appear to report feelings of disillusionment when work experience challenges their expectations of journalism's ability to provide a forum for creativity, change or new experiences. As one respondent in that research says:
"I remember my first day [on local radio]. We'd get in, and I was really excited, because I was like, "We're going to go out. We're going to find some original stories. We're right in the town. The town centre is just down the road. We can do something here." The guy running it, he basically just told us to get local newspaper website up, and take their stories and reword them, and then go and read them on air."
Equally, when academic work (rightly) explores problems such as concentration of ownership, media bias and under-representation, students could be forgiven for feeling gaslit:
"Recently, I've realised how messed up journalism is and how political it is. I didn't realise it was this bad. From all the lectures that you've been talking about, the gender and the racism and stuff like that … I didn't realise that there is this much in it. I didn't realise that we're actually … I'm not saying that we're screwed. There is always hope. I didn't realise … how censored it can be."
But just as solutions journalism engages audiences more by focusing on what can be done about a problem, rather than merely describing that problem, we can adopt a similar approach in our teaching and learning.
Yes, certain groups are misrepresented in the media — but where can we find successful examples of those groups being given a voice?
Yes, there is bias in the media — but how have journalists reported difficult stories with responsibly?
And yes, a few corporations dominate the media landscape — but what about the growth of grassroots publishing, hyperlocal journalism and new sources of funding?
What, in short, can I get passionate about?
Inviting guest speakers who can talk about those solutions and how they negotiated the system can help students see their path through it, and inspire them with the passion needed to negotiate it.
And given the role that news organisations themselves appear to have in the disillusionment of aspiring reporters, it may be that more can be done in partnership with employers: creating opportunities for students to work on initiating more creative projects or those that 'give a voice to the voiceless' and perform other civic roles.
The importance of mental health for passion
It's notable that two of the five tips in Poynter's article on staying "motivated" in the newsroom relate to mental health: having a life outside the newsroom; and taking good care of yourself:
"Eat well, sleep well and get exercise. It may sound obvious, but these are the basic things that help us stay sane and resilient, even as chaos descends upon us. One of the best pieces of advice I ever got was when I was in college. A family friend — a professor — told me that, yes, it's important to keep studying and learning over the years, but also to stay physically fit. Because that's going to help you keep attacking the hard work, even as you get older."
Anxiety and disillusionment can starve passion of energy. Providing trainee reporters with the toolkit they need to deal with the challenges of the job, and addressing common sources such as information overload and imposter syndrome, can help provide a better environment so that passion can grow.
As Austin R. Ramsey puts it:
"Reporters often blaze new ground. They take on amorphous tasks without clear guidelines or depictions of what success should look like. Without reassuring clues, many reporters drift from project to project not knowing how to understand their own achievements."
Advice from the article — from comparing yourself to yourself a year ago, to "actively try to define success" — is worth integrating into your teaching (I find it fits especially well during induction week when students set their expectations and habits for the rest of the course).
More broadly, Maslow's Hierarchy of Needs can provide a framework for considering motivations and obstacles to those. For example:
To what extent do students feel part of a wider journalism culture or team? (a sense of belonging)
Do they have opportunities for prestige and a feeling of accomplishment? (esteem needs)
Does activity include opportunities to be creative, overcome obstacles and achieve confidence? (self-actualisation)
Sharing examples that make people passionate
Among the things that make us passionate about journalism are stories that inspire us — investigations that changed people's lives; podcasts that made our jaws drop; interviews that held power to account and gave a voice to the voiceless.
Watching the film Spotlight makes me passionate about journalism; reading about the Sunday Times's Thalidomide investigation makes me passionate about it, too.
These are important to teach not only because they serve to inspire, but because they operate as cultural touchstones in the profession, too (Tell Me No Lies is one useful collection of these; Anya Schiffrin's anthologies of African Muckraking and Global Muckraking provide non-Anglocentric correctives).
But it's important to mix those cultural touchstones with inspiring examples that are closer both to the students' experiences and their current capabilities.
Even basic examples of the impact of routine reporting (tweets from readers; spikes in audience metrics) can inspire an appreciation of its importance.
And local, consumer and specialist reporters can often talk about experiences where they have been able to directly experience feedback from readers on the impact of their work.
@StevensBenW here's that image I was looking for – we show this to first year students as an example of the impact of journalism pic.twitter.com/FTFykUzbrV
— Paul Bradshaw (@paulbradshaw) February 22, 2017
HUGE thanks to the wonderful @DonnaLFerguson for talking to @BCUJournalism students today about the reporting that forced 2 banks to pay back over £300,000 to a scam victim. So great for students to see the real impact of journalism #bcujournos https://t.co/aEpC4xv2ct
— Paul Bradshaw (@paulbradshaw) October 26, 2020
From curiosity to passion and back again: how all 7 habits in this series work in tension with each other
With the final in this series of habits listed, it is worth looking back at how passion relates to each — and how each works in tension with the other.
A good journalist is passionately curious, but not credulous: their scepticism helps prevent this. But scepticism should be distinguished from cynicism: a sceptical journalist develops the habit of persistence in establishing whether something is true or not. Persistence is aided by empathy which helps you imagine the best places to look, and identify the best possible approach.
Journalism is absolutely a creative craft but the reporter should not be indulgent with respect to that creativity; they need discipline to employ that creativity towards a professional, journalistic objective, ensuring that their story is clear, coherent and appropriate to the target audience. But discipline does not mean rigidity: a modern journalist needs the tenacity and creativity to find a different approach. Above all they need passion to find new stories, give a voice to the voiceless, hold power to account, and keep going throughout it all.
Please get in touch if you've come across any useful resources on the habits explored in this series. The series will be updated with new examples, resources and research.
This entry was posted in online journalism and tagged 7 habits, Anya Schiffrin, Maslow's Hierarchy of Needs, Mental Health, passion, Spotlight on December 10, 2020 by Paul Bradshaw.
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1 thought on "From passion to disillusionment and back again — developing the 7th habit of successful journalists"
depatridge March 9, 2021 at 6:10 am
Reblogged this on The Searchlight. | {
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Wyndham teams up with Palladium to expand all-inclusive offering
Pax Global Media
TRS Yucatan Hotel – Riviera Maya, Mexico. (Supplied)
Wyndham Hotels & Resorts and Palladium Hotel Group have announced a commercial alliance that will add more than 6,500 rooms to Wyndham's Registry Collection.
The 14 all-inclusive TRS Hotels and Grand Palladium Hotels & Resorts will be managed by Palladium Hotel Group are located in Mexico, Dominican Republic, Jamaica, and Brazil and will join Wyndham's portfolio under a long-term agreement.
This aims to leverage Wyndham's extensive distribution and bring Wyndham's all-inclusive resort portfolio to 26 hotels, according to a news release issued Thursday (July 21).
"Expanding Registry Collection Hotels continues Wyndham's global growth in the luxury space and grants more travelers access to new, preeminent experiences in some of the most remarkable destinations," said Geoffrey A. Ballotti, president and chief executive officer of Wyndham Hotels & Resorts. "These unique, all-inclusive hotels are designed to ensure that guests - whether redeeming Wyndham Rewards points or booking directly – will enjoy an elevated vacation."
"A great pairing"
Jesús Sobrino, CEO of Palladium Hotel Group, said the agreement "is a great pairing for us," noting Wyndham's distribution capacities in the United States.
"This alliance is part of our commitment to the American market due to its proximity and great air connectivity with the Caribbean destinations where we operate," Sobrino stated. "We are also very pleased to add synergies between our Palladium Rewards program and Wyndham Rewards, recognized as one of the top loyalty programs in the industry. As a hotel management company, thanks to this agreement, we improve our value proposition to owners, as we add to it a great know-how and experience of operating all-inclusive resorts in the Caribbean."
Registry Collection Hotels is a selection of hotels handpicked for their experiences in destinations with an unmistakable individuality, elevated by design and world-class service.
As part of the agreement, Palladium Hotel Group's two all-inclusive luxury brands Grand Palladium Hotels & Resorts and TRS Hotels will join Wyndham's leading luxury Registry Collection brand.
Family All-Inclusive Vacations with Grand Palladium Hotels & Resorts: the hotels offer an all-inclusive experience with sports and leisure activities, entertainment, excellent dining options and extensive services and facilities for families, couples and business groups.
Adults-Only All-Inclusive Luxury Resorts with TRS Hotels: an adults-only experience with exclusive butler service and personalized attention for couples and single travellers around the world.
Through the alliance, both companies seek to strengthen their leadership position in the region and to continue expanding while delivering high-quality standards in key tourist destinations in the Caribbean.
By joining the Registry Collection, Palladium Hotel Group promises to maintain its "spirit and unique individuality" while tapping into the global scale of Wyndham Hotels & Resorts and their loyalty program.
The first four properties to join Registry Collection Hotels under this alliance include Grand Palladium Colonial Resort & Spa, Grand Palladium Kantenah Resort & Spa and Grand Palladium White Sand Resort & Spa in Riviera Maya (Mexico) and TRS Yucatan Hotel in Riviera Maya (Mexico).
Additional Grand Palladium Hotels & Resorts and TRS Hotels properties will be added to the Registry Collection this upcoming August and September.
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Unique Vacations: Bonus edition of "Sandals Resorts....in 10 Minutes"
Club Med: Buccaneer's Creek expansion and reno planned
"Resortainment" comes to Falcon's Resorts by Meliá
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PHOTOS: Four Seasons expands in Egypt with three luxury hotels, residential projects
Subject : Wyndham teams up with Palladium to expand all-inclusive offering
URL : https://news.paxeditions.com/news/hotel/wyndham-teams-palladium-expand-all-inclusive-offering | {
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the timeout value. Values less or equal to zero do not timeout the command.
the TimeUnit for the timeout. | {
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Servant of Two Masters
SKIN OF OUR TEETH
THE SERVANT OF TWO MASTERS
Written by Carlo Goldoni
Adapted by Constance Congdon
Translated by Christina Sibul
Further Adapted by Christopher Bayes and Steven Epp
Featuring Steven Epp
Directed by Christopher Bayes
The Company of nine includes: Liam Craig*, Aidan Eastwood, Steven Epp*, Allen Gilmore*, Andy Grotelueschen*, Eugene Ma*, Orlando Pabotoy*, Sam Urdang, Adina Verson*, Liz Wisan*, and Emily Young*.
Set Designer: Katherine Akiko Day; Costume Designer: Valérie Thérèse Bart; Lighting Designer: Chuan-Chi Chan; Sound Designers: Charles Coes and Nathan A. Roberts; Original Music: Christopher Curtis and Aaron Halva; Hair and Make-up Designer: Dave Bova; Properties Supervisor: Eric Reynolds; Fight Director: Rick Sordelet; Casting Director: Deborah Brown; Assistant Director: Gabriel Levey;
Assistant Stage Manager: Blake Kile*
Production Stage Manager: Sonja Thorson*
*Actors and Stage Managers appearing courtesy of Actors Equity Association.
CHRISTOPHER BAYES (Director) began his theater career with the Tony Award winning Theatre de la Jeune Lune where he worked for five years as an actor, director, composer, designer and artistic associate. In 1989 he joined the acting company of the Guthrie Theater where he appeared in over twenty productions, including his one-man show This Ridiculous Dreaming based on Heinrich Boll's novel The Clown.He has directed at the Juilliard School, New York University's Graduate Acting Program, HERE, Dixon Place, The Flea, the New York International Clown Festival, The Public Theater, Yale Rep, Shakespeare Theater, Guthrie Theater, Arts-Emerson, Seattle Rep, Intiman Theater, Berkeley Rep, Court Theater, the Idaho Shakespeare Festival, and Trinity Repertory Theater. He was part of the creative team for the Broadway and National Touring productions of THE 39 STEPS for which he created Movement/Choreography and served as Movement Director. He also created the Movement/Choreography for John Guare's Three Kinds of Exile at The Atlantic Theater. He is a 1999/2000 Fox Fellow. He has taught workshops internationally at The Beijing Center, Cirque Du Soliel, Williamstown Theatre Festival, the Big Apple Circus. He has served on the faculty of the Juilliard School, Actor's Center (founding faculty & Master Teacher of physical comedy/clown), the Public Theater's Shakespeare Lab, the Academy of Classical Acting, New York University's Graduate Acting Program, and Director of Movement and Physical Theater at the Brown/Trinity Consortium. He is currently Professor and Head of Physical Acting at the Yale School of Drama.
LIAM CRAIG (Brighella) was last seen at TFANA in The Killer. Broadway: Boeing Boeing (u/s, appeared). Off-Broadway: The Internationalist (Vineyard Theatre), Aunt Dan and Lemon (New Group), Two Noble Kinsmen (Public Theater), Juno and the Paycock (Roundabout). Regional: The Tempest The Government Inspector (Shakespeare Theatre Company); Accidental Death of an Anarchist, The Servant of Two Masters (Yale Rep); A Doctor in Spite of Himself (Berkeley Rep). TV: "Mozart in the Jungle," "Unforgettable," "Law & Order: SVU." Film: The Royal Tenenbaums. MFA: NYU.
STEVEN EPP (Truffaldino) was an actor, writer, director, and Co-Artistic Director at Theatre de la Jeune Lune (1983-2008), winner of the 2005 Tony Award for Best Regional Theatre. Steven is currently the co-Artistic Director of The Moving Company based in Minneapolis. Acting credits: title roles in Tartuffe, Crusoe, Hamlet, Gulliver, Figaro, The Miser, Man of La Mancha, The Servant of Two Masters, Accidental Death of an Anarchist, and Ruzante. Regional: Guthrie, La Jolla, Berkeley Rep, Trinity Rep, Spoleto Festival, ART, Alley, Intiman, CenterStage, The Shakespeare Theatre, PlayMakers, Seattle Rep, South Coast Rep, Yale Rep, New Victory, and TFANA. Co-Author or Adaptor of Children of Paradise, Fissures, Moliere's A Doctor in Spite of Himself, Goldoni's Il Campiello, Massoud, The Lion of Panjshir, The House Can't Stand, Come Hell and High Water. 1999 Fox Fellow. 2009 McKnight Playwrights Center Theatre Artist Fello. Beinecke Fellow. Numerous acting awards including Helen Hayes Award, Best Actor. He lives in Minneapolis with his wife and has three children.
ALLEN GILMORE (Pantelone) is delighted to make his TFANA debut in The Servant of Two Masters with his buddies in the cast. He recently performed as "Old Emile" in the world premiere of Man in the Ring by Michael Cristofer at Court Theater in Chicago. He has played "Pantalone" with this troupe at Guthrie Theater in Minneapolis, The Shakespeare Theater in D.C., Yale Rep in New Haven, ArtsEmerson in Boston, and Seattle Rep.
ANDY GROTELUESCHEN (Dottore). TFANA: Fiasco Theater's The Two Gentlemen of Verona (St. Clair Bayfield Award) and Cymbeline; Petruchio in The Taming of the Shrew (dir. Arin Arbus). With dir. Christopher Bayes: The Servant of Two Masters (Yale Rep/Shakespeare Theatre), The Moliere Impromptu (Trinity Rep), and Clowns (The Glass Contraption). Broadway: Cyrano de Bergerac. Off-Broadway: Into the Woods (Fiasco/Roundabout, Lucille Lortel nom.), The Odyssey (The Public Word, dir. Lear deBessonet), Measure for Measure (Fiasco/New Victory). Film: Coin Heist (Netflix), Geezer, Still on the Road (PBS). TV: "Elementary," "The Good Wife," "The Knick." Brown/Trinity MFA, and Fiasco Theater company member.
EUGENE MA (Silvio). TFANA debut. New York: La Mama, P.S. 122, Joe's Pub at The Public, JACK, the Ontological-Hysteric and the Ohio Theatre. Regional: Oregon Shakespeare Festival, Berkeley Rep, Yale Rep, Seattle Rep, Wallis Annenberg Center (with directors Mary Zimmerman, Christopher Bayes, and Stan Lai). Composing: Yale School of Drama, Joe's Pub, Urban Stages. A certified teacher of Clown and Commedia dell'Arte, Eugene completed his apprenticeship with Christopher Bayes at Yale and Juilliard. Eugene is currently a Resident Director at the Flea Theatre and holds a Drama Desk nomination. AEA. BFA: NYU. www.playwitheugene.com
ORLANDO PABOTOY (Florindo). TFANA debut. New York: The Public, New York Theater Workshop, The Met Opera, Ma-Yi Theater Company, Imua! Theatre Company, Lake Lucille, The Flying Machine Company, Division 13. Roles include Aufidius in Coriolanus; Puck, Lysander in A Midsummer Night's Dream; Azdak in Caucasian Chalk Circle; Petruchio, Grumio in The Taming of the Shrew; Prospero, Antonio, Stephano in The Tempest; Captain Jamy, French Lord in Henry V; Magno in The Romance of Magno Rubio; Genet in Jean Genet; and the Fool in King Lear. TV/Film: "Strangers with Candy" (Comedy Central), "JAG" (CBS), In the Weeds (Independent Feature), Blue Hour, "The Beat" (WB), and "Whoopi" (NBC). Regional/International: Old Globe, Yale Rep, and in Jedermann at the Salzburg Festival. Awards: Fox Fellow, John Houseman, OBIE. Graduate of Juilliard Drama Division.
ADINA VERSON (Clarice). Theatre credits include Indecent (Vineyard, Yale Rep, La Jolla), peerless (Barrington Stage Co.), As You Like It (STC, DC), Christopher Bayes's The Servant of Two Masters (Guthrie, Seattle Rep, ArtsEmerson), The Winter's Tale (Yale Rep), 4000 Miles (Cincinnati Playhouse), and Machine Makes Man (Amsterdam Fringe, Best Int'l Performance; National Arts Festival of South Africa, Cape Town Fringe), which she co-created with Michael McQuilken. TV: Miriam Setrakian on "The Strain" (FX). MFA: Yale School of Drama.
LIZ WISAN (Beatrice) performed as Beatrice and Smeraldina in Christopher Bayes's The Servant of Two Masters at Yale Rep, Shakespeare Theatre Company, Guthrie Theater, Seattle Rep, and ArtsEmerson. NY credits include: These Paper Bullets! (Atlantic Theater Company), Other Desert Cities (Broadway and LCT), The Tempest (La Mama). Regional: These Paper Bullets! and Caucasian Chalk Circle (Yale Rep), Baskerville (The Old Globe), Absurd Person Singular (Two River Theater), Intelligent Homosexual's Guide… (Berkeley Rep). TV/Film: "Elementary," Ready or Knot, and Bitches. MFA from Yale School of Drama. Member of New Neighborhood and The Actors Center. www.lizwisan.com
EMILY YOUNG (Smeraldina). Emily is a Fiasco Theater company member. Broadway: Bloody Bloody Andrew Jackson. Off-Broadway: The Two Gentlemen of Verona, TFANA; Into the Woods, Roundabout; Measure for Measure, New Victory; Cymbeline, TFANA/Barrow Street; Romeo and Juliet, TBTB. Regional: Old Globe, McCarter, Folger, NC Shakes, Illinois Shakes, Trinity Rep, Kennedy Center Millennium Stage, Sundance Theater Lab, New York Stage & Film. Film/TV: The Knick, God of Love (Oscar for Best Live Action Short 2011), Manhattan Melody (Telluride), Natives (SXSW). Training: Brown/Trinity; Brown University.
Offices: 154 Christopher Street Ste. 3D, NY, NY 10014
Polonsky Shakespeare Center: 262 Ashland Place, Brooklyn, NY 11217
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JOIN OUR E-LIST | {
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Q: Requesting help for a type of control I want to add a control in winforms that drops like combo box but can show data like datagridview. I have seen this control in an application but not able to find a way how to do it. (Pic attached)
Thanks
A: You can "emulate" it by showing a form without borders containing a DataGridView when user clics on that cell.
| {
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Home Uncategorized Thinking of voting?
I could go on about how democracy is the underpinning of our entire country's way of life. I could go on about how people fought and even died for the right to vote. I could tell you how, not so long ago, women were considered "property" and therefore not allowed to vote. Blacks were considered non-human and not allowed to vote. I could tell you how our latest government won by fewer votes than the number of people who didn't vote at all.
But you've heard all that.
Well, yes, in some respects they are. One party may have a line you like better than another's, but if you think about it, that's just your response to skilled marketing. They all promise stuff they can't deliver or have no intention of following up on. They hope you'll listen to their words and not do the math.
The various parties' track records speak for themselves, but they hope you won't remember that. They hope you'll listen to their cool promises and forget any damage they've done. They hope you won't do the research. They're betting on your ignorance.
Don't make your ignorance a sure thing.
My vote can't make a difference.
Did you really think it could? I can tell you how one vote makes a difference — the person at the top, the person with the power and the veto — that's the single vote that makes the difference. Where your vote comes in is deciding who to entrust with that power. One vote may not have much say, but no vote does NOT have no say; it gives your vote and your right to it to someone else.
Don't give your vote to someone else.
I'm too busy that day.
Oh, come ON. There are ways around that. Vote in an advance poll if you'll be away. Vote on your way to work. Vote after work. Home with your family? Take them with you; this is a teachable moment. It's one day, folks. So what if there's a lineup? It's probably a lot shorter than the lineup for those concert tickets or the latest iPhone. You'll enjoy the concert for a couple of hours, you'll have your new phone for a couple of years.
You'll be living with your government for four years.
I can't get to a polling place after work.
Your employer — all employers — are legally required to give you time off during work hours to vote, without loss of pay or other repercussions. So you can vote before work, or after work, or if it's practical, during work. If you need a ride, call your constituency office. Better yet, if you have the time, offer your services as a driver. There are also special advance polling places operated by Elections Canada where anyone can vote regardless of their residence.
It's too hard to vote.
It's not easy. Voting is work. You have to live in Canada, you have to be a Canadian citizen and you have to be able to prove that. Fair enough, right? The regulations are tighter now — a driver's license or other government-issued ID with your picture and address, or two other pieces of ID which can be chosen from a very long list, which you can find here.
If you're transgender, it gets harder, because you may not have ID that looks like you. Your license gender marker may not match who you are inside and the picture may be out of date in the same respect. All I can say is, gather everything you do have — ID, carry letters from your therapist, name change documents — whatever you do have, and take it with you when you vote. If you have to, muster your courage, put your pride away inside and prepare to look like your ID. But vote.
No, it's too hard not to vote.
Finally, please take thirteen minutes and watch The Right to be Heard . | {
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\section{Introduction}
\label{sec:introduction}
This paper concerns a problem of learning a classifier to optimize
approximate median significance (AMS), which was the goal of
the Higgs Boson Machine Learning Challenge (HiggsML), hosted by Kaggle
website (see \cite{AdamBourdarios_etal2014} for details on this contest
and description of the problem).
In particular, we are interested in an approach to optimize AMS, based
on the following two-stage procedure. First, a real-valued function
$f$ is learned by minimizing a surrogate loss for binary classification, such
as logistic loss function, on the training sample. In the second stage, given $f$, a threshold is
tuned on a separate ``validation'' sample, by direct optimization of AMS with respect to
a classifier obtained from $f$ by classifying all observations with value of $f$ above the threshold
as positive class (signal event), and all observations below the threshold as negative class (background event).
This approach became very popular among HiggsML challenge participants, mainly due to
the fact that its first stage, learning a classifier, does not exploit the task evaluation
metric (AMS) in any way and thus can employ without modifications any standard classification tools such as
logistic regression, LogitBoost, Stochastic Gradient Boosting, Random Forest, etc.
(see, e.g., \cite{FriedmanHastieTibshirani03}). Despite its simplicity,
this approach proved to be very effective in achieving high leaderboard score in HiggsML.
\footnote{See the HiggsML forum \url{https://www.kaggle.com/c/higgs-boson/forums} for discussions and
presentation of the top score solutions.}
The intuition behind this approach is clear: minimization of logistic loss results in estimation of
conditional probabilities of signal and background event,
and the AMS is assumed to be maximized by classifying
the events most likely to be signal as signal events.
This paper formalizes this intuition by showing that the approach described above constitutes
a consistent method of optimizing AMS. More specifically,
we use the notion of \emph{regret} with respect
to some evaluation metric, which is a difference between the performance of a given classifier
and the performance of the optimal classifier with respect to this metric.
Given a function $f$, and a classifier $h_{f,\hat{\theta}}$ obtained from $f$ by thresholding $f$ at
$\hat{\theta}$, we give a bound on the regret of $h_{f,\hat{\theta}}$ measured with respect to the squared AMS
by the regret of $f$ measured with respect to the
logistic loss, given that the threshold $\hat{\theta}$ is tuned by optimization of AMS among
all classifiers of the form $h_{f,\theta}$ for any threshold value $\theta$.
To our knowledge, this is the first regret bound of this form applicable to a non-decomposable
performance measure such as AMS. We also discuss generalization of our approach to different
performance measures and surrogate loss functions.
\paragraph{Related work.} The issue of consistent optimization of performance
measures which are functions of true positive and true negative rates
has received increasing attention recently in machine learning community
\citep{Narasimhan_etal2014,Natarajan_etal2014,Zhao_etal_2013}.
However, these works are mainly concerned with \emph{statistical consistency} also
known as \emph{calibration}, which determines whether convergence to the minimizer of a
surrogate loss implies convergence to the minimizer of the task performance measure as sample size
goes to infinity. Here we give a much stronger result which bounds the regret with respect
to squared AMS by the regret with respect to logistic loss. Our result is valid for all
finite sample sizes and informs about the rates of convergence.
Recently, \cite{MackeyBryan2014} proposed a classification cascade approach to optimize
AMS. Their method, based on the theory of Fenchel's duality, iteratively
alternates between solving a cost-sensitive binary classification problem and
updating misclassification costs. In contrast, the method described here requires
solving an ordinary binary classification problem just once.
\paragraph{Outline.} The paper is organized as follows. In Section \ref{sec:problem_setting},
we introduce basic concepts needed to state our main result presented
in Section \ref{sec:main_result} and proved in Section \ref{sec:proof}.
Section \ref{sec:generalization}
discusses generalization of our results beyond AMS and logistic loss.
\section{Problem Setting}
\label{sec:problem_setting}
\paragraph{Binary classifier.}
In binary classification, the goal is, given an input (feature vector) $x \in X$,
to accurately predict the output (label) $y \in \{-1,1\}$. We assume input-output
pairs $(x,y)$, which we call \emph{observations}, are generated i.i.d. according to $\Pr(x,y)$.\footnote{
The original HiggsML problem also involved observations' weights, but without loss
of generality, they can be incorporated into the distribution $\Pr(x,y)$.}
A \emph{classifier} is a mapping $h \colon X \to \{-1,1\}$. Given $h$, we define
the following two quantities:
\[
s(h) = \Pr(h(x) = 1, y=1),\qquad b(h) = \Pr(h(x) = 1,y=-1),
\]
which can be interpreted as true positive and false positive rates of $h$.
\paragraph{AMS and regret.}
Given a classifier $h$, define its \emph{approximate median significance} (AMS) score
\citep{Cowan_etal2011} as
$\mathrm{AMS}(h) = \mathrm{AMS}(s(h),b(h))$, where:\footnote{Comparing to the definition in \citep{AdamBourdarios_etal2014},
we skip the regularization term $b_{\mathrm{reg}}$. This comes without loss of generality, as
$b_{\mathrm{reg}}$ can be incorporated into $b$ and, since it affects all classifiers equally,
will vanish in the definition of regret.}
\[
\mathrm{AMS}(s,b) = \sqrt{2\left((s+b)\log\left(1 + \frac{s}{b}\right) - s\right)}.
\]
It is easier to deal with a \emph{squared} AMS, $\mathrm{AMS}^2(h)$, and this quantity
is used throughout the paper.
It is easy to verify that $\mathrm{AMS}^2(s,b)$ is increasing in $s$ and decreasing in $b$.
Moreover, $\mathrm{AMS}^2(s,b)$ is jointly convex with respect to $(s,b)$.
Let $h^*_{\mathrm{AMS}}$ be the classifier which maximizes the $\mathrm{AMS}^2$ over all possible
classifiers:
\[
h^*_{\mathrm{AMS}} = \argmax_{h \in \{-1,1\}^X} \mathrm{AMS}^2(h).
\]
Given $h$, we define its \emph{AMS regret} as the distance of $h$ from the optimal
classifier $h^*_{\mathrm{AMS}}$ measured by means of $\mathrm{AMS}^2$:
\[
R_{\mathrm{AMS}}(h) = \mathrm{AMS}^2(h^*_{\mathrm{AMS}}) - \mathrm{AMS}^2(h).
\]
\paragraph{Logistic loss and logistic regret.}
Given a real number $f$, and a label $y$, we define the logistic loss
$\ell_{\log} \colon \{-1,1\} \times \mathbb{R} \to \mathbb{R}_+$ as:
\[
\ell_{\log}(y,f) = \log\left(1+e^{-yf}\right).
\]
The logistic loss is a commonly used surrogate loss function for binary
classification, employed in various learning methods, such as logistic regression, LogitBoost
or Stochastic Gradient Boosting (see, e.g., \cite{FriedmanHastieTibshirani03}).
It is convex in $f$, so minimizing logistic
loss over the training sample becomes a convex optimization problem, which can be
solved efficiently. Another advantage of logistic loss is that the sigmoid transform of $f$,
$(1+e^{-f})^{-1}$, can be
used to obtain probability estimates $\Pr(y|x)$.
Given a real-valued function $f \colon X \to \mathbb{R}$, its expected logistic
loss $L_{\log}(f)$ is defined as:
\[
L_{\log}(f) = \mathbb{E}_{(x,y)}[\ell_{\log}(y,f(x))].
\]
Let $f^*_{\log} = \argmin_f L_{\log}(f)$ be the minimizer of $L_{\log}(f)$
among all functions $f \colon X \to \mathbb{R}$. We define the logistic \emph{regret}
of $f$ as:
\[
R_{\log}(f) = L_{\log}(f) - L_{\log}(f^*_{\log}).
\]
\section{Main Result}
\label{sec:main_result}
Any real-valued function $f \colon X \to \mathbb{R}$
can be turned into a classifier $h_{f,\theta} \colon X \to \{-1,1\}$, by thresholding at
some value $\theta$:
\[
h_{f,\theta}(x) = \mathrm{sgn}(f(x) - \theta),
\]
where $\mathrm{sgn}(x)$ is the sign function, and we use the convention that $\mathrm{sgn}(0)=1$.
The purpose of this paper is to address the following problem: given a function $f$
with logistic regret $R_{\log}(f)$, and a threshold $\theta$, what is the maximum
AMS regret of $h_{f,\theta}$? In other words, can we bound $R_{\mathrm{AMS}}(h_{f,\theta})$
in terms of $R_{\log}(f)$?
We give a positive answer to this question, which based on the following regret bound:
\begin{lemma}
\label{lemma:main}
There exists a threshold $\theta^*$, such that for any $f$,
\[
R_{\mathrm{AMS}}(h_{{f,\theta^*}}) \leq \frac{s(h^*_{\mathrm{AMS}})}{b(h^*_{\mathrm{AMS}})} \sqrt{\frac{1}{2} R_{\log}(f)}.
\]
\end{lemma}
The proof is quite long and hence is postponed to Section \ref{sec:proof}.
Interestingly, the proof goes by an intermediate bound of the AMS regret by
a cost-sensitive classification regret, with misclassification costs proportional
to the gradient coordinates of the AMS.
Lemma \ref{lemma:main} has the following interpretation.
If we are able to find a function $f$ with small logistic regret,
we are guaranteed that there exists a threshold $\theta^*$ such that $h_{f,\theta^*}$ has small AMS regret.
Note that the same threshold $\theta^*$ will work for any $f$, and the right hand side of the bound is \emph{independent} of $\theta^*$.
We are now ready to prove the main result of the paper:
\begin{theorem}
Given a real-valued function $f$, let $\hat{\theta} = \argmax_{\theta} \mathrm{AMS}(h_{f,\theta})$. Then:
\[
R_{\mathrm{AMS}}(h_{f,\hat{\theta}}) \leq \frac{s(h^*_{\mathrm{AMS}})}{b(h^*_{\mathrm{AMS}})} \sqrt{\frac{1}{2} R_{\log}(f)}.
\]
\label{thm:main}
\end{theorem}
\begin{proof}
The result follows immediately from Lemma \ref{lemma:main} by noticing that
solving $\max_{\theta}\mathrm{AMS}(h_{f,\theta})$ is equivalent to
solving $\min_\theta R_{\mathrm{AMS}}(h_{f,\theta})$, and that
$\min_\theta R_{\mathrm{AMS}}(h_{f,\theta}) \leq R_{\mathrm{AMS}}(h_{f,\theta^*})$.
\end{proof}
Theorem \ref{thm:main} motivates the following procedure for AMS maximization:
\begin{enumerate}
\item Find $f$ with small logistic regret, e.g. by
employing a learning algorithm minimizing logistic loss on the training sample.
\item Given $f$, solve $\hat{\theta} = \argmax_{\theta} \mathrm{AMS}(h_{f,\theta})$.
\end{enumerate}
Theorem \ref{thm:main} states that the AMS regret of the classifier obtained by
this procedure
is upperbounded by the logistic regret of the underlying real-valued function.
We now discuss how to approach step 2 of the procedure in practice.
In principle, this step requires maximizing AMS defined
by means of an unknown distribution $\Pr(x,y)$. However, it is sufficient to
optimize $\theta$ on the empirical counterpart of AMS calculated on a separate validation sample.
Due to space limit, we only give a sketch of the proof of this fact:
Step 2 involves optimization within a class of threshold functions
(since $f$ is fixed), which has VC-dimension equal to $2$ \citep{DevroyeGyorfiLugosi96}.
By convexity of $\mathrm{AMS}^2$,
\begin{equation}
\label{eq:deviation_of_empirical_AMS_from_AMS}
\mathrm{AMS}^2(s,b) - \mathrm{AMS}^2(\hat{s},\hat{b}) \leq
\left(\frac{\partial \mathrm{AMS}^2(s,b)}{\partial s}, \frac{\partial\mathrm{AMS}^2(s,b)}{\partial b} \right)^\top
(s - \hat{s}, b - \hat{b})
\end{equation}
(see, e.g. \cite{BoydVandenberghe2004}), where $\hat{s}$ and $\hat{b}$ are empirical counterparts of $s$ and $b$. By VC theory,
the deviations of $\hat{s}$ from $s$, and $\hat{b}$ from $b$ can be upperbounded
with high probability \emph{uniformly} over the class of all threshold functions by $O(1/\sqrt{m})$,
where $m$ is the validation sample size. This and (\ref{eq:deviation_of_empirical_AMS_from_AMS})
implies,
that $\mathrm{AMS}^2(s,b)$ of the empirical maximizer
is $O(1/\sqrt{m})$ close to the $\max_\theta \mathrm{AMS}^2(h_{f,\theta})$. Hence, step 2 can be performed
within $O(1/\sqrt{m})$ accuracy on a validation sample independent from the training sample.
\section{Proof of Lemma \ref{lemma:main}}
\label{sec:proof}
The proof consists of two steps. First, we bound the AMS regret of any classifier $h$ by
its cost-sensitive classification regret (introduced below).
Next, we show that there exists a threshold $\theta^*$, such that for any $f$,
the cost-sensitive classification regret of $h_{f,\theta^*}$
is upperbounded by the logistic regret of $f$.
\paragraph{Bounding AMS regret by cost-sensitive classification regret.}
Given a real number $c \in (0,1)$,
define a \emph{cost-sensitive classification loss}
$\ell_c \colon \{-1,1\} \times \{-1,1\} \to \mathbb{R}_+$ as:
\[
\ell_c(y,h) = c \mathds{1}[y=-1] \mathds{1}[h=1] + (1-c) \mathds{1}[y=1] \mathds{1}[h=-1],
\]
where $\mathds{1}[A]$ is the indicator function equal to $1$ if predicate $A$ is true, and $0$ otherwise.
The cost-sensitive loss assigns different costs of misclassification for positive and
negative labels.
Given classifier $h$, the expected cost-sensitive loss of $h$ is:
\[
L_c(h) = \mathbb{E}_{(x,y)}[\ell_c(y,h(x))]
= c b(h) + (1-c) (\Pr(y=1) - s(h)),
\]
where $s(h)$ and $b(h)$ are true positive and false positive
rates defined before. Let $h^*_c = \argmin_h L_c(h)$ be the minimizer of
the expected cost-sensitive loss among all classifiers.
Define the cost-sensitive classification regret as:
\[
R_c(h) = L_c(h) - L_c(h^*_c).
\]
Any convex and differentiable function $g(x)$ satisfies
$g(x) \geq g(y) + \nabla g(y)^\top(x-y)$ for any $x,y$ in its convex domain \citep{BoydVandenberghe2004}.
Applying this inequality to $\mathrm{AMS}^2(s,b)$ jointly convex in $(s,b)$,
we have for any $s,b,s^*,b^* \in [0,1]$:
\begin{equation}
\label{eq:ams_cost-sensitive_bound}
\mathrm{AMS}^2(s,b) \geq \mathrm{AMS}^2(s^*,b^*) +
\left(\frac{\partial \mathrm{AMS}^2(s^*,b^*)}{\partial s^*}, \frac{\partial\mathrm{AMS}^2(s^*,b^*)}{\partial b^*} \right)^\top
(s - s^*, b - b^*).
\end{equation}
Given classifier $h$, we set $s = s(h), b = b(h), s^*=s(h^*_{\mathrm{AMS}}), b^* = b(h^*_{\mathrm{AMS}})$,
and:
\[
C:= \frac{\partial \mathrm{AMS}^2(s^*,b^*)}{\partial s^*} -\frac{\partial \mathrm{AMS}^2(s^*,b^*)}{\partial b^*},
\qquad
c:= -\frac{1}{C} \frac{\partial \mathrm{AMS}^2(s^*,b^*)}{\partial b^*}.
\]
Since $\mathrm{AMS}^2(s,b)$ is increasing in $s$ and decreasing in $b$, both $\frac{\partial \mathrm{AMS}^2(s^*,b^*)}{\partial s^*}$ and $-\frac{\partial \mathrm{AMS}^2(s^*,b^*)}{\partial b^*}$
are positive, which implies $C > 0$ and $0 < c < 1$.
In this notation, (\ref{eq:ams_cost-sensitive_bound}) boils down to:
\begin{align*}
R_{\mathrm{AMS}}(h) = \mathrm{AMS}^2(h^*_{\mathrm{AMS}}) - \mathrm{AMS}^2(h) &\leq C \Big( c (b(h) - b(h^*_{\mathrm{AMS}})) + (1-c) (s(h^*_{\mathrm{AMS}})-s(h)) \Big)\\
&= C \big( L_c(h) - L_c(h^*_{\mathrm{AMS}}) \big)\\
&\leq C \big( L_c(h) - L_c(h^*_c) \big) = C R_c(h),
\end{align*}
where the last inequality follows from the definition of $h^*_c$. Thus, the AMS regret
is upperbounded by the cost-sensitive classification regret with costs proportional to the gradient coordinates
of $\mathrm{AMS}^2(s^*,b^*)$ at optimum $h^*_{\mathrm{AMS}}$.\footnote{Note that the gradient at optimum
does not vanish, as the optimum is with respect to $h$, not $(s,b)$.}
\paragraph{Bounding cost-sensitive classification regret by logistic regret.}
We first give a bound on
cost-sensitive classification regret by means of logistic regret \emph{conditioned} at a given $x$.
This part relies on the techniques used by \cite{Bartlett_etal06}.
Then, the final bound is obtained by taking expectation with respect to $x$, and applying
Jensen's inequality.
Given a label $h \in \{-1,1\}$, and $\eta \in [0,1]$, define \emph{conditional} cost-sensitive classification
loss as:
\[
\ell_c(\eta,h) = c(1-\eta) \mathds{1}[h=1] + (1-c) \eta \mathds{1}[h=-1].
\]
The reason this quantity is called ``conditional loss'' becomes clear if we note that for any classifier
$h$, $L_c(h) = \mathbb{E}_x [\ell_c(\eta(x),h(x))]$, where $\eta(x) = \Pr(y=1|x)$. In other words,
$\ell_c(\eta(x),h(x))$ is the loss of $h$ conditioned on $x$.
Given $\eta$, let $h^*_c = \argmin_{h \in \{-1,1\}} \ell_c(\eta,h)$.
It can be easily verified
that:
\[
h^*_c = \mathrm{sgn}\left(\eta - c \right),
\]
and $\ell_c(\eta,h^*_c) = \min\{c(1-\eta), (1-c) \eta \}$.
The conditional regret of $h$ is defined as $r_c(\eta,h) = \ell_c(\eta,h) - \ell_c(h^*_c)$.
Note that:
\[
r_c(\eta,h) =
\left\{
\begin{array}{ll}
0 & \quad \text{if~~} h = h^*_c,\\
\left|\eta - c \right|
& \quad \text{if~~} h \neq h^*_c.
\end{array}
\right.
\]
Given a real number $f$, and $\eta \in [0,1]$, define \emph{conditional} logistic loss as:
\[
\ell_{\log}(\eta,f) = (1-\eta) \log\left(1 + e^f\right) + \eta \log\left(1 + e^{-f}\right).
\]
Let $f^*_{\log} = \argmin_{f \in \mathbb{R}} \ell_{\log}(\eta,f)$. By differentiating
$\ell_{\log}(\eta,f)$ with respect to $f$, and setting the derivative to $0$, we get that:
\[
f^*_{\log} = \log \frac{\eta}{1-\eta},
\]
and $\ell_{\log}(\eta,f^*_{\log}) = -\eta \log \eta - (1-\eta) \log (1-\eta)$, the binary entropy of $\eta$.
The conditional logistic regret of $f$ is given by $r_{\log}(\eta,f) = \ell_{\log}(\eta,f) - \ell_{\log}(f^*_{\log})$.
The conditional regret has a particularly simple form
when $f$ is re-expressed as a probability estimate $\eta_f$:
\[
r_{\log}(\eta,f) = D(\eta \| \eta_f),
\qquad \text{where}
\quad \eta_f := \frac{1}{1+e^{-f}},
\]
and $D(\eta \| \eta_f) = \eta \log \frac{\eta}{\eta_f} + (1-\eta) \log \frac{1-\eta}{1-\eta_f}$
is the Kullback-Leibler divergence. By Pinsker's inequality,
\[
D(\eta \| \eta_f) \geq 2 (\eta - \eta_f)^2.
\]
Given real number $f$, define $h_{f,\theta^*} = \mathrm{sgn}(f-\theta^*)$,
where:
\[
\theta^* = \log \frac{c}{1-c}.
\]
We will now
bound the conditional cost-sensitive classification regret
$r_c(\eta,h_{f,\theta^*})$ in terms of conditional logistic regret $r_{\log}(\eta,f)$.
First note that:
\[
h_{f,\theta^*} = 1 ~\iff~ f \geq \theta^* = \log \frac{c}{1-c}
~\iff~ \frac{1}{1+e^{-f}} \geq c
~\iff~ \eta_f \geq c,
\]
so that we can equivalently write $h_{f,\theta^*} = \mathrm{sgn}(\eta_f - c)$.
Since $h^*_c = \mathrm{sgn}(\eta - c)$,
then whenever
$(\eta_f - c)(\eta - c) > 0$,
it holds $h_{f,\theta^*}= h^*_c$, and $r_c(\eta,h_{f,\theta^*}) = 0$.
On the other hand, when $(\eta_f - c)(\eta - c) \leq 0$,
it holds\footnote{
$r_c(\eta,h_{f,\theta^*}) = |\eta - c|$
if $(\eta_f - c)(\eta - c) < 0$, and can be either $0$ or
$|\eta - c|$ when $(\eta_f - c)(\eta - c) = 0$.
}
$r_c(\eta,h_{f,\theta^*}) \leq |\eta - c|$, whereas:
\begin{align*}
r_{\log}(\eta,f) &= D(\eta \| \eta_f) \stackrel{\mathrm{Pinsker's}}{\geq} 2(\eta - \eta_f)^2 = 2(\eta - c + c - \eta_f)^2 \\
&= 2(\eta-c)^2 + 4(\eta - c)(c - \eta_{f})
+ 2(c - \eta_f)^2 \\
&\geq 2 (\eta - c)^2
\geq 2 r^2_c(\eta,h_{f,\theta^*}),
\end{align*}
where the last but one inequality is implied by $(\eta_f - c)(\eta - c) \leq 0$. Taking both cases together, we get:
\[
r_c(\eta,h_{f,\theta^*}) \leq \sqrt{r_{\log}(\eta,f)/2}.
\]
Now, given any function $f$,
\begin{align*}
R_c(h_{f,\theta^*}) &= \mathbb{E}_x[r_c(\eta,h_{f,\theta^*})]
\leq \mathbb{E}_x\left[\sqrt{ r_{\log}(\eta,f)/2}\right]
\leq \sqrt{\mathbb{E}_x[r_{\log}(\eta,f)]/2}
= \sqrt{R_{\log}(f)/2},
\end{align*}
where the last inequality is from Jensen's inequality applied to the concave
function $x \mapsto \sqrt{x}$.
\paragraph{Finishing the proof.}
Combining the results from both parts, we get:
\[
R_{\mathrm{AMS}}(h_{f,\theta^*}) \leq C R_c(h_{f,\theta^*})
\leq C \sqrt{R_{\log}(f)/2},
\]
where $\theta^* = \log\frac{c}{1-c}$ is independent of $f$.
Recalling that
$C = \frac{\partial \mathrm{AMS}^2(s^*,b^*)}{\partial s^*} -\frac{\partial \mathrm{AMS}^2(s^*,b^*)}{\partial b^*}$, we calculate:
\[
C = \log\left(1 + \frac{s^*}{b^*}\right) - \left(\log\left( 1 + \frac{s^*}{b^*} \right) - \frac{s^*}{b^*} \right) = \frac{s^*}{b^*},
\]
where $s^* = s(h^*_{\mathrm{AMS}})$ and $b^* = b(h^*_{\mathrm{AMS}})$.
This finished the proof.
$\square$
Note that the proof actually specifies the exact value of the universal threshold $\theta^*$:
\[
\theta^* = \log \frac{c}{1-c}, \qquad \text{where~} c = 1 - \frac{b^*}{s^*} \log \left(1+\frac{s^*}{b^*}\right).
\]
\section{Generalization beyond AMS and logistic loss}
\label{sec:generalization}
Results of this paper can be generalized beyond AMS metric and logistic loss surrogate.
The AMS can be replaced by any other evaluation metric, which enjoys the following two properties:
1) is increasing in $s$, and decreasing in $b$; 2) is jointly convex in $s$ and $b$.
These were the only two properties of the AMS used in the proof of Lemma \ref{lemma:main}.
The logistic loss surrogate can be replaced by any other convex surrogate loss $\ell$,
such that the following property holds: There exists a threshold $\theta^*$ which is a function of the cost $c$, such that for all $f$,
\[
R_c(h_{f,\theta^*}) \leq \lambda \sqrt{R_{\ell}(f)},
\]
for some positive constant $\lambda$. This property is satisfied by, e.g., squared error loss $\ell_{\mathrm{sq}}(y,f) = (y-f)^2$ with $\lambda = 1$,
which can be verified by noticing that the logistic regret upperbounds the squared error regret by Pinsker's inequality.
We conjecture that all \emph{strongly proper composite losses} \citep{Agarwal2014} hold this property.
\acks{The author was supported by the Foundation For Polish Science Homing Plus grant, co-financed by the European Regional Development Fund.
The author would like to thank Krzysztof Dembczy{\'n}ski for interesting discussions and proofreading the paper.}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,308 |
Národní socialismus je označení různých ideologií, které se snaží spojit myšlenky socialismu a nacionalismu. Už koncem 19. století vznikaly strany označované jako národně-socialistické, se sociálně-demokratickým programem, které však odmítaly jeho internacionalismus a naopak zdůrazňovaly program národního sebeurčení. Tak se např. Česká strana národně sociální oddělila roku 1897 od Českoslovanské sociálně demokratické strany dělnické, pozdější ČSSD.
Od těchto demokratických stran se lišily strany či hnutí, které parlamentní demokracii odmítaly. Roku 1919 vznikla v Mnichově Německá dělnická strana (Deutsche Arbeiterpartei), která se od roku 1920 jmenovala Národně socialistická německá dělnická strana (NSDAP, Nationalsozialistische Deutsche Arbeiterpartei). Její program a politická praxe byly ovlivněny italským fašismem a ze zkratky jejího názvu vznikl pojem nacismus (nacionál-socialismus).
Politická filosofie
Směr se hlásí k myšlenkám, jakými jsou:
národní sebeurčení a význam národa vůbec jako nezbytného základu státu
potřeba národní jednoty
vymezení se vůči kapitalismu
preference zájmů celku (národa) nad zájmy jednotlivce (občana)
touha po "lidovém životě"
Seznam národně-socialistických stran
Národně socialistické strany, které existovaly před nacismem
Česká strana národně sociální, Rakousko-Uhersko, 1897, zakládající Alois Simonides a Josef Klečák.
Národně socialistická strana, Francie, 1903, zakladatel Pierre Biétry.
Sociálnědemokratická strana Slováků v Uhersku, 1905
Národně socialistická strana, Filipíny, 1930, zakladatel Emilio Aguinaldo.
Strany, které ovlivnily německý nacismus
Rakouský národní socialismus, Rakousko, 1903, zakladatel Franko Stein
Dělnická strana Německa, Německo, 1919, zakladatelé Anton Drexler, Karl Harrer, Gottfried Feder; předchůdkyně NSDAP s nevyjasněným programem
Strany, které mají blízko k nacistické ideologií
Bulharská strana národně socialistická, Bulharsko, 1920, zakladatel Hristo Kunčev.
Dánská národněsocialistická dělnická strana, Dánsko, 1930, zakladatel Cay Lembcke.
Dělnická strana, Česko, 2003, zakladatel Tomáš Vandas.
Dělnická strana sociální spravedlnosti, nástupnická po DS, Česko
Íránská strana národně socialistická, 1952, zakladatel Davud Monshizadeh.
Kotleba – Ľudová strana Naše Slovensko, Slovensko, 2010
Liberální národně socialistická strana zelených, Spojené státy, zakládající Craig Smith a Robert Lindstrom.
Maďarská strana národně socialistická, 1920, zakladatel Zoltán Böszörmény.
Modré Mongolsko
Moravská nacionální sociální strana, 1940, zakladatel Josef Miroslav Tichý
Národně socialistická asociace Tchaj-wanu, Tchaj-wan, 2006
Národní shromáždění, Norsko, 1933, zakládající Vidkun Quisling a Johan Bernhard.
Nacionální socialistická strana akce, Velká Británie, 1980, zakladatel Tony Malski.
Nacionálně socialistická česká dělnická a rolnická strana - Strana zeleného hákového kříže, Protektorát Čechy a Morava, po 15. březnu 1939,zakladatel František Mikuláš Mlčoch
Nacionálně socialistická nizozemská dělnická strana, Nizozemsko, 1941, zakladatel Ernst Herman van Rappard.
Nacionálně socialistická německá dělnická strana, 1920, zakladatel Adolf Hitler.
Nacionálně socialistická japonská dělnická a sociální strana, Japonsko, 1982.
Národní socialistická liga, Velká Británie, 1937, zakládající William Joyce, John Beckett a John Angus MacNab.
Národně socialistická liga, Spojené státy, 1974, zakladatel Russell VEH.
Národně socialistická strana Ameriky, Spojené státy, 1970, zakladatel Frank Collin.
Národní socialistická strana Nového Zélandu, Nový Zéland, 1969, zakladatel Colin King-Ansell.
Národně socialistická lidová strana Švédska, Švédsko, 1926, zakladatel Konrad Hallgren.
Národně socialistická bílá lidová strana, Spojené státy, 1966, zakladatel Harold Covington.
Národně socialistické dělnická strana, Španělsko.
Národně socialistické dělnická strana, Švédsko, 1933, zakladatel Sven Olof Lindholm.
Německá národně socialistická strana dělnická, Československo, 1919, zakládající Hans Knirsch, Hans Krebs, Adam Fahrner a Josef Patzel.
Ruská národně socialistická strana, Rusko, 1992, zakladatel Konstantin Kasimovský.
Ruská národní jednota, Rusko, 1990, zakladatel Alexander Barkašov.
Řecká národně socialistická strana, Řecko, 1932, zakladatel George S. Mercouris.
Říšská socialistická strana, Německo, 1949, zakladatel Fritz Dorls.
Strana národní jednoty, Kanada, 1934, zakladatel Adrien Arcand.
Sudetoněmecká strana, Československo, 1933, zakladatel Konrad Henlein .
Švédská národní socialistické strana rolníků a dělníků, Švédsko, 1924, zakladatel Birger Furugård.
Tsagaan Khass (Bílá Svastika), Mongolsko
Současné národně socialistické strany nesouvisející s nacismem
Národní strana práce, 1938
Baas, 1940, zakladatel Michel Aflak
Česká strana národně socialistická, 2005, zakladatel Jiří Stanislav
Rovnost a usmíření, 2007 , zakladatel Alain Soral
Eusko Alkartasuna, Baskicko - Španělsko, 1986
LEV 21 – Národní socialisté, 2011, zakladatel Jiří Paroubek
Národní revoluční socialistická strana, Indie, 1977.
Národní socialistická rada Nagalandu, Indie, 1980, zakládající Isak Chisi Swu a Thuingaleng Muivah.
Národní socialistická strana Tripura, Indie, 2003, zakladatel Hirendra Tripura.
Národní socialistická strana (Jatiyo Samajtantrik Dal), Bangladéš, 1972, zakladatel Abdul Jalil.
Sinn Féin, Irsko, 1905
Sociálně demokratická a dělnická strana, Severní Irsko, 1970
Strana národně socialistická, Jordánsko.
Strana práv občanů, 2009
Valencijské společenství, Esquerra Valenciana - Španělsko, 1998
VIZE – národní socialisté, 2020, zakladatel Jan Vondrouš
SPD Tomia Okamury - Hnutí má výrazné prvky národně socialistické, byť to otevřeně nezmiňují
Strany ovlivněné myšlenkami Otto Strassera
Černá fronta-strana revolučních národních socialistů,1930, zakladatel Otto Strasser,významný člen také Helmut Hirsch - strana která měla ekonomicky blíže k současným demokratickým socialistům
Odkazy
Reference
Literatura
Ottův slovník naučný nové doby, heslo Socialismus. Sv. 11, str. 48.
K. Žaloudek, Encyklopedie politiky. Praha: Libri 2004.
Související články
Nacismus
Externí odkazy
Politické ideologie
Nacionalismus
Socialismus
Krajní levice
Krajní pravice | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,638 |
Is Twitter just Noise or a Valid Business Tool?
This is a question that people ask me all the time. What does twitter have to do with business?
Isn't it just a waste of time? People thing it's just a bunch of self-obsessed people talking about what they are doing at that moment.
I'm here to tell you that Twitter is the real deal for businesses. As a marketer it is an awesome tool when used right.
The following post is a good explanation of how to approach twitter for business.
As with any successful marketing effort it takes strategy and planning. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,532 |
\section{Introduction}
Spin-orbit coupling (SOC) interwines the motional degrees of freedom of a system with its spin part, giving rise to many intriguing phenomena such as the atomic fine structure, the spin Hall effect~\cite{Kato2004, Konig2007} and topological insulators~\cite{Hasan2010}. In ultracold atom experiments, SOC has been realized using Raman laser dressing techniques~\cite{Lin2011, Goldman2013e, Zhai2015a}, where a two-photon Raman transition couples two different spin-momentum states. This optical method was successfully applied to many fermionic atom systems~\cite{Wang2012, Cheuk2012b, Song2016, Burdick2016} and recently extended to two-dimensions~\cite{Huang2016}, boosting the interest in exploring new exotic SOC-driven many-body phenomena~\cite{Zhai2015a,Cao2014, Xu2014}.
Alkaline-earth-like atoms with two valence electrons such as ytterbium and strontium provide a beneficial setting for studying SOC physics. Their transition linewidth is narrow in comparison to the hyperfine structure splitting, which is helpful to alleviate the unavoidable heating effect due to light-induced spontaneous scattering under the Raman dressing~\cite{Zhai2015a, Song2016} and also to generate spin-dependent optical coupling to the hyperfine ground states. Furthermore, as recently demonstrated with $^{173}$Yb atoms~\cite{Pagano2015,Hofer2015}, the interorbital interactions between the $^1S_0$ and $^3P_0$ states can be tuned via a so-called orbital Feshbach resonance~\cite{Zhang15}, which would broaden the research scope of the SOC physics with alkaline-earth-like atoms.
In this paper, we present momentum-resolved Raman spectra of a spin-polarized degenerate Fermi gas of $^{173}$Yb atoms, which are measured in the Raman laser configuration of the conventional SOC scheme. In particular, we measure the Raman spectra over a wide range of magnetic fields as well as laser intensities to investigate the interplay of multiple Raman transitions in the SOC scheme. We observe that two Raman transitions become simultaneously resonant at a certain magnetic field and a doublet structure develops in the spectrum for strong Raman laser intensities. We find that the spectral splitting at the double resonance is quantitatively accounted for by the Autler--Townes doublet effect~\cite{Autler1955}.
In the conventional SOC scheme, since one of the Raman laser beams has both $\sigma^+$ and $\sigma^-$ polarization components with respect to the quantization axis defined by the magnetic field, the Raman transition from one spin state to another, if any, can be made to impart momentum in either direction along the relative Raman beam propagation axis. In typical SOC experiments, the system parameters are set to make one of the transitions energetically unfavorable such that it can be ignored, but the double resonance observed in this work results from involving both of the Raman transitions. When all the Raman transitions are taken into account, the effect of the Raman laser fields is represented by a spatially oscillating effective magnetic field~\cite{Jimenez-Garcia2012}. We show that our measurement results are consistent with the spinful band structure of the Fermi gas under the effective magnetic field.
The paper is organized as follows. In Sec.~II, we describe our experimental apparatus and procedures for sample preparation and Raman spectroscopy. In Sec.~III, we present the Raman spectra measured for various conditions and the observation of the spectral doublet splitting at the double resonance. In Sec.~IV, we discuss the results in the perspective of the spinful band structure of the SO-coupled Fermi gas. Finally, a summary and outlooks are provided in Sec. V.
\section{Experiments}
Figure~1(a) shows the schematic diagram of our experimental apparatus for generating a degenerate Fermi gas of $^{173}$Yb atoms~\cite{Yb_machine}. We first collect ytterbium atoms with a Zeeman slower and a magneto optical trap (MOT). For the slowing light, we use a 399~nm laser beam that has a dark spot at its center to suppress the detrimental scattering effect on atoms in the MOT. The frequency modulation method is adopted for the 556~nm MOT beams to increase the trapping volume and capture velocity of the MOT~\cite{Fukuhara2007a}. As a result, more than $10^8$ atoms are collected in the MOT within 15~s. We transfer the atoms into an optical dipole trap (ODT) formed by a focused 1070~nm laser beam, where the transfer efficiency is $\approx 13\%$. Then, we transport the atoms by moving the ODT to a small appendant chamber which provides better optical access and allows high magnetic field application, and we generate a crossed ODT by superposing a focused 532~nm laser beam horizontally with the 1070~nm ODT.
After evaporation cooling, we obtain a quantum degenerate sample in the $F=5/2$ hyperfine ground state. For an equal mixture of the six spin components, the total atom number is $N\approx 1.0\times 10^5$ and the temperature is $T/T_F \approx 0.1$, where $T_F$ is the Fermi temperature of the trapped sample. The spin composition of the sample can be manipulated during evaporative cooling with optical pumping or removal of spin states by resonant light. For the case of a fully spin-polarized sample in the $m_F=-5/2$ state, $N\approx 1.2\times10^5$ and $T/T_F \approx 0.35$. The trapping frequencies of the crossed ODT are $(\omega_r, \omega_z) = 2\pi\times(52, 450)$~Hz at the end of the sample preparation.
\begin{figure}
\includegraphics[width=8.2cm]{figure1}
\caption{(a) Schematic diagram of the experimental apparatus. $^{173}$Yb atoms are collected with a Zeeman slower and a magneto-optical trap (MOT), loaded into a 1070~nm optical dipole trap (ODT), and transported to a small hexagonal chamber by moving the ODT. (b, c) Images of atoms after optical Stern-Gerlach spin separation~\cite{Taie2010, Stellmer2011}: equal mixture of the six spin components of the $F=5/2$ hyperfine ground state (b) and fully polarized sample in the $m_F=-5/2$ spin state (c).
}
\label{fig:sample_preparation}
\end{figure}
The setup for Raman spectroscopy is illustrated in Fig.~2(a). A pair of counter-propagating laser beams are irradiated on the sample in the $x$ direction and an external magnetic field $B$ is applied in the $z$ direction. The two laser beams are linearly polarized in the $y$ and $z$ directions, respectively. With respect to the quantization axis defined by the magnetic field in the $z$ direction, Raman beam 1 with linear $y$ polarization has both $\sigma^+$ and $\sigma^-$ components and Raman beam 2 with linear $z$ polarization has a $\pi$ component. Thus, a two-photon Raman process, e.g., imparting momentum of $+2\hbar k_R \hat{x}$ by absorbing a photon from Raman beam 1 and emitting a photon into Raman beam 2 changes the spin number by either $+1$ or $-1$, where $k_R$ is the wavenumber of the Raman beams. This is the conventional Raman laser configuration for SOC in cold atom experiments~\cite{Lin2011, Wang2012, Cheuk2012b, Burdick2016}.
The Raman lasers are blue-detuned by 1.97~GHz from the $|^1S_0, F = 5/2\rangle$ to $|^3P_1, F'=7/2\rangle$ transition~[Fig.~2(b)]. This laser detuning, set between the hyperfine states of $^3P_1$, is beneficial to induce spin-dependent transition strengths for the $F=5/2$ hyperfine spin states~\cite{Mancini2015, Mancini2015a}. The frequency difference of the two Raman beams is denoted by $\delta \omega$ [Fig.~2(a)]. The two beams are set to the same power $P$ and focused onto the sample. Their $1/e^2$ intensity radii are $\approx 150~\mu$m, which is much larger than the trapped sample size of 30~$\mu$m. We assume that the laser intensities are uniform over the sample.
\begin{figure}
\includegraphics[width=8.2cm]{figure2}
\caption{Raman spectroscopy of a degenerate Fermi gas (DFG). (a) Raman coupling setup with a pair of counter-propagating laser beams, whose frequency difference is denoted by $\delta\omega$. (b) Energy diagram of the $^3P_1$ state of $^{173}$Yb and the relative detuning of the Raman laser. (c, d) Examplary time-of-flight images of Fermi gases after applying a pulse of the Raman beams for $\delta \omega/2\pi = 14.8$~kHz (c) and $29.6$~kHz (d). The vertical dashed lines indicate the center of the unperturbed sample. (e, f) 1D momentum distributions $n(k_x)$ of the samples (yellow dashed curve) obtained by integrating the images along the $y$ direction. The normalized Raman spectra $S_R(k_x)$ (red solid curve) are measured as $S_R(k_x)=[n(k_x)-n_\textrm{ref}(k_x)]/n_\textrm{ref}(0)$, where $n_\textrm{ref}(k_x)$ is the reference distribution (blue dash-dotted curve) obtained without applying the Raman beam pulse.
}
\label{fig:raman_setup}
\end{figure}
Raman spectroscopy is performed by applying a pulse of the Raman beams and taking a time-of-flight absorption image of the sample. The image is taken at $B=0$~G along the $z$-axis with a linearly polarized probe beam resonant to the $^1S_0\rightarrow {^1}P_1$ transition. Two exemplary images are shown in Fig.~2(c) and 2(d), showing that atoms are scattered out of the original sample with different momenta for different $\delta\omega$. Since the expansion time $\tau$ is sufficiently long such that $\omega_r \tau \approx 5$, we interpret the time-of-flight image as the momentum distribution of the atoms. The 1D momentum distribution $n(k_x)$ is obtained by integrating the image along the $y$ direction [Fig.~2(e) and 2(f)], where $k_x=mx/(\hbar \tau)$ with $m$ being the atomic mass and $x$ the displacement from the center of mass of an unperturbed sample. In our imaging, the absorption coefficient for each spin state was found to vary slightly, within $\approx 10$\%~[Fig.~1(b)], which we ignored in the determination of $n(k_x)$.
The normalized Raman spectrum is measured as
\begin{equation}
S_R(k_x) = [n(k_x) - n_\mathrm{ref}(k_x)]/n_\mathrm{ref}(0),
\end{equation}
where $n_\mathrm{ref}$ is the reference distribution obtained without applying the Raman beams. In the spectrum, a momentum-imparting Raman transition appears as a pair of dip and peak, which correspond to the initial and final momenta of the transition, respectively. We observe that the spectral peaks and dips exhibit slightly asymmetric shapes, which we attribute to elastic collisions of atoms during the time-of-flight expansion~\cite{Veeravalli2008}. The Fermi momentum of the sample is $k_F/k_R\approx 1.2$ in units of the recoil momentum.
\section{Results}
The atomic state in an ideal Fermi gas is specified by wavenumber $k$ and spin number $m_F$, and its energy level is given by
\begin{equation} \label{eq:energy}
E(|k,m_F\rangle) = \frac{\hbar^2 k^2}{2m} + g_F \mu_B m_F B + E_S(m_F).
\end{equation}
The first term is the kinetic energy of the atom and the second term is the Zeeman energy due to the external magnetic field $B$, where $g_F$ is the Land\'{e} $g$-factor and $\mu_B$ is the Bohr magneton. The last term $E_S$ denotes the spin-dependent ac Stark shift induced by the Raman lasers. For a Raman transition from $|k_i, m_i\rangle$ to $|k_f=k_i+2r k_R, m_f=m_i+\Delta m_F\rangle$, which changes the momentum by $2 r \hbar k_R$ and the spin number by $\Delta m_F$, the energy conservation requires $E(|k_f,m_f\rangle)-E(|k_i,m_i\rangle)=r\hbar \delta \omega$, which gives the resonance condition for the initial wavenumber $k_i$ as
\begin{equation} \label{eq:resonance}
k_i = k_R \Big[ \frac{\hbar \delta\omega}{4E_R} - \frac{\Delta m_F}{r} \frac{B}{4B_R}-\frac{1}{r}\frac{\Delta E_S}{4E_R} - r \Big],
\end{equation}
where $E_R = (\hbar k_R)^2/2m = h \times 3.7$~kHz is the atomic recoil energy, $B_R=E_R/(g_F \mu_B)=17.9$~G and $\Delta E_S=E_S(m_f)-E_S(m_i)$. Here we neglect the quadractic Zeeman effect and the atomic interactions which are negligible in our experimental conditions.
\begin{figure}
\includegraphics[width=8.4cm]{figure3}
\caption{Raman spectra measured by scanning various experimental parameters including Raman beam pulse duration $t$, frequency difference $\delta \omega$, magnetic field $B$, and Raman beam power $P$: (a) $\delta\omega = 4E_R/\hbar$, $B=~16.6$~G, $P=0.47$~mW; (b) $t=2$~ms, $B=0$~G, $P=1.1$~mW; (c) $t=2$~ms, $\delta \omega= 13.4 E_R/\hbar$, $P=0.21$~mW; (d) $t=2$~ms, $\delta \omega = 13.4 E_R/\hbar$, $B=133$~G (see the text for details of the sample condition and the polarization configuration of the Raman beams). The dashed lines in (b) indicate $k=k_R(\frac{\hbar}{4E_R}\delta \omega -n)$ for $n=-2,-1,1$,and 2, and those in (c) and (d) are guides for the eyes having slopes of $\frac{dk}{d B}=-\frac{k_R}{4B_R}$ and $\frac{dk}{d P}=-0.3 k_R/$mW. Each spectrum was obtained by averaging more than three measurements.
}
\label{fig:Raman_Rabi}
\end{figure}
\begin{figure*}
\includegraphics[width=17.0cm]{figure4}
\caption{Double resonance of Raman transitions. (a--c) Raman spectra of an $m_F=-5/2$ spin-polarized sample as a function of the magnetic field $B$ for $\delta \omega =8 E_R/\hbar$ and various Raman beam powers (a) $P = 0.13$~mW, (b) 0.21~mW, and (c) 0.36~mW. As the Raman coupling strength increases with higher $P$, a spectral doublet splitting develops at $B=4 B_R\approx 72$~G where the $(r,\Delta m_F)=(1, 1)$ and $(2, 0)$ transitions are doubly resonant. The spectrum in (d) is the same of (c) with the guide lines (solid) indicating the resonant momentum positions for various Raman transitions, which are calculated from Eq.~(3) without including the ac Stark shift. The dashed lines are the corresponding final momentum positions. (e--h) Diagrams of the Raman transitions observed in the spectra.
}
\label{fig:highorder}
\end{figure*}
We first investigate the resonance condition of Eq.~(3) by measuring its dependence on various experimental parameters. Figure 3(a) shows a Raman spectrum measured by scanning the Raman beam pulse duration for $\delta \omega =4 E_R/\hbar$ at $B=16.6$~G. Spin-polarized samples were used and both of the Raman beams were set to linear $z$ polarization to make sure $\Delta m_F=0$. Momentum-dependent Rabi oscillations are clearly observed and the Rabi frequency is found to be well described by $\Omega(k) = \sqrt {\Omega_0^2 + (\hbar k_R k /m)^2 }$ with $\Omega_0 \approx 2\pi\times7$~kHz. The decoherence time is measured to be $\approx 1$~ms, which seems to be understandable with the characteristic time scale for momentum dephasing in the trap, $\pi/(2\omega_r) \approx 5$~ms. In the following, we set the pulse duration of the Raman beam to 2~ms, which is long enough to study the steady state of the system under the Raman laser dressing.
Figure~3(b) displays a spectrum of the equal mixture sample in the plane of wavenumber $k$ and frequency difference $\delta \omega$. Here, $B=0$~G and the Zeeman effect is absent in the measurement. The $r=1$ and $r=2$ transitions are identified in the spectrum with their spectral slope of $\frac{d k}{d \delta\omega}= \frac{\hbar k_R}{4E_R}$ and different offsets as predicted by Eq.~(3). The $(k,\delta \omega)\leftrightarrow (-k,-\delta \omega)$ symmetry of the spectrum indicates that the differential ac Stark shift is negligible in the measurement.
Figure~3(c) shows the Raman spectrum of the $m_F=-5/2$ spin-polarized sample over a range of magnetic fields from $B=100$~G to 195~G for $\delta \omega = 13.4 E_R/\hbar$. In the spectral plane of $k$ and $B$, the Raman transition with $(r,\Delta m_F) = (1, 1)$ appears as a line having the slope $\frac{dk}{dB}=-\frac{k_R}{4 B_R}$ as expected from Eq.~(3). A linear spectral shift is observed with increasing Raman beam power $P$~[Fig.~3(d)], which demonstrates the effect of the differential ac Stark shift $\Delta E_S$. In our experiment, $\Delta E_S = E_S(-3/2)-E_S(-5/2)\approx 1.2~E_R$ for $P = 1$~mW. This is in good a agreement with the Raman beam intensities estimated from the Rabi oscillation frequency $\Omega_+ \propto \sqrt{I_\sigma I_{\pi}}$, where $I_{\sigma,\pi}$ are the intensities of the Raman beam 1 and 2, respectively. The comparison of $\Delta E_s$ and $\Omega_+$ suggests $I_{\pi} = 0.6I_{\sigma}$, which we attribute to a slight mismatch of the beam waists.
Next we investigate a situation where one spin-momentum state is resonantly coupled to two final states simultaneously, which we refer to as a double resonance. When the two corresponding Raman processes are characterized with $(r_1, \Delta m_{F1})$ and $(r_2, \Delta m_{F2})$, we see from Eq.~(\ref{eq:resonance}), neglecting the small $\Delta E_S$ term, that the double resonance occurs when
\begin{equation} \label{eq:double}
\frac{B}{4 B_R} \frac{\Delta m_{F1}}{r_1} + r_1 = \frac{B}{4 B_R} \frac{\Delta m_{F2}}{r_2} + r_2.
\end{equation}
For the primary transition with $(r_1,\Delta m_{F1})=(1,1)$, the double resonance condition is satisfied at $B= \frac{4 r_2(r_2-1)}{r_2-\Delta m_{F2}} B_R$.
To observe the double resonance of the $(r,\Delta m_F)=(1,1)$ and $(2,0)$ transitions at $B=4B_R\approx 72$~G, we measure the Raman spectra of the spin-polarized sample in the $k$-$B$ plane over a range from $B=0$~G to 140~G [Fig.~4]. Here we set $\delta \omega =8 E_R /\hbar$ to have $k_x=0$ atoms on resonance for the $(2,0)$ transition, which is insensitive to $B$ for $\Delta m_F=0$. For low $P$, the $(1,1)$ transition appears with the spectral slope of $-\frac{k_R}{4B_R}$ as observed in Fig.~3(c) and the double resonance is indicated by a small signal at $(k,B)=(4 k_R, 4 B_R)$~[Fig.~4(a)]. This is understood as enhancement of the second-order Raman transition from $|k=0,-5/2\rangle$ to $|k=4 k_R, -5/2\rangle$ due to its intermediate state $|k=2k_R, -3/2\rangle$ being resonant. When the Raman beam power increases, we observe development of a spectral splitting at the resonance~[Figs.~4(b) and 4(c)]. The overall pattern of the high-$P$ spectrum shows the avoided crossing of the spectral lines corresponding to the two $(1,1)$ and $(2,0)$ transitions.
Near the double resonance, the system can be considered as a three-level system consisting of $|k,-5/2\rangle$, $|k+2k_R, -3/2\rangle$ and $|k+4k_R, -5/2\rangle$~[Fig.~5(a)]. For simplicity, we denote them by $|0\rangle$, $|1\rangle$, and $|2\rangle$, respectively. Since the Raman transition between $|0\rangle$ and $|1\rangle$ involves the $\sigma^+$ component of Raman beam 1 but that between $|1\rangle$ and $|2\rangle$ involves the $\sigma^-$ component, the coupling strengths $\Omega_+$ and $\Omega_-$ of the two transitions, respectively, can be different. In our case with $^{173}$Yb atoms in the $m_F=-5/2$ state, $\Omega_-=5.3~\Omega_+$. Since the coupling between $|1\rangle$ and $|2\rangle$ are much stronger than that between $|0\rangle$ and $|1\rangle$, the observed spectral splitting with increasing Raman beam intensity can be described as an Autler--Townes doublet~\cite{Autler1955}: two dressed states $|\alpha\rangle$ and $|\beta\rangle$ are formed with $|1\rangle$ and $|2\rangle$ under the strong coupling and their energy level splitting is probed via Raman transitions from the initial $|0\rangle$ state. In the rotating wave approximation, the energy levels of the two dressed states are given by $E_{\alpha, \beta} = \frac{1}{2}[E_1 + E_2 -3\hbar\delta\omega \pm \sqrt{(E_1 - E_2 + \hbar\delta\omega)^2 + (\hbar\Omega_-)^2}]$, where $E_{1,2}=E(|1,2\rangle)$. The resonant wavenumbers $k_{\alpha,\beta}$ of the initial state $|0\rangle$ are determined from $E(|0\rangle)=E_{\alpha,\beta}$ and for $\delta\omega=8E_R/\hbar$ and $B=4B_R$, we obtain $k_{\alpha,\beta}=\pm \frac{k_R}{8\sqrt{2}}\frac{\hbar \Omega_-}{E_R}$. We find our measurement results on the double resonance at $B=4B_R$ in good quantitative agreement with the estimation [Fig.~6(b)]. The coupling strength $\Omega_-$ was separately measured from the Rabi oscillation data of the $|0,-5/2\rangle \rightarrow |-2k_R,-3/2\rangle$ transition for $\delta \omega= -13.4 E_R/\hbar$ at $B=166$~G.
\begin{figure}
\includegraphics[width=8.0cm]{figure5}
\caption{Spectral splitting at double resonance. (a) Three atomic states involved in the double resonance at $B = 4B_R$. For a $^{173}$Yb atom in the $m_F=-5/2$ state, $\Omega_- = 5.3~\Omega_+$ and the upper two states are more strongly coupled.
(b) Raman spectrum for $\delta \omega = 8E_R/\hbar$ and $B= 4B_R$ as a function of the Raman coupling strength $\Omega_-$. The dashed lines are the theoretical prediction of $k_{\alpha,\beta}=\pm \frac{k_R}{8\sqrt{2}}\frac{\hbar \Omega_-}{E_R}$, which is calculated in the limit of $\Omega_+/\Omega_- \rightarrow 0$ (see text).
}
\label{fig:gap}
\end{figure}
The Raman spectra in Fig.~4 reveal another double resonance at $B=\frac{4}{3} B_R\approx 24$~G, where the $(r,\Delta m_F)=(2,0)$ line crosses the $(r,\Delta m_F)=(1,3)$ line. Although the $(1,3)$ transition is a third-order Raman transition, its spectral strength is observed to be higher than that of the (2,0) transition. In the intermediate region of $B\approx 35$~G, many Raman transitions are involved over the whole momentum space of the sample and the spectral structure for high Raman laser intensity shows interesting features which cannot be simply explained as crossing and avoided crossing of the spectral lines. It might be necessary to take into account the ac Stark shift effect and a further quantitative analysis of the Raman spectra will be discussed in future work.
\begin{figure}
\includegraphics[width=8.0cm]{figure6}
\caption{Energy band structures of a SO-coupled spin-1/2 atom under the Raman laser dressing: $\Omega_-=5.3~\Omega_+$ in (a--c) and $\Omega_-=\Omega_+$ in (d--e), and $\hbar\delta = 3 E_R$ in (a, d), $4 E_R$ in (b, e), and $5 E_R$ in (c, f). The color of the solid lines indicates the bare spin fractions of the energy eigenstates: blue for spin-up and red for spin-down. The dashed lines represent the energy spectrum for $\Omega_\pm=0$. The gray shadow areas in (a--c) indicate the region corresponding to the momentum states occupied by the sample in the experiment.
}
\label{fig:band}
\end{figure}
\section{Discussion}
In the Raman laser dressing scheme described in Fig.~2(a), two ways of couplings are allowed between the two spin states because the Raman beam that has linear polarization orthogonal to the magnetic field contains both $\sigma^+$ and $\sigma^-$ components. This means that a Raman transition from one spin state to the other spin state can occur while imparting momentum in either of the $x$ and $-x$ directions. In typical experimental conditions~\cite{Zhai2015a, Lin2011, Cheuk2012b, Wang2012}, one of the couplings is resonantly dominant over the other, giving rise to a form of SOC that has equal strengths of the Rashba and Dresselhaus contributions. However, when the Fermi sea of a sample covers a large momentum space, this approximation cannot be applied and it is necessary to include both of the Raman couplings for the full description of the system. Furthermore, as observed in the previous section, the two Raman couplings can be doubly resonant and play cooperative roles in the SOC physics of the system.
As an archetypal situation, we consider a spin-1/2 atom under the Raman dressing for $\delta \omega=0$. Here, the counterpropagating Raman beams form a stationary polarization lattice with spatial periodicity of $\pi/k_R$. Including all the allowed Raman transitions, the effective Hamiltonian of the system is given by
\begin{eqnarray}\label{eq:SOHamiltonian}
H&=&
\begin{bmatrix}
\frac{\hbar^2 k^2}{2m} + \frac{\hbar \delta}{2} & \frac{\hbar \Omega_+ }{2} e^{i2k_R x} + \frac{\hbar \Omega_- }{2} e^{-i2k_R x} \\
\frac{\hbar \Omega_+}{2} e^{-i 2k_R x} + \frac{\hbar \Omega_-}{2} e^{i 2k_R x} & \frac{\hbar^2 k^2 }{2m} -\frac{\hbar \delta}{2}
\end{bmatrix} \nonumber \\
&=&
\frac{\hbar^2 k^2}{2m} + \frac{\hbar\delta}{2} \sigma_z \nonumber \\
&&~~+ \frac{\hbar}{2}\Omega_x\cos(2k_R x)\sigma_x + \frac{\hbar}{2}\Omega_y \sin(2k_R x)\sigma_y,
\end{eqnarray}
where $\hbar \delta$ is the sum of the differential Zeeman and ac Stark shifts, $\sigma_i$ are the $2\times2$ Pauli matrices, and $\Omega_{x,y} = \Omega_{+}\pm\Omega_{-}$. The final form of $H$ shows that the Raman dressing is equivalent to an effective magnetic field ${\cal{B}}=(\delta,~\Omega_x \cos (2k_R x),~\Omega_y \sin (2k_R x))$, which has two parts: a bias field along the $z$ axis and a spatially oscillating field on the $x$-$y$ plane. Its chirality is determined by the sign of $\Omega_x \Omega_y = \Omega_+^2 - \Omega_-^2$. In the presence of the spatially oscillating magnetic field, the energy dispersion of the atom has a spinful band structure [Fig.~6].
Figure 6(b) displays a band structure for $\Omega_-=5.3~\Omega_+$ and $\delta=4E_R/\hbar$, which straightforwardly explains the observed spectral splitting at the double resonance. In the experiment, $\delta \omega=8E_R/\hbar$ and the polarization lattice of the Raman beams moves in the lab frame with velocity of $+2\hbar k_R /m \hat{x}$. Initially, the atoms in the trapped sample occupy the low quasi-momentum region of the second and third bands of the bare spin-down state, which is indicated by a gray region in Fig.~6(b), and they are projected to the eigenstates of the spinful band structure via the Raman spectroscopy process. The quasi-momentum separation between the gap opening positions, which is marked with $\Delta k$ in Fig.~6(b), is the spectral splitting observed in our Raman spectrum. We note that $\Delta k=0$ in the symmetric case of $\Omega_-=\Omega_+$ [Fig.~6(e)] and the spectral splitting would not occur in the Raman spectrum.
\section{Summary and outlook}
We have measured the Raman spectra of a spin-polarized degenerate Fermi gas of $^{173}$Yb atoms in the conventional SOC scheme and investigated the double resonance of Raman transitions. We observed the development of a spectral splitting at the double resonance of the $(r,\Delta m_F)=(1,1)$ and $(2,0)$ transitions and provided its quantiative explanation as the Autler--Townes doublet effect. Finally, we discussed our results in the context of the spinful energy band structure under the Raman laser dressing.
In general, when the system has multiple SOC paths in its spin-momentum space, a spinful energy band structure is formed because of the periodicity imposed by them. In previous experiments~\cite{Jimenez-Garcia2012, Cheuk2012b}, spinful band structures were designed and demonstrated by applying a RF field to the SO-coupled systems under the Raman laser dressing, where the role of the RF field was to open an additional coupling path between the two spin states. The results in this work highlight that the conventional Raman laser dressing scheme provides two ways of SOC and intrinsically generates a spinful band structure without the aid of an additional RF field. An interesting extention of this work is to investigate the magnetic ordering and properties of a Fermi gas in the spatially rotating magnetic field ${\cal{B}}$. In the $F=5/2$ $^{173}$Yb system, the chirality of ${\cal{B}}$ can be controlled to some extent by the choice of the two spin states that are coupled by the Raman lasers. If the $m_F = \pm1/2$ states are employed, $\Omega_y=0$ and ${\cal{B}}$ changes from an axial field to a alternating transverse field as a function of $\hbar\delta$. In particular, when $\hbar\delta=0$, ${\cal{B}}=0$ points are periodically placed, which might profoundly affect the magnetic properties of the system. It was discussed in Ref.~\cite{Cheuk2012b} to engineer a flat spinful band structure, which might be pursued via proper tuning of the parameters of our system.
\section{Acknowledgments}
This work was supported by IBS-R009-D1 and the National Research Foundation of Korea (Grant No. 2014-H1A8A1021987).
| {
"redpajama_set_name": "RedPajamaArXiv"
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Q: Question about u term in graph-related posts I am studying the solutions posted by Marko Riedel at https://math.stackexchange.com/questions/689526/how-many-connected-graphs-over-v-vertices-and-e-edges. I am porting some of the techniques listed there to another language (Python). I have two problems that are keeping me from succeeding.
1) When the code examples make use of the variable u, where (and how) is that variable defined? I see where it appears as part of a generating function, but I don't yet understand why it appears there. My immediate concern is that in the coding snippets, I don't see it defined anywhere. What should I do if I am attempting to port code that references u, such as gf3, gf4, and gf5?
2) In the function qq (accompanying the code for function gf5), there is a line that reads like this:
for p from max(0, k-1/2*(m+1)*m) to k-m do
I have a question about the k-1/2*(m+1)*m part. In languages like Python, where integer arithmetic is performed such that 1/2 = 0), should this be coded as something like k-((m+1)*m/2)?
Thanks a lot. As is probably obvious, I'm diving into something that's a little deep in math for me, and appreciate any help.
| {
"redpajama_set_name": "RedPajamaStackExchange"
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Phthiracarus falciformis är en kvalsterart som beskrevs av Morell och Subías 1991. Phthiracarus falciformis ingår i släktet Phthiracarus och familjen Phthiracaridae. Inga underarter finns listade i Catalogue of Life.
Källor
Spindeldjur
falciformis | {
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Robin Run is a dammed headwater major tributary of the Delaware River with a drainage area of 22.69 square miles that is 1.69 miles north 1.69 miles north of Mill Creek's Confluence with the Neshaminy Creek on the border of Buckingham and Wrightstown Townships), The headwaters originate in Buckingham Township, Bucks County, Pennsylvania and the stream flows generally southeast to its confluence with Mill Creek in Wrightstown Township.
Major tributaries of the Upper and Middle Neshaminy Creek include the West and North Branch of Neshaminy Creek, Pine Run, Cooks Run, Mill Creek, Lahaska Creek, Robin Run, Watson Creek, and Newtown Creek, all of which flow into the Main Stem Neshaminy Creek.
The Geographic Name Information System I.D. is 1185219, U.S. Department of the Interior Geological Survey I.D. is 02598.
Previously, the Neshaminy Water Resources Authority oversaw the dam, however, Bucks County assumed control and oversight of the dam(s). Robin Run in Buckingham Township is one of six county dams, including the Newtown Creek in Newtown Township, Nockamixon Dam in Pine Run in Doylestown, and, Core Creek in Middletown Township. For Robin Run, a regression analysis correlating existing discharges with drainage area was developed prior to construction of the dam.
In 2009, Bucks County workers performed an unauthorized valve repair that flushed "tens of thousands" of healthy fish out of the reservoir to suffer and die on the banks downstream. Bucks County was fined by the PA DEP for the illegal action, however, the fine was unpaid, or was paid with the proceeds not used to correct the damage and replace the fish that were killed. As a result, the lake has not been re-stocked with fish and remains a depressing place to go fishing.
Course
Robin Run rises a short distance northwest of Lower Mountain Road in Buckingham Township, flowing southeast flowing through a small unnamed pond then through Robin Run Lake, a dammed reservoir built by Bucks County in 1971, then passing into Wrightstown Township, where it meets its confluence with Mill Creek at 1.55 river mile. There are no other significant tributaries.
Geology
Robin Run begins in the Lockatong Formation, a sedimentary rock layer deposited in Pennsylvania, New Jersey, and New York during the Triassic Period. Mineralogy includes argillite, some shale, limestone, and calcareous shale named after the Lockatong Creek in Hunterdon County, New Jersey.
In 2018, the United States Geological Survey was awarded a $60,000 grant to conduct a dam assessment and watershed study of Robin Run.
The Lockatong is defined as a light to dark gray, greenish-gray, and black very fine grained sandstone, silty argillite, and laminated mudstone. In New Jersey, the cyclic nature of the formation is noted with hornfels near diabase and basalt flows.
The Lockatong is often described as lake or litoral sediments. The interfingering nature of the sediments with the surrounding Stockton Formation and Passaic Formation suggests that these littoral environments shifted as climate or as the dynamic terrane of the area developed. The deposition of calcitic sediments is indicative of a climate with high evaporation rates.
Appalachian Highlands Division
Piedmont Province
Gettysburg-Newark Lowland Section
Lockatong Formation
Brunswick Formation
It very quickly finds itself in the Brunswick Formation, a sedimentary rock layer deposited during the Jurassic and Triassic. Mineralogy includes shale, mudstone, siltstone, argillite, some hornfel.
Municipalities
Wrightstown Township
Buckingham Township
Crossings and Bridges
See also
List of rivers of Pennsylvania
List of rivers of the United States
List of Delaware River tributaries
References
Rivers of Pennsylvania
Rivers of Bucks County, Pennsylvania
Tributaries of the Neshaminy Creek | {
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People (23) Apply People filter
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Displaying 1 - 25 of 103 items.
David Starr Jordan (1851-1931)
David Starr Jordan studied fish and promoted eugenics in the US during the late nineteenth and early twentieth centuries. In his work, he embraced Charles Darwin s theory of evolution and described the importance of embryology in tracing phylogenic relationships. In 1891, he became the president of Stanford University in Stanford, California. Jordan condemned war and promoted conservationist causes for the California wilderness, and he advocated for the eugenic sterilization of thousands of Americans.
Subject: People, Reproduction
Abortion is the removal of the embryo or fetus from the womb, before birth can occur-either naturally or by induced labor. Prenatal development occurs in three stages: the zygote, or fertilized egg; the embryo, from post-conception to eight weeks; and the fetus, from eight weeks after conception until the baby is born. After abortion, the infant does not and cannot live. Spontaneous abortion is the loss of the infant naturally or accidentally, without the will of the mother. It is more commonly referred to as miscarriage.
Subject: Processes, Ethics, Reproduction
Charles Benedict Davenport (1866-1944)
Charles Benedict Davenport was an early twentieth-century experimental zoologist. Davenport founded both the Station for Experimental Evolution and the Eugenics Record Office at Cold Spring Harbor in New York. Though he was a talented statistician and skilled scientist, Davenport's scientific achievements are eclipsed by his lasting legacy as the scientific leader of the eugenics movement in the US.
The Baby Doe Rules (1984)
The Baby Doe Rules represent the first attempt by the US government to directly intervene in treatment options for neonates born with congenital defects. The name of the rule comes from the controversial 1982 case of a Bloomington, Indiana infant Baby Doe, a name coined by the media. The Baby Doe Rules mandate that, as a requirement for federal funding, hospitals and physicians must provide maximal care to any impaired infant, unless select exceptions are met. If a physician or parent chooses to withhold full treatment when the exceptions are not met, they are liable for medical neglect.
Adolescent Family Life Act (1981)
The 1981 Adolescent Family Life Act, or AFLA, is a US federal law that provides federal funding to public and nonprofit private organizations to counsel adolescents to abstain from sex until marriage. AFLA was included under the Omnibus Reconciliation Act of 1981, which the US Congress signed into law that same year. Through the AFLA, the US Department of Health and Human Services, or HHS, funded a variety of sex education programs for adolescents to address the social and economic ramifications associated with pregnancy and childbirth among unmarried adolescents.
Subject: Legal, Outreach, Ethics, Reproduction
"Abstinence Education: Assessing the Accuracy and Effectiveness of Federally Funded Programs" (2008), by Government Accountability Office
On 23 April 2008, the US Government Accountability Office, or GAO, released a report titled, "Abstinence Education: Assessing the Accuracy and Effectiveness of Federally Funded Programs," hereafter "Abstinence Education," in which it investigated the scientific accuracy and effectiveness of abstinence-only education programs sanctioned by individual states and the US Department of Health and Human Services, or HHS. GAO is a government agency whose role is to examine the use of public funds, evaluate federal programs and activities, and provide nonpartisan support to the US Congress.
Subject: Legal, Reproduction, Publications
John Hunter (1728–1793)
John Hunter studied human reproductive anatomy, and in eighteenth century England, performed one of the earliest described cases of artificial insemination. Hunter dissected thousands of animals and human cadavers to study the structures and functions of organ systems. Much of his anatomical studies focused on the circulatory, digestive, and reproductive systems. He helped to describe the exchange of blood between pregnant women and their fetuses. Hunter also housed various natural collections, as well as thousands of preserved specimens from greater than thirty years of anatomy work.
Progestin: Synthetic Progesterone
Progestin is a synthetic form of progesterone, a naturally occurring hormone, which plays an important role in the female reproductive cycle. During the 1950s two types of progestin that were later used in birth control pills were created, norethindrone and norethynodrel. In 1951 Carl Djerassi developed norethindrone at Syntex, S.A. laboratories located in Mexico City, receiving a patent on 1 May 1956. In 1953 Frank Colton developed norethynodrel at G.D. Searle and Company laboratories located in Chicago, receiving a patent on 29 November 1955.
Ethics of Designer Babies
A designer baby is a baby genetically engineered in vitro for specially selected traits, which can vary from lowered disease-risk to gender selection. Before the advent of genetic engineering and in vitro fertilization (IVF), designer babies were primarily a science fiction concept. However, the rapid advancement of technology before and after the turn of the twenty-first century makes designer babies an increasingly real possibility.
Subject: Ethics, Reproduction
Intracytoplasmic Sperm Injection
Intracytoplasmic Sperm Injection (ICSI) is an assisted reproductive technique (ART) initially developed by Dr. Gianpiero D. Palermo in 1993 to treat male infertility. It is most commonly used in conjunction with in vitro fertilization (IVF) or a less commonly used technique called zygote intrafallopian transfer (ZIFT). In natural fertilization, the sperm must penetrate the surface of the female egg, or oocyte.
Recombinant Gonadotropins Used in Fertility Treatments
First manufactured in 1988 by Serono laboratories, recombinant gonadotropins are synthetic hormones that can stimulate egg production in women for use in fertility treatments. Recombinant gonadotropins are artificially created using recombinant DNA technology, a technology that joins together DNA from different organisms. In vertebrates, naturally-occurring gonadotropins regulate the growth and function of the gonads, known as testes in males and ovaries in females.
Preimplantation genetic diagnosis (PGD) involves testing for specific genetic conditions prior to the implantation of an embryo in the uterine wall. This form of genetic screening has been made possible by the growth of in-vitro fertilization (IVF) technology, which allows for the early stages of development to occur in a laboratory dish rather than in vivo. The purpose of PGD is to identify what are considered to be abnormal embryos in order to select the most desirable embryos for implantation.
Social Implications of Non-Invasive Blood Tests to Determine the Sex of Fetuses
By 2011, researchers in the US had established that non-invasive blood tests can accurately determine the gender of a human fetus as early as seven weeks after fertilization. Experts predicted that this ability may encourage the use of prenatal sex screening tests by women interested to know the gender of their fetuses. As more people begin to use non-invasive blood tests that accurately determine the sex of the fetus at 7 weeks, many ethical questions pertaining to regulation, the consequences of gender-imbalanced societies, and altered meanings of the parent-child relationship.
Subject: Reproduction, Ethics, Legal
William Smellie (1697–1763)
William Smellie helped to incorporate scientific medicine into the process of childbirth in eighteenth century Britain. As a male physician practicing in childbirth and female reproductive health (man-midwife), Smellie developed and taught procedures to treat breech fetuses, which occur when a fetus fails to rotate its head towards the birth canal during delivery. Throughout his career, Smellie compiled a wealth of information about female anatomy in his writings. He modified medical technology such as the obstetrical forceps, an instrument used to maneuver the fetus during childbirth.
Endoscopy is a medical procedure that enables the viewing and biopsy of, and surgery on, internal tissues and organs. Endoscopic examinations are characterized by the introduction of a tube containing a series of lenses into the body through either an incision in the skin or a natural opening or cavity. During the mid-twentieth century, photographer Lennart Nilsson used endoscopes to capture the now-familiar images of embryos and fetuses.
Paretta v. Medical Offices for Human Reproduction [Brief] (2003)
The court decided a child of in vitro fertilization born with cystic fibrosis does not have the right to sue for wrongful life even in the presence of demonstrable acts of medical negligence because to allow such a case would grant the IVF child rights not possessed by naturally born children. The decision in Paretta has not been publicly tested in other jurisdictions.
Evangelium Vitae (1995), by Pope John Paul II
The encyclical entitled "Evangelium Vitae," meaning "The Gospel of Life," was promulgated on 25 March 1995 by Pope John Paul II in Rome, Italy. The document was written to reiterate the view of the Roman Catholic Church on the value of life and to warn against violating the sanctity of life. The document focuses on right to life issues including abortion, birth control, and euthanasia, but also touches on other concepts relevant to embryology, such as contraception, in vitro fertilization, sterilization, embryonic stem cell research, and fetal experimentation.
Subject: Religion, Reproduction
"Annual Research Review: Prenatal Stress and the Origins of Psychopathology: An Evolutionary Perspective" (2011), by Vivette Glover
In 2011, fetal researcher Vivette Glover published "Annual Research Review: Prenatal Stress and the Origins of Psychopathology: An Evolutionary Perspective," hereafter, "Prenatal Stress and the Origins of Psychopathology," in the Journal of Child Psychology and Psychiatry. In that article, Glover explained how an evolutionary perspective may be useful in understanding the effects of fetal programming. Fetal programming is a hypothesis that attempts to explain how factors during pregnancy can affect fetuses after birth.
Subject: Theories, Reproduction, Disorders
Mizuko Kuyo
Mizuko Kuyo is a Japanese Buddhist ceremony that focuses on a deceased fetus or stillborn child. This ceremony was originally developed to honor Jizo, a god believed to be responsible for transporting dead fetuses or children to the other world. The practice has become more popular in the last half century due to the growing number of abortions taking place and the particular views that Japanese Buddhists have about fetuses and abortion.
Zygote Intrafallopian Transfer
Zygote intrafallopian transfer (ZIFT) is an assisted reproductive technology (ART) first used in 1986 to help those who are infertile conceive a child. ZIFT is a hybrid technique derived from a combination of in vitro fertilization (IVF) and gamete intrafallopian transfer (GIFT) procedures. Despite a relatively high success rate close to that of IVF, it is not as common as its parent procedures due to its costs and more invasive techniques.
Francis Galton (1822-1911)
Sir Francis Galton was a British science writer and amateur researcher of the late nineteenth century. He contributed greatly to the fields of statistics, experimental psychology and biometry. In the history of biology, Galton is widely regarded as the originator of the early twentieth century eugenics movement. Galton published influential writings on nature versus nurture in human personality traits, developed a family study method to identify possible inherited traits, and devised laws of genetic inheritance prior to the rediscovery of Gregor Mendel's work.
The Time Has Come: A Catholic Doctor's Proposals to End the Battle over Birth Control (1963), by John Rock
In 1963, Roman Catholic fertility doctor John Rock published The Time Has Come: A Catholic Doctor's Proposals to End the Battle over Birth Control, a first-person treatise on the use of scientifically approved forms of birth control for Catholic couples. The first contraceptive pill, called Enovid, had been on the market since June 1960, and Rock was one of the leading researchers in its development. In The Time Has Come, Rock explicitly describes the arguments for and against the use of birth control from both a religious and a scientific perspective.
Subject: Publications, Religion, Reproduction | {
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Q: Graph Theory closest to be bipartite I have two graphs graphs
I know it isn't bipartite, but I need to calculate which one is closest to being bipartite, it is related with bipartivity in network.
To define a measure of network bipartivity is based on the concept of
closed walks.
the bipartivity is defined of G as:
bipartivity definition
is the sum of all closed walks of diferent lengths in the network starting and ending at each vertex of G.
My idea is to calculte te eigenvalues of each graph and do
bipartivity eq
I`m right?
thanks a lot.
A: With $|V(G)|=N$, The subgraph centralization measure can be defined as a weighted sum of
all closed-walks (CWs) of length $\ell$ in the network, $\mu_\ell$ , which is related to graph eigenvalues as follows :
$$ \langle SC \rangle = \frac{1}{N} \sum_\ell \frac{\mu_\ell}{\ell !} = \sum_{j=1}^N e^{\lambda_j}$$
This can be expressed as the sum of two contributions, one coming from odd and the other
from even CWs:
$$ \langle SC \rangle = \frac{1}{N} \sum_{j=1}^N \left(\cosh(\lambda_j) + \sinh(\lambda_j)\right) = \langle SC \rangle_{even} + \langle SC \rangle_{odd}$$
If $G$ is bipartite then it includes no odd closed walks, i.e. $\langle SC \rangle_{odd}=0$ and $$\langle SC \rangle =\langle SC \rangle_{even} = \frac{1}{N} \sum_{j=1}^N \cosh(\lambda_j)$$
Then indeed you can define a measure of the network bipartivity:
$$\beta(G) = \frac{\langle SC \rangle_{even}}{\langle SC \rangle} = \frac{\sum_{j=1}^N \cosh(\lambda_j)}{\sum_{j=1}^N e^{\lambda_j}}$$
You have, $\forall G$, $\beta(G)\leq 1$ with equality iif $G$ is bipartite. You also have a desired monotone property $\beta(G) \geq \beta(G+e)$
See this paper by Ernesto Estrada1 and Juan A. Rodríguez-Velázquez for a full description, and the adequate proofs.
| {
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{"url":"https:\/\/www.physicsforums.com\/threads\/forces-friction-question.657364\/","text":"Forces & Friction question!\n\n1. Dec 7, 2012\n\naoo\n\nA 5 kg block rests on a 30degree incline\nthe coefficient of static friction is between the block and incline is 0.2\nhow large a horizontal force must push on the block if the block is to be on the verge of sliding a)up the incline and b) down the incline\n\nSo far I have ma=Fapp-(m)mg-mgsin30\nwhere (m) is miu or friction co-efficient\n\nive been working on this for a long time but not sure how to get the answers\nany help will be much appreciated!\n\n2. Dec 7, 2012\n\nSimon Bridge\n\nWelcome to PF;\nWould you be able to do this if the block were not on an incline?\n\nSo far you have $ma=F_{app}-\\mu_kmg - mg\\sin{\\theta}$ (resist the urge to put the numbers in so soon).\n\nIs this for pushing the block up or down the slope?\nWhich direction does friction act?\n\nAccording to the description: what is the acceleration?","date":"2017-11-24 13:55:46","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5720159411430359, \"perplexity\": 1205.639443698362}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-47\/segments\/1510934808133.70\/warc\/CC-MAIN-20171124123222-20171124143222-00364.warc.gz\"}"} | null | null |
Lightweight metals leader Alcoa (NYSE:AA) today reported a third quarter profit as its value-add and upstream portfolios delivered solid results in the face of strong market and currency headwinds. Alcoa has been transforming by building its value-add portfolio for profitable growth and creating a globally competitive upstream business. The Company's successful multi-year transformation will culminate with the launch of two Fortune 500 companies, one value-add focused and the other upstream focused, expected in the second half of 2016. | {
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Q: video in html5 - movie not found in firefox I have a following code, which should show a video from file 139multiclub_mix.mp4. This file is in the same folder as file with html code. In browser Chrome all works fine. But in Firefox i see a default movie (from http://video-js.zencoder.com/oceans-clip.ogv) but my movie isn't shown. What i'm doing wrong?
My code:
<video id="sampleMovie" width="274" height="206" controls="controls" autoplay="autoplay" loop="loop" muted="" preload="auto">
<source src="139multiclub_mix.mp4" type="video/mp4; codecs="avc1.42E01E, mp4a.40.2"" />
<source src="http://video-js.zencoder.com/oceans-clip.ogv" type="video/ogg; codecs="theora, vorbis"" />
<source src="http://video-js.zencoder.com/oceans-clip.webm" type="video/webm; codecs="vp8, vorbis"" />
<object id="sampleMovie" width="274" height="206" data="../../../../plugins/forms/tiny_mce/plugins/media/moxieplayer.swf" type="application/x-shockwave-flash">
<param name="src" value="../../../../plugins/forms/tiny_mce/plugins/media/moxieplayer.swf" />
<param name="flashvars" value="url=139multiclub_mix.mp4&poster=/admin/box/edit-box/id/" />
<param name="allowfullscreen" value="true" /><param name="allowscriptaccess" value="true" /></object> </video>
A: I think Firefox still don't support MP4. Add ogv file for Firefox.
See: Media formats supported by the HTML audio and video elements
To avoid patent issues, support for MPEG 4, H.264, MP3 and AAC is not
built directly into Firefox.
| {
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{"url":"https:\/\/scicomp.stackexchange.com\/questions\/11515\/finding-null-vectors-of-a-parameter-dependent-matrix\/30670","text":"# Finding null vectors of a parameter-dependent matrix\n\nI have dense complex matrices $M(z)$ in which each element $M_{ij} = M_{ij}(z)$ depends on a complex parameter $z$. I need to find $z$ such that the matrix $M$ gets singular, i.e. I am looking for null vectors $\\vec v$ which satisfy $M(z) \\vec k = \\vec 0$.\n\nSo far, I use a Newton method. As the wanted $M(z)$ is singular, I need $M$'s determinant to vanish. Starting with an initial guess $z_0$, I iterate $$z_{i+1} = z_i - \\frac{g(z_i)}{g'(z_i)}$$ with $g(z) := \\det M(z)$. To avoid computing the determinant from definition, I use LU factorization. The iteration is stopped when $|z_{i+1} - z_{i}|$ is sufficiently small.\n\nUsing the matrix identity $\\frac{\\mathrm{d}}{\\mathrm{d}z}\\ln \\det M(z) = \\mathrm{trace}\\,(M^{-1}(z) M'(z))$, the reciprocal logarithmic derivative in the iteration formula can replaced to yield $$z_{i+1} = z_i - \\frac{1}{\\mathrm{trace}\\,(M^{-1}(z) M'(z))}\\,,$$ which is what I use in the end.\n\nA few details on steps involved: so far, I compute all derivatives $g'(z)$ from a forward finite difference scheme $g'(z) = \\frac{g(z + h) - g(z)}{h} - \\mathrm{i}\\frac{g(z + \\mathrm{i}h) - g(z)}{h}$, ($h$ real), likewise for derivatives of $M$. Also note I'm assuming that singular vectors have multiplicity $m = 1$ (although adaptation of the formulas above is possible using $g=(\\det M(z))^m$.\n\nThe matrix $M$ has no special structure in global (i.e. not Hermitian). Depending on the problem, it may be rectangular or square. In any case, it is dense and well-conditioned. Typical sizes range from ~ $100\\times100$ to $10k \\times 10k$. $M$'s origin is in the boundary element method. For open systems (that may have resonances), the null vectors I am looking are the resonance wavefunctions on the boundaries of the domains under consideration.\n\nI'd like to learn about alternative methods of adjusting the parameter $z$ such that $M$ gets singular. Do you have any other ideas? Or any comments on the method I described? Although the results are fine, I don't like the approach too much as both iterations described here involve a step that is essentially $O(N^3)$ (LU or the inverse). Furthermore, for the second scheme, I need to compute the inverse of near singular matrices.\n\n\u2022 Is M Hermitian? Could you something about where $M$ arises from? What are the typical dimensions of $M$? Further, do you run into problems as the iterations progress and the matrix becomes singular? \u2013\u00a0user2457602 May 5 '14 at 14:16\n\u2022 @user2457602, I have added more information to the question as you requested. I typically do not run into problems in the iteration. \u2013\u00a0AlexE May 5 '14 at 15:03\n\nEdit: Note that your formula for the finite difference approximation of $g'$ does not need the imaginary correction (just the first term is enough) if your function is analytic. If you can compute the derivatives analytically or automatically (with automatic differentiation), then that would be much better. However, I would guess this is not possible if your matrix comes from a BEM discretization.","date":"2021-03-02 20:52:49","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9227489829063416, \"perplexity\": 326.77555712043534}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178364764.57\/warc\/CC-MAIN-20210302190916-20210302220916-00440.warc.gz\"}"} | null | null |
In the last round of Steam sales, I bought a game at 50% off. A few days later, it went to 75% off. My plan on Steam games is increasingly to wait until I want to play it right now, and then keep waiting until there is a 50+% sale. I have been trained to avoid good sales because of potential buyer's remorse after great sales.
Buyer's remorse on items that go on sale (or a bigger sale) has always seemed like such an odd thing to me. It's like wanton regret. Unecessary. Either I was happy to purchase at the price when I did, or I wasn't.
What occurs afterwards is not only unimportant, but it's inevitable. Why would I do that to myself?
That's the spectre of deflation for you.
Yep, don't bother with 25 or 30% off deals unless you want to play the game immediately. It's bound to drop to 50% at some point, either during summer or winter sales or a daily deal, and depending on the publisher, has a good chance of going to 75% at some point. At 80-95% off, buy if you've the slightest interest in the game.
Ultimately, it's in Steam's long term interest to build a population of rabid Steam junkies capable of making such discerments, since we end up checking their store's homepage daily for deals, even if a developer doesn't make as much as they might have from a less discounted game.
I'm in Rog's camp, minus the gift to a friend bit. You bought it a price you thought was good, having it go for less later doesn't invalidate that original price being good. Also, in that case you had the game for some time, so it's not like that money was for nothing.
I reach the point of "must has NAO!" on Mass Effect 2 about a week before it went on Steam sale for half the price I'd paid at retail. But it was so friggin' good I didn't mind. I saved a fortune on other stuff, so I figure I came out way ahead for the month anyway.
Paid 2x for Divinity 2, 3x for CoH, and 3x for Global Agenda, all a matter of 4-6 weeks previous to their sales. It hurt but there are so many other games that Steam has practically given away I'd say its a wash in the long run.
Would be incapable of buying anything in a timely fashion if I waited just-in-case.
The next DAY, it went on sale for $1.99.
I even emailed Steam support. Silly, but worth a shot. First rule of Steam – don't buy until discounted below 75%.
I haven't bought anything from them since then. | {
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Un riproduttore di cassette (o anche lettore di cassette, mangianastri, mangiacassette, piastra a cassette e registratore a cassette se dotato di funzione di registrazione), nell'ambito degli apparecchi HI-FI, è un tipo di registratore a nastro per la riproduzione e/o la registrazione delle musicassette.
Storia
I primi apparecchi
Il riproduttore di cassette è stato introdotto dalla Philips nel 1963 ed è stato lanciato sul mercato nel 1965 come dispositivo per dettatura vocale progettato per uso portatile. In origine non era stato progettato per essere un sostituto del magnetofono. La Philips decise di includere entrambe le bobine del nastro in una piccola cassetta, eliminando la necessità di arrotolare il nastro attraverso bobine singole, e facendo sparire in questo modo molti inconvenienti dovuti alla struttura delle bobine stesse. La larghezza del nastro è di 3,81 millimetri (nominalmente 1⁄8 di pollice) e la velocità di scorrimento del nastro è di 4,76 centimetri (1,875 pollici) al secondo; questo significa che la qualità del suono registrato era appropriata soltanto per la riproduzione/registrazione di voce e dettatura, con risposta alle alte frequenze inferiore a 10 kHz e alto livello di rumore.
I primi registratori erano dispositivi portatili a batteria con controllo automatico delle prestazioni, adatti per la dettatura e la registrazione, ma dalla metà degli anni settanta i riproduttori di cassette con controlli manuali (dei livelli di tono, bilanciamento, ecc.) e indicatori VU (Volume Unit) divennero un componente diffuso degli stereo ad alta fedeltà casalinghi. Alla fine questi apparecchi sostituirono i registratori a bobine, che non raggiunsero una elevata diffusione nelle abitazioni in quanto più ingombranti, costosi e soggetti a particolari accorgimenti nell'inserire e nel riavvolgere il nastro: una cassetta, infatti, può essere estratta in qualunque momento, indipendentemente dalla quantità del nastro che deve ancora scorrere ed essere ascoltata, mentre prima di estrarre una bobina occorre riavvolgere il nastro. Le cassette possono inoltre essere usate in automobile e per applicazioni personali portatili (p.es. walkman); spesso chi le utilizza registra sul nastro delle canzoni in maniera tale da ottenere un "nastro da viaggio".
Nel 1971, la Advent Corporation combinò il sistema di riduzione del rumore e fruscio di fondo del nastro Dolby B con il nastro al diossido di cromo (CrO2) per creare l'Advent Model 201, il primo riproduttore di cassette ad alta fedeltà. Il sistema Dolby B aumentò i livelli degli acuti rispetto al livello di rumore, e li ridusse rispetto alla riproduzione, mentre il CrO2 usava diverse impostazioni per la distorsione e l'equalizzazione per ottenere gli stessi risultati, oltre ad estendere, per la prima volta, la risposta in frequenza in un livello di alta fedeltà di 15 kHz.
Questo riproduttore era basato su un meccanismo di caricamento del nastro dall'alto creato dalla Wollensak, una divisione della 3M che si occupa di applicazioni audio/video; tale meccanismo prevedeva (insolitamente) un solo indicatore VU che poteva essere smistato su uno o su entrambi i canali audio, e delle leve di controllo simili a quelle presenti nei meccanismi a bobina.
Molti produttori, inizialmente, adottarono un formato standard di caricamento dall'alto con controlli tramite pulsanti, doppi indicatori VU e controlli dello scivolamento. In seguito, verso la fine degli anni '70, prese piede lo standard ancora oggi diffuso, che prevede il caricamento del nastro sulla faccia anteriore, con il vano cassetta su un lato e gli indicatori VU sull'altro; successivamente furono introdotti formati con due vani per le cassette, con gli indicatori situati nel mezzo. I controlli meccanici furono sostituiti da pulsanti elettronici a solenoide, benché i modelli più economici mantenessero i controlli meccanici. Alcuni modelli più evoluti sono in grado di cercare e contare gli intervalli presenti tra una canzone e l'altra. I lettori di cassette introdussero l'attuale set di controlli (play, pause, stop, eject, forward, rewind e record) ed il "blocco" dei pulsanti di scorrimento veloce in avanti e all'indietro, di modo che se in precedenza per lo scorrimento veloce andava tenuto premuto il bottone apposito, ora è possibile schiacciare il bottone apposito una volta sola, e questo rimaneva abbassato fino a quando non veniva fermato.
La diffusione
I riproduttori di cassette divennero presto di largo uso e ne furono progettate diverse versioni per applicazioni professionali, impianti casalinghi, e anche per l'uso in automobile, come registratore portatile. Dalla metà degli anni settanta alla fine degli anni novanta, il lettore di cassette fu lo strumento preferito per l'ascolto di musica in auto: la sensibilità al moto del veicolo era bassa come per i nastri dello Stereo-8, rispetto ai quali era stata ridotta la vibrazione del nastro; a vantaggio delle cassette giocavano inoltre altri fattori, come la ridotta dimensione e la possibilità di scorrimento veloce nelle due direzioni.
Il miglioramento delle prestazioni
I riproduttori di cassette hanno raggiunto l'apice del loro rendimento a metà degli anni ottanta. I registratori delle ditte Nakamichi, Revox, e Tandberg includevano caratteristiche avanzate come le testine multinastro e la guida con due capstan, con motori separati per le bobine.
La tecnologia a tre testine usa testine separate per la registrazione e la riproduzione; grazie a questo meccanismo è possibile ascoltare ciò che intanto viene registrato sul nastro. Ciò era spesso possibile sui registratori a bobina, ma è più difficile per le cassette, che di per sé non forniscono aperture separate per le diverse testine dedicate alla registrazione e alla lettura, ma possiedono invece un design che prevede un'unica apertura centrale per la testina di registrazione/riproduzione, oltre a due aperture aggiuntive per la testina di cancellazione e per la guida del capstan. Alcuni modelli introdussero una testina di monitoraggio nell'area dedicata al capstan, mentre altri previdero sulla stessa testina spazi separati per la registrazione e la riproduzione.
Riproduttori comuni erano anche quelli venduti da Harman Kardon o da NAD e da ditte giapponesi come Aiwa, Akai, Denon, Pioneer, Sony, Teac, Technics e Yamaha, tutte in grado di offrire prodotti di alta qualità. Gli apparecchi migliori erano in grado di registrare e riprodurre l'intero spettro udibile dall'orecchio umano, da 20 Hz a 20 kHz, con vibrazione e distorsione inferiori allo 0,05% e livello di rumore molto basso.
Una registrazione dal vivo effettuata su cassetta poteva competere con il suono di un CD commerciale qualunque, sebbene la qualità delle cassette pre-registrate era generalmente inferiore rispetto a quella che poteva essere ottenuta da una registrazione casalinga di alta qualità. Le cassette rimangono popolari per le applicazioni audiovisive; alcuni registratori di CD includono un vano per le cassette per permettere di registrare su entrambi i formati l'audio di riunioni, sermoni e libri.
Il sistema di riduzione del rumore Dolby era la chiave per ottenere prestazioni con basso livello di rumore su nastri stretti e lenti; tale sistema funziona amplificando le alte frequenze in fase di registrazione per poi memorizzarle sul nastro così come sono state trasformate, abbassando inoltre la costante di rumore alle alte frequenze. Alcune versioni avanzate del Dolby includono i tipi C (inventato nel 1980) ed S, sebbene il sistema B sia l'unico standard supportato sulla maggior parte dei riproduttori da automobile ad alta fedeltà. Alcuni apparecchi possedevano microprocessori programmati per impostare automaticamente il bias del nastro. Nel 1982, in collaborazione con i Dolby Laboratories, la Bang & Olufsen sviluppò un sistema che fu chiamato HX-Pro per adattare dinamicamente il bias in funzione della frequenza da registrare, usato in molti apparecchi di fascia alta.
Negli anni successivi comparve la funzione "auto-reverse", che permetteva di riprodurre (e talvolta di registrare) su entrambi i lati della cassetta senza che fosse necessario rimuoverla, capovolgerla e reinserirla manualmente. Nei primi primi apparecchi dotati di auto-reverse, ed anche in molti apparecchi economici odierni, questo meccanismo è implementato con una testina bidirezionale che può riprodurre tutte e quattro le tracce presenti sul nastro (canali destro e sinistro del lato A, e canali destro e sinistro del lato B); di queste, solo due per volta erano effettivamente collegate all'impianto elettronico. Per quanto riguarda invece lo scorrimento del nastro, gli apparecchi sono forniti di due capstan e due rulli pressori, uno per ciascuna direzione. È comunque difficile allineare correttamente una testina bidirezionale per entrambe le direzioni. In molti degli apparecchi più costosi il meccanismo di auto-reverse opera sbloccando la testina, ruotandola di 180° e bloccandola nuovamente, fornendo le guide di allineamento per entrambe le direzioni. In un apparecchio Nakamichi, invece, lo scopo fu raggiunto con un meccanismo che estraeva la cassetta dal vano, la capovolgeva e la reinseriva. Gli apparecchi dotati di auto-reverse conobbero una maggior diffusione, poiché eliminavano la necessità di estrarre la cassetta manualmente per capovolgerla; tale caratteristica divenne poi uno standard per le autoradio installate in fabbrica.
Il declino
Le cassette analogiche cominciarono il loro declino con l'avvento del compact disc e di altre tecnologie digitali di registrazione quali Digital Audio Tape (DAT) e MiniDisc. La Philips promosse nel 1992 la Digital Compact Cassette (DCC), una versione digitale della musicassetta, caratterizzata dalla compatibilità in lettura con le audiocassette classiche, ma che fallì nel guadagnare una significativa quota di mercato e la produzione cessò abbastanza presto.
La TDK, la Sony, la Maxell e la Basf (con il marchio Emtec a partire dal 2000) sono tra le compagnie che hanno prodotto musicassette fino al 2012, mentre ad oggi esiste ancora qualche produttore che fabbrica musicassette in piccole quantità per usi professionali e per nicchie di mercato.
I computer possono facilmente produrre copie di interi CD o convertire tracce in MP3 o altri formati per essere ascoltate su dispositivi digitali portatili quali il popolare iPod; ciò ha permesso la copia e la distribuzione delle tracce audio in maniera più agevole rispetto al passato, in cui l'unica possibilità di copiare una traccia sonora in maniera semplice era quella di trasferirla su una musicassetta.
Nonostante il declino nella produzione di registratori a cassette, questi apparecchi godono ancora di reputazione ed utilizzo; in particolare dagli audiofili che ritengono che la tecnologia delle piastre a cassette e dei giradischi, a causa della loro natura analogica, fornisca registrazioni di qualità superiore alle attuali tecnologie digitali, quali il CD-R e il DAT.
Funzionamento
Il nastro viene raccolto su due bobine; rispetto al lato che si ascolta (o si registra), la bobina di destra è dedicata al riavvolgimento del nastro, mentre quella di sinistra contiene il nastro da svolgere. Una volta che la musicassetta viene inserita in un apparecchio, il nastro scorre su una testina, che viene a contatto con il nastro grazie a un'apertura centrale sul lato inferiore della cassetta. La testina riceve il segnale magnetico impresso sul nastro e lo converte in un segnale elettrico che darà origine al suono. Un'altra testina permette la registrazione del suono sul nastro attraverso un'altra apertura, posta più a sinistra (anche se alcuni modelli usano l'apertura centrale sia per la registrazione che per la riproduzione); in questo modo, un nastro può essere registrato e, subito dopo, riprodotto.
Il trascinamento del nastro avviene a una velocità costante di 4,76 cm/s (1 + 7/8 pollici al secondo), grazie alla rotazione di un piedino metallico, che è denominato capstan, che viene a contatto con il nastro grazie a un foro trasversale sulla cassetta in cui il capstan va ad entrare in un modo molto aderente. L'aderenza tra il capstan e il nastro è assicurata da un rullo pressore, ricoperto di gomma, che assicura il trascinamento e che va a premere il nastro sul capstan grazie ad un'apertura posta sulla destra del lato inferiore della cassetta. Il suo rullo pressore è parte integrante del riproduttore, a differenza di quanto avviene nel formato Stereo-8, in cui il rullo è parte integrante della cassetta. Inizialmente il cambio dal lato "A" al lato "B" avveniva manualmente, estraendo la cassetta dal lettore e capovolgendola. In seguito si diffusero riproduttori con doppia testina, in grado di invertire automaticamente la direzione di scorrimento e di lettura del nastro alla fine della riproduzione di ciascun lato (funzione di autoreverse). Le piste registrate e/o riprodotte per il lato A si trovano dalla parte opposta rispetto a quello che viene mostrato come lato A durante l'uso della cassetta; di conseguenza, quando l'utente riproduce il contenuto inciso su una facciata, la testina legge le piste rivolte verso l'interno dell'apparecchio; lo stesso vale in fase di registrazione.
Ad assicurare l'allineamento del nastro con il sistema di testine, capstan e rullo pressore, esistono apposite guide; due di queste si trovano direttamente nella cassetta, alle estremità del lato inferiore, mentre due guide fanno parte dell'apparecchio e vanno ad inserirsi in altrettanti fori trasversali situati sulla musicassetta. Il riavvolgimento del nastro avviene per mezzo del piatto di destra, il quale è frizionato, per far sì che il nastro non venga danneggiato né riavvolto velocemente se il rullo pressore è danneggiato; l'altro piatto, quello di sinistra, è invece dotato di un freno, in modo che il nastro sia sempre teso. Questi due piatti vengono inoltre usati per lo scorrimento veloce: quello di destra, in particolare, viene azionato a velocità elevata per lo scorrimento in avanti, quello di sinistra per lo scorrimento all'indietro. Nei lettori più economici, il piatto di sinistra può mancare, venendo sostituito da un semplice piedino avente la funzione di sostegno: in questo caso, a fare da freno alla bobina di destra è la forza d'attrito tra il piedino e la parte centrale della bobina stessa. Poiché in tali lettori manca il piatto di sinistra, la funzione di scorrimento veloce all'indietro è assente.
Il nastro, generalmente, possiede quattro piste longitudinali in cui viene registrato il suono, due per lato; per ciascuna facciata, abbiamo una pista per il canale sinistro ed una per il canale destro (che si fondono in un'unica pista per le registrazioni monofoniche). Le testine (sia di registrazione che di riproduzione), generalmente, sono fatte in modo da leggere o scrivere solo le piste che si trovano su un determinato lato, rendendo così necessaria la rotazione della cassetta o della testina per leggere le tracce dell'altro lato; i primi apparecchi dotati di autoreverse, invece, possedevano testine fisse che consistono in pratica in una doppia testina di lettura, una per ciascun lato. Alcuni sistemi professionali usano testine in grado di registrare e riprodurre più di due piste audio sullo stesso lato.
Riduzione dei rumori e fedeltà
Diversi schemi di riduzione del rumore sono stati usati per aumentare il livello di fedeltà, e tra questi il Dolby B è quello adottato in maniera quasi universale sia per i nastri pre-registrati che per le registrazioni casalinghe. Il Dolby B fu progettato per risolvere il problema del fruscio, e con l'aiuto delle migliorie nella composizione chimica del nastro contribuì a far accettare la cassetta come mezzo di alta fedeltà. In seguito gli apparecchi inclusero procedure automatiche di identificazione del tipo di nastro, attraverso l'introduzione di appositi fori sulla superficie superiore delle stesse cassette. Allo stesso tempo, il Dolby B garantiva prestazioni accettabili per le riproduzioni su apparecchi privi di circuiteria Dolby, dimostrando che non c'era ragione di non usarlo qualora fosse stato disponibile.
La principale alternativa al Dolby era il sistema di riduzione del rumore DBX, che faceva ottenere un elevato rapporto segnale/rumore, ma che risultava inascoltabile su apparecchi non dotati di circuiteria per la decodifica DBX. La Philips sviluppò un sistema alternativo di riduzione del rumore, il Dynamic Noise Limiter (DNL), che non richiedeva una apposita rielaborazione del segnale in fase di registrazione sul nastro; questo sistema è alla base del metodo di riduzione del rumore DNR.
In seguito, la Dolby introdusse i sistemi di riduzione del rumore Dolby C e Dolby S, che raggiunsero alti livelli di riduzione del rumore; il Dolby C divenne molto usato negli apparecchi ad alta fedeltà, mentre il Dolby S, lanciato quando già la vendita delle cassette aveva cominciato a diminuire, non raggiunse mai una diffusione notevole; fu utilizzato solo negli apparecchi di fascia più alta, come ad esempio quelli a tre testine.
Il sistema Dolby HX Pro fu un'altra invenzione della Dolby che garantiva una migliore risposta alle alte frequenze attraverso la riduzione del bias del nastro non udibile durante la registrazione di suoni potenti ad alta frequenza, che di per sé producevano un certo livello di bias. Sviluppato da Bang & Olufsen, non richiedeva un circuito di decodifica per la riproduzione.
Altri perfezionamenti per migliorare la performance delle cassette furono i sistemi DYNEQ della Tandberg, ADRES della Toshiba e Hi-Com della Telefunken, e, su alcuni apparecchi di fascia alta, registrazione automatica del bias, controllo fine del tono e, talvolta, dell'azimut delle testine.
Verso la fine degli anni ottanta, grazie ai miglioramenti dell'elettronica, del materiale del nastro e delle tecniche di fabbricazione, ed anche alle migliorie nella precisione della parte plastica della cassetta, delle testine e del trasporto meccanico, la fedeltà del suono sugli apparecchi dei produttori di vertice sorpassò i livelli attesi inizialmente. Sugli apparecchi audio adeguati, le cassette potevano garantire un ascolto molto piacevole. I migliori riproduttori casalinghi potevano raggiungere una risposta in frequenza tra i 20 Hz e i 20 kHz con wow and flutter inferiore allo 0,05%, e rapporto segnale/rumore di 70 dB col sistema Dolby C, inferiore a 80 dB con il Dolby S e di 90 dB con il DBX. Molti ascoltatori occasionali non percepivano la differenza tra la cassetta e il compact disc.
Dall'inizio degli anni '80, la fedeltà delle cassette preregistrate cominciò ad aumentare notevolmente. Mentre il Dolby-B era già largamente usato negli anni '70, i nastri preregistrati venivano duplicati su nastri di bassa qualità a velocità elevate, sì da non reggere il confronto con gli LP. Comunque, sistemi come l'XDR, contemporaneamente all'adozione di nastri di qualità superiore (come quelli al CrO2, sebbene registrati in maniera tale da essere riprodotti come nastri di tipo I, con un bias di 120 µs), e all'uso frequente della tecnologia Dolby HX Pro, fecero diventare le cassette un'alternativa ad alta fedeltà percorribile, che risultava più facilmente trasportabile e richiedeva una manutenzione minore rispetto ai dischi in vinile.
In aggiunta, la cover art, che generalmente era fino ad ora stata relegata alla creazione di una singola immagine della copertina degli LP con poche linee di testo, cominciò ad essere elaborata su misura per le cassette, con l'inserimento nelle custodie di fogli ripiegati o libretti contenenti i testi delle canzoni e di copertine ripiegate, che divennero presto di uso comune.
Negli anni '80 alcune compagnie, come la Mobile Fidelity, produssero cassette "audiofile", registrate su nastri di alto livello e duplicate in tempo reale a partire da un master digitale di prima generazione tramite attrezzature di prim'ordine. Diversamente dai vinili audiofili, che continuano ad avere un seguito di appassionati, le cassette audiofile furono messe da parte una volta che il compact disc conobbe una larga diffusione.
Uso in auto
Un elemento chiave del successo del registratore fu il suo uso nelle autoradio, grazie alle dimensioni della musicassetta, più piccola di una cartuccia dello Stereo 8. I registratori furono spesso integrati con delle radio, e l'insieme dei due apparecchi è stato denominato anche "stereo". I mangianastri per auto furono i primi a dotarsi di "auto-reverse", un dispositivo che permette, una volta terminata la riproduzione di un lato, di ascoltare l'altra parte del nastro senza dover estrarre la cassetta per girarla invertendo il senso di rotazione del nastro e ruotando la testina. In questo modo era possibile ascoltare un nastro senza interromperne mai la riproduzione. Dopo poco tempo anche alcuni registratori domestici si dotarono di questa caratteristica.
Ad oggi sono stati inventati anche nuovi dispositivi, denominati adattatori, che permettono l'ascolto, con un mangianastri, di supporti audio di tipo diverso (CD, MP3), utili in particolare in auto.
Manutenzione
L'apparecchiatura di un registratore necessita di una manutenzione regolare, poiché il nastro a cassetta è un mezzo magnetico che viene a contatto fisico con la testina e con altre parti metalliche del meccanismo riproduttore/registratore. Senza tale manutenzione, la risposta in alta frequenza dell'apparecchiatura del registratore sarà meno soddisfacente.
Un problema insorge quando particelle di ossido di ferro (o simili) già presenti sul nastro si depositano sulla testina destinata alla lettura (o anche su quella per la registrazione). Perciò, le testine richiederanno occasionalmente di essere ripulite. Il capstan ed il rullo pressore di gomma possono essere anch'essi sporcati con queste particelle, venendo dunque a causare un trascinamento irregolare del nastro sulle testine; ciò a sua volta può comportare un disallineamentro del nastro sopra la testina con conseguente variazione dell'angolo di azimut rispetto al traferro, anomalia che si traduce in un degrado di chiarezza nella riproduzione dei toni alti, come se la testina stessa fosse non allineata.
Inoltre, le testine ed altri componenti metallici nel percorso di nastro (quali il capstan e le guide) potrebbero venire magnetizzati e richiedere così la smagnetizzazione. Esistono in commercio appositi smagnetizzatori elettronici e kit di pulizia, questi ultimi sia a secco (nastro pulitore) che a umido (feltri e liquido); alcuni di essi hanno la forma di una musicassetta e come tali vanno utilizzati, seguendo le istruzioni allegate ad essi. Un metodo più accurato prevede invece l'uso di un bastoncino cotonato imbevuto di un liquido per la pulizia manuale delle singole componenti; tale liquido può essere l'alcool isopropilico, comunemente usato per pulire le testine del nastro e gli altri componenti, ad eccezione del rullo pressore, che con l'alcool tende ad indurirsi; in questo caso può andar bene l'acqua o, meglio ancora, l'uso di prodotti specifici validi anche per la pulizia delle altre componenti.
Note
Voci correlate
Magnetofono
Memoria di massa
Musicassetta
Walkman
Altri progetti
Apparecchi audio | {
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About Cedefop >
Public procurement >
Entrepreneurship competence in vocational education and training
AO/DSI/DKULSS/Entrepreneurship-competence-VET/004/20
The research to be carried out according to this contract will help better understand how entrepreneurship competence is embedded in VET in Europe, the related challenges and opportunities and how these vary between countries. The findings of the study will support policy makers, social partners, VET providers and other stakeholders in promoting entrepreneurship competence.
This call has been published in the Supplement to the Official Journal of the European Union 2020/S 141-347038 of 23/07/2020.
The procurement documents are available via the TED e-Tendering platform: https://etendering.ted.europa.eu/cft/cft-display.html?cftId=6894
Tenders must be submitted exclusively via the electronic submission system (e-Submission) available from the above link to the e-Tendering platform by: 21/09/2020 (15h59 Greek local time i.e. UTC+03:00).
Requests for additional information/clarification should be received by 11/09/2020, 23h59 (UTC+03:00 Greek local time). All questions and answers will be available in the above mentioned website.
Information about the public opening of tenders
With reference to the COVID-19 outbreak and the public health medical emergency as well as Cedefop's decision to close its premises as of 17 March, we kindly inform that the public opening of tenders may take place virtually and not in Cedefop's premises as initially announced. Tenderers that express their interest to participate in the virtual opening will receive further guidelines (details on how to express interest to attend the opening can be found in point 7 of the invitation to tender).
Tenderers that will not be present at the virtual opening session but wish to be provided with the information announced during the public opening may send their request to C4T-Services@cedefop.europa.eu .
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For any questions or clarifications, please read the frequently asked questions first, before contacting the procurement service by email mail
Privacy Statement on the protection of personal data in relation to Public Procurement | {
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{"url":"https:\/\/electronics.stackexchange.com\/questions\/548127\/can-i-run-2amp-1-76v-stepper-motor-on-drv8825-driver","text":"# Can I run 2amp\/1.76v stepper motor on drv8825 driver?\n\nI do electronics for fun and need help understanding the principles when coupling stepper motors and drivers!\n\nI have this type of stepper motor: kh56km2u027 and a drv8825 stepper driver, and was wondering if they would run safely together.\n\nIn the motor label, it's written 1.76V 2A and the driver can operate on 8.2V-45V 2.5A(peak)\n\nThere are few things that get me confused, like:\n\n\u2022 Labeled amps -> does it mean that it draws 2A in total whatever the case? (so I assume the driver can run it)\n\u2022 amps-per-phase -> what's the difference when it's labeled: 2A and 2A per-phase?\n\nOr is there anything else I'm missing here?\n\nThank you!\n\nEdit: Forgot to mention that I power the driver using 12V\/10A supply and the logic through Arduino board.\n\nFirst of all, the big question for stepper motors: it's bipolar, unipolar or configurable?\n\nIt depends on how many wires it has. A 5-wire motor can't be used with that driver. A 4-, 6- or (rarely)8- wire motor can. The KH56 is a 6-wire motor so it can be used in bipolar configuration.\n\nAs for the current rating: the stepper motor (especially in bipolar mode) is usually driven with a constant current, not with a given voltage. For that kind of motor you could either:\n\n\u2022 Simply apply the rated voltage (1.76V) to the winding, in which case it will consume about 2A, or\n\n\u2022 Apply a current of 2A and keep that regulated, which is the approach followed by drivers.\n\nThis is the per phase current rating.\n\nThe motor has two coils in quadrature so you will have a total of maximum 4A flowing in the windings. This is the whole motor current rating.\n\nHowever this doesn't mean that you need 4A on you 12V power rail.\n\nThe motor driver is more or less a switching converter. It only gives pulses of current to keep the magnetic field charged; you can see the current ripple in figure 9 in the datasheet, for example.\n\nIn short, you program the driver for a given current (with a resistor, in this case) and it regulates it but it's better if you think of it as power. About 2A at 2A are 4W, so it's an 8W motor. At 12V it's somewhat less of 700mA.\n\nBy the way circuit layout with the TI stepper drivers is quite critical; do it exactly as it is shown in the datasheet. Obey the recommendations on capacitors, they are quite important, too.\n\n\u2022 Thanks for your time Lorenzo! As I understand, the 2A rating stands for per-phase, but even tho it's a 2-phase motor, it never draws 2x2A at the same time, and so it's possible for the drv8825 to run the motor, right? As for wiring, I followed this diagram and set the Vref on driver using Vref = I(max) \/ 2 so Vref = 2A \/ 2 = 1. I tried this, it spins.. but driver gets hot after few minuets.. I jus don't know if it's safe to keep it spining like that or will it burn stuff!!! D: Feb 14, 2021 at 21:29\n\u2022 It will draw 2A per phase but the driver has 2A available per phase so in the end it works. Quoting the datasheet \"The DRV8825 is capable of driving up to 2.5 A of current from each output (with proper heat sinking, at 24 V and 25\u00b0C).\". See figure 4, 24V is its sweet point'' for Rdson, as 12V is slightly degraded. Proper heat sinking\" is the key word here. You should do some thermal calculation (section 7.4), maybe adding some heat sink or even lower the current if you are worried. Your board datasheet could have info on that. The motor will work at reduced current but with less torque Feb 15, 2021 at 7:21\n\u2022 I very much appreciate the info, as I couldn't wrap my head around all this by just combining bits from research! Thank you mate! I attached the heatsink (the small one that came with driver) and reduced the current draw a bit so now it doesn't get very hot and the torque is sufficient for my needs.. now only the speed is the issue :p.. have to figure that one out. Thanks again. Feb 16, 2021 at 8:41\n\u2022 I'm already using these drivers for positioning so I'm sharing the experience. Speed in steppers is a difficult issue because 1) they are simply not designed for speed 2) they have so called 'resonance point' where torque lowers a lot. If you can rise the supply voltage it will helps (due to inductance di\/dt limiting), it's not uncommon to see steppers powered at even 72V Feb 17, 2021 at 8:50","date":"2022-06-26 18:40:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.39237740635871887, \"perplexity\": 2111.1719622623086}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656103271763.15\/warc\/CC-MAIN-20220626161834-20220626191834-00306.warc.gz\"}"} | null | null |
{"url":"https:\/\/cob.silverchair.com\/jeb\/article\/209\/23\/4809\/16487\/Structure-and-sexual-dimorphism-of-the","text":"Electrocommunication signals of electric fish vary across species, sexes and individuals. The diversity of these signals and the relative simplicity of the neural circuits controlling them make them a model well-suited for studying the mechanisms, evolution and sexual differentiation of behavior. In most wave-type gymnotiform knifefishes, electric organ discharge (EOD)frequency and EOD modulations known as chirps are sexually dimorphic. In the most speciose gymnotiform family, the Apteronotidae, EOD frequency is higher in males than females in some species, but lower in males than females in others. Sex differences in EOD frequency and chirping, however, have been examined in only three apteronotid species in a single genus, Apteronotus. To understand the diversity of electrocommunication signals, we characterized these behaviors in another genus, Adontosternarchus. Electrocommunication signals of Adontosternarchus devenanzii differed from those of Apteronotus in several ways. Unlike in Apteronotus, EOD frequency was not sexually dimorphic in A. devenanzii. Furthermore,although A. devenanzii chirped in response to playbacks simulating conspecific EODs, the number of chirps did not vary with different stimulus frequencies. A. devenanzii chirps also differed in structure from Apteronotus chirps. Whereas Apteronotus species produce functionally distinct chirp types differing in frequency modulation (FM), A. devenanzii produced only high-frequency chirps that had either single or multiple frequency peaks. Males produced more multi-peaked chirps than females. Thus, the temporal structure of chirps, rather than the amount of FM, delineated chirp types in A. devenanzii. Our results demonstrate that the structure, function and sexual dimorphism of electrocommunication signals are evolutionary labile in apteronotids and may be useful for understanding the diversity of sexually dimorphic behavior.\n\nReproductive and agonistic communication signals are among the most conspicuous and diverse of animal behaviors. These signals vary both across and within species, are often highly sexually dimorphic and can therefore serve as models for understanding the evolution of behavioral diversity and the mechanisms that regulate sex differences in behavior.\n\nThe electrocommunication signals of weakly electric fish provide an opportunity to study the mechanisms and evolution of diversity in sexually dimorphic communication. Both African mormyriform and Neotropical gymnotiform fishes possess electric organs whose weak electrical discharges function in electrolocation and communication. The properties of electric organ discharges(EODs) differ between species and can also vary as a function of sex,reproductive condition and\/or social rank(Bass, 1986; Carlson et al., 2000; Dunlap and Larkins-Ford, 2003; Franchina et al., 2001; Hagedorn and Heiligenberg,1985; Hopkins,1988; Kramer et al.,1980; Zakon and Smith,2002). Each species produces one of two types of discharge:pulse-type or wave-type EODs. In pulse-type EODs, the duration of each discharge is much shorter than the time between discharges, whereas the duration of each discharge for wave-type EODs is approximately the same as the time between discharges, resulting in a quasi-sinusoidal signal (reviewed by Hopkins, 1988; Moller, 1995).\n\nIn species that produce wave-type EODs, the frequency of the discharge(i.e. number of discharges per second) often differs between the sexes. In most of the wave-type gymnotiform fish that have been studied, males emit lower frequency EODs than females (Dunlap and Zakon, 1998; Hagedorn and Heiligenberg, 1985; Hopkins,1974b). Interestingly, however, in the most speciose gymnotiform family, the Apteronotidae, sex differences in EOD frequency have been studied in only three species in a single genus, and the direction of sexual dimorphism differs between these species. In the black ghost knifefish(Apteronotus albifrons), males produce EODs at significantly lower frequencies than females, whereas in two closely related species commonly called brown ghost knifefish (Apteronotus leptorhynchus and Apteronotus rostratus), EOD frequency is higher in males than females(Dunlap et al., 1998; Hagedorn and Heiligenberg,1985; Kolodziejski et al.,2005; Meyer et al.,1987). Although the hormonal mechanisms underlying this reversal in the direction of sexual dimorphism in EOD frequency have been studied(Dunlap et al., 1998), the function of males having higher versus lower EOD frequency than females in apteronotids is not known.\n\nAnother type of electrocommunication signal, chirping, also differs across species and between sexes. Wave-type EODs are continuously emitted at precise frequencies that can indicate species, sex and\/or rank. When fish interact,however, they can also transiently modulate the frequency and\/or amplitude of their EODs to produce different types of signals known as chirps, gradual frequency rises (GFRs) and interruptions(Dye, 1987; Hagedorn and Heiligenberg,1985; Hopkins,1974b; Larimer and MacDonald,1968). In A. leptorhynchus, chirping is highly sexually dimorphic, with males chirping more than females(Dunlap et al., 1998; Kolodziejski et al., 2005; Zupanc and Maler, 1993). By contrast, the amount of chirping is not sexually dimorphic in A. albifrons (Dunlap and Larkins-Ford,2003; Dunlap et al.,1998; Kolodziejski et al.,2005).\n\nThe structure of chirps [i.e. the duration and degree of amplitude and frequency modulation (FM)] also varies between sexes and across species. Although A. leptorhynchus and A. albifrons both produce similar types of chirps, the chirps of A. albifrons are approximately 10 times longer in duration than comparable chirp types in A. leptorhynchus (Dunlap and Larkins-Ford, 2003; Kolodziejski et al., 2005). High-frequency chirps (i.e. chirps with more than 150 Hz of FM) are produced more often by males than females in both species, and the amount of FM and\/or duration of chirps is also sexually dimorphic(Dunlap and Larkins-Ford, 2003; Dunlap et al., 1998; Hagedorn and Heiligenberg,1985; Kolodziejski et al.,2005).\n\nThus, the closely related apteronotid species whose electrocommunication signals have been well-studied differ in the degree and\/or direction of sexual dimorphism in EOD frequency and chirping. Since more than 60 apteronotid species in 14 genera have been identified(Crampton and Albert, 2006) and because electrocommunication signals can be easily recorded and quantified,this family offers an unusual opportunity to investigate the evolution of sexually dimorphic communication. To take advantage of this species diversity,however, the communication signals of apteronotid fish in genera other than Apteronotus must be studied. We further characterized the diversity of electrocommunication signals by examining the structure of chirps and sex differences in EOD frequency and chirping in Adontosternarchus devenanzii, an apteronotid species in a genus with numerous derived characters, including intraspecific diversity in EOD waveform and the presence of accessory electric organs (Bennett,1971; Crampton and Albert,2006).\n\n### Subjects, housing and assessment of sex and reproductive condition\n\nAdontosternarchus devenanzii(Mago-Leccia et al., 1985) (11 males and 10 females) were purchased from a reputable commercial supplier(Rose Tropical Fish, Miami, FL, USA) and were housed in 65 l or 34 l tanks maintained at 26-27\u00b0C, pH 5.5-6.5 and conductivity of 100-500 S cm-1. Experiments complied with the National Institutes of Health Guide for the Care and Use of Laboratory Animals and protocols approved by the Indiana University Animal Care and Use Committee. A. devenanzii is not sexually dimorphic in body size or external morphology,and we were therefore unaware of the sex of each fish when its electrocommunication behavior was recorded. The sex of most fish was determined later by laparotomy. After behavioral testing was completed, fish were anesthetized with 0.075% 2-phenoxyethanol. A small incision was made in the ventral body wall, and the gonads were examined to determine the sex of the fish. The incision was sutured with 8-0 silk and sealed with Nexaband surgical tissue adhesive (Abbott Laboratories, North Chicago, IL, USA). One male and one female died after the study, and two males and two females were killed after their behavioral recordings for use in a separate immunohistochemical study. The sex of these fish was determined by post-mortem examination of the gonads. In these cases, gonads were removed and weighed,and reproductive condition was estimated by calculating the gonadosomatic index (GSI, gonad mass\u00d7100\/body mass). Some of the fish (five males and six females) were weighed to the nearest 0.1 g to test for sex differences in body mass and\/or correlations between size and EOD frequency or chirping.\n\n### Recording electrocommunication behavior\n\nThe EOD frequency of each fish was measured by placing a shielded pair of wires next to the tail, amplifying the voltage between those wires(100\u00d7; model P-55; Grass Instruments, W. Warwick, RI, USA) and using the frequency counter of a digital multimeter (Fluke model 187, Everett, WA, USA). The temperature of the water was also measured to the nearest 0.1\u00b0C, and a Q10 of 1.8 was used to correct each EOD frequency measurement to that expected at 26.0\u00b0C (Dunlap et al., 2000).\n\nEOD modulations were recorded and analyzed by using methods described previously (Kolodziejski et al.,2005). Briefly, fish were placed in a PVC tube with plastic mesh over both ends and a mesh-covered window midway down the length of the tube. The tube was placed in the center of a 37 l aquarium maintained at 25.8-27.0\u00b0C and at the conductivity and pH of the fish's home tank. The fish were allowed to acclimate to the recording tank for 1 h. A pair of carbon electrodes placed at the fish's head and tail recorded its EOD, and a second pair of electrodes on either side of the tube was used to present playback stimuli. The signal from the recording electrodes was band-pass filtered (0.1 Hz-10 kHz), amplified (100-1000\u00d7; Grass model P-55) and digitized at 44.1 kHz on the left channel of a sound card in a computer running Cool Edit Pro (Syntrillium, Phoenix, AZ, USA). Playback stimuli were sinusoidal voltage signals generated by a function generator (Model GFG-8216A or GFG-8219A;Instek, Chino, CA, USA) and calibrated to a root-meansquare field amplitude of 1.5 mV cm-1 parallel to the stimulating electrodes and midway between them. This amplitude approximates that of the EOD of a medium-sized A. devenanzii. A copy of the stimulus was digitized on the right channel of the sound card. A 4-min baseline recording was made from each fish without stimulation, and five recordings were made with different playback stimuli. Each recording consisted of a 1-min baseline period with no stimulation, two minutes of playback stimulation and 1 min post-stimulus. The frequencies of the playback stimuli were set relative to the fishes' own EOD frequencies: 150 Hz above and below the EOD frequency (\u00b1150 Hz), 20 Hz above and below the EOD frequency (\u00b120 Hz) and 5 Hz below the EOD frequency (-5 Hz). The playback frequencies spanned the species-typical range of EOD frequencies and were meant to simulate the presence of a conspecific fish in the recording tank. Based on results in other apteronotid fish, we expected the -5 Hz stimulus to evoke a jamming avoidance response (JAR)(Bullock et al., 1972). Stimuli were presented in random order and were separated by 10-min intervals without stimulation.\n\n### Analysis of EOD modulations\n\nWe used a customized procedure written by Brian Nelson (University of Oregon, Eugene, OR, USA) and running in Igor Pro (Wavemetrics, Lake Oswego,OR, USA) to calculate EOD frequency and to count and measure the parameters of EOD modulations (for details, see Kolodziejski et al., 2005). Briefly, any playback-induced contamination of the recording was removed by subtracting an appropriately scaled and phase-shifted copy of the playback signal. The fundamental frequency of the EOD was calculated by using an autocorrelation algorithm on 6 ms Hanning windows, advanced 2 ms per iteration. This process resulted in a temporal resolution of 2 ms and a frequency resolution of 0.5-3 Hz, depending on the signal-noise ratio of the recording. The Igor procedure used the mode of EOD frequency in sliding 2 s windows as a baseline frequency from which to detect EOD modulations. The procedure counted EOD modulations as any event in which EOD frequency exceeded this baseline frequency by more than 3 Hz for more than 10 ms and less than 60 s. The beginning and end of each EOD modulation was then defined as the time at which EOD frequency crossed a threshold of 1 Hz above or below the baseline frequency. The procedure then calculated the duration and peak frequency of each modulation. Each EOD modulation was also examined by the experimenter to confirm that the procedure accurately identified the EOD modulation and measured its parameters.\n\n### Statistics\n\nBody mass, EOD frequency and the numbers and parameters of different types of EOD modulations were compared between males and females by using unpaired t-tests. To avoid pseudoreplication, we calculated mean parameter values (FM and duration) for different EOD modulation types for each fish, and performed statistical analyses with the mean values for each individual as independent observations. Repeated measures analysis of variance (RM-ANOVA),with sex as an independent variable and stimulus frequency as the repeated measure, was used to determine whether the production of each type of EOD modulation was influenced by the frequency of the playback stimulus. Since all fish received the same set of stimuli and stimulus frequency did not affect the production of EOD modulations (see Results), we analyzed pooled data for all of the EOD modulations that each individual produced during all six 4-min recordings (five recordings with stimuli and one baseline recording). Pearson's correlations were used to test for correlations between body mass and EOD frequency and numbers of EOD modulations. Results of statistical tests were considered significant when P<0.05.\n\n### Reproductive condition and body mass\n\nThe fish in this study were sexually mature and had well-developed gonads. We were only able to measure the GSI in six fish that died or were killed for use in a separate pilot study. The GSI was 0.34\u00b10.14 (0.135-0.614, N=3) in the males and 1.47\u00b10.47 (0.725-2.35, N=3) in the females. These values are similar to those in previous studies that found sex differences in the electrocommunication signals of other apteronotid species (Dunlap et al., 1998; Kolodziejski et al., 2005). Visual inspection of the gonads in the laparotomized fish also demonstrated that the gonads were well-developed (e.g. yolked follicles were present in females) and that the reproductive condition of these fish was comparable to that of the fish whose GSI values were measured. All of the fish were adults,and total body length for the fish used in this study (180.1-219.0 mm) was at the top end of the range of lengths recorded in the holotype and paratypes of A. devenanzii (Mago-Leccia et al., 1985). The size of males and females overlapped considerably,and there was no sex difference in body mass(Table 1, t9=0.40, P=0.70).\n\nTable 1.\n\nSummary of sex differences in physical traits, EOD frequency and EOD modulations\n\nTraitMalesFemales1\nBody mass (g)\u00a034.8\u00b15.7\u00a032.2\u00b13.3\nEOD frequency(Hz)2\u00a01052.6\u00b122.7\u00a01101.4\u00b114.4\nEOD modulations (EODMs, chirps +GFRs)3\nTotal number4\u00a024.6\u00b14.5\u00a014.0\u00b13.3\nBy type3\nAll chirps (single- and multi-peaked)\nNumber produced4\u00a018.3\u00b14.2\u00a06.8\u00b12.85\n% of total EODMs\u00a068.3\u00b17.9\u00a040.0\u00b19.5*\nPositive FM (Hz)\u00a0170.7\u00b110.6\u00a0167.3\u00b112.2\nDuration (s)\u00a00.090\u00b10.011\u00a00.060\u00b10.021\nMulti-peaked chirps4\u00a06.0\u00b12.1\u00a00.4\u00b10.3*\nNumber produced4\u00a06.3\u00b11.2\u00a06.6\u00b11.5\n% of total EODMs\u00a031.7\u00b17.9\u00a060.0\u00b19.5*\nPositive FM (Hz)\u00a012.8\u00b12.4\u00a08.2\u00b11.2\nDuration (s)\u00a01.06\u00b10.33\u00a00.75\u00b10.48\nMulti-peaked GFRs4\u00a02.3\u00b10.5\u00a01.6\u00b10.5\nTraitMalesFemales1\nBody mass (g)\u00a034.8\u00b15.7\u00a032.2\u00b13.3\nEOD frequency(Hz)2\u00a01052.6\u00b122.7\u00a01101.4\u00b114.4\nEOD modulations (EODMs, chirps +GFRs)3\nTotal number4\u00a024.6\u00b14.5\u00a014.0\u00b13.3\nBy type3\nAll chirps (single- and multi-peaked)\nNumber produced4\u00a018.3\u00b14.2\u00a06.8\u00b12.85\n% of total EODMs\u00a068.3\u00b17.9\u00a040.0\u00b19.5*\nPositive FM (Hz)\u00a0170.7\u00b110.6\u00a0167.3\u00b112.2\nDuration (s)\u00a00.090\u00b10.011\u00a00.060\u00b10.021\nMulti-peaked chirps4\u00a06.0\u00b12.1\u00a00.4\u00b10.3*\nNumber produced4\u00a06.3\u00b11.2\u00a06.6\u00b11.5\n% of total EODMs\u00a031.7\u00b17.9\u00a060.0\u00b19.5*\nPositive FM (Hz)\u00a012.8\u00b12.4\u00a08.2\u00b11.2\nDuration (s)\u00a01.06\u00b10.33\u00a00.75\u00b10.48\nMulti-peaked GFRs4\u00a02.3\u00b10.5\u00a01.6\u00b10.5\n\nValues are means \u00b1 s.e.m.\n\n*\n\nStatistically significant sex difference; unpaired t-test, P<0.05.\n\n1\n\nFor body mass, N=5 males, 6 females. For EOD frequency, N=11 males, 10 females. For EOD modulations, N=11 males, 8 females.\n\n2\n\nTemperature compensated to that expected at 26.0\u00b0C (see Materials and methods).\n\n3\n\nSee Fig. 1, Materials and methods and Results for a description of EOD modulation types.\n\n4\n\nSince stimulus frequency did not affect the production of EODMs (see Fig. 4 and Results) and all fish received the same set of stimuli, data are pooled from all six 4-min recordings (five recordings with different stimuli and one baseline recording).\n\n5\n\nUnpaired t-test (males versus females), P=0.053.\n\n### EOD frequency\n\nEOD frequencies ranged from 917 to 1168 Hz at 26.0\u00b0C. In contrast to A. leptorhynchus and A. albifrons, in which EOD frequency is highly sexually dimorphic (Dunlap and Larkins-Ford, 2003; Kolodziejski et al., 2005; Meyer, 1983), EOD frequency in A. devenanzii did not differ significantly between males and females(t19=1.78, P=0.091), although males tended to have slightly lower EOD frequencies than females(Table 1). EOD frequency was not significantly correlated with body mass (r2=0.11, P=0.33).\n\n### Structure and types of EOD modulations\n\nAs in other apteronotids, A. devenanzii produced both chirps and GFRs (Fig. 1). The FM (increase in EOD frequency) of chirps ranged from 90 to 404 Hz, with most chirps having 100-250 Hz of FM. Chirp durations ranged from 18 ms to 2 s, although most chirps were 20-150 ms long. GFRs typically had much less FM (3-100 Hz,interquartile range 4.6-11.1 Hz) and longer and more variable duration (14 ms-15 s, interquartile range 32-264 ms). In A. leptorhynchus and A. albifrons, chirps can be unambiguously placed into two broad categories: high-frequency chirps with greater than 150 Hz of FM and low-frequency chirps with \u223c30-100 Hz of FM(Fig. 1B)(Bastian et al., 2001; Engler et al., 2000; Hagedorn and Heiligenberg,1985; Kolodziejski et al.,2005). The chirps of A. devenanzii could not be placed into clear categories based on the amount of FM. The FM of A. devenanzii chirps was most similar to that of the high-frequency chirps of A. leptorhynchus and A. albifrons (i.e. typically greater than 100 Hz of FM), and no low-frequency chirps were produced. Both the chirps and GFRs of A. devenanzii, however, did vary systematically in another parameter: the number of frequency peaks. Although some chirps and GFRs had a single frequency peak, similar to that in most chirps produced by A. leptorhynchus and A. albifrons, many A. devenanzii chirps (26.6%) and GFRs (33%) had multiple frequency peaks(Fig. 1A,E). Most multi-peaked chirps had 2-4 peaks, although a few had as many as nine peaks. Interestingly,although multi-peaked chirps typically had greater mean duration than single-peaked chirps, the duration of single-peaked chirps was more variable than that of multi-peaked chirps, and the longest chirps were single-peaked rather than multi-peaked (Fig. 1A,C,D). Unlike the high-frequency chirps of A. leptorhynchus, but similar to those in A. albifrons, the chirps of A. devenanzii lacked frequency undershoots (i.e. a decrease in EOD frequency below its baseline) at the end of the chirp. Similarly, chirps in both A. devenanzii and A. albifrons had durations several times longer than those of A. leptorhynchus(Fig. 1; Table 1)(Dunlap and Larkins-Ford, 2003; Kolodziejski et al.,2005).\n\n### Sex differences in EOD modulations\n\nEOD modulations were sexually dimorphic in A. devenanzii, but these sex differences were less pronounced than those in A. leptorhynchus (Table 1; Fig. 2). Although males tended to produce more chirps than females, this difference did not reach statistical significance (t17=2.08, P=0.053). The proportion of EOD modulations that were chirps (as opposed to GFRs), however, was significantly greater in males than females; on average, 68.3% of the male EOD modulations were chirps, compared with only 40% of the female EOD modulations(t17=2.30, P=0.035). Most of the multi-peaked chirps were produced by males; males produced more than 12 times as many multi-peaked chirps as females (t17=2.30, P=0.035). There were no sex differences in the number of GFRs, and neither the FM nor the duration of chirps or GFRs differed significantly between males and females (unpaired t-tests, P>0.15 for all).\n\nThe production of EOD modulations was not related to individual variation in size. Body mass was not significantly correlated with the number of total EOD modulations (r2<0.01, P=0.94), chirps(r2<0.01, P=0.84), GFRs(r2=0.07, P=0.42), multi-peaked chirps(r2=0.05, P=0.51) or multi-peaked GFRs(r2<0.01, P=0.78).\n\nFig. 1.\n\nEOD modulations in A. devenanzii (A,C-E) and A. leptorhynchus (B). (A) Scatter plot of the frequency modulation (FM, Hz)and duration (s) of 1286 EOD modulations recorded from 11 male and eight female A. devenanzii. Two types of EOD modulations could be distinguished based on the degree of FM: chirps (squares) and gradual frequency rises (GFRs, circles). Chirps and GFRs could have either single frequency peaks (open symbols) or multiple frequency peaks (grey symbols). Note that because so many chirps are plotted, many single-peaked chirps and GFRs are obscured by overlying multi-peaked chirps and GFRs. (B) A comparable plot for 7950 EOD modulations from A. leptorhynchus based on data from Kolodziejski et al. (Kolodziejski et al., 2005). A. leptorhynchus produces GFRs (circles) and two types of chirps: high-frequency chirps (squares) and low-frequency chirps(triangles). Note the absence of low-frequency chirps in Adontosternarchus and the lack of multi-peaked modulations in A. leptorhynchus. (C) Histogram of the duration of single-peaked chirps. Although most single-peaked chirps lasted 0.02-0.08 s, there is a tail' of longer duration chirps. (D) Histogram of the duration of multi-peaked chirps in A. devenanzii. (E) Examples of single- and multi-peaked chirps and GFRs produced by A. devenanzii. The top trace in each example tracks EOD frequency, and the bottom trace is the raw voltage record. Note the different time and frequency scales for chirps and GFRs.\n\nFig. 1.\n\nEOD modulations in A. devenanzii (A,C-E) and A. leptorhynchus (B). (A) Scatter plot of the frequency modulation (FM, Hz)and duration (s) of 1286 EOD modulations recorded from 11 male and eight female A. devenanzii. Two types of EOD modulations could be distinguished based on the degree of FM: chirps (squares) and gradual frequency rises (GFRs, circles). Chirps and GFRs could have either single frequency peaks (open symbols) or multiple frequency peaks (grey symbols). Note that because so many chirps are plotted, many single-peaked chirps and GFRs are obscured by overlying multi-peaked chirps and GFRs. (B) A comparable plot for 7950 EOD modulations from A. leptorhynchus based on data from Kolodziejski et al. (Kolodziejski et al., 2005). A. leptorhynchus produces GFRs (circles) and two types of chirps: high-frequency chirps (squares) and low-frequency chirps(triangles). Note the absence of low-frequency chirps in Adontosternarchus and the lack of multi-peaked modulations in A. leptorhynchus. (C) Histogram of the duration of single-peaked chirps. Although most single-peaked chirps lasted 0.02-0.08 s, there is a tail' of longer duration chirps. (D) Histogram of the duration of multi-peaked chirps in A. devenanzii. (E) Examples of single- and multi-peaked chirps and GFRs produced by A. devenanzii. The top trace in each example tracks EOD frequency, and the bottom trace is the raw voltage record. Note the different time and frequency scales for chirps and GFRs.\n\n### Effect of stimulus frequency on EOD modulations\n\nA. devenanzii did not chirp differently in response to different stimulus frequencies (Fig. 3). Neither the total number of chirps nor the number of multi-peaked chirps was affected by stimulus frequency (RM-ANOVA, main effect of stimulus frequency, F4,68=1.27 and 1.02, P>0.29). Consistent with the unpaired t-test demonstrating that males produced more multi-peaked chirps, the number of multi-peaked chirps was significantly affected by sex (RM-ANOVA, F1,17=5.15, P=0.037);but the interactions of stimulus frequency and sex on the total number of chirps or multi-peaked chirps were not significant (RM-ANOVA, F4,68=0.74 and 0.88, P>0.48). Neither sex nor stimulus frequency affected the number of GFRs or multi-peaked GFRs (RM-ANOVA, P>0.26 for all factors).\n\nFig. 2.\n\nNumbers of each type of EOD modulation produced (mean \u00b1 s.e.m.) by male (white bars) and female (grey bars) A. devenanzii. The P-values for significant or marginally insignificant sex differences(unpaired t-tests) are indicated above the bars.\n\nFig. 2.\n\nNumbers of each type of EOD modulation produced (mean \u00b1 s.e.m.) by male (white bars) and female (grey bars) A. devenanzii. The P-values for significant or marginally insignificant sex differences(unpaired t-tests) are indicated above the bars.\n\nFig. 3.\n\nEffect of stimulus difference frequency (i.e. the difference between the subject fish's EOD frequency and the stimulus frequency) on the number of chirps (A) and GFRs (B). The numbers of multi-peaked chirps and multi-peaked GFRs are indicated by broken lines, and the total numbers of chirps and GFRs by solid lines. Stimulus frequency did not significantly affect the number of any of these EOD modulations.\n\nFig. 3.\n\nEffect of stimulus difference frequency (i.e. the difference between the subject fish's EOD frequency and the stimulus frequency) on the number of chirps (A) and GFRs (B). The numbers of multi-peaked chirps and multi-peaked GFRs are indicated by broken lines, and the total numbers of chirps and GFRs by solid lines. Stimulus frequency did not significantly affect the number of any of these EOD modulations.\n\nAs in other apteronotid species (Bullock et al., 1972), the -5 Hz difference frequency stimulus evoked a JAR. The magnitude of the JAR (i.e. the sustained increase in EOD frequency caused by the -5 Hz difference frequency stimulus) was not sexually dimorphic(males 5.48\u00b10.76 Hz, females 5.09\u00b11.11 Hz; t16=0.30, P=0.77).\n\nThe electrocommunication signals of A. devenanzii differed from those in other apteronotid species. EOD frequency was not sexually dimorphic,and sex differences in chirping were less pronounced than in A. leptorhynchus. Chirps could not be separated into types based on the amount of FM, but both chirps and GFRs could have either single or multiple frequency peaks. Multi-peaked chirps were predominantly produced by males. Finally, unlike other apteronotids, A. devenanzii did not chirp differently in response to playbacks of different frequencies.\n\n### Species diversity in the structure of EOD modulations\n\nThe two other apteronotid species whose electrocommunication behavior has been extensively studied produce three comparable types of EOD modulations(Kolodziejski et al., 2005). Both A. leptorhynchus and A. albifrons produce high-frequency chirps, with more than 150 Hz of FM; low-frequency chirps, with approximately 20-150 Hz of FM; and GFRs, which have much less FM and more variable duration. The main difference between the EOD modulations of these two Apteronotus species is that the chirps of A. albifronslast 8-12 times longer than those of A. leptorhynchus(Dunlap and Larkins-Ford, 2003; Kolodziejski et al., 2005). A. devenanzii also produced chirps and GFRs, but the chirps of A. devenanzii could not be categorized as high- and low-frequency chirps. The FM of A. devenanzii chirps (90-404 Hz) was most similar to the FM of high-frequency chirps in Apteronotus. Unlike Apteronotus, A. devenanzii never produced chirps with less than 90 Hz of FM(Fig. 1A). A. devenanzii also differed from the other apteronotids in the production of multi-peaked EOD modulations. Many of the chirps and GFRs in A. devenanzii had multiple frequency peaks. Although A. albifronscan produce multi-peaked high-frequency chirps, and A. leptorhynchuscan produce extremely long-duration high-frequency chirps, these modulations are rare (Dunlap and Larkins-Ford,2003; Engler and Zupanc,2001). Similarly, although GFRs in all apteronotids are highly variable in duration and can have complex FM over time in A. albifrons (Serrano-Fernandez,2003), GFRs with multiple sharp frequency peaks like those in A. devenanzii (Fig. 1E) have not been reported in other apteronotids. The species differences in chirp parameters suggest that the structure of chirps and GFRs and the differentiation of chirps into categories have changed during apteronotid evolution (Fig. 4). In Apteronotus, distinct chirp types differ mainly in the degree of FM, whereas in Adontosternarchus chirp categories may be based on whether they are single- or multi-peaked. Previous studies have hypothesized distinct functions for different categories of chirps in A. leptorhynchus, with high-frequency chirps serving as courtship signals,low-frequency chirps as aggressive signals and GFRs as either submissive signals or victory cries' (Bastian et al.,2001; Dye and Heiligenberg,1987; Engler and Zupanc,2001; Hagedorn and Heiligenberg, 1985; Serrano-Fernandez, 2003). Additional comparative studies are needed to determine whether the function,as well as the structure, of different chirp types varies across apteronotid species.\n\nFig. 4.\n\nPartial phylogeny of wave-type gymnotiform fish, illustrating evolution of electrocommunication signals and their sexual dimorphism. Phylogeny based on Crampton and Albert (Crampton and Albert,2006). Comparison of electrocommunication signals based on this study and published reports (Hopkins,1974b; Hopkins,1974c; Meyer,1983; Hagedorn and Heiligenberg, 1985; Dye,1987; Zupanc and Maler,1993; Dunlap and Zakon,1998; Dunlap et al.,1998; Dunlap and Larkins-Ford,2003; Kolodziejski et al.,2005). X', presence of the trait; O', absence of the trait; ?',either the trait has not been investigated or the data are equivocal. 1Long- and short-duration interruptions in Eigenmannia may be analogous to high- and low-frequency chirps in Apteronotus(Hagedorn and Heiligenberg,1985; Hopkins,1974c). 2A. albifrons can produce GFRs and chirps that have complex spectro-temporal structure(Dunlap and Larkins-Ford, 2003; Serrano-Fernandez, 2003), but they are rare and are not similar to the multi-peaked chirps and GRFs of A. devenanzii. 3Hopkins recorded EOD modulations in only one female Sternopygus and it is thus unclear whether EOD modulations are sexually dimorphic in this genus(Hopkins, 1974b). 4Data from Hopkins (Hopkins,1974c) and Hagedorn and Heiligenberg(Hagedorn and Heiligenberg,1985) suggest sex differences in the number and\/or structure of interruptions and rises, but statistical analyses were not reported. A, lower EOD frequencies in males than females; B, gradual frequency rises (GFRs); C,EOD modulations differentially depend on EOD frequencies of other fish; D,chirping; E, distinct high- and low-frequency chirps; F, higher EOD frequencies in males than females; G, sex difference in number of EOD modulations; H, short-duration chirps; I, high-frequency chirps with frequency undershoots; J, loss of sex difference in EOD frequency; K, multi-peaked chirps and GFRs common; L, more multi-peaked chirps produced by males; M, loss of differential production of EOD modulations based on EOD frequencies of other fish.\n\nFig. 4.\n\nPartial phylogeny of wave-type gymnotiform fish, illustrating evolution of electrocommunication signals and their sexual dimorphism. Phylogeny based on Crampton and Albert (Crampton and Albert,2006). Comparison of electrocommunication signals based on this study and published reports (Hopkins,1974b; Hopkins,1974c; Meyer,1983; Hagedorn and Heiligenberg, 1985; Dye,1987; Zupanc and Maler,1993; Dunlap and Zakon,1998; Dunlap et al.,1998; Dunlap and Larkins-Ford,2003; Kolodziejski et al.,2005). X', presence of the trait; O', absence of the trait; `?',either the trait has not been investigated or the data are equivocal. 1Long- and short-duration interruptions in Eigenmannia may be analogous to high- and low-frequency chirps in Apteronotus(Hagedorn and Heiligenberg,1985; Hopkins,1974c). 2A. albifrons can produce GFRs and chirps that have complex spectro-temporal structure(Dunlap and Larkins-Ford, 2003; Serrano-Fernandez, 2003), but they are rare and are not similar to the multi-peaked chirps and GRFs of A. devenanzii. 3Hopkins recorded EOD modulations in only one female Sternopygus and it is thus unclear whether EOD modulations are sexually dimorphic in this genus(Hopkins, 1974b). 4Data from Hopkins (Hopkins,1974c) and Hagedorn and Heiligenberg(Hagedorn and Heiligenberg,1985) suggest sex differences in the number and\/or structure of interruptions and rises, but statistical analyses were not reported. A, lower EOD frequencies in males than females; B, gradual frequency rises (GFRs); C,EOD modulations differentially depend on EOD frequencies of other fish; D,chirping; E, distinct high- and low-frequency chirps; F, higher EOD frequencies in males than females; G, sex difference in number of EOD modulations; H, short-duration chirps; I, high-frequency chirps with frequency undershoots; J, loss of sex difference in EOD frequency; K, multi-peaked chirps and GFRs common; L, more multi-peaked chirps produced by males; M, loss of differential production of EOD modulations based on EOD frequencies of other fish.\n\nTwo chirp parameters in A. devenanzii were similar to those in A. albifrons, but differed from those in A. leptorhynchus. As in A. albifrons, A. devenanzii chirps lasted several times longer than most chirps in A. leptorhynchus. In addition, the high-frequency chirps of both A. devenanzii and A. albifrons lacked frequency undershoots. These results suggest that short-duration chirps and frequency undershoots are derived characters in A. leptorhynchus(Fig. 4). These two parameters could be mechanistically linked. Chirping is caused by glutamatergic excitation from the prepacemaker nucleus accelerating the firing rates of neurons in the pacemaker nucleus, the central pattern generator for the EOD. It is possible that the rapid removal of excitation needed to produce short-duration chirps in A. leptorhynchus results in rebound hyperpolarization in the pacemaker neurons, reducing their firing rate and leading to a frequency undershoot. Such rebound hyperpolarization might not occur if the removal of excitation is more gradual, as would be expected for the longer duration chirps of A. albifrons and A. devenanzii. Consistent with this hypothesis, A. leptorhynchusrarely produces extremely long duration high-frequency chirps, which also lack frequency undershoots [Type 4 chirps of Engler and Zupanc(Engler and Zupanc,2001)].\n\nFurther studies are needed to investigate the neural mechanisms underlying species differences in the structure of EOD modulations. In particular, what aspects of electromotor physiology allow the production of multi-peaked chirps and GFRs in Adontosternarchus but not Apteronotus? One possibility is that projection neurons in the prepacemaker nucleus in Adontosternarchus fire in bursts and excite the pacemaker nucleus in an oscillatory pattern during EOD modulations, whereas those in Apteronotus fire tonically. Alternatively, differences in chirp structure may result from species differences in postsynaptic responsiveness or intrinsic excitability of neurons in the pacemaker nucleus. For example,tonic glutamatergic excitation of pacemaker neurons by prepacemaker afferents might smoothly increase EOD frequency during chirps and GFRs in Apteronotus but cause oscillating FM in Adontosternarchus. Dunlap and Larkins-Ford similarly hypothesized that differences between A. leptorhynchus and A. albifrons in chirp duration might be mediated by postsynaptic mechanisms in the pacemaker nucleus(Dunlap and Larkins-Ford,2003). The ability to study neuronal excitability by using in vitro preparations of the pacemaker nucleus(Dye, 1988; Smith and Zakon, 2000) and prepacemaker nucleus (G.T.S. and J. A. Kolodziejski, unpublished observations)will allow these hypotheses to be tested.\n\nEOD waveform and frequency vary considerably across species and may be used to identify conspecifics (Hopkins,1974a; Kramer et al.,1980). The structure of EOD modulations, however, has been examined in relatively few species (Dunlap and Larkins-Ford, 2003; Hagedorn and Heiligenberg,1985; Hopkins,1974b; Hopkins,1974c; Kolodziejski et al.,2005). Our results and those of other studies suggest that the structure of EOD modulations may vary as much across species as EOD frequency,and thus may also convey species-identifying information.\n\n### Sex differences in EOD modulations\n\nSexual dimorphism of chirping varies across apteronotid species. Chirping is highly sexually dimorphic in A. leptorhynchus. Males chirp 20 to 40 times more than females, and high-frequency chirps are produced almost exclusively by males (Dunlap et al.,1998; Kolodziejski et al.,2005; Zupanc and Maler,1993). The number of chirps in A. albifrons is not sexually dimorphic, but chirp structure does differ between the sexes(Dunlap and Larkins-Ford, 2003; Dunlap et al., 1998; Kolodziejski et al., 2005). Male A. albifrons produce more high-frequency chirps than females,and male chirps last longer than those of females. As in A. albifrons, the total number of EOD modulations was not sexually dimorphic in A. devenanzii, but males and females did differ in the types of chirps produced. Unlike the Apteronotus species, A. devenanzii did not produce distinct high- and low-frequency chirps, and chirps similar to the high-frequency chirps of Apteronotus were produced by both sexes. Male A. devenanzii, however, produced more than 10 times as many multi-peaked chirps as females. Thus, multi-peaked chirps, the electrocommunication signals that are most unique to Adontosternarchus, are also the most sexually dimorphic signals in A. devenanzii. This raises the interesting possibility that different chirp parameters have been sexually selected in different apteronotid lineages. In Apteronotus, high-frequency chirps are largely male-specific signals, whereas in Adontosternarchus, multi-peaked chirps are predominantly produced by males. Future studies examining the behavioral responses of fish to different types of chirps could test the hypothesis that the different types of electrocommunication signals produced mostly by males (i.e. high-frequency chirps in Apteronotus and multi-peaked chirps in Adontosternarchus) have evolved similar functions (e.g. courtship). Sexual selection for different signal parameters in closely related lineages has also been reported for other reproductive communication signals. For example, different components of song have diversified through sexual selection in different congeneric songbird species and in different populations of a ring species(Irwin, 2000; Price and Lanyon, 2004).\n\n### Sex differences in EOD frequency\n\nWe found no significant sex difference in EOD frequency in A. devenanzii, even though, based on the GSI and the presence of yolked follicles in females, the fish in this study were sexually mature. EOD frequencies of males and females overlapped considerably. By contrast, EOD frequency in other apteronotid species differs markedly between males and females, with little or no overlap between the sexes(Dunlap et al., 1998; Hagedorn and Heiligenberg,1985; Kolodziejski et al.,2005; Meyer,1983). Thus, the four species of apteronotids in which sex differences in EOD frequency have been examined display three distinct patterns of sexual dimorphism: (1) males have higher EOD frequencies than females in A. leptorhynchus and A. rostratus; (2) males have lower EOD frequencies than females in A. albifrons; and (3) EOD frequency is not sexually dimorphic in A. devenanzii. The diversity in the pattern of sex differences in the EOD in the few apteronotid species studied demonstrates that the direction and magnitude of sexual dimorphism in EOD frequency is evolutionary labile in this family.\n\nIn both Sternopygus and Eigenmannia, non-apteronotid gymnotiforms that also produce wave-type EODs, EOD frequency is lower in males than females (Fig. 4)(Dunlap and Zakon, 1998; Hagedorn and Heiligenberg,1985; Hopkins,1974b; Zakon et al.,1991). It has thus been hypothesized that ancestral apteronotids also had males with lower EOD frequencies than females and that the reversal in the direction of sexual dimorphism of EOD frequency in A. leptorhynchus and A. rostratus is derived(Dunlap et al., 1998). Our results suggest that there may also have been a derived loss of sexual dimorphism in EOD frequency in the Adontosternarchus lineage(Fig. 4).\n\nThe interspecific variation not only in the presence or absence of sexual dimorphism of EOD frequency, but also in the direction of sex differences is unusual. Species differences in the magnitude of sexual dimorphism are common and may reflect differences in the relative strength of sexual and natural selection (Andersson, 1994). Although the direction of sexual dimorphism in body size also often varies(Fairbairn, 1997), species differences in the direction of sexual dimorphism in communication behavior,particularly in the absence of sex-role reversal, are rare. One example occurs in the parrot, Eclectus roratus, in which greater predation vulnerability in males and nest-site competition in females have favored females that are more brightly colored than males despite predominantly female parental care (Heinsohn et al.,2005).\n\nWhy does both the direction and degree of sexual dimorphism in EOD frequency vary across apteronotid species? In electric fish that produce pulse-type EODs, the waveform of the EOD is often sexually dimorphic. Furthermore, sexual dimorphism of EOD waveform is typically in the same direction: males have longer duration, higher amplitude and\/or more asymmetric EOD pulses than females (Hopkins,1999). Sex differences in EOD waveform may be driven by strong directional sexual selection because the long duration, high amplitude and asymmetric EOD pulses of males require more energy to produce and\/or make males more conspicuous to both females and electroreceptive predators(Hopkins, 1999; Stoddard, 1999; Stoddard, 2006). The relative costs and benefits of low-versus high-frequency EODs in wave-type electric fish are less clear. In fish that produce low-frequency EODs, each discharge lasts longer and thus may require more energy to produce(Mills and Zakon, 1987). However, because fish with low-frequency EODs produce fewer discharges per second, low-frequency EODs do not necessarily require more overall energy than high-frequency EODs (Hopkins,1999). Furthermore, because capacitative coupling in the neurogenic electric organ of apteronotids strongly attenuates the direct current (DC) components of the EOD that are detectable by ampullary electroreceptors (Bennett,1971), low-frequency EODs are unlikely to be any more or less conspicuous to electroreceptive predators than high-frequency EODs. If the costs and benefits of the EOD are not simply related to EOD frequency, the constraints underlying the directional sexual selection on the EOD in pulse-type electric fish may be relaxed in apteronotids, allowing diversification in the direction as well as the magnitude of sex differences in EOD frequency. Additional studies characterizing sex differences in EOD frequency in other gymnotiform species and determining whether the direction or magnitude of these sex differences is correlated with ecological factors(e.g. mating system, sociality, foraging ecology or predation) are needed to better understand the factors driving the evolution of sexually dimorphic EOD frequencies.\n\nThe physiological mechanisms underlying species diversity in electrocommunication behavior also require further investigation. The hormonal control of sex differences in EOD frequency has been characterized in Apteronotus. Consistent with the reversal in the direction of sexual dimorphism, androgens increase EOD frequency in A. leptorhynchus, but decrease EOD frequency in A. albifrons(Dunlap et al., 1998). The effects of hormones on electrocommunication signals in Adontosternarchus have not yet been studied, but one possible mechanism that could contribute to the lack of a sex difference in EOD frequency in this species would be an insensitivity of EOD frequency to gonadal steroids.\n\n### Sexual dimorphism in EOD frequency and differential responsiveness to playbacks\n\nSex differences in EOD frequency may be associated with responsiveness to playbacks of different frequencies. A. leptorhynchus produces more low-frequency chirps, which may function as agonistic signals, in response to playbacks of frequencies 5-10 Hz away from the fishes' own EOD than to more distant frequencies (Bastian et al.,2001; Engler and Zupanc,2001). Because EOD frequency is sexually dimorphic in A. leptorhynchus, more low-frequency chirps are thus produced in response to the EODs of same-sex than opposite-sex individuals. Furthermore, A. leptorhynchus males produce more high-frequency chirps, which may function in courtship, to playbacks with frequencies 50-200 Hz away from that of the male's own EOD (Bastian et al.,2001; Engler and Zupanc,2001). Because female EOD frequencies are typically 100-200 Hz lower than those of males, this behavior results in males producing high-frequency chirps mostly in response to female EODs. A. albifronsalso chirps differently in response to playbacks of different frequencies (J. A. Kolodziejski and G.T.S., unpublished observations). By contrast, we found no effect of stimulus frequency on the number or structure of chirps in A. devenanzii. The lack of an effect of stimulus frequency on chirping could be explained by the fact that EOD frequency in A. devenanzii is not sexually dimorphic and therefore does not convey information about sex. In both A. leptorhynchus and A. albifrons, sex differences in EOD frequency make it a reliable cue for directing chirps towards receivers of one sex or the other. By contrast, because EOD frequency does not differ between males and females in A. devenanzii, chirping differently to EODs of different frequencies would not necessarily direct chirps at individuals based on their sex. If A. devenanzii direct their chirps in a sex-specific manner, they may use cues other than EOD frequency to assess the sex of potential receivers.\n\n### Apteronotid electrocommunication as a model for studying the evolution of sexually dimorphic behavior\n\nThe results of this study and previous studies in other apteronotid species demonstrate abundant diversity in electrocommunication behavior(Fig. 4). Phylogenetic comparative methods to more thoroughly investigate the evolution of this diversity will require the characterization of sex differences in the electrocommunication behavior in additional species. The relative ease with which electrocommunication signals can be elicited and analyzed will facilitate this process. Furthermore, because electric fish respond robustly to playbacks with both electrical (e.g. chirping) and physical behaviors (e.g. by attacking electrodes or depositing eggs near electrodes playing back conspecific EODs), this system can be used to study the evolution of signal perception as well as production (Dye,1987; Hagedorn and Heiligenberg, 1985; Hopkins,1974c). Finally, the simplicity of neural circuits that control both the EOD and its modulations(Heiligenberg et al., 1996; Smith, 1999; Zakon and Smith, 2002) will allow comparative studies to investigate how sexually dimorphic behaviors and the physiological mechanisms that control them evolve together.\n\nThe authors thank Johanna Kolodziejski for technical assistance and Laura Hurley and an anonymous reviewer for comments on an earlier version of the manuscript. Supported by NIH MH 066960.\n\n1994\n).\nSexual Selection\n. Princeton, NJ: Princeton University Press.\nBass, A. H. (\n1986\n). A hormone-sensitive communication system in an electric fish.\nJ. Neurobiol.\n17\n,\n131\n-156.\nBastian, J., Schniederjan, S. and Nguyenkim, J.(\n2001\n). Arginine vasotocin modulates a sexually dimorphic communication behavior in the weakly electric fish Apteronotus leptorhynchus.\nJ. Exp. Biol.\n204\n,\n1909\n-1923.\nBennett, M. V. L. (\n1971\n). Electric organs. In\nFish Physiology\n. Vol.\n5\n(ed. 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\section{Introduction}\label{sec1}
The following question was posed to me by Shiri Artstein--Avidan: `Does every convex body $K$ in the plane have a point $z$ such that the union of $K$ and its reflection in $z$ is convex?' After some surprise about never having come across this simple question, and after some fruitless attempts to find counterexamples, this finally led to the following answer ({\cite{Sch16}). Here we call the point $z$ a {\em convexity point} of $K$ if $(K-z)\cup(z-K)$ is convex.
\begin{theorem}\label{Thm1}
A convex body in the plane which is not centrally symmetric has three affinely independent convexity points.
\end{theorem}
A triangle and a Reuleaux triangle are examples of convex bodies with precisely three convexity points. This then raises the question whether the existence of just three convexity points is `typical'. We recall the meaning of this terminology. The space ${\mathcal K}^2$ of convex bodies in the plane with the Hausdorff metric is a complete metric space and hence a Baire space, that is, a topological space in which any intersection of countably many dense open sets is still dense. A subset of a Baire space is called {\em comeager} or {\em residual} if its complement is a {\em meager} set, that is, a countable union of nowhere dense sets (also said to be of {\em first Baire category}). The intersection of countable many comeager sets in a Baire space is still dense, which is a good reason to consider comeager sets as `large'. Therefore, one says that `most' convex bodies in the plane have a certain property, or that a `typical' planar convex body has this property, if the set of bodies with this property is comeager in ${\mathcal K}^2$. With this definition, we prove the following.
\begin{theorem}\label{Thm2}
A typical convex body in the plane has infinitely many convexity points.
\end{theorem}
A result from which this one follows will be formulated at the end of the next section, after some preparations.
For surveys on Baire category results in convexity, we refer the reader to Gruber \cite{Gru85, Gru93} and Zamfirescu \cite{Zam91, Zam09}.
\section{The middle hedgehog}\label{sec2}
We work in the Euclidean plane ${\mathbb R}^2$, with scalar product $\langle\cdot\,,\cdot\rangle$, induced norm $\|\cdot\|$ and unit circle ${\mathbb S}^1$. The set of convex bodies (nonempty, compact, convex subsets) in ${\mathbb R}^2$ is denoted by ${\mathcal K}^2$. We use the Hausdorff metric $\delta$, which is defined on all nonempty compact subsets of ${\mathbb R}^2$ (for notions from convex geometry not explained here, we refer to \cite{Sch14}). Let $K\in{\mathcal K}^2$ and $u\in{\mathbb S}^1$. By $H(K,u)$ we denote the supporting line of $K$ with outer unit normal vector $u$, and we call the line
$$ M_K(u):=\frac{1}{2}[H(K,u)+H(K,-u)]$$
the {\em middle line} of $K$ with normal vector $u$ (hence, $M_K(u)=M_K(-u)$). With $F(K,u):= K\cap H(K,u)$, which is the face of $K$ with outer normal vector $u$, we call the convex set
$$ Z_K(u):=\frac{1}{2}[F(K,u)+F(K,-u)]$$
(either a singleton or a segment) the {\em middle set} of $K$ with normal vector $u$. If $F(K,u)$ is one-pointed, we write $F(K,u)=\{x_K(u)\}$, and if also $F(K,-u)$ is one-pointed, then $Z_K(u)=\{m_K(u)\}$ with
$$ m_K(u)= \frac{1}{2}[x_K(u)+x_K(-u)].$$
We call $m_K(u)$ a {\em middle point} of $K$. The set
$$ {\mathcal M}_K:= \bigcup_{u\in{\mathbb S}^1} Z_K(u)$$
is the {\em middle hedgehog} of $K$. It is a closed curve, the locus of all midpoints of affine diameters, that is, chords of $K$ connecting pairs of boundary points lying in distinct parallel support lines.
The following lemma, proved in \cite{Sch16}, was crucial for the proof of Theorem \ref{Thm1}.
\begin{lemma}\label{Lem1}
Suppose that $K\in{\mathcal K}^2$ has no pair of parallel edges. Then each exposed point of the convex hull of the middle hedgehog ${\mathcal M}_K$ is a convexity point of $K$.
\end{lemma}
We consider special examples of middle hedgehogs. First, let $K$ be a convex polygon with no pair of parallel edges. For each edge $F(K,u)$ of $K$ we have $F(K,-u)=\{x_K(-u)\}$ and $Z_K(u)= (1/2)[F(K,u)+x_K(-u)]$. (Each middle point belongs to some $Z_K(u)$ with suitable $u$.) The union ${\mathcal M}_K$ of these segments, over all unit normal vectors of the edges, is a closed polygonal curve.
Second, let $K\in{\mathcal K}^2$ be strictly convex. Then the support function of $K$, which we denote by $h(K,\cdot)$, is differentiable on ${\mathbb R}^2\setminus \{0\}$. To obtain a parametrization of ${\mathcal M}_K$, we choose an orthonormal basis $(e_1,e_2)$ of ${\mathbb R}^2$ and write
$$\mbox{\boldmath$u$}(\varphi):= (\cos\varphi)e_1+(\sin\varphi)e_2,\quad \varphi\in{\mathbb R};$$
then $(\mbox{\boldmath$u$}(\varphi),\mbox{\boldmath$u$}'(\varphi))$ is an orthonormal frame with the same orientation as $(e_1,e_2)$. We define
$$ {\bf x}(\varphi) := m_K(\mbox{\boldmath$u$}(\varphi))$$
and
\begin{equation}\label{2.1}
p(\varphi) := \frac{1}{2}\left[h_K(\mbox{\boldmath$u$}(\varphi))-h_K(-\mbox{\boldmath$u$}(\varphi))\right]
\end{equation}
for $\varphi\in[0,\pi]$. Note that ${\bf x}$ is a parametrized closed curve, since $m_K(u)=m_K(-u)$ for $u\in{\mathbb S}^1$. Since $h_K(u)=\langle x_K(u),u\rangle$, we have
\begin{equation}\label{2.2}
p(\varphi)=\langle {\bf x}(\varphi), \mbox{\boldmath$u$}(\varphi)\rangle.
\end{equation}
Differentiating (\ref{2.2}) and using that $x_K(u)= \nabla h_K(u)$ (where $\nabla$ denotes the gradient; see \cite{Sch14}, Corollary 1.7.3), we obtain
\begin{equation}\label{2.3}
p'(\varphi) =\langle {\bf x}(\varphi),\mbox{\boldmath$u$}'(\varphi)\rangle.
\end{equation}
The equations equations (\ref{2.2}) and (\ref{2.3}) together yield
$$ {\bf x}(\varphi) = p(\varphi) \mbox{\boldmath$u$}(\varphi) +p'(\varphi)\mbox{\boldmath$u$}'(\varphi),\quad \varphi\in[0,\pi].$$
This is a convenient parametrization of the middle hedgehog. The intersection point of the middle lines $M_K(\mbox{\boldmath$u$}(\varphi))$ and $M_K(\mbox{\boldmath$u$}(\varphi+\varepsilon))$ converges to $m_K(\mbox{\boldmath$u$}(\varphi))$ for $\varepsilon\to 0$, thus
${\bf x}$ is the envelope of the family of middle lines of $K$, suitably parametrized. We remark that generalized envelopes of more general line families were studied in \cite{HS53}.
We remark further that in the terminology of Martinez--Maure (see \cite{MM95, MM97}, for example, also \cite{MM99, MM06}), the curve ${\bf x}$ is a planar `projective hedgehog'. The set $\{{\bf x}(\varphi):\varphi\in[0,\pi)\}$ has been introduced and investigated as the `midpoint parallel tangent locus' in \cite{Hol01} and has been named the `area evolute' in \cite{Gib08}; a further study appears in \cite{Cra14}.
According to Lemma \ref{Lem1} and the fact that a typical convex body is strictly convex, Theorem \ref{Thm2} is a consequence of the following result.
\begin{theorem}\label{Thm3}
For a typical convex body in the plane, the convex hull of the middle hedgehog has infinitely many exposed points.\end{theorem}
\section{Proof of Theorem \ref{Thm3}}\label{sec3}
By ${\mathcal K}^2_*$ we denote the set of strictly convex convex bodies in ${\mathcal K}^2$. The set ${\mathcal K}^2_*$ is a dense $G_\delta$ set in ${\mathcal K}^2$ and hence is also a Baire space. Every set that is comeager in ${\mathcal K}^2_*$ is also comeager in ${\mathcal K}^2$.
To begin with the proof of Theorem \ref{Thm3}, we set
$$ {\mathcal A}:= \{K\in{\mathcal K}^2_*: {\rm conv}{\mathcal M}_K \mbox{ has only finitely many exposed points}\}$$
and, for $k\in {\mathbb N}$,
$$ {\mathcal A}_k:= \{K\in{\mathcal K}^2_*: {\rm conv}{\mathcal M}_K \mbox{ has at most $k$ exposed points}\}.$$
We shall prove the following facts.
\begin{lemma}\label{Lem2}
Each set ${\mathcal A}_k$ is closed in ${\mathcal K}^2_*$.
\end{lemma}
\begin{lemma}\label{Lem3}
Each set ${\mathcal A}_k$ is nowhere dense in ${\mathcal K}^2_*$.
\end{lemma}
When this has been proved, then we know that the set ${\mathcal A}= \bigcup_{k\in{\mathbb N}} {\mathcal A}_k$ is meager. Hence its complement, which is the set of all $K\in{\mathcal K}^2_*$ for which ${\rm conv}{\mathcal M}_K$ has infinitely many exposed points, is comeager in ${\mathcal K}^2_*$ and hence in ${\mathcal K}^2$. This is the assertion of Theorem \ref{Thm3}.
\vspace{2mm}
\noindent{\em Proof of Lemma} \ref{Lem2}. First we show that on ${\mathcal K}^2_*$, the mapping $K\mapsto{\mathcal M}_K$ is continuous (this would not be true if ${\mathcal K}^2_*$ were replaced by ${\mathcal K}^2$).
Let $(K_i)_{i\in{\mathbb N}}$ be a sequence in ${\mathcal K}^2_*$ converging to some $K\in {\mathcal K}^2_*$. To show that ${\mathcal M}_{K_i}\to {\mathcal M}_K$ in the Hausdorff metric for $i\to\infty$, we use Theorem 1.8.8 of \cite{Sch14} (it is formulated for convex bodies, but as its proof shows, it holds for connected compact sets---or see Theorems 12.2.2 and 12.3.4 in \cite{SW08}).
Let $x\in{\mathcal M}_{K}$. Then there is a vector $u\in{\mathbb S}^1$ with $x=(1/2)[x_K(u)+x_K(-u)]$. The sequence $(x_{K_i}(u))_{i\in{\mathbb N}}$ has a convergent subseqence, and its limit is a boundary point of $K$ with outer normal vector $u$, hence equal to $x_K(u)$. Since this holds for every convergent subsequence, the sequence $(x_{K_i}(u))_{i\in{\mathbb N}}$ itself converges to $x_K(u)$. Similarly, the sequence $(x_{K_i}(-u))_{i\in{\mathbb N}}$ converges to $x_K(-u)$. It follows that $m_{K_i}(u) = (1/2)[x_{K_i}(u)+x_{K_i}(-u)]\to (1/2)[x_{K}(u)+x_{K}(-u)]=x$ for $i\to\infty$, and here $m_{K_i}(u)\in {\mathcal M}_{K_i}$. Thus, each point in ${\mathcal M}_K$ is the limit of a sequence $(m_i)_{\in{\mathbb N}}$ with $m_i\in {\mathcal M}_{K_i}$ for $i\in{\mathbb N}$.
Let $x_{i(j)}\in{\mathcal M}_{K_{i(j)}}$ for a subsequence $(i(j))_{j\in{\mathbb N}}$, and suppose that $x_{i(j)}\to x$ for $j\to\infty$. Then $x_{i(j)}=(1/2)[x_{K_{i(j)}}(u_j)+x_{K_{i(j)}}(-u_j)]$ for suitable $u_j\in{\mathbb S}^1$ ($j\in{\mathbb N}$). There is a convergent subsequence of $(u_j)_{j\in{\mathbb N}}$, and we can assume that this is the sequence $(u_j)_{j\in{\mathbb N}}$ itself, say $u_j\to u$ for $j\to\infty$. Then $x_{K_{i(j)}}(u_j)\to x_K(u)$ and $x_{K_{i(j)}}(-u_j)\to x_K(-u)$, hence $x_{i(j)}\to (1/2)[x_K(u)+x_K(-u)]\in {\mathcal M}_K$. It follows that $x\in{\mathcal M}_K$. This completes the continuity proof for the mapping $K\mapsto{\mathcal M}_K$.
To show that ${\mathcal A}_k$ is closed in ${\mathcal K}^2_*$, let $(K_i)_{i\in{\mathbb N}}$ be a sequence in ${\mathcal A}_k$ converging to some $K\in {\mathcal K}^2_*$. As just shown, we have ${\mathcal M}_{K_i}\to {\mathcal M}_K$ and hence also ${\rm conv}{\mathcal M}_{K_i}\to {\rm conv}{\mathcal M}_K$ for $i\to\infty$, since the convex hull mapping is continuous (even Lipschitz, see \cite{Sch14}, p. 64). Since each ${\rm conv}{\mathcal M}_{K_i}$ is a convex polygon with at most $k$ vertices, also ${\rm conv}{\mathcal M}_{K}$ is a convex polygon with at most $k$ vertices, thus $K\in {\mathcal A}_k$. This completes the proof of Lemma \ref{Lem2}. \qed
To prepare the proof of Lemma \ref{Lem3}, we need to have a closer look at the middle hedgehog ${\mathcal M}_P$ of a convex polygon $P$. We assume in the following that $P$ has interior points and has no pair of parallel edges.
First, the unoriented normal directions of the edges of $P$ have a natural cyclic order. We may assume, without loss of generality, that no edge of $P$ is parallel to the basis vector $e_1$. Then there are angles $-\pi/2 < \varphi_1 < \varphi_2<\dots< \varphi_k< \pi/2$ such that, for each $i \in \{1,\dots,k\}$, either $\mbox{\boldmath$u$}(\varphi_i)$ or $-\mbox{\boldmath$u$}(\varphi_i)$ is an outer normal vector of an edge of $P$ (not both, since $P$ does not have a pair of parallel edges), and all unit normal vectors of the edges of $P$ are obtained in this way. We denote by $E_i$ the edge of $P$ that is orthogonal to $\mbox{\boldmath$u$}(\varphi_i)$. We call the pair $(E_i, E_{i+1})$ {\em consecutive} (where $E_{k+1} := E_1$; this convention is also followed below), and in addition we call it {\em adjacent} if $E_i\cap E_{i+1}$ is a vertex of $P$. For an angle $\psi\in[-\pi/2,\pi/2)$ we say that $\psi$ is {\em between} $\varphi_i$ and $\varphi_{i+1}$ if either $i\in \{1,\dots,k-1\}$ and $\varphi_i<\psi<\varphi_{i+1}$, or $i=k$ and either $-\pi/2 <\psi<\varphi_1$ or $\varphi_k<\psi\le \pi/2$. Let $(E_i, E_{i+1})$ be a consecutive pair. The following facts, to be used below, follow immediately from the definitions. If $\psi$ is between $\varphi_i$ and $\varphi_{i+1}$, then $\mbox{\boldmath$u$}(\psi)$ is not a normal vector of an edge of $P$. Suppose that, say, $\mbox{\boldmath$u$}(\varphi_i)$ is the outer normal vector of $E_i$. If $(E_i,E_{i+1})$ is adjacent, then $\mbox{\boldmath$u$}(\varphi_{i+1})$ is the outer normal vector of $E_{i+1}$. If $(E_i,E_{i+1})$ is not adjacent, then $\mbox{\boldmath$u$}(\varphi_{i+1})$ is the inner normal vector of $E_{i+1}$. These definitions of $E_i$ and $\varphi_i$ will be used in the rest of this note.
Now let $p$ and $q$ be {\em opposite} vertices of $P$, that is, vertices with $H(P,\mbox{\boldmath$u$}(\psi))\cap P=\{p\}$ and $H(P,-\mbox{\boldmath$u$}(\psi))\cap P=\{q\}$ for some $\psi$. After interchanging $p$ and $q$, if necessary, we can assume that $\psi\in[-\pi/2,\pi/2)$. Then there is a unique index $i\in\{1,\dots,k\}$ such that $\psi$ is between $\varphi_i$ and $\varphi_{i+1}$. The middle sets $Z_P(\mbox{\boldmath$u$}(\varphi_i))$ and $Z_P(\mbox{\boldmath$u$}(\varphi_{i+1}))$ have the midpoint $x=(p+q)/2$ in common. We say that $x$ is a {\em weak corner} of the middle hedgehog ${\mathcal M}_P$ if the pair $(E_i,E_{i+1})$ is adjacent, and $x$ is a {\em strong corner} of ${\mathcal M}_P$ if $(E_i,E_{i+1})$ is not adjacent. If $x$ is a weak corner, then the middle sets $Z_P(\mbox{\boldmath$u$}(\varphi_i))$ and $Z_P(\mbox{\boldmath$u$}(\varphi_{i+1}))$ lie on different sides of the line through $p$ and $q$, and if $x$ is a strong corner, then $Z_P(\mbox{\boldmath$u$}(\varphi_i))$ and $Z_P(\mbox{\boldmath$u$}(\varphi_{i+1}))$ lie on the same side of this line.
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\rput(8.8,0.5){$E_1$
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\rput(1.05,6.1){$E_4$
\rput(13.1,5.5){$E_5$
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\end{center}
\vspace{-2mm}
\noindent Figure 1: The middle hedgehog has one weak corner and seven strong corners, five of which are vertices of the convex hull.
\vspace{5mm}
\begin{lemma}\label{Lem4}
A weak corner of the middle hedgehog ${\mathcal M}_P$ is not a vertex of ${\rm conv}{\mathcal M}_P$.
\end{lemma}
\begin{proof}
We begin with an arbitrary vertex $x$ of ${\rm conv}{\mathcal M}_P$. Since ${\mathcal M}_P$ is the union of the finitely many middle sets $Z_P(\mbox{\boldmath$u$}(\varphi_i))$ (with $\varphi_i$ as above), the point $x$ must be one of the endpoints of these segments, thus $x$ is either a weak or a strong corner of ${\mathcal M}_P$.
We need to recall some facts from the proof of Lemma 6 in \cite{Sch16}. As there, we may assume, without loss of generality (after applying a rigid motion to $P$), that $x=0$ and that the orthonormal basis $(e_1,e_2)$ of ${\mathbb R}^2$ is such that
\begin{equation}\label{3.1}
\langle y,e_2\rangle >0 \quad \mbox{for each } y\in {\rm conv}{\mathcal M}_P \setminus\{0\}.
\end{equation}
Let $L$ be the line through $0$ that is spanned by $e_1$. For $\varphi\in (-\pi/2,\pi/2)$, the middle line $M_P(\mbox{\boldmath$u$}(\varphi))$ intersects the line $L$ in a point which we write as $f(\varphi)e_1$, thus defining a continuous function $f:(-\pi/2,\pi/2)\to{\mathbb R}$. It was shown in \cite{Sch16} that
$$ f(\varphi) =\frac{p(\varphi)}{\cos\varphi}.$$
At almost all $\varphi$, the functions $\varphi\mapsto h(P,\mbox{\boldmath$u$}(\varphi))$ and $\varphi\mapsto h(P,-\mbox{\boldmath$u$}(\varphi))$ are differentiable, hence the same holds for the function $f$, and where this holds, we have
\begin{equation}\label{3.2}
f'(\varphi)= \frac{\langle m_P(\mbox{\boldmath$u$}(\varphi)),e_2\rangle}{\cos^2\varphi},
\end{equation}
as shown in \cite{Sch16}.
We now first recall the rest of the proof of Lemma 6 in \cite{Sch16}, in a slightly simplified version. The claim to be proved is that
\begin{equation}\label{3.3}
0 \in M_P(\mbox{\boldmath$u$}(\varphi)) \mbox{ for some }\varphi\in(-\pi/2,\pi/2)\quad \Longrightarrow \quad 0\in Z_P(\mbox{\boldmath$u$}(\varphi)).
\end{equation}
By (\ref{3.2}) and (\ref{3.1}) we have $f'(\varphi)\ge 0$ for almost every $\varphi\in(-\pi/2,\pi/2)$. We conclude that the function $f$ (which is locally Lipschitz and hence the integral of its derivative) is weakly increasing on $(-\pi/2, \pi/2)$. Therefore, the set $I:=\{\varphi\in(-\pi/2, \pi/2): f(\varphi)=0\}$ is a closed interval (possibly one-pointed). Since $0\in{\mathcal M}_P$, there is some $\varphi_0\in (-\pi/2,\pi/2)$ with $0\in Z_P(\mbox{\boldmath$u$}(\varphi_0))$. If $I$ is one-pointed, then $I=\{\varphi_0\}$, and $0\notin M_P(\mbox{\boldmath$u$}(\varphi))$ for $\varphi\not=\varphi_0$. Thus, (\ref{3.3}) holds in this case. If $I$ is not one-pointed, then $f'(\varphi)=0$ for $\varphi \in {\rm relint}\,I$ and hence, by (\ref{3.2}) and (\ref{3.1}), $m_P(\mbox{\boldmath$u$}(\varphi))=0$ for $\varphi \in {\rm relint}\,I$. By continuity, we have $0\in Z_P(\mbox{\boldmath$u$}(\varphi))$ for all $\varphi\in I$. This shows that (\ref{3.3}) holds generally.
Now we can finish the proof of Lemma \ref{Lem4}. Suppose, to the contrary, that $0$ is a weak corner of ${\mathcal M}_P$. Then there is a consecutive, adjacent pair $(E_i,E_{i+1})$ of edges of $P$ such that $E_i\cap E_{i+1}=\{p\}$ for a vertex $p$ of $P$ and the line $H(P,\mbox{\boldmath$u$}(-\pi/2))$ supports $P$ at $p$. This is only possible if $(E_i,E_{i+1})= (E_k,E_{k+1})$. In this case, all the middle lines $M_P(\psi)$ with $\psi$ between $\varphi_k$ and $\varphi_{k+1}=\varphi_1$ pass through $0$. This means that the function $f$ defined above satisfies $f(\varphi)=0$ for $-\pi/2<\varphi \le \varphi_1$ and for $\varphi_k\le\varphi<\pi/2$. But since $f$ is increasing, it must then vanish identically, which is a contradiction, since $P$ is not centrally symmetric. This contradiction completes the proof of Lemma \ref{Lem4}.
\end{proof}
\vspace{2mm}
\noindent{\em Proof of Lemma} \ref{Lem3}. Let $k\in{\mathbb N}$. Since ${\mathcal A}_k$ is closed by Lemma \ref{Lem2}, the proof that ${\mathcal A}_k$ is nowhere dense amounts to showing that ${\mathcal A}_k$ has empty interior in ${\mathcal K}^2_*$. For this, let $K\in {\mathcal A}_k$ and $\varepsilon>0$ be given. We show that the $\varepsilon$-neighborhood of $K$ contains an element of ${\mathcal K}^2_*\setminus {\mathcal A}_k$.
In a first step, we choose a convex polygon $P$ with
\begin{equation}\label{3.0}
K\subset {\rm int} P,\quad P \subset {\rm int}(K+\varepsilon B^2),
\end{equation}
where $B^2$ denotes the closed unit disc of ${\mathbb R}^2$. We can do this in such a way that $P$ satisfies the following assumptions. First, $P$ has no pair of parallel edges. Second, $P$ has no `long' edge, by which we mean an edge the endpoints of which are opposite points of $P$. The goal of the following is to perform small changes on the polygon $P$ so that the number of vertices of ${\rm conv}{\mathcal M}_P$ is increased.
Let $x$ be a vertex of ${\rm conv}{\mathcal M}_P$. It is a corner of ${\mathcal M}_P$, and by Lemma \ref{Lem4} a strong corner. Therefore, there is a consecutive, non-adjacent edge pair $(E_i,E_{i+1})$ of $P$ and there are an endpoint $p$ of $E_i$ and an endpoint $q$ of $E_{i+1}$ such that $x=(p+q)/2$.
We position $P$ and choose the orthonormal basis $(e_1,e_2)$ in such a way that $x=0$, that $e_1$ is a positive multiple of $q$, and that $\langle y,e_2\rangle\ge 0$ for all $y\in E_i\cup E_{i+1}$ (note that $E_i$ and $E_{i+1}$ lie on the same side of the line through $p$ and $q$, since $0$ is a strong corner of ${\mathcal M}_P$).
We may assume (the other case is treated similarly) that $\mbox{\boldmath$u$}(\varphi_i)$ is the inner normal vector of $E_i$; then $\mbox{\boldmath$u$}(\varphi_{i+1})$ is the outer normal vector of $E_{i+1}$. Let $E_j\not= E_i$ be the other edge of $P$ with endpoint $p$, and let $E_m\not= E_{i+1}$ be the other edge of $P$ with endpoint $q$. The edges $E_j$ and $E_m$ do not lie in the line through $p$ and $q$, since $P$ has no long edge. We have $\varphi_m<\varphi_i< \varphi_{i+1} < \varphi_j$, since $\mbox{\boldmath$u$}(\psi)$ with $\psi$ between $\varphi_i$ and $\varphi_{i+1}$ is not a normal vector of an edge of $P$.
\vspace{-1.2cm}
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\rput(10.4,4.6){$L_q$}
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\vspace{-8mm}
\noindent Figure 2: The vertices $p$ and $q$ are cut off by new edges, in the figure with endpoints $p+t_1, p+t_2$, respectively $q+s_1,q+s_2$.
\vspace{5mm}
By assumption, $0$ is a vertex of ${\rm conv}{\mathcal M}_P$. Therefore, there is a support line $S$ of ${\rm conv}{\mathcal M}_P$ which has intersection $\{0\}$ with ${\rm conv}{\mathcal M}_P$. Since $S$ supports also the convex hull of $Z_P(\mbox{\boldmath$u$}(\varphi_i))$ and $Z_P(\mbox{\boldmath$u$}(\varphi_{i+1}))$, the (with respect to ${\rm conv}{\mathcal M}_P$) outer unit normal vector $\mbox{\boldmath$u$}(\alpha)$ of the support line $S$ has an angle $\alpha$ that satisfies either $-\pi/2 \le \alpha <\varphi_i$ or $\varphi_{i+1}-\pi/2<\alpha < -\pi/2$. We assume that $-\pi/2 \le \alpha <\varphi_i$; the other case is treated analogously, with the roles of $p,E_i,E_j$ and $q,E_{i+1},E_m$ interchanged.
In the following, $t_1$ and $t_2$ denote vectors such that $p+t_1\in E_j$ and $p+t_2\in E_i$. For such vectors, let $\psi_p=\psi_p(t_1,t_2)$ with $\varphi_i < \psi_p < \varphi_{i+1}$ be the angle for which $\mbox{\boldmath$u$}(\psi_p)$ is orthogonal to the line through $p+t_1$ and $p+t_2$. Trivially, there are a constant $c>0$ and a continuous function $\gamma: [\varphi_i, \varphi_{i+1}] \to {\mathbb R}^+$ with $\lim_{\psi\to \varphi_i}\gamma(\psi) =0$ such that
\begin{align}\label{3.6}
\|t_1\| <c\|t_2\|\quad &\Longrightarrow \quad \psi_p(t_1,t_2) <\varphi_{i+1},\\
\label{3.7}
\psi\in [\varphi_i, \varphi_{i+1}]\mbox{ and } \|t_1\| >\gamma(\psi) \|t_2\|\quad & \Longrightarrow \quad \psi_p(t_1,t_2)> \psi.
\end{align}
Let $L_q$ be a line parallel to $E_i$ and strongly separating $q$ from the other endpoints of $E_{i+1}$ and $E_m$. This line intersects $E_m$ in a point $q+s_1$, and it intersects $E_{i+1}$ in a point $q+s$. We choose the line $L_q$ so close to $q$ that the vector $t:= s-s_1$ satisfies $p+t\in E_i$.
Let $0<\tau<1$, and let $\sigma>1$ be such that $q+\sigma s\in E_{i+1}$. The line through the point $q+s_1+\tau t$ parallel to the support line $S$ and the line through $q+\sigma s$ parallel to $E_j$ intersect in a point $q+\sigma s+t_1$. This defines a vector function $t_1=t_1(\tau,\sigma)$, with the property that $\|t_1\|$ is strictly increasing in $\sigma$. For $\tau,\sigma\to 1$ we have $\|t_1\|\to 0$; in particular, $p+t_1\in E_j$ if $\tau,\sigma$ are sufficiently close to $1$. Therefore, we can fix $t_2=\tau t $ (so that $t_1$ now depends only on $\sigma$) and choose $\sigma_0>1$ such that
$$ p+t_1(\sigma)\in E_j\quad\mbox{and}\quad\|t_1(\sigma)\|<c\|t_2\|\quad\mbox{for } 1<\sigma\le \sigma_0.$$
Let $\psi_q(\sigma)\in(\varphi_i,\varphi_{i+1})$ be the angle for which $\mbox{\boldmath$u$}(\psi_q)$ is orthogonal to the line through $q+s_1$ and $q+\sigma s$. We choose $\sigma_1$ with $1<\sigma_1<\sigma_0$ so close to $1$ that
$$ \gamma(\psi_q(\sigma_1)) < \|t_1(\sigma_1)\|/\|t_2\|,$$
which is possible because of $\lim_{\psi\to \varphi_i}\gamma(\psi) =0$ and $\lim_{\sigma\to 1}\|t_1(\sigma)\|>0$. For $\sigma\in (\sigma_1,\sigma_0]$ sufficiently close to $\sigma_1$ we then have
$$ \gamma(\psi_q(\sigma)) < \|t_1(\sigma)\|/\|t_2\| \le \|t_1(\sigma_0)\|/\|t_2\| <c.$$
Therefore, by (\ref{3.6}) and (\ref{3.7}), the angles $\psi_q=\psi_q(\sigma)$ and $\psi_p=\psi_p(t_1(\sigma), t_2)$ satisfy
\begin{equation}\label{3.9}
\varphi_i <\psi_q <\psi_p<\varphi_{i+1}.
\end{equation}
In the following we write $t_1(\sigma)=t_1$ and $\sigma s= s_2$.
Now we choose a number $0<\lambda<1$ and replace $p+t_1, p+t_2, q+s_1, q+s_2$ respectively by $p+\lambda t_1, p+ \lambda t_2, q+\lambda s_1, q+\lambda s_2$. This does not change the angles $\psi_p,\psi_q$. We replace $P$ by the polygon $P_\lambda$ that is the convex hull of the points $p+\lambda t_1, p+ \lambda t_2, q+\lambda s_1, q+\lambda s_2$ and of the vertices of $P$ different from $p$ and $q$. By choosing $\lambda$ sufficiently small, we can achieve that still
$$ K\subset {\rm int} P_\lambda.$$
Note that $P_\lambda \subset {\rm int}(K+\varepsilon B^2)$ holds trivially.
By decreasing $\lambda$ further, if necessary, we can also achieve that ${\rm conv}{\mathcal M}_{P_\lambda}$ has more vertices than ${\rm conv}{\mathcal M}_P$, as we now show. First we notice that the inequalities (\ref{3.9}) imply that $p+\lambda t_2$ and $q+\lambda s_1$ are opposite vertices of $P_\lambda$ and that also $p+\lambda t_1$ and $q+\lambda s_2$ are opposite vertices of $P_\lambda$. Hence,
$$\frac{1}{2}(p+\lambda t_2 + q+\lambda s_1) = \frac{\lambda}{2}(s_1+t_2)=:y_\lambda$$
and
$$ \frac{1}{2}(p+\lambda t_1 +q+\lambda s_2)= \frac{\lambda}{2}(s_2+t_1)=: z_\lambda$$
are strong corners of ${\mathcal M}_{P_\lambda}$. By construction,
\begin{equation}\label{3.4}
z_\lambda-y_\lambda = \frac{\lambda}{2}[(q+s_2+t_1)-(q+s_1+t_2)] \quad\mbox{is parallel to } S.
\end{equation}
Since the non-zero vectors $s_1-s_2$ and $t_1-t_2$ have different directions, we have $y_\lambda\not= z_\lambda$.
Let $v_0=0,v_1,\dots,v_r$ be the vertices of ${\rm conv}{\mathcal M}_P$. They are corner points of ${\mathcal M}_P$. To each $i\in\{0,\dots,r\}$ we choose a line $L_i$ that strongly separates $v_i$ from the other vertices; the particular line $L_0$ is chosen parallel to the support line $S$. We can choose a number $\eta>0$ such that, for any $\bar v_0,\dots,\bar v_r \in {\mathbb R}^2$ with $\|\bar v_i- v_i\| < \eta$ for $i=0,\dots,r$, the line $L_i$ strongly separates $\bar v_i$ from the points $\bar v_j\not= \bar v_i$. Then we can further decrease $\lambda$ so that the $\eta$-neighborhood of each $v_i$, $i=1,\dots,r$, contains at least one corner point of ${\mathcal M}_{P_\lambda}$, and that the $\eta$-neighborhood of $0$ contains the points $y_\lambda$ and $z_\lambda$. Since $L_0$ is parallel to $S$, it follows from (\ref{3.4}) that $y_\lambda$ and $z_\lambda$ are both vertices of ${\rm conv}{\mathcal M}_{P_\lambda}$. Thus, ${\rm conv}{\mathcal M}_{P_\lambda}$ has more vertices than ${\rm conv}{\mathcal M}_P$.
Since (\ref{3.0}) with $P$ replaced by $P_\lambda$ still holds, we can repeat the procedure. After finitely many steps, we obtain a polygon $Q$ with
\begin{equation}\label{3.5}
K\subset {\rm int}\, Q,\quad Q \subset {\rm int}(K+\varepsilon B^2)
\end{equation}
for which ${\mathcal M}_{Q}$ has more than $k$ vertices. Finally, we replace $Q$ by a strictly convex body $M$, by replacing each edge of $Q$ by a circular arc of large positive radius $R$. If $R$ is large enough, then (\ref{3.5}), with $Q$ replaced by $M$, still holds, and the number of vertices of ${\rm conv}{\mathcal M}_M$ is the same as for ${\rm conv}{\mathcal M}_{Q}$. Thus, in the $\varepsilon$-neighborhood of $K$ we have found an element of ${\mathcal K}^2_*\setminus {\mathcal A}_k$. \qed
| {
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{"url":"https:\/\/irmar.univ-rennes1.fr\/seminaire\/geometrie-et-algebre-effectives\/isabella-panaccione","text":"# The Power Error Locating Pair algorithm\n\nIn this talk, we will focus on particular decoding algorithms for Reed Solomon codes. It is known that for these codes, it is possible to correct an amount of errors equal to or even superior than half the minimum distance. In general though, beyond this bound, the uniqueness of the solution is no longer entailed. Hence, given a vector $y\\in\\mathbb{F}_q^n$, it is possible to either use list decoding algorithms, like Sudan algorithm and Guruswami-Sudan algorithm, which return the list of all the closest codewords to $y$, or unique decoding algorithms, like the Power Decoding algorithm, which return the closest codeword to $y$ if unique at the prize of some failure cases.\n\nIn this scenario, I will present a new unique decoding algorithm, that is the Power Error Locating Pairs algorithm. Based on Pellikaan's Error Correcting Pairs algorithm, it has for Reed Solomon codes the same decoding radius as Sudan algorithm and Power Decoding algorithm, but with an improvement in terms of complexity. Moreover, like the Error Correcting Pairs algorithm, it can be performed on all codes which dispose from a special structure (among them, Reed Solomon codes, algebraic geometry codes and cyclic codes).","date":"2020-01-18 02:22:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.40353456139564514, \"perplexity\": 549.328847368499}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-05\/segments\/1579250591431.4\/warc\/CC-MAIN-20200117234621-20200118022621-00094.warc.gz\"}"} | null | null |
Can The Trade Desk Inc. (NASDAQ: TTD) Still Be Considered A Loss When It's Down -16.05% YTD?
During the recent session, The Trade Desk Inc. (NASDAQ:TTD)'s traded shares were 2.44 million, with the beta value of the company hitting 2.26. At the last check today, the stock's price was $79.09, reflecting an intraday gain of 2.81% or $2.16. The 52-week high for the TTD share is $114.09, that puts it down -44.25 from that peak though still a striking 40.94% gain since the share price plummeted to a 52-week low of $46.71. The company's market capitalization is $38.04B, and the average trade volume was 4.98 million shares over the past three months.
The Trade Desk Inc. (TTD) received a consensus recommendation of a Buy from analysts. That translates to a mean rating of 2.30. TTD has a Sell rating from 0 analyst(s) out of 19 analysts who have looked at this stock. 4 analyst(s) recommend to Hold the stock while 1 suggest Overweight, and 14 recommend a Buy rating for it. 0 analyst(s) has rated the stock Underweight. Company's earnings per share (EPS) for the current quarter are expected to be $0.27.
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The Trade Desk Inc. (TTD) registered a 2.81% upside in the last session and has traded in the red over the past 5 sessions. The stock spiked 2.81% in intraday trading to $79.09 this Thursday, 01/06/22, hitting a weekly high. The stock's 5-day price performance is -18.39%, and it has moved by -19.63% in 30 days. Based on these gigs, the overall price performance for the year is 0.82%.
The consensus price target of analysts on Wall Street is $103.27, which implies an increase of 23.41% to the stock's current value. The extremes of the forecast give a target low and a target high price of $60.00 and $120.00 respectively. As a result, TTD is trading at a discount of -51.73% off the target high and 24.14% off the low.
The Trade Desk Inc. (TTD) estimates and forecasts
Statistics show that The Trade Desk Inc. has outperformed its competitors in share price, compared to the industry in which it operates. The Trade Desk Inc. (TTD) shares have gone down -1.31% during the last six months, with a year-to-date growth rate more than the industry average at 14.49% against 6.40. Yet analysts are ramping up their growth forecast for the fiscal year 2022. Revenue is predicted to shrink -27.00% this quarter and then drop -21.40% in the quarter after that. In the rating firms' projections, revenue will increase 42.40% compared to the previous financial year.
Revenue for the current quarter is expected to be $389.2 million as predicted by 17 analyst(s). Meanwhile, a consensus of 12 analyst(s) estimates revenue growth to $287.84 million by the end of Mar 2022. As per earnings report from last fiscal year's results, sales for the corresponding quarters totaled $319.9 million and $219.81 million respectively. In this case, analysts expect current quarter sales to grow by 21.70% and then jump by 30.90% in the coming quarter.
An analysis of the company's performance over the past 5 years shows that the company's earnings grew an estimated 78.80%. While earnings are projected to return 118.30% in 2022, the next five years will return 32.00% per annum.
TTD Dividends
The Trade Desk Inc. is due to release its next quarterly earnings between February 16 and February 21. However, it is important to remember that the dividend yield ratio is merely an indicator meant to only serve as guidance.
The Trade Desk Inc. (NASDAQ:TTD)'s Major holders
The Trade Desk Inc. insiders own 0.63% of total outstanding shares while institutional holders control 67.64%, with the float percentage being 68.07%. Baillie Gifford and Company is the largest shareholder of the company, while 997 institutions own stock in it. As of Sep 29, 2021, the company held over 52.08 million shares (or 11.94% of all shares), a total value of $3.66 billion in shares.
The next largest institutional holding, with 39.51 million shares, is of Vanguard Group, Inc. (The)'s that is approximately 9.06% of outstanding shares. At the market price on Sep 29, 2021, these shares were valued at $2.78 billion.
Also, the Mutual Funds coming in first place with the largest holdings of The Trade Desk Inc. (TTD) shares are Vanguard Total Stock Market Index Fund and Vanguard U.S. Growth Fund. Data provided on Sep 29, 2021 indicates that Vanguard Total Stock Market Index Fund owns about 12.28 million shares. This amounts to just over 2.82 percent of the company's overall shares, with a $863.2 million market value. The same data shows that the other fund manager holds slightly less at 11.59 million, or about 2.66% of the stock, which is worth about $928.07 million.
US STOCKS-S&P 500, Nasdaq dip; labor market seen tight despite weak payrolls report
Sonos Inc. (NASDAQ: SONO) Stock Could Be An Option To Consider | {
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Villa Cusona è un edificio storico del comune di San Gimignano, situato nell'omonima località de La Cusona.
Storia e descrizione
La tenuta è citata già in un documento del 994. Appartenne alle famiglie Strozzi-Guicciardini, i cui discendenti posseggono tuttora il complesso, e vi tengono una pregiata produzione vinicola.
L'edificio padronale storico ha una forma pressoché cubica, nata dall'accorpamento di più corpi di fabbrica e incentrata attorno al cortile interno. A circa quindici metri si profondità si trovano le storiche cantine, che hanno la forma di un criptoportico quadrangolare, interrato sotto l'impluvium del cortile. A sud poi si trova la villa "nuova", separata dal resto del fabbricato ed eretta all'inizio del XIX secolo, come ricorda un'iscrizione sulla facciata. Tra i due edifici principali si trovano i giardini all'italiana, composti in aiuole geometriche delimitate dal bosso, con affaccio panoramico verso est. Ancora più a sud si trova un altro livello del giardino, dove un lungo doppio filare di cipressi divide il boschetto, a ovest, dal labirinto di bosso e cipressi.
Altri progetti
Collegamenti esterni
Sito ufficiale
architetture di San Gimignano
Cusona | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,475 |
System Shuts down...and i'd like it to stop.
Have you tried adding extra fans targeted onto the graphics card to see if it's a heat issue? I know it's obvious sorry, just reaching at straws for you.
Lol thanks man Crazy - that's the same thing that was happening to me on clocked settings after it had been on for ages. Maybe try leaving it off for an hour and then trying again.
I just updated my thread - but to hit the high points: After installing a new hiper 580w type-r power supply this afternoon it's been running 3d games for about 9 hours and it hasn't shut down yet. I'm hopeful but having managed to get it to run for 4 hours on the Tagan without shutting down i'm going to leave it looping on 3dmark06 overnight. If it survives till the morning then i would say problem cured. Crazy nate did you actually manage to get your friend's psu into your pc or have you not tried a different psu yet?
So my hiper 580w type-r finally arrived today. Installed it at lunchtime - restored my clocked settings which were basically the bleeding edge OCZ suggestions. I have had world of warcraft running in the background since 1pm - then switched to some Age of Empires 3 at 6pm and then have been looping 3dmark06 since 8pm. It's now 9pm and there is no sign of a shut down. It's too early to properly tell whether it's cured it but the signs certainly appear to be hopeful. I'll update tomorrow and let you know if it has successfully survived the night of 3dmark06 looping - which the tagan never managed. Fingers crossed that it was the power supply then.
I've noticed that mine seems to be able to survive for up to 4 hours if i use it from cold - i.e. after having left it off for 2 hours. I'm still hoping that it's a psu issue and not a temp one. I mean what can heat up over 4 hours and then crash .. and then cools down in 2 hours? Like you crazy my voltages and temps all appear to be in normal operating tolerances. Really weird. Have you replaced the psu or tried with a different one?
hey mate, thanks for the reply :-) i just replaced my old memory with brand spanking new ocz gold 2x1gb and it still does it. ocz is memtesting for hours no problems. the only things i havent replaced in the system are psu and cpu. I bit the bullet and ordered a hiper 580w type-r - but unfortunately no delivery on easter weekend Discouragingly someone with a similiar problem on another thread just had to rma his cpu.. if i did that it would mean i've built an entirely new system from the one that i had originally purchased. I'll update the thread when it arrives and let y'all know if it fixes it or not.
That's seriously bad news mate. I have the same shutdown problem as you - I've literally replaced every single element of the system bar the psu and cpu. I broke down this afternoon and ordered a hiper 580w type-r. (OCZ Powerstream was my first choice but I couldnt find a decent uk supplier that I knew and trusted). Tagan have been VERY disappointing in their RMA process. I emailed on multiple occasions, starting from a week ago today and the best result i have had from them was from their sales department who said they have forwarded my rma request to their uk reseller - but still no RMA number or response!? No phone numbers or contact details at all - very frustrating. Anyway I hope that the psu cures all my ills but unfortunately I secretly harbour the feeling that it will still shutdown - just as yours had. That's seriously bad news, I'm sorry to hear it both for your sakes and mine. Keep us informed. Thanks and all the best of luck in a fast rma and working cpu.
I have this same shutdown problem and it's driving me crazy. See this thread: http://www.dfi-street.com/forum/showthread.php?t=47598 I have replaced the mobo, gfx card and memory.. removed all peripherals and reinstalled into a clean build. All of which produced various fluctuations in the frequency of and duration before shutdown - but none cured the issue. The latest theory as stated in that thread is that it's a bad PSU problem. I notice you have an Antec - if i remember correctly people have discovered problems with them and certain boards - might be worth a search on the forums to see if you can validate that theory. Another option is to stick some extra fans in and see if that helps, if it does then it's a heat issue. Good luck.
Win64 or Win32 while waiting for Vista?
I installed x64 to try and solve a pc problem i'm struggling with.. anyway so far it's all been ok except: no nvidia wdm drivers for vivo - which means no webcam. azureus turns off my second monitor whenever it's launched.. weird lol saitek x45 joystick 64bit drivers don't allow calibration - which renders them unuseable with custom software and custom button settings. other than those niggles it's sweet as a nut. the only thing of course is that I don't really see any gain or benefit from using winxp x64.. so am sort of considering returning to 32bit if i can get my problem sorted and dont mind another reformat. so in conclusion, it's fun to be on x64, apart from some specialised niggles you might discover for your particular config but ultimately is there really that much to gain from it?
The problem had originally started on winxp sp2 - i upgraded to winxp x64 with clean drivers and minimal install in the hope it might cure it. 3dMark01, 3dMark05 and 3dMark06 all cause shutdowns. Prime95 hasn't caused any shutdowns. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,482 |
package ch.liquidmind.deflector.processing;
import java.lang.reflect.Constructor;
import java.lang.reflect.Method;
import java.util.ArrayList;
import java.util.HashSet;
import java.util.List;
import java.util.Set;
import ch.liquidmind.deflector.DeflectorConfig;
import ch.liquidmind.deflector.reflection.ConstructorBehavior;
import ch.liquidmind.deflector.reflection.MethodBehavior;
public abstract class ClassProcessor extends Processor
{
private Class< ? > sourceClass;
public ClassProcessor( Class< ? > sourceClass )
{
super( null );
this.sourceClass = sourceClass;
}
public ClassProcessor( Class< ? > sourceClass, DeflectorWriter writer )
{
super( writer );
this.sourceClass = sourceClass;
}
protected void createTargetClass()
{
writeClassDeclaration();
getWriter().println( "{" );
handleCheckedExceptionClass();
handleBehaviors();
handleInnerClasses();
getWriter().println( "}" );
getWriter().println();
}
private void handleCheckedExceptionClass()
{
if ( !isCheckedExceptionClass( getSourceClass() ) )
return;
getWriter().indent();
getWriter().println( "private static final long serialVersionUID = 1L;" );
getWriter().println();
String sourceClass = typeToString( getSourceClass(), getShowTypeVariableReference(), false, true, false );
getWriter().println( "public " + getTargetClassSimpleName() + "( " + sourceClass + " e )" );
getWriter().println( "{" );
getWriter().indent();
getWriter().println( "super( e );" );
getWriter().deindent();
getWriter().println( "}" );
getWriter().deindent();
}
private void handleBehaviors()
{
for ( Constructor< ? > constructor : getDeclaredConstructors() )
handleBehavior( new ConstructorProcessor( new ConstructorBehavior( constructor ), getWriter() ) );
for ( Method method : getDeclaredMethods() )
handleBehavior( new MethodProcessor( new MethodBehavior( method ), getWriter() ) );
}
private List< Constructor< ? > > getDeclaredConstructors()
{
List< Constructor< ? > > declaredConstructors = new ArrayList< Constructor< ? > >();
for ( Constructor< ? > constructor : sourceClass.getDeclaredConstructors() )
if ( !constructor.isSynthetic() && !isAbstract( sourceClass.getModifiers() ) )
declaredConstructors.add( constructor );
return declaredConstructors;
}
private List< Method > getDeclaredMethods()
{
List< Method > declaredMethods = new ArrayList< Method >();
for ( Method constructor : sourceClass.getDeclaredMethods() )
if ( !constructor.isSynthetic() )
declaredMethods.add( constructor );
return declaredMethods;
}
private void handleBehavior( BehaviorProcessor processor )
{
getWriter().indent();
processor.process();
getWriter().deindent();
}
// TODO Introduce inheritance between deflector classes. This allows (static) super class methods
// to be invoked on sub classes, e.g. org.eclipse.jetty.util.component.AbstractLifeCycle.start()
// from org.eclipse.jetty.server.Server.start().
private void writeClassDeclaration()
{
String staticModifer = ( isTargetClassStatic() ? "static " : "" );
String extendsString = ( Throwable.class.isAssignableFrom( getSourceClass() ) ? " extends " + RuntimeException.class.getName() : "" );
if ( DeflectorConfig.getJavaMajorVersion() >= 1 && DeflectorConfig.getJavaMinorVersion() >= 5 )
getWriter().println( "@SuppressWarnings(\"deprecation\")" );
getWriter().println( "public " + staticModifer + "final class " + getTargetClassSimpleName() + extendsString );
}
private String getTargetClassSimpleName()
{
return DeflectorConfig.getClassPrefix() + getSourceClass().getSimpleName();
}
protected boolean isTargetClassStatic()
{
return false;
}
private void handleInnerClasses()
{
Set< Class< ? > > innerClasses = getProcessableInnerClasses();
for ( Class< ? > innerClass : innerClasses )
{
getWriter().indent();
InnerClassProcessor innerClassProcessor = new InnerClassProcessor( innerClass, getWriter() );
innerClassProcessor.process();
getWriter().deindent();
}
}
private Set< Class< ? > > getProcessableInnerClasses()
{
Set< Class< ? > > processableClasses = new HashSet< Class< ? > >();
for ( Class< ? > declaredClass : sourceClass.getDeclaredClasses() )
if ( isProcessableInnerClass( declaredClass ) )
processableClasses.add( declaredClass );
return processableClasses;
}
protected Class< ? > getSourceClass()
{
return sourceClass;
}
protected static boolean isProcessableRootClass( Class< ? > theClass )
{
boolean isProcessableClass = isPublic( theClass.getModifiers() );
// If the class is an inner class --> not processable (non-local inner classes are
// processed as part of non-inner classes).
if ( theClass.getEnclosingClass() != null )
isProcessableClass = false;
return isProcessableClass;
}
protected static boolean isProcessableInnerClass( Class< ? > theClass )
{
return isPublic( theClass.getModifiers() );
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,518 |
Tropidion persimile is a species of beetle in the family Cerambycidae. It was described by Martins in 1960.
References
Tropidion
Beetles described in 1960 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,087 |
Q: Assembly fails to load in restricted AppDomain I am trying to load an Assembly into a restricted AppDomain. If I do not specify any restrictions, the Assembly will load correctly:
var permissionSet = new PermissionSet(System.Security.Permissions.PermissionState.Unrestricted);
AppDomain targetAppDomain = AppDomain.CreateDomain("LockedDomain" + Guid.NewGuid().ToString("N"),null,domainSetup,permissionSet,null);
var instance = (IRemoteFilterClass) targetAppDomain.CreateInstanceFromAndUnwrap(tempAssemblyPath, "CompiledCode.CompiledClass");
Howevere I want to lock down the created AppDomain as completely as possible, i.e. only give permissions which are absolutely necessary. If I specify a PermissionSet to restrict Permissions, the Assembly fails to load:
var permissionSet = new PermissionSet(System.Security.Permissions.PermissionState.None);
permissionSet.AddPermission(new FileIOPermission(FileIOPermissionAccess.Read, tempAssemblyPath));
permissionSet.AddPermission(new FileIOPermission(FileIOPermissionAccess.PathDiscovery, tempAssemblyPath));
permissionSet.AddPermission(new FileIOPermission(FileIOPermissionAccess.Read, Assembly.GetExecutingAssembly().Location));
permissionSet.AddPermission(new FileIOPermission(FileIOPermissionAccess.PathDiscovery, Assembly.GetExecutingAssembly().Location));
permissionSet.AddPermission(new ReflectionPermission(PermissionState.Unrestricted)); //Not sure if this is necessary, but does not work anyway
AppDomain targetAppDomain = AppDomain.CreateDomain("LockedDomain" + Guid.NewGuid().ToString("N"),null,domainSetup,permissionSet,null);
var instance = (IRemoteFilterClass) targetAppDomain.CreateInstanceFromAndUnwrap(tempAssemblyPath, "CompiledCode.CompiledClass");
The following exception is thrown:
System.IO.FileLoadException: Could not load file or assembly '5e1a72b7c5584f7c92c18ea9b221222f, Version=0.0.0.0, Culture=neutral, PublicKeyToken=null' or one of its dependencies. Failed to grant permission to execute. (Exception from HRESULT: 0x80131418) ---> System.Security.Policy.PolicyException: Execution permission cannot be acquired.
at System.Security.CodeAccessSecurityEngine.ResolveGrantSet(Evidence evidence, Int32& specialFlags, Boolean checkExecutionPermission)
--- End of inner exception stack trace ---
at System.Reflection.RuntimeAssembly._nLoad(AssemblyName fileName, String codeBase, Evidence assemblySecurity, RuntimeAssembly locationHint, StackCrawlMark& stackMark, IntPtr pPrivHostBinder, Boolean throwOnFileNotFound, Boolean forIntrospection, Boolean suppressSecurityChecks)
at System.Reflection.RuntimeAssembly.nLoad(AssemblyName fileName, String codeBase, Evidence assemblySecurity, RuntimeAssembly locationHint, StackCrawlMark& stackMark, IntPtr pPrivHostBinder, Boolean throwOnFileNotFound, Boolean forIntrospection, Boolean suppressSecurityChecks)
at System.Reflection.RuntimeAssembly.InternalLoadAssemblyName(AssemblyName assemblyRef, Evidence assemblySecurity, RuntimeAssembly reqAssembly, StackCrawlMark& stackMark, IntPtr pPrivHostBinder, Boolean throwOnFileNotFound, Boolean forIntrospection, Boolean suppressSecurityChecks)
at System.Reflection.RuntimeAssembly.InternalLoadFrom(String assemblyFile, Evidence securityEvidence, Byte[] hashValue, AssemblyHashAlgorithm hashAlgorithm, Boolean forIntrospection, Boolean suppressSecurityChecks, StackCrawlMark& stackMark)
at System.Reflection.Assembly.LoadFrom(String assemblyFile, Evidence securityEvidence)
at System.Activator.CreateInstanceFromInternal(String assemblyFile, String typeName, Boolean ignoreCase, BindingFlags bindingAttr, Binder binder, Object[] args, CultureInfo culture, Object[] activationAttributes, Evidence securityInfo)
at System.AppDomain.CreateInstanceFrom(String assemblyFile, String typeName)
at System.AppDomain.CreateInstanceFromAndUnwrap(String assemblyName, String typeName)
at System.AppDomain.CreateInstanceFromAndUnwrap(String assemblyName, String typeName)
It seems as if there are still Permissions missing, but I have no idea which ones are missing.
A: It is necessary to add a SecurityPermission with the SecurityPermissionFlag.Execution set.
Here is the working code:
var permissionSet = new PermissionSet(System.Security.Permissions.PermissionState.None);
permissionSet.AddPermission(new FileIOPermission(FileIOPermissionAccess.Read, tempAssemblyPath));
permissionSet.AddPermission(new FileIOPermission(FileIOPermissionAccess.PathDiscovery, tempAssemblyPath));
//The following line fixed the code
permissionSet.AddPermission(new SecurityPermission(SecurityPermissionFlag.Execution));
AppDomain targetAppDomain = AppDomain.CreateDomain("LockedDomain" + Guid.NewGuid().ToString("N"),null,domainSetup,permissionSet,null);
var instance = (IRemoteFilterClass) targetAppDomain.CreateInstanceFromAndUnwrap(tempAssemblyPath, "CompiledCode.CompiledClass");
(Source: https://social.msdn.microsoft.com/Forums/vstudio/en-US/23a9197e-3581-4a28-912d-968004488773/how-to-change-permissions-of-appdomain?forum=clr)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,518 |
\section{Introduction}
Density Functional Theory (DFT) \cite{yang_parr_book,dreiz_book} is the most used
computational method for electronic structure calculations of molecular and extended systems,
providing the highest accuracy/computational cost ratio.
In the conventional DFT formalism, the Kohn-Sham (KS) scheme \cite{kohn1965self}, the ground-state electronic density
$n(\R)$ is determined from a set of auxiliary KS orbitals ($\phi_i(\R)$): the KS-DFT method is exact but for the
approximations of the exchange-correlation (XC) functional.
However, for large scale calculations, the computational cost of KS-DFT
becomes unaffordable, as one needs to compute all the occupied KS
orbitals in order to construct the density as $n(\R)=\sum_i^{occ.} f_i |\phi_i(\R)|^2$, where $f_i$ is the occupation number (2,
for closed-shell systems).
Among other linear scaling methods \cite{yang91,god99,bowler02,kuss13}, two DFT methods are attracting strong
interest:
i) In the orbital-free version of DFT (OF-DFT) \cite{sma94,wang2002orbital,gavini07,wesobook,chen16}, $n(\R)$ can be
computed directly via the
Euler equation \cite{yang_parr_book}, without the need of KS orbtials;
ii) In the subsystem version of DFT (Sub-DFT) \cite{cortona,wesorev,pavanello15,wesochemrev}, also known as
Frozen-Density-Embedding (FDE),
$n(\R)$ is computed as the sum of the electronic densities of several (smaller) subsystems
in which the total system is partitioned, which can be computed simultaneously.
Both approaches allow in principle calculations of large systems, but the
final accuracy depends directly on the approximations of the non-interacting
kinetic energy (KE) functional $T_s$ (which
are definitely more important than the ones for the XC energy, that are also present in standard KS
calculations).
We recall that the exact KS KE functional is:
\begin{equation}
T_s^{exact}=\frac{1}{2}\sum_i^{occ.} \int f_i |\nabla \phi_i({\bf r})|^2 {\rm d^3}{\bf r} .
\end{equation}
Thus the KE is explicitly known only as a function of $\phi_i$ but not as a
functional of $n$.
On the other hand, in Sub-DFT the interaction between the subsystems is taken into account via the so called embedding
potentials \cite{wesorev,pavanello15,wesochemrev},
which depends on the non-additive-KE: in the case of just two subsystems (A and B) it is
$T_s^{nadd}[n_A;n_B]=T_s[n_A+n_B]-T_s[n_A]-T_s[n_B]$.
The development of an accurate approximation of $T_s[n]$ (and/or
$T_s^{nadd}[n_A;n_B]$) is one of the biggest DFT
challenges \cite{kara12,carter12,lastra08}.
Nowadays, the most sophisticated KE approximations have been constructed to be exact for the linear response of
jellium model, by incorporating the Lindhard function in their fully non-local expressions
\cite{wang2002orbital,wang1998orbital,ho2008analytic,garcia1996nonlocal,garcia1998nonlocal,garcia2008approach}.
We recall that the Lindhard function
\cite{lindhard1954properties,wang2002orbital}
\begin{equation}
F^{Lind}=\left( \frac{1}{2}+\frac{1-\eta^2}{4\eta}\ln \left| \frac{1+\eta}{1-\eta} \right| \right)^{-1},
\label{eq1}
\end{equation}
where $\eta=k/(2k_F)$ is the dimensionless momentum ($k_F=(3\pi^2 n)^{1/3}$ being the Fermi
wave vector of the jellium model with the constant density $n$),
is related to the Jellium density response $\chi^{Jell}$ via\cite{wang2002orbital}
\begin{equation}
\label{chif}
-\frac{1}{\chi^{Jell.}}=\frac{\pi^2}{k_F} F^{Lind} \, .
\end{equation}
The non-local KE functionals based on the Lindhard function are accurate for simple metals
where the nearly-free electron gas is an excellent model
but they can not describe well
semiconductors and insulators, where the density response function behaves as
\cite{pick1970microscopic,huang2010nonlocal}
\begin{equation}
-\frac{1}{\chi^{Semic.}(k)} \underset{k\rightarrow 0}{\longrightarrow} \frac{b}{k^2},
\label{eqllw1}
\end{equation}
with $b$ being positive and material-dependent. Several KE functionals have
been constructed
to improve the description of semiconductors \cite{huang2010nonlocal,shin2014enhanced}, but Eq. (\ref{eqllw1}) has not been
explicitly used in their expressions due to the lack of a sophisticated analytical form that can recover both
the Lindhard function and Eq. (\ref{eqllw1}).
In this article, we will investigate the generalization
of the Lindhard function for the jellium-with-gap model which satisfies Eq. (\ref{eqllw1}).
The jellium-with-gap model\cite{rey1998virtual}, was developed outside the KS framework,
using perturbation theory to take into account the band
gap energy. This model was used to have qualitative and quantitative insight for semiconductors
\cite{callaway1959correlation,penn1962wave,srinivasan1969microscopic,levine1982new,tsolakidis2004effect},
to develop an XC kernel for the optical properties of materials
\cite{trevisanutto2013optical}, and to construct accurate correlation energy functionals for the ground-state DFT
\cite{rey1998virtual,krieger1999electron,krieger2001density,toulouse2002validation,
toulouse2002new,fabiano2014generalized}.
We will show that the Lindhard function for the jellium-with-gap model ($F^{GAP}$), previously
introduced by Levine and Louie \cite{levine1982new} in a different
context (dielectric constant and XC potential),
may be seen as a sophisticated analytical form suitable for KE approximations.
The article is organized as follow:
In Section II we discuss the properties of $F^{GAP}$, we derive its (band-gap-dependent) KE gradient expansion,
and we assess it for large atoms. By using a local gap model, we propose
a simple KE gradient expansion that it is very accurate for the semiclassical atom theory.
In Section III we discuss the implications of this result in
DFT by constructing a simple KE functional at the
Generalized Gradient Approximation (GGA) level of theory based on the gradient expansion of
the jellium-with-gap model.
GGA KE functionals are computationally very efficient and play a key
role for the simulation of large systems.
We mention that the development of semilocal KE functionals is nowadays an active field
\cite{constantin2011semiclassical,laricchia2011generalized,kara13,laricchia2013laplacian,
borgo2013density,borgo14,kinairy14,alpha,cancioJCP16,smiga2017}.
Finally, in Section IV we summarize our results.
\section{Theory}
\subsection{Properties and gradient expansions for the jellium model}
For the conventional infinite jellium model, the Lindhard function behaves as:
\begin{eqnarray}
\label{eq2}
F^{Lind}& \rightarrow & 1+ \frac{1}{3}\eta^2+\frac{8}{45}\eta^4+\mathcal{O}(\eta^6),\;\;\rm{for}\;\;
\eta\rightarrow 0, \\
\label{eq3}
F^{Lind} & \rightarrow & 3\eta^2 -\frac{3}{5}-\frac{24}{175}\frac{1}{\eta^2}+\mathcal{O}(\eta^{-4}),
\;\;\rm{for}\;\;
\eta\rightarrow \infty.
\end{eqnarray}
Equation (\ref{eq2}) contains important physics that has been used in the construction
of semilocal
KE density functionals \cite{wang2002orbital}.
Thus,
the KE gradient expansion which recovers the first three terms in the right hand side of Eq. (\ref{eq2})
can be easily derived \cite{wang2002orbital} (see also Eqs. (\ref{gap4exp}) and
(\ref{eq14}) in section II-C and the corresponding discussion). It is
\begin{equation}
T_s^{Lind4}[n]=\int d\R\, \tau^{TF}\left(1+\frac{5}{27}s^2+\frac{8}{81}q^2\right),
\label{eq4}
\end{equation}
where $\tau^{TF}=\frac{3}{10}(3\pi^{2})^{2/3}n^{5/3}$ is the Thomas-Fermi KE density
\cite{thomas1927calculation,fermi1927metodo}, which is exact for the jellium model, and
$s=|\nabla n|/[2 k_F n]$, $q=\nabla n^{2}/[4(3\pi^{2})^{2/3}n^{5/3}]$ are the reduced
gradient and Laplacian, respectively.
Equation (\ref{eq4}) resembles the second-order gradient expansion \cite{Kirz57} (GE2)
$$T_s^{GE2}[n]=\int d\R \tau^{TF}(1+\frac{5}{27}s^2)$$
(derived also within the linear response of the jellium model),
as well as the fourth-order gradient expansion
\cite{hodges1973quantum,brack1976extended,laricchia2013laplacian}
of the KE
$$T_s^{GE4}[n]=\int d\R \tau^{TF}(1+\frac{5}{27}s^2+\frac{8}{81}q^2-\frac{1}{9}s^2 q +\frac{8}{243}s^4),$$
with the exception of the terms $\propto s^2 q$, $\propto s^4$, which are beyond the linear response.
Note that $F^{Lind}(\eta=0)=1$ is the leading term in the expansion of
Eq. (5) and
it corresponds to the
Thomas-Fermi local density approximation, whose linear response in the wave vector space is just the Fourier
transform of the second-functional derivative, i.e. $\delta^2 T_s^{TF}/\delta n(\R)\delta n(\R')\sim
k_F^{-1}\delta(\R-\R')$.
We recall that the limit $\eta=0$ is very powerful, being also used in the construction of the adiabatic local
density approximation (ALDA) XC kernel of
the linear response time-dependent DFT \cite{burke2005time,constantin2007simple}.
\subsection{Properties of the Lindhard function for the jellium-with-gap model}
Levine and Louie \cite{levine1982new} proposed
the density-response function $\chi^{GAP}(k,\omega)$ of the jellium-with-gap model, and the corresponding [i.e. from
Eq. (\ref{chif})]
Lindhard function for jellium-with-gap model is
\begin{eqnarray}
&& 1/F^{GAP}=\frac{1}{2}-\frac{\Delta(\arctan(\frac{4\eta+4\eta^2}{\Delta})+
\arctan(\frac{4\eta-4\eta^2}{\Delta}))}{8\eta}\nonumber+ \\
&& \;\;\; + (\frac{\Delta^2}{128\eta^3}+\frac{1}{8\eta}-\frac{\eta}{8})\ln(\frac{\Delta^2+(4\eta+4\eta^2)^2}
{\Delta^2+(4\eta-4\eta^2)^2}),
\label{eq6}
\end{eqnarray}
where $\Delta=2E_g/k_F^2$ and $E_g$ is the gap.
For a given $\Delta$, a series expansion of $F^{GAP}$ for $\eta\rightarrow 0$ gives:
\begin{eqnarray}
&& F^{GAP}\longrightarrow \frac{3\Delta^2}{16\eta^2}+\frac{9}{5}+
\frac{3}{175}\frac{175\Delta^2-192}{\Delta^2}\eta^2+ \nonumber \\
&&\;\;\; -\frac{64}{875}\frac{525\Delta^2-368}{\Delta^4}\eta^4+
\mathcal{O}(\eta^6)\;\;\rm{when}\;\;\eta\rightarrow 0 \; .
\label{eqe0}
\end{eqnarray}
Thus, for any system with $\Delta >0$ we have that $F^{GAP}\propto \Delta^2 \eta^{-2}$.
This term is correct (see Eq. (4)) and it has been also used in the jellium-with-gap XC kernel
\cite{trevisanutto2013optical},
which gives accurate optical absorption spectra of semiconductors and insulators.
On the other hand, if we first perform a series expansion for $\Delta\rightarrow 0$, and then
a series expansion for $\eta\rightarrow 0$ we obtain:
\begin{eqnarray}
&& F^{GAP}\longrightarrow \left[1+\frac{1}{3}\eta^2+\frac{8}{45}\eta^4+...\right]+ \nonumber\\
&& \Delta \left[\frac{\pi}{8}\frac{1}{\eta}+\frac{\pi}{12}\eta+\frac{7\pi}{120}\eta^3+...\right]+
\Delta^2 \Bigg[\frac{\pi^2-4}{64}\frac{1}{\eta^2}+\frac{3\pi^2-16}{192}+ \nonumber\\
&& \left(\frac{-17}{180}+\frac{13\pi^2}{960}\right)\eta^2+
\left(\frac{-383}{3780}+\frac{683\pi^2}{60480}\right)\eta^4+...\Bigg]+... \; .
\label{eqd0}
\end{eqnarray}
Equation (\ref{eqd0}) confirms that, by construction, we have
\begin{equation}
F^{GAP}= F^{Lind}, \;\;\rm{when}\;\;\Delta=0\ .
\label{eq9}
\end{equation}
Inspection of Eqs. (\ref{eqe0}) and (\ref{eqd0}) clearly shows that
\begin{eqnarray}
\lim_{\Delta\rightarrow0}\lim_{\eta\rightarrow0}F^{GAP} & = & \infty\ ,
\label{eq10} \\
\lim_{\eta\rightarrow0}\lim_{\Delta\rightarrow0}F^{GAP} & = &1\ , \label{eq10a}
\end{eqnarray}
meaning that $F^{GAP}$ has an ``order of limits problem''.
Such a situation is common in DFT. For example, we recall
that several meta-GGA XC functionals (e.g. TPSS \cite{TPSS}, revTPSS
\cite{perdew2009workhorse,perdew2011erratum}, BLOC \cite{bloc,blochole}, SA-TPSS \cite{constantin2016semilocal}, VT\{8,4\}
\cite{del2012new}) suffer of such a
order of limits problem. Nonetheless, they are accurate for many systems and properties,
showing realistic system-averaged XC hole models \cite{blochole}.
In the opposite limit, i.e. for $\eta\rightarrow \infty$, we have
\begin{eqnarray}
&& F^{GAP}\rightarrow
3\eta^2-\frac{3}{5}+ \nonumber\\
&& (-\frac{24}{175}+\frac{3}{16}\Delta^2)\frac{1}{\eta^2}+
\mathcal{O}(\frac{1}{\eta^4}) \,.
\label{eq8}
\end{eqnarray}
Therefore, in this limit, $F^{GAP}$ always behaves as $F^{Lind}$ for $\Delta=0$.
In the upper panel of Fig. \ref{f1}, we show $1/F^{GAP}$ for several values of $\Delta$.
The plots are all smooth.
At large $\eta$, $F^{GAP}$ recovers the Lindhard function [see Eq. (\ref{eq8})], while at small $\eta$
it is driven by the term $\propto \eta^{-2}$.
The plot of the linear response of $T_s^{Lind4}[n]$ (Eq. (\ref{eq2})) is also given for comparison.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{fig1.eps}
\caption{Upper panel:$1/F^{GAP}$ versus $\eta$ for various values of $\Delta$. Also shown is the
small-$\eta$ expansion of Eq. (\ref{eq2}).
Lower panel: Comparison between $F^{GAP}$ (solid-lines) and the
expansion (dashed-lines) of Eq. (\ref{eqd0}), for $\Delta=0.1$, 0.5, and 1, respectively.
}
\label{f1}
\end{figure}
In the lower panel of Fig. \ref{f1} we report the accuracy of Eq. (\ref{eqd0}), considering only the terms
explicitly indicated in the equation, for $\Delta=0.1, 0.5,$ and $1$.
Even for the case $\Delta=1$, this expansion is still very accurate for $\eta\leq 1$.
\subsection{Kinetic energy gradient expansions from the linear-response of the jellium-with-gap model}
\label{subll1}
Next we proceed to build the linear-response jellium-with-gap KE gradient expansion, that should
recover Eq. (\ref{eq4}) when $\Delta=0$. To this purpose, we consider the GAP4 expansion,
with the general form of the KE fourth-order gradient expansion
\begin{eqnarray}
&& T_s^{GAP4}[n]=\int d\R \tau^{TF}(\frac{a_1}{s^2}+\frac{a_2}{s}+a_3+a_4 s+ a_5 q+ \nonumber\\
&&+a_6 s^2+a_7 s q+a_8 s^3+a_{9}s^4+a_{10}q^2+a_{11} s^2 q) \; .
\label{gap4exp}
\end{eqnarray}
Performing the linear response of such a functional
\begin{equation}
F(\eta)=\frac{k_F}{\pi^2}\mathcal{F} \left( \frac{\delta^2 T_s[n]}{\delta n(\R)\delta n(\R')}|_{n_0}\right),
\label{eq14}
\end{equation}
where $\mathcal{F}$ represents the Fourier transform, we can find the coefficients $a_{i}$, by
comparing term-by-term with Eq. (\ref{eqd0}). Nevertheless, the straightforward calculation of
Eq. (\ref{gap4exp}) requires a tedious and long algebra \cite{nazarov2011optics,karimi2014three}.
Instead, a more elegant and simpler way to obtain the linear response of a given semilocal functional has
been proposed in Ref. \onlinecite{tao2008nonempirical}: consider a small pertubation in density at
$\R=\mathbf{0}$, of the form $n=n_0+n_ke^{i\mathbf{k}\R}$, such that $\nabla n=n_k i
\mathbf{k}e^{i\mathbf{k}\R}$, and $\nabla^2 n=-n_k k^2e^{i\mathbf{k}\R}$, with $n_k\ll n_0$. Thus, at
$\R=\mathbf{0}$, these expressions are simply $n=n_0+n_k$, $\nabla n=n_k i \mathbf{k}$, and $\nabla^2 n=-n_k k^2$.
Inserting them in the functional expression, the linear response is obtained as twice the
second-order coefficient of the series expansion with respect to $n_k/n_0$. After some algebra,
the KE gradient expansion which gives the linear response of Eq. (\ref{eqd0}) is found to be
\begin{eqnarray}
&& T_s^{GAP4}[n]=\int d\R \tau^{TF}[\Delta^2 \frac{27}{91} \frac{\pi^2-4}{64}\frac{1}{s^2}+
\Delta\frac{5\pi}{72}\frac{1}{s}+
1+ \nonumber \\
&& \Delta^2(\frac{\pi^2}{64}-\frac{1}{12})+
\Delta\frac{5\pi}{36} s+
(\frac{5}{27}+\Delta^2(\frac{-17}{324}+\frac{13\pi^2}{1728}))s^2+ \nonumber \\
&& \Delta\frac{-7\pi}{216} sq+
(\frac{8}{81}+\Delta^2(\frac{-383}{6804}+\frac{683\pi^2}{108864}))q^2].
\label{gap4}
\end{eqnarray}
The terms $\propto s^{-2}$ and $\propto s^{-1}$ account for the terms $\propto \eta^{-2}$ and
$\propto \eta^{-1}$ of Eq. (\ref{eqd0}). These terms contribute only for a non-zero gap,
i.e. in semiconductors and insulators, but not in
metals. At $\Delta=0$, $T_s^{GAP4}[n]$ correctly recovers $T_s^{Lind4}[n]$.
To test $T_s^{GAP4}[n]$, we perform calculations for noble atoms, up to $Z=290$ electrons,
using LDA orbitals and densities, in the Engel code \cite{engel1993accurate,engel2003orbital}.
We consider $\Delta=2E_g/k_F^2(\R)$ with $E_g$ being the KS band gap of the atoms. Because the gradient
expansion is well defined only at small gradients and small-$\Delta$, we perform all the integrations
over the volume $V$ defined by the conditions $-1\leq q \leq 1$ and $\Delta\leq 1$,
in a similar manner as in Ref. \onlinecite{constantin2016semiclassical}.
\begin{table*}
\begin{center}
\caption{\label{ta1} Comparison of several linear-response KE gradient expansions.
All integrations are performed over the volume $V$, defined by $-1\leq q \leq 1$ and $\Delta\leq 1$.
We show the exact KE ($T_s^{exact}$) and the errors $E_s^{approx}=T_s^{approx}-T_s^{exact}$ (in Hartree).
The GAP4 and LGAP functionals are defined in Eq. (\ref{gap4}) and Eq. (\ref{lgap}), respectively.
The best result of each line is shown in bold style. We use LDA orbitals and densities.}
\begin{ruledtabular}
\begin{tabular}{lrrrrr}
atom & $T_s^{exact}$ & $E_s^{GE2}$ & $E_s^{Lind4}$ & $E_s^{GAP4}$ & $E_s^{LGAP-GE}$ \\%& $E_s^{mGAP3}$ & $E_s^{mGAP4}$
\\
\hline
Ne & 125.8 & {\bf -0.2} & 1.2 & 4.0 & 1.9 \\%& 2.8 & 5.9 \\
Ar & 512.2 & {\bf 4.3} & 8.1 & 12.0 & 11.4 \\%& 14.5 & 23.0 \\
Kr & 2742.3 & -21.1 & -6.7 & {\bf 1.0} & 10.3 \\%& 22.7 & 49.9\\
Xe & 7214.4 & -50.7 & -19.6 & {\bf -8.9} & 23.2 \\%& 50.4 & 104.0 \\
Rn & 21829.6 & -146.5 & -72.4 & -56.8 & {\bf 48.9} \\%& 114.2 & 227.4 \\
Uuo & 46259.6 & -298.4 & -162.3 & -139.6 & {\bf 81.1} \\%& 198.2 & 388.4 \\
168 $e^{-}$ & 106907.1 & -636.4 & -369.8 & -336.9 & {\bf 158.9} \\%& 379.5 & 717.9 \\
218 $e^{-}$ & 198077.5 & -1065.9 & -622.5 & -579.7 & {\bf 318.9} \\%& 673.8 & 1210.4 \\
290 $e^{-}$ & 389072.0 & -1888.9 & -1114.7 & -1056.4 & {\bf 630.1} \\%& 1208.6 & 2070.1\\
\end{tabular}
\end{ruledtabular}
\end{center}
\end{table*}
The results are reported in Table \ref{ta1}. For small atoms (Ne and Ar), the GE2 is
more accurate than $T_s^{Lind4}[n]$ and $T_s^{GAP4}[n]$. However, we recall that in the case of a
small number of electrons, the semiclassical and statistical concepts beyond the gradient
expansions do not hold. In fact, for larger atoms (Kr to
the noble atom with 290 electrons), both $T_s^{GAP4}$ and $T_s^{Lind4}$ outperform
GE2. In particular, $T_s^{GAP4}$ shows the best performance,
improving over $T_s^{Lind4}[n]$ and proving that, due to the inclusion of the gap,
$F^{GAP4}$ contains important physics beyond $F^{Lind4}$.
\subsection{Local band-gap}
In order to use Eq.~(\ref{gap4}) in semilocal DFT, we need to replace the true band gap $E_g$, with a density
dependent
local band gap. There are several models for the local band gap \cite{fabiano2014generalized,krieger1999electron},
constructed from the exponentially decaying density behavior \cite{krieger1999electron} or from conditions
of the correlation energy \cite{fabiano2014generalized}. In the slowly-varying density limit, they behave as
$E_g\sim |\nabla n|^m$, with $m\ge 2$. However, none of them can be considered accurate in this density regime.
On the other hand, under a uniform density scaling $n_\lambda(\R)=\lambda^3 n(\lambda \R)$,
the local band gap should behave as $E_g\sim \lambda^2$.
This condition is fulfilled by the general formula
\begin{equation}
E_g(\R)=a|\nabla n(\R)|^m/n(\R)^{2(2m-1)/3},\;\;\; m\ge 0, \;\;\; a\ge 0.
\label{eq16}
\end{equation}
Because other exact conditions of the local gap in the slowly varying density limit are not known,
we use Eq. (\ref{eq16}) in the expression of $T_s^{GAP4}$, considering the case with $m=2$.
We fix the parameter $a$ requiring that the gradient expansion should recover the first two terms of the
kinetic energy asymptotic expansion for the large, neutral atom
\cite{thomas1927calculation,fermi1927metodo,scottLEDPMJS52,schwingerPRA80,schwingerPRA81,englertPRA84,englertPRA85,elliottPRL08,
lee2009condition,fabiano2013relevance}
\begin{equation}
T_s=c_0 Z^{7/3}+c_1 Z^2+c_2 Z^{5/3}+...,
\label{eq17}
\end{equation}
where $Z$ is the number of electrons. The first term in Eq. (\ref{eq17}) is the Thomas-Fermi one
\cite{thomas1927calculation,fermi1927metodo}, the second
is the Scott correction due to the atomic inner core \cite{scottLEDPMJS52},
and the last term accounts for quantum
oscillations \cite{schwingerPRA80,schwingerPRA81,englertPRA84,englertPRA85}.
The exact coefficients are shown in the first line of Table
\ref{ta2}. As in Ref. \onlinecite{lee2009condition}, we assume that any gradient expansion
that is exact for the uniform electron gas, should
have the exact $c_0$ coefficient. The calculation of $c_1$ and $c_2$ has been done using
the method proposed in Ref. \onlinecite{lee2009condition}.
We recall that the semiclassical atom theory has been often used in
the development of exchange functionals
\cite{constantin2016semiclassical,constantin2011semiclassical,sunPRL15,sunPNAS15,b88,cancio2012laplacian}
and occasionally also for kinetic energy functionals \cite{laricchia2011generalized}.
Finally, we mention that
these gradient expansions are models for the total KE, and not for the KE density,
where the use of the reduced Laplacian $q$ (which is not
present in linear response of the jellium model) is essential
\cite{cancioJCP16,cancio2016visualizing,kinairy14,smiga2015subsystem}.
\begin{table}
\begin{center}
\caption{\label{ta2} The coefficients $c_0$, $c_1$, and $c_2$ of the large-$Z$ expansion of
the kinetic energy (see Eq. (\ref{eq17})).}
\begin{ruledtabular}
\begin{tabular}{lrrr}
& $c_0$ & $c_1$ & $c_2$ \\
\hline
Exact & 0.768745 & -0.500 & 0.270 \\
GE2 & 0.768745 & -0.536 & 0.336 \\
LGAP-GE & 0.768745 & -0.500 & 0.283 \\
\end{tabular}
\end{ruledtabular}
\end{center}
\end{table}
Using the procedure described above, we find $a=0.0075$, and we obtain the following gradient expansion (denoted as LGAP-GE)
\begin{eqnarray}
& T_s^{LGAP-GE}= \int d\R \tau^{TF}[1+a \frac{5\pi}{72}s+(\frac{5}{27}+a^2 \frac{27}{91}
\frac{\pi^2-4}{64})s^2+ \nonumber \\
& a \frac{5\pi}{36}s^3+\mathcal{O}(|\nabla n|^4)] \nonumber \\
& = \int d\R \tau^{TF}[1+0.0131 s+ 0.18528 s^2+0.0262 s^3].
\label{lgap}
\end{eqnarray}
Note that in Eq. (\ref{lgap}) only terms to up $s^3$ are considered (terms in Eq. (\ref{gap4}) proportional to $q$ or
$q^2$ are neglected, as these terms will correspond to $s^4$).
As shown in Table \ref{ta2}, LGAP-GE gives a very accurate large-$Z$ expansion, having the $c_2$ coefficient
close to exact.
The results for noble atoms are reported in Table \ref{ta1}. LGAP-GE is reasonably accurate
for all atoms and, as expected due to the inclusion of the semiclassical atom theory,
the accuracy increases with the number of electrons.
One additional
observation is that
LGAP-GE contains odd powers of the reduced gradient, in contrast with $F^{Lind4}$.
Nevertheless, Ou-Yang and Mel Levy have already shown that
using non-uniform coordinates scaling requirements
\cite{ou1990nonuniform}, the GE4 terms in the KE gradient expansion can be replaced by an $s$-only
dependent term \cite{ou1991approximate}, whose
coefficient must be positive (and was fitted to the Xe atom).
The resulting simple KE functional, that behaves better
than GE4 for the non-uniform density scaling, has the following enhancement factor
($F_s=\tau^{approx}/\tau^{TF}$):
\begin{equation}
F_s^{OL1}=1+\frac{5}{27}s^2+c s,
\label{eq19}
\end{equation}
with $c=0.01459$ being slightly bigger than its LGAP-GE counterpart.
Anyway, we need to acknowledge that, since
the kinetic potential of a GGA functional (with the enhancement factor $F_s$) has the general form
\begin{equation}
\frac{\delta T_s}{\delta n}=\frac{\partial \tau^{TF}}{\partial n}F_s(s)+\tau^{TF}\frac{\partial F_s}{\partial s}\frac{\partial
s}{\partial n}-\nabla\cdot[\frac{1}{s}\frac{\partial F_s}{\partial s}\cdot\frac{\nabla n}{n^{8/3}}],
\label{eq20}
\end{equation}
a necessary condition for it to be well defined is $|\frac{1}{s}\frac{\partial F_s}{\partial s}|< \infty$.
This is not satisfied by
the LGAP-GE (and OL1). Thus, the term $\propto s$ gives a diverging
kinetic potential ($\delta T_s/\delta n\rightarrow \infty$) at $s=0$.
This is due to the high non-locality of Eq.
(\ref{gap4}), which was not fully suppressed by the local gap model of Eq. (\ref{eq16}) with $m=2$.
Note that this divergence is a direct consequence of the jellium-with-gap theory.
Nevertheless, for molecular systems $s=0$ only at the middle of bonds, and it has been
found that this divergence is not important in real calculations of weakly-bounded molecular systems
\cite{gotz2009performance}. In fact, the same problem is shared by other well-known KE functionals
\cite{ou1991approximate,thakkar1992comparison,borgo2013density}.
\section{Kinetic energy functional constructed from the LGAP gradient expansion}
\subsection{The LGAP GGA}
To show the importance of the LGAP-GE, we construct a simple GGA functional (named LGAP-GGA or simply LGAP)
that recovers the LGAP-GE in the slowly-varying density regime.
We consider the RPBE exchange enhancement factor form
\cite{hammer1999improved}, $F_x^{RPBE}=1+\kappa(1-e^{-\mu s^2/\kappa})$, and we fix $\kappa=0.8$ from the Lieb-Oxford bound
\cite{lieb1981improved}, using the approximate link between the kinetic and exchange energies
(i.e. \textit{the conjointness conjecture} \cite{lee1991conjoint,march1991non,constantin2011semiclassical}).
Note that, to our knowledge, the RPBE functional form has not been
yet used in the development of kinetic functionals.
The LGAP kinetic enhancement factor is therefore defined as
\begin{equation}
F^{LGAP}_s=1 + \kappa \Big( 1 - e^{-\mu_1 s-\mu_2 s^2-\mu_3 s^3} \Big),
\label{eq23}
\end{equation}
where $\mu_1=b_1/\kappa$, $\mu_2=b_2/\kappa+\mu_1^2/2$,
and $\mu_3=b_3/\kappa+\mu_1 \mu_2-\mu_1^3/6$,
such that it recovers the LGAP-GE in the slowly-varying density limit.
Here $b_1=0.0131$, $b_2=0.18528$, and $b_3=0.0262$ [see Eq. (\ref{lgap})].
\begin{figure}[t]
\includegraphics[width=\columnwidth]{fig2.eps}
\caption{Comparison of kinetic enhancement factors }
\label{fs}
\end{figure}
\begin{table*}
\begin{center}
\caption{\label{ta3} Mean absolute relative errors (MARE) of the non-self-consistent benchmark tests,
and mean absolute errors (MAE in mHa) of FDE self-consistent tests, given by several KE functionals.
The best result of each group is highlighted in bold style.
}
\begin{ruledtabular}
\begin{tabular}{lrrrrr}
& GE2 & revAPBEk & OL1 & LC94 & LGAP \\
\hline
\multicolumn{6}{c}{Total KE (non-self-consistent calculations)}\\
Atoms and ions & 1.1 & 1.2 & 1.1& \bf{0.8} & 1.1 \\%& 1.41 & 1.22 \\
Jellium clusters & 1.0 & \bf{0.8} & 1.0 & 0.9 & \bf{0.8} \\%& \bf{0.8} & 0.9 \\
Jellium slabs & 0.6 & 0.5 & 0.5& 0.5 & \bf{0.4} \\%& \bf{0.4} & 0.5 \\
Molecules & 0.9 & 0.4 & 0.7 & 0.5 & \bf{0.2} \\%& 0.6 & 0.4 \\
\multicolumn{6}{c}{KE differences (non-self-consistent calculations)}\\
Jellium cluster DKE & 27.2 & 23.1 & 28.9 & \bf{21.3} & 22.6 \\%& 23.8 & 23.4 \\
Jellium surfaces & 3.3 & 3.6 & 3.4& 3.8 & \bf{3.1} \\%& \bf{3.1} & 3.4 \\
Jellium slabs dKE & 5.0 & 3.5 & 4.7 & 4.1 & \bf{3.4} \\%& \bf{3.2} & 3.9 \\
Molecules AKE & 184 & 155 & 185 & \bf{154} & 159 \\%& 156 & 157 \\
\hline
\multicolumn{6}{c}{FDE results for molecular systems \footnotemark[1] (self-consistent calculations)}\\
Weak-interactions (WI) & 2.46 & \textbf{0.13} & 2.49& 0.36 & 0.21 \\%& 0.20 & 0.21 \\
Dipole interactions (DI) & 6.48 & 0.48 & 6.59& 0.67 & \textbf{0.45} \\%& 0.47 & \bf{0.40} \\
Hydrogen bonds (HB) & 10.68 & \textbf{1.27} & 10.90& 1.34 & 1.69 \\%& 1.33 & 1.66 \\
Dihydrogen bonds (DHB) & 4.39 & 3.08 & 4.50& 2.92 & \textbf{2.58} \\%& 3.03 & \bf{2.59} \\
Charge transfer (CT) & 5.05 & 2.61 & 6.94& 2.79 & \textbf{1.95} \\%& 2.42 & \bf{1.85} \\
MAE FDE & 5.66 & 1.72 & 6.31 & 1.82 & \textbf{1.50} \\%& 1.67 & \bf{1.46} \\
\end{tabular}
\end{ruledtabular}
\footnotetext[1]{Embedding energy errors
$\Delta E = E^{\text{FDE}} -E^{\text{KS}} $ (mHa) for different KE functionals and complexes.
In the last line, the mean absolute error (MAE) is reported.}
\end{center}
\end{table*}
\subsection{The kinetic energy benchmark}
In order to assess the LGAP KE functional, we consider several known tests.
For {\bf total KE}:
\begin{itemize}
\item {The benchmark set of atoms and
ions \cite{perdew2007laplacian,laricchia2013laplacian,laricchia2011generalized}.
All calculations employed analytic Hartree-Fock orbitals and
densities \cite{CR74}};
\item {The Na jellium clusters ($r_s=3.93$) set with magic
electron numbers 2, 8, 18, 20, 34, 40, 58, 92, and 106, used in Refs.
\onlinecite{perdew2007laplacian,laricchia2013laplacian,laricchia2011generalized}.
We use exact exchange orbitals and densities};
\item {The set of two interacting jellium slabs at different
distances \cite{laricchia2013laplacian}.
Each jellium slab has $r_s=3$ and a thickness of $2\lambda_F$. Here $\lambda_F=2\pi/k_F$ is
the Fermi wavelength. The calculations were performed using the orbitals and
densities resulting from numerical Kohn-Sham calculations within the
local density approximation \cite{kohn1965self} for the XC
functional};
\item {The set of molecules
(H$_2$, HF, H$_2$O, CH$_4$, NH$_3$, CO, F$_2$, HCN, N$_2$, CN, NO, and O$_2$) used in Refs.
\onlinecite{iyengar2001challenge,perdew2007laplacian,laricchia2013laplacian}.
The noninteracting kinetic energies of test molecules
were calculated using the PROAIMV code \cite{KBT82}.
The required Kohn-Sham orbitals were obtained by Kohn-Sham calculations
performed with the uncontracted 6-311+G(3df,2p) basis set, the
Becke 1988 exchange functional \cite{b88}, and Perdew-Wang
correlation functional \cite{PW91}}.
\end{itemize}
For {\bf KE differences}:
\begin{itemize}
\item The disintegration kinetic energy (DKE) of a jellium cluster \cite{constantin2009kinetic,laricchia2013laplacian};
\item The jellium surfaces test with bulk parameter $r_s$=2, 4, and 6 into the liquid drop model (LDM), as in Refs.
\onlinecite{perdew2007laplacian,laricchia2013laplacian,laricchia2011generalized};
\item The dissociation KE (dKE) of a jellium slab into two pieces (as in Ref. \onlinecite{laricchia2013laplacian});
\item The atomization KE (AKE) of molecules \cite{iyengar2001challenge,perdew2007laplacian,laricchia2013laplacian}.
\end{itemize}
For {\bf non-additive KE}:
\vspace{0.2cm}
We employ the LGAP functional in subsystem DFT
calculations, using the
TURBOMOLE \cite{turbomole} program, together with FDE script \cite{FDElar}.
The FDE calculations have been performed with a supermolecular def2-TZVPPD \cite{def2tzvpp,furchepol} basis set and the Perdew-Burke-Ernzerhof\cite{pbe} XC functional.
Five weakly interacting groups of molecular complexes are considered as a benchmark
\cite{laricchia2011generalized,laricchia2013laplacian,Laricchia2011114,savio2,savio3, smiga2015subsystem}:
\begin{itemize}
\item[{\bf WI}:] weak interaction (He-Ne, He-Ar, Ne$_2$, Ne-Ar, CH$_4$-Ne,
C$_6$H$_6$-Ne, (CH$_4$)$_2$);
\item[{\bf DI}:] dipole-dipole interaction ((H$_2$S)$_2$, (HCl)$_2$, H$_2$S-HCl, CH$_3$Cl-HCl,CH$_3$SH-HCN, CH$_3$SH-HCl);
\item[{\bf HB}:] hydrogen bond ((NH$_3$)$_2$, (HF)$_2$, (H$_2$O)$_2$, HF-HCN,
(HCONH$_2$)$_2$, (HCOOH)$_2$);
\item[{\bf DHB}:] double hydrogen bond (AlH-HCl, AlH-HF, LiH-HCl, LiH-HF,
MgH$_2$-HCl, MgH$_2$-HF, BeH$_2$-HCl, BeH$_2$-HF);
\item[{\bf CT}:] charge transfer (NF$_3$-HCN,C$_2$H$_4$-F$_2$,NF$_3$-HCN,
C$_2$H$_4$-Cl$_2$, NH$_3$-F$_2$, NH$_3$-ClF, NF$_3$-HF, C$_2$H$_2$-ClF,
HCN-ClF, NH$_3$-Cl$_2$, H$_2$O-ClF, NH$_3$-ClF).
\end{itemize}
\subsection{Results}
We compare our results with revAPBEk \cite{constantin2011semiclassical} and LC94 \cite{lc94} GGAs,
which are considered state-of-the-art KE functionals for FDE \cite{laricchia2011generalized},
as well as with GE2 \cite{Kirz57,brack1976extended} and OL1 \cite{ou1991approximate}.
The KE enhancement factors of the considered functionals are reported in Fig. \ref{fs}.
In the inset of Fig. \ref{fs}, we show that LGAP and LGAP-GE are identical
(by construction) at relatively small values of
the reduced gradient ($0\leq s \leq 0.5$), both differing significantly from the GE2 behavior.
Consequently, LGAP shows a bigger enhancement factor than both LC94 and revAPBEk (i.e.
$F_s^{LGAP}\ge\sim F_s^{revAPBEk}\ge\sim F_s^{LC94}$) when $s\leq 2.5$. Such a
feature has been proved to be essential for jellium surfaces \cite{airy3}.
On the other hand, the LGAP enhancement
factor recovers its maximum value $F_s\rightarrow 1+\kappa$ at $s\approx 3$, faster than revAPBEk.
In Table \ref{ta3} we report the numerical results of all the tests.
For total KE tests, LGAP gives the best overall performance, among the considered functionals,
being the best for jellium clusters, jellium slabs and molecules.
For KE differences LGAP is the most accurate for jellium surfaces and dissociation KE
of jellium slabs.
We also mention that LGAP performs reasonably well for all the other tests,
being in line with revAPBEk.
Finally, LGAP outperforms the other functionals for the FDE theory, being especially
accurate for dipole-dipole, dihydrogen bond and charge transfer interactions.
These latter results show that, in agreement with the finding of
Ref. \onlinecite{gotz2009performance}, the divergence at $s=0$ of the
LGAP kinetic potential is not important for calculations of weakly-bounded molecular systems.
Moreover, results indicate that the LGAP-GE gradient expansion
can be successfully used in the kinetic energy functional construction, which perform relatively well in FDE theory.
\section{Conclusions}
In conclusion, we have investigated the linear response of the jellium-with-gap model,
in the context of semilocal kinetic functionals. We have shown that the Levine and Louie \cite{levine1982new}
analytical generalization of the Lindhard function ($F^{GAP}$) contains important physics beyond jellium model,
and in particular we mention the following properties:
$(i)$ $F^{GAP}$ recovers the Lindhard function when the band gap is zero (i.e. $E_g=0$);
$(ii)$ $F^{GAP}$ has the correct behavior (see Eq. (4)) at small wave vectors, expressing the material-dependent
constant $b$ in terms of the band gap;
$(iii)$ In the regime of small band gap energy (i.e. $E_g \leq E_F$, with $E_F$ being the Fermi energy),
$F^{GAP}$ gives the GAP4 gradient expansion of the kinetic energy (see Eq. (17)), which is band-gap-dependent, and
performs remarkably well in the atomic regions where the density varies slowly, improving
over $T_s^{Lind4}$ of Eq. (7) (see Table I).
These features show that $F^{GAP}$ should be further
investigated and exploited in the field of non-local kinetic functionals
\cite{wang2002orbital,huang2010nonlocal,wang1998orbital,shin2014enhanced,ho2008analytic,
alonso1978nonlocal,garcia1996nonlocal,chacon1985nonlocal,garcia1998nonlocal,garcia2008approach,
karasiev2014progress,karasiev2014finite,karasiev2015chapter,cangi2010leading,ribeiro2016leading}, and we will like to
address this important issue in further work.
Finally, by considering a local band gap model, and a simple enhancement factor form, we
have constructed the
non-empirical
LGAP GGA kinetic energy functional, derived from the linear response of the jellium-with-gap model (a.i. the GAP4
gradient expansion). This functional
showed the best performance in the context of FDE theory. Thus, it can be further used in real applications.
\newline
\section*{Acknowledgements}
This work was partially supported by the National Science Center under
Grant No.
DEC-2013/11/B/ST4/00771 and DEC-2016/21/D/ST4/00903.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,691 |
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