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Joan Martí i Cantó (Barcelona, abril del 1829 – Barcelona, febrer del 1887) va ser prevere, escriptor i músic.
Biografia
Treballà com a impressor i llibreter i més tard estudià per capellà a Vic i fou ordenat sacerdot el 1855. Fou vicari al Bruc, a Sant Cugat del Vallès, Santa Maria de Gràcia, i a Barcelona a Betlem, ecònom de Sant Pau (el 1858) i Sant Cugat, i de Sant Miquel de la Barceloneta, i el del Sant Àngel Custodi, d'Hostafrancs, d'on obtingué la propietat del càrrec i en fou primer rector, càrrec que ocupà del 1868 al 1881. L'església patí dos incendis intencionats (1868 i 1872) i l'esfondrament del sostre (1877). Feu construir la primera capella, que va desaparèixer arran de la construcció del nou temple (1891). Va morir essent rector de Santa Madrona.
Hi ha documentades prop de 50 publicacions. Algunes d'elles degueren tenir força èxit perquè se'n troben diverses edicions. Així de títols diferents n'hi ha una trentena llarga, on cal afegir una traducció i la publicació de tres revistes. Totes les obres són de caràcter religiós, molt sovint de devoció mariana. Gairebé totes són en castellà, tret d'uns goigs en català («Goigs en alabansa del gloriós sant Ángel Custodi de Barcelona») del 1879, i una obra en llatí.
Fou molt actiu en la divulgació de la devoció mariana, essent un dels fundadors, l'any 1879, de la Confraria de la Mare de Déu de Lourdes, a Santa Madrona però en particular de la Mare de Déu de Montserrat, publicà una guia de Montserrat de la que se'n feren cinc edicions i també tres edicions del «Mes lírico de Maria, o los cancioneros de Montserrat» aquesta darrera obra va rebre l'aprovació explícita del Papa Pius IX.
Tingué un especial interès en la música com a eina pastoral i publicà diverses obres de caràcter musical: l'esmentada «Mes lírico de Maria, o los cancioneros de Montserrat», «Cantos religiosos puestos en música fácil y agradable», «Armonías angélicas a la Inmaculada Concepción de Maria Santísima» i dos llibres de música religiosa a cant pla. Martí i Cantó fou l'autor de moltes de les músiques i també de les lletres de les seves publicacions musicals, tot i que també hi col·laboraren diversos músics: Bernat Calvó Puig i Capdevila, Francesc Andreví i Castellar, Càndid Candi i Casanovas, Josep Marraco i Xauxas, Josep Marraco i Ferrer, Josep Barba i Bendad, Bartomeu Blanch i Castells, Manuel Borrell, Josep Casals, Manuel Dordal, Domènec Ferrer, Pere Gassull, Carles Isern, Nicolau Manent i Puig, Miquel Masramon, D.N. Monserrat, Antoni Oller i Biosca, Jaume Puig i Torrens, Josep Rosés, Josep Sabatés, Mateu Sabatés, Rafael Selleras, Josep Saltó, Joan Tolosa i Noguera, Martí Valls i Miquel Vila, P. Ametller.
Les lletres les signen, a més del mateix Martí i Cantó els següents poetes: Jaume Agustí i Milà, Miquel Campderós, Francesc Crusellas, Ramon Garcia, Gaspar Gasch, Pere Muntañà, C. Orriols i C, Narcis Planas i Gispert, D.J. Puigdemasa, Joaquim Roca i Cornet, Jaume Roig, Ferran de Segarra, Josep Tarrí
Obres
1856.- Mes Lirico de Maria o Los Cancioneros de Montserrat. Conforme se practica en la Iglesia Parroquial de los santos Justo y Pastor de Barcelona (1a edició), Imprempta de Jose Gorgas, Barcelona. 289 pp.
1857.- Armonías angélicas a la Inmaculada Concepción de Maria Santísima. Novenario. Barcelona. V. Magriña, 44, 8 p.
1858.- Flores del alma consagradas a la Virgen Santísima durante el mes de mayo, según el mes lírico de María. Barcelona. Librería de J. Subirana. 287 p
1859.- Novenario que la Archicofradía de San Luis Gonzaga, establecida en la Iglesia de N.S. de Belén, consagra a su protector / arreglada por Juan Martí y Cantó. Barcelona : Imprenta de Vicente Magriná, 32 pp.
1863.- Mes lírico de María ó Los cancioneros de Montserrat. 2a edición considerablemente aumentada, Impremta de Juan Magriñá y Subirana, Barcelona, 234 pp.
1863.- Apariciones de Ntro. Sr. Jesucristo resucitado á su madre santísima : práctica de piedad / que ofrece a las almas cristianas el presbítero Juan Martí y Cantó. Barcelona : Imprenta de Magriñá y Subirana. 16 pp
1863.- Historia de la imagen y santuario de Ntra. Sra. de Montserrat. Barcelona : Impr. de Magriñá y Subirana. 128 p
1864.- Historia de la imagen y santuario de Ntra. Sra. de Montserrat. Barcelona 3a edició: Impr. de Magriñá y Subirana. 160 p
1865.- Los Dolores de María. Puestos a la consideración del cristiano durante los siete viernes de cuaresma. Imp. Magriñá y Subirana. Barcelona 168 pp.
1865.- Ramillete de flores celestiales nacidas en el vergel de María, y consagradas a la Santísima Virgen de las Mercedes, gloria de Barcelona, durante el mes de mayo. Imp. Magriñá y Subirana. Barcelona 480 pp.
1866.- Aroma de la infancia. Devocionario de los niños, utilísimo para regalar a los del uno y del otro sexo; para aguinaldos, premios de examenes y de doctrina en las Parroquias, Colegios, Primeras Comuniones, Sociedades Catequísticas, etc., etc.
1868.- Historia de la imagen y santuario de Ntra. Sra de Montserrat y viaje pintoresco a sus cuevas subterraneas 4º edi Viuda Bassas, Imp. del Porvenir. 256 pp.
1869.- Cantos religiosos puestos en música fácil y agradable, con acompañamiento de piano, armónium u órgano propios para todas las solemnidades y épocas del año, misiones, tríduos, novenarios, etc. Barcelona : Imprenta y librería del heredero de D. Pablo Riera. 490 pp
1870.- Mes Lirico de Maria, o los Cancioneros de Montserrat, etc. Tercera Edicion Barcelona. 239 pp.
1870.- El Pan nuestro de cada dia, que ofrece á sus queridos hijos los cristianos la mas tiérna de las madres, Maria Santísima. Devocionario completísimo para todos los dias y épocas del año inclusos el tiempo de Adviento, Natividad, Cuaresma, Semana Santa y Pascua; Novenarios de la Inmaculada Concepcion, y de las santas Almas; el mes de Maria; y un sin número de otras prácticas piadosas. 2a ed. Barcelona. Librería de la Viuda é Hijos de J. Subirana, 668 pp.
1871.- Misas á canto llano. Imprenta de los herederos de la viuda Pla. Barcelona. 69p
1871.- Novenario a la purísima reina de los cielos María Santísima, patrona de España, en el misterio de su Inmaculada Concepción. Barcelona : Imp. de El Porvenir, a cargo de J. Medina. 120 pp.
1872.- Intróitos, graduales, ofertorios y comunios á canto llano. Imprenta de los herederos de la viuda Pla. Barcelona. 91 pp.
1877.- Historia de la imagen y santuario de Ntra. Sra. de Montserrat, y viaje pintoresco a sus cuevas subterráneas. 5a edición. Barcelona, Juan Roca y Bros, 252 pp.
1877.- Viaje pintoresco a las cuevas de Montserrat, en Cataluña. Juan Roca y Bros, 224 pp.
1878.- El dia grande del alma cristiana. Reflexiones oraciones y meditaciones, propias para preparar con fruto a los niños y niñas para el acto solemne de su primera comunion. Juan Roca y Bros. 121 pag + indices.
1878.- Trilogía ascética consagrada a la Sagrada Familia : mes de junio dedicado al sacratisimo corazón de Jesús. Barcelona: Libr. Montserrat de Sucessor de Roca y Bros. 225 pp [any aproximat]
1879.- Goigs en alabansa del gloriós sant Ángel Custodi de Barcelona. Barcelona : Estamp. de Magriñá, 1879. Text a quatre columnes separades per filets. Caixa: 300 x 200 mm Goigs cantat a Barcelona (Barcelonès). Imatge de l'advocació flanquejada per dos imatges al·legòriques i orla amb motius tipogràfics. ''Tornada / Puig per consol nostre os feu, / Lo Senyor, Custódi amat, / Protegiu, Angel de Déu, / Vostra parròquia y Ciutat.''
1879.- Sacro trimestre consagrado á la benditísima Trinidad y familia de Nazaret : 2.º el mes de mayo santificado en honor y gloria de María Santísima. Barcelona : Juan Roca y Bros. 310 p
1881.- Historia de las mercedes de la Inmaculada en Lourdes. Juan Roca y Bros, Editor. 214 pag + indices.
1882.- Brevis collectio ex rituali romani : ad parochorum commodum eorunque vicariorum in Sacramentorum administratione, in infirmorum cura et eorum interitu, et alia utilissima. Barcinone : Typis. Bibliothecae Religiosae, 256 p.
1883.- El báculo del alma cristiana. Para sostenerla durante su travesia por el destierrode la vida actual. Imprenta de Luis Tasso y Serra. 634 pag
1883.- El cielo en la tierra : cuatro palabritas que dirige a sus tres hermanas monjas y a todas las religiosas en general. Barcelona : Juan Roca y Bros, 235 p., [1] f
1883.- Trisagio Mariano de la peregrinación catalana de 1883 á Lourdes / compuesto por el presidente de la misma D. Juan Martí y Cantó. Barcelona : Litografia de Y. Arce, 7 p.
1885.- Más allá de la tumba, o sea, El amor cristiano enjugando las lágrimas de las benditas almas del purgatorio : preces, oraciones, sufragios, indulgencias y demás obras buenas que la iglesia pone en nuestras manos para que las utilicemos en favor de los difuntos. Barcelona : Juan Roca y Bros, 712 p
1887.- El Romero de Montserrat : utilizando cuanto ve y admira en su peregrinación para dar gloria a Dios y honrar a su madre santisima Barcelona : J. Roca Bros, 459 p.
1887.- Historia de la imagen y santuario de Ntra. Sra de Montserrat y viaje pintoresco á sus cuevas subterráneas 5 ed. Barcelona : Juan Roca y Bros, (Tip. de Casanovas) 252 p., 1 f.; 12.5 cm
1889.- Sacro trimestre consagrado a la benditísima Trinidad y familia de Nazaret. J. Roca y Bros. Barcelona. 1889
1890.- Sacro trimestre consagrado a la benditísima Trinidad y familia de Nazaret : 2º el mes de mayo santificado en honor y gloria de María Santísima. Barcelona : Imp. y Lib. de Ntra. Sra. de Montserrat. 310 p
1892.- El Dia grande del alma cristiana : reflexiones, oraciones y meditaciones propias para preparar con fruto á los niños y niñas para el acto solemne de su primera comunión. Barcelona : Librería de Montserrat, de Juan Roca y Bros. 121, [6] p
1896.- Sacro trimestre consagrado a la benditisima trinidad y familia de Nazaret. 1º el mes de marzo dedicado al castisimo esposo de la Inmaculada Virgen San Jose. Edit. Juan Roca y Bros. 1896-1879 247 + XXXII pag. 1a edición.
1898.- Sacro trimestre consagrado a la benditisima trinidad y familia de Nazaret 1º el mes de marzo 1898, Barcelona, Libreria de Montserrat, primera edicion, 275 pag.
S.D.- El Angel del peregrino cristiano
S.D.- Mes de Maria. Oraciones, meditaciones, ejemplos y flores espirituales para celebrar digna y santamente el Mes de Mayo segun el Mes lírico de Maria'' S.D.- Modo de hacer con fruto una peregrinación ó Romeria á Ntra. Sra. De Montserrat en su célebre monasterio. S.D.- Historia completa de la imagen y santuario de Nuestra Señora de Montserrat y viaje pintoresco a sus cuevas subterraneas 5a edición.
S.D.- Manual de meditaciones. En caracteres grandes para las personas de vista cansada. Traduccions
1855.- Orsini, Mathieu, 1802-1875. Flores del cielo : imitación de las santas / escrita porel Abate Orsini; traducida de la segunda edición francesa por D. Juan Marti y Cantó. Barcelona : Imprenta de José Gorgas, 1855 462 p., [1] p.; 19 cm
Revistes
1881.- Ecos del amor a la Inmaculada Virgen y a los santos ángeles revista mensual. Barcelona : [s.n, 1881-18--]
1867-1878.- Ecos del amor de María : publicacion quincenal destinada á difundir las glorias y la devocion á la Inmaculada Reina delos cielos María Santísima. Barcelona. Imprenta y Librería del Heredero de Pablo Riera, [12] vols.
S.D.- Los Santos Angeles. Revista mensual. ''
Referències
Compositors barcelonins contemporanis
Escriptors barcelonins contemporanis
Religiosos barcelonins contemporanis
Impressors barcelonins
Llibreters barcelonins
Morts a Barcelona
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was the daimyō of Suō Province and the head of the Ōuchi clan, succeeding Ōuchi Yoshioki.
In 1522, he fought the Amago clan along with his father, Yoshioki, to win the control of Aki Province. Upon Yoshioki's death in 1528, Yoshitaka became the head of Ōuchi clan. In the 1530s, he led a military actions in the northern Kyūshū, defeating Shōni clan to win control of the area. With his back then secure, in 1540 he again started combating the Amago clan and by 1541, managed to completely control the Aki province.
However, in 1542, an invasion into Izumo Province ended in a disaster, with Yoshitaka losing his adopted son Ōuchi Harumochi along with large number of troops against Amago Haruhisa. His 1542–43 Siege of Toda Castle ended in failure. He completely lost his ambitions of expanding his domains and devoted his energy to the arts and culture. His retainers split into two factions. Those led by Sagara Taketō wanted the Ōuchi clan to simply do nothing more than maintain the control of their current domains, while those led by Sue Harukata wanted to continue expanding. Yoshitaka sided with the former.
Under the patronage of Yoshitaka, foreign trade and the arts flourished, and the Ōuchi home city Yamaguchi prospered greatly. In addition, Yoshitaka also attracted the Portuguese missionary Francis Xavier, and allowed him to proselytize while he was in Yamaguchi. At the same time, Yoshitaka fostered a close relationship with Emperor Go-Nara in Kyoto, and sponsored many imperial rites that the imperial court could not have afforded otherwise. On March 27, 1551, the embattled emperor appointed Ōuchi Yoshitaka as Acting Governor of Yamashiro (山城権守), the home province where the imperial capital Kyoto was located, in a bid to leverage the Ōuchi against the ravages of the warlord Miyoshi Nagayoshi, who occupied the capital. Yoshitaka, as Acting Governor of Yamashiro and, by extension, the protector of the court, embarked on a daring plan to relocate the emperor and the court to Yamaguchi. High-ranking courtiers and performers of imperial rites moved to Yamaguchi, including dignitaries such as former regent (kampaku) Nijō Tadafusa and retired Grand Minister (Sadaijin) Sanjō Kin'yori (三条公頼; father-in-law of Takeda Shingen). By the end of the eighth month of 1551, nearly the whole court, save for the emperor himself and the palace ladies, was in Yamaguchi.
The military establishment of the Ōuchi resented Yoshitaka's apparent "weakness" and his plan to settle the imperial court in Yamaguchi — such a move would see privileges accorded to the courtiers and undermine their own standing within the Ōuchi clan. In September 1551, the faction led by Sue Harukata revolted and attempted to take over the Ōuchi clan. With the control of troops in Harukata's hand, it was over in few days—the courtiers and ministers were massacred and Yoshitaka was forced to perform seppuku at the Tainei-ji Temple (大寧寺) in Nagato Province after composing his death poem:
Both the victor
and the vanquished are
but drops of dew,
but bolts of lightning –
thus should we view the world.
References
1507 births
1551 deaths
Daimyo
Suicides by seppuku
Ōuchi clan
People from Yamaguchi (city) | {
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{"url":"https:\/\/cdkharris.github.io\/posts\/2019\/08\/load-remote-tecplot-data\/","text":"Published:\n\nThe following are directions I wrote up for my research group which explain how to use tecplot\u2019s szlserver tool to preview remote data before downloading it. Our simulation results are often many GB per file, so this is an important capability for reviewing data before committing to a lengthy download. This post also accompanies my pytecplot-accessories repo.\n\nTecplot has the capability to show and manipulate data that is stored on a remote machine. We can show Tecplot data that is saved on NASA Pleiades from the GUI on our university desktops. There are a few limitations and caveats:\n\n\u2022 The remote data should be on a linux operating system (as are most supercomputers).\n\u2022 The remote machine should have the szlserver tool installed (this ships with Tecplot and can be installed in the User\u2019s home directory).\n\u2022 The remote data must be in the SZL file format (files can be converted easily with tecplot\u2019s batch processing command line tool).\n\nAfter the initial set up there are a couple extra steps, but once remote data is loaded it can be manipulated and inspected just as if the data was stored on the client computer. It promises to be very convenient for inspecting simulations in progress before committing to a possibly lengthy download.\n\nThis tutorial assumes working knowledge of the command line or terminal, bash, and several command line tools.\n\n## Setting up\n\n### Install szlserver\n\nLog in to the remote machine via SSH. The remote machine should have a current installation of Tecplot. Run the installation script:\n\n$bash \/path\/to\/tecplot\/2018r1\/360ex_2018r1\/szlserver\/tecplotszlserver2018r1_linux64.sh This path will certainly change as new versions of Tecplot come out. Press CTRL-C to skip the license (or read it) and type y to continue. Unless you have root privileges you probably will not be able to install the server tool in the default location, so install it in your home directory. For me this could be: \/home\/cdkharris\/tecplotszlserver2018r1 Then add the tool to your path. If you use bash then put the following line in your bash profile: PATH=$PATH:$HOME\/tecplotszlserver2018r1\/bin Do source .profile and try which szlserver to be sure that the tool is installed. ### Convert data to tecplot szl file format Still on the remote machine, navigate to where your data is saved. Try which tec360 to be sure that Tecplot\u2019s batch processing tool is installed. Run the following command to convert a single plt file to SZL format: $ tec360 data.plt -o output.szplt\n\n\nIf your simulation results span many .plt files you can combine the multiple zones (e.g. for multiple time steps or iterations) into a single .szplt file.\n\n$tec360 z=0_mhd_*.plt -o output.szplt ## Loading remote data On your local machine open an instance of Tecplot. Select File > Load Remote Data... to open the Remote Data Load Options dialog. Select Manual Connection and from there select Connect. This will open the Waiting for Server Connection dialog. The dialog will provide you with a command that will look something like: $ szlserver -m 141.212.196.111 -p 58232 -k 744704943\n\n\nOnce the connection is established, the Waiting\u2026 dialog on your local machine should close and the Remote\u2026 dialog should permit you to select data from the remote machine. Find the .szplt data and load it.","date":"2020-08-12 12:28:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.31588655710220337, \"perplexity\": 3896.037708434816}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-34\/segments\/1596439738892.21\/warc\/CC-MAIN-20200812112531-20200812142531-00138.warc.gz\"}"} | null | null |
(ethoxymethoxy)cyclododecane
EU Method A.4 (Vapour Pressure)
0.296 Pa
Summary of vapour pressure data
Log10 of vapour pressure at 25 °C
Vapour pressure at 25 °C in Pa
The vapour pressure of the test substance was 0.296 Pa at 25 °C using the vapour pressure balance method.
The vapour pressure of the test substance registered was determined under GLP using a vapour pressure balance in accordance with EU Method A.4. A sample of the test substance was placed in the balance and a sequence of ten test runs was started after the sample had been under vacuum for about 4.5 hours. Each run was tested over a temperature range from 25 to 35 °C. Linear regression analysis was used to determine the vapour pressure of the substance at 25 °C. The average vapour pressure of the substance at 25 °C as determined in ten runs was 0.296 Pa. The test substance did not change in appearance under the test conditions.
0.296 Pa at 25 °C
The vapour pressure of the substance was 0.296 Pa at 25 °C was determined in accordance with OECD 104 using the vapour pressure balance method. | {
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Ionization alters water in two significant and measurable ways: pH and ORP. These alterations to water are what make it very different from other waters you may drink.
pH stands for "potential hydrogen" and is a measurement that provides an indication of the level of hydrogen in a substance. It is measured by the pH scale. Proper body pH is an important factor in good health.
The other way an ionizer alters the water is in ORP. This stands for Oxidation Reduction Potential (also known as "Redox") Most leading water researchers from Asia agree that in ionized water the elevated pH is good, but that ORP is more important. Alteration to the ORP is what causes the micro-clustering, antioxidant and oxygenating effects.
ORP is a "potential" energy that is stored and ready to be put to work. It's not necessarily working, but we know that the energy is there and we can measure it. Another way to look at this potential might be to look at pressure. If you blow up a balloon, there is air pressure inside. As long as the balloon is closed, the pressure remains and can be measured. When released, this Potential energy becomes kinetic energy.
In electrical terms, potential energy can be measured. When we use the term "potential" in describing ORP, we are actually talking about electrical potential as expressed in millivolts. This potential is measured in water with an ORP meter. What you measure is the very slight voltage in water. We are actually measuring the presence of oxidizing or reducing agents by their specific electrical charge, thus Oxidation-Reduction "Potential". High pH water has More "reducing" agents (-ORP) and low pH water has more oxidizing agents (+ORP).
Oxidation is what turns an apple brown after it is cut or causes metal to rust. Rust weakens metal and signifies the deterioration of the apple. The process of oxidation "steals" electrons from the surface that is oxidized. When we measure a something's oxidizing potential, it is expressed in +ORP and measures the concentration of ions or oxidizing agents.
A "reducing" agent is simply something that inhibits or slows the process of oxidation. The reducing agent does this by "donating" an electron. When we measure a solution's oxidation-reduction potential, it is expressed in terms of ORP and measures the concentration of ions or reducing agents. In its most basic form, a reducing agent is an "antioxidant" ~ reducing oxidation. Follow this link to read more detailed info about the science of pH and ORP.
The ORP of most tap water in North America is between +200 to +600mv, ie. these waters are oxidizing agents. High pH ionized water demonstrates a –ORP and so is a reducing agent or "antioxidant". Most bottled waters are acidic (low pH) –many are quite acidic — and also have higher ORPs (over +400mv).
An ionizer works primarily on the mineral content in the water. It is the dissolved mineral content (referred to as TDS) which creates the pathway for the "ionization" (or more correctly electrolysis) to occur. Water without mineral content or TDS, like reverse osmosis or distilled water, will not conduct the current and therefore can not be "ionized". This first variable is the most crucial to performance. Tap waters vary widely in the dissolved mineral content. The higher the mineral content ("harder" water) the higher the levels of pH and ORP alteration an ionizer can achieve; the lower the mineral content ("softer water") the lower levels of pH and ORP alteration. The importance of this variable can not be emphasized enough.
The heart of an ionizer is the water cell which contains the electrodes. The electrodes are what deliver the current and creates the "ionization". We control the voltage conducted through the electrodes and then to the water by selecting the different "Alkaline" settings on an ionizer. The higher the Alkaline setting (or voltage), the more alteration you will achieve in pH and ORP. Effective conductivity is the primary determinant – not electrode size – of effective delivery of the current or voltage into the water needed to create electrolysis. Do not be fooled by the claim some manufacturers make those larger electrodes will necessarily deliver better performance. Generally, the larger electrodes have poorer conductivity – so they have to be larger.
On most ionizers, you can only adjust the flow rate by using your faucet or tap. If your faucet is all the way "on" the water will process very fast through the machine. If your faucet is just barely on this reduces the flow and the water will process for much longer. With a fast flow rate you may only achieve slight alteration in pH and ORP, slow it down and you will get higher pH and better ORP. Simply put, speed it up, you get a less alteration; slow it down and you'll get more.
To illustrate this principle lets look at two very different tap waters and their effect on performance. Remember the crucial variable is the dissolved mineral content or TDS (total dissolved solids) which is measured in parts per million. This creates the pathway for the ionization to occur. In California the tap water tests at 385 – 501ppm of total dissolved solids. The tap water in Seattle tests at approximately 40 – 47ppm. You could test water from an ionizer in at a given setting and flow rate and you would get a certain result. You could test the exact same ionizer in another setting, without altering the setting or flow rate and you would get dramatically different results. Is it the ionizer? No. It is the water as the main variable in performance. There is much less pathway in Seattle's water. To further illustrate variability, you could alter the voltage or flow rates through the ionizer in either Carlsbad or Seattle and you would get different results again.
Lastly, comparing ORP is a tricky business. Stating absolute values is impossible. Anyone who really knows and understands ionizers/ORP would agree. Anyone who states absolutes in performance reveals ignorance of the science. Further, pH and ORP are not tied to another. In other words, you can measure ORP in two pH9 waters and get two very different readings. Another factor to consider when comparing ORP is the level of pH you will drink.
Water with a pH over about pH10 does not taste good to the vast majority of people. Japanese research states that the ideal range for drinking alkaline water is between pH8.5 and pH9.5. Given this, testing ORP at those levels is where the real bang for the buck is; ORP at a pH level one would actually drink. Therefore, the only salient way to compare ORP in ionizers is side-by-side, with the same source water and each machine set to achieve the same drinkable level of pH. If you drink pH9 then the ORP you get at pH9 is the effective ORP in the ionizer. Not some "absolute" or even extraordinarily high ORP. | {
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{"url":"https:\/\/www.itwissen.info\/en\/octet-106128.html","text":"# octet\n\nAn octet is a group of eight bits that are transmitted, conveyed or processed together (e.g. PCM data word), but which do not necessarily have to have anything to do with each other.\n\nThe term octet originates from CCITT and is used synonymously with the byte, although a byte can also differ from eight bits, whereas the octet always consists of eight bits. In addition, the bits of a byte form a unit, for example an ASCII character. The byte is mainly used for memory sizes. The octet is used in long-distance data transmission and describes frames and data packet formats which, in addition to the user data, also include the start bits, stop bits and check bits.\n\nOctets represent 256 (2^8) values that can be written in binary, hexadecimal or decimal notation. Likewise, octets can be divided into two nibbles of four bits each.\n\nInformations:\n Englisch: octet Updated at: 22.01.2022 #Words: 142 Links: phase change material (PCM), data, word, Comit\u00e9 Consultatif International T\u00e9l\u00e9graphique et T\u00e9l\u00e9phonique (CCITT), byte (B) Translations: DE Sharing:","date":"2022-11-26 11:39:47","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\": 0.24193637073040009, \"perplexity\": 3680.962655519631}, \"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\/1669446706291.88\/warc\/CC-MAIN-20221126112341-20221126142341-00117.warc.gz\"}"} | null | null |
{"url":"https:\/\/jira.lsstcorp.org\/browse\/DM-21016?focusedCommentId=241475&page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel","text":"# Handle DECam instrument signature data in gen3\n\nXMLWordPrintable\n\n#### Details\n\n\u2022 Type: Improvement\n\u2022 Status: Done\n\u2022 Resolution: Done\n\u2022 Fix Version\/s: None\n\u2022 Component\/s:\n\u2022 Labels:\n\u2022 Story Points:\n8\n\u2022 Sprint:\nAP F19-6 (November), AP S20-1 (December), AP S20-2 (January), AP S20-3 (February), AP S20-4 (March)\n\u2022 Team:\n\n#### Description\n\nDM-20763 dealt with ingesting DECam data, but punted on managing the instrument signature data. This will require at-minimum implementing writeCuratedCalibrations.\n\nI've linked a variety of DECam calibration-related tickets as \"related', since we can maybe close some of those as won't fix once this is done (since they might be gen2-specific).\n\n#### Activity\n\nHide\nJohn Swinbank added a comment -\n\nFrom a pure EV perspective, yes, if this work is being handed off from AP to Architecture, let's do that by writing a brief summary of what happened on this ticket, marking it as done, and then using another ticket for Arch activities. (It's not obvious from the description here whether that other ticket is DM-23976, but I have no horse in that race.)\n\nShow\nJohn Swinbank added a comment - From a pure EV perspective, yes, if this work is being handed off from AP to Architecture, let's do that by writing a brief summary of what happened on this ticket, marking it as done, and then using another ticket for Arch activities. (It's not obvious from the description here whether that other ticket is DM-23976 , but I have no horse in that race.)\nHide\nTim Jenness added a comment -\n\nI think I will put renaming of instrument classes (removal of gen3) onto a new ticket since it doesn't affect any of the other discussions. I absolutely agree that now is the time to change names and class paths.\n\nShow\nTim Jenness added a comment - I think I will put renaming of instrument classes (removal of gen3) onto a new ticket since it doesn't affect any of the other discussions. I absolutely agree that now is the time to change names and class paths.\nHide\nTim Jenness added a comment -\n\nDM-23980 has been created for the instrument class renames.\n\nShow\nTim Jenness added a comment - DM-23980 has been created for the instrument class renames.\nHide\nJohn Parejko added a comment -\n\nShould we close this as either Invalid or Done, given DM-23976?\n\nShow\nJohn Parejko added a comment - Should we close this as either Invalid or Done, given DM-23976 ?\nHide\nJohn Parejko added a comment -\n\nDM-23976 has implemented writeCuratedCalibrations for obs_decam, so defects and camera are now taken care of. DM-23980 took care of removing the gen3 bit from the module hierarchies. The other cleanups and renames on this ticket are probably not worth trying to sort out in what would likely be an ugly rebase, and many of them depend on the not yet finished plan for how calibrations will be handled in general.\n\nI will close the four PRs on this ticket, and am marking the ticket as \"done\" because at least defects are now being managed per the ticket description.\n\nShow\nJohn Parejko added a comment - DM-23976 has implemented writeCuratedCalibrations for obs_decam, so defects and camera are now taken care of. DM-23980 took care of removing the gen3 bit from the module hierarchies. The other cleanups and renames on this ticket are probably not worth trying to sort out in what would likely be an ugly rebase, and many of them depend on the not yet finished plan for how calibrations will be handled in general. I will close the four PRs on this ticket, and am marking the ticket as \"done\" because at least defects are now being managed per the ticket description.\n\n#### People\n\nAssignee:\nJohn Parejko\nReporter:\nJohn Parejko\nWatchers:\nChristopher Waters, Eric Bellm, Jim Bosch, John Parejko, John Swinbank, Krzysztof Findeisen, Meredith Rawls, Simon Krughoff, Tim Jenness","date":"2021-01-21 21:55:48","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.4320175349712372, \"perplexity\": 4683.564082531294}, \"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-2021-04\/segments\/1610703527850.55\/warc\/CC-MAIN-20210121194330-20210121224330-00571.warc.gz\"}"} | null | null |
Configure your /etc/hosts file with your docker IP address. The project hostname is "project.loc"
Start docker environment :
$ make start
Stop docker environment :
$ make stop
Start provisioning :
$ make up
Run tests :
$ make tests
| {
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Q: Hilbert modular forms and Hecke operators over Q Let F be a totally real field. We know that we can define a Hecke operator $T_\mathfrak{m}$ on the space of Hilbert modular forms over $F$, say with some level structure, for any ideal $\mathfrak{m}$ that is coprime to the level.
Let's say we have two eigenforms $f_1$ and $f_2$ whose Hecke eigenvalues agree on $T_p$ for all rational primes prime to the level. Does it follow that the two forms are the say by some sort of multiplicity theorem?
A: No, it doesn't follow. Take e.g. the case where F is real quadratic. The space of Hilbert modular forms has an involution coming from the action of the nontrivial automorphism $\sigma \in \operatorname{Gal}(F/\mathbf{Q})$, and this interchanges the Hecke eigenvalues at $\mathfrak{m}$ and $\mathfrak{m}^\sigma$. So the Hecke eigenvalues at rational primes cannot distinguish $f$ from its image $f^\sigma$ under this involution.
| {
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Michel Biyoghé (ur. 9 grudnia 1970) – gaboński piłkarz grający na pozycji obrońcy. W swojej karierze rozegrał 20 meczów w reprezentacji Gabonu.
Kariera klubowa
Swoją karierę piłkarską Biyoghé rozpoczął w klubie Petrosport FC, w którym zadebiutował w 1994 roku. W 1995 roku przeszedł do US Bitam, w którym grał do 2006 roku. Wraz z US Bitam wywalczył mistrzostwo Gabonu w 2003 roku oraz zdobył trzy Puchary Gabonu w latach 1999, 2003 i 2006. W latach 2006-2008 grał w klubie Delta Téléstar.
Kariera reprezentacyjna
W reprezentacji Gabonu Biyoghé zadebiutował 7 listopada 1993 w przegranym 1:2 towarzyskim meczu z Maltą, rozegranym w Tunisie. W 1994 roku został powołany do kadry na Puchar Narodów Afryki 1994. Nie rozegrał na nim żadnego spotkania. Od 1993 do 2003 rozegrał w kadrze narodowej 20 meczów.
Bibliografia
Reprezentanci Gabonu w piłce nożnej
Piłkarze US Bitam
Uczestnicy Pucharu Narodów Afryki 1994
Urodzeni w 1970 | {
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\section{Introduction}\label{sec:Intro}
The Cucker-Smale system is a prototypical individual-based model of collective behavior.
It was introduced in the seminal papers~\cite{CS1, CS2}, originally as a model for language evolution.
Later the interpretation as a model for flocking in animals (birds) prevailed.
The model describes a group of $N\in\mathbb{N}$ autonomous agents
located in the physical space $\mathbb{R}^d$, $d\geq 1$.
The agents are described by their phase-space
coordinates $(x_i(t), v_i(t))\in\mathbb{R}^{2d}$, $i=1,2,\dots,N$,
where $x_i(t)$ denotes the position and $v_i(t)$ the velocity of the $i$-th agent.
The agents are subject to the following collective dynamics,
\(
\tot{x_i(t)}{t} &=& v_i(t), \label{eq:CS001} \\
\tot{v_i(t)}{t} &=& \frac{1}{N-1} \sum_{j=1}^N \psi(|x_j(t)-x_i(t)|) (v_j(t)-v_i(t)), \label{eq:CS002}
\)
for $t>0$ and $i\in[N]$, where here and in the sequel we denote $[N] := \{1, \ldots, N\}$.
The nonnegative real function $\psi:[0,\infty)\to [0,\infty)$,
called \emph{influence function}, measures the intensity of the influence between
agents depending on their distance.
Typically, it is assumed to be globally bounded, so that, by an eventual rescaling of time,
one has $0\leq\psi\leq 1$. We shall adapt this assumption in our paper.
The terms $x_j(t)$ and $v_j(t)$ appearing in the right-hand side of equation \eqref{eq:CS002}
reflect the implicit modelling assumption that each agent receives information
about the current phase-space co-ordinates of all other agents immediately, without any time lags
(or, that the eventual time lags are negligibly small with respect to the typical time scales
relevant for the system).
However, for certain applications in biology \cite{Camazine, Smith, Vicsek}, socio-economics \cite{Krugman, Naldi}
or engineering (e.g., swarm robotics, \cite{Hamman, Jadbabaie, E3B, Valentini}), delays in communication between agents
caused by finite speed of information propagation may be relevant.
For instance, in radio communication between satellites on the orbit or in outer space,
where the distances are not negligible with respect to the speed of light,
or in swarm robotics with acoustic communication between agents (i.e., underwater robots).
This motivates us to introduce a modification of the Cucker-Smale model \eqref{eq:CS001}--\eqref{eq:CS002}
where information propagates with a constant finite speed ${\mathfrak{c}}>0$, called \emph{propagation speed} in the sequel.
Then, agent $i$ located at $x_i=x_i(t)$ at time $t>0$ observes the
phase-space co-ordinates of agent $j$ at time $t-\ta{i}{j}(t)$, where $\ta{i}{j}=\ta{i}{j}(t)$ solves
\( \label{eq:tau}
{\mathfrak{c}} \ta{i}{j}(t) = |x_i(t) - x_j(t-\ta{i}{j}(t))|.
\)
In other words, $\ta{i}{j}(t)$ is the time that information (light, sound) needs to travel from location $x_j(t-\ta{i}{j}(t))$
to location $x_i(t)$.
In general it is neither guaranteed that a solution $\ta{i}{j}(t)$ of \eqref{eq:tau}
exists nor that it is unique. This issue is of course related
to the possibility of agents traveling faster than the propagation speed ${\mathfrak{c}}$.
We shall formulate sufficient conditions for the well-posedness
of the model in the course of our analysis.
For the time being, let us assume that \eqref{eq:tau} is uniquely solvable with solution $\ta{i}{j}(t)\geq 0$
for all $t\geq 0$ and $i,j\in[N]$, and introduce the following notation
\[
\wx{j}{i} := x_j(t - \ta{i}{j}(t)), \qquad \wv{j}{i} := v_j(t - \ta{i}{j}(t)).
\]
We also introduce the formal notation $\ta{i}{i}:=0$ and $\wx{i}{i}:=x_i(t)$, $\wv{i}{i}:=v_i(t)$,
and, if no danger of confusion, we shall usually drop the explicit time dependence,
writing just $x_i$ for $x_i(t)$ etc.
With this notation, the system that we study in this paper is written as
\(
\dot x_i &=& v_i, \label{eq:CS1} \\
\dot v_i &=& \frac{1}{N-1} \sum_{j=1}^N \psi(|\wx{j}{i} - x_i|) \left( \wv{j}{i} - v_i \right), \label{eq:CS2}
\)
for $t>0$ and $i\in[N]$.
We shall also frequently use the shorthand notation for the \emph{communication rates}
\[
\widetilde\psi_{ij} := \psi\left( \left| \wx{j}{i} - x_i \right| \right).
\]
The system \eqref{eq:CS1}--\eqref{eq:CS2} is equipped with the initial datum
\( \label{IC:0}
x_i(t) = x_i^0(t), \quad v_i(t) = v_i^0(t) \qquad\mbox{for } i\in[N],\quad t\leq 0,
\)
where $x_i^0=x_i^0(t)$, $v_i^0=v_i^0(t)$ are continuous paths on $(-\infty,0]$.
We shall impose the physically relevant assumption that
\( \label{IC:comp}
x^0_i(t) = x^0_i(0) + \int_0^t v^0_i(s) \mathrm{d} s \qquad\mbox{for } i\in[N],\quad t\leq 0,
\)
so that we in fact only prescribe $v_i^0=v_i^0(t)$ for $t\leq 0$ and $x_i^0(0)$.
We note that the results provided in this paper would not lose their validity
even if we dropped the "compatibility" assumption \eqref{IC:comp},
however, it would lead to unnecessary technicalities in the proofs.
Moreover, in order to construct the solution of \eqref{eq:CS1}--\eqref{eq:CS2}
on a bounded time interval $[0,T]$, the values of the initial datum are only relevant
on a bounded time interval $[-S(T),0]$, with some $S(T)>0$ depending on ${\mathfrak{c}}$, ${\mathfrak{s}}$
and the configuration of the system at time $t=0$.
We shall make this dependence explicit later.
From the mathematical point of view, the system \eqref{eq:tau}--\eqref{eq:CS2}
is a system of functional differential equations with state-dependent delay,
which stems from the fact that the delay $\tau_{ij}(t)$ in \eqref{eq:CS2}
depends on the configuration of the system
in a nontrivial (even implicit) way through \eqref{eq:tau}.
This poses new analytical challenges: in particular,
the standard well-posedness theory for ODE systems
(the classical theorems of Peano and Picard-Lindel\"of)
does not apply to \eqref{eq:tau}--\eqref{eq:CS2}.
Therefore, the first goal of this paper, addressed in Section \ref{sec:ex},
is to establish global existence and uniqueness of solutions for \eqref{eq:tau}--\eqref{eq:CS2}.
The second goal of this paper is to study the asymptotic
behavior of solutions of the system \eqref{eq:tau}--\eqref{eq:CS2}.
In particular, the usual question asked in the context
of the Cucker-Smale model is under which conditions
its solutions exhibit the \emph{(asymptotic) flocking behavior},
defined as follows.
\begin{definition}[Asymptotic flocking]\label{def:flocking}
We say that the system with particle positions $x_i(t)$ and velocities $v_i(t)$,
$i\in[N]$ and $t\geq 0$, exhibits \emph{(asymptotic) flocking} if
\( \label{flocking}
\sup_{t\geq 0} {d_{\solx}}(t) < \infty,\qquad \lim_{t\to\infty} {d_{\solv}}(t) = 0,
\)
where we denoted the spatial and, resp., velocity diameters
\( \label{dXdV}
{d_{\solx}}(t) := \max_{1 \leq i,j \leq N}|x_i(t) - x_j(t)| \quad \mbox{and} \quad {d_{\solv}}(t) := \max_{1 \leq i,j \leq N}|v_i(t) - v_j(t)|.
\)
\end{definition}
We shall derive sufficient conditions for flocking in the system \eqref{eq:tau}--\eqref{eq:CS2}
in terms of the propagation speed ${\mathfrak{c}}$, the decay properties of the influence function $\psi$
and certain properties of the initial datum \eqref{IC:0}. The presence of the state-dependent delay
poses severe challenges for the analysis. In particular, the Lyapunov functional-type approaches,
that were developed for Cucker-Smale-type systems with a-priori given (i.e., state-independent)
delay, fail. Consequently, new methods need to be developed to study the asymptotic flocking
for the system \eqref{eq:tau}--\eqref{eq:CS2}. This goal will be addressed in Section \ref{sec:flocking}.
Finally, the third goal of this paper, addressed in Section \ref{sec:MF},
is to study the mean-field limit of \eqref{eq:tau}--\eqref{eq:CS2}
as $N\to\infty$. As is well known, mean-field limits of systems of interacting particles
are typically described in terms of a Fokker-Planck equation that governs the evolution
of a time-dependent particle density in the phase space.
However, for certain types of systems involving delay, such a description may not be available.
For a Cucker-Smale system with constant delay, this issue was discussed in \cite[Section 4]{HasMar}.
It is not a-priori clear whether the system \eqref{eq:tau}--\eqref{eq:CS2} with state-dependent delay
admits a Fokker-Planck-type description in the mean-field limit.
We approach the problem by first deriving a mean-field limit formulated
in terms of probability measures on the space of Lipschitz continuous curves.
In Section \ref{sec:MF} we prove the well-posedness of such a description
and, using a stability result, show that it is indeed obtained from the discrete system
\eqref{eq:tau}--\eqref{eq:CS2} in the limit as $N\to\infty$.
Many of the analytical techniques applied in Section \ref{sec:MF} are adaptations
of the methods developed in \cite{CCR} to the setting
of probability measures on the space of Lipschitz continuous curves.
Finally, we argue that the mean-field limit in fact \emph{cannot} be formulated
as a (standard) Fokker-Planck equation.
Cucker-Smale-type systems with delay and their flocking behavior have been studied in a series of recent papers.
However, to our best knowledge, all the previous works
\cite{Cartabia, ChoiH1, ChoiH2, Choi-Pignotti, EHS, H:SIADS, HasMar, HasMar2, Liu-Wu, Pignotti-Reche1, Pignotti-Reche2, Pignotti-Trelat, E6}
assume the delay to be state-independent, i.e., given a-priori either as a constant,
time-dependent function or probability distribution.
State-dependent delay induced by finite speed of information propagation
was considered in \cite{Has:sdHK} for a Hegselmann-Krause-type model
of consensus formation \cite{HK}. This model can be seen as a first-order version
of \eqref{eq:tau}--\eqref{eq:CS2}. However, the mathematical challenges
stemming from the second-order model are significantly different.
Consequently, analysis of the system \eqref{eq:tau}--\eqref{eq:CS2}
requires development of new techniques, which is the main focus of this paper.
The paper is organized as follows: In Section \ref{sec:overview}
we present an overview of our main results on well-posedness,
flocking behavior and mean-field limit of the system \eqref{eq:tau}--\eqref{eq:CS2}.
In Section \ref{sec:ex} we provide the proof of well-posedness
(global existence and uniqueness) of solutions of the discrete system.
In Section \ref{sec:flocking} we provide the proof of our result
on asymptotic flocking behavior.
Finally, in Section \ref{sec:MF} we study the
mean-field limit $N\to\infty$.
\section{Overview of main results}\label{sec:overview}
\textbf{Notation.}
In the sequel we shall denote $\mathbf{x} = (x_1, \ldots, x_N)\in \mathbb{R}^{Nd}$ the vector
of position trajectories and $\mathbf{v} = (v_1, \ldots, v_N)\in \mathbb{R}^{Nd}$ the vector of velocity trajectories.
By $C(\mathcal{I}; \mathbb{R}^d)$ we denote the space of continuous functions on the interval $\mathcal{I}$ with values in $\mathbb{R}^d$,
and by $C^1_b(\mathcal{I}; \mathbb{R}^d)$ the space of continuous uniformly bounded functions on $\mathcal{I}$ with
continuous first-order derivative.
By $C_{\mathfrak{s}}(\mathcal{I}; \mathbb{R}^d)$ we denote the set of Lipschitz continuous functions on $\mathcal{I}$
with Lipschitz constant ${\mathfrak{s}}\geq 0$; we shall also often use the abbreviated expression
``${\mathfrak{s}}$-Lipschitz continuous functions on $\mathcal{I}$''.
The notation $\mathbf{v}\in C_{\mathfrak{s}}(\mathcal{I}; \mathbb{R}^{Nd})$ is understood
as the space of $N$ trajectories in the $d$-dimensional space, such that $v_i\in C_{\mathfrak{s}}(\mathcal{I}; \mathbb{R}^d)$
for all $i\in[N]$, where $[N]:= \{1,\ldots,N\}$.
\subsection{Existence and uniqueness of solutions}
We start with the observation that, in general, \eqref{eq:tau} can only be uniquely solvable if the agents
move with speeds $|v_i|$ strictly less than the propagation speed ${\mathfrak{c}}$.
This motivates us, for $0 < {\mathfrak{s}} < {\mathfrak{c}}$ and $T\geq 0$, to introduce the space
\( \label{def:Vs}
\mathbb{V}_{\mathfrak{s}}^T := \left\{ \mathbf{v}\in C((-\infty,T];\mathbb{R}^{Nd});\, |v_i(t)| \leq {\mathfrak{s}} \mbox{ for all } t\leq T,\, i\in [N] \right\},
\)
where we use the notation $\mathbf{v} = (v_1, \ldots, v_N)\in \mathbb{R}^{Nd}$.
Then, it is natural to require that the initial datum $\mathbf{v}^0 = (v^0_1, \ldots, v^0_N)$ in \eqref{IC:0} is an element of $\mathbb{V}_{\mathfrak{s}}^0$
for some ${\mathfrak{s}}<{\mathfrak{c}}$.
Our first result provides existence and uniqueness of global solutions of the system \eqref{eq:tau}--\eqref{IC:comp}.
\begin{theorem}[Existence and uniqueness of global solutions] \label{thm:ex}
Let the influence function $0 \leq \psi \leq 1$ be uniformly Lipschitz continuous on $[0,\infty)$.
Let $0<{\mathfrak{s}}<{\mathfrak{c}}$ be fixed and let the initial datum $\mathbf{v}^0 = (v^0_1, \ldots, v^0_N) \in \mathbb{V}_{\mathfrak{s}}^0$.
Then, for any $T>0$ the system \eqref{eq:tau}--\eqref{IC:comp} admits a unique global solution in $\mathbb{V}_{\mathfrak{s}}^T$.
Moreover, the initial datum $\mathbf{v}^0$ only needs to be prescribed on the compact interval $\left[-S(T), 0\right]$, with
\( \label{ST}
S(T) := \frac{{d_{\solx}}(0) + [{\mathfrak{c}}-3{\mathfrak{s}}]^- T}{{\mathfrak{c}}-{\mathfrak{s}}},
\)
with $[a]^- := \max\{-a,0\}$ denotes the negative part of $a$.
\end{theorem}
\subsection{Flocking behavior}
Our second result provides sufficient conditions for asymptotic flocking behavior of the system \eqref{eq:tau}--\eqref{IC:comp} in sense of Definition \ref{def:flocking}.
For this, we adopt the additional assumption that the initial datum $\mathbf{v}^0$ is constant,
while $\mathbf{x}^0$ verifies \eqref{IC:comp}. This assumption may be seen restrictive, although we do not consider
it unrealistic. In fact, a significant part of the proof of the flocking result in Section \ref{sec:flocking}
applies to the general setting $\mathbf{v}^0 \in \mathbb{V}_{\mathfrak{s}}^0$ and does not require $\mathbf{v}^0$ to be constant.
Only in the last step (Lemma \ref{lem:notsocrazy}), a solution of a nonlinear system of algebraic equations
has to be found, which only seems to be achievable analytically
for constant $\mathbf{v}^0$.
For the general case, one would need to resort to numerical approaches to confirm solvability of the algebraic system
and so derive a sufficient condition for flocking in terms of the parameter values; see Remark \ref{rem:v0} for details.
\begin{theorem}[Critical propagation speed for flocking] \label{thm:flocking}
Let the influence function $0 \leq \psi \leq 1$ be uniformly Lipschitz continuous on $[0,\infty)$.
Let ${\mathfrak{s}}>0$ be fixed and let the initial datum
$\mathbf{v}^0 = (v^0_1, \ldots, v^0_N)$ be constant on $(-\infty,0]$ with $\max_{i\in[N]} |v_i^0| \leq {\mathfrak{s}}$.
Assume that there exists $\eta>0$ such that
\( \label{ass:bb}
\psi\left({d_{\solx}}(0) + \frac{{d_{\solv}}(0)}{\eta}\right) > \eta.
\)
Then there exists ${\c^\ast}>{\mathfrak{s}}$, calculable from the values of ${d_{\solx}}(0)$, ${d_{\solv}}(0)$, ${\mathfrak{s}}$ and $\eta$,
such that if ${\mathfrak{c}}\geq {\c^\ast}$, the system \eqref{eq:tau}--\eqref{IC:comp}
exhibits flocking in the sense of Definition \ref{def:flocking}.
\end{theorem}
Let us note that the assumption \eqref{ass:bb} is very natural
in the context of the Cucker-Smale model,
where slow enough decay of the influence function
is necessary (and sufficient) condition for (unconditional) flocking.
In particular, we have the following result.
\begin{proposition}\label{corr:CS}
Let the influence function $\psi=\psi(s)$ be such that
\( \label{corr:CS:cond}
\liminf_{s\to\infty} \frac{\psi(s)}{s^\alpha} > 0
\)
for some $\alpha>-1$.
Then for any ${d_{\solx}}(0)$, ${d_{\solv}}(0)\geq 0$ there exists $\eta>0$ such that \eqref{ass:bb} holds.
\end{proposition}
Condition \eqref{corr:CS:cond} is almost equivalent to the unconditional flocking
condition $\int^{\infty} \psi(s) \mathrm{d} s = \infty$ for the classical Cucker-Smale model,
which is known to be sharp \cite{CS1, CS2, Tadmor-Ha}.
Indeed, the nonintegrability of $\psi$ at infinity is implied by \eqref{corr:CS:cond}
with any $\alpha \geq -1$, so that Proposition \ref{corr:CS} only excludes
influence functions that decay like $s^{-1}$.
In particular, for the generic choice of $\psi$ introduced in~\cite{CS1, CS2}
and considered in most of the subsequent papers,
\( \label{CS:psi}
\psi(s) = \frac{1}{(1+s^2)^\beta},
\)
assumption \eqref{corr:CS:cond} of Proposition \ref{corr:CS}, and thus assumption \eqref{ass:bb}
of Theorem \ref{thm:flocking}, is equivalent to $\beta < 1/2$.
This is precisely the condition for {unconditional flocking} formulated in \cite{CS1, CS2}
for the classical Cucker-Smale model.
\subsection{Mean-field limit}
For fixed ${\mathfrak{s}}>0$ and $T\geq 0$ we introduce the set
\( \label{def:OmegasT}
\Omega_{\mathfrak{s}}^T :=
\Bigl\{ \gamma\in C_b^1((-\infty, T];\mathbb{R}^d) \cap C_{\mathfrak{s}}((-\infty,T]; \mathbb{R}^d); \; \dot\gamma|_{[0,T]} \in C_{2{\mathfrak{s}}}([0,T]; \mathbb{R}^d) \Bigr\},
\)
where we recall that $C_b^1((-\infty, T];\mathbb{R}^d)$ denotes the space of continous bounded functions on $(-\infty,T]$
with continuous first-order derivative, and $C_{\mathfrak{s}}((-\infty,T]; \mathbb{R}^d)$ denotes the space of (globally) Lipschitz continuous functions on $(-\infty,T]$
with Lipschitz constant ${\mathfrak{s}}$.
We equip the set $\Omega_{\mathfrak{s}}^T$ with the topology induced by the norm $\Norm{\cdot}_{\Omega_{\mathfrak{s}}^T}$,
\( \label{def:norm}
\Norm{\gamma}_{\Omega_{\mathfrak{s}}^T}:= \Norm{\gamma}_{L^\infty(-\infty,T)} + \Norm{\dot \gamma}_{L^\infty(0,T)} \qquad
\mbox{for } \gamma\in \Omega_{\mathfrak{s}}^T.
\)
We note that with the particular choice $T:=0$ in \eqref{def:OmegasT} we obtain the space $\Omega_{\mathfrak{s}}^0 = C_b^1((-\infty, 0];\mathbb{R}^d) \cap C_{\mathfrak{s}}((-\infty,0]; \mathbb{R}^d)$,
and \eqref{def:norm} reduces to $\Norm{\gamma}_{\Omega_{\mathfrak{s}}^0}= \Norm{\gamma}_{L^\infty(-\infty,0)}$.
We denote ${\mathcal{P}(\Omega_\s^T})$ the space of probability measures on $\Omega_{\mathfrak{s}}^T$ with finite first-order moment, i.e.,
\[
\int_{\Omega_{\mathfrak{s}}^T} \Norm{\gamma}_{\Omega_{\mathfrak{s}}^T} \mathrm{d}\rho(\gamma) < +\infty \qquad\mbox{for any } \rho\in{\mathcal{P}(\Omega_\s^T}).
\]
In Section \ref{sec:MF} we will show that the mean-field limit as $N\to\infty$ of the discrete system \eqref{eq:tau}--\eqref{IC:comp} is represented
by the measure $\rho\in{\mathcal{P}(\Omega_\s^T})$,
\( \label{eq:law}
\rho=\mbox{law}(x),
\)
with
\( \label{eq:MF}
\dot x(t) = v(t), \qquad \dot v(t) = F_t[\rho](x(t),v(t))
\)
for $t\in (0,T)$,
with the operator $F_t[\rho]: \mathbb{R}^d\times\mathbb{R}^d \to \mathbb{R}^d$,
\( \label{def:F}
F_t[\rho](x,v) := \int_{\Omega^T_{\mathfrak{s}}} \psi\left( \left|\Gamma_{t,x}[\gamma] - x \right|\right) \left(\Pi_{t,x}[\gamma] - v \right) \mathrm{d}\rho(\gamma).
\)
The mapping $\Gamma_{t,x}: \Omega^T_{\mathfrak{s}} \mapsto \mathbb{R}^d$ is defined, for $t\in\mathbb{R}$ and $x\in\mathbb{R}^d$, as
\( \label{def:Gamma}
\Gamma_{t,x}[\gamma] := \gamma(t-\tau_{t,x}[\gamma])
\)
with $\tau_{t,x}[\gamma]:=\tau$ the unique solution of
\( \label{eq:tau_gamma}
{\mathfrak{c}}\tau = |x - \gamma(t-\tau)|.
\)
As we shall prove below, the existence and uniqueness of $\tau$ is guaranteed by the ${\mathfrak{s}}$-Lipschitz continuity of $\gamma\in\Omega^T_{\mathfrak{s}}$.
The mapping $\Pi_{t,x}: \Omega^T_{\mathfrak{s}} \mapsto \mathbb{R}^d$ is defined as
\( \label{def:Pi}
\Pi_{t,x}[\gamma] := \dot\gamma(t-\tau_{t,x}[\gamma]).
\)
The system \eqref{eq:law}--\eqref{def:F} is equipped with the initial datum $\rho^0\in \mathcal{P}(\Omega_{\mathfrak{s}}^0)$,
which is imposed in terms of the push-forward identity
\( \label{MF:IC}
\mathbb{I}\#\rho = \rho^0,
\)
with the mapping $\mathbb{I}: \Omega_{\mathfrak{s}}^T \to \Omega_{\mathfrak{s}}^0$ given by
\( \label{def:I}
\mathbb{I} : \gamma \mapsto \gamma|_{(-\infty,0]},
\)
and we recall that the push-forward measure $\mathbb{I}\#\rho\in \mathcal{P}(\Omega_{\mathfrak{s}}^0)$ is defined by $\mathbb{I}\#\rho(B):= \rho(\mathbb{I}^{-1}(B))$,
where $B$ is any measurable subset of $\Omega_{\mathfrak{s}}^0$.
Then, the initial conditions $x(0)$ and $v(0)$ for \eqref{eq:MF} are distributed according to $X_0\#\rho^0$
and, resp., $V_0\#\rho^0$, with the mappings
\( \label{XtVt}
X_t[\gamma] := \gamma(t), \qquad V_t[\gamma] := \dot\gamma(t).
\)
In Section \ref{sec:MF} we shall provide a proof of the following result
on existence and uniqueness of solutions of the mean-field system
\eqref{eq:law}--\eqref{MF:IC}.
\begin{theorem}[Existence and uniqueness of measure solutions] \label{thm:MF}
Fix $T>0$ and ${\mathfrak{s}}<{\mathfrak{c}}$.
Then, for any $\rho^0\in \mathcal{P}(\Omega_{\mathfrak{s}}^0)$ there exists a unique probability measure
$\rho\in{\mathcal{P}(\Omega_\s^T})$ that verifies \eqref{eq:law}--\eqref{MF:IC}.
\end{theorem}
Moreover, we shall provide a stability result, controlling a suitable notion of distance of solutions
by the distance of the initial data. For this purpose, we equip the space ${\mathcal{P}(\Omega_\s^T})$ with the Monge--Kantorovich--Rubinstein distance
\( \label{def:MKR}
\mathcal{W}_T(\rho,\nu) := \inf_{\pi\in\Lambda(\rho,\nu)} \iint_{\Omega_{\mathfrak{s}}^T \times \Omega_{\mathfrak{s}}^T} \Norm{\gamma-\xi}_{\Omega_{\mathfrak{s}}^T} \mathrm{d}\pi(\gamma,\xi),
\)
with $\Norm{\cdot}_{\Omega_{\mathfrak{s}}^T}$ given by \eqref{def:norm} and $\Lambda(\rho,\nu)$ denoting the set of transference plans between the measures $\rho$ and $\nu$,
i.e., probability measures on the product space $\Omega_{\mathfrak{s}}^T \times \Omega_{\mathfrak{s}}^T$ with first and second marginals
$\rho$ and $\nu$, respectively.
We then have the following stability result.
\begin{theorem}[Stability] \label{thm:stability}
For each $T>0$ there exists a constant $M_T>0$ such that
for any $\rho^0$, $\nu^0 \in {\mathcal{P}(\Omega_\s^0})$,
\( \label{stability}
\mathcal{W}_T(\rho,\nu) \leq M_T \mathcal{W}_0(\rho^0, \nu^0),
\)
where $\rho\in{\mathcal{P}(\Omega_\s^T})$, and, resp., $\nu\in{\mathcal{P}(\Omega_\s^T})$ are the unique solutions of \eqref{eq:law}--\eqref{MF:IC}
constructed in Theorem \ref{thm:MF} subject to the initial data $\rho^0$ and, resp. $\nu^0$.
\end{theorem}
The above stability theorem justifies the approximation of the system \eqref{eq:law}--\eqref{MF:IC}
by finite sets of discrete particles satisfying \eqref{eq:tau}--\eqref{IC:comp}.
Indeed, let us consider $(\mathbf{x}, \mathbf{v})$ a solution of \eqref{eq:tau}--\eqref{IC:comp}
with a given $N\in\mathbb{N}$, and define $\rho_N\in{\mathcal{P}(\Omega_\s^T})$ the atomic measure being concentrated on the set of trajectories
$\mathbf{x} = (x_1, \ldots, x_N)$, i.e.,
\( \label{atomic}
\rho_N(\gamma) := \frac{1}{N} \sum_{j=1}^N \delta(\gamma - x_j),
\)
with $\delta$ denoting the Dirac measure on $\Omega_{\mathfrak{s}}^T$, concentrated on the constant zero function $\gamma\equiv 0$.
Then, from \eqref{def:F} we have
\[
F_t[\rho_N](x,v) = \frac{1}{N} \sum_{j=1}^N \psi\left( \left|x_j(t-\tau_{t,x}[x_j]) - x \right|\right) \left(\dot x_j(t-\tau_{t,x}[x_j]) - v \right),
\]
where $\tau_{t,x}[x_j]$ is the unique solution of \eqref{eq:tau_gamma} with $\gamma:=x_j$.
Consequently, \eqref{eq:tau}--\eqref{eq:CS2} can be rewritten as
\[
\dot x_i(t) = v_i(t), \qquad \dot v_i(t) = \frac{N}{N-1} F_t[\rho_N](x_i(t),v_i(t)).
\]
Note that $\rho_N = \mbox{law}(\{x_i\}_{i\in[N]})$. Consequently,
up to the factor $\frac{N}{N-1}$ that can be removed by rescaling of time,
$\rho_N\in {\mathcal{P}(\Omega_\s^T})$ given by \eqref{atomic}
is a solution of \eqref{eq:law}--\eqref{MF:IC} subject to the initial datum
\( \label{IC:atomic}
\rho^0_N(\gamma) := \frac{1}{N} \sum_{j=1}^N \delta(\gamma - x^0_j).
\)
Therefore, the important consequence of Theorem \ref{thm:stability} is that it provides
a method to derive the measure-valued mean-field description \eqref{eq:law}--\eqref{MF:IC}
as the limit $N\to\infty$ of the particle approximations \eqref{eq:tau}--\eqref{IC:comp}.
\begin{corollary}[Convergence of the particle method] \label{cor:convergence}
Given $\rho^0\in \mathcal{P}(\Omega_{\mathfrak{s}}^0)$, take a sequence $\rho^0_N\in \mathcal{P}(\Omega_{\mathfrak{s}}^0)$
of atomic measures of the form \eqref{IC:atomic} such that
\[
\lim_{N\to\infty} \mathcal{W}_0(\rho^0_N, \rho^0) = 0.
\]
Let $\rho^N\in{\mathcal{P}(\Omega_\s^T})$ be given by \eqref{atomic} with $(\mathbf{x}, \mathbf{v})$
a solution of \eqref{eq:tau}--\eqref{IC:comp}.
Then, for any $T>0$,
\[
\lim_{N\to\infty} \mathcal{W}_T(\rho_N, \rho) = 0,
\]
where $\rho^N\in{\mathcal{P}(\Omega_\s^T})$ is the unique solution of \eqref{eq:law}--\eqref{MF:IC}
subject to the initial datum $\rho^0$.
\end{corollary}
Finally, in Section \ref{subsec:FP} we discuss the question
whether one can express the mean-field limit problem \eqref{eq:law}--\eqref{MF:IC}
in terms of a Fokker-Planck equation for some time dependent phase-space particle density $g_t \in \mathcal{P}(\mathbb{R}^d\times\mathbb{R}^d)$, $t\geq 0$.
A short consideration leads to the conclusion that no such description is possible,
i.e., one indeed has to resort to the formulation in terms of probability measures
on the space of time-dependent trajectories.
\section{Global existence and uniqueness of solutions - proof of Theorem \ref{thm:ex}} \label{sec:ex}
We start by proving two auxiliary results establishing unique solvability of the equation \eqref{eq:tau}
and a bound on the delay $\tau_{ij}$.
\begin{lemma} \label{lem:tau}
For some $t\in\mathbb{R}$, let $x\in C_{\mathfrak{s}}((-\infty,t]; \mathbb{R}^d)$ with the Lipschitz constant ${\mathfrak{s}}<{\mathfrak{c}}$.
Then, for each $z\in\mathbb{R}^d$, the equation
\( \label{tauxz}
{\mathfrak{c}} \tau = |z-x(t-\tau)|
\)
is uniquely solvable in $\tau\geq 0$.
\end{lemma}
\begin{proof}
See the proof of \cite[Lemma 2.2]{Has:sdHK}.
\end{proof}
\begin{lemma}\label{lem:tauij}
For some $t\in\mathbb{R}$, let $\mathbf{x} = (x_1,\ldots, x_N) \in C_{\mathfrak{s}}((-\infty,t]; \mathbb{R}^{Nd})$ with all trajectories $x_i$ uniformly Lipschitz continuous on $(-\infty,t]$
with Lipschitz constant ${\mathfrak{s}}<{\mathfrak{c}}$.
Then for all $i, j \in [N]$ we have
\( \label{est:tauij}
\tau_{ij}(t) \leq \frac{{d_{\solx}}(t)}{{\mathfrak{c}}-{\mathfrak{s}}},
\)
where $\tau_{ij}=\tau_{ij}(t)$ is the unique solution of \eqref{eq:tau} and ${d_{\solx}}={d_{\solx}}(t)$ defined in \eqref{dXdV}.
\end{lemma}
\begin{proof}
By \eqref{eq:tau} we have
\[
{\mathfrak{c}} \tau_{ij} = |\wx{j}{i} - x_i|.
\]
On the other hand, due to the ${\mathfrak{s}}$-Lipschitz continuity of $x_j$,
\( \label{est:wxjixj}
|\wx{j}{i} - x_j| = |x_j(t-\tau_{ij}) - x_j(t)| \leq {\mathfrak{s}} \tau_{ij}.
\)
Therefore, by the triangle inequality,
\[
{\mathfrak{c}} \tau_{ij} = |\wx{j}{i} - x_i| \leq |x_i-x_j| + |\wx{j}{i}-x_j| \leq {d_{\solx}}(t) + {\mathfrak{s}}\tau_{ij},
\]
and \eqref{est:tauij} follows.
\end{proof}
Next, we prove local in time existence and uniqueness of solutions of the system \eqref{eq:tau}--\eqref{IC:comp}.
The proof is an adaptation of the Picard-Lindel\"of theorem, based on construction of a contraction mapping.
\begin{lemma}\label{lem:local}
Let the influence function $0 \leq \psi \leq 1$ be uniformly Lipschitz continuous on $[0,\infty)$.
Let $0<{\mathfrak{s}}<{\mathfrak{c}}$ be fixed and let the initial datum $\mathbf{v}^0 = (v^0_1, \ldots, v^0_N) \in \mathbb{V}_{\mathfrak{s}}^0$,
Then the system \eqref{eq:tau}--\eqref{IC:comp} admits unique local solutions
in the class of Lipschitz continuous velocity trajectories.
\end{lemma}
\def\Upsilon{\Upsilon}
\begin{proof}
Let us fix any ${\mathfrak{m}}$ such that
\[
{\mathfrak{s}} < {\mathfrak{m}} < {\mathfrak{c}}
\]
and for some $T>0$ to be specified later, define the set $\mathbb{W}_{\mathfrak{m}}^T$
of continuous velocity trajectories on $(-\infty,T]$,
which coincide with the initial datum $\mathbf{v}^0$ on the interval $(-\infty, 0]$,
and are uniformly bounded by ${\mathfrak{m}}$
and $2{\mathfrak{m}}$-Lipschitz continuous on the interval $[0, T]$, i.e.,
\[
\mathbb{W}_{\mathfrak{m}}^T &:=& \Bigl\{ \mathbf{v}\in C((-\infty, T];\mathbb{R}^{Nd});\, \mathbf{v}|_{(-\infty, 0]} \equiv \mathbf{v}^0, \; \mathbf{v}|_{[0,T]}\in C_{2{\mathfrak{m}}}([0,T]; R^{Nd}), \Bigr. \\
&& \qquad\qquad\qquad\qquad\qquad\qquad\qquad \Bigl. |v_i(t)| \leq {\mathfrak{m}} \mbox{ for all } t\in [0,T],\, i\in [N]\Bigr\},
\]
where we again use the notation $\mathbf{v} = (v_1, \ldots, v_N)\in \mathbb{R}^{Nd}$
and $C_{2{\mathfrak{m}}}([0,T]; R^{Nd})$ denotes the set of $2{\mathfrak{m}}$-Lipschitz continuous
trajectories on $[0,T]$. Let us note that since the initial datum $\mathbf{v}^0$ verifies
$|v^0_i(t)|\leq {\mathfrak{s}} < {\mathfrak{m}}$ for all $t\leq 0$ and $i\in [N]$, the set $\mathbb{W}_{\mathfrak{m}}^T$ is nonempty.
We equip the set $\mathbb{W}_{\mathfrak{m}}^T$ with the topology of uniform convergence,
i.e., with the $L^\infty$-norm on $(0,T)$.
Then $\mathbb{W}_{\mathfrak{m}}^T$ is a complete metric space.
We construct a unique local solution of \eqref{eq:tau}--\eqref{IC:comp} by finding a unique fixed point
of the Picard operator $\Upsilon: \mathbb{W}_{\mathfrak{m}}^T \to \mathbb{W}_{\mathfrak{m}}^T$,
where the $i$-th component is the $d$-dimensional vector-valued function $\Upsilon_i[\omega](t)$,
given for $t\in [0,T]$ by
\( \label{Picard}
\Upsilon_i[\omega](t) := v_i^0(0) + \frac{1}{N-1} \int_0^t \sum_{j\neq i} \psi(|\xi_j(s-\tau_{ij}(s))-\xi_i(s)|) (\omega_j(s-\tau_{ij}(s))-\omega_i(s)) \mathrm{d} s
\)
with
\( \label{def:xi}
\xi_i(t) = x_i^0(0) + \int_0^t \omega_i(s) \mathrm{d} s \qquad\mbox{for }i\in[N], \quad t\leq T,
\)
and we set $\Upsilon[\omega]$ equal to $\mathbf{v}^0$ on $(-\infty,0]$.
Clearly, with $\omega\in \mathbb{W}_{\mathfrak{m}}^T$, the trajectories $\xi_i$ defined by \eqref{def:xi} are Lipschitz continuous with Lipschitz constant ${\mathfrak{m}} < {\mathfrak{c}}$.
Then, Lemma \ref{lem:tau} gives unique solutions $\tau_{ij}=\tau_{ij}(t)$ of the equation
\( \label{Picard:tau}
{\mathfrak{c}}\tau_{ij}(t) = \left|\xi_j(t-\tau_{ij}(t)) - \xi_i(t) \right|,
\)
for all $i,j\in [N]$ and $t\leq T$.
We show that for sufficiently small $T>0$ the Picard operator $\Upsilon$ defined by \eqref{Picard}
maps the space $\mathbb{W}_{\mathfrak{m}}^T$ into itself and is a contraction on $\mathbb{W}_{\mathfrak{m}}^T$
with respect to the $L^\infty(0,T)$-norm.
With the global bound $\psi\leq 1$ we have for any $\omega\in \mathbb{W}_{\mathfrak{m}}^T$,
\[
\left| \Upsilon_i[\omega](t) \right| &\leq& \left| v_i^0(0) \right| + \frac{1}{N-1} \int_0^t \sum_{j\neq i} \left| \omega_j(s-\tau_{ij}(s)) \right| + \left| \omega_i(s)) \right| \mathrm{d} s \\
&\leq& {\mathfrak{s}} + 2T{\mathfrak{m}},
\]
so that choosing $T \leq \frac{{\mathfrak{m}} - {\mathfrak{s}}}{2{\mathfrak{m}}}$ gives
\[
\left| \Upsilon_i[\omega](t) \right| \leq {\mathfrak{m}}
\]
for all $i\in [N]$ and $t \in [0,T]$.
Moreover, for any $0\leq t_1 < t_2 \leq T$ we have
\[
\left| \Upsilon_i[\omega](t_1) - \Upsilon_i[\omega](t_2) \right| &\leq&
\frac{1}{N-1} \int_{t_1}^{t_2} \sum_{j\neq i} \psi(|\xi_j(s-\tau_{ij}(s))-\xi_i(s)|) |\omega_j(s-\tau_{ij}(s))-\omega_i(s)| \mathrm{d} s \\
&\leq&
2{\mathfrak{m}} |t_1-t_2|,
\]
so that $\Upsilon[\omega]$ is $2{\mathfrak{m}}$-Lipschitz continuous.
Consequently, for small enough $T>0$, $\Upsilon$ maps the space $\mathbb{W}_{\mathfrak{m}}^T$ into itself.
To prove contractivity of $\Upsilon$, let us pick $\omega, \kappa \in \mathbb{W}_{\mathfrak{m}}^T$
and calculate, for any fixed $i\in [N]$,
\(
\label{integrand}
\left|\Upsilon_i[\omega](t) - \Upsilon_i[\kappa](t)\right| &\leq& \frac{1}{N-1} \sum_{j\neq i}
\int_0^t \bigl| \psi(|\xi_j(s-\tau^\xi_{ij}(s))-\xi_i(s)|)(\omega_j(s-\tau^\xi_{ij}(s))-\omega_i(s)) \bigr. \\
&&\qquad\qquad \bigl. - \psi(|\eta_j(s-\tau^\eta_{ij}(s))-\eta_i(s)|)(\kappa_j(s-\tau^\eta_{ij}(s)) - \kappa_i(s)) \bigr| \mathrm{d} s,
\nonumber
\)
where
\[
\xi_i(t) = x_i^0(0) + \int_0^t \omega_i(s) \mathrm{d} s, \qquad
\eta_i(t) = x_i^0(0) + \int_0^t \kappa_i(s) \mathrm{d} s,
\]
and $\tau_{ij}^\xi$ and, resp., $\tau_{ij}^\eta$ are solutions of \eqref{Picard:tau} with $\xi$, resp., $\eta$.
Note that $\xi_i \equiv \eta_i$ on $(-\infty,0]$ for all $i\in[N]$ and
\( \label{est:xi-eta}
\Norm{\xi_i - \eta_i}_{L^\infty(0,T)} \leq \max_{t\in [0,T]} \left| \int_0^t \omega_i(s) - \kappa_i(s) \mathrm{d} s \right|
\leq T \Norm{\omega_i-\kappa_i}_{L^\infty(0,T)}.
\)
We now estimate, for $s\in [0,T]$, the integrand in \eqref{integrand} by
\( \nonumber
\bigl| \psi(|\xi_j(s-\tau^\xi_{ij}(s))-\xi_i(s)|)(\omega_j(s-\tau^\xi_{ij}(s))-\omega_i(s))
- \psi(|\eta_j(s-\tau^\eta_{ij}(s))-\eta_i(s)|)(\kappa_j(s-\tau^\eta_{ij}(s))) - \kappa_i(s) \bigr| \\
\label{est:integrand}
\leq
\bigl| \psi(|\xi_j(s-\tau^\xi_{ij}(s))-\xi_i(s)|) - \psi(|\eta_j(s-\tau^\eta_{ij}(s))-\eta_i(s)|) \bigr|
\bigl| \omega_j(s-\tau^\xi_{ij}(s))-\omega_i(s) \bigr| \\
\nonumber
+ \psi(|\eta_j(s-\tau^\eta_{ij}(s))-\eta_i(s)|)
\bigl| \omega_j(s-\tau^\xi_{ij}(s)) - \kappa_j(s-\tau^\eta_{ij}(s)) - \omega_i(s) + \kappa_i(s) \bigr|.
\)
To estimate the first term we use the assumed uniform Lipschitz continuity of $\psi$ with constant $L_\psi$
and the uniform boundedness $|\omega_i| \leq {\mathfrak{m}}$,
\[
\bigl| \psi(|\xi_j(s-\tau^\xi_{ij}(s))-\xi_i(s)|) - \psi(|\eta_j(s-\tau^\eta_{ij}(s))-\eta_i(s)|) \bigr|
\bigl| \omega_j(s-\tau^\xi_{ij}(s))-\omega_i(s) \bigr| \\
\leq
2{\mathfrak{m}} L_\psi \bigl| |\xi_j(s-\tau^\xi_{ij}(s))-\xi_i(s)| - |\eta_j(s-\tau^\eta_{ij}(s))-\eta_i(s)| \bigr|.
\]
By the triangle inequality we have
\[
\bigl| |\xi_j(s-\tau_{ij}^\xi(s))-\xi_i(s)| - |\eta_j(s-\tau_{ij}^\eta(s)) - \eta_i(s)| \bigr| &\leq&
\Norm{\xi_i - \eta_i}_{L^\infty(0,T)} + |\xi_j(s-\tau_{ij}^\xi(s)) - \eta_j(s-\tau_{ij}^\eta(s))| \\
&\leq&
2 \Norm{\xi - \eta}_{L^\infty(0,T)} + |\eta_j(s-\tau_{ij}^\xi(s)) - \eta_j(s-\tau_{ij}^\eta(s))| \\
&\leq&
2 \Norm{\xi - \eta}_{L^\infty(0,T)} + {\mathfrak{m}} |\tau_{ij}^\xi(s) - \tau_{ij}^\eta(s)|,
\]
where we used the ${\mathfrak{m}}$-Lipschitz continuity of $\eta_j$ for the last inequality.
Now, with \eqref{Picard:tau} we have
\( \label{est:tautau}
|\tau_{ij}^\xi(s) - \tau_{ij}^\eta(s)| = {\mathfrak{c}}^{-1} \bigl| |\xi_j(s-\tau_{ij}^\xi(s)) - \xi_i(s)|
- |\eta_j(s-\tau_{ij}^\eta(s)) - \eta_i(s)| \bigr|,
\)
so that
\[
\bigl| |\xi_j(s-\tau_{ij}^\xi(s))-\xi_i(s)| - |\eta_j(s-\tau_{ij}^\eta(s)) - \eta_i(s)| \bigr| &\leq&
2 \Norm{\xi - \eta}_{L^\infty(0,T)}
\\ &+& {\mathfrak{m}}{\mathfrak{c}}^{-1} \bigl| |\xi_j(s-\tau_{ij}^\xi(s)) - \xi_i(s)|
- |\eta_j(s-\tau_{ij}^\eta(s)) - \eta_i(s)| \bigr|,
\]
which immediately gives
\( \nonumber
\bigl| |\xi_j(s-\tau_{ij}^\xi(s))-\xi_i(s)| - |\eta_j(s-\tau_{ij}^\eta(s)) - \eta_i(s)| \bigr|
&\leq& 2 \left( 1 - {\mathfrak{m}}{\mathfrak{c}}^{-1} \right)^{-1} \Norm{\xi - \eta}_{L^\infty(0,T)} \\
&\leq& 2 \left( 1 - {\mathfrak{m}}{\mathfrak{c}}^{-1} \right)^{-1} T \Norm{\omega - \kappa}_{L^\infty(0,T)},
\label{est:tautautau}
\)
where we used \eqref{est:xi-eta} for the second inequality.
Consequently, the first term in the right-hand side of \eqref{est:integrand} is bounded by the expression
\[
4{\mathfrak{m}} L_\psi \left( 1 - {\mathfrak{m}}{\mathfrak{c}}^{-1} \right)^{-1} T \Norm{\omega - \kappa}_{L^\infty(0,T)}.
\]
The second term of the right-hand side in \eqref{est:integrand} is estimated as follows, using again $\psi\leq 1$,
\[
\psi(|\eta_j(s-\tau^\eta_{ij}(s))-\eta_i(s)|)
\bigl| \omega_j(s-\tau^\xi_{ij}(s)) - \kappa_j(s-\tau^\eta_{ij}(s)) - \omega_i(s) + \kappa_i(s) \bigr| \\
\leq
\Norm{\omega - \kappa}_{L^\infty(0,T)}
+ \bigl| \omega_j(s-\tau^\eta_{ij}(s)) - \kappa_j(s-\tau^\eta_{ij}(s)) \bigr|
+ \bigl| \omega_j(s-\tau^\xi_{ij}(s)) - \omega_j(s-\tau^\eta_{ij}(s)) \bigr| \\
\leq 2\Norm{\omega - \kappa}_{L^\infty(0,T)}
+ 2{\mathfrak{m}} |\tau_{ij}^\xi(s) - \tau_{ij}^\eta(s)|,
\]
where we used the $2{\mathfrak{m}}$-Lipschitz continuity of $\omega$ in the second inequality.
Then, using \eqref{est:tautau} and \eqref{est:tautautau}, we conclude that
the second term in the right-hand side of \eqref{est:integrand} is bounded by the expression
\[
2 \left( 1 + 2 {\mathfrak{m}}{\mathfrak{c}}^{-1} \left( 1 - {\mathfrak{m}}{\mathfrak{c}}^{-1} \right)^{-1} T \right) \Norm{\omega - \kappa}_{L^\infty(0,T)}.
\]
Therefore, with \eqref{integrand} we finally arrive at
\( \label{contractivity}
\Norm{\Upsilon_i[\omega] - \Upsilon_i[\kappa]}_{L^\infty(0,T)} \leq
2 T \left( 1 + 2{\mathfrak{m}} (L_\psi + {\mathfrak{c}}^{-1}) \left( 1 - {\mathfrak{m}}{\mathfrak{c}}^{-1} \right)^{-1} T \right)
\Norm{\omega - \kappa}_{L^\infty(0,T)}
\)
and choosing $T>0$ sufficiently small, the claim follows.
Finally, let us note that choosing ${\mathfrak{m}}_1\neq {\mathfrak{m}}_2$ may lead to different values
of $T_1$, $T_2$, however, due to the contraction property \eqref{contractivity},
the solution remains unique on $[0, \min\{T_1, T_2\}]$.
\end{proof}
With Lemma \ref{lem:local} we constructed a unique solution of \eqref{eq:tau}--\eqref{IC:comp}
on a sufficiently short time interval $[0,T]$.
In the next step we prove that, assuming $\mathbf{v}^0 \in \mathbb{V}_{\mathfrak{s}}^0$,
the velocity trajectories remain uniformly bounded by ${\mathfrak{s}}$,
which implies that the solution is in fact global in time.
For this purpose, let us define the velocity radius of the agent group,
\( \label{def:Rv}
R_v(t) := \max_{i\in[N]} |v_i(t)|.
\)
\begin{lemma}\label{lem:Rvbound}
Let the initial datum $\mathbf{v}^0 \in \mathbb{V}_{\mathfrak{s}}^0$ with ${\mathfrak{s}}<{\mathfrak{c}}$.
Then, along the solutions of \eqref{eq:tau}--\eqref{IC:comp}, the diameter $R_v$ defined in \eqref{def:Rv} satisfies
\( \label{v-bound}
R_v(t) \leq {\mathfrak{s}} \qquad\mbox{for all } t\geq 0.
\)
\end{lemma}
\begin{proof}
Since ${\mathfrak{s}}<{\mathfrak{c}}$, we may fix $\varepsilon >0$ such that ${\mathfrak{s}} + \varepsilon < {\mathfrak{c}}$.
We shall prove that for all $t\geq 0$
\( \label{R-eps}
R_v(t) < {\mathfrak{s}} + \varepsilon.
\)
By continuity of the velocity trajectories,
\eqref{R-eps} holds on the maximal interval $[0,T)$ for some $T>0$.
For contradiction, let us assume that $T<+\infty$.
Then we have $R_v(T) = {\mathfrak{s}} + \varepsilon$.
Since ${\mathfrak{s}} + \varepsilon < {\mathfrak{c}}$, Lemma \ref{lem:local} allows us to extend the solution
$(\mathbf{x}, \mathbf{v})$ past $T$. I.e., the solution exists on some right neighborhood of $T$
and we have
\( \label{R-cont}
\tot{}{t+} R_v(T)^2 \geq 0,
\)
where $\tot{}{t+} R_v(T)^2$ denotes the right-hand side derivative of $R_v^2$ at $t=T$.
Moreover, by continuity of the velocity trajectories,
there exists an index $i\in[N]$ such that $R_v(t) \equiv |v_i(t)|$
for $t\in [T,T+\delta)$ for some $\delta>0$.
From \eqref{eq:CS2} we then have
\[
\tot{}{t+} R_v(T)^2 = \tot{}{t+} |v_i(T)|^2 &=& \frac{2}{N-1} \sum_{j=1}^N \widetilde\psi_{ij} \left[v_j(T-\tau_{ij}(T))- v_i(T)\right]\cdot v_i(T) \\
&=& \frac{2}{N-1} \sum_{j=1}^N \widetilde\psi_{ij} \left[ v_j(T-\tau_{ij}(T))\cdot v_i(T) - |v_i(T)|^2\right].
\]
We now distinguish two cases:
\begin{itemize}
\item
If $\tau_{ij}(T) = 0$ for all $j\in[N]$, then \eqref{eq:tau} implies $x_j(T) = x_i(T)$ for all $j\in[N]$.
Moreover, since by definition $|v_j(T)| \leq R_v(T) = |v_i(T)|$, we have by the Cauchy-Schwarz inequality
\( \nonumber
v_j(T-\tau_{ij}(T))\cdot v_i(T) - |v_i(T)|^2 &=& v_j(T)\cdot v_i(T) - |v_i(T)|^2 \\
&\leq& \left( |v_j(T)| - |v_i(T)| \right) |v_i(T)| \leq 0,
\label{CauchySchwarz1}
\)
so that
\[
\tot{}{t+} R_v(T)^2 = \tot{}{t+} |v_i(T)|^2
\leq \frac{2}{N-1} \sum_{j=1}^N \widetilde\psi_{ij} \left( |v_j(T)| - |v_i(T)| \right) |v_i(T)| \leq 0.
\]
Then \eqref{R-cont} implies $\tot{}{t+} R_v(T)^2 = 0$, which means that equality takes place
in the Cauchy-Schwarz inequality \eqref{CauchySchwarz1}, i.e.,
\[
v_j(T)\cdot v_i(T) = |v_i(T)|^2 \qquad\mbox{for all } j\in [N].
\]
That means that $v_j(T) = v_i(T)$ for all $j\in[N]$, and since also $x_j(T) = x_i(T)$ for all $j\in[N]$,
the system reached equilibrium at time $T$ and does not evolve further
(note that Lemma \ref{lem:local} provides local uniqueness,
so the constant equilibrium solution is the only possible continuation
past the time $T$).
This trivially implies that \eqref{R-eps} holds for all $t>0$.
\item
If there exists at least one $j\in[N]$ such that $\tau_{ij}(T)>0$, then again by the Cauchy-Schwarz inequality we have
\[
v_j(T-\tau_{ij}(T))\cdot v_i(T) - |v_i(T)|^2 \leq \left( |v_j(T-\tau_{ij}(T))| - |v_i(T)| \right) |v_i(T)| < 0,
\]
since $|v_j(T-\tau_{ij}(T))| \leq R_v(T-\tau_{ij}(T)) < {\mathfrak{s}} + \varepsilon$
and $|v_i(T)| = R_v(T) = {\mathfrak{s}} + \varepsilon > 0$.
Consequently,
\[
\tot{}{t+} R_v(T)^2 < 0,
\]
which is a contradiction to \eqref{R-cont}.
\end{itemize}
The uniform bound \eqref{v-bound} is obtained by taking the limit $\varepsilon\to 0$.
\end{proof}
Combination of the local uniqueness and existence result of Lemma \ref{lem:local}
with the uniform bound on the velocity diameter directly implies
the global existence and uniqueness claim of Theorem \ref{thm:ex}.
It remains to observe that, by Lemma \ref{lem:tauij}, we have
\[
t - \tau_{ij}(t) \geq t - \frac{{d_{\solx}}(t)}{{\mathfrak{c}}-{\mathfrak{s}}},
\]
and from \eqref{eq:CS1},
\[
{d_{\solx}}(t) \leq {d_{\solx}}(0) + \int_0^t {d_{\solv}}(s) \mathrm{d} s \leq {d_{\solx}}(0) + 2{\mathfrak{s}} t,
\]
where we used the bound ${d_{\solv}}(t) \leq 2R_v(t) \leq 2{\mathfrak{s}}$ provided by \eqref{v-bound}.
Consequently,
\[
t - \tau_{ij}(t) \geq \frac{({\mathfrak{c}}-3{\mathfrak{s}})t - {d_{\solx}}(0)}{{\mathfrak{c}}-{\mathfrak{s}}},
\]
and
\[
\min_{t\in [0,T]} \left( t - \tau_{ij}(t) \right) \geq - \frac{{d_{\solx}}(0) + [{\mathfrak{c}}-3{\mathfrak{s}}]^- T}{{\mathfrak{c}}-{\mathfrak{s}}}.
\]
Therefore, the initial datum is only relevant on the bounded interval $\left[- S(T), 0\right]$,
with $S(T)$ given by \eqref{ST}.
\section{Flocking - proof of Theorem \ref{thm:flocking}}\label{sec:flocking}
In this Section we shall assume the influence function $\psi$ to be a nonincreasing function of its argument on $[0,\infty)$.
This is without loss of generality, since if $\psi$ was not monotone,
it could be replaced by its nonincreasing rearrangement $\Psi$ in the below proofs,
\( \label{Psi}
\Psi(u) := \min_{s\in[0,u]} \psi(s) \qquad\mbox{for } u\geq 0,
\)
without affecting their validity.
\begin{lemma}\label{lem:shrinkage}
Along the solutions of \eqref{eq:tau}--\eqref{IC:comp} we have
\( \label{claim:shrinkage}
\tot{}{t} {d_{\solv}}(t) \leq - \frac{N}{N-1}\underline{\psi}(t) {d_{\solv}}(t) + 2D(t) \qquad\mbox{for almost all } t>0,
\)
with ${d_{\solv}}={d_{\solv}}(t)$ given by \eqref{dXdV},
\( \label{def:upsi}
\underline{\psi}(t) := \min_{i,j \in [N]} \widetilde\psi_{ij} (t),
\)
and
\( \label{def:D}
D(t) := \max_{i\in [N]} \frac{1}{N-1} \sum_{j\neq i} \widetilde\psi_{ij} |\wv{j}{i} - v_j|.
\)
\end{lemma}
\begin{proof}
Due to the continuity of the solution trajectories $x_i(t)$,
there is an at most countable system of open, mutually disjoint
intervals $\{\mathcal{I}_\sigma\}_{\sigma\in\mathbb{N}}$ such that
\[
\bigcup_{\sigma\in\mathbb{N}} \overline{\mathcal{I}_\sigma} = [0,\infty)
\]
and for each ${\sigma\in\mathbb{N}}$ there exist indices $i(\sigma)$, $k(\sigma)$
such that
\[
{d_{\solv}}(t) = |v_{i(\sigma)}(t) - v_{k(\sigma)}(t)| \quad\mbox{for } t\in \mathcal{I}_\sigma.
\]
Then, using the abbreviated notation $i:=i(\sigma)$, $k:=k(\sigma)$,
we have for every $t\in \mathcal{I}_\sigma$,
\( \label{shrinkage:1}
\frac12 \tot{}{t} {d_{\solv}}(t)^2 &=& \frac12 \tot{}{t} |v_i - v_k|^2 \\
&=& \frac{1}{N-1} \sum_{j\neq i} \widetilde\psi_{ij} \bigl(\wv{j}{i} - v_i\bigr)\cdot (v_i-v_k)
- \frac{1}{N-1} \sum_{j\neq k} \widetilde\psi_{kj} \bigl(\wv{j}{k} - v_k\bigr)\cdot (v_i-v_k). \nonumber
\)
Let us work on the first term of the right-hand side. We have for any $j\in [N]$,
\( \label{shrinkage:2}
\widetilde\psi_{ij} \left(\wv{j}{i} - v_i\right)\cdot (v_i-v_k) = \widetilde\psi_{ij} \left(\wv{j}{i} - v_j\right)\cdot (v_i-v_k) + \widetilde\psi_{ij} \left(v_j- v_i\right)\cdot (v_i-v_k).
\)
By the Cauchy-Schwarz inequality and \eqref{def:D} we have
\[
\frac{1}{N-1} \sum_{j\neq i} \widetilde\psi_{ij} \left(\wv{j}{i} - v_j\right)\cdot (v_i-v_k) \leq
\frac{1}{N-1} \sum_{j\neq i} \widetilde\psi_{ij} \left|\wv{j}{i} - v_j\right| \left| v_i-v_k \right| \leq D(t) {d_{\solv}}(t).
\]
For the second term in \eqref{shrinkage:2}, we observe, using the Cauchy-Schwarz inequality,
\[
(v_j- v_i)\cdot (v_i-v_k) &=& (v_j-v_k)\cdot(v_i-v_k) - |v_i-v_k|^2 \\
&\leq& |v_i-v_k| \bigl( |v_j-v_k| - |v_i-v_k| \bigr) \leq 0,
\]
since, by definition, $|v_j-v_k| \leq {d_{\solx}} = |v_i-v_k|$.
Moreover, with \eqref{def:upsi} we have
\[
\widetilde \psi_{ij} \left(v_j- v_i\right)\cdot (v_i-v_k) \leq \underline{\psi} \, (v_j- v_i)\cdot (v_i-v_k).
\]
Carrying out analogous steps for the second term of the right-hand side of \eqref{shrinkage:1},
we finally obtain
\[
\frac12 \tot{}{t} {d_{\solv}}^2 &\leq& 2{d_{\solv}} D + \frac{\underline{\psi}}{N-1} \sum_{j=1}^N
\bigl[(v_j- v_i) - (v_j-v_k)\bigr]\cdot (v_i-v_k) \\
&=& \left[ 2D - \frac{N}{N-1} \underline{\psi} {d_{\solv}} \right] {d_{\solv}}.
\]
This immediately gives the statement.
\end{proof}
For $t\geq 0$ let us define the maximal delay at time $t$,
\( \label{otau}
\overline{\tau}(t) := \max_{i,j\in [N]} \tau_{ij}(t).
\)
\begin{lemma} \label{lem:D}
Let the initial datum $\mathbf{v}^0\in C((-\infty,0]; \mathbb{R}^{Nd})$ be Lipschitz continuous
on $(-\infty,0]$, with Lipschitz constant $L_{\mathbf{v}}^0\geq 0$.
Then, along the solutions of \eqref{eq:tau}--\eqref{IC:comp}, we have
\( \label{D}
D(t) \leq L_{\mathbf{v}}^0 \, [t-\overline{\tau}(t)]^- + \int_{[t-\overline{\tau}(t)]^+}^t D(s) \mathrm{d} s + \int_{[t-\overline{\tau}(t)]^+}^t {d_{\solv}}(s) \mathrm{d} s \qquad\mbox{for } t\geq 0,
\)
where we denoted $[a]^+ := \max\{a,0\}$, $[a]^- := \max\{-a,0\}$,
the quantity $D=D(t)$ is defined in \eqref{def:D} and ${d_{\solv}}={d_{\solv}}(t)$ given by \eqref{dXdV}.
\end{lemma}
\begin{proof}
For any $i, j\in [N]$ with $i\neq j$ and $t\geq 0$ we have,
\[
|\wv{j}{i} - v_j| = |v_j(t-\tau_{ij}) - v_j(t)| &\leq& \int_{t-\tau_{ij}}^t \left| \dot v_j(s) \right| \mathrm{d} s \\
&\leq& \int_{t-\overline{\tau}}^{[t-\overline{\tau}]^+} \left| \dot v_j(s) \right| \mathrm{d} s + \int_{[t-\overline{\tau}]^+}^t \left| \dot v_j(s) \right| \mathrm{d} s.
\]
For the first integral of the right-hand side we have
\[
\int_{t-\overline{\tau}}^{[t-\overline{\tau}]^+} \left| \dot v_j(s) \right| \mathrm{d} s \leq L_{\mathbf{v}}^0 \, [t-\overline{\tau}]^-,
\]
while for the second we use \eqref{eq:CS2},
\[
\int_{[t-\overline{\tau}]^+}^t \left| \dot v_j(s) \right| \mathrm{d} s
&\leq& \frac{1}{N-1} \int_{[t-\overline{\tau}]^+}^t \sum_{\ell\neq j} \widetilde{\psi}_{j\ell} \left| \widetilde v^j_\ell(s) - v_j(s) \right| \mathrm{d} s \\
&\leq& \frac{1}{N-1} \int_{[t-\overline{\tau}]^+}^t \sum_{\ell\neq j} \widetilde{\psi}_{j\ell} \left( \left| \widetilde v^j_\ell(s) - v_\ell(s) \right| + \left| v_\ell(s) - v_j(s) \right| \right) \mathrm{d} s \\
&\leq& \int_{[t-\overline{\tau}]^+}^t D(s) \mathrm{d} s + \int_{[t-\overline{\tau}]^+}^t {d_{\solv}}(s) \mathrm{d} s,
\]
where we used the estimate $\frac{1}{N-1} \sum_{\ell\neq j} \widetilde{\psi}_{j\ell} \leq 1$ implied by universal bound $\psi\leq 1$.
Consequently,
\[
D(t) &=& \max_{i\in [N]} \frac{1}{N-1} \sum_{j\neq i} \widetilde\psi_{ij} |\wv{j}{i} - v_j| \\
&\leq& \max_{i\in [N]} \frac{1}{N-1} \sum_{j\neq i} \widetilde\psi_{ij} \left( L_{\mathbf{v}}^0 \, [t-\overline{\tau}]^- + \int_{[t-\overline{\tau}]^+}^t D(s) \mathrm{d} s + \int_{[t-\overline{\tau}]^+}^t {d_{\solv}}(s) \mathrm{d} s \right) \\
&\leq& L_{\mathbf{v}}^0 \, [t-\overline{\tau}(t)]^- + \int_{[t-\overline{\tau}]^+}^t D(s) \mathrm{d} s + \int_{[t-\overline{\tau}]^+}^t {d_{\solv}}(s) \mathrm{d} s.
\]
\end{proof}
\begin{lemma} \label{lem:crazy}
Fix ${d_{\solx}}(0)$, ${d_{\solv}}(0)$, $L_{\mathbf{v}}^0 \geq 0$, ${\mathfrak{s}}>0$, and $\eta>0$ provided by assumption \eqref{ass:bb}.
Assume that there exist ${\mathfrak{c}}>{\mathfrak{s}}$, $\kappa> D(0)$ and $\sigma > {d_{\solv}}(0)$ such that
\( \label{crazy:1}
L_{\mathbf{v}}^0\tau^\ast + (\kappa+\sigma) \frac{e^{\eta\tau^\ast} - 1}{\eta} \leq \kappa \qquad\mbox{with } \tau^\ast:= ({\mathfrak{c}} - {\mathfrak{s}})^{-1} \left( {d_{\solx}}(0) + \frac{\sigma}{\eta} \right),
\)
and
\( \label{crazy:2}
\frac{2\kappa}{\psi^\ast-\eta} \leq \sigma - {d_{\solv}}(0)
\qquad\mbox{with }
\psi^\ast := \psi({\mathfrak{c}}\tau^\ast) > \eta.
\)
Then
\( \label{claim:crazy}
{d_{\solv}}(t) < \sigma e^{-\eta t} \mbox{ and } D(t) < \kappa e^{-\eta t}
\)
for all $t>0$.
\end{lemma}
\begin{proof}
Let us define the set
\( \label{setS}
\mathcal{S} := \left\{ t>0; \; D(t) < \kappa e^{-\eta t} \quad\mbox{and}\quad {d_{\solv}}(t) < \sigma e^{-\eta t} \right\}.
\)
Since, by assumption, $\kappa>D(0)$ and $\sigma > {d_{\solv}}(0)$, the set $\mathcal{S}$ is nonempty, so that we may define
$T:=\sup\mathcal{S}$.
We aim at showing that $T=+\infty$. For contradiction, let us assume that $T<+\infty$.
By continuity of the functions ${d_{\solv}}={d_{\solv}}(t)$ and $D=D(t)$ we then have
\( \label{for_contr}
D(T) = \kappa e^{-\eta T} \qquad\mbox{or}\qquad {d_{\solv}}(T) = \sigma e^{-\eta T}.
\)
By the definition \eqref{dXdV} of the diameters ${d_{\solx}}$ and ${d_{\solv}}$ we have
\[
{d_{\solx}}(t) \leq {d_{\solx}}(0) + \int_0^t {d_{\solv}}(s) \mathrm{d} s,
\]
and from the definition \eqref{setS} of the set $\mathcal{S}$ it follows for $t<T$,
\[
\int_0^t {d_{\solv}}(s) \mathrm{d} s < \sigma \int_0^t e^{-\eta s} \mathrm{d} s < \frac{\sigma}{\eta},
\]
so that
\( \label{est:dx}
{d_{\solx}}(t) < {d_{\solx}}(0) + \frac{\sigma}{\eta}.
\)
An application of Lemma \eqref{lem:tauij} then gives
\( \label{est:otau}
\overline{\tau}(t) = \max_{i,j\in [N]} \tau_{ij}(t) \leq \frac{{d_{\solx}}(t)}{{\mathfrak{c}}-{\mathfrak{s}}} < \frac{1}{{\mathfrak{c}}-{\mathfrak{s}}} \left( {d_{\solx}}(0) + \frac{\sigma}{\eta} \right) = \tau^\ast
\)
for $t<T$.
Using \eqref{D} and \eqref{setS}, we have for $t\leq T$,
\[
D(t) &\leq& L_{\mathbf{v}}^0 \, [t-\overline{\tau}(t)]^- + \int_{[t-\overline{\tau}(t)]^+}^t D(s) \mathrm{d} s + \int_{[t-\overline{\tau}(t)]^+}^t {d_{\solv}}(s) \mathrm{d} s \\
&<& L_{\mathbf{v}}^0 \, [t-\tau^\ast]^- + (\kappa + \sigma) \int_{[t-\tau^\ast]^+}^t e^{-\eta s} \mathrm{d} s \\
&\leq& \left( L_{\mathbf{v}}^0 \, [t-\tau^\ast]^- e^{\eta t} + (\kappa + \sigma) \frac{e^{\eta\tau^\ast} - 1}{\eta} \right) e^{-\eta t}.
\]
An elementary calculation, using the fact that $\eta<1$ and realizing that \eqref{crazy:1} implies $\tau^\ast<1$, gives
\[
[t-\tau^\ast]^- e^{\eta t} \leq \tau^\ast \qquad\mbox{for all } t\geq 0.
\]
Consequently,
\[
D(T) < \left( L_{\mathbf{v}}^0 \, \tau^\ast + (\kappa + \sigma) \frac{e^{\eta\tau^\ast} - 1}{\eta} \right) e^{-\eta T},
\]
and assumption \eqref{crazy:1} gives $D(T) < \kappa e^{-\eta T}$,
so that the first alternative in \eqref{for_contr} is excluded.
To exclude the second alternative in \eqref{for_contr}, we recall that $\widetilde\psi_{ij}(t) = \psi(|\wx{j}{i} - x_i|)$
and apply the triangle inequality, \eqref{est:wxjixj} and \eqref{est:tauij}, which yields
\[
|\wx{j}{i} - x_i| \leq |\wx{j}{i} - x_j| + |x_i - x_j| \leq {\mathfrak{s}} \tau_{ij} + {d_{\solx}}(t) \leq \frac{{\mathfrak{c}}}{{\mathfrak{c}}-{\mathfrak{s}}} {d_{\solx}}(t).
\]
Since, by assumption, $\psi$ is a non-increasing function, we have for all $t\geq 0$,
\[
\widetilde\psi_{ij}(t) \geq \psi\left( \frac{{\mathfrak{c}}}{{\mathfrak{c}}-{\mathfrak{s}}} {d_{\solx}}(t) \right).
\]
Combining this with \eqref{est:dx}, we obtain $\widetilde\psi_{ij}(t) \geq \psi^\ast$ for all $t<T$ and all $i,j\in [N]$,
with $\psi^\ast$ defined in \eqref{crazy:2}.
Then, Lemma \ref{lem:shrinkage} gives
\[
\tot{}{t} {d_{\solv}}(t) &\leq& - \frac{N}{N-1}\psi^\ast {d_{\solv}}(t) + 2D(t) \\
&<& - \psi^\ast {d_{\solv}}(t) + 2 \kappa e^{-\eta t}
\]
for almost all $t<T$. An integration on the interval $(0,T)$ then yields
\[
{d_{\solv}}(T) &\leq& {d_{\solv}}(0) e^{-\psi^\ast T} + 2 \kappa \frac{e^{-\eta T} - e^{-\psi^\ast T}}{\psi^\ast - \eta} \\
&<& \left( {d_{\solv}}(0) + \frac{2\kappa}{\psi^\ast-\eta} \right) e^{-\eta T},
\]
where we used the fact that $\eta<\psi^\ast$ in the second inequality.
Assumption \eqref{crazy:2} gives ${d_{\solv}}(T) < \sigma e^{-\eta T}$,
and, consequently, the second alternative in \eqref{for_contr} is excluded.
We conclude that $T=+\infty$ which finishes the proof.
\end{proof}
Obviously, \eqref{claim:crazy} implies flocking in the sense of Definition \ref{def:flocking},
noting that the uniform boundedness of ${d_{\solx}}={d_{\solx}}(t)$ follows from
\[
{d_{\solx}}(t) \leq {d_{\solx}}(0) + \int_0^t {d_{\solv}}(s) \mathrm{d} s < {d_{\solx}}(0) + \sigma \int_0^t e^{-\eta s} \mathrm{d} s < {d_{\solx}}(0) + \frac{\sigma}{\eta}
\qquad\mbox{for all } t\geq 0.
\]
Clearly, the crucial step of the proof of Theorem \ref{thm:flocking}
is to find values of ${\mathfrak{c}}>{\mathfrak{s}}$, $\sigma>{d_{\solv}}(0)$ and $\kappa>D(0)$
such that the assumptions of Lemma \ref{lem:crazy} are satisfied.
This essentially means to resolve the nonlinear algebraic system \eqref{crazy:1}--\eqref{crazy:2}.
We were only able to find an analytical solution under the additional assumption
that the initial velocity trajectories $\mathbf{v}^0$ are constant on $(-\infty,0]$.
Then $L_{\mathbf{v}}^0 = 0$ and $D(0)=0$, which facilitates the following result.
\begin{lemma} \label{lem:notsocrazy}
Assume that $L_{\mathbf{v}}^0 = 0$ and $D(0)=0$.
For any fixed ${d_{\solx}}(0)$, ${d_{\solv}}(0)\geq 0$, ${\mathfrak{s}}>0$, and $\eta>0$
provided by assumption \eqref{ass:bb}, there exist ${\c^\ast}>{\mathfrak{s}}$,
$\sigma>{d_{\solv}}(0)$ and $\kappa>0$ such that for any ${\mathfrak{c}}\geq {\c^\ast}$
the conditions \eqref{crazy:1} and \eqref{crazy:2} are verified.
\end{lemma}
\begin{proof}
Assumption \eqref{ass:bb} and continuity of $\psi$ implies that, for fixed $\eta>0$, there exist
$\varepsilon>0$ and $\sigma > {d_{\solv}}(0)$ such that
\( \label{crazy:4}
\psi\left( (1+\varepsilon) \left({d_{\solx}}(0) + \frac{\sigma}{\eta} \right) \right) > \eta.
\)
We set
\( \label{crazy:5}
\kappa := \frac{\sigma-{d_{\solv}}(0)}{2} \left[ \psi\left( (1+\varepsilon) \left({d_{\solx}}(0) + \frac{\sigma}{\eta} \right) \right) - \eta \right].
\)
Now, with the values of ${\mathfrak{s}}>0$, $\eta>0$, $\sigma > {d_{\solv}}(0)$ and $\kappa>0$ fixed, the equation
\( \label{crazy:3}
\frac{1}{\eta} \ln\left( \frac{\eta\kappa}{\kappa+\sigma} + 1 \right) = \frac{1}{{\mathfrak{c}}-{\mathfrak{s}}} \left({d_{\solx}}(0) + \frac{\sigma}{\eta} \right)
\)
is uniquely solvable in ${\mathfrak{c}}>{\mathfrak{s}}$, since $\frac{1}{{\mathfrak{c}}-{\mathfrak{s}}}$ is a monotonically decreasing function of ${\mathfrak{c}}>{\mathfrak{s}}$
with values in $(0,\infty)$.
We denote this unique solution ${\mathfrak{c}}_1$ and set
\( \label{cst}
{\c^\ast} := \max\left\{ {\mathfrak{c}}_1, \frac{1+\varepsilon}{\varepsilon} {\mathfrak{s}} \right\}.
\)
We claim that the above choice of $\sigma$, $\kappa$ and ${\c^\ast}$
verifies the assumptions of Lemma \ref{lem:crazy},
in particular, the conditions \eqref{crazy:1} and \eqref{crazy:2}.
Indeed, recalling the definition of $\tau^\ast$ given by \eqref{crazy:1},
\( \label{taust}
\tau^\ast = \frac{1}{{\c^\ast} - {\mathfrak{s}}} \left( {d_{\solx}}(0) + \frac{\sigma}{\eta} \right),
\)
equation \eqref{crazy:3} gives
\[
\frac{1}{\eta} \ln\left( \frac{\eta\kappa}{\kappa+\sigma} + 1 \right) \leq \tau^\ast,
\]
which is equivalent to
\[
(\kappa+\sigma) \frac{e^{\eta\tau^\ast} - 1}{\eta} \leq \kappa.
\]
This proves \eqref{crazy:1} with $L_{\mathbf{v}}^0=0$.
By \eqref{cst} we have $\frac{{\c^\ast}}{{\c^\ast}-{\mathfrak{s}}} \leq 1+\varepsilon$, which implies
\[
{\c^\ast} \tau^\ast
\leq (1+\varepsilon) \left( {d_{\solx}}(0) + \frac{\sigma}{\eta} \right).
\]
Due to the monotonicity of $\psi$, we then have for $\psi^\ast := \psi({\c^\ast} \tau^\ast)$
\[
\psi^\ast \geq \psi\left( (1+\varepsilon) \left( {d_{\solx}}(0) + \frac{\sigma}{\eta} \right) \right),
\]
and, due to \eqref{crazy:4}, $\psi^\ast > \eta$.
Moreover,
\[
\frac{2\kappa}{\psi^\ast-\eta} \leq 2\kappa \left[ \psi\left( (1+\varepsilon) \left({d_{\solx}}(0) + \frac{\sigma}{\eta} \right) \right) - \eta \right]^{-1},
\]
and substituting for $\kappa$ from \eqref{crazy:5} gives
\[
\frac{2\kappa}{\psi^\ast-\eta} \leq \sigma - {\mathfrak{s}},
\]
which is \eqref{crazy:2}.
Finally, let us note that for any ${\mathfrak{c}}\geq {\c^\ast}$ we have
\[
\frac{1}{{\mathfrak{c}} - {\mathfrak{s}}} \left( {d_{\solx}}(0) + \frac{\sigma}{\eta} \right) \leq \tau^\ast.
\]
Consequently, \eqref{crazy:1} remains valid with ${\c^\ast}$ replaced by ${\mathfrak{c}}$.
Moreover,
\[
{\mathfrak{c}} \tau := \frac{{\mathfrak{c}}}{{\mathfrak{c}} - {\mathfrak{s}}} \left( {d_{\solx}}(0) + \frac{\sigma}{\eta} \right) \leq \frac{{\c^\ast}}{{\c^\ast} - {\mathfrak{s}}} \left( {d_{\solx}}(0) + \frac{\sigma}{\eta} \right) = {\c^\ast}\tau^\ast,
\]
since $\frac{{\mathfrak{c}}}{{\mathfrak{c}} - {\mathfrak{s}}}$ is a decreasing function of ${\mathfrak{c}}>{\mathfrak{s}}$.
Consequently, $\psi({\mathfrak{c}}\tau) \geq \psi^\ast$ and \eqref{crazy:2} keeps its validity as well.
\end{proof}
\begin{remark} \label{rem:v0}
Note that for a given influence function $\psi$ and values of $\eta$, ${d_{\solv}}(0$), ${d_{\solx}}(0)$ and ${\mathfrak{s}}$,
the proof of Lemma \ref{lem:notsocrazy} provides a recipe for calculating the value of ${\c^\ast}$ explicitly
(although it may not provide the optimal value).
On the other hand, admitting nonconstant initial datum $\mathbf{v}^0$
makes it necessary to decide the solvability of the algebraic system \eqref{crazy:1}--\eqref{crazy:2},
which does not seem to be achievable analytically.
However, the problem is well approachable by numerical methods.
\end{remark}
Finally, we provide the proof of Proposition \ref{corr:CS}.
Clearly, \eqref{ass:bb} is satisfiable whenever $\inf_{s\in(0,\infty)} \psi(s) > 0$.
Therefore, let us consider $\psi=\psi(s)$ such that $\lim_{s\to\infty} \psi(s) = 0$.
Moreover, recall that we assume $\psi=\psi(s)$ to be monotone
(otherwise we can replace it by its nonincreasing rearrangement).
Assumption \eqref{corr:CS:cond} implies that $\psi(s) \geq c s^\alpha$ for some constant $c>0$ and all $s>0$ large enough.
Denoting $\Psi(s):=c s^\alpha$, we readily calculate for $\alpha>-1$,
\[
\lim_{\eta\to 0+} \Psi\left({d_{\solx}}(0) + \frac{{d_{\solv}}(0)}{\eta}\right) - \eta = 0,
\qquad
\tot{}{\eta} \left[ \Psi\left({d_{\solx}}(0) + \frac{{d_{\solv}}(0)}{\eta}\right) - \eta \right]_{\eta=0+} = -\infty.
\]
Consequently, for $\eta>0$ small enough
\[
\psi\left({d_{\solx}}(0) + \frac{{d_{\solv}}(0)}{\eta}\right) - \eta \geq \Psi\left({d_{\solx}}(0) + \frac{{d_{\solv}}(0)}{\eta}\right) - \eta > 0,
\]
and the claim of Proposition \ref{corr:CS} follows.
\section{Mean field limit} \label{sec:MF}
Let us fix $T>0$ and $0<{\mathfrak{s}}<{\mathfrak{c}}$ and recall the definition \eqref{def:OmegasT} of the set
\[
\Omega_{\mathfrak{s}}^T :=
\Bigl\{ \gamma\in C_b^1((-\infty, T];\mathbb{R}^d) \cap C_{\mathfrak{s}}((-\infty,T]; \mathbb{R}^d); \;
\dot\gamma|_{[0,T]} \in C_{2{\mathfrak{s}}}([0,T]; \mathbb{R}^d) \Bigr\},
\]
equipped with the norm $\Norm{\cdot}_{\Omega_{\mathfrak{s}}^T}$ defined in \eqref{def:norm},
\[
\Norm{\gamma}_{\Omega_{\mathfrak{s}}^T}:= \Norm{\gamma}_{L^\infty(-\infty,T)} + \Norm{\dot \gamma}_{L^\infty(0,T)} \qquad
\mbox{for } \gamma\in \Omega_{\mathfrak{s}}^T.
\]
Obviously, due to the uniform boundedness of $\gamma\in\Omega_{\mathfrak{s}}^T$ on $(-\infty,T)$ and the uniform Lipschitz continuity of $\dot\gamma$ on $(0,T)$,
we have $\Norm{\gamma}_{\Omega_{\mathfrak{s}}^T} < +\infty$ for all $\gamma\in\Omega_{\mathfrak{s}}^T$.
We also recall that ${\mathcal{P}(\Omega_\s^T})$ denotes the space of probability measures on $\Omega_{\mathfrak{s}}^T$
with finite first-order moment
and it is equipped with the Monge--Kantorovich--Rubinstein distance $\mathcal{W}_T$
defined in \eqref{def:MKR}.
For any $\rho$, $\nu\in{\mathcal{P}(\Omega_\s^T})$ we have $\mathcal{W}_T(\rho,\nu) < +\infty$ since
\[
\mathcal{W}_T(\rho,\nu) &\leq& \iint_{\Omega_{\mathfrak{s}}^T \times \Omega_{\mathfrak{s}}^T} \Norm{\gamma-\xi}_{\Omega_{\mathfrak{s}}^T} \mathrm{d}\rho(\gamma)\mathrm{d}\nu(\xi) \\
&\leq& \int_{\Omega_{\mathfrak{s}}^T} \Norm{\gamma}_{\Omega_{\mathfrak{s}}^T} \mathrm{d}\rho(\gamma) + \int_{\Omega_{\mathfrak{s}}^T} \Norm{\xi}_{\Omega_{\mathfrak{s}}^T} \mathrm{d}\nu(\xi),
\]
and the measures $\rho$, $\nu\in{\mathcal{P}(\Omega_\s^T})$ have, by definition, finite first-order moment.
The set ${\mathcal{P}(\Omega_\s^T})$ endowed with the distance $\mathcal{W}_T$ is a complete metric space.
We consider the problem
\(
\rho &=& \mbox{law}(x), \label{eq:MF0} \\
x(t) &=& x(0) + \int_0^t v(s) \mathrm{d} s, \label{eq:MF1} \\
v(t) &=& v(0) + \int_0^t F_s[\rho](x(s),v(s)) \mathrm{d} s \label{eq:MF2}
\)
with the operator $F_t[\rho]: \mathbb{R}^d\times\mathbb{R}^d \to \mathbb{R}^d$ given by \eqref{def:F}, i.e.,
\[
F_t[\rho](x,v) := \int_{\Omega^T_{\mathfrak{s}}} \psi\left( \left|\Gamma_{t,x}[\gamma] - x \right|\right) \left(\Pi_{t,x}[\gamma] - v \right) \mathrm{d}\rho(\gamma).
\]
Note that $F_t[\rho]$ is well defined since with the universal bound $\psi\leq 1$ and $\rho\in{\mathcal{P}(\Omega_\s^T})$ we have
\[
\left| \int_{\Omega^T_{\mathfrak{s}}} \psi\left( \left|\Gamma_{t,x}[\gamma] - x \right|\right) \left(\Pi_{t,x}[\gamma] - v \right) \mathrm{d}\rho(\gamma) \right|
\leq
\int_{\Omega^T_{\mathfrak{s}}} \left( \left| \Pi_{t,x}[\gamma] \right| + |v| \right) \mathrm{d}\rho(\gamma) \leq {\mathfrak{s}} + |v|.
\]
We recall that the initial datum $\rho^0\in \mathcal{P}(\Omega_{\mathfrak{s}}^0)$ for the mean-field problem
is imposed in terms of the push-forward identity $\mathbb{I}\#\rho = \rho^0$,
with the mapping $\mathbb{I}: \Omega_{\mathfrak{s}}^T \to \Omega_{\mathfrak{s}}^0$ given by \eqref{def:I}.
Then, $x(0)$ and, resp., $v(0)$ in \eqref{eq:MF1}-\eqref{eq:MF2} are distributed according to $X_0\#\rho^0$
and, resp., $V_0\#\rho^0$.
\subsection{Auxiliary results}
For any $\rho\in {\mathcal{P}(\Omega_\s^T})$ we define the mapping $Z[\rho]: \Omega_{\mathfrak{s}}^0 \to C^1((-\infty,T])$,
\( \label{def:Z}
Z[\rho]: \gamma^0 \mapsto \gamma,
\)
where $\gamma\in C^1((-\infty,T])$ is identical to $\gamma^0$ on $(-\infty,0]$
and on $(0,T]$ it is the unique solution $x=x(t)$ of the system \eqref{eq:MF1}--\eqref{eq:MF2}
subject to the initial datum $x(0)=\gamma^0(0)$ and $v(0)=\dot\gamma^0(0)$.
The existence and uniqueness of solutions of \eqref{eq:MF1}--\eqref{eq:MF2}
is established by a slight modification of the arguments carried out in Section \ref{sec:ex}.
\begin{lemma}\label{lem:Z}
For any $T>0$ and $\rho\in {\mathcal{P}(\Omega_\s^T})$, the mapping $Z[\rho]$ given by \eqref{def:Z} maps the set $\Omega^0_{\mathfrak{s}}$
into $\Omega^T_{\mathfrak{s}}$.
\end{lemma}
\begin{proof}
Let us fix $\gamma^0\in \Omega^0_{\mathfrak{s}}$ and denote $x:= Z[\rho](\gamma^0)$, $v:=\dot x$.
We first prove that $x$ is ${\mathfrak{s}}$-Lipschitz continuous on $(-\infty, T]$.
On the interval $(-\infty,0]$ this follows directly from the definition.
For $t\geq 0$ we have from \eqref{eq:MF2}--\eqref{def:F},
\[
\frac12 \tot{}{t} |v(t)|^2 &=& \int_{\Omega^T_{\mathfrak{s}}} \psi\left( \left|\Gamma_{t,x(t)}[\gamma] - x(t) \right|\right) \left( \Pi_{t,x(t)}[\gamma] - v(t) \right) \cdot v(t) \, \mathrm{d}\rho(\gamma) \\
&\leq& \int_{\Omega^T_{\mathfrak{s}}} \left( \left| \Pi_{t,x(t)}[\gamma] \right| - |v(t)| \right) |v(t)| \, \mathrm{d}\rho(\gamma),
\]
where we used the universal bound $\psi\leq 1$.
Moreover, by the definition \eqref{def:OmegasT}, we have $|\Pi_{t,x}[\gamma]| = |\dot\gamma(t-\tau_{t,x}[\gamma])| \leq {\mathfrak{s}}$
for all $t\in (-\infty,T]$ and $\gamma\in\Omega^T_{\mathfrak{s}}$.
Consequently, for almost all $t>0$,
\[
\tot{}{t} |v(t)| \leq {\mathfrak{s}} - |v(t)|,
\]
with $|v(0)| \leq {\mathfrak{s}}$.
This implies $|\dot x(t)| = |v(t)| \leq {\mathfrak{s}}$ for all $t>0$.
Since, by definition, $Z[\rho](\gamma^0) \equiv x$ on $(0,T]$,
we obtain the ${\mathfrak{s}}$-Lipschitz continuity of $Z[\rho](\gamma^0)$ on the interval $[0,T]$.
Moreover, with the universal bound $\psi\leq 1$, $|v(t)| \leq {\mathfrak{s}}$ and $\left| \Pi_{t,x}[\gamma] \right| \leq {\mathfrak{s}}$, we have for $t\in [0,T]$
\[
|\dot v(t)| = |F_s[\rho](x(t),v(t))| \leq
\int_{\Omega^T_{\mathfrak{s}}} \psi\left( \left|\Gamma_{t,x}[\gamma] - x(t) \right|\right) \left| \Pi_{t,x}[\gamma] - v(t) \right| \mathrm{d}\rho(\gamma)
\leq 2{\mathfrak{s}}.
\]
This gives the uniform $2{\mathfrak{s}}$-Lipschitz continuity of $v= \tot{}{t} Z[\rho](\gamma^0)$ on $[0,T]$
and we finally conclude that $Z[\rho](\gamma^0)\in \Omega^T_{\mathfrak{s}}$.
\end{proof}
\begin{lemma} \label{lem:gamma-gamma}
For any $\gamma$, $\xi\in \Omega_{\mathfrak{s}}^T$ and $x$, $y\in\mathbb{R}^d$, $t\in [0,T]$ we have
\( \label{GammaGamma}
\left|\Gamma_{t,x}[\gamma] - \Gamma_{t,y}[\xi] \right| \leq
\frac{{\mathfrak{c}}}{{\mathfrak{c}}-{\mathfrak{s}}} \Bigl( \Norm{ \gamma - \xi }_{L^\infty(0,t)} + {\mathfrak{s}}{\mathfrak{c}}^{-1} |x-y| \Bigr),
\)
where the mapping $\Gamma_{t,x}[\gamma]$ is defined in \eqref{def:Gamma}, and
\( \label{PiPi}
\left|\Pi_{t,x}[\gamma] - \Pi_{t,y}[\xi] \right| \leq
\Norm{ \dot\gamma - \dot\xi }_{L^\infty(0,t)} +
\frac{2{\mathfrak{s}}}{{\mathfrak{c}}-{\mathfrak{s}}} \Bigl( \Norm{ \gamma - \xi }_{L^\infty(0,t)} + |x-y| \Bigr),
\)
with $\Pi_{t,x}[\gamma]$ defined in \eqref{def:Pi}.
\end{lemma}
\begin{proof}
Fix $t\in [0,T]$.
By definition of $\Gamma_{t,x}$, we have
\[
\left|\Gamma_{t,x}[\gamma] - \Gamma_{t,y}[\xi] \right| =
\left| \gamma(t-\tau_{t,x}[\gamma]) - \xi(t-\tau_{t,y}[\xi]) \right|,
\]
with
\( \label{taugammaxi}
{\mathfrak{c}}\tau_{t,x}[\gamma] = \bigl| x - \gamma(t-\tau_{t,x}[\gamma] \bigr|, \qquad
{\mathfrak{c}}\tau_{t,y}[\xi] = \bigl| y - \xi(t-\tau_{t,y}[\xi] \bigr|.
\)
The triangle inequality gives
\[
\bigl| \gamma(t-\tau_{t,x}[\gamma]) - \xi(t-\tau_{t,y}[\xi]) \bigr| &\leq&
\bigl| \gamma(t-\tau_{t,x}[\gamma]) - \xi(t-\tau_{t,x}[\gamma]) \bigr|
+ \bigl| \xi(t-\tau_{t,x}[\gamma]) - \xi(t-\tau_{t,y}[\xi]) \bigr| \\
&\leq&
\Norm{ \gamma - \xi }_{L^\infty(0,t)} + {\mathfrak{s}} \bigl| \tau_{t,x}[\gamma] - \tau_{t,y}[\xi] \bigr|,
\]
where we used the ${\mathfrak{s}}$-Lipschitz continuity of $\xi$ for the last inequality.
Now, with \eqref{taugammaxi} we have
\[
\bigl| \tau_{t,x}[\gamma] - \tau_{t,y}[\xi] \bigr|
&=& {\mathfrak{c}}^{-1} \bigl| |\gamma(t-\tau_{t,x}[\gamma]) - x| - |\xi(t-\tau_{t,y}[\xi]) - y| \bigr| \\
&\leq& {\mathfrak{c}}^{-1} \Bigl( |x-y| + \bigl| \gamma(t-\tau_{t,x}[\gamma]) - \xi(t-\tau_{t,y}[\xi]) \bigr| \Bigr),
\]
so that
\[
\bigl| \gamma(t-\tau_{t,x}[\gamma]) - \xi(t-\tau_{t,y}[\xi]) \bigr| &\leq&
\Norm{ \gamma - \xi }_{L^\infty(0,t)} + {\mathfrak{s}}{\mathfrak{c}}^{-1} \Bigl( |x-y| + \bigl| \gamma(t-\tau_{t,x}[\gamma]) - \xi(t-\tau_{t,y}[\xi]) \bigr| \Bigr),
\]
which immediately gives
\[
\bigl| \gamma(t-\tau_{t,x}[\gamma]) - \xi(t-\tau_{t,y}[\xi]) \bigr| \leq
\frac{{\mathfrak{c}}}{{\mathfrak{c}}-{\mathfrak{s}}} \Bigl( \Norm{ \gamma - \xi }_{L^\infty(0,t)} + {\mathfrak{s}}{\mathfrak{c}}^{-1} |x-y| \Bigr).
\]
This proves \eqref{GammaGamma}.
Moreover,
\[
\left|\Pi_{t,x}[\gamma] - \Pi_{t,y}[\xi] \right| &=&
\left| \dot\gamma(t-\tau_{t,x}[\gamma]) - \dot\xi(t-\tau_{t,y}[\xi]) \right| \\
&\leq&
\left| \dot\gamma(t-\tau_{t,x}[\gamma]) - \dot\xi(t-\tau_{t,x}[\gamma]) \right|
+ \left| \dot\xi(t-\tau_{t,x}[\gamma]) - \dot\xi(t-\tau_{t,y}[\xi]) \right| \\
&\leq&
\Norm{ \dot\gamma - \dot\xi }_{L^\infty(0,t)} +
2{\mathfrak{s}} \bigl| \tau_{t,x}[\gamma] - \tau_{t,y}[\xi] \bigr|,
\]
where we used the fact that $\dot\xi$ is $2{\mathfrak{s}}$-Lipschitz continuous on $[0,T]$.
Combining the above estimates, we readily obtain
\[
\bigl| \tau_{t,x}[\gamma] - \tau_{t,y}[\xi] \bigr| \leq \frac{1}{{\mathfrak{c}}-{\mathfrak{s}}} \bigl( \Norm{ \gamma - \xi }_{L^\infty(0,t)} + |x-y| \bigr),
\]
which gives \eqref{PiPi}.
\end{proof}
\begin{lemma}\label{lem:LipschZ}
For any $T>0$ there exists a constant $L_Z^T>0$ such that, for any $\nu\in{\mathcal{P}(\Omega_\s^T})$,
\( \label{LipschZ}
\Norm{(Z[\nu](\eta_0) - Z[\nu](\xi_0)}_{\Omega_{\mathfrak{s}}^T} \leq L_Z^T \Norm{\eta_0-\xi_0}_{\Omega_{\mathfrak{s}}^0}
\qquad\mbox{for all }\eta_0, \xi_0 \in \Omega_{\mathfrak{s}}^0.
\)
The value of the constant $L_Z^T$ is explicitly calculable and depends only on ${\mathfrak{c}}$, ${\mathfrak{s}}$, $T$ and the Lipschitz constant $L_\psi$
of the influence function $\psi$.
\end{lemma}
\begin{proof}
Let us fix $\nu\in{\mathcal{P}(\Omega_\s^T})$ and $\eta_0$, $\xi_0\in \Omega_{\mathfrak{s}}^0$
and denote $x:=Z[\nu](\eta_0)|_{[0,T]}$, $v:=\dot x$, and $y:=Z[\nu](\xi_0)|_{[0,T]}$, $w:=\dot y$.
Then we have, by the definition \eqref{def:norm} of $\Norm{\cdot}_{\Omega_{\mathfrak{s}}^T}$,
\[
\Norm{(Z[\nu](\eta_0) - Z[\nu](\xi_0)}_{\Omega_{\mathfrak{s}}^T} =
\Norm{\eta_0-\xi_0}_{L^\infty(-\infty,0)} + \Norm{x-y}_{L^\infty(0,T)} + \Norm{v-w}_{L^\infty(0,T)}.
\]
With
\[
\Norm{x-y}_{L^\infty(0,T)} &\leq& |x(0) - y(0)| + T\Norm{v-w}_{L^\infty(0,T)} \\
&\leq& \Norm{\eta_0-\xi_0}_{\Omega_{\mathfrak{s}}^0} + T\Norm{v-w}_{L^\infty(0,T)},
\]
we have
\( \label{goingback}
\Norm{(Z[\nu](\eta_0) - Z[\nu](\xi_0)}_{\Omega_{\mathfrak{s}}^T} \leq 2 \Norm{\eta_0-\xi_0}_{\Omega_{\mathfrak{s}}^0} + (1+T) \Norm{v-w}_{L^\infty(0,T)}.
\)
We thus need to estimate the term $\Norm{v-w}_{L^\infty(0,T)}$.
For any $t\in (0,T)$ we calculate, using \eqref{eq:MF2},
\[
|v(t) - w(t)| &\leq& \int_0^t |F_s[\nu](x(s),v(s)) - F_s[\nu](y(s),w(s))| \, \mathrm{d} s \\
&\leq&
\int_0^t
\int_{\Omega^T_{\mathfrak{s}}} \left| \psi\left( \left|\Gamma_{s,x(s)}[\gamma] - x(s) \right|\right) \left(\Pi_{s,x(s)}[\gamma] - v(s) \right)
\right. \\
&& \qquad\qquad\qquad \left.
- \psi\left( \left|\Gamma_{s,y(s)}[\gamma] - y(s) \right|\right) \left(\Pi_{s,y(s)}[\gamma] - w(s) \right) \right| \mathrm{d}\nu(\gamma) \mathrm{d} s.
\]
We estimate the integrand as
\(
\left| \psi\left( \left|\Gamma_{s,x(s)}[\gamma] - x(s) \right|\right) \left(\Pi_{s,x(s)}[\gamma] - v(s) \right)
- \psi\left( \left|\Gamma_{s,y(s)}[\gamma] - y(s) \right|\right) \left(\Pi_{s,y(s)}[\gamma] - w(s) \right) \right| \nonumber \\
\leq
\left| \psi\left( \left|\Gamma_{s,x(s)}[\gamma] - x(s) \right|\right) - \psi\left( \left|\Gamma_{s,y(s)}[\gamma] - y(s) \right|\right) \right|
\left| \Pi_{s,x(s)}[\gamma] - v(s) \right| \label{est:ZZZ} \\
+ \left| \Pi_{s,x(s)}[\gamma] - \Pi_{s,y(s)}[\gamma] \right| + |v(s) - w(s)|, \nonumber
\)
where we used the universal bound $\psi\leq 1$.
For the first term of the right hand side we apply the bound $|v(s)|\leq{\mathfrak{s}}$ provided by Lemma \ref{lem:Z}
and the bound $|\Pi_{s,x(s)}[\gamma]| \leq {\mathfrak{s}}$ implied by the fact that $\gamma\in\Omega_{\mathfrak{s}}^T$, so that we have
\[
\left| \psi\left( \left|\Gamma_{s,x(s)}[\gamma] - x(s) \right|\right) - \psi\left( \left|\Gamma_{s,y(s)}[\gamma] - y(s) \right|\right) \right|
\left| \Pi_{s,x(s)}[\gamma] - v(s) \right| \\
\leq
2{\mathfrak{s}} L_\psi \left( \left| \Gamma_{s,x(s)}[\gamma] - \Gamma_{s,y(s)}[\gamma] \right| + |x(s)-y(s)| \right),
\]
where $L_\psi$ is the Lipschitz constant of the influence function $\psi$.
The estimate \eqref{GammaGamma} of Lemma \ref{lem:gamma-gamma} gives
\[
\left| \Gamma_{s,x(s)}[\gamma] - \Gamma_{s,y(s)}[\gamma] \right| \leq \frac{{\mathfrak{s}}}{{\mathfrak{c}}-{\mathfrak{s}}} |x(s) - y(s)|,
\]
se that the first term of the right hand side in \eqref{est:ZZZ} is bounded by
\[
\frac{2{\mathfrak{s}}{\mathfrak{c}} L_\psi}{{\mathfrak{c}}-{\mathfrak{s}}} |x(s)-y(s)|.
\]
For the second term of the right-hand side of \eqref{est:ZZZ} we apply the estimate \eqref{PiPi} of Lemma \ref{lem:gamma-gamma},
\[
\left| \Pi_{s,x(s)}[\gamma] - \Pi_{s,y(s)}[\gamma] \right| \leq \frac{2{\mathfrak{s}}}{{\mathfrak{c}}-{\mathfrak{s}}} |x(s)-y(s)|.
\]
Combining the above estimates, we arrive at
\[
|v(t) - w(t)| &\leq& \int_0^t \frac{2{\mathfrak{s}}}{{\mathfrak{c}}-{\mathfrak{s}}} \left( {\mathfrak{c}} L_\psi + 1 \right) |x(s)-y(s)| + |v(s) - w(s)| \mathrm{d} s \\
&\leq& \frac{2{\mathfrak{s}} T}{{\mathfrak{c}}-{\mathfrak{s}}} \left( {\mathfrak{c}} L_\psi + 1 \right) \left( |x(0)-y(0)| + \int_0^t |v(s)-w(s)| \mathrm{d} s \right) + \int_0^t |v(s) - w(s)| \mathrm{d} s \\
&\leq& \frac{2{\mathfrak{s}} T}{{\mathfrak{c}}-{\mathfrak{s}}}\left( {\mathfrak{c}} L_\psi + 1 \right) |x(0)-y(0)| + \left( \frac{2{\mathfrak{s}} T}{{\mathfrak{c}}-{\mathfrak{s}}} \left( {\mathfrak{c}} L_\psi + 1 \right) + 1 \right) \int_0^t |v(s)-w(s)|,
\]
for all $t\in (0,T)$, where we used the estimate
\[
\int_0^t |x(s)-y(s)| \mathrm{d} s &\leq& T |x(0)-y(0)| + \int_0^t \int_0^s |v(\sigma)-w(\sigma)| \mathrm{d}\sigma \mathrm{d} s \\
&\leq& T |x(0)-y(0)| + T \int_0^t |v(s)-w(s)| \mathrm{d} s.
\]
Denoting
\[
K_T := \frac{2{\mathfrak{s}} T}{{\mathfrak{c}}-{\mathfrak{s}}}\left( {\mathfrak{c}} L_\psi + 1 \right) \exp\left( \frac{2{\mathfrak{s}} T^2}{{\mathfrak{c}}-{\mathfrak{s}}} \left( {\mathfrak{c}} L_\psi + 1 \right) + T \right),
\]
an application of the Gronwall lemma on $(0,T)$ yields
\[
\Norm{v-w}_{L^\infty(0,T)} \leq K_T |x(0)-y(0)| \leq K_T \Norm{\eta_0-\xi_0}_{\Omega_{\mathfrak{s}}^0}.
\]
Recalling \eqref{goingback}, we finally obtain
\[
\Norm{(Z[\nu](\eta_0) - Z[\nu](\xi_0)}_{\Omega_{\mathfrak{s}}^T} \leq \bigl[ 2 + (1+T) K_T \bigr] \Norm{\eta_0-\xi_0}_{\Omega_{\mathfrak{s}}^0},
\]
so that the claim \eqref{LipschZ} follows with the choice $L_Z^T := 2 + (1+T) K_T$.
Obviously, the constant $L_Z^T$ does not depend on the choice of $\nu\in{\mathcal{P}(\Omega_\s^T})$.
\end{proof}
\begin{lemma}\label{lem:LipschZZ}
Fix $T>0$. Then for any $\nu\in{\mathcal{P}(\Omega_\s^T})$ we have
\( \label{LipschZZ}
\mathcal{W}_T(Z[\nu]\#\rho^0, Z(\nu)\#\mu^0) \leq L_Z^T \mathcal{W}_0(\rho^0, \mu^0)
\qquad\mbox{for all }\rho^0, \mu^0 \in {\mathcal{P}(\Omega_\s^0}),
\)
where $L_Z^T$ is the constant provided by Lemma \ref{lem:LipschZ}.
\end{lemma}
\begin{proof}
A straightforward adaptation of the proof of \cite[Lemma 3.13]{CCR},
using the Lipschitz continuity result \eqref{LipschZ} of Lemma \ref{lem:LipschZ}.
\end{proof}
\begin{lemma} \label{lem:Wdist}
For any pair of Borel measurable mappings $Y_1, Y_2: \Omega_{\mathfrak{s}}^0 \to \Omega_{\mathfrak{s}}^T$ and $\mu^0\in {\mathcal{P}(\Omega_\s^0})$, we have
\(
\mathcal{W}_T(Y_1\#\mu^0, Y_2\#\mu^0) \leq
\sup_{\gamma^0\in \Omega_{\mathfrak{s}}^0} \Norm{Y_1(\gamma^0)-Y_2(\gamma^0)}_{\Omega_{\mathfrak{s}}^T}.
\)
\end{lemma}
\begin{proof}
See the proof of \cite[Lemma 3.11]{CCR}.
\end{proof}
\begin{lemma}\label{lem:Zgronwall}
There exists a continuous positive function $r:\mathbb{R}^+\to\mathbb{R}^+$ such that for any $\mu^0\in {\mathcal{P}(\Omega_\s^0})$
and any $\rho$, $\nu\in{\mathcal{P}(\Omega_\s^T})$,
\( \label{Zgronwall}
\mathcal{W}_T(Z[\rho]\#\mu^0, Z(\nu)\#\mu^0) \leq r(T) \mathcal{W}_T(\rho, \nu).
\)
The function $r=r(T)$ is explicitly calculable and depends only on the values of ${\mathfrak{s}}$, ${\mathfrak{c}}$
and the Lipschitz constant $L_\psi$ of the influence function $\psi$.
Moreover, it has the property
\( \label{lim-r}
\lim_{T\to 0} r(T) = 0.
\)
\end{lemma}
\begin{proof}
Let us fix $\rho$, $\nu\in {\mathcal{P}(\Omega_\s^T})$
and apply Lemma \ref{lem:Wdist},
\[
\mathcal{W}_T(Z[\rho]\#\mu^0, Z[\nu]\#\mu^0)
\leq \sup_{\gamma^0\in \Omega_{\mathfrak{s}}^0} \Norm{(Z[\rho](\gamma^0) - Z[\nu](\gamma^0)}_{\Omega_{\mathfrak{s}}^T}.
\]
Let us now fix $\gamma^0\in \Omega_{\mathfrak{s}}^0$
and denote $x:=Z[\rho](\gamma^0)|_{[0,T]}$, $v:=\dot x$ and $y:=Z[\nu](\gamma^0)|_{[0,T]}$, $w:=\dot y$.
Then, according to the definition \eqref{def:norm} of the norm $\Norm{\cdot}_{\Omega_{\mathfrak{s}}^T}$, we have
\[
\Norm{(Z[\rho](\gamma^0) - Z[\nu](\gamma^0)}_{\Omega_{\mathfrak{s}}^T} =
\Norm{x-y}_{L^\infty(0,T)} + \Norm{v-w}_{L^\infty(0,T)},
\]
where we used the fact that, by definition, $Z[\rho](\gamma^0)$ and $Z[\nu](\gamma^0)$
coincide on $(-\infty,0]$.
Moreover, with $|x(t)-y(t)| \leq \int_0^t |\dot x(s) - \dot y(s)| \mathrm{d} s = \int_0^t |v(s) - w(s)| \mathrm{d} s$, we have
\[
\Norm{(Z[\rho](\gamma^0) - Z[\nu](\gamma^0)}_{\Omega_{\mathfrak{s}}^T} \leq (1+T) \Norm{v-w}_{L^\infty(0,T)}.
\]
Let us fix $t\in (0,T)$ and calculate, using \eqref{eq:MF2},
\[
|v(t) - w(t)| &\leq& \int_0^t |F_s[\rho](x(s),v(s)) - F_s[\nu](y(s),w(s))| \, \mathrm{d} s \\
&=&
\int_0^t
\left|
\int_{\Omega^s_{\mathfrak{s}}} \psi\left( \left|\Gamma_{s,x(s)}[\gamma] - x(s) \right|\right) \left(\Pi_{s,x(s)}[\gamma] - v(s) \right) \mathrm{d}\rho(\gamma)
\right. \\
&& \qquad\qquad\qquad \left.
- \int_{\Omega^s_{\mathfrak{s}}} \psi\left( \left|\Gamma_{s,y(s)}[\xi] - y(s) \right|\right) \left(\Pi_{s,y(s)}[\xi] - w(s) \right) \mathrm{d}\nu(\xi)
\right| \mathrm{d} s.
\]
Let $\pi$ be an optimal transference plan between the measures $\rho$ and $\nu$
(for the proof of existence of the optimal transference plan, see, e.g., \cite[Theorem 4.1]{old-new}).
Then, using the fact that $\pi$ has marginals $\rho$ and $\nu$, we have
\[
&& \int_{\Omega^s_{\mathfrak{s}}} \psi\left( \left|\Gamma_{s,x(s)}[\gamma] - x(s) \right|\right) \left(\Pi_{s,x(s)}[\gamma] - v(s) \right) \mathrm{d}\rho(\gamma) \\
&& \qquad
- \int_{\Omega^s_{\mathfrak{s}}} \psi\left( \left|\Gamma_{s,y(s)}[\xi] - y(s) \right|\right) \left(\Pi_{s,y(s)}[\xi] - w(s) \right) \mathrm{d}\nu(\xi) \\
&=& \iint_{\Omega^s_{\mathfrak{s}} \times \Omega^s_{\mathfrak{s}}}
\Bigl[ \psi\left( \left|\Gamma_{s,x(s)}[\gamma] - x(s) \right|\right) \left(\Pi_{s,x(s)}[\gamma] - v(s) \right)
\Bigr. \\
&& \qquad\qquad\qquad \Bigl.
- \psi\left( \left|\Gamma_{s,y(s)}[\xi] - y(s) \right|\right) \left(\Pi_{s,y(s)}[\xi] - w(s) \right)
\Bigr] \mathrm{d}\pi(\gamma,\xi).
\]
We write the integrand as
\(
\label{integrand:MF}
\begin{aligned}
& \Bigl[ \psi\left( \left|\Gamma_{s,x(s)}[\gamma] - x(s) \right|\right) - \psi\left( \left|\Gamma_{s,y(s)}[\xi] - y(s) \right|\right) \Bigr]
\left(\Pi_{s,x(s)}[\gamma] - v(s) \right) \\
&+ \psi\left( \left|\Gamma_{s,y(s)}[\xi] - y(s) \right|\right)
\Bigl[ \left(\Pi_{s,x(s)}[\gamma] - v(s) \right) - \left(\Pi_{s,y(s)}[\xi] - w(s) \right) \Bigr].
\end{aligned}
\)
The first line in \eqref{integrand:MF} is estimated from above by
\[
&& \Bigl| \psi\left( \left|\Gamma_{s,x(s)}[\gamma] - x(s) \right|\right) - \psi\left( \left|\Gamma_{s,y(s)}[\xi] - y(s) \right|\right) \Bigr|
\left|\Pi_{s,x(s)}[\gamma] - v(s) \right| \\
&\leq&
2{\mathfrak{s}} L_\psi \Bigl| \left|\Gamma_{s,x(s)}[\gamma] - x(s) \right| - \left|\Gamma_{s,y(s)}[\xi] - y(s) \right| \Bigr| \\
&\leq&
2{\mathfrak{s}} L_\psi \Bigl( \left|\Gamma_{s,x(s)}[\gamma] - \Gamma_{s,y(s)}[\xi] \right| + \left|x(s) - y(s) \right| \Bigr) \\
&\leq&
2{\mathfrak{s}} L_\psi \left[ \frac{{\mathfrak{c}}}{{\mathfrak{c}}-{\mathfrak{s}}} \Bigl( \Norm{ \gamma - \xi }_{L^\infty(0,s)} + {\mathfrak{s}}{\mathfrak{c}}^{-1} |x(s)-y(s)| \Bigr) + \left|x(s) - y(s) \right| \right] \\
&=&
\frac{2{\mathfrak{s}}{\mathfrak{c}} L_\psi}{{\mathfrak{c}}-{\mathfrak{s}}} \left[ \Norm{ \gamma - \xi }_{L^\infty(0,s)} + \left|x(s) - y(s) \right| \right],
\]
where we used the Lipschitz continuity of the influence function $\psi$
with constant $L_\psi$ and the estimate $\left|\Pi_{s,x(s)}[\gamma] - v(s) \right| \leq 2{\mathfrak{s}}$ for the first inequality,
and estimate \eqref{GammaGamma} of Lemma \ref{lem:gamma-gamma} for the last inequality.
The second line in \eqref{integrand:MF} is estimated by
\[
&& \psi\left( \left|\Gamma_{s,y(s)}[\xi] - y(s) \right|\right)
\Bigl| \left(\Pi_{s,x(s)}[\gamma] - v(s) \right) - \left(\Pi_{s,y(s)}[\xi] - w(s) \right) \Bigr| \\
&\leq&
|v(s)-w(s)| + \Norm{ \dot\gamma - \dot\xi }_{L^\infty(0,s)} +
\frac{2{\mathfrak{s}}}{{\mathfrak{c}}-{\mathfrak{s}}} \Bigl( \Norm{ \gamma - \xi }_{L^\infty(0,s)} + |x(s)-y(s)| \Bigr),
\]
where we used the universal bound $\psi\leq 1$ and the estimate \eqref{PiPi} of Lemma \ref{lem:gamma-gamma}.
Combining the above estimates, we arrive at
\[
|v(t)-w(t)| &\leq& \int_0^t \iint_{\Omega^s_{\mathfrak{s}} \times \Omega^s_{\mathfrak{s}}}
\left[ \frac{2{\mathfrak{s}}}{{\mathfrak{c}}-{\mathfrak{s}}}\left( {\mathfrak{c}} L_\psi + 1\right) \Norm{ \gamma - \xi }_{L^\infty(0,s)} + \Norm{\dot\gamma - \dot\xi }_{L^\infty(0,s)} \right]
\mathrm{d}\pi(\gamma,\xi) \mathrm{d} s + \\
&& \qquad\qquad\qquad +\, \int_0^t |v(s)-w(s)| + \frac{2{\mathfrak{s}}}{{\mathfrak{c}}-{\mathfrak{s}}}\left( {\mathfrak{c}} L_\psi + 1 \right) |x(s)-y(s)| \mathrm{d} s.
\]
Recalling that $x(0)=y(0)$, we have
\[
\int_0^t |x(s)-y(s)| \mathrm{d} s \leq \int_0^t \int_0^s |v(\sigma)-w(\sigma)| \mathrm{d}\sigma \mathrm{d} s \leq t \int_0^t |v(s)-w(s)| \mathrm{d} s,
\]
which gives
\[
|v(t)-w(t)| &\leq& \max\left\{ 1, \frac{2{\mathfrak{s}}}{{\mathfrak{c}}-{\mathfrak{s}}}\left( {\mathfrak{c}} L_\psi + 1\right) \right\} \int_0^t
\iint_{\Omega^s_{\mathfrak{s}} \times \Omega^s_{\mathfrak{s}}} \Norm{\gamma-\xi}_{\Omega_{\mathfrak{s}}^s} \mathrm{d}\pi(\gamma,\xi) \mathrm{d} s \\
&&\qquad\qquad\qquad +
\left( \frac{2{\mathfrak{s}} t}{{\mathfrak{c}}-{\mathfrak{s}}}\left( {\mathfrak{c}} L_\psi + 1 \right) + 1 \right) \int_0^t |v(s)-w(s)| \mathrm{d} s.
\]
Since $\pi$ is an optimal transference plan, we have $\iint_{\Omega^s_{\mathfrak{s}} \times \Omega^s_{\mathfrak{s}}} \Norm{\gamma-\xi}_{\Omega_{\mathfrak{s}}^s} \mathrm{d}\pi(\gamma,\xi) = \mathcal{W}_s(\rho,\nu)$,
so that
\[
|v(t)-w(t)| \leq \max\left\{ 1, \frac{2{\mathfrak{s}}}{{\mathfrak{c}}-{\mathfrak{s}}}\left( {\mathfrak{c}} L_\psi + 1\right) \right\} \int_0^t \mathcal{W}_s(\rho,\nu) \mathrm{d} s
+ \left( \frac{2{\mathfrak{s}} t}{{\mathfrak{c}}-{\mathfrak{s}}}\left( {\mathfrak{c}} L_\psi + 1 \right) + 1 \right) \int_0^t |v(s)-w(s)| \mathrm{d} s.
\]
An application of the Gronwall lemma gives then
\[
|v(t)-w(t)| \leq \bar r(t) \int_0^t \mathcal{W}_s(\rho,\nu) \mathrm{d} s
\]
with
\( \label{barr}
\bar r(t) := \max\left\{ 1, \frac{2{\mathfrak{s}}}{{\mathfrak{c}}-{\mathfrak{s}}}\left( {\mathfrak{c}} L_\psi + 1\right) \right\} \exp\left(\frac{2{\mathfrak{s}} t}{{\mathfrak{c}}-{\mathfrak{s}}}\left( {\mathfrak{c}} L_\psi + 1 \right) + 1 \right).
\)
We finally conclude that
\(
\Norm{(Z[\rho](\gamma^0) - Z[\nu](\gamma^0)}_{\Omega_{\mathfrak{s}}^T} &\leq& (1+T) \Norm{v-w}_{L^\infty(0,T)} \nonumber \\
&\leq& (1+T) \bar r(T) \int_0^T \mathcal{W}_s(\rho,\nu) \mathrm{d} s \label{ZZZZ} \\
&\leq& (1+T) T \bar r(T) \mathcal{W}_T(\rho,\nu), \nonumber
\)
where for the last inequality we used the fact that, by definition, $\mathcal{W}_s(\rho,\nu)$
is a nondecreasing function of $s\in[0,T]$.
Consequently, \eqref{Zgronwall} is verified with $r(T):= (1+T) T \bar r(T)$,
and \eqref{lim-r} holds since $\bar r(T)$ is bounded as $T\to 0$.
\end{proof}
\subsection{Existence and uniqueness of solutions}
For a fixed initial datum $\rho^0\in {\mathcal{P}(\Omega_\s^0})$ we define the set ${\mathcal{P}_\s^T[\rho^0]} \subset {\mathcal{P}(\Omega_\s^T})$,
\( \label{def:POsTI}
{\mathcal{P}_\s^T[\rho^0]} := \left\{ \rho\in {\mathcal{P}(\Omega_\s^T});\; \mathbb{I}\#\rho = \rho^0 \right\}.
\)
The solution $\rho=\mbox{law}(x)$ of the mean-field problem \eqref{eq:MF1}--\eqref{eq:MF2}
shall be constructed as the unique fixed point of the mapping
$\mathcal{Z}[\rho^0]: {\mathcal{P}_\s^T[\rho^0]} \to {\mathcal{P}_\s^T[\rho^0]}$,
\( \label{def:ZZ}
\mathcal{Z}[\rho^0](\rho) := Z[\rho]\#\rho^0,
\)
with the mapping $Z: \Omega^0_{\mathfrak{s}} \to \Omega^T_{\mathfrak{s}}$ defined in \eqref{def:Z}.
\begin{lemma}\label{lem:ZZ}
Fix $\rho^0\in {\mathcal{P}(\Omega_\s^0})$.
For small enough $T>0$, the mapping $\mathcal{Z}[\rho^0]$ given by \eqref{def:ZZ} is a contraction on ${\mathcal{P}_\s^T[\rho^0]}$ in the topology
induced by the Monge-Kantorowich-Rubinstein distance $\mathcal{W}_T$ defined in \eqref{def:MKR}.
\end{lemma}
\begin{proof}
We apply the Banach contraction theorem on the complete metric space $({\mathcal{P}_\s^T[\rho^0]}, \mathcal{W}_T)$.
Clearly, by Lemma \ref{lem:Z}, $\mathcal{Z}[\rho^0]$ maps ${\mathcal{P}_\s^T[\rho^0]}$ into itself.
From Lemma \ref{lem:Zgronwall} we have
\[
\mathcal{W}_T(\mathcal{Z}[\rho^0](\rho), \mathcal{Z}[\rho^0](\nu)) = \mathcal{W}_T(Z[\rho]\#\rho^0, Z[\nu]\#\rho^0)
\leq r(T) \mathcal{W}_T(\rho, \nu),
\]
where $r=r(T)$ is a continuous function with $\lim_{T\to 0} r(T) = 0$.
Consequently, $\mathcal{Z}[\rho^0]$ is a contraction on ${\mathcal{P}_\s^T[\rho^0]}$ for small enough $T>0$.
\end{proof}
The solution $\rho=\mathcal{Z}[\rho^0](\rho)$ constructed in Lemma \ref{lem:ZZ}
for short times $T>0$ can be extended
as long as the characteristics given by \eqref{eq:MF1}--\eqref{eq:MF2}
remain ${\mathfrak{s}}$-Lipschitz continuous with $2{\mathfrak{s}}$-Lipschitz continuous derivative.
This property is guaranteed by Lemma \ref{lem:Z} for any $T>0$.
Consequently, the solutions are global in time, which concludes the proof of Theorem \ref{thm:MF}.
\subsection{Stability}
\begin{theorem}\label{thm:stability}
Fix $T>0$. Then there exists a constant $M_T>0$ such that
for any $\rho^0$, $\nu^0 \in {\mathcal{P}(\Omega_\s^0})$,
\( \label{stability}
\mathcal{W}_T(\rho,\nu) \leq M_T \mathcal{W}_0(\rho^0, \nu^0).
\)
where $\rho=Z[\rho]\#\rho^0$ and $\nu=Z[\nu]\#\nu^0$ are the unique fixed points
of the mapping $\mathcal{Z}$ provided by Lemma \ref{lem:ZZ}.
\end{theorem}
\begin{proof}
Using the triangle inequality, we have for any $t\in (0,T)$,
\[
\mathcal{W}_t(\rho, \nu) &=& \mathcal{W}_t(Z[\rho]\#\rho^0, Z(\nu)\#\nu^0) \\
&\leq&
\mathcal{W}_t(Z[\rho]\#\rho^0, Z(\nu)\#\rho^0) + \mathcal{W}_t(Z[\nu]\#\rho^0, Z(\nu)\#\nu^0).
\]
The first term of the right-hand side is estimated using Lemma \ref{lem:Wdist} and \eqref{ZZZZ},
\[
\mathcal{W}_t(Z[\rho]\#\rho^0, Z[\nu]\#\rho^0)
&\leq& \sup_{\gamma^0\in \Omega_{\mathfrak{s}}^0} \Norm{(Z[\rho](\gamma^0) - Z[\nu](\gamma^0)}_{\Omega_{\mathfrak{s}}^t} \\
&\leq& (1+t) \bar r(t) \int_0^t \mathcal{W}_s(\rho,\nu) \mathrm{d} s
\]
with $\bar r(t)$ defined in \eqref{barr}.
For the second term we use the estimate \eqref{LipschZZ} of Lemma \ref{lem:LipschZZ}.
We thus arrive at
\[
\mathcal{W}_t(\rho, \nu) \leq (1+t) \bar r(t) \int_0^t \mathcal{W}_s(\rho,\nu) \mathrm{d} s + L_Z^t \mathcal{W}_0(\rho^0, \nu^0),
\]
where $L_Z^t$ is the constant provided by Lemma \ref{lem:LipschZ}.
We conclude by an application of the Gronwall lemma for $t\in [0,T]$, noting that both $(1+t)\bar r(t)$ and $L_Z^t$
are bounded on compact intervals.
\end{proof}
\subsection{Remark about Fokker-Planck description}\label{subsec:FP}
A natural question to ask is whether one can express the mean-field limit of the system
\eqref{eq:tau}--\eqref{eq:CS2} in terms of a Fokker-Planck equation
for some time dependent phase-space particle density $g_t \in \mathcal{P}(\mathbb{R}^d\times\mathbb{R}^d)$, $t\geq 0$.
An obvious candidate for such particle density is the push-forward measure $g_t := (X_t,V_t)\#\rho$,
where the maps $X(t)$ and $V(t)$ were defined in \eqref{XtVt},
and $\rho\in{\mathcal{P}(\Omega_\s^T})$ is a solution of the mean-field problem constructed in Theorem \ref{thm:MF}.
Then, taking any test function $\varphi=\varphi(x,v)$ and applying formally
the change-of-coordinates formula for the push-forward $g_t := (X_t,V_t)\#\rho$, we have
\[
\tot{}{t} \iint_{\mathbb{R}^d\times\mathbb{R}^d} \varphi(x,v) \, \mathrm{d} g_t(x,v) &=& \tot{}{t} \int_{\Omega_{\mathfrak{s}}^T} \varphi(X_t[\gamma], V_t[\gamma]) \, \mathrm{d}\rho(\gamma) \\
&=& \int_{\Omega_{\mathfrak{s}}^T} \nabla_x\varphi\cdot \dot X_t[\gamma] + \nabla_v\varphi\cdot \dot V_t[\gamma] \,\mathrm{d}\rho(\gamma) \\
&=& \int_{\Omega_{\mathfrak{s}}^T} \nabla_x\varphi\cdot V_t[\gamma] + \nabla_v\varphi\cdot F_t[\rho](X_t[\gamma], V_t[\gamma]) \, \mathrm{d}\rho(\gamma),
\]
where we used the characteristic system \eqref{eq:MF1}--\eqref{eq:MF2} in the last equality.
Now, if we were able to express $F_t[\rho]$ in terms of $g$ as $G_t[g]$,
we could reverse the change-of-coordinates formula to obtain
\[
\int_{\Omega_{\mathfrak{s}}^T} \nabla_x\varphi\cdot V_t[\gamma] + \nabla_v\varphi\cdot F_t[\rho](X_t[\gamma], V_t[\gamma]) \, \mathrm{d}\rho(\gamma)
= \iint_{\mathbb{R}^d\times\mathbb{R}^d} \nabla_x\varphi\cdot v + \nabla_v\varphi\cdot G_t[g](x,v) \, \mathrm{d} g_t(x,v).
\]
Then we could formally describe the mean-field limit by the Fokker-Planck equation
\( \label{eq:FP}
\partial_t g_t + v\cdot\nabla_x g_t + \nabla_v\cdot (G_t[g] g_t) = 0.
\)
Note that with \eqref{def:F} we have
\[
F_t[\rho](x,v) &=&
\int_{\Omega_{\mathfrak{s}}^T} \psi\left( \left|\Gamma_{t,x}[\gamma] - x \right|\right) \left(\Pi_{t,x}[\gamma] - v \right) \mathrm{d}\rho(\gamma) \\
&=&
\int_{\Omega_{\mathfrak{s}}^T} \psi\left( \left|\gamma(t-\tau_{t,x}[\gamma])- x \right|\right) \left(\dot \gamma(t-\tau_{t,x}[\gamma]) - v \right) \mathrm{d}\rho(\gamma) \\
&=&
\int_{\Omega_{\mathfrak{s}}^T} \psi\left( \left|X_{t-\tau_{t,x}[\gamma]}[\gamma]- x \right|\right) \left(V_{t-\tau_{t,x}[\gamma]}[\gamma] - v \right) \mathrm{d}\rho(\gamma),
\]
where $\tau_{t,x}[\gamma]:=\tau$ is the unique solution of \eqref{eq:tau_gamma}, i.e.,
\[
{\mathfrak{c}}\tau = | x - X_{t-\tau}[\gamma]|.
\]
However, the change-of-variables formula corresponding
to the push-forward $g_t = (X_t,V_t)\#\rho$ cannot be applied here since the evaluation times
in $X_{t-\tau_{t,x}[\gamma]}$ and $V_{t-\tau_{t,x}[\gamma]}$
depend on $\gamma$. Consequently, the operator $F_t[\rho]$ does not seem
to admit an equivalent form in terms of the push-forward measure $g_t = (X_t,V_t)\#\rho$.
In particular, one may be tempted to believe that the mean-field limit of \eqref{eq:tau}--\eqref{eq:CS2}
should be given by \eqref{eq:FP} with the operator $G_t[f]$ taking the form
\[
G_t[g](x,v) = \iint_{\mathbb{R}^d\times\mathbb{R}^d} \psi(|y-x|) (w-v) \, g(t-{\mathfrak{c}}^{-1}|x-y|, y, w) \, \mathrm{d} y \,\mathrm{d} w.
\]
However, apart from the obvious difficulties with giving a meaning to the expression $g(t-{\mathfrak{c}}^{-1}|x-y|, y, w) \, \mathrm{d} y \,\mathrm{d} w$
for measure-valued $g$, the above argument indicates that such an intuitive expectation is wrong.
We therefore conclude that the mean-field limit does not admit a description in terms of the (classical) Fokker-Planck equation \eqref{eq:FP},
and one has indeed to resort to the formulation with probability measures on the space of time-dependent trajectories,
as we did in this paper.
\section*{Acknowledgment}
The author acknowledges the support of the KAUST baseline funds.
He also acknowledges the fruitful discussions with Oliver Tse that have taken place during his visit of TU Eindhoven,
and with Jan Vyb\'\i ral during his visit of Czech Technical University in Prague,
which helped to initiate and develop some ideas presented in this paper.
| {
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{"url":"https:\/\/homework.cpm.org\/category\/CC\/textbook\/cc2\/chapter\/9\/lesson\/9.2.1\/problem\/9-58","text":"### Home > CC2 > Chapter 9 > Lesson 9.2.1 > Problem9-58\n\n9-58.\n1. Ellen is building a scale model of the space shuttle. A space shuttle is approximately 122 feet long and has a wingspan of 78 feet. Homework Help \u270e\n\n1. How many inches long is the space shuttle?\n\n2. Ellen plans to build her model so that 1 cm on the model represents 10 inches on the space shuttle. Write a proportion to show this relationship for the length of her model. Find how long will her model be (in centimeters)?\n\n3. Write and solve a proportion to find her model\u2019s wingspan (in centimeters)?\n\n4. Remember that 1 inch is approximately equal to 2.54 cm. How many inches long will her model be?\n\n1 foot = 12 inches\n\n$\\frac{1\\text{ cm}}{10\\text{ inches}}$\n$\\text{Multiply your answer from part (b) by this ratio: }\\frac{2.54\\text{ cm}}{1\\text{ inch}}$","date":"2019-10-23 02:54:37","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 2, \"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\": 0.7239726185798645, \"perplexity\": 1797.6173286427252}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570987828425.99\/warc\/CC-MAIN-20191023015841-20191023043341-00452.warc.gz\"}"} | null | null |
Article by Roberta Capozucca – Communication Officer of Materahub
I If the future is uncertain and gloom, we can imagine a new one
II Magma Fest – Videos Playlist
If the future is uncertain and gloom, we can imagine a new one
Now it is time to really work together.
This is the key I chose to tell you what happened during the first edition of the Magma Fest and what it has been to me the real outcome of this fantastically dense two days.
The new gathering launched by Materahub, as a way to promote the dialogue between disciplines and to inspire new models of cooperation in South of Italy, made my brain tremble for the so unconventional position brought to the stage by the incredible guests we had the chance to host in Matera during the 1st and the 2nd of Decemeber.
Yes I call it gathering, you read it right, because more than a conference or a festival what we had the chance to experience was a safe space for exchanging opinion, an honest interest in listening to each others' opinion and the will to find a way to collaborate.
If this contemporary times threats us with urgent and complex challenges, we are all expected to do our part. With this background Magma had the ambition to create a thriving context for the design of new non-disciplinary models, in which each vision, competence and approach can contribuite to imagine unexpected solutions to face the urgent and complex challenges.
So what shall we prefer? Interdisciplinary, transdisciplinarity, multidisciplinarity or cross-disciplinarities? The point here is that it should not really matter, because realities is much more complex than the disciplines we invented, which in a capitalist educative dimensions leads to the prominence of the ego and of the self. Yet now we need to act as a whole.
Like the magma is a heterogeneous fluid that moves under our feet in constant need of stabilization, so do different disciplines that together need to seek and invent new models to collaborate for design a new model of development based on social, economic and environmental sustainability.
Going up to the surface, in this first edition of the Magma Fest we attempted to define understand how creativity and cross-fertilization change the way organizations create valuable relationships with the natural environment, the technological transformation, the political context and the local communities.
But it is not all. We also attempted do agree on the meaning of words which populates the interdisciplinary and innovative ecosystem we are trying to define. A process of contamination and negotiation, which gave life to our brand new Augmented Vocabulary.
Magma Fest – Videos Playlist
Newlsetter January 2023
Materahub newsletter - January 2023
Radio Deus
Radio Deus is a pop-up radio station airing online for one day, targeting the creative sector, professionals, students and policy makers.
Focus on Prime Minister, the Italian school of politics and civic activism for young women founded in 2019 in southern Italy. | {
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} | 2,453 |
Q: case statement query based on date I posted this query earlier. Posting it again with more details to help better understand my problem
Original Dataset
Name currency lcfeerate effectivestartdate
Institution1 USD 0.0029 7/9/2009
Institution1 CAD 0.0029 7/9/2009
Institution1 USD 0.0034 4/3/2017
Institution2 CAD 0.0029 7/9/2009
Institution2 USD 0.0029 7/9/2009
Institution3 CAD 0.0029 7/9/2009
Institution3 USD 0.0029 7/9/2009
Institution3 USD 0.0034 4/3/2017
Institution3 CAD 0.0034 4/3/2017
I need to run query such as to return one row each corresponding to each institution and corresponding currency. i.e. Institution1 will have 2 rows, 1 each for USD and CAD. Similarly Institutions 2 and 3 will have 2 rows each. Hence the final result is a table of 6 rows.
The rule to filter the table is that for each institution and currency, lcfeerate is chosen based on the effectivestartdate.
When the effectivestartdate lies between declared startdate and enddate then the feerate is chosen for that effectivestartdate. When there is no effectivestartdate between the declared startdate and enddate then it checks for the previous maximum effectivestartdate.
Here are two examples of the output required.
Example 1
Start date- 1/1/2017
End date- 3/31/2017
Name currency lcfeerate effectivestartdate
Institution1 USD 0.0029 7/9/2009
Institution1 CAD 0.0029 7/9/2009
Institution2 CAD 0.0029 7/9/2009
Institution2 USD 0.0029 7/9/2009
Institution3 CAD 0.0029 7/9/2009
Institution3 USD 0.0029 7/9/2009
Since there are no effective startdates between declared start date and end date, it chooses the next available effective startdate of 7/9/2009 and provides lcfeerate corresponding to those dates for each institution and currencies USD and CAD.
Example 2
Start date- 4/1/2017
End date- 5/31/2017
Name currency lcfeerate effectivestartdate
Institution1 CAD 0.0029 7/9/2009
Institution1 USD 0.0034 4/3/2017
Institution2 CAD 0.0029 7/9/2009
Institution2 USD 0.0029 7/9/2009
Institution3 USD 0.0034 4/3/2017
Institution3 CAD 0.0034 4/3/2017
In this case, since for Institution 3 effectivestartdate of 4/3/2017 lies between declared startdate and enddate it provides new lcfeerates for it. For institution1, USD rate has effectivestartdate between the declared dates so that one is provided and rest 3 rows do not have effectivestartdate between the declared dates hence the previouse effectivestartdates are chosen to provide lcfeerate.
It is very easy to sequence the table as per the effectivestartdate and group by name and currency to get the highest value but the condition I have I am not sure how to write this query.
The query I have tried is here:
declare @startdate as datetime = '1-Jan-2017';
declare @enddate as datetime = '31-Mar-2017';
select bankname, lcfeerate
,case when effectivestartdate between @startdate and @enddate then lcfeerate
when effectivestartdate not between @startdate and @enddate
then (select *
from (
select *, row_number()
over (partition by name, currency order by effectivestartdate desc) as seqnum
from table1
) t1
where seqnum = 1)end as lcfeerate
from table1
I get the following error:
Only one expression can be specified in the select list when the
subquery is not introduced with EXISTS.
A: You are pretty close, but you got the ranking wrong. First ranking criterion should be whether the date is in the date range, second is the date in descending order.
select name, currency, lcfeerate, effectivestartdate
from
(
select
name, currency, lcfeerate, effectivestartdate,
row_number() over (partition by name, currency order by
case when effectivestartdate between @startdate and @enddate the 1 else 2 end,
effectivestartdate desc) as rn
from table1
) ranked;
A: declare @startdate as datetime = '1-Jan-2017';
declare @enddate as datetime = '31-Mar-2017';
select bankname, lcfeerate
,case when effectivestartdate between @startdate and @enddate then lcfeerate
when effectivestartdate not between @startdate and @enddate
then (select top 1 lcfeerate
from (
select lcfeerate, row_number()
over (partition by name, currency order by effectivestartdate desc) as seqnum
from table1
) t1
where seqnum = 1)end as lcfeerate
from table1
I hope it is helpful for you.
| {
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{"url":"http:\/\/mathhelpforum.com\/math-topics\/121076-really-hard-fraction-question-help.html","text":"# Thread: really hard fraction question HELP\n\n1. ## really hard fraction question HELP\n\n$\\frac{8}{2 - \\sqrt{2}}$\n\nis this right, $\\frac{8}{2 - \\sqrt{2}} X \\frac{2 + \\sqrt{2}}{2 + \\sqrt{2}}$ = 16+ $8\\sqrt{2}$, as the numerator and 6 as the dinominator?\n\n2. Originally Posted by andyboy179\n$\\frac{8}{2 - \\sqrt{2}}$\n\nis this right, $\\frac{8}{2 - \\sqrt{2}} \\times \\frac{2 + \\sqrt{2}}{2 + \\sqrt{2}}$ = $\\frac{16}+ 3\\sqrt{2}}$\n$= \\frac {16 + 8\\sqrt{2}}{2^2 - (\\sqrt{2})^2} = . . .$\n\nmultiply it term by term and then simplify\n\n3. just did it but i can't work out how to write it so it comes up how i want it\n\n4. is what i put for the answer right? 16+, as the numerator and 6 as the denominator?\n\n5. Originally Posted by andyboy179\nis what i put for the answer right? 16+, as the numerator and 6 as the denominator?\nthe denominator is $2^2 - (\\sqrt{2})^2 = . . .$\n\n6. Originally Posted by dedust\nthe denominator is $2^2 - (\\sqrt{2})^2 = . . .$\n\nhow? could you please explain it?\n\n7. Originally Posted by andyboy179\nhow? could you please explain it?\n$\\frac{8}{2 - \\sqrt{2}} \\times \\frac{2 + \\sqrt{2}}{2 + \\sqrt{2}} = \\frac{8(2 + \\sqrt{2})}{(2 - \\sqrt{2})(2 + \\sqrt{2})}$\n$= \\frac{(16 + 8\\sqrt{2})}{(2^2 + 2\\sqrt{2} - 2\\sqrt{2} + \\sqrt{2}^2)} = \\frac{(16 + 8\\sqrt{2})}{(4-2)} = \\frac{(16 + 8\\sqrt{2})}{2} = 8 + 4\\sqrt{2}$\n\n8. i get told to do this at school for the denominator,\n\n9. Originally Posted by andyboy179\ni get told to do this at school for the denominator,\n\nthe last entry in the 2nd column should be $-2$, because\n$(-\\sqrt{2}) \\times (+\\sqrt{2}) = -{2}$\n\n10. thankyou so much","date":"2013-12-13 08:52:07","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 13, \"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\": 0.9623803496360779, \"perplexity\": 1592.8386917881737}, \"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-2013-48\/segments\/1386164920565\/warc\/CC-MAIN-20131204134840-00025-ip-10-33-133-15.ec2.internal.warc.gz\"}"} | null | null |
Q: New Firebase Version - Android - Facebook login won't work So here is what I'm stuck with for more than one day now.
I'm trying to implement the new Firebase specs to login users using Facebook into my Android project.
I really thank in advance anyone who might be able to help me figuring out what's wrong.
Error I get again and again
The activity is indeed in the manifest, I can launch it, but when I click on the login button, I get the following error (What's weird is that it mentions FacebookActivity and not FacebookLoginActivity, which is the name of my activity):
06-10 11:38:49.058 7210-7210/com.yatoo E/AndroidRuntime: FATAL EXCEPTION: main
Process: com.yatoo, PID: 7210
Log in attempt failed: FacebookActivity could not be started. Please make sure you added FacebookActivity to the AndroidManifest.
at com.facebook.login.LoginManager.startLogin(LoginManager.java:369)
at com.facebook.login.LoginManager.logInWithReadPermissions(LoginManager.java:263)
at com.facebook.login.widget.LoginButton$LoginClickListener.onClick(LoginButton.java:737)
at com.facebook.FacebookButtonBase$1.onClick(FacebookButtonBase.java:359)
at android.view.View.performClick(View.java:5697)
at android.widget.TextView.performClick(TextView.java:10814)
at android.view.View$PerformClick.run(View.java:22526)
at android.os.Handler.handleCallback(Handler.java:739)
at android.os.Handler.dispatchMessage(Handler.java:95)
at android.os.Looper.loop(Looper.java:158)
at android.app.ActivityThread.main(ActivityThread.java:7224)
at java.lang.reflect.Method.invoke(Native Method)
at com.android.internal.os.ZygoteInit$MethodAndArgsCaller.run(ZygoteInit.java:1230)
at com.android.internal.os.ZygoteInit.main(ZygoteInit.java:1120)
Double-checked, in the manifest
<meta-data
android:name="com.facebook.sdk.ApplicationId"
android:value="@string/facebook_app_id" />
<activity
android:name=".activity.FacebookLoginActivity" />
The FacebookLoginActivity code
Which is totally similar to the example providen by Google and that can be found here: Google Link to Facebook Login integration using the new Firebase version
public class FacebookLoginActivity extends AppCompatActivity implements
View.OnClickListener {
private static final String TAG = "FacebookLogin";
private TextView mStatusTextView;
private TextView mDetailTextView;
// [START declare_auth]
private FirebaseAuth mAuth;
// [END declare_auth]
// [START declare_auth_listener]
private FirebaseAuth.AuthStateListener mAuthListener;
// [END declare_auth_listener]
private CallbackManager mCallbackManager;
@Override
public void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
FacebookSdk.sdkInitialize(getApplicationContext());
setContentView(R.layout.activity_facebook_login);
// Views
mStatusTextView = (TextView) findViewById(R.id.status);
mDetailTextView = (TextView) findViewById(R.id.detail);
findViewById(R.id.button_facebook_signout).setOnClickListener(this);
// [START initialize_auth]
// Initialize Firebase Auth
mAuth = FirebaseAuth.getInstance();
// [END initialize_auth]
// [START auth_state_listener]
mAuthListener = new FirebaseAuth.AuthStateListener() {
@Override
public void onAuthStateChanged(@NonNull FirebaseAuth firebaseAuth) {
FirebaseUser user = firebaseAuth.getCurrentUser();
if (user != null) {
// User is signed in
Log.d(TAG, "onAuthStateChanged:signed_in:" + user.getUid());
} else {
// User is signed out
Log.d(TAG, "onAuthStateChanged:signed_out");
}
// [START_EXCLUDE]
updateUI(user);
// [END_EXCLUDE]
}
};
// [END auth_state_listener]
// [START initialize_fblogin]
// Initialize Facebook Login button
mCallbackManager = CallbackManager.Factory.create();
LoginButton loginButton = (LoginButton) findViewById(R.id.button_facebook_login);
loginButton.setReadPermissions("email", "public_profile");
loginButton.registerCallback(mCallbackManager, new FacebookCallback<LoginResult>() {
@Override
public void onSuccess(LoginResult loginResult) {
Log.d(TAG, "facebook:onSuccess:" + loginResult);
handleFacebookAccessToken(loginResult.getAccessToken());
}
@Override
public void onCancel() {
Log.d(TAG, "facebook:onCancel");
// [START_EXCLUDE]
updateUI(null);
// [END_EXCLUDE]
}
@Override
public void onError(FacebookException error) {
Log.d(TAG, "facebook:onError", error);
// [START_EXCLUDE]
updateUI(null);
// [END_EXCLUDE]
}
});
// [END initialize_fblogin]
}
// [START on_start_add_listener]
@Override
public void onStart() {
super.onStart();
mAuth.addAuthStateListener(mAuthListener);
}
// [END on_start_add_listener]
// [START on_stop_remove_listener]
@Override
public void onStop() {
super.onStop();
if (mAuthListener != null) {
mAuth.removeAuthStateListener(mAuthListener);
}
}
// [END on_stop_remove_listener]
@Override
protected void onActivityResult(int requestCode, int resultCode, Intent data) {
super.onActivityResult(requestCode, resultCode, data);
mCallbackManager.onActivityResult(requestCode, resultCode, data);
}
// [START auth_with_facebook]
private void handleFacebookAccessToken(AccessToken token) {
Log.d(TAG, "handleFacebookAccessToken:" + token);
// [START_EXCLUDE silent]
// [END_EXCLUDE]
AuthCredential credential = FacebookAuthProvider.getCredential(token.getToken());
mAuth.signInWithCredential(credential)
.addOnCompleteListener(this, new OnCompleteListener<AuthResult>() {
@Override
public void onComplete(@NonNull Task<AuthResult> task) {
Log.d(TAG, "signInWithCredential:onComplete:" + task.isSuccessful());
// If sign in fails, display a message to the user. If sign in succeeds
// the auth state listener will be notified and logic to handle the
// signed in user can be handled in the listener.
if (!task.isSuccessful()) {
Log.w(TAG, "signInWithCredential", task.getException());
Toast.makeText(FacebookLoginActivity.this, "Authentication failed.",
Toast.LENGTH_SHORT).show();
}
// [START_EXCLUDE]
// [END_EXCLUDE]
}
});
}
// [END auth_with_facebook]
public void signOut() {
mAuth.signOut();
LoginManager.getInstance().logOut();
updateUI(null);
}
private void updateUI(FirebaseUser user) {
if (user != null) {
mStatusTextView.setText(getString(R.string.facebook_status_fmt, user.getDisplayName()));
mDetailTextView.setText(getString(R.string.firebase_status_fmt, user.getUid()));
findViewById(R.id.button_facebook_login).setVisibility(View.GONE);
findViewById(R.id.button_facebook_signout).setVisibility(View.VISIBLE);
} else {
mStatusTextView.setText(R.string.signed_out);
mDetailTextView.setText(null);
findViewById(R.id.button_facebook_login).setVisibility(View.VISIBLE);
findViewById(R.id.button_facebook_signout).setVisibility(View.GONE);
}
}
@Override
public void onClick(View v) {
switch (v.getId()) {
case R.id.button_facebook_signout:
signOut();
break;
}
}
}
A: You should also define default FacebookActivity in your manifest like
<activity
android:name="com.facebook.FacebookActivity"
android:configChanges="keyboard|keyboardHidden|screenLayout|screenSize"
android:label="@string/app_name"/>
A: You may also want to go over the Facebook quickstart guide for Android here https://developers.facebook.com/quickstarts/?platform=android
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,060 |
<?php
namespace Rebond\Services;
class View
{
/**
* View list of linked items
* @param array $items
* @param array $allItems = []
* @return string
*/
public static function viewForeignKeyLink(array $items, array $allItems = [])
{
if (empty($items)) {
return '';
}
$html = '<div class="rb-form-item rb-check-list">';
if (empty($allItems)) {
foreach ($items as $item) {
$html .= '<label class="check">' . $item . '</label>';
}
} else {
foreach ($allItems as $item) {
$selected = (in_array($item, $items)) ? ' highlight' : '';
$html .= '<label class="check' . $selected . '">' . $item . '</label>';
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$html .= '</div>';
return $html;
}
/**
* View linked item
* @param string $item
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{
$html = '<div class="rb-form-item">';
$html .= '<div class="check">' . $item . '</div>';
$html .= '</div>';
return $html;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,601 |
Témoin indésirable () est un roman policier d'Agatha Christie publié le au Royaume-Uni. Il est publié en 1959 aux États-Unis et en France.
Considéré par la critique comme l'un de ses meilleurs ouvrages tardifs et qualifié par Agatha Christie elle-même comme l'une de ses deux œuvres préférées, avec La Maison biscornue (1949), il s'agit également de l'une de ses œuvres les plus noires, proposant une vision approfondie de la psychologie de l'innocence et de sa démonstration.
Résumé
Alors qu'il purge sa peine pour le meurtre de sa mère, meurtre qu'il nie en bloc, Jackie Argyle décède en prison d'une pneumonie. Deux ans après cet événement, le témoin qui aurait dû lui servir d'alibi apparaît soudainement et veut prouver son innocence. La famille du détenu doit alors faire face à la suspicion, une suspicion qui se pose sur chacun d'eux, puisque le vrai meurtrier est encore parmi eux.
Personnages
Personnages principaux
Arthur Calgary (alias Dr Calgary) : personnage principal et témoin
Jackie Argyle (alias Jacko) : fils adoptif de Leo et Rachel et accusé de meurtre, décédé
Rachel Argyle : (ex-)femme de Leo, décédée
Leo Argyle : père de Jackie et veuf de Rachel Argyle
Marry Durrant (alias Polly) : fille adoptive de Leo et Rachel
Philip Durrant : mari de Marry Durrant, décédé
Hester Argyle : fille adoptive de Leo et Rachel
Tina Argyle : fille adoptive de Leo et Rachel
Micky Argyle : fils adoptif de Leo et Rachel
Kirsten Lindstorm (alias Kirsty) : gouvernante et infirmière
Gwenda Vaughan : secrétaire de Leo Argyle
Personnages secondaires
Maureen Clegg : veuve de Jackie Argyle
Huish : superintendant de la police
Donald Craig (alias Dr Craig ou Don) : médecin et fiancé d'Hester
M. Marshall : avocat de la famille Argyle
M. Clegg : époux actuel de Maureen
Dédicace
Le livre est dédicacé à Billy Collins : . C'est lui qui convainquit Christie de quitter Bodley Head, l'éditeur de ses six premiers livres, pour obtenir un contrat plus intéressant chez William Collins, Sons en 1926.
Éditions
Témoin indésirable ( Jean-Marc Mendel), dans :
Adaptations
Au cinéma
1984 : Témoin indésirable (), film britannique de Desmond Davis
À la télévision
2004 : Témoin indésirable (), téléfilm de la série britannique Miss Marple d'ITV (épisode 3.02), avec l'ajout du personnage de Miss Marple joué par Geraldine McEwan
2009 : Am Stram Gram, téléfilm de la série française Les Petits Meurtres d'Agatha Christie de France 2, avec l'ajout du duo d'enquêteurs Larosière-Lampion joués par Antoine Duléry et Marius Colucci
2018 : Témoin indésirable (), mini-série britannique de Sandra Goldbacher diffusé en 4 épisodes sur Canal+ (3 épisodes sur la BBC)
En bande dessinée
2006 : Témoin indésirable, bande dessinée française de la collection Agatha Christie de Chandre (scénario et dessin)
À la radio
2014 : , feuilleton radiophonique de BBC Radio 4.
Notes et références
Voir aussi
Lien externe
Roman britannique paru en 1958
Roman policier d'Agatha Christie
1958 en littérature policière
Roman britannique adapté au cinéma
Roman britannique adapté à la télévision
Roman britannique adapté en bande dessinée
Roman britannique adapté à la radio
Ouvrage publié dans la collection Le Masque | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,990 |
Thank you for downloading this Howard Books eBook.
* * *
Join our mailing list and get updates on new releases, deals, bonus content and other great books from Howard Books and Simon & Schuster.
CLICK HERE TO SIGN UP
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## CONTENTS
Epigraph
Foreword by Beth Clark
Introducing the Duck Commander Women
Kay Robertson
Lisa Robertson
Melissa "Missy" Louise West Robertson
Korie Robertson
Jessica Robertson
Part One
* * *
HOW I BECAME A ROBERTSON
1: Introduction: A Message from the Wives
2: The Cheerleader and the Quarterback: Miss Kay
3: It Started with a Crush: Lisa
4: The Best Joke I Ever Played: Missy
5: Young Love: Korie
6: Fancy Meeting You Here: Jessica
Part Two
* * *
NO ORDINARY IN-LAWS
7: Introduction: A Message from the Daughters-in-Law
8: Oh, Kay!
9: Getting Our Phil
10: What a Team!
11: Passing It On
Part Three
* * *
HAPPILY EVER AFTER CAN TAKE A WHILE
12: Introduction: Miss Kay
13: I Found Out What It Means to Fight: Miss Kay
14: Have Hope: Lisa
Part Four
* * *
TALKIN' ABOUT MY GENERATIONS
15: Introduction: A Message from the Wives
16: Ain't Life Grand?: Miss Kay
17: A Legacy of Love: Jessica
18: Just Call Me Mam: Lisa
19: A Heritage of Faith: Korie
20: Generational Blessings: Missy
Part Five
* * *
SOMETIMES MIRACLES HIDE
21: Introduction: Jessica
22: Tough Love: Miss Kay
23: A Miracle Named Mia: Missy
Part Six
* * *
LIFE IN THE LIMELIGHT
24: Introduction: A Message from the Wives
25: Living in a Glass House: Lisa
26: It's Not Just for Us: Missy
27: We've Come a Long Way, Baby!: Miss Kay
28: We're Happy No Matter What: Jessica
29: Great Rewards: Korie
Part Seven
* * *
INQUIRING MINDS WANT TO KNOW
30: Introduction: Korie
31: Answers to the Twenty Questions People Ask Us Most
Photographs
About Kay Robertson, Korie Robertson, Missy Robertson, Jessica Robertson and Lisa Robertson
Dedication
We would like to dedicate this book to the main men in our lives: Phil, Alan, Jase, Willie, and Jep. You love us, support us, and keep us laughing. We love you, respect you, and feel blessed to be your soul mates. Life with you is never boring.
We also give thanks and honor to our Lord Jesus Christ for giving us this opportunity to share our faith with the world. Without His sacrifice on the cross we would not have the hope of eternal life.
The simple story of Jesus.
Love, Miss Kay
Television is what we do; faith is who we are.
—The Robertson wives
## FOREWORD
The distinct sound of a duck call pierced the quiet of the commuter jet cabin just before we landed in Monroe, Louisiana. At least I know I'm in the right place, I thought as I joined my fellow passengers in applause. I don't know how many people's travels that day would have anything to do with Duck Dynasty or the Robertson family, but everyone on the flight seemed excited to have finally arrived in the lush green bayou country the Robertsons call home.
As I made my way toward West Monroe to meet the Robertson women for the first time, I had some of the same questions much of America has about them: Are they really who they seem to be—down-to-earth, sincere, commonsense, loving wives and mothers? Do the women really get along and support each other? Do they really love their family as much as the show portrays? Are their children truly as respectful as they come across? And is their sense of humor what the show would have an audience think? In a word, yes. Definitely yes.
Within a few minutes of sitting down for a meal with these women, I began to see that they are the real deal—warm, friendly, gracious, humble, fun, passionate about God, devoted to family, and altogether genuine. There's no pretense in the Robertson wives. Each one is strong, smart, talented, and enormously capable in her own right, and each keeps the family fame in perspective. While they appreciate the chance to be on a television show that breaks ratings records and makes history, they're not impressed with themselves; they are focused on the things that really matter. They are committed to timeless values, and they are wholly true to a strong set of unshakable convictions, convictions based on the ancient, proven truth of Scripture. They have learned the hard way that a life of faith is not always easy, but it is always good.
In this book, the Robertson wives are happy to share the stories, insights, and experiences that have made them who they are. With transparency, humility, and forthrightness, they write about the things that have shaped their lives not as television stars, but as human beings—as women who know the same longings and cares, joys and sorrows, victories and struggles as women everywhere. If you have ever wondered why they are the way they are, you'll find answers in these pages. If you want to move toward a greater commitment to faith, family, and the things that make life truly rich, you will find direction here.
This behind-the-scenes look into the Robertson women's world is not just made of words; it's made of influence. Its stories and lessons have the power to shape your life in the most positive ways and the potential to lead you and others into the kind of good life and values-based living that is the foundation of the Robertsons' success.
The Robertson wives are not clamoring for the spotlight but cooperating with the spotlight, because they understand that for a moment in television history they are part of something that calls millions of people back to the basic goodness of life—and they share that goodness with humor and self-effacing honesty. They don't pay too much attention to what the world says is important. Their must-haves are not designer clothes or luxury cars. Their must-haves are love, loyalty, kindness, forgiveness, generosity, respect, honest hard work, care for others, and a host of other virtues. The unprecedented popularity of Duck Dynasty makes me wonder if, at the end of the day, these simple qualities are what the heart of America really longs for.
The whole Duck Dynasty phenomenon is a bit of a revolution. It's certainly turned the world of entertainment on its ear and challenged what many media insiders think America wants to watch. They're finding out that clean humor, marital fidelity, and prayer before a meal can trump sex, violence, and bad language. The funny thing is, the revolution of Duck Dynasty is not leading us into anything new; it's simply taking us back to some things our culture is in danger of losing—love, laughter, family, and faith. In the Robertson women these things run deeper than any once-a-week television show could ever convey; they are the very fabric of their lives.
Beth Clark
Nashville, Tennessee
## Introducing
## THE DUCK COMMANDER WOMEN
### Kay Robertson
(OTHERWISE KNOWN AS "MISS KAY")
BORN AND RAISED IN: Born in Vivian, Louisiana; raised in Ida, Louisiana
HUSBAND'S NAME: Phil Alexander Robertson
CHILDREN'S NAMES AND BIRTH YEARS:
Alan (born 1965)
Jason (we call him Jase; born 1969)
Willie (born 1972)
Jeptha (we call him Jep; born 1978)
HUSBAND'S BEST QUALITY: His passionate faith and charismatic personality
FAVORITE DUCK DISH: Duck wraps, though I like pretty much everything if it's made of duck
LEAST-FAVORITE FOOD: I haven't yet met a food I didn't like
TALENT NOT VERY MANY PEOPLE KNOW ABOUT: I can draw and paint
FAVORITE COLOR: Blue
FAVORITE VACATION DESTINATION: Gulf Shores, Alabama
FAVORITE SCRIPTURE: Philippians 4:13, NKJV: "I can do all things through Christ who strengthens me."
FAVORITE SONG: "Amazing Grace," the old version and the new one
THE BEST THING ABOUT BEING A ROBERTSON IS: We have a heritage of faith
THREE THINGS I'M THANKFUL FOR:
1. Family: my husband, children, grandchildren, and great-grandchildren
2. My grandmother, who shared her faith with me, and my sister, Ann, who is one of my best friends
3. Living in America
MORNING PERSON OR NIGHT PERSON? Night person
ONE THING THAT REALLY GETS ON MY NERVES: Disrespectful children and adults
ONE THING ON MY BUCKET LIST: To have a great seat at a New Orleans Saints football game and to visit with the team
ONE PERSON I REALLY ADMIRE AND WHY: My son Alan. He did not get to have a childhood or do many of the things little boys often get to enjoy doing. He never resented the problems and struggles he had to deal with for ten years during a difficult time in our family's life.
HOW I FEEL ABOUT HOUSEWORK: I'm not fond of it. I feel blessed to be able to have a housekeeper. Here's a quick story to make my point. A little girl and her mother came to see me one day. When the girl looked around my house, she said, "Miss Kay, somebody done come here and messed up your how-se!" Her mother was so embarrassed. I wasn't. I just smiled and said, "Look, we live in this house. We do the best we can, and we have a housekeeper." I guess the housekeeper had not been there that day. I absolutely loved that child's honesty.
TOTAL NUMBER OF DAYS SPENT IN A DUCK BLIND: One
BEST ADVICE ANYONE EVER GAVE ME: My grandmother told me, "You're going to have to fight for your marriage."
WORST ADVICE ANYONE EVER GAVE ME: "Leave Phil" (when times were tough)
MY GRANDMOTHER ALWAYS SAID: "Pretty is as pretty does."
I WOULD NEVER LEAVE THE HOUSE WITHOUT: My sense of humor and my American Express card
MY FAVORITE THING TO DO DURING DUCK SEASON IS: Sleep in and go shopping
IF I COULD DO ONE THING TO CHANGE THE WORLD TODAY, I WOULD: Love all the people who have no one to love them and feed as many as I could
WHEN I WAS LITTLE, I WANTED TO GROW UP AND BE: A somewhat modern pioneer woman who became a great wife and mother, had a lot of animals to love, and was the best cook around. I always wanted to have a smile on my face and to be pleasant at all times.
### Lisa Robertson
BORN AND RAISED IN: West Monroe, Louisiana
HUSBAND'S NAME: Marshal Alan Robertson
CHILDREN'S NAMES AND BIRTH YEARS:
Anna (born 1986)
Alex (born 1987)
GRANDCHILDREN: Carley Elizabeth Stone (born 2005), Bailey Kay Stone (born 2007), Corban Marshal Mancuso, to be named after Alan (due March 2014), and another on the way
HUSBAND'S BEST QUALITY: His forgiving heart
FAVORITE DUCK DISH: Duck and dressing
LEAST-FAVORITE FOOD: Crawfish. It's a long story.
TALENT NOT VERY MANY PEOPLE KNOW ABOUT: My Mexican corn bread is better than Miss Kay's (and even Miss Kay agrees)!
FAVORITE COLOR: Purple
FAVORITE VACATION DESTINATION: Anywhere on the beach
FAVORITE SCRIPTURE: Proverbs 24:26, NIV 2011: "An honest answer is like a kiss on the lips."
FAVORITE SONG: "Someone Like You" by Adele
THE BEST THING ABOUT BEING A ROBERTSON IS: Being married to Alan
THREE THINGS I'M THANKFUL FOR:
1. Salvation
2. Love
3. Family
MORNING PERSON OR NIGHT PERSON? I'm a midday person, or a "pretty much anytime" person
ONE THING THAT REALLY GETS ON MY NERVES: People who make fun of those who are disabled or mentally challenged
ONE THING ON MY BUCKET LIST: To live to see my great-great-grandchildren
ONE PERSON I REALLY ADMIRE AND WHY: Lynda Hammitt, a friend and coworker at Duck Commander. She has been to hell and back, and lived to witness about it. Her positive attitude is amazing!
HOW I FEEL ABOUT HOUSEWORK: I don't mind it. I love a clean house.
TOTAL NUMBER OF DAYS SPENT IN A DUCK BLIND: One
BEST ADVICE ANYONE EVER GAVE ME: "Forgive and move on."
WORST ADVICE ANYONE EVER GAVE ME: "Use cloth diapers."
MY GRANDMOTHER ALWAYS SAID: "Eat what's on your plate. There are starving kids who wish they could have what you leave."
I WOULD NEVER LEAVE THE HOUSE WITHOUT: Lipstick
MY FAVORITE THING TO DO DURING DUCK SEASON IS: Sleep in
IF I COULD DO ONE THING TO CHANGE THE WORLD TODAY, I WOULD: Tell everyone about Jesus and His love, grace, and salvation
WHEN I WAS LITTLE, I WANTED TO GROW UP AND BE: A nurse. I wanted to be a nurse until I became a mom—and then I knew I couldn't handle it.
### Melissa "Missy" Louise West Robertson
BORN AND RAISED IN: Lubbock, Texas, but moved to West Monroe, Louisiana, when I was six months old
HUSBAND'S NAME: Jason Silas "Jase" Robertson
CHILDREN'S NAMES AND BIRTH YEARS:
Reed Silas (born 1995)
Cole Foster (born 1997)
Mia Elaine (born 2003)
HUSBAND'S BEST QUALITY: Forgiveness
FAVORITE DUCK DISH: Duck wraps
LEAST-FAVORITE FOOD: Bologna sandwiches. I ate them every day for my school lunch when I was a kid. I can hardly even stomach the smell of them today.
TALENT NOT VERY MANY PEOPLE KNOW ABOUT: Not many people know I sing, have been trained to sing, and have sung my entire life.
FAVORITE COLOR: Green
FAVORITE VACATION DESTINATION: So far? Hawaii, but I plan on exploring this more in the future.
FAVORITE SCRIPTURE: Philippians 2:14: "Do everything without complaining or arguing." I don't always live by it, but I try. I want to be a shining star for Him!
FAVORITE SONG: I love music. I always have. I love eighties pop. I love today's country. I love Christian music. I love big orchestral, classical pieces. I even like some opera. My life has always been full of music.
THE BEST THING ABOUT BEING A ROBERTSON IS: The many people who love and accept me for who I am, faults and all
THREE THINGS I'M THANKFUL FOR:
1. Jesus' sacrifice for me and His consistent love and forgiveness
2. My husband's dedication to me and our family
3. The blessing of family
MORNING PERSON OR NIGHT PERSON? Neither. I require a lot of sleep.
ONE THING THAT REALLY GETS ON MY NERVES: Mumblers. It doesn't matter what people say if no one can understand them!
ONE THING ON MY BUCKET LIST: I really want to visit the Holy Land and see where Jesus walked on the earth
ONE PERSON I REALLY ADMIRE AND WHY: I greatly admire my sister-in-law Lisa. She lived a life of bad choices for a long time, but when she decided to change her life, she truly repented. She is a godly, caring, loving woman, and I am proud to call her my sister.
HOW I FEEL ABOUT HOUSEWORK: I love a clean house! I just hate to clean it. In the past, the first thing I'd splurge on when Jase and I had a little extra money was hiring a housecleaner. I now have it worked into the budget.
TOTAL NUMBER OF DAYS SPENT IN A DUCK BLIND: In twenty-five years with Jase? A grand total of three days. The first time was when Jase and I dated, just to see what he was so obsessed with. The second time was for our first television show on Outdoor Channel. I killed two ducks on that hunt and winged another one. The third time I'll write about later in the book.
BEST ADVICE ANYONE EVER GAVE ME: "The best thing you can do for your kids is to love their daddy."
WORST ADVICE ANYONE EVER GAVE ME: "Go ahead, take a bite. It tastes just like chicken."
MY MOTHER ALWAYS SAID: "If your friends jumped off the Empire State Building, would you?"
I WOULD NEVER LEAVE THE HOUSE WITHOUT: My cell phone
MY FAVORITE THING TO DO DURING DUCK SEASON IS: Eat duck wraps!
WHEN I WAS LITTLE, I WANTED TO GROW UP AND BE: A successful, single woman living in New York City. Then I met Jase.
### Korie Robertson
BORN AND RAISED IN: West Monroe, Louisiana
HUSBAND'S NAME: Willie Jess Robertson
CHILDREN'S NAMES AND BIRTH YEARS:
John Luke (born 1995)
Sadie (born 1997)
Will (born 2001)
Bella (born 2002)
Rebecca (born 1988; came to live with us in 2005)
HUSBAND'S BEST QUALITY: Besides always being able to make me laugh, Willie is a great leader for our family and to others. Even as a kid, he always loved to take the underdogs and lead them to victory. And most important is his spiritual leadership in our home. He loves God with all his heart, and it shows in how he lives. Plus he's a great cook! Okay, maybe those are his three best qualities.
FAVORITE DUCK DISH: Duck wraps: duck, jalapeño pepper, and cream cheese wrapped in bacon
LEAST-FAVORITE FOOD: Coconut. I don't like the texture.
TALENT NOT VERY MANY PEOPLE KNOW ABOUT: I can do a backflip on the trampoline and off the diving board.
FAVORITE COLOR: Green
FAVORITE VACATION DESTINATION: Snow skiing, anywhere
FAVORITE SCRIPTURE: Micah 6:8: "He has showed you, O man, what is good. And what does the Lord require of you? To act justly and to love mercy and to walk humbly with your God."
FAVORITE SONG: "Boondocks" by Little Big Town. Sounds a lot like the Robertsons!
THE BEST THING ABOUT BEING A ROBERTSON IS: It's never boring! We laugh a lot. But most important, everyone, and I mean everyone, puts God first—and when you do that it makes your relationships so much better.
THREE THINGS I'M THANKFUL FOR:
1. My family
2. My home
3. Getting to travel
MORNING PERSON OR NIGHT PERSON? Definitely night
ONE THING THAT REALLY GETS ON MY NERVES: Mosquitoes
ONE THING ON MY BUCKET LIST: A visit to Africa
ONE PERSON I REALLY ADMIRE AND WHY: My mom. She is hands-down the most unselfish person I know. She is an awesome example of a godly wife, mom, and friend. She's strong but kind, and she's busy but always has enough time for you. She puts her family first but has big goals and dreams of her own. She has never let me down. Plus, I wish I had her energy. She never stops!
HOW I FEEL ABOUT HOUSEWORK: I'm a disaster at housework. I like to organize, clean out closets, etc., but nothing ever stays that way. It will look great for about two days and then it's right back where we started. We are not good at cleaning up as we go along in our house. I like to enjoy life, then when it gets to a certain level, we have what I call "Family Cleanup Time." I feel like if I'm cleaning, someone else should be too! Plus, it gets done a lot faster if everyone pitches in.
TOTAL NUMBER OF DAYS SPENT IN A DUCK BLIND: One
BEST ADVICE ANYONE EVER GAVE ME: I heard or read somewhere: "A woman can do everything she wants in life, just not all at once." I was a stay-at-home mom when our kids were little and I loved that; wouldn't change it for the world. Now I'm a working mom and I love that too. It's all about balance and patience. You've got a lifetime to reach your goals; don't rush it and put pressure on yourself. You'll just make yourself and everyone around you stressed.
WORST ADVICE ANYONE EVER GAVE ME: To get that short haircut in the eighties. I have a big head and skinny legs. I looked like a Q-tip.
MY MOTHER ALWAYS SAID: "If you can't say something nice, don't say anything at all." This was usually said in reference to how I talked to my siblings or how they talked to me, but the Bible says one should be quick to listen, slow to speak, and slow to become angry, so I think she had something there.
I WOULD NEVER LEAVE THE HOUSE WITHOUT: Lip gloss. I feel naked without something on my lips.
MY FAVORITE THING TO DO DURING DUCK SEASON IS: Go back to sleep after Willie crawls out of bed at four A.M. on those cold, rainy mornings. I've also been known to take a few shopping trips to New York City with the girls during this time!
IF I COULD DO ONE THING TO CHANGE THE WORLD TODAY, I WOULD: Give all the orphans homes with a loving family.
WHEN I WAS LITTLE, I WANTED TO GROW UP AND BE: A stockbroker (not sure why), an art teacher, and a model—but mostly a mom.
### Jessica Robertson
BORN AND RAISED IN: Born in Bossier, Louisiana; raised in West Monroe, Louisiana
HUSBAND: Jules Jeptha (Jep) Robertson
CHILDREN'S NAMES AND BIRTH YEARS:
Lillian (Lily) Mae (born 2002)
Merritt Decatur (born 2004)
Priscilla June (born 2006)
River Alexander (born 2008)
HUSBAND'S BEST QUALITY: Jep has such a big heart, a heart after God
FAVORITE DUCK DISH: Duck wraps
LEAST-FAVORITE FOOD: Barbecue
TALENT NOT MANY PEOPLE KNOW ABOUT: I can sew really well
FAVORITE COLOR: Red
FAVORITE VACATION ACTIVITY: Snow skiing
FAVORITE SCRIPTURE: 3 John 4: "I have no greater joy than to hear that my children are walking in the truth."
FAVORITE SONG: "(Sitting On) The Dock of the Bay" by Otis Redding
THE BEST THING ABOUT BEING A ROBERTSON IS: The abundant love and all the encouragement we share
THREE THINGS I'M THANKFUL FOR:
1. The death, burial, and resurrection of Jesus Christ
2. My husband and my children
3. Family, friends, and other church family
MORNING PERSON OR NIGHT PERSON: Although I grew up a morning person (I was usually in bed by nine P.M.), Jep turned me into a night person. My junior and senior years, I took seven A.M. classes (that's how much of a morning person I was). Now I can't seem to get up before seven.
ONE THING THAT REALLY GETS ON MY NERVES: Smacking!!
ONE THING ON MY BUCKET LIST: A vacation in Italy
ONE PERSON I REALLY ADMIRE AND WHY: My mamaw Nellie Fincher. I have learned so much from her about commitment, love, Jesus, selflessness, and true joy. She loves Jesus Christ more than anything or anyone else. She is the most influential woman in my life.
HOW I FEEL ABOUT HOUSEWORK: I really don't like housework; I hate it! I am a good cleaner; I just hate to do it. I like organization and no clutter.
TOTAL NUMBER OF DAYS SPENT IN A DUCK BLIND: Two, but more to come. I really like hunting. I grew up going with my dad, but it's hard to go when you have four babies all two years apart.
BEST ADVICE ANYONE EVER GAVE ME: "Keep God first."
WORST ADVICE ANYONE EVER GAVE ME: "God would never want you to be unhappy."
MY MOTHER ALWAYS SAID: "Be nice to everyone, no matter where they come from, how much money they have, or the color of their skin."
I WOULD NEVER LEAVE THE HOUSE WITHOUT: My lipstick or ChapStick
MY FAVORITE THING TO DO DURING DUCK SEASON IS: Plan for Christmas
IF I COULD DO ONE THING TO CHANGE THE WORLD TODAY, I WOULD: Protect children from abusive situations
WHEN I WAS LITTLE, I WANTED TO GROW UP AND BE: An attorney. I now know I would have made a terrible lawyer.
# Part One
# HOW I BECAME A ROBERTSON
"Wherever you go, I will go; wherever you live, I will live. Your people will be my people, and your God will be my God. Wherever you die, I will die, and there I will be buried. May the Lord punish me severely if I allow anything but death to separate us!"
RUTH 1:16–17, NLT
## 1
## INTRODUCTION
## A Message from the Wives
All of us wives agree that being a Robertson is a privilege and a joy. We are a happy family; we love God, and we love each other. We help and support one another, and we each have a passion to see the others succeed. We work together, play together, pray together, and laugh together.
When people see us around the table at the end of each Duck Dynasty episode, they get a good glimpse into who we are because they can tell we are people of faith. But they do not get a complete picture of what makes our family work, and that is one reason we wrote this book.
We all came into the Robertson family the same way: we fell in love with a Robertson man and believed we could build a good life with him, a life based on faith and family. Of course, we also knew we might spend a good bit of time apart from him during duck season! We all value different aspects of the family and appreciate being part of this remarkable group for different reasons, but we all agree that our family is amazing and wonderful.
### Miss Kay: I LOVED THEM BEFORE I KNEW THEM
For as long as I can remember, all I really wanted to do with my life was be the best wife and mother I could possibly be. As my boys began to grow up, I realized I also wanted to be the best mother-in-law I could be. Long before the boys met the women they eventually married, I began to pray for those girls. I prayed for them for years! I asked the Lord to give the boys godly wives who would love Him first and love them second. I don't believe any of my daughters-in-law ended up in the family by happenstance; God sent each one and each one is perfect for the Robertson man she married.
I believe the two most important decisions and vows we make are: first, to make Jesus Christ the Lord of our lives, and second, to choose godly mates and make lifelong commitments to them. I always knew the wives my sons chose would determine how their faith would grow and develop, and I prayed they would marry women who would help and encourage them spiritually. I am happy and thankful to say God has answered those prayers.
Before my boys started dating, I made up my mind that I would love their wives. I never was one of those women who thought no young lady could be good enough for my son, and I never wanted to be a mother-in-law who competed with a daughter-in-law. I always had a heart to be kind and loving to whomever my sons chose, to be supportive of them and to embrace each one like my own daughter.
Lisa, Missy, Korie, and Jessica all know they can come to me for anything. If they have a fuss with their husbands, they know I will not automatically side with my boys. I will judge a situation according to what is right and wrong, based on what God says in His Word. I am not the kind of person who defends bad behavior just because one of my sons does it. If someone does something wrong, even if it's one of the boys, I will call it what it is!
I am glad to have such close relationships with my daughters-in-law and to be part of a great family of faith with them. Each one has a different personality and different ways of doing things. But each one is special to me and I love them all dearly!
### Jessica: LET'S GET ONE THING STRAIGHT
Somehow, because of Duck Dynasty, people often brand us Robertson wives as "gold diggers." That might be offensive to us if we were thin-skinned or if it were true. But it is not. In fact, it's so not true that we always get a good laugh when we hear the latest rumor about how we married our husbands because of their fortunes. People know Miss Kay didn't marry Phil for his money, and they don't necessarily see Lisa as a gold digger because they know Alan worked for years as a pastor before joining Duck Commander. But Korie, Missy, and me—people seem to think we plotted and schemed to capture wealthy, long-haired hunters, determined to marry them for their riches. Let me set the record straight.
When Missy married Jase, hardly anyone outside the hunting world had ever heard of him. Missy worked full-time in an administrative role at a local medical clinic to help support the two of them. She never dreamed he would one day become duck call royalty!
Korie and Willie married while they were in college and they laugh now about having to go to a friend's house to do their laundry. They were on such a tight budget that their favorite mealtime splurge was chicken strips and macaroni and cheese out of a box. Willie has always been an industrious person and a hard worker. All of his life, he has been happy to pick up odd jobs for extra money, but in the early years of his and Korie's marriage, they were both in college and money was tight. When their children were young, Korie worked in a paid position as children's minister at our church. She also used her skills as a fine artist to paint detailed pictures on duck calls and sold them at hunting shows.
When I married Jep and for several years afterward, I had numerous jobs, including making hand-sewn heirloom dresses and smocked children's clothes to sell to boutiques, working as a Realtor, and being a sales representative for a clothing company. Though Jep has always been the head of our home and the leader of our family, I have always tried to contribute financially as best I could. When we first got married, several years before Duck Dynasty started, we lived in a little trailer and could barely make ends meet. We just wanted to be able to pay our bills, buy groceries, and put gas in our cars.
I can say with complete confidence that each one of us wives married her husband because she loved and respected him; we knew they were men we could honor and trust. All of us value and appreciate these men because of who they are on the inside, not because they are now TV stars. We all married them when they had very little, and we were happy. We struggled financially, just as many other couples do—and we were happy. Now God has blessed us with more resources than we have ever had, and we're still happy. But I can assure you that all of us would rather have strong, solid marriages with godly men than have all the riches this world could offer. And you can take that to the bank!
## 2
## THE CHEERLEADER AND THE QUARTERBACK
## Miss Kay
I cannot remember a time in my life when I did not want to grow up, get married, and raise a family. I can remember a couple of times when I wanted to be an airline stewardess or a teacher, but those did not last very long. Being a wife and mother was always my dream, and it was never far from my mind. As I mentioned earlier, I not only wanted to be a wife and mother, I wanted to be the best wife and mother on earth. I believed that if I could do that, I would have a happy family and a great life.
As a child, the best role models I had were my grandparents. I loved my mom and dad, too, but my mother did some strange things I later found out were due to alcohol use, and my father, whom I adored, died when I was only fourteen years old. Before he passed away, my parents ran the local store in our community, a business that was in our family for seventy-five years. Taking care of the store took most of their time, so I spent a lot of time either at the store visiting with customers or with my grandparents.
### A POWERFUL INFLUENCE
My grandmother, whom I called Nannie, loved to tell me stories from the old days and had a powerful influence on me. She married at the age of fifteen. She went to school for a short time before she had to quit but learned a lot about life because she survived some very hard times. She was also a woman of wisdom, because she was a woman of great faith, and faith and wisdom usually go together. She was a great cook and spent a lot of time working in the kitchen, like many women of her generation—before microwaves and the other gadgets that make cooking easy these days. She had a big, full garden and taught me all about gardening. At her house I had the best time picking vegetables, shelling peas, and shucking corn with her in preparation for a meal. She seemed to be cooking all the time and taught me to cook at a young age. One of my very favorite ways to spend time with Nannie was to stay right beside her and cook with her. We developed a very strong bond in the kitchen together, whether I was helping her cook her daily meals for the workers at my parents' store or helping her prepare smaller meals for our family. I definitely got my now-famous kitchen abilities from my grandmother, and Phil says he will always love her for that!
One of my favorite things to do when I was young was to spend my evenings after supper sitting and talking with Nannie in the big swing that sat in her yard. We spent hours and hours just swaying back and forth, talking about all kinds of things and waving to every car that drove by. I have wonderful memories of those times.
I also remember watching Nannie and my grandfather together. In one room of their house were two maple rocking chairs with a gas heater between them, and they sat in those chairs every day as Nannie read the Bible to my grandfather. In their bedroom were two double beds with a nightstand between them. When I spent the night with my grandparents, I slept with Nannie while my grandfather slept in the other bed, which I guess he did every night. Even though they did not sleep in the same bed, they both reached out and held hands across the nightstand, and I always thought that was so sweet. When we woke up the next morning, Nannie never failed to take my grandfather a cup of coffee, very early. That daily act of kindness touched me and I never forgot it. My grandmother had a true servant's heart, and she deeply loved her husband.
Because my grandmother was such an important part of my life and such a good example to me, her words carried a lot of weight. We never talked about anything intimate when it came to boys or men because people simply did not do that in her day, but she always told me that marriage is "one man, one wife—for life." I believed that as a child, and I still believe it today. I knew from a very young age that when I met the man I wanted to marry, I would stick with him!
### A PERFECT MATCH
Phil says he and I have always been a perfect match. We started going together when I was in the ninth grade and he was in the tenth grade. He was the star quarterback of our high school football team, and I was a cheerleader. I was so happy to be dating him once we got together! We took a little break when hunting season started just after Christmas that year because Phil wanted to spend his free time hunting and decided he did not have time for a girlfriend. But then my dad died suddenly and Phil came to his funeral. I think he was trying to send the message that he cared about me—and it worked. I was glad he showed up, even though we did not have a chance to talk.
My dad had been my rock and my protector. I always felt safe with him and wanted to marry someone with qualities similar to his. I knew Phil was the same type of strong, protective person my dad was, so in many ways Phil became like a replacement for my dad. He gave me back some of what I lost when my dad passed away. In addition, he was everything I wanted in a man. I had always wanted someone outdoorsy and someone who was strong and courageous. That was Phil!
A few weeks after my dad died, Phil asked me out and, of course, I went. We have basically been together ever since. My mother also started dating not too long after my dad passed away, so within several months of his death, she and I were both dating at the same time. That was not always good. I needed her help and guidance because I had not been in a relationship with a guy before, but she was not available. Plus, she was not in favor of my dating Phil because his family did not have much money. That was a lot more important to her than it was to me. During that time, I spent less time at home with my mother and more time at the Robertsons' house, surrounded by a happy, loving family.
As Phil and I grew closer, we faced the same temptations most young people deal with when they are in love. I did not know how to handle those feelings. My mother was not available to talk to and I felt I could not mention something like sex to my grandmother. I had always been a "good" girl. I was not the slightest bit adventurous as a child. I obeyed and I behaved. As a teenager I had to figure out a lot of things for myself. I was on my own when it came to knowing what to do about a physical relationship with Phil.
I always believed people should only sleep with one person, and I knew I wanted to marry Phil someday, so he would be that person for me. We did have sex before we got married and I did get pregnant. I know that these days, with Duck Dynasty on TV, some people think Phil and I are way too open about our sex life. In fact, some people in our own family are still shocked by the way we talk about it! I just have two things to say about that. First, we're married, and married people have sex. Second, I wish someone had been open with me and talked about sex when I was dating Phil. I desperately needed somebody to teach me what it was all about and help me understand how to deal with my young, intense feelings for him.
### A WHOLE NEW LIFE
When Phil graduated high school, he received a football scholarship to Louisiana Tech University. I was pregnant with Alan, so of course I went to Ruston, Louisiana, with Phil and we began our lives together in a little student-housing apartment. During our first year together I finished high school and had a baby, while Phil impressed coaches, players, and fans with his talents on the football field and went hunting in his spare time.
That was just the beginning of my life as part of the Robertson family. In a later chapter, I will write about some of the challenges Phil and I faced and about how we got through them. As of this writing, Phil and I have now been married nearly fifty years. I'm still in love with him, and I cannot imagine being part of any other family.
## 3
## IT STARTED WITH A CRUSH
## Lisa
In 1977, when I was in sixth grade at Pinecrest Elementary School in my hometown of West Monroe, Louisiana, I saw the cutest boy I had ever laid eyes on. He was new to our school, and I quickly found out his name was Alan Robertson. I was popular in school and people seemed to like me, but no matter how I tried, that cute boy did not seem to know I was alive. Maybe that's because he was in eighth grade and did not have time for younger girls like me. That did not stop me from following him around school, though—during every recess, fire drill, and class change. Sometimes when I speak publicly about this now, I say I could have been on fire and he would not have noticed. At least that's what I thought; he says he was vaguely aware that he had a sixth-grade stalker with braces.
By the end of that school year, Alan had become a favorite among his peers. He was even elected "Mr. Pinecrest," which he says is the only title he has ever really held. Because our school only went to eighth grade, I knew Alan would not be at Pinecrest the following year. Our paths did not cross again until he was the cool senior at our local high school and I was "kid" sophomore.
### NOT WHAT I EXPECTED
I was excited to see Alan again when I got to high school but soon realized he was not the same "nice boy" I remembered. He was spending time with one of my cousins, drinking and smoking pot on a regular basis. Many of the teenagers in our community often hung out at our local McDonald's, and Alan happened to notice me there one night. He thought I was attractive (actually, he says he thought I was a babe), so right then and there he asked me out for the following weekend.
I had been dreaming of a date with Alan Robertson since sixth grade. When I finally went out with him for the first time, it was horrible! He did not pick me up; he asked me to meet him. When he showed up, he had two of my cousins with him. Who wants to go on a date with their cousins? Our evening consisted of nothing more than cruising around in a car while my cousins got drunk and high. At the end of the date, even Alan was completely passed out. He did not turn out to be the gentleman I was hoping for, at least not that night.
People might think I would drop a guy who acted that way like a hot rock, but I did not. I kept seeing him and soon began smoking and drinking with him. I really cared about him and was willing to do anything to please him. When I say "anything," I mean anything.
Since Alan was such a looker, girls were always interested in him. I decided to make sure none of those girls got their meat hooks into the man I had been dreaming of. (I have a lot more redneck in me than my sisters-in-law. They are far more like "yuppie girls," as Phil would say.) On one occasion, one of Alan's former girlfriends tried to make her move on him and I had to show her how a country girl hangs on to her man! We had a good "cussin' catfight," and even though I was pretty scrawny back then, I held my own and neutralized the threat. I think Alan was impressed, and we laugh about it to this day. I learned way back then that someone or something you really care about is worth fighting for.
### WHAT? YOU'RE LEAVING?
With Alan, I really thought I was in dating heaven. I was finally in a relationship with the man I had loved since sixth grade. I knew the way we were living was not the way love was supposed to go, but the fact that we were intimate convinced me we could be "in love" forever, no matter what happened—and things were definitely happening. Alan's behavior had deteriorated to the point that Phil and Miss Kay kicked him out of their house. He decided to go live with his aunt, Phil's sister, in New Orleans, a city that probably did not provide the atmosphere Phil and Miss Kay were hoping for. I hated to see him leave but truly believed our "love" could stand the longdistance test. After all, I loved him and had given myself to him. Surely that meant as much to him as it meant to me.
Alan now says he had no intention of continuing his relationship with me once he got to New Orleans. He also freely admits he did not have the character or integrity to inform me of that decision. All I knew was that I loved him, I missed him, and with every passing week, I was having a harder time reaching him by phone.
One weekend I finally decided to go with the Robertsons to visit Alan. He knew we were coming, so I was shocked when we arrived and discovered he was out on a date with someone else. That was his way of breaking up with me. I was devastated! I cried the whole way home from New Orleans to West Monroe and for a week after that.
### ALAN COMES HOME
My broken heart sent me into a complete free fall. For the next year and a half I dedicated my life to finding the love I had lost with Alan. My life was a complete disaster. Alan's life in New Orleans was also a mess until he finally had an epiphany on the wrong end of a jealous husband's crowbar (Phil tells that story in detail in his book, Happy, Happy, Happy). After that incident, Alan came home to West Monroe and called me, and we went out again, this time under much better circumstances.
We knew that resisting the temptation to be physically intimate would be a problem while we were dating, so after we had dated about six months, Alan asked me to marry him. I was so happy! Of course, I said yes. He did not believe in long engagements so we were married the following Friday. This was not met with enthusiasm on my parents' part, but Alan and I were determined to begin our lives together immediately.
When we first married, we moved in with Phil's parents, Granny and Pa, who lived in a small house next door to Phil and Kay's larger house—not like typical next-door neighbors, but in a small house on Phil and Miss Kay's property, just down the hill from them. After about six months, Alan and I were able to buy a little camp house beside Miss Kay and Phil. I enjoyed being around them, and around Granny and Pa. I learned so much in those early years from these godly people—including how to cook, from none other than the kitchen commander herself, Miss Kay! Alan was long-suffering with me as I learned the ways of his mother's cooking, but in the end it paid off. I can cook most of what Miss Kay does and Alan will go so far as to say that some of my cooking is better than Miss Kay's, if you want to believe that.
My journey with Alan, and Alan's journey with me, has not been easy. It has not even been pretty at times. But we have learned to love God and to love each other, in that order. Being a Robertson has been a blessing to me for many years. I will always thank God for the chance to be Alan's wife and to be part of such a loving, supportive, forgiving family.
## 4
## THE BEST JOKE I EVER PLAYED
## Missy
I was sixteen and Jason (known on TV as Jase) was eighteen when we started dating. One of my friends—we'll call her Christy—was actually interested in him, and the two of them had started seeing each other. Jase did not know Christy was already dating someone else and had been for quite some time. He found this out at her house one Sunday afternoon when she ran down the stairs telling him he had to leave immediately. About that time, he heard the screeching of tires from the front of her house. Her boyfriend had arrived. The boyfriend (we'll call him Greg) was obviously not happy with the current arrangement and was there to set things straight with Jason. He told Jason he wanted to talk inside his truck. Jase ended up getting into Greg's vehicle, which he quickly regretted, and Greg proceeded to drive to an undisclosed location to fight it out. Quickly, Jase realized the situation and told Greg that if all of this was over Christy, he could have her. She was not worth it to him.
Since Greg did not seem to respond to this direction in the conversation, Jase switched gears and started preaching to him. He proceeded to tell Greg that Jesus died for him and for all the rotten things he had done in his life. He told him God would forgive him if he would turn his life over to Jesus, be baptized for his sins, and start living a life that reflected Jesus' love for him.
Since Greg did not seem to respond to this dialogue either, Jase told him simply, "Just don't hit me in the face." Greg stopped the truck, dragged Jase out, roughed him up a bit, and left him at the end of a dead-end road. Jason never threw one punch. Obviously, the relationship between Jason and Christy was officially over.
### WE THOUGHT IT WAS JUST FOR FUN
News of what happened between Jason and Greg quickly hit our church youth group as well as Christy's and my friends at school. Jason and I started talking about the situation and, as a ruse, decided to go on a mock date with each other in order to make Christy jealous. (Remember, we were sixteen and eighteen at the time—not the most mature individuals.) I took him as my date to a school function, even though he had already graduated from another area school, to make sure Christy saw us together. We made sure she witnessed the fun we were having talking and laughing and flirting. I soon realized we really were talking, laughing, and flirting with each other, and I did not want the night to end. At some point during the evening, I think we both forgot about Christy. We actually have not thought much about her since.
I must share one more aspect of this story before I continue with what happened with Jase and me. A week after our date, during Sunday morning church services, someone tapped Jason on the shoulder and told him to look in the back of the auditorium. He turned around and saw Greg standing there. He thought, Oh no, round two! After services concluded, Greg approached Jase and told him that he actually listened when Jase was telling him about Jesus' love for him. He asked Jase to tell him again about Jesus and why He had died for him. He apologized to Jason for the one-sided fight and asked for his forgiveness. After hearing the message of Jesus again from the youth director, Greg was baptized that night. Today, Greg is a pilot with his own plane and has recently flown Jase and me to events where Jase shares that same message of Christ with many audiences.
### THINGS GET SERIOUS
Now back to my story about how Jase and I ended up together. We started dating in the fall of 1987. Our first kiss was October 8, 1987. At first I was drawn to Jase because of his good looks, of course, but I soon noticed how self-confident he was. Not cocky, but self-confident; there is a big difference. He was not like any other boys I had dated, who tried to look and act cool in front of their friends. You know, those boys who would call and talk to you on the phone for hours one evening and then act like they did not know you in front of their friends the next day at school. I liked the fact that he was not that way.
My friends were not convinced Jason was right for me. I mean, he was a redneck from West Monroe, for goodness' sakes! I was a student at a private school in Monroe. Some of my friends were very well-off in terms of money and material possessions. They drove nice cars, lived in manicured neighborhoods, and, well, they didn't shoot their dinner!
Jason's family had nothing in common with my friends' families. The Robertsons had multiple cars, but he had to take a gamble on which one would work when he needed to drive. They had a basketball hoop nailed to a tree in their front yard, so there was absolutely no grass. Since they did not have a concrete driveway, they just pulled the vehicles up "in the yard," as they said. Their house was a two-bedroom, one-bath camp house with a laundry room addition and a metal roof off the back. It was twenty miles outside of town, in the middle of nowhere. And it was the busiest home I had ever seen in my life. The smells that came from that tiny kitchen were to die for! Miss Kay's corn bread and banana pudding were some of the first things I remember eating in that house.
The family welcomed me just like they welcomed every other soul who made the drive up "in the yard." They treated everybody the same, no matter how much money they had (or did not have), no matter what color skin they had, no matter who they voted for for president, and no matter what god they served. Everybody was welcome, and everybody was going to hear about Jesus from Phil. That family was like a magnet. They attracted any and all, and people found it very hard to tear themselves away.
Jason was true to who he was and what he believed. I was hooked immediately. The more we dated, the more I realized he could help me get to heaven. He was funny, confident, smart, driven, and super cute. We dated for a year and two months before he asked me to marry him. It went something like this.
It was Christmas of 1988, and we were exchanging our Christmas gifts alone at his grandparents' house, next to the family home. I gave him his gifts: an LSU sweatshirt and cap. And he gave me his gift: a small potted plant. When I gave him a look that said, "Are you kidding me?" he grinned and told me to dig in the dirt. Reluctantly, I dug into the little clay pot until I hit something. I uncovered a little felt box, pulled it out, and opened it to find a beautiful engagement ring! I looked at Jase with eyes wide, and he said words I will never forget: "Well, you're gonna marry me, aren't ya?" Obviously, I said yes. And that's my engagement story.
### ONLY GOD COULD LOVE HIM MORE
Jason and I were married August 10, 1990, in our home church, the same church we still attend together with all our family. Our wedding was completely perfect. My parents did not have much money (my dad is a preacher, and my mom was a teacher at our private Christian school). My dress was the exact duplicate of a beautiful gown I had spotted in our local bridal store, thanks to my mom's best friend, who was a wonderful seamstress. It had all the details of the designer gown: beads, lace, tulle, and more. It also had big puffy sleeves—it was the end of the eighties era, you know. I sang a song entitled "Only God Could Love You More," which I prerecorded and played as I walked down the aisle. That song title was exactly how I felt about Jase, and I knew he felt the same way about me.
Oh, remember Christy? Her dad was our preacher at the time, and he officiated the wedding. Needless to say, we were all back to being friends. All of Jase's brothers were in our wedding, as well as Lisa, Alan's wife. I was the second woman Jase's family brought into the Robertson fold. I was nineteen.
## 5
## YOUNG LOVE
## Korie
The first time I met Phil, he asked a friend and me, "Have you girls met my sons Jason Silas and Willie Jess?"
We nervously answered, "Yes, sir."
Then he went on to say, "They'll make good husbands someday."
Keep in mind, my friend and I were only in the fifth grade when this conversation happened. He went on to tell us that his sons would be good providers because they were good hunters and fishermen. This did not really matter to me at the time; I just thought Willie was cute.
Now, I have to tell you, Phil is a little intimidating when you first meet him, so this was all a little embarrassing and shocking to me. Before this meeting, I had only been around Willie one other time. Let me explain.
### WORTH WRITING ABOUT
As a child, one of the highlights of my year was going to summer camp. When I was in the third grade, my whole life changed at that camp, though I did not know it at the time. There, as an eight-year-old girl, I had my first crush—on the person who would eventually become my husband.
I fell for Willie immediately. In all my young life I had never seen such cute dimples and such a great smile. I was smitten. Like a lot of girls my age I had a diary, but I did not write in it much. But Willie was worth an entry, which read: "I met a boy at summer camp and he was so cute. He asked me on the moonlight hike, and I said yes!"
Each year at our camp, the girls wondered which young man would ask them to walk with him on the hike. There was only one boy I wanted to go with, and I was thrilled when he asked me. If a moonlight hike at summer camp counts as a date, then Willie and I had our first date that year and it was a success!
Even though Willie and I lived in the same fairly small city, we did not see each other for quite some time after camp that summer. We did not attend the same school, nor did our families go to the same church. The Robertsons went to a small country church, while my family worshipped at White's Ferry Road Church.
### I NEVER FORGOT HIM
I ended up at Phil and Miss Kay's house a few years later because I was friends with the daughter of the preacher at our church. Her name was Rachel. Her father had struck up a friendship with Phil and was trying to convince him to bring his family to White's Ferry Road. I was beyond excited when Rachel asked me to join her family at a fish fry at the Robertsons' house so the families could get to know each other and the Robertsons could learn more about the church. Even though I had not seen Willie in two years, I had not forgotten him.
When we arrived at the Robertsons', I was nervous, but I remember a lot about that night. Strangely, I can still picture exactly what I was wearing. It was the eighties, so of course my jeans were tight-rolled, and I wore a black waffle-weave shirt with a fluorescent-green rope belt and fluorescent-pink dangly earrings. I also remember thinking Willie was the funniest guy I had ever met. He worked hard to impress me by blowing on his thumb and blowing up his muscle, and showing me all his "Vulcan death grips." In addition to those things, he ate sardines straight out of a can. This was all pretty impressive to a ten-year-old!
Two things about that visit to Willie's house caught my attention. First was the fact that Phil told me what a good husband he would be as soon as I walked in the door. Second was that Phil and Miss Kay had a sign on their bedroom door that read HONEYMOON SUITE. Of course I did not say anything, but even then I was surprised at how blatant they were about their honeymooning. Now, because of Duck Dynasty, millions of people know that Phil and Kay freely discuss that aspect of their lives.
After that night, I did not see Willie again for another two years. Seeing the guy who had captured my heart once every two years was hard on my love-struck young self. Thankfully, the Robertsons finally joined our church when I was in seventh grade. All the girls were immediately interested in Jase and Willie, not only because they were new but also because they were good-looking and they were genuinely nice guys.
Willie finally asked me out on our first real date—if we don't count the moonlight hike—when I was in the eleventh grade. At that time, because he had not been nice to one of my friends a few years earlier, I didn't go out with him. She had given him money to buy her a soft drink on a church trip, and he bought baseball cards instead. Of all the nerve! Willie went on to date other people and so did I.
Not long after Christmas during my senior year in high school, in 1990, Willie and I saw each other at the mall. Without ever saying anything, we both seemed to know we would see each other again. Things had changed. I called him a couple of days later, knowing I needed to be the one to reach out to him since I was the one who had rejected him previously. I had to leave a message for him and was thrilled when he returned my call the next day. We went to lunch at Bonanza that day, and by the end of January 1991, we were definitely dating. And we were serious.
### I WASN'T GOING TO MISS THIS OPPORTUNITY
In the fall of 1991, I was preparing to go to college at Harding University in Searcy, Arkansas. I was hoping to convince Willie to join me, but he was attending seminary school at our church and was not the least bit interested in going to college or leaving West Monroe. He did not want me to go either, but I had wanted to go to Harding since I was a little girl. Both my parents had gone there, and I had an academic scholarship. This was a tough decision, but I decided it was an opportunity I did not want to pass up, so Willie and I broke up before I left for school.
A few weeks later, in September, Willie called me and said he wanted to get back together. I knew in my heart that I loved him and wanted to get back together, too, but I was not quite ready to tell him, so I said I would call him back the next day. When I did, I simply said, "Let's get back together." That was the last time we ever broke up. About a month later, we decided we were ready to get married.
My engagement to Willie did not go over well with my parents. They had nothing against Willie, but they had a lot against our getting married so young. I was barely eighteen! A huge argument between Willie and my parents, complete with shouting on both sides, took place at Alan and Lisa's house, where Willie was living at the time.
My parents are not yellers. I actually do not remember an argument like that with my parents before or since that night. I am glad I wasn't there to witness it, but I realize that tempers were high because everyone involved really loved me and wanted the best for me. They simply did not agree on what "the best" was.
The argument did not end well. Willie called to tell me just how badly it went. And then something surprising happened: My parents called me and said, "If you're determined to do this, we're going to support you." I was, they did, and they've never stopped.
My parents threw Willie and me a big, beautiful wedding on January 11, 1992. We had ice sculptures and white trees, which made the place look like a winter wonderland. Since Willie and I were both born and raised in the West Monroe area, lots of family and friends came to our wedding—about eight hundred of them. It was such a happy time. And it was enough of a big deal to Phil that he wore his dress-up clothes: corduroy pants and a button-down shirt!
### ADVENTUROUS BEGINNINGS
Willie did end up moving to Searcy, Arkansas, with me. In fact, we moved the day after we got married and settled into a tiny one-bedroom apartment. Not long after that, with my parents' help, we bought a very small house where we lived while going to school at Harding. We were on a super-tight budget, and some of our biggest fights happened at the grocery store when we had a few extra dollars and I wanted to buy a People magazine while Willie wanted to buy baseball cards. We certainly could not afford both.
One of the great opportunities Willie and I had during college was the chance to spend a summer in Florence, Italy. This was part of a study-abroad program, and we were so excited. We had never been away from home together until we took that trip. As I mentioned, we moved to Arkansas immediately after our wedding, so we did not take a honeymoon trip. My parents did take us to Hawaii the following summer, and we all had a wonderful time, but it was more of a family vacation than a romantic getaway.
Willie and I had all kinds of new and exciting experiences in Italy. We wanted to see Europe, but we did not have money for hotel rooms, so we traveled by train at night from one country to another. We boarded the train in one country, went to sleep, and woke up somewhere else. We had all kinds of adventures in Italy and in various other European countries. We were glad to get home when we returned to the States that fall, but we were also happy and grateful for all the fun we had and for everything we learned while we were away.
I was born into a wonderful, godly family, and I am so blessed to have also become a Robertson. Phil was exactly right when he told me many years ago that Willie would make a good husband. Willie's a great husband and father, and I am so glad I married him. I love being his wife, and I love being part of such a fun and loving family—but I still find myself surprised at times when Phil and Kay talk as openly as they do about their honeymooning!
## 6
## FANCY MEETING YOU HERE
## Jessica
I first met the man I married at a hair salon. I was going out the door; Jep was going in—for a haircut. Seriously. Nowadays, most of the Robertson men don't get haircuts, but Jep did back then. When our paths crossed that day, we said nothing more than "hi" to each other, just one word.
Jep and I both grew up in West Monroe, Louisiana, and he is two years older than I am. We went to different high schools, but because we lived in a close community, we had heard about each other. He knew who I was, and I knew who he was—and I thought he had a cool name. I had heard good things about him, including, "He's a dream." When our paths crossed at the hair salon and we simply said hello, I had no way of knowing the hairdresser would tell Jep all about me as she cut his hair that day. Both of us had gone to her for years, so she knew us pretty well, and she said really nice things about me to Jep. In fact, she takes credit for getting us together! After we were married I found out that when he left the hair salon that day, he went home and told his best friend, "I just met the girl I'm going to marry."
"What's her name?" his friend asked.
"Jessica," Jep responded. He only knew this because the hairdresser had told him.
"Jessica who?" his friend asked. "What's her last name?"
"I don't know," Jep admitted.
I love the fact that Jep knew he would marry me after only seeing me once. Maybe he did not know my last name, but the next time he saw me, he made sure to find out a little more about me.
### WHAT EXACTLY IS A DUCK COMMANDER?
A few weeks after our encounter at the hair salon, Jep and I both attended a concert in our hometown and saw each other again. He walked up to me and said, "I'm Jeptha Robertson. You're Jessica, right?"
We went on to have a conversation that went something like this.
Jep said, "My dad is Phil Robertson."
I was unimpressed. Even though I come from a family of hunters, I had never heard of Phil Robertson, and that seemed obvious to Jep, so he continued. "You know, the Duck Commander."
I'd heard of Daffy Duck; Donald Duck; and Duck, Duck Goose, but I had never heard of a Duck Commander. "What is a duck commander?" I asked, not sure if it would be a person, a job title, a tool, or what.
Jep was totally shocked by my ignorance but very sweet about it. I thought his disbelief was cute. Our interaction did not go much further beyond the fact that I did not know who Phil was, but it was pleasant.
The next time I saw Jep was at a Chili's restaurant. He was there with his friends and I was there with my family. He and his friends were going home, to a house he rented with one of his buddies, to play a game of dominoes. Jep invited me to join them. At that time, I was a homebody and an introvert. I was not adventurous in any way and felt uneasy about playing games with people I did not really know. But there was just something intriguing about Jep that caused me to step out of my comfort zone and go with Jep and his buddies.
Not long after the night we first played dominoes, Jep invited me to a Bible study he held with his friends several nights a week. Those guys preached the Gospel to me, and soon after I started attending the study, Jep baptized me one night in a muddy pond in a local neighborhood. By the time we got there, it was already dark, so we used Jep's Jeep headlights to help us see so we could wade in and out of the water.
### MEETING THE ROBERTSONS
Soon after Jep baptized me, I met his family. I have always thought of myself as an old soul, and I felt an immediate connection with Miss Kay. From the time we met, we have been close. That's my perspective of course, but if you ask her, she will tell you the same thing!
Jep, too, has always been close to his mom. After we married she told me that when we were dating, but before I met her, she asked him one day why he liked me so much and why he thought I was the one for him. I have always thought his answer was so sweet. He said to her, "Momma, of all the girls I've dated, this one is the most like you."
In June 2000, Jep and I admitted we really liked each other and began dating. In September, we broke up. I was miserable without Jep, and I think he was miserable without me. We finally said to each other, "I missed you," and got back together after only a couple of weeks apart. Later that month, we were studying the Bible together one evening at Phil and Miss Kay's house and Jep said, "We should get married." There was no getting down on one knee, no long profession of his undying love for me, just that matter-of-fact observation. I knew by that time that God had brought us together for a reason and that Jep loved me and would love me forever, so I did not need any of that; I simply agreed: yes, we should get married.
### JOINING THE FAMILY
On Sunday, October 7, 2001, just two weeks after Jep and I agreed to marry, we had a beautiful wedding in the backyard of a family friend. It was just what we wanted—outside on a gorgeous fall day, surrounded by nature and people we loved. Alan performed the ceremony, and Missy, a talented vocalist, sang. It was not elaborate because we did not have time to plan a big wedding. It all happened quickly, but it was perfect in our eyes.
The day we married we went to church in the morning, then Jep told me he needed to go somewhere and would see me later. I had no idea what he was going to do but found out he went straight from church to the mall, which opened at noon, to buy me a ring. I now know Miss Kay had also bought me a ring, at Walmart I think, because she wanted to make sure I had one and knew Jep did not have much money. I will always be grateful to her for being so thoughtful about that. But it was important to Jep to buy me a ring himself, so he picked a simple, pretty wedding band out and paid for it over time. It's the ring I still wear with love and pride. No big diamond necessary! By two o'clock that afternoon, I was wearing a dress I had bought off the sale rack at Dillard's and Jep was wearing a borrowed outfit consisting of a pair of slacks, a button-down shirt that was slightly large, and a tie. Both of us were ready to commit the rest of our lives to each other.
One thing I noticed early on about Jep—and part of what made me fall in love with him—is that he is so kind, generous, and eager to help anyone. When I became part of the Robertson family, I quickly realized one thing about them: Robertsons are some of the most loving people I have ever known. They have open doors and open arms to everyone. Phil and Miss Kay are incredibly generous people, and the rest of the family has followed their example. In fact, I would go so far as to say that part of what it means to be a Robertson is to love and serve others, no matter their background, skin color, or status in life. I can remember times when Phil and Kay had nothing to offer financially, but they shared their love and faith with everyone. They have always helped anyone they can, in any way they can. When I think of becoming a Robertson, it has meant so much more to me and to all the wives than simply a change of last name. When we became part of the family, we became part of everything Phil and Kay represent and taught their sons to represent. I am so thankful to belong to a family so deeply rooted in love for God and love for others.
# Part Two
# NO ORDINARY IN-LAWS
Two people are better off than one, for they can help each other succeed.
ECCLESIASTES 4:9, NLT
## 7
## INTRODUCTION
## A Message from the Daughters-in-Law
Missy's first visit to the Robertson home took place as a teenager when she went to their house for a Sunday night church group meeting. She remembers that when she got there, she said to herself, "These people are different from anyone I've ever met." A lot of people would probably say the same about Miss Kay and Phil. They are different from anyone most of us encounter on a regular basis, and they are different in all the best ways. Not much about them is ordinary, and they certainly do not fit the traditional stereotypes of in-laws. We could not be happier about that, and we could not love and appreciate them any more than we do. They are extraordinary individuals and extraordinary in-laws.
## 8
## OH, KAY!
## A Message from the Daughters-in-Law
As Miss Kay's daughters-in-law, we feel blessed with the best mother-in-law on earth. She means something different to each of us, and we enjoy our own special relationships with her. But we all immediately respond with some of the same answers when people ask us about her. We say she is an absolute joy to be around and that she is one of the most fun people we know. And we tell them that none of us feels like an in-law; she loves each one of us as she loves her own children.
We have all had experiences when we turned to her for good advice during disagreements with our husbands, and we've learned to trust her to be fair and objective, never taking the boys' side just because they are her sons. She is not quick to find fault with anyone but listens to all sides of a situation and judges it based on what is right or wrong according to God's Word. She will tell us—or the guys—when we are wrong. We can count on her for wisdom, comfort, encouragement, and a big dose of love. Most of all, Miss Kay has an enormous, forgiving heart. Years ago, that put our family on track to become who we are today and it remains part of what keeps us strong.
### Missy: GREAT MOTHER-IN-LAW, GREAT FRIEND
Miss Kay is a great mother-in-law because she is a great friend. I don't think of her as my mother-in-law first. She is a friend first whom I am fortunate to have as my husband's mother. When I am with girlfriends and the subject turns to mother-in-law problems, which it sometimes does, I sit very quietly. I honestly have nothing to contribute. I hear my friends talk about how their mothers-in-law try to control them, try to tell them how to parent their kids, make snide comments about their choices of clothes, and so forth. I do not have any of those experiences with Kay, and usually, before the conversation is over, someone makes a comment about how I can't relate because I have Miss Kay as my mother-in-law. It is definitely apparent to everyone that I have the best mother-in-law in the world. Bar none. I tend to agree.
I learned about 90 percent of what I know about cooking from watching Kay. At my wedding shower, I received a recipe card set. I took that set of blank cards and headed straight for Miss Kay's kitchen. I pulled them out, took the first one, got a pen, and asked her to start giving me recipes for the things Jase liked to eat best. She happily obliged.
There was only one problem. Miss Kay had no idea what any measurement was for any of her ingredients. She would say, "One shake of this," or "Two scoops of that." Since I had no knowledge of cooking, I was looking for exact measurements. I did not want to mess up Jase's favorite recipes. I had some big shoes to fill, for goodness' sake!
Miss Kay tried to give me her best directions while she was busy around the house. At that time she didn't understand how little I knew, and we both became frustrated. One example of this was when she told me how to make mashed potatoes. She said to cut up four or five large potatoes and boil them. I asked, "How long do you boil them?"
She replied, "Until they're done."
"How many minutes does that take?" I asked, thinking I could set a timer.
She said, "You can't go by time."
"Then how do you know when they're done?"
"They're done when they're soft," she answered.
Thinking about how much I did not want to stick my hands in boiling water to see when they turned soft, I asked, "How do you know when they are soft?"
At that point, Miss Kay had become completely frustrated at this whole ridiculous line of questioning on my part. She said rather abruptly, "You stick a fork in them!"
I apologized for my ignorance, and Miss Kay realized I needed special attention. She then pulled up a chair, put her hand on my arm, and said, "Okay, let's start from the beginning." The next few minutes consisted of her gently instructing me in the ways of heating canned corn in a skillet, browning hamburger meat for her homemade spaghetti, making her famous homemade white sauce, and creating many other dishes I still make for my family on an almost daily basis.
One of my most special memories of Miss Kay is what she did when we found out our daughter, Mia, would be born with special needs. Miss Kay bought a new white baby bed (Mia was the first girl in the family for Jase and me), lots of antique knickknacks, and a beautiful antique baby doll carriage for her room. Jase and I did not have much extra money back then for a baby room makeover and neither did Phil and Miss Kay, so she also helped arrange the makeover as a surprise for me.
I worked at Duck Commander at that time, and the company was still run out of Phil and Miss Kay's house, so she distracted me by keeping me busy at work until late in the day. I came home to a completely redone baby room for my daughter. Jason's cousin Melissa, Korie, and some of my friends had spent the entire day getting it ready. It was absolutely beautiful, with mint-colored walls, white furniture (all of which I owned but had been repainted to match a girl's room), my great-great-grandmother's rocking chair reupholstered, new pink and white bedding, and even old children's books Kay found at garage sales. It was very emotional for me.
Miss Kay knew she could not change the outcome of this baby's being born with problems, so she did what she does best: she gave everything she could to provide comfort. I was so very proud of that room and, except for the baby bed, kept it exactly the same when we moved into the home we have now. I didn't change it until Mia's eighth birthday.
### Lisa: A FRIENDSHIP THAT WEATHERED THE STORM
Miss Kay and I are only eighteen years apart, so even when I was a young newly married woman, she and I got along well. During those early years we developed a friendship that has stood the test of time and of great difficulty, a friendship that has survived what would have devastated many mother-in-law–daughter-in-law relationships. But Miss Kay is no ordinary mother-in-law, and the Robertson women as a whole are not ordinary women. We do fight for what we love and believe in. Miss Kay and I both believe a marriage is one of those relationships to love, to believe in, and to fight for, second only to a person's relationship with Christ.
When I first married Alan, I did not have a great relationship with my parents. They were not thrilled about my union with Alan. In their defense, I will say that way back then, he was not the fine upstanding man he is today. Miss Kay helped to mold me during those days, and I will be eternally grateful to her always for being there for me when I needed that guidance.
Kay's and my friendship just seems to thicken with time and with every obstacle Satan tries to use to knock us down. One of the things Kay and I try to do is to encourage our girls (my younger sisters-in-law) to face problems head-on. We have learned not to sweep problems under the rug because one day they will raise their ugly heads and take a bite out of you! I think when Kay and I work together like this, it makes our bond even stronger. I am truly a blessed woman to have the Robertson family, but I am doubly blessed to have a wise and loving woman like Kay as my mother-in-law.
As each daughter-in-law has joined our family, I have seen Kay work hard to establish a special relationship with her, just as each of her sons' wives has worked to build a special relationship with Miss Kay. She is not jealous of our relationships with her sons; she wants close relationships with us too. She knows that, down the road, interfering will only bring heartache for the ones she loves.
Miss Kay's best quality is her gentle, encouraging spirit. No matter what a person is going through, she will say she understands, but she is also quick to remind us that God can heal any situation and use it to His glory. I have learned from Miss Kay how important it is not to write people off just because they are dealing with a particular struggle. She has a remarkable ability to see the good in people and to give them opportunities to realize their potential. She definitely showed me how to believe in my children, love them, and show them the way to the Lord even when I may not have agreed with their ways of thinking. Scripture tells us to train our children in the way of the Lord and that when they are older, they will not depart from it (Proverbs 22:6). Kay has lived that verse and made it part of her legacy to us.
### Korie: SHE KEEPS THE FAMILY FUN
Miss Kay has such a fun spirit. She is lighthearted, she loves to laugh, and she has a great sense of humor. She looks for the fun in everything, and if she cannot readily find it, she makes it. Even as grown men, her sons can be very playful with her. They love to tickle her or walk up behind her in the kitchen and put an ice cube down the back of her shirt. She never gets upset with them, even if she is trying to get a meal on the table. In fact, she loves it! She thinks what they do is so cute and funny, and there's no doubt that her boys have inherited her playful spirit. She has fun with all of us and we have fun with her. Duck Dynasty shows the boys pulling pranks and being silly, and in real life, they do the same. Life around Miss Kay and her boys is lots of fun because Miss Kay is a fun person.
Miss Kay is also genuinely kind. One of the sweetest things she does for friends and family is send cards to them. Often she includes an encouraging note or some little message that really makes the recipient feel special. But other times, especially for our birthdays or other holidays, she gives us totally random cards, maybe because she likes the picture or the sentiment they express. Sometimes the sentiment doesn't even match the holiday or occasion! Getting a "Happy Valentine's Day" card in October with the word Valentine's scratched out and Birthday written over it is not uncommon. Miss Kay's philosophy is to buy the card she likes and then alter it to fit the celebration. No matter what the occasion, we all laugh at our mismatched cards.
Miss Kay has many "favorite" things, such as her dogs and her cookbooks, but there's not a doubt in anyone's mind that Miss Kay's pride and joy is her family. She loves it when we are all at her house, and she misses us when she does not see us for a couple of days. Miss Kay has spent her lifetime showing her love for others by cooking huge meals. But just as much as she loves the actual cooking, she loves her family being around her while she's cooking. She makes everyone feel loved and important just for being in her home.
Since we began Duck Dynasty, all of our schedules have gotten incredibly busy, but Miss Kay is determined to continue her Bible-study groups and the many things she does to care for others. Many weeks, the ladies in the group have to change the time or the day of their Bible study to accommodate Miss Kay's filming schedule, but they love her so much they're willing to work it out. I really respect the fact that she doesn't choose to slow down when everyone would understand if she did. She is always thinking of what she can do to help someone else. I've seen her cook extra food for families who are in need. I've watched her stop just to buy a card for a friend who needed a word of encouragement. Between filming scenes for Duck Dynasty, I've even heard her on the phone counseling young women who are having marriage trouble. Miss Kay will do whatever it takes to help those she loves.
Miss Kay has taught me not to take my family or my marriage for granted. This is something anyone around her can see because she lives it every day. She works on her relationships. I don't think Miss Kay even realizes she is teaching this lesson because it comes so naturally to her, but I see how she does it. She continuously makes sure she gives her marriage the attention it needs to grow and thrive. She purposefully thinks about ways to make it better and to make Phil happy, happy, happy. She does the same for the rest of the family. She thinks of ways to bring us together and notices if one person needs a little more attention at a certain time or is struggling with something, and then she works to encourage the person or fix the problem. These valuable life lessons are now a part of my life, and I hope my children and husband can see me doing the same.
### Jessica: THE BIGGEST HEART I KNOW
The first time I met Miss Kay, she gave me a big bear hug. Since that day, as long as I have known her, I have felt she embraced me as her very own, and the two of us have had a very strong relationship. When Jep and I first got together, I was the new girl at church dating a guy most girls thought was really good-looking (and he was). That is not usually a good way to make friends with other young women, so for quite a while I felt that Miss Kay was the only friend I had.
The fact that Miss Kay embraced me as she did is not really surprising. She has a bigger heart for people than anyone I have ever known. She is a friend to all—seriously, to everyone she meets. She picks up friends like nobody I have ever seen, but she also has a lot of long-standing relationships because she is so loyal to her friends. When you are Miss Kay's friend, it means that when you are discouraged or disappointed, she will help lift your spirits. When you are sad, she will comfort you. And when you make a mistake, she will never tear you down or try to make you feel bad about yourself.
Miss Kay is a pure delight to be around. She expresses her affection freely and has taught me by example so much about how to love and how to forgive. Jep and I did not date for very long before we married, so in many ways we had to learn about each other after our wedding. Miss Kay was so helpful and loving toward me during that time, as she helped me know how to be an encouraging and supportive wife. Those lessons helped me do my part to lay a firm foundation for a good marriage.
Kay and I were both blessed with very close relationships with our grandmothers. Her mother's mother, Nannie, had a major influence on her life and Miss Kay learned so much from her, especially about cooking. Kay loved her grandmother greatly and valued the things her grandmother told her. Likewise, I love my mamaw Nellie so much, and the two of us have had a strong bond with each other ever since I can remember. I still treasure the things she has said to me and the lessons she has taught me all these years. She is truly one of the most godly women I have ever known.
Miss Kay and I also have similar tastes. We both love classic movies, especially Doris Day movies and films like Pride and Prejudice. We also both enjoy old-timey music and have a love for hats. Some people even call us "the hat ladies." Miss Kay and I share an appreciation of antiques and have taken many girl trips to a well-known antiques area near Dallas, Texas. She and I do dinner-and-a-movie nights together and have been in many women's Bible-study groups with each other. Every time I am with her, I have fun and feel blessed to have her as my mother-in-law.
## 9
## GETTING OUR PHIL
## A Message from the Daughters-in-Law
One thing we can say with total confidence and unity about Phil is that he is a man of the Word—God's Word, that is. If there is anything he loves more than a good duck hunt, it is studying the Bible and sharing God's love with people. We honor him for the hard choices he has made in his life as he has chosen to follow God and for his steadfast, unshakable faith.
Phil is definitely not concerned about what other people think of him; he has very strong convictions and he lives his life according to them. Phil is a wise and loving man and a great leader in our family. We are thankful to have such a great father-in-law.
### Korie: A WISE AND HUMBLE MAN
I know it won't be breaking news to anyone for me to say that Phil is blunt. He just "tells it like it is" all the time—no sugarcoating, no diplomacy, no punches pulled. But Phil is also incredibly wise. That catches some people by surprise, but everyone in our family recognizes and respects the wisdom in this man. In many ways, he has had a tough life and he has learned a lot through his experiences. Any time my children have a chance to be around him—whether it's in a duck blind or a fishing boat, out working on his land, or around the table—I want them to be.
One of the things I value most about Phil is his passion for God. He knows the Bible backward and forward. He was baptized almost forty years ago but still thirsts like a new Christian to learn and grow and share what God has done. He studies God's Word all the time; I don't know that he ever reads anything else.
Even though Phil is very wise, holds a master's degree, invented a famous duck call, and played some pretty good football in his day, he is also very humble. It would be easy for Phil, especially as he's gotten older, to sit back and rest on what he has accomplished in life, but he doesn't. He is always open to growing and changing and even to being proven wrong, if someone could do that. When he is not sure about something, he'll say, "Hey, why not? Let's try it." Not many men in Phil's position are willing to be challenged like he is.
Sometimes Phil can come across as hardheaded and intimidating, and in some ways that might be true. But what many people don't realize about Phil is that he truly loves people. He really wants the best for them. That is his motivation for doing what he does, especially when it comes to sharing Jesus with others. His love for his fellow man drives him to open his home to complete strangers to tell them the good news of Jesus.
In many families, when control of a family business is passed from one generation to another, the older generation struggles to let go. That has never been the case with Phil. He has always shown respect and support for Willie as Willie has taken over the company his dad started and ran for many years. Again, this is a rare trait in someone with Phil's experience, but it sure has made Willie's job a lot easier!
### Jessica: HE MAKES US WANT TO BE LIKE HIM
When I first had a chance to be around Phil, I thought he was rather quiet. The more time I spent in his presence, the more I realized he is a great storyteller, and he definitely has some great stories to tell. I also found out quickly that he is a man who truly loves God and is not afraid for anyone to know it. He genuinely cares about people's souls. He is so good at forgiving people and not holding anything against anyone.
I am so grateful for the way Phil understands and shows true forgiveness, because he taught Jep to do the same. When Jep and I first met, I was going through something a lot of twenty-year-olds go through. I had "baggage." Jep later told me that when he talked to Phil about it, Phil said, "Since she is a child of God Almighty, she is forgiven." He reminded Jep that my sins were washed clean by the blood of Christ.
I think Jep has always valued his parents' thoughts and opinions. He always took to heart the scripture "honor your father and mother" (Exodus 20:12). The fact that Phil spoke such affirming words to Jep about me gave him the courage to take the next step in our relationship. I will always appreciate that more than I can say.
A lot of parents tell their children and grandchildren, "Do as I say, not as I do." Phil never needs to say anything like that. He lives in such a way that he inspires us to want to do what he does. He "walks the walk"; he does not just "talk the talk." He is deeply committed to knowing God through His Word and when we look at Phil's life, we want to be the same way. His faithfulness and constant willingness to learn more about God have set an awesome example for all of our family.
### Missy: HE'S HAPPY TO SHARE
To me, Phil's best quality is his love for the Lord and his thirst for the knowledge of God. When I picture Phil at home, I picture him laid back in his recliner with one leg over the arm of his chair, his reading glasses on, his Bible open, with a notebook and a pen on his lap. This is a scene I have witnessed too many times to count. Sometimes he acknowledges my presence, sometimes he doesn't, depending on how deep he is into his study. Matthew 5:6 describes Phil: "Blessed are those who hunger and thirst for righteousness, for they will be filled." He truly has a longing for God's Word, and he loves to share his knowledge whenever he gets a chance.
Phil has taught me to never judge a book by its cover. From the beginning of my relationship with Jase, I saw this firsthand in Phil. He accepted all people into his home. Everyone who entered his house got the same treatment. They were told about Jesus and asked how their lives were going. Some people were honest with Phil; some were dishonest. It didn't matter. Every person was treated like the one before and the one after. They got a meal, a bed (or couch) if needed, and a Bible study.
Phil does not get upset or excited about much. He does not get out of his chair to shake your hand if you are wealthy or famous, because he does not get out of his chair if you aren't. No one is better than anyone else in his eyes. Without Jesus, everyone is lost. That pretty much puts everyone on an even playing field. Phil looks at everyone the same way. His attitude is, "That person needs Jesus, and I am more than happy to share Him with them." I love that about Phil.
### Lisa: STAND BY . . .
Phil has always been the big, burly, totally straightforward guy he is today. After all, he was once a star quarterback! Even with his imposing physical presence, I never had any reservations about speaking my mind in front of him or in front of Miss Kay. That was not always a good thing, but they understood my immaturity and did not keep a record of my wrongs.
One of the most important lessons I have learned in life I learned from Phil. It is this: whether or not you agree with a decision your child has made, you still stand by that child (even if he or she is an adult) and you stand by that decision.
When Alan and I had our relationship problems in the late 1990s (more about this later), Phil told Alan that he did not think we should get back together. Alan made the decision to accept me back into our home and into his life. When that happened, Phil told Alan that even though he thought I would just hurt him again, he would respect Alan's decision and he would treat me with respect. That is exactly what he has done since that time, and he has never once questioned our decision. He even defended Alan's choice when others questioned him and tried to put stipulations on the relationship. Phil has acted out 1 Corinthians 13:5, which says that love keeps no record of wrongs.
## 10
## WHAT A TEAM!
## A Message from the Daughters-in-Law
Sometimes people who have been married as long as Phil and Miss Kay reach a point where they lead comfortable but somewhat separate lives. This is definitely not true for Phil and Miss Kay. Although they are not "joined at the hip," and they do not do everything together every day (Miss Kay would not want to spend that much time in a duck blind), they have a great relationship and an exceptionally strong marriage. They know how to work together, play together, serve God together, and love our family together. They are both strong, gifted individuals, and they make a terrific team.
### Jessica: WHO COULD ASK FOR ANYTHING MORE?
Growing up, my mother got along very well with my father's parents, and my father got along very well with hers. They both loved each other's families and enjoyed spending time with them. We spent many summers as a family at my dad's parents' fishing camp and many weekends and holidays during fall and winter hunting seasons at my mom's parents' hunting camp. Sometimes, both sets of grandparents ended up at the hunting camp, and all of us had a great time together.
People sometimes tell horror stories about their in-laws, but I never saw anything other than love and respect in my biological family, so when I married Jep, I expected to have great relationships with Phil and Miss Kay, and that's exactly what has happened.
I don't think anyone could ask for better in-laws than Phil and Miss Kay; they are everything I could have dreamed of, and I feel so blessed to have them. When Jep and I were first dating, I was going through a lot in my life, and they showed me love and forgiveness. Kay has so much love in her heart, and Phil is one of the wisest men I know.
I love the relationship Miss Kay and Phil have with each other. She respects and adores him. When Phil looks at Miss Kay or speaks to her, you can see in his eyes and hear in his voice that he thinks she is the best wife in the entire world. What I have observed since becoming part of the Robertson family is that Miss Kay has always taken care of Phil, and he has loved and cherished her. Even after many years together, they still laugh, they still love, and they still hug and kiss each other (even in front of the family and other people). Miss Kay still crawls into Phil's lap while he sits in his big recliner, and when he walks close to her, he tickles her or gives her a pinch on the rear end.
Miss Kay and Phil have an amazing love for God and for people. They believe not only in giving second chances but also in giving third and fourth chances. When they need a little extra work done on their property, they often hire people who need help and try to help them as best they can. They spend hours and hours, several nights a week, counseling people and sharing their wisdom with them. If Jep and I can do even half of the good things Phil and Miss Kay have done, we will consider ourselves to have truly succeeded and made a difference in the world.
### Missy: A TEAM OF SERVANTS
Phil and Miss Kay are definitely not typical in-laws. So many good qualities set Phil and Miss Kay apart from other people, but I think their best quality as a couple is their teamwork in serving others. They host many people in their home; they have always done this. Many of these people need wisdom, advice, salvation, or simply the knowledge that someone really cares about them.
Miss Kay used to cook all the meals and Phil used to do all the preaching and counseling. Over the years, they have realized that both of them have both of those strengths. Now Miss Kay and Phil both do the cooking, and they both sit down after the meal to talk with people, whether they are counseling a couple in a marriage crisis or sharing the message of Jesus with someone. They have grown together over the years and have recognized and embraced each other's strengths. This makes for a strong and godly marriage.
### Korie: THEY JUST KEEP ON LOVING
Anyone who knows Miss Kay and Phil would have to say they are some of the most hospitable people anyone could ever find, anywhere. When I first met them, they did not have much money, but their home was always open. They lived in the same house they currently live in, but it was about half the size it is now. The kitchen was incredibly small, with no dishwasher, but that did not stop them from having guests regularly. They cooked huge, delicious meals out of that tiny kitchen, and everyone just gathered around standing or sitting on every available surface to enjoy the meal and laugh and talk together. They understood the true value of good conversation, good friends, good food, and family togetherness. Those things mattered more to them than anything else.
It's important to understand that the warm, fun, loving environment Miss Kay and Phil have created in their home did not happen automatically. In fact, it was years in the making. Miss Kay forgave Phil when a lot of people thought she had good reason not to, and she loved him when he was unlovable. The same dynamic happened to him that often happens when we truly understand that God forgives us and loves us when we are unlovable: it makes us want to do better and to be better. Miss Kay offered this to Phil, and he has done the same for her.
I'm sure there were times after Phil came to Christ that he still messed up, when he still sinned and fell short. Miss Kay had to forgive him, not just that one moment when he showed up outside her workplace begging her forgiveness, but many, many times after that. Forgiveness has to be offered over and over again in any marriage, and that was certainly true for Miss Kay and Phil. As in any husband-wife relationship, each person sometimes has to be the one who needs forgiving and sometimes has to be the one who does the forgiving.
Miss Kay showed God's love to Phil through her love and forgiveness, and he "got" it. Phil truly changed his life, and not just a little bit. He's not a lukewarm Christian by any means. He turned his life around totally and completely, and he now uses the gifts and talents God gave him for God's purposes rather than for his own selfish desires.
Now, with their most difficult years far behind them, Phil and Miss Kay show love continuously. They are playful with each other and laugh together all the time. They have weathered a lot of storms in their life together and are being rewarded for their faithfulness with a great marriage. It's so exciting for us to watch this couple who have been together for nearly fifty years continue to grow in their love for each other.
At any point either one of them could have given up. Had they done that, the rest of us would not be here today. People would not be watching our family on television, and lots of lives would have never been touched—not by us, at least. I know God is raising up others to do His work and touch many people, but I am proud that for this moment, for this time, He is using our family, and we will not take that for granted. We are grateful for the difficult choices Phil and Miss Kay made so we could all enjoy the close family and the good lives we have now.
### Lisa: SO IN LOVE
Miss Kay and Phil know the pain I caused their son (more about that later), and they still chose to support us as a couple and chose to love and care for me. They have helped Alan and me in every way they possibly could through the years, whether that involved babysitting (Kay was the babysitter; Phil was the one who got credit for it on Duck Dynasty), financial support, or sharing their wisdom with us. Alan played an important role in the Duck Commander business during its early years, and they give him a lot of credit for that. This makes me proud for him and proud of him.
Miss Kay has always done a lot around the house for Phil. She has such a giving heart and is a natural caretaker, so she keeps an eye out for anything he needs or wants, and she accommodates him whenever possible. She is not one to grumble or complain about anything. She is a model of commitment, as everyone knows. She likes to bless other people, starting with Phil.
Miss Kay is also respectful by nature. The respect with which she treats Phil made a deep impression on me when I first got to know them and continues to impress me today. One simple way Miss Kay demonstrates respect for Phil is to keep in touch with him if she changes her plans throughout the day. She respects the fact that he needs to know where she is in case he should need to reach her for some really important reason (like asking her to stop at the store to pick up snuff). Especially if her schedule changes or if she is running late, she gives him a quick call and keeps him up-to-date on where she is. That's the kind of communication that keeps a relationship running smoothly. On top of all the good things Miss Kay does for Phil, she is totally, completely head over heels in love with him, as he is with her!
I know that Christ lives in Miss Kay and Phil, and they pattern their lives after His teachings. They truly practice what they preach. Together, they impact a lot of people in the most positive ways. Phil has a straightforward, up-front personality, while Miss Kay is understanding and comforting. That's a great combination. When the two of them counsel people who need help or guidance, especially couples who are struggling in their marriages, the people they help leave believing they can conquer the world together.
## 11
## PASSING IT ON
## A Message from the Daughters-in-Law
Miss Kay and Phil are terrific grandparents to our children. We are always glad when the children—whether they are very young or whether they are young adults—can be with them because they always learn something valuable and because they come away knowing they are loved in very special ways. Miss Kay and Phil have established a powerful legacy of love and faith from which all of our children can benefit. They have so many good qualities to pass on to their grandchildren, and we look forward to seeing how these qualities manifest in their lives in the future.
### Lisa: GREAT RELATIONSHIPS
My children are grown, but my daughters have a great relationship with Miss Kay and Phil. In fact, Miss Kay is both of my girls' best friend. Both Anna and Alex have kind, loving, gentle hearts, and I know they learned these qualities from Miss Kay. I was not always the mother I needed to be for my children, but Miss Kay took up the slack, and they are awesome women today because of her.
Both girls have spent a lot of time with Miss Kay and Phil over the years. Anna started working for Miss Kay as a teenager, and although she now works full-time at Duck Commander, she also does special projects for Miss Kay and travels with her as much as possible. When asked about her relationship with Kay, Anna says, "I have been close to Mamaw Kay all my life, but even more so since I started working for her when I was fourteen or fifteen. Some people have grandmas they cannot open up to, but I can talk about everything with Mamaw Kay. I always turn to her when I need someone to talk to, and she gives really good advice. She knows about so many things because she has been there; she has lived it. And I can trust her. If I need to talk to her about something I do not want her to share, she will not tell it.
"Plus, Mamaw Kay is great when it comes to fashion. She knows what's in style and stays up on trends, and I like that about her. But most of all, I have noticed in my life that some grandparents are quick to say 'I love you,' but they do not always do much to show it. With Mamaw Kay, I have always felt loved, and I know I always will. Every time we get together, when it's time to go, we give each other a big hug and say, 'I love you!' I know Mamaw Kay really means those words when she says them to me, and I really mean them when I say them to her."
Phil has always had a special place for my girls in his heart, but when they were young, he did not always know how to show it. As the years have mellowed him, he now has a great, loving relationship with my granddaughters. They adore Papaw Phil. My daughters are old enough to understand and appreciate all of Phil and Miss Kay's good qualities. I believe they will inherit all of those attitudes and characteristics (they already demonstrate a lot of them), and I pray they will pass them down to their own children.
### Missy: A LOT TO LEARN
We all have so much to learn from Miss Kay and Phil. My prayer and hope is that my children have learned the valuable life lessons they have to share. I hope my children have learned from Phil and Miss Kay to look at everyone they meet with a nonjudgmental heart. I hope they have learned and will continue to learn that judgment belongs to God and we are here only to guide people to Him. I hope they have learned to laugh at the little things and the big things and to know that God has given them this life as a gift.
I hope my kids have learned a strong work ethic by watching their papaw Phil work on the land and in the yard cleaning fish, deer, and ducks. I hope they learn to be generous in feeding family, friends, and even strangers a good, home-cooked meal and that having an open-door policy is much more fulfilling than a closed, locked door with a security system. Most of all, I hope they have watched their grandparents stay real in an unreal world, learned that they can turn to God for all of their problems, and come to understand that they can love people enough to share that same God with everyone they meet.
### Korie: STRENGTH, LOVE, AND COMPASSION
The three traits I have tried to instill in my children since they were born are strength, love, and compassion. Both Phil and Miss Kay exemplify these traits, even though they express them in different ways. I hope my children will continue to develop strength based on a deep knowledge of whose they are (God's) and understand that His power moves mountains. I pray they will always love others but love God above all else. I also hope and pray they will consistently demonstrate compassion by treating others with kindness and humility. They certainly see these qualities in their grandparents. I can already see these same traits unfolding in my children, and I look forward to seeing these good characteristics mature in them over the years to come.
### Jessica: LEAVING A LEGACY
Phil and Miss Kay will definitely leave a legacy of love for God, love for each other, and love for people in general. They both truly love and care for others, regardless of a person's skin color or economic status, and will help anyone in any way they can. They are not very technological; they value quality time with people and good conversations over text messages and e-mail. They want everyone with whom they come in contact to know God. Both of them are dedicated to family and to the togetherness that comes from the family table, and I hope these things will continue in my children and throughout the generations of Robertsons to come.
I hope my children will always love to read and love to cook, as Miss Kay does. I pray they will inherit the way Miss Kay cheers for the underdog, lifts the brokenhearted, and gives to anyone in need. I hope they will also notice and learn from the fact that she loves and respects Phil and is a loyal, committed wife and friend to him.
I feel that my children have a rare opportunity to glean from Phil some things that have been lost in modern society. In many ways, he is "old school." He still works his land and takes care of it, which is important for them to see. He is hardworking, industrious, and resourceful. He could not care less about the things of this world; he is totally nonmaterialistic but is a man who hungers and thirsts to know God. Sometimes I am amazed to see him studying his Bible in his recliner because he already knows it so well. But he continues to study diligently. His desire to walk in truth is deep, and his faithfulness to God is remarkable. I hope my children absorb these things from Phil and incorporate them into their lives to the greatest possible degree.
I also want our children, boy and girls, to have a love for the outdoors like Phil and Miss Kay have. Miss Kay loves nature and has a special affection for all kinds of animals. Phil loves having the ability to provide for his family through growing vegetables, fishing, frogging, and hunting deer and ducks. All of that knowledge from both of them has been passed to us, and it is so important that the generations who come after us not lose the ability to live off the land. Jep and I are actively teaching our children survival skills, in addition to the skills they observe and learn from Phil and Miss Kay.
I don't mean to imply that we never buy meat or vegetables at the grocery store; we simply have the skills and ability to provide for our family from what God has created, and we have a love and appreciation for the outdoors, which Jep and I both got from our families.
I am aware that some people don't like "meat eaters" because they think we are killing animals for the sake of killing, but that is not true. We are providing food for our family. One of my favorite meats to eat is deer. You can't get a leaner, more healthy meat. I like certain parts of the deer better than others, but we do not waste anything. We often take deer meat and ducks to local neighborhoods in which people may not have the money to buy meat, and we just give it to them. We also share meat with widows in our church. The looks on their faces when we do this are priceless.
This combination of living off the land, being resourceful, not wasting anything, and sharing with others comes from Phil and Miss Kay. It's the kind of thing they do, it's what Jep and I do, and it's what we are teaching our children to do.
# Part Three
# HAPPILY EVER AFTER CAN TAKE A WHILE
We also rejoice in our sufferings, because we know that suffering produces perseverance; perseverance, character; and character, hope.
ROMANS 5:3–4
## 12
## INTRODUCTION
## Miss Kay
I just love to read. I especially love to read to my grandchildren. In fact, at my house, I set up a little library for them. They can go in there, look through the books, choose some to read, and then sign their names on a piece of paper to "check out" the books like I used to do before so many libraries got computers. Like any good librarian, I want to make sure I get my books back so other children can enjoy them!
When I was a child, my favorite stories were ones that ended with the words "and they lived happily ever after." My grandchildren like that kind of ending too. But what I know, and what they have not yet learned, is that "happily ever after" sometimes takes a while.
I learned the hard way how long a happy ending can take and how difficult it can be. So did my daughter-in-law Lisa. Both of us want to share our stories with you. We do not necessarily enjoy talking about the heartache and struggles we have been through, but we want to talk about them because we want you to have hope for any disappointing or devastating situation in your life and to know that God is always in the business of healing and restoration, no matter how bad the circumstances might be.
## 13
## I FOUND OUT WHAT IT MEANS TO FIGHT
## Miss Kay
My grandmother once told me, "You'll have to fight for your marriage." When she said those words, I did not understand them. I had no idea what she meant. I never really saw her fight for her marriage because she had a good relationship with my grandfather. I would not say they were lovey-dovey all the time, but they treated each other with respect and there was peace in their home.
In the early years of my marriage to Phil, I did everything I could think of to be a good wife and a good mother to Alan. I had all kinds of dreams about a happy marriage and a loving family, and I honestly believed if I worked hard enough, those dreams would come true. They didn't, no matter how hard I tried—at least not for a long, long time.
I was pregnant with Alan when Phil and I moved to Ruston, Louisiana, for him to attend college and play football at Louisiana Tech. Phil was really good! In fact, when he left the team a couple of years later, his replacement was a guy who was also really good, but not as good as Phil. That second-string quarterback was named Terry Bradshaw, and he went on to become a very famous football player.
### BAD COMBINATIONS
The football team and everything that went along with being a player did not provide a good environment for Phil. After spring training of his first year, he had to spend some time living in a dorm with his teammates—a bunch of single guys out from under their parents' watchful eyes for the first time. They enjoyed drinking and partying, and because Phil was the star quarterback, they always wanted him to join them. He was young, like the other guys on the football team, and some of them told him he was really missing something because he had never had his "wild time." I guess he believed them, because he got wild pretty fast and started drinking with his buddies. When that happened, I tried to be with him without getting involved in all the things he was doing. I went to some parties, but when the drinking started, all I could think about was my mother and what alcohol had done to her. Besides, I had enough sense to know that drinking and being pregnant did not go together. So Phil started sowing his wild oats, while I stayed sober and scared of what was happening to him.
During this time, Phil and I did not go to church. He didn't want to. He did not have his own faith at that point, and neither did I. Both of us grew up attending church, but once we were out on our own we were free to choose whether we would continue or not. Part of me wanted to go, but I believed Phil was the head of our household and I needed to do what he wanted, which was to stay home on Sunday mornings. Another part of me did not want to go because I was young and pregnant and unmarried, and I felt embarrassed. I kept thinking we would go to church later, in a year or two when we got married and things settled down. I had no idea how bad our lives would get before we finally did.
After Alan was born, I had my hands full. I was very young, and of course I had never had a baby before. I did not have much help or support, but I was determined to be a good mother and that took a lot of my time and energy. I could not stay out late at night partying with Phil and I certainly was not going to get drunk.
One night, I had a major reality check. Phil and I were at a party and had taken Alan. One of Phil's good friends from home was also in school at Tech. His wife was a good friend of mine and they lived close to us. We were all together at a party one weekend and my friend suggested I check on Alan. He was throwing up. I wrapped him in a blanket, found Phil, and said, "We have to go home. The baby's sick." Phil would not leave the party, so I took Alan home alone. There's no telling when Phil showed up. That night was the end of my party time. From then on, Phil partied and drank, but I did not go with him. I remember being so torn inside because I really wanted to be with Phil. At the same time, when I thought about those parties and everything that went on during them, all I could say to myself was, "I can't do this. I just can't do it." I didn't; I quit all of that, but Phil kept on.
### THINGS WILL GET BETTER
I truly believed Phil would leave his wild ways behind once he finished college, got away from his football buddies, and started working. Even though he didn't make a priority of his studies and eventually left the football team so he could spend more time hunting and fishing, Phil graduated with both a bachelor's and a master's degree. He was well prepared to be a teacher and a coach. I was so excited about the next stage of our lives, convinced things would be better and that the happy home life I always dreamed of would finally come true.
Phil got a job soon after graduation. A man named Al Bolen recruited him as a teacher in Junction City, Arkansas. I was so happy when I realized the school was going to provide us with a little house and I could work as the school secretary. Finally, just as I had hoped, everything was shaping up just as I wanted it to. We even lived across the street from a sweet elderly couple, an old preacher and his wife, who was blind. They took an interest in Alan as soon as they met him. They loved him, and he loved them. They took him to church every Sunday, starting when he was five years old. The preacher and Alan really had an amazing relationship, and I was so thankful for that.
After Jase was born, the preacher and his wife took him to church, too, and sometimes I went with them. Phil still was not interested in church at that point, except when his parents, Granny and Pa, came to visit. When his parents were with us, we all went to church, but Phil was miserable. Phil has never been a person to pretend. Everything is black or white, good or bad, with him. If he likes something, he lets it be known. If he doesn't like something, he does not keep it to himself. He did not like church, and everyone knew it.
Maybe one reason he did not like church was that he had not left behind his drinking and partying when we moved from Ruston. Al Bolen turned out to be a big fisherman and duck hunter, just like Phil, but he also had a drinking problem, which was the last thing Phil needed to be around. Instead of finally being able to live my dream in Arkansas, I was right back in the same old nightmare.
### MOVING ON
By the time Willie was born, people in the community were aware of Phil's behavior. He did not drink every day; he could go days or even weeks without taking a sip. He was a party guy. When he got around other people and started drinking, he did not stop until he was good and drunk. In a small town, word travels fast when people do things like that, especially when those people are schoolteachers. The school administration and students' parents finally began to lose their patience with Phil, and he knew he would soon be fired. He decided to get another job and move our family out of town before that happened. With my Pollyanna attitude and my firm belief that everything would get better if I could just be the perfect wife and mother, I hoped a new start would be exactly what we needed. It wasn't.
Phil's personal situation, our marriage, and our family life got worse. Phil decided to leave teaching and coaching, and, as he put it, "make some money." Without even mentioning to me what he wanted to do, he leased a bar in a rural area of Arkansas! We lived in a trailer next door to it. All I could think was, Seriously? You are going to run a bar in the middle of nowhere? What am I supposed to do, take my kids to a bar every night? I knew I would have to help Phil in this new business, but I did not know anything about running a bar. I didn't even drink!
By this time, I had thought a lot about our marriage and family. It was not turning out the way I dreamed it would, not even close. I was disappointed, of course, but more than that, I was at a total loss about what to do. I knew that if I talked to my sister, she would tell me to leave Phil. One of Phil's brothers had already said I needed to leave. But I couldn't get my grandmother's advice out of my head: "You're going to have to fight for your marriage." I was finding out what those words meant, and the fight was a whole lot worse than I ever thought it could be.
Although some people thought I had a good reason to leave Phil when we got ready to move out of town so he could run the bar, I decided to go with him. I knew he could end up in big, big trouble—and I thought I could protect him. If not, at least I could keep an eye on him and at least our three little boys would have both parents in the home with them.
Once Phil started operating the bar, I went to work as a barmaid. It was the only way I knew to keep up with what he was doing. The local people who visited the bar knew immediately that I did not belong there. They kept telling me I needed to be in church, not waiting tables in a bar. I got the feeling they would have fought to the death for me. They thought I was a "nice lady"; they really respected me and refused to let anybody say anything bad about me. They did not understand why I worked in the bar when it seemed so out of character for me, but they also did not understand it was about much more than serving drinks for tips every night: I was fighting with all my might to save my marriage.
During this time, I found an elderly lady who babysat the boys while Phil and I worked, so thankfully they were not exposed to many of the things I saw and experienced. So many unsafe things happened, and I spent a lot of time frightened and anxious about what we should do. There were times when Phil started drinking and simply disappeared for a few days, leaving me alone with three boys and a run-down bar. When that happened, an elderly man who lived in the area ran the bar, while I kept serving drinks and wondering when Phil would come home.
### OUR DARKEST DAYS
Phil became cold and harsh during those days. He was mean and threatening to me, and I was terrified of what would happen to my boys and me. Even though I had not been in church because I was so embarrassed about everything that was happening and about Phil's behavior, I did make sure the boys got there every Sunday and I never forgot my Christian upbringing. One day, I began to pray with all my heart, "God, just get us out of here."
Somehow, we made it through the first year of Phil's lease on the bar, even though it was a terrible time. Three or four months later, the landlords showed up one day and cussed out Phil, saying they did not like the way he was running the place and were raising his rent. They were rough people, and I think what they really didn't like was that Phil had turned the place into a profitable business and was making good money off it.
Those people were not smart. They had no idea that trying to push Phil Robertson around and cuss him out would lead to disaster. He got so angry with them that he beat them up—both of them. By the time I got to the bar to try to figure out what was happening, all I saw were people being loaded into an ambulance.
During all the confusion, Phil did the only thing he knew to do: run away. He told me quickly that he was leaving and he would be gone for a while, just before he slipped out a back door. I knew he would be in trouble with the law and the only way to avoid that would be for him to hide out. He told me to handle the situation the best I could and then leave. I faced five police cars that night and enough questions to make my head spin. They wanted Phil, and I could honestly tell them I had no idea where they might find him.
So there I was: no source of income, husband on the run, three little boys, in the middle of Arkansas. Phil had made a huge mess of our lives and had left me to clean it up. I felt completely helpless and hopeless.
When the couple who owned the bar got out of the hospital, they put up a barricade around my trailer. The boys and I were trapped! I couldn't move the trailer and I couldn't leave. One day they said they wanted to meet with me, and I had no choice but to talk to them. They offered to drop the charges against Phil if I would pay them a certain amount of money. It was extortion, but I paid them because it was the only way I knew to clear Phil's name and get the boys and myself away from them. I gave them almost all the money we had except a little bit that was in a lockbox, and they gave me the trailer. I also had some things stored in another building on their property—some keepsakes and things that held special memories for me—and a washer and dryer. They would not let me get near any of those things, so I had to leave them all behind.
### FROM BAD TO WORSE
I hired a moving company to move the trailer from Arkansas to Bayou D'Arbonne Lake, near Farmerville, Louisiana. I had told Phil in a phone call where he could find us, and he soon came out of the swamp and joined us. He was so relieved when I told him he would not be arrested over the incident with the bar owners. While he was on the run, he had found a job in an offshore oil field, but we still needed money, so I went to work at a local chicken place and made just enough to pay our electric bill.
Phil was drinking worse than ever by that time, and I began to get seriously depressed. Not only were my hopes and dreams shattered, I couldn't even figure out how to make anything in our lives work. Everything was falling apart.
I soon got a new job as an insurance clerk in Monroe, Louisiana, a little less than thirty miles from our new home on the lake. The company that hired me was Howard Brothers Discount Stores, the family business of my daughter-in-law Korie, though she was only a few years old at the time and I had no idea who she was. It's a good thing I had a decent job, because not long after I got it, Phil was hurt offshore and had to stay home. I was afraid to leave the boys with him, not knowing what he might do if he drank too much, so I put them in day care.
As the situation continued, I grew more and more depressed. I worked with two Christian men at Howard Brothers, and every day, one of them would give me a Bible verse, just to try to encourage me. Those verses gave me the strength I needed to get through this terrible time in my life.
One rainy night, I had car trouble and was late picking up the boys from day care. When we finally got home, Phil accused me of having an affair! It was ridiculous. When was I going to find time to have an affair—between working full-time and changing diapers? I had always told him I would never cheat, and I would not have. I never believed in being unfaithful; it's just not in my character.
That cheating accusation was the last straw for me. I hit rock bottom. I have never felt as totally hopeless as I did that night. I simply could not see any way out of a terrible situation for the boys and me. I finally accepted the fact that I could not fix our lives and had no one to help. So I did what a lot of women do when they need to be alone: I went into the bathroom and locked the door. I cried and cried, and finally realized I just wanted to go to sleep for a long time. I did not consciously want to kill myself; I just wanted to take enough Tylenol (because that's all we had) to have a nice, long rest. And I wanted to scare the daylights out of Phil. I wanted to punish him for everything he had put me through. I told myself I didn't care if I slept forever, but deep down I don't think I really wanted to die.
In the midst of that low place, the darkest place I have ever been emotionally, with thoughts of sleep and rest filling my mind, through my sobs I heard the scurry of little feet headed toward the bathroom door. I could tell all three boys, in their house shoes, were coming to talk to me. Alan spoke first: "Mom, don't cry. Don't cry anymore. God will take care of us." I was silent for a moment. Then I heard Jase ask, "Did she quit crying?" And I could hear Willie doing something he did often, making smacking noises while sucking on two of his fingers.
In an instant, it was like a lightbulb came on for me. "What am I doing?" I asked myself. "I have three little boys. I can't leave them with a drunk."
I spoke to my sons through the door: "I'm okay. I love y'all. I'll be out in a minute."
I then got on my knees and prayed. "God, help me. Just help me. I don't want to leave these kids. I don't know what to do or where to find You. Just lead me to somebody who can help me."
### PEACE, HOPE, AND LOVE
The next day, I saw a television commercial for some kind of religious TV show. The ad said something like, "Do you want peace, hope, and a reason for living? Do you want someone to love you and never let you go?" I turned up the volume just as the announcer said, "Then call this number."
So I called that number and told the woman who answered, "I want to speak with that man who is on TV talking about peace, hope, and love. It's an emergency. I need to talk to him right away."
The man's name was Bill Smith. When I saw him, I knew exactly who he was. Phil's sister was a member of his church, and months earlier she had brought him to the bar hoping he could talk some sense into Phil, but Phil would not listen to him. I didn't care whether Phil wanted to hear what Bill Smith had to say or not. I did.
When I got to his church, White's Ferry Road Church, and met with him, the first question he asked me was if I thought I would go to heaven when I died.
"Of course I'd go to heaven. You have no idea what I have been through and what I have put up with from my husband." Then I told him how hard I had fought for my marriage and how faithful I had been, even though Phil had done terrible things.
The preacher asked me if I thought I had earned my way to heaven. I certainly did!
He then asked me if I had any peace or hope in my life. That was my problem. My peace and hope had run out years earlier. I now see what a disconnect was going on in my mind. I thought I had earned my way to God, but I wasn't at peace and I had lost all hope.
He then shared the gospel with me, and I realized two things. First, I realized I never really had my own faith. For many years, I'd lived off my grandmother's faith, but faith was not deeply personal for me. Second, I saw that I really was a good person, but I was a good person without Jesus Christ, and I desperately needed Him. That very day, before I left the church, I confessed to Jesus, made Him Lord of my life, and was baptized. Needless to say, I felt so much better! I had peace in my heart, and best of all, I had hope again.
Everyone in the church that day was so happy for me. The janitor, the housekeeper, and the church secretary all gave me big hugs. "You're part of our family now! We'll be there for you, and you can be there for us," they said.
That's nice, I thought, but my husband is a drunk.
### AT LEAST ONE OF US CHANGED
Preacher Smith was a very wise man. He knew that no matter what had happened for me that day, nothing at all had happened for Phil. He gave me a clear warning before I went home, telling me that even though I had become a Christian and I would never be alone and God would never leave me, that didn't mean Phil would act any better. He would still be the same person I had struggled with for years. He would still get drunk, be mean, and do the same things he had always done.
I continued to stay with Phil because I knew God would help me. I prayed and prayed for him; the boys did too. I would invite the preacher over to talk to Phil and Phil would slip out the back door as the preacher came in the front, and he sometimes stayed gone for days. But I was still determined to fight for my marriage. Then one night I was late coming home from work and Phil again accused me of running around on him. He yelled at me, saying he was sick of me. He said I was bad to live with before, but now I was a holy roller and a goody-two-shoes. According to Phil at the time, I thought I was "the judge of the world." I did not think that at all. I was just trying to stay sane and keep my boys safe.
At that point, Phil said angrily, "You are messing up my life. I can't live with you. I want you and your kids to get out."
"I have fought for this marriage," I replied, "and you are kicking me out?"
Yes he was. He wanted me gone from his life. When I tell this story, I make a point to emphasize the fact that I did not leave. I got thrown out, and I was heartbroken.
The boys and I went to a relative's house, and I was hoping we could stay there for a while. But even though this man was a close relative on Phil's side of the family, he would only allow us in his home for one night. He was just as afraid of Phil as I was, maybe more so. He was terrified of what Phil might do to him and his family if Phil knew we were staying with them.
### MY CHURCH BECOMES MY FAMILY
The only people I knew to turn to was my church family. I knew the people at White's Ferry Road would help me. Someone there helped me arrange an income-based apartment. When I took a relative to the place I had been living with Phil so we could get the things the boys and I needed, she saw how hurt I was, and she was angry with Phil for the way he had treated the boys and me. She suggested we destroy everything that belonged to him. I had one answer for that: "I don't retaliate." As much as Phil had hurt me, hurting him in return went against my nature and, by then, against my Christian beliefs.
In our little apartment, the boys and I had a very small television. We'd had a larger one when we lived with Phil, but he'd kept it and the boys really missed it. I told them I didn't care what kind of TV we had; we needed to be focused on studying the Bible. So that's what we did; we studied the Bible and we all got on our knees and prayed for Phil, every day.
Even though Phil had treated me badly, there was a hole in my heart after he kicked out the boys and me. I so desperately wanted God to change him. I prayed, the boys prayed, and I got everybody I knew to pray with us. My friends at work prayed for our family; I took Bible classes at church and asked everyone in every class to pray for my husband. I even remember standing in line at Walmart one day and asking the woman behind me, "Do you pray?" When she said yes, I told her about Phil. I knew only God could change things for us and that the way to get to Him was through prayer.
About three months later, I went to lunch one day with a friend from work. When we returned to the Howard Brothers offices, I saw Phil's old truck in the parking lot. My friend asked me if I wanted her to call the police, and I said, "No, I'll go talk to him. Just watch me through the window. If anything happens, then call them." As I walked toward the truck and saw Phil bent over the steering wheel, I assumed he was drunk. He was not; he was crying. I opened the door of the truck and for the first time in my life saw huge tears flowing down his face. I'll never forget what he said: "I can't sleep. I can't eat. I want my family back, and I am never going to drink again."
My first thought was, This is the man I want. This one, right here. But I had enough sense not to say that right away.
"Phil, you can't do it by yourself," I told him. "You need help. You really need help."
"Are you talking about God?" he asked.
"Yep, that's it," I answered.
"I don't know how to find Him," said Phil.
"Well, I do," I responded. "You be back in this parking lot at five o'clock and follow me home. I'll have someone there to talk to you."
Phil agreed. Back in my office, I called Bill Smith, told him what happened, and asked him to come to my apartment at five fifteen that evening to talk to Phil. He said he would have to check his calendar.
"Check your calendar?" I said, almost in disbelief. "What on earth could be more important than this lost soul?"
He must have realized I was right, because he immediately said, "I'll be there."
### THE TURNAROUND BEGINS
When Phil walked into our apartment that night, the boys were so happy. The first thing they wanted to know was whether he'd brought back the big TV. All Phil could say was, "I didn't know I was supposed to do that." He looked around the sparse room where we had been living and said, "You should have gotten more stuff." It never was about stuff to me. The last thing we needed during that time was more stuff.
Bill Smith and his wife, Margaret, arrived right at five fifteen. Phil looked at him and immediately said, "I don't trust people."
Smith held up his copy of the Bible and asked, "Do you trust this?"
"Yes," said Phil. "And I am going to check out everything you say."
The preacher said to me, "Get a pencil so Phil can write everything down."
Margaret and I took the boys into a little back room and we prayed and prayed while Phil talked with the preacher.
When their visit was over, Phil said, "I'm not going to do any of this until I check it out."
Bill Smith came back and helped Phil study the Bible the next night and the night after that. I let Phil stay in the apartment with us, and he was so humble. He loved the boys—and that made all of us happy. The change in him was like night and day. The fourth night, I believe, I got home from work one evening, expecting to find Bill Smith and Phil studying the Bible, but I didn't see them anywhere. Our apartment was so small I didn't have to look very far. I can't remember now whether they did not leave a note or whether I just didn't see it, but I had no idea where they were, so I went to the church to look for them. When I got there, Phil was getting baptized!
That was just like Phil—to make up his mind to do something and then not even tell me or wait for me to get there after I had prayed so long and hard for that moment. It was okay, though. As long as he made Jesus Christ his Lord and Savior, I was happy.
Things did not change for us overnight, but they did change over time. Phil stopped drinking very quickly, and once he started studying the Bible, he never stopped. At times, as God was changing him, he had to suffer the consequences of some of the things he had done, but he has thoroughly and completely changed from the man he used to be. After a lot of hurt and disappointment, and a lot of prayer, God really did change him. He is now the kindest, most loving man I have ever known, and he is fearless about sharing his faith because he knows how much God changed his life. I can honestly say, after those terrible times in our early years, Phil truly became the man of my dreams.
## 14
## HAVE HOPE
## Lisa
As I mentioned earlier, I fell for Alan when I was in the sixth grade. When we finally got married, after some drama in both of our lives individually and in our relationship, I was thrilled! In our early days of marriage, I thought Alan walked on water. He was wonderful! My thinking he was too wonderful got our marriage off to a bad start, but I did not know that for several years. I did not realize I not only loved Alan, I actually worshipped him. He was more important to me than God was, and I had a greater love for him than I did for God. Having a terrific husband is a blessing, but when a wife gives her husband the worship God deserves, trouble is on its way.
Within the first five years of our marriage, we had two beautiful daughters. I wish I could say we lived happily ever after, and ultimately we did. But we went through a painful process to get there, a process that began when I was just a little girl.
### I SHOULD HAVE BEEN SAFE
I was the youngest of three children in my family. My brother was twelve years older than I, and my sister was seven years older. Because I was so much younger than my siblings and my mom worked outside the home, I spent most of my time at my grandmother's house. Until I started school, I was at her house five days a week. After I started school, I stayed with her every day during the summertime.
One of my favorite things about being at my grandmother's house was that she served eggs, bacon, biscuits, and sweet tea for breakfast. I do have some pleasant memories of being there, as a lot of people do, but I have a lot that are not so pleasant because something tragic happened to me at my grandmother's house, something that damaged me deeply and haunted my life for years.
One of the things Miss Kay and I have in common is that both of us understand firsthand how dangerous alcohol can be to a family. Miss Kay dealt with her mother's alcohol use and later, to a much greater degree, with Phil's problem. A number of people in my family abused alcohol, and both of my siblings eventually died young because of it.
As a little girl, I had an extended family member who had major drug and alcohol problems. Unfortunately, that person lived with my grandparents, so I had to see him often. Because I spent so much time at my grandparents' house, I was easy prey for him. My earliest memory of being molested was at the age of seven when he started to do things to me, things that made me feel bad and dirty. I don't remember how he threatened me (every abuser threatens) if I ever told anyone about it, but whatever he said worked. Chances are, he probably told me my dad would be upset with me if he found out. I adored my dad, and I would never have done anything to upset him. The abuser knew that.
One morning at my grandmother's house, I became very frightened and called my mother at work, begging her to come get me and take me home. I would not tell her why I wanted to leave, so she probably thought I had gotten into trouble with my grandparents. I don't remember what triggered my fear, but my mother did not come get me that day. I never told her or my dad about the abuse.
The abuser continued to abuse me whenever I visited my grandmother's house until I was about fourteen years old, when my grandfather died. During a big family gathering at my grandmother's house after the funeral, he got me alone. He was planning to continue his actions right after we'd buried my grandfather! But that time was different. I had had enough. I finally stood up for myself and told him that if he ever touched me again I would tell my dad and my dad would kill him. I am positive I was right about that.
After many years, I did tell my mother about the abuse because she babysat my grandchildren and I wanted to make sure they were protected. I also forgave the abuser eventually, but not for a long time—and not without first being personally devastated, not without deeply wounding the people I love most, and not without ultimately going through a powerful experience of having someone forgive me.
### UNFORTUNATELY, IT'S NOT UNCOMMON
As I thought about writing this book, I wanted to share about the abuse for a couple of reasons. First, what happened to me happens to many, many people. It's tragic, but it's common. Second, I want all abuse survivors to know they have hope. They can have hope for complete healing, hope for great relationships, and hope for a wonderful life, free from the lingering effects of the trauma they have suffered. I say this with complete confidence because after a lot of help from special people and a lot of healing from God, it happened for me.
Sexual abuse is always physically damaging; it's also emotionally damaging, and most people recognize that. I would like to add that for me, it was mentally damaging. Let me explain. Being abused did something to the way I thought about myself and about men in general. It set in motion some unhealthy thought processes that took root in my mind without my knowledge—and certainly without my agreement—and stayed in place for many years. To put it simply, I unconsciously began to believe my purpose in life was to please men. Therefore, because the abuse happened at such a young age, I never developed a healthy sense of identity and purpose for myself. This helps explain a lot of the bad choices I made while I was dating. Please understand that I take full responsibility for the wrong life choices I made. I did those things to the ones I loved. I committed those sins. But I do want you to understand that the sins of others can damage our thought processes.
### A CRISIS POINT AND A TURNING POINT
When I married Alan, I did not understand how faulty my thinking was, and I had no idea it would eventually lead to trouble in our relationship. In the late 1990s, for various reasons, I had an affair. It lasted fourteen months. Alan was devastated and eventually, so was I. When he first began to suspect something, I denied it. When Alan discovered hard evidence of my behavior, I finally broke down and told him everything. He asked me to leave our home and to tell our two daughters why I was leaving.
Alan was pastor of a large church at this time and though he did not ask me to do it, I decided to face the congregation one Sunday morning and tell everyone what I had done. I knew that in order to break the power of my behavior and to truly have strength to change my ways, I had to be accountable, and I chose to be accountable to a lot of people that day.
Our church loved me and surrounded me, although many of them were angry with me because of how I had hurt Alan. The Robertson family had every reason to be furious with me, and they were deeply hurt over the way I had treated Alan, but after some time, they handled the situation in a godly way. I always tell people Miss Kay is a gentle person, and I know that maybe better than anyone. Even though I caused her and the whole family great pain, she always treated me with gentleness during that terrible time.
In the aftermath of my confession, a special group of women spent time with me every evening and we studied the Bible and prayed for a new path for my life. One night after one of our group meetings, I was baptized in a friend's pool and felt relief and renewal and hope for the first time in a long, long time. The period of separation from Alan gave me a time of reflection that forced me to finally turn everything over to my Lord.
While Alan and I were apart, I lived with our next-door neighbors, who also happened to be our closest friends at the time and remain that way today. I went to their house so I could give Alan the space he needed and still take care of our home and our girls. Alan was not mean or ugly to me, but at the same time, he would not touch me or talk to me about anything except matters concerning the girls. I desperately wanted him to know what God was doing in my life, but I knew he did not trust me, so I kept a respectful distance.
### TOGETHER AGAIN
One night Alan called me and asked me to meet him. I was afraid and excited at the same time. We talked a long time about reconciliation and what it would take to survive all the trauma our marriage had endured and to renew our relationship. By the end of the conversation, I was filled with hope, but I also knew that because Jesus had by then replaced Alan as the Lord of my life, I could survive no matter what happened with us.
At that time, I was seeing a wonderful counselor who was helping me through so much. After Alan and I talked that night, we started meeting with her together. I was learning that I had to forgive all of those who had sinned against me in my past (like the family member who abused me), and I was learning to build honesty and integrity into every decision I made. I learned that I did not exist to please men, but that God had created me with a unique and special purpose to please Him. I also learned to love God's Word in a whole new way, especially the Psalms and the Proverbs, which were my food during this deep look at myself. One of my new favorite verses was Proverbs 24:26, which says, "An honest answer is like a kiss on the lips." I determined that honesty with God, with myself, and with Alan was going to be one of my new character traits!
After about two months of being separated, Alan and I were reconciled. We renewed our vows to each other and bought new rings to signify our new beginning. We did not do any of this publicly, but just between ourselves, because we wanted it to be about God and about our starting something new and special. We continued our counseling together until we felt ready to begin a newly healed life and marriage.
### STILL GOING STRONG
Alan and I are still going strong; we are more in love than ever and we have done our best to use the pain we have been through to help as many other people as we can. Our whole story is much longer and more complicated than I can share in this book. What I really want to communicate is that with God's help, a commitment to live by His Word, a loving family, a support system of fellow Christians, good counseling, and enough time, anything a person has been through can be healed, restored, and redeemed.
I have learned many valuable lessons during my life and in my journey as a Robertson—many of them the hard way through pain and difficulty. I can never say enough about how grateful I am that Alan and his family chose to forgive me, but one of the most important lessons I had to learn was to forgive myself. Without that, the forgiveness Alan, our daughters, our family, and our friends have extended to me would have been compromised. Had I never been able to forgive myself, I would have stayed stuck in my pain while others were moving on.
### THERE IS HOPE FOR YOU
To people who have been through situations similar to mine, especially those who have experienced abuse or unfaithfulness on any level, I would say you can always come home to Christ. He is big enough to handle anything you have done or anything that has been done to you. His sacrifice on the cross is enough to cover anything.
Human beings can do a lot of bad things, but nothing you can do will cause Him to turn His back on you. No matter what has happened, refuse to live one more day as a victim. Know that Christ forgives you, and let that empower you to forgive yourself for even the biggest things.
I could spend every day of the rest of my life lamenting how much my past actions hurt the people I love. I refuse to do that because life is too short and because I know that God and the people around me want me to be healed and whole and strong, using everything that has happened in my life to encourage and support other people in their difficult times and to be the best wife, mother, and grandmother I can be.
I spent years doing the work I needed to do, with God's help and the support of great people, and now I can live in total freedom from the past. When talking about things like this, Miss Kay has a short but powerful piece of advice: "Confess it, own it, and move on." Sometimes that takes a while. Getting to the point where you have the courage to confess may not happen overnight. Learning to own your thoughts, words, and actions may be a process. And you may have a few false starts before you get enough momentum to really move on. But even if it takes a while to work through your stuff, it's worth it. You, too, can ultimately end up in your very own happily ever after.
# Part Four
# TALKIN' ABOUT MY GENERATIONS
Lord, You have been our dwelling place in all generations.
PSALM 90:1, NKJV
## 15
## INTRODUCTION
## A Message from the Wives
Our husbands grew up with the same parents and grandparents, but obviously, all of us became Robertsons later in life. We come from a variety of family experiences, all of which helped determine our values, shape our characters, and make us who we are today. Some of us have always enjoyed the kind of family closeness that characterizes the Robertsons; for others, coming into such a tight-knit family has been a blessing we did not have a chance to experience in our younger days.
We all understand the importance of generations and generational blessings. When we get together with Phil and Miss Kay and with our children and Lisa's grandchildren, we have four generations under the same roof. That family line is a powerful thread running through all of us as Robertsons. But intertwined with that thread is another thread for each of us, the generational heritage we bring from our own mothers and fathers, our grandparents, and other members of our extended families. It's amazing how all of this works together to weave a tapestry of love and faith in our lives and in the lives of our children.
## 16
## AIN'T LIFE GRAND?
## Miss Kay
When I think about the old saying "Ain't life grand?" I can't help but think about my grandmother, Nannie, and my grandchildren. All of these people have made my life grand indeed.
As a child, I lived near my grandparents, but not quite close enough to walk to their house easily. My family lived just down the road from my grandparents, and our family store was almost exactly halfway between the two houses. Our little town of about three hundred people, Ida, Louisiana, was located on the highway that runs between Texarkana and Shreveport, right about the halfway point, so a lot of traffic came through each day. The traffic was good for business in the store, but it meant no one would let me walk to my grandparents' house alone. Someone always had to take me or, when I got a little older, watch me cross the highway.
I spent a lot of time at my grandparents' house and loved every minute of it. It was a safe and happy place for me, and I look back on the years I spent with my grandparents as some of the best years of my life. One of my biggest regrets is that I did not record or write down many of the things my grandmother told me—the stories of her childhood, stories about her faith, and the simple words of wisdom she spoke as far back as I can remember. Like most young people, there was a time I thought I would never forget those things. There was also a time when I thought I would always have my grandparents. I could not imagine ever being without them, especially without my grandmother—but my grandfather died when I was twelve. Those people are gone now, and some of the memories are already getting a little fuzzy. I would give anything to be able to pull out a tape or a journal and relive them. I try to encourage people I know to take time to record or write down the things that are important to them, especially family stories and special memories, because a time will come when they will want to read or listen to them.
### NANNIE AND ME
When I was growing up, in a small Southern town, the local store was not anything like the grocery stores or supermarkets of today. It was a gathering place. Life seemed to move more slowly than it does today, and people came to my daddy's store not only to buy their groceries but also to visit with their neighbors, talk about the news, and catch up on what was happening around town.
For as long as I can remember, I have loved and valued older people. Maybe that started during my days in the store. Especially in the wintertime, old women came to the store to sit around the heater. They were lonely (and cold), and the store was a hub of life and activity for them. I was a very social child, and I totally enjoyed visiting with the older people in our town. They called me "Little Katie Carroway," and I thought that was cute. I did my best to engage them in conversation or to entertain them with stories about my animals.
But there were no old people I loved like I loved my grandmother. In addition to sitting on the swing, waving at cars, and talking every evening, we did all kinds of other things together. For example, I had the cutest little black-and-white Shetland pony at my grandparents' house. His name was Tony, and he was so patient with me! I would ride him while my grandfather plowed, then Nannie would tie him to a tree and watch me do tricks while he just stood there and let me do whatever I wanted to do to him. In those days, I wanted to be just like Annie Oakley. I had a cowgirl outfit, and I would turn flips off of Tony. I had the time of my life with that little horse, and I think Nannie enjoyed it too.
Nannie had a pet too—a little fox terrier named Lady who really could do tricks. She could jump up in the air and do all kinds of things, even flips. Nannie and I were just crazy about that little dog. We never knew exactly where she came from but decided she must have been traveling with a family who stopped at the store and accidentally left her there. I cannot imagine anyone would have left behind such a good pet on purpose, especially in those days, but at least she went to a good home. Nannie kept Lady for the rest of that dog's life.
Nannie taught me all kinds of things about nature, such as the fact that there are "nice" caterpillars and "bad" caterpillars. I learned at an early age to tell the difference between the two and I collected as many nice ones as I could find. But I did not just collect them; I made a circus out of them! I had all kinds of little circuses when I was a child, using any kind of animal I could corral—earthworms, turtles, nice caterpillars, whatever I could find. Daddy always told me to stay away from skunks, and he would not let me touch squirrels, so most of my circus animals were small, but I just loved them.
One day, I put an entire circus of caterpillars in a box and took it to my daddy's store. The old people who sat around visiting with each other thought the circus and I were great entertainment. They thoroughly enjoyed it until all the caterpillars got loose. That was the end of my traveling circus, but I kept a circus of various types of animals at home and at my grandparents' house for years.
When I was not playing circus, I liked to play beauty shop, and Nannie never minded being my model. She had very thin hair, which she never cut—ever, not once in her whole life. It grew down her back to a certain point and then stopped. I remember many times when she sat patiently while I teased it, braided it, or put it up on top of her head. She also enjoyed sitting and watching me try on her hats, scarves, and jewelry while parading in front of the mirror.
One thing I really liked about my grandmother was that she was definitely not a neat freak. I am pretty sure I inherited that from her, and I think all my family would agree. No one could ever rightly accuse me of being too neat. I loved my grandmother's relaxed approach to everything and the fact that her house was often a little messy, because that made me feel at home. I never was afraid I would lose or mess up anything. I felt free to explore Nannie's house, play with her things, and let my imagination run wild.
My aunt Georgie visited my grandmother a lot and even though she was actually Nannie's sister, she called Nannie "Mother." She was a neat freak, and she had a habit of getting onto Nannie and me when I played with Nannie's nice things. I cannot even count the times I heard her say to Nannie, "Don't let Katie play with that!"
Nannie had a set of beautiful demitasse cups I used to play tea party. No one was there, of course, but I liked to pretend I had a group of friends around me and all of us were drinking tea from Nannie's special cups. That especially got on Aunt Georgie's nerves. One day, as usual, she saw me playing tea party with the cups and said to Nannie, "Don't let her play with those. They are expensive and they are nice, and she will break them."
I knew the cups were special, and I was very careful with them. I felt so good that day when Nannie responded to Aunt Georgie, "Number one, she would only break something by accident. She would not do it on purpose. And number two, those cups are just things, and people are more important than things." Nannie knew how much fun I had at my tea parties and my having a good time was much more important to her than making sure nothing got broken.
I am so glad my grandmother felt the way she did about her things. She always kept her possessions in perspective, and I had hours of enjoyment at her house because she was that way. I try to be the same way with my grandchildren. I do not want them coming to my house and being afraid I will get upset if they break something. Over the years, society has put so much emphasis on stuff—having nice stuff and getting more stuff. We have become a culture of accumulation and "What about me?" I never heard those kinds of things growing up. People worked hard to get what they needed and wanted; they took care of what belonged to them and they shared with others in need. People were not so focused on getting more and more stuff.
I wish we had not lost that attitude and generous spirit in our country because it was a very good way to live. In recent years, with 9/11, we have seen that many people still have the ability to reach out, pull together, share with others, and think about others. A tragedy will almost always help people return to the things that really matter. But I am afraid we have lost our ability to be content and to care for others, and I wish we could get it back.
### SOMEONE'S IN THE KITCHEN WITH NANNIE
As I mentioned earlier, my grandmother always seemed to be cooking. She was often the first person in our community to take food to people who had sickness or a death in their family. I saw her do that often while I was growing up, and other ladies in the community did the same. If a family lost a loved one or was dealing with an illness, they hardly ever had to think about what they would eat. The women of the town provided for them. Taking food to people in need was a way of showing them you cared about them and about what they were going through. That doesn't seem to happen much anymore; people do not reach out to others as much as they used to. Back then, food was a way of showing love, and for me, it still is.
These days I try to teach my grandchildren about cooking and to pass on my love for cooking to them. Cooking is not something I do because our family has to eat; it's something I do because I thoroughly enjoy it. My love for cooking started in childhood in my grandmother's kitchen, and now I want my grandchildren to learn to love cooking in my kitchen. They love my biscuits, and if they don't learn anything else, they will learn how to make those! If you want to learn to make them, too, the recipe is in the section of this book called "Inquiring Minds Want to Know" and also in my cookbook entitled Miss Kay's Duck Commander Kitchen.
### FUN TIMES TOGETHER
My grandchildren are always eager to hear stories about my childhood, whether I talk about being in the kitchen with Nannie, doing tricks on Tony, or loving the old people at the store. They are interested in stories about their fathers when they were young. I sometimes rely on Alan to talk about those days because he has a great memory and remembers a lot of things I have forgotten. They also love to hear about the things Nannie and I used to do together. I try to do with them some of the same things I did with her years ago, and we also do new things I have thought of recently.
I love to laugh with my grandchildren. I want them to have fun with me, and I try to make just about everything an adventure. One day I decided to take them to visit Phil's sister, who lives just over the hill and through the woods from Phil and me. There is a well-worn trail Phil and I normally take to her house, but that day I thought I would give the children a little adventure by taking a different route. I led the way down an overgrown path that was completely covered with vines. We almost felt like we were pioneering through an African jungle. The children loved it, and I thought it was fun, too, until I got so tangled up in a bunch of vines I literally could not move. I didn't want to frighten them, so I started laughing and crying, "Help me! Help me!" very dramatically. To this day, the children have no idea I was really stuck. I truly could not get out of those vines. I was laughing so hard they thought I was kidding. Thankfully, someone came to my house while I was all tied up, heard us laughing in the distance, and came to my rescue. Now that was an adventure, and the kids just love to laugh and retell it.
My adventurous spirit is something that has grown in me over the years. I certainly was not born with it. My sister seemed to get all the adventure genes in the family, while I was more like Chicken Little. Except for doing tricks on my pony, which some people would say takes bravery, the most adventurous thing I ever did as a child was to skip one class while others skipped a whole day of school. They thought playing hooky would be so much fun, but I was afraid to do it. While they were out having fun, I decided to hide behind the Coke machine during one class period. It was no fun at all! I was bored and scared I would get caught. Over the years, maybe from living with a man as adventurous as Phil, I have learned to look for adventure and embrace it, especially when the grandchildren are around.
Sometimes, when we have had enough adventures for a while, we play a quiet little game called the Listening Game. I say to the children, "Close your eyes, and just listen." The children sit for a few seconds and say they do not hear anything. I tell them to keep listening. Eventually the children hear different kinds of birds, frogs, squirrels, and other animals—even a woodpecker pecking on a nearby tree. By the time the game is over, they are delighted with the sounds of the outdoors.
### THE GREATEST STORY OF ALL
I love being able to spend time with my grandchildren and having fun with them. I am so thankful for the things we can do together and the good times we enjoy. I am glad they are interested in the stories I tell, but there is one story I want to make sure they know better than all the rest. It's the story that determines everything else about their lives, the story of Jesus.
When I talk to the children about Jesus, I use a simple illustration of symbols across a page. You can see it in my own handwriting on the dedication page of this book. A downward-pointing arrow means, "Jesus came." In other words, He gave up everything wonderful about heaven and came to live on earth, where everything is not always wonderful. He came as a regular boy and grew to be a man. The next symbol looks like a large "plus" sign, but it's really a cross and it means, "Jesus died." He died for all of our sins. The next symbol looks kind of like an extended letter "n," and it represents the tomb where Jesus' body lay for three days after His death. The next symbol is an upward-pointing arrow, which means Jesus ascended to heaven. The final symbol is another downward-pointing arrow, representing the promise that Jesus will one day return to earth and take us back to heaven with Him if we have trusted Him as our Lord and Savior.
When my boys were little, I made sure each one knew this story as soon as he was old enough to understand it. Now I do the same with my grandchildren. Sometimes, just to reinforce the power of the story, I talk to them about Jesus' crucifixion and say, "All those people treated Jesus so well. They took Him out to eat and bought His dinner . . ." Someone, usually Mia, quickly interrupts with, "No, Mamaw Kay! They didn't! They put real nails and thorns in His hands—and thorns hurt."
I like to explain to the children that no one gets buried in a tomb today, at least not where we live. As I talk about that, the children like to envision what a dark, musty tomb would have been like. When we talk about the tomb, we talk about dying and going to heaven. Then we talk about how a person's spirit goes to heaven when he or she dies and then gets reunited with the body when Jesus comes back. And we talk about the fact that while we are living here on earth, we can talk to God any time we want, knowing He is always listening when we pray.
When Phil is around and I am reminding the children of the Gospel story, he waits for me to get to the part about Jesus' ascending back to heaven and inserts, very intensely, "He probably went up just like a rocket ship!" They are fascinated by that thought. I don't know if it was really like a rocket ship or not, but it makes a great visual for little ones. I love to talk to my grandchildren about Jesus and to ask them questions to make sure they understand this awesome story. They do.
I am confident they know and believe the truth of the Gospel, and I am thankful they all have parents who also make a priority of telling this story in their homes and of taking the children to church so they can learn even more about God.
The grandchildren also like to hear other stories from the Bible, and I enjoy reminding them of these stories that so many generations of people, including me, grew up on. Their favorites are the ones about God's miracles or other things that really stand out, such as Jonah and the whale or Balaam and the talking donkey. Of course, all of us Robertsons love animals, so we tell the story of Noah and the ark. I take every opportunity I have to remind the children of these stories because I want to make sure they never lose sight of everything God has done and everything God can do. I know the children understand because I sometimes hear them retelling the stories in their own words, and their versions are so sweet.
I love the innocence and faith the younger grandchildren have, and I hope they never lose it. I also love the boldness and faith the older ones are developing. They just amaze me. When they have opportunities to speak, they do not shy away from sharing their faith. They are bold about it, and I am so proud of them for standing up for what they believe.
### USE YOUR IMAGINATION
When I was a child, there were no such things as cell phones or electronic games. I always wanted to be outside with Tony the pony or with my circus, so I was not the type of person to sit inside and move little things around on a screen. I would not have sat in a chair texting people because I wanted to be at the store interacting personally with the people of our community. I understand how helpful technology can be, but I do not want it taking over my time with my grandchildren, so the little ones know they cannot bring their phones or latest gadgets to my house. When they're with me, we do artsy-crafty projects, we collect interesting things from nature, and we talk about how unique they are and how God created them.
One day when we were outside, Lily caught a granddaddy longlegs. It was harmless of course, but where we live we also have black widows and brown recluses, which are not harmless! I asked her to let the granddaddy longlegs go, but she did not really want to. I explained that the younger children in the family were watching her and even though she knew the difference between harmless and dangerous spiders, they did not. She understood right away and did not want to set a bad example for them. That did not stop her from catching a garden snake a few weeks later, though!
I will never forget my circus and when I think about it these days, I realize being able to do that and to have some encouragement from my family in it really helped me develop my own curiosity and imagination. I try to revisit that with my granddaughters and River because I think imagination is a lost art in modern society. Any time I can encourage the children to be creative and to use their imaginations, I do.
One of my favorite little happy things to do is to give the children prizes—not anything expensive, just something that communicates, "I was thinking of you." Sometimes, River loses his prize while he is at my house. When I find it, I give it back to him, and he thinks it's new. I am not sure how long that will last (at the time of this writing he is only five years old), but I am going to make the most of it while it does.
### LICKETY SPLIT
Phil and I have several small buildings on our property, many of which were used for Duck Commander at one time. In one of those buildings close to our house, I made my granddaughters a little playhouse called the Lickety Split. It's just for the girls because little girls have certain things they really like to do. It's just precious what they are into. For example, my granddaughters like to have a play office and a place to do little performances. In 2013, I hired some decorators to improve the Lickety Split. They created a stage and place to sit, and put in a karaoke machine, a keyboard, and a microphone. They also installed a play kitchen they call a bistro. In another part of the room, there is an old dresser where the girls can do their hair and play dress-up. The ladies did a great job with it; it's darling.
As much as I appreciated the decorators' talents, I was most excited about the fact that once they finished their work, the girls took over. They made it their own. They do some things the decorators might call "messing up their work," but that's not the case. The girls are simply creative, and like most children, they want to put their stamp on things. I was happy to help create the Lickety Split for them and do my best to make it something they can enjoy, but I was even happier to see them use their imaginations and make it even more of what they wanted it to be.
### I LOVE THEM ALL
All of my grandchildren are a blessing. They are all different and all special. They seem to trust me, and I treasure that. I would not trade their trust for anything in this whole wide world. I enjoy them so much, and one of these days, say, when they are my age, I want them to really know and understand just how much I loved them. I hope they will always think of me as a fun grandmother who loved to laugh and as a positive influence in their lives. I want to be someone who teaches them about life and faith and about right and wrong. I want to be to them everything my grandmother was to me, and more. I'm working on it every day. I do not want them to think of me as a cranky ol' grumbling grandmother; I like to say I want to be a sunshine grandma.
I know Korie's grandmother, and I know Jessica's grandmother, both lovely women. When I look at Korie and Jessica, they both have so many of the qualities I see in their grandmothers, especially their sweetness and humility. In years to come, I hope people who have known me will know my grandchildren and be able to say, "She got that from Miss Kay," or "He's that way because of what Miss Kay taught him." Most of all, I want them to grow up to be people who love and obey God, love their families, and know the difference between right and wrong. I pray I am a godly influence and a blessing in their lives, just as they are in mine.
## 17
## A LEGACY OF LOVE
## Jessica
I grew up as part of a large extended family, and a lot of our family gatherings took place at my grandparents' house. I have such happy memories of being at their house with my aunts, uncles, and cousins. As a child, I was a tomboy, and I climbed trees and rode four-wheelers with my cousins. I also remember being so excited to get to go with my mamaw Nellie and papaw Ted (my mom's parents) to their garden and pick vegetables. Some of my favorite times were when our entire family gathered for a big feast after church on Sundays, the same kind of large meals Miss Kay prepares for the Robertson clan today.
These memories, and the fact that I grew up around such hardworking, faith-filled, wonderful people, are part of the reason I grew up loving my family. For years, I never really thought people lived any other way. Not spending time with family was never an option for us. Now I am sad when I think about people who did not have that kind of upbringing or children who are not able to experience the blessings of a close-knit family. Don't get me wrong, no one is perfect and no one's life is perfect. My family had struggles just like any other family, but we loved each other unconditionally—and unconditional love never gives up on anyone. We stood by each other, and when one member of the family felt broken, the others helped lift that one up. The same is true in the Robertson family; no one views himself or herself as better than anyone else. We all love each other and want each other to succeed.
### I WAS MEANT TO BE A ROBERTSON
Sometimes I think I was always destined to be a Robertson because I come from a family full of hunters and fishermen—on both sides. One of my grandfathers, Papaw Ted, worked in a chicken plant to make a living and had a hunting camp as a hobby.
My other grandfather, Papaw James, and grandmother, Mamaw Lola, had a fishing camp. I remember lots of lazy summer days when the men in our family went fishing early in the mornings and the rest of us swam and soaked ourselves in the sun until they got back. When they returned, we all took the boat out again for an afternoon of water skiing, tubing, and knee-boarding. At night, we fried the morning's catch for dinner and always had delicious meals because both of my grandmothers were great cooks.
When I think back to my growing-up days with my extended family, I also remember being so excited as a little girl, at early elementary school age, when I went hunting with my dad, my uncles, and my grandfather. On those cold mornings, they let me drink coffee with lots of cream and sugar. That was a real treat, and I loved it!
Both of my grandfathers, my uncles, and my dad were all outdoorsmen, so part of my generational legacy is part of Jep's legacy too. We both come from people who know how to bait hooks and load rifles—and how to clean and eat what they kill. Both of our families include careful, responsible hunters and fishermen who understand nature and the life cycles that take place in it. They know how to respect and work within the natural order of things.
I also come from a line of women who know how to garden and how to cook—not just cook, but how to cook really well. In the South, that is important! I love and appreciate my family so much, and I am thankful for the great family experiences I had growing up.
Like me, Jep also loves all of his family and is very loyal to them. He had especially good relationships with Granny and Pa. Because they lived so close to Phil and Miss Kay when Jep was a little boy, he stayed with them a lot while Miss Kay was busy working. In fact, Granny taught Jep how to bait a fishhook and how to play hearts and gin rummy. Miss Kay likes to talk about the fact that Granny and Pa taught the boys how to play dominoes, which really helped them with their math skills. Now she is a lot like they were because she loves combining play and fun things with learning opportunities. Both Granny and Pa taught Jep and his brothers a lot of important lessons about life, which they still remember and are passing on to their own children.
Our daughter Merritt is named after Granny. When all the grandchildren were born, Granny made each of them an afghan. When she found out Merritt was her namesake, she made Merritt two!
### THE VALUE OF FAMILY
Jep and I are doing our best to teach our children the importance of spending time with family. We understand the value of the generations and want our children to know and love their grandparents the way we knew and loved ours, so we try to make sure they see their grandparents several times a week. I sometimes hear about people who do not get along well with their in-laws, and they hinder their children's relationships with their grandparents. This always makes me sad because I believe grandparents add so much to children's lives and that young people suffer when they cannot be around their grandparents. The love of a grandmother is different from the love of a mother; the same is true for fathers and grandfathers, and children can really benefit from all the generations before them.
Most Wednesdays Miss Kay takes the girls to Outback or Cracker Barrel and then to Hobby Lobby for little prizes or trinkets. She also takes them to libraries and reads to them. And, of course, she lets them check out books from "their" library at her house.
I want my children to spend as much time as possible with all of their grandparents. My parents have something special to offer them, just as Phil and Miss Kay do. I believe the children need to hear stories about Miss Kay's hard times and they need to learn from Phil how to get along in the woods. They even need to know Uncle Si went to Vietnam and risked his life for the freedom they now enjoy.
### OUR DREAMS COME TRUE
The children also need time with Jep and me and with each other. Jep and I really make an effort to have family night with our children at least once a week. Jep and I both always wanted a large family; that was something we talked about and dreamed of before we married. Now we have been blessed with four awesome children.
Lily, our oldest daughter, has always been so sweet and loves everyone. She was born in 2002 and went through a lot of illness during the first few years of her life, including a weeklong hospital stay for rotavirus when she was two years old. Through all her illnesses, she remained the sweetest little girl, not crying or fussing very much, even though I know she was uncomfortable.
Lily has always been quiet, and she gets that from Jep, not from me. I am definitely a talker! Lily is shy, but she is also smart and hardworking, and she makes good grades in school. She loves babies and music. In fact, not long before I started writing this book, she started taking mandolin lessons. She also has good athletic abilities, and whether she is playing a sport, doing her homework, or doing something else, she gives 100 percent. I love Lily's sweet spirit and the way she loves everyone.
Merritt is our second child and second daughter. She was my hardest baby and literally cried almost all the time; but she is a surprisingly easy child. She is strong, spunky, and independent. Like her dad, she does not feel the need for other people's approval (I am much more of a people pleaser). Because Merritt does not make decisions based on what other people think, she is really good at choosing friends. She decides whether or not to be friends with people based on their character and personality, not based on whether they are considered "popular." I love the confidence I see in her. Merritt is a talented singer and pianist, and she took up the guitar not long before I started writing this book. She is also a pretty good golfer, and Jep is really glad about that!
Priscilla is our third child and third daughter. Of all our children, Priscilla looks the most like Jep. Her baby pictures are almost identical to his. She is definitely our sassiest child! She is extremely competitive, yet she is also very sweet, and she loves to hug and give kisses. Physically speaking, she is tough, and she does not recognize her own strength. Because of her strength and toughness, I will not be surprised if she excels in sports someday. Priscilla is also very girly. She loves to wear dresses, curl her hair, and play with baby dolls. She also plays the piano and takes fiddle lessons. Jep says I baby her more than the others, and I just say, "Well, she will always be my baby girl."
Our fourth child and only boy is River. He is the last of the Robertson grandchildren and the last of Jep's and my children, so I have to admit he gets a little spoiled. He is such a cutie, with Jep's dark hair and blue eyes, and those famous Robertson dimples! At the time of this writing, River is five years old. I know he will have another leading lady in his life someday, but for now I am enjoying this time when he wants his mama more than anyone else. He is "a lover and a fighter." He loves on me all the time, but he also gets physical with his sisters over something on a daily basis. He can be a little stinker in that way, but he really loves his sisters. I know that being around all these girls will help him be a better husband one day.
### THE PRIORITY OF FAMILY TIME
With our busy schedules, family time together is not always easy for us, but it is important, so we try to prioritize it even if we have to sacrifice other things. "Family night" means different things at different times; the only requirement is that we spend time together as a family. Sometimes, we all cook and eat outside together. Then Jep and I watch the children play and light the fire pit so we can end the evening with s'mores. Sometimes we play games or watch good, clean, wholesome movies.
On some family nights, Jep and I teach the children a Bible lesson because we know how valuable God's Word is and how much it can help them in life. We want them to know what it says and to be able to understand what it means so they can apply it in their lives, even at their young ages.
As often as possible, whether it's family night or not, Jep and I cook dinner for the children. Well, actually, Jep prepares the meals and I clean up. I absolutely love the fact that my man can cook! Two of our favorite family activities are playing Monopoly and making the kids' all-time favorite dessert, homemade snow cones. Several times a week, Jep takes the kids outside to hit golf balls and they have so much fun doing it. Jep has fun, too, but I am pretty sure he secretly wants one of them to turn pro!
Recently, we came up with an idea for an activity that has had such a positive impact on our family. We give everyone a chance to draw a word out of a hat. Every word in the hat comes from Galatians 5:22–23, a scripture passage that refers to "the fruit of the Spirit," meaning the qualities of a person whose life is filled with and led by God's Holy Spirit. It says: "The fruit of the Spirit is love, joy, peace, patience, kindness, goodness, faithfulness, gentleness, and self-control."
We have nine words in the hat, and we draw one each week. I post the word on the refrigerator, and it becomes the character quality we all work on for the next seven days. Let me tell you, my children do not forget what we are working on that week, because when one of them does not display that particular fruit, the others will let him or her know. I can't count the times I've heard someone say, "You are not being patient with me," or "Remember, we're supposed to be working on kindness this week, and you are not being kind!"
Like any mom, I would really like for each person to work on his or her own fruit before pointing out someone else's shortcoming, but I have to say: it works!
In our family, we make it a point to pray together as much as we can. I believe some of the sweetest prayers ever prayed come from children, because they are so pure-hearted, so innocent, and so full of faith. As a mom, I just love to hear them talk to God, and of course, I love it when they see the answers to their prayers.
Family—and everything family means to us—is the legacy Jep and I want to leave our children. A big part of that legacy is a commitment to faith and love. My parents have lived that way and so have my grandparents. That's what I have seen in them, and that's what I want our children to see in Jep and me. If they do, we will have succeeded. And if Jep and I can raise children who live according to the fruit of the Spirit, know and obey God's Word, pray, and love their family, we will have accomplished something very fulfilling, and we will have given them the foundation they need in order to build great lives in the future.
## 18
## JUST CALL ME MAM
## Lisa
I am so glad Miss Kay and my sisters-in-law have happy memories with their grandparents. My memories are less pleasant, simply because my grandparents' house was the place where a member of my extended family abused me for years. Even though I spent a lot of time there, much of it was filled with fear and pain. But my grandmother was a very sweet lady, and my mother tells me I received some of my physical attributes from her. If you wonder what I mean, let me get specific: hips and rear!
My grandmother and I spent a lot of time together during my childhood, and when I stayed with her, she took me fishing about three times a week. That is how I learned to spit, potty in the woods, and bait a hook. Those times with her were also where I gained my love for fishing with a cane pole. She could slap that thing down in the water like nobody's business! When she took me fishing, we caught bream and then went home to scrape 'em, clean 'em, and eat 'em.
My grandmother also let me help her in her garden, and I still love to shell peas and shuck corn because of her influence. I spent a lot of time with my grandmother; I do not fault her for the choices my relative made, but my memories sometimes get a little skewed when I think of my childhood with her.
My biological family has endured a lot of sadness and hardship over the years, so sometimes looking back is difficult. I lost my father to cancer, my sister and brother to alcoholism, a nephew to a motorcycle accident, and many aunts, uncles, and cousins for different reasons. I am the only child left to my mother. She has really been through a lot of heartache, but she has endured, and we are held together by our mother-daughter bond and Christ's love. She is very strong and opinionated, and Alan will quickly say I received that trait from her.
I don't blame my parents for what happened in my childhood. They didn't know! When I did tell my mother about it, she confronted our relative, but he denied it or said he didn't "remember doing that." I have told her it has been dealt with and Christ has healed me. She was glad for that news, at least.
My mother has found joy in keeping my grandchildren while Alan and I work, and they share a close relationship with her. Even though thinking about my past is difficult, thinking about the family I have with Alan and looking forward—now that's a different story. When I think about the generations in our family, the people who bring me the most joy are my daughters and grandchildren. I believe our daughters, our sons-in-law, and our grandchildren have a bright future built on a strong relationship with God, and that gives me great hope for the days ahead.
Earlier in the book I wrote about the troubles Alan and I faced at one period in our marriage. There was a time when we both wondered whether we would still be together all these years later, and we thank God we are. I want to pick up that story now where I left off.
I was only eighteen years old when Alan and I married, and as I mentioned, we spent the early days of our marriage living with Granny and Pa on Phil and Miss Kay's property. As a young wife, I had great role models in Granny and Miss Kay. In fact, Miss Kay taught me some lessons in those early years that saved my marriage to Alan later on. I learned from her to fight for my marriage. Though her troubles with Phil were different from the problems Alan and I went through, she had a level of commitment to her husband and her marriage I had never seen before. I knew she had been through extremely difficult times and refused to let them get the best of her. I also noticed how genuinely loving and respectful she was toward Phil. She never held anything against him. When she forgave, she forgave completely, and that laid the groundwork for the wonderful marriage she and Phil have today. She taught me how to love, especially how to love Alan, and even though I seemed to abandon those lessons for a while, I now put them to good use every day. As a young bride, I had no idea how powerful those things I learned from her would be for me, but I thank God I learned them.
### THE TABLE: MORE THAN JUST A PIECE OF FURNITURE
One of the best things about those years was the time our family spent together around the table. Robertsons are people who value the family table, not just because it's filled with some of the best food on earth, but because it provides a place for conversation, storytelling, talking about the day, and sometimes a spirited disagreement—which all help build strong family relationships. When Alan and I lived on Phil and Miss Kay's property, the whole family ate together almost every night. Miss Kay usually did most of the cooking, but Granny and I pitched in a little. Granny and I took our food up the hill to Phil and Miss Kay's house, and we all sat down to eat together. Occasionally, when Miss Kay was too busy to cook, Granny prepared our meals and we all went to her and Pa's house to eat.
In many parts of our society today, the family table has disappeared and a lot of people have lost something vital in family relationships because of it. When I was young, and even after I was married, everyone ate what was served and we ate when it was ready. Children did not eat alone in front of the television or in their rooms with a fork in one hand and a cell phone in the other. Families actually sat together, prayed together, and talked as they shared their meal. Around the table, people knew they had a place where they belonged, a place to disagree and still be loved, a place to talk about what was going on in their lives, and a safe place to learn relational skills. I believe that concept of the family table is one of many values of the past that contributed to strong families, which in turn made strong communities and built a strong nation. In our family, we still value the table, and I know many others do too. I am thankful for that, because I believe it is so important. It certainly is for us, and it always has been. It is one part of what makes us Robertsons who we are.
### A NEW GENERATION OF ROBERTSONS
Within the first five years of my marriage to Alan, we had two daughters, Anna and Alex. Anna was born prematurely, at twenty-nine weeks. She was twelve inches long and weighed one pound, fifteen ounces. The first several months of Anna's life, she lived in the neonatal intensive care unit and underwent serious heart surgery. Alan and I were filled with fear and uncertainty, not sure whether she would live or not.
Thankfully, Anna survived and soon became a happy baby. Miss Kay had taught me how to cook when Alan and I were first married, and she taught me how to be a mother after Anna was born. I don't know what I would have done without her and Granny!
After raising four boys, Miss Kay was so excited to have a little girl in the family. When Anna was born, Miss Kay bought paper dolls, Barbies, hair bows, dress-up clothes—anything she could think of that was girlie. Even though she had to wait a while before Anna was old enough to play with some of those things, Miss Kay had them ready. She and Anna are still extremely close. I learned early in their relationship something I missed in my own childhood—the legacy of having grandmothers and being able to learn from them is amazing.
Anna was strong-willed as a child but became very mild-mannered and compliant as a teenager. Some people might say that could never happen, but it did. The opposite happened with our daughter Alex. She was an easygoing child but followed a more typical pattern when she reached her teenage years and became rebellious and stubborn.
At age eighteen, in a decision that really disappointed Alan and me, Alex was doing some things we did not approve of and decided to move away from home. Alan and I loved her but did not think she had made a wise decision. However, we did not see any wisdom in trying to force her to do what we wanted her to do. We refused to take away her power of choice. We hoped and prayed she would come to realize her mistakes on her own and choose to do differently. We were always there for her when she wanted guidance and we gave her good advice. Then we waited for her to decide what to do with that advice and to figure out how she wanted to live her life.
Several years later, I received the following note from Alex.
. . . I just wanted to tell you how much I love you and admire you. Sometimes when I was younger, we didn't get along because I didn't understand why you wouldn't just let me do what I wanted to do! But now I see that you did everything because you love me and you wanted me to stay out of trouble! I am glad you have become not only my mom, but one of my very best friends as well. When I think of the perfect mom, I think of you. I have so much respect for you now that I am old (and wise) enough to see all of the great, wonderful qualities in you. My admiration for you tripled when I heard your full testimony in Africa. To be able to go through all the things you have gone through and still come out a loving, godly, sparkling, shining example of what a woman should be is very inspiring. I hope I wind up just like you some day! I love you and Daddy with all my heart!
Today, Alex has graduated from culinary school, and she is a fabulous cook. She and her husband, Vinny, live close to Alan and me in West Monroe, and our relationship with her is completely restored.
### THE SPECIAL GIFT OF GRANDCHILDREN
Because of choices I made when our daughters were young and what they lived through during the time Alan and I were apart, I have told them, "I may not have been the best mother and I'm sorry, but I promise I am going to be a great grandmother to your children." I work toward that goal every day.
I have learned a lot about being a grandmother from Miss Kay, and she is exactly the kind of grandmother I want to be. She is just great with all of her grandchildren. They all think she is so much fun and they love her dearly. She is such an encourager to everyone, and she definitely encourages her grandsons and granddaughters in every way she can. She makes an effort to spend as much time with them as possible, and that has not been easy since Duck Dynasty started. But she works hard to arrange time with them. She teaches the girls how to cook, she plays Barbies, she sings in the car, and she has created an outright fun masterpiece with the Lickety Split.
At the time of this writing, Alan and I have two granddaughters. Anna and her husband, Jay, are the happy parents of Carley, who was born in 2005, and Bailey, born in 2007. Alex and her husband, Vinny, are expecting our third grandchild. It will be our first grandson in March.
Our grandchildren are such an awesome blessing. Being a grandparent is completely different from being a parent, and Alan and I love it! Of course, we think our granddaughters are adorable and fun. Like many grandparents, we think our granddaughters are the cutest little girls who ever lived. And they are. But for Alan and me, the granddaughters are so much more than cute. They are an amazing symbol of God's restoration and healing power. When we think back on our years of doubt, heartache, and possible divorce, we realize that our lives—and the lives of our children and grandchildren—would be totally different had we not made the choices we made. Had we decided not to stay together, we would not be able to do any of the fun things we do together now. We would not take our granddaughters on trips (which we do as often as possible), and we would not enjoy having them spend the night with us (sometimes multiple nights in a row) or special holiday times together. Our grandchildren would have to visit us in separate homes and miss the togetherness and security of being surrounded by the extended family into which they were born. Personally, I realize I would also have missed the joy of knowing my Robertson nieces and nephews. Had Alan and I divorced, generations of Robertsons would have suffered. Because God gave us the grace to tough it out, and because Alan forgave me and the Robertsons embraced me again after I had failed, our whole family is happy together. It was not easy, but was definitely worth it. I believe our grandchildren are God's gift to Alan and me because we did not give up. We love spending time with them. We deeply love both of our daughters, but we both agree—as most grandparents would—that God has given us something special in our grandchildren.
When the time came for Alan and me to decide what we wanted our grandchildren to call us, we landed on Mam and Pap. The parents in a book I read years ago used these names and I always liked them, so I started calling myself Mam, and it stuck. When our granddaughters were learning to say "yes, ma'am" and "no, ma'am," "yes, sir" and "no, sir," they did fine with Alan, but when they spoke to me, they had to say, "Yes, ma'am, Mam." They still say that, and it is so cute.
As much as Alan and I love our grandchildren, we want our daughters and sons-in-law to be their parents. We do not want to overstep parental boundaries. We want to influence in the right ways, but we do not want to take on roles that are not appropriate for us. When the grandchildren make a mess at our house, we insist they clean it up, but we do not discipline them in ways that are better left to their parents.
Alan and I also let our grandchildren know they can tell us anything. I stress this point because of my past. I tell them there are no secrets between them and Mam. I even go so far as to talk about appropriate touches and who should touch them, and where it is okay for them to be touched. I also tell them that no matter what they tell Pap and me, we will never stop loving them. We want them to know, as we know and our daughters know, that we are family and we do not give up on one another. Because of our story, that really means something.
## 19
## A HERITAGE OF FAITH
## Korie
I come from a long line of people who value faith and family, just as the Robertsons do. From both my dad's side, the Howards, and my mom's side, the Shackelfords, my children and I have a generational heritage of dedicated Christians who loved the Lord, loved others, and made a big difference for God in the world around them. I am thankful for that legacy and enjoy seeing specific traits that have been passed down to our children and are helping shape them into the individuals they are today and will become in the future.
### OUR OWN LITTLE FAMILY
John Luke is our firstborn. When I think of him, the first word that comes to mind is "adventurer." He loves to meet new people and will stop by a tent revival or county fair on the side of the road in a heartbeat. He is the first to stop and pull a truck out of a ditch or give money to someone in need. When he was a little boy, he often made elaborate maps of the woods around our house and loved to take off on his own little explorations. Whatever he is doing and in whatever circumstance, he is happy. For as long as I can remember, every time I ask him how his day was his response has always been, "Great!"
John Luke sees the good in life and in others. Like his dad, he is fun to be around and is quite the romantic. He loves to make others smile and plan little surprises that will make the ones he loves happy—whether that's a friend, his girlfriend, or even his mom.
John Luke is a passionate follower of Christ. I could not have been more proud than when I looked at the "Notes" section on his iPhone and found page after page of Bible studies, favorite verses, and notes he had taken from lessons at church. John Luke believes he can change the world. He doesn't make the excuse that since he is a teenager he is supposed to "sow his wild oats" and rebel against his parents. He's out to make a difference, and he will. Not that he's never irresponsible or carefree; trust me, he's still a teenager. He has fun, is the first to jump up and get the party started, is always planning his next practical joke, and sometimes borrows Willie's truck and brings it back on empty.
Sadie is our second-born. She is funny and kind and thoughtful. When she was a child Willie nicknamed her "the Original." He said there has never been one like her and never will be.
Sadie keeps us laughing with her imitations of people in the family, her friends, and her teachers. She loves to sing and dance and is great at sports. I really haven't found anything she can't do. She thinks about others' feelings and seems to always know when someone needs a little encouragement. She is always there to give it through a note, a hug, or a kind word. She does certain things with each of her siblings that make them feel special, and I could not be more proud of the way our children love each other and care for one another. Sadie makes sure this happens.
Sadie loves God with all her heart, and this was evident even when she was a young child. We have a video of her "preaching" at about five years old. Standing on the coffee table in our living room, she says, "It doesn't matter if you are a policeman or a jail person, God loves you. He wants you to be in heaven with Him."
We watched this video recently and were amazed at the insight Sadie had at such a young age. She even went on to say, "Even if I become famous one day, I will not just love myself. I will not forget about the Lord, 'cause I know He loves me and He is there for me." Wow! What wisdom and thoughtfulness from a five-year-old! She finished her speech with a cheer—"Let's give it up for God!"—in the cutest singsong voice. Yep, she's original, all right.
Will came next. He may not have grown in my stomach, but he certainly grew in my heart. Willie and I dreamed of and prayed for him from the time we were dating and knew we wanted to adopt a child someday. As soon as we saw a picture of this precious baby boy, we knew he was ours.
Will has a way of capturing the heart of everyone he meets. He makes friends in the toy section of Walmart, on the beach playing in the sand, and in the video game room at the pizza place. He has a great smile and a contagious laugh, and he remembers the name of everyone he meets. Even as a little boy he would say hi to the older ladies at church and call them by name. They were so impressed that he knew their names; they would gush over just how special he is, and I would agree.
Will is a people person and a family guy. He loves his siblings, his cousins, and, of course, his mom and dad, and he gives the best hugs to all. He's also got some special talents; he can beat-box and dance "Gangnam Style" like nobody's business. He was born with a beat in his head. As a baby he played the drums on his high chair and now can sit down at his drum set and pick out a beat at the drop of a hat. He also has a beautiful singing voice, which I know he will someday use for God. I could not be more proud of the man Will is becoming and can't wait to see what God has in store for him in the future.
Bella is the baby of the family, but just barely. She was born when Will was only ten months old. Willie and I had always planned on having four children, so we knew we wanted one more baby after we adopted Will, but we were surprised when she came so soon. And Bella's been surprising us ever since. She has a fun, quirky, strong spirit. At three weeks old, she contracted salmonella. She was in the hospital for a week, but it took months before her little digestive system recovered. She was a tough little cookie then and she still is now.
I have always said Bella was born knowing what she wants and where she is going. From the moment she could walk and talk, she would just take off, and you knew you'd better keep up! At just three years old, she loved to place her own order at the drive-through. She would stick her head out the window and say in her cute little-girl voice, "I'll have chicken nuggets and a Dr Pepper, with no ice." When she was about five years old she sat on Santa's lap and told him she wanted Dr Pepper for Christmas. I think she knew she was being funny, but she said it in all seriousness, and guess what she had under the tree on Christmas morning: Dr Pepper.
Bella is my little partner. She doesn't care to watch TV or really ever sit down; she likes to be busy. She's always by my side, "helping" me work. I call her my personal assistant, and you would be surprised at what she can do at ten years old (her age at the time of this writing). We've been making television shows since she was a little girl, and she always makes friends with the film crew. She keeps them laughing behind the scenes. Her most recent nickname is "Barbara," because she's a little Barbara Walters when meeting new people. She can carry on a conversation with any adult by asking questions about their lives. She'll find out how old you are, how long you've been married, how many kids you have, and where you work. She is interested in other people and other people are interested in her. If you meet her once, you will look forward to the next time. She's quick-witted and a fun kid to be around. It's been a joy to be her mom, and I'm so excited to see how God will use her talents and specific traits for His work. He's got big plans for her.
Rebecca came to our family a little late, but we are sure glad she did. She was sixteen when we greeted her at the airport with WELCOME TO AMERICA signs and excitement in our hearts. She didn't speak much English, but we loved her immediately. Bella took hold of her hand and wouldn't let go. Sadie talked her ear off (even though Rebecca didn't have any idea what she was saying), and the boys tried to win her over by wrestling with her and playing hide-and-seek. Rebecca, for her part, was scared to death, but she was a trouper. I can't imagine being sixteen years old, moving to a new country, and living with a family I've never met. She was equal parts brave and sweet—and she still is.
We laugh a lot in our family, and Rebecca fits right in. She's got an easy spirit, and she can laugh at herself, which is an important trait for a Robertson. She's incredibly talented and creative and goes for what she wants in life. Because she came to America in her junior year of school, she didn't have enough high school credits to actually get her diploma. She received a certificate of completion from the high school she attended and then had to get her GED to get into college. Despite all of this, she then went on to graduate from Louisiana State University in just four years with a degree in fashion design, right on schedule—something lots of Americans can't do!
Rebecca does have a mom in Taiwan, so I've always been careful to respect that. We are her American family, though, and if I ever make the mistake of saying I have only four kids, the rest of our children are quick to point out I have five. She is an awesome big sister and has become a beautiful, talented woman of God. We claim her as our own and are so thankful she claims us too!
### MAMAW AND PAPAW HOWARD
When Willie was just a little boy, Miss Kay worked at Howard Brothers Discount Stores, one of my grandfather's companies. When she went to work there, I am sure neither she nor my papaw Howard, my dad's father, had any idea our lives would be forever intertwined. God's plan for my family and Willie's did not start with us. He has used the two families to bless each other as far back as two generations.
My papaw Howard was an exuberant man and a tireless entrepreneur. He had a strong faith in God and a sense of determination that would not take no for an answer. If he wanted to do something and believed in it, he went for it. He was not afraid to take a risk; some of those risks failed, but others took off in very big ways.
When Papaw Howard wasn't working, he had all kinds of fun hobbies, one of which was taking me fishing. We had many good times together baiting hooks and casting lines. He was a man of many talents who never stopped dreaming and creating. For example, he wrote songs, and my siblings and I often sang along with him while he played the piano. At other times, we followed him as he marched around his living room singing at the top of his lungs for "exercise." He was a wonderful host and often held get-togethers at his house, which everyone enjoyed. Every New Year's Eve, he had a party at his home, and we all rang in the New Year singing praises to God.
Of all my papaw's ventures, one of the most fun (for me at least) was a singing group he started. The "stars" were four sisters who were friends of our family. They sang country music and called themselves the Steffin Sisters. Papaw Howard wrote their songs, started promoting them, and even made a couple of music videos I got to be in! The Steffin Sisters went on a European tour when I was thirteen years old, and I went along as the babysitter. We traveled to Sweden, Denmark, Norway, Ireland, and England for six weeks one summer, going from one music festival to another, and I loved every minute of it.
In addition to being a great businessman and providing so much fun and entertainment, Papaw Howard gave me a strong legacy of faith. His commitment to Christ has outlived him; he is still helping lots of people through the ministries he helped start, such as the Christian camp Camp Ch-Yo-Ca, where my parents met and where I first laid eyes on that cute Willie Robertson when I was in third grade. He also played a huge role in founding the Christian school I attended, where all the Robertson children go today. He was instrumental in helping establish the many ministries our church still supports today, as well, including World Radio and Relief Ministries. In addition, he founded a Christian publishing company called Howard Publishing (now called Howard Books) and wrote many songs that churches all around the world still sing. He was a man who lived for God and did everything he could do to impact this world in positive ways. I think he did an amazing job!
Papaw's wife was my mamaw Howard. One of my favorite memories of her is that she often took me to a local nursing home and we passed out Little Debbie Nutty Bars to the residents. She had the sweetest, kindest spirit, and I cannot imagine her ever saying a bad word about anyone. She also had the gift of hospitality and invited someone from church over every Sunday for lunch.
Mamaw was also a prayer warrior. The Bible teaches us to "pray without ceasing" (1 Thessalonians 5:17, NKJV), and she did! She always seemed to be reading her Bible and often read verses to my cousins and me. She recited some of her favorite verses every night, and everyone who spent the night with her got the verses read over them in the beautiful old King James Version. One of her favorites was Isaiah 41:10: "Fear thou not; for I am with thee: be not dismayed; for I am thy God: I will strengthen thee; yea, I will help thee; yea, I will uphold thee with the right hand of my righteousness." I can still hear her sweet voice reading those words today and they still bring me comfort when I think of them.
Not long before I started writing this book, my mamaw passed away. At her funeral I heard stories I had never heard before about her helping people. I knew she had helped many, many people in many ways, but at her funeral I realized that her legacy of caring for people was even more powerful than I knew and went far beyond the stories I knew.
Mamaw Howard was a kind, sweet, lovely lady, but she was a strong woman too. She had a lot of struggles in her life, but she overcame them all. She always looked to God for help and strength and always praised Him in everything she did.
Papaw Howard was a good provider for Mamaw and their family. They were very well-off by the world's standards, but that was not important in either of their lives. Mamaw could have had any jewels, fancy cars, or anything else she wanted, but those things did not matter to her. Helping others and putting God first was what her heart was after. My papaw called her "Queenie." He loved and cherished her, and she did him, all the days of their lives.
### MAMAW AND PAPAW SHACK
My mamaw Jo Shackelford, my mom's mom, is always stylish and beautiful but quick to point out that it's what's on the inside of a person that counts. One of her sayings is "Pretty is as pretty does." She is a living example of that generations-old truth.
Mamaw Jo can do anything. When my mom was growing up, Mamaw Jo sewed all of her clothes. She is quite a seamstress, and she cooks delicious meals, but her abilities go beyond her domestic skills. Now in her mideighties, she still owns and runs her own real estate business, as she has for years.
She and my grandfather, whom we called "Papaw Shack," lived in Shreveport when I was young, and I always enjoyed going to visit them. When I went, she made sure to have all of my favorite things. I love ice cream, and she made me homemade hot fudge sauce that was to die for.
I also remember her teaching me old-timey songs and playing "the Song Game," which I still play with my own kids. Here's how it goes: One person says a word and everyone else has to think of a song with that word in it. The first person unable to sing a song containing that word loses. Mamaw rarely loses at this game. She knows a lot of songs!
Mamaw Jo always made things fun and magical. She used to tell us she could make the traffic light change with magic. When we pulled up to a red light, she would recite this poem: "Rotten tomatoes and old tin cans, light on the corner of Main Street [or whatever street we were on], turn green, shazam!" And as soon as she said, "shazam!" the light would magically turn green!
Mamaw Jo has always been a strong lady. She's definitely not the kind of grandmother who will let you win at a game just because you are a kid; she wants you to earn the victory. She's a competitive Scrabble player, and when I was growing up, she and I played games all the time, everything from checkers to Old Maid. She beat me often! Now, carrying on the family tradition, Bella loves to go over to Mamaw Jo's house for late-night games of Uno, and they keep a running tally of who has won the most. She is so proud of each of her grandchildren and great-grandchildren and makes sure we all know it!
Mamaw's husband, Papaw Shack, always had a hug and a kind word for everyone. He passed away in 2008, and I still often meet people who tell me how much my papaw Shack meant to them. He worked hard all his life; his dad left his family when he was a young boy, and he worked as a janitor in his school when he was just in middle school. He said they were "so poor they couldn't pay attention." He loved his family and everything he did was for them.
My cousins and I know Papaw was our biggest supporter. He loved basketball. In fact, during World War II, he was on a team for the marines, a group that played exhibition games to boost the morale of the troops. That turned out to boost his morale, too, because he met my mamaw when he was a basketball player and she was a cheerleader. Papaw attended every one of his grandchildren's basketball games, but he also loved to hear about everything else we had going on. He wrote sweet notes to me, often encouraging me and telling me how proud he was of me. His family and his faith were his life.
Mamaw and Papaw Shack loved each other dearly. Anyone could tell each was the other's best friend. They worked together in their real estate business and enjoyed the simple blessings of life, such as waking up early, having coffee, and reading the paper together. When Mamaw Jo and Papaw Shack had people over for a meal, she was the cook, but Papaw always helped her with the dishes afterward. They were a great team, and I am thankful to have had that example of a husband and wife working together. It's something Willie and I have emulated, only he's the one doing the cooking, and I'm the one doing the dishes. I am so blessed to have had my papaw as my grandfather and to still have Mamaw Jo in my life today.
### GRANNY AND PA ROBERTSON
I treasure who my grandparents were and who my mamaw Jo still is; I also appreciate having the chance to know Willie's grandparents, Granny and Pa. They lived close to Phil and Miss Kay for many years when Willie and I were dating, but they had to move away because their little house on the river suffered major flood damage. The house was old before the flood, and after the water got in it, it simply was not worth saving. Granny and Pa moved to Shreveport to live with Phil's sister, and except for holidays we did not see them nearly as much as we had in years past.
Granny really enjoyed making quilts, and not long after Willie and I married, she offered to make me one. This was during the time when the "country Americana" theme was popular and I found a design I wanted. It included red, blue, and black-and-white-checked fabric. She said the black-and-white check nearly caused her to go blind, but she made the quilt for us, along with a dust ruffle and pillows to go with it. I loved it and used it on our bed for the first couple of years of our married life.
Willie and I had a special visit from Granny the year Bella was born—on Granny's ninety-second birthday. Granny loved the thought of sharing a birthday with a great-grandchild, and she offered to come "help" with the new baby. Now, remember, she was ninety-two, so she wasn't much help, but I will cherish the memories of that time forever. She gave me advice on taking care of a newborn (even though I already had three babies) and helped as much as she could. We sat and talked and loved on little Bella those first few days of her life. Granny had made Sadie a quilt when she was born, and she crocheted a blanket for Bella.
Pa passed away not long after Willie and I married, so he did not ever know our children, but I will never forget his funeral. It was in the small country church he and Granny attended when Phil was young, and the congregation that day was full of bearded men! We sang a lot of old songs and hymns, and all the men sang at the tops of their lungs. Many kind words were said about Pa, and since the Robertsons can always laugh at themselves, even the funeral included funny stories. That day, I felt taken back in time to a different era, and I could sense the strength of the community that Granny and Pa enjoyed and the hard work it took for them to keep their family together.
Through it all, Granny and Pa stuck together, through good times and bad, for richer or poorer. When I attended Pa's funeral, I could sense the roots of the Robertson family and the foundation that made Phil who he is, which in turn made Willie the man he has become. That's how generations work, so I believe those same good qualities—along with characteristics of the Howards and Shackelfords—are in my children too.
### MY MOM AND DAD
Willie and I both have deep roots in Louisiana. Both sets of my grandparents have lived in the state for years, just as Granny and Pa did. Willie and I currently live on some property my grandfather once owned. Members of my extended family have lived close together, on the same street, for generations. Now my parents, Johnny and Chrys Howard, live right next door to Willie and me—and we did that on purpose! I love living so close to them, and I do not take that blessing for granted. My parents are always there for me, and with the busy lives and unusual schedules Willie and I have, that is so helpful. Besides Willie and me, they are our children's biggest cheerleaders, and they never—I mean never—miss an activity our kids are involved in. They are such an integral part of our children's lives, and all our kids know they can count on them or turn to them for help, no matter what they want or need.
The kids—mine and Willie's, those who belong to everyone else in the family, and even close friends—call my parents "Two-Mama" and "Two-Papa." There's a great story behind those names.
John Luke and Sadie are only twenty months apart in age. When I was twenty-six weeks pregnant with Sadie, I went into premature labor. The doctors were able to stop the labor but placed many restrictions on what I could do. The same thing happened again when I was about thirty-one weeks along, so I went on bed rest for about five weeks. At that time, John Luke was only a year and a half old and was busy, busy, busy. My mom had to help out a lot, and he loved being with her.
At only about eighteen months old, John Luke was not saying many words, but he could definitely say "Mama," so during this time he called both my mom and me "Mama." We tried to figure out another name for him to call Mom, but nothing stuck.
One day, after Sadie was born, I was driving with him in the car and he kept saying he wanted mama. I said, "Mama is right here," and he said, "No, Two-Mama." I realized he was asking for my mom. He has called her Two-Mama ever since, so we dubbed my dad Two-Papa. They are the best grandparents you can imagine. All of our kids, plus the other Robertson kids and many close friends of the family, love Two-Mama and Two-Papa.
God has really blessed me with an awesome family, not just with Willie and our kids, but throughout many generations. Our kids are not all old enough to fully understand and articulate the amazing legacy they have, but they definitely get it and appreciate it. I pray and believe they will carry on our family's faith, good qualities, and ability to make a difference for God for the rest of their lives.
## 20
## GENERATIONAL BLESSINGS
## Missy
When I think of generations, I think about the legacy and influence the people in each generation pass to the next one. Jase and I started dating when we were very young and had the opportunity to basically grow into adulthood together. Because of this, we have shared many "firsts" together. When our dating relationship started to become more serious, we made a commitment to stay sexually pure until our wedding night. Each of us had this goal before we started dating, but when we fell in love, that goal became one for each other as well. I knew that God expected this purity from His children, and I trusted God enough, even at my young age, to understand that His way was the best way. Jase and I reached our goal after dating two years, ten months, and two days. But who's counting? We were! Whew! We made it!
That night was the first sexual experience either of us had ever had, and we have only known each other since then. Being pure and faithful to each other and to God is a top priority for us to this day. Our decision to remain pure is something we have not been silent to our children about. The older we get and the older our children get, the more we realize how hard accomplishing that was and still is for kids today. We built our relationship on a spiritual foundation many years ago, and we feel a great responsibility to pass that spiritual foundation on to our children. At the time of this writing, our oldest child, Reed, is eighteen; Cole is sixteen; and Mia is ten.
### PREPARING FOR A NEW GENERATION
When Jase and I married, we decided to wait a few years before having children. We wanted to spend this time together, just the two of us, before starting a family. We also wanted to prepare as best we could before starting to raise another human being. We felt like this was a huge responsibility. Once we began contemplating starting our family, I went to as many Christian parenting classes as I could find. We are blessed with many qualified and talented speakers in our church, and I was there every time the doors were open.
During one of those classes, a lady I still admire greatly said, "The best gift you can give your children is to love your husband." I first heard this when I was pregnant with Reed but have kept it close to my heart for the past eighteen years. I can honestly say my kids are confident in the fact that their mom and dad are completely committed to each other and to God, no matter what circumstance we face.
Since we have been in the limelight of Duck Dynasty, many women have approached Jase in person and on social media. However, because of his commitment to his Creator and to me, our family has become even stronger. Jase tells me almost every day how beautiful I am. He tells our teenage boys in front of me, "Your mama is one hot-lookin' woman!" They just laugh. No matter how difficult a situation may become, neither Jase nor I is going anywhere.
Our kids are also confident that their parents try to make decisions from a spiritual point of view. This doesn't mean we succeed every time, but our kids know, without a doubt, that we love God more than anything else in the world. When we fail, we have a Savior who forgives us and encourages us to try again. We try to do the same with our kids. When they fail, we are disappointed, but we try to show them that they are forgiven and encourage them to get back up and keep going. Living a spiritual life with God at the forefront is top priority for me, and passing that on to my children is my ultimate goal. I thank God for the previous generations who have influenced both Jase and me in this regard.
### MY EARLY YEARS
My parents moved to West Monroe when I was six months old. They grew up in New Mexico and Texas, so they were leaving all family behind when they made the move. They had no friends and knew no one; they simply moved on faith when my dad accepted a minister position at a church, the same church we still attend today. Because we did not live close to any family, I saw each set of my grandparents only once or twice a year. Because of this, I never formed a close bond with my grandparents. My dad was an only child, and his parents were not pleased with his decision to move away from them; this made their relationship very strained.
My mom was the oldest of five children, so she grew up in a very busy household. Each was expected to pull his or her weight. When my mom was fifteen years old, her baby sister, Bonny, was born. Five years later, my mom married my dad, and when she was twenty-three she had me. Since my aunt Bonny is only eight years older than I am (the same age difference between Reed and Mia), she became like my older sister.
Going to Grandma and Grandpa's house in Austin, Texas, meant I could see Aunt Bonny. For a while, we fought like cats and dogs. I was just far enough behind her in age to really bug her. I thought she was the coolest girl I ever knew. She grew to tolerate me over the years, and now we are very close. At the time of this writing she is working for Jase and me, assisting us with business affairs.
Since we traveled to Grandma and Grandpa's house mostly on holidays, that meant all the other aunts, uncles, and cousins would be there too. Their house was full of people for the entire week. It was so much fun! My mom's parents remind me so much of Phil and Miss Kay. Their house was always open to anyone who needed a place to stay. They supported missionaries who lived abroad and always helped students who were being trained in seminaries. They believed in God and have always been active in their church. I remember one Sunday—I think I was about twelve years old—when the entire family was together at their church; we got up onstage and sang the "Hallelujah Chorus"! Not many families can say that.
### THAT ROBERTSON GENEROSITY
When Jase and I started dating, he introduced me to Granny and Pa. Not too long after we married, the flood forced them to move to Phil's sister's house about an hour away. Pa was getting feeble and sick and was soon moved to a nursing home. We didn't get to see them much after that.
Our daughter, Mia, was born with a cleft lip and palate that required multiple surgeries and procedures during the first few months after her birth (I'll tell her story in greater detail later in the book). On the way to her first surgery, we stopped to see Granny. Granny pulled Jase and me into her bedroom and offered to give us money for Mia's initial operation. Even though we didn't have nearly enough to cover our costs, we knew Granny sure didn't have it either. We hugged her and told her we were fine, that we didn't need anything. She made us promise that if we did need something, we would ask for her help. Of course we agreed. It was such a selfless act of generosity on her part, and I will never forget it.
Granny also made all of our children homemade afghans when they were born. Mia has had that blanket beside her through every surgical procedure. She's ten years old and still sleeps with what's left of it today.
### A MUSICAL FAMILY
All of my aunts and uncles are musical, and my mom has a degree in music education. I was in Mom's choruses and special music projects until I graduated high school. I didn't know how good she was until I went to college and was in the performing choir. I've sung in several different groups and choruses throughout my life, but no director has ever compared to my mom. She is a perfectionist when it comes to music. She is never pleased with good enough. I have adopted that same mentality when it comes to music, and I work very hard at it.
Reed, Cole, and Mia all have a musical ear and have great singing abilities. Currently, Mia takes piano lessons. While I would have loved for the boys to take lessons when they were younger, our budget did not allow it. However, over the last few years, both Reed and Cole have taught themselves how to play the piano by ear and by watching how-to videos on the Internet. They have also taught themselves to play the guitar, ukulele, cajón drums, harmonica, and any other musical instrument they can get their hands on. We love singing together as a family and have started adding that to our family appearances. Actually, nothing gives me more joy than to stand side by side with my kids and sing praises to our God and Savior.
My mom is a big supporter of my kids' musical endeavors. She comes to all their performances and recitals and is happy to give advice, if they so desire. Music is and has always been a big part of my life. I'm glad my mom gets to share it with my kids and me.
### SURROUNDED BY GENERATIONS
Unlike me, my kids have grown up with both sets of grandparents living in the same town where they live. What a blessing it has been! Over the years, the grandparents have been a great help to Jase and me. When we go out of town, somebody is at home to see to the kids, take them to school and pick them up, feed them, take them to practices and games, and take care of other things they need.
My dad calls me often and says, "Can I help you with the kids today?" He has an old Model A car he sometimes picks them up from school in. When the boys were little, they would call it the hot rod. Now that they are older, they don't think it's so cool. But they had many fun times in that car with my dad with the top down, going to Sonic and ordering ice cream and cheese sticks after school. He never let me eat ice cream in his car when I was a kid! My dad would spoil them rotten if I let him. Funny how grandparents do things for their grandkids that they would never do for their children. I guess Jase and I will experience that one day too.
One tradition my dad has with my kids is that every time they spend the night at my parents' house, usually on weekends, my dad takes them for doughnuts the next morning. If they keep them on a school night, he still takes them for doughnuts. Since school starts at seven forty-five A.M., this means they are at the doughnut shop before seven in the morning. But they love it! My dad shows my kids how special they are by spending time with them. He did the same for my brother and me. My parents are right beside Jase and me at every one of my kids' ball games, track meets, performances, recitals, and birthday parties. They wouldn't want to be anywhere else, and I am very grateful for their help, support, and unconditional love.
Making the most of generational influence means grasping the good from those who came before you and doing your best to weed out the bad. We all want to pass down the best to our children, and we all hope they forgive and forget what's not so good. That's a consistent trait throughout all generations. We want our kids to have it better than we did.
# Part Five
# SOMETIMES MIRACLES HIDE
And we know that God causes everything to work together for the good of those who love God and are called according to his purpose for them.
ROMANS 8:28, NLT
## 21
## INTRODUCTION
## Jessica
Bruce Carroll sang a song that was popular in the early 1990s, when I was just a preteen. But its message never grows old, and its title is simple: "Sometimes Miracles Hide." When we're in the midst of downright difficult circumstances, we can't always see that God is doing something great in our lives. Sometimes challenges and hard times can be so heartbreaking that all we can do is survive. But then, once we've survived, we see that God was working a miracle right in the middle of our hardships.
I have seen Missy and Jase walk this kind of journey since 2003, when their daughter, Mia, was born with special needs. Missy will write more about it in this section of the book, but I just want to say that they have been amazing as they have gone through situations not many parents have to go through. I remember being with Missy in the ultrasound room when she first wondered if there was something different about her baby. From that day until now, she and Jase have dealt with some unique challenges, but they have met them all with faith and trust in God. Now everyone in our family adores Mia, and we love having her in our lives.
I am also aware that Miss Kay suffered some very hard times with Alan and Jep as teenagers. As Jep's wife, I cannot even begin to thank his brothers and parents enough for the way they intervened in his life when he was in trouble. Sometimes I still get tears in my eyes when I think about how much Jep's brothers loved him when they first told Phil and Miss Kay he needed help. Had the brothers not cared enough about Jep to plan a family meeting to confront him—and had Phil and Kay not practiced tough love—I might not have the husband and family I have today. I will always be in awe of how much Jep's family loved him and grateful for the way they handled that situation.
Both Kay's situation and Missy's seemed overwhelming at times. Missy and Jase still face challenges. But both circumstances have proven to our family in up-close, personal ways that God can do miracles when it looks like nothing good can happen. When the world would refer to a situation as "bad," not only can God work it for good, He can do something miraculous.
## 22
## TOUGH LOVE
## Miss Kay
I love my boys! In the early days, the three oldest ones and I had to stick together just to survive. By the time Jep came along, Phil was an entirely different person than he was when Alan, Jase, and Willie were little. But in those early days, the oldest sons and I endured some lean, frightening, and difficult times.
Once Phil got his life straightened out by God's grace, I hoped my hardest days were behind me, and in many ways they were. But I did face two more situations that completely tore me up. In their teenage years, both Alan and Jep strayed from the way they were raised and did things they should not have done. These two boys reminded me of the Bible story about the Prodigal Son (Luke 15:11–32). They both went their own ways for about two years, and those experiences were very hard on our family. One thing that was very difficult for me was that they were both drinking, and I especially hated that because I had such bad memories of what alcohol had done to my mother and to Phil.
Today, I have a true passion to help people fight for their marriages whenever I can. I also come into contact with a lot of women whose children are far from their families and far from the Lord. Sometimes, when people only know me from Duck Dynasty, they think I have spent my whole life in the kitchen, happily feeding my tight-knit family. They do not think I could ever truly understand real pain, especially the depth of the pain of a broken marriage or a wayward child. I do. From years of personal experience with the pain, disappointment, and devastation, I really do understand.
I read the Bible a lot, and it has some great stories about miracles. One time, God completely dried up an entire sea so a whole nation of people could cross it on dry land and escape an army that was trying to kill them. We do not see many miracles like that one today, but I know God is still able to do them and that He does all kinds of other miracles every day. I have seen them, and I have even experienced them in my own life.
My loving, happy marriage with Phil is a miracle. Duck Dynasty is a miracle, not because our family has a television show, but because our family is together. Most people have no idea how close Phil and I came to breaking up years ago. We are only with each other right now because God did something miraculous in enabling us to forgive the pain of the past and to make a new start. The fact that all our boys are serving God today is also a miracle. Marriages can be healed, and prodigals can come home. I want everyone who is suffering in these kinds of situations to know that, and I hope this book will bring hope and encouragement to all who are going through what I went through years ago.
### AMAZING LITTLE BOY, AMAZING MAN
Alan was a remarkable little boy. I truly do not know what I would have done without him during the years when Phil and I had so many problems. I did not live close to anyone in my biological family, and I was not involved in church during that time, so I had very little help or support. But when we ran into problems when Phil was drinking, Alan always stepped up to the plate. I did not have to ask him to help; he just did what needed to be done. As early as seven years old, he could feed and bathe babies almost as well as I could!
At a time when most boys his age were playing Little League baseball, Alan was taking care of his little brothers Willie and Jase. He did not get to do a lot of the fun things many children do; he helped me and definitely became my "main man." He was a much better caretaker for the younger boys than anyone I could have hired; he was totally trustworthy and dependable.
One time, Phil got more drunk than usual. It was scary, and I had to deal with a lot that night. In the middle of it all, Alan said to me, "It's okay, Momma. I fed the baby [Willie], and I burped him and changed his diaper. I fed Jason too. We're good. Don't worry about us."
Willie and Jase were so young at that time, they could not begin to understand everything Alan did and everything he sacrificed so they could stay alive and healthy. They will never know what all he did for them and for me. There were times I did not think I could go on, but Alan made me feel like I could.
Alan always loved his family. One of my favorite stories about him is that as a teenager, he bought his own car. It was old, but it ran, and he paid for it with money he earned working at a grocery store and doing other odd jobs. He also helped Phil and me a lot at Duck Commander, but at that time we could not pay him anything.
Phil's truck broke down one day, and we could not afford to fix it. We really needed that vehicle and didn't know what to do. Without saying a word to us, Alan sold his car and gave Phil the money to get a new truck. Then Alan started over saving money to buy himself another car. That's the kind of thoughtful, generous person he is.
Alan was an amazing little boy, and he is an amazing man now, a man of strong character and integrity. He had more opportunities to be around my grandmother Nannie than the other boys, and he caught some of his heart from her. As I mentioned earlier, he also had a close relationship with a retired preacher, one of our neighbors in Arkansas, and I think that man had a powerful influence on him too.
Alan grew up to be a big help to Phil in the early days of Duck Commander and then to become a great preacher for many years. He still preaches from time to time, but now, he and his wife, Lisa, both work at Duck Commander. Sometimes, his work as a preacher makes people nervous. I am always sorry to hear about this, because he is the one who enabled the others in our family to be where they are today. Even though he has not been seen on Duck Dynasty as much as the other boys, our family would not be complete without him. Everyone loves and respects him so much, and he is a great blessing to all of us.
### A SAD SURPRISE
In between being an amazing little boy and becoming an amazing man, Alan had some rough times. I was totally shocked when he began to change during his high school years. I will not write the whole story because it is Alan's to tell. Lisa has already shared some of it in this book, and Phil writes about it in greater detail in his book, Happy, Happy, Happy. I just want to say that I was surprised and heartbroken to find out about the choices Alan had made and the way he was living.
I only knew Alan was drinking because Jase told me. At first I thought there was no way it could be true—that Alan would never drink or get drunk because of everything he saw Phil go through. But it was true. One time when Alan and his friends got caught misbehaving, Phil told Alan he was disappointed in him and reminded Alan of what he (Phil) had put me through when Alan was young. I still remember hearing that lecture and Phil's saying at the end, "I hope you'll learn." Alan admits today that he did not learn.
Alan's behavior did not get any better. When he finished high school, Phil and I had to make a really hard decision. We told Alan he could not keep living in our house and do the things he was doing. We gave him a choice: he could straighten up, or he could go live with Phil's sister in New Orleans. He chose New Orleans! We hated to see him go but felt it was the best thing for our family as a whole. We did visit him a few times, but mostly we prayed for him.
Alan had a run-in with the police one Sunday morning while he was in New Orleans and as best he can recall, one of the officers said to him, "Let me talk to you. What are your mom and dad doing right now?"
"They're in church, where they always go," Alan answered.
"I knew," said the officer, "that you were raised different." In other words, the policeman could tell Alan was not what some people might call a "common criminal." The officer went on to speak some very strong words: "You have just done something really bad. Whatever you're doing here, pack it up. Go home and live like your mom and dad; go live like you were raised. I don't know your parents, but I have a feeling they will welcome you back like the Prodigal Son."
Phil and I had not been able to get through to Alan or influence him to change his ways while he was living with us, but that policeman in New Orleans sure got through to him. Sometimes we wonder if that policeman was an angel. Whether he was or was not, God definitely used him to get Alan back where he needed to be.
Alan left "the Big Easy" right away and came back to us. He started walking with God again; he reconnected with Lisa. He and Phil began studying the Bible together; Phil baptized him in the river by our house, and he has been a totally different person ever since.
### IT HAPPENED AGAIN
Jep only knows about the struggles his brothers, Phil, and I went through during our hard times because we have told him. Thankfully, he did not live through them. He only knows the new Phil, the man made new in Christ. Jep often says, "I'm glad I didn't know that old Phil. I like the one we have now!" Jep has always had a righteous, godly father who loves the Lord and loves the Scriptures.
Jep's experiences have been totally different from his brothers', but when he was old enough to make some important choices about his life, he went down a path very similar to the one Alan had taken years earlier. This situation, too, broke my heart, because until that time Jep had been nothing but a joy in my life. Phil and I had never had any major problems with him. Phil tells this story in his book, but it's important to me to tell it here from my perspective.
Phil and I were better off financially by the time Jep came along than we were when the other boys were young. We wanted Jep to have a biblically based education and to have the influence of going to chapel with his friends, so when he reached high school age, we sent him to a private Christian school, even though we struggled to pay for it. We hoped sending him to that school would mean he would have a good group of Christian boys as his friends and all of them would be good influences on one another.
For a while Jep did have a great group of friends and even a couple of nice Christian girlfriends. During his senior year, he got hurt playing sports and he went through a bad breakup with a girl. As a mom, I believe he started thinking his dreams were not going to come true, and he began hanging around with a new group of boys who were not good influences on him. I now know that when Jep went to visit one of those boys at home, some of the family was drinking and Jep drank with them.
After graduation, Jep moved into an apartment with his cousin. He did not live far from anyone in the family in terms of distance, but we did not see him as much as we had before. I knew Jep was not living as he had been raised, but I was not fully aware of all his struggles. I wrote him a lot of letters during that time, letting him know how much I loved him and how much God loved him.
### THE CHOICE
My boys look out for one another. Jase is the one who told Phil and me when Alan had problems, and Willie is the one who let us know about Jep's bad behavior. Both Willie and Jase were typical boys and through the years they did the kinds of things boys do. But they did not get involved in the same activities Alan and Jep did, and when they saw their brothers in trouble, they were quick to run to their rescue and make sure they got help.
After Willie told us what was going on with Jep, the whole family decided to confront him about it. I don't think we called it an "intervention" back then, but that's the term that would be used today. We all got together at our house, and when Jep came over that night, he knew something was going on.
Basically, Phil said to him, "Son, this is what was reported to me." And he told Jep what he had heard. Jep could not deny it. His brothers were all standing nearby, and he knew they knew the truth. They were not there to condemn him; they were there because they loved him so much.
Phil continued: "We don't support this. You can be on your own, and you'll be without my money and without a truck. Or you can live at home under house arrest. If you do that, you will have to live by my rules because we are not going to continue to do what we have been doing."
Jep hung his head and started confessing everything he had been involved in. Phil was crying; I was crying; Jep's brothers were crying. We all assured Jep that God would forgive him and we would too.
The next thing Jep said to Phil was heart wrenching: "Dad, why did it take you so long to rescue me?"
Phil spoke gently and humbly to him: "I'm sorry I waited so long. We just didn't want to believe what we were hearing."
After that, Jep told us he wanted house arrest. He wanted to clean up his life and live with us, according to the standards we set. When our family meeting was over that night, Alan, who was married by then, left our house and headed back to his home in West Monroe. He stopped along the way, got out of his car, went into a field, and fell to his knees weeping. He knew Jep could have died if his brothers had not told us what he was doing. He understood, in ways none of the rest of us ever could have, what it meant for us to reach out to Jep and call him to account for his actions. The Bible says God will go after one lost sheep and rejoice when that one is found, even if there are ninety-nine others in the flock (Matthew 18:12–14). Alan understood, like no one else, what a powerful event had taken place in our home that day.
Eventually, after Jep had been home with Phil and me for a while, we allowed him to go live with a friend. We knew the friend's parents and they agreed to monitor his behavior. We saw Jep often during that time because he soon started studying the Bible with Phil and eventually brought his friends with him. One night he showed up at our house with fifteen people, wanting Phil to share the gospel with them.
Later, talking about his wild days, Jep told me, "Mom, I might have done all that stuff, but I always felt guilty about it. Your letters really meant a lot to me."
### LOOKING BACK
As I think back now on those painful days years ago, I do not question the way Phil and I handled the situations with Alan and Jep. We used tough love with both of them, but because each of the boys is an individual, we could not handle both situations the same way. We felt we had to ask Alan to leave our home for a season, while we believed we could let Jep choose for himself whether to stay with us or to move out. Sometimes I wonder what would have happened if we had not done anything. What if we had not been willing to face the truth and confront our sons about it? Where would they be today? We cannot say. We are just thankful they did not continue on the paths down which they were headed.
No parent wants to go through situations like these, but plenty of parents do. And plenty of good parents do. All parents have to make their own decisions about how to deal with their children when these things happen. I would not presume to tell anyone what to do, but I do want everyone to know that just because a young person strays from his or her upbringing does not mean he or she is a lost cause. Sometimes, as in Jep's case, the love of family will bring people back to their senses and cause them to start replacing their bad choices with good ones. Sometimes, as happened with Alan, God will use a total stranger to help a person see the truth.
I like to say that all of us are imperfect people following a perfect Christ. That's just as true for people who get themselves in trouble as it is for those who don't. All of us have flaws and weaknesses. The point is that no matter what we do or how far we stray, we can always call on the perfect Christ to help us. In my own life, and in the lives of many other people, I have seen Him do this in the most amazing ways.
## 23
## A MIRACLE NAMED MIA
## Missy
Behold, children are a heritage from the Lord, the fruit of the womb is a reward.
—PSALM 127:3, NKJV
Every child is a miracle. Jase and I have three of them, one of whom is our daughter, Mia. She is smart, strong, self-confident, happy, and a good friend to everyone she knows. On top of that, she has overcome a lot in her young life, and she is a continual reminder to all of our family that God will never leave us or forsake us. Deuteronomy 31:6 says, "Be strong and courageous. Do not be afraid or terrified because of them, for the Lord your God goes with you; he will never leave you nor forsake you."
### IT'S A GIRL
Jase and I had two boys, Reed and Cole, when I got pregnant again. That became a terrible experience with a tubal pregnancy, and I went through a very scary and dramatic situation when the surgery to remove that pregnancy went wrong, and I ended up losing the entire tube in a second operation. That time in our lives was sad and difficult, to say the least.
Not long after I healed physically from my surgeries, we were overjoyed when we found out I was pregnant again. A phone call from the doctor a few weeks later gave us a harsh reality check when she told me there was a problem with my blood work. To make a complicated situation easy to understand: my blood type is A-negative, and because I did not receive a Rhogam shot after my tubal pregnancy, I would be at risk for another miscarriage if this baby's blood type were positive. Without Rhogam, my body would see the baby as a foreign object and try to dispose of it until it succeeded.
When I asked what the chances were for this baby to have a negative blood type and not be at risk, I was told that since Jase and I both have negative blood, 99 percent of all our pregnancies would result in a baby with positive blood, but there was always that 1 percent chance I could have a baby with negative blood. Both of my boys are positive and our lost baby had also been positive. The doctor told me our goal for this pregnancy was to make it to twenty-six weeks before taking the baby by cesarean section, because it would have a better chance of surviving outside my body than inside. This sobering news drove Jase and me to pray—a lot, much more than usual—about our baby.
There was no way to test the baby's blood in utero, so a specialist monitored my condition using markers in the blood work. If they began to go up, I would not be able to carry the baby to term. Miraculously, they did not! Because of that, we concluded that, statistically, our baby was one in one hundred. We have now confirmed that; that baby was our daughter, Mia, and she has negative blood.
With the crisis involving the blood type behind us, Jase and I quickly moved beyond the anxiety it had caused and regained our previous sense of joy about our new baby. Since we had every reason to believe the child was healthy, we could hardly wait to find out whether we were having a boy or a girl. We learned at twenty weeks that she would be a girl, and we were thrilled. We loved having two boys, but we wanted a girl in our family too.
### TECHNOLOGY BECOMES A BLESSING
At that time, in 2003, four-dimensional ultrasounds were new, and people in the medical field were very pleased with the quality of the images they provided. So at thirty-one weeks, I scheduled that scan "just for fun," to see the new baby in a more detailed way. Jase had been present for my other ultrasound, but he was not with me that day. I did have several spectators, though, including Miss Kay, Lisa, Reed, Cole, and Jessica. We were all so excited!
Our excitement soon turned to concern when I asked the tech, "Does her nose look a little bit smushed?"
The look on her face told me she thought something was wrong.
"I need to get the doctor," she said as she turned to leave the room.
Miss Kay and Jessica took Reed and Cole out of the ultrasound room, and Lisa stayed with me. When the doctor saw the ultrasound, she confirmed what the tech had feared: the baby had a cleft lip. She could not tell how severe the problem was nor whether the palate was involved, but she did tell us clefts are often associated with a variety of other physical problems and syndromes. All she could say was, "We'll have to wait and see."
Miss Kay called Jase, and he came to the doctor's office so we could talk to the doctor together. While waiting on the doctor in the waiting room, Jase put his arm around me and said, "Well, we'll just have to teach her that beauty is on the inside." This is definitely not what I wanted to hear at the time. I wanted to hear that it was all a mistake, that we would get a second opinion, that this little girl we had waited on for so long was going to be born perfect. I didn't know this child yet, but I did know I already loved her. I also knew I didn't want her to suffer in any way, physically or emotionally, and I knew this condition would cause both.
Needless to say, our families rallied around us for encouragement and support, with a lot of tears. After a few days of grief and disbelief, I went into work mode. I learned all I could about this condition and started trying to find out where we could get her the best medical care. As hard as it was to find out my baby was going to be born with problems, I am very thankful for that 4-D ultrasound technology. Because we knew ahead of time, we had a chance to prepare ourselves and to arrange the care and services we would need for her. I cannot imagine how difficult it would have been had we been faced with trying to do all that and make major decisions for her in the moments and weeks right after delivery.
### WE ONLY WANTED THE BEST
Knowledgeable people recommended two very good facilities to us, but both were teaching hospitals associated with medical schools. I decided I did not want my child to be a learning opportunity for some future doctor. I wanted the best, most experienced physician in the world. While we looked for that person, we prayed to the Great Physician and asked God to heal her in the womb.
The following Sunday morning, Jase and I shared our situation with our church family so they could pray for us. After the service, a couple approached us and said they had a client whose grandchild was born with a cleft palate. "Could we contact that family for you?" they asked.
"Yes!" we said, so relieved to think we might be able to speak with someone who could help us.
Later that night, I received a phone call from the mother of the child the couple had told us about. After we talked for a few minutes, she said confidently, "We have found the team to handle this problem." She went on to explain that their son was born with a cleft lip and palate six months earlier. They had researched surgeons who specialize in this area—and they located hundreds. When her husband narrowed the search and began asking which physicians had been published, that field narrowed to only three—one in Los Angeles, one in Pennsylvania, and one in Dallas. They decided to contact the doctor in Dallas, Dr. Kenneth Salyer, the world-renowned craniofacial surgeon who separated a pair of conjoined twins in 2003. They were more than pleased with his team of doctors and were happy to share this information with us. We felt this was an answer to our prayers and that God had led us to this family, who ultimately became a large part of our support system.
At about thirty-four weeks, I communicated with Dr. Salyer's office and made arrangements for Mia to become his patient. They told us to call them when she was born, and they would see her one week later. Jase and I knew bringing this baby home would be a much different experience than we'd had with Reed and Cole, but we were comforted by the knowledge that Dr. Salyer was the right physician for us and the fact that plans were in place to see him so quickly after Mia's birth. So we did what all expectant parents do when the time of birth draws near: we waited for her to arrive.
### SHE'S HERE!
I hoped to have a normal birth with Mia because when each of our boys was born, something was unusual. Reed was ten days late and faceup; he also got stuck in my pelvis during delivery. Cole decided to come three weeks early and was breech, requiring a C-section. I really wanted a nondramatic delivery with Mia!
Mia was due at the end of September, and I went to the doctor on September 11 for a normally scheduled appointment. My blood pressure was very high, so the doctor wanted to do a C-section right away. For various reasons—including the fact that my mom was out of town and I wanted her around for the birth—I asked if we could wait until the next day. The doctor agreed, as long as I promised to stay in bed and not move.
Jase and I headed to the hospital the next morning, and Mia was born later that day. As C-sections go, everything was fairly normal. Did God heal her in the womb? No. We know He could have, but He chose not to. All of the tissue from the top of her mouth to her nose was present; it just looked like someone had cut it with a pair of scissors because it was not fused together. We had wondered whether her palate would be affected or just her lip. Yes, we soon learned, the palate was cleft.
Mia weighed six pounds, nine ounces, so she was not unusually small. But she was born with a condition called wet lungs. Because of that and her cleft, the doctors sent her to the neonatal intensive care unit (NICU). When the specialists examined her there, they found that in spite of the wet lungs, her breathing was normal, and everything else they checked was fine. But once a baby goes into the NICU, that child cannot be released until he or she passes certain thresholds, one of which is volume of formula intake. She needed to be able to drink one ounce of formula in one sitting and keep it down before they would let her go. When she was born, she drank only a few milliliters at a time and had difficulty keeping it down. Getting to one ounce took her six days. We were so happy when she reached that point because we could take her home.
### FIRST STEPS IN THE RIGHT DIRECTION
Dr. Salyer was not available as soon as we had hoped to see him because he was doing mission work overseas. Our first visit with him took place when Mia was seventeen days old. Of course, Jase and I both went on that trip to Dallas. My mom and dad, Miss Kay, and my aunt Bonny also went with us. We needed a lot of moral support! That first visit was grueling, and it literally lasted all day. Everything Dr. Salyer and his craniofacial team did had to be done, but much of it was uncomfortable for little Mia and all of it was stressful for Jase and me. By the time all the examinations were complete and all the test results were in, these experts were able to tell us with some confidence that, as far as they could tell at that point, Mia had none of the conditions that often accompany a cleft palate. They told us this was great news. We were so exhausted and so fried, we did not know what to think. We now know that it was indeed the best news we could have hoped for that day.
At that time, we thought Mia's condition could be "fixed." The doctors talked to us about a years-long schedule of surgeries. With so many procedures, we assumed, surely they could eventually make her mouth, palate, jaws, and other affected areas just like those of other children. That is not the case. Because Mia was born with her condition, the way her bones grow—or don't grow—impacts everything else. We have come to realize this condition cannot be "fixed" but will be managed for the rest of her life. We have been on a medical journey with her since before she was born, and we will be on it for years to come.
### SHE'S AMAZING
At the time of this writing, Mia is almost ten years old. She's an amazing little girl—confident, secure, well liked, and quite sassy and spunky. I could go on and on listing her good qualities. She is a leader among her girl cousins; whenever a group gets together, they want to know if Mia will be there. They love and support her, and she loves and supports them. Mia's brothers adore her and would take up for her in a split second if anyone ever gave her trouble, and she gets a kick out of hearing them say that to her.
Because of issues with Mia's bone structure and tissue, she is unable to form words the way most people do. But she taught herself to compensate for what she is missing physically and she speaks very clearly. She is not self-conscious; she interacts with others well and has fun wherever she goes.
People seem to love Mia. We have not had any serious experiences with people being unkind to her. One time, when Mia was about six years old, I heard a little girl ask, "What's wrong with your lip?"
Mia responded matter-of-factly, "I had surgery. You ever had surgery? It's really cool. I got ice cream." And that was the end of it. Her difference is not a deficiency, and that's because she knows how to handle it and she knows how to deal with other people. She does not let it stop her, slow her down, or keep her from doing anything.
Mia has suffered a lot as she has undergone her surgeries, but she bounces back well from each one. At the time of this writing, she wears headgear. It's different from what most people think, as it does not wrap around the bottom of her face (like regular headgear for braces). It is specially designed to help align her top jaw with her bottom jaw, because the top one stopped growing due to scar tissue from a previous surgery.
She has to wear the headgear twelve to fourteen hours each day and was told she could not do any physical activity while she wore it. At first, the process of getting it off and on was difficult and painful since her mouth was so sore from the newness of all the attached metal. The doctor told her she did not have to wear it to school but would need to put it on as soon as she got home in order to get in all the hours. The reality of this additional life change for my child brought on a truckload of emotions for me. Since sleep only took care of nine and a half of the required number of hours, she and I both realized she still had at least three to four hours left each day. I was so sad to think she would have to give up riding her scooter, swimming, or jumping on the trampoline with Bella. However, Mia quickly figured out that if she wore the headgear at school, she could resume her normal playtime activities at home. This decision shocked all of her doctors and therapists. They told us that in all their years of practice, they had never had one patient wear it to school because of its visual effect.
Mia is a good patient, but she's an exceptional kid. We spend many hours sitting in waiting rooms with other families who are going through the same thing we are. We share stories and advice, and we marvel at how these kids cope with their conditions. Jase and I have seen firsthand that all of these kids are exceptional because of the suffering they endure. Romans 5:3–4 says, "But we also rejoice in our sufferings, because we know that suffering produces perseverance; perseverance, character; and character, hope." According to this scripture, character is not something we are born with; it is produced by persevering through suffering. I still pray that Mia will suffer as little as possible in this life, but at the same time I am grateful for what she has gone through. Without her suffering, she would not have the character she has today, the intense love and acceptance she has toward other people, and the enormous generosity that spills from her heart. Because of her character, she gives hope to other families going through similar hardships.
Jase and I are well aware that Mia will have many more challenges in the future. But with the character she has developed, with the faith we have as a family, with our trust that God will never leave us, we are confident she will not only handle them but overcome them and continue to amaze us all.
# Part Six
# LIFE IN THE LIMELIGHT
Let your light shine before others, that they may see your good deeds and glorify your Father in heaven.
MATTHEW 5:16, NIV 2011
## 24
## INTRODUCTION
## A Message from the Wives
We are biased, of course, but we honestly believe we have the greatest fans in the entire world. We love them, appreciate them, and try to interact with them as much as possible. Having fans is one of many things we never dreamed would happen to us before Duck Dynasty went on television and became a hit. The visibility the show has brought to our family has changed our lives in some ways but not in others. For example, the show has given us lots of exposure, and some people say it has made us famous, but it has not changed who we are as human beings. It's relieved some financial pressure, but it has not changed our value system. It's put our family in the spotlight in ways that can be intense, but it has not affected the love and support we have for one another. It has definitely brought us closer together and hopefully it has brought love, laughter, faith, family, and ducks to millions of people around the world.
### Korie: WE'RE ALL IN THIS TOGETHER
I think one of the best things about the increased visibility Duck Dynasty has brought to our lives is that we all experience it together. We feel blessed to be able to do the show as a family. It's not like one person suddenly has the attention of the world while the rest of us sit in the shadows. We have heard about families in which one person becomes famous, and it changes their family dynamic completely. The whole family moves to Hollywood and that does not always go well; sometimes the struggles and problems that result are almost insurmountable. When everything centers around one person, it can disrupt the family life.
In our family, we are all in the spotlight together. Fame has happened to all of us at the same time, and that keeps us normal and grounded. As my sister-in-law Jessica says, "We don't let anyone get too big for their britches." We see the fame and celebrity for what it is. We appreciate it, but we are not impressed with ourselves because of it. We are able to share both the burdens and the blessings of this unexpected and amazing opportunity, and that helps all of us deal with it better than any of us would on his or her own.
## 25
## LIVING IN A GLASS HOUSE
## Lisa
Alan and I did not appear on Duck Dynasty until season four, so we have not dealt with the spotlight in the same way others in our family have. We learned about it in an entirely different way that was sometimes just as intense. We were in the ministry for many years, and I found out soon after we started that a minister and his wife live in a glass house. Somehow, it seems that everyone sees, hears, and knows everything that goes on in your life! We quickly learned that we needed to be totally transparent. Now our lives are open books. If people need to know something about us, we want them to hear it from us in the most straightforward way, not through the grapevine after it has been embellished or misinterpreted.
We also want people to know it was God's power that helped us deal with all the difficulties we have faced through the years. Typically the church is a place where people find hope and healing, but in any church, there are some people who want to know all the "juicy stuff" about the minister and his family. This goes on everywhere. We are all human beings; some of us want to find the positive attributes of a ministry family, and others, unfortunately, want to find out the flaws and less-than-desirable qualities of church leaders. During our years in ministry, Alan and I learned to speak openly about our flaws and to accentuate what God has accomplished through us in spite of them.
I think one of the saddest things about living in a glass house, especially in the ministry, is that the children of the minister suffer so much during their growing-up years. Some people think the minister's sons or daughters should be perfect. These people do not take into account that they are children, and they will make mistakes—as all children do. There is no "special potion" we as ministers get to rub on our kids to make them act a certain way! We rub Jesus all over them with our love, devotion, and acceptance of them, but they still make mistakes, just like their minister parents! Okay, enough preaching. I think I might have started meddling. Whether a family lives in the spotlight because of a ministry position, a television show, or some other reason that brings visibility, there are special challenges that come with the increased visibility.
### GOD PREPARES PEOPLE
When Alan and I look back on our years in ministry, we have a lot to be thankful for. We worked with a truly great church, we had some exciting opportunities to travel and do mission work overseas, and we served God alongside some incredibly faithful and really good people. We were always aware of those things. But we were not aware at the time that God had something else in store for us and that He was not only using our time in the ministry to help people in the present but He was also using it to prepare and train us for our future. Without the experiences we had and the lessons we learned back then, we would not have been prepared for the life we have today.
We often hear about people, especially young people, who are desperate to become "famous." We see people who think everything in their lives would be wonderful if they could just make it big and become a star. Many of them do not realize there is a price for fame and that there are both positive and negative aspects to living in the public eye. Without a well-grounded life and a strong support system, being a star has the potential to devastate individuals and their families.
We believe God prepared everyone in our family in different ways for the visibility we now have. For Alan and me, living in the glass house of ministry gave us a very firm foundation for the limelight of the entertainment world. God always knows what's ahead for all of us. In His love for us and His desire for us to be blessed, He prepares us. When Alan and I worked at the church, we never dreamed we would someday be so involved in a TV show. God trained us in the relative safety of a church setting for everything we would deal with in the realm of television. He let us grow and make mistakes in front of a small, loving audience before He put our family in front of the whole world. We are very thankful He gave us the training we needed over a period of time, among family and friends, instead of just letting us wake up one morning with our last name as a household word.
I always want to tell people who have big dreams that we just never know what's ahead. We have to embrace every season God takes us through, trusting that He will use each one for our good and for His glory. We have to believe that if He wants to put us in the spotlight, He will prepare us for it and do it in such a way that we can handle it well.
I cannot imagine what Duck Dynasty would have done to our family had we not been ready for all the changes it brought and had we not been grounded in our faith and our family. A lot of people crave overnight success, but we are glad that did not happen to us. We see the wisdom and love of God in our lives as He spent years with us behind the scenes making us ready for the high level of exposure our family has today.
### THE BEARDLESS BROTHER
One thing that sets Alan apart from the Robertson men who frequently appear on Duck Dynasty is that he does not have a beard. In fact, when the media heard he would appear on the show starting in season four, lots of headlines and articles identified him as "the one without the beard."
By the time a large audience began to hear about Alan and about the two of us as part of the Robertson family, we were well established in knowing that very little matters in life except pleasing God, pleasing each other as husband and wife, and honoring our family. We had learned the hard way that we could not please everyone around us, so we do not spend much energy trying. We do our best to live our lives with integrity and to love and serve the people around us. We make plenty of mistakes, but we recognize them, own them, and make our apologies. None of us has any room in his or her life to judge the others. Rather, we work to love and forgive each other. Our family has hearts that seek after God; I can say that with total confidence about each and every person in our family, and we all do the best we can to fulfill our responsibilities and enjoy the life God has given us.
### WE WANT TO BE GOOD EXAMPLES
Because of the publicity our family receives, we are responsible for holding our standards high. We want to be good examples. That starts at home long before it shows up in front of the camera. Though I do not often appear on Duck Dynasty, I do participate in plenty of casual conversations about the show with my sisters-in-law and Miss Kay. We talk about the fact that the impact of everything they do is magnified because they are in the limelight. The way they dress, the way they talk to their husbands and children, the way their children talk to them, the way they control their actions, and all kinds of other things make statements about who they are. Alan's and my children are grown, but we have grandchildren who have to learn these lessons too. We must teach each generation about these truths. Once a show is filmed and goes on the air, they do not get a do-over and they do not get to go back and tell the world what they really meant. An audience takes things at face value and draws their own conclusions, which means first impressions are extremely important.
First impressions are just that—first impressions. You only get to make that first impression one time. If a person gets to know you and interact with you, a bad first impression can be changed. But with television, people do not get to interact with us and see us at our best. Our best has to be what we put in front of the camera every time it's on. We hope and pray our family is making an excellent impression on the world through Duck Dynasty, giving people a lot of good laughs. But even more important than the laughs, we want to be true to the fact that we love God, we love each other, and we desire—through the show—to honor Him, to honor one another, and to affirm the goodness of faith, family, and the true love of Jesus.
## 26
## IT'S NOT JUST FOR US
## Missy
As Jessica mentioned earlier in the book, I also have to laugh when I hear people talk about Korie, Jessica, and me as gold diggers. I think, Oh please! Jase was skinning raccoons for extra money when Cole (our second child) was born! The only gold I was interested in was Jason's old gold-colored Chevy, because it had a bench seat and I could snuggle up to him while he drove.
When Duck Dynasty started, we lived paycheck to paycheck. Most of us in the family did not really have anything extra. Jason and I often heard our friends talking about financial planning for their children's college careers, but that seemed almost impossible to us; we simply hoped our car kept running.
I will never forget the days, not so long ago, when instead of traveling around making live appearances for large audiences on weekends, Jason traveled to small towns to preach in churches and typically brought home some kind of honorarium that provided us with a little extra money. Financially speaking, we did not have much, but we were happy, and we had a good life. We were blessed before we ever dreamed of being on television. Now we are overwhelmingly blessed. We were blessed to have enough for many years; now we have been blessed with abundance.
### GENEROSITY COMES FROM THE HEART
I have never been affluent; Jase hasn't either. Spending a lot of money on ourselves and "building barns and bigger barns" is not in our personalities. We have always tried to be generous, and Duck Dynasty has not changed that. Now we simply have a little more to be generous with.
I once heard a saying that went something like "If you won't share a peanut butter and jelly sandwich when you are poor, you won't share a steak when you are rich." Generosity comes from the heart, not from your bank account. In 1997, Jase and I built a house in the country. For six years before that, we had lived in a 1,099-square-foot two-bedroom, one-bath house in the middle of town. That was where we brought home our first child, Reed, in 1995. For six years, we hosted Sunday-night church fellowships in our house with anywhere from fifty to eighty people in our carport turned living room. Finally Jase said, "I just can't live here anymore. I really need to move to the country."
At that time, Duck Commander was still being run out of Phil and Miss Kay's house, and Jase was Phil's only help when it came to making duck calls. Since Phil and Miss Kay lived so far from town, we decided to try to move closer to them. Phil's good friend Mac had three acres next to his house and offered it to us for a great price. We built a four-bedroom, two-bath house on that property, where we lived for ten years and brought two more babies home from the hospital.
Since we were on a tight budget, our contractor offered many different ways for us to save money. He said, "Every corner we build is an added expense." So I said, "Then I want only four corners."
We made our front bedroom into a playroom with a glass-pane door in order to accommodate the many church groups and Bible studies we planned to have in our home. This way, kids could use that room quietly while the adults watched them during our Bible studies. It worked wondrously, and people could come to the gatherings without having to hire a babysitter.
I learned from watching Phil and Miss Kay what it means to use all of your resources in ways that help others. Jase and I understand that the material blessings we have are gifts from God, not something we earned ourselves. If you think you deserve something or have something because of your own deeds, it will be far more difficult to share it. Living a life of gratefulness and generosity is far more rewarding than counting your silver coins every night.
Now God has blessed us with a platform. The visibility and resources we now have were given to us directly by Him, and we know God has given them to us for a reason. That reason, I believe, is not to please ourselves; it's to help others. We have a beautiful home and nice vehicles, which we enjoy immensely. But they are only things. They are not nearly as important to us as people and our relationships with them.
### WE RECOGNIZE OUR RESPONSIBILITIES
People who watch Duck Dynasty do not always realize that we have a family business—not just for television but in real life. We work hard, just like lots of other people in America. For some reason, God has seen fit to use us in the world of entertainment, and we believe He has a purpose and a plan in doing so.
All of us feel that the opportunities and resources we have been given are big responsibilities for the Robertson family as a whole and for each of us individually. This is something God has given us that does not ordinarily happen to many other families, and we want to be good faithful stewards of it. Jase and I talk to our children a lot about the parable of the talents. In Matthew 25:14–30, Jesus told a parable about a man who left on a long journey. Before he left he gave to one servant five talents (a type of money in New Testament times), to another two, and to another one. The one with five talents put his money to work and gained five more. The person with two talents also doubled his money. But the servant to whom he gave one talent buried it. When the man returned, he was very proud of the first two servants but very disappointed in the last one. Because the third servant had not used the talent he'd been given—but had buried it—he took that servant's only talent away from him. In other words, God gives what He feels we are responsible enough to use, and if we aren't responsible with it, He will take it away. In our family, we want to use what He gives us in all the right ways and never misuse it or fail to appreciate it. Will we always make the right decision? Sadly, no. We will make mistakes, misjudgments, and downright selfish decisions sometimes. But thanks to Jesus, we are given many more chances to get things right the next time.
### JUST REGULAR FOLKS
Jase and I are very thankful to have our children in a school where teachers, coaches, and classmates treat them the same way they treat everyone else. They definitely don't get the star treatment, and that's the way we want it. Our boys have to work just as hard as their peers to make a sports team, and they will get benched as quickly as anyone else if they do not play well. I love that they do not get treated any differently from their fellow students.
My parents, along with four other couples, founded Ouachita Christian School in 1974, so in a way it feels like home to me. Korie, Jep, and I all went to school there, and Phil taught there for a while before he started Duck Commander. Now all the Robertson cousins go there. It's a school founded on Christian principles, and Jase and I know it is a place that affirms the values we have for our family. Everyone there treats us as though we are regular moms and dads—which, of course, we are—so we appreciate that. We can attend Reed's home football games and sit right there in the stands with the other fans and families, and no one bothers us. Even on the road at away games, our friends help keep us as anonymous as possible so we can enjoy being just parents. Sometimes it works; sometimes it doesn't. We try to accommodate as many autograph and picture requests as possible before and after those games and during halftime. But when Reed steps onto the field, we are there in full support of him and his teammates, and autographs have to wait.
### OUR SACRED SPACE
One thing that really surprised me, and probably our whole family, after our show started was that people we did not even know came to our church to try to see us or to ask for autographs. I have sung on our church's praise team almost every Sunday for more than twenty-two years, and I love doing it. One Sunday after a very prayerful song, someone yelled from the audience, "Go, Missy!" I turned three shades of red, I'm sure. As our praise leader went right into the next song, a couple of men made their way over to where that person was sitting in order to defuse any potential problems. Nothing else happened, but obviously, the somber moment was lost, and the mood became very awkward and a bit tense.
We appreciate the interest in our family, but being able to attend services without distractions is important to us. We go to church to worship God and for fellowship with our longtime friends—people who knew us and loved us before our faces were ever seen on TV. We hope everyone who visits our church is blessed by being there, and we really appreciate all the people who respect us and respect our church family by allowing us to attend church uninterrupted.
We know this is only a season in our lives. It will not always be like this. One day, maybe sooner than we think, we will go back to being regular church members in a regular church on a regular Sunday morning. We might even miss being asked to take a picture during "meet and greet." Well, maybe "miss" is too strong of a word. Our church family is very special. Our services are incredible and our a cappella singing is inspiringly beautiful. People are completely uplifted when they visit our congregation, and it honestly has nothing to do with us. We are no more important than anyone else there. Every member makes it special. That's what is so great about God's family. Every part has a purpose—television star or not.
## 27
## WE'VE COME A LONG WAY, BABY!
## Miss Kay
After the way I grew up and all the hard times Phil and I have been through, who would have ever dreamed I would write about what it's like to live in the limelight? Back in the days when I lived in a trailer with three little boys, barely able to keep the lights on, who would have imagined I would ever be talking about being famous? Who would have thought I would have trouble finding a parking space at the Duck Commander office because so many fans had come to see it? Not me!
I appreciate everything Duck Dynasty has enabled us to do, but I have to say, there are both good and bad parts to it. Some things in our lives have gotten easier because of it, but others have gotten harder. For example, I love the fact that some of my grandchildren have opportunities to speak and can share their faith with large audiences. Also, I'm so glad my children are able to pay for their homes—that makes me really proud. But I am sorry for the freedom I have lost in my schedule and the freedom I have lost to be an ordinary, unknown person who just loves to cook and enjoy her family. I don't get to do family dinners the way I want to anymore. They have become much more structured, and a lot of times, we do them with a full television crew in my kitchen!
### SO MANY VISITORS
I remember the carefree days when I could open my door on a beautiful morning and go outside in my pajamas without anyone seeing me. That almost never happens anymore, and I miss those times! Oh, I still go outside in my pajamas, but there's often someone standing nearby ready to snap my photo the second I go out the door. I have to pay a lot more attention to the way I look than I used to. I have to do my hair and my makeup, and I have to wear decent clothes, because every time I walk out, there's a good chance I will have my picture taken many times before I get back home.
I was raised on the Golden Rule: "So in everything, do to others what you would have them do to you" (Matthew 7:12). I do my best to live by those words, so when I get ready to say or do something that involves others, I try to ask myself if I would want them to do to me what I am about to do. I just wonder if the people who stand in my front yard with cameras flashing would want me standing in their yards taking pictures of them in their pajamas. Maybe only if I had a plate of hot biscuits to share!
I have to draw a line when I go out with my grandchildren to do something special. I have to tell people nicely that I cannot stop for a photo right then. Again, if they were out trying to have a special day with their grandchildren, I do not think they would want strangers trying to take pictures.
### WE NEVER DREAMED . . .
When Phil and I moved to the river many years ago, we never in our wildest dreams thought we would have to put up a fence or a gate just to maintain a little privacy. We live so far out of town that we were totally shocked when people first started coming to our house trying to catch a glimpse of us. They came early in the morning, sometimes before I was even out of bed, and they came as late as eleven or twelve o'clock at night. Some of them even knocked on our door!
We eventually had to get a fence and a gate; this really went against our nature, because we have always wanted our home to be a warm and welcoming place for all our friends and family. We love to have people over, but when people we'd never even met started inviting themselves and showing up at all hours, we had to do something in order to maintain the privacy we have left. I need my home to be a safe haven, just like most other women do. And, like most people, I want to live in a place where my grandchildren and my family can be safe. I don't think that's too much to ask.
If we ever forget, in the midst of the fame, where we have come from, then I pray God will take it all away from us and move us right back to where we were in the beginning. I really do. I think a lot of us in the family feel that way. We do not want everything the show has done for us to change us in a bad way. Occasionally, Alan gets us together for a reality check. He talks to us about our lives, about how we are doing, and about not allowing our "stardom" to affect us negatively.
I am thankful for the great opportunities we have been given because of the show. Sometimes we get to stay in really nice hotels and when that happens, I try as hard as I can to be friendly and to talk enthusiastically to everyone. I do not care if a person is a housekeeper, a maintenance worker, a concierge, or the owner of the whole hotel; I want to give everyone the same kind of attention and kindness. Everywhere we go, we see people who are working hard. I know what many of them are going through. I have not forgotten what it's like to struggle to make a living—and I pray I never do forget.
### DON'T MAKE MOMMA MAD
There are only a few things in life that make me really, really angry. One of them is when people struggle in their marriages and refuse to fight for them, but I have already mentioned that. Another thing that infuriates me—and embarrasses me so much for the people who do it—is when women nearly fall all over my sons flirting with them. They try some of the most disgraceful things to catch Willie's, Jase's, or Jep's attention. Some of the behavior I have seen toward my sons—and even toward Phil and Si—is just shameful! I don't understand how people can let themselves act that way, and as a woman, I really am humiliated when other women do such things.
I realize all the boys are good-looking, and I know what great men they are, but they're taken. A lot of people don't respect the vows and commitments of marriage anymore and simply do not have any self-respect. They do not seem to have any reservation at all about flirting with men they know to be married. When people don't honor the fact that each of my sons already has the woman he has chosen, I want to say, "Come on! These boys are happily married men. Go find your own duck hunter!"
This kind of thing did not happen before we went on television, and I hate to see it happening now. As much as I enjoy interacting with our fans and hearing stories about the positive impact Duck Dynasty has had on so many people, I will never be okay with women chasing after my sons.
### A GREAT WAY TO SHARE OUR FAITH
Though I don't like the flirtatiousness the boys have to deal with at times, I am grateful that the show gives us a way to share our faith and our values with millions of people. All that really happened to us is that some people at a television network saw something in us, liked us, thought we were funny, and realized we could make money for them. The entertainment industry is a very secular environment, but when we had a chance to get involved in it, we did and used it to share our faith.
I think about Phil's journey to faith in Christ. For many years, his journey through life was painful for him, for me, and for the boys. We prayed and prayed for him, and when he repented and got baptized, we were so happy we could hardly stand it. That man had a 100 percent turnaround in his life. And now, he tells everyone about it. He is the most courageous Christian I have ever known. He will share his faith with absolutely anyone. I am so glad God has given him—and all of us—a chance to do that in such a big way. The inconveniences and the sacrifices we make in order to have this opportunity are worth it.
## 28
## WE'RE HAPPY NO MATTER WHAT
## Jessica
I appreciate the fact that Duck Dynasty gives our entire family a chance to show the world who we are. When people ask me what makes us Robertsons the kind of family we are or what we do in order to have the kind of family life we enjoy, I have two words: "Robertsons love." That's the bottom line. To me, the best thing about the limelight and the exposure we have is that we get to present to the world the love we have for God and the love we have for one another. We also get to do a lot of good, whether that means calling attention to a good cause, donating money to charity (we have done events and given all the proceeds to charity), or just taking the opportunity to share love and kindness with the world. We have not always had those opportunities, and we are blessed to have them now.
When I think about what it means to enjoy some celebrity status and about how Duck Dynasty has changed our lives, I also have to think about how it has not changed us. For one thing, we are still the same happy, loving people we have always been. Jep and I have always found our joy in God and in our family. We have never been impressed by fame or fortune, and we still aren't.
By the time Jep came into the Robertson family, some of Phil and Miss Kay's major financial challenges were behind them. While Alan, Jase, and Willie remember times when their family struggled to buy groceries, Jep remembers that Miss Kay bought him a new G.I. Joe almost every time one came out. My upbringing was similar. Our family always had enough, and we took care of our belongings. I remember, as a fifth or sixth grader, when my parents bought us a trampoline. I thought we were rich. For all I knew, it might as well have been a yacht! I share those stories to make the point that Jep and I have always felt taken care of; we did not feel we lacked anything we needed. Now we want our children to feel the same way—whether or not we are on television, whether we have $500 in our checking account or $50,000.
### TEACHING OUR CHILDREN
Children today face temptations that Jep and I never encountered. Technology alone has totally changed the way many children think about money and possessions. While I grew up having to save my money for something like a new bicycle and knowing I would not get another one for several more years, a lot of children we know have the latest electronics handed to them and then get new ones as soon as updates are available. We try hard to help our children understand what it means to earn their own money and make their own purchases and to appreciate and take care of what they have. We want them to realize they do not need the latest, greatest gadget. I did not even have a cell phone until I met Jep. I use my phone and laptop for many purposes, but I have no need to upgrade them every time I turn around.
I could hardly believe my ears the day one of our children told me we needed more square footage in our home. I didn't know she had ever even heard of square footage! Apparently, someone at school had been talking about moving into a larger house, and our daughter thought we should do the same. I said to her, "You know what? I am glad that family can have a bigger home, but square footage or having the biggest and best of anything does not make anyone happy. I'm happy. All my children are healthy; we have a nice home and food on our table. All our needs are met. We have a peaceful, happy life. We don't fight. And we're all together. If we get a bigger house someday, that would be great. But if we don't, I will still be the same happy person I am right here in this house. Having my dream house, a new car, or the nicest clothes will not make or break my happiness." I think she got the message!
Joy does not come from what we have; it comes from knowing God. And God has blessed our family with so many gifts no amount of money could ever buy. I was content when Jep and I hardly had anything at all, and I am just as content now. We can truly say we are rich in love, and that is what matters most to us. It's more important for us to raise our children knowing Jesus Christ than it is to be wealthy from the world's perspective.
### KEEPING OUR BALANCE
Jep and I are determined to stay happy and balanced, no matter what happens in our lives. Like everyone involved in Duck Dynasty, we had to make major adjustments to our schedule when it started. We suddenly had to squeeze filming, travel, speaking engagements, and other live appearances into an already busy life with four young children. The filming alone takes much longer than most people think. Even filming the short dinner scenes at the end of most of the episodes often takes one and a half to two hours. We have to add to that the time it takes for hair and makeup, which we do every time we film anything. And sometimes, the dinner scene is not filmed at a normal mealtime, so that throws off our schedules a little, but at least we can enjoy time with our family and have a good meal.
When Duck Dynasty started, Jep and I learned quickly we would have to monitor our lives closely and set firm boundaries in order to keep time from getting away from us and to keep ourselves from getting worn down. It did not take long for us to see how fast the scales of our lives could tip out of balance, causing us to feel burned out and empty. For me, especially, the travel can be grueling. Sometimes, when I'm on the road, I can get so tired and miss my kids so much that I want to cry. Every so often our priorities get out of balance, and we have to stop and get everything back in order. The great thing is that Jep and I are on the same page, whether the issue is raising the kids or our views on life in general. When I overextend myself, he can see it and he will let me know. I do the same for him, because we love each other and are on the same team; we are helpmates to each other.
Prayer is a big part of our lives, and Jep and I pray regularly about our commitments and our schedules. Is it important to be paid for a television show so we can have nice things? It's not nearly as important as providing our children with happy childhoods and not being so exhausted that we won't remember this time in our lives ten or twenty years from now. God blessed Jep and me with four sweet children, and it is our responsibility to teach them how to grow to be godly people, to make the right choices in life, and to treat others well. We continually ask ourselves what is most important in our lives. We realize our children are the youngest of the Robertson grandchildren, which means they might possibly spend more of their lives in the spotlight, starting at younger ages, than their cousins. We do not want them, or ourselves, to lose anything that is truly valuable in life just because we are on TV.
### MAKING WISE CHOICES
One thing we have had to be very diligent about is teaching our children to choose their friends wisely. Our children have experienced a few instances in which others have been nice to them because of their last name. One of our daughters, Lily, is very shy when she is not around family. She did not have many friends during her first few years in school, but when Duck Dynasty started, she suddenly got a flood of invitations to do things with other children. We cannot judge anyone's intentions, but we did notice the dramatic increase in people who wanted to be her friend. We wanted to make sure the people who reached out to her were truly interested in her, not in anything else.
In helping our children choose good friends, we have taught them to pay attention to the way people treat others as much as they pay attention to the way people treat them. We have encouraged them to see whether the children who want to be their friends treat their teachers with respect and whether they are nice and kind to everyone. We have taught them to notice whether other children obey their parents, teachers, and others in authority. These are important values to us, and we want our children to choose friends who share them.
We have laughed at times because once Duck Dynasty went on the air, little boys started liking our daughter Merritt—who is nine years old at the time of this writing. She is a cute little girl, so I can understand this, but it could also be that they just want to meet Jep. Either way, it's cute! We might be a little concerned if she were really into boys, but she's not into them right now, thank the Lord!
We felt good about our youngest daughter, Priscilla, one day when we heard that Willie asked her, teasingly, "Do you want to be a TV star when you grow up?" She looked at him and answered humbly but matter-of-factly, "I already am a TV star." It was almost as if she were thinking, Been there, done that. What's next? At this point, she is unfazed by the whole television situation. It's part of her life, but it's not her whole life.
Our son, River, is the youngest member of the family and was only two years old when we started filming Duck Dynasty. He may get a few things other little boys do not have, and he will likely have some opportunities others may not, but Jep and I do not want him to have anything handed to him. We want him to know what it means to work hard and save and wait for what he wants. Our job as parents is to teach him those lessons.
### GENERATIONAL LESSONS
When Jep was very young, his parents taught him the same kinds of lessons we want our children to learn. He tells a story about a time when he was a little boy and went with Miss Kay to take food to someone who lived in an underprivileged neighborhood. The person could not afford a very nice house and none of the interior rooms had doors; they only had sheets hanging from the door frames. When Jep walked in, he said, "I love this! No doors!" Then he went flying through the sheets pretending to be a superhero.
Jep did not see the person's lack. He saw something fun. He saw value in what some people would have viewed as a deficiency. He is still that way, and I try to be too. My grandparents have never had a lot, materially speaking, but they have worked hard all their lives, and they are wonderful people whom I dearly love and am so close to. Jep and I make a concentrated effort not to pay attention to economic differences, even when they are very obvious. We want to value and appreciate each person we come in contact with, and we want to teach our children to do the same.
I still remember when Jep and I spent our weekends at outdoor shows, selling duck calls to make extra money. Even in those days, people lined up to get Phil's autograph. Now people ask us for our autographs, and sometimes we can hardly believe it.
Being in the limelight has brought us both blessings and challenges, but if we lose everything tomorrow, we will still be happy. We can be confident of that because we were happy before we had it. We believe that the more we have, the more we can share, and that's what we are determined to do.
Of course, the television show has made some things easier. We still live in the same house we have lived in for years, but we did finally get a new vehicle. I am still a serious bargain shopper, and I still go through the clearance racks, but we can now afford a few more groceries, and we can have people over to our house more often. We can definitely serve people more effectively and meet more of the needs around us. We do not have as much time as we used to, but we make the most of what we do have. I am diligent about being at home with the kids in the afternoons, and we still make a priority of family time together. In the end, that is what's important to us and that's what we most want to preserve.
## 29
## GREAT REWARDS
## Korie
I married Willie thinking he was clean-cut and preppy. When we got married, he had short hair and no beard. And he did not dip. He was the "city boy" among the Robertson men. On the other hand, Jase was outdoorsy. He spent his time in the woods or on the river—hunting, fishing, catching frogs, or doing other things connected to outdoor life—like Phil. The Willie I married was headed in a totally different direction from Phil and Jase. He wanted to do his own thing, chart his own course. If I could only tell you about all the businesses Willie has dreamed of starting, all the jobs he was going to have, from professional golfer to karaoke deejay! Willie has always thought big. He has also always enjoyed hunting, but early in our marriage he thought he would do something different from his dad. I had no idea that all these years later, he would become so much like Phil he would practically turn into his dad and that our family would appear each week on a television show centered around duck calls and hunting.
Phil and Willie are a lot alike. Even Miss Kay says that. They are both strong-minded, visionary, and entrepreneurial. Phil started Duck Commander and grew it to a certain point with a lot of hard work and determination. When Willie got involved in the business, he saw all kinds of ways to expand it. He and I eventually took a leap of faith and bought a portion of the business from Phil, never dreaming it would take our family to prime time.
The Robertson men have been involved in media for years, much longer than some people realize. They started with hunting videos that appealed primarily to serious hunters. Because of those, Phil especially became well known among certain audiences, but nothing like he is now. Ten years ago, he and Willie could have walked through Times Square and no one would have noticed them—well, except for Phil's beard and rough look. That would not be the case today!
Being on the show has given us so many opportunities we never thought we would have. As a family, we enjoy doing live appearances together, and we have been involved in some really fun things. Sometimes, when we all pack up to travel to a live appearance, we feel like we are going on a field trip. Those times often provide our children with chances to do things Willie and I never imagined they would be able to do. For example, we had a great time when we all participated in the 2013 Country Music Association Awards. John Luke and Sadie even got to introduce Dierks Bentley, which was fun for all of us to see.
### GREAT PEOPLE, GOOD NEWS
One of the best blessings of the visibility Duck Dynasty has provided us is the chance to interact with so many great people. Our fans are terrific and we enjoy getting a minute to chat with them whenever we can. We also have opportunities to meet talented, dedicated people who are doing some amazing work, mostly projects or initiatives to serve others by relieving suffering, offering tangible hope, transforming communities, or finding other ways to make an impact in this world. We are often invited to grand openings or fund-raisers for these kinds of activities, and I love seeing how creative, passionate, and determined some people are to do good and to bless others.
Unfortunately, the news these days is full of negativity, sad stories, and flat-out bad reporting. Because our show has given us a platform to travel and meet people, we have learned that there is also quite a bit of good news in the world. Many people we meet are doing interesting and wonderful things that do not receive any news coverage. Once we started traveling, we realized quickly that the positive things happening in the world are highly underreported!
We cannot possibly say yes to every invitation we receive; there simply are not enough hours in the day. But we try to do as much as we can to help various charities. One of my favorite stories involves an organization that was trying to build a park with a playground for children with special needs. They needed to raise one hundred thousand dollars to build the park. We attended their fund-raiser, and when it was over, they were shocked and excited to have hit their mark in one night! We were so glad our presence provoked so much generosity and so happy to know the children were going to have a nice place to play. We really enjoy being involved in things like this.
### 'DUCK DYNASTY' IS MAKING A DIFFERENCE
Doing what we do does take a lot out of us at times, but it's worth it. Our busy schedules and frequent travel can be tiring, but they provide us with so much fulfillment. What is most fulfilling for me is hearing stories from fans about the positive impact Duck Dynasty and our family have had on their lives and their families. We have had people tell us with tears in their eyes that the show has brought their family closer together, and we love that—because close-knit family is what we're all about. One woman told us she and her husband had not been able to find a single television show to watch together in more than fifteen years—until Duck Dynasty. People also tell us their pastors preach about the show on Sundays, using lessons from the show to help people grow in their faith.
Maybe the most meaningful thing of all to us is that families have started praying together as a result of the dinner scenes at the conclusion of most Duck Dynasty episodes. We have even heard of one group of fifth graders who call themselves the "Duck Dynasty Club." They sit together during lunch at school every day and pray before their meal. We feel so blessed to be part of these good things that are happening because of one TV show. We had no idea it would make such an impact.
When we first started doing Duck Dynasty, several people asked us if our faith would play a part in it. We thought that was kind of an odd question, because we do not know how to separate our faith from everything else we do. Faith is how we live; it's who we are. No one can take it out of us. If anyone is going to do a show about us, they will also get our faith.
So much about today's entertainment industry is based on shock value. Every season, it seems like networks make an effort to push the envelope a little more in terms of sex, violence, and language. When we went on television, we presented an alternative. People can find disrespectful spouses, disobedient children, and bad language on other shows; they will not get it from us because that's not the way we live. Some people have been surprised at our success; others are simply grateful to have a family-centered show that keeps the language clean, affirms traditional values, and honors God in every episode.
### WE CAN'T WORRY ABOUT WHAT PEOPLE THINK
One of the most important lessons I have learned about living a very public life is that we cannot pay too much attention to what people think or say about us. I actually learned this lesson a long time ago, because everyone deals with this throughout his or her life to some extent. But the issue is magnified when you have a hit television show. Still, it's a lesson for everyone. Whether it happens in school, at church, or in the workplace, people are going to be critical of the things you do, what you wear, the choices you make, or something else. You cannot control that; all you can control is what you do and how you live. And if you are living to please God, that's all that matters.
Our family cannot control the press we get, but when it's inaccurate or when it simply does not tell the whole story about us, we do not have to let it upset us. I remember a specific incident that took this thought to a whole new level for John Luke and made it personal for all of us.
John Luke was out with his friends, a group of guys, in Willie's truck. I should have known when they left our house to send a girl with them; there was way too much testosterone in that vehicle! While they were out, they saw an old, beat-up boat on the riverbank (it even had bullet holes in it). It had been there for years, but for some reason it caught their attention that day and John Luke decided to bring it home. It was waterlogged and full of junk, so the boys spent hours pulling it out of the river. They then loaded it on a trailer, not realizing there is a limit to how much weight a trailer can haul.
Feeling pretty good about themselves, the boys climbed into the truck, rolled down the windows, and headed home. They were living the life, just having fun. As John Luke rounded a corner, the weight of the boat was too much for the truck and the trailer. The truck flipped four times. To look at the wreckage, a person would think no one survived. But thank God, no one was seriously hurt. John Luke crawled out of that truck without a scratch on him, and the boys in the backseat—who were not even wearing their seat belts—had only a couple of bumps and bruises. It was a miracle, and we still thank God for protecting those guys.
When Willie and I heard about the accident, like any parents, we rushed to the scene to make sure they were all okay. When people passing by noticed us, they stopped, and soon a crowd had gathered. The police eventually had to ask us to leave. We hated to do that, but we understood we were attracting too much attention. The police had a hard time doing their job with so many people around.
I didn't even notice at the time because I was too focused on the boys being okay, but the policemen pointed out that the boat had been full of empty beer cans that were now strewn across the road. We all knew the boys had not been drinking, but the policeman was concerned for us that people might jump to the wrong conclusion and that the news might pick that up and report it.
I used those circumstances to reinforce a lesson Willie and I have been teaching our children for years. We tell them not to ever worry about what people think, especially when those people do not even know them. We help them understand that people often make judgments based on wrong or incomplete information, so we cannot worry about that either. All we need to care about is what God thinks and what the people who know us and love us think. We do our best to live lives that are pleasing to God, and we do not pay much attention to the rest.
When I was John Luke's or Sadie's age, no one had ever heard of social media. But now, while many of the responses we get through social media or the press are positive, there are times when I'm appalled by what people post about our family. A lot of it is not true, and some of it is downright hateful, so we just delete it and do not give it a second thought. It's a lot easier for me to say this at my age than it is for my children to believe it as teenagers, so I make sure to reinforce this continuously. Social media and the press can work the other way too; you can read all the great things people say about you and get a little too big for your britches fast, if you are not careful. I can say for sure that ever since Duck Dynasty started, all of us Robertsons have gotten really good at ignoring what certain people think or say! We focus on honoring God and living godly lives with our family and friends. At the end of the day, when the spotlight turns away from us and on to the next thing, those are the people who will be there for us. We will always appreciate our fans, but our family will be the ones we turn to when we need wisdom, comfort, or just plain old help.
### ALL GOOD THINGS MUST COME TO AN END
I try to look at this Duck Dynasty time in our lives as one season of many seasons to come. I am enjoying this one, but I look forward to different seasons in the future, which we can enjoy for different reasons. I know we will not live in the limelight forever. Television shows all have a life span. Even television's most popular, longest-running shows eventually come to an end. We know ours will too. Maybe, when that time comes, the guys will shave their beards and cut their hair, and we'll ride off into the sunset.
# Part Seven
# INQUIRING MINDS WANT TO KNOW
Your word is a lamp to my feet and a light to my path.
PSALM 119:105, NKJV
## 30
## INTRODUCTION
## Korie
Everywhere we go, people ask us questions. Sometimes the questions—and their answers—are really funny; sometimes they are very serious. They want to know everything from whether we like our husbands' beards to how we juggle busy schedules to whether they can have Miss Kay's biscuit recipe (which is included in this section). They ask us how we keep our children grounded and whether or not Uncle Si really is everything he seems to be. They also ask about our faith and about what it's like to have such a large family. Sometimes, girls ask me if they can marry John Luke or guys ask me if they can have Sadie's phone number. I'll go ahead and answer those now: no and no.
Before we finish this book, we want to address some of the most common questions we receive. Some of those questions are directed to a specific person; others are ones we all want to comment on. I hope you will get to know us even better through these last pages, and if you have a question we don't answer, maybe we'll see you in our travels one of these days and you can ask us then!
## 31
## ANSWERS TO THE TWENTY QUESTIONS PEOPLE ASK US MOST
1. Do you like the beards?
Miss Kay: If Phil ever shaved his beard, I'd think I was committing adultery.
Korie: When I married Willie, he was clean-shaven and had short hair. Boy, how things change! Over the years, I've really come to like the look he has now, including the beard.
Missy: I love Jase. I don't like the beard. I miss the days of scratch-free kisses. Besides, he's just too cute under there!
Jessica: Yes! Although Jep is really cute under all that hair, and although he does have the Robertson dimples, I still prefer the beard. I think sometime over the course of our marriage I transitioned to loving the beard. I do make him trim the mustache every once in a while for better kisses! I also feel safer with the beard; I know no one is going to mess with us because the beard kind of scares people. For some reason, I think they think he's a madman!
Lisa: Alan is often referred to as "the Robertson without a beard," and I like it that way!
2. Tell us the truth about Uncle Si.
Lisa: Si is one of the most gracious people in our family. He is willing to do almost everything I ask him to do. He has a soft spot for kids. He is a kind, generous, and loving man. He loves his wife, his children, and his eight grandsons. Si is a little eccentric, but he is a lot of fun to be in a family with.
Korie: When Willie and I travel and get to interact with our fans, someone almost always asks us, "Where's Si?" People just love him! I usually give the same answer every time: "Taking a nap." I figure if I say that at any given time, there's a very good chance it's true!
Jessica: Si is one of the sweetest men I have ever known. He is such a great uncle to the kids in our family, and in fact, all kids seem to love him. I have never heard him say a negative word about anyone—and I mean anyone. He has the biggest heart!
Missy: The specific question people most often ask me is "Is Si really as crazy as he seems?" Yes.
3. Miss Kay, you seem like a very wise woman. What's your secret?
Miss Kay: All I know to do is just live by what the Bible says is right and wrong. It's that simple. To me, the only way to raise a family is to do what the Bible says to do. People often want me to talk about family values or say that Duck Dynasty is based on family values. But, plain and simple, family values come from the Bible. My parents and grandparents taught my sister and me how to behave, about respecting other people, about forgiveness, and about discipline when we were very young. All those things come from the Bible, and I'm so thankful I was raised to live by what the Bible says. I did the same for my boys.
I love God's Word and I have a lot of favorite Bible verses. The one I call my "clutch verse"—because of the way I hung on to it through the bad years—is Philippians 4:13: "I can do all things through Christ who strengthens me" (NKJV). Another one of my favorites is Proverbs 22:6: "Train up a child in the way he should go, and when he is old he will not depart from it" (NKJV). I did my best to teach my boys what God says when they were very young. Even though a couple of them had rebellious seasons, they all came back to their roots of faith. Now that they are older, they have not departed from the lessons they learned as children.
4. Korie, is Miss Kay's cooking really that good, and is yours really that bad?
Korie: The answer is yes and yes. I have never eaten anything from Miss Kay's table that I didn't like. She really enjoys cooking, and she loves watching people enjoy what she cooks. She is the kind of mom who gets her feelings hurt if you come to her house not hungry. If you show up there at dinnertime, you'd better at least take a bite and let her know how good it is!
I, on the other hand, always seem to have too many things going on to be a good cook. I really tried in our early years of marriage, but I would get distracted and burn the bread or put apple cinnamon muffin mix in the corn casserole instead of corn bread mix (true story) and ruin the entire dish. Corn and cinnamon do not go together. I've learned to accept that there are other things I'm good at and leave the cooking to Willie in our house.
5. Jessica, you have four young children. How do you juggle it all?
Jessica: Well, lots of prayer. God is my strength! And with being blessed with such a large family, we lean on each other for support. As a family, we help each other out. When I am gone, I can call on any one of my sisters-in-law or on Miss Kay to help, and they know they can call on me any time they need me. I also have great parents who help out a lot. And we have one babysitter who has been with us since Lily (our oldest child) was little. She is more like a family member than a babysitter; the kids view her as a big sister.
6. Missy, who's the better shot: you or Jase?
Missy: The quick and obvious answer is Jase. He is well known for his accuracy (even though Si claims he shoots all the ducks). However, I love to share this story: On the last day of duck season 2012, Jase took Mia and me on a late-afternoon duck hunt. Mia was shooting BBs at the decoys and Jase and I were waiting for the last run of ducks to come through for the year when two ducks came flying over from right to left. We both fired when they got in front of us. He aimed for the front one, and I aimed for the back one, and the back one fell. Jase missed. It was a glorious hunt.
7. Lisa, does the entire Robertson family really vacation together?
Lisa: Well, almost everyone. We call it EBP, "Everybody but Phil." He stays home because he simply does not understand what we love about the beach. The things he doesn't like, we love—such as sand and sun. There are thirty of us, and we try to rent a large house or a duplex when we vacation. We lead very busy lives, and even though we film together, we also love to go and just let our hair down and have fun together. The guys golf almost every day, while the ladies sun and shop.
8. Are the Robertson kids really as respectful and obedient as they seem?
All: Yes. We have raised them to respect and obey their elders. They are all typical kids, but respect is one of the core values of our family and they truly are respectful kids.
9. Korie, what's your favorite episode of Duck Dynasty?
Korie: This is a hard one because there are a lot of episodes I love, but I think my favorite is "Driving Miss Sadie." It's fun to see what happens when I'm not around, like hearing the driving advice Si gave Sadie. Plus, it's like having really great home videos of milestones in our children's lives.
10. Lisa, why do the boys call their parents "Kay" and "Phil" instead of "Mom" and "Dad"?
Lisa: I am not sure anyone remembers exactly how the boys started calling their parents by their first names, but I have a couple of ideas.
For one thing, during the boys' formative years, when Granny and Pa lived next door to Phil and Miss Kay, the entire family worked together, ate together, and enjoyed very close relationships. Phil called Granny "Mom" and Pa "Dad," so maybe having two people called "Mom" and two called "Dad" was too confusing. Granny and Pa called Phil and Kay by their first names, so maybe the boys picked it up from them.
The other reason it may have started is that for a lot of years, Phil and Miss Kay ran their own business. The boys helped them in all kinds of ways. When dealing with customers and vendors, using first names for every employee is more professional than calling the boss "Dad." They would not have said to someone who called, "Yes, you may speak to my mom," but "Yes, you may speak to Kay."
Sometimes people think the boys are being disrespectful by calling Phil and Kay by their first names. The important thing is that Phil and Miss Kay do not think it's disrespectful at all. They know how much the boys respect them, and they do not need to be called by any particular terms to prove it. Miss Kay likes to say, "I'm fine with them calling me 'Kay' as long as they call me!"
11. Jessica, what's it like being married to a serious hunter?
Jessica: It's definitely not like having a husband with a nine-to-five job! A lot of marriages and families have to adjust to whatever a spouse does for a living. People who are married to doctors, soldiers, firefighters, or others all have to find a way to make life work in the midst of unusual schedules. One difference between us and some of our friends is that hunting is part of the job for Robertson men. Sure, they enjoy it, but it's also how they make a living. There were times when duck hunting and the things that go with it were the only way they made a living. So, hunting is not a hobby in our family; it is part of our livelihood.
All of us wives know we will not see as much of our husbands during duck season as we do at other times of the year. For example, during duck season, Jep gets up and leaves the house at about four o'clock in the morning and does not return until about six o'clock at night. When he gets home, he is exhausted. His day may have included not only hunting but also filming, editing, and producing hunting videos, because those are also part of his job responsibilities.
During duck season, I have had to learn to let him rest when he gets home and to find creative things to do with our children while he is away. We all miss him when he's gone so much of the time, so he and I have had to find ways to make the most of the time we do have together during hunting season, which isn't much. Thankfully, my sisters-in-law and Miss Kay understand this well. We're all in the same boat!
12. Missy, did you ever think your life would turn out like this?
Missy: Never in my wildest dreams. I knew I wanted to be used by God in big ways. I always prayed He would trust me enough to use me to make a difference in His Kingdom, but I never dreamed it would be through a cable television show, the number one cable television show in A&E network history, as of this writing! Ephesians 3:20–21 best describes how I feel: "Now to him who is able to do immeasurably more than all we ask or imagine, according to his power that is at work within us, to him be glory in the church and in Christ Jesus throughout all generations, for ever and ever! Amen." It is not because of any power or wisdom we possess that this happened. It is all because of His power, His power working through us. What a dream come true!
13. Miss Kay, may I have your biscuit recipe?
Miss Kay: Sure! Here it is, and you can find more great recipes of mine in my cookbook, Miss Kay's Duck Commander Kitchen.
Homemade Biscuits
2 cups Pioneer Original Biscuit and Baking Mix, plus a little extra
1 cup sour cream
1/2 cup Sprite or 7 Up
1 stick of butter
Mix all ingredients except butter with a pie blender (pastry blender).
Pour a little biscuit mix on wax paper.
Put a cup of dough in the middle of the biscuit mix.
Form a ball and pat it down so you can cut the biscuits with a glass.
Cut just enough to fill a skillet or cake pan with biscuits (I use a skillet).
Melt a stick of butter in a skillet or cake pan.
Roll biscuits in butter and place them in the pan.
Bake at 375 degrees until brown.
14. Lisa, what are some of the things for which you have heard the boys really got in trouble when they were younger?
Lisa: I have always heard the boys got in big trouble for three things: disrespecting their mother, tearing up good equipment (whether that was a fishing pole, a rifle, or something in the house), and coming to blows with each other. They did not get punished for disagreeing with each other, and disagreements were common I'm sure. But if a conflict escalated to the point that fists started swinging, then they got punished. Standard punishment in the Robertson household was three licks.
When people comment on the fact that they got in trouble for disrespecting their mother but not for disrespecting their father, Alan likes to say with a chuckle, "Who would want to disrespect Phil, as good a shot as he is?"
15. What is Christmas like for the Robertson family?
Miss Kay: It's a Cajun seafood feast!
Korie: One of the most fun things about Christmas with the Robertsons is that Miss Kay makes us laugh with her gifts. She loves to buy gag gifts, and she puts cards on them saying these gifts are from the dogs or from someone famous. For example, she would buy a gigantic bra and give it to one of us with a card saying, "Merry Christmas from Dolly Parton." She once had a dog named Doogie Howser, and one year she gave Missy a plastic pile of dog poop from Doogie. The funniest thing about her gag gifts, though, is when people open a present and get a quizzical look on their face because they can't figure out the joke (not all of the gag gifts are as obvious as Dolly Parton's), and Miss Kay bursts out laughing because she can't remember why she thought it was funny. We think that is even funnier than her gifts!
16. Missy, how do you keep your kids grounded?
Missy: Jase and I have what I like to call "Come to Jesus Meetings" whenever we feel the need. Jase is the kind of person who does not like to let things go without being settled. I was raised quite differently. Neither my parents nor I are confrontational people, so we usually just waited for things to get better on their own or work themselves out. Sometimes they did; sometimes they didn't. Not so with a Robertson.
Jase will analyze each behavior, good or bad, until the issue is completely resolved. This is a great quality when it comes to parenting, and even though it was against my nature, I gladly jumped on board, especially when I saw the end results. When we see bad behavior or potential bad behavior with our kids, we sit down and figure out the problem. Many nights have been spent in our living room with Bibles open and notebooks in hand while we hash out what's going on. Bad behavior doesn't just happen on its own. Something unresolved is going on in their lives that fuels that bad behavior.
With our teenage boys, most of the issues have revolved around bad choices in their friendships. Teenage boys love to be admired and given positive attention. Teenage girls have figured this out, godly girls and ungodly girls. It's our job as parents to make sure our boys are associating with people who want what's best for them, not what's best for themselves. And when your kids are stars on a national television show, there is definitely a big difference. I have to be very skeptical about new friendships in order to protect them.
17. Jessica, why weren't you and Jep on the show more during its early seasons?
Jessica: Well, it really wasn't up to us. I appreciate and love all the time we are all together as a family filming Duck Dynasty, and I think everything happens for a reason, in God's perfect timing. As our kids get a little older, juggling filming and the demands of raising four children gets a little easier! We have to keep our priorities straight.
18. Korie, is Sadie really that bad of a driver?
Korie: She went to driver's ed, and now she's actually really good. I think she's better than John Luke!
19. Does everyone in the family really get along as well as you appear to get along on Duck Dynasty?
All: We get so many questions about the way we live our lives. People wonder if we really are nice to each other and if we really can get through a day or a week without someone shouting or erupting in anger. We do not have very sophisticated answers. All we can say is that we do our best to live with integrity, character, and a strong value system—the way God wants us to live. We take our instructions from the Good Book, the Word of God. We do not just read it; we seek to apply its principles to every area of our everyday lives—doing business, raising children, building marriages, dealing with interpersonal relationships, managing our time and money, and more. So far, it's worked pretty well for us!
20. How long will Duck Dynasty last?
All: We don't know, but we are enjoying it while it does, and we hope it will go on for as long as it is part of God's plan for our lives. We love and appreciate all of our fans and hope every episode will bring love, laughter, and a renewed sense of faith and family to everyone who watches.
Write your own story here . . .
Miss Kay
One of my favorite outfits as a child—my jeans, boots, and cowgirl hat.
Me with my older sister, Ann, and our father.
My grandfather with his plow, behind Tony and me.
Nannie's mother, my great-grandmother.
Even as a little girl, I was obviously comfortable and happy at the table!
My papaw Carroway, my dad's father.
When Phil and I met, he was a football player and I was a cheerleader. Here I am in my uniform.
This sign commemorates my daddy's store in Ida, Louisiana.
A picture from my beauty pageant days.
Phil, about the time I met him. Is it any wonder I fell for him?
No matter what I was going through, I always took good care of my boys. Here I am with Alan (right) and Jase (left).
Phil and me with Alan (left) and Jase (right), in our early days.
I love being a grandmother. Here are my two oldest grandchildren, Anna (left) and Alex (right), with me at the beach.
The Duck Commander and me. We've been through a lot!
Korie
Here I am, accessorized with hat and necklace!
Willie and me at our college club banquet. One of the few pics of Willie in a tie.
This was at our engagement party. Willie and I were so excited to be starting our life together.
We were young, but we were happy—and we still are! Willie and me on our wedding day, January 11, 1992.
Willie and me, with the city of Florence, Italy, in the background, during our college trip to Europe. Great memories!
Look at that cutie! Willie and me at the Atlanta Olympics, 1996.
Two people for whom I'm extremely grateful: my parents, Johnny and Chrys Howard. Looking good!
Four generations! Mamaw Shack, Mom, me, and John Luke.
Our first little vacation after Bella was born. We went to Hot Springs, Arkansas. Our hands and hearts were full! From left: Will, John Luke, me, Sadie, Willie, Bella.
Mamaw and Papaw Shack with John Luke and Sadie. Thankful for their love and example.
So thankful for a godly heritage. Sadie with Mamaw and Papaw Howard.
Ski trip with my parents to Steamboat, Colorado, in 2012. Front row: Bella. Middle row: Sadie, John Luke, Will. Back row: Two-Mama (my mom), Two-Papa (my dad), Willie, me.
Love our family! Front row: Bella. Back row: Will, Sadie, Willie, me, Rebecca, John Luke.
Missy
Me as a child. Obviously, it was school picture day!
"Go, Eagles!" Me in my cheerleading uniform during my freshman year at Ouachita Christian School.
Jase and me, before we had children, in our 1992 church pictorial directory. See? I told you he is really handsome underneath that beard!
Beardless Robertson boys with Miss Kay. From left: Willie, Alan, Miss Kay, Jase, Jep.
Mia as a newborn. She has overcome so much!
Mia, at approximately three weeks old, before her first surgery.
A beautiful room for our baby girl. This is the room Miss Kay and others surprised me with before Mia was born.
My family with my parents and Miss Kay. From left: Miss Kay, Jase, Cole, Mia, Reed, me, my mom, and my dad at the Ouachita Christian School Homecoming presentation in 2011.
My parents and me, in front of the Ouachita Christian School flag. They helped found the school with four other couples in 1974.
My aunt, Bonny, who is like my sister, and me at Sadie's Sweet Sixteen birthday party in 2013.
Miss Kay, a great mother-in-law and friend, and me one Mother's Day.
Remember the duck that Jase missed and I shot? Here it is, with Mia and me.
Family vacation. This group goes to the beach together every year. Standing next to Miss Kay is her sister, Ann.
Jessica
My sister, Stacy, and me one Christmas.
My wonderful new in-laws! Miss Kay, Phil, Jep, and me right after our wedding.
Jep with his brothers on our wedding day. From left: Jase, Alan, Jep, Willie.
Granny with Merritt, her namesake. Granny was so proud to have a great-grandchild named after her.
Lily as a little cowgirl.
Miss Kay and me in front of Christmas decorations.
My parents, Terry and Kathy Strickland, with Jep, me, and the kids.
Jep and me with the girls before River was born. Front row from left: Merritt, Priscilla, Lily.
The Robertson women beachside. From left: Lisa, Korie, Miss Kay, me, Missy.
Our girls, in beautiful dresses, with River. Back row: Priscilla. Front row: Lily, River, Merritt.
Jep and me with our children at Miss Kay and Phil's vow renewal in 2013. Back row: Jep, me. Front row: Lily, Merritt, River (making a face!), Priscilla.
The kids share a loving moment. Back row: Lily, Merritt, Priscilla. Front row: River.
Lisa
A school photograph.
Happy couple! Alan and me at our wedding reception.
Anna as a newborn. She was born at twenty-nine weeks.
Two women who taught me so much. Granny and Miss Kay with Anna and Alex.
A powerful picture of God's restoration and healing power. Alan and me with our granddaughters, Carley (left) and Bailey (right).
Miss Kay is both of my girls' best friend. Here are (from left) Alex and Anna with their mamaw Kay.
Our family at Anna's wedding. From left: me, Alan, Anna, Jay, Alex.
Our daughter Anna with some of the special men in her life.
A special moment for Alex. Alan officiated her wedding ceremony, so Papaw Phil walked her down the aisle.
Our daughter Alex and her husband, Vinny, at Alex's wedding, with the Robertson men and Miss Kay. From left: Si, Jase, Alan, Willie, Phil, Miss Kay, Jep, Alex, Vinny.
From left: me, my sister, Barbara, and my mom, Maudie.
They call me Mam. My granddaughters and me at a pumpkin patch.
Alan and me with Phil and Miss Kay. I have such love and respect for my in-laws.
Still going strong. Alan and me at the beach.
Kay Robertson is the revered matriarch of the Robertson family and star of A's Duck Dynasty. She and Phil have been together since she was fifteen and Phil was sixteen, and since then she's been keeping him and her boys from spending too much time in the woods by bringing them back to civilization each night with a home-cooked meal.
Korie Robertson and her husband, Willie, star on A's hit show Duck Dynasty and own and operate Duck Commander, a family-operated business that creates products for duck hunters, and Buck Commander, which creates products for deer hunters. Korie is an art education major and is involved in her local church, teaching Bible classes to children and working in the church's summer camp program. She lives in West Monroe, Louisiana, and has five children.
Missy Robertson is Jase's wife and one of the stars of A's Duck Dynasty. Having grown up in a minister's home, Missy is a talented singer and has participated in the church praise team for the past twenty years; she has also led many children's choruses. Missy is involved in the children's teaching program at her local church and volunteers in the church's summer camp program. Missy is the happy mother of three—Cole, Reed, and Mia.
Jessica Robertson is the wife of Jep, the youngest Robertson brother. Jessica is a busy mom with four children ranging from five to ten years old. Jessica and Jep also star on Duck Dynasty and speak around the country about their faith and the family.
Lisa Robertson is the wife of Alan, the eldest Robertson brother. Lisa worked with Al at the church for several different ministries before joining him at Duck Commander. She is a committed mother and grandmother, and guides her family in the ways of Christ.
MEET THE AUTHORS, WATCH VIDEOS AND MORE AT
SimonandSchuster.com
authors.simonandschuster.com/Kay-Robertson
authors.simonandschuster.com/Korie-Robertson
authors.simonandschuster.com/Missy-Robertson
authors.simonandschuster.com/Jessica-Robertson
authors.simonandschuster.com/Lisa-Robertson
Facebook.com/HowardBooksSimonandSchuster
@Howard_Books
OTHER BOOKS BY THE ROBERTSON FAMILY
The Duck Commander Family
Happy, Happy, Happy
Si-cology 1
The Duck Commander Devotional (also available in pink camo and leather-touch)
Miss Kay's Duck Commander Kitchen
Everything's Better with a Beard
We hope you enjoyed reading this Howard Books eBook.
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Howard Books
A Division of Simon & Schuster, Inc.
1230 Avenue of the Americas
New York, NY 10020
www.SimonandSchuster.com
Copyright © 2014 by Kay Robertson, Korie Robertson, Missy Robertson, Jessica Robertson, and Lisa Robertson
Unless otherwise noted, Scripture quotations in this book are from The Holy Bible, New International Version. © 1973, 1978, 1984, International Bible Society. Used by permission of Zondervan Bible Publishers. All rights reserved.
Scripture quotation marked NIV 2011 is from THE HOLY BIBLE, NEW INTERNATIONAL VERSION®, NIV®. Copyright © 1973, 1978, 1984, 2011 by Biblica, Inc.® Used by permission. All rights reserved worldwide.
Scripture quotations marked NKJV are taken from the New King James Version. © 1982 by Thomas Nelson, Inc. Used by permission. All rights reserved.
Scripture quotations marked NLT are taken from the Holy Bible, New Living Translation, copyright © 1996, 2004. Used by permission of Tyndale House Publishers, Inc., Wheaton, IL 60189. All rights reserved.
All rights reserved, including the right to reproduce this book or portions thereof in any form whatsoever. For information address Howard Books Subsidiary Rights Department, 1230 Avenue of the Americas, New York, NY 10020.
First Howard Books hardcover edition April 2014
HOWARD and colophon are trademarks of Simon & Schuster, Inc.
The Simon & Schuster Speakers Bureau can bring authors to your live event. For more information or to book an event, contact the Simon & Schuster Speakers Bureau at 1-866-248-3049 or visit our website at www.simonspeakers.com.
Interior design by Jaime Putorti
Jacket design by Bruce Gore
Jacket photographs © Steven Palowsky
Library of Congress Cataloging-in-Publication Data
Robertson, Kay.
The women of Duck Commander : surprising insights from the women behind the beards about what makes this family work / Kay Robertson, Korie Robertson, Missy Robertson, Jessica Robertson, and Lisa Robertson ; with Beth Clark. — First Howard Books hardcover edition.
pages cm
1. Robertson, Kay. 2. Robertson, Kay—Family. 3. Robertson, Korie. 4. Robertson, Korie—Family. 5. Robertson, Missy. 6. Robertson, Missy—Family. 7. Robertson, Jessica. 8. Robertson, Jessica—Family. 9. Robertson, Lisa. 10. Robertson, Lisa—Family. 11. Television personalities—United States—Biography. 12. Duck dynasty (Television program) I. Robertson, Korie. II. Robertson, Missy. III. Robertson, Jessica. IV. Robertson, Lisa. V. Clark, Beth. VI. Title.
PN1992.4.R534A3 2014
791.4502'809252—dc23
[B]
2013039886
ISBN 978-1-4767-6330-9
ISBN 978-1-4767-6357-6 (ebook)
| {
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Q: Delete old files in recycle bin with powershell Ok, I have a script I am writing in powershell that will delete old files in the recycle bin. I want it to delete all files from the recycle bin that were deleted more than 2 days ago. I have done lots of research on this and have not found a suitable answer.
This is what I have so far(found the script online, i don't know much powershell):
$Path = 'C' + ':\$Recycle.Bin'
Get-ChildItem $Path -Force -Recurse -ErrorAction SilentlyContinue |
#Where-Object { $_.LastWriteTime -lt (Get-Date).AddDays(-3) } |
Remove-Item -Recurse -exclude *.ini -ErrorAction SilentlyContinue
It is working great with one exception, it checks the file parameter "LastWriteTime". That is awesome if the user deletes the file they same day they modify it. Otherwise it fails.
How can I modify this code so that it will check when the file was deleted, not when it was written.
-On a side note, if I run this script from an administrator account on Microsoft Server 2008 will it work for all users recycle bins or just mine?
Answer:
the code that worked for me is:
$Shell = New-Object -ComObject Shell.Application
$Global:Recycler = $Shell.NameSpace(0xa)
foreach($item in $Recycler.Items())
{
$DeletedDate = $Recycler.GetDetailsOf($item,2) -replace "\u200f|\u200e",""
$dtDeletedDate = get-date $DeletedDate
If($dtDeletedDate -lt (Get-Date).AddDays(-3))
{
Remove-Item -Path $item.Path -Confirm:$false -Force -Recurse
}#EndIF
}#EndForeach item
It works awesome for me, however 2 questions remain...How do I do this with multiple drives? and Will this apply to all users or just me?
A: WMF 5 includes the new "Clear-RecycleBin" cmdlet.
PS > Clear-RecycleBin -DriveLetter C:\
A: These two lines will empty all the files recycle bin:
$Recycler = (New-Object -ComObject Shell.Application).NameSpace(0xa)
$Recycler.items() | foreach { rm $_.path -force -recurse }
A: This article has answers to all your questions
http://baldwin-ps.blogspot.be/2013/07/empty-recycle-bin-with-retention-time.html
Code for posterity:
# -----------------------------------------------------------------------
#
# Author : Baldwin D.
# Description : Empty Recycle Bin with Retention (Logoff Script)
#
# -----------------------------------------------------------------------
$Global:Collection = @()
$Shell = New-Object -ComObject Shell.Application
$Global:Recycler = $Shell.NameSpace(0xa)
$csvfile = "\\YourNetworkShare\RecycleBin.txt"
$LogFailed = "\\YourNetworkShare\RecycleBinFailed.txt"
function Get-recyclebin
{
[CmdletBinding()]
Param
(
$RetentionTime = "7",
[Switch]$DeleteItems
)
$User = $env:USERNAME
$Computer = $env:COMPUTERNAME
$DateRun = Get-Date
foreach($item in $Recycler.Items())
{
$DeletedDate = $Recycler.GetDetailsOf($item,2) -replace "\u200f|\u200e","" #Invisible Unicode Characters
$DeletedDate_datetime = get-date $DeletedDate
[Int]$DeletedDays = (New-TimeSpan -Start $DeletedDate_datetime -End $(Get-Date)).Days
If($DeletedDays -ge $RetentionTime)
{
$Size = $Recycler.GetDetailsOf($item,3)
$SizeArray = $Size -split " "
$Decimal = $SizeArray[0] -replace ",","."
If ($SizeArray[1] -contains "bytes") { $Size = [int]$Decimal /1024 }
If ($SizeArray[1] -contains "KB") { $Size = [int]$Decimal }
If ($SizeArray[1] -contains "MB") { $Size = [int]$Decimal * 1024 }
If ($SizeArray[1] -contains "GB") { $Size = [int]$Decimal *1024 *1024 }
$Object = New-Object Psobject -Property @{
Computer = $computer
User = $User
DateRun = $DateRun
Name = $item.Name
Type = $item.Type
SizeKb = $Size
Path = $item.path
"Deleted Date" = $DeletedDate_datetime
"Deleted Days" = $DeletedDays }
$Object
If ($DeleteItems)
{
Remove-Item -Path $item.Path -Confirm:$false -Force -Recurse
if ($?)
{
$Global:Collection += @($object)
}
else
{
Add-Content -Path $LogFailed -Value $error[0]
}
}#EndIf $DeleteItems
}#EndIf($DeletedDays -ge $RetentionTime)
}#EndForeach item
}#EndFunction
Get-recyclebin -RetentionTime 7 #-DeleteItems #Remove the comment if you wish to actually delete the content
if (@($collection).count -gt "0")
{
$Collection = $Collection | Select-Object "Computer","User","DateRun","Name","Type","Path","SizeKb","Deleted Days","Deleted Date"
$CsvData = $Collection | ConvertTo-Csv -NoTypeInformation
$Null, $Data = $CsvData
Add-Content -Path $csvfile -Value $Data
}
[System.Runtime.Interopservices.Marshal]::ReleaseComObject($shell)
#ScriptEnd
A: Had to do a bit of research on this myself, the recycle bin contains two files for every file deleted on every drive in win 10 (in win 7 files are as is so this script is too much and needs to be cut down, especially for powershell 2.0, win 8 untested), an info file created at time of deletion $I (perfect for ascertaining the date of deletion) and the original file $R, i found the com object method would ignore more files than i liked but on the up side had info i was interested in about the original file deleted, so after a bit of exploring i found a simple get-content of the info files included the original file location, after cleaning it up with a bit of regex and came up with this:
# Refresh Desktop Ability
$definition = @'
[System.Runtime.InteropServices.DllImport("Shell32.dll")]
private static extern int SHChangeNotify(int eventId, int flags, IntPtr item1, IntPtr item2);
public static void Refresh() {
SHChangeNotify(0x8000000, 0x1000, IntPtr.Zero, IntPtr.Zero);
}
'@
Add-Type -MemberDefinition $definition -Namespace WinAPI -Name Explorer
# Set Safe within deleted days and get physical drive letters
$ignoreDeletedWithinDays = 2
$drives = (gwmi -Class Win32_LogicalDisk | ? {$_.drivetype -eq 3}).deviceid
# Process discovered drives
$drives | % {$drive = $_
gci -Path ($drive+'\$Recycle.Bin\*\$I*') -Recurse -Force | ? {($_.LastWriteTime -lt [datetime]::Now.AddDays(-$ignoreDeletedWithinDays)) -and ($_.name -like "`$*.*")} | % {
# Just a few calcs
$infoFile = $_
$originalFile = gi ($drive+"\`$Recycle.Bin\*\`$R$($infoFile.Name.Substring(2))") -Force
$originalLocation = [regex]::match([string](gc $infoFile.FullName -Force -Encoding Unicode),($drive+'[^<>:"/|?*]+\.[\w\-_\+]+')).Value
$deletedDate = $infoFile.LastWriteTime
$sid = $infoFile.FullName.split('\') | ? {$_ -like "S-1-5*"}
$user = try{(gpv "HKLM:\Software\Microsoft\Windows NT\CurrentVersion\ProfileList\$($sid)" -Name ProfileImagePath).replace("$(gpv 'HKLM:\Software\Microsoft\Windows NT\CurrentVersion\ProfileList' -Name ProfilesDirectory)\",'')}catch{$Sid}
#' Various info
$originalLocation
$deletedDate
$user
$sid
$infoFile.Fullname
((gi $infoFile -force).length / 1mb).ToString('0.00MB')
$originalFile.fullname
((gi $originalFile -force).length / 1mb).ToString('0.00MB')
""
# Blow it all Away
#ri $InfoFile -Recurse -Force -Confirm:$false -WhatIf
#ri $OriginalFile -Recurse -Force -Confirm:$false- WhatIf
# remove comment before two lines above and the '-WhatIf' statement to delete files
}
}
# Refresh desktop icons
[WinAPI.Explorer]::Refresh()
A: This works well also as a script with the task scheduler.
Clear-RecycleBin -Force
| {
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} | 673 |
Simmons Hardware Store is a heritage-listed commercial building located at 226 George Street, Windsor, City of Hawkesbury, New South Wales, Australia. It is also known as Peter O'Hara's General Store and Simmons Store. It was added to the New South Wales State Heritage Register on 2 April 1999.
History
226 George Street was built in the mid-nineteenth century. It housed a general store run by Peter O'Hara, a prominent local businessman. It weathered the great fire that ravaged Windsor in December 1874. The fire engulfed the building where the packing straw for O'Hara's goods of tin, earthenware and china was quickly fanned into flames and is believed to have destroyed all but the brick walls. A new store, or rebuilding of the earlier store, was completed shortly after. Although the detail of the building has undergone fabric manipulation and change over the 140 years it remains externally a surviving example of an early Victorian shop in an evolving streetscape.
More recently, a new single-storey storage outbuilding was built in the rear yard and was subject to an archaeological monitoring report by Siobhan Lavelle, c.1999-2000. The front facade was also changed from a suspended flat awning to a concave verandah with timber posts as recommended within the Windsor Streetscape Study 1986, which had been prepared by Noel Bell Ridley Smith and Partners.
Description
226 George Street is an early Victorian two-storey commercial streetscape building constructed of sandstock brick in the mid-19th century. Gable end walls leading to prominent chimneys, symmetrical at the front with a single storey street verandah/awning supported by cantilevered steel pipe brackets off the wall and with timber posts (probably non-load-bearing). Below this verandah the walls have been cement-rendered, with ashlar markings, providing a marked change in appearance, together with two timber-framed shopfront picture windows and central doorway. The remainder of the street facade has three 12-pane windows along its second storey.
To the rear is an attached "L" shaped skillion roof building linked to the ground floor rooms. The ground floor has a side entrance leading to the first floor staircase and office accommodation. The resultant external character of the original sandstock brick building has been changed by it being painted entirely.
There is evidence of painted historic advertising signage is readily evident within the upper south-facing gable wall with the word "Castrol" showing through the painted layers in a curved format.
Although the detail of the building has undergone fabric manipulation and change over the 140 years it remains externally a surviving example of an early Victorian shop in an evolving streetscape.
Heritage listing
Simmons Hardware Store was listed on the New South Wales State Heritage Register on 2 April 1999.
See also
References
Bibliography
Attribution
New South Wales State Heritage Register
Windsor, New South Wales
Commercial buildings in New South Wales
Retail buildings in New South Wales
Articles incorporating text from the New South Wales State Heritage Register | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,011 |
Q. Can you describe your range of emotions right now from winning the game but also losing KD and Kevon once again?
STEVE KERR: I don't think that I can, honestly. I just told the team I didn't know what to say because on the one hand I'm so proud of them, just the amazing heart and grit that they showed, and on the other I'm just devastated for Kevin. So it's a bizarre feeling that we all have right now.
An incredible win and a horrible loss at the same time.
Q. How many times have you seen the Steph and Klay show just take over a game and deliver when it's all on the line?
STEVE KERR: I've been here five years, so I don't know, a hundred-plus games times five years, I've seen it an awful lot. It doesn't happen every night but it seems to happen most nights.
They're amazing. They're amazing competitors, great shooters. Mark Jackson said it years ago, they're the best shooting back court of all time. But maybe what people don't know is how competitive they are, and I thought that showed tonight.
Q. We obviously got some of the information regarding Kevin and Kevon with his injuries but is there any new information to report as well?
STEVE KERR: I believe Bob Myers is going to come in and speak on the health front. I don't really know much. Bob was back there during the game, he'll have more information.
Q. Is there any regrets about bringing Kevin back into this series, now with what happened?
STEVE KERR: Again, I'm going to leave that to Bob.
Q. It looked like DeMarcus was not going to be in the rotation tonight and then Kevin gets hurt. How critical were those four minutes from DeMarcus and kind of getting everybody back into the game after such a shocking thing?
STEVE KERR: Yeah, I thought DeMarcus was fantastic tonight. He stayed ready. He didn't get the first call for that second-quarter run. We went to Bogut and then with the injury we knew we needed his scoring and he stayed ready and played a brilliant game.
So very happy for him and he's been through an awful lot himself over the last year plus with his own injuries. So this was a great night for him individually and very happy for him.
Q. Kawhi Leonard had a huge stretch in that fourth quarter, he took over the game, they regained the lead, the Raptors did, and the crowd was going crazy. You guys could have folded but you guys regained the lead and then pulled it out. How do you think your guys were able to do that and overcome that?
STEVE KERR: I think we went down six, if I'm not mistaken, and maybe it was five, I don't remember, was it five? Six? But Steph and Klay hit back-to-back threes, I believe, we got stops. Our defense was bending down the stretch but we didn't break and the last stop was tremendous. Amazing defense on that last play from all five guys. Draymond's block, he covered so much ground on Kyle's shot from the corner.
So our guys just stayed with it and they stayed poised and just an amazing job finishing the game.
Q. You talked about resiliency and overcoming these injuries, what have you seen from your team throughout this postseason run when it comes to overcoming those types of obstacles?
STEVE KERR: I've seen it over and over again. So it's not really surprising, this is who they are. They have accomplished so much over the years, and that doesn't just happen and it doesn't just happen with talent. There has to be more that goes into it and it's that fight, that competitive desire that I talked about and that ability to stay poised under pressure. It was brilliant to watch.
Q. We talked yesterday about locking in and staying focused on the moment. How unique of a challenge was that after KD left the floor, knowing that you guys had to dig this win out but what obviously KD was probably going through?
STEVE KERR: Yeah, I mean, it had made it difficult, especially with the start that we got off to and Kevin was playing so well, so it was a real shock when he went down. So I give our guys credit, they hung in there and expanded the lead and kept us in a pretty good spot going into halftime.
Q. The whole air went out of the building when Kevin went down. Everyone seemed shook. How do you and the leaders from your team try to go back out there and fight after experiencing something like that what did you say to them, what was that moment like?
STEVE KERR: I didn't say anything. There was nothing to be said. The Raptors players were telling the crowd to be quiet, out of respect, which I appreciated. Some of the fans were cheering when it happened, and I think the Raptors players understood how serious it was and they sort of quieted the crowd. There was just a couple minutes there where it all seemed so eerie and strange, and it took maybe a little bit for both teams to collect themselves. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,607 |
\section{Introduction}
In spite of many existing classes of proposed topological insulators (TIs), the search for new classes with more favorable properties continues. One important factor is that most existing TIs are defective in the interior, thus are not insulating enough in the bulk to allow study and potential application of their surface bands.
One recipe for finding new TIs is to look for small gap insulators that have valence and conduction bands with opposite parity and with small or negative gaps, and a different chemistry that might promote stoichiometry. For very small gap materials, the bands may be inverted by spin orbit-coupling (SOC), leading to insulating gaps that house topological states.\cite{Kane2005} Ideally, the band gap (without SOC) must be small enough for band inversion, while large enough to inhibit bulk conductivity and enable application at room temperature. The other possibility is to have a band overlap semimetal (before SOC), with a gap opened at the Fermi level by SOC. In both cases the strength of SOC governs the magnitude of gap that can be obtained. This realization has focused attention on heavy atoms with large SOC.
Thermoelectric properties, based foremost on a large Seebeck coefficient, also in essence require small band gaps, since one wants a large derivative of the electronic density of states at the Fermi level, $dN(E)/dE|_{E_F}$, with strong particle-hole asymmetry and a low carrier density to minimize electronic thermal conduction. Unlike topological properties which depend sensitively to the momentum $\vec k$ dispersion of bands on either side of the gap, for thermoelectric properties the energy dependence [$N(E)$, and the square velocity $v^2(E)$ of carriers at energy $E$] is the focus. At this level of discussion any connection between topological and thermoelectric properties not evident, aside from the importance of a small or possibly zero gap.
Compounds containing heavy atoms become of special interest. First, if gapped, the gap often is small, favorable for both thermoelectric and topological properties. Of course SOC is large, increasing the likelihood of band inverson and topological character. However, a connection, though somewhat indirect, has been predicted and demonstrated by Singh and collaborators.\cite{singh,singh2} The key variable is the strength of SOC, which is large for heavy atoms, and the observation that in highly itinerant materials this strength can be varied quasi-continuously by isovalent doping. Substituting, say, Te with Se decreases the SOC strength, thereby tuning the band structure near the gap and adjusting both topological and thermoelectric properties. The predicted effect was verified by contrasting Bi$_2$Te$_3$ with isostructural and isovalent Bi$_2$Te$_2$Se.\cite{singh2}
As just pointed out, heavy elements such as bismuth (Bi) and tellurium (Te) are favorable for TIs. A handful of materials, viz. Bi$_2$Se$_3$ family \cite{Zhang2009,Zhang2010,Yazyev2010} and Bi- and Te-based perovskites,\cite{Jin2013,Yang2015} have been discovered to possess non-trivial topological characteristics.
Perovskites, primarily oxide-based ones, are one of the more prevalent structures to be explored and designed, due to their cubic structure and multitude of members.
Their electronic structures often display a narrow gap with band minimum and maximum at high symmetry points, \cite{Sun2010,Jin2012,Hsieh2014} implying that these materials may have potential in thermoelectric applications. \cite{Bilal2015} Doubling the structures will affect the dispersion and have been suggested in certain cases to increase the topological insulator's bulk energy gap.\cite{Pi2017,Lee2017}
In this paper, we study the topological nature and the electronic structure based thermoelectric properties of several pnictide-based double antiperovskites, based on our previous study on single antiperovskite compounds \cite{Goh2018}. Our methods of calculations are described in Sec. II, followed in Sec. III by a description of the structures. In Sec. IV the calculated band gaps and inversion energies are provided for the $X_6AA'B_2$ class of compounds, where $X$ is one of the divalent alkaline earths Ca, Sr, and Ba; $AA'$ are pairs of heavy pnictides SbAs, BiAs,and BiSb; $B$ is one of the lighter pnictides N, P, and As. The electronic structures of two of the compounds are illustrated in Sec. IV. In Sec. V the possibilities for topological properties are outlined, including effects of uniaxial strain. The thermoelectric coefficients of two illustrative compounds are presented in Sec. VI, and Sec. VII provides a brief summary.
\section{Methods}
The electronic structure calculations are done with the full-potential
local orbital (FPLO) code \cite{Koepernik1999},
using the generalized gradient approximation (GGA) exchange-correlation
of Perdew, Burke, and Ernzerhof \cite{Perdew1996}.
A dense $20 \times 20 \times 20$ $k$-mesh was used for self-consistency because
of the delicate band overlap near $\Gamma$.
Spin-orbit coupling (SOC), in fact all relativistic effects, was included precisely by using the
fully relativistic four component Kohn-Sham-Dirac equation
implemented in FPLO, without resorting to the customary intermediate scalar relativistic approximation.
The calculations of thermoelectric properties are done in BoltzTraP \cite{Madsen2006} by solving the Boltzmann equation via band interpolation scheme
based on the band energy obtained from WIEN2k calculation \cite{Blaha2001}.
A dense k-point mesh of 125000 (3107 in IBZ) are used and interpolated onto a mesh of 20 times denser.
Bands within 4 eV around the Fermi level are used in the integration, with a fine energy mesh of 0.5 meV.
\section{Crystal Structure}
\begin{figure}
\centering
\includegraphics[width=0.3\textwidth]{figure1.png}
\caption{Crystal structure of a double antiperovskite with $A$ (purple) $\neq$ $A'$ (blue) and $B = B'$ (small yellow spheres). Cations (green) form octahedra surrounding the $B$ anions.}
\label{crystal-doubleperov}
\end{figure}
The fcc double antiperovskite structure in Fig. \ref{crystal-doubleperov}
contains two alternating unit cells of a single antiperovskite cube,
denoted in generality by $X_6AA'BB'$, where $X$ is an alkaline earth element
and $A,A',B,B'$ are pnictides elements on the A and B sites of the
perovskite structure. Since the $B$ and $B'$ anions are surrounded by
$X_6$ octahedra, if $B\neq B'$ then $X$ is not required by symmetry to lie
midway between them. However, $A\neq A'$ but $B=B'$, $X$ does still reside
on the special site midway between $X$ ions. With distinct $A$-site atoms but
$B=B'$, the spacegroup is $Fm\bar{3}m$ (\#225). Both $A$ and $A'$ have a
site symmetry of $m\bar{3}m$, while $B$ and $X$ have a site symmetry of
$\bar{4}3m$ and $mmm$ respectively. The optimized lattice constants of
the double aPVs are close to twice of that of the single aPVs.
We are interested in $A\neq A'$ and $B=B'$ because the $A$ site pnictide provides the upper valence bands at the gap (or band overlap). Thus modulation on the $A$ site is of particular interest.Based on our previous study on the topological characteristics of antiperovskites, 21 double antiperovskites $X_6AA'B_2$ involving heavy elements, where $X=$Ca, Sr, Ba, $A$ = Sb or Bi, $A'$ = Sb or As, and $B$ = N, P, and As,
were selected for the study of potential topological characteristics.
\section{Electronic Structure}
\begin{figure}
\centering
\begin{subfigure}[b]{4in}
\includegraphics[width=0.95\textwidth]{figure2a.png}
\caption{$\mathrm{Ca_6BiAsN_2}$ vs $\mathrm{Ca_6Sb_2N_2}$.}
\label{n}
\end{subfigure}
\
\begin{subfigure}[b]{4in}
\includegraphics[width=0.95\textwidth]{figure2b.png}
\caption{$\mathrm{Ca_6BiAsP_2}$ vs $\mathrm{Ca_6Sb_2P_2}$.}
\label{p}
\end{subfigure}
\caption{Band structure of double antiperovskites, to compare and contrast $\mathrm{Ca_6BiAsN_2}$ and $\mathrm{Ca_6Sb_2N_2}$ (above), and $\mathrm{Ca_6BiAsP_2}$ and $\mathrm{Ca_6Sb_2P_2}$ (below). Spin-orbit coupling included in all plots.
The fat bands show even parity $s$-character of the heavy A atoms (As, Sb, Bi).}
\label{bandstruct-doubleperov}
\end{figure}
Near the gap (or band overlap) at $\Gamma$, which is the region of interest, these antiperovskites have valence bands contributed from the A site pnictide anion $p$ bands, while the conduction bands derive from the alkaline-earth cation $d$ bands. Doubling to $B~-~B'$ pairs leaves the direct (positive or negative) gap at the $\Gamma$ point, with examples shown in Fig. \ref{bandstruct-doubleperov}. Studying the eigenvalues at the $\Gamma$ point then gives an idea of which candidates are most likely to be topological insulators.
As pointed out previously,\cite{Goh2018} the bands disperse quadratically from $\Gamma$ except for a single high velocity band which can become nearly linear if it approaches a valence band with small or negative gap, a peculiarity of this antiperovskite structure.
From the projected density of states (DOS) this unusual band has even parity $s$-character on both cation and anion sites, so we refer to this band as an extended $s$-like state (ext-$S$).
Since this ext-$S$ character has opposite parity to the occupied bands, inverting it with the odd parity valence band maximum state at the $\Gamma$ point transforms the system into a topological state.
A convenient way to determine if the system is in a topological insulating state is by looking at the band ordering at $\Gamma$, since band orderings at the other time reversal invariant momenta (high symmetry zone edge points) are widely gapped and never change in these systems. It is common, especially with heavy atoms, that SOC closes the band gap and inverts the band ordering. The band structure of $\mathrm{Ca_6BiAsP_2}$ with SOC shown in Fig. \ref{p} depicts a zero gap semiconducting state. At $\Gamma$ the band at the Fermi level has a two-fold degeneracy (excluding spin degeneracy) and pushes the even parity ext-$S$-state into the occupied region. This band inversion leaves a topological zero-gap insulating state, with a $Z_2$ invariant of 1;(000).
It is instructive to compare the electronic structure of double antiperovskites to a reference single antiperovskites in double perovskite supercell structure, especially when the reference has a B atom that is midway between the two in the double aPV.
In Fig. \ref{bandstruct-doubleperov}, where SOC is included, the band gaps of $\mathrm{Ca_6BiAsN_2}$ and $\mathrm{Ca_6Sb_2N_2}$ are equal. Changing the cation to Sr gives a different result: $\mathrm{Sr_6BiAsN_2}$ is a zero gap semiconductor while $\mathrm{Sr_6Sb_2N_2}$ has about 0.5 eV energy gap. The former is a topological zero gap semiconductor, the latter is a trivial semiconductor. The difference is the position of the ext-$S$ state at $\Gamma$.
The difference does however indicate that double aPVs are more likely to undergo band inversion than single aPVs.
\begin{table*}[]
\caption{This table provides the effect of mBJ gap correction to the GGA+SOC band energies, for the inversion energy (see text) and the band gap, of selected double antiperovskites. Results are presented for B site pnictogen smaller than the A, A' atoms, because these are the only energeticaly stable combinations. }
\label{table1-2}
\resizebox{\textwidth}{!}{
\begin{tabular}{|c|l|l|l|l|l|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{$\mathrm{X_6AA'B_2}$} & B & \multicolumn{3}{c|}{N} & \multicolumn{3}{c|}{P} & \multicolumn{3}{c|}{As} \\ \hline
AA' & X & & SOC & SOC+mBJ & mBJ effect & SOC & SOC+mBJ & mBJ effect & SOC & SOC+mBJ & mBJ effect \\ \hline
\multirow{6}{*}{SbAs} & \multirow{2}{*}{Ca} & Inversion Energy & 1.28 & 2.39 & 1.11 & 0.30 & 1.08 & 0.78 & \multicolumn{3}{l|}{\multirow{12}{*}{}} \\ \cline{3-9}
& & Band Gap & 0.51 & 1.15 & 0.64 & 0.30 & 1.08 & 0.78 & \multicolumn{3}{l|}{} \\ \cline{2-9}
& \multirow{2}{*}{Sr} & Inversion Energy & 0.63 & 1.71 & 1.08 & 0.03 & 0.82 & 0.78 & \multicolumn{3}{l|}{} \\ \cline{3-9}
& & Band Gap & 0.38 & 1.03 & 0.66 & 0.03 & 0.82 & 0.78 & \multicolumn{3}{l|}{} \\ \cline{2-9}
& \multirow{2}{*}{Ba} & Inversion Energy & 0.49 & 1.45 & 0.96 & 0.15 & 0.92 & 0.77 & \multicolumn{3}{l|}{} \\ \cline{3-9}
& & Band Gap & 0 & 0.53 & 0.53 & 0.15 & 0.85 & 0.70 & \multicolumn{3}{l|}{} \\ \cline{1-9}
\multirow{6}{*}{BiAs} & \multirow{2}{*}{Ca} & Inversion Energy & 0.82 & 1.92 & 1.10 & -0.07 & 0.69 & 0.76 & \multicolumn{3}{l|}{} \\ \cline{3-9}
& & Band Gap & 0.40 & 1.05 & 0.65 & 0 & 0.69 & 0.69 & \multicolumn{3}{l|}{} \\ \cline{2-9}
& \multirow{2}{*}{Sr} & Inversion Energy & 0.29 & 1.36 & 1.07 & -0.29 & 0.48 & 0.76 & \multicolumn{3}{l|}{} \\ \cline{3-9}
& & Band Gap & 0.29 & 1.00 & 0.70 & 0 & 0.48 & 0.48 & \multicolumn{3}{l|}{} \\ \cline{2-9}
& \multirow{2}{*}{Ba} & Inversion Energy & 0.31 & 1.29 & 0.97 & -0.12 & 0.63 & 0.75 & \multicolumn{3}{l|}{} \\ \cline{3-9}
& & Band Gap & 0 & 0.56 & 0.56 & 0 & 0.63 & 0.63 & \multicolumn{3}{l|}{} \\ \hline
\multirow{6}{*}{BiSb} & \multirow{2}{*}{Ca} & Inversion Energy & 1.07 & 2.09 & 1.02 & 0.10 & 0.84 & 0.74 & -0.17 & 0.56 & 0.73 \\ \cline{3-12}
& & Band Gap & 0.21 & 0.81 & 0.61 & 0.10 & 0.84 & 0.74 & 0 & 0.56 & 0.56 \\ \cline{2-12}
& \multirow{2}{*}{Sr} & Inversion Energy & 0.51 & 1.52 & 1.01 & -0.14 & 0.62 & 0.76 & -0.35 & 0.39 & 0.74 \\ \cline{3-12}
& & Band Gap & 0.18 & 0.82 & 0.64 & 0 & 0.62 & 0.62 & 0 & 0.39 & 0.39 \\ \cline{2-12}
& \multirow{2}{*}{Ba} & Inversion Energy & 0.53 & 1.43 & 0.90 & 0 & 0.75 & 0.74 & -0.18 & 0.55 & 0.73 \\ \cline{3-12}
& & Band Gap & 0 & 0.46 & 0.46 & 0 & 0.72 & 0.72 & 0 & 0.55 & 0.554 \\ \hline
\end{tabular}
}
\end{table*}
We have surveyed the isovalent class of Ae-Pn double aPV compounds.
Changing the $B$ atom to a heavier one, for example, from $\mathrm{Ca_6BiAsN_2}$ to $\mathrm{Ca_6BiAsP_2}$ lowers the position of the ext-$S$ band, giving smaller inversion energy and more likely topological character (see Table \ref{table1-2}). Similar conclusions result when changing the size of $A$ and $A'$ atoms, although sometimes the volume change affects this trend.
In summary, heavy elements favor band inversion but volume effects are relevant.
There are cases for which the band inversion can arise without the help of SOC.
For example, in $\mathrm{Sr_6BiAsP_2}$ where the interplay of element size and volume of the system has reached the ideal point, band inversion occurs without SOC.
\section{Topological States}
The inversion energy is defined as the eigenvalue of the even ext-$S$-state minus that of the odd valence band p-state,
thus negative inversion energies will promote a non-trivial $Z_2$ index. From Table \ref{table1-2}, $\mathrm{\{Ca,Sr,Ba\}_6BiAsP_2}$, $\mathrm{Sr_6BiSbP_2}$ and $\mathrm{\{Ca,Sr,Ba\}_6BiSbAs_2}$ show
negative inversion energy. Since band calculations with GGA exchange-correlation often underestimate the experimental band gap and give spurious inversion results, it is common to include the modified Becke-Johnson (mBJ) exchange-correlation potential \cite{Becke2006,Tran2007,Tran2009}, which provides a self-energy-like correction to the eigenvalues. On average, the GGA inversion energy is in the range of 0.3 eV while mBJ splits the valence and conduction bands by about 0.7 eV. This increase in gap is detrimental in obtaining topological states.
\begin{figure}
\centering
\begin{subfigure}[b]{4in}
\includegraphics[width=\textwidth]{figure3a.png}
\caption{}
\end{subfigure}
\
\begin{subfigure}[b]{4in}
\includegraphics[width=\textwidth]{figure3b.png}
\caption{}
\end{subfigure}
\caption{GGA+mBJ band structure of uniaxial (001) strained $\mathrm{Ca_6BiAsP_2}$, resulting in a topological insulator for 5\% compressive strain (top panel) or Dirac semimetal for 5\% tensile strain (bottom panel).}
\label{ti-dirac-doubleperov}
\end{figure}
However, strain can open up a gap by symmetry-allowed hybridization, while promoting the inverted band ordering with non-trivial topological invariant. Without SOC, for example, (001) strain splits the $p$ triplet at $\Gamma$ into a singlet and a doublet, with one {\it rising} in energy (narrowing the gap or inverting the bands) and the other being lowered, depending on the sign of strain. Similarly, SOC splits the doublet, with the eigenvalue that is displaced upward tending to decrease or invert the gap. For $\mathrm{Ca_6BiAsP_2}$ for example, the maximum gap achieved is 70 meV with compressive strain of $6\%$, but 110 meV in $\mathrm{Sr_6BiAsP_2}$ with compressive strain of $8\%$.
Fig. \ref{ti-dirac-doubleperov} shows the band structures of $5\%$ compressive and tensile strain.
Both signs of strain result in topological states.
Due to the band inversion at the $\Gamma$ point for the compressive case, the $Z_2$ invariant is [1;000], a strong topological insulator state results. The topological surface bands crossing within the bulk band gap will give rise to a Dirac point. Tensile strain, on the other hand, produces a Dirac semimetal, with bands gapped everywhere except near $\Gamma$ along $\Gamma$-$Z$, thus two symmetry related Dirac points. The dispersion around this Dirac point is highly anisotropic: the velocity toward $\Gamma$ is extremely small.
Fig. \ref{ti-dirac-doubleperov}(b) shows the corresponding band structures for these strains. For $5\%$ tensile strained $\mathrm{Ca_6BiAsP_2}$ along $\Gamma$ - Z, the two double spin-degenerate bands with $j_z = \frac{1}{2}$ and $j_z = \frac{3}{2}$ cross, producing a four-fold degenerate Dirac point which, as mentioned, is also orbitally degenerate due to the mirror symmetry with respect to the $x-y$ plane. Strain therefore interpolates between surface and bulk topological behavior in Ca$_6$BiAsP$_2$.
The nearly flat band emerging from this Dirac point toward $\Gamma$ may have implications for the transport properties.
Table \ref{table1-2} shows the band gap and inversion energy, including SOC, of selected double aPVs with and without the mBJ gap correction. Note that the inversion energy is defined as the energy difference between the``ext-S'' and the valence band maximum. Some compounds show negative inversion energy; the band orderings have been inverted by SOC. These inversion energies are at least half the value of that added to the gap by the mBJ shift, in which case the band ordering is no longer inverted. The band gaps and the inversion energies show a range of values; the mBJ shifts are consistent across the $B$ = (N, P, As) column. In general, compounds with nitrogen have a larger mBJ shift than the compounds with P or As.
\section{Thermoelectric Properties}
\subsection{Formalism}
The semiconducting band structures in these compounds suggest possible candidates for thermoelectric applications, with the possibly of learning more about the connection of thermoelectric properties to electronic structure.\cite{singh} Unlike the topological properties just discussed, which depend critically on positions in energy and dispersions in certain regions of the zone, the thermoelectric functions, viz. Seebeck coefficient $S$,
power factor $P$, and figure of merit $ZT$ depend almost entirely on the distribution of available states -- the density of states N(E). As commonly done, we treat the elastic scattering time $\tau$ as energy independent and isotropic, the latter being particularly good in cubic materials such as those we have treated here. For several properties $\tau$ tne cancels out.
The relations necessary to identify the origin of the calculated behavior are the following.
\begin{eqnarray}
S(T)&=& \nu(T)/\sigma(T),\nonumber \\
\frac{\sigma(E)}{\tau}&=&\frac{e^2}{3} N(E)v^2(E) \nonumber \\
\frac{\sigma(T)}{\tau}&=& \int dE \frac{\sigma(E)}{\tau} [-\frac{df(E-\mu;T)}{dE}] \nonumber \\
\frac{\nu(T)}{\tau} &=&\frac{1}{eT} \int dE (E-\mu)\frac{\sigma(E,T)}{\tau} [-\frac{df(E-\mu;T)}{dE}] \nonumber \\
\frac{\kappa_e(T)}{\tau}&=&\frac{1}{e^2T} \int dE (E-\mu)^2\frac{\sigma(E,T)}{\tau} [-\frac{df(E-\mu;T)}{dE}] \nonumber \\
P(T)&=&S(T)\sigma(T)/\tau \nonumber \\
Z(T)T&=&S^2(T)\frac{\sigma(T)/\tau}{\kappa_e/\tau}.
\end{eqnarray}
$\kappa_e$ is the electronic contribution to the thermal conductivity. The actual figure of merit $Z$ includes a lattice contribution in the above equation as well as the electron one $\kappa_e$, so $ZT$ as given above, and displayed below, is an `electronic figure of merit.' $v^2(E)$ is the mean squared velocity averaged over the surfaces of constant energy $E$, and $f(E-\mu;T)$ is the Fermi-Dirac thermal distribution function. The Seebeck coefficient $S$ is the ratio of the thermal and electric conductivity response functions, $\nu$ and $\sigma$ respectively. The chemical potential $\mu(T)$ is determined at each temperature $T$ to conserve particle number: ${\cal N}=\int N(E)f(E-\mu)dE$. Under the stated conditions, the quantities above are independent of $\tau$. Note that the integrands in $\sigma$, $\nu$, and $\kappa_e$ involve increasingly higher powers of the carrier energy $E$ with respect to $\mu$, causing them to reflect more detail of the dispersion around the gap.
We caution that the lattice thermal conductivity is not included in our results. It will lower the value of the total $ZT$, but realistic calculations in line with these electronic contributions are challenging, and require additional algorithms and codes. The options, as outlined by Stackhouse and Stixrude,\cite{Stixrude2010} are: the Green-Kubo thermal Green's function method, which at its most basic level a fully quantum-mechanical approach; non-equilibrium first principles molecular dynamics; transient non-equilibrium molecular dynamics, a version of the method just above requiring additional algorithms. Each of these requires special codes and is far beyond the scope of this paper and of most works evaluating the electronic contribution to ZT. Differences between oxides (where more work has been done) and nitrides will be mentioned in the Summary.
Another point is that the lattice thermal conductivity is dominated, especially at the lower temperature end, by acoustic modes. The compounds we propose have very heavy atoms (Bi or Sb) that will give acoustic modes low group velocities. The velocity enters the thermal conductivity as the square (see Stackhouse and Stixrude\cite{Stixrude2010}), hence as 1/M (mass). In addition, these heavy atoms reside in the large A site of the perovskite cell and may be expected to "rattle" (have large amplitude and be strongly anharmonic), leading to a short relaxation time that will further reduce the lattice thermal conductivity. The correction to our `electronic figure of merit' thus may be expected to be small.
The examples of the electronic thermoelectric behavior with temperature versus chemical potential $\mu$ ({\it i.e.} doping level) are plotted in Fig. \ref{thermo} and will be discussed below. The ``independent variable'' $\mu$ in these plots account for both temperature and coping dependence. T-dependence is determined by the particle conservation mentioned just above. Doping dependence is treated in the rigid band approximation, which is reliable in itinerant systems in the low carrier density regime (where changes in band structure due to carriers is negligible). One then has the commonly quoted result that in the low temperature limit $S(T)\propto dN(E)/dE|_{E=\mu(0)}T$. Thus having $\mu(0)$ near a band edge or near other sharp structure in $N(E)$ leads to large values of $S(T)/T$.
\subsection{Examples}
Our discussion now will focus on two representative cases, the small gap (0.4 eV) semiconducting compound Ca$_6$BiAsN$_2$ (CBAN) and zero gap semiconductor Ca$_6$BiAsP$_2$ (CBAP). For both types of spectra,
$\kappa_e(\mu)$ is not shown, as its behavior is simple: it has a minimum at the intrinsic chemical potential and rises quadratically to rather high doping levels, and the temperature dependence is unremarkable. The commonly studied electronic conductivity $\sigma$ (also not shown) is a thermally broadened version of transport product $N(E)v^2(E)$ which is a much more slowly varying function of $E$ than $N(E)$ and $v^2(E)$ separately.
\begin{figure*}
\centering
\begin{subfigure}[b]{3in}
\includegraphics[width=\textwidth]{figure4a.png}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{3in}
\includegraphics[width=\textwidth]{figure4b.png}
\caption{}
\end{subfigure}
\
\begin{subfigure}[b]{3in}
\includegraphics[width=\textwidth]{figure4c.png}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{3in}
\includegraphics[width=\textwidth]{figure4d.png}
\caption{}
\end{subfigure}
\
\begin{subfigure}[b]{3in}
\includegraphics[width=\textwidth]{figure4e.png}
\caption{}
\end{subfigure}
\begin{subfigure}[b]{3in}
\includegraphics[width=\textwidth]{figure4f.png}
\caption{}
\end{subfigure}
\caption{The Seebeck coefficient (top panels), thermopower (middle panels), and figure of merit $ZT$ (lower panels) of $\mathrm{Ca_6BiAsN_2}$ (left column) with small gap and $\mathrm{Ca_6BiAsP_2}$ (right column) with zero gap, versus chemical potential, in the temperature range $100\mathrm{K}-800\mathrm{K}$.
}
\label{thermo}
\end{figure*}
The Seebeck coefficient shows the usual (for low doping) positive sign for hole doping $\mu<0$ and negative sign for electron doping $\mu>0$. The calculated value reaches nearly 2000 $\mu$V/K at 100 K for CBAN, dropping to around 700 $\mu$V/K at room temperature. In the more relevant coefficients, this high value is tempered by other energy-dependent factors. Unfortunately, the thermopower $P(T)$ does not make good use of the large values of $S(T)$ very near the gap, because the conductivity is low there. The theromopower becomes very low for $\mu$ within the gap, due to the extremely low conductivity there. Unlike the Seebeck coefficient which is nearly symmetric between electrons and holes, the thermopower reveals asymmetry, being nearly 40\% lower for the holes at comparable doping levels.
For zero gapped CBAP, the Seebeck coefficient is an order of magnitude smaller than for CBAN, and the particle hole asymmetry is evident. This asymmetry of the effective masses is visible in Fig.~2. The conduction band minimum of CBAP at the $\Gamma$ point has very small mass followed by higher velocities, due to the steep dip in conduction band at $\Gamma$ also evident in Fig. 2. Since its valence band maximum remains free-electron-like, the Seebeck coefficient is asymmetric and somewhat shifted towards negative chemical potential. The calculated electric conductivity and electronic thermal conductivity over relaxation time (not shown) of CBAN and CBAP are similar; they are further from the chemical potential and larger at higher temperature due to the enhancement of carrier concentration.
The thermopower and figure of merit $ZT$ quantify the potential efficiency in thermoelectric applications. The power factor of CBAN shows two peaks around $\mu=0$, while that of CBAP shows a higher peak at positive chemical potential, indicating that better thermoelectric properties can be realized by electron doping. With rising temperature, the value of the power increases by five times and eight times for CBAN and CBAP respectively, from room temperature to 800 K.
At 100 K, $ZT$ of CBAN is calculated to be unity for low doping levels of either sign. This dimensionless thermoelectic device efficiency decreases slowly with rising temperature, dropping by only 5$\%$ at room temperature. This small decrease indicates that this material has the potential to produce thermoelectic power effectively at low temperature.
For zero gap CBAP, the value is half that of the small gap semiconductor. Nevertheless, these values are among the highest found in the literature,\cite{Li2017,Bhamu2017}
including these promising thermoelectric double perovskites \cite{Aguirre2019,Sahnoun2017,Takahashi2012,Roy2016,Arribi2016}.
Our results indicate that these pnictide-based double antiperovskites have the potential to be competitive thermoelectric materials.
\section{Summary}
The topological characteristics and thermoelectric properties of representative pnictide-based double antiperovskites have been studied.
Doubling the perovskite structure offers a larger bulk insulating gap than single perovskite provides, plus they are more likely to be inverted by SOC.
Based on the GGA exchange-correlation functional,
uniaxial compressive strain opens up an energy gap at the $\Gamma$ point producing a topological insulator with a large bulk band gap.
Tensile strain along $z$ direction, on the other hand, gives a Dirac semimetal with Dirac band crossing along $\Gamma$ - Z.
However, an uncertainty remains whether these topological materials can be realized, as the corrections to the band energy calculation with mBJ potential suggest trivial insulators.
Nonetheless, they are potential candidates for thermoelectric applications due to the high values of Seebeck coefficient, thermoelectric power and figure of merit.
\section{Acknowledgments} W.F.G. was supported by NSF Grant DMR 1534719, while W.E.P. had support from DOE grant DE-FG02-04ER46111.
\vskip 5mm
\section{Bibliography}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,882 |
\section{Introduction}
\label{sec:intro}
Facial animation plays a major role in computer generated imagery because the face is the primary outlet of information. The problem of generating realistic talking heads is multifaceted, requiring high-quality faces, lip movements synchronized with the audio, and plausible facial expressions. This is especially challenging because humans are adept at picking up subtle abnormalities in facial motion and audio-visual synchronization.
Of particular interest is speech-driven facial animation since speech acoustics are highly correlated with facial movements \cite{Yehia1998QuantitativeBehavior}. These systems could simplify the film animation process through automatic generation from the voice acting. They can also be applied in movie dubbing to achieve better lip-syncing results. Moreover, they can be used to generate parts of the face that are occluded or missing in a scene. Finally, this technology can improve band-limited visual telecommunications by either generating the entire visual content based on the audio or filling in dropped frames.
The majority of research in this domain has focused on mapping audio features (e.g. MFCCs) to visual features (e.g. landmarks, visemes) and using computer graphics (CG) methods to generate realistic faces \cite{Karras2017}. Some methods avoid the use of CG by selecting frames from a person-specific database and combining them to form a video \cite{Bregler1997,Suwajanakorn2017}. Regardless of which approach is used these methods are subject dependent and are often associated with a considerable overhead when transferring to new speakers.
Subject independent approaches have been proposed that transform audio features to video frames \cite{Chung2017} but there is still no method to directly transform raw audio to video. Furthermore, many methods restrict the problem to generating only the mouth. Even techniques that generate the entire face are primarily focused on obtaining realistic lip movements, and typically neglect the importance of generating natural facial expressions.
Some methods generate frames based solely on present information \cite{Chung2017}, without taking into account the facial dynamics. This makes generating natural sequences, which are characterized by a seamless transition between frames, challenging. Some video generation methods have dealt with this problem by generating the entire sequence at once \cite{Vondrick2016} or in small batches \cite{Saito2016}. However, this introduces a lag in the generation process, prohibiting their use in real-time applications and requiring fixed length sequences for training.
We propose a temporal generative adversarial network (GAN), capable of generating a video of a talking head from an audio signal and a single still image (see \figref{fig:summary} ). First, our model captures the dynamics of the entire face producing not only synchronized mouth movements but also natural facial expressions, such as eyebrow raises, frowns and blinks. This is due to the use of an RNN-based generator and sequence discriminator, which also gives us the added advantage of handling variable length sequences. Natural facial expressions play a crucial role in achieving truly realistic results and their absence is often a clear tell-tale sign of generated videos. This is exploited by methods such as the one proposed in \cite{LiInBlinking}, which detects synthesized videos based on the existence of blinks.
Secondly, our method is subject independent, does not rely on handcrafted audio or visual features, and requires no post-processing. To the best of our knowledge, this is the first end-to-end technique that generates talking faces directly from the raw audio waveform.
The third contribution of this paper is a comprehensive assessment of the performance of the proposed method. An ablation study is performed on the model in order to quantify the effect of each component in the system. We measure the image quality using popular reconstruction and sharpness metrics, and compare it to a non-temporal approach. Additionally, we propose using lip reading techniques to verify the accuracy of the spoken words and face verification to ensure that the identity of the speaker is maintained throughout the sequence. Evaluation is performed in a subject independent way on the GRID \cite{Cooke2006} and TCD TIMIT \cite{Harte2015} datasets, where our model achieves truly natural results. Finally, the realism of the videos is assessed through an online Turing test, where users are shown videos and asked to identify them as either real or generated.
\section{Related Work}
\label{sec:related}
\subsection{Speech-Driven Facial Animation}
\label{sec:speech_driven_animation}
The problem of speech-driven video synthesis is not new in computer vision and has been the subject of interest for decades. Yehia {\em et al.} \cite{Yehia1998QuantitativeBehavior} first examined the relationship between acoustics, vocal-tract and facial motion, showing a strong correlation between visual and audio features and a weak coupling between head motion and the fundamental frequency of the speech signal \cite{Yehia2002LinkingAcoustics}.
Some of the earliest methods for facial animation relied on hidden Markov models (HMMs) to capture the dynamics of the video and speech sequences. Simons and Cox \cite{Simons1990} used vector quantization to achieve a compact representation of video and audio features, which were used as the states for their HMM. The HMM was used to recover the most likely mouth shapes for a speech signal. A similar approach is used in \cite{Yamamoto1998} to estimate the sequence of lip parameters. Finally, the {\em Video Rewrite} \cite{Bregler1997} method relies on the same principals to obtain a sequence of triphones that are used to look up mouth images from a database.
Although HMMs were initially preferred to neural networks due to their explicit breakdown of speech into intuitive states, recent advances in deep learning have resulted in neural networks being used in most modern approaches. Like past attempts, most of these methods aim at performing a feature-to-feature translation. A typical example of this is \cite{Taylor2017}, which uses a deep neural network (DNN) to transform a phoneme sequence into a sequence of shapes for the lower half of the face. Using phonemes instead of raw audio ensures that the method is subject independent. Similar deep architectures based on recurrent neural networks (RNNs) have been proposed in \cite{Fan2015,Suwajanakorn2017}, producing realistic results but are subject dependent and require retraining or re-targeting steps to adapt to new faces.
Convolutional neural networks (CNN) are used in \cite{Karras2017} to transform audio features to 3D meshes of a specific person. This system is conceptually broken into sub-networks responsible for capturing articulation dynamics and estimating the 3D points of the mesh. Finally, Chung {\em et al.} \cite{Chung2017} proposed a CNN applied on Mel-frequency cepstral coefficients (MFCCs) that generates subject independent videos from an audio clip and a still frame. The method uses an $L_1$ loss at the pixel level resulting in blurry frames, which is why a deblurring step is also required. Secondly, this loss at the pixel level penalizes any deviation from the target video during training, providing no incentive for the model to produce spontaneous expressions and resulting in faces that are mostly static except for the mouth.
\subsection{GAN-Based Video Synthesis}
\label{sec:gans}
The recent introduction of GANs in \cite{Goodfellow2014} has shifted the focus of the machine learning community to generative modelling. GANs consist of two competing networks: a generative network and a discriminative network. The generator's goal is to produce realistic samples and the discriminator's goal is to distinguish between the real and generated samples. This competition eventually drives the generator to produce highly realistic samples. GANs are typically associated with image generation since the adversarial loss produces sharper, more detailed images compared to $L_1$ and $L_2$ losses. However, GANs are not limited to these applications and can be extended to handle videos \cite{Mathieu2015,Li2017,Vondrick2016,Tulyakov2017}.
Straight-forward adaptations of GANs for videos are proposed in \cite{Vondrick2016, Saito2016}, replacing the 2D convolutional layers with 3D convolutional layers. Using 3D convolutions in the generator and discriminator networks is able to capture temporal dependencies but requires fixed length videos. This limitation was overcome in \cite{Saito2016} but constraints need to be imposed in the latent space to generate consistent videos.
The {\em MoCoGAN} system proposed in \cite{Tulyakov2017} uses an RNN-based generator, with separate latent spaces for motion and content. This relies on the empirical evidence shown in \cite{Radford2015} that GANs perform better when the latent space is disentangled. {\em MoCoGAN} uses a 2D and 3D CNN discriminator to judge frames and sequences respectively. A sliding window approach is used so that the 3D CNN discriminator can handle variable length sequences.
GANs have also been used in a variety of cross-modal applications, including text-to-video and audio-to-video. The text-to-video model proposed in \cite{Li2017} uses a combination of variational auto encoders (VAE) and GANs in its generating network and a 3D CNN as a sequence discriminator. Finally, Chen {\em et al.} \cite{Chen1998} propose a GAN-based encoder-decoder architecture that uses CNNs in order to convert audio spectrograms to frames and vice versa.
\section{End-to-End Speech-Driven Facial Synthesis}
\label{sec:model}
The proposed architecture for speech-driven facial synthesis is shown in \figref{fig:model}. The system is made up of a generator and 2 discriminators, each of which evaluates the generated sequence from a different perspective. The capability of the generator to capture various aspects of natural sequences is directly proportional to the ability of each discriminator to discern videos based on them.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.99\textwidth]{images/model_block.pdf}
\end{center}
\caption{The deep model for speech-driven facial synthesis. This uses 2 discriminators to incorporate the different aspects of a realistic video. Details about the architecture are presented in the supplementary material.}
\label{fig:model}
\end{figure}
\subsection{Generator}
\label{sec:generator}
The inputs to the generator networks consist of a single image and an audio signal, which is divided into overlapping frames each corresponding to $0.16$ seconds. The generator network in this architecture can be conceptually divided into subnetworks as shown in \figref{fig:gendisc}. Using an RNN-based generator allows us to synthesize videos frame-by-frame, which is necessary for real-time applications.
\subsubsection{Identity Encoder}
\label{sec:id_encoder}
The speaker's identity is encoded using a 6 layer CNN. Each layer uses strided 2D convolutions, followed by batch normalization and ReLU activation functions. The {\em Identity Encoder} network reduces the input image to a $50$ dimensional encoding $z_{id}$.
\subsubsection{Context Encoder}
\label{sec:context_encoder}
Audio frames are encoded using a network comprising of 1D convolutions followed by batch normalization and ReLU activations. The initial convolutional layer starts with a large kernel, as recommended in \cite{Dai2016}, which helps limit the depth of the network while ensuring that the low-level features are meaningful. Subsequent layers use smaller kernels until an embedding of the desired size is achieved. The audio frame encodings are input into a 2 layer GRU, which produces a context encoding $z_c$ with $256$ elements.
\subsubsection{Frame Decoder}
\label{sec:frame_decoder}
The identity encoding $z_{id}$ is concatenated to the context encoding $z_c$ and a noise component $z_n$ to form the latent representation. The 10-dimensional $z_n$ vector is obtained from a {\em Noise Generator}, which is a 1-layer GRU that takes Gaussian noise as input. The {\em Frame Decoder} is a CNN that uses strided transposed convolutions to produce the video frames from the latent representation. A U-Net \cite{Ronneberger2015} architecture is used with skip connections between the {\em Identity Encoder} and the {\em Frame Decoder} to help preserve the identity of the subject.
\begin{figure}[ht]
\centering
\begin{subfigure}[b]{0.5\linewidth}
\centering\includegraphics[width=0.97\textwidth]{images/generator.pdf}
\caption{\label{fig:gendisc_gen} Generator}
\end{subfigure}%
\begin{subfigure}[b]{0.5\linewidth}
\centering\includegraphics[width=0.97\textwidth]{images/conditional_seq_disc2.pdf}
\caption{\label{fig:gendisc_disc} Sequence Discriminator}
\end{subfigure}
\caption{The architecture of the (a) Generator which consists of a {\em Context Encoder} (audio encoder and RNN), an {\em Identity Encoder}, a {\em Frame Decoder} and {\em Noise Generator} (b) Sequence Discriminator, consisting of an audio encoder, an image encoder, GRUs and a small classifier.}
\label{fig:gendisc}
\end{figure}
\subsection{Discriminators}
\label{sec:discriminators}
Our system has two different types of discriminator. The {\em Frame Discriminator} helps achieve a high-quality reconstruction of the speakers' face throughout the video. The {\em Sequence Discriminator} ensures that the frames form a cohesive video which exhibits natural movements and is synchronized with the audio.
\subsubsection{Frame Discriminator}
\label{sec:frame_disc}
The {\em Frame Discriminator} is a 6-layer CNN that determines whether a frame is real or not. Adversarial training with this discriminator ensures that the generated frames are realistic. The original still frame is used as a condition in this network, concatenated channel-wise to the target frame to form the input as shown in \figref{fig:gendisc}. This enforces the person's identity on the frame.
\subsubsection{Sequence Discriminator}
\label{sec:seq_disc}
The {\em Sequence Discriminator} presented in \figref{fig:gendisc} distinguishes between real and synthetic videos. The discriminator receives a frame at every time step, which is encoded using a CNN and then fed into a 2-layer GRU. A small (2-layer) classifier is used at the end of the sequence to determine if the sequence is real or not. The audio is added as a conditional input to the network, allowing this discriminator to classify speech-video pairs.
\subsection{Training}
\label{sec:training}
The {\em Frame discriminator} ($D_{img}$) is trained on frames that are sampled uniformly from a video $x$ using a sampling function $S(x)$. The {\em Sequence discriminator} ($D_{seq}$), classifies based on the entire sequence $x$ and audio $a$. The loss of our GAN is an aggregate of the losses associated with each discriminator as shown in \eref{eq:adv_loss}.
\begin{equation}
\begin{split}
\pazocal{L}_{adv}(D_{img}, D_{Seq}, G) = & \EX_{x \sim P_d}[\log D_{img}(S(x), x_1)] + \EX_{z \sim P_z}[\log (1- D_{img}(S(G(z)), x_1))] + \\
& \EX_{x \sim P_d}[\log D_{seq}(x, a)] + \EX_{z \sim P_z}[\log (1- D_{seq}(G(z), a))]
\end{split}
\label{eq:adv_loss}
\end{equation}
An $L_1$ reconstruction loss is also used to improve the synchronization of the mouth movements. However we only apply the reconstruction loss to the lower half of the image since it discourages the generation of facial expressions. For a ground truth frame $F$ and a generated frame $G$ with dimensions $W \times H$ the reconstruction loss at the pixel level is:
\begin{equation}
\pazocal{L}_{L_1} = \sum_{p \in [ 0, W] \times [ \frac{H}{2}, H ] }|F_{p} - G_{p}|
\label{eq:reconstruction_loss}
\end{equation}
The final objective is to obtain the optimal generator $G^*$, which satisfies \eref{eq:loss}. The model is trained until no improvement is observed on the reconstruction metrics on the validation set for 10 epochs. The $\lambda$ hyperparameter controls the contribution of each loss factor and was set to $400$ following a tuning procedure on the validation set.
\begin{equation}
\begin{split}
\arg \min_{G} \max_{D}\pazocal{L}_{adv} + \lambda \pazocal{L}_{L_1}
\end{split}
\label{eq:loss}
\end{equation}
We used Adam \cite{Kingma2014} for all the networks with a learning rate of $0.0002$ for the {\em Generator} and $0.001$ {\em Frame Discriminator} which decay after epoch 20 with a rate of $10\%$. The {\em Sequence Discriminator} uses a smaller fixed learning rate of $5\cdot 10^{-5}$.
\section{Experiments}
\label{sec:experiments}
Our model is implemented in PyTorch and takes approximately a week to train using an Nvidia GeForce GTX 1080 Ti GPU. During inference the average generation time per frame is 7ms on the GPU, permitting the use of our method use in real time applications. A sequence of 75 frames can be synthesized in 0.5s. The frame and sequence generation times increase to 1s and 15s respectively when processing is done on the CPU.
\subsection{Datasets}
\label{sec:datasets}
The GRID dataset has 33 speakers each uttering 1000 short phrases, containing 6 words taken from a limited dictionary. The TCD TIMIT dataset has 59 speakers uttering approximately 100 phonetically rich sentences each. We use the recommended data split for the TCD TIMIT dataset but exclude some of the test speakers and use them as a validation set. For the GRID dataset speakers are divided into training, validation and test sets with a $50\% - 20\% - 30\%$ split respectively. As part of our preprocessing all faces are aligned to the canonical face and images are normalized. We increase the size of the training set by mirroring the training videos. The amount of data used for training and testing is presented in \tabref{tab:Subjects}.
\begin{table}[h!]
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
Dataset & Samples (Tr) & Hours (Tr) & Samples (V)& Hours (V)& Samples (T)& Hours (T) \\
\hline\hline
GRID & 31639 & 26.36 & 6999 & 5.83 & 9976 & 8.31 \\
TCD & 8218 & 9.13 & 686 & 0.76 &977 & 1.09 \\
\hline
\end{tabular}
\end{center}
\caption{The samples and hours of video in the training (Tr), validation (V) and test (T) sets.}
\label{tab:Subjects}
\end{table}
\subsection{Metrics}
\label{sec:metrics}
We use common reconstruction metrics such as the peak signal-to-noise ratio (PSNR) and the structural similarity (SSIM) index to evaluate the generated videos. During the evaluation it is important to take into account the fact that reconstruction metrics penalize videos for any spontaneous expression. The frame sharpness is evaluated using the cumulative probability blur detection (CPBD) measure \cite{Narvekar2009}, which determines blur based on the presence of edges in the image and the frequency domain blurriness measure (FDBM) proposed in \cite{De2013}, which is based on the spectrum of the image. For these metrics larger values imply better quality.
The content of the videos is evaluated based on how well the video captures identity of the target and on the accuracy of the spoken words. We verify the identity of the speaker using the average content distance (ACD) \cite{Tulyakov2017}, which measures the average Euclidean distance of the still image representation, obtained using OpenFace \cite{Amos2016}, from the representation of the generated frames. The accuracy of the spoken message is measured using the word error rate (WER) achieved by a pre-trained lip-reading model. We use the LipNet model \cite{Assael2016}, which surpasses the performance of human lipreaders on the GRID dataset. For both content metrics lower values indicate better accuracy.
\subsection{Ablation Study}
\label{sec:ablation}
In order to quantify the effect of each component of our system we perform an ablation study on the GRID dataset (see \tabref{tab:ablation}). We use the metrics from section \ref{sec:metrics} and a pre-trained LipNet model which achieves a WER of $21.4 \%$ on the ground truth videos. The average value of the ACD for ground truth videos of the same person is $0.74 \cdot 10^{-4}$ whereas for different speakers it is $1.4 \cdot 10^{-3}$. The $L_1$ loss achieves slightly better PSNR and SSIM results, which is expected as it does not generate spontaneous expressions, which are penalized by these metrics unless they happen to coincide with those in ground truth videos. This variation introduced when generating expressions is likely the reason for the small increase in ACD. The blurriness is minimized when using the adversarial loss as indicated by the higher FDBM and CPBD scores and \figref{fig:bluriness}. Finally, the effect of the sequence discriminator is shown in the lip-reading result achieving a low WER.
\begin{table}[h!]
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
\multicolumn{1}{|l|}{Method} & \multicolumn{1}{c|}{PSNR} & \multicolumn{1}{c|}{SSIM} & \multicolumn{1}{c|}{FDBM} & \multicolumn{1}{c|}{CPBD} & \multicolumn{1}{c|}{ACD} & \multicolumn{1}{c|}{WER}\\
\hline\hline
Ground Truth Videos & N/A & N/A & 0.121 & 0.281 & 0.74 $\cdot 10^{-4}$ & 21.40\% \\
$L_1$ loss & \textbf{28.47} & \textbf{0.859} & 0.101 & 0.210 & \textbf{ 0.90 $\cdot 10^{-4}$} & 27.90\% \\
$L_1 + Adv_{img}$ & 27.71 & 0.840 & 0.114 & 0.274 & 1.04 $\cdot 10^{-4}$& 27.94\% \\
$L_1 + Adv_{img} + Adv_{seq}$ & 27.98 & 0.844 & \textbf{0.114} & \textbf{0.277} & 1.02 $\cdot 10^{-4}$ & \textbf{25.45}\% \\
\hline
\end{tabular}
\end{center}
\caption{Assessment of each model in the ablation study performed on the GRID dataset}
\label{tab:ablation}
\end{table}
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.5\linewidth}
\centering\includegraphics[width=0.99\textwidth]{images/blur_exampleb.pdf}
\caption{\label{fig:bluriness_l1} $L_1$ loss on entire frame}
\end{subfigure}%
\begin{subfigure}[b]{0.5\linewidth}
\centering\includegraphics[width=0.99\textwidth]{images/blur_examplea.pdf}
\caption{\label{fig:bluriness_model} Proposed loss on frames}
\end{subfigure}
\caption{Frames using (a) only an $L_1$ loss on the entire face compared to (b) frames produced using the proposed method. Frames generated using an $L_1$ loss on the entire face (a) are blurrier than those produced from the proposed method (b).}
\label{fig:bluriness}
\end{figure}
\subsection{Qualitative Results}
\label{sec:qualitative}
Our method is capable of producing realistic videos of previously unseen faces and audio clips taken from the test set. The examples in \figref{fig:diff_audio} show the same face animated using sentences from different subjects (male and female). The same audio used on different identities is shown in \figref{fig:diffface}. From visual inspection it is evident that the lips are consistently moving similarly to the ground truth video. Our method not only produces accurate lip movements but also natural videos that display characteristic human expressions such as frowns and blinks, examples of which are shown in \figref{fig:expressions}.
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.49\linewidth}
\centering\includegraphics[width=0.99\textwidth]{images/diff_audio.pdf}
\caption{\label{fig:diff_audioa} Female voice uttering the word ``bin''}
\end{subfigure}
\begin{subfigure}[b]{0.49\linewidth}
\centering\includegraphics[width=0.99\textwidth]{images/diff_audiob.pdf}
\caption{\label{fig:diff_audiob} Male voice uttering the word ``white''}
\end{subfigure}
\caption{Generated sequences for (a) the word ``bin'' (b) the word ``white'' from the GRID test set. Coarticulation is evident in (a) where ``bin'' is followed by the word ``blue''.}
\label{fig:diff_audio}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.99\textwidth]{images/diffface.pdf}\\
\end{center}
\caption{Animation of different faces using the same audio. The movement of the mouth is similar for both faces as well as for the ground truth sequence. Both audio and still image are unseen during training.}
\label{fig:diffface}
\end{figure}
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.49\linewidth}
\centering\includegraphics[width=0.99\textwidth]{images/frown.pdf}
\caption{\label{fig:frown} Example of generated frown}
\end{subfigure}
\begin{subfigure}[b]{0.49\linewidth}
\centering\includegraphics[width=0.99\textwidth]{images/blink.pdf}
\caption{\label{fig:blink} Example of generated blink}
\end{subfigure}
\caption{Facial expressions generated using our framework include (a) frowns and (b) blinks.}
\label{fig:expressions}
\end{figure}
The works that are closest to ours are those proposed in \cite{Suwajanakorn2017} and \cite{Chung2017}. The former method is subject dependent and requires a large amount of data for a specific person to generate videos. For the latter method there is no publicly available implementation so we compare our model to a static method that produces video frames using a sliding window of audio samples like that used in \cite{Chung2017}. This is a GAN-based method that uses a combination of an $L_1$ loss and an adversarial loss on individual frames. We will also use this method as the baseline for our quantitative assessment in the following section. This baseline produces less coherent sequences, characterized by jitter, which becomes worse in cases where the audio is silent (e.g. pauses between words). This is likely due to the fact that there are multiple mouth shapes that correspond to silence and since the model has no knowledge of its past state generates them at random. \figref{fig:jitterface} shows a comparison between our approach and the baseline in such cases.
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.49\linewidth}
\centering\includegraphics[width=0.99\textwidth]{images/jittermouth.pdf}
\caption{\label{fig:jitterface_silence} Audio-visual inconsistency during silence}
\end{subfigure}
\begin{subfigure}[b]{0.49\linewidth}
\centering\includegraphics[width=0.99\textwidth]{images/jitterface.pdf}
\caption{\label{fig:jitterface_cont_break} Extraneous frames that break continuity}
\end{subfigure}
\caption{Examples of consecutive frames showcasing the failures of the static baseline including (a) opening the mouth when words are not spoken (b) producing irrelevant frames that do not take into account the previous face state, thus breaking the sequence continuity.}
\label{fig:jitterface}
\end{figure}
\vspace{-5pt}
\subsection{Quantitative Results}
\label{sec:quantitative}
We measure the performance of our model on the GRID and TCD TIMIT datasets using the metrics proposed in section \ref{sec:metrics} and compare it to the static baseline. Additionally, we present the results of a 30-person survey, where users were shown 30 videos from each method and were asked to pick the more natural ones. The results in \tabref{tab:quantitative} show that our method outperforms the static baseline in both frame quality and content accuracy. Although the difference in performance is slight for frame-based measures (e.g. PSNR, ACD) it is substantial in terms of user preference and lipreading WER, where temporal smoothness of the video and natural expressions play a significant role.
\begin{table}[h!]
\centering
\begin{tabular}{|c|l|r|r|r|r|r|r|r|}
\hline
& \multicolumn{1}{c|}{Method}& \multicolumn{1}{c|}{PSNR} & \multicolumn{1}{c|}{SSIM} & \multicolumn{1}{c|}{FDBM} & \multicolumn{1}{c|}{CPBD} & \multicolumn{1}{c|}{ACD}& \multicolumn{1}{c|}{User} & \multicolumn{1}{c|}{WER}\\
\hline\hline
\small \parbox[t]{2mm}{\multirow{2}{*}{\rotatebox[origin=c]{90}{GRID}}}
& Proposed Model & \textbf{27.98} & \textbf{0.844} & \textbf{0.114} & 0.277 & \textbf{1.02} $\cdot 10^{-4}$ & \textbf{79.77}\% & \textbf{25.4}\% \\
& Baseline & 27.39 & 0.831 & 0.113 & \textbf{0.280} & 1.07 $\cdot 10^{-4}$ & 20.22\% & 37.2\% \\
\hline
\small \parbox[t]{2mm}{\multirow{2}{*}{\rotatebox[origin=c]{90}{TCD }}}
& Proposed Model & \textbf{23.54} & \textbf{0.697} & \textbf{0.102} & \textbf{0.253} & \textbf{2.06} $\cdot 10^{-4}$ &\textbf{77.03\%}&N/A\\
& Baseline & 23.01 & 0.654 & 0.097 & 0.252 & 2.29 $\cdot 10^{-4}$ & 22.97\%&N/A\\
\hline
\end{tabular}
\caption{Performance comparison of the proposed method against the static baseline. The pretrained LipNet model is not available for the TCD TIMIT so the WER metric is omitted.}
\label{tab:quantitative}
\end{table}
We further evaluate the realism of the generated videos through an online Turing test. In this test users are shown 10 videos, which were chosen at random from GRID and TIMIT consisting of 6 fake videos and 4 real ones. Users are shown the videos in sequence and are asked to label them as real or fake. Responses from 153 users were collected with the average user labeling correctly 63\% of the videos. The distribution of user scores is shown in \figref{fig:response_chart}.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.99\textwidth]{images/chart.pdf}\\
\end{center}
\vspace{-20pt}
\caption{Distribution of user scores for the Turing test.}
\label{fig:response_chart}
\end{figure}
\section{Conclusion and Future Work}
\label{sec:conclusion}
In this work we have presented an end-to-end model using temporal GANs for speech-driven facial animation. Our model is capable of producing highly detailed frames scoring high in terms of PSNR, SSIM and in terms of the sharpness measures on both datasets. According to our ablation study this can be mainly attributed to the use of a {\em Frame Discriminator}.
Furthermore, our method produces more coherent sequences and more accurate mouth movements compared to the static approach, as demonstrated by a resounding user preference and the difference in the WER. We believe that these improvements are not only a result of using a temporal generator but also due to the use of the conditional {\em Sequence Discriminator}. Unlike previous approaches \cite{Chung2017} that prohibit the generation of facial expressions, the adversarial loss on the entire sequence encourages spontaneous facial gestures. This has been demonstrated with examples of blinks and frowns. All of the above factors make the videos generated using our approach difficult to separate from real videos as revealed from the Turing test results, with the average user scoring only slightly better than chance. It is also noteworthy that no user was able to perfectly classify the videos.
This model has shown promising results in generating lifelike videos. Moving forward, we believe that different architectures for the sequence discriminator could help produce more natural sequences. Finally, at the moment expressions are generated randomly by the model so a natural extension of this method would attempt to also capture the mood of the speaker from his voice and reflect it in the facial expressions.
\section{Acknowledgements}
This work has been funded by the European Community Horizon 2020 under grant agreement
no. 645094 (SEWA).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,537 |
<?php
require(CRYSTAL_ROOT_DIR . 'Validation' . DIRECTORY_SEPARATOR . 'MinLength.php');
class TestOfMinLengthValidation extends UnitTestCase
{
function TestStringParam()
{
$alpha = new Crystal_Validation_MinLength('test');
$this->assertFalse($alpha->result, 'MinLength validation with string should be false');
}
function TestArrayParam()
{
$alpha = new Crystal_Validation_MinLength(array('test'));
$this->assertFalse($alpha->result, 'MinLength validation with array should be false');
}
function TestEmptyParam()
{
$alpha = new Crystal_Validation_MinLength();
$this->assertFalse($alpha->result, 'MinLength validation with no params should be false');
}
function TestValidParam()
{
$alpha = new Crystal_Validation_MinLength('category',5);
$this->assertTrue($alpha->result, 'MinLength validation with MinLength should be true');
}
function TestInValidParam()
{
$alpha = new Crystal_Validation_MinLength('cat',5);
$this->assertFalse($alpha->result, 'MinLength validation with MinLength should be false');
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,336 |
Q: Question over Nodes of Runge-Kutta methods I have (hopefully) an easy question.
Suppose I have the following system (derived from an integration)
$I'(t)= f(t), \, t \in [0,\pi]\\
I(0)=0$
My goal is to find $I(\pi)= \int_{0}^{\pi}f(t) \text{d}t$ using a RK method on the differential setting.
Since $f$ does not depend on $I$, a RK method of size $m \geq1$ can be described only through the vectors $b,c \in \mathbb{R}^{m}$.
Suppose we divide $[0,\pi]$ into a collection of equidistant $N_{t}>1$ points. We then have $\Delta t = \frac{\pi}{N_{t}-1}$ This means $t_{k}=k \Delta t $ for $k \in \lbrace 0, \ldots, N_{t}-1\rbrace$.
A single step of Runge Kutta reads:
\begin{align}
I^{k+1}-I^{k}= \Delta t \sum_{i=1}^{m} b_{i} f(t_{k}+c_{i} \Delta t)
\end{align}
summing over all $N_{t}$ points we get
\begin{align}
I^{N_{t}}-I^{0} = \sum_{k=0}^{N_{t}-1}(I^{k+1}-I^{k}) =\sum_{k=0}^{N_{t}-1} \Delta t \sum_{i=1}^{m} b_{i} f(t_{k}+c_{i} \Delta t)
\end{align}
By assumption $I(0)=0$ so that we obtain \begin{align}
I^{N_{t}} = \Delta t \sum_{k=0}^{N_{t}-1} \sum_{i=1}^{m} b_{i} f(t_{k}+c_{i} \Delta t)
\end{align}
Up to here everything is reasonable in my mind. But then I encounter problems when a $c$ contains an entry bigger or equal than one.
For example here's Heun's method $c =(0, \, 1); b = (\frac{1}{2}, \, \frac{1}{2})$.
We then have for calculating $I^{N_{t}}$
\begin{align}
I^{N_{t}} = \frac{\Delta t}{2} \sum_{k=0}^{N_{t}-1} [ f(t_{k})+f(t_{k+1})]
\end{align}
consider now only the last summand that is $$f(t_{N_{t}-1})+f(t_{N_{t}}).$$
Here's my problem: While $t_{N_{t}-1}$ is a well defined point in our $t-$grid, $t_{N_{t}}$ is not.
Now since $f$ is given and can be computed even on nodes outside of those considered, this should not arise any questions. Anyway I was wondering if having nodes outside of the "chosen" ones is a problem or not.
I apologize for any language related problems, unfortunately English is not my mother tongue.
Thanks a lot in advance for any help!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,866 |
Rhamphostomella peristomata is een mosdiertjessoort uit de familie van de Umbonulidae. De wetenschappelijke naam van de soort is voor het eerst geldig gepubliceerd in 1993 door Gontar.
Umbonulidae | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,198 |
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How SASB is Working to Reform Sustainability Data Disclosure
Words by Raz Godelnik
If you go these days to sustainable business conferences and events you will find out pretty quickly that this field is mainly occupied with two main themes it struggles with: storytelling and metrics. Last week, for example, it was all about metrics in an event organized by the Sustainability Accounting Standards Board (SASB).
SASB is a non-profit organization engaged in "establishing an understanding of material sustainability issues facing industries and creating sustainability accounting standards suitable for disclosure in standard filings such as the Form 10-K and 20-F." Its goal is to set standards for 10 sectors and 88 until the second quarter of 2015, starting with the Healthcare sector and moving on to the Financial sector, which was at the center of the NY event.
After last week's event I can add another title to SASB: a potential game changer.
I'm not sure if all the 300 participants at the event hosted by Bloomberg in New York shared this sentiment. Yet, I'm quite positive they recognized the importance of SASB's efforts to create a standardized framework that will help companies and investors identify and evaluate material ESG issues. After all, as Jean Rogers, SASB Executive Director commented, the world needs better performance and to get better performance we need better measurement. And this is where SASB enters the picture.
Actually, if we want to be accurate (after all, we're talking about accounting here) SASB enters one or two steps before the measurements, as it helps companies and investors to understand what should be measured and where would be the most effective place to present this information.
And it all begins with materiality. No matter what industry or sector we talk about, the starting point of the standardization process is a preparation of a materiality map in order to understand the relative materiality of issues in the industry. Each map, Harvard Business School Professors Bob Eccles and George Serafeim explain in their article "The Performance Frontier" prioritizes 43 ESG issues, ranking their materiality for a given industry from a scale from 0.5 to 5, with 5 being most material.
But while it is no brainer that materiality is a fundamental concept in sustainability reporting, it is still a somewhat elusive concept that can be interpreted in different ways. The first panel in the event was looking into this issue trying to better understand what do we talk about when we talk about materiality in the sustainability universe.
The main reference presented in this debate was the Supreme Court definition of materiality, which was presented in 1976 in the case of TSC Industries v. Northway, Inc. Back then, the court defined materiality as "a substantial likelihood that the disclosure of the omitted fact would have been viewed by the reasonable investor as having significantly altered the 'total mix' of the information made available."
This definition is used for example by the SEC in its rules on climate change disclosure, but not everyone in the panel agreed with it. Bob Herz, former Chairman of FASB for example said "could" would be a better fit than "would" and very quickly the whole panel was involved in a lively discussion on how broad the definition should be – whether sustainability information "would," "could," or "should" have been viewed by the reasonable investor as having significantly altered the total mix of information made available.
The discussion wasn't only focusing on linguistics. Other issues such as the characteristics of the reasonable investor and fiduciary duty were brought up as well. In addition Prof. Serafeim, who participated in the panel mentioned that we need to think about the SASB disclosure framework also as a managerial tool, reflecting the notion in his and Eccle's article that in order to create sustainable strategies that simultaneously boost both financial and ESG performance companies need first and foremost to focus strategically on the most material ESG issues.
The second panel was focusing on another important issue – where should a company disclose the material data? SASB's standards are designed for integrated reporting and are appropriate for disclosure in the SEC form 10-K and 20-F, in the MD&A section, and SASB hopes that the SEC will adopt eventually this approach and require companies to do it.
But is Form 10-K really the best place to disclose this sort of information from stakeholders' point of view in general and investors in particular? The panel seemed to think that it is, or at least that it's better than a regular CSR report. Dan Hanson, Managing Director, Portfolio Manager at Jarislowsky Fraser said it is important to have this information on Form 10-K or the annual report because from an investor's point of view if this data not on Form 10-K or the annual report it indicates that these issues are not vital for the management.
It makes a lot of sense but this would be quite a challenge considering the starting point, at least in the Financial sector. Half of the 63 companies surveyed by SASB in this sector said they currently don't provide any disclosure on ESG issues in their 10-K filings, with another 14 percent offering only boilerplate statements and 30 percent reporting on industry-specific issues. Only 6 percent said they disclose ESG metrics.
While it remains to be seen if SASB's standards will be widely accepted in the sustainable business universe, right now it looks like the best chance we have to move ahead with sustainability metrics. Now we need to see how we reform storytelling.
[Image credit: SASB]
Raz Godelnik is the co-founder of Eco-Libris and an adjunct faculty at the University of Delaware's Business School, CUNY SPS and Parsons the New School for Design, teaching courses in green business, sustainable design and new product development. You can follow Raz on Twitter.
Raz Godelnik
Raz Godelnik is an Assistant Professor and the Co-Director of the MS in Strategic Design & Management program at Parsons School of Design in New York. Currently, his research projects focus on the impact of the sharing economy on traditional business, the sharing economy and cities' resilience, the future of design thinking, and the integration of sustainability into Millennials' lifestyles. Raz is the co-founder of two green startups – Hemper Jeans and Eco-Libris and holds an MBA from Tel Aviv University.
Read more stories by Raz Godelnik
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© 2020 3BL Media & TriplePundit. All rights reserved. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 742 |
EDITOR'S NOTE: Members of the media may contact Jeannette Rea-Keywood, Rutgers 4-H Youth Development by phone at 609-827-0199 or by email at reakeywood@njaes.rutgers.edu.
New Jersey 4‑H'ers will join six million boys and girls nationwide in observing National 4‑H Week, October 7-13.
"4‑H is a community of young people across America who are learning leadership, citizenship and life skills. Building on this, 4‑H'ers will continue to be involved in serving their communities. Through 4‑H programs, young people combine their concerns with practical skills so they can make a difference. Working together and having a good time through 4‑H encourages youth to make volunteering a lifelong habit", says Jeannette Rea-Keywood, State 4‑H Agent, Department of 4‑H Youth Development, Rutgers University.
In New Jersey, there are over 1,500 4‑H volunteers working with nearly 40,000 youth involved in 4‑H clubs and educational programs. These volunteers are among the over 500,000 volunteers nationwide who each contributed about 220 hours of service last year.
The 4‑H Youth Development Program of Rutgers Cooperative Extension, a unit of Rutgers New Jersey Agricultural Experiment Station, is open to boys and girls in kindergarten through one year out of high school wherever they live. Programs are conducted in the over 3,100 counties of the United States, the District of Columbia and six territories.
Since its beginning over 100 years ago (1902), about 60 million Americans from all walks of life have been involved in 4‑H. In addition, 50 countries around the world have youth programs similar to 4‑H reaching one million young people. The 4‑H Youth Development Program thrives because of a unique partnership with the public and private sectors who contribute at the local, state and national level. Not only does this partnership include financial support, but many business people volunteer their time and talents to boys and girls. Support at the local level is received through the help of many businesses, financial institutions, civic organizations and service groups. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,011 |
Jennifer Brandt Discusses the Richard Simmons Lawsuit on Fox 29
Jennifer Brandt, co-chair of Cozen O'Connor's Family Law practice, was featured on Good Day Philadelphia to discuss the Richard Simmons lawsuit. "Since early 2014, Mr. Simmons has taken a leave of absence from the media spotlight in order to retreat from his 40-year career in television, fitness and other arenas of entertainment. Starting from around May 2015, Mauro Oliveira, an individual who has blackmailed, extorted and stalked Mr. Simmons for several years with the intention of destroying the career and reputation of Mr. Simmons, contacted several press outlets, including the National Enquirer and Radar, and offered information on Mr. Simmons's disappearance in exchange for a fee," lawyers for Simmons allege in the court filing. The suit also claims that the Enquirer and Radar "knew and acted in reckless disregard for the fact that the information provided by Mr. Oliveira was false and that he was not a credible or reliable source."
Family Law Focus
Updates and insights on divorce, custody, support and other family law developments. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 311 |
\section{Historical review and methodology.}
The one of the question which naturaly appears in all algebraic studies, is
the question of the classification (up to isomorphism) of the algebraic
objects from some class. One of the classical example of this kind of
results is the classification of the semisimple finite dimensional
associative algebras over fields. Also we have the very easely observed
classification of the finitely generated abelian groups. Both of these
classification were achieved many years ago. About 60 years ago, the
classification of the simple Lie algebras over $%
\mathbb{C}
$ was achieved. The classification of the finite simple groups, is a newer
result of this kind. This result requested a huge effort of many
mathematicians.
The natural next step after the classification of the finitely generated
abelian groups, is the classification of the finitely generated nilpotent
class $2$ groups, in particular the classification of the finitely generated
torsion free nilpotent class $2$ groups. In the 1970s the research in this
area was very active. We can remember \cite{GrSch}, \cite{GrSeSt}, auxiliary
technical work \cite{Sch} and others. Achievements were summarized in \cite%
{GrSe}. A full classification up to isomorphism was established only for
finitely generated torsion free nilpotent class $2$ groups of Hirsh length $%
6 $. Such modest results indicate the complication of the problem. This
complication follows, as it will be seen from our survey, first of all, from
the complication of the wild matrix problem.
Also in the 1970s, the problem of classification of the nilpotent class $2$ $%
p$-groups up to isomorphism was considered in \cite{Serg1}. It was proved
that this problem can be reduced to the wild matrix problem even when the
rank of the center of the groups is equal to $2$.
It is known that for every nilpotent torsion free group $G$ there is a
Maltsev completion $\sqrt{G}$ - the minimal group, such that $G\subset \sqrt{%
G}$ and for every $x\in \sqrt{G}$ and every $n\in \mathbf{%
\mathbb{N}
}$ there exists $x^{\frac{1}{n}}\in \sqrt{G}$, such that $\left( x^{\frac{1}{%
n}}\right) ^{n}=x$. The element $x^{\frac{1}{n}}\in \sqrt{G}$ is uniquely
defined by $x\in \sqrt{G}$ and $n\in \mathbf{%
\mathbb{N}
}$. The $\sqrt{G}$ is the nilpotent group of the same class of nilpotence as
$G$. It is clear that if two nilpotent torsion free groups are isomorphic,
then their Maltsev completions are isomorphic too. The inverse, of course,
is false. So the classification of the complete (in the Maltsev sense)
nilpotent torsion free groups of the finite rank up to isomorphism is a
simpler problem than the classification up to isomorphism of the arbitrary
finitely generated nilpotent class $2$ groups. But even in this simpler
problem we have (see \cite{GrSe}) a solution, only in the case when the rank
of the center of the groups is not greater than $2$.
If the problem of the classification up to isomorphism is so complicated, it
is natural to consider a less delicate classification. The notion of the
geometric equivalence of universal algebras (see \cite{Pl1}), was
investigated in 1995 by B. Plotkin. By \cite{Pl2} two finitely generated
universal algebras $A_{1}$ and $A_{2}$ from some variety $\Theta $ are
geometrically equivalent if and only if, the first one can be embedded into
some direct power of the second and vice versa (we denote\ $A_{1}\sim A_{2}$%
). So, the classification up to geometric equivalence is less delicate then
the classification up to isomorphism.
Classification of the nilpotent groups up to geometric equivalence is
especialy interesting because in the case of the nilpotent groups geometric
equivalence is closely connected with the logic proprieties of groups: two
nilpotent groups\ are geometrically equivalent if and only if, they have the
same quasi-identities (see \cite{Ts}). So classification of nilpotent
groups, up to geometric equivalence is an equivalent to the classification
of the quasi-varieties generated by a single nilpotent group.
The classification of the finitely generated abelian groups up to geometric
equivalence was achieved in \cite{Be}. It was proved that two abelian groups
are geometrically equivalent if and only if, for every prime number $p$ the
exponents of their corresponding $p$-Sylow subgroups coincide, and if one of
these group is not periodic, then the second group is not periodic either.
So the classification of finitely generated abelian groups up to geometric
equivalence, is in principal, simpler than the classification of these
groups up to isomorphism. The classification of the torsion free abelian
groups up to geometric equivalence is trivial: all these groups are
geometrically equivalent.
In the case of the nilpotent class $2$ groups we have a different situation.
Even classification of the finitely generated torsion free nilpotent class $%
2 $ groups up to geometric equivalence is a very complicated problem.
By \cite[Theorem 1]{Ts} every finitely generated torsion free nilpotent
class $2$ group geometrically equivalent to it's Maltsev completion. So for
resolving of our problem it is enough to classify up to geometric
equivalence the nilpotent class $2$ finite rank torsion free complete groups.
It is well known (see for example \cite[Chapter 8]{Ba}) that in every
nilpotent Lie $\mathbf{%
\mathbb{Q}
}$ -algebra $L$, we can define multiplication by Campbell-Hausdorff formula.
With this multiplication $L$ will be a group, which we denote $L^{\circ }$.
The group $L^{\circ }$ will be torsion free and complete. It has the same
class of nilpotency as algebra $L$. Conversely for every complete nilpotent
torsion free group $A$, there is nilpotent Lie $\mathbf{%
\mathbb{Q}
}$-algebra $L$, such that $A\cong L^{\circ }$. Algebra $L$ has the some
class of nilpotency as group $A$. The homomorphisms (epimorphisms,
monomorphisms, isomorphisms) of the nilpotent Lie $\mathbf{%
\mathbb{Q}
}$-algebras coincide with the homomorphisms (epimorphisms, monomorphisms,
isomorphisms) of the corresponding groups and vice versa. In other words,
the functor $\Gamma :L\rightarrow L^{\circ },\left( \lambda
:L_{1}\rightarrow L_{2}\right) \rightarrow \left( \lambda :L_{1}^{\circ
}\rightarrow L_{2}^{\circ }\right) $ provides an isomorphism from the
category of the nilpotent class $s$ Lie $\mathbf{%
\mathbb{Q}
}$-algebras to the category of the nilpotent class $s$ torsion free complete
groups. By this isomorphism to the finite dimensional Lie $\mathbf{%
\mathbb{Q}
}$ -algebra corresponds the nilpotent group of the finite rank and vice
versa.
So two complete nilpotent torsion free finite rank groups $%
A_{1}=L_{1}^{\circ }$ and $A_{2}=L_{2}^{0}$, are isomorphic if and only if,
the Lie $\mathbf{%
\mathbb{Q}
}$-algebras $L_{1}$ and $L_{2}$ are isomorphic. And two complete nilpotent
torsion free finite rank groups $A_{1}=L_{1}^{\circ }$ and $%
A_{2}=L_{2}^{\circ }$ are geometrically equivalent if and only if, the Lie $%
\mathbf{%
\mathbb{Q}
}$-algebras $L_{1}$ and $L_{2}$ are geometrically equivalent, i.e. as it was
stated above, if and only if the algebra $L_{1}$ can be embedded into some
direct power of the algebra $L_{2}$ and vice versa. So for researching our
problem, we can concentrate on the geometric equivalence of the finite
dimension nilpotent class $2$ Lie $\mathbf{%
\mathbb{Q}
}$-algebras. It will be proved in this paper, that the problem of
classification of the finite dimension nilpotent class $2$ Lie $\mathbf{%
\mathbb{Q}
}$-algebras up to the geometric equivalence, is equivalent to the problem of
classification of these algebras up to the isomorphism. It means that the
problem of the classification of the nilpotent class $2$ finite rank torsion
free complete groups up to the geometric equivalence, is equivalent to the
problem of classification of these groups up to the isomorphism.
The problem of classification of the finite dimension nilpotent class $2$
Lie algebras over an algebraic closed field up to the isomorphism was
considered in \cite{Serg2} and \cite{BLS}. In \cite{Serg2} this problem was
resolved when the dimension of the center of the algebra is not great then $%
2 $. In \cite{BLS} it was proved that the problem of classification of the
finite dimension nilpotent class $2$ Lie algebras over an algebraic closed
field up to the isomorphism when the dimension of the center of the algebra
is great then $2$ is equivalent to the wild problem.
In \cite{Be} it was proved that if $A_{1}\sim A_{2}$ and $B_{1}\sim B_{2}$ ($%
A_{1},A_{2},B_{1},B_{2}$ are arbitrary universal algebras from some variety $%
\Theta $) then $A_{1}\oplus B_{1}\sim A_{2}\oplus B_{2}$. So for
classification up to the geometric equivalence it is enough to consider the
algebras which are can not be decomposed to the direct sum. In our
situation, we can consider only Lie $\mathbf{%
\mathbb{Q}
}$-algebras $L$ which fulfill
\begin{equation}
\left[ L,L\right] =Z\left( L\right) , \label{com_cond}
\end{equation}%
where $Z\left( L\right) $ is a center of the algebra $L$.
The Lie $\mathbf{%
\mathbb{Q}
}$-algebra $L$, which fulfills this condition, can be considered as the
direct sum of the $\mathbf{%
\mathbb{Q}
}$-linear spaces $L=V\oplus W$ (form this place and below we considered the
direct sum only in the category of the $\mathbf{%
\mathbb{Q}
}$-linear spaces), where $W=Z\left( L\right) $, $V\cong L/Z\left( L\right) $%
. The Lie brackets in $L$, define the skew symmetric non singular bilinear
mapping $\omega _{L}:V\times V\ni \left( v_{1},v_{2}\right) \rightarrow %
\left[ v_{1},v_{2}\right] \in W$. (For arbitrary skew symmetric bilinear
mapping $\omega :V\times V\rightarrow W$ we denote $\ker \omega =\left\{
x\in V\mid \forall v\in V\left( \omega \left( x,v\right) =0\right) \right\} $
and we say that $\omega $ is singular if $\ker \omega \neq \left\{ 0\right\}
$; other skew symmetric bilinear mappings we call non singular.)
Contrariwise, if we have two $\mathbf{%
\mathbb{Q}
}$-linear spaces $V$ and $W$ and the skew symmetric bilinear mapping $\omega
:V\times V\rightarrow W$, then in the direct sum $L=V\oplus W$ we can define
the Lie brackets: $\left[ v_{1}+w_{1},v_{2}+w_{2}\right] =\omega \left(
v_{1},v_{2}\right) $ ($v_{1},v_{2}\in V$, $w_{1},w_{2}\in W$). If $\omega $
is a non singular, then $Z\left( L\right) =W$ and condition (\ref{com_cond})
is an equivalent to the condition%
\begin{equation}
\omega \left( V,V\right) =W. \label{im_cond}
\end{equation}%
If $L=V\oplus W$ and $\dim V=n$, $\dim W=m$, $\left\{ v_{1},\ldots
,v_{n}\right\} $ is a basis of $V$, $\left\{ w_{1},\ldots ,w_{m}\right\} $
is a basis of $W$, then the skew symmetric bilinear mapping $\omega $ is
defined by $m$ skew symmetric matrices of the size $n\times n$: $A^{\left(
1\right) },\ldots ,A^{\left( m\right) }$, such that $\left[ v_{i},v_{j}%
\right] =\sum\limits_{k=1}^{m}a_{i,j}^{\left( k\right) }w_{k}$ ($1\leq
i,j\leq n$). There is a homomorphism of the Lie algebras with condition (\ref%
{com_cond}) $\lambda :L=V_{L}\oplus W_{L}\rightarrow S=V_{S}\oplus W_{S}$ if
and only if, there is a pair of the linear mappings $\left( \varphi ,\psi
\right) $ such that $\varphi :V_{L}\rightarrow V_{S}$, $\psi
:W_{L}\rightarrow W_{S}$ and $\omega _{S}\left( \varphi \left( v_{1}\right)
,\varphi \left( v_{2}\right) \right) =\psi \omega _{L}\left(
v_{1},v_{2}\right) $ for every $v_{1},v_{2}\in V_{L}$. $\lambda =\varphi
\oplus \psi $ holds.
By using this approach in \cite{Ts} (and by using the isomorphism from the
category of the nilpotent class $2$ Lie $\mathbf{%
\mathbb{Q}
}$ -algebras to the category of the nilpotent class $2$ torsion free
complete groups) the problem of the classification of the nilpotent class $2$
finite rank torsion free groups, whose centers have rank no more then $2$,
up to the geometric equivalence was deeply researched. It was proved two
theorems:
\begin{enumerate}
\item Theorem 3. Two nilpotent torsion free class $2$ finitely generated
groups $G_{1}$ and $G_{2}$ with the cyclic center are geometrically
equivalent ($G_{1}\sim G_{2}$) if and only if their Maltsev completions are
isomorphic: $\sqrt{G_{1}}\cong \sqrt{G_{2}}$.
\item Theorem 4. Let $G_{1},G_{2}$ two nilpotent torsion free class $2$
finitely generated groups, whose centers have rank $2$. Then $G_{1}\sim
G_{2} $ if and only if, or there is a nilpotent torsion free class $2$
finitely generated group with the cyclic center $N$, such that $G_{1}\sim
N\sim G_{2}$, or $\sqrt{G_{1}}\cong \sqrt{G_{2}}$.
\end{enumerate}
Also Proposition 1 and Proposition 2 from \cite{Ts}, which formulated by the
language of the properties of the skew symmetric bilinear forms, provide us
tools for finding when for two nilpotent torsion free class $2$ finitely
generated groups $G_{1}$ and $G_{2}$, whose centers have rank $2$, fulfills
the first or the second condition of the Theorem 4.
\section{New results.}
Bellow the word "algebra" means: nilpotent class $2$ finite dimension Lie $%
\mathbf{%
\mathbb{Q}
}$-algebra.
\begin{definition}
\label{decomp}We say that the algebra is \textit{geometrically decomposable}
if it is geometrically equivalent to the direct product of some of its
nontrivial subalgebras. Other algebras we call \textit{geometrically
indecomposable}.
\end{definition}
\begin{proposition}
\label{equivalent_isom}\textit{If two geometrically indecomposable algebras
are geometrically equivalent, then they are isomorphic.}
\end{proposition}
\begin{proof}
We assume that $L$ and $S$ are geometrically indecomposable and $L$
geometrically equivalent to $S$. There is a family of
homomorphisms\linebreak\ $\left\{ \lambda _{i}:L\rightarrow S\mid i\in
I\right\} $ such that $\bigcap\limits_{i\in I}\ker \lambda _{i}=\left\{
0\right\} $. Also exists a family of homomorphisms $\left\{ \sigma
_{j}:S\rightarrow L\mid j\in J\right\} $ such that $\bigcap\limits_{j\in
J}\ker \sigma _{j}=\left\{ 0\right\} $. If $L=\left\{ 0\right\} $ then $%
S=\left\{ 0\right\} $ and vice versa, so we can assume that $L,S\neq \left\{
0\right\} $.
We assume that $\ker \lambda _{i}\neq \left\{ 0\right\} $ for every $i\in I$%
. We consider the family of endomorphisms $\left\{ \sigma _{j}\lambda
_{i}:L\rightarrow L\mid j\in J,i\in I\right\} $. $\bigcap\limits_{\substack{ %
j\in J \\ i\in I}}\ker \sigma _{j}\lambda _{i}=\left\{ 0\right\} $, so
there exists an embedding $L\hookrightarrow \prod\limits_{\substack{ j\in J
\\ i\in I}}\mathrm{im}\sigma _{j}\lambda _{i}$. $\ker \sigma _{j}\lambda
_{i}\supset \ker \lambda _{i}\neq \left\{ 0\right\} $, so, by reason of
dimensions, $\mathrm{im}\sigma _{j}\lambda _{i}$ is not equal to $L$. We
have that $L$ is geometrically equivalent to $\prod\limits_{\substack{ j\in
J \\ i\in I}}\mathrm{im}\sigma _{j}\lambda _{i}$. If $L\neq \left\{
0\right\} $ then there is a nonzero factor in this product. This is a
contradiction.
By symmetry we achieve a contradiction when we assume that $\ker \sigma
_{j}\neq \left\{ 0\right\} $ for every $j\in J$.
So there exist $i\in I$, such that $\ker \lambda _{i}=\left\{ 0\right\} $
and $j\in J$ , such that $\ker \sigma _{j}=\left\{ 0\right\} $. By reason of
dimensions, $L$ and $S$ are isomorphic.
\end{proof}
\begin{proposition}
\label{decompos_cond}\textit{If the algebra }$L=V\oplus W$\textit{\ is
geometrically decomposable then there exists a family of linear mappings }$%
\left\{ \psi _{i}:W\rightarrow W\mid i\in I\right\} $\textit{\ such that all
skew symmetric bilinear mappings }$\psi _{i}\omega $\textit{\ are singular
and }$\bigcap\limits_{i\in I}\ker \psi _{i}=\left\{ 0\right\} $ ($\omega
:V\times V\rightarrow W$ is a skew symmetric bilinear mapping defined by the
Lie brackets of the \textit{algebra }$L$)\textit{.}
\end{proposition}
\begin{proof}
Let the algebra $L$ be geometrically decomposable. Then $L$ is geometrically
equivalent to $\prod\limits_{i\in I_{0}}L_{i}$, where $L_{i}$ ($i\in I_{0}$)
is nontrivial subalgebras of $L$. So there exists an embedding $%
L\hookrightarrow \left( \prod\limits_{i\in I_{0}}L_{i}\right) ^{J}$. But we
can write $\left( \prod\limits_{i\in I_{0}}L_{i}\right)
^{J}=\prod\limits_{i\in I}L_{i}$,where $I=I_{0}\times J$ and $L_{i}$ ($i\in
I $) is also nontrivial subalgebras of $L$. By Remak theorem there exists a
family of endomorphisms $\left\{ \widetilde{\lambda }_{i}:L\rightarrow
L_{i}\mid i\in I\right\} $, such that $\bigcap\limits_{i\in I}\ker
\widetilde{\lambda }_{i}=\left\{ 0\right\} $. Denote $\iota
_{i}:L_{i}\hookrightarrow L$ the embedding and $\lambda _{i}=\iota _{i}%
\widetilde{\lambda }_{i}$. We have $\lambda _{i}:L\rightarrow L$ and\ $%
\bigcap\limits_{i\in I}\ker \lambda _{i}=\left\{ 0\right\} $. $\mathrm{im}%
\lambda _{i}\leq L_{i}$, $L_{i}$ is a nontrivial subgroup of $L$, so $\dim
\mathrm{im}\lambda _{i}<\dim L$ for every $i\in I$. By reason of dimension, $%
\ker \lambda _{i}\neq \left\{ 0\right\} $ for every $i\in I$.
Let $\lambda _{i}=\varphi _{i}\oplus \psi _{i}$, where $\varphi
_{i}:V\rightarrow V$, $\psi _{i}:W\rightarrow W$. If $\ker \psi _{i}=\left\{
0\right\} $ then $\ker \lambda _{i}=\left\{ 0\right\} $ (the intersection of
a nontrivial normal subgroup of the nilpotent group with the center of group
is nontrivial - \cite[16.2.5]{KM}, similar theorem for nilpotent class Lie
algebras can be easy proved). If $\ker \varphi _{i}=\left\{ 0\right\} $ then
$\dim \psi _{i}\left( W\right) =\dim \psi _{i}\omega \left( V,V\right) =\dim
\omega \left( \varphi _{i}\left( V\right) ,\varphi _{i}\left( V\right)
\right) =\dim \omega \left( V,V\right) =\dim W$. Hence $\ker \psi
_{i}=\left\{ 0\right\} $. It is a contradiction. So, if $\ker \lambda
_{i}\neq \left\{ 0\right\} $, then $\ker \psi _{i}\neq \left\{ 0\right\} $
and $\ker \varphi _{i}\neq \left\{ 0\right\} $. If $x\in \ker \varphi _{i}$,
then for every $v\in V$ we have $\psi _{i}\omega \left( x,v\right) =\omega
\left( \varphi _{i}\left( x\right) ,\varphi _{i}\left( v\right) \right) =0$.
So $x\in \ker \psi _{i}\omega $, i.e., skew symmetric bilinear mapping $\psi
_{i}\omega $\ is singular. $\bigcap\limits_{i\in I}\ker \psi _{i}\subset $ $%
\bigcap\limits_{i\in I}\ker \lambda _{i}=\left\{ 0\right\} $.
\end{proof}
Now for every algebra $L$ we will construct a specific geometrically
indecomposable algebra $E\left( L\right) $, such that $L\subset E\left(
L\right) $. Let $L=V\oplus W$, $\left\{ v_{1},\ldots ,v_{n}\right\} $ be a
basis of $V$, $\left\{ w_{1},\ldots ,w_{m}\right\} $ be a basis of $W$.
Then, we construct the $E\left( L\right) $ this way: $Z\left( E\left(
L\right) \right) =W\oplus T$, where $T=\mathrm{Sp}\left\{ t\right\} $ is $1$%
-dimensional $\mathbf{%
\mathbb{Q}
}$-linear spaces, $E\left( L\right) /Z\left( E\left( L\right) \right) \cong
U\oplus V$, where $U$ is a $n$-dimensional $\mathbf{%
\mathbb{Q}
}$-linear spaces with the basis $\left\{ u_{1},\ldots ,u_{n}\right\} $. If
the skew symmetric bilinear mapping $\omega _{L}$ is defined by $m$ skew
symmetric matrices of the size $n\times n$:
\begin{equation*}
A^{\left( 1\right) },\ldots ,A^{\left( m\right) }
\end{equation*}%
then the skew symmetric bilinear mapping $\omega _{E\left( L\right) }$
define by $m+1$ skew symmetric matrices of the size $2n\times 2n$:%
\begin{equation*}
\left(
\begin{array}{cc}
0 & I_{n} \\
-I_{n} & 0%
\end{array}%
\right) ,\left(
\begin{array}{cc}
0 & 0 \\
0 & A^{\left( 1\right) }%
\end{array}%
\right) ,\ldots ,\left(
\begin{array}{cc}
0 & 0 \\
0 & A^{\left( m\right) }%
\end{array}%
\right)
\end{equation*}%
i.e.
\begin{equation}
\left[ u_{i},u_{j}\right] =0,\left[ u_{i},v_{j}\right] =-\left[ v_{j},u_{i}%
\right] =\delta _{ij}t\text{ (}1\leq i,j\leq n\text{).} \label{basis_calc}
\end{equation}
\begin{proposition}
\label{extension1}For every algebra $L$ algebra $E\left( L\right) $ is
\textit{geometrically indecomposable.}
\end{proposition}
\begin{proof}
Let $\omega =\omega _{E\left( L\right) }:\left( U\oplus V\right) \times
\left( U\oplus V\right) \rightarrow W\oplus T$. We assume that $\psi
:W\oplus T\rightarrow W\oplus T$ is a linear mapping such that $\psi \left(
t\right) =z\neq 0$ and $x\in \ker \psi \omega $. Denote $x=\sum%
\limits_{i=1}^{n}x_{i}u_{i}+\sum\limits_{i=1}^{n}x_{n+i}v_{i}$. $\psi \omega
\left( x,u_{j}\right) =\psi \left( -x_{n+j}t\right) =-x_{n+j}z=0$, so $%
x_{n+j}=0$ for every $j\in \left\{ 1,\ldots ,n\right\} $. Hence $%
x=\sum\limits_{i=1}^{n}x_{i}u_{i}$ and $\psi \omega \left( x,v_{j}\right)
=x_{j}z=0$, so $x_{j}=0$ for every $j\in \left\{ 1,\ldots ,n\right\} $.
Therefore $x=0$ and $\ker \psi \omega =0$. So for every linear mapping $\psi
:W\oplus T\rightarrow W\oplus T$, for which we have $\ker \psi \omega \neq 0$%
, we also have $\psi \left( t\right) =0$. By Proposition \ref{decompos_cond}
$E\left( L\right) $ is geometrically indecomposable.
\end{proof}
$U\oplus T=H$ is an ideal of the algebra $E\left( L\right) $, $\dim \left(
H\cap Z\left( E\left( L\right) \right) \right) =1$, $\dim H/\left( H\cap
Z\left( E\left( L\right) \right) \right) =n$, $E\left( L\right) /H\cong
V\oplus W\cong L$.
\begin{theorem}
\label{main}Let $L_{1}=V_{1}\oplus W_{1}$\textit{\ and }$L_{2}=V_{2}\oplus
W_{2}$ are \textit{algebras, }$\dim V_{1}=\dim V_{2}=n$, $\dim W_{1}=\dim
W_{2}=m$,\textit{. Then }$E\left( L_{1}\right) \cong E\left( L_{2}\right) $%
\textit{, if and only if }$L_{1}\cong L_{2}$.
\end{theorem}
\begin{proof}
We denote $E\left( L_{i}\right) =H_{i}\oplus L_{i}$, $H_{i}=U_{i}\oplus
T_{i} $, $T_{i}=\mathrm{Sp}\left\{ t^{\left( i\right) }\right\} $, $\left\{
v_{1}^{\left( i\right) },\ldots ,v_{n}^{\left( i\right) }\right\} $ - basis
of $V_{i}$, $\left\{ u_{1}^{\left( i\right) },\ldots ,u_{n}^{\left( i\right)
}\right\} $ - basis of $U_{i}$ ($i=1,2$).
We assume that there is an isomorphism of algebras Lie $\alpha :E\left(
L_{1}\right) \rightarrow E\left( L_{2}\right) $. $\alpha \left( H_{1}\right)
$ is an ideal of the algebra $E\left( L_{2}\right) $. $\dim \left( H_{1}\cap
Z\left( E\left( L_{1}\right) \right) \right) =1$, so\linebreak\ $\dim \left(
\alpha \left( H_{1}\right) \cap Z\left( E\left( L_{2}\right) \right) \right)
=1$; $H_{1}\nsubseteq Z\left( E\left( L_{1}\right) \right) $ so $\alpha
\left( H_{1}\right) \nsubseteq Z\left( E\left( L_{2}\right) \right) $. First
of all, we shall prove that $\alpha \left( H_{1}\right) \subset U_{2}\oplus
Z\left( E\left( L_{2}\right) \right) $. Let $l=u+v+z\in \alpha \left(
H_{1}\right) $ ($u\in U_{2}$, $v\in V_{2}$, $z\in Z\left( E\left(
L_{2}\right) \right) $). If $v\neq 0$, then $v=\sum%
\limits_{i=1}^{n}b_{i}v_{i}^{\left( 2\right) }$, where $b_{1},\ldots
,b_{n}\in
\mathbb{Q}
$, and exists $j\in \left\{ 1,\ldots ,n\right\} $ such that $b_{j}\neq 0$.
Then $\left[ l,u_{j}^{\left( 2\right) }\right] =-b_{j}t^{\left( 2\right) }$
by (\ref{basis_calc}) and $T_{2}\subset \alpha \left( H_{1}\right) $. Also
there exists $v_{0}\in V$, such that $\left[ v,v_{0}\right] \in
W\smallsetminus \left\{ 0\right\} $, because the skew symmetric bilinear
mapping $\omega _{L_{2}}$ is a non singular. Therefore $\left[ l,v_{0}\right]
=\left[ u,v_{0}\right] +\left[ v,v_{0}\right] \notin T_{2}$ \ and $\dim
\left( \alpha \left( H_{1}\right) \cap Z\left( E\left( L\right) \right)
\right) >1$. By this contradiction we have that $v=0$ and $\alpha \left(
H_{1}\right) \subset U_{2}\oplus Z\left( E\left( L_{2}\right) \right) $.
$\alpha \left( H_{1}\right) \nsubseteq Z\left( E\left( L_{2}\right) \right) $
so there exists $l=u+z\in \alpha \left( H_{1}\right) $ ($u\in
U_{2}\smallsetminus \left\{ 0\right\} $, $z\in Z\left( E\left( L_{2}\right)
\right) $). Because $u\neq 0$, we have as above that there is $j\in \left\{
1,\ldots ,n\right\} $ such that $\left[ l,v_{j}^{\left( 2\right) }\right]
\in T_{2}\smallsetminus \left\{ 0\right\} $. But $\dim \left( \alpha \left(
H_{1}\right) \cap Z\left( E\left( L_{2}\right) \right) \right) =1$, so $%
\alpha \left( H_{1}\right) \cap Z\left( E\left( L_{2}\right) \right) =T_{2}$.
It is clear that $\left( U_{2}\oplus Z\left( E\left( L_{2}\right) \right)
\right) \cap L_{2}=\left( U_{2}\oplus W_{2}\oplus T_{2}\right) \cap \left(
V_{2}\oplus W_{2}\right) =W_{2}$, so $\alpha \left( H_{1}\right) \cap
L_{2}\subset W_{2}\subset Z\left( E\left( L_{2}\right) \right) $ and $\alpha
\left( H_{1}\right) \cap L_{2}\subset W_{2}\cap Z\left( E\left( L_{2}\right)
\right) \cap \alpha \left( H_{1}\right) =W_{2}\cap T_{2}=\left\{ 0\right\} $%
. By arguments of dimensions we have $E\left( L_{2}\right) =\alpha \left(
H_{1}\right) \oplus L_{2}$ and, because $\alpha \left( H_{1}\right) $ is an
ideal the linear mapping $E\left( L_{2}\right) /\alpha \left( H_{1}\right)
\rightarrow L_{2}$ is an isomorphism of algebras. So we have that $%
L_{1}\cong E\left( L_{1}\right) /H_{1}\cong \alpha \left( E\left(
L_{1}\right) \right) /\alpha \left( H_{1}\right) \cong E\left( L_{2}\right)
/\alpha \left( H_{1}\right) \cong L_{2}$.
Let $\lambda =\varphi \oplus \psi :L_{1}=V_{1}\oplus W_{1}\rightarrow
L_{2}=V_{2}\oplus W_{2}$ is an isomorphism of algebras. The linear mapping $%
\varphi $ is a bijection, so in the referred above bases of $V_{1}$ and $%
V_{2}$ it is presented by the invertible matrix $F=\left( f_{ij}\right)
_{i,j=1}^{n}\in GL_{n}\left( \mathbf{%
\mathbb{Q}
}\right) $. We take the matrix $G=\left( g_{ij}\right) _{i,j=1}^{n}=\left(
F^{-1}\right) ^{t}$ and by this matrix and by referred above bases of $U_{1}$
and $U_{2}$ define the linear mapping $\gamma :U_{1}\rightarrow U_{2}$.
Also, we define the linear mapping $\tau :T_{1}\ni t^{\left( 1\right)
}\rightarrow t^{\left( 2\right) }\in T_{2}$. We will prove that the linear
mapping $\gamma \oplus \varphi \oplus \tau \oplus \psi :E\left( L_{1}\right)
=U_{1}\oplus V_{1}\oplus T_{1}\oplus W_{1}\rightarrow U_{2}\oplus
V_{2}\oplus T_{2}\oplus W_{2}=E\left( L_{2}\right) $ is an isomorphism of
algebras. This mapping is a bijection, so it is necessary to prove that for
every $u^{\prime },u^{\prime \prime }\in U_{1}$ and every $v^{\prime
},v^{\prime \prime }\in V_{1}$ that the $\left( \tau \oplus \psi \right) %
\left[ u^{\prime }+v^{\prime },u^{\prime \prime }+v^{\prime \prime }\right] =%
\left[ \left( \gamma \oplus \varphi \right) \left( u^{\prime }+v^{\prime
}\right) ,\left( \gamma \oplus \varphi \right) \left( u^{\prime \prime
}+v^{\prime \prime }\right) \right] $ fulfills. But it is enough to prove
this, for basis elements of $U_{1}\oplus V_{1}$. For $1\leq i,j\leq n$ we
have $\left[ \varphi \left( v_{i}^{\left( 1\right) }\right) ,\varphi \left(
v_{j}^{\left( 1\right) }\right) \right] =\psi \left[ v_{i}^{\left( 1\right)
},v_{j}^{\left( 1\right) }\right] $, because $\varphi \oplus \psi $ is an
isomorphism of algebras, $\left[ \gamma \left( u_{i}^{\left( 1\right)
}\right) ,\varphi \left( v_{j}^{\left( 1\right) }\right) \right] =\left[
\sum\limits_{k=1}^{n}g_{ki}u_{k}^{\left( 2\right)
},\sum\limits_{s=1}^{n}f_{sj}v_{s}^{\left( 2\right) }\right]
=\sum\limits_{s=1}^{n}\sum\limits_{k=1}^{n}g_{ki}f_{sj}\delta _{ks}t^{\left(
2\right) }=\delta _{ij}t^{\left( 2\right) }=\tau \left[ u_{i}^{\left(
1\right) },v_{j}^{\left( 1\right) }\right] $ by (\ref{basis_calc}), and $%
\left[ \gamma \left( u_{i}^{\left( 1\right) }\right) ,\gamma \left(
u_{j}^{\left( 1\right) }\right) \right] =0=\tau \left[ u_{i}^{\left(
1\right) },u_{j}^{\left( 1\right) }\right] $, because $\gamma \left(
u_{i}^{\left( 1\right) }\right) ,\gamma \left( u_{j}^{\left( 1\right)
}\right) \in U_{2}$. So $E\left( L_{1}\right) \cong E\left( L_{2}\right) $.
\end{proof}
So if one can resolve the problem of the classification of the nilpotent
class $2$\ finite dimension nilpotent class $2$ Lie $\mathbf{%
\mathbb{Q}
}$-algebras up to geometric equivalence, then (by Proposition \ref%
{equivalent_isom} and Proposition \ref{extension1}) he can classify up to
isomorphism the algebras $E\left( L\right) $, where $L$\ is an arbitrary Lie
nilpotent class $2$ finite dimension $\mathbf{%
\mathbb{Q}
}$-algebra ($\dim Z\left( \left( E\left( L\right) \right) \right) =\dim
Z\left( L\right) +1$) and by Theorem \ref{main} he can classified up to
isomorphism all nilpotent class $2$ Lie finite dimension $\mathbf{%
\mathbb{Q}
}$-algebras and all nilpotent class $2$ finite rank torsion free complete
groups.
\section{Acknowledgements.}
This research was motivated by Prof. B. Plotkin. I would like to express my
gratitude to him and to Prof. S. Margolis for their constant attention to
this work. Conversations with Prof. E. Rips, Prof. Z. Sela, Prof. E.
Hrushovski, Prof. A. Mann and Dr. E. Plotkin were very useful. After
discussions with Prof. D. Kazhdan and Prof. Yu. Drozd, I paid my attention
to the researches of Prof. V. Sergeichuk and his collaborators (\cite{Serg1}%
, \cite{Serg2}, \cite{BLS}). The debates about this problem with Dr. R.
Lipyanski led to the major break in this research, and I would like to
express my sincere gratitude. I appreciate all the authors of the paper \cite%
{BLS}, which was very contributory to this research.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 457 |
Oblicze Dnia ('The Face of the Day') was a Polish weekly literary and political newspaper published from Warsaw. The periodical was launched in and published during 1936. It was directed towards the intelligentsia and was inspired by the Popular Front victories in Spain and France. For a short period of time, the publication attracted various prominent cultural figures. Collaborators of the newspaper included Adam Strug, Zofia Nałkowska, Maria Dąbrowska, Romain Rolland, Louis Aragon and Paul Langevin. The editors of the newspaper belonged to the Communist Party of Poland and the left-wing tendency of the Polish Socialist Party (PPS).
Wanda Wasilewska served as the editor-in-chief, and the periodical was named after one of her novels, published in 1934. The book dealt with a worker strike in Kraków and had obtained cult status amongst Polish left intellectuals. Wasilewska was a member of the Socialist Party but was considered a close ally of the Communist Party. At the time, negotiations on forming a Popular Front were taking place between the two parties. However, the launching of Oblicze Dnia led to the breakdown of these talks, as the Socialist Party felt that the Communist Party had launched this newspaper under the supposed cover of a PPS sympathizer. The publication remained close to the Communist Party; in particular, it was associated with the communist front organ Czerwona Pomoc ("Red Aid"). As with other publications linked to the Communist Party, Oblicze Dnia was confiscated by the state authorities from time to time.
References
1936 establishments in Poland
Defunct newspapers published in Poland
Defunct weekly newspapers
Newspapers published in Warsaw
Weekly newspapers published in Poland
Polish-language newspapers
Publications established in 1936 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,584 |
Q: Connecting to MySQL container from Mysqlworkbench I built a MySQL container from an image, found its IP address but unable to connect from command line or mysqlworkbench. Here's my configuration.
Dockerfile
FROM mysql
ENV MYSQL_DATABASE=test
ENV MYSQL_ROOT_PASSWORD=password
COPY ./schema.sql /docker-entrypoint-initdb.d/
Command
docker build -t mysql-image .
&& docker run
-p 6603:3306
--name mysql-container
mysql-image
I can see that the container is running from docker container ls. I found its IP address using this command with the help of this answer.
docker inspect -f '{{range .NetworkSettings.Networks}}{{.IPAddress}}{{end}}' mysql-container
It gave me this ip -> 172.17.0.2
Error
Edit
The linked duplicate tag isn't the solution, the problem lies with the following command that misled me.
docker inspect -f '{{range .NetworkSettings.Networks}}{{.IPAddress}}{{end}}' mysql-container
A: You need to set the allow login from any ip with MYSQL_ROOT_HOST=%
Full docker command line as follows:
>docker run --name mysql-server -e MYSQL_ROOT_HOST=% -e MYSQL_ROOT_PASSWORD=12345 -p 3306:3306 -d mysql/mysql-server
--name means that it's container name
-d means that docker will run in deattach mode. Final it's mysql docker image
When docker status is Up 3 minutes (healthy) you can easy connect to DB via mysql Workbench
Hope to help you!
A: You need to use localhost (127.0.0.1) instead the container IP (in your case 172.17.0.2) at hostname
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,355 |
Sudan's army has stepped in to oust President Omar al-Bashir, in an effort to put an end to the months of mass protests and strikes calling for the ouster of his regime.
Awad Ibn Auf, the minister of defence and deputy president, announced Thursday that the military had arrested al-Bashir, who seized power in a coup in 1989, suspended the constitution, shut border crossings and closed the country's airspace for the next 24 hours.
He declared a three-month state of emergency, putting the country under military rule, and said that the army would oversee a two-year transitional period leading up to elections. Political prisoners would be released, he claimed.
The military coup follows four months of social unrest triggered by a government decision that tripled the price of bread. The spontaneous protests quickly developed into nationwide anti-government demonstrations calling for al-Bashir to step down. The movement drew in ever broader sections of the population with nationwide strikes of workers, including at Port Sudan on the Red Sea, and several work stoppages and protests at major telecom providers and other corporations.
Al-Bashir responded with brutal measures aimed at crushing resistance to the government, including the use of live ammunition by snipers, tear gas and baton charges. At least 60 people have been killed, including children and medics, some of whom died in prison as a result of torture.
Security forces arrested hundreds of demonstrators, with at least 800 sentenced to lengthy terms of imprisonment. Women were sentenced to floggings. There have been numerous arrests of oppositionists, including leaders of the main opposition Umma Party and the Sudan Communist Party (SCP).
In February, al-Bashir announced a year-long state of emergency, making mass demonstrations illegal, and dismissed his cabinet and all the 18 provincial governments, replacing the governors with military and security officers.
This did little to curb the widespread unrest over unemployment, soaring inflation and controls on accessing foreign currency and cash that have made living conditions intolerable. There is enormous popular hatred of al-Bashir's regime for its never-ending wars in different parts of the country, brutal repression, corruption and indifference to endemic poverty.
The regime has suppressed all opposition to its policies over the last 30 years and waged war against its own people in South Sudan and Darfur, with armed conflicts still ongoing in South Kordofan and Blue Nile provinces, in what has become known as Sudan's third civil war.
Al-Bashir announced his resignation from the ruling National Congress Party (NCP), appointing his close associate Ahmad Harun as deputy head of the NCP, who called for a "national dialogue." But this was understood as a manoeuvre to win over some elements of the bourgeois opposition and maintain NCP rule via stage-managed elections in 2020—where Harun or Bashir would run.
The powerful movement of the Sudanese working class is part of a growing movement of strikes and demonstrations by workers across North Africa—in Algeria, Tunisia and Morocco—and around the world.
Sudan's rallies have been led by a coalition that includes the Sudanese Professional Association (SPA) of doctors, lawyers and teachers, the National Consensus Forces (NCF), Sudan Call, the Unionist Gathering and the Umma Party.
Masses of workers and youth have come out onto the streets, not for a military coup or political reshuffle at the top, but rather a fundamental transformation of the entire social order. Since Saturday, the anniversary of the military coup that forced Jaafar Nimeiri to step down in 1985 after 16 years in power following massive protests, there have been mass demonstrations outside the military's headquarters in Sudan's capital, Khartoum. At least 800,000 people took part Saturday, with the number swelling to 2 million the next day, the biggest protest against the government in Sudan's tumultuous history.
There were reports of some soldiers intervening to protect demonstrators after security forces tried to disperse a mass sit-in outside the defence ministry, with al-Bashir's gunmen killing at least 20 people.
The coup was clearly green-lighted by the United States and the UK, the former colonial power in Sudan, along with Sudan's neighbour, the Egyptian dictator General Abdel Fattah el-Sisi, who made his second visit to the White House earlier this week. El-Sisi's discussions with President Donald Trump were held against the backdrop of a similar attempt to neuter anti-regime protests in Algeria—with the military announcing the resignation of President Abdelaziz Bouteflika—as well as the raging civil war in Libya.
But attempts to present a civilian transitional government as capable of producing a flourishing democracy that would resolve the enormous social and economic problems confronting Sudanese workers are no less fraudulent than a military transitional council. Either way, the country remains dominated by a small, wealthy clique. The only way to establish a democratic regime in Sudan is through a struggle led by the working class to take power and expropriate the ill-gotten wealth of the entire ruling class, in the context of a broad international struggle of the working class against capitalism and for the building of socialism. | {
"redpajama_set_name": "RedPajamaC4"
} | 532 |
\section{Green's functions tool for solving differential equations}
Green's functions \cite{Ohtaka,Don,Koo,Frie}
are encountered as response functions, time-ordered
expectation values, certain solutions of boundary-value problems or
resolvent kernels. This introduction to Green's functions is based on their
role as kernels of differential equations. The procedures to construct
solutions to a differential equation with an external source or with an
inhomogeneity term are put together to {\it derive} the Dyson equation for the
Green's function of the inhomogeneous system. Very different areas of physics
such as, for example, electrodynamics (see section II and \cite{Chic})
and quantum transport (see section III and \cite{Sche98,Cue}), can profit from
such Green's function formalisms.
\subsection{Introduction}
Green's function formalisms do not present a cure-all for solving differential
equations, because essentially the problem of finding the solution of the
differential equation is shifted to that of finding the corresponding Green's
function. This can, however, be a simplification and even give access to the
solution of a more general class of problems. Starting from ordinary
differential equations, this short review is meant to introduce how the
corresponding Green's functions are defined and how they are involved in
constructing the solutions for different types of differential equations.
Although bearing similarities, the Green's function formalism can go
beyond perturbation theory. Special emphasize is laid here upon the parallels
between homogeneous and inhomogeneous ordinary differential and Green's
functions equations.
\subsection{Homogeneous equation}
The starting point is a homogeneous differential equation
\begin{equation}
D_{\xi}\; \phi_0(\xi) = 0
\end{equation}
which we suppose is exactly solvable, although $\phi_0(\xi)$ will not
explicitly be needed. $D_{\xi}$ is some differential operator which may include
multiplication with a constant or even another function of $\xi$. $\xi$ is
either a space or time variable. What we need to know is the solution $g$
of the corresponding Green's functions equation
\begin{equation}
D_{\xi}\; g(\xi,\xi') = \delta(\xi-\xi')
\end{equation}
$g$ is a tensor-like function of two arguments, $D_{\xi}$ only acting on
the first of them. Like $g$ replaces $\phi$ on the left, the zero on the
right side of (1) is replaced by a $\delta$-distribution in (2). There is no
general recipe, but knowing $\phi_0(\xi)$ can help to get $g(\xi,\xi')$.
\subsection{Source term}
Having $g$, the construction of a solution
\begin{equation}
\phi_q(\xi)=\phi_0(\xi)+\int d\xi' \; g(\xi,\xi') \; Q(\xi')
\end{equation}
of the differential equation with a source term $Q$ on the right side
\begin{equation}
D_{\xi} \; \phi_q(\xi) = Q(\xi)
\end{equation}
is straight forward. Of course, a solution $\phi_0$ of (1) can be added
independently of $Q$, so we only need to proove that the integral term from (3)
satisfies (4):
\begin{eqnarray}
D_{\xi} \int d\xi' \; g(\xi,\xi') \; Q(\xi') &=&
\int d\xi' \; [D_{\xi} \; g(\xi,\xi')] \; Q(\xi') \nonumber \\ &=&
\int d\xi' \; \delta(\xi-\xi') \; Q(\xi') = Q(\xi) \nonumber
\end{eqnarray}
(All integrals are understood to range over the entire $\xi$-space.)
\subsection{Inhomogeneity}
Instead of a source term, the differential equation can contain a potential
term which we shall call an inhomogeneity.
\begin{equation}
D_{\xi} \; \phi_{ih}(\xi) - V(\xi) \; \phi_{ih}(\xi) = 0
\end{equation}
that is a $\xi$-dependent function $V$ multiplied with $\phi$. $V(\xi)$ is
not included in $D_{\xi}$, because we assume that it so much complicates the
equation that a standard solution is no longer known. The minus sign is a
useful convention. We put the inhomogeneity term on the right
\renewcommand{\theequation}{\arabic{equation}a}
\setcounter{equation}{4}
\begin{equation}
D_{\xi} \; \phi_{ih}(\xi) = V(\xi) \; \phi_{ih}(\xi)
\end{equation}
\renewcommand{\theequation}{\arabic{equation}}
to make (5) formally look like (4). With $V\phi$ playing the role of $Q$ a
formal solution is constructed analogously to (3):
\begin{equation}
\phi_{ih}(\xi) = \phi_0(\xi) + \int d\xi' \; g(\xi,\xi') \; V(\xi') \; \phi_{ih}(\xi')
\end{equation}
This presents an implicit equation for $\phi_{ih}$, the so-called
{\it Lippmann-Schwinger} equation. We have replaced the differential
equation by an integral equation. Inserting the solution of the homogeneous
equation $\phi_0$ also for $\phi_{ih}$ in the integral on the right side of (6)
would give the {\it Born} approximation $\phi_{ih}(\xi)\approx \phi_0(\xi)+
\int d\xi' \; g(\xi,\xi') \; V(\xi') \; \phi_0(\xi')$,
which is appropriate if $V\phi$ is
a small perturbation compared to $D_{\xi}\phi$. However, it is the virtue of
the Green's functions method that in contrast to perturbation theory the
inhomogeneity need {\it not} be a small deviation. Although not yet providing
an explicit solution for $\phi_{ih}$, one can make use of
(6) in numerical calculations (see section II and \cite{DerGir}).
As (2) is the corresponding Green's function equation with a point source to
the homogeneous equation (1), the equation defining the Green's function $G$
for the inhomogeneous case derived from (5) reads:
\begin{equation}
[D_{\xi}-V(\xi)] \; G(\xi,\xi') = \delta(\xi-\xi')
\end{equation}
However, while supposing that we know or can easily guess $g(\xi,\xi')$,
there is no hint yet how to calculate $G(\xi,\xi')$.
\subsection{General case}
Finally we have to treat the most general case with source and inhomogeneity
term:
\begin{equation}
D_{\xi} \; \phi_{ihq}(\xi) - V(\xi) \; \phi_{ihq}(\xi) = Q(\xi)
\end{equation}
In analogy to (3) and (4) a yet formal solution can be written down using (7):
\begin{equation}
\phi_{ihq}(\xi) = \phi_{ih}(\xi) + \int d\xi' \; G(\xi,\xi') \; Q(\xi')
\end{equation}
Here we have the freedom to add any solution of (6) to the integral.
But let us start again solving the problem directly from (8) which we rewrite
as
\renewcommand{\theequation}{\arabic{equation}a}
\setcounter{equation}{7}
\begin{equation}
D_{\xi} \; \phi_{ihq}(\xi) - Q(\xi) = V(\xi) \; \phi_{ihq}(\xi)
\end{equation}
\renewcommand{\theequation}{\arabic{equation}}
\setcounter{equation}{9}
We compare this with (5a) and construct a solution of the same form as (6).
$\phi_q$ takes the role of $\phi_0$, because it would satisfy (8a) if the
right side were zero. $\phi_{ih}$ has to be replaced by $\phi_{ihq}$. Using
(3) for $\phi_q$ we obtain
\begin{equation}
\phi_{ihq}(\xi) = \underbrace{ \phi_0(\xi) + \int d\xi' \; g(\xi,\xi') \;
Q(\xi') }_{\phi_q} + \int d\xi'' \; g(\xi,\xi'') \; V(\xi'') \;
\phi_{ihq}(\xi'')
\end{equation}
(10) could also have been established by a different line of thought. In (8)
putting everything that differs from the homogeneous equation (1) on the
right side gives
\renewcommand{\theequation}{\arabic{equation}a}
\setcounter{equation}{7}
\begin{equation}
D_{\xi} \; \phi_{ihq} = Q(\xi) + V(\xi) \; \phi_{ihq}(\xi)
\end{equation}
\renewcommand{\theequation}{\arabic{equation}}
\setcounter{equation}{10}
If it were not for the inhomegeneity term $V\phi$, the solution would be (3).
And if it were not for the source term $Q$, we could use (6). Although this is,
of course, not a correct way to solve non-homogeneous differential equations,
we can understand (10) as an ansatz adding these two contributions. The
$\phi_0$-part does not have to be written twice. And $\phi_{ihq}$ has to
appear instead of $\phi_{ih}$ in the integral taken from (6). This still
unknown $\phi_{ihq}$ leaves the neccessary freedom to somehow counterbalance
the $Q$-term
not present in (5) and (6), which justifies (10) as an ansatz.
Whichever way it was obtained, (10) is an implicit integral equation for
$\phi_{ihq}$ as (6) is for $\phi_{ih}$.
\subsection{Dyson equation}
Now we shall profit from the fact
that with (9) we have a second representation of $\phi_{ihq}$. Insert the
expression from (9) for $\phi_{ihq}$ both on the left and on the right side in
(10):
\begin{eqnarray}
\phi_{ih}(\xi) &+& \int d\xi' \; G(\xi,\xi') \; Q(\xi') =
\phi_0(\xi) + \int d\xi' \; g(\xi,\xi') \; Q(\xi') + \nonumber \\ &&
\int d\xi'' g(\xi,\xi'') \; V(\xi'') \;
[\; \phi_{ih}(\xi'') + \int d\xi' \; G(\xi'',\xi') \; Q(\xi') \; ]
\nonumber \end{eqnarray}
$\phi_{ih}$ on the left cancels with $\phi_0$ and one integral term on the
right according to (6) and we are left with
\begin{eqnarray}
\int d\xi' \; G(\xi,\xi') \; Q(\xi') &=& \int d\xi' \; g(\xi,\xi') \; Q(\xi')
+ \nonumber \\ &&
\int d\xi'' \int d\xi' \; g(\xi,\xi'') \; V(\xi'') \; G(\xi'',\xi') \; Q(\xi')
\nonumber \end{eqnarray}
This is an implicit equation for the unknown Green's function $G$. But as $G$
by definition (7) does not depend on any source term, (11) must be valid for
arbitrary $Q$. The special choice $Q(\xi')=\delta(\xi'-\xi_0)$ gives
\begin{eqnarray}
\int d\xi' \; G(\xi,\xi') \; \delta(\xi'-\xi_0) &=& \int d\xi' \; g(\xi,\xi') \;
\delta(\xi'-\xi_0) + \nonumber \\ &&
\int d\xi'' \int d\xi' \; g(\xi,\xi'') \; V(\xi'') \;
G(\xi'',\xi') \; \delta(\xi'-\xi_0)
\nonumber \end{eqnarray}
which after carrying out the $\xi'$-integrals and then renaming $\xi_0$ to
$\xi'$ becomes
\begin{equation}
G(\xi,\xi') = g(\xi,\xi') + \int d\xi' \; g(\xi,\xi'') \; V(\xi'') \;
G(\xi'',\xi')
\end{equation}
(11) is the general implicit integral equation for the inhomogeneous system's
Green's function $G$ and called {\it Dyson} equation.
We could have derived (11) by doing all calculations from (8) on with only a
point source $\delta(\xi-\xi_0)$ instead of $Q(\xi)$. Putting in the solutions
(3) and (9) for this case would just have looked a bit awkward.
(11) is not an explicit solution for $G$, but still an implicit equation,
and even though it is often written as $G=g+gVG$ one must not forget that
there is a convolution-like integration over the inner $\xi$-index behind
the sequence of factors $gVG$. The
integration can either be transformed into a discrete finite sum, and thus (11) into
a linear system of equations solvable by a matrix inversion
(see section II and \cite{EPJ}), or the
convolution can be replaced by a multiplication going to frequency space by
a Fourier transformation (see section III and \cite{Cue}).
(10) and the preparation for (11) would have looked much more elegant leaving
out $\phi_0$ and the $\phi_{ih}$-terms right from the start. Indeed, one could
argue that in (3) one is only interested in the part different from the
trivial homogenous solution, if no further boundary conditions have to be
accounted for. With a similar argument, the $\phi_{ih}$-part could have been
dropped in (9). The identities
\renewcommand{\theequation}{\arabic{equation}*}
\setcounter{equation}{2}
\begin{equation}
\phi_q^*(\xi) = \int d\xi' \; g(\xi,\xi') \; Q(\xi')
\end{equation}
\renewcommand{\theequation}{\arabic{equation}*}
\setcounter{equation}{5}
\begin{equation}
\phi_{ih}^*(\xi) = \int d\xi' \; g(\xi,\xi') \; V(\xi') \; \phi_{ih}^*(\xi')
\end{equation}
\renewcommand{\theequation}{\arabic{equation}*}
\setcounter{equation}{8}
\begin{equation}
\phi_{ihq}^*(\xi) = \int d\xi' \; G(\xi,\xi') \; Q(\xi')
\end{equation}
\renewcommand{\theequation}{\arabic{equation}*}
\setcounter{equation}{9}
\begin{equation}
\phi_{ihq}^*(\xi) = \int d\xi' \; g(\xi,\xi') \; Q(\xi') +
\int d\xi'' \; g(\xi,\xi'') \; V(\xi'') \; \phi_{ihq}^*(\xi'')
\end{equation}
\renewcommand{\theequation}{\arabic{equation}}
\setcounter{equation}{11}
would obviously also have given (11). Leaving out $\phi_0$ in (6) would be
precarious (see below). However, dropping it when
constructing (11) along our second line of thought causes no problem.
The term with $Q$ will ensure that $\phi_{ihq}^*$ cannot simply be zero.
\subsection{Expansions}
There is some subtlety about the $\phi_0$ contribution in (6).
(6*) has the trivial solution $\phi_{ih}^*(\xi)=0$ and not necessarily
another one. Surely, $\phi_{ih}(\xi)=0$ satisfies (5), but it is not what
we want. $\phi_0$ is the background excitation replacing boundary conditions
in this kind of problem, which shall become clear when discussing
electrodynamics (see following article). $\phi_0$ acts as a source term which -
instead of being put in $Q$ - can be more appropriately and conveniently set
as a fixed part of the $\phi$ we are looking for.
Inserting (6) into itself ever
and ever again, $\phi_{ih}$ is developed into a series in powers of $gV$:
\begin{eqnarray}
\phi_{ih} &=& \phi_0 + \int gV\phi_{ih} \nonumber \\ &=&
\phi_0 + \int gV (\phi_0 + \int gV (\phi_0 + \int gV ( \ldots \; \ldots \;
\ldots ))) \nonumber \\ &=& \phi_0 + gV\phi_0 + gVgV\phi_0 + \ldots =
\sum_{n=0}^{\infty} (gV)^n \phi_0
\end{eqnarray}
Just for shorthand notation we dropped the arguments and integration variables
and in the third line also the integral signs. The resulting series is
recognized from perturbation theory summing
interactions to zeroth, first, second, etc. order. If there were no
contribution $\phi_0$ in $\phi_{ih}$, only the contribution of $gV$ to
power infinity would exist with nothing to multiply to at the end. In other
words we would have no basis on which to develop $\phi_{ih}$. The
Green's functions formalism as we use it aims at solving (6), (10) or
(11) in a closed form, {\it not} cutting the series in (12) at some finite
order.
The development (12) is only shown here precisely to demonstrate that
interactions are included to all orders as well as to explain the importance
of the homogeneous background contribution.
The Dyson equation (11) can be expanded in an analogous manner to the
Lippmann-Schwinger equation, by the way prooving the equivalence to its complementary
form $G=g+GVg$:
\begin{equation}
G = g + \int gVG = g \sum_{n=0}^{\infty} (Vg)^n
= \sum_{n=0}^{\infty} (gV)^n \; g = g + \int GVg
\end{equation}
(Integration over inner arguments is understood in all contributions to
the sums.)
Either formally or as a matrix calculation in finite discrete $\xi$-space,
(11) is often solved as
\begin{equation}
G=({\bf 1}-gV)^{-1}g
\end{equation}
Developing the $(\ldots)^{-1}$-factor in (14) into a geometric series just
results in the infinite sums written in (13).
Putting together (12) and (13) we can get an alternative representation to
(6) for the solution of the inhomogeneous differential equation:
\begin{equation}
\phi_{ih}(\xi)=\phi_0(\xi)+\int d\xi' \; G(\xi,\xi') \; V(\xi') \; \phi_0(\xi')
\end{equation}
Therefore in the case that should it be easier to get the Green's function $G$
than to solve the implicit equation (6) for $\phi_{ih}$, we see that $G$ can
be useful also for treating the equation without an external source $Q$.
Nevertheless, (15) again illustrates that $V\phi_0$ plays the role of the
source.
\subsection{Conclusions and Outlook for section I}
Using Green's functions it has been shown how differential equations can be
treated that differ from easily solvable ones by an additional potential
term or an arbitrary source term. The formalism as presented here is for
open-boundary in contrast to boundary-value problems \cite{Koo,Jac}.
From constructing
the solutions of the differential equations, we also obtained the constituting
relation for the inhomogeneous system's Green's function, which characterizes
the response to a point source and includes all-order interactions.
Examples of applications will be given in the two following sections.
\newpage
\setcounter{equation}{0}
\section{Non-boundary value problems in near-field optics}
The fundamental homogeneous-medium Green's tensor of electrodynamics is
deduced from the field of a dipole. Based upon that a numerical procedure
is presented to solve the wave-equation for the near-field in a scattering
setup for arbitrary material distributions. The full inhomogeneous system's
Green's function is not explicitly needed to get the fields, although it can
be obtained by a very similar calculation and in optics can be interpreted as
a density of states.
\subsection{Introduction}
The typical problem in nano-optics \cite{Gre97,DerChem} is the situation that some tiny structures
are illuminated by an extended source, a plane wave for example, and then
one is interested in the field distribution that arises from multiple
scattering \cite{scat}, especially to identify places where the field intensity gets
considerably enhanced \cite{Martinres,Ebbfluo}.
A theory can be based on Green's functions, however, their implication differs
slightly from the standardly taught cases of fixed boundary field values
\cite{Jac_1} or
located sources. Modern optical scanning microscopes make it possible to
probe and map directly even different quantities of the near-field
\cite{interpret}, such as
the electric and magnetic field intensities \cite{Eloise}
or the density of states \cite{Chicel}.
Applications of tayloring nano-structures with respect to optical properties
include resonant particles \cite{dicro,partweb} and cavities
\cite {Krenncav,Perney}, squeezed fields \cite{squeeze},
wave guides and their adressing \cite{adres,MarLith,Quidwave}
as well as transmission apertures \cite{Perney,Thio} and lithography masks
\cite{MarLith}.
We here present the Green's functions formalism that forms the bases of a
finite-element quite effective numerical algorithm used in current research
\cite{girneu,EPJel,coral}.
This treatise is also given as an application example of the general methods
to solve differential equations with certain perturbations presented in the
preceeding paper.
\subsection{Problem}
The discussion can be reduced to monochromatic light, that is a single
frequency $\omega$ and thus time dependence $e^{-i\omega t}$ for the fields.
With non-magnetic materials the problem is to find the solution
$\vec{E}(\vec{r})$ of the wave equation
\begin{equation}
-\vec{\nabla} \times \vec{\nabla} \times \vec{E}(\vec{r}) + \frac{\omega^2}{
c^2} \; \varepsilon(\vec{r}) \; \vec{E}(\vec{r})=0
\end{equation}
(1) is a differential equation of the type (5) from section I.
There is no source term on the right hand side of (1). If the source were,
for example, a dipole located at some point, a source term with its oscillation
strength and direction would have to be put on the right as
$\vec{Q}\delta(\vec{r}-\vec{r}_0)$.
However, we shall see that a plane-wave source can be and is better included
in (1) as it is. For simplicity we shall assume that the background medium,
in which objects with different permittivities $\varepsilon$ are located (Fig.1),
is vacuum with permittivity $\varepsilon_0$. For another embedding medium, its
dielectric constant $\varepsilon_b$ would take the role of $\varepsilon_0$.
In (1) $\varepsilon(\vec{r})$ of the material distribution designates the
dimensionless relative permittivity with respect to vacuum or the background
medium. To separate (1) into a part representing a homogeneous differential
equation with known solution and an inhomogeneity write its as
\renewcommand{\theequation}{\arabic{equation}a}
\setcounter{equation}{0}
\begin{equation}
-\vec{\nabla} \times \vec{\nabla} \times \vec{E}(\vec{r}) + \frac{\omega^2}{
c^2} \; \vec{E}(\vec{r}) + \frac{\omega^2}{c^2} \; (\varepsilon(\vec{r})-1) \;
\vec{E}(\vec{r}) = 0
\end{equation}
\renewcommand{\theequation}{\arabic{equation}}
The correspondances to the quantities of the general formalism given in
section I are
$$ \xi \leftrightarrow \vec{r}, \;
D_{\xi} \ldots \leftrightarrow -\vec{\nabla} \times \vec{\nabla} \times
\ldots+\frac{\omega^2}{c^2}\cdot \ldots, \;
V \leftrightarrow -\frac{\omega^2}{c^2}(\varepsilon-1), \;
\phi_{ih} \leftrightarrow \vec{E}$$
and $\phi_0$ will become $\vec{E}_b$. We can write down the solution
following section I after having prepared the background Green's function in the
next section.
\begin{figure}
\includegraphics[width=10cm,angle=270]{elek1.jpg}
\caption{Objects distributed in space and discretized into cubes of
equal size. A plane in space discretized into a quadratic mesh is also shown.
Arrows indicate only some possible scattering paths to a location marked by
the cross.}
\end{figure}
\subsection{Background Green's function}
In any case we need the Green's function $g(\vec{r},\vec{r'})$ of the
homogeneous problem satisfying
\begin{equation}
-\vec{\nabla}_{\vec{r}} \times \vec{\nabla}_{\vec{r}} \times g(\vec{r},\vec{r'})
+k^2g(\vec{r},\vec{r'})={\bf 1} \; \delta(\vec{r}-\vec{r'})
\end{equation}
with $k=\omega /c$.
$g$ will be a $3\otimes 3$ tensor or matrix here. More commonly \cite{Chew}
the small letter $g$ is used for the scalar function
\begin{eqnarray}
g_{scal}(\vec{r},\vec{r'})=-\frac{e^{ik\vert \vec{r}-\vec{r'} \vert}}{
4\pi \vert \vec{r}-\vec{r'} \vert} \quad {\rm fulfilling} \nonumber \\
\triangle_{\vec{r}} \; g_{scal}(\vec{r},\vec{r'})+k^2 g_{scal}(\vec{r},\vec{r'})
=\delta(\vec{r}-\vec{r'}) \end{eqnarray}
with $k=\omega /c$. The Green's function $g$ from (2) is then named $G^h$
with index $h$ for {\it homogeneous}. There are several ways to obtain $g$.
One is based on the knowledge that if we have a scalar function $\Psi(\vec{r})$
solving $\triangle\Psi+k^2\; \Psi=0$, then $\vec{F}_1=\vec{\nabla} \times
(\vec{a}\Psi)$ and $\vec{F}_2=\vec{\nabla} \times \vec{\nabla} \times (\vec{
a}\Psi)$ with a constant but arbitrary pivot vector $\vec{a}$ will both solve
the vectorial equation $-\vec{\nabla} \times \vec{\nabla} \times \vec{F} +
k^2 \vec{F}=0$. The tensor $g$ looked for in (2) can be constructed out of
$\vec{F}_1(\vec{r})$, $\vec{F}_1(\vec{r'})$, $\vec{F}_2(\vec{r})$,
$\vec{F}_2(\vec{r'})$ together with $\vec{F}_3(\vec{r})=\vec{\nabla}\Psi(\vec{r})$
and $\vec{F}_3(\vec{r'})$.
We shall not enter into the details of this mathematically slightly precarious
approach \cite{AD}. A second recipe just mentioned here for completeness is
given by the following statement \cite{Chew}:
If $g_{scal}(\vec{r},\vec{r'})$ satisfies (3), then
\begin{equation}
g(\vec{r},\vec{r'})=({\bf 1}+\frac{\vec{\nabla}_{\vec{r}} \otimes
\vec{\nabla}_{\vec{r}}}{k^2}) \; g_{scal}(\vec{r},\vec{r'})
\end{equation}
is the tensor defined by (2). Of course, because of being in homogeneous space
$g_{scal}$ and $g$ effectively are functions of $\vec{R}=\vec{r}-\vec{r'}$
alone. ${\bf 1}$ is the unit matrix in 3 by 3 cartesian coordinate space and
\newline
$\vec{\nabla} \otimes \vec{\nabla}$ means building a matrix out of derivatives
$\begin{pmatrix}
\partial^2_x & \partial_x \partial_y & \partial_x \partial_z \cr
\partial_y \partial_x & \partial^2_y & \partial_y \partial_z \cr
\partial_z \partial_x & \partial_z \partial_y & \partial^2_z \cr\end{pmatrix}$.
We shall deduce $g$ from a physical reasoning.
From standard electrodynamics \cite{Jac_2}
one has the electric field of an oscillating dipole
\begin{equation}
\vec{E}(\vec{R})=\frac{k^2e^{ikR}}{4\pi\varepsilon_0R}\left(
\frac{\vec{p}R^2-\vec{R}(\vec{R}\vec{p})}{R^2}+\frac{3\vec{R}(\vec{R}\vec{p})
-\vec{p}R^2}{R^2}\left(\frac{1}{k^2R^2}-\frac{i}{kR}\right)\right)
\end{equation}
It is important to take the exact formula here including retardation in
contrast to common near- or far-field approximations. (5) gives the space
part, the time dependence is just $e^{-i\omega t}$ everywhere. To get the
Green's tensor, we have to evaluate from (5) what field components in
x-, y- and z-direction a dipole $\vec{p}$ at $\vec{r'}$ oriented along x
would produce at $\vec{r}$, what components a dipole oriented along y would
produce and what components a dipole along z would produce and assemble all
these in a matrix. A point dipole is the elementary excitation corresponding
to the $\delta$ on the right side of (2). The physical meaning of $g$ is to
tell us what field any such dipole would have. That is shown formally in
the first matrix in (5a). Decomposing any $\vec{p}$ into its cartesian
components, (5) can be rewritten as
$$\vec{E}(\vec{R})=\begin{pmatrix}
E_x(\vec{r}) \leftarrow p_x(\vec{r'}) & \phantom{e} &
E_x(\vec{r}) \leftarrow p_y(\vec{r'}) & \phantom{e} &
E_x(\vec{r}) \leftarrow p_z(\vec{r'}) \cr
E_y(\vec{r}) \leftarrow p_x(\vec{r'}) & \phantom{e} &
E_y(\vec{r}) \leftarrow p_y(\vec{r'}) & \phantom{e} &
E_y(\vec{r}) \leftarrow p_z(\vec{r'}) \cr
E_z(\vec{r}) \leftarrow p_x(\vec{r'}) & \phantom{e} &
E_z(\vec{r}) \leftarrow p_y(\vec{r'}) & \phantom{e} &
E_z(\vec{r}) \leftarrow p_z(\vec{r'}) \cr
\end{pmatrix} \begin{pmatrix} p_x \cr p_y \cr p_z \cr \end{pmatrix}$$
\begin{center} ${\displaystyle =
\frac{k^2e^{ikR}}{4\pi\varepsilon_0R}\left( \left( 1-\frac{1-ikR}{k^2R^2} \right)
\begin{pmatrix} 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr \end{pmatrix}
\begin{pmatrix} p_x \cr
p_y \cr p_z \cr \end{pmatrix} - \frac{-3+3ikR+k^2R^2}{
k^2R^4} \begin{pmatrix} X^2 & XY & XZ \cr YX & Y^2 & YZ \cr ZX & ZY & Z^2 \cr \end{pmatrix}
\begin{pmatrix} p_x \cr p_y \cr p_z \cr \end{pmatrix} \right)
}$ \end{center}
$$ \eqno (5a) $$
from which we easily see that the matrix to be multiplied with $\vec{p}$
to produce $\vec{E}(\vec{R})$ is
$$
\frac{k^2e^{ikR}}{4\pi\epsilon_0R}\left( {\bf 1} \; \left( 1-\frac{1-ikR}{
k^2R^2}\right) - \vec{R} \otimes \vec{R} \; \frac{-3+3ikR+k^2R^2}{k^2R^4} \right)
$$
$R$ without arrow means the absolut value $\vert \vec{R} \vert = \sqrt{
X^2+Y^2+Z^2}$ and $X$, $Y$, $Z$ stand for $x-x'$, $y-y'$ and $z-z'$,
respectively. $\vec{R}\otimes \vec{R}$ is the matrix written with $X$, $Y$
and $Z$ from (5a). The terms from (5) vectorially oriented along $\vec{p}$
cause the diagonal matrix contribution, those stemming from terms
with $\vec{R}(\vec{R}\vec{p})$ the full matrix in (5a). Compared to the
above expression $g(\vec{R})$ from (4) has a minus sign and misses a factor
$k^2/\varepsilon_0$. As will be discussed later, the source to put into
equation (1) corresponding to an oscillating dipole is not the dipole moment
$\vec{p}$ itself, but $-\mu_0\omega^2$ times $\vec{p}$. And because
$-\frac{k^2}{\varepsilon_0}\frac{1}{\mu_0\omega^2}=-1$ we have
\begin{equation} g(\vec{R})=
-\frac{e^{ikR}}{4\pi R}\left( {\bf 1} \; \left( 1-\frac{1-ikR}{
k^2R^2}\right) - \vec{R} \otimes \vec{R} \; \frac{-3+3ikR+k^2R^2}{k^2R^4} \right)
\end{equation}
The formula (6) fails for $\vec{r}=\vec{r'}$ or $\vec{R}=0$. $g(\vec{R})$
including the case $\vec{R}=0$ can be represented using the principal volume
method \cite{Chew}. In practice, working with finite elements, the value to put for
$g$ of its two arguments the same place can be derived from the polarization
of a dielectric body. The discussion of $g(\vec{r},\vec{r})$ is postponed
to the next section.
\subsection{Solution for the field}
The starting point to find a solution for the electric field with the objects
present is eq. (6) of section I, which rewritten in the variables of our
problem here reads
\begin{equation}
\vec{E}(\vec{r})=\vec{E}_b(\vec{r})+\int d^3\vec{r'} \; g(\vec{r},\vec{r'}) \;
V(\vec{r'}) \;\vec{E}(\vec{r'})
\end{equation}
where $\vec{E}_b(\vec{r})$ is a solution of $-\vec{\nabla} \times \vec{\nabla}
\times \vec{E}_b(\vec{r}) +\frac{\omega^2}{c^2}\vec{E}_b(\vec{r})=0$
or already assumed to be the space part of a linearly polarized plane wave
$\vec{E}_b(\vec{r})=\vec{E}_0e^{i\vec{k}\vec{r}}$ with a fixed amplitude
vector $\vec{E}_0$. The time dependence $e^{-i\omega t}$ can be omitted in
$\vec{E}_b(\vec{r})$ as well as in $\vec{E}(\vec{r})$.
From (7) a numerical procedure can be deduced if the objects
with $\varepsilon(\vec{r})\ne \varepsilon_0$ only occupy fractions of
space rather small on the scale of the wavelength, not at all principally
necessarily
much smaller than $\lambda$, though. Then we divide them up into finite
elements (Fig.1) of volume $\Delta v$, which we enumerate $i=1,\ldots,N$ and
associate permittivities $\varepsilon(\vec{r}_i)$ or perturbances
$V(\vec{r}_i)=-\frac{\omega^2}{c^2}(\varepsilon(\vec{r}_i)-1)$ and local
fields $\vec{E}(\vec{r}_i)$ uniform over $\Delta v$. The linear sizes of
the object cells should not exceed about $\lambda/10$. They need not at all
be placed on a regular grid. And the only reason for demanding small enough
objects is not to get too many elements $N$.
The integrand in (7) only exists
at places where $V$ does not vanish, and to evaluate the field
$\vec{E}(\vec{r})$ also at such places $\vec{r} \in \{ \vec{r}_i \}$, no
values outside the objects appear in the equation. Changing to finite
elements we thus get a linear system of equations for the fields in the object
cells
\begin{equation}
\vec{E}(\vec{r}_i)=\vec{E}_b(\vec{r}_i)+\sum_{j=1}^N \Delta v\; g(\vec{r}_i,
\vec{r}_j) \; V(\vec{r}_j)\; \vec{E}(\vec{r}_j)
\end{equation}
which can be solved by a matrix inversion. To evaluate the resulting field
at any other place, that is outside the objects, the $\vec{E}(\vec{r}_i)$ just have
to be inserted into the finite-element version of (7):
\begin{equation}
\vec{E}(\vec{r})=\vec{E}_b(\vec{r})+\sum_{j=1}^N \Delta v\; g(\vec{r},
\vec{r}_j) \; V(\vec{r}_j)\; \vec{E}(\vec{r}_j) \qquad
\vec{r} \not\in \{ \vec{r}_i\}
\end{equation}
Of course, we already needed $g(\vec{r}_i,\vec{r}_j)$ with
$\vec{r}_i=\vec{r}_j$ to set up the system (8). Let us suppose that there
is only a single cell with an $\varepsilon$ differing from the background
$\varepsilon_0$. This is placed into a homogeneous field $\vec{E}_b$.
If the cell has the shape of a sphere, the local field throughout its inside
is aligned in the direction of $\vec{E}_b$ and its value is
$\vec{E}_{loc}=\frac{3}{2+\varepsilon}\vec{E}_b$ \cite{Kop,Jac_3}. No retardation
effects have to be considered here as the size of the cell can in principle
be made arbitrarily small. In (8) only keeping the term of the sum with
$i=j$, setting $\vec{E}(\vec{r}_i)=\vec{E}_{loc}$ and
$V(\vec{r}_i)=-k^2(\varepsilon-1)$ gives
$$\frac{3}{2+\varepsilon}\; \vec{E}_b=\vec{E}_b-\Delta v\;
g(\vec{r}_i,\vec{r}_i) \; k^2(\varepsilon-1) \; \frac{3}{2+\varepsilon}\;
\vec{E}_b$$
from which follows that
\begin{equation}
g(\vec{r}_i,\vec{r}_i)=\frac{1}{3k^2\Delta v}\; {\bf 1}
\end{equation}
The factor 1/3 is also valid for cubic elementary cells, however,
other shapes require different depolarization factors \cite{Kop,Chew}.
The background and the resulting field at each point are already 3-vectors.
Nevertheless, in order to solve (8), imagine the $\vec{E}$s for the object
cells assembled into long or ''double'' vectors of $N$ times 3 components
$$ \vec{\vec{\rm E}} \quad {\rm with} \; (\vec{\vec{\rm E}})_i=\vec{E}(\vec{r}_i)
\qquad {\rm and} \qquad \vec{\vec{\rm E}}_b \quad {\rm with} \quad
(\vec{\vec{\rm E}}_{b})_{i}=\vec{E}_b(\vec{r}_i).$$
With further the big $3N\times 3N$ matrix ${\rm M}$ consisting of 3x3 blocks
$$ M_{ij}={\bf 1}\; \delta_{ij} - \Delta v \; g(\vec{r}_i,\vec{r}_j) \;
V(\vec{r}_j)$$
(8) then reads
\renewcommand{\theequation}{\arabic{equation}a}
\setcounter{equation}{7} \begin{equation}
{\rm M} \vec{\vec{\rm E}} = \vec{\vec{\rm E}}_b \quad {\rm or} \quad
\vec{\vec{\rm E}}={\rm M}^{-1} \vec{\vec{\rm E}}_b \end{equation}
\renewcommand{\theequation}{\arabic{equation}}
\setcounter{equation}{10}
${\rm M}$ can be inverted using the procedure described in the appendix,
however, as ${\rm M}$ is a full matrix and the complete invers ${\rm M}^{-1}$
is needed, that is of no advantage and a standard inversion algorithm will
do as well.
For the problems of a few small scatterers here by introducing finite elements
the implicit integral equation (7) for the electric field has been turned into
a linear system of equations that is easily solvable. Even shying this effort,
the coarsest, so-called Born approximation consists in replacing $\vec{E}$ in
the integral in (7) or in the sums in (8) or (9) by $\vec{E}_b$. Keeping the
summation over finite elements to estimate the integral we directly get
\begin{equation}
\vec{E}(\vec{r})\approx \vec{E}_b(\vec{r})+\sum_{j=1}^N \Delta v \; g(\vec{r}_i,
\vec{r}_j) \; V(\vec{r}_j) \; \vec{E}_b(\vec{r}_j)
\end{equation}
No distinction between places $\vec{r}$ in or outside the object cells is
necessary in (11). The Born approximation only takes into account first-order
scattering off every object and thus can only be good for weak scatterers
with distances between them rather large on the scale of the wavelength.
Producing a clearly different field pattern from the exact solution including
all scattering orders, Fig.2 demonstrates that the Born approximation is
likely to be insufficient to model near-field optics setups.
\begin{figure}
\includegraphics[width=5.5cm,angle=270]{elek2a.jpg}
\includegraphics[width=6.2cm,angle=270]{elek2b.jpg}
\includegraphics[width=6.2cm,angle=270]{elek2c.jpg}
\caption{Three cubes of 25nm side length and $\varepsilon=-16.0+1.0i$ in
a background of $\varepsilon=1$ illuminated by a p-polarized
plane wave of unit amplitude
with $\lambda=500{\rm nm}$ from the front and under an angle of $30^{\sf o}$
from beneath the plane
defined by the cubes. Setup (a) and electric field intensity $\vert \vec{E}
\vert^2$ calculated via the Born approximation (b) and the exact Green's
functions method (c). Besides subtle differences in the near-field pattern
remark that in (b) the grey-scale is from 0.978 to 1.022 whereas in (c) it is
from 0.995 to 1.005. The intensity map is taken at a height 25nm above the
cubes and given for an area of $3\mu {\rm m} \times 3 \mu {\rm m}$.
}\end{figure}
\subsection{System Green's function and density of states}
Adding an arbitrary source term to our original wave equation (1) changes it into
\begin{equation}
-\vec{\nabla}\times\vec{\nabla}\times \vec{E}(\vec{r})+\frac{\omega^2}{c^2}
\varepsilon(\vec{r})\vec{E}(\vec{r})=\vec{Q}(\vec{r}).
\end{equation}
If now we know a tensor function $G(\vec{r},\vec{r'})$ satisfying
\begin{eqnarray}
-\vec{\nabla}_{\vec{r}}\times\vec{\nabla}_{\vec{r}}\times G(\vec{r},\vec{r'})
+\frac{\omega^2}{c^2}\varepsilon(\vec{r})G(\vec{r},\vec{r'}) = {\bf 1}\;
\delta(\vec{r}-\vec{r'}) \quad {\rm or} \\
-\vec{\nabla}_{\vec{r}}\times\vec{\nabla}_{\vec{r}}\times G(\vec{r},\vec{r'})
+k^2 G(\vec{r},\vec{r'}) -V(\vec{r}) G(\vec{r},\vec{r'})
= {\bf 1}\; \delta(\vec{r}-\vec{r'}) \nonumber
\end{eqnarray}
then obviously
\begin{equation}
\vec{E}(\vec{r})=\int d^3\vec{r'} \; G(\vec{r},\vec{r'}) \; \vec{Q}(\vec{r'})
\end{equation}
would give a special solution of (12). Any solution of (1) could be added.
As generally deduced in section I the implicit relation to get $G$ from is the
Dyson equation
\begin{equation}
G(\vec{r},\vec{r'})=g(\vec{r},\vec{r'})+\int d^3\vec{r''} \; g(\vec{r},
\vec{r''}) \; V(\vec{r''}) \; G(\vec{r''},\vec{r'})
\end{equation}
$G(\vec{r},\vec{r'})$ for both arguments $\vec{r}$ and $\vec{r'}$ covering all
space is too much information to display at once and usually much more than
what one is interested in.
The imaginary part of $G(\vec{r},\vec{r})$ is proportional to the density of
states $\rho$ \cite{coral,AD}. The deduction of this statement found in quantum mechanics
book \cite{Calla}, however, rather argues with a system of energy eigenstates and the
variation of the Green's function as well as the density of states with
energy. No real $\vec{r}$-space is explicitly mentioned. Our interest lies in
the spatial dependence of the density of states at fixed light frequency
$\omega$.
Even if described in terms of fields, concepts like reactance and work known
from electrical circuits may be applied \cite{Jac_4,circtip}. The time average of the
work done by the fields is given by
\begin{equation} {\rm Re} \; \frac{1}{2} \int d^3\vec{r'} \;
\vec{J}^*(\vec{r'}) \; \vec{E}(\vec{r'}) \end{equation}
With no other imposed fields, charges or currents than an oscillating point
dipole, the latter will present the only external current $\vec{J}$, which
will thus be located as $\delta(\vec{r'}-\vec{r})$. If the dipole moment
oscillates as $\vec{p}(\vec{r'},t)=\vec{p}_0e^{-i\omega t}
\delta(\vec{r'}-\vec{r})$, the corresponding current is $\vec{J}(\vec{r'},t)
=-i\omega \vec{p}_0e^{-i\omega t} \delta(\vec{r'}-\vec{r})$. Deducing the
wave equation for time harmonic fields (in vacuum for simplicity here) from
Maxwells equations with current term
\begin{equation}
\vec{\nabla}\times \vec{E}-i\omega \mu_0\vec{H}=0 \quad {\rm and} \quad
\vec{\nabla}\times \mu_0 \vec{H}+\varepsilon_0\mu_0 i\omega \vec{E}=\mu_0
\vec{J} \nonumber \end{equation}
leads to \begin{equation}
-\vec{\nabla}\times\vec{\nabla}\times \vec{E}+\frac{\omega^2}{c^2}\vec{E}=
-i\omega\mu_0\vec{J} \end{equation}
from which we see that the source term for the dipole has to be set as
$\vec{Q}(\vec{r'})=-i\omega \mu_0 \vec{J}(\vec{r'})=-\omega^2\mu_0\vec{p}_0
e^{-i\omega t} \delta(\vec{r'}-\vec{r})$. The integral (16) reduces to the
value of $\vec{J}^*\vec{E}$ at $\vec{r}$. The electric field we get from (14):
\begin{equation}
\vec{E}(\vec{r})=-\omega^2\mu_0 e^{-i\omega t} \int d^3\vec{r'} \;
G(\vec{r},\vec{r'}) \; \delta(\vec{r'}-\vec{r}) \; \vec{p}_0 = -\omega^2
\mu_0 e^{-i\omega t} G(\vec{r},\vec{r}) \vec{p}_0
\nonumber\end{equation}
Inserting $\vec{J}^*$ and $\vec{E}$ into (16) the time factors cancel as
expected for a time average and but for a factor $\omega^3\mu_0/2$ we get
\begin{equation}
{\rm Re} \; i \; \vec{p}_0 \; G(\vec{r},\vec{r}) \; \vec{p}_0 =
-\; {\rm Im} \; \vec{p}_0 \; G(\vec{r},\vec{r}) \; \vec{p}_0
\end{equation}
Choosing unit vectors along the coordinate axis for the probe dipole $\vec{p}_0$,
(20) will filter out the trace elements of the matrix $G(\vec{r},\vec{r})$.
We associate $\rho_x(\vec{r})\propto -{\rm Im}G_{xx}(\vec{r},\vec{r})$,
$\rho_y(\vec{r})\propto -{\rm Im}G_{yy}(\vec{r},\vec{r})$,
$\rho_z(\vec{r})\propto -{\rm Im}G_{zz}(\vec{r},\vec{r})$ and a total
$\rho(\vec{r})=\rho_x(\vec{r})+\rho_y(\vec{r})+\rho_z(\vec{r})$.
A motivation for taking the negative imaginary part of $G(\vec{r},\vec{r})$
as a measure for the presence of modes can also be obtained by comparison to
the energy resonance of a forced oscillator \cite{Alonso}. For optimal excitation
from the energy point of view - in contrast to amplitude resonance - the
force has to be $\pi/2$ ahead of the elongation or in phase with the
velocity of the oscillator. $G(\vec{r},\vec{r})$ describes the field caused
by backaction of the system at the place of the probe dipole moment (taken as
reference phase zero), and therefore $-{\rm Im}G(\vec{r},\vec{r})$ is the part
that can in a resonant manner further enhance the dipole oscillation.
(In reality radiation out of the system will provide strong damping.)
We now intend to evaluate a map of $G(\vec{r},\vec{r})$ on, for example, a
horizontal plane. The plane may lie above or below object cells or even cut
some. Like the objects, the, of course, finite area of interest on the plane
is divided into cells. Just depending on the desired resolution of the map the
unit cell length of this mesh may well differ from the cell size chosen to
discretize the objects (Fig.1). The list of object cell midpoints $\{\vec{r}_i\}$
from the last section, which shall be called region $A$, is extended by all
cell midpoints from the map in the plane, which shall be called region $B$
and is now understood to to be included in counting $i$ from 1 to a new $N$.
Analogously to (8) the integral in (15) is replaced by a sum:
\begin{equation}
G(\vec{r}_i,\vec{r}_k)=g(\vec{r}_i,\vec{r}_k)+\sum_{j=1}^N\Delta v\;
g(\vec{r}_i,\vec{r}_j) \; V(\vec{r}_j) \; G(\vec{r}_j,\vec{r}_k)
\end{equation}
It does not matter that the map $B$ has a different mesh from $\Delta v$ as
for the objects $A$, as $V(\vec{r}_j)=0$ for $\vec{r}_j$ in $B$, anyway. (Spatial
overlap of cells from $A$ and $B$ and even coincidence of midpoints is no
problem; a place can be counted with $V(\vec{r}_j)$ in $A$ and without in $B$.)
Although only $G(\vec{r}_i,\vec{r}_i)$ with $\vec{r}_i$ in region $B$ is
wanted as a final result, (21) has to be set up as an equation for a matrix of
all $G$ with each of its arguments any cell in $A$ or $B$, schematically
sketched as
$\begin{pmatrix}\vbox{\halign{\strut # & \vrule \ # \cr
AA & AB \cr \noalign{\hrule} BA & BB \cr }}\end{pmatrix}$.
To solve (21) for $G$ we have to invert the same kind of matrix as in (8), the
only difference being that $j$ now also runs over the plane cells in addition
to the object cells.
\renewcommand{\theequation}{\arabic{equation}a}
\setcounter{equation}{20}
\begin{equation} G={\rm M}^{-1}\; g \end{equation}
\renewcommand{\theequation}{\arabic{equation}}
\setcounter{equation}{21}
$G$ and $g$ are themselves matrices on contrast to vectors $\vec{\vec{E}}$
and $\vec{\vec{E}}_b$. $G$ consists
of 3x3-blocks $G(\vec{r}_j,\vec{r}_k)$, $g$ is made of blocks $g(\vec{r}_i,\vec{r}_k)$
and the 3x3-block at position $(i,j)$ in ${\rm M}$ given by ${\bf 1}\delta_{ij}-
\Delta v\; g(\vec{r}_i,\vec{r}_j) \; V(\vec{r}_j)$. The $g$ in (21a) and ${\rm M}$ like
the big $G$-matrix have the structure
$\begin{pmatrix}\vbox{\halign{\strut # & \vrule \ # \cr
AA & AB \cr \noalign{\hrule} BA & BB \cr }}\end{pmatrix}$.
One could invert ${\rm M}$ as given, for example by the procedure from
appendix A. The matrix $M={\bf 1}-{\rm M}$ there is initialized with $gV$.
Its $AB$ and $BB$ quadrants are zero and will stay zero throughout the procedure,
$M=\begin{pmatrix}\vbox{\halign{\strut # & \vrule \ # \cr
AA & 0 \cr \noalign{\hrule} BA & 0 \cr }}\end{pmatrix}$. This is no contradiction,
as it is not ${\rm M}$ that is singular. Quadrant $BA$ will be needed for
multiplication with $g$ in (21a).
However there is an even more efficient algorithm to get $G$ that
already includes the multiplication by $g$. It directly calculates
$G=({\bf 1}-gV)^{-1}g$, which is the compact way to write (21a) as the
solution of (15), also denoted $G=g+gVG$ for short. The technical details
can be found in appendix B.
\begin{figure}
\includegraphics[width=6.8cm,angle=270]{stadstrukt.jpg}
\vspace{-1.2cm}
\includegraphics[width=7.1cm,angle=270]{rhox.jpg}
\vspace{-1.2cm}
\includegraphics[width=7.1cm,angle=270]{rhoy.jpg}
\vspace{-1.2cm}
\includegraphics[width=7.1cm,angle=270]{rhoz.jpg}
\end{figure}
\setcounter{figure}{2}
\begin{figure}
\caption{(a) Top view of the structure: gold pads (90x90x30nm)
on a glass subtrate. Calculated densities of states $\rho_{xx}$ (b)
$\rho_{yy}$ (c) and $\rho_{zz}$ (d) in a plane 100nm above the substrate
for wavelength $\lambda=543{\rm nm}$ [reproduced after \cite{Chicel}].
The grey scale is from -50 to 35 in (b), -45 to 30 in (c), -7.5 to 15 in (d)
and higher or lower values are white or black, respectively.
}\end{figure}
Trace components of $G(\vec{r},\vec{r})$ meaning densities of states for the
three polarization directions (Fig.3)
above an optical coral \cite{coral} in analogy to a quantum coral \cite{Eigler}
have been measured \cite{Chicel} in a so called forbidden-light near-field
optical microscope \cite{forbidden}. The sample consists of a stadium arrangement of gold
particles on a glass surface. The forbidden-light setup prevents detecting
light emitted from the fiber tip that has not passed through surface modes
that make up the density of states for this system.
Like for antinodal and nodal points in a resonator, more energy can go into
the system when the excitation is placed at a point of high density of states
than when coupling is bad where the density is low.
\subsection{Remarks on the source terms and alternative solutions}
In section 4 we saw that it is convenient to start from a solution for the
field in the form (7) if the excitation comes, for example, from a background
field belonging to a plane wave. Though the matrix to invert bore a certain
similarity to the evaluation of the Green's tensor in section 5, with (8) and (9) we
directly calculated the field. In contrast, more adapted to localized sources,
there is (14) as a solution of (12). If there is no additional background field
to cause any excitation, no solution of the equation (1) with zero right
side is to be
added as further contribution and (14) is the field distribution to be
observed. (14) has to be rewritten in terms of finite elements in order to be
used in a numerical calculation. In the same way as the objects the source
$\vec{Q}$ has to be devided into discrete cells or elementary dipoles. To
distinguish their $L$ locations from those of the objects we shall enumerate
them as $\vec{\rho}_{\alpha}$, $\alpha=1,2,\ldots,L$. The place $\vec{r}$ to
evaluate the field $\vec{E}$ may be anywhere outside or inside the objects as
well as beside or even at a source location. For the following development
the Dyson equation for the Green's tensor is needed in a discretized form for
both its variants $G=g+gVG$ and $G=g+GVg$.
\setcounter{equation}{0}
\renewcommand{\theequation}{22\alph{equation}}
\begin{eqnarray}
\vec{E}(\vec{r})&=& \sum_{\alpha=1}^L G(\vec{r},\vec{\rho}_{\alpha})
\vec{Q}(\vec{\rho}_{\alpha}) \\
&=& \sum_{\alpha=1}^L g(\vec{r},\vec{\rho}_{\alpha}) \vec{Q}(\vec{\rho}_{
\alpha}) +\sum_{\alpha=1}^L \sum_{j=1}^N G(\vec{r},\vec{r}_j) V(\vec{r}_j)
g(\vec{r}_j,\vec{\rho}_{\alpha}) \vec{Q}(\vec{\rho}_{\alpha}) \\
&=& \sum_{\alpha=1}^L g(\vec{r},\vec{\rho}_{\alpha}) \vec{Q}(\vec{\rho}_{
\alpha}) + \sum_{\alpha=1}^L \sum_{j=1}^N g(\vec{r},\vec{r}_j) V(\vec{r}_j)
g(\vec{r}_j,\vec{\rho}_{\alpha}) \vec{Q}(\vec{\rho}_{\alpha}) + \nonumber \\
&& \sum_{\alpha=1}^L \sum_{j=1}^N \sum_{i=1}^N g(\vec{r},\vec{r}_i) V(\vec{r}_i)
G(\vec{r}_i,\vec{r}_j) V(\vec{r}_j) g(\vec{r}_j,\vec{\rho}_{\alpha})
\vec{Q}(\vec{\rho}_{\alpha}) \\
&=& \sum_{\alpha=1}^L g(\vec{r},\vec{\rho}_{\alpha})
\vec{Q}(\vec{\rho}_{\alpha}) +\sum_{\alpha=1}^L
\sum_{j=1}^N g(\vec{r},\vec{r}_j) V(\vec{r}_j) G(\vec{r}_j,\vec{\rho}_{\alpha})
\vec{Q}(\vec{\rho}_{\alpha}) \nonumber \\
&=& \sum_{\alpha=1}^L g(\vec{r},\vec{\rho}_{\alpha})\vec{Q}(\vec{\rho}_{\alpha})
+\sum_{j=1}^N g(\vec{r},\vec{r}_j) V(\vec{r}_j) \vec{E}(\vec{r}_j)
\end{eqnarray}
\setcounter{equation}{22}
\renewcommand{\theequation}{\arabic{equation}}
Having in mind a region where and a resolution with which $\vec{E}(\vec{r})$
is to be evaluated like the discretized plane $B$ from the last section, it
would be possible to supply $G$ for all needed combinations of arguments
$(\vec{r},\vec{\rho}_{\alpha})$ and calculate $\vec{E}(\vec{r})$ as the
single sum from (22a). To weave in the influence of the objects, $G$ would have
to be set up as a big matrix like in the last section over all combinations
of three regions $A$, $S$ and $B$ here, the objects, the source and the map.
Having calculated $G$ in the $A$-$B$-scheme from the last section, one
could evaluate
$\vec{E}(\vec{r})$ with $G(\vec{r},\vec{r}_j)$ as written in (22b). However,
the most efficient way is given in (22c). $G(\vec{r}_i,\vec{r}_j)$ is merely
needed for $\vec{r}_i$ and $\vec{r}_j$ from the set of object cells, keeping
a matrix to be inverted as small as possible, namely of $(AA)$-type. Choosing
$\vec{r}_i$ and $\vec{r}_k$ in (21) in the object set $A$, instead of in $A$ or
$B$ as the equation was originally set up for, we see that (21) presents a
closed system of equations for all such $G(\vec{r}_i,\vec{r}_k)$.
Then for (22c) more
summations over products with $g$-functions, which are analytically known for
any pair of arguments, can be considered less demanding in computing time than
the inversion of large matrices.
In the transformation from (22c) to (22d) after swapping index names $i$ and
$j$ in the last sum, $G=g+GVg$ and (14) have been exploited.
Using (22d) for $\vec{E}(\vec{r})$
renders an implicit equation for the field in the form (10) or (10*) from
section I.
$\vec{E}(\vec{r})$ is the equivalent of $\phi_{ihq}$ and as stated earlier,
we assume that physically there is no background field $\phi_0$ that could
initiate an additional field distribution $\phi_{ih}$.
Although not very convenient, a plane wave as exciting field could be
understood as stemming from a sufficiently long and dense array of Huygens
elementary dipole sources $\vec{Q}$ reasonably far away from the objects.
The other way round,
for a single dipole source or a number of dipole sources distributed in space
the field they would produce at any location $\vec{r}$ in homogeneous space
is the superposition of their individual fields, and putting
$\vec{E}_b(\vec{r})=\sum_{\alpha=1}^L g(\vec{r},\vec{\rho}_{\alpha})
\vec{Q}(\vec{\rho}_{\alpha})$ the ansatz (7) can be used also for this case.
Like the field $\vec{E}$ anywhere was obtained as a straight-forward summation
once having its values at the places of the object cells, finally an alternative
way to the procedure from the last section to get the Green's tensor $G$ shall
be given, also requiring only the inversion of a matrix with size the number
of object cells. Series expansion is used to rewrite the solution of the
Dyson equation:
\setcounter{equation}{0}
\renewcommand{\theequation}{23\alph{equation}}
\begin{eqnarray}
G&=& ({\bf 1}-gV)^{-1}g = [{\bf 1}+\sum_{n=1}^{\infty} (gV)^n]g
= g+gV(\sum_{n=0}^{\infty} (gV)^n)g \nonumber \\
&=& g+gV({\bf 1}-gV)^{-1}g \\
&=& g+gVg+gV(\sum_{n=0}^{\infty} (gV)^n)gVg =
g+gVg+gV\; ({\bf 1}-gV)^{-1}g\; Vg \nonumber \\
&=& g+gVg+gVGVg
\end{eqnarray}
Designating regions the spatial arguments belong to on (23a) we get for
$G_{BB}$: \setcounter{equation}{0}
\renewcommand{\theequation}{23\alph{equation}'}
\begin{equation}
G_{BB}=g_{BB}+g_{BA}\; V_A\; ({\bf 1}-gV)^{-1}_{AA}\; g_{AB},
\end{equation}
As $V$ does not vanish only in region $A$, the first index of
$({\bf 1}-gV)^{-1}$ obviously must be $A$. $({\bf 1}-gV)^{-1}$ being
$\sum_{n=0}^{\infty} (gV)^n$, for power zero the second region index automatically
is the same as the first and all other powers ending with $V$ imply second
index $A$.
It is sufficient to set up the matrix ${\bf 1}-gV$ as an $(AA)$-block
and invert that. Should one prefer to evaluate a complete $G$, which
differs from the above inverted matrix by a factor $g$, line
(23b) like (22c) shows that it is in principle only necessary to get $G$ from
some self-consistent implicit equation in the object region $A$.
$G_{AA}$ can be constructed applying the procedure described in appendix B to
a matrix set up as $(AA)$-block only. The inversion has to be completed in this
case, though. Going through the diagonal elements, all lines and columns
have to be updated in each step, including the ones above and to the left of as well as
the ones the respective diagonal element is in. Writing (23b)
\setcounter{equation}{1}
\renewcommand{\theequation}{23\alph{equation}'}
\begin{equation}
G_{BB}=g_{BB}+g_{BA}\; V_A\; g_{AB}+g_{BA}\, V_A\, G_{AA}\; V_A\; g_{AB}
\end{equation}
as summation over discrete elements ready for use in a calculation then reads:
\setcounter{equation}{1}
\renewcommand{\theequation}{23\alph{equation}''}
\begin{eqnarray}
G(\vec{r},\vec{r'})=g(\vec{r},\vec{r'})&+&\sum_{j=1}^N g(\vec{r},\vec{r}_j)
\; V(\vec{r}_j) \; g(\vec{r}_j,\vec{r'}) \nonumber \\
&+& \sum_{j=1}^N \sum_{i=1}^N g(\vec{r},\vec{r}_j) \; V(\vec{r}_j) \;
G(\vec{r}_j,\vec{r}_i) \; V(\vec{r}_i) \; g(\vec{r}_i,\vec{r'})
\end{eqnarray}
Of course, summations run over all object cells here. No numerical advantage
can be drawn out of $\vec{r'}=\vec{r}$ in (23b''). With the same effort of making
$G_{AA}$ it can be used to evaluate maps of $G(\vec{r},\vec{r})$ as well as
plots of $G(\vec{r},\vec{r'})$ with $\vec{r'}$ fixed or even some function of
$\vec{r}$.
\setcounter{equation}{23}
\renewcommand{\theequation}{\arabic{equation}}
\subsection{Conclusions and Outlook for section II}
We have presented a method to solve the problem of scattering of electromagnetic
waves off an arbitrary distribution of dielectric objects, that is the exact
evaluation of the field, especially in the near zone where higher-order
multiple reflections can become important. Besides the field distribution
we have obtained the Green's tensor characterizing the system independently
from the form of the excitation. It represents the response function and
also the density of states for supported electric fields.
For a methodical introduction we have restricted our considerations to
dielectric materials and the electric field. Without magnetic susceptibilities
the magnetic field distribution can be calculated once having the electric
field inside the objets by a formula like (9) with the magnetic background
field and replacing $g$ by a tensor including the conversion from the electric
to the magnetic field by taking the rotation \cite{Girard}. It is further possible to treat
non-uniform magnetic permeabilities and even mixed systems with dielectric
and magnetic objects \cite{EPJel}. The electric Green's tensor presented above is then
paired by a magnetic counterpart and genuine mixed response functions also
exist. Whereas the calculation of the field distributions even for mixed
systems is quite straight forward, the construction of the Green's tensor
is more involved. It lives of the idea of handling one kind of objects first
and then considering this setup as the background to include the other kind.
There is no approximation or ranking in importance in this procedure.
The discussion here has only considered finite objects in a homogeneous
background as well as cartesian coordinates where vector and tensor components
have been written out. Cylindrical and spherical coordinates are also commonly
used \cite{Chew,ellipse}
and the Green's functins formalism has been developped for layered media
\cite{MarLith,Chew,stratified}.
Besides wave-guide applications the use for modelling typical near-field
optics experiments, where the microstructures to investigate are prepared on
a substrate surface, lies in putting the influence of this surface into a
background Green's tensor \cite{Girard,AD} which is then implied the way we used $g$ here.
Details of applications of the Green's functions technique in electrodynamics
to more complicated situations as well as beautiful results of corresponding
experiments can be found in the given references. This text focussed on
calculation techniques and further intended to give an overview of slightly
different formal ways to calculate Green's tensors and fields of which either
may be optimal for a specific problem.
\subsection*{Appendix A: an unusual matrix inversion}
Suppose a complex quadratical matrix to invert is already given in the form
${\rm M} ={\bf 1}-M={\bf 1}-(m)_{ij}$ or if it is not, we rewrite it like that.
There is no restriction on the values of the numbers $m_{ij}$. To get the
inverted matrix proceed as follows:
Of matrix $M$ one by one take the diagonal elements $m_{ii}$ and to all elements
$m_{ab}$ add $m_{ai}(1-m_{ii})^{-1}m_{ib}$. After having worked through the
matrix for one such $m_{ii}$, the changed matrix values have to be taken to do
so for the next, also already changed, diagonal element. Obviously $N$ such
steps are required for an $N\times N$-matrix. This will yield $({\bf 1}-M)^{-1
}-{\bf 1}$, such that in the end 1 has to be added to all diagonal elements in
order to obtain $({\bf 1}-M)^{-1}$. For clearness we write out the first two
transformation steps of the matrix:
\begin{eqnarray}
\begin{pmatrix} m_{11} & m_{22} & \cr m_{21} & m_{22} & \cr & & \ddots \cr
\end{pmatrix} \rightarrow \begin{pmatrix}
m_{11}+m_{11}(1-m_{11})^{-1}m_{11} & m_{12}+m_{11}(1-m_{11})^{-1}m_{12} \cr
m_{21}+m_{21}(1-m_{11})^{-1}m_{11} & m_{22}+m_{21}(1-m_{11})^{-1}m_{12} \cr
\end{pmatrix} \nonumber \\
\rightarrow \begin{pmatrix}
m_{11}+m_{11}(1-m_{11})^{-1}m_{11} + & m_{12}+m_{11}(1-m_{11})^{-1}m_{12} +\cr
(m_{12}+m_{11}(1-m_{11})^{-1}m_{12})\cdot & (m_{12}+m_{11}(1-m_{11})^{-1}m_{12})\cdot \cr
(1-m_{22}-m_{21}(1-m_{11})^{-1}m_{12})^{-1} & (1-m_{22}-m_{21}(1-m_{11})^{-1}m_{12})^{-1} \cr
\cdot (m_{21}+m_{21}(1-m_{11})^{-1}m_{11}) & \cdot (m_{22}+m_{21}(1-m_{11})^{-1}m_{12}) \cr
& \cr
m_{21}+m_{21}(1-m_{11})^{-1}m_{11} + & m_{22}+m_{21}(1-m_{11})^{-1}m_{12} +\cr
(m_{22}+m_{21}(1-m_{11})^{-1}m_{12})\cdot & (m_{22}+m_{21}(1-m_{11})^{-1}m_{12})\cdot \cr
(1-m_{22}-m_{21}(1-m_{11})^{-1}m_{12})^{-1} & (1-m_{22}-m_{21}(1-m_{11})^{-1}m_{12})^{-1} \cr
\cdot (m_{21}+m_{21}(1-m_{11})^{-1}m_{11}) & \cdot (m_{22}+m_{21}(1-m_{11})^{-1}m_{12}) \cr
\end{pmatrix} \rightarrow \ldots \nonumber \\
\end{eqnarray}
The inverted matrix $({\bf 1}-M)^{-1}$ can be represented as a geometric series:
\begin{equation}
({\bf 1}-M)^{-1}={\bf 1}+M+M^2+\ldots
\end{equation}
Truncating and using the sum from the right side is only possible if the series
converges whereas the closed form on the left is valid in any case.
In contrast to the infinite sum on the right side of (25), our inversion procedure
consists in a finite number of steps of adding contributions to the matrix
elements. Nevertheless, (25) tells us that the invers $({\bf 1}-M)^{-1}$ is the
sum of all powers of $M$ and thus each element in row $a$ and column $b$ must be
the sum of all possible products $m_{an_1}m_{n_1n_2}m_{n_2n_3}\ldots m_{n_{j-1
}n_j}m_{n_jb}$ with any number $j$ of inner indices including none. (Diagonal
elements get an extra +1.) There are $N$ different indices $n_i$ and they may
repeat, of course. Considering that $(1-m_{ii})^{-1}$ can also be written as
$$
(1-m_{ii})^{-1}=1+m_{ii}+m_{ii}^2+\ldots =1+m_{ii}+m_{ii}m_{ii}+\ldots
$$
we see that the first step in (24) adds to each matrix element $m_{ab}$ the sum
of all products $m_{a1}m_{11}m_{11}\ldots m_{11}m_{1b}$. In these at
least one pair of indices 1 is squeezed between $a$ and $b$ as in $m_{a1}m_{1b}$,
the contribution $m_{ab}$ was already there. In the second step all products
$m_{a\nu_1}m_{\nu_1\nu_2} \ldots m_{\nu_{\alpha}2}m_{2\nu_{\beta}} \ldots
m_{\nu_{j-1}\nu_j}m_{\nu_jb}$ with
$\nu_1\nu_2\ldots \nu_{j-1}\nu_j$ every possible sequence of 1s and 2s will be
added. The products with only indices 1 between $a$ and $b$ were there before.
In the third step every sequence of indices 1, 2 and 3 with at least one
3-link is added. And so on until in the end at each
matrix position between outer indices $a$ and $b$ we have created all possible
sequences of an endless game of dominos $m_{ij}$ with numbers $i$ and $j$ from
1 to $N$. This argument was to proove that the result of (24) indeed gives
$({\bf 1}-M)^{-1}-{\bf 1}$. We calculate a finite number of $(1-m_{ii})^{-1}$
or products $m_{ai}(1-m_{ii})^{-1}m_{ib}$. The sequence on the right side of (25)
need not converge and the original entries in $M$ need not at all be small
compared to 1 in their absolute values. There is no approximation in the sense
of a perturbation theory. If the matrix ${\bf 1}-M$ is degenerate, the failure
of the inversion will be noticed when a value $1-m_{ii}$ becomes zero at some
step. Not to confuse notation, remark that in (24) and in
products in the text like $m_{a1}m_{11}\ldots m_{1b}$ letters $m$ meant the
original matrix entries whereas in expressions $1-m_{ii}$, $(1-m_{ii})^{-1}$ and
$m_{ai}(1-m_{ii})^{-1}m_{ib}$ we referred to the entries at the respective step
of the matrix transformation.
In the application from the main text $i=1,\ldots ,N$ enumerates the object cells.
At position $(i,j)$ in $({\bf 1}-gV)^{-1}$ when expanded into a series having
every possible sequence
$g(\vec{r}_{n_1},\vec{r}_{n_2})V(\vec{r}_{n_2})g(\vec{r}_{n_2},\vec{r}_{n_3})
V(\vec{r}_{n_3})g \ldots g(\vec{r}_{n_{j-1}},\vec{r}_{n_j})V(\vec{r}_{n_j})$
shows that the resulting field at any place (inside or outside the objects)
is the interference of the background field and the fields reradiated by all
the object dipoles having undergone every possible scattering path between
the objects (Fig.1).
A complication in the electrodynamics application at this stage is the fact that each
matrix element $m$ actually in itself is a 3x3 matrix indicating the effect
of three field components at one place onto three field components at a another place.
$g$ on the discrete space of the objects can be written blockwise with a
scheme of quadruple indices
\begin{center}
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\{ \vec{r}_1,x;\vec{r}_1,z\} \cr &&\cr
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\{ \vec{r}_1,y;\vec{r}_1,y\} &
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$V(\vec{r}_k)$ in the product $g(\vec{r}_i,\vec{r}_k)V(\vec{r}_k)$
just multiplies the respective column.
One could use the inversion procedure working off the diagonal elements
marked by ovals, requiring $3N$ steps then. However, the process is equally
applicable to 3x3 blocks as marked by the dashed rectangles,
since by its dimension the whole matrix can be
divided up into 3x3-blocks. Then $({\bf 1}-m_{ii})^{-1}$ means the inversion of
a 3x3 matrix and $m_{ai}({\bf 1}-m_{ii})^{-1}m_{ib}$ to update the blocks
means the product of three 3x3 matrices. These operations should be
programmed as elementary procedures.
The given procedure to invert a matrix can become of advantage if for sparse
matrices conventional routines run into numerical difficulties because of
many zero values. Besides that, it can be adapted to become quite efficient
if only parts of the inverted matrix are needed or for symmetry reasons it is
known that blocks or patterns of matrix elements vanish and will stay zero
throughout the inversion. Although for the calculation of the Green's tensor
a slightly modified procedure is applied that directly optimizes the numerical
solution of the Dyson equation (see appendix B), the matrix inversion was
discussed here, because it may be used for more general purposes and in other
contexts as well.
\subsection*{Appendix B: Calculating the Green's tensor}
In the following instructions are given how to calculate the Green's tensor
\setcounter{equation}{20}
\begin{equation}
G=({\bf 1}-gV)^{-1}g=g+gVg+gVgVg+\ldots
\end{equation}
\setcounter{equation}{25}
being efficient in the way that finally only $G(\vec{r},\vec{r})$ of equal
arguments on the mesh points of map $B$ need to have the correct values
\cite{priv,EPJel}. For a start nevertheless consider that $G(\vec{r}_i,\vec{r}_k)$
with arguments $\vec{r}_i$ and $\vec{r}_k$ from the big set of all object
and map cell midpoints will have to be the sum of all products
$$
g(\vec{r}_i,\vec{r}_{n_1})V(\vec{r}_{n_1})g(\vec{r}_{n_1},\vec{r}_{n_2})
V(\vec{r}_{n_2})g(\vec{r}_{n_2},\vec{r}_{n_3}) \ldots V(\vec{r}_{n_{j-1}},
\vec{r}_{n_j})g(\vec{r}_{n_j},\vec{r}_k).
$$
$\vec{r}_{n_1}$, $\vec{r}_{n_2}$, $\ldots$, $\vec{r}_{n_j}$ can only be object cells and
any sequence of them has to be created, including the empty one with no $V$
giving the term $g(\vec{r}_i,\vec{r}_k)$. Initiate a matrix
$\begin{pmatrix}
\vbox{\halign{\strut # & \vrule \ # \cr
AA & AB \cr \noalign{\hrule} BA & {\tiny
\vbox{\halign{\strut # & # & # \cr
BB & & \cr & $\ddots$ & \cr & & BB \cr }} }
\cr }} \end{pmatrix}$
- for simplicity call it $G$ from the beginning - with line and column
arguments running over all object cells in $A$ and all map cells in $B$ with
$g(\vec{r}_i,\vec{r}_k)$ in each 3x3-subblock. In the $BB$-quadrant only
diagonal blocks $g(\vec{r}_i,\vec{r}_i)$ will be needed, however.
Work off the diagonal subblocks through the $AA$-quadrant.
For the $n$th one prepare the inverted 3x3-matrix $({\bf 1}-G(\vec{r}_n,
\vec{r}_n)V(\vec{r}_n))^{-1}$ and to all entries from line $n+1$ and column
$n+1$ on add the product given below.
\begin{equation}
G(\vec{r}_i,\vec{r}_k) \rightarrow G(\vec{r}_i,\vec{r}_k) +
G(\vec{r}_i,\vec{r}_n) \; V(\vec{r}_n)
\; ({\bf 1}-G(\vec{r}_n,\vec{r}_n)V(\vec{r}_n))^{-1}
\; G(\vec{r}_n,\vec{r}_k)
\end{equation}
In the following step use the updated entries in the above recipe. In the
$n$th step you need not update entries in line $n$ or above or in column $n$
or to the left of it, because these will not be needed as multiplication
factors $G(\vec{r}_i,\vec{r}_m)$ and $G(\vec{r}_m,\vec{r}_k)$ for $m>n$
any more. The parts of quadrants $AB$ and $BA$ remaining after these
restrictions have to be changed by (26) for every $n$.
Merely diagonal 3x3 blocks have to be done in the $BB$-quadrant.
The first step adds all products consisting of any non-zero number of
factors $V(\vec{r}_1)$ between $g$s, the second step all products eventually
containing $V(\vec{r}_1)$ and one up to any number of $V(\vec{r}_2)$, and
so on. The procedure is finished after $\vec{r}_n$ has run down the
diagonal of the $AA$-quadrant. All sequences of multiple scattering from
the objects are then included.
We could have filled
the whole $BB$-quadrant with initial $g_{BB}$ 3x3-blocks and updated the
entire $BB$-quadrant in each step. Like in the matrix inversion procedure
from appendix A we should further have worked down the complete diagonal and
done (26) for every $\vec{r}_n$ from region $B$ as well. There will, however,
be no additions as $V(\vec{r}_n)=0$ for $\vec{r}_n$ in $B$ (even if a map
cell accidently coincides with an object cell). This argument also reveals
why non-diagonal elements in $BB$ do not have to be evaluated. They can
never appear as multiplication factors $G(\vec{r}_i,\vec{r}_n)$ or
$G(\vec{r}_n,\vec{r}_k)$. The $BB$ diagonal is the resulting $G(\vec{r},
\vec{r})$-map we wanted.
Initiating the matrix by $g$ and introducing the $V$ on treating the
respective diagonal element makes the outcome of the procedure directly
$({\bf 1}-gV)^{-1}g$ compared to initiating the matrix with $gV$ and as an
intermediate step obtaining $({\bf 1}-gV)^{-1}$.
\newpage
\setcounter{equation}{0}
\section{All-order quantum transport}
It is demonstrated how the transport problem for two open free-electron gas
reservoirs with arbitrary coupling can be solved by finding the system's
Green's function. In this sense the article is an introduction on Green's
functions for treating interaction.
A very detailed discussion of the current formula is given on an elementary
basis.
Despite formal resemblances the
stationary transport situation, however, differs in its nature from
introducing coupling between energy levels in a closed system where then
the interest lies in modified eigenvalues and eigenstates.
\subsection{Introduction}
By modern lithography techniques so-called point contacts \cite{Agrait} can be arranged
between conductors. These have sufficiently small dimensions such that
electronic modes get quantized. However, coupling across such constrictions
need not be so weak as to be described by a small tunnel probability, but
can be influenced by coherent interference of multiple reflections. Point
contacts can be obtained by indenting STM-tips into some material \cite{RubioSTM},
electromigration \cite{herre} or the break-junction technique \cite{break}. Whereas for constrictions
imposed by gate electrodes to a two-dimensional electron gas in semiconductors
\cite{vanWees}
one observes quantized conductance values in the sense of fully transmitting
or totally switched-off modes, the application in mind behind this work is
the type of connection like the single-atom contact, characterized by an
ensemble of channels \cite{buett}, which can also have intermediate transmission amplitudes
between zero and one \cite{Sche97}. Even with some fully transmitting modes the contact
bears a resistance in the order of the quantum resistances $R_k=h/e^2=26k\Omega$
\cite{SCT}, such
that viewing the system as a left and a right side with some interaction is
an appropriate picture. Furthermore contributions from several channels just
add in the current. Although a transmitting channel in a point contact is the
application in mind, this article shows how to set up a general procedure
to solve the problem of transport for two open reservoirs with more or less
strong coupling between them, and thus how the Green's functions formalism
from section I is implied in an area of current research
\cite{aktuel_1,aktuel_2}.
\subsection{Green's functions formalism}
As a preparation, consider systems like, for example, the bulk material on the
left or the right side (Fig.1a) without coupling. For these we suppose that we
know the Hamiltonian $H^0$ and the wave function $\Psi^0$ at any given energy
$\hbar\omega$ satisfying the Schr\"odinger equation. Denoting the Hamiltonians
$H_{LL}=H_{RR}=H^0$ (doubling the index makes sense later), the wave functions
$\Psi_L^0=\Psi_R^0=\Psi^0$ and the time derivative as $\partial_{\tau}$, the
Schr\"odinger equations for both sides - for the moment just formally put
into matrix form - are
\begin{equation}
\begin{pmatrix} i\hbar\partial_{\tau}-H_{LL} & 0 \cr 0 & i\hbar\partial_{\tau}-H_{RR} \cr\end{pmatrix}
\begin{pmatrix}\Psi^0_L(\tau) \cr \Psi^0_R(\tau)\cr\end{pmatrix}=\begin{pmatrix}0 \cr 0\cr \end{pmatrix}
\end{equation}
Corresponding to this differential equation we have the Green's functions
equation
\begin{eqnarray}
\begin{pmatrix} i\hbar\partial_{\tau}-H_{LL} & 0 \cr 0 & i\hbar\partial_{\tau}-H_{RR} \cr\end{pmatrix}
\begin{pmatrix} g_{LL}(\tau,\tau') & 0 \cr 0 & g_{RR}(\tau,\tau')\cr \end{pmatrix}=
\begin{pmatrix} \delta(\tau-\tau') & 0 \cr 0 & \delta(\tau-\tau')\cr \end{pmatrix}
\nonumber \\ \end{eqnarray}
If we knew $g=g_{LL}=g_{RR}$, we could immediately also give a solution to (1)
with a source term added
\begin{equation}
\begin{pmatrix} i\hbar\partial_{\tau}-H_{LL} & 0 \cr 0 & i\hbar\partial_{\tau}-H_{RR} \cr \end{pmatrix}
\begin{pmatrix} \Psi^q_L(\tau) \cr \Psi^q_R(\tau)\cr \end{pmatrix}=
\begin{pmatrix} Q_L(\tau) \cr Q_R(\tau)\cr \end{pmatrix}
\end{equation}
namely
\begin{equation}
\begin{pmatrix} \Psi^q_L(\tau) \cr \Psi^q_R(\tau)\cr \end{pmatrix}=
\int d\tau' \; \begin{pmatrix} g_{LL}(\tau,\tau') & 0 \cr 0 & g_{RR}(\tau,\tau')\end{pmatrix} \;
\begin{pmatrix} Q_L(\tau') \cr Q_R(\tau')\cr \end{pmatrix}
\end{equation}
However, we are more interested in the solution when an interaction between
left and right is present, expressed through coupling Hamiltonians $H_{LR}$
and $H_{RL}$,
\begin{equation}
\begin{pmatrix} i\hbar\partial_{\tau}-H_{LL} & -H_{LR} \cr -H_{RL} & i\hbar\partial_{\tau}-H_{RR} \cr\end{pmatrix}
\begin{pmatrix} \Psi_L(\tau) \cr \Psi_R(\tau)\cr\end{pmatrix}=\begin{pmatrix} 0 \cr 0\cr\end{pmatrix}
\end{equation}
and we shall refer to this case by $\Psi$ without upper index. Introducing
the coupling, of course, was the motivation for writing the Schr\"odinger
equation in matrix form over the site space consisting of
L(left) and R(right). Putting the coupling terms on the
right side of the equation, they mimic a source
\begin{equation}
\begin{pmatrix} i\hbar\partial_{\tau}-H_{LL} & 0 \cr 0 & i\hbar\partial_{\tau}-H_{RR} \cr\end{pmatrix}
\begin{pmatrix} \Psi_L(\tau) \cr \Psi_R(\tau)\cr\end{pmatrix}=
\begin{pmatrix} 0 & H_{LR} \cr H_{RL} & 0 \cr\end{pmatrix}
\begin{pmatrix} \Psi_L(\tau) \cr \Psi_R(\tau)\cr\end{pmatrix}
\end{equation}
and following (4) the solution can formally be written as
\begin{eqnarray}
\begin{pmatrix} \Psi_L(\tau) \cr \Psi_R(\tau)\cr\end{pmatrix}=\int d\tau'\;
\begin{pmatrix} g_{LL}(\tau,\tau') & 0 \cr 0 & g_{RR}(\tau,\tau')\end{pmatrix}
\begin{pmatrix} 0 & H_{LR} \cr H_{RL} & 0 \cr\end{pmatrix}
\begin{pmatrix} \Psi_L(\tau') \cr \Psi_R(\tau')\cr\end{pmatrix} \nonumber \\
\end{eqnarray}
obtaining the implicit Lippmann-Schwinger equation for the wave functions.
Although for the coupled system we do not expect the eigenvectors of
$\begin{pmatrix} H_{LL} & H_{LR} \cr H_{RL} & H_{RR} \cr\end{pmatrix}$ to be one with only an
upper component localized on the left and one with only a lower component
localized on the right, it is convenient to denote these solutions as vectors
$\begin{pmatrix} \Psi_L \cr \Psi_R \cr \end{pmatrix}$, keeping indices
L and R. In contrast
to section II, where the Lippmann-Schwinger equation was solved in a discretized
form to obtain field distributions, for our purposes here (7) will merely
serve as a formal step in the derivation of the Green's function. What we are
precisely interested in from a physical point of view is the current that will
flow between the left and the right side, and not necessarily explicitly
evaluating the wave functions.
The Green's function $G$ of the coupled system is inferred from an equation
analogous to (5), namely
\begin{eqnarray}
\begin{pmatrix} i\hbar\partial_{\tau}-H_{LL} & -H_{LR} \cr -H_{RL} & i\hbar
\partial_{\tau}-H_{RR} \cr\end{pmatrix} \begin{pmatrix} G_{LL}(\tau,\tau') & G_{LR}(\tau,\tau') \cr
G_{RL}(\tau,\tau') & G_{RR}(\tau,\tau') \cr\end{pmatrix} = \begin{pmatrix} \delta(\tau-\tau') &
0 \cr 0 & \delta(\tau-\tau')\cr\end{pmatrix} \nonumber \\
\end{eqnarray}
$G$ like the Hamiltonian is a full matrix in site space. With $G$, a solution of
\begin{equation}
\begin{pmatrix} i\hbar\partial_{\tau}-H_{LL} & -H_{LR} \cr -H_{RL} & i\hbar
\partial_{\tau}-H_{RR} \cr \end{pmatrix} \begin{pmatrix} \Psi^{cq}_L(\tau) \cr \Psi^{cq}_R(\tau)\cr\end{pmatrix} =
\begin{pmatrix} Q_L(\tau) \cr Q_R(\tau) \cr\end{pmatrix}
\end{equation}
would read
\begin{equation}
\begin{pmatrix} \Psi^{cq}_L(\tau) \cr \Psi^{cq}_R(\tau)\cr\end{pmatrix} = \int d\tau' \;
\begin{pmatrix} G_{LL}(\tau,\tau') & G_{LR}(\tau,\tau') \cr G_{RL}(\tau,\tau') &
G_{RR}(\tau,\tau') \cr\end{pmatrix} \begin{pmatrix} Q_L(\tau') \cr Q_R(\tau')\cr\end{pmatrix}
\end{equation}
where the index cq stands for coupling and source. As explained in section I,
we could also have set up a Lippmann-Schwinger equation for $\Psi^{cq}$ as
\begin{eqnarray}
\begin{pmatrix} \Psi^{cq}_L(\tau) \cr \Psi^{cq}_R(\tau) \cr\end{pmatrix} &=&
\int d\tau' \; \begin{pmatrix} g_{LL}(\tau,\tau') & 0 \cr 0 & g_{RR}(\tau,\tau')\end{pmatrix}
\begin{pmatrix} Q_L(\tau') \cr Q_R(\tau')\cr\end{pmatrix} + \nonumber \\ && \int d\tau' \;
\begin{pmatrix} g_{LL}(\tau,\tau') & 0 \cr 0 & g_{RR}(\tau,\tau')\end{pmatrix}
\begin{pmatrix} 0 & H_{LR} \cr H_{RL} & 0 \cr\end{pmatrix} \begin{pmatrix} \Psi^{cq}_L(\tau') \cr
\Psi^{cq}_R(\tau')\end{pmatrix} \nonumber \\
\end{eqnarray}
Now from inserting (10) into (11) and for $Q$ choosing some $\delta(\tau'-
\tau_0)$ either in the L- or in the R-component the Dyson equation for $G$
is obtained:
\begin{eqnarray}
&&\begin{pmatrix} G_{LL}(\tau,\tau') & G_{LR}(\tau,\tau') \cr
G_{RL}(\tau,\tau') & G_{RR}(\tau,\tau') \cr\end{pmatrix}=
\begin{pmatrix} g_{LL}(\tau,\tau') & 0 \cr 0 & g_{RR}(\tau,\tau') \cr\end{pmatrix}
+ \nonumber \\ && \int d\tau'' \;
\begin{pmatrix} g_{LL}(\tau,\tau'') & 0 \cr 0 & g_{RR}(\tau,\tau'') \cr\end{pmatrix}
\underbrace{
\begin{pmatrix} 0 & H_{LR} \cr H_{RL} & 0\cr\end{pmatrix}}_{\sigma}
\begin{pmatrix} G_{LL}(\tau'',\tau') & G_{LR}(\tau'',\tau') \cr
G_{RL}(\tau'',\tau') & G_{RR}(\tau'',\tau') \cr\end{pmatrix} \nonumber \\
\end{eqnarray}
\subsection{Explicit Green's functions in time and frequency domain}
\begin{figure}
\includegraphics[width=5cm]{qucouple.jpg}
\includegraphics[width=5.5cm]{circuit.jpg}
\includegraphics[width=5.5cm]{fermis.jpg}
\caption{A left and a right bulk reservoir, uncoupled (a1) and coupled (a2).
(b) Contact as embedded in a circuit.
(c) Fermi levels left and right with applied voltage. The dotted line
marks a fixed energy over both sides.}
\end{figure}
The reservoirs on the left and right being bulk metal, the differential operator
of the homogeneous differential equation (1) is given by the free-partical
Hamiltonian
\begin{equation}
i\hbar\partial_{\tau}-H^0=i\hbar\partial_{\tau}+\frac{\hbar^2}{2m}\Delta
\end{equation}
and the corresponding wave functions are $\Psi^0=e^{\pm i\vec{k}\vec{r}-
i\omega \tau}$ according to the dispersion relation $E=\hbar\omega=
\frac{\hbar^2}{2m}\vec{k}^2$. Looking for a Green's function at given
frequency $\omega$, in (2) we can replace $H^0$ by $\hbar\omega$ and thus
have to solve
\begin{equation}
(i\hbar\partial_{\tau}-\hbar\omega)\; g(\tau,\tau',\omega)=\delta(\tau-\tau')
\end{equation}
We need an expression, the derivative of which produces $\delta(\tau-\tau')$.
One easily verifies that (14) is satisfied by either
\renewcommand{\theequation}{15\alph{equation}}
\setcounter{equation}{0}
\begin{eqnarray}
g^r(\tau,\tau',\omega)&=&-\frac{i}{\hbar}\; \theta(\tau-\tau')\; e^{-i\omega
(\tau-\tau')} \\ g^a(\tau,\tau',\omega)&=& \frac{i}{\hbar}\;
\theta(\tau'-\tau)\; e^{-i\omega (\tau-\tau')}
\end{eqnarray}
The retarded function $g^r$ only exists for $\tau \ge \tau'$ and the advanced
function $g^a$ for $\tau \le \tau'$. In (14) for simplicity we restricted
ourselves to a single energy and therefore (15) still contain $\omega$ as
a parameter. For complete Green's functions in the time domain these terms
have to be multiplied by the density of states $\mathcal{D} (\omega)$ and integrated
over energy. Later in transport we shall be interested in the amount of charge
transferred, not resolving any more, which energy levels contributions came
from. Thinking physically of a small contact between two metallic leads one might
argue that in the constriction transverse $k$-vectors are quantized and a
one-dimensional continuous density of states remains. Nevertheless, for not
too high voltages only a certain energy range around the Fermi energy will play
a role in transport and thus setting the density of states equal to its value
at the Fermi energy is a good approximation. Anyway, a constant density of
states $\mathcal{D}(\omega)=\mathcal{D}$ with occupied states below and empty ones
above the Fermi level for each side, left and right, shall just enter our
model here as an assumption. We are led to the following representations of
the retarded and advanced Green's functions:
\setcounter{equation}{0}
\renewcommand{\theequation}{16\alph{equation}}
\begin{eqnarray}
g^r(\tau,\tau')\; &=&-\frac{i}{\hbar}\; \theta(\tau-\tau') \int d\omega \;
\frac{\mathcal{D}(\omega)}{2\pi} \; e^{-i\omega(\tau-\tau')} \nonumber \\
&\widehat{=}& \frac{\mathcal{D}}{
h} \int d\omega \; g^r(\omega)\; e^{-i\omega(\tau-\tau')} \quad
{\rm with} \quad g^r(\omega)=-i \\
g^a(\tau,\tau') \; &\widehat{=}& \frac{\mathcal{D}}{
h} \int d\omega \; g^a(\omega)\; e^{-i\omega(\tau-\tau')} \quad
{\rm with} \quad g^a(\omega)=i
\end{eqnarray}
\setcounter{equation}{16}
\renewcommand{\theequation}{\arabic{equation}}
Taking out the factor $\mathcal{D}/h$, we get the dimensionless functions
$g^r(\omega)=-i$ and $g^a(\omega)=i$ in frequency space. Remark that their
deduction here did not consist in calculating a Fourier transformation. The
phase factor $e^{-i\omega(\tau-\tau')}$ was already there in (15).
The $\omega$-integrals from (16) will be implemented as a useful representation
of $g^{r/a}$ still in the time domain.
The $\theta$-functions have been deliberately skipped after the
$\widehat{=}$-signs. We shall however see that $g^r$ and $g^a$ finally only
appear with the correct relation between their first and second time argument.
The $\omega$-integral should indeed be understood as summing over all energies
and, even if $g^{r/a}(\omega)$ is a constant, on no account be interpreted
as this constant times $2\pi\delta(\tau-\tau')$.
$e^{-i\omega(\tau-\tau')}=e^{-i\omega\tau} \cdot (e^{-i\omega\tau'})^*
=\vert\Psi^0(\tau)><\Psi^0(\tau')\vert$ is the
projection-like conversion of the phase unit vector
of an oscillation from time $\tau'$ to time $\tau$.
Left(L) and right(R) are distinguished as the origins of
wave functions constituting our basis of states, but the junction is considered
point-like, that is $\vec{r}=0$ for both left and right and thus $e^{\pm i
\vec{k}\vec{r}}=1$ drops out in an alike combination of $\Psi^0s$ even from
L and R. The two-fold time argument suggests to take (16) formally even as
double Fourier transform (however with different signs in the exponentials with
$\tau$ and $\tau'$)
\begin{equation}
\frac{h}{\mathcal{D}} \; g^{r/a}(\tau,\tau')=\int d\omega_1
\; \int d\omega_2 \; g^{r/a}(\omega_1,\omega_2) \; e^{-i\omega_1\tau} \;
e^{i\omega_2\tau'}
\end{equation}
with $g^{r/a}(\omega_1,\omega_2)=\mp i \delta(\omega_1-\omega_2)$, however,
because different energies stay independent. (More or less guessing the Green's
functions $g$ was easy in our normal-conducting simplest model here. Generally,
it has to be found as the solution to an equation like (14) with the differential
operator from the uncoupled system's Schr\"odinger equation and an elementary
perturbation $\delta$. In the superconducting state, for example, the
two-particle interaction in the Hamiltonian forces the Green's function to
include Andreev reflection \cite{ketterson}, and the result is such that $g(\omega)$ does not
just vanish in the gap of the quasi-particle density of states.)
Before we can solve (12) for the coupled system's Green's function $G$, we
have to specify the coupling parts of the Hamiltonian $H_{LR}$ and $H_{RL}$.
Like with the density of states the simplest model will assume that the
coupling is energy-independent and described by a constant (real) interaction
energy $W$. The reservoirs are unaltered by transport between them. Due to the
applied voltage (Fig.1b) incoming charge carriers (from the left on the right) are led
away and outgoing ones (on the left to the right) get replaced. In the contact
we do not allow relaxation or other energy-changing processes. We choose the
repective Fermi levels as zeros of energy on either side (Fig.1c). An
electron going from right to left has to strip off its phase $e^{-i\omega_R\tau}$
and aquire $e^{-i\omega_L\tau}$ to fit in on the left. An analogous argument
holds for transitions from left to right, and therefore the coupling terms are
\setcounter{equation}{0}
\renewcommand{\theequation}{18\alph{equation}}
\begin{eqnarray}
H_{LR}&=&\sigma_{LR}(\tau)=W\; e^{-i(\omega_L-\omega_R)\tau}=W\; e^{ieV\tau/\hbar} \\
H_{RL}&=&\sigma_{RL}(\tau)=W\; e^{-i(\omega_R-\omega_L)\tau}=W\; e^{-ieV\tau/\hbar}
\end{eqnarray}
\setcounter{equation}{18}
\renewcommand{\theequation}{\arabic{equation}}
The phase factor gives a time-dependence to $H_{LR}$ and $H_{RL}$, but
$\omega_R-\omega_L=eV/\hbar$ is indeed independent of energy. For $G$ we make
an ansatz like (17) as a two-fold Fourier representation:
\begin{equation}
\frac{h}{\mathcal{D}} \; G(\tau,\tau') = \int d\omega_1 \;
\int d\omega_2 \; G(\omega_1,\omega_2) \; e^{-i\omega_1\tau} \;
e^{i\omega_2\tau'}
\end{equation}
From (12), which is valid for either advanced or retarded functions, as an
example, we pick the upper left component of the 2x2 matrix in LR-space and
insert (17), (18) and (19): (Multiple integral signs are skipped from now on.)
\begin{eqnarray}
G_{LL}(\tau,\tau')&=&g_{LL}(\tau,\tau')+\int d\tau'' \; g_{LL}(\tau,\tau'') \;
H_{LR}(\tau'') \; G_{RL}(\tau'',\tau') \quad \Longleftrightarrow \nonumber \\ &&
\frac{\mathcal{D}}{h} \int d\omega_{L1} \; d\omega_{L2} \;
G_{LL}(\omega_{L1},\omega_{L2}) \; e^{-i\omega_{L1}\tau} \;
e^{i\omega_{L2}\tau'} = \nonumber \\ &&
\frac{\mathcal{D}}{h} \int d\omega_{L1} \; d\omega_{L2} \;
g_{LL}(\omega_{L1},\omega_{L2}) \; e^{-i\omega_{L1}\tau}\;
e^{i\omega_{L2}\tau'} + \nonumber \\ &&
\int d\tau'' \; d\omega_{L1} \; d\omega_{L3} \; d\omega_{R} \; d\omega_{L2} \;
\frac{\mathcal{D}}{h} \; g_{LL}(\omega_{L1},\omega_{L3})\;
e^{-i\omega_{L1}\tau} \; e^{i\omega_{L3}\tau''}\; \cdot \nonumber \\ &&
\cdot \; W \; e^{ieV\tau''/\hbar}\;
\frac{\mathcal{D}}{h}\; G_{RL}(\omega_R,\omega_{L2}) \;
e^{-i\omega_R\tau''}\; e^{i\omega_{L2}\tau'}
\end{eqnarray}
The integral over $\tau''$ produces $2\pi\delta(\omega_{L3}+\frac{eV}{\hbar}-
\omega_R)$ and in $g(\omega_{L1},\omega_{L3})$ there is $\delta(\omega_{L1}-
\omega_{L3})$ anyway, such that the last term of (20) becomes
$$\frac{\mathcal{D}}{h} \int d\omega_{L1} \; d\omega_{L2} \; g(\omega_{L1}) \; t \;
G_{RL}(\omega_{L1}+\frac{eV}{\hbar},\omega_{L2}) \; e^{-i\omega_{L1} \tau} \;
e^{i\omega_{L2} \tau'} $$
with $t=W\mathcal{D}/\hbar$.
Strictly speaking, if (20) were for the retarded function, in the last term
there would be $\theta(\tau-\tau'')$ from $g^r_{LL}$ and $\theta(\tau''-\tau')$
from $G^r_{RL}$, and if it were for the advanced function, $\theta(\tau''-\tau)$
and $\theta(\tau'-\tau'')$, such that the integral over $\tau''$ only exists
between $\tau$ and $\tau'$ instead of having minus and plus infinity as limits.
However, with $\omega_{L3}$ and $\omega_R$ running over any value, one can argue
that even with a finite $\tau''$-integral the only remaining contribution
stems from $\omega_{L3}+\frac{eV}{\hbar}-\omega_R=0$. A discussion of time
ordering will again appear in section 5.
$\mathcal{D}$ is the density of states per frequency interval, dividing by $\hbar$
makes it the number of states per energy interval. $W$ is an energy. $t$ can
be understood as a dimensionless transmission amplitude. Now we set up the
convention that all frequency arguments of $g$ and $G$ are written with respect
to the left zero level and for the case that they correspond to the right,
that is an R-index, it is understood that $eV/\hbar$ is added. (20) has
to hold for any $\tau$ and $\tau'$ and from comparing Fourier coefficients we get
\setcounter{equation}{19}
\renewcommand{\theequation}{\arabic{equation}a}
\begin{equation}
G_{LL}(\omega_{L1},\omega_{L2})=g_{LL}(\omega_{L1}) \; \delta(\omega_{L1}-
\omega_{L2}) +
g_{LL}(\omega_{L1}) \; t \; G_{RL}(\omega_{L1},\omega_{L2})
\end{equation}
We could have inserted $G_{RL}=g_{RR}H_{RL}G_{LL}$ and again replaced
$G_{LL}=g_{LL}+g_{LL}H_{LR}G_{RL}$ and so on. Instead of an implicit equation
for $G$ this would have led to an infinite series (see section I):
\setcounter{equation}{20}
\renewcommand{\theequation}{\arabic{equation}}
\begin{equation}
G=g\sum_{n=0}^{\infty} (\sigma g)^n=\left( \sum_{n=0}^{\infty} (g\sigma)^n
\right) g =
g+g\sigma g+g\sigma g \sigma g + \ldots
\end{equation}
(21) is written for whole matrices in LR-space, calculations like (20) can be
done analogously for $G_{LR}$, $G_{RL}$ and $G_{RR}$. (21) can be
read as equation in the time domain. Then $\sigma$ is the matrix consisting of
$H_{LR}$ and $H_{RL}$ and each multiplication of two following $g$s with
$\sigma$ in between means an integration over time. However, (21) is as well
valid as relation in frequency space. In this case $\sigma$ is the 2x2 matrix
with just $t$ as off-diagonal elements. Like we have seen through evaluating
the $\tau''$-integral in (20), in (21) each connection $H_{LR/RL}$ from
$\sigma$ passes the frequency argument from the $g$ in front to the $g$ behind.
And as $g$ can only have two identical frequency arguments, no $\omega$
different from the first can ever appear, such that $\omega_{L1}=\omega_{L2}=
\omega$ and $G$ also effectively is a function of only one frequency argument:
$G(\omega_1,\omega_2)=\delta(\omega_1-\omega_2)G(\omega_1)$. With
Green's functions of a single frequency argument (20a) and its analogues for the
other three components in LR-space become a simple algebraic equation:
\begin{equation}
\begin{pmatrix} G_{LL}(\omega) & G_{LR}(\omega) \cr G_{RL}(\omega) & G_{RR}(\omega)\cr\end{pmatrix}=
\begin{pmatrix} g_{LL}(\omega) & 0 \cr 0 & g_{RR}(\omega) \cr \end{pmatrix}
+\begin{pmatrix} g_{LL}(\omega) & 0 \cr 0 & g_{RR}(\omega) \cr \end{pmatrix}
\begin{pmatrix} 0 & t\cr t & 0\cr\end{pmatrix}
\begin{pmatrix} G_{LL}(\omega) & G_{LR}(\omega) \cr G_{RL}(\omega) & G_{RR}(\omega)\cr\end{pmatrix}
\end{equation}
(The fact that even the Green's function of the coupled system turns out to
be a function of a single frequency argument is a special feature of our
simple model for the normal conducting case. In the extension of this model
to superconducting reservoirs and transmission processes including Andreev
reflection, $G$ becomes a function of two frequency arguments, the second,
however, restricted to values differing from the first by an integer multiple
of $eV/\hbar$, such that effectively there is one continuous and one discrete
frequency parameter \cite{Cuequ}.) Here, with $g^{r/a}(\omega)=\mp i$, (22) is easily
solved and $G(\omega)$ comes out independent of frequency, too:
\begin{eqnarray}
\begin{pmatrix} G_{LL} & G_{LR} \cr G_{RL} & G_{RR} \cr\end{pmatrix}^{r/a}(\omega)&=&
\left[ \begin{pmatrix} 1 & 0 \cr 0 & 1 \cr\end{pmatrix} -
\begin{pmatrix} \mp i & 0 \cr 0 & \mp i \cr\end{pmatrix}
\begin{pmatrix} 0 & t \cr t & 0 \cr\end{pmatrix} \right]^{-1}
\begin{pmatrix} \mp i & 0 \cr 0 & \mp i \cr\end{pmatrix}
\nonumber \\
&=&\frac{1}{1+t^2}\begin{pmatrix} \mp i & -t \cr -t & \mp i \cr\end{pmatrix}
\end{eqnarray}
The same result could have been obtained from (21) by writing out a few more
of the matrix multiplications and using the formula for the geometric series
in each element. We shall need two further types of Green's functions. $T$
will be introduced in the next section and $g^{+-}$ and $G^{+-}$ when
calculating the current.
\subsection{Transfer Green's functions}
In (21) there was a sum of products of arbitrary many factors $g$ and $\sigma$
with outer factors $g$. ''Product'', of course, except with the Green's functions
taken of a single frequency parameter, in the time domain or with two-fold
frequency dependence still meant a convolution-type integration over inner
arguments. In analogy we define the sum of products with outer factors
$\sigma$ (as integrals in the time domain or just algebraically with
$g(\omega)$ and $\sigma=\begin{pmatrix} 0 & t \cr t & 0\cr\end{pmatrix}$):
\begin{equation}
T=\sigma+\sigma g \sigma +\sigma g\sigma g\sigma + \ldots =
\sigma \sum_{n=0}^{\infty}(g\sigma)^n=\left( \sum_{n=0}^{\infty} (\sigma g)^n
\right) \sigma
\end{equation}
Whereas all contributions to $G$ in (21) began and ended with staying some time
in a reservoir, described by $g$ - at L from $\tau$ to $\tau''$ for a start
in the last term of (20), for example - each term of $T$ in (24) begins and
ends with a transition $\sigma$ and further contains at least one such hopping
across the junction (which $g$ does not). Therefore we call $T$ the {\it
transfer} Green's function.
The same as for $G$, the relation between $T$ as a function of times and as
a function of frequency is given by
\begin{equation}
\frac{\mathcal{D}}{\hbar}\; T_{JK}(\tau,\tau')=\frac{1}{2\pi}\int d\omega \;
T_{JK}(\omega)\; e^{-i\omega_J\tau} \; e^{i\omega_K\tau}
\end{equation}
where $\omega_J=\omega$ if J=L and $\omega_J=\omega+eV/\hbar$ if J=R and the
same for $\omega_K$.
(Taking out $2\pi$ of $\mathcal{D}$ in (16) was a convention.
The prefactor of the $\omega$-integral for $T$ follows from consistency
requirements. (19) for $G$ was in complete analogy to (16) for $g$.
$T$ with two time arguments, however, has a little different character from
$\sigma$ with just one time parameter.)
Alternatively to (24) $T$ could be defined through its link to $G$
\begin{equation}
\sigma G=Tg \quad {\rm or} \quad G\sigma =gT
\end{equation}
Be careful that replacing one by the other can introduce another internal
time integration as, for example,
$\sigma(\tau)\; G(\tau,\tau')=\int d\tau''\; T(\tau,\tau'')\; g(\tau'',\tau')$.
(24) and (26) hold for retarded and advanced functions.
From (24) it is immediately seen that $T$ like $G$
satisfies a Dyson equation
\begin{equation}
G=g+g\sigma G \quad {\rm and} \quad T=\sigma+\sigma g T
\end{equation}
and even the complementary forms
\begin{equation}
G=g+G\sigma g \quad {\rm and} \quad T=\sigma+Tg\sigma
\end{equation}
are analogues. In Fourier space, like (22) the $T$-equation (27) is an
algebraic equation and the solution like
\begin{equation}
G=({\bf 1}-g\sigma )^{-1}g \quad {\rm is} \quad T=({\bf 1}-\sigma g)^{-1}\sigma.
\end{equation}
Inserting $g$ and $\sigma$ explicitly for our model we get
\begin{eqnarray}
\begin{pmatrix} T_{LL} & T_{LR}\cr T_{RL} & T_{RR}\cr\end{pmatrix}^{r/a}(\omega)&=&
\left[ \begin{pmatrix} 1 & 0\cr 0 & 1\cr\end{pmatrix}-\begin{pmatrix}0 & t\cr t & 0\cr\end{pmatrix}
\begin{pmatrix} \mp i & 0 \cr 0 & \mp i\cr\end{pmatrix} \right]^{-1}
\begin{pmatrix} 0 & t\cr t & 0\cr\end{pmatrix} \nonumber\\
&=& \frac{1}{1+t^2}\begin{pmatrix} \mp it^2 & t \cr t & \mp it^2\cr\end{pmatrix}
\end{eqnarray}
especially
\begin{equation}
T_{LR}^{r/a}=T_{RL}^{r/a}=\frac{t}{1+t^2}=t-t^3+t^5-\ldots
\end{equation}
\begin{figure} \begin{center}
\includegraphics[width=5cm]{order.jpg} \end{center}
\caption{Transfer processes from left to right of different order.}
\end{figure}
Whereas $t$ is the single hopping amplitude, $T_{LR/RL}$ is a renormalized
transfer amplitude. One may wonder why a model for transport could not have
been set up adding amplitudes for transfer processes of all orders, as the
interaction (18) seems to be introduced the way it is just in order to result
in powers of $t$. However, $g^{r/a}(\omega)=\mp i$ deduced from the
Schr\"odinger equation is decisive for the signs in (30) and (31). One may
wonder that multiple reflections are not added as $t+t^3+t^5+\ldots=
\frac{t}{1-t^2}$. Processes of different order (Fig.2) are not independent,
but interfere. $T_{RL}$ is the transfer amplitude per single electron supplied
on the left by the voltage source. But for every electron that goes over to
the right in an $n$th order process ($n$ odd)
with weight $t^2$ there is one that has hopped
once more to the right and back ($n+2$ order process) and thus is not to be
newly supplied, but to be again sent through the junction.
The amplitude $t$ is renormalized by $1+t^2$ as denominator.
However, such interpretations of quantum mechanical amplitudes are precarious,
and the full conversion of $t$ to a transmission probability will be
established later.
It is quite instructive to solve (27) in a slightly different way than done in
(30). Firstly, for the four components in LR-space we have
\renewcommand{\theequation}{\arabic{equation}a}
\setcounter{equation}{26}
\begin{eqnarray}
T_{LL}=\phantom{\sigma_{RL}+} \sigma_{LR}g_{RR}T_{RL} \qquad
T_{LR}=\sigma_{LR}+\sigma_{LR}g_{RR}T_{RR} \nonumber \\
T_{RL}=\sigma_{RL}+\sigma_{RL}g_{LL}T_{LL} \qquad
T_{RR}=\phantom{\sigma_{RL}+} \sigma_{RL}g_{LL}T_{LR}
\end{eqnarray}
\renewcommand{\theequation}{\arabic{equation}}
\setcounter{equation}{31}
Inserting these into each other, for example, an equation for $T_{LR}$ alone
is obtained:
\begin{equation}
T_{LR}=\sigma_{LR}+\sigma_{LR}g_{RR}\sigma_{RL}g_{LL}T_{LR}
\end{equation}
This implicit equation is the basis for calulating the transfer Green's
function in more complicated cases than discussed here
\cite{Cuequ,my06}, like for example
the superconducting junction. In our model, inserting $\sigma_{LR}=
\sigma_{RL}=t$ and $g^{r/a}_{LL}(\omega)=g^{r/a}_{RR}(\omega)=\mp i$ into (32)
immediately also leads to $T_{LR}^{r/a}(\omega)=\frac{t}{1+t^2}$.
\begin{figure} \begin{center}
\includegraphics[width=7cm,angle=270]{Dyson.jpg} \end{center}
\caption{Illustration of the Dyson equation for the transfer Green's function.}
\end{figure}
More easily than the Dyson equation for the ordinary Green's function
$G$, the one for the transfer Green's function $T$ is illustrated as is
done for the LR-component in Fig.3. (Normally indices are read from right
to left such that $T_{LR}$ is considered a transition from right to left,
but it does not really matter whether they are interpreted the other way
round as in Fig.3. The actual sequence of what is earlier or later in time
will be discussed when calculating the current in the next section.)
Fig.3 demonstrates the implicitness of the Dyson equation: Any transition
from left to right is either a single transfer or an electron hopping to the
right and back followed by any process beginning on the left and ending on
the right, no matter what happens in between. This last part by definition
is the sane as the other side of the equation, namely $T_{LR}$.
\subsection{Calculating the current}
From the Heisenberg picture of quantum mechanics we know that the time derivative
of a not explicitly time-dependent operator $A$ is given by the commutator
with the Hamiltonian \cite{CoTan_2}:
\begin{equation}
\frac{d}{d\tau}A=\frac{i}{\hbar}[H,A]
\end{equation}
The operator of interest here is the projector on either side of the junction
\begin{equation}
\rho_L=\frac{\vert \Psi^0_L><\Psi^0_L\vert}{\vert \Psi^0_L \vert^2} \widehat{=}
\begin{pmatrix} 1 & 0\cr 0 & 0\cr\end{pmatrix} \quad {\rm or} \quad
\rho_R=\frac{\vert \Psi^0_R><\Psi^0_R\vert}{\vert \Psi^0_R \vert^2} \widehat{=}
\begin{pmatrix} 0 & 0\cr 0 & 1\cr\end{pmatrix}
\end{equation}
As explained earlier, with the junction coupling left and right together, the
solution $\begin{pmatrix}\Psi_L \cr \Psi_R \cr\end{pmatrix}$ is not limited to one side, however,
the projectors take out the respective part:
\setcounter{equation}{0}
\renewcommand{\theequation}{34\alph{equation}}
\begin{equation}
\begin{pmatrix} 1 & 0\cr 0 & 0\cr\end{pmatrix}
\begin{pmatrix} \Psi_L \cr \Psi_R \cr\end{pmatrix}=
\begin{pmatrix} \Psi_L \cr 0 \cr\end{pmatrix} \quad {\rm and} \quad
\begin{pmatrix} 0 & 0\cr 0 & 1\cr\end{pmatrix} \begin{pmatrix}\Psi_L \cr \Psi_R \cr\end{pmatrix}=
\begin{pmatrix} 0 \cr \Psi_R\cr\end{pmatrix}
\end{equation}
\renewcommand{\theequation}{\arabic{equation}}
\setcounter{equation}{34}
$\rho_L$ and $\rho_R$ are proportional to the amount of charge on the left and
on the right side. Their time derivatives have equal absolute values, but
opposite sign and represent the current.
\begin{equation}
I=-e<\frac{d\rho_L}{d\tau}>=e<\frac{d\rho_R}{d\tau}>
\end{equation}
($<\; \vert$ and $\vert\; >$ are used for {\tt bra}- and {\tt ket}-states. Here $<\; >$ means
the expectation value, of course.)
$e$ is the charge of an electron and the sign of $I$ can be defined
arbitrarily. We choose to do the calculation with $\rho_L$ and evaluate
the commutator with the Hamiltonian:
\begin{equation}
[H,\rho_L]=\left[ \begin{pmatrix} H_{LL} & H_{LR} \cr H_{RL} & H_{RR} \cr\end{pmatrix},
\begin{pmatrix} 1 & 0 \cr 0 & 0 \cr\end{pmatrix} \right] = \underbrace{
\begin{pmatrix} 0 & -H_{LR} \cr H_{RL} & 0 \cr\end{pmatrix}}_{\sigma_c}
\end{equation}
Putting together (33) and (35) we obviously need the expectation value of
the operator $[H,\rho_L]$. $[H,\rho_L]$ shall be called $\sigma_c$.
Even if (33) stems from the Heisenberg picture, it is written in such a way,
that the right hand side is to be evaluated in the Schr\"odinger system with
time dependent states, and we shall here change to the interaction picture
\cite{Mahan}
for the calculation. With $^{\dagger}$ standing for complex conjugation
as well as transposition from column to line vector, the value of
${\displaystyle \frac{d\rho_L}{dt}}$ in state $\begin{pmatrix}
\Psi_L\cr \Psi_R\cr\end{pmatrix} $ at time $\tau$ is given by
\begin{eqnarray} &&
\begin{pmatrix} \Psi_L(\tau)\cr \Psi_R(\tau)\end{pmatrix}^{\dagger} \;
\frac{i}{\hbar} \; \begin{pmatrix} 0 &
-H_{LR}(\tau) \cr H_{RL}(\tau) & 0 \end{pmatrix} \;
\begin{pmatrix} \Psi_L(\tau) \cr
\Psi_R(\tau)\end{pmatrix} = \nonumber \\ &&
<\Psi^0(\tau_0) \vert \widetilde{T} \; \exp (\frac{1}{\hbar}\int_{\tau_0}^{\tau} i \; \sigma_{
\mathcal{H}}(\tau') \; d\tau') \; i[H,\rho_L]_{\mathcal{H}} \; \widehat{T} \;
\exp (\frac{1}{\hbar}\int_{\tau_0}^{\tau} (-i) \; \sigma_{\mathcal{H}}(\tau') \; d\tau') \vert
\Psi^0(\tau_0)> \nonumber \\
\end{eqnarray}
$H_{LR}(\tau)$ and $H_{RL}(\tau)$ denote the time occurence of $\tau$ from (18)
which has to be considered as still belonging to the Schr\"odinger picture
in our case here. $\widehat{T}$ means time ordering \cite{Mahan} and $\widetilde{T}$
anti-time ordering. $\Psi^0$ in the uncoupled system, of course, also stands
for a two-vector with left and right component.
Replacing $\begin{pmatrix} \Psi_L(\tau)\cr \Psi_R(\tau)\end{pmatrix}$ by $\Psi^0(\tau_0)$ in (37) we took out both
the coupling as well as the time dependence from the states. Accordingly the
$\sigma_{\mathcal{H}}$ in the integrals, in contrast to the
Schr\"odinger-picture $\sigma$ used in the preceding sections, in a
Heisenberg way have to include the time dependence of the uncoupled states.
$\sigma(\tau)$ could already be written as
$$ \sigma(\tau)=\begin{pmatrix} 0 & H_{LR}(\tau) \cr H_{RL}(\tau) & 0 \cr\end{pmatrix}=
\begin{pmatrix} e^{-i\omega_L\tau} & 0\cr 0 & e^{-i\omega_R\tau}\cr\end{pmatrix}
\begin{pmatrix} 0 & W\cr W & 0\cr\end{pmatrix}
\begin{pmatrix} e^{i\omega_L\tau} & 0\cr 0 & e^{i\omega_R\tau}\cr\end{pmatrix}$$
or $\sigma(\tau)=\vert\Psi^0(\tau)>W<\Psi^0(\tau)\vert$ for short, where
{\tt bra}, {\tt ket} and $W$ still mean the respective matrices. However,
this decomposition might rather be confusing and will not be used, anyway.
The translation to the interaction picture is the following:
\begin{eqnarray}
\sigma_{\mathcal{H}}&=&\begin{pmatrix} e^{\frac{i}{\hbar}H_{LL}(\tau-\tau_0)} & 0 \cr 0 &
e^{\frac{i}{\hbar}H_{RR}(\tau-\tau_0)}\cr\end{pmatrix} \begin{pmatrix}
0 & H_{LR}(\tau) \cr H_{RL}(\tau) & 0 \cr\end{pmatrix}
\begin{pmatrix} e^{-\frac{i}{\hbar}H_{LL}(\tau-\tau_0)} & 0 \cr 0 &
e^{-\frac{i}{\hbar}H_{RR}(\tau-\tau_0)}\cr\end{pmatrix}\nonumber \\ &=&
\begin{pmatrix} e^{-i\omega_L\tau_0} & 0 \cr 0 & e^{-i\omega_R\tau_0}\cr\end{pmatrix}
\begin{pmatrix} e^{i\omega_L\tau} & 0 \cr 0 & e^{i\omega_R\tau}\cr\end{pmatrix} \;
\begin{pmatrix} 0 & H_{LR}(\tau) \cr H_{RL}(\tau) & 0 \cr\end{pmatrix} \cdot \nonumber \\ &&
\qquad \qquad \qquad \qquad \qquad \qquad
\begin{pmatrix} e^{-i\omega_L\tau} & 0 \cr 0 & e^{-i\omega_R\tau}\cr\end{pmatrix}
\begin{pmatrix} e^{i\omega_L\tau_0} & 0 \cr 0 & e^{i\omega_R\tau_0}\cr\end{pmatrix}
\end{eqnarray}
or $\sigma_{\mathcal{H}}(\tau)=\vert\Psi^0(\tau_0)><\Psi^0(\tau)\vert\;
\sigma(\tau)\; \vert\Psi^0(\tau)><\Psi(\tau_0)\vert$. The Heisenberg picture
always refers the operator back to the undeveloped state at $\tau_0$.
\begin{figure}\begin{center}
\includegraphics[width=8cm,angle=270]{Keldysh.jpg}
\end{center}
\caption{The Keldysh contour illustrating the development
out of the uncoupled system's states of the {\tt bra} $<\Psi(\tau)\vert$ on
the minus and the {\tt ket} $\vert \Psi(\tau)>$ on the plus branch.
This view picks out the single transition at time $\tau$ and visualizes
the calculation of $<\Psi(\tau)\vert \sigma(\tau) \vert \Psi(\tau)>$.
Each point on the contour represents an LR- or RL-transition described by
$\sigma$, each line segment the development of the wave-function phase
which is given $g$.}
\end{figure}
Analogously $\sigma_c$ will have to be extended by the time-dependence of the
uncoupled states.
The meaning of the time-integral over $\sigma_{\mathcal{H}}$ as an exponential
is best explained by writing explicitly:
\begin{eqnarray}
\widehat{T}\; \exp (\frac{1}{\hbar} \int_{\tau_0}^{\tau}(-i)\;
\sigma_{\mathcal{H}}(\tau')\; d\tau') = \sum_n
\frac{1}{\hbar} \int_{\tau_2}^{\tau} d\tau_1 \ldots
\frac{1}{\hbar} \int_{\tau_{j+1}}^{\tau_{j-1}} d\tau_j \ldots
\frac{1}{\hbar} \int_{\tau_0}^{\tau_{n-1}} d\tau_n \cdot \nonumber \\
(-i) \sigma_{\mathcal{H}}(\tau_1) \ldots (-i) \sigma_{\mathcal{H}}(\tau_j)
\ldots (-i) \sigma_{\mathcal{H}}(\tau_n)
\end{eqnarray}
where all arguments $\tau_1,\ldots ,\tau_j,\ldots ,\tau_n$ have to lie between
$\tau_0$ and $\tau$ and all products from none to arbitrarily many factors
$\sigma_{\mathcal{H}}$ have to be added. With time ordering, furthermore
$\tau_1\ge\ldots\ge\tau_j\ge\ldots\ge\tau_n$ is imposed. Without $\widehat{T}$
the scheme for the exponential series $e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}$
in (39) would produce $\sum_n \frac{1}{n!} \; \frac{1}{\hbar}
\int_{\tau_0}^{\tau}d\tau_1 \ldots
\frac{1}{\hbar} \int_{\tau_0}^{\tau}d\tau_j \ldots
\frac{1}{\hbar} \int_{\tau_0}^{\tau}d\tau_n \ldots$ .
But then for every fixed set $\{ \tau_1,\ldots ,\tau_j,\ldots ,\tau_n\}$
there are $n!$ permutations for the time values to occur such that time
ordering and restricting the arguments to mutually exclusive intervals as
done in (39) cancels the factorial denominators.
Writing out the anti-time ordered part in the same way as shown for the
time-ordered part, the expression from (37) becomes
\renewcommand{\theequation}{37\alph{equation}}
\setcounter{equation}{0}
\begin{eqnarray} &&
{\bf <\Psi^0(\tau_0)\vert} \quad \Psi^0(\tau_0)> \frac{i}{\hbar} <\Psi^0(\tau_m)\vert
\sigma(\tau_m)\vert \Psi^0(\tau_m)><\Psi^0(\tau_0)\vert \; \ldots
\nonumber \\ &&
\ldots \vert \Psi^0(\tau_0)> \frac{i}{\hbar} <\Psi^0(\tau_k)\vert\sigma(\tau_k)\vert
\Psi^0(\tau_k)><\Psi^0(\tau_0)\vert \quad
{\it \Psi^0(\tau_0)> \frac{i}{\hbar} <\Psi^0(\tau_{k-1})\vert}
\nonumber \\ &&
{\it \sigma(\tau_{k-1})\vert}\ldots
\vert\sigma(\tau_{1a})\vert\Psi^0(\tau_{1a})><\Psi^0(\tau_0)\vert
\quad {\bf \Psi^0(\tau_0)> \frac{i}{\hbar} <\Psi^0(\tau)\vert\sigma_c(\tau)\vert}
\nonumber \\ &&
{\bf \Psi^0(\tau)><\Psi^0(\tau_0)\vert}\quad
\Psi^0(\tau_0)>(\frac{-i}{\hbar})<\Psi^0(\tau_{1r})\vert\sigma(\tau_{1r})\vert \ldots
\nonumber \\ &&
{\it \ldots \vert\sigma(\tau_{j-1})\vert\Psi^0(\tau_{j-1})><\Psi^0(\tau_0)}
\quad \vert \Psi^0(\tau_0)>(\frac{-i}{\hbar})<\Psi^0(\tau_j)\vert\sigma(\tau_j)\vert
\Psi^0(\tau_j)>
\nonumber \\ &&
<\Psi^0(\tau_0)\vert
\ldots \vert\sigma(\tau_n)\vert\Psi^0(\tau_n)><\Psi^0(\tau_0) \quad
{\bf \vert \Psi^0(\tau_0)>}
\end{eqnarray}
where $\tau_0\le\tau_m\le\ldots\le\tau_k\le\ldots\tau_{1a}\le\tau$,
$\tau_0\le\tau_n\le\ldots\le\tau_j\le\ldots\le\tau_{1r}\le\tau$
and integration over all time arguments except $\tau$ and $\tau_0$ is
understood. Like in (39) we mean the sum of all products with arbitrary
many inner development factors like $\Psi^0$ of $\tau_k$ and
$\Psi^0$ of $\tau_j$. Spaces and different font types are just used in (37a)
to recognize sequential $\sigma_{\mathcal{H}}$-parts like (38).
{\it Inner} development factors like $\Psi^0$ of $\tau_k$ and
$\tau_j$ as well as $\Psi^0(\tau)$ and the inner $\Psi^0(\tau_0)$
have to sum over space or the basis of states and therefore should be
thought of as matrices as in (38).
We have not yet decided which components {\it outer}
{\tt bra} and {\tt ket} with ${\bf \Psi^0(\tau_0)}$ will project out.
All $<\Psi^0(\tau_0)\vert\Psi^0(\tau_0)>$ are the unit matrix and drop out
($\Psi^0$ being given by a non-normalizable $e$-function does not pose
a problem here).
Then $\vert\Psi^0(\tau_k)>\frac{i}{\hbar}<\Psi^0(\tau_{k-1})\vert$
with $\tau_k\le\tau_{k-1}$ is recognized as $g^a(\tau_k,\tau_{k-1})$ and
$\vert\Psi^0(\tau_{j-1})>\frac{(-i)}{\hbar}<\Psi^0(\tau_j)\vert$
with $\tau_{j-1}\ge\tau_j$ as $g^r(\tau_{j-1},\tau_j)$. These $g^{a/r}$
are 2x2 diagonal matrices.
With (21) in (37a) the whole sequence from $\vert\Psi^0(\tau_m)>$ to
$<\Psi^0(\tau)\vert$ can be replaced by $G^a(\tau_m,\tau)$ and the long part
from $\vert\Psi^0(\tau)>$ up to $<\Psi^0(\tau_n)\vert$ is just $G^r(\tau,\tau_n)$.
The complete operator between the outermost ${\bf <\Psi^0(\tau_0)}\vert$ and
${\bf \vert\Psi^0(\tau_0)>}$ in (37a) therefore becomes
\begin{eqnarray}
\vert\Psi^0(\tau_0)>\frac{i}{\hbar}<\Psi^0(\tau_m)\vert
\left[ {\bf 1}\; \delta(\tau_m-\tau)+\sigma(\tau_m)G^a(\tau_m,\tau)\right]
\sigma_c(\tau) \nonumber \\
\left[ {\bf 1}\; \delta(\tau_n-\tau)+G^r(\tau,\tau_n)\sigma(\tau_n)\right]
\vert\Psi^0(\tau_n)><\Psi^0(\tau_0)\vert
\end{eqnarray}
The ${\bf 1}$-contributions stem from the cases where there are no
factors with $\Psi^0(\tau_k)$ or $\Psi^0(\tau_j)$.
We regard the coupled system as having developed out of the uncoupled system,
but we are looking for a stationary state. To achieve this, the coupling has
to have been turned on infinitely long ago, thus we let $\tau_0\rightarrow
-\infty$. The sequence of time arguments is usually represented on the
so-called Keldysh contor (Fig.4). We still have to take the expectation
value of (37) or (37a). This means summing over the basis of states for the
outer ${\bf \Psi^0(\tau_0)}$. The uncoupled basis consists of $\begin{pmatrix}
1\cr 0\cr\end{pmatrix}$
and $\begin{pmatrix} 0\cr 1\cr\end{pmatrix}$ for a state on the left and a state on the right
for each frequency $\omega$. We shall discuss the occupation or emptiness of
states shortly after having worked off some further more formal points.
Summing over $\begin{pmatrix} 1\cr 0\cr\end{pmatrix}$ and $\begin{pmatrix}
0\cr 1\cr\end{pmatrix}$
for the outer ${\bf \Psi^0(\tau_0)}$ will return the trace of the matrix $M$
given in (37b).
$$ \begin{pmatrix} 1\cr 0\cr\end{pmatrix}^{T} \underbrace{\begin{pmatrix}
m_{11} & m_{12}\cr m_{21} & m_{22}\cr\end{pmatrix}}_M \begin{pmatrix}
1\cr 0\cr\end{pmatrix} + \begin{pmatrix}0\cr 1\cr\end{pmatrix}^{T}
\underbrace{\begin{pmatrix} m_{11} & m_{12}\cr m_{21} & m_{22}\cr\end{pmatrix}}_M
\begin{pmatrix} 0\cr 1\cr\end{pmatrix} = {\rm Tr}\; M =m_{11}+m_{22}$$
Only being interested in the trace as a result, in a matrix multiplication
the order of factor matrices can be changed cyclically. As in (37b) every
{\tt bra} and every {\tt ket} as well as each operator part between the two
$\vert$ is a matrix we can rotate factors to obtain
\begin{eqnarray}
\sigma_c(\tau)
\left[ {\bf 1}\; \delta(\tau_n-\tau)+G^r(\tau,\tau_n)\sigma(\tau_n)\right]
\quad \vert\Psi^0(\tau_n)>\; \cdot\nonumber \\
<\Psi^0(\tau_0)\vert \Psi^0(\tau_0)>\frac{i}{\hbar}<\Psi^0(\tau_m)\vert \;
\left[ {\bf 1}\; \delta(\tau_m-\tau)+\sigma(\tau_m)G^a(\tau_m,\tau)\right]
\end{eqnarray}
\renewcommand{\theequation}{\arabic{equation}}
\setcounter{equation}{39}
$<\Psi^0(\tau_0)\vert\Psi^0(\tau_0)>={\bf 1}$ again drops out.
$\vert\Psi^0(\tau_n)>\frac{i}{\hbar}<\Psi^0(\tau_m)\vert$ is of the structure
$g$, however, with no restriction as to which argument $\tau_n$ or $\tau_m$
is earlier or later in time. We define this new type of Green's function as
\begin{equation}
g^{+-}(\tau,\tau',\omega)=\frac{i}{\hbar}e^{-i\omega(\tau-\tau')}=
\begin{cases} -g^r(\tau,\tau',\omega) \quad {\rm if} \; \tau>\tau' \\
\phantom{-} g^a(\tau,\tau',\omega) \quad {\rm if} \; \tau<\tau' \end{cases}
\end{equation}
(An eventually ill-defined single point $\tau=\tau'$
is irrelevant for later integrations.)
As the cases $\tau>\tau'$ and $\tau<\tau'$ are mutually exclusive
the function can also be given by $g^{+-}(\tau,\tau')=g^a(\tau,\tau')-
g^r(\tau,\tau')$ (conclusions on $g^{+-}(\omega)$ from that are risky,
to my opinion, though). We have thus deduced the current formula
\begin{eqnarray}
I(\tau)&=&-e\; {\rm Tr}\; \{ \; \sigma_c(\tau) \int_{-\infty}^{\tau} d\tau_n
\int_{-\infty}^{\tau} d\tau_m \; \left[ ({\bf 1}\delta(\tau_n-\tau)+
G^r(\tau,\tau_n)\sigma(\tau_n)\right] \; \cdot\nonumber \\ && \cdot \;
g^{+-}(\tau_n,\tau_m) \; \left[ {\bf 1}\delta(\tau_m-\tau)+\sigma(\tau_m)
G^a(\tau_m,\tau) \right] \; \}
\end{eqnarray}
(like discussed in (20) it is rather irrelevant whether the upper integration
limits are set as $\tau$ or $\infty$)
or
\renewcommand{\theequation}{\arabic{equation}a}
\setcounter{equation}{40}
\begin{equation}
I(\tau)=-e\; {\rm Tr}\; \left\{ \sigma_c(\tau)\; G^{+-}(\tau,\tau) \right\}
\end{equation}
\renewcommand{\theequation}{\arabic{equation}}
\setcounter{equation}{41}
if $G^{+-}$ is defined as
\begin{eqnarray}
G^{+-}(\tau,\tau')=\int d\tau_1 \int d\tau_2 \; && [{\bf 1}\delta(\tau_1-\tau)
+G^r(\tau,\tau_1)\sigma(\tau_1)] \; g^{+-}(\tau_1,\tau_2) \cdot \nonumber \\
&& [{\bf 1}\delta(\tau_2-\tau)+\sigma(\tau_2)G^a(\tau_2,\tau')]
\end{eqnarray}
or
\renewcommand{\theequation}{\arabic{equation}a}
\setcounter{equation}{41}
\begin{equation}
G^{+-}=({\bf 1}+G^r\sigma)\; g^{+-}\; ({\bf 1}+\sigma G^a)
\end{equation}
\renewcommand{\theequation}{\arabic{equation}}
\setcounter{equation}{42}
in short notation.
The current only needs $G^{+-}$ of two identical time arguments.
We shall need the Fourier representation of $g^{+-}(\omega)$, but none such for $G^{+-}$.
Although in our simple model for the normal-conducting contact the current
will come out the same for any $\tau$, $I(\tau)$ generally does depend on time
(our expectation value is an ensemble, not a time average). For the
superconducting case the ac parts \cite{Cuequ} like the Josephson current are included in
the expression (41), although one is mostly interested in the contribution
in $I$ that is independent of $\tau$ and gives the dc part.
Using $g^{+-}(\tau_n,\tau_m)$ without a third argument $\omega$ in (41)
we understood integration over frequency like in (16). But this point
requires more care as it has not been taken into account so far whether
states are occupied or empty at $\tau_0\rightarrow -\infty$. Writing out the
trace from (41a) in LR-components reveals four contributions
\begin{eqnarray}
-I/e &=& \sigma_{c,LR}\; (1+G^r_{RL}\sigma_{LR})\; g^{+-}_{RR} \;
\sigma_{RL}G^a_{LL}\nonumber \\
&+& \sigma_{c,LR} \; G^r_{RR}\sigma_{RL}\; g^{+-}_{LL}\;
(1+\sigma_{LR}G^a_{RL}) \nonumber \\
&+& \sigma_{c,RL}\; G^r_{LL}\sigma_{LR}\; g^{+-}_{RR}\;
(1+\sigma_{RL}G^a_{LR})\nonumber \\
&+& \sigma_{c,RL}\; (1+G^r_{LR}\sigma_{LR})\; g^{+-}_{LL}\;
\sigma_{LR}G^a_{RR}
\end{eqnarray}
It is clear that in contrast to the other appearing $\sigma$, in $\sigma_c$
from (36) there is an additional relativ minus sign between the LR- and the
RL-component. Obviously the net current is the difference between the
current from left to right and the current from right to left. In the first
- as well as the second - term in (43) a transition from R to L at $\tau$,
the time argument of $\sigma_c$, is picked out to be counted for the current
(see also Fig.4). With $g^{+-}$ at R, the charge carrier is supplied from a
state originally located on the right. It is an electron if the energy lies
below the right-side Fermi level. Until $\tau$ the state has evolved to
again be on the right. The originally left behind empty state at the same
energy below the Fermi level must have evolved to be on the left at $\tau$
such that by the R$\rightarrow$L transfer the electron can go into it
(Fig.5a). Or you might say that the plus and the minus branch of the Keldysh
contor represent two possible parts in the evolution of an original wave
function $\Psi^0_R$, between which there is a non-vanishing matrix element
of the operator $\sigma_c=[H,\rho_L]$. However, still regarding the first
line of (43), there is the further possibility that an empty state from
above the Fermi level on the right evolves to be at R at $\tau$ again, but
its left-behind complement (a negative charge) has evolved to be at L.
$\sigma_{LR}(\tau)$ in this case means the transition of an empty state or
positively charged particle from right to left (Fig.5b). Although one does not
usually introduce the concept of holes with transport in normal
conducting metals, it makes sense here to call unoccupied states simply
''holes''. In this way the model already includes the dual nature of charge
carriers needed for the superconducting case. In the normal conducting
case states do not change in nature (electron or hole) or energy during
their evolution (little zigzags are drawn in Fig.5 only to make the
multiple hoppings visible). At the superconducting junction, Andreev
reflection can be interpreted as changing an electron into a hole or vice
versa and mirroring its energy at the Fermi level \cite{my06} (see Fig.9 in section 6).
To have a charge carrier at a certain energy level at time $\tau$ to make a
certain transition, it is important that there was one at the corresponding
energy in the original uncoupled system. The coupling may have changed the
distribution compared to the occupation in uncoupled bulk reservoirs. And
the applied voltage imposes a non-equilibrium situation, anyway.
For the evolution of a state from the right as shown in Fig.5 it does not
matter whether the state at the respective energy on the left is occupied
or empty. If, for example the regarded energy lies below the Fermi levels
both left and right, there will be two states evolving as an electron on the
plus and a hole on the minus branch of the Keldysh contor, one having originated
at R and the other at L. These original, uncoupled and independent states are
our basis, especially for calculating an expectation value as trace. They
do not interfere. Terms with $g^{+-}_{RR}$ and $g^{+-}_{LL}$ are simply
added in (43). Schemes analogous to Fig.5 could be drawn for the terms from
the last three lines of (43) as well. The conclusion of the whole argumentation
of how to let the original Fermi occupation function for the reservoirs left
and right enter the current calculation is that $g^{+-}_{LL}$ has to change
sign at the left Fermi energy and $g^{+-}_{RR}$ at the right Fermi energy.
Let us note $g^{+-}$ like $g^r$ and $g^a$ for any bulk reservoir with Fermi
level at $\omega=0$. We shall keep $g^{+-}$ corresponding to $g^r$ and $g^a$
as in (40) for occupied electron states below the Fermi level and change the
sign for empty states above it.
\begin{equation}
g^{+-}(\tau,\tau')=\frac{\mathcal{D}}{h}\int d\omega \; g^{+-}(\omega) \;
e^{-i\omega(\tau-\tau')} \; {\rm with} \; g^{+-}(\omega)=\begin{cases}
\phantom{-}i \; {\rm for}\; \omega<0 \\ -i \; {\rm for}\; \omega>0\end{cases}
\end{equation}
Although it might be practical to use (19), (23), (44), (18) and (36) in (41)
to quite directly produce an expression that calculates the current finally
as an integral over frequency and in our simple model can even be analytically
evaluated, in parallel to \cite{Cuequ} we shall use the transfer Green's functions
here. The current is best translated into an expression of the transfer
functions from the form already resolved into LR-components (43). Furthermore
eliminate $\sigma_c$ through $\sigma_{c,LR}=-\sigma_{LR}$ and
$\sigma_{c,RL}=\sigma_{RL}$. L- and R-indices follow logically from (26), for
example $G_{LL}\sigma_{LR}=g_{LL}T_{LR}$.
\renewcommand{\theequation}{43\alph{equation}}
\setcounter{equation}{0}
\begin{eqnarray}
I/e&=&\sigma_{LR}\; (1+g^r_{RR}T^r_{RR}) \; g^{+-}_{RR} \; T^a_{RL}g^a_{LL}
\nonumber \\
&+& \sigma_{LR} \; g^r_{RR}T^r_{RL} \; g^{+-}_{LL} \; (1+T^a_{LL}g^a_{LL})
\nonumber \\
&-& \sigma_{RL} \; g^r_{LL}T^r_{LR} \; g^{+-}_{RR} \; (1+T^a_{RR}g^a_{RR})
\nonumber \\
&-& \sigma_{RL} \; (1+g^r_{LL}T^r_{LL}) \; g^{+-}_{LL} \; T^a_{LR}g^a_{RR}
\end{eqnarray}
As quantities here are no longer matrices, but simply functions of time or
frequency, the leading $\sigma_{LR}$ and $\sigma_{RL}$ in the second and third
line of (43a) can be moved to the ends of the products. From (41) we remember
that their argument is $\tau$, the same as the second argument of the last
factor to the right. Then using relations from (27a) and complementary forms
the current formula simplifies to
\begin{eqnarray}
I/e&=& T^r_{LR} g^{+-}_{RR} T^a_{RL}g^a_{LL} \; + \;
g^r_{RR} T^r_{RL} g^{+-}_{LL} T^a_{LR} \nonumber \\
&-& g^r_{LL} T^r_{LR} g^{+-}_{RR} T^a_{RL} \; - \;
T^r_{RL} g^{+-}_{LL} T^a_{LR} g^a_{RR}
\end{eqnarray}
\renewcommand{\theequation}{\arabic{equation}}
\setcounter{equation}{44}
The terms with the factors 1 from the brackets have elegantly been made to
vanish. Now we use the Fourier representations for all functions in (43b).
As all terms follow the same scheme, the second is treated in an exemplary way
here (all integrals run from $-\infty$ to $+\infty$):
\begin{eqnarray}
\int d\tau_1\; d\tau_2\; d\tau_3\; d\omega_1\; d\omega\; d\omega_2\; d\omega_3\;
\frac{\mathcal{D}}{h} g^r_{RR}(\omega_1) e^{-i\omega_1\tau} e^{i\omega_1\tau_1}
\frac{\hbar}{2\pi\mathcal{D}}T^r_{RL}(\omega_2)\nonumber \\
e^{-i(\omega_2+eV/\hbar)\tau_1} e^{i\omega_2\tau_2} \;
\frac{\mathcal{D}}{h} g^{+-}_{LL}(\omega) e^{-i\omega\tau_2} e^{i\omega\tau_3}
\frac{\hbar}{2\pi\mathcal{D}} T^a_{LR}(\omega_3)
e^{-i\omega_3\tau_3} e^{i(\omega_3+eV/\hbar)\tau} \nonumber
\end{eqnarray}
Doing the $\tau_1$,$\tau_2$- and $\tau_3$-integrals gives $(2\pi)^3\delta(
\omega_1-\omega_2-eV/\hbar)\delta(\omega_2-\omega)\delta(\omega-\omega_3)$.
Then even the exponentials with $\tau$ cancel and the term simplifies to
$$ \frac{1}{2\pi}\int d\omega\; g^r_{RR}(\omega+eV/\hbar)\; T^r_{RL}(\omega)\;
g^{+-}_{LL}(\omega)\; T^a_{LR}(\omega) $$
In our case (31) tells us that $T^{r/a}_{LR/RL}(\omega)$ are all identical and
real, $g^r_{LL/RR}(\omega)=-i$ is the complex conjugate of $g^a_{LL/RR}(\omega)
=i$ and $-g^{+-}_{LL/RR}$ is the complex conjugate of
$g^{+-}_{LL/RR}$ of the same $\omega$ as $g^{+-}(\omega)$ is purely imaginary,
too. Thus the last two terms in (43b) are the complex conjugates of the first
two and thus twice the real part of these first two can be taken for $I/e$.
In the superconducting version of the model, where $T$ and $g$ actually are
$\omega$-dependent, complex conjugate relations \cite{my06} also exist between
$T$s as well as $g$s, and the current formula can be reduced in the same way.
Just to note the quite general formula \cite{Cuequ} in short form:
\begin{equation}
I/e=2\; {\rm Re}\; \left\{ T^r_{LR} g^{+-}_{RR} T^a_{RL} g^a_{LL} +
g^r_{RR} T^r_{RL} g^{+-}_{LL} T^a_{LR} \right\}
\end{equation}
For the integrand from the second term in our model we get
$$ g^r_{RR}(\omega+eV/\hbar) T^r_{RL}(\omega) g^{+-}_{LL}(\omega) T^a_{LR}(
\omega)=(-i) \frac{t}{1+t^2} (\mp i) \frac{t}{1+t^2} =
\mp \frac{t^2}{(1+t^2)^2} $$
where the signs refer to $\omega$ greater or less than zero and come out reversed
for the first term with $g^{+-}_{RR}$, because there is $g^a$ instead of $g^r$.
Care has to be taken with the reference point for $\omega$ in both terms.
This may easily be overlooked in the normal conducting case here in contrast
to the superconducting case where $T$ indeed is $\omega$-dependent and like
$G$ as a function of only one argument always referred to the same Fermi
level (the left, for example). If we call the argument of $g^{+-}_{LL}$
from the second term $\omega$, the one for $g^{+-}_{RR}$ in the first term
is $\omega+eV/\hbar$. On an $\omega_{(L)}$-axis, the second term
changes sign at zero, however, the first jumps at $-eV/\hbar$ (Fig.6). A shift
of the integration parameter cannot be made independently for both terms.
Thus, for the normal conducting model here
\renewcommand{\theequation}{45\alph{equation}}
\setcounter{equation}{0}
\begin{eqnarray}
I/e=2\; {\rm Re} \; \frac{1}{2\pi} \int d\omega \; \{ \; T^r_{LR}(\omega)
g^{+-}_{RR}(\omega+\frac{eV}{\hbar}) T^a_{RL}(\omega) g^a_{LL}(\omega)
\nonumber \\ + \;
g^r_{RR}(\omega+\frac{eV}{\hbar}) T^r_{RL}(\omega) g^{+-}_{LL}(\omega)
T^a_{LR}(\omega) \; \} \end{eqnarray}
\renewcommand{\theequation}{\arabic{equation}}
\setcounter{equation}{45}
In principle the convention is needed, that the $T$-argument always refers
to the left side, but for $g$ general formula like (44) for a single bulk
with Fermi level at zero frequency are applied. A more involved situation
where choosing integration intervals consistently for all contributing current
terms is crucial can be found in \cite{my06}.
\begin{figure} \begin{center}
\includegraphics[width=12cm]{evolution.jpg}
\end{center}
\caption{(a) Evolution of an electron state from R such that at $\tau$ the
charge is transferred to the left. (b) Evolution of an unoccupied state above
the Fermi level from R such that at $\tau$ it gets filled by a charge from
the left or the hole state shifts to the left. The lower pictures sketch the
effect of the original uncoupled state for the current. Sending electrons
from the right to the left will be compensated by electrons sent from left
to right. But with levels as in Fig.1c for holes sent from right to left there
will be a range between the two Fermi levels where there are no counterbalancing
holes going the other way. (For the left, electrons going to the right are not
outweighed in this range.)}
\end{figure}
\begin{figure} \begin{center}
\includegraphics[width=8cm,angle=270]{integrand.jpg} \end{center}
\caption{The two terms under the integral from equation (45). They cancel
except on an interval of length $eV/\hbar$.}
\end{figure}
From Fig.6 it is easily seen that that the integral is twice the constant
$\frac{t^2}{(1+t^2)^2}$ integrated over an interval of length $eV/\hbar$,
and outside that interval contributions cancel. With the other factor 2 from
twice the real part and the prefactor $1/2\pi$
the result for the current finally is
\begin{eqnarray} &&
I/e=2\cdot \frac{1}{2\pi}\cdot 2\cdot \frac{t^2}{(1+t^2)^2}\cdot\frac{eV}{\hbar}
\quad\quad \Longleftrightarrow \nonumber \\ &&
\frac{V}{I}=\frac{h}{e^2}\left(\frac{4t^2}{(1+t^2)^2}\right)^{-1}
\qquad \Longleftrightarrow \qquad
\frac{I}{V}=\frac{e^2}{h} \frac{4t^2}{(1+t^2)^2}
\end{eqnarray}
The factor $\theta=4t^2/(1+t^2)^2$ is the conductance in units of the quantum
conductance or its inverse the resistance in units of the quantum resistance
\cite{Cuequ}.
(The conductance of a channel doubles if two spin states are allowed. Then
$h/2e^2=13k\Omega$ should be taken as resistance unit.)
$\theta$ is the transmission probability of the conductance channel through
the junction we regarded. The result that in the normal-conducting case
the current is proportional to the voltage is not at all surprising, of course.
The non-trivial result is the conversion of the quantum mechanical transmission
amplitude $t$ to the measurable transmission probability $\theta$.
$\theta=0$ if $t=0$. And a totally open channel with $t=1$ has
transmission probability $\theta=1$.
It may seem a contradiction on the one hand calculating the current from a
changing amount of charge on one side and on the other hand saying that missing
charges are replaced and superfluous ones led away by the voltage source.
A slightly different viewpoint may help to get convinced that the calculated
quantity is indeed the current in the stationary, but non-equilibrium system.
The crucial point was putting on to evaluate $d\rho_L/dt$ in state $\Psi(t)$.
Without further ado we could not tell whether this state of the coupled system
was occupied or not. The Schr\"odinger equation (5) set up the left and right
material properties as well as the coupling across the junction, however,
did not take any account of the effect of the voltage source. Without need
to specify real locations for the division, just principally view our structure
as consisting of a junction region and leads. The $\Psi(t)$ defined through (5)
describes states in the junction region. But think of them as offered by the
system and following their time development whether occupied or not. The leads
are always occupied exactly up to their Fermi levels. Electrons freshly
supplied by the voltage source need not be in phase with present ones. A
random phase is most easily modelled by assuming the left lead wave function
at any time $\begin{pmatrix} 1 \cr 0\cr\end{pmatrix}$ without a phase in contrast to the
assumed $\begin{pmatrix} \exp(-i\omega_L\tau)\cr 0\cr\end{pmatrix}$ for the left side of the
junction region. Which states to which extent actually get occupied in the
left side of the junction region, that is $\rho_L(t)$, is determined by the
overlap of $\Psi(t)$ with the phaseless left lead wave function
$\begin{pmatrix} 1\cr 0\cr\end{pmatrix}$,
which we recognize as $\Psi_L^0(-\infty)$.\ $\rho_L=\vert <
\begin{pmatrix}1\cr 0\cr\end{pmatrix}\vert\begin{pmatrix}\Psi_L(t)\cr 0\cr\end{pmatrix}>\vert^2$. The current
then is the change in time of $\rho_L$ due to charge flow through the junction,
that is processes inside the junction region only, corresponding to what our
Hamiltonian was set up for. A similar line of thought with state overlaps
can be applied to the passing of charges out of the junction region into
the (right) lead; it may be helpful to view this as putting empty states or
holes into the junction region, though. Of course, only the L-part of $\Psi(t)$
can overlap with $\begin{pmatrix} 1\cr 0\cr\end{pmatrix}$. $\begin{pmatrix}
\Psi_L(t)\cr 0\cr\end{pmatrix}$ can thus
be replaced by $\Psi(t)$ in our new expression for $\rho_L$. Recalling the
operator definition (34) the time derivative of $\rho_L$
$$
\frac{d}{dt}\; \rho_L=\frac{d}{dt}\; \vert < \begin{pmatrix}1\cr 0\cr\end{pmatrix}\vert \Psi(t)>
\vert^2=\frac{d}{dt} \left\{ <\Psi(t) \vert \begin{pmatrix}1\cr 0\cr\end{pmatrix}><
\begin{pmatrix} 1\cr 0\cr\end{pmatrix} \vert \Psi(t) > \right\}
$$
is identical to the ansatz made by putting together (33) and (34) in (37).
\begin{figure}\begin{center}
\includegraphics[width=8cm,angle=270]{koppel.jpg}
\end{center}
\caption{Hypothetical division of the structure into the actual junction
region and bulk leads.}
\end{figure}
\subsection{Comment on the two-level system}
At first glance the problem posed by the Schr\"odinger equations without and
with coupling, (1) and (5), especially if we regard a single energy level on
each side with corresponding $\omega_L$ and $\omega_R$ as in Fig.1c, looks
like the two-level system known from quantum mechanics textbooks \cite{CoTan}.
By the coupling the two energy levels are shifted. The new eigenstates lie
further apart from one another. If the system is initiated in one of the
uncoupled states, it will oscillate harmonically between the two levels,
and both both the period of the oscillation as well as the maximum
transition probability to the other state depend on the energy difference of
the original states.
\begin{figure}\begin{center}
\includegraphics[width=7cm]{twolevel.jpg}
\end{center}
\caption{Attempt to adjust energy levels from left and right to same
reference.}
\end{figure}
Even if the time dependence of the interaction (18) can be got rid of by
changing to the interaction picture, it makes no sense to numbly calculate
eigenvalues and eigenvectors of a Hamiltonian $\begin{pmatrix} \hbar\omega_L & W\cr
W & \hbar\omega_R\cr\end{pmatrix}$. It is not clear from which reference levels such
eigenenergies $E_{\pm}$ should be counted as zero levels are different for
$\omega_L$ and $\omega_R$. If one tries to refer the left and right energies
already of the uncoupled system to the same reference level, $E_F(R)+\frac{eV}{2}$
for example (Fig.8), $\Psi^0_L$ (and $\Psi^0_R$) cannot be taken as eigenstates any
more, because from $i\hbar\partial_{\tau}e^{-i\omega_L\tau}=\hbar\omega_L
e^{-i\omega_L\tau}$ with $\omega_L=\omega_L'+\frac{eV}{2}$ it does {\it not}
follow that $i\hbar\partial_{\tau}e^{-i(\omega_L'+eV/2)\tau}$ equals
$\hbar\omega_L'e^{-i(\omega_L'+eV/2)\tau}$. The standard treatment of the
two-level system is not applicable to the non-equilibrium situation.
\subsection{Superconducting junction}
Despite having been the simplest example to introduce the Green's functions
scheme, applying the formalism to the normal conducting junction to get out that
the current is proportional to the applied voltage was breaking a butterfly
on the wheel. Although repeatedly mentioned, fully developing the extension
to the superconducting case is beyond the scope of this presentation. But the
resulting current-voltage characteristics shall be shown as a plea for the
usefulness of the method. A different approach based on matching wave
functions \cite{AveBar} leads to identical results, though.
\begin{figure} \begin{center}
\includegraphics[width=8cm,angle=270]{andreev.jpg}\end{center}
\caption{Illustration of Andreev reflection as mirroring at the Fermi levels
for electron-hole conversion and vice versa. For either side the incoming and
reflected levels can lie inside or outside the gap. Only the beginning and end
of a complete multiple-reflection process have to be in the electron or hole
reservoir below or above the gap, respectively.}
\end{figure}
Some formula shall be listed, because they are not necessarily written out
in complete analogy to the presentation here in \cite{Cuequ} and other literature.
Working in the quasiparticle picture, in the superconducting case each entry
in LR-space of a Green's or transfer function expands into another 2x2 matrix
in Nambu space over electrons and holes. There is \cite{ketterson_2}:
\begin{equation}
g^{r/a}_{LL/RR}(\omega)=\begin{pmatrix} g^{ee} & g^{eh} \cr g^{he} & g^{hh} \cr\end{pmatrix}^{
r/a}_{LL/RR}(\omega)=\frac{1}{\sqrt{(\Delta/\hbar)^2-(\omega\pm i\eta)^2}}
\begin{pmatrix} -\omega\mp i\eta & \Delta/\hbar \cr -\Delta/\hbar & \omega\pm i\eta\cr\end{pmatrix}
\end{equation}
$g$ is the same for LL and RR. Different signs refer to the retarded and advanced
function. $\Delta$ is half the gap of the superconductor. The small imaginary
part $\eta$ is added to get the correct root besides slightly smoothening
singularities. $g(\tau,\tau') =\mathcal{D}/h\; \int d\omega \; g(\omega)\;
e^{-i\omega\tau} \; e^{i\omega\tau'}$ still holds.
\begin{equation}
\sigma_{LR/RL}=\begin{pmatrix} \sigma^{ee} & 0 \cr 0 & \sigma^{hh}\end{pmatrix}_{LR/RL}=
\frac{\hbar}{\mathcal{D}}\;
\begin{pmatrix} t\; e^{\pm i eV\tau/\hbar} & 0 \cr 0 & -t\; e^{\mp ieV\tau/\hbar}\end{pmatrix}
\end{equation}
Here signs refer to the LR- and RL-direction, respectively. Remark the
reversed signs for holes in the exponential compared to electrons.
There is no electron-hole conversion during a single hopping. The $eh$-
and $he$-elements of $g$ introduce so-called Andreev reflection.
(Cooper-pair tunneling \cite{Urbina} is not included.) $T$ is a full matrix in $eh$-space
and has to be set up as
\begin{equation}
\frac{\mathcal{D}}{\hbar} \; T(\tau,\tau') =
\frac{1}{2\pi} \sum_n \int d\omega \; T_n(\omega)\;
e^{-i\omega\tau} \; e^{ineV\tau/\hbar}\; e^{i\omega\tau'}
\end{equation}
We skipped the distinction of reference $\omega_{J/K}$ as in (25), because in
practice only one type, e.g. $T_{LR}$, has to be calculated, as then $T_{RL}$
follows from complex conjugation. $n$ runs over all integers. A Dyson equation
like (32) leads to a recursion which connects $T_n$ to $T_{n-2}$ and $T_{n+2}$.
It can be solved by reasonably truncating the $n$-range. Then, just also
summing over $n$, the established current formula (45) can be used.
Expressions like (32) have to sum over all possible combinations of $e$- and
$h$-indices. For example:
\begin{eqnarray}
T^{ee}_{LR}=\sigma_{LR}^{ee}&+&
\sigma_{LR}^{ee}g^{ee}_{RR}\sigma_{RL}^{ee}g^{ee}_{LL}T^{ee}_{LR}+
\sigma_{LR}^{ee}g^{eh}_{RR}\sigma_{RL}^{hh}g^{he}_{LL}T^{ee}_{LR}\nonumber \\ &+&
\sigma_{LR}^{ee}g^{ee}_{RR}\sigma_{RL}^{ee}g^{eh}_{LL}T^{he}_{LR}+
\sigma_{LR}^{ee}g^{eh}_{RR}\sigma_{RL}^{hh}g^{hh}_{LL}T^{he}_{LR}
\end{eqnarray}
\begin{figure}\begin{center}
\includegraphics[width=7cm,angle=270]{cuevkurv.jpg}\end{center}
\caption{Calculated current-voltage curves for transport channels of three
different transmissions (normalized to $\theta$) [reproduced after \cite{Cuequ}].
The normal conducting $IV$ is added for comparison (keep $\Delta$ just for units).}
\end{figure}
To point out the Andreev reflection in the model, we look at a term
$g\sigma g\sigma g$ with alternating $e$- and $h$-indices from the
development (21). Such a sequence will also be contained as parts in
higher-order terms from (21) or (24). Contracting inner time arguments
results in
\begin{eqnarray} &&
\int d\tau_1\; d\tau_2\; g^{(h)e}_{RR}(\tau,\tau_1) \; \sigma^{ee}_{RL}(
\tau_1)\; g^{eh}_{LL}(\tau_1,\tau_2)\; \sigma^{hh}_{LR}(\tau_2) \;
g^{h(e)}_{RR}(\tau_2,\tau') \nonumber \\ &=&
\int d\tau_1\; d\tau_2\; d\omega_2 \; d\omega_1\; d\omega\;
\frac{\mathcal{D}}{h} g^{(h)e}_{RR}(\omega_2)e^{-i\omega_2\tau}e^{i\omega_2\tau_1}\;
\frac{\hbar}{\mathcal{D}} t e^{-ieV\tau_1/\hbar} \; \cdot
\nonumber \\ && \phantom{++}
\frac{\mathcal{D}}{h} g^{eh}_{LL}(\omega_1)e^{-i\omega_1\tau_1}e^{i\omega_1\tau_2}\;
\frac{\hbar}{\mathcal{D}} (-t) e^{-ieV\tau_2/\hbar} \;
\frac{\mathcal{D}}{h} g^{h(e)}_{RR}(\omega) e^{-i\omega\tau_2}e^{i\omega\tau'}
\nonumber \\ &=& (-t^2)\; \frac{\mathcal{D}}{h} \int d\omega \; d\omega_1\; d\omega_2\;
\delta(\omega_2-\frac{eV}{\hbar}-\omega_1) \;
\delta(\omega_1-\frac{eV}{\hbar}-\omega) \; \cdot \nonumber \\ && \phantom{+++++}
g^{(h)e}_{RR}(\omega_2) \; g^{eh}_{LL}(\omega_1) \; g^{h(e)}_{RR}(\omega) \;
e^{-i\omega_2\tau}\; e^{i\omega\tau'} \nonumber \\ &=&
-t^2\; \frac{\mathcal{D}}{h}\int d\omega\; g^{(h)e}_{RR}(\omega+2\frac{eV}{
\hbar})\; g^{eh}_{LL}(\omega+\frac{eV}{\hbar}) \; g^{h(e)}_{RR}(\omega)\;
e^{-i(\omega+2eV/\hbar)\tau}\; e^{i\omega\tau'} \nonumber \\ && \phantom{Zeile}
\end{eqnarray}
because the $\tau_1$ and $\tau_2$-integration give $4\pi^2\delta(\omega_2-
eV/\hbar-\omega_1)\delta(\omega_1-eV/\hbar-\omega)$.
In Fig.9 for holes the energy axis has to be reversed as compared to electrons.
Therefore the initial $\omega$ is negative. For the argument here, it does
not matter if the beginning factor in (51) is $g^{he}_{RR}$ or $g^{ee}_{RR}$.
If it were $h$, then the Andreev reflection marked by the dashed arrow would
follow. The last index of the last $g_{RR}$ could also be $h$ instead of $e$.
Whereas in the normal conducting case in multiple reflections the charge
carrying particle always came back to the same energy with regard to the
Fermi level when coming back to the same side of the junction (Fig.1), in
(51) the frequency argument of the first $g_{RR}$ is shifted by twice the
applied voltage equivalent with respect to the second $g_{RR}$
(Fig.9). Such shifts enable transport even when the gaps of the left and
right side are still overlapping, that is for voltages below $2\Delta/e$
(Fig.10). In the $IV$-curves steps towards lower voltages at fractions of
$2\Delta$ are associated to ever higher-order Andreev reflections. Of course,
these current contributions
become the more prominent the greater the channel transmission.
Besides the step heights and positions the curvature on each step is an
important signature. The model agrees excellently with experimental results
\cite{Sche97}.
The interference of transport processes of different order is correctly
taken into account. Simpler models \cite{Bratus} that add up tranfer probabilities
proportional to $\theta^{order}$ cannot reproduce these characteristics and
are only valid for low transmissions $\theta<<1$.
\section{Conclusions and Outlook for section III}
It has been explained on a quite basic level how quantum transport between
two reservoirs in a stationary non-equilibrium state can be modelled. The
purpose was to present a Green's functions technique for handling coupling
in the context of a field of current research interest, namely transport
through point contacts. Although besides the general formalism (also see section I)
requiring the development of quite some more subject specific mathematical
framework such as the transfer Green's functions, the transport through a
contact with arbitrary transmission is a suitable example to illustrate the
inclusion of interaction up to all orders in the implicit Dyson equations.
Calculations
have been carried through in every detail for the normal-conducting single
junction. Basic formula and results have been given for the superconducting
junction. The decomposition into transport channels (eigenmodes) of a point
contact can be inferred from its superconducting transport characteristics
which can be taken like a PIN-code \cite{Sche97}. The presented Green's functions formalism
has a great potential for extension. Systems of two more or less coherently
linked junctions can be modelled \cite{my06}.
Models for transport through molecules \cite{Heurich,Roland,Wohltat} or
atomic chains \cite{Mertig} so far mainly rely on ab initio calculations of
the density of states. Time dependent density functional theory for
non-equilibrium situations is also developed \cite{Kurth}.
\begin{acknowledgements}
The author thanks Alain Dereux for learning about Green's functions in
electrodynamics, Juan-Carlos Cuevas for through his former work introducing
her to Green's functions in quantum transport, her supervisor Elke Scheer
for the freedom to work on this subject as well as Klaus-Ulrich Neumann
for the suggestion to write a paper of this kind starting from some
lecture notes.
\end{acknowledgements}
| {
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function (error) {
//stay here it didn't work
vm.deleting = false;
console.log("delete error:", error);
});
});
}
//Don't touch this guy, hes a nice little fella' and we don't want to hurt him or break this controller by deleting him.
}//You're safe with me little curly bracket.
//Safe and sound.
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VWR supports Notre Dame Research to detect Low-Quality Medicines in Kenya
Author: Joya Helmuth
In partnership with the Notre Dame Initiative for Global Development (NDIGD), Professor of Chemistry Marya Lieberman and her research team have received support from VWR , a leading global independent provider of product and service solutions to laboratory and production customers. Through VWR's longstanding agreement with Notre Dame, VWR will donate $10,000 of lab chemicals and consumables to the Paper Analytical Device (PAD) Project annually for at least the next three years.
Professor Marya Lieberman at the University of Notre Dame, researchers at Purdue University, and Moi Teaching and Referral Hospital are working on an inexpensive, easy-to-use method to detect low quality medicine in Kenya and other countries in East Africa. Samples are collected by covert shoppers, and rapidly screened using a paper analytical device, or PAD. Confirmatory testing by HPLC is being conducted in several locations through the Distributed Pharmaceutical Analysis Laboratory (DPAL).
"We are thrilled to get this donation to increase the capacity of the PAD project for detecting fake pharmaceuticals," says Dr. Lieberman. "My lab at Notre Dame has three refrigerators full of suspicious samples from the PAD project that need to be assayed. The Distributed Pharmaceutical Analysis Lab (DPAL) matches those suspicious samples with educational institutions that have analytical instrumentation, and hundreds of students at the partner institutions analyze these samples in their analytical lab classes. Now we will be able to develop PADs for screening new types of pharmaceuticals, and provide our DPAL partners with lab supplies from VWR. This support will help us expose fake medicines that are sold in the developing world."
Within the Keough School of Global Affairs, NDIGD builds relationships with private sector partners to further Notre Dame research and promote human development in developing countries around the globe. Other current partnerships include the Mandela Washington Fellowship for Young African Leaders, supported by IBM and Coca-Cola; and The Connectivity, Electricity, and Education for Entrepreneurship (CE3) Project, supported by Accenture and Lenovo. More information on working with NDIGD can be found here.
The PAD project, started by Dr. Lieberman and her research team in partnership with the University of Notre Dame Eck Institute for Global Health, has received numerous research grants, including most recently a USAID Development Innovation Ventures (DIV) grant.
Contact: Joya Helmuth, NDIGD Outreach and Training Associate Director, jhelmuth@nd.edu
VWR (NASDAQ: VWR), headquartered in Radnor, Pennsylvania, is the leading, global, independent provider of product and service solutions to laboratory and production customers. With sales in excess of $4.3 billion in 2015, VWR enables science for customers in the pharmaceutical, biotechnology, industrial, education, government and healthcare industries. With more than 160 years of experience, VWR has cultivated a value proposition delivering product choice, operational excellence and differentiated services to improve our customers' productivity from research to production. VWR's differentiated services provide innovative, flexible and customized solutions from scientific research services to custom-manufactured chemical blends. Their dedicated team of more than 9,300 associates is focused on supporting scientists, medical professionals and production engineers to achieve their goals. | {
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import("packages")
function main(...)
for plat, pkgs in pairs(packages()) do
cprint("${magenta}%s${clear}:", plat)
for _, pkg in ipairs(pkgs) do
if pkg.generic then
cprint(" ${yellow}->${clear} %s", pkg.name)
else
cprint(" ${yellow}->${clear} %s (%s)", pkg.name, table.concat(pkg.archs, ", "))
end
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,222 |
Q: Algebra confusion: $A^2 + B^4$? So i am currently trying to solve a practice question and i have hit a stump.
The first term of a GP is $1$ and the sum of the third and fifth term is $90$. Find the common ratio.
Using $T_n = a \cdot r^{n-1}$ I got that $a = 1$
Hence
$ar^2 + ar^4 = 90 \implies r^2 + r^4 = 90$
But this is where i got stumped. The answer is given as $ \pm \;3$
Any help is highly appreciated.
A: All of your calculations are right. We find ourselves needing to solve $r^2+r^4=90$.
Rearranging gives $r^4+r^2-90=0$. This can be thought of as a quadratic in $r^2$. Let $y=r^2$ then $y^2=r^4$ and so $r^4+r^2-90=0$ becomes $y^2+y-90=0$.
It turns out that $y^2+y-90$ factorises to gives $(y-9)(y+10)$.
Since $y$ was shorthand for $r^2$ we see that $r^4+r^2-90 \equiv (r^2-9)(r^2+10)$.
The solutions to $(r^2-9)(r^2+10) =0$ are $r^2=9$ and $r^2 = -10$. The equation $r^2=9$ has solutions $r=\pm 3$, while the equation $r^2=-10$ has no real solutions.
A: You have the equation $R^2+R^4=90$ If we define $x=R^2$ this becomes $x+x^2=90$, which is a quadratic and you solve it by the usual techniques. But then you have to take the square root of $x$ to get $R$ and it can be either sign.
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{"url":"https:\/\/www.physicsforums.com\/threads\/textpad-plain-text-processor.58934\/","text":"# TextPad - Plain text processor.\n\n1. Jan 7, 2005\n\n### Staff: Mentor\n\nTextPad\u00ae 4.7.3 is a powerful, general purpose editor for plain text files.\n\nA 'free' trial copy is available, but it is inexpensive, and quite powerful, and it is available in several languages.\n\nI have used a paid copy for over 7 years. It handles text files over 100 MB without choking. One copy blocks of text or single columns, sort, create macros. Saves files in PC, UNIX, Mac formats.\n\n2. Jan 8, 2005\n\n### poolwin2001\n\nI use it to compile and run my Java programs.TextPad Rocks!\n\n3. Jan 8, 2005\n\n### dduardo\n\nStaff Emeritus\nI've gotten used to vim that I don't like to use anything else.\n\n4. Jan 10, 2005\n\n### ramollari\n\nIt's good that TextPad demo version is fully functional except that it reminds you once in a while to purchase it.\n\n5. Jan 10, 2005\n\n### gnome\n\nFor a good free Windows editor that doesn't nag you for \\$ try Notepad2.\n\n6. Jan 11, 2005\n\n### franznietzsche\n\nI love note pad. People in my class don't understand why. I jsut tell them it isn't loaded with the extraneous crap that word has. Word is useful for some things, but not for writing code. But a context sensitive notepad? Greatness.\n\n7. Jan 13, 2005\n\n### ramollari\n\nWord is out of question. As soon as you start writing code it will start auto-corrections and underlining of errors. The simpler the better. Personally I hate tools overloaded with features. They are slow, difficult to use, and may crash. Notepad would do for me to write any kind of source file.\n\n8. Jan 18, 2005\n\n### poolwin2001\n\nYou can allways disable the features you dont like.\n\nLast edited: Jan 18, 2005","date":"2017-06-26 02:09:54","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\": 0.2449633628129959, \"perplexity\": 7214.739078439461}, \"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-2017-26\/segments\/1498128320666.34\/warc\/CC-MAIN-20170626013946-20170626033946-00698.warc.gz\"}"} | null | null |
Andrés Jiménez Fernández (Carmona, provincia de Sevilla, 6 de junio de 1962), fue un jugador de baloncesto español de los años 80 y 90. Destacó como jugador del F.C. Barcelona, con el que ganó siete ligas ACB y cuatro Copas del Rey, entre otros títulos, y como integrante de la Selección de baloncesto de España, con la que disputó 186 partidos, participando en la consecución de la medalla de plata en los Juegos Olímpicos de Los Ángeles 1984, y en la del Eurobasket de Nantes de 1983. Jugaba en la posición de ala-pívot, y medía 2,05 metros de altura.
Se retiró como jugador en activo en junio de 1998. El 13 de septiembre de 1998 el F.C. Barcelona le hizo un homenaje retirándole la camiseta con su número 4, y colgándola en el Palau Blaugrana, en reconocimiento a sus méritos y dedicación al club.
Carrera
El Cotonificio de Badalona de Aíto, Héctor Perotas, Agustín Cuesta, Jiménez, Joaquim Costa y compañía es una especie de leyenda del básquet español. Con un presupuesto ridículo se instaló en la élite del básquet nacional firmando varias campañas de enorme mérito llegando a ser juez de la liga 77/78 derrotando al Madrid y entregándole el título al Joventut de Badalona de Zoran Slavnic. Son varios los detalles que hacían de aquel equipo algo inhabitual en la época: entrenaban mañana y tarde todos los días (no era lo habitual), realizaban una defensa que fue bautizada como karate pres (se empezó a usar la defensa de "saltar y cambiar" dos años antes de que Díaz-Miguel la usara en la selección) y sobre todo, por jugar con Andrés Jiménez por fuera, algo que Aíto empezó a usar en el Coto y que luego pulió en el Joventut de Badalona y en el Barcelona siempre con el mismo jugador como protagonista.
La progresión de Jiménez fue constante y se convirtió en internacional en el 80, formando parte de la gloriosa generación baloncestística que logró las platas de Los Ángeles y Nantes.
Cuando Aíto ficha por el Joventut en la temporada 83/84 se lleva consigo a su jugador-talismán y juntos revitalizan a la Penya, que venía de quedar séptimo, firmando dos temporadas con un tercer puesto (84/85) y una final de liga perdida contra el Madrid pese a la portentosa exhibición de los Jimix (sobrenombre con el que Andrés firma los cómics que dibuja), Jordi Villacampa, José Antonio Montero y compañía en el primer partido de la serie en Madrid.
En la temporada 85/86, el camino de Jiménez se separa por primera vez del de Aíto, al fichar éste por el Barcelona. Su sustituto en el Joventut de Badalona será Miquel Nolis y el equipo badalonés quedará tercero en la clasificación mientras el Barcelona perderá la final contra el Real Madrid.
A comienzos de la temporada 86/87, Aíto reúne en el Barcelona al triángulo mágico del Cotonificio Badalona (Costa, Jimix y él mismo), que unidos a los Epi, Solozábal, Sibilio y compañía dan como resultado un equipo inabordable, más aún si tenemos en cuenta que el Joventut de Badalona queda debilitado por la ausencia de Jiménez y el Madrid por la de Martín (en su fugaz paso por el banquillo de Portland). Ese equipo logra todos los títulos en juego (Liga, Copa y Copa Korac) y, con el importantísimo añadido de Audie Norris, inicia la era de mayor dominio que ha conocido la ACB y parte totalmente esencial de dicho dominio hay que otorgársela a la figura de Andrés Jiménez, que por entonces ya juega casi siempre de tres, siendo prácticamente indefendible.
Su más que decente tiro de 4-5 metros le permite ser una amenaza para sus defensores en esas distancias y su velocidad y notable coordinación propician que sea capaz de dejarles atrás con relativa facilidad. Cerca del aro, sus movimientos son veloces y pulidos, lo que unido a su altura le permite rendir con solvencia también ahí. Su categoría de "hombre que marca la diferencia" se pone de manifiesto al comprobar la frecuencia con la que Lolo Sainz, entrenador del Real Madrid, cambia de alero americano en sus vanos intentos por pararle (ni Larry Spriggs, ni Wendell Alexis, ni Linton Townes, ni Johnny Rogers, ni Anthony Frederick lo conseguirán en los momentos decisivos). En esa época sólo tiene un lunar: los tiros libres. Su porcentaje suele ser más que discreto y es algo que no logró corregir hasta que su carrera tocaba a su fin.
En el verano del 87 forma parte de la selección española que queda cuarta en el Europeo de Atenas y su sensacional trabajo es premiado con la inclusión en el mejor quinteto del campeonato.
Tras cuatro ligas consecutivas y toparse dos veces en la Final Four con la Jugoplastika, Aíto deja su puesto de entrenador al serbio Bozidar Maljkovic y en esa temporada Jiménez sufre la lesión más importante de su carrera que le tendrá más de media temporada lejos de las canchas, no pudiendo disputar ni la final de la Copa de Europa ni la de liga (ambas perdidas, respectivamente, ante Jugoplastika y Joventut). Es el prólogo a 2 años de sequía para el F.C. Barcelona, que incluirán la peor temporada ACB de su historia (92/93, séptimos).
La progresión de Jiménez va un poco más allá con Bozidar Maljkovic y con el regreso de Aíto y empieza a probar suerte desde más allá de la línea de 6,25, quizá fruto de los años que le hacen perder velocidad hasta el punto de que en sus últimos años ya prácticamente nunca se le ve en sus clásicas acciones fuera-dentro, centrándose en el tiro exterior y en jugar al poste bajo, jugando con la inteligencia que siempre le caracterizó, supliendo su cuesta abajo física producto de la edad.
Jiménez cierra su carrera con tres ligas más consecutivas, la última de ellas conquistada en la pista del Madrid en un extraordinario quinto partido en la temporada 96/97 en el cual tiene un papel más que digno pese a sus 35 años. Con motivo de su retirada, fue objeto de diversos homenajes, tanto por parte de su último club (que le organizó un desconcertante partido de homenaje en el que no jugó y le retiró la camiseta) como por parte de la ACB al ser el último representante de la generación que triunfó en Los Ángeles en retirarse.
Tuvo un hermano menor también baloncestista, Paco Jiménez (n. 1967), que al igual que Andrés inició su carrera en el Club Cotonificio de Badalona.
Clubs
Cotonificio Badalona: 1978-1983.
Club Joventut de Badalona: 1983-1986.
F.C. Barcelona: 1986-1998.
Selección española
186 veces internacional con la selección nacional absoluta.
Debuta con la selección nacional absoluta en partido amistoso el 22/07/82 frente a la selección de Cuba en Palma de Mallorca (España 97 - Cuba 93 ).
17 veces internacional con la selección nacional Júnior.
19 veces internacional con la selección nacional Juvenil.
Palmarés
Con el F.C. Barcelona:
7 Liga ACB: 1987, 1988, 1989, 1990, 1995, 1996, 1997.
4 Copas del Rey: 1987, 1988, 1991, 1994.
1 Copa Príncipe de Asturias: 1987-1988.
1 Copa Korac: 1987.
Con la selección española:
Medalla de plata en los Juegos Olímpicos de Los Ángeles 1984.
Medalla de plata en el Eurobasket de Nantes 1983.
Medalla de bronce con la selección Juvenil en el Eurobasket juvenil de Damasco 1979.
Consideraciones personales
Nominado Mejor Jugador Español de la temporada 1984-85 por el diario El Mundo Deportivo.
Integrante del Mejor Quinteto del Campeonato de Europa de Atenas-87.
Nominado Gigante de Leyenda en la temporada 1997-98 por la revista "Gigantes del Basket".
El 25/09/98 recibe la Medalla de Oro al Mérito Deportivo.
Participante en el ACB All-Star Vigo-87.
Referencias
Enlaces externos
Ficha en ACB.COM
Baloncestistas de España
Baloncestistas de España en los Juegos Olímpicos de Los Ángeles 1984
Baloncestistas de España en los Juegos Olímpicos de Barcelona 1992
Medallistas olímpicos de plata de España
Baloncestistas del Fútbol Club Barcelona
Baloncestistas de la Liga ACB
Baloncestistas del Círcol Catòlic de Badalona
Baloncestistas del Club Joventut de Badalona
Hermanos baloncestistas
Deportistas de la provincia de Sevilla
Nacidos en Carmona | {
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{"url":"https:\/\/stats.meta.stackexchange.com\/questions\/4721\/unexpected-reset-of-the-daily-login-count-towards-enthusiast-badge","text":"I am logging into the site everyday, for the \"Enthusiast\", without fail. However today the login count was reset from 22 to 1 ! What should I do?\n\nEDIT While going through the activity log and votes log I found another bug where the number of votes is different in the same profile.\n\n\u2022 Are you tracking your days according to GMT? (Local time is irrelevant for this badge.) \u2013\u00a0whuber Apr 5 '17 at 14:04\n\u2022 Yes, with GMT tracking, still count got reset \u2013\u00a0user12065 Apr 5 '17 at 14:05\n\u2022 Your record shows little activity on our site. Although the details of what kinds of activity qualify for this badge are not publicly known, it is known that merely logging in does not count as being active on the site. The point of this badge is to encourage people into constructive interaction: editing, voting, commenting, asking, answering. If you do those, the badges will take care of themselves. \u2013\u00a0whuber Apr 5 '17 at 14:09\n\u2022 @whuber I understand what you are saying in the spirit of badge, but in the letter there is a bug which causes the reset, there is some programming bug, some data corruption, that needs to be taken care of... \u2013\u00a0user12065 Apr 5 '17 at 14:15\n\u2022 I'm not seeing any evidence of such a bug. As I wrote above, I find very little evidence of any activity by you on our main site. I'm surprised you actually got up to 22 days. \u2013\u00a0whuber Apr 5 '17 at 14:18\n\u2022 @whuber I am not sure what kind of evidence you are looking for here. Should I have 21 different questions asked on consecutive days to provide evidence ? If you want to ignore the bug its all right. \u2013\u00a0user12065 Apr 5 '17 at 14:24\n\u2022 I am not advocating ignoring a bug--I'm just saying there's nothing here anyone can take any action about, because there's no evidence of any bug at all: stats.stackexchange.com\/users\/12065\/sarvex?tab=activity. (As a moderator I have access to far more detailed information than that--but it doesn't show anything more.) \u2013\u00a0whuber Apr 5 '17 at 14:26\n\u2022 These are my votes stats.stackexchange.com\/users\/12065\/sarvex?tab=votes They show a regular activity apart for April 3 which is counted as missed \u2013\u00a0user12065 Apr 5 '17 at 14:28\n\u2022 It looks like this is [status-bydesign], as @whuber said. \u2013\u00a0amoeba says Reinstate Monica Apr 5 '17 at 15:31\n\u2022 Might want to consider posting the votes issue as a separate bug report. It is probably just a caching issue, but still better to report two separate issues in two separate posts nevertheless. \u2013\u00a0JNat Apr 18 '17 at 9:59\n\nJust go to the user tab profile and click on link visited x days, y consecutive to see if any 'day' was missed. For example, I missed day 15th in March, here in CV Meta :(.","date":"2020-01-27 15:02:54","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\": 0.24172785878181458, \"perplexity\": 1865.927465111308}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-05\/segments\/1579251700988.64\/warc\/CC-MAIN-20200127143516-20200127173516-00537.warc.gz\"}"} | null | null |
Dragons defenders of berk season 2 episode 1 dailymotion
Dragons defenders of berk season 2 episode 1 dailymotion. List of DreamWorks Dragons episodes 2019-01-24
Thursday, January 24, 2019 7:50:22 AM Kerry
Watch Dragons: Defenders of Berk Season 2 Online
Sullivan Jack Thomas February 16, 2018 2018-02-16 In order to complete the Dragon Eye and discover the location of the King of Dragons, Johann needs more lenses while Hiccup still needs gemstones to complete his Dragon Eye 2. Fishlegs believes he has found the perfect candidate, but it turns out to be home to Slitherwings, venomous serpentine dragons that poison Stormfly as she defends Garff to trigger strange wild behavior. The series was announced by Cartoon Network on October 12, 2010. They eventually learn the Rumblehorn has been attempting to warn everyone of an approaching tsunami and chase them off before the wave would wipe them all out. Sullivan Jack Thomas February 16, 2018 2018-02-16 After a battle with the Flyers, the Riders discover remnants of Deathsong amber underneath a Singetail's wing, leading them to deduce that Krogan is using a Deathsong to lure and capture Singetails for his Flyers.
Stoick and Astrid try to convince Hiccup that the Singetails must also be treated as the enemy and potentially sacrificed for their victory, but Hiccup remains firm on their innocence and is left lost while the rest of Berk prepares for war. When he returns to say goodbye, however, he discovers that in reality, Hookfang has been helping the female protect her eggs from a Titan Wing Monstrous Nightmare. The sixth and final season of DreamWorks Dragons: Race to the Edge was released on February 16, 2018. Hiccup, Toothless and the dragon riders discover an injured Scauldron on the shores of Changewing Island. Despite their competition through the three challenges, Ruffnut decides to save Tuffnut instead of completing the final challenge, which Gruffnut reveals was the secret, fourth challenge: putting family above everything else.
Sullivan John Tellegen February 17, 2017 2017-02-17 The riders attend Berk's 400 year anniversary, but there is little chance for celebration when they learn Viggo has placed a bounty on Hiccup's head. Dagur is giving Alvin the support of his vast fleet in exchange for the Skrill, but the truce is uneasy. The dragons bond and fly away, allowing the riders to return home. Tuffnut learns to put the needs of the team before his own and saves the others by sacrificing his Macey and freeing the others. They launch their attack and rescue the others, and both sides prepare for their next encounter. Hiccup is captured by everyone from two dim-witted brothers to Savage to the masked man from Viggo's auction, leading the riders and Stoick on a wild goose chase to find him. Using copied notes from the Dragon Eye, the riders locate a living Buffalord, but it refuses to leave the island where it lives.
The riders escape with the help of the disposal Hotburple named Grump who Gobber bonds with and send the hunters running. During the trip, they are ambushed by hunters, but the twins rescue Hiccup and Johann and deduce that the only way the hunters knew they'd be there is if there was a mole within their group. Mala is finally convinced that the riders are allies of the dragons and the two groups align to defeat the hunters for good. Fishlegs manages to free the Quakens just as their poundings cause the island to sink and have them destroy the Hunters' stone base. The other riders interrogate the hunters for information on their whereabouts and trick them into sending word to the arena and following the Terror to their location. The three trust the Fire Terrors to take the Eruptodon egg and it safely reaches the nesting ground where it is bathed in the lava.
Buy Dragons: Defenders of Berk, Season 2
Once out of the cave, the Riders destroy the Flyers' facilities and free the Singetails. Though they realize that they won't have enough time to save them all, the Dramillions learn the technique to destroy the shackles themselves and are able to free each other. The dragons bond and fly away, allowing the riders to return home. . They deliver the axe late only to discover a fight broke out and the wedding was cancelled.
Snotlout and Hookfang join forces to gain dominance over the Titan Wing and drive it away, saving the eggs and their mother. Using a Skrill as a diversion, Hiccup and Viggo enter their headquarters where Viggo betrays Hiccup to earn favor with Johann and Krogan once more. After much delaying and 'keeping Gustav busy' by the riders, Gustav takes the Dragon Eye to go on a treasure hunt. Feeling he may be denying his sister a chance at lifelong happiness, Tuffnut uses a visiting Changewing on migration to fake his death and allow Ruffnut to go with Throk, but Ruffnut refuses to move on by painting her brother's image on the belly of a Changewing she names Snuffnut and Throk refuses to leave without her and attempts to tame Snuffnut on his own. When the riders find Ryker, he tells them that Viggo is setting them up and he can give them the Dragon Eye as the Berserkers are attacked by the Shell Fire, a giant sea dragon equipped with Hunter weaponry. To make things worse, the ingredients needed to cure the illness are in Toothless' satchel and all but Fishlegs are caught sick.
List of DreamWorks Dragons episodes
Though Hiccup feels guilty for driving Viggo to make such a move, Stoick expresses confidence that he and the others will always be able to stop him. Snotlout uses the Smothering Smokebreaths to get it back and the riders help the Armorwing when they pick apart its hide, earning its trust. Jones Jack Thomas June 24, 2016 2016-06-24 With Dagur's information and a map of the location stolen by Johann, the riders send Snotlout and Gobber undercover with Berk's reluctantly lent gold to the hunters' latest dragon auction to rescue the dragons and cripple Viggo's business. But, when Dagur the Deranged escapes from the prison on Outcast Island, he decides to take his revenge on Hiccup. Astrid is cured and the riders offer words for the deceased sailors. He reveals that Ryker has taken charge of the Hunters and offers to surrender the Dragon Eye in return for Hiccup's help in stopping Ryker, but escapes during an attack on the Defenders of the Wing, leaving the riders with the baby Eruptodon in their care. Eventually the Riders save Throk, Tuffnut reveals his lie, and the Changewings are diverted away, and Throk decides to leave understanding that the Thorston twins are better together.
Despite Johann's attempts, Ruffnut and Tuffnut insist on accompanying them. After the twins throw everyone in the 'dungeon' they realise they need the other riders to stop the island becoming a pile of ash. With no metal, and therefore no weapons to protect their village, Hiccup and the others must find a way to get rid of the rogue dragons, only to discover that the Smokebreaths' appearance was planned as an attack by Dagur the Deranged. Hiccup also discovers a new series of lenses for the 'Dragon Eye'. In the task you have to cross the island of dragons from one end to the other, at night, with no dragons, and 1 weapon. Heather meanwhile tracks down Dagur alone but is captured by his new armada. Towards the end of her mission, Astrid learns and confirms Heather's true motives for helping the Outcasts-freeing her parents.
They discover that the village has been overrun by Speed Stingers, a non-flying dragon that move at blindingly fast speeds and can paralyze their victims with their tails. Just as they leave, however, the volcano erupts. DreamWorks Dragons features the voice talents of , , , Julie Marcus, , , , and. However, he states that they must compete against each other, as only one can prevail. After much effort they finally get it open and Hiccup is able to retrieve the device from a distracted Dagur. Hiccup plans to use the Flightmare's mist to paralyze the hunters, but Viggo's advanced tactics allow him to escape with the Flightmare, Heather and Windshear as captives, leaving Hiccup to wonder about their dangerous new opponent.
Hiccup manages to match Viggo's wit and free the Flightmare and Heather with unexpected help from Dagur, but once again Viggo turns the tables and manages to snatch away the Dragon Eye. Despite the fact that nobody trusts him, Alvin's warnings allow them to defeat Dagur's invasion, but before they can destroy Dagur he captures Stoick; Dagur flees back to Outcast Island with his captive, demanding Hiccup and Toothless in exchange for Stoick's life. Meanwhile, the Hunters, now co-led by the masked man Krogan and a still-living Viggo, make preparations for their next move. Worried, the Riders quickly check on Garffiljorg while the others take the injured Singetail back to the Edge for recovery. Adventure takes flight with all-new episodes of Dragons: Defenders of Berk Part 2! Stoick decides to leave, but Hiccup helps him see things from the Jorgensons' point of view and convinces him to return, but when more Singetails arrive, Spitelout decides to abandon the island, stating they will build another storehouse elsewhere. Thanks to Toothless, Hiccup develops a plan to divide and trap the newly christened 'Shadow Wings' one by one, reminding him of the importance of teamwork and strategy, and returns to Dragon's Edge to join the riders in a well-deserved nap in the Edge's basement levels. Though Viggo manages to get away with Berk's gold, the dragons are still rescued, which Gobber assures Hiccup counts for something. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,656 |
\section{Introduction}
The detection of a line with an energy between 3.50 -- 3.57 keV (hereafter indicated as ``the 3.5 keV line'' for brevity) in the X-ray data from individual and stacked observations of clusters of galaxies \citep{Bulbul:2014sua}, from the Galactic center \citep{Jeltema:2014qfa} and, tentatively, from M31 \citep{Boyarsky:2014jta} \citep[see however][]{Jeltema:2014qfa, reply} has triggered widespread interest: the line might be associated with a two-body radiative decay including one photon of a dark matter particle with a mass of around 7 keV and a lifetime of about $6-8\times10^{27}$ sec. Such a particle has a natural theoretical counterpart in sterile neutrino models, a class of dark matter candidates whose motivation goes beyond that of explaining the missing non-baryonic matter in the universe \citep[see e.g.][for a review]{Boyarsky:2009ix}.
\cite{Jeltema:2014qfa} pointed out early on that atomic de-excitation lines from He-like Potassium ions (K XVIII) are a plausible counterpart to the 3.5 keV line both in clusters of galaxies and in the Milky Way. This possibility was initially discarded by \cite{Bulbul:2014sua} based on estimates of the required K abundance that relied on photospheric K solar abundances, and on multi-temperature models biased towards high temperatures. The latter, as demonstrated in \cite{reply}, artificially suppress the brightness of the K XVIII de-excitation lines by up to more than one order of magnitude. Additionally, coronal K abundances are larger by about one order of magnitude than photospheric K solar abundances, as recently pointed out by \cite{Phillips:2015wla}. As a result, the case for K XVIII as the culprit for the 3.5 keV line appears at present quite plausible.
Additional circumstantial evidence against a dark matter decay origin for the 3.5 keV line has also emerged. \cite{Malyshev:2014xqa} searched for the line in stacked, archival XMM observations of dwarf spheroidal galaxies, reporting a null result that highly constrained a dark matter decay origin for the line. \cite{Anderson:2014tza} analyzed stacked observations of galaxies and galaxy groups, systems where the thermal emission would be too faint to produce a detectable line from e.g. K XVIII, and also failed to find any evidence for a 3.5 keV line. \cite{Urban:2014yda} studied Suzaku data from X-ray-bright clusters, confirming that the 3.5 keV signal could naturally be ascribed to K, and questioning the compatibility of the line morphology with the dark matter decay hypothesis. Finally, \cite{Carlson:2014lla} studied in detail the morphology of the 3.5 keV emission from the Perseus cluster of galaxies and from the Galactic center, finding a notable correlation with the morphology of bright elemental emission lines, and excluding a dark matter decay origin even for cored Galactic dark matter density profiles. A recent study of charge exchange processes indicates that an additional possibility is that the 3.5 keV line originates from a set of high-$n$ S XVI transitions (populated by charge transfer between bare sulfur ions and neutral hydrogen) to the ground state \citep{Gu:2015gqm}.
\begin{figure*}
\begin{centering}
\includegraphics[width=0.93\columnwidth]{mos_binned.ps}\qquad
\includegraphics[width=0.93\columnwidth]{pn_binned.ps}
\end{centering}
\caption{\small {\em Left}: Combined MOS spectrum and residuals in the 2.5-5.0 keV energy range fit to an unfolded, single power law. For visual effect here the spectrum has been binned by a factor of five. {\em Right:} Combined PN spectrum and residuals (with a similar factor of five binning) in the 2.5-5.0 keV energy range fit to an unfolded single power law plus an instrumental line due to Ti K$\alpha$ emission at 4.51 keV. A weaker Ti K$\beta$ line can be seen at 4.93 keV but has no effect on the 3.5 keV line constraints.}
\label{fig:spec}
\end{figure*}
It is important to acknowledge that null results obtained so far are still compatible with a non-standard origin for the 3.5 keV line. Notably, axion-like particle conversion in magnetic fields \citep{Cicoli:2014bfa, Alvarez:2014gua} could reproduce the morphology of the 3.5 keV line in Perseus reported in \cite{Carlson:2014lla}; other possibilities include, for example, inelastic excited dark matter \citep{Finkbeiner:2014sja}. In all such instances, the signal strength scales non-trivially with the integrated dark matter mass along the line of sight, or it depends sensitively on astrophysical conditions such as the magnetic field strength.
\cite{Lovell:2014lea} used N-body simulations from the Aquarius project \citep{aquarius} to estimate the flux ratio for a standard dark matter decay process across different targets, including for the Draco dwarf spheroidal galaxy (dSph) and the Galactic center (GC). This ratio has a certain statistical distribution, which depends on the choice of the placement of the observer. The central finding of that study is that a 1.3 Ms long XMM-Newton observation of the Draco dSph would enable the discovery or exclusion at the 3$\sigma$ level of a dark matter decay interpretation of the 3.5 keV signal.
Here, we utilize recent, deep archival XMM-Newton observations of the Draco dSph to test a dark matter decay origin for the 3.5 keV line. We find no evidence of a line in either the MOS or PN data, and we are able to rule out a dark matter decay origin at greater than the 99\% confidence level.
The remainder of this manuscript has the following structure: we describe the XMM observations and data reduction in the following section \ref{sec:data}; we then describe our flux calculation and compare with the flux limits from the XMM MOS and PN data in section \ref{sec:analysis}, and we present our conclusions in the final section \ref{sec:conclusions}.
\section{XMM Data Analysis}\label{sec:data}
Draco was observed by XMM-Newton in 31 separate observations, 5 in 2009 (PI Dhuga) and 26 in 2015 (PI Boyarsky), with individual exposure times ranging from 17 to 87 ksec and a total time in all observations of 1.66 Msec. We reprocessed all 31 observations using standard procedures and utilizing the XMM SAS\footnote{http://xmm.esac.esa.int/sas/} and ESAS \citep{esas,esas2} software packages. Starting from the Observation Data Files, the raw EPIC data was pipeline-processed with the {\tt emchain} and {\tt epchain} tasks. Flare filtering was carried out with the ESAS tasks {\tt mos-filter} and {\tt pn-filter}; these time periods of increased particle background due to soft protons can lead to background levels elevated by two orders of magnitude and are thus removed from the data. Unfortunately, in the case of Draco particle background flaring was significant in many of the observations. For the two MOS detectors, two observations (ObsID 0603190401 and 0770190601) were almost entirely contaminated by flaring, and we removed these from our final data set; the other observations had reduced usable exposure times. The net exposure time after filtering was a little over one Msec for each MOS detector with a total time for both detectors of 2.1 Msec. The PN detector is typically more effected by particle flaring than the MOS detectors, and we found that only 20 of 31 observations had flares satisfactorily removed by {\tt pn-filter}; for these observations the net usable exposure time for PN was 0.58 Msec.
\begin{figure*}
\begin{centering}
\includegraphics[width=0.97\columnwidth]{MOS.eps}\qquad
\includegraphics[width=0.97\columnwidth]{PN.eps}
\end{centering}
\caption{\small {\em Left}: Limits on the flux of a line in the energy range between 3.3 and 3.7 keV from MOS observations of the Draco dSph, at the 68\%, 90\% and 99\% C.L. (green, red and black lines, respectively) and predictions for the flux of a 3.5 keV line assuming a dark matter decay origin for the line detected at that energy from stacked clusters of galaxies and from the Milky Way center (see text for details). The horizontal black lines indicate the 1$\sigma$ energy range for the line position as inferred by Boyarsky et al. 2014 for Perseus ($3.50\pm0.04$ keV) and for M31 ($3.53\pm0.03$ keV) and by Bulbul et al. 2014 from cluster observations ($3.57\pm 0.02$ and $3.51\pm0.03$ keV for their ``full sample'' MOS and PN results, resepctively); {\em Right}: same, for PN observations (note the difference in vertical scale).}
\label{fig:flux}
\end{figure*}
Point sources were detected and removed separately from each observation using the ESAS task {\tt cheese}; point source detection was run on broad-band images (0.4-7.2 keV) with a flux limit of $10^{-14}$ erg cm$^{-2}$ s$^{-1}$ and a minimum separation of 10 arcsec. Low exposure regions are likewise masked by {\tt cheese}. Spectra were extracted from the full field-of-view from each detector in each flare-filtered observation; however, for the MOS1 detector CCDs 3 and 6 were excluded due to micrometeroid damage. Spectra and corresponding redistribution matrix
files (RMF) and ancillary response files (ARF) for the 0.4-7.2 keV range were created using {\tt mos-spectra} and {\tt pn-spectra} in the ESAS package. The individual spectra and response files were co-added using the routines {\tt mathpha}, {\tt addrmf}, and {\tt addarf} in the {\tt FTOOLS} package \citep{ftools}. Combined RMF and ARF files were weighed by the relative contribution of each observation to the total exposure time. The spectra and responses for the MOS1 and MOS2 cameras were combined in to a single summed MOS spectrum, while the spectra and responses for the PN detector were combined separately.
Spectral modeling employed the energy range between 2.5 keV and 5 keV. This energy range was chosen to exclude strong instrumental emission lines while being much, much broader than the energy resolution of the detectors ($\sim 100$ eV). At these energies, the X-ray background is dominated by the quiescent particle background \citep{esas2} which we model with an unfolded, power law (no vignetting) in {\tt XSPEC} \citep[version 12.8.1p, ][]{xspec}. As shown in Fig.~\ref{fig:spec}, the combined MOS spectrum in the 2.5-5 keV range is well fit by an unfolded, single power law alone with reduced $\chi^2 = 0.96$ ($\chi^2=475/497$ degrees of freedom). Adding a Gaussian line between 3.4 and 3.6 keV gives no improvement to the fit, and a line at these energies with a flux greater than $\sim 10^{-6}$ photons cm$^{-2}$ s$^{-1}$ is excluded, as shown in Fig.~\ref{fig:flux}. The confidence contours in Fig.~\ref{fig:flux} are determined based on the change in the fit statistic when stepping over the relevant parameters using the {\tt steppar} command in {\tt XSPEC}. The combined PN spectrum is well fit by an unfolded power law plus an instrumental line due to Ti K$\alpha$ emission (4.51 keV), which we model as a narrow Gaussian (Fig.~\ref{fig:spec}, right). The reduced $\chi^2$ for this fit is 0.99 ($\chi^2=490/495$ degrees of freedom). Again, adding a Gaussian line between 3.4 and 3.6 keV gives no significant improvement to the fit. The fit is somewhat improved by adding a second instrumental line, Ti K$\beta$, at 4.93 keV, but this feature has no effect on the 3.5 keV line constraints. As can be seen from Fig.~\ref{fig:flux}, right the upper limit on the flux of a line near 3.5 keV from the PN data is weaker than from the MOS data given the shorter usable exposure time but does serve as additional confirmation of the lack of a 3.5 keV line from Draco.
\section{Flux Limits and Constraints on Dark Matter Decay}\label{sec:analysis}
We utilize three distinct predictions for the 3.5 keV line flux that should have been observed with the XMM observations described above for a dark matter decay origin . The first one makes use of the results of \cite{Lovell:2014lea}, which calculated the flux expected from a 14 arcmin angular region around Draco given the brightness of the 3.5 keV line observed from the Galactic Center and the ratio of the flux from Draco-like halos and from the Galactic center as extrapolated from the Aquarius simulation. The resulting distribution in predictions is bracketed by the range
$$
F=(1.0-5.2)\times10^{-6}\ {\rm cts}\ {\rm cm}^{-2}\ {\rm s}^{-1},
$$
where the lower and upper values bracket 95\% of the predictions, and with the most-probable value being $F=2.3\times 10^{-6}\ {\rm cts}\ {\rm cm}^{-2}\ {\rm s}^{-1}$ (see especially their Appendix C3 for additional details on assumptions and method). We calculated that the point source masking we adopt and the non-uniform coverage (e.g. from the lost MOS CCDs and chip gaps) described in the previous section suppress the predicted flux to 77\% of its un-masked value (we neglect the additional signal from the annulus between 14 and 15 arcmin). We calculated the fraction of masked signal employing the dark matter density profile and distance to Draco quoted in \cite{Abdo:2010ex}. We verified that the impact on the masking fraction of varying the parameters in the dark matter density profile and the distance within their 2-$\sigma$ range is in all cases smaller than 0.1\%. We show the resulting range of expected fluxes in Fig.~\ref{fig:flux} with the vertical blue bar, and indicate the central value with a full blue square.
We additionally considered two alternate predictions for the flux expected in the data we analyze for the decaying dark matter scenario. We followed the procedure outlined in \cite{Malyshev:2014xqa}. There, two alternate determinations of the mass within half-light radius were adopted to estimate the integrated line of sight dark matter column density for Draco, based on the results of the analyses of \cite{wolfetal} and \cite{geringer-smithetal}. \cite{Malyshev:2014xqa} then proceeded to add, for the direction of Draco, the flux from dark matter within the Milky Way, and added to it the prediction for the component from the Draco dSph itself. For the former, \cite{Malyshev:2014xqa} presents a ``mean'' flux based on the ``favored NFW'' Milky Way dark matter halo of \cite{Klypin:2001xu}, as well as a very conservative ``minimal'' model based on a ``maximal disk'' halo structure. We choose to show our predictions for the flux in Fig.~2 for the mean flux from the Milky Way, but it is a straightforward exercise (and one that does not impact our results or conclusions qualitatively) to rescale them for the minimal Milky Way model. The predicted flux was normalized, as in \cite{Malyshev:2014xqa} and in \cite{Lovell:2014lea}, to the parameters corresponding to the best-fit point of the cluster observations of \cite{Bulbul:2014sua}. We show the predictions for the Draco halo determination in \cite{wolfetal} with brown diamonds and \cite{geringer-smithetal} with purple triangles. The uncertainties we show in the figure reflect the uncertainties from the determination of the halo parameters as in \cite{Malyshev:2014xqa}. As we did for the predictions from \cite{Lovell:2014lea}, we corrected the predicted flux quoted in \cite{Malyshev:2014xqa} for masking in our observations, and for the larger, in this case, angular region we utilize compared to the flux predictions in \cite{Malyshev:2014xqa}.
We illustrate our results in Fig.~\ref{fig:flux}. We indicate with green, red and black lines the 68\%, 90\% and 99\% Confidence Level (C.L.) limits on the maximal allowed flux associated with a line at the energy indicated by the x-axis. We also show the predictions for the line flux described above as well as the range of energies for the line reported from cluster observations as described in \cite{Bulbul:2014sua} and the range obtained by \cite{Boyarsky:2014jta} from observations of the Perseus cluster and of M31.
As can be seen in Fig.~\ref{fig:flux}, the lack of a detected line in the MOS data rule out at higher than 99\% confidence level a line with even the most conservative predicted fluxes based on a conservative range of possible density profiles for the Draco dwarf. Specifically, the lower limit on the predicted Draco line flux of \cite{Lovell:2014lea} based on the brightness of the line observed in the Galactic Center is excluded at 99.1\%; the lower limit on the predicted flux given the observed stacked cluster line flux are excluded at better than 99.999\% for either the \cite{wolfetal} or the \cite{geringer-smithetal} profiles. Therefore, a generic dark matter decay origin of the 3.5 keV line feature is highly unlikely.
In Fig.~\ref{fig:limits}, we show constraints on the sterile neutrino parameter space in terms of the particle's mass $m_s$ and mixing angle with active neutrinos $\theta$ given the line flux limits from the MOS Draco observations, in the relevant mass range for a dark matter decay interpretation of the 3.5 keV line. The cyan shaded region is excluded at the 2$\sigma$ level, and assumes the central value for the \cite{geringer-smithetal} dark matter halo parameters for Draco, and the most conservative Milky Way halo considered in \cite{Malyshev:2014xqa} (corresponding to the ``maximal disk model'' of \cite{Klypin:2001xu}). The blue shaded region, instead, adopts the default ``favored NFW'' Milky Way dark matter halo density profile \citep{Klypin:2001xu}.
Taking the most conservative possible assumptions both for the flux from Draco and from the Milky Way Galactic halo (corresponding to the predictions of \cite{wolfetal} for the flux from Draco and the most conservative Milky Way halo of \cite{Klypin:2001xu}), we are able to set a lower limit on the lifetime of a 7 keV sterile neutrino decaying into a 3.5 keV line of $\tau>2.7\times 10^{28}$ s (95\% C.L.), corresponding to a mixing angle $\sin^2(2\theta)<1.6\times 10^{-11}$ (95\% C.L.). Our most conservative limits are thus more than a factor 4 below the favored mixing angle predicated by a dark matter decay interpretation of the 3.5 keV line signal \citep[$\sin^2(2\theta)\approx7\times 10^{-11}$, ][]{Bulbul:2014sua}.
After our paper appeared, \cite{ruchayskiy15} analyzed essentially the same data set as this work. They come to a similar conclusion that no 3.5 keV line is detected in Draco, though their limit on the flux of the line is less stringent than ours. The main difference appears to stem from the fact that they find a mild excess in the PN data which we do not see. Here we comment briefly on some of the differences in the two analyses. \cite{ruchayskiy15} make three comments on our analysis. 1) We do not include an extragalactic background power law in addition to the particle background; 2) the $\sim 1 \sigma$ excesses in the MOS spectrum near 3.35 keV and 3.7 keV might be due to un-modeled weak instrumental lines (from K K$\alpha$ and Ca K$\alpha$ at 3.31 and 3.69 keV); 3) we do not jointly fit the MOS and PN data. We have now tried all three of these variations and find that they all have a negligible effect on our results. Adding additional lines or background components as in points 1) and 2) is not warranted by the data nor does it significantly improve our fits when done. In fact, we find that when we do add these components our flux limit on a line near 3.5 keV in the MOS data is actually {\em strengthened}, albeit slightly, lowering the flux limit by 10\%. We note that when adding an additional power law for the extragalactic background in addition to the unfolded power law for the particle background, we expanded the energy range to 2.5-7 keV and added lines for the strong Cr, Mn, and Fe instrumental lines in this range, but again we find no excess near 3.5 keV. Here the power law components are free to vary; the best fit photon indices are 1.5 for the extragalactic power law and 0.33 for the unfolded power law with a reduced $\chi^2$ of 0.99 ($\chi^2=876/885$ degrees of freedom).
Results from combined fits to the MOS and PN data must be interpreted with care as there are known offsets in the relative calibration of the two camera types. In particular, fits to a stack of bright sources show an offset of 5-8\% between MOS and PN \citep{read14} above 3 keV, which is similar to or larger than the ratio of the predicted 3.5 keV line flux to the neighboring continuum. In performing a joint fit to the MOS and PN spectra, we separately fit the continuum models for the two instruments, but jointly fit the 3.5 keV line energy and flux. We again find no excess near 3.5 keV. Our limit on the flux of a line near 3.5 keV is weakened only slightly (by 25-30\%) from the MOS-only limit shown in the left panel of Fig.~\ref{fig:flux}, and we still exclude the most conservative predictions from \cite{Lovell:2014lea} at the 99\% confidence level.
The primary reason that \cite{ruchayskiy15} quote a weaker limit on the flux of a line from Draco appears to be due to the fact that they find a $\sim 2 \sigma$ excess in the PN which is not present in our analysis. It is unclear why they find an excess where we do not, but we note that in their spectral fits they include data up to 10 keV where the effective area is both rapidly dropping and a factor of $\sim 4$ lower than at 3 keV for PN, and more than an order of magnitude lower for MOS. In addition, their background model, given the large energy range, includes 13 line components and two power laws compared to our single power law.
\section{Discussion and Conclusions}\label{sec:conclusions}
Using $\sim1.6$ Msec observations of the Draco dSph with XMM-Newton we were able to obtain one of the most stringent constraints on a dark matter decay origin for the 3.5 keV line observed from clusters of galaxies and from the Milky Way center. Our results rule out a dark matter decay interpretation with greater than 99\% C.L., and, under very conservative assumptions on the relevant dark matter density profiles, imply a lower limit on the dark matter lifetime of $\tau>2.7\times 10^{28}$ s at 95\% C.L. for a dark matter mass of 7 keV radiatively decaying to a two-body final state with one photon.
In view of the results presented here, and in view of the recent re-assessment of the potassium abundance \citep{Phillips:2015wla}, we conclude that the most probable counterpart to the 3.5 keV line observed towards the Milky Way center and from individual and stacked observations of clusters of galaxies are atomic de-excitation lines of the K XVIII ion. Charge-exchange processes might also provide an alternate astrophysical explanation \citep{Gu:2015gqm}. Scenarios advocating new physics where a 3.5 keV signal is suppressed in dwarf galaxies, such as an axion-like particle conversion to 3.5 keV photons in the presence of a magnetic field, are not ruled out unless Occam's razor is advocated.
Future observations of clusters and of the Galactic center with Astro-H remain a priority to pinpoint the physical origin and the nature of the 3.5 keV line, while, in view of our results, additional deep observations of local dwarf galaxies with current or future telescopes are unlikely to advance our understanding of this particular feature.
\begin{figure}
\begin{centering}
\includegraphics[width=0.9\columnwidth]{parameterspace.eps}
\end{centering}
\caption{\small Constraints on the parameter space of sterile neutrinos, defined by the particle's mass $m_s$ and mixing angle with active neutrinos, $\theta$. The cyan-shaded region is excluded, at the 2$\sigma$ level ($\sim$95\% C.L.), by Draco MOS observations, using the most conservative Milky Way dark matter density profile considered in Malyshev et al. (2014), while the blue-shaded region employs the nominal ``favored NFW'' profile, which we also use for Fig.~\ref{fig:flux}.}
\label{fig:limits}
\end{figure}
\section*{Acknowledgments}
\noindent We would like to thank the referee for their valuable comments on our paper. TJ is partly supported by NSF AST 1517545. SP is partly supported by the US Department of Energy, Contract DE-SC0010107-001.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,428 |
post_title: dcos node diagnostics download
menu_order: 3
---
# Description
View the details of diagnostics bundles.
# Usage
```bash
dcos node diagnostics download <bundle> [OPTION]
```
# Options
| Name, shorthand | Default | Description |
|---------|-------------|-------------|
| `--location=<location>` | Current directory | Download the diagnostics bundle to a specific location. If not set, the default location is your current working directory. |
# Positional arguments
| Name, shorthand | Default | Description |
|---------|-------------|-------------|
| `<bundle>` | | The bundle filename. For example, `bundle-2017-02-01T00:33:48-110930856.zip`. |
# Parent command
| Command | Description |
|---------|-------------|
| [dcos node](/docs/1.10/cli/command-reference/dcos-node/) | View DC/OS node information. |
<!-- # Examples -->
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,226 |
Tom Moylan
Boehringer Ingelheim Roxane Inc.
Ven-A-Care
Abbott Laboratories Inc.
B. Braun Medical Inc.
Roxane Laboratories
whistle-blower
California court opinions
California courts
California Official Reports
Electronic Advance Sheet
Kevin LaCroix
SEC Actions
Thomas Gorman
Washington court opinions
Abbott, B. Braun, Boehringer Ingelheim Settle Drug-Pricing Cases For $421M
BOSTON -- (Mealey's) Drug and medical supply makers Abbott Laboratories Inc., B. Braun Medical Inc. and Roxane Laboratories (now known as Boehringer Ingelheim Roxane Inc.) have agreed to pay a total of $421 million to settle False Claims Act allegations that they published inflated prices and caused the federal government to pay higher costs, the U.S. Justice Department announced Dec. 7.
Boehringer Ingelheim Roxane will pay $280 million for reporting false prices for azathioprine, diclofenac sodium, furosemide, hydromorphone, ipratropium bromide, Oramorph SR, Roxanol, Roxicodone and sodium polystyrene sulfonate, the Justice Department said.
Abbott will pay $126.5 million with respect to its reported prices for dextrose solutions, sodium chloride solutions and a combination product of sterile water and the antibiotic vancomycin.
B. Braun Medical, a U.S. subsidiary of B. Braun Melsungen AG of Germany, will pay $14,744,000 for inflating the prices of 49 drug products, including water-based intravenous solutions such as dextrose, sodium chloride and sterile water and lactated ringers solutions. Also included are intravenous nutritional solutions and other intravenous drugs, the government said.
The government said a False Claims Act complaint was filed on behalf of the government against Roxane and that the government intervened in the lawsuit on Jan. 18, 2007. Two qui tam lawsuits were filed against Abbott and the government intervened in one in May 2006.
The whistle-blower suits were filed by Ven-A-Care of the Florida Keys Inc., a home infusion company, and its principals, Mark T. Jones, Luis Cobo and Dr. John M. Lockwood. The relators will receive about $88.4 million of the settlement as their statutory share of the recovery, the government said.
The Ven-A-Care lawsuits were filed in the U.S. District Court for the Southern District of Florida and were later transferred to the District of Massachusetts as In Re: Ven-A-Care Cases (No. 06-11337, D. Mass.) where other average wholesale price litigation is centralized before Judge Patti B. Saris.
[Editor's Note: Full coverage will be in the Dec. 16 issue of Mealey's Emerging Drugs & Devices. For all of your legal news needs, please visit www.lexisnexis.com/mealeys.]
For more information, call editor Tom Moylan at 215-988-7739, or e-mail him at tom.moylan@lexisnexis.com. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,568 |
Bari Sardo est une commune de la province de Nuoro en Sardaigne (Italie).
Administration
Hameaux
Communes limitrophes
Cardedu, Ilbono, Lanusei, Loceri, Tortolì
Évolution démographique
Commune dans la province de Nuoro | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,173 |
{"url":"https:\/\/lukemartinlogan.github.io\/professional-website\/projects\/PotholeDetectives\/","text":"# Pothole Detectives (PhDs)\n\n## Overview\n\nPothole damage costs US drivers \\$3 billion annually. In order to combat this problem, driverless pothole-filling robots will become reality. In order to prepare for this, our team designed a routing algorithm to decide which blocks in Chicago to visit in order to fill potholes effectively, a machine-learning algorithm to predict the number of potholes that would be formed in a ward of Chicago over a period of time (which can be used to determine the number of robots that would be necessary to counteract more potholes from forming), and a website to visualize the results of the routing algorithm and the pothole count predictor.\n\nIn this project, I was responsible for the design and implementation of the routing algorithm. The algorithm is implemented in Python and uses OSMNX to get graphs of Chicago, Geopandas to divide the graph of Chicago into wards using ward boundaries, and the Chicago Data Portal to get active pothole locations. The algorithm divides the graph of wards of Chicago into blocks, assigns potholes to blocks, and then ranks each block based off of the number of pothole reports in the block, the amount of time it would take to fill every pothole in the block, and the distance the block is from the facility a robot is located in. The route is decided using a combination of the Traveling Salesman Problem (TSP), Binary Programming, and the Chinese Postman Problem (CPP).\n\n## Scenario\n\nWe have a weighted, directed multigraph of a ward in Chicago. The edges of this graph are roads and the vertices are intersections. The ward has potholes as well as robots to fill those potholes. Each robot can travel a certain maximum speed and are assigned to designated facilities in the ward. Ie, each robot is assigned to a vertex in the graph of the ward. The robot can be outside for a certain period of time before traffic becomes an issue, but the robots cannot travel faster than the speed limit on any particular road. Furthermore, it takes time to fill each pothole in the ward (we assume it's a constant). We want the robots to fill the potholes in the ward for as long as it can be outside without disturbing traffic. Furthermore, we want the robot to return to the facility where it came from when it is done filling potholes.\n\n## Precomputation\n\n1. ### Step 1: Make the ward graph strongly connected\n\nBy default, the graph of a ward is a directed multigraph, and it is not necessarily strongly connected. This means it's possible for the robot to never be able to return to the facility where it came from.\n\nIn order to make the graph of the region strongly connected, we follow these steps:\n\n1. Find the set of strongly connected components (SCCs) in the graph\n2. For each SCC:\n1. Find the SCC that is nearest to it\n2. Find the smallest road connecting the two SCCs and make it bidirectional\n2. ### Step 2: Divide the ward into blocks\n\nA block is the smallest geographic area that is encompassed entirely by streets. Thus, the set of blocks can be found by computing the cycle basis of the ward graph without considering the direction of the streets. After this, we determine whether or not a cycle is strongly connected while considering the direction of the edges. If it is, we consider the cycle to be a block. Any edges that were not apart of a strongly connected cycle in the cycle basis will be considered blocks.\n\n3. ### Step 3: Chinese Postman Problem\n\nIf a robot visits a block, we will assume that the robot travels across every single road in the block at least once in order to find potholes that may not have been reported. We will also assume the robot starts and stops at the same spot in the block. This is known as the Chinese Postman Problem. The solution to this problem is to augment the graph of the ward to be Eulerian by adding edges to the graph of the ward. Typically, edges represent single roads, and thus it doesn't make sense to just fabricate new edges. However, edges can also represent a path of connected roads. Thus, we fabricate edges that are equally weighted to the minimum-weight path connecting the two vertices in the edge. The weight of an edge is considered to be its length in meters.\n\n4. ### Step 4: Find the time it takes to travel the roads\n\nEach robot can travel a certain maximum speed; however, the robot may not be able to travel that fast due to speed limits. To factor the speed limit, we go to each road in the graph of a ward, take the minimum of the speed limit and the speed of the robot, and then divide the length of the road by that speed. This will be the amount of time it takes for the robot to travel across the road. If the robots have different maximum speeds, then this calculation is performed for each unique maximum speed.\n\n5. ### Step 5: Map potholes to blocks\n\nWe obtain the set of potholes from the Chicago Data Portal using Pandas. Each pothole has a latitude and longitude in this dataset. We find the vertex that is nearest to this pothole using OSMNX and then find the block that the vertex is contained in. After this, we compute the number of pothole reports in each block and estimate the amount of time it would take to fill all potholes in the block at once. We assume the amount of time it takes to fill a single pothole is a constant (which we set to 15 minutes by default). We also consider the priority of filling potholes in the block to be the number of potholes assigned to that block.\n\n## Routing Algorithm\n\n1. ### Step 1: Rank blocks\n\nEach block has a priority of being visited; however, we cannot simply visit the highest-priority blocks first. This is because high-priority blocks may be distant from the robot, which would end up wasting a lot of time if we visit them in that order. It's more efficient to visit some lower-priority blocks on the way to higher-priority blocks. Thus, we rank the blocks as a function of priority, the distance the center of the block is from the starting point of the robot, and the amount of time it would take to travel a block. The amount of time to travel a block is the sum of the time it takes to travel every road in the block and the time it takes to fill every pothole in the block.\n\nThe rank is computed as follows:\n\n$$\\begin{equation*} \\text{rank}(priority, dist, TTT) = \\begin{cases} \\frac{priority}{\\sqrt{TTT}} & dist \\leq 500\\text{m} \\\\ \\frac{priority}{\\sqrt{TTT*\\frac{dist}{500}}} & dist > 500\\text{m}\\\\ \\end{cases} \\end{equation*}$$\n\n2. ### Step 2: Select blocks\n\nEach block now has a rank. We want to maximize the total rank of blocks selected under the constraint the robot is not outside for too long. This optimization problem can be phrased in terms of a Binary Integer Program, which we implemented using ConvexPY (CVXPY).\n\nThe optimization problem is as follows:\n\n\u2022 $$X$$ is a vector of bits that represents whether or not a block has been selected\n\u2022 $$R$$ is a vector of positive real numbers that represent the rank of a block\n\u2022 $$TTT$$ is a vector of positive real numbers that represent the amount of time it takes to travel across a block\n\u2022 $$MaxTTT$$ is the maximum amount of time the robot can be outside\n\n$$\\text{Maximize}\\: X*R^T \\:\\text{such that}$$ $$X*TTT^T \\leq MaxTTT$$\n\n3. ### Step 3: Order blocks\n\nAfter we select the blocks, we must decide the order with which to visit the blocks. First, imagine that we fabricate roads between the blocks. The roads connect the center of each block to the centers of every other block. The length of this road is the Euclidean distance between the centers of the two blocks. We want to find the subset of these fabricated roads to travel across in order to visit every block at least once. This is equivalent to the Traveling Salesman Problem (TSP). To do this, we used a genetic algorithm from a TSP package in Python. Note, we do not actually fabricate those roads, we just compute the distance between the centers of two blocks whenever they get selected by the genetic algorithm. This prevents a substantial memory overhead.\n\n4. ### Step 4: Connect blocks\n\nNow that we have the order with which to travel the blocks, we find the path that takes the least amount of time for the robot to travel (considering speed limits). Initially, the robot starts in a facility. Using OSMNX, we find the vertex that is nearest to the facility in the graph of the next block the robot will travel to. We then use Dijkstra's Shortest Path Algorithm to find the minimum-weight path connecting the two vertices. We use travel time as the weight instead of distance. We do this for each block until the last block is connected to the block where the robot facility is located.\n\n5. ### Step 5: Remove blocks\n\nAt this point, we have a route for a single robot to take. This route contains a set of blocks with potholes to fill. We do not want multiple robots visiting the same block, so we remove the set of blocks that this robot will visit from the pool of all blocks. This will guarantee that no two robots attempt to fill potholes in the same block.","date":"2022-11-28 04:10:23","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 2, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4180888831615448, \"perplexity\": 421.4647253582619}, \"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-2022-49\/segments\/1669446710473.38\/warc\/CC-MAIN-20221128034307-20221128064307-00064.warc.gz\"}"} | null | null |
\section{Introduction} \label{sec:Intr}
Estimating moments of central values in families of $L$-functions is an active topic of number theory.
There are now well-established conjectures for these moments,
thanks to the work of
Keating and Snaith \cite{KS00a,KS00b},
with subsequently works of
Diaconu, Goldfeld and Hoffstein \cite{DGH03} and Conrey, Farmer, Keating, Rubinstein and Snaith \cite{CFKRS05}.
In \cite{RS05,RS06}, Rudnick and Soundararajan developed a method for establishing conjectured order lower bounds for moments of central values of families of $L$-functions, provided a little more than the first moment of this family of $L$-functions can be computed.
This method was extended by Radziwi{\l\l} and Soundararajan \cite{RS13} and Heap and Soundararajan \cite{HS20} to all positive real moments.
Recently, the authors \cite{HH21} proved an asymptotic formula for the twisted first moment of central values of quadratic twisted $\GL(3)$ $L$-functions.
In this paper, we work out the conjectured order lower bound for the $k$-th moment of central values of this family of $L$-functions for all $k\geq 1$, based on \cite{HS20} and \cite{HH21}.
Let $\phi$ be a self-dual Hecke--Maass cusp form of type $(\nu,\nu)$ for $\SL(3,\mathbb{Z})$ with the normalized Fourier coefficients
$A(m,n)$.
We have the conjugation relation $A(m,n)=\overline{A(n,m)}=A(n,m)$, see Goldfeld \cite[Theorem 9.3.11]{GB06}.
For an automorphic representation $\pi$ of $\GL(3,\mathbb{A_Q})$,
the symmetric square lift $L$-function $L(s,\sym^2\pi)$ has a simple pole at $s=1$ if and only if $\pi$ is the Gelbart--Jacquet lift of an automorphic representation on $\GL(2,\mathbb{A_Q})$ with trivial central character, see \cite{GRS99}. That is, $\pi$ is a self-contragredient cuspidal automorphic representation, see \cite{GJ76} and \cite{RM14}.
So $\phi$ is self-dual if and only if $L(s,\sym^2 \phi)$ has a simple pole at $s=1$. Note that when the $\GL(3)$ cusp form is self-dual, its Fourier coefficients are all real.
We denote $\theta_3$ be the least common upper bound of power of $p$ for $|A(p,1)|$, i.e., $|A(p,1)|\leq 3p^{\theta_3}$ for all prime $p$.
The Generalized Ramanujan Conjecture implies that $\theta_3=0$, and from Kim--Sarnak \cite[Appendix 2]{Kim2003} we know $\theta_3\leq\frac{5}{14}$.
Any real primitive character
must be of the form $\chi_d(n)=(\frac{d}{n})$ where $d$ is a fundamental discriminant \cite[Theorem 9.13]{MV07}, i.e., a product of pairwise coprime integers of the form $-4$, $\pm 8$, $(-1)^{\frac{p-1}{2}}p$ where $p$ is an odd prime.
There are two primitive characters to the modulus $q$ if $8\parallel q$ and only one otherwise.
For $\Re s$ sufficiently large we define the twisted $L$-function $L(s,\phi\otimes \chi_{8d})$ to be
$$L(s,\phi\otimes \chi_{8d})=\sum_{n=1}^\infty \frac{A(n,1)\chi_{8d}(n)}{n^s}.$$
It has an analytic continuation to the entire complex plane and satisfies the functional equation (see \cite[Theorem 7.1.3]{GB06})
\begin{equation*}
\Lambda (s,\phi\otimes \chi_{8d}) := (8d)^{s/2}\pi^ {-3s/2}\prod_{i=1}^3\Gamma\left(\frac{s-\gamma_i}{2}\right)L(s,\phi\otimes \chi_{8d})
= \Lambda (1-s,\phi\otimes \chi_{8d})
\end{equation*}
when $d$ is positive and $\phi$ is self-dual, where $\gamma_1=1-3\nu$, $\gamma_2=0$, and $\gamma_3=-1+3\nu$ are the Langlands parameters of $\phi$.
Recently, we \cite{HH21} proved asymptotic formulas for the twisted first moment of quadratic twisted $\GL(3)$ central $L$-values.
In this paper,
we only consider the self-dual case and prove the conjectured lower bound for the $k$-th moment for quadratic twisted self-dual $\GL(3)$ central $L$-values for all $k\geq 1$, by using our twisted first moment result.
We will use the lower bounds principle of Heap and Soundararajan \cite{HS20}, where
they achieved lower bounds of moments of Riemann zeta function on the critical line,
which is a case touch to unitary group.
Recently, Gao and Zhao \cite{GZ21} estiblished the analogous result on lower bounds for moments of central values of quadratic twisted modular form $L$-functions.
We extend their method to the $\GL(3)$ case.
Our main result is as follows.
\begin{theorem}\label{thm1}
Let $\phi$ be a self-dual Hecke--Maass cusp form of $\SL(3,\mathbb{Z})$, and $\Phi$ a smooth nonnegative Schwarz class function supported in interval $(1,2)$.
For any $k\geq 1$ we have
\begin{equation}\label{eq1}
\mathop{\sum\nolimits^\flat}_{2\nmid d} \left| L\left(\frac{1}{2},\phi\otimes \chi_{8d}\right)\right|^k \Phi\left(\frac{d}{X}\right)
\gg_{k,\Phi} X(\log X)^{\frac{k(k+1)}{2}},
\end{equation}
where $\sum_{2\nmid d}\nolimits^\flat$ means summing over positive odd square-free $d$.
\end{theorem}
Our result shows that the family of quadratic twisted $L$-functions $L(s,\phi\otimes \chi_{8d})$ with $\phi$ self-dual is a symplectic family, which is the same as the family of quadratic Dirichlet $L$-functions $L(s,\chi_{8d})$.
Note that the family of quadratic twists of a given $\GL(2)$ newform $f$, $L(s, f\otimes \chi_{8d})$, is an orthogonal family as proved in \cite{RS05,RS06}.
If $\phi$ is a non self-dual Hecke--Maass cusp form of $\SL(3,\mathbb{Z})$, then our result in \cite[Theorem 1.3]{HH21} implies that the family of quadratic twisted $L$-functions $L(s,\phi\otimes \chi_{8d})$ is an orthogonal family as the $\GL(2)$ case.
\section{Preliminaries}
Let $N,M$ be sufficiently large natural numbers depending on $k$ only.
Let $\{\ell_j\}|_{j=1}^R$ be a sequence of decreasing even integers, where $\ell_1= 2\lceil N \log \log X\rceil$ and $\ell_{j+1} = 2 \lceil N \log \ell_j \rceil$ for $j\geq 1$, and $R$ is the largest integer satisfying $\ell_R >10^M$.
Assume that $M$ is large enough so that $\ell_{j-1} >\ell_{j}^2$ for all $2 \leq j \leq R$.
We will use the following two inequalities
\begin{equation*}
R \ll \log \log \ell_1 \quad \mbox{and} \quad \sum^R_{j=1}\frac 1{\ell_j} \leq \frac 2{\ell_R}.
\end{equation*}
Let ${P}_1$ be the set of primes in interval $[3, X^{1/\ell_1^2}]$ and
${P_j}$ be the set of primes in interval $(X^{1/\ell_{j-1}^2}, X^{1/\ell_j^2}]$ for $2\le j\le R$.
We define
\begin{equation*}
{\mathcal P}_j(d)=\sum_{p\in P_j}\frac{A(p,1)}{\sqrt{p}}\chi_{8d}(p),
\end{equation*}
$E_{\ell}(x)=\sum_{j=0}^{\ell} \frac{x^{j}}{j!}$,
${\mathcal N}_j(d,\alpha)=E_{\ell_j}(\alpha{\mathcal P}_j(d))$,
and
\begin{equation*}
\mathcal{N}(d,\alpha)=\prod_{j=1}^{R}
{\mathcal N}_j(d,\alpha)
=\prod_{j=1}^{R}E_{\ell_j}(\alpha{\mathcal P}_j(d)), \quad(\alpha\in \mathbb{R}).
\end{equation*}
Note that ${\mathcal P}_j(d)$
are all real for self-dual form $\phi$, since $A(p,1)\in\mathbb{R}$.
For $l> 0$ even and $x\in \mathbb{R}$,
then we can choose $x_0\neq 0$ which is a local minimum of $E_l$ because $E_l(\pm \infty)=+\infty$ and $E_l'(0)=E_{l-1}(0)=1$,
thus $E_l'(x_0)=E_{l-1}(x_0)=0$,
and we know $E_l(x_0)=\frac{x_0^l}{l!}+E_{l-1}(x_0)>0$.
Together with $E_l(0)=1$,
we have $E_l(x)$ must be positive for any even integer $l\geq 0$ and $x\in \mathbb{R}$.
This is a part of \cite[Lemma 1]{RS15}.
Hence we know ${\mathcal N}_j(d,\alpha)$ is always positive in our case.
The case $k=1$ in Theorem \ref{thm1} is a simple consequence of our Lemma \ref{lm:mysum} below with $l=1$ which we proved in \cite{HH21}. From now on, we assume $k>1$.
To prove Theorem \ref{thm1}, we need the following propositions.
By the H\"older's inequality, we know
\begin{multline*}
\sum_{2\nmid d}\nolimits^\flat L(\frac{1}{2},\phi\otimes\chi_{8d}){\mathcal N}(d,k-1)\Phi(\frac{d}{X}) \\
\leq(\sum_{2\nmid d}\nolimits^\flat |L(\frac{1}{2},\phi\otimes \chi_{8d})|^{k}\Phi(\frac{d}{X}))
^{\frac{1}{k}}(\sum_{2\nmid d}\nolimits^\flat {\mathcal N}(d,k-1)^{\frac{k}{k-1}}\Phi(\frac{d}{X}))
^{\frac{k-1}{k}}
\end{multline*}
when $k>1$. So by Propositions \ref{pp1} and \ref{pp2} we prove Theorem \ref{thm1}.
\begin{proposition}\label{pp1}
When $k>1$, we have
\begin{equation*}
\sum_{2\nmid d}\nolimits^\flat L(\frac{1}{2},\phi\otimes \chi_{8d}){\mathcal N}(d,k-1)\Phi(\frac{d}{X})\gg X(\log X)^{\frac{k^2+1}{2}}.
\end{equation*}
\end{proposition}
\begin{proposition}\label{pp2}
When $k>1$, we have
\begin{equation*}
\sum_{2\nmid d}\nolimits^\flat {\mathcal N}(d,k-1)^{\frac{k}{k-1}}\Phi(\frac{d}{X})
\ll X(\log X)^{\frac{k^2}{2}}.
\end{equation*}
\end{proposition}
To prove the above two propositions we will use the following Lemma.
\begin{lemma}[\cite{LWY05}]\label{lm:sum}
For normalized self-dual Hecke--Maass cusp form $\phi$ for $\SL(3,\mathbb{Z})$ with Fourier coefficients $A(m,n)$,
we have
\begin{equation*}
\sum_{p\leq x}\frac{A(p,1)^2}{p}=\log\log x+O_\phi(1).
\end{equation*}
\end{lemma}
\section{Proof of Proposition \ref{pp1}}
For convenience we write an integer $n=(n)_1(n)_2^2$ with $(n)_1$ is square-free.
We would prove Proposition \ref{pp1} with the help of the following lemma.
\begin{lemma}[\cite{HH21}]\label{lm:mysum}
Let $\phi$ be a self-dual Hecke--Maass cusp form of $\operatorname{SL}(3,\mathbb{Z})$ with normalized Fourier coefficients $A(m,n)$.
For sufficiently large $X>0$ and odd $l\ll X^{\frac{1}{10}-\varepsilon}$ with arbitrarily small $\varepsilon >0$, and any smooth nonnegative Schwarz class function $\Phi$ supported in the interval $(1,2)$, we have
\begin{multline*}
\sum_{2\nmid d}\nolimits^\flat L(\frac{1}{2},\phi\otimes \chi_{8d})\chi_{8d}(l)\Phi(\frac{d}{X})
= c\check{\Phi}(0)\frac{\lim_{s\to 1} (s-1)L^{\{2\}}(s,\sym^2 \phi)}
{\sqrt{(l)_1}}
\\
\times X(G(l)(\log \frac{X}{(l)_1 ^{\frac{2}{3}}}+c_1)+H(l))
+O_{\Phi}(l^{\frac{3}{4}+\varepsilon}
X^{\frac{19}{20}+\varepsilon})
\end{multline*}
where $\hat{\Phi}(x)=\int_{-\infty}^\infty \Phi(t)e(-xt)dt$, $\Phi_{(3)}=\max_{0\leq j \leq 3}\int_1^2|\Phi^{(j)}(t)|dt$,
$$G(l)=\prod_{\textrm{odd prime }p}G_p(l)$$
with
\begin{equation*}
G_p(p^a M)=G_p(p^a)=\left\{\begin{aligned}
&1+O(p_2^{-2+2\theta_3+\varepsilon}),\quad &2\mid a,\\
&A(p_1,1)+O(p^{-1+2\theta_3+\varepsilon}),\quad
&2\nmid a,
\end{aligned}
\right.
\end{equation*}
for $(M,p)=1$, with effective $O$-constant no more than 35.
$$H(l)=O\Big( \prod_{p|l_1}
(|A(p_1,1)|+p^{-1+2\theta_3+\varepsilon})
+\sum_{p\mid l_1}
p^{-1+2\theta_3}\log p
\prod_{\substack{
p_1|l_1\\ p_1\neq p}}
(|A(p_1,1)|+p^{-1+2\theta_3+\varepsilon}) \Big)
,$$
with effective $O$-constant.
And $c,c_1$ are constants depending on $\phi$ and $\Phi$.
\end{lemma}
Let $g(n)$ be a multiplicative function with $g(p^m)=m!$.
Let $\Omega(n)$ be the total number of prime factors of $n$.
Let indicator function $b_j(n)=1$ when all prime factors of $n$ are in $P_j$ and $\Omega(n)\leq \ell_j$, and $b_j(n)=0$ otherwise.
Let $A(m)$ be a completely multiplicative function with $A(p)=A(p,1)$ when $p$ is prime,
so $A(m)=A(m,1)$ when $m$ is square-free.
Thus
\begin{equation*}
{\mathcal N}_j(d,k-1)= \sum_{n_j}\frac{A(n_j)}{\sqrt{n_j}} \frac{(k-1)^{\Omega(n_j)}}{g(n_j)}
b_j(n_j)\chi_{8d}(n_j)
\end{equation*}
for $1\leq j\leq R$.
From Lemma \ref{lm:mysum} we know
\begin{multline*}
\sum_{2\nmid d}\nolimits^\flat L(\frac{1}{2},\phi\otimes \chi_{8d}){\mathcal N}(d,k-1)\Phi(\frac{d}{X})\\
\gg X\sum_{n_1,\dots,n_R}\prod_{j=1}^R\frac{A(n_j)
b_j(n_j)} {\sqrt{n_j(n_j)_1}}\frac{(k-1)^{\Omega(n_j)}}{g(n_j)}\\
\times (G(\prod_{j=1}^R n_j)(\log \frac{X}{((n_1)_1\dots(n_R)_1)^{\frac{2}{3}}}
+c_1)+H(\prod_{j=1}^R n_j)).
\end{multline*}
Here we only concentrate on the terms involving $G(\prod_{j=1}^R n_j)\log \frac{X}{((n_1)_1\dots(n_R)_1)^{\frac{2}{3}}}$.
By using the same argument in the following discussion,
the terms involving $G(\prod_{j=1}^R n_j)c_1,H(n_1\dots n_R)$ contribute as
$O(X(\log X)^{\frac{k^2+1}{2}-1+\varepsilon})$,
then
\begin{equation*}
\begin{split}
&\sum_{2\nmid d}\nolimits^\flat L(\frac{1}{2},\phi\otimes \chi_{8d}){\mathcal N}(d,k-1)\Phi(\frac{d}{X})\\
&\hskip 30pt \gg X\log X\sum_{n_1,\dots,n_R}G(\prod_{j=1}^R n_j)\prod_{j=1}^R\frac
{A(n_j)b_j(n_j)} {\sqrt{n_j(n_j)_1}}\frac{(k-1)^{\Omega(n_j)}}
{g(n_j)}\\
&\hskip 60pt -\frac{2}{3}X\sum_{n_1,\dots,n_R}G(\prod_{j=1}^R n_j)
(\prod_{j=1}^R\frac{A(n_j)b_j(n_j)} {\sqrt{n_j(n_j)_1}}\frac{(k-1)^{\Omega(n_j)}}
{g(n_j)}\log (\prod_{j=1}^R(n_j)_1)\\
&\hskip 90pt +O(X(\log X)^{\frac{k^2+1}{2}-1+2\theta_3+\varepsilon})\\
&\hskip 30pt =S_1-\frac{2}{3}S_2
+O(X(\log X)^{\frac{k^2+1}{2}-1+2\theta_3+\varepsilon}).
\end{split}
\end{equation*}
Evidently all $n_j$ in $S_1$ satisfy that $\Omega(n_j)\leq \ell_j$.
Removing the restriction for $\Omega(n_j)$ we get
\begin{equation*}
\begin{split}
S_1\geq X\log X\prod_{j=1}^R\Big (\prod_{p_j\in P_j}
(&\sum_{i=0}^{\infty}\frac{A(p_j)^{2i}G_{p_j}
(p_j^{2i})} {p^{i}}\frac{(k-1)^{2i}}{(2i)!}
+\sum_{i=0}^{\infty}\frac{A(p_j)^{2i+1}G_{p_j}
(p_j^{2i+1})} {p^{i+1}}\frac{(k-1)^{2i+1}}{(2i+1)!
})\\
& -\sum_{n_j}\frac{|A(n_j)\prod_{p_j\in P_j}G_{p_j}(n_j)|}
{\sqrt{n_j(n_j)_1}}
\frac{(k-1)^{\Omega(n_j)}}{g(n_j)} 2^{\Omega(n_j)-\ell_j}\Big )\prod_{p>X^{\frac{1}{\ell_R^2}}}G_p(1).
\end{split}
\end{equation*}
Note that $2^{\Omega(n_j)-\ell_j}> 1$ when $\Omega(n_j)>\ell_j$.
Hence we have
\begin{equation*}
\begin{split}
S_1\geq &\frac{X\log X}{\zeta(2-2\theta_3-0.01)^{35}} \prod_{j=1}^R\Bigg(\prod_{p_j\in P_j}
\Big(1-(\frac{(k-1)^2}{2}+k-1)\frac{A(p_j)^2}{p_j}
\Big)
^{-1}(1+O(\frac{|A(p_j)|+1}{p_j^{2-2\theta_3}}))\\
&\hskip 30pt -2^{-\ell_j}\prod_{p_j\in P_j}\Big(1+\sum_{i=1}^{\infty}\frac{A(p_j)^{2i}
G_{p_j}(p_j^{2i})} {p_j^{i}}\frac{(k-1)^{2i}2^{2i}}{(2i)!}
\\
&\hskip 60pt
+\sum_{i=0}^{\infty}\frac{A(p_j)^{2i+1}G_{p_j}(p_j^{2i+1})} {p_j^{i+1}}\frac{(k-1)^{2i+1}2^{2i+1}}{(2i+1)!
}\Big)\Bigg )\\
&\geq \frac{X\log X}{\zeta(2-2\theta_3-0.01)^{35}} \prod_{j=1}^R\Bigg (\exp\Big(\frac{k^2-1}{2}
\sum_{p_j\in P_j}(\frac{A(p_j,1)^2}{p_j}+O(\frac{|A(p_j,1)|}
{p_j^{2-2\theta_3}}))\Big)\\
&\hskip 30pt -2^{-\ell_j}\exp\Big((2(k-1)^2+2(k-1))\sum_{p_j\in P_j}(\frac{A(p_j,1)^2}{p_j}+O(\frac{|A(p_j,1)|}
{p_j^{2-2\theta_3}}))\Big) \Bigg ),
\end{split}
\end{equation*}
because $1+x\leq \exp(x)$ for $x\in\mathbb{R}$.
In our setting $M$ is large enough such that any $\ell_j$ is large,
and $N$ is large enough such that
\begin{equation}\label{doueq}
\frac{\ell_j}{4N}\leq \sum_{p_j\in P_j}\frac{A(p_j,1)^2}{p_j}\leq \frac{2\ell_j}{N}
\end{equation}
for every $j$ from Lemma \ref{lm:sum}.
Thus
\begin{multline*}
2^{-\ell_j}\exp\Big((2(k-1)^2+2(k-1))\sum_{p_j\in P_j}(\frac{A(p_j,1)^2}{p_j}+O(\frac{|A(p_j,1)|}
{p_j^{2-2\theta_3}}))\Big)\\
\leq 2^{-\frac{\ell_j}{2}}\exp\Big(\frac{k^2-1}{2}
\sum_{p_j\in P_j}(\frac{A(p_j,1)^2}{p_j}+O(\frac{|A(p_j,1)|}
{p_j^{2-2\theta_3}}))\Big).
\end{multline*}
So we get
\begin{equation}\label{eqn:S1>}
S_1\geq \frac{X\log X}{\zeta(2-2\theta_3-0.01)^{35}} \prod_{j=1}^R\Bigg ((1-2^{-\frac{\ell_j}{2}})\exp\Big(\frac{k^2-1}
{2}\sum_{p_j\in P_j}(\frac{A(p_j,1)^2}{p_j}+O(\frac{|A(p_j,1)|}
{p_j^{2-2\theta_3}}))\Big)\Bigg ).
\end{equation}
Now we estimate $S_2$.
Consider the primes dividing $\prod_{i=1}^R (n_i)_1$.
Let indicator function $\tilde{b}_{i,l}(n_i)=b_i(n_ip^l)$ when $b_i(p)\neq 0$ and $\tilde{b}_{i,l}(n_i)=b_i(n_i)$ otherwise, we have
\begin{equation*}
\begin{split}
S_2&\leq X\sum_{p\in \cup P_j}(\sum_{l\geq 0}\frac{A(p)^{2l+1}G_p(p^{2l+1})\log p}{p^{l+1}}\frac{(k-1)^{2l+1}}{(2l+1)!})\\
& \hskip 30pt \times
\prod_{i=1}^R (\sum_{(n_i,p)=1}\frac{A(n_i)\prod_{p\neq p_i\in R_i} G_{p_i}(n_i)}
{\sqrt{n_i(n_i)_1}}
\frac{(k-1)^{\Omega(n_i)}\tilde{b}_{i,l}(n_i)}
{g(n_i)})\prod_{p>X^{\frac{1}{\ell_R^2}}}G_p(1)\\
&=:X\prod_{p>X^{\frac{1}{\ell_R^2}}}G_p(1)
\sum_{p\in \cup P_j}S_p.
\end{split}
\end{equation*}
When $p\in P_{i_0}$, we know
\begin{equation*}
\begin{split}
S_p=&\prod_{\substack{i=1\\ i\neq i_0}}^{R}
(\sum_{n_i}\frac{A(n_i)b_i(n_i)\prod_{p_i\in P_i}G_{p_i}(n_i)}
{\sqrt{n_i(n_i)_1}}
\frac{(k-1)^{\Omega(n_i)}}{g(n_i)})
(\sum_{l\geq 0}\frac{A(p)^{2l+1}G(p^{2l+1})\log p}{p^{l+1}}\frac{(k-1)^{2l+1}}{(2l+1)!})\\
&\hskip 30pt \times
(\sum_{(n_{i_0},p)=1}\frac{A(n_{i_0})
\tilde{b}_{i_0,l}(n_{i_0})\prod_{p\neq p_i\in P_{i_0}}G_{p_i}(n_{i_0})}
{\sqrt{n_{i_0}(n_{i_0})_1}}
\frac{(k-1)^{\Omega(n_{i_0})}}
{g(n_{i_0})}).
\end{split}
\end{equation*}
Removing the restriction for $\tilde{b}_{i,l}$ on $\Omega(n_i)$, just as our discussion above, we have
\begin{equation*}
\begin{split}
&\sum_{(n_i,p)=1}\frac{A(n_i)b_i(n_i)
\prod_{p_i\in P_i}G_{p_i}(n_i)}
{\sqrt{n_i(n_i)_1}}
\frac{(k-1)^{\Omega(n_i)}}{g(n_i)}\\
&\leq \prod_{\substack{q\in P_i\\(p,q)=1}}\Big(\sum_{m=0}^{\infty}
\frac{A(q)^{2m}G_q(q^{2m})}{q^m}
\frac{(k-1)^{2m}}{(2m)!}
+\sum_{m=0}^{\infty}\frac{A(q)^{2m+1}G_q(q^{2m+1})} {q^{m+1}}\frac{(k-1)^{2m+1}}{(2m+1)!}\Big)\\
&\leq \exp\Big(\frac{k^2-1}{2}
\sum_{q\in P_i}(\frac{A(q,1)^2}{q}+
O(\frac{|A(q,1)|}{q^{2-2\theta_3}}))\Big),
\end{split}
\end{equation*}
and the error is no more than
\begin{multline*}
\sum_{n_j}\frac{A(n_i)\prod_{p\neq q\in P_i}G_q(n_i)}
{\sqrt{n_i(n_i)_1}}
\frac{(k-1)^{\Omega(n_i)}
2^{l\delta_{i=i_0}+\Omega(n_i)-\ell_i}}{g(n_i)}
\\
\leq 2^{l\delta_{i=i_0}-\frac{\ell_i}{2}}
\exp\Big(\frac{k^2-1}{2}
\sum_{q\in P_i}(\frac{A(q,1)^2}{q}+
O(\frac{|A(q,1)|}{q^{2-2\theta_3}}))\Big).
\end{multline*}
From $|A(p,1)|\leq 3p^{\theta_3}$ it shows that
\begin{equation*}
\begin{split}
S_p&\leq \prod_{i=1}^{R}\exp\Big(\frac{k^2-1}{2}
\sum_{q\in P_i}(\frac{A(q,1)^2}{q}+
O(\frac{|A(q,1)|}{q^{2-2\theta_3}}))\Big)\\
&\hskip 30pt\times
\prod_{\substack{i=1\\ i\neq i_0}}^{R}
(1+2^{-\frac{\ell_i}{2}})
(\sum_{l\geq 0}
\frac{A(p)^{2l+1}G(p^{2l+1})\log p}{p^{l+1}}\frac{(k-1)^{2l+1}}{(2l+1)!}
(1+2^{l-\frac{\ell_i}{2}}))\\
&\leq \prod_{i=1}^{R}\exp\Big(\frac{k^2-1}{2}
\sum_{q\in P_i}(\frac{A(q,1)^2}{q}+
O(\frac{|A(q,1)|}{q^{2-2\theta_3}}))\Big)\\
&\hskip 30pt\times
\prod_{\substack{i=1\\ i\neq i_0}}^{R}
(1+2^{-\frac{\ell_i}{2}})
(\sum_{l\geq 0}\frac{A(p)^{2l+1}G_p(p^{2l+1})\log p}{p^{l+1}}\frac{(k-1)^{2l+1}}{(2l+1)!}
(1+2^{l-\frac{\ell_i}{2}}))\\
&\leq B\frac{A(p)^2\log p}{p}\prod_{i=1}^{R}\exp \Big(\frac{k^2-1}{2}\sum_{q\in P_i}\frac{A(q,1)^2}{q}\Big),
\end{split}
\end{equation*}
where $B$ is a constant depending only on $k$.
Thus
\begin{equation}\label{eqn:S2<}
\begin{split}
S_2&\leq \zeta(2-2\theta_3-0.01)^{35}BX\prod_{i=1}^{R}\exp\Big(\frac{k^2-1}{2}
\sum_{q\in P_i}\frac{A(q,1)^2}{q}\Big)
\sum_{p\in \cup P_j}\frac{A(p,1)^2\log p}{p}\\
&\leq \zeta(2-2\theta_3-0.01)^{35}BX(\frac{\log X}{10^{2M}}+O(1))\exp\Big(\frac{k^2-1}{2}\sum_{q\in P_i}\frac{A(q,1)^2}{q}\Big).
\end{split}
\end{equation}
For $M$ large enough, by \eqref{eqn:S1>} and \eqref{eqn:S2<}, we obtain
\begin{equation*}
S_1-\frac{2}{3}S_2\gg X\log X\prod_{i=1}^{R}\exp\Big((\frac{k^2-1}{2}
)\sum_{q\in P_i}\frac{A(q,1)^2}{q}\Big)\gg X(\log X)^{\frac{k^2+1}{2}},
\end{equation*}
which proves Proposition \ref{pp1}.
\section{Proof of Proposition \ref{pp2}}
In this section we will prove Proposition \ref{pp2} with the help of the following lemmas.
\begin{lemma}[\cite{SD00}]\label{prsum}
For any odd $n>0$, we have
\begin{equation*}
\sum_{2\nmid d}\nolimits^\flat \chi_{8d}(n)\Phi(\frac{d}{X})=
\delta_{n=\square}{\hat\Phi}(0) \frac{2X}{3\zeta(2)}\prod_{p|n}
(\frac{p}{p+1})+
O(X^{1/2+\varepsilon}\sqrt{n}),
\end{equation*}
where $\delta_{n=\square}=1$ if $n$ is a square and $\delta_{n=\square}=0$ otherwise.
\end{lemma}
\begin{lemma}\label{lmNj}
For any $1\leq j\leq R$, we have
\begin{equation*}
{\mathcal N}_j(d,k-1)^{\frac{k}{k-1}}\leq {\mathcal N}_j(d,k)\frac{(1+e^{-\ell_j})^{\frac{k}{k-1}}}
{(1-e^{-\ell_j})}+(\frac{12k^2{\mathcal P}_j(d)}{\ell_j})^{2r_k\ell_j}.
\end{equation*}
where $r_k=\lceil \frac{k}{2k-2}\rceil$.
\end{lemma}
\begin{proof}
We will prove this lemma analogous to \cite[Lemma 3.4]{GZ21} and \cite[Lemma 1]{HS20}.
For $|x|\leq \frac{K}{10}$, we have
\begin{equation*}
|\sum_{r=0}^{K}\frac{x^r}{r!}-e^x|\leq \frac{|x|^K}{K!}\leq (\frac{e}{10})^K
\leq e^{-|x|-K},
\end{equation*}
then
\begin{equation*}
\sum_{r=0}^{K}\frac{x^r}{r!}=e^x(1+O(
e^{-K}))
\end{equation*}
and the $O$-constant could takes $1$.
For $x=\alpha{\mathcal P}_j(d)$ and $K=\ell_j$, with $\alpha>0$ and $|\alpha{\mathcal P}_j(d)|\leq \ell_j /10$, we have
\begin{equation*}
{\mathcal N}_j(d,\alpha)= \exp(\alpha{\mathcal P}_j(d))(1+O(e^{-\ell_j})).
\end{equation*}
Thus when $|{\mathcal P}_j(d)|\leq \frac{\ell_j}{10k}$, we have
\begin{equation}\label{eqn:N<1}
{\mathcal N}_j(d,k-1)^{\frac{k}{k-1}}\leq
\exp(k{\mathcal P}_j(d))(1+e^{-\ell_j})^{\frac{k}{k-1}}\leq
|{\mathcal N}_j(d,k)|(1+e^{-\ell_j})^{\frac{k}{k-1}}
(1-e^{-\ell_j})^{-1}.
\end{equation}
When $|{\mathcal P}_j(d)|\geq \frac{\ell_j}{10k}$, we know
\begin{equation}\label{eqn:N<2}
|{\mathcal N}_j(d,k-1)|\leq
\sum_{r=0}^{\ell_j}\frac{|(k-1){\mathcal P}_j(d))|^r}{r!}\leq |k{\mathcal P}_j(d))|^{\ell_j}\sum_{r=0}^{\ell_j}
(\frac{10k}{\ell_j})^{\ell_j-r}\frac{1}{r!}
\leq(\frac{12k^2|{\mathcal P}_j(d))|}{\ell_j})^{\ell_j}.
\end{equation}
By \eqref{eqn:N<1} and \eqref{eqn:N<2}, we complete the proof of the lemma.
\end{proof}
From $\prod_{j=1}^{R}\max(\frac{(1+e^{-\ell_j})^{\frac{k}{k-1}}}
{(1-e^{-\ell_j})},1)\ll 1$, we have
\begin{equation}\label{eqpp2}
\begin{split}
\sum_{2\nmid d}\nolimits^\flat {\mathcal N}(d,k-1)^{\frac{k}{k-1}}
\Phi(\frac{d}{X})&\ll
\sum_{2\nmid d}\nolimits^\flat \prod_{j=1}^{R}({\mathcal N}_j(d,k)+(\frac{12k^2{\mathcal P}_j(d)}{\ell_j})^{2r_k\ell_j})
\Phi(\frac{d}{X})\\
&=:\sum_{2\nmid d}\nolimits^\flat \prod_{j=1}^{R}(\sum_{n_j\leq X^{\frac{2r_k}{\ell_j}}}\frac{a_{n_j}b_j(n_j)}
{\sqrt{n_j}}\chi_{8d}(n_j))\Phi(\frac{d}{X}).
\end{split}
\end{equation}
Recall that
\begin{equation*}
{\mathcal N}_j(d,k)=\sum_{n_j}\frac{A(n_j)}
{\sqrt{n_j}}\frac{k^{\Omega(n_j)}}
{g(n_j)}b_j(n_j)\chi_{8d}(n_j),
\end{equation*}
and
\begin{equation*}
{\mathcal P}_j(d)^{2r_k\ell_j}=\sum_{\substack{
\Omega(n_j)=2r_k\ell_j\\p\mid n_j\Rightarrow p\in P_j}}\frac{A(n_j)}{\sqrt{n_j}}
\frac{(2r_k\ell_j)!}{g(n_j)}\chi_{8d}(n_j),
\end{equation*}
where $r_k=\lceil \frac{k}{2k-2}\rceil$.
From
\begin{equation}\label{n!eq}
(\frac{n}{e})^n\leq n!\leq n(\frac{n}{e})^n,
\end{equation}
we know
\begin{equation*}
(\frac{12k^2}{\ell_j})^{2r_k\ell_j}
(2r_k\ell_j)!\leq 2r_k\ell_j(\frac{24k^2}{e})^{2r_k\ell_j}.
\end{equation*}
By setting $M$ large enough so that $\ell_R>12r_k$,
we have $|a_{n_j}|\leq B_k^{\ell_j}n_j^{\theta_3}$ with some constant $B_k$ depending only on $k$.
Using Lemma \ref{prsum} on the right side in \eqref{eqpp2}, the contribution from error terms in Theorem \ref{prsum} are
\begin{equation*}
\ll X^{\frac{1}{2}+\varepsilon}
\prod_{j=1}^{\ell_j} B_k^{\ell_j}
X^{(1+\theta_3)\frac{2r_k}{\ell_j}}\ll
B_k^{R\ell_1}X^{\frac{1}{2}
+\frac{4(1+\theta_3)r_k}{\ell_R}+\varepsilon}
\ll X^{1-\varepsilon}.
\end{equation*}
The main term of the right side in \eqref{eqpp2} is a scalar multiple of
\begin{multline*}
X\times\sum_{\prod_{j=1}^R n_j=\square}(\prod_{j=1}^{R}
\frac{a_{n_j}b_j(n_j)}
{\sqrt{n_j}}\prod_{p\mid n_j}\frac{p}{p+1})
=
X\times\prod_{j=1}^{R}
\sum_{n_j=\square}\frac{a_{n_j}b_j(n_j)}
{\sqrt{n_j}}\prod_{p\mid n_j}\frac{p}{p+1}
\\
=X\times\prod_{j=1}^{R}
\Big (
\sum_{n_j=\square}\frac{A(n_j)}{\sqrt{n_j}}
\frac{k^{\Omega(n_j)}}{g(n_j)}b_j(n_j)
\prod_{p\mid n_j}\frac{p}{p+1}
\\
+(\frac{12k^2}{\ell_j})^{2r_k\ell_j}
(2r_k\ell_j)!\sum_{\substack{n_j=\square \\
\Omega(n_j)=2r_k\ell_j\\p\mid n_j\Rightarrow p\in P_j}}
\frac{A(n_j)}{\sqrt{n_j}}
\frac{1}{g(n_j)}
\prod_{p\mid n_j}\frac{p}{p+1}
\Big ).
\end{multline*}
From $|A(p,1)|\leq 3p^{\theta_3}$ and the Taylor expansion of $\exp(x)$ we can see
\begin{equation*}
\begin{split}
\sum_{n_j=\square}\frac{A(n_j)}{\sqrt{n_j}}
\frac{k^{\Omega(n_j)}}{g(n_j)}b_j(n_j)
\prod_{p\mid n_j}\frac{p}{p+1}
&\leq \prod_{p\in P_j}\Big(1+\sum_{1\leq i\leq \lceil \frac{\ell_i}{2}\rceil}\frac{A(p,1)^{2i}k^{2i}}{p^{i}(2i)!}
\frac{p}{p+1}\Big)
\\
&\leq \prod_{p\in P_j}(1+\frac{A(p,1)^2k^2}{2p}\frac{p}{p+1}
(1+\sum_{i\geq 1}\frac{(3k)^{2i}}{p^{(1-2\theta_3)i}(2i)!}))\\
&\leq \exp\Big(\frac{k^2}{2}\sum_{p\in P_j}\frac{A(p,1)^2}{p}(1+
\frac{e^{3k}}{p^{1-2\theta_3}})\Big)\\
&\ll_k \exp(\frac{k^2}{2}\sum_{p\in P_j}\frac{A(p,1)^2}{p}).
\end{split}
\end{equation*}
From \eqref{doueq} and \eqref{n!eq} we know
\begin{equation*}
\begin{split}
&(\frac{12k^2}{\ell_j})^{2r_k\ell_j}
(2r_k\ell_j)!\sum_{\substack{n_j=\square\\
\Omega(n_j)=2r_k\ell_j\\p\mid n_j\Rightarrow p\in P_j}}
\frac{A(n_j)}{\sqrt{n_j}}
\frac{1}{g(n_j)}
\prod_{p\mid n_j}\frac{p}{p+1}\\
&
\leq
(\frac{12k^2}{\ell_j})^{2r_k\ell_j}
\frac{(2r_k\ell_j)!}{(r_k\ell_j)!}
(\sum_{p\in P_j}
\frac{A(p,1)^2}{p})^{r_k\ell_j}
\\
&
\leq
2r_k\ell_j(\frac{576k^4r_k}{e\ell_j})
^{r_k\ell_j}(\sum_{p\in P_j}
\frac{A(p,1)^2}{p})^{r_k\ell_j}\\
&
\leq
2r_k\ell_j(\frac{1152k^4r_k}{eN})
^{r_k\ell_j}\\
&
\ll e^{-\ell_j}\exp\Big(\frac{k^2}{2}\sum_{p\in P_j}\frac{A(p,1)^2}{p}\Big).
\end{split}
\end{equation*}
Then we have
\begin{equation*}
\begin{split}
\sum_{2\nmid d}\nolimits^\flat {\mathcal N}(d,k-1)^{\frac{k}{k-1}}
\Phi(\frac{d}{X})&\ll
X\prod_{j=1}^{R}(1+e^{-\ell_j})
\exp\Big(\frac{k^2}{2}\sum_{p\in P_j}\frac{A(p,1)^2}{p}\Big)\ll X(\log X)^{\frac{k^2}{2}}.
\end{split}
\end{equation*}
This proves Proposition \ref{pp2}.
\section*{Acknowledgements}
The authors would like to thank referees for their careful reading and nice comments.
| {
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{"url":"https:\/\/bora.uib.no\/bora-xmlui\/handle\/1956\/19642\/browse?type=subject&value=ATLAS","text":"Viser treff 1-3 av 3\n\n\u2022 A Setup for Testing of Silicon Pixel Modules for the ITk Tracker in the ATLAS Experiment at CERN \ufeff\n\n(Master thesis, 2021-06-14)\nNew pixel sensors will be installed to the new Inner Tracker upgrade at the ATLAS detector. These sensor will have higher radiation tolerance than the ones currently on the Inner Detector. This thesis will concentrate on ...\n\u2022 Search for Dark Matter produced in association with a Higgs boson decaying to tau leptons at $\\sqrt{\\textrm{s}}$= 13 TeV with the ATLAS detector \ufeff\n\n(Master thesis, 2020-12-09)\nThis thesis presents a search for Dark Matter produced in association with the Standard Model Higgs boson decaying into two tau leptons at $\\sqrt{s}=13$ TeV using Run 2 data from the ATLAS detector corresponding to an ...\n\u2022 Search for squarks in events with jets, hadronically decaying \u03c4 -lepton, and missing transverse momentum in the final state in proton-proton collision at \u221as = 13 TeV with the ATLAS detector \ufeff\n\n(Master thesis, 2021-09-01)\nSignal regions have been designed for search of squarks in events with jets, at least one hadronically decaying \u03c4 -lepton, and missing transverse momentum in the final state. The analysis is based on the LHC Run 2 dataset ...","date":"2021-12-05 14:50:48","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4483410716056824, \"perplexity\": 2493.7598830253296}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964363189.92\/warc\/CC-MAIN-20211205130619-20211205160619-00108.warc.gz\"}"} | null | null |
ACCEPTED
#### According to
International Plant Names Index
#### Published in
null
#### Original name
Viscum platyphyllum Spreng.
### Remarks
null | {
"redpajama_set_name": "RedPajamaGithub"
} | 8,001 |
San Gregorio de Polanco ist eine Stadt in Uruguay.
Geographie
Sie liegt im Süden des Departamento Tacuarembó am rechtsseitigen Ufer des Río Negro.
Geschichte
Am 15. Oktober 1963 wurde San Gregorio de Polanco durch das Gesetz Nr. 11.689 in die Kategorie "Villa" eingestuft.
Einwohner
Bei der Volkszählung 1996 wurden 3.101 Einwohner gezählt. 2004 betrug die Einwohnerzahl von San Gregorio de Polanco 3.673 (Stand: 2004). Bei der Volkszählung 2011 waren 3.415 Einwohner in der Stadt registriert, davon 1.702 männliche und 1.713 weibliche. Damit ist sie die drittgrößte Stadt des Departamentos.
Quelle: Instituto Nacional de Estadística de Uruguay
Infrastruktur
Verkehr
Durch die Stadt führt die Ruta 43.
Stadtverwaltung
Bürgermeister (Alcalde) von San Gregorio de Polanco ist Sergio Texeira.
Söhne und Töchter der Stadt
Julio Alpuy (* 1919), Künstler
Cacho Labandera, Musiker
Weblinks
Stadtplan von San Gregorio de Polanco (PDF; 114 kB)
Einzelnachweise
Ort im Departamento Tacuarembó | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,259 |
Q: ASP.NET MVC How many levels deep should a view or URL be? I am still learning ASP.NET MVC. With webforms, I would create a new folder let's call it admin. In there I might have many pages for create_product, edit_product, etc. So the URL might look like http://somesite.com/admin/create_product.aspx.
But with MVC it is a little different. I am trying to see what would be the best way to do this.
Would doing http://somesite.com/admin/product/create be right? Or should it just be http://somesite.com/product/create? If I do it as the first way, do I put everything in the "admin" controller or should it be separated into a "product" controller?
I know this is probably subjective or personal choice, but I would like to get some advise.
Thanks.
A: Part of the benefit of ASP.NET MVC (and more generally, the URL Routing Engine common to all of ASP.NET in .NET 3.5 SP1) is that the URLs can be flexibly configured to map to any folder / file structure you prefer. That means it's much easier than it was in the days of WebForms to modify your URLs after you've started building your project.
To your specific questions:
*
*One Admin Controller vs. Product Controller - In general, the guidance is to keep controllers focused so that they are easier to test and maintain. For that reason, I would suggest using a single controller per object type (like Product) with your CRUD actions. Examples in your case:
/admin/product/create
/admin/product/edit/34 or /admin/product/edit/red-shoes (if name is unique)
In either case, the Create, Edit, Deatils actions will all be in the ProductController. You may just have custom routes for the "admin actions" (like Create and Edit) that limit their usage (and add the "admin" text to the URL), and then the Details action would be usable by all visitors to your site.
*Securing Admin Views - One important fact to remember with MVC: all requests go directly to controllers, not views. That means the old "secure a directory with web.config" does not apply (usually) to MVC for securing your Admin. Instead, you should now apply security directly to the controllers. This can easily be achieved by using attributes to Controller classes like:
*
*[Authorize] - Just checks that the user is logged-in
*[Authorize(Roles = "Admin")] - Limit to specific user roles
*[Authorize(Users = "Joe")] - Limit to specific users
You can even create a custom route for "Admin" views in your site and limit access to those views by enforcing your authorization check in the URL routing, like this:
routes.MapRoute(
"Admin",
"Admin/{controller}/{action}",
new { controller = "Product", action = "Index" },
new { authenticated= new AuthenticatedConstraint()}
);
Where AuthenticatedConstraint looks something like:
using System.Web;
using System.Web.Routing;
public class AuthenticatedConstraint : IRouteConstraint
{
public bool Match(HttpContextBase httpContext, Route route, string parameterName, RouteValueDictionary values, RouteDirection routeDirection)
{
return httpContext.Request.IsAuthenticated;
}
}
Good details on Stephen Walther's blog:
ASP.NET MVC Tip #30 – Create Custom Route Constraints
A: For admin stuff, just mark with [Authorize] attribute. To ensure only admins can use it, do something like [Authorize(Roles = "Admin")]. Check out this question
Also, /product/create is most common, I think :)
A: I3Dx definitely has the right guidance for the Authorize attribute, this is essential for keeping controller secure, you can apply to a controller or individual actions.
As far as the URL depth, I would not worry about the depth, I would be more concerned that the route made logical sense for example:
domain.com/admin/products/edit/1
domain.com/admin/groups/edit/1
domain.com/products/view/1
domain.com/groups/view/1
This way you know what is happening with each route. it is obvious that one is an admin and one is an end user.
The easyest way to check is to get someone to read your URL and ask them what they would expect to see.
Hope this helps.
OH and one last thing, for client side routes we often use "slugs" rather than ids so that it is more readable. So when someone creates a product we slugify the name so it can be used in the route such as:
domain.com/products/view/big-red-bucket
rather than
domain.com/products/view/1
| {
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namespace mozc {
namespace dictionary {
struct Token {
typedef uint8 AttributesBitfield;
enum Attribute {
NONE = 0,
SPELLING_CORRECTION = 1,
LABEL_SIZE = 2,
// * CAUTION *
// If you are going to add new attributes, make sure that they have larger
// values than LABEL_SIZE!! The attributes having less values than it are
// tightly integrated with the system dictionary codec.
// The following attribute is not stored in the system dictionary but is
// added by dictionary modules when looking up from user dictionary.
USER_DICTIONARY = 1 << 7,
};
Token() : cost(0), lid(0), rid(0), attributes(NONE) {}
string key;
string value;
int cost;
int lid;
int rid;
AttributesBitfield attributes;
};
} // namespace dictionary
} // namespace mozc
#endif // MOZC_DICTIONARY_DICTIONARY_TOKEN_H_
| {
"redpajama_set_name": "RedPajamaGithub"
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{"url":"https:\/\/www.ademcetinkaya.com\/2022\/10\/how-is-machine-learning-used-in-trading_13.html","text":"Efficient Market Hypothesis (EMH) is the cornerstone of the modern financial theory and it states that it is impossible to predict the price of any stock using any trend, fundamental or technical analysis. Stock trading is one of the most important activities in the world of finance. Stock price prediction has been an age-old problem and many researchers from academia and business have tried to solve it using many techniques ranging from basic statistics to machine learning using relevant information such as news sentiment and historical prices. We evaluate S&P\/BMV IPC Index prediction models with Modular Neural Network (CNN Layer) and Multiple Regression1,2,3,4 and conclude that the S&P\/BMV IPC Index stock is predictable in the short\/long term. According to price forecasts for (n+8 weeks) period: The dominant strategy among neural network is to Hold S&P\/BMV IPC Index stock.\n\nKeywords: S&P\/BMV IPC Index, S&P\/BMV IPC Index, stock forecast, machine learning based prediction, risk rating, buy-sell behaviour, stock analysis, target price analysis, options and futures.\n\n## Key Points\n\n1. Game Theory\n3. Why do we need predictive models?\n\n## S&P\/BMV IPC Index Target Price Prediction Modeling Methodology\n\nIn this paper a Bayesian regularized artificial neural network is proposed as a novel method to forecast financial market behavior. Daily market prices and financial technical indicators are utilized as inputs to predict the one day future closing price of individual stocks. The prediction of stock price movement is generally considered to be a challenging and important task for financial time series analysis. We consider S&P\/BMV IPC Index Stock Decision Process with Multiple Regression where A is the set of discrete actions of S&P\/BMV IPC Index stock holders, F is the set of discrete states, P : S \u00d7 F \u00d7 S \u2192 R is the transition probability distribution, R : S \u00d7 F \u2192 R is the reaction function, and \u03b3 \u2208 [0, 1] is a move factor for expectation.1,2,3,4\n\nF(Multiple Regression)5,6,7= $\\begin{array}{cccc}{p}_{a1}& {p}_{a2}& \\dots & {p}_{1n}\\\\ & \u22ee\\\\ {p}_{j1}& {p}_{j2}& \\dots & {p}_{jn}\\\\ & \u22ee\\\\ {p}_{k1}& {p}_{k2}& \\dots & {p}_{kn}\\\\ & \u22ee\\\\ {p}_{n1}& {p}_{n2}& \\dots & {p}_{nn}\\end{array}$ X R(Modular Neural Network (CNN Layer)) X S(n):\u2192 (n+8 weeks) $\u2211 i = 1 n s i$\n\nn:Time series to forecast\n\np:Price signals of S&P\/BMV IPC Index stock\n\nj:Nash equilibria\n\nk:Dominated move\n\na:Best response for target price\n\nFor further technical information as per how our model work we invite you to visit the article below:\n\nHow do AC Investment Research machine learning (predictive) algorithms actually work?\n\n## S&P\/BMV IPC Index Stock Forecast (Buy or Sell) for (n+8 weeks)\n\nSample Set: Neural Network\nStock\/Index: S&P\/BMV IPC Index S&P\/BMV IPC Index\nTime series to forecast n: 14 Oct 2022 for (n+8 weeks)\n\nAccording to price forecasts for (n+8 weeks) period: The dominant strategy among neural network is to Hold S&P\/BMV IPC Index stock.\n\nX axis: *Likelihood% (The higher the percentage value, the more likely the event will occur.)\n\nY axis: *Potential Impact% (The higher the percentage value, the more likely the price will deviate.)\n\nZ axis (Yellow to Green): *Technical Analysis%\n\n## Conclusions\n\nS&P\/BMV IPC Index assigned short-term Ba3 & long-term B1 forecasted stock rating. We evaluate the prediction models Modular Neural Network (CNN Layer) with Multiple Regression1,2,3,4 and conclude that the S&P\/BMV IPC Index stock is predictable in the short\/long term. According to price forecasts for (n+8 weeks) period: The dominant strategy among neural network is to Hold S&P\/BMV IPC Index stock.\n\n### Financial State Forecast for S&P\/BMV IPC Index Stock Options & Futures\n\nRating Short-Term Long-Term Senior\nOutlook*Ba3B1\nOperational Risk 6685\nMarket Risk7838\nTechnical Analysis3361\nFundamental Analysis9047\nRisk Unsystematic6159\n\n### Prediction Confidence Score\n\nTrust metric by Neural Network: 79 out of 100 with 803 signals.\n\n## References\n\n1. V. Konda and J. Tsitsiklis. Actor-Critic algorithms. In Proceedings of Advances in Neural Information Processing Systems 12, pages 1008\u20131014, 2000\n2. Dimakopoulou M, Zhou Z, Athey S, Imbens G. 2018. Balanced linear contextual bandits. arXiv:1812.06227 [cs.LG]\n3. Breusch, T. S. (1978), \"Testing for autocorrelation in dynamic linear models,\" Australian Economic Papers, 17, 334\u2013355.\n4. Firth JR. 1957. A synopsis of linguistic theory 1930\u20131955. In Studies in Linguistic Analysis (Special Volume of the Philological Society), ed. JR Firth, pp. 1\u201332. Oxford, UK: Blackwell\n5. Swaminathan A, Joachims T. 2015. Batch learning from logged bandit feedback through counterfactual risk minimization. J. Mach. Learn. Res. 16:1731\u201355\n6. Chernozhukov V, Demirer M, Duflo E, Fernandez-Val I. 2018b. Generic machine learning inference on heteroge- nous treatment effects in randomized experiments. NBER Work. Pap. 24678\n7. Chernozhukov V, Newey W, Robins J. 2018c. Double\/de-biased machine learning using regularized Riesz representers. arXiv:1802.08667 [stat.ML]\nFrequently Asked QuestionsQ: What is the prediction methodology for S&P\/BMV IPC Index stock?\nA: S&P\/BMV IPC Index stock prediction methodology: We evaluate the prediction models Modular Neural Network (CNN Layer) and Multiple Regression\nQ: Is S&P\/BMV IPC Index stock a buy or sell?\nA: The dominant strategy among neural network is to Hold S&P\/BMV IPC Index Stock.\nQ: Is S&P\/BMV IPC Index stock a good investment?\nA: The consensus rating for S&P\/BMV IPC Index is Hold and assigned short-term Ba3 & long-term B1 forecasted stock rating.\nQ: What is the consensus rating of S&P\/BMV IPC Index stock?\nA: The consensus rating for S&P\/BMV IPC Index is Hold.\nQ: What is the prediction period for S&P\/BMV IPC Index stock?\nA: The prediction period for S&P\/BMV IPC Index is (n+8 weeks)","date":"2022-12-05 00:58:46","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 2, \"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\": 0.6147044897079468, \"perplexity\": 13830.625169319906}, \"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-2022-49\/segments\/1669446711001.28\/warc\/CC-MAIN-20221205000525-20221205030525-00198.warc.gz\"}"} | null | null |
If you want to live in a tiny town in the northwestern part of Colorado, you'll love Yampa. This Routt County getaway has fewer than 1,000 residents, which makes it the kind of place where you can get to know your neighbors and make new friends. For those who are searching for a tight-knit community on the outskirts of Steamboat Springs, you'll enjoy becoming a part of this mountainside town.
Here's more information about life and real estate in Yampa, so you can decide whether or not to make this municipality your home.
This small town, which used to be a hunting camp, has grown into a place where visitors in Colorado can go on their way to the famed Flat Topped Mountains. Because it was once home to numerous yampa plants, it got its name from being a leading food source in the area. Yampa then became a shipping center over a century ago, providing residents of the Steamboat Springs area with food that was transported by rail into Wolcott.
Even though the city has grown out of its role of connecting these settlements to the outside world, it remains a popular place to grow food for the cattle and sheep living on nearby farms.
Given that Yampa is one of the more rural retreats in this area, it's a popular place for visitors and residents alike to enjoy some peace and quiet. The flourishing natural surroundings offer countless opportunities for recreation, where you can camp, boat and fish. With attractions like the Stillwater Reservoir near the town, each and every resident can keep busy while spending quality time outside with friends and family.
You can also enjoy attractions like the Routt National Forest and the Flat Tops Wilderness Area, where you'll find hiking trails, stunning views and more in and around the large mountain range.
Yampa is an appealing place to settle down if you want to surround yourself with nature and spend time outside all year long. Because the area is home to pastures, lakes, and forests, you can choose to look at lots and land with easy access to these sights. You'll find a range of single-family residences on acreages, ranches, and equestrian properties, therefore, which make Yampa suitable for every kind of buyer. Once you have a good idea of your budget for investing in a home, you can hone in on the type of house and land you want to make the most of living in this remote part of Colorado with your family.
For those who are hoping to take in the state's natural beauty at every turn while at home in Colorado, Yampa is a great place to start your house hunt. This small town lets you become apart of the friendly community, where you can also enjoy your own open spaces year-round. You'll have access to tourist destinations as well, which gives property owners the best of both worlds in this part of the Rocky Mountains.
If you have any questions about life and real estate in Yampa, please don't hesitate to contact us for more information today. | {
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Experience the fun of hometown parades, summer picnics and fireworks with Americana, a collection from Pebbles. Images of stars, stripes, and florals are presented in a truly traditional palette of cherry red, brilliant white and navy blue.
This element pack includes 60 elements in PNG format. | {
"redpajama_set_name": "RedPajamaC4"
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Q: cyclic bayesian network i have several elements A,B,C,AB,ABC,.. (see image below) where each element either exists or not. the rule that governs this system is as follows: if AB exists, then A and B must also exist. generally speaking if a tupel exists, all smaller tupels which are subsets of this tupel must also exist. furthermore if a tupel does not exist, all tupels which make up a superset of this tupel do not exist.
http://i.stack.imgur.com/8fNl6.gif
Example:
Given ABC exists then A, B, C, AB, AC, BC exist too. Given BC does not exist then ABC,BCD,ABCD do not exist either.
now what i struggle with is, how do i calculate e.g. P(AB|A,B,!ABC) which means the probability that AB exists, given A exists, B exists and ABC does not exist. foreach element i have a basic starting probability p(X) which tells me how likely it is for X to exists given NO constraints. and usually i check the existence of A,B,C,D,ABCD beforehand so the system has boundaries.
my problem is that this is a cyclic network. i would be very grateful for any help as i tried solving this problem for the last couple of weeks without success. i only want to calculate the probability that one element exists, given any situation/constraint. note that elements like AB and !BD are not independent.
A: if you want to use bayesian network, then first you need to add directions in your chart, which would be from bottoms up.
Then draw the DAG with your p values using MSBNx, which you can get from: http://research.microsoft.com/en-us/um/redmond/groups/adapt/msbnx/ and run Bayesian inference, you should have no problem doing Probability queries on it.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,658 |
\section{Introduction}
The quantum electrodynamics (QED) is the most precisely experimentally tested theory in today's fundamental theories. In usual perturbation theory of QED, the expansion parameter is the fine structure constant $\alpha \sim \frac{1}{137}$. However, in strong external field, since the interaction is correspondingly strong, the perturbation theory breaks down. From the result of relativistic quantum mechanics, the vacuum around an atom with large atomic number $Z\gtrsim 137$ is expected to collapse.
In non relativistic quantum mechanics, in the region where the potential is larger than the energy, the wave function falls off exponentially. Namely, all the incoming particles are reflected; that is, the reflection rate is $R=1$ and transmision rate is $T=0$. On the other hand, in relativistic quantum mechanics, when height of the potential $V_0$ is larger than twice of particle mass $2m$, the reflection rate becomes larger than unity ($R>1$), which is called the Klein tunneling\cite{Klein:1929zz,Bjorken:1964}. This mechanism originates from the fact that the Dirac equation has both the positive and negative energy solutions as opposed to the Schr\"dingier equation. The same mechanism also prevents the electron to form bound states in a very strong attractive potential.
In particular, an electron around a nuclei with a sufficiently large atomic number $Z$ does not form a bound state due to the strong Coulomb potential, and fall into the nuclei. This phenomenon is called the atomic collapse, and has been known theoretically for a long time. However, since the atom with $Z\gtrsim 137$ can be created for only a short time in heavy ion collision experiment, it is difficult to observe the phenomena experimentally at the quantitive level \cite{Schweppe:1983yv,Cowan:1985cn}.
The situation has changed since the discovery of the graphene in 2004\cite{Novoselov:2004}. The electric structure of the graphene at low energy is known to be the same as that of the massless Dirac fermion. In addition, the effective fine structure constant $\alpha $ is about $300$ times as large as that in the Quantum Electro Dynamics (QED). Due to this property, the essential point of the physics in the strongly coupled QED can be tested in the experiment using the graphene. Putting charged impurities on the graphene, one can realize a system similar to the large $Z$ atom system, which enable us to observe the ``atomic collapse''.
The graphene is very thin, very light, very strong, and has very high electron conductivity and made from carbon atoms which is a ubiquitous element on earth. Therefore the graphene is expected to serve as ideal device in future. In this point of view, understanding the response of the electron to the charged impurity in graphene is a very important problem and is studied actively. This system is well studied in one body quantum mechanics as the system of the two dimensional massless electron in Coulomb potential \cite{Pereira:2007,Shytov:2007a,Nishida:2014}. It is predicted that when the charge of the impurity exceeds a critical value $Z_{cr}$, the wave function drastically changes. The massless fermion forms infinite number of quasibound states with negative energy, and the characteristic resonances appear in the local density of states (LDOS) of the electron\cite{Shytov:2007b}. Inspired by these theoretical studies, the scanning tunneling microscope (STM) experiment was carried out and a characteristic peak in LDOS was measured\cite{Wang:2013}.
However, the above theoretical studies do not take into account the many body effect which involve electron-positron pair creation. In the graphene case, since the pair creation can occur with no cost of extra energy, the many body effect should not be neglected, which should be treated in the quantum field theory. Moreover, because of the large coupling perturbative approximation cannot be valid.
Thus, this problem should be studied in some nonperturbative way.
We analyze the field theory of 2+1 dimensional Dirac massless fermion around an external charge using the bosonization technique. In two dimensional theory, the fermion theory is converted to the boson theory\cite{Coleman:1974bu,Mandelstam:1975hb}. It is known that a part of quantum effect of the fermion theory can be extracted from the classical boson theory. The bosonization method has been used to analyze the system with the fermion around monopole assuming that the classical boson theory captures the essential features of the quantum effect of the original fermion theory\cite{Callan:1982au,Harvey:1983tp}. The bosonization method is applied also to the atomic collapse problem in 3+1 dimensions\cite{Hirata:1986yt}. We apply this method to the atomic collapse problem in 2+1 dimensions.
Following the studies in 3+1 dimensions mentioned above, first restricting the gauge and the fermion field to s-wave field, we reduce the theory to 1+1 dimensional fermion effective theory. Next, we map the two dimensional fermion theory to the two dimensional boson theory. Then we solve the classical equation of motion for the boson field. As a result, we find the vacuum structure including the charge screening of the impurity charge.
This paper is organized as follows. In section 2, the result of foregoing analysis in one body theory for the Coulomb impurity problem on graphene is briefly reviewed. In section 3, we will explain the s-wave approximation and the bosonization formalism proposed in Ref. \cite{Hirata:1986yt}. In section 4, we will show the details about our study of vacuum solution and the result of our numerical analysis. Section 5 is devoted to summary and discussion.
\section{Review on Coulomb impurity on graphene}
\label{sec:model}
In this section, we review the Coulomb impurity problem on graphene. The electronic properties of the graphene are described by the tight-binding model where interactions between different orbits are neglected. And it is assumed that the electron can hop to only the nearest neighbor site. In momentum space, the energy of electron becomes zero at two points ($K$ and $K'$). The low energy effective theory is obtained by expanding the equation which the electron obeys around these points. It is known that the effective Hamiltonian takes the same form as that of massless Dirac fermion. That is, the fermionic low energy excitation obeys the Dirac equation
\begin{align}
-v_F\left(
\begin{array}{cc}
0 & \hat{p_x}-i\hat{p_y} \\
\hat{p_x}+i\hat{p_y} & 0
\end{array}
\right) \psi =\varepsilon \psi , \label{Dirac equation}
\end{align}
and has the linear dispersion relation
\begin{align}
\varepsilon =\pm v_F\sqrt{p_x^2+p_y^2}, \label{linear dispersion relation}
\end{align}
at low energy. The parameter $v_F$ in the above equation is the Fermi velocity which is roughly estimated as $v_F\sim \frac{c}{300}$. Since $v_F$ plays the similar role as the speed of light $c$ in quantum electrodynamics, the effective fine structure constant for the fermionic excitations on graphene is $\alpha _{\rm eff}\sim \frac{300}{137}$. This means that the massless Dirac fermion on graphene is strongly coupled.
The behavior of electron in a hydrogen like atom is studied in relativistic quantum mechanics. It is known that the bound state of electron and a point charge $Ze$ cannot exist when $Z\alpha \ge 1$. For such a strongly coupled system, it is expected that the strong electric field makes the vacuum unstable since the strong Coulomb potential causes particle-hole pair creations. Such a phenomenon is called the ``atomic collapse'' and has been discussed for a long time. In the experimental side, the atomic collapse has been tested in heavy-ion collision. However the instability of large atomic number nuclei makes it difficult to observe the phenomenon clearly.
In the graphene case, such a situation can be easily set up due to the large value of the effective coupling $\alpha \sim \frac{300}{137}$ of the Dirac fermion. Recently, Wang and his collaborators studied the graphene system with Coulomb impurities with STM and observed the resonance like the quasibound state\cite{Wang:2013}. They put Ca dimers as impurity, and measured the local density of states (LDOS) of electron around the impurity. They showed that the peak appears in energy dependence of LDOS. The peak point is below the Dirac point when 5 Ca dimers are put. According to them, this is the quasi-bound state expected in one body theory. The quasi-bound state spatially spread through about 10 nm around the center of Ca dimers in this experiment.
In view of this STM experiment, it is now very important to study the graphene system with Coulomb impurities theoretically. In one body theory, the solution of the Dirac equation with Coulomb potential by a charged impurity can be exactly obtained\cite{Pereira:2007,Shytov:2007a}. The behavior of the solution drastically changes when $Z\alpha >1/2$. Because the electrons in graphene are massless, they do not seem to make bound state even in small $Z\alpha $. However, by introducing graphene lattice cutoff, the quasi-stable bound state is predicted to appear in strong coupling case.
In Ref. \cite{Shytov:2007b}, the existence of the quasi-bound state is semiclassically discussed. Here, we briefly review their discussion. The Hamiltonian for 2 dimensional massless fermion in Coulomb potential is
\begin{align}
H={\boldsymbol \sigma }\cdot {\bf p}-\frac{Z\alpha }{r}. \label{2DDhamiltonianiC}
\end{align}
When we write the square of momentum in terms of the radial momentum $p_r$ and the angular momentum $j$
\begin{align}
p^2=p_r^2+j^2/r^2,
\end{align}
the Hamiltonian (\ref{2DDhamiltonianiC}) leads to
\begin{align}
p^2_r=\left( \varepsilon +\frac{Z\alpha +j}{r}\right) \left( \varepsilon +\frac{Z\alpha -j}{r}\right) ,
\end{align}
where $\varepsilon $ is energy eigenvalue. The classically forbidden region where $p_r^2<0$ corresponds to
\begin{align}
r_1\equiv \frac{Z\alpha -j}{|\varepsilon |}<r<\frac{Z\alpha +j}{|\varepsilon |}\equiv r_2.
\end{align}
Notice that if $Z\alpha >j$, there exist classically allowed region inside; that is, $r<r_1$. Therefore in strongly coupled case, quasibound states can be found by imposing the Bohr-Sommerfeld quantization condition
\begin{align}
\int _{r_0}^{r_1}p_r dr=n\pi ,
\end{align}
where $r_0$ is the lattice cutoff.
In one particle theory, the interesting feature mentioned above can be found and LDOS can be calculated. However, since the atomic collapse is a phenomenon which comes from pair creation effect, it should be analyzed in a way which contain nonperturbative multi body effects. In the following section, we will show the 2+1 dimensional massless fermion version of the bosonization formulation proposed in Ref. \cite{Hirata:1986yt}.
\section{Approximation and Formalism}
\label{sec:formalism}
In this section, we study the vacuum structure of the massless Dirac fermion system in 2+1 dimensions around a Coulomb impurity. In order to analyze the system nonperturbatively, we employ the method proposed in Ref. \cite{Hirata:1986yt} for the atomic collapse QED in 3+1 dimensions. We firs restrict the theory with s-wave electromagnetic field and the lowest partial wave electron field. Under this approximation, the theory is reduced to 1+1 dimensional effective theory with time and radial degrees of freedom. We then bosonize the effective 1+1 dimensional fermion theory. Since it is known that the bosonized theory captures important part of the nonperturbative effect of the original fermion theory even at the classical level, we study the nonperturbative vacuum structure by constructing the classical solution of the bosonized theory.
\subsection{1+1D Effective Theory}
Since the gauge field is in 3+1 dimensions, we start from the following gauge action
\begin{align}
S_g=\int d^4 x \left[ -\frac{1}{4}F_{\mu \nu }F^{\mu \nu }-Ze\rho (x)A_0\right] , \label{starting gauge action}
\end{align}
where the charge density of impurity is spherically symmetric $\rho (x)=\rho (r,t)$, and normalized as $\int d^3x \rho (x)=1$. The s-wave electromagnetic field takes following form
\begin{align}
A_0(x)=a_0(r,t),\ A_i(x)=\hat{r}_ia_1(r,t), \label{s-waveEMF}
\end{align}
where $\hat{r}_i=r_i/r$ is $i$th component of the unit vector in radial direction. In this approximation, the gauge action becomes
\begin{align}
S_g=\int drdt\left[ 2\pi r^2(\partial _0a_1-\partial _ra_0)^2-4\pi Zer^2\rho (r,t)a_0\right] \label{Gaction}.
\end{align}
When the graphene is on $z=0$ surface and the electron is trapped on this surface, the action for the electron coupled with the gauge field is
\begin{align}
S_f&=\int d^4x \left[ \overline{\psi } (i \Slash{\partial } +e\Slash{A})\psi \right] \delta (z) \label{S3Df}\\
&=\int d^4x \left[ \psi ^\dag \gamma ^0(i\gamma ^0\partial _0+i\gamma ^i\partial _i+e\gamma ^0A_0+e\gamma ^iA_i)\psi \right] \delta (z).\nonumber
\end{align}
where $\psi $ is 2 component Weyl spinor. We take gamma matrices as
\begin{align}
\gamma ^0=\sigma _3,\ \gamma ^1 =i\sigma _2,\ \gamma ^2=-i\sigma _1. \label{gammaM}
\end{align}
Because we are considering $z=0$ surface and using the s-wave approximation (\ref{s-waveEMF}), $\gamma ^3$ disappears from (\ref{S3Df}). From now on, $i$ runs from $1$ to $2$. The fermion action becomes
\begin{align}
S_f=\int d^2xdt\psi ^\dag \left[ (i\partial _0+ea_0)+\sigma ^i(i\partial _i+e\hat{r}_ia_1)\right] \psi .
\end{align}
We expand the fermion field $\psi $ as
\begin{align}
\psi =\frac{1}{\sqrt{r}}\sum_{m,\sigma }v_{m,\sigma }(r,t)\Psi _{m,\sigma }(\varphi ),
\end{align}
where $\sigma =\pm1$ and $m$ is half integer, and
\begin{align}
\Psi _{m,\sigma }=\frac{1}{\sqrt{4\pi }}\left(
\begin{array}{c}
e^{i(m-1/2)\varphi }\\
\sigma e^{i(m+1/2)\varphi }
\end{array}
\right) . \label{definition of Psi}
\end{align}
is normalized as
\begin{align}
\int d\varphi \Psi_{m'\sigma '}^\dag \Psi_{m,\sigma }=\delta _{m,m'}\delta _{\sigma ,\sigma '}.
\end{align}
Using the relation
\begin{align}
\sigma _i\hat{r}_i\Psi _{m,\sigma }=\sigma \Psi _{m,\sigma },
\end{align}
we get
\begin{align}
\sigma _i\partial _i\psi =\frac{1}{\sqrt{r}}\sum _{m,\sigma }\sigma \left( \partial _rv_{m,\sigma }(r,t)\Psi _{m,\sigma }(\varphi )+\frac{m}{r}\Psi _{m,-\sigma }(\varphi )\right) .
\end{align}
Therefore the action for fermion becomes
\begin{align}
S_f=\int drdt \sum_{m,\sigma }\left[ v_{m,\sigma }^*\left\{ i\partial _0+ea_0+\sigma (i\partial _r+ea_1)\right\} v_{m,\sigma }-i\sigma v_{m,\sigma }^*\frac{m}{r}v_{m,-\sigma }\right] .
\end{align}
We restrict ourself to consider only the lowest ($j=1/2$) partial wave, and define 1+1 dimensional fermion
\begin{align}
u_m&:=\left( \frac{1+i}{2}+\frac{1-i}{2}\sigma _3\right) \left(
\begin{array}{c}
v_{m,+}\\
{\rm sign} (m) v_{m,-}
\end{array}
\right) \nonumber \\
&=\left(
\begin{array}{c}
v_{m,+}\\
{\rm sign}(m)i v_{m,-}
\end{array}
\right) ,
\end{align}
where $m=\pm 1/2$. From now on, we take
\begin{align}
\gamma ^0=\sigma _2,\ \gamma ^1=i\sigma _1,\ \gamma _5 =\gamma ^0\gamma ^1=\sigma _3
\end{align}
as 2 dimensional gamma matrices. Then we can rewrite 2 dimensional fermion action as
\begin{align}
S_f=\int drdt\sum_{m=\pm 1/2}\left[ \overline{u}_m\left\{ \gamma ^0(i\partial _0+ea_0)+\gamma ^1(i\partial _r+ea_1)\right\} u_m+i\frac{1}{2r}\overline{u}_m\gamma ^5u_m\right] \label{2DFactionIG}.
\end{align}
The last term represents centrifugal force. Unlike that of Ref. \cite{Hirata:1986yt},we have a different coefficient of centrifugal force term and no mass term.
We have to set the boundary condition for fermion field $u_{m}$ at $r=0$ by requiring no singularity at $r=0$. From (\ref{definition of Psi}),
\begin{align}
\Psi _{1/2,+}-\Psi _{1/2,-},\ \Psi _{-1/2,+}+\Psi _{-1/2,-}
\end{align}
has $\varphi $ dependence. If the coefficients of these are finite value at $r=0$, the singularity arises. So, we set the boundary condition
\begin{align}
v_{m,+}(0,t)-{\rm sign}(m)v_{m,-}(0,t)=0.
\end{align}
Written in 2D fermion $u_m$,
\begin{align}
(1-\gamma ^0)u_m(0,t)=0. \label{2dfboundary}
\end{align}
On the other hand, since
\begin{align}
\Psi _{1/2,+}+\Psi _{1/2,-},\ \Psi _{-1/2,+}-\Psi _{-1/2,-}
\end{align}
don't have $\varphi $ dependence, the coefficient of these can be finite at $r=0$. Therefore we can also use the same boundary condition as Ref. \cite{Hirata:1986yt}.
By the way, in one body theory, the boundary condition is set not at $r=0$, but at $r=r_0$\cite{Pereira:2007, Shytov:2007a}, which is lattice cut off size of graphene. And the cut off plays very important role to discuss the drastic change of wave function and quasi-bound state in strong coupling region. In our case, however, even if we set the boundary condition at $r=r_0$, we get the same result for $r=0$. Therefore, here we set the boundary condition at $r=0$.
\subsection{Bosonization}
We apply bosonization to this theory. Regarding interaction term as perturbation, we bosonize free fermion field to free boson field,
\begin{align}
u_m(r,t)=\left( \frac{\mu }{2\pi }\right) ^{1/2}\left(
\begin{array}{c}
-iN_\mu \exp [i\sqrt{\pi }(\phi _m(r,t)+\tilde{\phi }_m(r,t))]\\
N_\mu \exp [i\sqrt{\pi }(-\phi _m(r,t)+\tilde{\phi }_m(r,t))]
\end{array}
\right) ,
\end{align}
where
\begin{align}
\tilde{\phi }(x)=\lim_{\epsilon \to 0}\int_r^\infty dse^{-\epsilon s}\dot{\phi }(s,t), \label{phitilde}
\end{align}
and $N_\mu $ represents normal ordering at IR mass scale $\mu $. From now on, we use the overdot and prime for time and spatial derivative, respectively. Because the action and the boundary condition are almost the same as Ref. \cite{Hirata:1986yt}, we can bosonize this theory following the same calculation.
In this case, we should impose the boundary condition on the boson field. The boundary condition (\ref{2dfboundary}) is rewritten in boson field as
\begin{align}
\phi _m(0,t)=0. \label{BboundaryC}
\end{align}
Free boson field can be expanded in plane wave as
\begin{align}
\phi (x,t)=\int_{k>0} \frac{dk}{2\pi }\left[ \bar{a}^\dag (k)e^{ik(x+t)}+a(k)e^{ik(x-t)}+\bar{a}(k)e^{-ik(x+t)}+a^\dag (k)e^{-ik(x-t)}\right],
\end{align}
where $a,\ \bar{a},\ a^\dag ,\ \bar{a}^\dag $ are creation-annihilation operators satisfying appropriate commutation relations. While $a(k),\bar{a}(k)$ are independent operators without the boundary condition, with the boundary condition (\ref{BboundaryC})
\begin{align}
0&=\phi (0,t)\nonumber \\
&=\int_{k>0} \frac{dk}{2\pi }\left[ (\bar{a}^\dag (k)+a^\dag (k))e^{ikt}+(a(k)+\bar{a}(k))e^{-ikt}\right] ,
\end{align}
these are dependent on each other
\begin{align}
\bar{a}(k)=-a(k).
\end{align}
Then the boson field $\phi $ and $\tilde{\phi }$ can be written as
\begin{align}
\phi (r,t)=\int _{k>0} \frac{dk}{2\pi }\left[ a(k)(e^{ikr}-e^{-ikr})e^{-ikt}+a^\dag (k)(e^{-ikr}-e^{ikr})e^{ikt}\right] ,
\end{align}
and
\begin{align}
\tilde{\phi }(r,t)=\int _{k>0}\frac{dk}{2\pi }\left[ \left( e^{ikr}+e^{-ikr}\right)a(k)e^{-ikt}+\left( e^{ikr}+e^{-ikr}\right)a^\dag (k)e^{ikt} \right] .
\end{align}
We split these into
\begin{align}
\phi ^{(+)}(r,t)=\int _{k>0} \frac{dk}{2\pi }a(k)\left( e^{ikr}-e^{-ikr}\right) e^{-ikt},
\end{align}
\begin{align}
\phi ^{(-)}(r,t)=\int _{k>0} \frac{dk}{2\pi }a^\dag (k)\left( e^{-ikr}-e^{ikr}\right) e^{ikt},
\end{align}
\begin{align}
\tilde{\phi }^{(+)}(r,t)=\int_{k>0}\frac{dk}{2\pi }a(k)\left( e^{ikr}+e^{-ikr}\right) e^{-ikt},
\end{align}
\begin{align}
\tilde{\phi }^{(-)}(r,t)=\int _{k>0}\frac{dk}{2\pi }a^\dag (k)\left( e^{ikr}+e^{-ikr}\right)e^{ikt}.
\end{align}
From the commutation relation $[a(k),a^\dag (k)]=2\pi \frac{1}{2k}\delta (k-k')$, we get the relation
\begin{align}
&[\tilde{\phi }^{(+)}(r,t)+\eta \phi ^{(+)}(r,t),\tilde{\phi }^{(-)}(r',t')+\eta '\phi ^{(-)}(r',t')]\nonumber \\
&\qquad =-\frac{1}{4\pi }[(1-\eta )(1-\eta ')A_++(1+\eta )(1+\eta ')A_-+(1-\eta )(1+\eta ')B_++(1+\eta )(1-\eta ')B_-], \label{BcommutationR}
\end{align}
where
\begin{align}
A_\pm (r,t;r',t')&\equiv -\int_{k>0}dk\frac{1}{k}e^{ik(\mp (r-r')-(t-t'))}\nonumber \\
&=\lim_{\epsilon \to 0}\ln \left( i\mu [ t-t'\pm (r-r')-i\epsilon \right] ) \label{A},
\end{align}
\begin{align}
B_\pm (r,t;r',t')&\equiv -\int_{k>0}dk\frac{1}{k}e^{ik(\mp (r+r')-(t-t'))}\nonumber \\
&=\lim_{\epsilon \to 0}\ln \left( i\mu [ t-t'\pm (r+r')-i\epsilon \right] ) \label{B},
\end{align}
are renormalized at IR mass scale $\mu $. $B_{+,-}$ arise from the boundary condition.
Using the commutation relation (\ref{BcommutationR}), we rewrite the interaction terms in fermion theory in terms of boson field. After some point splitting procedure, we get
\begin{align}
\overline{u}\gamma ^\mu u=-\frac{1}{\sqrt{\pi }}\epsilon ^{\mu \nu }\partial _\nu \phi \label{interactionT},
\end{align}
\begin{align}
\overline{u}\gamma _5u=-\frac{i}{2\pi r}N_\mu \cos (2\sqrt{\pi }\phi ) \label{centrifugalT},
\end{align}
where $\epsilon $ is anti-symmetric symbol with $\epsilon ^{10}=1$.
We rewrite the fermion action (\ref{2DFactionIG}) in terms of the above boson operators. In $a_1=0$ gauge,
\begin{align}
S_f=\int drdt\sum_{m=\pm 1/2}\left[ \frac{1}{2}\partial ^\mu \phi _m\partial _\mu \phi _m-\frac{e}{\sqrt{\pi }}a_0'\phi _m+\frac{1}{4\pi r^2}\cos (2\sqrt{\pi }\phi _m)\right] ,
\end{align}
where the second term is integrated by parts. Therefore we get the total action
\begin{align}
S=\int drdt\left[ 2\pi r^2a_0'^2+e\Phi (r,t)a_0'+\sum_{m=\pm 1/2}\left( \frac{1}{2}\partial ^\mu \phi _m\partial _\mu \phi _m-\frac{e}{\sqrt{\pi }}a_0'\phi _m+\frac{1}{4\pi r^2}\cos (2\sqrt{\pi }\phi _m)\right) \right] , \label{BTaction}
\end{align}
where the $\Phi (r,t)$ is defined by
\begin{align}
\Phi '(r,t)=4\pi Zr^2\rho (r,t).
\end{align}
From the action (\ref{BTaction}), we notice that $a_0$ has no dynamical degrees of freedom. Using the equation of motion for $a'_0$
\begin{align*}
4\pi r^2a_0'+e\Phi (r,t)-\sum_{m=\pm 1/2}\frac{e}{\sqrt{\pi }}\phi _m=0,
\end{align*}
we can eliminate $a_0$. Therefore the Hamiltonian becomes
\begin{align}
H=\int dr\left[ \sum_{m=\pm 1/2}\left\{ \frac{1}{2}\left( \pi _m ^2+\phi _m'^2\right)-\frac{1}{4\pi r^2}\cos (2\sqrt{\pi }\phi _m)\right\} +\frac{e^2}{8\pi r^2}\left( \Phi (r,t)-\frac{1}{\sqrt{\pi }}\sum_m \phi _m\right) ^2\right] .
\end{align}
Adding the c-number to the Hamiltonian,
\begin{align}
H&=\int dr\left[ \sum_{m=\pm 1/2}\left\{ \frac{1}{2}\left( \pi _m ^2+\phi _m'^2\right)+\frac{1}{4\pi r^2}\left( 1-\cos (2\sqrt{\pi }\phi _m)\right) \right\} \right. \nonumber \\
&\qquad \left. +\frac{e^2}{8\pi r^2}\left\{ \left( \Phi (r,t)-\frac{1}{\sqrt{\pi }}\sum_m \phi _m\right) ^2-\Phi (r,t)^2\right\} \right] \label{hamiltonian},
\end{align}
we shift the energy so that the energy becomes zero when $\phi _m=0$ which is vacuum configuration with $Z=0$. In the next section, we numerically calculate the solution which minimize this Hamiltonian.
\section{Study of Vacuum Solution}
\label{sec:Study of Vacuum Solution}
In this section we find the classical solution which minimizes the bosonized Hamiltonian in the previous section. For this purpose, we have to solve the Euler-Lagrange equations for boson fields under the appropriate boundary conditions. The boundary condition at $r=0$ is determined by eq.(\ref{BboundaryC}). The solution is characterized by the boundary condition at $r=\infty $.
Eq.(\ref{interactionT}) indicates that the density of electron $\rho _e(r)$ can be written in terms of the boson field as
\begin{align}
\rho _e(r)&\equiv \psi ^\dag \psi =\sum_m \frac{1}{\sqrt{\pi }}\partial _r\phi _m(r).
\end{align}
Therefore, we get the spatial distribution of induced electron density corresponding to the solution. Total induced charge which screens the impurity charge is given by
\begin{align}
Q_{\mathrm{EM}}&\equiv -e\int dr \rho _e(r)=-\frac{e}{\sqrt{\pi }}\sum_m\phi _m(\infty ).
\end{align}
In order to study the vacuum structure, we consider only static solution $\pi _m=0$. We rewrite the Hamiltonian in terms of the new variable
\begin{align}
\phi _\pm =\frac{1}{\sqrt 2}(\phi _{+1/2}\pm \phi _{-1/2}).
\end{align}
Using the formula
\begin{align}
\cos [\sqrt{2\pi }(\phi _++\phi _-)]+\cos [\sqrt{2 \pi }(\phi _+-\phi _-)]=2\cos (\sqrt{2\pi }\phi _+)\cos (\sqrt{2\pi }\phi _-),
\end{align}
the Hamiltonian becomes
\begin{align}
H&=\int _0^\infty dr \left[ \frac{1}{2}(\phi '^2_++\phi _-'^2)+\frac{1}{2\pi r^2}\{ 1-\cos (\sqrt{2\pi }\phi _+)\cos (\sqrt{2\pi }\phi _-)\} \right. \nonumber \\
&\qquad \left. +\frac{\alpha }{\pi r^2}\left\{ \left( \phi _+-\sqrt{\frac{\pi }{2}}\Phi (r,t)\right) ^2-\frac{\pi }{2}\Phi (r,t)^2\right\} \right] ,
\end{align}
where $\alpha =\frac{e^2}{4\pi }$. The Euler-Lagrange equations for $\phi _+, \phi _-$ are given by
\begin{align}
\phi _+''-\frac{1}{\sqrt{2\pi }r^2}\sin (\sqrt{2\pi }\phi _+)\cos (\sqrt{2\pi }\phi _-)-\frac{2\alpha }{\pi r^2}\left( \phi _+-\sqrt{\frac{\pi }{2}}\Phi (r)\right) =0, \label{eq:phi+}
\end{align}
\begin{align}
\phi _-''-\frac{1}{\sqrt{2\pi }r^2}\cos (\sqrt{2\pi }\phi _+)\sin (\sqrt{2\pi }\phi _-)=0, \label{eq:Eqphi-}
\end{align}
respectively. Since it satisfies Eq.(\ref{eq:Eqphi-}) we can take the symmetric ansatz $\phi _-=0$. Then Eq.(\ref{eq:phi+}) reduces to
\begin{align}
\phi _+''-\frac{1}{\sqrt{2\pi }r^2}\sin (\sqrt{2\pi }\phi _+)-\frac{2\alpha }{\pi r^2}\left( \phi _+-\sqrt{\frac{\pi }{2}}\Phi (r)\right) =0. \label{ELEPhi+}
\end{align}
We assume that the impurity charge is spherically spread over radius $R$:
\begin{align}
\rho (r)=\frac{3}{4\pi R^3}\theta (R-r).
\end{align}
The corresponding $\Phi (r)$ is
\begin{align}
\Phi (r)=\left\{
\begin{array}{cc}
Z\left( \frac{r}{R}\right) ^3&\quad (r<R)\\
Z&\quad (r>R).
\end{array}
\right.
\end{align}
Since Eq.(\ref{ELEPhi+}) is a second order differential equation, in addition to the boundary condition at the origin we need to impose another boundary condition at $r=\infty $. For finiteness of total energy, the boson field should asymptotically be constant ($\phi_+\rightarrow \phi _*$) at large $r$. Substituting $\phi _+=\phi _*$ into the Euler-Lagrange equation at large $r$, we find that $\phi _*$ should be the solution of the following equation:
\begin{align}
\sin (\sqrt{2\pi }\phi _*)=-2\alpha \left( \sqrt{\frac{2}{\pi }}\phi _*-Z\right) . \label{ELEinfinity}
\end{align}
Notice that the asymptotic value $\sqrt{\frac{2}{ \pi }}\phi _*$ can take non-integer value. Charge screening with non-integer charge may seem counter intuitive if one tries to interpret the phenomena as particle hole pair creation. One should interpret such screening as the polarization effect.
In fact, it is known that the screening of non-integer charge actually occurs in massless Schwinger model \cite{Callan:1982au, Rubakov:1983vn,Iso:1988zi}.
In the following subsections, we show the detailed numerical analysis and its results.
\subsection{Numerical Analysis}
\subsubsection{Strategy}
Our numerical analysis is done in various parameters $\alpha , Z$, according to the following steps:
\begin{description}
\item[(i)]\mbox{}Find the solution of Eq.(\ref{ELEinfinity}) and obtain the asymptotic form at large $r$.
\item[(ii)]\mbox{}Solve the Euler-Lagrange equation (\ref{ELEPhi+}) with the boundary condition at large $r$ (\ref{asymptoticS}) with various $A$.
\item[(iii)]\mbox{}Find $A$ with which the solution satisfies the boundary condition at $r=0$ (\ref{BboundaryC}).
\end{description}
\subsubsection{Asymptotic form}
In order to numerically solve the Euler-Lagrange equation (\ref{ELEPhi+}), we should find the asymptotic form at large $r$.
To do so, we parameterize $\phi_+(r)$ by introducing a function $f$ which describes the deviation of $\phi _+(r)$ and $\phi_*$ at large $r$ as
\begin{align}
\phi _+(r)=\phi _* - f(r). \label{asymptoticf}
\end{align}
where $\phi _*$ is the solution of Eq.(\ref{ELEinfinity}).
Substituting Eq.(\ref{asymptoticf}) into Eq.(\ref{ELEPhi+}), and expanding it up to linear order in $f$, we obtain
\begin{eqnarray}
f^{\prime\prime} -\frac{1}{r^2}\left( \cos(\sqrt{2\pi} \phi_*) + \frac{2\alpha}{\pi} \right) f+ O(f^2)= 0
\label{eq:eq_f}
\end{eqnarray}
Assuming that the solution for $f$ can take the form
\begin{align}
f(r) \approx \frac{A}{r^\lambda }, \label{asymptotic_sol_f}
\end{align}
at large $r$ with $A$ being a constant and substituting it into Eq.(\ref{eq:eq_f}), we find that the power $\lambda$ satisfies the following equation:
\begin{eqnarray}
\lambda^2 + \lambda -\left(\cos(\sqrt{2\pi} \phi_*) + \frac{2\alpha}{\pi} \right) = 0 .
\label{eq:quad_lambda}
\end{eqnarray}
From Eq.(\ref{eq:quad_lambda}), $\lambda $ should be
\begin{align}
\lambda =\frac{1}{2}\left[ -1+\sqrt{1+4\cos (\sqrt{2\pi }\phi _*)+\frac{8\alpha }{\pi }}\right] . \label{eq:lambda}
\end{align}
\begin{figure}[htbp]
\centering
\includegraphics[width=8cm]{sin_linear.pdf}
\caption{The green line is l.h.s. of eq.(\ref{ELEinfinity}), and blue, red, yellow lines are r.h.s. of eq.(\ref{ELEinfinity}) with $\alpha =0.1$, $\alpha =0.2$, $\alpha =0.4$, respectively.}
\label{sin_linear}
\end{figure}
We show some examples of $Z=4$ case. In this case, there are three screening patterns depending on the value of $\alpha $ as shown in Fig. \ref{sin_linear}. In $\alpha \lesssim 0.14$ case, there are five values of $\phi_*$. However, when $\phi _*$ is equal to the second or fourth smallest value, based on Eq.(\ref{eq:lambda}), $\lambda $ becomes imaginary. Only the solutions with the real positive values of $\lambda $ make sense. So, there are three possibilities. For $0.14\lesssim \alpha \lesssim 0.34$ case, there are three values of $\phi_*$. Similarly the second smallest value of $\phi _*$ is not a physical solution. So, there are two possibilities. And in $0.34\lesssim \alpha $ case, there is only value for $\phi_*$ which corresponds to the full screening solution.
In Fig. \ref{Number}, we show the number of possible asymptotic solutions at large $r$ for each set of values of ($\alpha$, $Z$).
\begin{figure}[b]
\centering
\includegraphics[width=12cm]{number.pdf}
\caption{Number of possible asymptotic solutions at large $r$ for each set of values of ($\alpha$, $Z$). Crosses are the parameter points where we solved Eq. (\ref{ELEPhi+})}
\label{Number}
\end{figure}
\subsubsection{Example of the solution}
Starting from the asymptotic solutions and solving the differential equation numerically, we can obtain the full solution.
Taking the following asymptotic from
\begin{align}
\phi _+(r) \approx \phi _*-\frac{A}{r^\lambda }, \label{asymptoticS}
\end{align}
at large $r$ and varying $A$, we can search for the physical solution which satisfies the boundary condition at $r=0$.
Practically, we solve Eq. (\ref{ELEPhi+}) from $r=0.001R$ to $r=100000R$, setting the boundary condition at large $r=100000R$ with various values of $\alpha, Z$.
For illustration, we show the example for $Z=4$ and $\alpha=0.2$.
In this case, there are two asymptotic solutions, but only the solution which realizes the smallest value of $\phi_*$ can satisfy appropriate boundary condition.
The full solutions from the other asymptotic forms do not satisfy the boundary condition at $r=0$ but end up have positive values no matter how we choose the value of $A$.
We show the solution of $\phi_+(r)$ in Fig. \ref{sol024}. In the other case, the shapes of solutions are qualitatively similar to the solution in this case. The induced electron density is depicted in Fig. \ref{density}. We notice that most of the induced electrons fall into the impurity.
\begin{figure}[htbp]
\begin{minipage}{0.5\hsize}
\centering
\includegraphics[width=7cm]{sol024.pdf}
\caption{The solution for $\alpha =0.2, Z=4$.}
\label{sol024}
\end{minipage}
\begin{minipage}{0.5\hsize}
\centering
\includegraphics[width=7cm]{density.pdf}
\caption{Induced electron density with $\alpha =0.1$ (blue line), $\alpha =0.2$ (red line), $\alpha =0.4$ (yellow line).}
\label{density}
\end{minipage}
\end{figure}
\subsection{Result}
\subsubsection{Phase structure}
We looked for the solution for various set of parameters of ($\alpha ,Z$), where the parameter set is given in Fig. \ref{Number} .
We found that only the solution with the smallest value of $\phi_*$ can satisfy correct boundary condition at $r=0$ in all cases.
From this fact, we reach the conjecture that the magnitude of screening can be determined by the smallest intersection of $\sin (\sqrt{2\pi }\phi _*)$ and $-2\alpha ( \sqrt{2/\pi }\phi _*-Z) $. According to this conjecture, we get effective impurity charge seen from infinitely separated point,
\begin{align}
Z_{\rm eff}=Z-\sqrt{\frac{2}{\pi }}\phi _*,
\end{align}
which is screened by induced charge Fig. \ref{Zeff}. Notice that when $\alpha \gtrsim 0.2$, the effective impurity charge $Z_{\rm eff}$ in any odd $Z$ case is the same one as in $Z=1$ case. Also when $\alpha \gtrsim 0.14$, $Z_{\rm eff}$ in any even $Z$ case is the same one as in $Z=2$ case. From this result, a phase diagram of screening is described as in Fig. \ref{phase}. In larger $Z$ case, more branches appear in small $\alpha $ regime.
\begin{figure}[htbp]
\centering
\includegraphics[width=16cm]{Zeff.pdf}
\caption{$Z_{\rm eff}$ for each $Z$. Dashed lines in $Z=3,4$ cases describe $Z_{\rm eff}$ in $Z=1,2$ cases, respectively.}
\label{Zeff}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=12cm]{phase.pdf}
\caption{Phase diagram of screening. Crosses are the parameter points where we solved Eq. (\ref{ELEPhi+})}
\label{phase}
\end{figure}
\subsubsection{Scaling law}
The induced 2D electron density can be obtained as
\begin{align}
n(r)=\frac{\rho _e(r)}{2\pi r}.
\end{align}
If graphene sheet can be treated as perfect metal, the scaling law is calculated as in Ref. \cite{Fogler:2007}:
\begin{align}
n(r)\propto r^{-3},
\end{align}
in the range of distances $1\ll r/R \ll 2\alpha ^2Z$. In our calculation, we fit the scaling law
\begin{align}
n(r)\propto r^{-\gamma }
\end{align}
in the range of distances $1\ll r/R \ll 10$. The scaling exponent $\gamma $ depends on parameters $\alpha ,Z$ as shown in Fig.\ref{scaling}. In small screening regime, we get $\gamma \sim 2.7$, independently of $\alpha $. Near the value of $\alpha $ where magnitude of screening jumps, $\gamma $ drastically decreases. In larger $\alpha $ regime, $\gamma $ increases and becomes close to the value calculated in perfect metal approximation.
\begin{figure}[htbp]
\centering
\includegraphics[width=16cm]{scaling.pdf}
\caption{Crosses describe the scaling exponent in each $Z$ case. Dashed line describes one in perfect metal approximation.}
\label{scaling}
\end{figure}
\section{Summary and Discussion}
\label{sec:discussion}
In this paper, we studied quantum field theory with the 2+1 dimensional massless fermion around an external Coulomb field. We reduced the theory to a two dimensional fermion theory, where the higher partial waves are neglected. Bosonizing the theory, we have found the static solution of classical equation of motion for the boson field. The magnitude of screening is determined only by the asymptotic equation of motion. Which of these asymptotic solutions satisfies the boundary condition at $r=0$ is determined by dynamics.
Through the study of several examples, we have concluded that the realized solution is always the smallest screening one. As a result, we have found patterns of screening depending on the coupling $\alpha $ and the impurity charge $Z$. The screening charge undergoes a drastic change as we change the value of $\alpha $ at some critical values.
We also obtained the phase diagram characterized by the patterns of screening.
By solving the equation of motion in full spatial regime, we have obtained the spatial distribution of density of the induced electron. The radial profile of the two dimensional induced charge density can be fitted by negative power in $r$ which is the distance from the impurity. In weak coupling regime, scaling exponent $\gamma $ is independent of $\alpha $ and $Z$; $\gamma \approx 2.7$. Near the screening jumping point, $\gamma $ decreases. This means that the induced fermion is widely spread near the screening jumping point. And in larger $\alpha $ regime, $\gamma $ become close to the value of the perfect metal approximation; $\gamma \approx 3$.
The validity of the approximation to neglect higher partial wave can be discussed somewhat in semi classical theory mentioned in section 2. According to the semi classical theory, only $Z\alpha >j$ wave can form quasi-bound states. So, the fermion mode whose angular momentum $j$ is higher than $Z\alpha $ is irrelevant to anomalous behavior of the electron in strong Coulomb potential. When $Z\alpha >3/2$, the next to lowest partial wave $j=3/2$ should be relevant to this problem. Therefore our approximation should be valid only when $Z\alpha <3/2$.
To compare our analysis with the result of one particle theory or the experiment, many things remain to be done. Validity of classical treatment for boson theory should be confirmed quantitatively. In Ref. \cite{Hirata:1989px}, the bosonized atomic collapse problem in 3+1 dimensions is treated within small fluctuation approximation. They show the existence of meta stable states in supercritical phase. In the same way it may be possible to show the existence of the meta stable states in our 2+1 dimensional massless fermion case.
The contribution of higher momentum partial wave should be evaluated for understanding larger $\alpha ,Z$ case. Furthermore to understand the behavior in the regime closer to the impurity, the effect of graphene lattice should be considered. For that purpose, the simulation by lattice gauge theory is important.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,923 |
U 1224 war ein deutsches Unterseeboot vom Typ IX C/40, welches im Zweiten Weltkrieg von Adolf Hitler als Geschenk der Kaiserlich Japanischen Marine (Nihon Kaigun) übergeben und dort als RO 501 in Dienst gestellt wurde.
Geschichte
Das Boot wurde am 25. August 1941 bei der Deutschen Werft AG in Hamburg in Auftrag gegeben. Die Kiellegung erfolgte am 30. November 1942 mit der Baunummer 387. Der Stapellauf fand am 7. Juli 1943 statt. Kapitänleutnant Georg Preuss stellte U 1224 am 20. Oktober 1943 in Dienst. Das Boot wurde der 31. U-Flottille unterstellt, einer Ausbildungsflottille, die in Hamburg stationiert war. Während der Ausbildung wurde U 1224 am 15. Februar 1944 einer japanischen Besatzung übergeben, die zuvor an Bord des japanischen U-Boots I-8 von Japan nach Europa gebracht worden war, und in RO 501 umbenannt. Die ehemalige deutsche Besatzung stellte im April das Typ IX D2 Boot U 875 in Dienst. Der neue Kommandant von RO 501, Kapitänleutnant Norita Sadatoshi gab dem Boot ein Turmemblem: Die deutsche Seekriegsflagge, welche sich hinter der japanischen Seekriegsflagge befand, als Zeichen der Verbundenheit beider Länder.
Marco Polo II
U 1224 war das zweite deutsche U-Boot das dem japanischen Kaiser zum Geschenk gemacht wurde. Im September 1943 hatte die Kriegsmarine U 511, ebenfalls vom Typ IX C, unter dem Codenamen Marco Polo an die Nihon Kaigun übergeben. Das Boot war dann unter der Bezeichnung RO 500 in Dienst gestellt worden. Entsprechend wurde U 1224 mit dem Codenamen Marco Polo II bezeichnet.
Verlust
Am 30. März 1944 lief RO 501 aus Kiel in Richtung Japan aus. Durch Dechiffrierung des japanischen Funkverkehrs waren die Alliierten über den Kurs des Bootes informiert. Auf Höhe der Kapverdischen Inseln wurde RO 501 von einer US-amerikanischen U-Boot-Jagdgruppe um den Geleitflugzeugträger USS Bogue erwartet. Die Gruppe bestand aus den Geleitzerstörern USS Haverfield (DE-393), USS Janssen (DE-396), USS Willis (DE-395), USS Francis M. Robinson (DE-220) und USS Wilhoite (DE-397). Die Francis M Robinson erzielte am 13. Mai 1944 eine Sonarortung, woraufhin Kommandant Johansen einen Hedgehog-Angriff befahl. Bei diesem Angriff wurde RO 501 mit seiner gesamten 51 Mann starken Besatzung versenkt.
Weblinks
U 1224 auf ubootarchiv.de
Literatur
Rainer Busch, Hans-Joachim Röll: Der U-Boot-Krieg 1939–1945. Band 1: Die deutschen U-Boot-Kommandanten. E. S. Mittler und Sohn, Hamburg u. a. 1996, ISBN 3-8132-0490-1.
Rainer Busch, Hans-Joachim Röll: Der U-Boot-Krieg 1939–1945. Band 2: Der U-Boot-Bau auf deutschen Werften. E. S. Mittler und Sohn, Hamburg u. a. 1997, ISBN 3-8132-0512-6.
Einzelnachweise
Yanagi-Mission
U-Boot-Klasse IX
Schiffsverlust 1944
Schiffsverlust im Zweiten Weltkrieg
Deutsche Werft
U1224 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,910 |
package org.forwoods.docuwiki.documentationWiki.db;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
import com.mongodb.MongoClient;
import io.dropwizard.lifecycle.Managed;
public class MongoManaged implements Managed {
private static final Logger LOGGER = LoggerFactory.getLogger(MongoManaged.class);
MongoClient mongo;
public MongoManaged(MongoClient mongo) {
super();
this.mongo = mongo;
}
@Override
public void start() throws Exception {
}
@Override
public void stop() throws Exception {
LOGGER.info("Stopping mongo client");
mongo.close();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,610 |
<?php
namespace Cygnite\Cache\Storage;
use Cygnite\Cache\Exceptions\InvalidCacheDirectoryException;
use Cygnite\Cache\StorageInterface;
use Cygnite\Helpers\Config;
if (!defined('CF_SYSTEM')) {
exit('External script access not allowed');
}
/**
* Cygnite File Cache
*
* @author Sanjoy Dey <dey.sanjoy0@gmail.com>
*
*/
class File implements StorageInterface
{
/**
* The path to the cache file folder
*
* @var string
*/
private $cachePath;
/**
* The name of the default cache file
*
* @var string
*/
private $cacheName = 'default';
/**
* The cache file extension
*
* @var string
*/
private $extension = '.tmp';
private $where = false;
/**
* Constructor of File Cache
* We will initialize file cache
*/
public function __construct()
{
$config = Config::get('global.config', 'cache');
$data = [
'name' => $config['file']['name'],
'path' => $config['file']['directory'],
'extension' => $config['file']['extension']
];
if ($data['path'] == "") {
throw new InvalidCacheDirectoryException('You must define cache directory to use cache.');
}
$this->setup($data);
}
/**
* @param array $config
* @return $this
*/
public function setup($config = [])
{
$path = toPath($config['path']);
if (is_array($config)) {
$this->setCache($config['name']);
$this->setPath(CYGNITE_BASE . DS . $path . DS);
$this->setCacheExtension($config['extension']);
return $this;
}
$this->setCache($config);
return $this;
}
/**
* @param $name
* @return $this
* @return $this
*/
public function setCache($name)
{
$this->cacheName = $name;
return $this;
}
/**
* @param $pathUrl
* @return $this
*/
public function setPath($pathUrl)
{
$this->cachePath = $pathUrl;
return $this;
}
/**
* @param $ext
* @return $this
*/
public function setCacheExtension($ext)
{
$this->extension = $ext;
return $this;
}
/**
* @param $key
* @return bool
*/
public function isCached($key)
{
if ($this->getCache() != false) {
$cached = $this->getCache();
return isset($cached[$key]['data']);
}
}
/**
* @return bool|mixed
*/
private function getCache()
{
if (file_exists($this->getDirectory())) {
return json_decode(
file_get_contents(
$this->getDirectory()
),
true
);
}
return false;
}
/**
* @return string
*/
private function getDirectory()
{
if ($this->hasDirectory() === true) {
$fileName = $this->getCacheName();
$fileName = preg_replace('/[^0-9a-z\.\_\-]/i', '', strtolower($fileName));
return $this->getPath() . md5($fileName) . $this->getCacheExtension();
}
}
/**
* @return bool
* @throws \Exception
*/
public function hasDirectory()
{
if (!is_dir($this->getPath()) && !mkdir($this->getPath(), 0775, true)) {
throw new InvalidCacheDirectoryException('Unable to create cache directory ');
} elseif (
!is_readable($this->getPath()) ||
!is_writable($this->getPath())
) {
if (!chmod($this->getPath(), 0775)) {
throw new InvalidCacheDirectoryException(
'Cache Path Error ' . $this->getPath() . ' directory must be writable'
);
}
}
return true;
}
/**
* @return string
*/
private function getPath()
{
return toPath($this->cachePath);
}
/**
* get cache name
*
* @return string
*/
public function getCacheName()
{
return $this->cacheName;
}
/**
* Cache file extension Getter
*
* @return string
*/
public function getCacheExtension()
{
return $this->extension;
}
/**
* @return int
*/
public function getTimeout()
{
return (int)ini_get('session.gc_maxlifetime');
}
/**
* Save data into cache
*
* @false string
* @false mixed
* @false integer [optional]
* @param $key
* @param $value
* @param int $expiration
* @return object
*/
public function store($key, $value, $expiration = 0)
{
// $this->getTimeout(); Do delete based on the session time out
$data = array(
'time' => time(),
'expire' => $expiration,
'data' => $value
);
if ($this->where == true) {
$this->setCache($key)
->setPath(CYGNITE_BASE . DS . toPath('public.storage.cache') . DS);
}
if (is_array($this->getCache())) {
$array = $this->getCache();
$array[$key] = $data;
} else {
$array = [$key => $data];
}
$cacheData = json_encode($array);
if ($this->getDirectory() == true) {
@file_put_contents($this->getDirectory(), $cacheData);
}
return $this;
}
/**
* Checking cache existence
*
* @param $key
* @param $key
* @return bool
*/
public function has($key)
{
if ($this->where == true) {
$this->setCache($key)->setPath(CYGNITE_BASE . DS . toPath('public.storage.cache') . DS);
}
$cached = $this->getCache();
return !empty($cached[$key]) ? true : false;
}
/**
* Retrieve cache value from file by key
*
* @false string
* @false boolean [optional]
* @param $key
* @param bool $timestamp
* @return string
*/
public function get($key, $timestamp = false)
{
if ($this->where == true) {
$this->setCache($key)->setPath(CYGNITE_BASE . DS . toPath('public.storage.cache') . DS);
}
$cached = [];
$cached = $this->getCache();
if ($timestamp === false) {
return $cached[$key]['data'];
} else {
return $cached[$key]['time'];
}
}
/**
* @param $name
* @return $this
*/
public function where($name)
{
$this->where = true;
return $this->setCache($name);
}
/**
* @param $method
* @param $arguments
* @throws \BadMethodCallException
* @return mixed
*/
public function __call($method, $arguments)
{
if ($method == 'as') {
return call_user_func_array([$this, 'where'], [$arguments]);
}
throw new \BadMethodCallException("Invalid method called File::$method");
}
/**
* We will destroy expired cache from the directory
*
* @return int
*/
public function destroyExpiredCache()
{
$cacheData = $this->getCache();
if (true === is_array($cacheData)) {
$counter = 0;
foreach ($cacheData as $key => $entry) {
if (true === $this->isExpired($entry['time'], $entry['expire'])) {
unset($cacheData[$key]);
$counter++;
}
}
if ($counter > 0) {
$cacheData = json_encode($cacheData);
@file_put_contents($this->getDirectory(), $cacheData);
}
return $counter;
}
}
/**
* @param $timestamp
* @param $expiration
* @return bool
*/
private function isExpired($timestamp, $expiration)
{
$result = false;
if ($expiration !== 0) {
$timeDiff = time() - $timestamp;
$result = ($timeDiff > $expiration) ? true : false;
}
return $result;
}
public function destroy($key)
{
}
/**
* Erase all cached entries
*
* @return object
*/
public function destroyAll()
{
$cacheDir = $this->getDirectory();
if (file_exists($cacheDir)) {
$cacheFile = fopen($cacheDir, 'w');
fclose($cacheFile);
}
return $this;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,704 |
Sri Lanka tour…
One-Day
Gareth Copley/Getty Images Sport
Sri Lanka tour 'ridiculous', according to former England captain
@CharlieSmith118
Former England captain Nasser Hussain has slammed England's current tour of Sri Lanka, claiming the timing has been 'ridiculous', Sky Sports report.
All six of England's limited-overs fixtures so far have ben hit by the weather, with the most recent one coming on Saturday, as Eoin Morgan's side claimed the series with a DLS victory.
The ECB recently released a statement, defending their decision to tour Sri Lanka in monsoon season, claiming they had little 'wriggle room'.
The series win for England was their ninth one-day international series win in a row, as they prepare for the World Cup next year.
However, Hussain, who captained England's Test side for four years between 1999-2003, has slammed the ECB's decision to squeeze in the tour before the end of the year, claiming the cricketing authorities need to think about 'quality over quantity'.
It's ridiculous as these guys are professional cricketers. It's ridiculous for the people who have come out here to follow the side.
Moeen Ali has faced one ball, Jos Buttler has faced 26. It's not on sending professional cricketers around the world to sit in hotel rooms to watch it rain.
Saying 'we've got a month, we've got to fill it with cricket', we have got to get away from that. It is quality, not quantity. This is not good enough.
The fifth and final ODI is set to take place in Colombo on Tuesday, before the Test series gets underway at the beginning of November.
Read more from Charlie Smith
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Lille assess Liverpool's Pepe move
Read Liverpool • By Abdinasir Ali-Hassan
Chelsea determined to have Rudiger fit for Wolves clash
Read Chelsea • By Luke Osman
West Ham receive €9m offer from Sassuolo for Obiang
Read West Ham • By Luke Osman
Arsenal in for Seri – would he be a good signing?
Read Arsenal • By Luke Osman | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,733 |
BioWare loves their community, a press release.
Company News - posted by Saint_Proverbius on Tue 21 January 2003, 21:36:30
Tags: Neverwinter Nights
Here you go! A nice, long press release full of hype from BioWare!
BioWare fan community an unqualified success
Recent initiatives push BioWare community numbers to record levels.
EDMONTON, Alberta, Canada - January 20th, 2003 - BioWare Corp. today released new growth statistics for the official BioWare Community as well as new download statistics for Witch's Wake I: The Fields of Battle.
Since October 2001, the official BioWare Community has experienced a tremendous growth rate, recently passing through the three-quarters of a million member milestone (768,787 members currently). "We're happy that our fans find value in the close contact our community site provides. We believe we can extend the value of our game and improve our game design through the community's active involvement in our games, as will for example occur with our upcoming game, Star Wars: Knights of the Old Republic for Xbox and PC," said Dr. Greg Zeschuk, Joint CEO of BioWare Corp.
The official BioWare Community website supports fans of BioWare's games with community forums, downloads, and technical support. For Neverwinter Nights, the BioWare Community website also offers new game modules, tutorials, as well as custom support that provide each guild with the ability to operate their own meeting space, complete with their own discussion forums, news system, calendar system, and membership management tools.
"We're very proud of our fan community's many achievements. The fact that over 2000 community-created Neverwinter Nights modules are now available for free download is a testament to the creativity of the Neverwinter Nights mod-making community, and also indicates the degree of support that BioWare's Community/Live Team is providing to our fans - they're very important to us," said Dr. Ray Muzyka, Joint CEO of BioWare Corp. "BioWare's Community/Live team has been actively monitoring fan feedback from Neverwinter Nights and I think our fans will be pleased to see some of this feedback reflected in the two upcoming Neverwinter Nights expansion packs, the first of which is entitled Neverwinter Nights: Shadows of Undrentide," he concluded.
Additionally, BioWare's latest Neverwinter Nights module, Witch's Wake I: Fields of Battle, recently passed the 200,000 download mark. The Witch's Wake modules are a downloadable series of BioWare developed Neverwinter Nights story modules, released as a multi-part serial adventure over several months. Additional downloadable adventure series are in the works at BioWare as well, and more details will be provided about these in the future.
"It's gratifying to see this many downloads of the first part of Witch's Wake," said Dr. Greg Zeschuk, Joint CEO of BioWare Corp. "Our goal with the Witch's Wake module series is to continue to support Neverwinter Nights fans through the development of exciting new adventures, tutorials and tools by the BioWare Live/Community Team. We're planning to support our future games in a similar manner," he concluded.
I'd be proud too if I could do a half assed job and then get someone else to make up for my work for free.
Spotted at RPG Dot.
There are 7 comments on BioWare loves their community, a press release.
mon5shot0004 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,730 |
Q: Initialization of EntityManager class object, how to do it? I have following abstract dao class with me:
I am using find(Long primaryKey) method for getting data.
like
public abstract class AbstractDao<T> {
static final Logger logger = Logger.getLogger(AbstractDao.class);
@PersistenceContext
private EntityManager entityManager;
protected EntityManager getEntityManager() {
return this.entityManager;
}
public T find(Long primaryKey) {
//Here entityManager is null therefore I am getting null pointer exception
return entityManager.find(entityClass,primaryKey);
}
}
Please suggest some techniques to intitalize entityManager object.
A: You can remove PersistenceContext annotation from EntityManager and create next abstract method
public abstract void setEntityManager(EntityManager entityManager);
In this way you can put next method in the main class
@PersistenceContext(unitName = "HERE YOU HAVE TO PUT NAME OF ENTITY MANAGER FACTORY")
public void setEntityManager(EntityManager entityManager) {
this.entityManager = entityManager;
}
and all will be works ;) I have it in my owner DAO and all works
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,966 |
#ifndef _TXMPP_EVENT_H_
#define _TXMPP_EVENT_H_
#ifndef NO_CONFIG_H
#include "config.h"
#endif
#if defined(WIN32)
#include "win32.h" // NOLINT: consider this a system header.
#elif defined(POSIX)
#include <pthread.h>
#else
#error "Must define either WIN32 or POSIX."
#endif
#include "basictypes.h"
namespace txmpp {
class Event {
public:
Event(bool manual_reset, bool initially_signaled);
~Event();
bool Set();
bool Reset();
bool Wait(int cms);
private:
bool EnsureInitialized();
bool is_manual_reset_;
#if defined(WIN32)
bool is_initially_signaled_;
HANDLE event_handle_;
#elif defined(POSIX)
bool event_status_;
bool event_mutex_initialized_;
pthread_mutex_t event_mutex_;
bool event_cond_initialized_;
pthread_cond_t event_cond_;
#endif
};
} // namespace txmpp
#endif // _TXMPP_EVENT_H_
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,226 |
Blog / Apr. 16
Italian Retail Executives Share Best Practices, Exclusive Insights on Retail Reinvention
Patrizia Calvia
On April 4, the "who's who" of Italian fashion retailers attended Aptos' Thinking Retail Forum Milano. The event was held at the Lindt Chocolate Factory, managed by Lindt partner SelecTTrade.
In this colorful and "sweet" atmosphere, the Thinking Retail Forum welcomed C-level executives and managers from the highest-profile luxury and fashion brands in Italy, replicating the success of the Thinking Retail Forums that were held in Paris, Berlin, Shanghai and New York City in 2018.
Under the theme "Retail Reinvention: How to Improve the Customer Journey at Each Step," Aptos' Alberto Riva delivered the keynote speech that discussed the opportunities – and challenges – Italian retailers face as more digital-ready consumers embrace multichannel shopping.
Alberto Rivas of Aptos kicks off the day with an analysis of the evolving customer journey
"Delivering a differentiated customer experience is about finding new ways to interact, but also about getting product, assortment and stock just right," noted Riva. "An end-to-end approach, from POS to PLM and planning, can help retailers unlock hidden omnichannel margins by ensuring localized assortments and integrated experiences."
Following Riva on stage was Edoardo Bulgheroni, President of SelecTTrade, the Exclusive Italian Retail Partner of the Lindt Group. Bulgheroni shared with delegates his deep expertise and understanding of the Italian retail market. He also described the experience of SelecTTrade in managing the Lindt Italian retail adventure, developing and operating a network of over 50 Lindt stores throughout the country.
Edoardo Bulgheroni, President of SelecTTrade, describes their success with Lindt chocolates
During Bulgheroni's session, it became clear that the confectionary sector shared many similarities with fashion, including vast assortments, peak periods, different retail formats, and the number one priority across all retailers – delighting customers.
"Our emphasis is on engaging consumers through great in-store experiences to drive brand equity and loyalty," stated Bulgheroni. "Our Promise is that every Customer will leave our Shops with a Chocolate Smile and a Bag full of Lindt Enjoyment".
The next speaker was Francesco Pinto, Chairman of Inticom SpA, part of Pianoforte Group. Pinto spoke in depth about the PLM journey the company has been undertaking with Aptos across its brands – Carpisa, Yamamay and Jaked.
Pinto highlighted how important it is to establish collaborative and streamlined processes inside the company and how PLM can ultimately help in managing these flows.
"Our products are the result of a collective effort and intelligence," Pinto stated. "Different people from different departments, in different moments, contribute to the realization of a style. The product is not a static entity, but evolves within a structured, collaborative and monitored workflow of ideas and information."
Following Pinto's inspiring presentation, Massimo Degan, COO of Original Marines, captivated the audience by describing how the popular kidswear retailer had successfully made the transition to omnichannel.
"Retailers have to change their answers because the questions have changed," noted Degan. "Modern retailing has evolved from being manufacturing-driven to being completely customer-driven, and this calls for new business and operational models. At Original Marines, we have undertaken several programs over the years that brought our business closer to our customers and their needs and made our processes more efficient and sustainable. The Aptos Planning implementation is an important component of this evolution and strategy."
Degan also presented a compelling history of Original Marines, a company that recently celebrated its 35th anniversary.
In addition to the insights and real-world examples provided by Degan, Pinto and Bulgheroni, the Thinking Retail Forum included presentations from Aptos experts who provided live demos of Aptos solutions for retail planning and PLM, in addition to a demo of Aptos ONE, a microservices-based SaaS platform.
Aptos PLM Product Manager, Luca Ferraris takes the stage to talk on the future of collaboration
With an eye to the future, two additional presentations were dedicated to augmented reality (AR); one session described innovative use cases of AR technology within PLM, while the other, led by Aptos partner Sense, provided attendees with the fascinating experience of an AR-enabled catwalk.
Executives from Sense helped attendees try A/R product design on their own photos
In addition to an agenda rich in content, Italian retail leaders participated in robust networking throughout the day, demonstrating just how valuable it is for members of the retail community to meet face to face in a noncompetitive environment to discuss new ideas and share best practices that can be applied back to their businesses.
We can't thank enough all of the speakers and delegates who participated in this event and contributed to its great success. Grazie to everyone who attended, particularly our dynamic retailer speakers Massimo Degan, Francesco Pinto and Edoardo Bulgheroni.
If you are interested in attending an upcoming Aptos event, click here to see where our retail and technology experts will be next. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,399 |
Q: YamlDotNet throwing exception when deserializing object So, I am new to Yaml and YamlDotNet. I wrote the following code to parse a yaml file I am using for configuration of an client API...
public bool TryGet(string path, out DiagnosticScannerConfig config)
{
var deserializer = new DeserializerBuilder()
.WithNamingConvention(new HyphenatedNamingConvention())
.Build();
try
{
using (var reader = File.OpenText(path))
{
var deserializedConfig = deserializer.Deserialize<InternalDiagnosticScannerConfig>(reader);
config = new DiagnosticScannerConfigImpl(deserializedConfig);
return true;
}
}
catch (Exception e)
{
config = DiagnosticAnalyzerConfigCache.Default;
return true;
}
}
public interface DiagnosticScannerConfig
{
DiagnosticAnalyzerConfig Analyzer { get; }
}
public interface DiagnosticAnalyzerConfig
{
uint HighClosureRateWarningThreshold { get; }
uint HighCreationRateWarningThreshold { get; }
uint QueueHighFlowThreshold { get; }
uint QueueLowFlowThreshold { get; }
decimal MessageRedeliveryCoefficient { get; }
decimal SocketUsageCoefficient { get; }
decimal RuntimeProcessUsageCoefficient { get; }
decimal FileDescriptorUsageWarningCoefficient { get; }
decimal ConsumerUtilizationWarningCoefficient { get; }
}
class DiagnosticScannerConfigImpl : DiagnosticScannerConfig
{
public DiagnosticScannerConfigImpl(InternalDiagnosticScannerConfig config)
{
Analyzer = new DiagnosticAnalyzerConfigImpl(config.Analyzer);
}
class DiagnosticAnalyzerConfigImpl : DiagnosticAnalyzerConfig
{
public DiagnosticAnalyzerConfigImpl(Analyzer config)
{
HighClosureRateWarningThreshold = config.HighClosureRateWarningThreshold;
HighCreationRateWarningThreshold = config.HighCreationRateWarningThreshold;
QueueHighFlowThreshold = config.QueueHighFlowThreshold;
QueueLowFlowThreshold = config.QueueLowFlowThreshold;
MessageRedeliveryCoefficient = config.MessageRedeliveryCoefficient;
SocketUsageCoefficient = config.SocketUsageCoefficient;
RuntimeProcessUsageCoefficient = config.RuntimeProcessUsageCoefficient;
FileDescriptorUsageWarningCoefficient = config.FileDescriptorUsageWarningCoefficient;
ConsumerUtilizationWarningCoefficient = config.ConsumerUtilizationWarningCoefficient;
}
public uint HighClosureRateWarningThreshold { get; }
public uint HighCreationRateWarningThreshold { get; }
public uint QueueHighFlowThreshold { get; }
public uint QueueLowFlowThreshold { get; }
public decimal MessageRedeliveryCoefficient { get; }
public decimal SocketUsageCoefficient { get; }
public decimal RuntimeProcessUsageCoefficient { get; }
public decimal FileDescriptorUsageWarningCoefficient { get; }
public decimal ConsumerUtilizationWarningCoefficient { get; }
}
public DiagnosticAnalyzerConfig Analyzer { get; }
}
public class Analyzer
{
[YamlMember(Alias = "high-closure-rate-warning-threshold")]
public uint HighClosureRateWarningThreshold { get; set; }
[YamlMember(Alias = "high-creation-rate-warning-threshold")]
public uint HighCreationRateWarningThreshold { get; set; }
[YamlMember(Alias = "queue-high-flow-threshold")]
public uint QueueHighFlowThreshold { get; set; }
[YamlMember(Alias = "queue-low-flow-threshold")]
public uint QueueLowFlowThreshold { get; set; }
[YamlMember(Alias = "message-redelivery-coefficient")]
public decimal MessageRedeliveryCoefficient { get; set; }
[YamlMember(Alias = "socket-usage-coefficient")]
public decimal SocketUsageCoefficient { get; set; }
[YamlMember(Alias = "runtime-process-usage-coefficient")]
public decimal RuntimeProcessUsageCoefficient { get; set; }
[YamlMember(Alias = "file-descriptor-usage-warning-coefficient")]
public decimal FileDescriptorUsageWarningCoefficient { get; set; }
[YamlMember(Alias = "consumer-utilization-warning-coefficient")]
public decimal ConsumerUtilizationWarningCoefficient { get; set; }
}
public class InternalDiagnosticScannerConfig
{
public Analyzer Analyzer { get; }
}
I trying to parse the following yaml file:
---
high-closure-rate-warning-threshold: 90
high-creation-rate-warning-threshold: 60
queue-high-flow-threshold: 90
queue-low-flow-threshold: 10
message-redelivery-coefficient: 0.60
socket-usage-coefficient: 0.60
runtime-process-usage-coefficient: 0.65
file-descriptor-usage-warning-coefficient: 0.65
consumer-utilization-warning-coefficient: 0.65
...
When I execute the above code I am getting the following error:
YamlDotNet.Core.YamlException: (Line: 2, Col: 5, Idx: 8) - (Line: 2, Col: 5, Idx: 8): Exception during deserialization ---> System.Runtime.Serialization.SerializationException: Property 'high-closure-rate-warning-threshold' not found on type 'HareDu.Diagnostics.Configuration.InternalDiagnosticScannerConfig'.
at YamlDotNet.Serialization.TypeInspectors.TypeInspectorSkeleton.GetProperty(Type type, Object container, String name, Boolean ignoreUnmatched)
at YamlDotNet.Serialization.NodeDeserializers.ObjectNodeDeserializer.YamlDotNet.Serialization.INodeDeserializer.Deserialize(IParser parser, Type expectedType, Func`3 nestedObjectDeserializer, Object& value)
at YamlDotNet.Serialization.ValueDeserializers.NodeValueDeserializer.DeserializeValue(IParser parser, Type expectedType, SerializerState state, IValueDeserializer nestedObjectDeserializer)
--- End of inner exception stack trace ---
at YamlDotNet.Serialization.ValueDeserializers.NodeValueDeserializer.DeserializeValue(IParser parser, Type expectedType, SerializerState state, IValueDeserializer nestedObjectDeserializer)
at YamlDotNet.Serialization.ValueDeserializers.AliasValueDeserializer.DeserializeValue(IParser parser, Type expectedType, SerializerState state, IValueDeserializer nestedObjectDeserializer)
at YamlDotNet.Serialization.Deserializer.Deserialize(IParser parser, Type type)
at YamlDotNet.Serialization.Deserializer.Deserialize[T](IParser parser)
at YamlDotNet.Serialization.Deserializer.Deserialize[T](TextReader input)
at HareDu.Diagnostics.Configuration.DiagnosticScannerConfigProvider.TryGet(String path, DiagnosticScannerConfig& config) in /Users/albert/Documents/Git/HareDu2/src/HareDu.Diagnostics/Configuration/DiagnosticScannerConfigProvider.cs:line 59
A: The exception message is telling you the problem: your type InternalDiagnosticScannerConfig does not have a 'high-closure-rate-warning-threshold' property. Indeed on your code that property is defined in the Analyzer class. You should either deserialize to that type, or update your YAML so that it's structure reflects your classes.
In practice, either use this:
deserializer.Deserialize<InternalDiagnosticScannerConfig>(reader);
Or this:
---
analyzer:
high-closure-rate-warning-threshold: 90
high-creation-rate-warning-threshold: 60
queue-high-flow-threshold: 90
queue-low-flow-threshold: 10
message-redelivery-coefficient: 0.60
socket-usage-coefficient: 0.60
runtime-process-usage-coefficient: 0.65
file-descriptor-usage-warning-coefficient: 0.65
consumer-utilization-warning-coefficient: 0.65
...
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,180 |
The Painful Decision To Cancel Weddings, Funerals, And Major Life Events Because Of The Coronavirus
Around the world, restrictions on gatherings of large crowds have forced people to make agonizing decisions about major life events.
by Brianna Sacks
by Amber Jamieson
Posted on March 13, 2020, 11:23 pm
Courtesy of Nils Van den Bossche., Courtesy of Nils Van den Bossche
Sons who can't properly bury their fathers, couples calling off their weddings, and friends missing joyous events such as bachelorette weekends, bat mitzvahs, and 18th birthdays. Around the world, the coronavirus has upended people's lives in unfathomable ways, forcing them to make agonizing decisions to cancel meaningful and monumental life events.
"It's bullshit he can't have a funeral," said Nils Van den Bossche of his late father. "I understand the severity of the virus, but at the same time I think it's hypocritical to let people go on working, get sick there, but a gathering of a 100 friends family is not allowed?"
On March 3, Gerd Van den Bossche, 51, died suddenly after collapsing in his office from a heart attack, according to his son. For the past 10 days, the 24-year-old's mother and sister have been in overdrive planning the funeral. It was going to be a large memorial, with about 200 people who wanted to come pay their respects.
But then coronavirus fears started "lighting up" in Belgium with more than 500 infections so far. The Belgian government, like so many other countries, enacted swift, drastic precautionary measures to try to thwart the spread.
On Friday night, Nils's family was dealt another painful blow: Less than 24 hours before the service was slated to start, they got the news that their memorial was officially canceled due to the restrictions on social gatherings.
It's painful and infuriating for Nils to not be able to honor his father the way he should be, with his friends, coworkers, and members of the community. Gerd had worked at the same company fixing wires on streetlights for 18 years and also worked with a nearby soccer club helping injured players.
"It's quite miserable," his son said. "Especially when you know your father was loved by a lot of people."
Courtesy of Nils Van den Bossche
Now, due to new rules on social gatherings, only 10 close friends and family members can come pay their respects to Gerd's life. The Saturday ceremony will last about 15 minutes, then the procession will head to the graveyard to bury the body. After that, the small group will disband and go home.
"There's nothing else to do," Nils said. "We aren't even allowed to be together at this moment."
Nils joked that it's probably what his father would have wanted.
"He'd say, 'This is too much money, so either we cut costs or cancel it entirely,'" Nils chuckled. "I am just trying to stay as positive as possible as I have to call all these people and notify them they can't come."
One of the hardest parts of the tragedy is that the two had just started to become friends again. Nils said he'd had a rocky relationship with his father for about 10 years, since he became a teenager. But, the last few months, they had begun to repair it, grow "a lot closer and we started having an actual father–son relationship."
He wrote a letter to his father, which he had hoped to read at his funeral. Now, he said, he will publish it online.
"I want people who couldn't attend to see one final time how much he meant to me," he said. "He was always so proud of his family, his children. I want him to know I am proud of him, in the end."
Maja Szot
Thousands of miles away in Opoczno, Poland, Maja Szot had just finished telling the 40 or so teens who had RSVP'd to her 18th birthday party that the whole thing was off.
She'd been planning this day for a year — buying decorations, finding the perfect venue, locking down a DJ, creating the playlist, and nailing down her "really, really cool cake, which looks like Shrek."
"In Poland, the18th birthday is, like, a huge thing, many teenagers prepare for this day like a year or two before," the high schooler told BuzzFeed News. "To be honest, I was scared at first I didn't wanted to celebrate the fact I am going to be an adult, but my friends convinced me somehow and I really liked the idea."
Like scores of students across the globe, Szot's school is on lockdown for at least two weeks, putting their academic and social lives on pause. She's been spending a lot of time watching Netflix and connecting with friends on Instagram and TikTok, but it's been lonely. She and her friends were "really excited" to finally see each other on April 3, the day of her birthday party.
But Friday night, her dad called and told her they had to call it off.
"I got really sad and angry at the same time," she said. "It feels a little depressing, but I know it's for the better, so it's OK."
Instead, to celebrate she said a few friends will come over March 31, the day she turns 18, and watch a movie. It's not how she imagined ringing in the moment she "became an adult," but she knows she's not alone. Other teens in her grade are also preparing to cancel their big days too.
There's one thing, though, that's she's not canceling: the massive Shrek cake she ordered. "I will eat it all by myself," she said.
Courtesy of Kristin Godsell
As the coronavirus continues to cause panic and bring daily life to a grinding halt, couples are being forced to make the painful decision to cancel or postpone weddings and bachelor and bachelorette parties.
Kristin Godsell was supposed to celebrate finishing the New York City Half Marathon and her upcoming May wedding with eight of her best friends in Boston next weekend for her bachelorette party. Now the 29-year-old won't be doing either.
After officials shut down the race, she had a feeling she'd be forced to do the same for her bachelorette weekend, which she'd been planning since October.
"It's this stressful waiting game for things to get canceled," she said. "It's a fear if we don't cancel that means I could be putting people's health in jeopardy, and I didn't feel comfortable with that."
With the help and support of her friends, she switched gears, called off the trip, and invited anyone who felt comfortable to her home in Norwalk, Connecticut, for a more low-key, intimate gathering.
As "upset," as she was, she said the experience gave her a deeper perspective of and appreciation for what it really means to get married.
"You learn who your friends are quickly. People who reached out to say, 'How can we help?'" she said. "I talked to my fiancé a lot and we realized you lose hindsight about the goal of these weekends. It's about being together. These times show you what you are capable of doing and remind you why you did this to begin with, and you roll with it. I am choosing to be on the other side."
Courtesy of Ashley Boyle
In just a few hours on Thursday, bride-to-be Ashley Boyle went from buying a deep burgundy lipstick for her wedding this Saturday to canceling the whole event.
After 40 guests pulled out within 24 hours due to coronavirus concerns, she said she and her fiancé of six years, Steve Schiraldi, knew they didn't have a choice, even if it was less than two days before their ceremony.
"I got home around 1 p.m. and basically just dissolved," she said. "From that point, we knew we either have a wedding we didn't want or we were postponing."
The 29-year-old had taken Thursday off from her job as a first-grade teacher in Washington, DC, to spend the day relaxing and running last-minute errands for the wedding, but her phone kept ringing. Guest after guest — many of whom lived in New York City, DC, Seattle, and Los Angeles, all areas struggling with the virus outbreak — kept calling and canceling.
"A lot of people who were canceling were younger," she said. "And they were actually expressing that they didn't want to risk that they were carrying it and passing it to our older family and relatives, which was hard to hear but commendable."
Then, Virginia Gov. Ralph Northam declared a state of emergency and told people to avoid large gatherings. Ten minutes later, Boyle's mother called, informing her that very close family members from upstate New York had contacted her, letting her know they didn't feel comfortable traveling.
"To get that one fell swoop of these seven people who are so dear to your heart can't come, I dropped the phone," she said. "I can't imagine turning to walk down that aisle and not having some of my closest relatives there to see it. I just can't do it."
They reached out to their venue, John Marshall Ballrooms in downtown Richmond, which immediately offered to reschedule the wedding at no additional cost. They booked Sept. 6 as a tentative date — the only weekend available for the rest of the year.
"It was incredible," said Boyle. "So far every single one of our vendors said, 'No problem, no fee, we will see you in September.'"
But Boyle said she knows that since she is one of the first weddings to cancel due to the pandemic, future brides may not be so lucky with rescheduling.
She also acknowledged that in recent days, she'd been playing down the risk of the coronavirus and community spread to herself. "I was honestly trying to block it out and was in denial," said Boyle. "As soon as we made the choice, it was, This is obviously what we have to do."
All the roughly 200 invited guests have been reinvited to the wedding, which will now be in September.
As for her wedding day this Saturday, Boyle still plans on walking down an aisle. Since the couple had already gotten their marriage license, which needs to be used within 60 days, they are still getting married. Their parents, siblings, and few close friends will be in attendance and Boyle will carry a bouquet, since the wedding flowers were ready and waiting.
And then in six months they hope they'll do it again — just as she had always dreamed.
"The great thing that has helped it to be tolerable for us," said Boyle, "is we've just been inundated with messages with people saying, 'This is the worst thing in the world but we are going to celebrate you and you are going to have your day.'"
The Teens Are Doing Coronavirus Lockdown Right Stephanie McNeal · March 13, 2020
College Students Are Scrambling To Make Last-Minute Arrangements As Coronavirus Fears Close Dorms Brianna Sacks · March 12, 2020
Brianna Sacks
Brianna Sacks is a reporter for BuzzFeed News and is based in Los Angeles.
Contact Brianna Sacks at brianna.sacks@buzzfeed.com.
Amber Jamieson
Amber Jamieson is an editor for BuzzFeed News and is based in New York.
Contact Amber Jamieson at amber.jamieson@buzzfeed.com. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 733 |
Q: GCP Cloud CDN will not compress content when set to "Automatic" GCP Cloud CDN does not compress any responses when the strategy for compression is set to AUTOMATIC (as it should per the docs) (UI of the CDN in question).
No compression takes place, and no content-encoding header is sent, even tho an accept-encoding header (gzip, deflate, br) is sent.
A prime example of an ~300kb file not being compressed appropriately can be found behind a CDN here: https://himmer.software/main.6a971e1e28a0da9a.js (as of 30.11.2022). The object in question in the connected backend bucket: Object
I feel I must be overseeing something, the mime type and request headers are correct, so having the compression mode set to automatic should return a compressed version of the object.
I set compression mode to AUTOMATIC, set back to DISABLED and back to AUTOMATIC, ran v1.compute.urlMaps.invalidateCache with /* on the load balancer (invalidated all records AFAICT), but still nothing.
A: Are you using the new Google Global Load Balancer or the Classic Load Balancer? Dynamic compression is not currently supported on the new GCLB option, only the Classic GCLB at this time.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,353 |
Q: Coproduct of metrizable vector spaces Does the arbitrary coproduct, i.e. the arbitrary direct sum of metrizable vector spaces exist? I know already, that for metric spaces the coproduct doesn't exists, because one considers short maps as morphisms, which ultimately leads to the failure.
My own idea was to sum up chosen metrics of the $X_i$ in a suitable way: Let $\{X_i\}$ be a family of metrizable vector spaces and choose a family of metrics $\{d_i\}$ generating the topology of $X_i$. For the direct sum we take the direct sum as with vector spaces, so that we have elements of the form
\begin{align}
x=\sum\limits_ix_i
\end{align}
with $x_i\in X_i$ and only finitely many nonzero. For $x,y\in \oplus_i X_i$ we set
\begin{align}
d(x,y)=\sum\limits_id(x_i,y_i).
\end{align}
However, I do not see, how the arising topology is independent of the choice of metrics (which should be the case, for the coproduct to be unique up to topological isomorphism). Also, I am in particularly interested in the setting of complete locally convex metrizable vector spaces, i.e. Fréchet spaces. For locally convex spaces, I think one could choose one seminorm out of every space $X_i$ and write a similar sum as above. Varying over all possible seminorms, this yields a family of seminorms on $\oplus_i X_i$. However, in the setting of Fréchet spaces, this family needs to stay countable. Could it be, that this is just not possible for Fréchet spaces, and that an arbitrary direct sum of Fréchet spaces need not be a Fréchet space?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,864 |
The investment funds offered by Ethos saw their score increase this year during their certification by the German Forum for Sustainable Investments: three of them obtained the maximum of three stars and a fourth fund obtained two stars.
Ethos publishes its positive impact methodology
While it is more necessary than ever to redirect capital towards sustainable activities, it is not always easy for investors to identify companies that offer products and services whose environmental and social impact can be seen as positive. Objective criteria are essential and Ethos has therefore developed its own positive impact methodology. Published in a summarised version, it groups together in ten themes the activities that play a key role in the transition to a more sustainable economy.
Corporate climate reporting: Ethos' position on the new legislation put forward by the Swiss Federal Council
The Ethos Foundation publishes today its position on the consultation on the ordinance on climate reporting by large Swiss companies. While Ethos welcomes the Federal Council's desire to legislate in the field of companies' non-financial and climate reporting, Ethos regrets that the current project does not oblige them to publish all their greenhouse gas emissions - including indirect emissions - nor to have the information verified by an independent third party.
Ethos members and shareholders validate the climate strategy
The annual general meetings of the Ethos Foundation and Ethos Services SA were held this morning in Bern. All the items on the agenda were approved, starting with the climate action plan which should enable Ethos to achieve its goal of net zero CO2 emissions by 2050. This afternoon, a conference dedicated to the sustainable and inclusive economy will will bring together a panel of experts who and will be broadcast live from 1:45 p.m. on the Ethos Foundation website (in French or German).
EU Taxonomy: Ethos opposes the integration of nuclear power and natural gas
The Foundation Ethos is concerned about a decision that could go against the very original purpose of the taxonomy, which is to direct investments towards activities that have a favorable impact on the environment.
Three Ethos investment funds obtain the FNG label
The FNG label is only awarded to funds which meet the quality standards for sustainable investments as defined by the Nachhaltige Geldanlagen Forum and which have been verified by an independent audit of the University of Hamburg. This certification contributes to the recognition of Ethos' 25 years of expertise in sustainable finance.
Natural capital: a new study estimates the cost of environmental neutrality for the non-financial companies of the Swiss Market Index at CHF 28 billion per year
As states from around the world gather in Glasgow for COP26 to try to find solutions to the climate crisis, the authors of a study published on Tuesday measure for the first time the investments required for the largest industrial companies listed in Switzerland to reduce their net greenhouse gas emissions (GHG) to zero and to continue to operate while preserving the planet's water and cropland reserves. Investments which should also allow them to ultimately save CHF 34 billion per year.
Human rights and Environment: Ethos' position on the new legislation put forward by the Federal Council
Following the rejection of the Responsible Business Initiative in November 2020, the Swiss Federal Council has been working on implementing the indirect counter-project adopted by the Swiss parliament. As the open consultation on a draft ordinance on the duties of diligence and transparency in the fields of minerals and metals from conflict zones and child labor (ODiTr) comes to an end, the Ethos Foundation regrets the lack of ambition of the new bill.
Climate change: substantial risks for Swiss pension funds
Two weeks before a key vote on the revision of the Swiss CO2 Act, the Ethos Foundation publishes a report dedicated to climate risks for pension funds and their policyholders. While there are various means for pension funds to manage these risks, a regulatory framework remains more necessary than ever to supervise and encourage sustainable and responsible investments. For this reason, the Ethos Foundation supports the revised CO2 Act. In addition, a public online conference is being organised on 10 June to discuss these questions.
Ethos Swiss Sustainable Equities fund launched successfully
First step in the new partnership between Ethos and BCV, the investment fund has been launched on 19 November. Intended for both private and institutional investors, it has currently CHF 160 million in assets under management.
Subscribe to Responsible investment | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,191 |
\section{Introduction}
\label{sec:introduction}
Understanding the full semantics of rich visual scenes is a complex task that involves detecting individual entities, as well as reasoning about the combination of entities and the relations between them. To represent entities and their relations jointly, it is natural to view them as a graph, where nodes are entities and edges represent relations. Such representations are often called \textit{Scene Graphs} (SGs) \cite{johnson2015image}. Because SGs allow to explicitly reason about images, substantial efforts have been made to infer them from raw images \cite{img_retriev_using_sg, johnson2015image, sg_generation_msg_pass, support_relations, neural_motifs, mapping_to_imgs, InterpretableModel_nips18}.
\begin{figure}[t!]
\begin{center}
\includegraphics[trim=75 0 190 0, clip,width=\linewidth]{Figures/DSG_Figure1-intro_2019-07-26.pdf}
\vspace{-15pt}
\caption{{\bf Differentiable Scene Graphs}: An intermediate ``graph-like" representation that provides a distributed representation for each entity and pair of entities in an image. Differentiable scene graphs can be learned with gradient descent in an end-to-end manner, only using supervision about a downstream visual reasoning task (referring relationships here).}
\label{fig:teaser_fig}
\end{center}
\end{figure}
While scene graphs\xspace have been shown to be useful for various tasks \cite{img_retriev_using_sg, johnson2015image,johnson2018image}, using them as a component in a visual reasoning system is challenging: (a) Because scene graphs\xspace are discrete representations, it is difficult to learn them in an end-to-end fashion from a downstream task. (b) The alternative is to pre-train SG predictors separately from supervised data, but this requires arduous and prohibitive manual annotation. Moreover, pre-trained SG predictors have \emph{low coverage}, because the set of labels they are pre-trained on rarely fits the needs of a downstream task. For example, given an image of a parade and a question \nl{point to the officer on the black horse}, that horse might not be a node in the graph, and the term ``officer" might not be in the vocabulary. Given these limitations, it is an open question how to make scene graphs\xspace useful for visual reasoning applications.
In this work, we describe \textit{Differentiable Scene-Graphs} (DSG), which address the above challenges (Figure~\ref{fig:teaser_fig}). DSGs are an \textbf{intermediate representation trained end-to-end from the supervision for a downstream reasoning task}. The key idea is to relax the discrete properties of scene graphs such that each entity and relation is described with a dense differentiable descriptor.
We demonstrate the benefits of DSGs in the task of resolving \emph{referring relationships} (RR) \cite{krishna2018referring} (see Figure 1). Here, given an image and a triplet query \tripleq{subject}{relation}{object}, a model has to find the bounding boxes of the subject and object that participate in the relation.
We train an RR model with DSGs as an intermediate component. As such, DSGs are not trained with direct supervision about entities and relations, but using several supervision signals about the downstream RR task. We evaluate our approach on three standard RR datasets: Visual Genome \cite{krishnavisualgenome}, VRD \cite{lang_prior} and CLEVR \cite{clevr}, and find that DSGs substantially improve performance compared to state-of-the-art approaches \cite{lang_prior,krishna2018referring}.
To conclude, our novel contributions are: (1) A new \textit{Differentiable Scene-Graph} representation for visual reasoning, which captures information about multiple entities in an image and their relations. We describe how DSGs can be trained end-to-end with a downstream visual reasoning task without direct supervision of manually annotated scene-graphs. (2) A new architecture for the task of referring relationships, using DSGs as its central component. (3) New state-of-the-art results on the task of referring relationships on the Visual Genome, VRD and CLEVR datasets.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=\linewidth]{Figures/figure2_arch9_iccv_RH.jpg}
\vspace{-25pt}
\caption{\textbf{The proposed architecture}. The input consists of an image and a relationship query triplet \tripleq{subject}{relation}{object}. (1) A detector produces a set of bounding box proposals. (2) An \textit{ROI-Align} layer extracts object features from the backbone using the boxes. In parallel, every pair of box proposals is used for computing a union box, and pairwise features are extracted in the same way as object features. (3) These features are used as inputs to a Differentiable Scene-Graph Generator Module which outputs the Differential Scene Graph, a set of node and edge features that result from applying a graph convolutional network to the input features. (4) The DSG is used for both refining the original box proposals, as well as a Referring Relationships Classifier, which classifies each bounding box proposal as either \texttt{Subject}, \texttt{Object}, \texttt{Other} or \texttt{Background}. The ground-truth label of a proposal box will be \texttt{Other} if this proposal appears in another query relationship for this image. Otherwise the ground truth label will be \texttt{Background}.
}
\vspace{-10pt}
\label{fig:latent_graph}
\end{center}
\end{figure*}
\section{Referring Relationship: The Learning Setup}
\label{sec:setup}
In the referring relationship task \cite{krishna2018referring} we are given an image $I$ and a subject-relation-object query $q=\langle s,r,o \rangle$. The goal is to output a bounding box $\mathcal{B}_s$ for the subject, and another bounding box $\mathcal{B}_o$ for the object. In practice there are sometimes several boxes for each.
See \figref{fig:teaser_fig} for a sample query and expected output.
Following \cite{krishna2018referring}, we focus on training a referring relationship predictor from labeled data. Namely, we use a training set consisting of images, queries and the correct boxes for these queries. We denote these by $\{(I_j, q_j, (\mathcal{B}^s_j, \mathcal{B}^o_j)\}_{j=1}^N$. As in \cite{krishna2018referring}, we assume that the vocabulary of query components (subject, object and relation) is fixed.
In our model, we break this task into two components that we optimize in parallel. We fine-tune the position of bounding boxes such that they cover entities tightly, and we also label each box as one of the following four possible labels. The labels ``Subject'' and ``Object'' correspond to the 's' and 'o' entities in the query. The label ``Other'' corresponds to boxes that contain entities (e.g., person or any other category that can appear as a subject or object in queries) that are not the subject or the object of the query. Finally, the label ``Background'' corresponds to boxes that do not contain an entity. We refer to the above two modules as \textbf{Box Refiner{}} and \textbf{Referring Relationships Classifier{}}.
\section{Differentiable Scene Graphs}
\label{sec:dsgs}
We begin by discussing the motivation and potential advantages of using intermediate scene-graph-like representations, as compared to standard scene graphs. We then explain how DSGs fit into the full architecture of our model.
\subsection{Why use intermediate DSG layers?}
A ``perfect'' scene graph (representing all entities and relations) captures most of the information needed for visual reasoning, and thus should be useful as an intermediate representation.
Such a SG can then be used by downstream reasoning algorithms, using the predicted SG as an input. Unfortunately, learning to predict ``perfect'' scene graphs for any downstream task is unlikely. First, there is rarely enough data to train good SG predictors, and second, learning to predict SGs in a way that is independent of the downstream task, tends to yield less relevant SGs.
Instead, we propose an intermediate representation, which we refer to as a ``Differentiable Scene Graph'' layer (DSG). A DSG captures the relational information as in a scene graph but can be trained end-to-end in a task-specific manner (\figref{fig:latent_graph}).
Like SGs, a DSG keeps descriptors for visual entities and their relations. Unlike SGs, whose nodes and edges are annotated by discrete values (labels), a DSG contains a dense distributed representation vector for each detected entity (referred to as a \emph{node descriptor}) and each pair of entities (referred to as an \emph{edge descriptor}). These representations are themselves learned functions of the input image (see supplementary for more details). Like SGs, a DSG only describes candidate boxes which cover entities of interest and their relations.
Unlike SGs, each DSG descriptor captures not only the local information about a node, but also information about its context.
Most importantly, because DSGs are differentiable, they are used as input to downstream visual-reasoning modules (in our case, a referring relationships module).
DGSs provide several computational and modelling advantages:
\noindent {\bf Differentiability.} Because node and edge descriptors are differentiable functions of detected boxes, and are fed into a differentiable reasoning module, the entire pipeline can be trained with gradient descent.
\noindent {\bf Dense descriptors.} By keeping dense descriptors for nodes and edges, the DSG keeps more information about possible semantics of nodes and edges, instead of committing too early to hard sparse representations. This allows it to better fit downstream tasks.
\noindent {\bf Supervision using downstream tasks.} Collecting supervised labels for training scene graphs is hard and costly. DGSs can be trained using training data that is available for downstream tasks, saving costly labeling efforts. On the other hand, when labeled scene graphs are available for given images, that data can be used when training the DSG, using an additional loss component.
\noindent {\bf Holistic representation.} DSG descriptors are computed by integrating global information from the entire image using graph neural networks (see supplemental materials). Combining information across the image increases the accuracy of object and relation descriptors.
\subsection{The DSG Model for Referring relationships}
\label{subsec:model_components}
We now describe how DSGs can be combined with other modules to solve a visual reasoning task. The architecture of the model is illustrated in \figref{fig:latent_graph}. First, the model extracts bounding boxes for entities and relations in the image. Next, it creates a differentiable scene-graph over these bounding boxes.
Then, DSG features are used by two output modules, aimed at answering a referring-relationship query: a Box Refiner{} module that refines the bounding box of the relevant entities, and a Referring Relationships Classifier{} module that classifies each box as \texttt{Subject}, \texttt{Object}, \texttt{Other} or \texttt{Background}. We now describe these components in more detail.
\textbf{Object Detector.} We detect candidate entities using a standard region proposal network (RPN) \cite{faster_rcnn}, and denote their bounding boxes by $\boldsymbol{b}{}_1,\ldots,\boldsymbol{b}{}_B$ ($B$ may vary between images). We also extract a feature vector $\boldsymbol{f}{}_i$ for each box and concatenate it with the box coordinates, yielding $\boldsymbol{z}_i = [\boldsymbol{f}{}_i;\boldsymbol{b}_i{}]$. See details in the supplemental material
\textbf{Relation Feature Extractor.} Given any two bounding boxes $\boldsymbol{b}{}_i$ and $\boldsymbol{b}{}_j$ we consider the smallest box that contains the two boxes (their ``union'' box). We denote this ``relation box'' by $\boldsymbol{b}{}_{i,j}$ and its features by $\boldsymbol{f}{}_{i,j}$.
Finally, we denote the concatenation of the features $\boldsymbol{f}{}_{i,j}$ and box coordinates $\boldsymbol{b}_{i,j}{}$ by $\boldsymbol{z}_{i,j} = [\boldsymbol{f}{}_{i,j};\boldsymbol{b}_{i,j}]$.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.85\linewidth]{Figures/refinening_figure_wacv.jpg}
\vspace{-2em}
\caption{The effect of box refinement and RR classification. (a) The DSG network is applied to an input image. (b) The \textit{object detector} component generates box proposals for entities in the image. (c) The \textit{RR classifier} component uses information from the DSG to label candidate boxes as \texttt{object} or \texttt{subject} entities. Then, the \textit{box refinement} component also uses DSG information, this time to improve box locations for those boxes labeled as entities by RR classifier. Here, boxes are tuned to focus on the most relevant entities in the image: the two ``men'', the ``surfboard'', the ``sky'' and the ``ocean''. (d) Once the RR classifier labels entity boxes, it can correctly refer to the entities in the query \tripleq{cloud}{in}{sky} (sky in green, clouds in violet). (e) Examples of candidate boxes classified as \texttt{background}.}
\label{fig:boxes}
\end{center}
\vspace{-1em}
\end{figure*}
\textbf{Differentiable Scene-Graph Generator.}
As discussed above, the goal of the DSG Generator is to transform the above features $\boldsymbol{z}_i$ and $\boldsymbol{z}_{i,j}$ into differentiable representations of the underlying scene graph. Namely, map these features into a new set of dense vectors $\boldsymbol{z}`_i$ and $\boldsymbol{z}`_{i,j}$ representing entities and relations.
This mapping is intended to incorporate the relevant context of each feature vector. Namely, the representation $z'_i$ contains information about the $i^{th}$ entity, together with its image-wide context.
There are various possible approaches to achieve this mapping. Here we use the model proposed by \cite{mapping_to_imgs}, which uses a graph neural network for this transformation (see supplemental material).
\hspace{10pt}
\textbf{Multi-task objective}. In many domains, training with multi-task objectives can improve the accuracy of individual tasks, because auxiliary tasks operate as regularizers, pushing internal representations away from overfitting and towards capturing useful properties of the input. We follow this idea here and define a multi-task objective that has three components:
(a) a Referring Relationships Classifier{} matches boxes to subject and object query terms. (b) A Box Refiner{} predicts accurate tight bounding boxes. (c) A Box Labeler recognizes visual entities in boxes if relevant ground truth is available.
\figref{fig:boxes} illustrates the effect of the first two components, and how they operate together to refine the bounding boxes and match them to the query terms. Specifically, \figref{fig:boxes}c, shows how box-refinement produces boxes that are tight around objects and subjects, and \figref{fig:boxes}d shows how RR classification matches boxes to query terms.
\textbf{(A) Referring Relationships Classifier{}}. Given a DSG representation, we use it for answering referring relationship queries. Recall that the output of an RR query \tripleq{subject}{relation}{object} should be bounding boxes $\mathcal{B}_s, \mathcal{B}_o$ containing subjects and objects that participate in the query relation.
Our model has already computed $B$ bounding boxes $\boldsymbol{b}{}_i$, as well as representations $\boldsymbol{z}'_i$ for each box. We next use a prediction model $F_{RRC}(\boldsymbol{z}'_i,q)$ that takes as input features describing the bounding box and the query, and outputs one of four labels $\{$\texttt{Subject}, \texttt{Object}, \texttt{Other}, \texttt{Background}$\}$ (see \secref{sec:setup}).
Denote the logits generated by this classifier for the $i^{th}$ box by $\boldsymbol{r}_i\in\mathbb{R}^4$. The output set $\mathcal{B}_s$ (or $\mathcal{B}_o$) is simply the set of bounding boxes classified as \texttt{Subject} (or \texttt{Object}). See supplemental materials for further implementation details.
\textbf{(B) Box Refiner{}}. The DSG is also used for further refinement of the bounding-boxes generated by the RPN network. The idea is that additional knowledge about image context can be used to improve the coordinates of a given entity. This is done via a network $F_{BR}(\boldsymbol{b}_i, \boldsymbol{z}'_i)$ that takes as input the RPN box coordinates and a differentiable representation $\boldsymbol{z}'_i$ for box $i$, and outputs new bounding box coordinates. See \figref{fig:boxes} for an illustration of box refinement, and the supplemental material for further details.
\textbf{(C) Optional auxiliary losses: Scene-Graph Labeling}.
In addition to the \textit{Box Refiner{}} and \textit{Referring Relationships Classifier{}} modules described above, one can also use supervision about labels of entities and relations if these are available at training time.
Specifically, we train an object-recognition classifier operating on boxes, which predicts the label of every box for which a label is available. This classifier is trained as an auxiliary loss, in a multi-task fashion, and is described in detail below.
\section{Training with Multiple Losses}
\label{sec:train}
We next explain how our model is trained for the RR task, and how we can also use the RR training data for supervising the DSG component. We train with a weighted sum of three losses:
(1) Referring Relationships Classifier{} (2) Box Refiner{} (3) Optional Scene-Graph Labeling loss.
We now describe each of these components. Additional details are provided in the supplemental material.
\subsection{Referring Relationship Classification Loss}
\label{sec:train_RRC}
The \textit{Referring Relationships Classifier{}} (\secref{subsec:model_components}) outputs logits $\boldsymbol{r}_i$ for each box, corresponding to its prediction (\texttt{subject}, \texttt{object}, etc.). To train these logits, we need to extract their ground-truth values from the training data. Recall that a given image in the training data may have multiple queries, and so may have multiple boxes that have been tagged as subject or object for the corresponding queries. To obtain the ground-truth for box $i$ and query $q=\langle s,r,o \rangle$ we take the following steps. First, we find the ground-truth box that has maximal overlap with box $i$. If this box is either a subject or object for the query $q$, we set $\boldsymbol{r}^{gt}_i$ to be \texttt{Subject} or \texttt{Object} respectively. Otherwise, if the overlap with a ground-truth box for a different image-query is greater than $0.5$, we set $\boldsymbol{r}^{gt}_i=\texttt{Other}$ (since it means there is some other entity in the box), and we set $\boldsymbol{r}^{gt}_i=\texttt{Background}$ if the overlap is less than $0.3$. If the overlap is in $[0.3,0.5]$ we do not use the box for training. For instance, given a query \tripleq{woman}{feeding}{giraffe} with ground-truth boxes for \nl{woman} and \nl{giraffe},
consider the box in the RPN that is closest to the ground-truth box for ``woman''. Assume the index of this box is $7$. Similarly, assume that the box closest to the ground-truth for ``giraffe' has index $5$. We would have $\boldsymbol{r}_7^{gt}={\texttt{Subject}}$, $\boldsymbol{r}_5^{gt}={\texttt{Object}}$ and the rest of the $\boldsymbol{r}_i^{gt}$ values would be either \texttt{Other} or \texttt{Background}. Given these ground-truth values, the Referring Relationship Classifier Loss is simply the sum of cross entropies between the logits $\boldsymbol{r}_i$ and the one-hot vectors corresponding to $\boldsymbol{r}_i^{gt}$.
\subsection{Box Refiner{} Loss}
\label{sec:train_BBR}
To train the Box Refiner{}, we use a smooth $L_1$ loss between the coordinates of the refined (predicted) boxes and their ground truth ones.
\subsection{Scene-Graph Labeling Loss}
\label{sec:train_SGL}
When ground-truth data about entity labels is available, we can use it as an additional source of supervision to train the DSG. Specifically, we train two classifiers. A classifier from features of entity boxes $\boldsymbol{z}'_{i}$ to the set of entity labels, and a classifier from features of relation boxes $\boldsymbol{z}'_{i,j}$ to relation labels. We then add a loss to maximize the accuracy of these classifiers with respect to the ground truth box labels.
\ignore{Denote by $E$ the number of entity values (e.g., \nl{horse}, \nl{cat}, \nl{cow} etc.) and $R$ the number of relation values (e.g., \nl{holding}, \nl{kicking} etc.). We construct two classifiers. The first is a linear classifier that takes as inputs a vector $\boldsymbol{z}'_i$ and outputs logits $\vv_i \in \mathbb{R}^E$. The second is a linear classifier that takes as input a vector $\boldsymbol{z}'_{i,j}$ output logits $\vv_{i,j} \in \mathbb{R}^R$. For each query, we have one triplet ground-truth for the above logits. Thus, our DSG Labeling Loss is a sum of cross-entropy loss for the three ground-truth labels (one for the subject, one for the object and one for the relation).
}
\subsection{Tuning the Object Detector}
In addition to training the DSG and its downstream visual-reasoning predictors, the object detector RPN is also trained.
The output of the RPN is a set of bounding boxes. The ground-truth contains boxes that are known to contain entities. The goal of this loss is to encourage the RPN to include these boxes as proposals. Concretely, we use a sum of two losses: First, an RPN classification loss, which is a cross entropy over RPN anchors where proposals of 0.8 overlap or higher with the ground truth boxes were considered as positive. Second, an RPN box regression loss which is a smooth L1 loss between the ground-truth boxes and proposal boxes.
\begin{figure*}[t!]
\begin{center}
\includegraphics[trim=0 0 770 0, clip,width=0.49\linewidth]{Figures/half_page_update_iccv4.jpg}
~
\includegraphics[width=0.49\linewidth]{Figures/DSG_error_analaysis-2019-07-26.pdf}
\vspace{-10pt}
\caption{Qualitative examples demonstrating successful predictions of the DSG model (six left panels) and errors (six right panels).
The right panels illustrate common failure cases for each error type. \textbf{a.} \textit{Missed detection:} the detector missed the glasses on the table. \textbf{b,c.} \textit{Misclassified object:}, the cake is detected but classified as a background. \textbf{d.} \textit{Misclassified relation:} The box classified as \textit{Subject} is indeed a man but it is not the man that has the required relation with the skate. \textbf{e,f.} \textit{Multiplicity}, Either too few or too many GT boxes are classified as \textit{Subject} or \textit{Object}.
}
\label{fig:win_examples}
\end{center}
\end{figure*}
\begin{table}
\small
\setlength\tabcolsep{4.5 pt}
\begin{tabular}{lcccccc}
\multicolumn{1}{c}{} & \multicolumn{6}{c}{Average IOU}\\
\multicolumn{1}{c}{} & \multicolumn{2}{c}{Visual Genome} & \multicolumn{2}{c}{VRD} & \multicolumn{2}{c}{CLEVR} \\
& subject & object & subject & object & subject & object \\
\midrule
\scriptsize{\textsc{SS} \cite{shift}} & 0.399 & 0.469 & 0.320 & 0.371 & 0.740 & 0.740 \\
\scriptsize{\textsc{CO}} \cite{cooccur2008} & 0.414 & 0.490 & 0.347 & 0.389 & 0.691 & 0.691 \\
\scriptsize{\textsc{VRD} \cite{lang_prior}} & 0.417 & 0.480 & 0.345 & 0.387 & 0.734 & 0.732 \\
\scriptsize{\textsc{SASS} \cite{krishna2018referring}} & 0.421 & 0.482 & 0.369 & 0.410 & 0.778 & 0.778\\
\hline
\scriptsize{\textsc{no-DSG}} & 0.412 & 0.47 & 0.333 & 0.366 & 0.937 & 0.937 \\
\scriptsize{\textsc{DSG}} & \textbf{0.489} & \textbf{0.539} & \textbf{0.4} & \textbf{0.435} & \textbf{0.963} & \textbf{0.963} \\
\bottomrule
\hline
\end{tabular}
\caption{\textbf{Comparison with baselines.} Test-set mean IOU in the referring relationship task for the baselines in \secref{sec:baselines} and the Differentiable Scene Graph (DSG) model. Results are also reported for a \textsc{no-DSG} model (see \secref{sec:ablations}) which classifies the referring relationship directly from the RPN output.}
\hfill
\vspace{-1em}
\end{table}
\begin{table}
\vspace{-1em}
\begin{center}
\begin{tabular}{lcc}
\multicolumn{1}{l}{} & \multicolumn{2}{c}{Average IOU} \\
& subject & object \\
\midrule
\scriptsize{\textsc{Two Step}} & $0.430 \pm 0.0014$ & $0.491 \pm 0.0014$ \\
\scriptsize{\textsc{no-DSG}} & $0.405 \pm 0.0013$ & $0.461 \pm 0.0013$ \\
\scriptsize{\textsc{DSG -SGL}} & $0.455 \pm 0.0014$ & $0.511 \pm 0.0013$ \\
\scriptsize{\textsc{DSG -BR}} & $0.469 \pm 0.0014$ & $0.519 \pm 0.0014$ \\
\scriptsize{\textsc{DSG}} & $\textbf{0.477} \pm 0.0014$ & $\textbf{0.528} \pm 0.0014$\\
\bottomrule
\hline
\end{tabular}
\caption{{\bf Model ablations}: Results (including standard errors) for DSG variants on the validation set of the Visual Genome dataset. DSG values slightly differ from Table 1 which reports IOU on the test set. The various models are described in \secref{sec:ablations}.}
\label{results}
\end{center}
\vspace{-1em}
\end{table}
\section{Experiments}
\label{sec:experiments}
In the following sections we provide details about the datasets, training, baselines models, evaluation metrics, model ablations and results. Implementation details of the model are provided in the supplemental material.
\subsection{Datasets}
\label{datasets}
We evaluate the model in the task of referring relationships on three datasets, each exhibiting a unique set of characteristics and challenges.
\newline
\textbf{CLEVR \cite{clevr}.} A synthetic dataset generated from scene-graphs with four spatial relations: ``left'', ``right'', ``front'' and ``behind'', and 48 entity categories. It has over 5M relationships where 33\% are ambiguous entities (multiple entities of the same type in an image).
\newline
\textbf{VRD \cite{lang_prior}.} The Visual Relationship Detection dataset contains 5,000 images with 100 entity categories and 70 relation categories. In total, VRD contains 37,993 relationship annotations with 6,672 unique relationship types and 24.25 relations per entity category. 60.3\% of these relationships refer to ambiguous entities.
\newline
\textbf{Visual Genome \cite{krishnavisualgenome}.} VG is the largest public corpus for visual relationships in real images, with 108,077 images annotated with bounding boxes, entities and relations. On average, images have 12 entities and 7 relations per image. In total, there are over 2.3M relationships where 61\% of those refer to ambiguous entities.
For a proper comparison with previous results \cite{krishna2018referring}, we used the data from \cite{krishna2018referring} including the same entity and relation categories, query relationships and data splits.
\subsection{Evaluation Metrics}
We compare our model to previous work using the average IOU for subjects and for objects. To compute the average subject IOU, we first generate two $L \times L$ binary attention maps: one that includes all the ground truth boxes labeled as \texttt{Subject} (recall that few entities might be labeled as \texttt{Subject}) and the other includes all the box proposals predicted as \texttt{Subject}. If no box is predicted as \texttt{Subject}, the box with the highest score for the label \texttt{Subject} is included in the predicted attention map. We then compute the Intersection-Over-Union between the binary attention maps. For a proper comparison with previous work \cite{krishna2018referring}, we use $L=14$. The object boxes are evaluated similarly.
\subsection{Baselines}
\label{sec:baselines}
The Referring Relationship task was introduced recently \cite{krishna2018referring}, and the SSAS model was proposed as a possible approach (see below). We report the results for the baseline models in \cite{krishna2018referring}.
When evaluating our Differentiable Scene-Graph model, we use exactly the evaluation setting as in \cite{krishna2018referring} (i.e., same data splits, entity and relation categories). The baselines reported are:
\ignore{
\textbf{(1) \textsc{Symmetric Stacked Attention Shifting (SSAS) \cite{krishna2018referring}}}: An iterative model that localizes the relationship entities using attention shift component learned for each relation.
\textbf{(2) \textsc{Spatial Shifts \cite{shift}}}: Same as SSAS, but with no iterations and by replacing the shift attention mechanism with a simple statistical spatial shift for each relation label.
\textbf{(3) \textsc{Co-Occurrence \cite{cooccur2008}}}: Uses an embedding of the subject and object pair for attending over the image features.
\textbf{(4) \textsc{Visual Relationship Detection (VRD) \cite{lang_prior}}}: Similar to Co-Occurrences model, but with an additional relationship embedding.
}
\begin{enumerate}
\setlength{\itemsep}{0pt}%
\setlength{\parskip}{0pt}%
\item \textsc{Symmetric Stacked Attention Shifting (SSAS):} \cite{krishna2018referring} An iterative model that localizes the relationship entities using attention shift component learned for each relation.
\item \textsc{Spatial Shifts \cite{shift}}: Same as SSAS, but with no iterations and by replacing the shift attention mechanism with a simple statistical spatial shift for each relation label.
\item \textsc{Co-Occurrence \cite{cooccur2008}}: Uses an embedding of the subject and object pair for attending over the image features.
\item \textsc{Visual Relationship Detection (VRD) \cite{lang_prior}:} Similar to Co-Occurrences model, but with an additional relationship embedding.
\end{enumerate}
\section{Results}
Table 1 provides average IOU for \texttt{Subject} and \texttt{Object} over the three datasets described in \secref{datasets}. We compare our model to four baselines described in \secref{sec:baselines}. Our Differentiable Scene-Graph approach outperforms all baselines in terms of the average IOU.
Our results for the CLEVR dataset are significantly better than those in \cite{krishna2018referring}. Because CLEVR objects have a small set of distinct colors (Fig \ref{clevr_example}), object detection in CLEVR is much easier than in natural images, making it easier to achieve high IOU. The baseline model without the DSG layer (\textsc{no-DSG}) is an end-to-end model with a two-stage detector in contrast to \cite{krishna2018referring} and already improves strongly over prior work with 93.7\%, and our novel DSG approach further improves to 96.3\% (reducing error by 50\%).
\begin{figure}
\begin{center}
\includegraphics[width=0.8\linewidth]{Figures/clevr-example-iccv.jpg}
\caption{A typical image from the CLEVR \cite{clevr} dataset. The image was trimmed to focus on areas with visual content.}
\label{clevr_example}
\end{center}
\end{figure}
\subsection{Analysis of Success and Failure Cases.}
\figref{fig:win_examples} shows example of success cases and failure cases. We further analyzed the types of common mistakes and their distribution. Since DSGs depend on box proposals, they are sensitive to the quality of the object detectors. Manual inspection of images revealed four main error types: \textbf{(1)} 30\%: Detector failed: the relevant box is missing from the box proposal list. \textbf{(2)} 23.3\% \textit{Subject} or \textit{Object} detected but classified as \textit{Other} or as \textit{Background}. \textbf{(3)} 16.6\%: Relation misclassified. The entities classified as \textit{Subject} and \textit{Object} match the query, but without the required relation. \textbf{(4)} 16.6\%: \emph{Multiplicity}. Either too few or too many of the GT boxes are classified as \textit{Subject} or \textit{Object}. \textbf{(5)} 13.3\%: Other, including incorrect GT, and hard-to-decide cases.
\subsection{Model Ablations}
\label{sec:ablations}
To gain further insight into the performance of the DSG model we performed the following ablations. First, since the model is trained with three loss components, we quantify the contribution of the Box Refinement loss and the Scene-Graph Labeling loss (it is not possible to omit the Referring Relationships Classifier{} loss). We further evaluate the contribution of the DSG compared with a two-step approach which first predicts an SG, and then reasons over it.
We compare the following models:
\ignore{
\textbf{(1) \textsc{DSG}}: The Differentiable Scene Graph model described in \secref{subsec:model_components} and trained as described in \secref{sec:train}.
\textbf{(2) \textsc{Two steps}}: A two-step model that first predicts a scene-graph, and then matches the query with the SG. The SG predictor consists of the same components used in the DSG: A box detector, DSG dense descriptors, and an SG labeler. It is trained with the same set of SG labels used for training the DSG.
\textbf{(3) \textsc{DSG -SGL}}: DSG without the Scene-Graph Labeling component described in \secref{sec:train_SGL}).
\textbf{(4) \textsc{DSG -BR}}: DSG where the \textit{Box Refiner{}} component of Section \ref{sec:train_BBR} is replaced with fine tuning the coordinates of the box proposal using the visual features $\boldsymbol{f}^{}_i$ extracted by the Object Detector.
\textbf{(5) \textsc{no-DSG}}: A baseline model that does not use the DSG representation. Instead, the model includes only an Object Detector and a RR classifier. The RR classifier uses the $\boldsymbol{f}_i$ features extracted by the Object Detector instead of the $\boldsymbol{z}'_i$ features. This model allows us to quantify the benefit of the differentiable scene representation for RR classification.
}
\begin{enumerate}
\setlength{\itemsep}{2pt}%
\setlength{\parskip}{2pt}%
\item \textsc{DSG}: The Differentiable Scene-Graph model described in \secref{subsec:model_components} and trained as described in \secref{sec:train}.
%
\item \textsc{Two steps}: Two-step model. We first predict a scene-graph, and then match the query with the SG. The SG predictor consists of the same components used in the DSG: A box detector, DSG dense descriptors, and an SG labeler. It is trained with the same set of SG labels used for training the DSG. Details in the supplemental material.
\item \textsc{DSG -SGL}: DSG without the Scene-Graph Labeling component described in \secref{sec:train_SGL}).
\item \textsc{DSG -BR}: DSG where the \textit{Box Refiner{}} component of Section \ref{sec:train_BBR} is replaced with fine tuning the coordinates of the box proposal using the visual features $\boldsymbol{f}^{}_i$ extracted by the Object Detector. This variant allows us to quantify the benefit of refining the box proposals based on the differentiable representation of the scene.
\item \textsc{no-DSG}: A baseline model that does not use the DSG representations. Instead, the model includes only an Object Detector and a RR classifier. The RR classifier uses the $\boldsymbol{f}_i$ features extracted by the Object Detector instead of the $\boldsymbol{z}'_i$ features. This model allows us to quantify the benefit of the differentiable scene representation for RR classification
\end{enumerate}
Table 2 provides results of ablation experiments for the Visual Genome dataset \cite{krishnavisualgenome} on the validation set.
All model variants based on scene representation perform better than the model that does not use the DSG representation (i.e., \textsc{DSG -SG}), demonstrating the power of contextualized scene representation. The \textsc{DSG} model outperforms all model ablations, illustrating the improvements achieved by using partial supervision for training the differentiable scene-graph.
\figref{fig:ablation_examples} illustrates the effect of ablating various components of the model.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.99\linewidth]{Figures/DSG_horse_and_sg-2019-07-26.png}
\caption{Inferring a Scene Graph from a DSG. Applying the RPN to this image results in 28 boxes. In (a) we show five of these, which received the largest weight in the attention model (details in the supplemental material.) within the DSG generator (\secref{subsec:model_components}). As mentioned in \secref{sec:train_SGL} in ``Scene-Graph Labeling Loss'' we can use the DSG for generating a labeled scene graph, corresponding to a fixed set of entities and relations. (b) shows this scene graph (i.e., the output of the classifiers predicting entity labels and relations), restricted to the largest confidence relations. It can be seen that most relations are correct, despite not having trained this model on complete scene graphs.}
\label{fig:horse}
\end{center}
\end{figure}
\subsection{Inferring SGs from DSGs}
The DSG is designed as a dense representation of objects and relations in the scene. It is thus natural to use it to predict these. This is easy to do in our context, since in \secref{sec:train_SGL} we in fact train such classifiers as an auxiliary task. Thus, for a given image we can construct a scene-graph out of the outputs of these objects and relation classifiers.
\figref{fig:horse} illustrates the result of this process, showing a Scene-Graph inferred from the DSG. The predicted graph is indeed largely correct, even though it was not directly trained for this task (but rather from partial supervision). We further analyzed the accuracy of predicted SGs by comparing to ground-truth SG on visual genome (complete SGs were not used for training, only for analysis). SGs decoded from DSGs achieve accuracy of $76\%$ for object labels and $70\%$ for relations (calculated for proposals with IOU $\ge 0.8$).
\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{Figures/ablations_wacv_small.jpg}
\caption{Comparing failures of ablations models with \textsc{DSG} predictions. The top row shows \textsc{DSG} results, while the bottom row shows results from different ablations models as specified in ~\secref{sec:ablations}. In the first column, \textsc{Two Step} model the SG did not include the shirt of one of the the men, therefore this "subject" prediction was missed. In the second column, the \textsc{DSG -SGL} predicted failed to distinct between few entity classes `woman'' and ``child''. In the third column, the \textsc{DSG} refine the box of ``sky'' to cover all of the sky area. In the last column, the \textsc{NO-DSG} didn't classify the "object" box correctly.
}
\label{fig:ablation_examples}
\end{center}
\vspace{-1em}
\end{figure}
\section{Related Work}
\label{sec:related_work}
\textbf{Graph Neural Networks.} Recently, major progress has been made in constructing graph neural networks (GNN). These refer to a class of neural networks that operate directly on graph-structured data by passing local messages \cite{gilmer2017neural, Li2015GatedGS}. Variants of GNNs have been shown to be highly effective at relational reasoning tasks \cite{nn_for_relational}, classification of graphs \cite{bruna2013invariant, dai_disc_embs, niepert16, DefferrardBV16} and classification of nodes in large graphs \cite{graph_conv, inductive_repr_large_grphs}. The expressive power of GNNs has also been studied in \cite{mapping_to_imgs, deep_sets}. GNNs have also been applied to visual understanding in \cite{mapping_to_imgs, graph_rcnn, Wang_videogcnECCV2018, herzig2019STAG} and control \cite{Gonzalez18, relational_rl_deepmind18}. Similar aggregation schemes have also been applied to object detection \cite{hu2018relation}. Our goal here is to generate DSG such that each object descriptor encompasses not only the local information about the object, but also information about its context within the scene. To achieve this we use the GNN proposed by \cite{mapping_to_imgs}.
\textbf{Visual Relationships and Scene Graphs.} Earlier work aimed to leverage visual relationships for improving detection \cite{VisualPhrases}, action recognition \cite{herzig2019STAG}, few shot \cite{chen2019scene, dornadula2019visual}, pose estimation \cite{DesaiR12}, semantic image segmentation \cite{GuptaD08} or detection of human-object interactions \cite{yang2017support, plummerPLCLC2017, Li_2019_CVPR}. Lu {\em et al.} \cite{lang_prior} were the first to formulate detection of visual relationships as a separate task. They learn a likelihood function that uses a language prior based on word embeddings for scoring visual relationships and constructing SGs. SGs provide a compact representation of the semantics of an image. Previous SG prediction works used attention \cite{Qi_2019_CVPR, Gkanatsios_2019_ICCV} or neural message passing \cite{sg_generation_msg_pass}. \cite{pixels_to_graph} suggested to predict graphs directly from pixels in an end-to-end manner. \cite{neural_motifs} considers global context using an RNN by reading sequentially the independent predictions for each entity and relation and then refines those predictions.
SGs have been shown to be useful for semantic-level interpretation and reasoning about a visual scene \cite{johnson2018image, ashual2019specifying, herzig2019canonical, Schroeder_2019_ICCV}. Extracting SGs from images provides a semantic representation that can later be used for reasoning, question answering \cite{yi2018neural, hu2019language, Liang_2019_ICCV}, and image retrieval \cite{img_retriev_using_sg, entangled_scene}. Using SGs for reasoning tasks is challenging. Instead, we propose an intermediate representation which captures the relational information as in SGs but can be trained end-to-end in a task-specific manner.
\textbf{Referring Relationships.} The RR task is closely related to the task of referring expressions, where an entity in an image needs to be identified given a natural language expression. Several recent works considered using context for this task \cite{kazemzadeh2014referitgame, krahmer2012computational, Chen_2019_ICCV, Tanaka_2019_ICCV, Yang_2019_ICCV, Liu_2019_ICCV}. \cite{mao2016generation} described a model that has two parts: one for generating expressions that point to an entity in a discriminative fashion and a second for understanding these expressions and detecting the referred entity. \cite{yu2016modeling} explored the role of context and visual comparison with other entities in referring expressions. Modelling context was also the focus of \cite{nagaraja2016modeling}, using a multi-instance-learning objective. RR \cite{krishna2018referring} as opposed to referring expression, focuses on the vision side rather than the language side by forming a simple structured query that requires modeling interactions between the image entities. \cite{krishna2018referring} also introduce an explicit iterative model that localizes the two entities in the RR task, conditioned on one another. We use the RR task to demonstrate the power of our semantic latent representation, resulting in a new state of the art results on three vision datasets that contains visual relationships.
\section{Conclusion}
\label{sec:conclusion}
This work is motivated by the assumption that accurate reasoning about images may require access to a detailed representation of the image. While scene graphs provide a natural structure for representing relational information, it is hard to train very dense SGs in a fully supervised manner, and for any given image, the resulting SGs may not be appropriate for downstream reasoning tasks. Here we advocate DSGs, an alternative representation that captures the information in SGs, which is continuous and can be trained jointly with downstream tasks. Our results, both qualitative (Fig \ref{fig:win_examples} ) and quantitative (Table 1,2), suggest that DSGs effectively capture scene structure, and that this can be used for down-stream tasks such as referring relationships.
One natural next step is to study such representations in additional downstream tasks that require integrating information across the image. Some examples are caption generation and visual question answering. DSGs can be particularly useful for VQA, since many questions are easily answerable by scene graphs (e.g., counting questions and questions about relations). Another important extension to DSGs would be a model that captures high-order interactions, as in a hyper-graph. Finally, it will be interesting to explore other approaches to training the DSG, and in particular finding ways for using unlabeled data for this task.
\newline
\newline
{\bf Acknowledgments:} This project was funded by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant ERC HOLI 819080).
{\small
\bibliographystyle{ieee}
\section*{Supplementary Material}
This supplementary material includes: (1) Model implementation details. (2) Details about the reasoning component in two steps ablation module.
\section{Model Details}
\label{apendix:Supplementary}
The model in Sec. 3.2 is implemented as follows.
\textbf{Object Detector and Relation Feature Extractor.} For object detection, we used Faster-RCNN with a 101-layers ResNet backbone. The RPN was trained with anchor scales of $\{4, 8, 16, 32\}$ and aspect ratios $\{0.5, 1, 2\}$. RPN proposals were filtered by non-maximum suppression with IOU-threshold of 0.5 and score higher than 0.8. We use at most 32 proposals per image. Both the entity features $\boldsymbol{f}^{}_i$ and the relation features $\boldsymbol{f}^{}_{i,j}$ are first extracted from the convolutional network feature map by the ROI-Align layer as $7 \times 7 \times 2048$ features. They are then reduced to a $7 \times 7 \times 512$ by convolution layer of size $1 \times 1$ and finally reduced to $1 \times 512$ by an average pooling layer.
\textbf{Referring Relationship Classifier.} The referring relationship classifier $F_{RRC}$ is a fully-connected network with two layers of 512 hidden units each.
\textbf{Bounding Box Refinement.} The box refinement model applies a linear function to $\boldsymbol{z}_i$ to obtain four outputs $[dx, dy, dw, dh]$.
Denote the RPN box by $ [x, y, w, h]$. The refined box is then: $[dx \cdot w + x, dy \cdot h + y, e^{dw} \cdot w, e^{dh} \cdot h]$ (as in the correction used by Faster-RCNN).
\textbf{Computational Estimation.} Our model creates a graph with $n$ nodes for objects and $n^2$ edges for relations. In the datasets we analyzed, using $n=32$ objects within an image is sufficient. Adding the DSG component has a limited effect on complexity and run time. Specifically, as shown in \tabref{analysis}, the DSG component adds 4M parameters and up to 1.5G operations (when $n=32$) which is \textbf{only 10\% of the parameters} and number of operations of the backbone network ($\sim$40M parameters and $\sim$15G operations). Adding DSG \textbf{increases training time by only 15\%}. This is largely thanks to the fact that all $n^2$ relations can parallelized.
\begin{table}[h]
\vspace{-1em}
\begin{center}
\begin{tabular}{L{3.1CM}C{2.4CM}C{1.5CM}}
\hline
& DSG Generator & Resnet101 \\
\midrule
Trainable parameters & $< 4M$ & $> 40M$ \\
Number of operations & $< 1.5G$ & $> 15G$\\
\bottomrule
\hline
\end{tabular}
\begin{tabular}{L{3.1CM}C{2.4CM}C{1.5CM}}
\hline
& \textsc{DSG} & \textsc{DSG -SG}\\
\midrule
Running time [sec] & 0.054 & 0.045 \\
Training time [sec] & 0.19 & 0.165 \\
\bottomrule
\hline
\end{tabular}
\caption{Analysis of running/training time and computational resources of DSGs.}
\label{analysis}
\end{center}
\vspace{-1em}
\end{table}
\textbf{Differentiable Scene Graph Generator.} We next describe the module that takes as input features $\boldsymbol{z}_i$ and $\boldsymbol{z}_{i,j}$ extracted by the RPN and outputs a set of vectors $\boldsymbol{z}'_i,\boldsymbol{z}'_{ij}$ corresponding to Differentiable Scene-Graph over entities and relationships in the image. For this model, we use the Graph Permutation Invariant (GPI) architecture introduced in \cite{mapping_to_imgs}. A key property of this architecture is that it is invariant to permutations of the input that do not affect the labels.
The GPI transformation is defined as follows. First, the set of all input features is summarized via a permutation-invariant transformation into a single vector $\gg$:
\begin{equation}\label{sgp_eq}
\gg = \sum_{i=1}^n \boldsymbol{\alpha} (\boldsymbol{z}_i, \sum_{j \neq i} \boldsymbol{\phi}(\boldsymbol{z}_{i}, \boldsymbol{z}_{i,j}, \boldsymbol{z}_{j}))
\end{equation}
Here $\boldsymbol{\alpha}$ and $\boldsymbol{\phi}$ are fully connected networks. Then the new representations for entities and relations are computed via:
\begin{equation}
\boldsymbol{z}'_{k} = {\boldsymbol\rho}^{entity}(\boldsymbol{z}_{k},\gg) \ , \
\boldsymbol{z}'_{k,l} = {\boldsymbol\rho}^{relation}(\boldsymbol{z}_{k,l},\gg)
\end{equation}
where ${\boldsymbol{\rho}}$ above are fully connected networks.
The three networks $\boldsymbol{\phi}$, $\boldsymbol{\alpha}$ and $\boldsymbol{\rho}$, described in GPI architecture are two fully-connected layers with 512 hidden units. The output size of $\phi$ and $\alpha$ is 512, and of $\boldsymbol{\rho}$ is 1024. We used the version with integrated attention mechanism replacing the sum operations in equation \ref{sgp_eq}.
\section{Model Ablations}
Additional details about the \textsc{Two Step} model:
Recall that in Two-step model a scene graph is first predicted, followed by a reasoning module. The reasoning module gets as an input a query \tripleq{subject}{relation}{object} and a scene graph and outputs the nodes that represents the subject and the nodes that represents the object. In case the triplet \tripleq{subject}{relation}{object} exists in the scene graph, the reasoning module simply returns the involved nodes. Otherwise, it selects the triplet in the scene graph that has the highest probability to be the required triplet according to the probabilities provided by the scene graph.
\section{Introduction}
\label{sec:introduction}
Understanding the full semantics of rich visual scenes is a complex task that involves detecting individual entities, as well as reasoning about the joint combination of the entities and the relations between them.
To represent entities and their relations jointly, it is natural to view them as a graph, where nodes are entities and edges represent relations. Such representations are often called
\textit{Scene Graphs} (SGs) \cite{johnson2015image}. Because SGs allow to explicitly reason about images, substantial efforts have been made to infer them from raw images \cite{img_retriev_using_sg, johnson2015image, sg_generation_msg_pass, support_relations, neural_motifs, mapping_to_imgs, InterpretableModel_nips18}.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=\linewidth]{Figures/figure1_wacv.jpg}
\caption{{\bf Differentiable Scene Graphs}:
an intermediate ``graph-like" representation that provides a distributed representation for each entity and pair of entities in an image. Differentiable scene graphs can be learned with gradient descent in an end-to-end manner from the supervision for a downstream visual reasoning task only (referring relations here). \gal{Add visuals stressing where the losses are pushed. There is no supervision about the DSG }}
\label{fig:teaser_fig}
\end{center}
\end{figure}
While scene graphs\xspace have been shown to be useful for some tasks \cite{img_retriev_using_sg, johnson2015image,johnson2018image},
using them as a component in a visual reasoning system is challenging: (a) Because scene graphs\xspace are
discrete and non-differentiable, it is difficult to learn them end-to-end from a downstream task.
(b) The alternative is to pre-train SG predictors separately from supervised data,
but this requires arduous and prohibitive manual annotation.
Moreover, pre-trained SG predictors have \emph{low coverage},
because the set of labels they are pre-trained on rarely fits the needs of a downstream task.
For example, given an image of a parade and a question \nl{point to the officer on the black horse}, that horse might not be a node in the graph, and the term ``officer" might not be in the vocabulary.
Given these limitations,
it is an open question how to make scene graphs\xspace useful for visual reasoning applications.
In this work, we describe \textit{Differentiable Scene-Graphs} (DSG), which address the above challenges (Figure~\ref{fig:teaser_fig}). DSGs are an \textbf{intermediate representation trained end-to-end from the supervision for a downstream reasoning task}. The key idea is to relax the discrete properties of scene graphs such that each entity and relation is described with a dense differentiable descriptor.
To evaluate DSGs, we tackle the task of resolving \emph{referring relationships} (RR) \cite{krishna2018referring},
where given an image and a \tripleq{subject}{relation}{object} query, a model must find the subject and object bounding boxes that participate in the specified relation.
The advantage of RR is it requires very little \emph{language understanding} compared to other visual reasoning tasks (e.g., VQA \cite{DBLP:journals/corr/AntolALMBZP15} or referring expressions \cite{kazemzadeh2014referitgame,krahmer2012computational}), focusing on the visual aspect.
We train an RR model with DSGs as the central component.
The latent DSG representation is trained using a multi-task objective that includes
(a) the end-to-end referring relationship objective, as well as (b) auxiliary objectives that are based on labels for a small subset of the objects and relations, given as part of the RR dataset. We evaluate our approach on three standard RR datasets: Visual Genome \cite{krishnavisualgenome}, VRD \cite{lang_prior} and CLEVR \cite{clevr}, and find that DSGs substantially improve performance compared to state-of-the-art approaches \cite{lang_prior,krishna2018referring}. Moreover, we find that our proposed multi-task objective, which explicitly considers the partial supervision provided in the training set, clearly contributes to the improved performance.
To conclude, our novel contributions are: (1) A new \textit{Differentiable Scene-Graph} representation for visual reasoning, which captures information about multiple entities in an image and their relations. We describe how DSGs can be trained end-to-end with a downstream visual reasoning task without direct supervision of pre-collected scene-graphs. (2) A new architecture for the task of referring relationships, using a DSG as its central component. (3) New state-of-the-art results on the task of referring relationships on the Visual Genome, VRD and CLEVR datasets.
\ignore{
is a fundamental problem in machine perception. It requires understanding the context \gal{see my comment about context above} of the scene such as "demonstration", recognizing multiple objects in a scene such as "man", "police officer" and a "horse" together with many object attributes such as "angry" , "tired" and "blue". In addition, it requires understating many object-object interactions such as "police officer riding a horse". One option is to represent it by a scene-graph. Scene-Graph which introduce by \cite{johnson2015image} capture high level semantic of the scene and contains object classes (nodes) and object-object relations (edges). Such representation can be useful for semantic-level interpretation of visual scene.
Scene-Graphs have been used in the literature before for an improved understanding of the visual scene \cite{img_retriev_using_sg, johnson2015image}. Different works as \cite{img_retriev_using_sg, johnson2015image, sg_generation_msg_pass, support_relations, neural_motifs, mapping_to_imgs} trying to predict and generate scene graphs\xspace for a predefined tasks, such as Scene-Graph Prediction and Scene-Graph Generation. Lately, there was a first try to do the opposite direction as \cite{johnson2018image}, which tries to generate image from a Scene-Graph.
While scene-graph gives great benefits for few machine-understanding tasks \gal{I'm not convinced this is true} with the compact representation and built in object-object interactions, many tasks will find the scene-graph representation rigid and limited. The task of inferring graphs from images is limited to a fixed set of object and relation classes. For example: it will be impossible to correctly answer the question ""Which vegetable is on the plate?"" if ""broccoli"" is not a member in the scene-graph object classes or to answer the question ""What color are the towels?"" if color attributes are not included in the scene-graph. Defining a satisfactory set of object and relation classes for the challenging machine understanding tasks might not scalable for the general task .
Therefore, it still remains an open question whether the general machine understanding task can benefit from scene graphs\xspace \gal{This statement contradicts the above statement that SG give great benefits ;)}. In order to exploit the advantages of a scene-graph compact scene representation and built in object-object interactions, but still enjoy from rich, flexible and scalable features, we suggest latent scene-graph representation. In latent scene-graph, each of nodes and edges will hold a feature vector extracted from their bounding-boxes. The feature selection for those vectors learned in an unsupervised manner and will allow it to hold any feature relevant for the task.
In this work we used latent Scene-Graph model as a contextualize representation for Referring Relationships task \cite{krishna2018referring} and we show the resulting model achieves a new state of the art results on three vision datasets that contain visual relationships: CLEVR \cite{clevr}, VRD \cite{lang_prior} and Visual Genome \cite{krishnavisualgenome}.
In summary, the novel contributions of this paper are: a) Design an architecture that captures image latent contextualize scene representation useful for image understanding task. b) Empirically demonstrating the benefit of a latent Scene-Graph in an end to end model. c) Developing a state-of-the-art model for Referring Relationships task on three vision datasets that contain visual relationships.
}
\ignore{
Understanding the semantics of a complex visual scene is a fundamental problem in machine perception. It requires recognizing multiple objects in a scene, together with their spatial and functional relations. This set of objects and relations is sometimes represented as a graph, connecting objects (nodes) with their relations (edges) and is known as a Scene-Graph.
scene graphs\xspace provide a compact representation of the semantics of an image, and can be useful for semantic-level interpretation and reasoning about a visual scene.
While scene-graph gives great benefits for few machine-understanding tasks with the compact representation and built in object-object interactions, many tasks will find the scene-graph representation rigid and limited. The task of inferring graphs from images is limited to a fixed set of object and relation classes. For example: it will be impossible to correctly answer the question "Which vegetable is on the plate?" if "broccoli" is not a member in the scene-graph object classes or to answer the question "What color are the towels?" if color attributes are not included in the scene-graph representation. Defining a satisfactory set of object and relation classes for the challenging machine understanding tasks is hard and might even infeasible.
The set of features required for those tasks, must be rich and flexible.
In order to benefit from compact scene representation and built in object-object interactions, but still enjoy from rich and flexible features, we suggest latent scene-graph representation. In latent scene-graph, each of nodes and edges will hold a feature vector extracted from their bounding-boxes. Those vectors could hold any feature relevant for the task. In our work, we demonstrate that (a) latent scene-graph provide rich and flexible features which allows to outperform existing approaches on "referring-relationship" task (b)
training the latent scene-graph for multiple tasks simultaneously such referring-relationships and scene-graph prediction enrich the latent features, accelerating the learning process and helps to generalize better.}
\section{Differentiable Scene Graphs}
A scene graph represents entities and relations in an image as a set of nodes and edges. A ``perfect'' scene graph (representing all entities and relations) captures most of the information needed for visual reasoning, and thus should be useful as an intermediate representation. However, learning to predict ``perfect'' scene graphs for any downstream task is unlikely due to the aforementioned challenges.
Instead, we propose an intermediate representation, termed ``Differentiable Scene Graph'' Layer (DSG), which captures the information in a scene graph but can be trained end-to-end in a task-specific manner (\figref{fig:latent_graph}). A DSG contains a dense distributed representation vector for each detected entity (termed \emph{node descriptor}) and each pair of entities (termed \emph{edge descriptor}). These representations are themselves learned functions of the input image, as we explain in \secref{subsec:sgp}. The DSG is then used as input to a visual-reasoning module (e.g., referring relations in our case).
The advantages of our approach are:
\noindent
{\bf Differentiability:} Because node and edge descriptors are differentiable functions of detected boxes, and are fed into a differentiable reasoning module, the entire pipeline can be trained with gradient descent.
\newline
{\bf Dense Coverage:}
By keeping dense descriptors for nodes and edges, the DSG succeeds to keep dense information about possible semantics of nodes and edges. Instead of committing too early to hard sparse representations, it keeps a more flexible representation that better fits a series of downstream tasks.
\newline
{\bf Indirect supervision:} Training data that is available for downstream tasks often contains partial annotation of an image, such as the label for some of the entities. We show that such information can be used to train the DSG in a multi-task setting.
\newline
{\bf Holistic Representation:} DSG descriptors are computed by integrating global information from the entire image using graph neural networks (see Sec \ref{subsec:sgp}), which increases the accuracy of each descriptor.
\section{Referring Relationship: The Learning Setup}
\label{sec:setup}
In the referring relationship task \cite{krishna2018referring} we are given an image $I$ and a subject-relation-object query $\langle s,r,o \rangle$. The goal is to output two sets of bounding boxes $\mathcal{B}_s$ and $\mathcal{B}_o$, where each bounding box $\boldsymbol{b}_s \in \mathcal{B}_s$ corresponds to a subject entity $s$ and each bounding box $\boldsymbol{b}_o \in \mathcal{B}_o$ corresponds to an object entity $o$ that participates in the ordered relation $r$ in the image $I$.\footnote{There can be multiple boxes, because an image may contain multiple instances of ``man kicking ball''.} See \figref{fig:teaser_fig} for a sample query and expected output.
Following \cite{krishna2018referring}, we will focus on training a referring relationship predictor from labeled data. Namely, we will use
a training set consisting of images, queries and the correct boxes for these queries. We denote these by $\{(I_j, q_j, (\mathcal{B}^s_j, \mathcal{B}^o_j)\}_{j=1}^N$. Also following \cite{krishna2018referring}, we assume that the vocabulary of the query components (subject, object and relation) is fixed.
We next formally describe the architecture of our model.
\section{Model}
\label{sec:model}
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=\linewidth]{Figures/figure2_arch9_iccv_RH.jpg}
\vspace{-2em}
\caption{\textbf{The proposed architecture}. The input consists of an image and a relationship query triplet \tripleq{subject}{relation}{object}. (1) A detector produces a set of bounding box proposals. (2) An \textit{RoiAlign} layer extracts object features from the backbone using the boxes. In parallel, every pair of box proposals is used for computing a union box, and pairwise features extracted in the same way as object features. (3) These features are used as inputs to a Differentiable Scene-Graph Generator Module which outputs the Differential Scene Graph, a new and improved set of node and edge features. (4) The DSG is used for both refining the original box proposals, as well as a Referring Relationships Classifier, which classifies each bounding box proposal as either \texttt{Subject}, \texttt{Object}, \texttt{Other} or \texttt{Background}. The ground-truth label of a proposal box will be \texttt{Other} if this proposal is involved in another query relationship over this image. Otherwise the ground truth label will be \texttt{Background}. \gal{Lets clarify where the loss is pushed back. We should make it clear that no direct supervision is given about the DSG}
}
\label{fig:latent_graph}
\end{center}
\end{figure*}
The key element in our approach is the use of a differentiable scene-graph representation for answering visual-reasoning tasks. In what follows we describe how this representation is calculated, and how it is trained using partial supervision with a multi-task objective.
\subsection{Model Components}
\label{subsec:model_components}
At a high level, the model works as follows (\figref{fig:latent_graph}). It first extracts bounding boxes from the image. Next, it creates a scene-graph over these bounding boxes. A scene-graph here refers to a set of distributed representations for all the bounding boxes, as well as representations of relations for each pair of bounding boxes. Finally, the scene-graph representation is used as input for a model that answers the query. Namely, the model labels each box as \texttt{Subject}, \texttt{Object},
\texttt{Other} and \texttt{Background}.\footnote{The label ``Other'' refers to a case where we know a box contains an entity that is not the given subject or object. The label ``Background'' refers to a case where we cannot determine if the box describes an entity at all.}
\textbf{Object Detector} - The first stage is to detect candidate entities in the image using a standard region proposal network (RPN) \cite{faster_rcnn}. The RPN output is a set of bounding boxes which are assumed to contain entities of interest. We denote the bounding boxes for these by $\boldsymbol{b}{}_1,\ldots,\boldsymbol{b}{}_B$.\footnote{The number of boxes may vary between images, but we use the same $B$ for all for simplicity.} For each bounding box, we also extract the corresponding feature vector and denote it by $\boldsymbol{f}{}_i$. Finally, we denote the concatenation of $\boldsymbol{f}{}_i$ and $\boldsymbol{b}_i{}$ by $\boldsymbol{z}_i = [\boldsymbol{f}{}_i;\boldsymbol{b}_i{}]$
\textbf{Relation Feature Extractor} - Next, we extract features that will be useful for representing relations between entities in two bounding boxes. Consider two bounding boxes $\boldsymbol{b}{}_i$ and $\boldsymbol{b}{}_j$. In order to reason about the relation between these, it is useful to consider the smallest box that contains both these boxes. We denote this ``relation box'' box by $\boldsymbol{b}{}_{i,j}$. Since $\boldsymbol{b}{}_{i,j}$ is also in the set of regions considered by the RPN, we have a corresponding feature vector $\boldsymbol{f}{}_{i,j}$. Finally, we denote the concatenation of $\boldsymbol{f}{}_{i,j}$ and $\boldsymbol{b}_{i,j}{}$ by $\boldsymbol{z}_{i,j}$.
\textbf{Differentiable Scene-Graph Generator} - The goal of the Differentiable Scene-Graph Generator model is to transform the above features $\boldsymbol{z}$ into a representation of the underlying scene graph. Namely, into a set of vectors representing
the entities and relations in the image, and taking into account image context.\footnote{See description of ``Scene-Graph Labeling Loss'' in \secref{train} for information on how to extract a scene graph from these vectors.} We denote the vector for the $i^{th}$ entity by $\boldsymbol{z}'_i$ and for the relation between entity $i$ and entity $j$ by $\boldsymbol{z}'_{i,j}$. To obtain these vectors, we can use any architecture for scene-graph prediction that outputs a representation for entities and relations. Here we use the model proposed by \cite{mapping_to_imgs}, which takes as input initial entity representations $\boldsymbol{z}_i$ and initial relation representations $\boldsymbol{z}_{i,j}$ and uses a graph neural network to transform these into the desired $\boldsymbol{z}'$ representation. See \secref{subsec:sgp} for details on this network.
\textbf{Referring Relationship Classifier} -
Given a scene-graph representation, we can use it for answering referring relationship queries. Recall that the output of an RR query \tripleq{subject}{relation}{object} should be bounding boxes $\mathcal{B}_s, \mathcal{B}_o$ containing subjects and objects that participate in the query relation.
Our model has already computed $B$ bounding boxes $\boldsymbol{b}{}_i$, as well as representations $\boldsymbol{z}'_i$ for each box. We next use a prediction model $F_{RRC}(\boldsymbol{z}'_i,q)$ that takes as input features describing a bounding box and the query, and outputs one of four labels $\{$\texttt{Subject}, \texttt{Object}, \texttt{Other}, \texttt{Background}$\}$ where \texttt{Other} refers to a bounding box which is not the query \texttt{Subject} or \texttt{Object} and \texttt{Background} refers to a false entity proposal. Denote the logits generated by this classifier for the $i^{th}$ box by $\boldsymbol{r}_i\in\mathbb{R}^4$. The output set $\mathcal{B}_s$ (or $\mathcal{B}_o$) is simply the set of bounding boxes classified as \texttt{Subject} (or \texttt{Object}). See supplemental materials for further implementation details.
\textbf{Bounding Box Refinement} -
The Differentiable Scene-Graph can also be used for further refinement of the bounding-boxes generated by the RPN network. The idea is that additional knowledge about image context may help to improve the coordinates of a given object. This is done
via a network $F_{BR}(\boldsymbol{b}_i, \boldsymbol{z}'_i)$ that takes as input the RPN box coordinates and a differentiable representation $\boldsymbol{z}'_i$ for box $i$, and outputs new bounding box coordinates. See Fig. \ref{fig:boxes} for an illustration of box refinement and see supplemental materials for further implementation details.
\ignore{
\begin{figure}[t!]
\begin{center}
\includegraphics[width=\linewidth]{Figures/perm_inv.png}
\caption{A schematic representation of the GPI Module in \hyperref[graph_permutation_form] {Theorem \ref{graph_permutation_form}}. \textbf{(a)} First, the features $\zz_{i,j}$ are processed element-wise by $\phi$. \textbf{(b)} Features are summed to create a vector $\boldsymbol{s}_i$, which is concatenated with $\zz_i$. \textbf{(c)} A representation of the entire graph is created by applying $\boldsymbol{\alpha}$ $n$ times and summing the created vector. \textbf{(d)} The graph representation is then finally processed by $\rho$ together with $\zz_k$. }
\label{fig:f_gpi}
\end{center}
\end{figure}
}
\subsection{The Scene Graph Generator}
\label{subsec:sgp}
\gal{We should not use the terms SG generator. We are not generating SGs, we generate DSGs.}
We next describe the module that takes as input features $\boldsymbol{z}_i$ and $\boldsymbol{z}_{i,j}$ extracted by the RPN and outputs a set of vectors $\boldsymbol{z}'_i,\boldsymbol{z}'_{ij}$ corresponding to Differentiable Scene-Graph over entities and relationships in the image.
For this model, we use the Graph Permutation Invariant (GPI) architecture introduced in \cite{mapping_to_imgs}. A key property of this architecture is that it is invariant to permutations of the input that do not affect the labels.
The GPI transformation is defined as follows. First, the set of all input features is summarized via a permutation-invariant transformation into a single vector $\gg$:
\begin{equation}\label{sgp_eq}
\gg = \sum_{i=1}^n \boldsymbol{\alpha} (\boldsymbol{z}_i, \sum_{j \neq i} \boldsymbol{\phi}(\boldsymbol{z}_{i}, \boldsymbol{z}_{i,j}, \boldsymbol{z}_{j}))
\end{equation}
Here $\boldsymbol{\alpha}$ and $\boldsymbol{\phi}$ are fully connected networks. Then the new representations for entities and relations are computed via:
\begin{equation}
\boldsymbol{z}'_{k} = {\boldsymbol\rho}^{entity}(\boldsymbol{z}_{k},\gg) \ , \
\boldsymbol{z}'_{k,l} = {\boldsymbol\rho}^{relation}(\boldsymbol{z}_{k,l},\gg)
\end{equation}
where ${\boldsymbol{\rho}}$ above are fully connected networks.
See supplemental materials for further implementation details.
\section{Training}
\label{train}
We next explain how our model is trained for the RR task, and how we can also use the RR training data for supervising the DSG component. The training loss will be a sum of the following losses. Some of these correspond directly to the referring relationship task, and some to the auxiliary task of DSG labeling.
\gal{Roei, Moshiko, can we provide examples showing what happens to teh DSG and to the results if we "ablate" each of the losses? }
{\bf Referring Relationship Classifier Loss} -
The \textit{Referring Relationship Classifier} (\secref{sec:model}) outputs logits $\boldsymbol{r}_i$ for each box, corresponding to its prediction (\texttt{subject}, \texttt{object}, etc.). To train these logits, we need to extract their ground-truth values from the training data.
Recall that a given image in the training data may have multiple queries, and so may have multiple boxes that have been tagged as subject or object for the corresponding queries. To obtain the ground-truth for box $i$ and query $q=\langle s,r,o \rangle$ we take the following steps. First, we find the ground-truth box that has maximal overlap with box $i$. If this box is either a subject or object for the query $q$, we set $\boldsymbol{r}^{gt}_i$ to be
\texttt{Subject} or \texttt{Object} respectively. Otherwise, if the overlap with a ground-truth box for a different image-query is greater than $0.5$, we set $\boldsymbol{r}^{gt}_i=\texttt{Other}$ (since it means there is some other entity in the box), and we set $\boldsymbol{r}^{gt}_i=\texttt{Background}$ if the overlap is less than $0.3$. If the overlap is in $[0.3,0.5]$ we do not use the box for training. For instance, given a query \tripleq{woman}{feeding}{giraffe} with ground-truth boxes for \nl{woman} and \nl{giraffe},
consider the box in the RPN that is closest to the ground-truth box for ``woman''. Assume the index of this box is $7$. Similarly, assume that the box closest to the ground-truth for ``giraffe' has index $5$.
We would have $\boldsymbol{r}_7^{gt}={\texttt{Subject}}$, $\boldsymbol{r}_5^{gt}={\texttt{Object}}$ and the rest of the $\boldsymbol{r}_i^{gt}$ values would be either \texttt{Other} or \texttt{Background}. Given these ground truth values, the Referring Relationship Classifier Loss is simply the sum of cross entropies between the logits $\boldsymbol{r}_i$ and the one-hot vectors corresponding to $\boldsymbol{r}_i^{gt}$.
{\bf Scene-Graph Labeling Loss} -
As mentioned earlier, we can also use the ground-truth data as partial labels for training the DSG.
Consider again the query \tripleq{woman}{feeding}{giraffe}, where the index for the box in the RPN that is closest to the ground-truth box for ``woman'' is 7, and the index of the box closest to the ground-truth for ``giraffe' has index $5$.
The box corresponding to \nl{woman} has a DSG representation vector $\boldsymbol{z}'_7$. It makes sense to require that the label \nl{woman} should be inferred from the vector $\boldsymbol{z}'_7$. To enforce this, we train a classifier from $\boldsymbol{z}'$ to the set of entity labels, and add a loss to maximize its accuracy with respect to the ground truth. Similarly, recall that we have a representation $\boldsymbol{z}'_{i,j}$ for relations in the DSG. Thus we would want the label \nl{feeding} to be inferred from the vector $\boldsymbol{z}'_{7,5}$.
The above approach is implemented as follows. Denote by $E$ the number of entity values (e.g., \nl{horse}, \nl{cat}, \nl{cow} etc.) and $R$ the number of relation values (e.g., \nl{holding}, \nl{kicking} etc.). We construct two classifiers. The first is a linear classifier that takes as inputs a vector $\boldsymbol{z}'_i$ and outputs logits $\vv_i \in \mathbb{R}^E$. The second is a linear classifier that takes as input a vector $\boldsymbol{z}'_{i,j}$ output logits $\vv_{i,j} \in \mathbb{R}^R$. For each query, we have three ground-truths for the above logits. Thus, our DSG Labeling Loss is a sum of cross-entropy loss for the three ground-truth labels (one for the subject, one for the object and one for the relation).
\figref{fig:detailed_example} shows an example of a Scene-Graph produced by our model, trained using this loss. It can be seen that the predicted graph is indeed largely correct despite the fact that it was trained from partial supervision only. \roeih{We emphasize that our goal is not to predict SGs. Rather, we use DSGs as an intermediate representation, and add an auxiliary loss to decode nodes included in the RR labels. However, it is pleasing that the DSG intermediate representation can often be decoded into high quality SGs. This is shown in \figref{fig:detailed_example}. We further analyzed SG accuracy by comparing to ground-truth SG on visual genome (complete SGs were not used for training, only for analysis in this rebuttal). SGs decoded from DSGs achieve accuracy of $76\%$ for object labels and $70\%$ for relations (calculated for proposals with IOU $\ge 0.8$).}
{\bf Object Detector Loss} -
The output of the RPN is a set of bounding boxes. The ground-truth contains boxes that are known to contain objects. The goal of this loss is to encourage the RPN to include these boxes as proposals. Concretely, we use a sum of two losses: First, an RPN classification loss, which is a cross entropy over RPN anchors where proposals of 0.8 overlap or higher with the ground truth boxes were considered as positive. Second, an RPN box regression loss which is a smooth L1 loss between the ground-truth boxes and proposal boxes.
{\bf Box Refinement Loss} -
Recall that we have a \textit{Bounding Box Refinement} component. As with the Object Detector Loss above, we add a smooth L1 loss between the refined boxes and the ground truth ones. Thus, we have two loss terms for the bounding boxes generated in the network: the first one for the boxes generated by the RPN, and the second one for the boxes generated by the refinement. Our ablation study shows that using the refined boxes does improve performance.
\ignore{
\item{\bf DSG Labeling Loss}
Recall from \secref{sec:lsgl} that we use the DSG to predict logits $\vv_i$ for entities, and logits $\vv_{ij}$ for relations. For each query with ground-truth, we have three ground-truths for these logits.
Thus, we construct a loss that is a cross entropy between the predicted logits and these three ground-truths. See ilustration in \figref{fig:lsgl_loss_pic}
}
\section{Experiments}
\label{sec:experiments}
In the following sections we provide details about the datasets, training, baselines models, evaluation metrics, model ablations and results. Due to space consideration, the implementation details of the model are provided in a supplement.
\subsection{Datasets}
\label{datasets}
We evaluate the model in the task of referring relationships across three datasets, each exhibiting a unique set of characteristics and challenges.
\newline
\textbf{CLEVR \cite{clevr}:} A synthetic dataset generated from scene-graphs with four spatial relations: ``left'', ``right'', ``front'' and ``behind'', and 48 entity categories. It has over 5M relationships where 33\% are ambiguous entities (namely cases where there are multiple entities of the same type in an image).
\newline
\textbf{VRD:} \cite{lang_prior} The Visual Relationship Detection dataset contains 5,000 images with 100 entity categories and 70 relation categories. In total, VRD contains 37,993 relationship annotations with 6,672 unique relationship types and 24.25 relations per entity category. 60.3\% of these relationships refer to ambiguous entities.
\newline
\textbf{Visual Genome:} \cite{krishnavisualgenome} VG is the largest public corpus for visual relationships in real images, with 108,077 images annotated with bounding boxes, entities and relations. On average, images have 12 entities and 7 relations per image. In total, there are over 2.3M relationships where 61\% of those refer to ambiguous entities.
For a proper comparison with previous results \cite{krishna2018referring}, we used the data from \cite{krishna2018referring} including the same entity and relation categories, query relationships and data splits.
\subsection{Training Details}
Our model was trained using SGD with momentum 0.9 and learning rate 0.01 decaying by a factor of 0.5 every two epochs over Visual Genome and CLEVR and every ten epochs over VRD. The model was trained for 20 epochs over Visual Genome and CLEVR and for 50 epochs over VRD. The detector weights were initialized with an ImageNet pre-trained detector. Each batch includes all the query relationships in a single image. The total loss was set to be a weighted sum of the losses of the following components (see \secref{train}): Object Detector, Referring Relationship Classifier, Box Refinement and Differentiable Scene-Graph Labeling. The weights were set to 1, 0.2, 1 and 0.01 respectively. In order to be comparable to previous works described in \secref{baselines}, only relationships annotated for the referring relationship task were considered in the losses.
\subsection{Evaluation Metrics}
We compare our model to previous work using the average IOU for subjects and for objects. To compute the average subject IOU, we first generate two $L \times L$ binary attention maps: one that includes all the ground truth boxes labeled as \texttt{Subject} (recall that few entities might be labeled as \texttt{Subject}) and the other includes all the box proposals predicted as \texttt{Subject}. If no box is predicted as \texttt{Subject}, the box with the highest score for the label \texttt{Subject} is included in the predicted attention map. We then compute the Intersection-Over-Union between the binary attention maps. For a proper comparison with previous work \cite{krishna2018referring}, we use $L=14$. The object boxes are evaluated in the exact same manner.
\ignore{
For further evaluation of the referring relationship classifier, we also computed precision, recall and F1 score of the bounding box proposals for all model variants. Proposals with 0.3 overlap or lower with ground-truth objects were labeled as background. Proposals with overlap of 0.5 or higher with the ground-truth boxes were labeled according to their query relationship as either \textit{subject}, \textit{object} or \textit{other}.
}
\subsection{Model Ablations}\label{ablations}
We explored the power of our model through model ablations, comparing the following models:\\
(1) \textsc{DSG}: The Differentiable Scene-Graph model described in \secref{subsec:model_components} and trained as described in \secref{train}\\
(2) \textsc{DSG -DSGL}: This model is a variant of DSG but without the Differentiable Scene-Graph Labeling Loss (see \secref{train}). This ablation allows us to study the effect of training the Differentiable Scene-Graph component using partial labels.\\
(3) \textsc{DSG -BR}: This model is a variant of DSG but replacing the \textit{Box Refinement} component with fine tuning the coordinates of the box proposal using the visual features $\boldsymbol{f}^{}_i$. This variant allows us to quantify the benefit of refining the box proposals based on the differentiable representation of the scene.\\
\ignore{
(3) \textsc{Scene-Graph-Model} - Exploring the option of predicting referring relationship class from scene-graph instead from differentiable scene-graph.
In this model we predicting referring relationship class in two steps. First, predicting scene-graph by a Scene-Graph Labeling that gets the entity features $\boldsymbol{f}^{}_i$ and relation features $\boldsymbol{f}^{}_{i,j}$ and outputs entity class predictions scores $N_{Ent-Pred}$ and relation class prediction scores $N_{Rel-Pred}$. Then the Differentiable Scene-Graph Generator gets as an input $N_{Ent-Pred}$ and $N_{Rel-Pred}$ instead of the entity features $\boldsymbol{f}^{}_i$ and relation features $\boldsymbol{f}^{}_{i,j}$. No change in Referring Relationship Classifier and Box Refinement.\\
}
(4) \textsc{DSG -SG}: A baseline model that does not use the DSG representations at all. Instead, the model includes only Object Detector and a referring relationship classifier. The referring relationship classifier uses the $\boldsymbol{f}_i$ features extracted by the Object Detector instead of the $\boldsymbol{z}'_i$ features. This model allows us to quantify the benefit of the differentiable scene representation for referring relationship classification
\ignore{
Comparing the \textsc{Scene-Graph-Model} / \textsc{No-Scene-Graph-Labeling-Model} to \textsc{Baseline-Model} allows to evaluate the benefit of scene representation and explore whether referring relationship classification of each object benefit from the semantic-level interpretation or visual features of the scene.
Comparing \textsc{No-Scene-Graph-Labeling-Model} to \textsc{Scene-Graph-Model} allows to evaluate fixed high level semantic scene representation against differentiable one. This comparison is possible for tasks such as referring relationship classification where all the required information covered in a semantic-level scene-graph.
However, in the more probable scenario, the semantic scene-graph representation will be incomplete giving significant advantage to differentiable representation.
Comparing \textsc{Main-Model} to \textsc{No-Scene-Graph-Labeling-Model} allows to evaluate the benefit of training the scene representation for multiple tasks.
Comparing \textsc{Main-Model} to \textsc{Scene-Graph-Model} allows to evaluate the power of flexible scene representation including both high level semantic features and visual features.
}
\roeih{Our success criteria is \textbf{both} classification and regression (as in object recognition tasks), therefore, we can only ablate the following: (1) w/o DSG, (2) w/o box refinement, and (3) w/o the SG Labeling Loss. From those ablations we can see the learned DSG latent representations are important for: the success of this task (1) and better tight boxes (2). Moreover, it can be seen the labeling of scene graph loss plays an important role. We cannot ablate the RR classifier, because that is our target task.}
\subsection{Baselines}
\label{baselines}
The Referring Relationship task was introduced recently \cite{krishna2018referring}, and the SSAS model was proposed as a possible approach (see below). We report the results for the baseline models in \cite{krishna2018referring}.
When evaluating our Differentiable Scene-Graph model, we use exactly the evaluation setting as in \cite{krishna2018referring} (i.e., same data splits, entity and relation categories). The baselines reported are:
\begin{enumerate}
\item \textsc{Symmetric Stacked Attention Shifting (SSAS):} \cite{krishna2018referring} An iterative model that localizes the relationship entities using attention shift component learned for each relation.
\item \textsc{Spatial Shifts \cite{shift}}: Same as SSAS, but with no iterations and by replacing the shift attention mechanism with statistically learned shift per relation that ignores the semantic meaning of entities.
\item \textsc{Co-Occurrence \cite{cooccur2008}}: Uses an embedding of the subject and object pair for attending over the image features.
\item \textsc{Visual Relationship Detection (VRD) \cite{lang_prior}:} Similar to Co-Occurrences model, but with an additional relationship embedding.
\end{enumerate}
\begin{table}
\small
\setlength\tabcolsep{4.5 pt}
\begin{tabular}{lcccccc}
\multicolumn{1}{c}{} & \multicolumn{6}{c}{Average IOU}\\
\multicolumn{1}{c}{} & \multicolumn{2}{c}{Visual Genome} & \multicolumn{2}{c}{VRD} & \multicolumn{2}{c}{CLEVR} \\
& subject & object & subject & object & subject & object \\
\midrule
\scriptsize{\textsc{SS} \cite{shift}} & 0.399 & 0.469 & 0.320 & 0.371 & 0.740 & 0.740 \\
\scriptsize{\textsc{CO}} \cite{cooccur2008} & 0.414 & 0.490 & 0.347 & 0.389 & 0.691 & 0.691 \\
\scriptsize{\textsc{VRD} \cite{lang_prior}} & 0.417 & 0.480 & 0.345 & 0.387 & 0.734 & 0.732 \\
\scriptsize{\textsc{SASS} \cite{krishna2018referring}} & 0.421 & 0.482 & 0.369 & 0.410 & 0.778 & 0.778\\
\hline
\scriptsize{\textsc{DSG -SG}} & 0.412 & 0.47 & 0.333 & 0.366 & 0.937 & 0.937 \\
\scriptsize{\textsc{DSG}} & \textbf{0.489} & \textbf{0.539} & \textbf{0.4} & \textbf{0.435} & \textbf{0.963} & \textbf{0.963} \\
\bottomrule
\hline
\end{tabular}
\caption{Test-set mean IOU in the referring relationship task for the baselines in \secref{baselines} and the Differentiable Scene Graph (DSG) model. Additionally, results are reported on a \textsc{DSG -SG} model (see \secref{ablations}) which classifies the referring relationship directly from the RPN output.
}
\hfill
\vspace{-1em}
\end{table}
\begin{table}
\vspace{-1em}
\begin{center}
\begin{tabular}{lcc}
\multicolumn{1}{l}{} & \multicolumn{2}{c}{Average IOU} \\
& subject & object \\
\midrule
\scriptsize{\textsc{DSG -SG}} & $0.405 \pm 0.0013$ & $0.461 \pm 0.00013$ \\
\scriptsize{\textsc{DSG -DSGL}} & $0.455 \pm 0.0014$ & $0.511 \pm 0.0013$ \\
\scriptsize{\textsc{DSG -BR}} & $0.469 \pm 0.0014$ & $0.519 \pm 0.0014$ \\
\scriptsize{\textsc{DSG}} & $\textbf{0.477} \pm 0.0014$ & $\textbf{0.528} \pm 0.0014$\\
\bottomrule
\hline
\end{tabular}
\caption{{\bf Model ablations}: Results, including standard error for DSG variants on the validation set of Visual Genome dataset. The different models are described in \secref{ablations}.}
\label{results}
\end{center}
\vspace{-1em}
\end{table}
\subsection{Results}
Table 1 provides average IOU for \texttt{Subject} and \texttt{Object} over the three datasets described in \ref{datasets}. We compare our model to four baselines described in \ref{baselines}. Our Differentiable Scene-Graph approach
outperforms all baselines in terms of the average IOU.
\begin{figure}
\begin{center}
\includegraphics[width=0.7\linewidth]{Figures/clevr-example-iccv.jpg}
\caption{Typical example in CLEVR \cite{clevr} dataset. \gal{Explain what makes this typical? } \roeih{I think we can remove this figure. I don't see what new info. it brings to table.} \mr{It was used to show that CLEVR is very simple which explains our very high results}}
\label{clevr_example}
\end{center}
\vspace{-1em}
\end{figure}
Our results for the CLEVR dataset are significantly better than those in \cite{krishna2018referring}. Because CLEVR objects have a small set of distinct colors (Fig \ref{clevr_example}), object detection in CLEVR is much easier than in natural images, making it easier to achieve high IOU. The baseline model without the DSG layer (DSG-SG) is an end-to-end model with a two-stage detector in contrast to \gal{improve this sentence} \cite{krishna2018referring} and already improves strongly over prior work with 93.7\%, and our novel DSG approach further improves to 96.3\% (reducing error by 50\%).
Table 2 provides results of ablation experiments for the Visual Genome dataset \cite{krishnavisualgenome}.\footnote{The numbers are different from Table 1 because Table 2 uses the validation set.}\gal{avoid footnotes} All model variants based on scene representation perform better than the model that does not use scene-graphs (i.e., \textsc{DSG -SG}) in terms of average IOU over subject and object, demonstrating the power of contextualized scene representation.
\ignore{
As expected, the precision for \textit{Other-entity} and the recall for \textit{background} are relatively low since only the entities involved in the query relationship are annotated, and the remaining entities could not be compared to baseline work \gal{why?}. Also, the precision for \textit{subject} and \textit{object} are relatively low since not all the entities involved in the query relationship are annotated\gal{again, its worth explaining more}.
}
\noindent \textbf{Analysis of failure cases.}\roeih{Its now in the figure and text both.} We analyzed the types of common mistakes and their distribution, and provide examples in Figure~\ref{fig:failure}. Since DSGs depend on box proposals, our 2-step approach is sensitive to the quality of the object detectors in the first step. Manual inspection of \gal{Can we increase this to 50 or 100?} 30 images revealed the following main error types: \textbf{(1)} 30\%: Detector failed and the relevant box is not in the box proposal list. \textbf{(2)} 23.3\% \textit{Subject} or \textit{Object} detected but classified as \textit{Other} or as \textit{Background}. \textbf{(3)} 16.6\%: Relation misclassified. \textbf{(4)} 16.6\%: \emph{Multiplicity}. Either too few or too many of the GT boxes are classified as \textit{Subject} or \textit{Object}. \textbf{(5)} 13.3\%: Other, including incorrect GT, and hard-to-decide cases.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=\linewidth]{Figures/error_analaysis_wacv.jpg}
\vspace{-3pt}
\caption{Common failure cases for each error type. \textbf{a.} \textit{Failed detection}(30\%), the detector missed the glasses on the table. \textbf{b,c.} \textit{Misclassified object}(23\%), the cake is detected but classified as a background. \textbf{d.} \textit{misclassified relation}(16.6\%). \gal{EXPLAIN}. \textbf{e,f.} \textit{Multiplicity}(16.6\%), Either too few or too many GT boxes are classified as \textit{Subject} or \textit{Object}.}
\label{fig:failure}
\end{center}
\end{figure}
The full \textsc{DSG} model outperforms all model ablations, illustrating the improvements achieved by using partial supervision for training the differentiable scene-graph. Finally, \figref{fig:qualitative_results_neg} shows some success and failure cases for our DSG model, and \figref{fig:boxes} shows that the box refinement step is indeed effective.
\begin{figure*}
\begin{center}
\includegraphics[width=0.9\linewidth]{Figures/refinening_figure_wacv.jpg}
\vspace{-2em}
\caption{The effect of box refinement and RR-classification on refining bounding boxes. (a) The DSG network is applied to an input image. (b) The \textit{object detector component} generates box proposals for entities in the image. (c) The \textit{referring relationship classifier} uses information from DSG to label candidate boxes as \texttt{object} or \texttt{subject} entities. Then, the \textit{box refinement component} also uses DSG information, this time to improve box locations for those boxes labeled as entities by RR. Here, boxes are tuned to focus on the most relevant entities in the image: the two men, the surfboard, the sky and the ocean. (d) Once the RR classifier labeled entity boxes, it can correctly refer to the entities in the query \tripleq{cloud}{in}{sky} (sky in green, clouds in violet). (e) Examples of candidate boxes classified by RR as \texttt{background} (non-entity), allowing to skip them when answering queries.
}
\label{fig:boxes}
\end{center}
\vspace{-1em}
\end{figure*}
\section{Related Work}
\label{sec:related_work}
\textbf{Graph Neural Networks:} Recently, major progress has been made in constructing graph neural networks (GNN). These refer to a class of neural networks that operate directly on graph-structured data by passing local messages \cite{gilmer2017neural, Li2015GatedGS}. Variants of GNNs have been shown to be highly effective at relational reasoning tasks \cite{nn_for_relational}, classification of graphs \cite{bruna2013invariant, dai_disc_embs, niepert16, DefferrardBV16}, and classification of nodes in large graphs \cite{graph_conv, inductive_repr_large_grphs}. The expressive power of GNNs has also been studied in \cite{mapping_to_imgs, deep_sets}. GNNs have also been applied to visual understanding in \cite{mapping_to_imgs, graph_rcnn, Wang_videogcnECCV2018, herzig2019STAG} and control \cite{Gonzalez18, relational_rl_deepmind18}. Similar aggregation schemes have also been applied to object detection \cite{hu2018relation}.
\textbf{Visual Relationships:} Earlier work aimed to leverage visual relationships for improving detection \cite{VisualPhrases}, action recognition and pose estimation \cite{DesaiR12}, semantic image segmentation \cite{GuptaD08} or detection of human-object interactions \cite{yang2017support, plummerPLCLC2017}. Lu {\em et al.} \cite{lang_prior} were the first to formulate detection of visual relationships as a separate task. They learn a likelihood function that uses a language prior based on word embeddings for scoring visual relationships and constructing scene graphs.
\textbf{Scene Graphs:}
Scene graphs provide a compact representation of the semantics of an image, and have been shown to be useful for semantic-level interpretation and reasoning about a visual scene \cite{johnson2018image}. Extracting scene graphs from images provides a semantic representation that can later be used for reasoning, question answering \cite{yi2018neural}, and image retrieval \cite{img_retriev_using_sg, entangled_scene}.
Another work on scene graph predictions used neural message passing algorithms \cite{sg_generation_msg_pass}, while \cite{pixels_to_graph} suggested to predict graphs directly from pixels in an end-to-end manner. \text{NeuralMotif} \cite{neural_motifs} employs an RNN that provides global context by reading sequentially the independent predictions for each entity and relation and then refines those predictions.
\textbf{Referring Relationships:}
Several recent studies looked into the task of detecting an entity based on a referring expression \cite{kazemzadeh2014referitgame,krahmer2012computational}, while taking context into account. \cite{mao2016generation} described a model that has two parts: one for generating expressions that point to an entity in a discriminative fashion and a second for understanding these expressions and detecting the referred entity. \cite{yu2016modeling} explored the role of context and visual comparison with other entities in referring expressions.
Modelling context was also the focus of \cite{nagaraja2016modeling}, using a multi-instance-learning objective.
Recently, \cite{krishna2018referring} introduced an explicit iterative model that localizes the two entities in the referring relationship task, conditioned on one another using attention from one entity to another. However, in contrast to this work, we show an implicit model that uses latent scene context, resulting in new state of the art results on three vision datasets that contain visual relationships.
\section{Conclusion}
\label{sec:conclusion}
Our motivation for this work is the assumption that accurate image-understanding requires a detailed representation of many entities in an image and their relations. A scene-graph is a natural structure for representing this information, but it is hard to train these in a fully supervised manner. Thus, here we advocate using a scene-graph representation that is continuous and that can also be trained from partial supervision. Our results, both qualitative (e.g., \figref{fig:detailed_example}) and quantitative suggest that the differentiable scene-graph does encode scene structure, and that this can be used for down-stream tasks such as referring relationships.
One natural next step is to study such representations in other downstream tasks that require integrating information across the image. Some examples are caption generation and visual question answering. In the latter in particular a scene-graph could be very useful, as many questions of interest are easily answerable by scene graphs (e.g., counting questions and questions about relations).
Another important extension is to scene-graphs that also model higher order interactions (i.e., hyper-graphs).
Finally, it will be interesting to explore other approaches to training the scene-graph component, and in particular finding ways of using unlabeled data for this task.
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\noindent
We describe a system for zero-g frobnication. This
system is new because it handles the following cases:
A, B. Previous systems [Zeus et al. 1968] didn't
handle case B properly. Ours handles it by including
a foo term in the bar integral.
...
The proposed system was integrated with the Apollo
lunar lander, and went all the way to the moon, don't
you know. It displayed the following behaviours
which show how well we solved cases A and B: ...
\end{quotation}
As you can see, the above text follows standard scientific convention,
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written by Zeus \etal, but cannot make any decision based on that guess.
He or she would have to be sure that no other authors could have been
contracted to solve problem B.
FAQ: Are acknowledgements OK? No. Leave them for the final copy.
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\label{fig:long}
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\subsection{Miscellaneous}
\noindent
Compare the following:\\
\begin{tabular}{ll}
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\end{tabular}\\
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Frobnication has been trendy lately.
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\begin{table}
\begin{center}
\begin{tabular}{|l|c|}
\hline
Method & Frobnability \\
\hline\hline
Theirs & Frumpy \\
Yours & Frobbly \\
Ours & Makes one's heart Frob\\
\hline
\end{tabular}
\end{center}
\caption{Results. Ours is better.}
\end{table}
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{\small
\bibliographystyle{ieee}
\subsection{The Differentiable Scene Graph Generator}
\label{subsec:sgp}
\gal{We should not use the terms SG generator. We are not generating SGs, we generate DSGs.}
We next describe the module that takes as input features $\boldsymbol{z}_i$ and $\boldsymbol{z}_{i,j}$ extracted by the RPN and outputs a set of vectors $\boldsymbol{z}'_i,\boldsymbol{z}'_{ij}$ corresponding to Differentiable Scene-Graph over entities and relationships in the image. For this model, we use the Graph Permutation Invariant (GPI) architecture introduced in \cite{mapping_to_imgs}. A key property of this architecture is that it is invariant to permutations of the input that do not affect the labels.
The GPI transformation is defined as follows. First, the set of all input features is summarized via a permutation-invariant transformation into a single vector $\gg$:
\begin{equation}\label{sgp_eq}
\gg = \sum_{i=1}^n \boldsymbol{\alpha} (\boldsymbol{z}_i, \sum_{j \neq i} \boldsymbol{\phi}(\boldsymbol{z}_{i}, \boldsymbol{z}_{i,j}, \boldsymbol{z}_{j}))
\end{equation}
Here $\boldsymbol{\alpha}$ and $\boldsymbol{\phi}$ are fully connected networks. Then the new representations for entities and relations are computed via:
\begin{equation}
\boldsymbol{z}'_{k} = {\boldsymbol\rho}^{entity}(\boldsymbol{z}_{k},\gg) \ , \
\boldsymbol{z}'_{k,l} = {\boldsymbol\rho}^{relation}(\boldsymbol{z}_{k,l},\gg)
\end{equation}
where ${\boldsymbol{\rho}}$ above are fully connected networks.
See supplemental materials for further implementation details.
\begin{table*}
\label{ablations_results}
\begin{tabular}{p{2cm}cccccccccccccc}
\multicolumn{1}{|p|}{} & \multicolumn{2}{c|}{Average IOU} & \multicolumn{4}{c|}{Recall} & \multicolumn{4}{c|}{Precision} & \multicolumn{4}{c|}{F1} \\
& S & O & S & O & E & BG & S & O & E & BG & S & O & E & BG\\
\midrule
\scriptsize{\textsc{DSG -SG}} & 0.405 & 0.461 & 0.552 & 0.654 & 0.778 & 0.444 & 0.501 & 0.498 & 0.354 & 0.921 & 0.525 & 0.565 & 0.487 & 0.599 \\
\ignore{
\textsc{SG} & 0.441 & 0.493 & \textbf{0.665} & \textbf{0.703} & \textbf{0.814} & 0.273 & 0.487 & 0.485 & 0.304 & 0.937 & 0.562 & 0.574 & 0.443 & 0.422 \\
}
\scriptsize{\textsc{DSG -DSGL}} & 0.455 & 0.511 & 0.621 & 0.67 & 0.77 & \textbf{0.522} & 0.527 & 0.528 & \textbf{0.386} & 0.915 & 0.57 & 0.591 & \textbf{0.514} & \textbf{0.665} \\
\scriptsize{\textsc{DSG -BR}} & 0.469 & 0.519 & \textbf{0.658} & \textbf{0.711} & 0.789 & 0.495 & 0.533 & 0.534 & 0.365 & \textbf{0.939} & \textbf{0.589} & \textbf{0.61} & 0.499 & 0.648 \\
\scriptsize{\textsc{DSG}} & \textbf{0.477} & \textbf{0.528} & 0.628 & 0.686 & \textbf{0.801} & 0.494 & \textbf{0.539} & \textbf{0.543} & 0.366 & 0.938 & 0.58 & 0.606 & 0.503 & 0.647 \\
\bottomrule
\hline
\end{tabular}
\caption{Model ablations \ref{ablations} results over Visual Genome validation set. The table includes: 1) average IoU for subject (S) and object (O) 2) recall for subject (S), object (O), other entity (E) and background (BG) 3) precision for subject (S), object (O), other entity (E) and background (BG) 4) F1 score for subject (S), object (O), other entity (E) and background (BG)}
\label{results}
\end{table*}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,888 |
{"url":"https:\/\/math.stackexchange.com\/questions\/340978\/change-of-basis-matrix-to-convert-standard-basis-to-another-basis\/340991","text":"# Change of basis matrix to convert standard basis to another basis\n\nConsider the basis $$B=\\left\\{\\begin{pmatrix} -1 \\\\ 1 \\\\0 \\end{pmatrix}\\begin{pmatrix} -1 \\\\ 0 \\\\1 \\end{pmatrix}\\begin{pmatrix} 1 \\\\ 1 \\\\1 \\end{pmatrix} \\right\\}$$ for $$\\mathbb{R}^3$$.\n\nA) Find the change of basis matrix for converting from the standard basis to the basis B.\n\nI have never done anything like this and the only examples I can find online basically tell me how to do the change of basis for \"change-of-coordinates matrix from B to C\".\n\nB) Write the vector $$\\begin{pmatrix} 1 \\\\ 0 \\\\0 \\end{pmatrix}$$ in B-coordinates.\n\nObviously I can't do this if I can't complete part A.\n\nCan someone either give me a hint, or preferably guide me towards an example of this type of problem?\n\nThe absolute only thing I can think to do is take an augmented matrix $$[B E]$$ (note - E in this case is the standard basis, because I don't know the correct notation) and row reduce until B is now the standard matrix. This is basically finding the inverse, so I doubt this is correct.\n\n\u2022 In which basis is your B defined? I see that it is a collection of 3 vectors $B = [\\vec B_1 \\vec B_2 \\vec B_3]$ whereas each of the $B_n$ is a vector of coordinates. Coordinates must be specified wrt to some another basis (or with B itself?). What is that basis?\n\u2013\u00a0Val\nJun 3, 2014 at 11:59\n\nDenote $E$ the canonical basis of $\\mathbb{R}^3$.\n\nA) These three column vectors define a $3\\times 3$ matrix $$P=\\left(\\matrix{-1&-1&1\\\\1&0&1\\\\0&1&1}\\right)$$ which is the matrix of the linear map $$Id:(\\mathbb{R}^3,B)\\longrightarrow (\\mathbb{R}^3,E).$$ This means in particular that whenever you right multiply it by a column vector $(x_1,x_2,x_3)$ where $x_j$ are the coordinates of a vector $x=x_1B_1+x_2B_2+x_3B_3$ with the respect to the basis $B$, you obtain the coordinates of $x$ in the canonical basis $E$.\n\nWhat you want is the matrix of $$Id:(\\mathbb{R}^3,E)\\longrightarrow (\\mathbb{R}^3,B).$$ That is $P^{-1}$, the inverse of the matrix above. This will transform, by right multiplication, the coordinates of a vector with respect to $E$ into its coordinates with respect to $B$. That's the change of basis matrix you need.\n\nB) As explained above, you just have to right multiply the change of basis matrix $P^{-1}$ by this column vector.\n\n$$P^{-1}=\\left(\\matrix{-1\/3&2\/3&-1\/3\\\\-1\/3&-1\/3&2\/3\\\\1\/3&1\/3&1\/3} \\right)$$ $$\\left(\\matrix{-1\/3&2\/3&-1\/3\\\\-1\/3&-1\/3&2\/3\\\\1\/3&1\/3&1\/3} \\right)\\left(\\matrix{1\\\\0\\\\0}\\right)=\\left(\\matrix{-1\/3\\\\-1\/3\\\\1\/3}\\right).$$\n\n\u2022 What is Id? Is it identity matrix? If so, how can it convert anything?\n\u2013\u00a0Val\nJun 3, 2014 at 11:54\n\u2022 It's the name of the function I assume Jun 4, 2014 at 13:07\n\u2022 Id is an identity mapping, since in general there are three steps: (1) represent the input vector in the input basis, (2) do a mapping, which in this case is the idneity mapping, ie do nothing, and then (3) represent the result in the output basis. Feb 5, 2017 at 17:58\n\nBy definition change of base matrix contains the coordinates of the new base in respect to old base as it's columns. So by definition $$B$$ is the change of base matrix. Key to solution is equation $$v = Bv'$$ where $$v$$ has coordinates in old basis and $$v'$$ has coordinates in the new basis (new basis is B-s cols) suppose we know that in old basis $$v$$ has coords $$(1,0,0)$$ (as a column) (which is by the way just an old base vector) and we want to know $$v'$$ (the old base vector coordinates in terms of new base) then from the above equation we get $$B^{-1}v = B^{-1}Bv' \\Rightarrow B^{-1}v = v'$$\n\nAs a side-node, sometimes we want to ask how does that change of base matrix B act if we look at it as linear transformation, that is given vector v in old base $$v=(v_1,...,v_n)$$, what is the vector $$Bv$$? In general it is a vector whith i-th coordinate bi1*v1+...+bin*vn (dot product of i-th row of $$B$$ with $$v$$). But in particular if we consider v to be an old base vector having coordinates (0...1...0) (coordinates in respect the old base) where 1 is in the j-th position, then we get $$Bv = (b_{1j},...,b_{nj})$$ which is the j-th column of B, which is the j-th base vector of the new base. Thus we may say that B viewed as linear transformation takes old base to new base.\n\n\u2022 As another side-note, if both old basis and new basis are orthonormal then finding inverse of B is simple: inverse(B) = transpose(B), but this is not applicable to the B in the original post, because it is not orthonormal. Mar 27, 2018 at 12:10\n\nJust to clarify 1015 answer for myself\n\nWe have\n\n$$B = [\\vec b_1 \\vec b_2 \\vec b_3] = E \\left[\\matrix{-1&-1&1\\\\1&0&1\\\\0&1&1}\\right] = E [B]_E = EP$$\n\nIt says that $$P = [B]_E$$ consists of columns of $$b_n$$, the basis vectors $$b_n$$ in basis standard $$E = [\\vec e_1, \\vec e_2, \\vec e_3]$$, so that\n\n$$\\vec b_1 = [\\vec e_1 \\vec e_2 \\vec e_3] \\left[\\matrix{-1\\\\1\\\\0}\\right].$$\n\nNow, we can represent any vector in basis E as well in basis B\n\n$$\\vec v = E [\\vec v]_E = B [\\vec v]_B = E P [\\vec v]_B$$\n\nor\n\n$$[\\vec v]_E = P [\\vec v]_B$$\n\nWe see that P translates vector B-coordinates into E-coordinates.\n\nIn problem A), we have P, coordinates $$[\\vec v]_E$$ of vector $$\\vec v$$ basis E, and wish to compute them into $$[\\vec v]_B$$. That is easy from the last equation,\n\n$$[\\vec v]_B = P^{-1}[\\vec v]_E.$$\n\nYou see, $$P^{-1}$$ does the conversion. I call it inverse of change of basis matrix. 1015 has already computed it for your convenience. I just wanted to explain why.\n\nFor the problem B), just plug $$[\\vec v]_E = \\left[\\matrix{1\\\\0\\\\0}\\right].$$ I assume the standard basis, though I want to know why. Similarly, I want to know why don't you specify the basis for the components of B.\n\nIt must be noted though that textbooks normally have $$\\vec v = E[\\vec v]_e = EPP^{-1}[\\vec v]_e = B [\\vec v]_b$$ so that basis is translated by right-multiplying with change of basis matrix $$P$$, $$B = EP,$$ and coordinates are translated contravariantly, $$[\\vec v]_b = P^{-1} [\\vec v]_e$$.\n\nFor some reason 1015 has chosen the inverse $$P^{-1}$$, used to translate the coordinates, to be the change of basis matrix.","date":"2022-08-19 20:43:04","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\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 36, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8709482550621033, \"perplexity\": 270.76653847207365}, \"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-2022-33\/segments\/1659882573760.75\/warc\/CC-MAIN-20220819191655-20220819221655-00179.warc.gz\"}"} | null | null |
\section{Introduction}
\label{IntroSec}
Three natural possible definitions of local injectivity of a homomorphism $f$ from an \emph{input} oriented graph $G$ to a \emph{target} oriented graph $H$ are: for every vertex $x \in V(G)$, the function $f$ is injective when restricted to:
\begin{enumerate}
\item the in-neighbourhood $N^-(x) $; or
\label{PropIO}
\item $N^-(x)$ and $N^+(x) $ separately; or
\label{PropIOS}
\item the union $N^-(x) \cup N^+(x)$.
\label{PropIOT}
\end{enumerate}
When $H$ is \emph{reflexive}, that is, has a loop at every vertex, the three definitions are different. When $H$ is \emph{irreflexive}, that is, has no loops, definitions \ref{PropIOS} and \ref{PropIOT} coincide. Each of these five situations leads naturally to a notion of locally-injective oriented $k$-colouring.
Locally-injective homomorphisms (as in possible definition \ref{PropIO}) and colourings of oriented graphs were first introduced as an example in monadic second order logic \cite{courcelle}. Consequently, by Courcelle's Theorem, these problems are all solvable in polynomial time when the input has bounded treewidth. The same holds for the other possible definitions above.
Possible definition \ref{PropIO} has been studied in previous papers for both irreflexive and reflexive targets \cite{mrs,mrs2,mrs3,mrs1,cobusthesis}. A fairly complete theory has been developed. When the target, $H$, is reflexive there is a dichotomy theorem characterizing the oriented graphs $H$ for which the problem of deciding the existence of a homomorphism to $H$ is Polynomial, and those for which it is NP-complete. When $H$ is irreflexive the complexity has been determined when $H$ has maximum in-degree $\Delta^- \geq 3$ or $\Delta^- \leq 1$; when $\Delta^-=2$ the situation is as rich as that for all digraph homomorphism problems, and hence all constraint satisfaction problems \cite{mrs1}.
Possible definitions \ref{PropIOS} and \ref{PropIOT} have been studied in \cite{russell, CCM}. Obstructions to
(subgraphs that prevent the existence of) homomorphisms to small tournaments are the focus of \cite{CCM}. Both definitions
are considered.
Possible definition \ref{PropIOT} is the main focus of \cite{russell}.
Locally-injective colourings of undirected graphs were first explicitly studied by Hahn, Kratochvil, Si\v{r}an and Sotteau \cite{gena}. Subsequent papers have considered chordal graphs \cite{hrs08}, planar graphs (see \cite{KS}) and other graph classes, as well as list versions \cite{fk06}. The complexity of locally-injective homomorphisms has been extensively studied by Fiala, Kratochvil, and others (e.g. see \cite{fkp08,fpt08}).
The purpose of this paper is to contribute to the theory of locally-injective homomorphisms and colourings under possible definitions \ref{PropIOS} and \ref{PropIOT} above. In each of the three cases that arise, the complexity of deciding the existence of a homomorphism to $H$ is determined for the four tournaments on at most three vertices. These results appear in Sections 3, 4, and 5. Later, in Section 6, these results are then used to determine the complexity of the associated locally-injective oriented colouring problems.
We conclude this section by noting that the complexity of deciding whether a given directed graph $G$ has a homomorphism to a tournament $H$ has been studied \cite{BHM}.
There is a dichotomy theorem: the problem is Polynomial when $H$ has at most one directed cycle, and NP-complete when $H$ has at least two directed cycles.
The results reported in this paper are first steps towards finding a similar theorem for locally-injective homomorphisms.
\section{Notation and terminology}
\label{Sec2}
An \emph{oriented graph} is a directed graph $G$ with the property that for any two different vertices $x$ and $y$, at most one of the arcs $xy, yx$ belongs to $E(G)$.
An oriented graph $G$ can be viewed as arising from a simple graph $H$ by assigning a direction, or \emph{orientation}, to each edge.
The graph $H$ is called the \emph{underlying graph} of $G$, and $G$ is referred to as \emph{an orientation} of $H$.
The \emph{converse} of an oriented graph $G$ is the oriented graph $G^c$ with the same vertex set as $G$, and arc set $\{yx: xy\in E(G)\}$.
An oriented graph is \emph{reflexive} if it has a loop at each vertex, and \emph{irreflexive} if it has no loops. The superscript ``$r$'', as in $C_3^r$, indicates that the oriented graph under consideration is reflexive.
Oriented graphs without this superscript, as in $G$, are irreflexive.
We use $P_n, C_n$, and $T_n$ to denote the directed path on $n$ vertices, the directed cycle on $n$ vertices, and the transitive tournament on $n$ vertices, respectively, $n \geq 1$. It will be assumed throughout that $C_3$ has vertex set $\{c_1, c_2, c_3\}$ and arc set $\{c_1c_2, c_2c_3, c_3c_1\}$, and that $T_n$ has vertex set $\{t_0, t_1, \ldots, $ $t_{n-1}\}$ and arc set $\{t_it_j: i < j\}$.
A \emph{homomorphism} of an oriented graph $G$ to an oriented graph $H$ is a function $f: V(G) \to V(H)$ such that $f(x)f(y) \in E(H)$ whenever $xy \in E(G)$.
When $H$ has a loop, any directed graph has a homomorphism to $H$: map all vertices of $G$ to a vertex of $H$ with a loop.
Thus, when loops are present, the existence of a homomorphism is a non-trivial question only in the presence of some side condition like selecting the image of each vertex from a list of possible images, or local injectivity.
The book \cite{hn_book} contains a wealth of information about homomorphism of graphs and digraphs.
We call a homomorphism $f$ of an oriented graph $G$ to an oriented graph $H$:
\begin{itemize}
\item \emph{ios-injective} if, for every vertex $x$ of $G$, the restriction of $f$ to $N^-(x)$ is injective, as is the restriction of $f$ to $N^+(x)$; and
\item \emph{iot-injective} if, for every vertex $x$ of $G$, the restriction of $f$ to $N^-(x) \cup N^+(x)$ is injective.
\end{itemize}
These two concepts are the same when $H$ is an irreflexive oriented graph, and different when $H$ is a reflexive oriented graph.
The designations ``ios'' and ``iot'' arise from the local injectivity being on \underline{{\bf i}}n-neighbourhoods and \underline{{\bf o}}ut-neighbourhoods \underline{{\bf s}}eparately, and on \underline{{\bf i}}n-neighbourhoods and \underline{{\bf o}}ut-neighbourhoods \underline{{\bf t}}ogether. In introducing the designations ``ios'' and ``iot'', the qualifier ``locally'' has been dropped as it is part of the definition.
It is easy to see that the composition of two ios-injective homomorphisms is an ios-injective homomorphism, and similarly for iot-injective homomorphisms.
The following structure and its converse will be particularly useful.
We define the \textit{hat} $H_3$ to be the oriented graph with vertex set $V(H_3) = \{v_0, v_1, v_2\}$
and edge set $E(H_3) = \{v_0v_1, v_2v_1\}$. The vertices $v_0$ and $v_2$ will be referred to as the \emph{ends} of $H_3$ or $H_3^c$. Whether or not $H$ is reflexive, in an ios-injective or iot-injective homomorphism of $H_3$ or $H_3^c$ to $H$, the vertices $v_0$ and $v_2$ must have different images.
\begin{figure}[htbp]
\begin{center}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=vertex] (0) at (-5, 1) {};
\node [style=vertex] (1) at (-4.5, 2) {};
\node [style=vertex] (2) at (-4, 1) {};
\node [style=vertex] (3) at (-2, 1) {};
\node [style=vertex] (4) at (-1.5, 2) {};
\node [style=vertex] (5) at (-1, 1) {};
\node [style=vertex] (6) at (-4.5, 2) {};
\node [style=box] (7) at (-4.5, 0.25) {$H_3$};
\node [style=box] (8) at (-1.5, 0.25) {$H_3^c$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=arc] (0) to (1);
\draw [style=arc] (2) to (1);
\draw [style=arc] (4) to (3);
\draw [style=arc] (4) to (5);
\end{pgfonlayer}
\end{tikzpicture}
\caption{The hat and its converse.}
\label{fig1}
\end{center}
\end{figure}
\section{Irreflexive targets}
In this section we show that, if $T$ is an irreflexive tournament on at most 3 vertices, then the problem of deciding whether a
given oriented graph has an ios-injective (and hence also iot-injective) homomorphism to $T$ is Polynomial.
A given oriented graph has an ios-injective homomorphism to $T_1$ if and only if it has no edges,
and has an ios-injective homomorphism to $T_2$ if and only if it is a disjoint union of copies of $T_1$ and $T_2$.
A given oriented graph, $G$, has an ios-injective homomorphism to $C_3$ if and only if it has maximum in-degree 1, maximum out-degree 1, and has a homomorphism to $C_3$. It follows that $G$ has an ios-injective homomorphism to $C_3$ if and only if it is a disjoint union of directed
paths, and directed cycles of length a multiple of 3. These conditions are easy to check in polynomial time. It remains to consider ios-injective homomorphisms to the transitive triple.
\begin{prop}
The problem of deciding whether a given oriented graph has an ios-injective homomorphism to $T_3$ is Polynomial.
\end{prop}
\noindent \textbf{Proof.}
Let $G$ be a given digraph.
If the underlying graph of $G$ has a vertex of degree 3 or more, then $G$ has no ios-injective homomorphism to $T_3$.
Hence assume that $G$ is an orientation of a graph with maximum degree at most 2.
Therefore the underlying graph of $G$ is a disjoint union of paths and cycles, and hence has treewidth at most 2.
Since ios-injective homomorphism is expressible in monadic second-order logic, the statement now follows from
Courcelle's Theorem.
\hfill $\square$
\section{ios-injective homomorphisms to small reflexive targets}
In this section, we determine the complexity of deciding whether there exists an ios-injective homomorphism from a given oriented graph $G$ to the fixed oriented graph $H$ when $H$ is one of the four reflexive tournaments on at most three vertices.
It is clear that an oriented graph $G$ has an ios-injective homomorphism to $T_1^r$ if and only if it has maximum in-degree at most one and maximum out-degree at most one, that is, if and only if neither $H_3$ nor $H_3^c$ is a subgraph of $G$. Consequently, the only oriented graphs which have an ios-injective homomorphism to $T_1^r$ are disjoint unions of directed paths and directed cycles.
\begin{prop}
The problem of deciding whether a given oriented graph has an ios-injective homomorphism to $T_2^r$ is Polynomial.
\label{Prop:iosT_2^r}
\end{prop}
\noindent \textbf{Proof.}
We describe a reduction to 2-SAT.
Associate the vertices $t_0$ and $t_1$ of $T_2^r$ with false and true, respectively.
Given an oriented graph $G$, the corresponding instance of 2-SAT has the set of variables $\{x_v: v \in V(G)\}$.
Since no oriented graph with a vertex of in-degree at least 3, or a vertex of out-degree at least 3, has an
ios-injective homomorphism to $T_2^r$, we can assume that $\Delta^+(G) \leq 2$ and $\Delta^-(G) \leq 2$.
The set of clauses is constructed as follows.
\begin{itemize}
\item[(i)] If $\mathrm{deg}^+(v) = 2$, then $\neg x_v$ is a clause.
\item[(ii)] If $\mathrm{deg}^-(v) = 2$, then $x_v$ is a clause.
\item[(iii)] If $vw \in E$, then $\neg x_v \vee x_w$ is a clause.
\item[(iv)] If $v$ and $w$ are the ends of a copy of $H_3$ or $H_3^c$, then $ x_v \vee x_w$ and $\neg x_v \vee \neg x_w$ are clauses.
\end{itemize}
All clauses in groups (i) and (ii) are satisfied if and only if the image of any vertex of out-degree 2 is $t_0$ and the image of any vertex of in-degree 2 is $t_1$.
All clauses in group (iii) are satisfied if and only if the mapping corresponding to the truth assignment preserves arcs.
And finally, all clauses in group (iv) are satisfied if and only if the ends of a copy of $H_3$ or $H_3^c$ are assigned different images.
It follows that there is an ios-injective homomorphism of $G$ to $T_2^r$ if and only all clauses are satisfied.
\hfill $\square$
We now show that the problem of deciding the existence of an ios-injective homomorphism to $C_3^r$ is NP-complete.
Some ``gadget'' oriented graphs which map to $C_3^r$ only in special ways will be used in the NP-completeness proof.
For an integer $d \geq 1$, the oriented graph $D_d$ is constructed from a directed cycle $v_1, v_2, \ldots, v_{6d}, v_1$ by adding the vertices $x_1, x_2, \ldots, x_{3d}$ and arcs $v_{2t}x_t, x_tv_{2t-1},$ $t = 1, 2, \ldots, 3d$.
\begin{lem}
In an ios-injective homomorphism of $D_d$ to $C_3^r$ the vertices $x_1, x_4,$ $\ldots,$ $ x_{3d-2}$ all have the same image.
\label{LemmaD_d}
\end{lem}
\noindent \textbf{Proof.}
Let $f$ be an ios-injective homomorphism of $D_d$ to $C_3^r$. Without loss of generality, suppose $f(v_1) = c_1$. Then $f(v_2)$ is either $c_1$ or $c_2$.
Suppose first that $f(v_2) = c_1$. Then, observing that an ios-injective homomorphism of an irreflexive directed 3-cycle to $C_3^r$ either assigns every vertex the same image, or assigns no two vertices the same image, it must be that $f(x_1) = c_1$. By injectivity $f(v_3) \not= f(x_1)$, so $f(v_3) = c_2$, the only other out-neighbour of $c_1$. It follows that $f(x_2) = c_2$. Similarly, $f(v_4) \neq c_3$, so that $f(v_4) = f(x_2) = c_2$. Continuing in this way, the vertices $v_1, v_2, \ldots, v_{6d}$ map to $c_1, c_1, c_2, c_2, c_3, c_3, c_1, c_1, \ldots, c_3, c_3$, respectively, and the vertices $x_1, x_2, \ldots, x_{3d}$ map to $c_1, c_2, c_3, c_1, \ldots, c_3$, respectively.
Now suppose that $f(v_2) = c_2$. By our observation regarding homomorphisms of irreflexive directed 3-cycles, it must be that $f(x_1) = c_3$. Arguing as in the previous paragraph, ios-injectivity implies $f(v_3) = c_2$, and $f(x_2) = c_1$, which in turn implies $f(v_4) = c_3$. Continuing in this way, the vertices $v_1, v_2, \ldots, v_{6d}$ map to $c_1, c_2, c_2, c_3, c_3, c_1, c_1, \ldots, c_3, c_3, c_1$, respectively, and the vertices $x_1, x_2, \ldots,$ $x_{3d}$ map to $c_3, c_1, c_2, c_3, c_1, \ldots$, $c_2$, respectively.
\hfill $\square$
\medskip
For $d \geq 2$, let $X_d$ be the oriented graph constructed from $D_d$ by adding $d$ new vertices $n_1, n_2, \ldots, n_d$ and the arcs belonging to $\{x_{3i-2}n_i, n_ix_{3i+1}: i = 1, 2, \ldots, d,\}$, where addition is modulo $3d$. The following is a consequence of Lemma \ref{LemmaD_d}.
\begin{cor}
In an ios-injective homomorphism of $X_d$ to $C_3^r$, the vertices of the directed cycle $x_1, n_1, x_4,$ $n_2, \ldots, n_d, x_1$ must all be assigned the same image. Futher, any partial mapping in which these vertices are all assigned the same image can be extended to an ios-injective homomorphism of $X_d$ to $C_3^r$.
\end{cor}
\begin{thm}
The problem of deciding if a given oriented graph $G$ has an ios-injective homomorphism to $C_3^r$ is NP-complete.
\label{C_3^rNPc}
\end{thm}
\noindent \textbf{Proof.}
The transformation is from 3-colouring of graphs with minimum degree at least 3. Suppose a graph $G$ is given. For each vertex $x \in V(G)$, regard the edges incident with $x$ as being in 1--1 correspondence with the integers $1, 2, \ldots, \mathrm{deg}_G(x)$ so that it is meaningful to talk about the $i^{\textup{th}}$ edge incident with $x$. Construct a digraph $G^\prime$ as follows. For each vertex $x \in V(G)$ there is a copy of $X_{\mathrm{deg}_G(x)}$. (Note that $\mathrm{deg}_G(x) \geq 3$.) Each edge of $G$ is replaced by an oriented path on three vertices. Suppose $wz \in E(G)$ is the $i^{\textup{th}}$ edge incident with $w$ and the $j^{\textup{th}}$ edge incident with $z$. Add a new vertex $u_{wz}$ and arcs from vertex $n_i$ of the copy of $X_{\mathrm{deg}_G(w)}$ corresponding to $w$, and from vertex $n_j$ of the copy of $X_{\mathrm{deg}_G(z)}$ corresponding to $z$, to $u_{wz}$. The transformation can be accomplished in polynomial time. We will show that $G$ is 3-colourable if and only if there is an ios-injective homomorphism of $G^\prime$ to $C_3^r$.
Suppose that $G$ is 3-colourable, and fix a 3-colouring using the colours $c_1, c_2, c_3$. If the colour of $x$ is $c_i$, then map vertices $x_1, n_1, x_4, n_2, \ldots, n_d, x_1$ of the copy of $X_{\mathrm{deg}_G(x)}$ corresponding to $x$ to $c_i$ and extend this to an ios-injective homomorphism to $C_3^r$. The ends of each oriented path that replaced an edge of $G$ are now assigned different images, and the mapping so far can be extended to the remaining vertex of each oriented path that replaced an edge of $G$.
Suppose $G^\prime$ has an ios-injective homomorphism to $C_3^r$. Then, in each copy of $X_{\mathrm{deg}_G(x)}$, all vertices of the directed cycle $x_1, n_1, x_4, n_2, \ldots, n_{\mathrm{deg}_G(w)}, x_1$ are assigned the same image. Assign this colour to $x$. By the construction of $G^\prime$ and ios-injectivity, adjacent vertices of $G$ are assigned different colours.
\hfill $\square$
\medskip
We conclude this section by showing that the problem of deciding whether a given oriented graph $G$ has an ios-injective homomorphism to $T_3^r$ is NP-complete. A useful technical lemma is established first.
\begin{figure}[htbp]
\begin{center}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=vertex] (0) at (0, 4.25) [label=above:$d$] {};
\node [style=vertex] (1) at (3.5, 3.5) [label=above left:$t_2$]{};
\node [style=vertex] (2) at (4.25, 0.5) [label=right:$c$]{};
\node [style=vertex] (3) at (3.5, -2.5) [label=above right:$t_0$]{};
\node [style=vertex] (4) at (0, -3.25) [label=below:$d$] {};
\node [style=vertex] (5) at (-3.5, -2.5) [label=above left:$t_2$] {};
\node [style=vertex] (6) at (-4.25, 0.5) [label=left:$c$]{};
\node [style=vertex] (7) at (-3.5, 3.5) [label=above right:$t_0$] {};
\node [style=vertex] (8) at (4.25, 4.25) {};
\node [style=blackvertex] (9) at (4.25, -3.25) [label=right:$v$] {};
\node [style=vertex] (10) at (-4.25, -3.25) {};
\node [style=blackvertex] (11) at (-4.25, 4.25) [label=left:$u$] {};
\node [style=vertex] (12) at (0, 3.25) [label=above right:$t_1$] {};
\node [style=vertex] (13) at (1.5, 1.25) [label=right:$t_0$]{};
\node [style=vertex] (14) at (2, 0.5) {};
\node [style=vertex] (15) at (1, 0.5) {};
\node [style=vertex] (16) at (1.5, -0.25) [label=right:$t_2$] {};
\node [style=vertex] (17) at (1.5, 2.75) [label=above right:$e$] {};
\node [style=vertex] (18) at (1.5, -1.75) [label=below right:$e$] {};
\node [style=vertex] (19) at (0, -2.25) [label=below left:$t_1$] {};
\node [style=vertex] (20) at (-1.5, -1.75) [label=below left:$f$]{};
\node [style=vertex] (21) at (-1.5, -0.25) [label=left:$t_2$] {};
\node [style=vertex] (22) at (-1, 0.5) {};
\node [style=vertex] (23) at (-2, 0.5) {};
\node [style=vertex] (24) at (-1.5, 1.25) [label=left:$t_0$] {};
\node [style=vertex] (25) at (-1.5, 2.75) [label=above left:$f$] {};
\node [style=vertex] (26) at (-3, 0.5) [label=below left:$t_1$] {};
\node [style=vertex] (27) at (-2.25, 2) [label=above:$a$] {};
\node [style=vertex] (28) at (-0.75, 2) [label=above:$t_2$] {};
\node [style=vertex] (29) at (0, 2.5) {};
\node [style=vertex] (30) at (0, 1.5) {};
\node [style=vertex] (31) at (0.75, 2) [label=above:$t_0$] {};
\node [style=vertex] (32) at (2.25, 2) [label=above:$a$] {};
\node [style=vertex] (33) at (3, 0.5) [label=below right:$t_1$] {};
\node [style=vertex] (34) at (2.25, -1) [label=below:$b$] {};
\node [style=vertex] (35) at (0.75, -1) [label=below:$t_0$] {};
\node [style=vertex] (36) at (0, -0.5) {};
\node [style=vertex] (37) at (0, -1.5) {};
\node [style=vertex] (38) at (-0.75, -1) [label=below:$t_2$] {};
\node [style=vertex] (39) at (-2.25, -1) [label=below:$b$] {};
\end{pgfonlayer}[latex arrows]
\begin{pgfonlayer}{edgelayer}
\draw [style=arc] (7) to (0);
\draw [style=arc] (0) to (1);
\draw [style=arc] (2) to (1);
\draw [style=arc] (3) to (2);
\draw [style=arc] (3) to (4);
\draw [style=arc] (4) to (5);
\draw [style=arc] (6) to (5);
\draw [style=arc] (7) to (6);
\draw [style=arc] (7) to (11);
\draw [style=arc] (8) to (1);
\draw [style=arc] (3) to (9);
\draw [style=arc] (10) to (5);
\draw [style=arc] (0) to (12);
\draw [style=arc] (12) to (17);
\draw [style=arc] (13) to (17);
\draw [style=arc] (13) to (15);
\draw [style=arc] (13) to (14);
\draw [style=arc] (15) to (16);
\draw [style=arc] (14) to (16);
\draw [style=arc] (18) to (16);
\draw [style=arc] (20) to (21);
\draw [style=arc] (22) to (21);
\draw [style=arc] (23) to (21);
\draw [style=arc] (24) to (23);
\draw [style=arc] (24) to (22);
\draw [style=arc] (24) to (25);
\draw [style=arc] (25) to (12);
\draw [style=arc] (4) to (19);
\draw [style=arc] (26) to (6);
\draw [style=arc] (26) to (27);
\draw [style=arc] (27) to (28);
\draw [style=arc] (29) to (28);
\draw [style=arc] (30) to (28);
\draw [style=arc] (31) to (29);
\draw [style=arc] (31) to (30);
\draw [style=arc] (31) to (32);
\draw [style=arc] (33) to (32);
\draw [style=arc] (33) to (2);
\draw [style=arc] (34) to (33);
\draw [style=arc] (35) to (34);
\draw [style=arc] (35) to (36);
\draw [style=arc] (35) to (37);
\draw [style=arc] (36) to (38);
\draw [style=arc] (37) to (38);
\draw [style=arc] (39) to (38);
\draw [style=arc] (39) to (26);
\draw [style=arc] (20) to (19);
\draw [style=arc] (19) to (18);
\end{pgfonlayer}
\end{tikzpicture}
\caption{The oriented graph $F$ in Lemma \ref{ios-gadget}.}
\label{figF}
\end{center}
\end{figure}
\begin{lem}
Let $F$ be the oriented graph in Figure \ref{figF}. Then
for $x \in \{t_0, t_1, t_2\}$, there exists an ios-injective homomorphism of $F$ to $T_3^r$ that maps $u$ to $x$, and any such homomorphism also maps $v$ to $x$.
\label{ios-gadget}
\end{lem}
\noindent \textbf{Proof.}
We sketch the proof that in an ios-injective homomorphism of $F$ to $T_3^r$ that maps $u$ to $t_1$, the vertex $v$ also maps to $t_1$.
Referring to Figure \ref{figF}, it is straightforward to check that in any ios-injective homomorphism of $F$ to $T_3^r$,
the vertices labelled $t_0, t_2$ must map to $t_0, t_2$, respectively.
It is also easy to check that the vertices labelled $a$ must have the same image,
and similarly for the vertices labelled $b, e$ and $f$.
It will follow from the argument below that the vertices labelled $c$ must have the
same image, and similarly for the vertices labelled $d$.
We show that the vertices labelled $t_1$ must map to $t_1$.
Suppose $u$ maps to $t_1$.
Then by injectivity its out-neighbour labelled $c$ maps to $t_0$ or $t_2$.
Since $c$ has in-degree 2, it must map to $t_2$.
Therefore $d$ maps to $t_0$.
The in-neighbour of $c$ labelled $t_1$ must map to $t_1$ or $t_2$.
But its in-neighbour labelled $b$ has an out-neighbour labelled $t_2$, so
the in-neighbour of $c$ labelled $t_1$ must map to $t_1$.
By injectivity, the out-neighbour labelled $a$ of this vertex must map to $t_1$,
so the symmetrically located vertex labelled $a$ must also map
to $t_1$, and its in-neighbour labelled $t_1$ must map to $t_1$.
A similar argument shows that the other vertices labelled $t_1$
must map to $t_1$.
We now show that $v$ maps to $t_1$.
By the above argument and injectivity, the vertex labelled $c$ on the right of the figure maps to $t_2$.
A symmetric argument shows that the vertex labelled $d$ on the bottom of the figure
must map to $t_0$.
Now, by injectivity, $v$ maps to $t_1$, as wanted.
Similar arguments show that if $u$ maps to $t_0$ then so does $v$, and if $u$ maps to $t_2$ then so does $v$.
\hfill $\square$
\medskip
\begin{thm}
The problem of deciding if a given oriented graph has an ios-injective homomorphism to $T_3^r$ is NP-complete.
\label{iosT_3^r}
\end{thm}
\noindent \textbf{Proof.}
The transformation is from 3-edge colouring of cubic graphs \cite{holyer}. Suppose such a graph $G$ is given. Construct a graph $G^\prime$ as follows.
For each $x \in V(G)$, regard the edges incident with $x$ as being in 1--1 correspondence with the integers $1, 2, 3$ so that it is meaningful to talk about the $i^{\textup{th}}$ edge incident with $x$.
Let $H_4$ denote the orientation of $K_{1, 3}$ in which there is a vertex of in-degree 3. In the sequel we refer to $H_4$ as an \emph{in-star}.
Start with a collection of $|V(G)|$ disjoint copies of $H_4$. Let $S_x$ denote the copy of $H_4$ corresponding to vertex $x$. Regard the leaves of each oriented graph $S_x$ to be in 1--1 correspondence with $\{1, 2, 3\}$.
Suppose $xy \in E(G)$ is the $i^{\textup{th}}$ edge incident with $x$ and the $j^{\textup{th}}$ edge incident with $y$. Add a new copy of the oriented graph $F$ shown in Figure \ref{figF} and identify the vertices labelled $u$ and $v$ having in-degree
one with the $i^{\textup{th}}$ leaf of $S_x$ and the $j^{\textup{th}}$ leaf of $S_y$. The transformation may be accomplished in polynomial time. We claim that $G$ is $3$-edge-colourable if and only if $G^\prime$ has an ios-injective homomorphism to $T_3^r$.
Suppose $G$ has a $3$-edge-colouring $f:E(G)\rightarrow \{t_0, t_1, t_2\}$ (the colours are the vertices of $T_3^r$). For any edge $xy$ of $G$, map the vertices labelled $u$ and $v$ in the corresponding copy of $F$ to $f(xy)$. Finally, map the centre of each in-star of $G^\prime$ to its only possible image, $t_2$.
Conversely, suppose $G^\prime$ has an ios-injective homomorphism to $T_3$. For each edge $xy$ of $G$, the vertices labelled $u$ and $v$ in the corresponding copy of $F$ in $G^\prime$ must have the same image. Use this for the colour of $xy$. The resulting assignment is a 3-edge-colouring because the leaves of each in-star $S_x$ in $G^\prime$ must have different images.
\hfill $\square$
\section{iot-injective homomorphisms to small reflexive targets}
In this section we consider the complexity of deciding whether there exists an iot-injective homomorphism from a given oriented graph $G$ to the fixed oriented graph $H$, when $H$ is one of the four reflexive tournaments on at most three vertices.
It is clear that an oriented graph has an iot-injective homomorphism to $T_1^r$ if and only if it contains no oriented path on three vertices, that is, if and only if it is a disjoint union of copies of $T_1$ and $T_2$.
We now turn our attention to $T_2^r$. No orientation of a graph with a vertex of degree three has an iot-injective homomorphism to $T_2^r$.
Thus, if $G$ admits an iot-injective homomorphism to $T_2^r$, then the underlying graph of $G$ is a disjoint union of paths and cycles.
The following proposition can be proved using a reduction to 2-SAT, or by an appeal to Courcelle's Theorem.
\begin{prop}
The problem of deciding whether a given oriented graph has an iot-injective homomorphism to $T_2^r$ is Polynomial.
\end{prop}
We next consider iot-injective homomorphism to $C_3^r$.
Consider the family of oriented cycles $\mathcal{B}$ such that
each $B \in \mathcal{B}$ is comprised of two disjoint perfect matchings oriented in opposite directions; that is, $V(B) = \{v_0, v_1, \ldots, v_{2k-1}\}$
and $E(B) = \{v_0v_1, v_2v_3, \ldots, v_{2k-2}v_{2k-1}\}
\cup \{v_0v_{2k-1}, v_2v_1,$ $ \ldots, v_{2k-2}v_{2k-3}\}$.
\begin{lem}
Let $B \in \mathcal{B}$ have order $n$. Then
(i) $B$ has an iot-injective homomorphism to $C_3^r$ if and only if $n \equiv 0$ (mod 6), and
(ii) $B$ has an iot-injective homomorphism to $T_3^r$ if and only if $n \equiv 0$ (mod 4).
\label{hmm}
\end{lem}
\noindent{\bf Proof.}
Let $B \in \mathcal{B}$.
We first consider iot-injective homomorphism of $B$ to $C_3^r$.
Suppose $B$ has $n$ vertices.
Let $x$ be a vertex of out-degree two.
Without loss of generality $x$ maps to $c_1$.
Then its out-neighbours map to $c_1$ and $c_2$.
Let $y$ be the out-neighbour that maps to $c_1$.
Its out-neighbour must map to $c_2$.
Continuing in this way, starting from $x$, the images of
consecutive vertices are $c_1, c_1, c_2, c_2, c_3, c_3, c_1, c_1, \ldots$.
Therefore an iot-injective homomorphism exists if and only
if f $n \equiv 0$ (mod 6).
We now consider iot-injective homomorphism of $B$ to $T_3^r$.
Suppose $B$ has $n$ vertices.
Let $x$ be a vertex of out-degree two.
Then $x$ maps to $t_0$ or $t_1$.
Suppose first that $x$ maps to $t_0$.
Let $v$ be an out-neighbour of $x$.
If $v$ were mapped to $t_0$, then its other in-neighbour must also map to $t_0$,
in violation of injectivity.
Therefore, the out-neighbours of $x$ map to $t_1$ and $t_2$.
Let $y$ be the out-neighbour that maps to $t_2$.
Then $y$'s other in-neighbour, $z$, must map to $t_1$
and $z$'s other out-neighbour, $a$, must also map to $t_1$.
The vertex $a$ has another in-neighbour, $b$.
By injectivity, $b$ maps to $t_0$.
Continuing in this way, starting from $x$, the images of consecutive vertices
are $t_0, t_2, t_1, t_1, t_0, \ldots t_0, t_2, t_1, t_1, t_0$.
Therefore $n \equiv 0$ (mod 4).
Now suppose $x$ maps to $t_1$.
As above, the out-neighbours of $x$ map to $t_1$ and $t_2$.
Let $y$ be the out-neighbour that maps to $t_1$.
The vertex $y$ has another in-neighbour, $z$, which by injectivity maps to $t_0$.
Now, following the same argument as in the previous paragraph
we have that, starting from $x$, the images of consecutive vertices
are $t_1, t_1, t_0, t_2, t_1, \ldots t_1, t_1, t_0, t_2, t_1$.
Again, $n \equiv 0$ (mod 4).
It now follows that an iot-injective homomorphism exists if and only
if $n \equiv 0$ (mod 4).
\hfill$\Box$
\begin{cor}
For $t \geq 1$, let $B_{6t} \in \mathcal{B}$ have $6t$ vertices. In any iot-injective homomorphism $f$ of $B_{6t}$ to $C_3^r$ we have $f(v_i)=f(v_j)$, when $i\equiv j$ (mod 6).
\label{CorollaryB_18}
\end{cor}
\noindent{\bf Proof.}
This follows from the argument in Lemma \ref{hmm}.
\hfill$\Box$
\begin{thm}
The problem of deciding whether an oriented graph has an iot-injective homomorphism to $C_3^r$ is NP-complete.
\label{thm:C3}
\end{thm}
\noindent \textbf{Proof.}
The transformation is from 3-colouring of connected graphs \cite{Garey}.
Suppose such a graph $G$ is given.
Construct a graph $G^\prime$ as follows.
For each $x \in V(G)$, regard the edges incident with $x$ as being in 1--1 correspondence with the integers $1, 2, \ldots, \mathrm{deg}(x)$ so that it is meaningful to talk about the $i^{\textup{th}}$ edge incident with $x$.
Replace every vertex $x\in V(G)$ with a copy $R_x$ of $B_{6 \cdot \mathrm{deg}(x)}$ where,
without loss of generality, the vertices $x_{6i} \in V(R_x),\ 0 \leq i \leq \mathrm{deg}(x)-1$ have
in-degree 2.
Suppose $xy$ is the $i^{\text{th}}$ edge incident with $x$ and the $j^{\text{th}}$ edge incident with $y$.
Construct an oriented path $P_{xy}$ by adding a new vertex $t_{xy}$ and joining each of
$x_{6(i-1)}\in V(R_x)$ and $y_{6(j-1)}\in V(R_y)$ to it by adding a directed path of length two (the midpoint of each such directed path is a new vertex).
The transformation can be carried out in polynomial time. We claim that $G$ is $3$-colourable if and only if $G'$ has an iot-injective homomorphism to $C_3^r$.
Suppose $G$ has a $3$-colouring $f:V(G) \rightarrow \{c_1,c_2,c_3\}$.
For each vertex $x$, map the vertices $x_{0},x_{6},\ldots x_{6 \cdot \mathrm{deg}(x)}$ of $R_x$ to $f(x)$.
By Corollary \ref{CorollaryB_18}, this partial mapping extends to an iot-injective homomorphism of $R_x$ to $C_3^r$.
We claim that this mapping of the oriented cycles $R_x$ extends to the oriented paths $P_{xy}$, where $xy \in E(G)$. Since adjacent vertices in $G$ must receive different colours, this mapping of the copies of $B_{6 \cdot \mathrm{deg}(x)}$ assigns the vertices $v_{0},v_{6},\ldots v_{6 \cdot \mathrm{deg}(x)}$ of $R_x$ a different image than it assigns the corresponding vertices of $R_y$. Suppose, without loss of generality, that the vertices $v_{0},v_{6},\ldots, v_{6 \cdot \mathrm{deg}(x)}$ of $R_x$ are mapped to $c_1$ and the corresponding vertices of $R_y$ are mapped to $c_2$. The in-neighbours of the vertices in $R_x$ are mapped to $c_1$ and $c_3$, while the neighbours of the corresponding vertices in $R_y$ are mapped to $c_2$ and $c_1$. The vertex $t_{xy}$ can be mapped to $c_3$ and the assignment extended to an iot-injective homomorphism of $P_{xy}$ to $C_3^r$. This proves the claim, and completes the proof of the implication.
On the other hand, suppose $G^\prime$ has an iot-injective homomorphism to $C_3^r$. Fix such a mapping.
Then, for each $v \in V(G)$,
the vertices $v_{0},v_{6},\ldots v_{6 \cdot \mathrm{deg}(v)} \in V(R_v)$ all have the same image; assign this to be the colour of vertex $v$ of $G$.
We claim that vertices $x$ and $y$ that are adjacent in $G$ are assigned different colours.
Suppose not.
By symmetry of $C_3^r$, assume both are assigned $c_1$.
Suppose also that $xy$ is the $i$-th edge incident with $x$ and the $j$-th edge incident with $y$.
Then, the vertices $x_{6(i-1)} \in V(R_x)$ and $y_{6(j-1)} \in V(R_y)$ both
map to $c_1$.
By construction, $x_{6(i-1)}$ has two in-neighbours in $R_x$ and one out-neighbour on the directed path to $t_{xy}$,
and similarly for $y_{6(j-1)}$.
Since both $x_{6(i-1)}$ and $y_{6(j-1)}$ map to $c_1$, in each case their in-neighbours must map to $c_1$ and $c_3$.
By injectivity, in each case their out-neighbour on the directed path to $t_{xy}$ must map to $c_2$.
Therefore $t_{xy}$ has 2 in-neighbours that map to $c_2$, which violates injectivity.
This proves the claim, and completes the proof.
\hfill $\square$
\medskip
Finally, we consider iot-injective homomorphism to $T_3^r$.
The following lemma can be proved similarly to Lemma \ref{ios-gadget}.
The proof of Lemma \ref{ios-gadget} relies only on injectivity on
in-neighbourhoods or out-neighbourhoods, and never both
at the same vertex.
\begin{lem}
Let $F$ be the oriented graph in Figure \ref{figF}. Then
For $x \in \{t_0, t_1, t_2\}$, there exists an iot-injective homomorphism of $F$ to $T_3^r$ that maps $u$ to $x$, and any such homomorphism also maps $v$ to $x$.
\label{iot-gadget}
\end{lem}
The proof of the following theorem is similar to that of Theorem \ref{iosT_3^r} and is omitted. For details, see \cite{russell}.
\begin{thm}
The problem of deciding whether an oriented graph has an iot-injective homomorphism to $T_3^r$ is
NP-complete.
\end{thm}
\section{Colourings}
Recall that a (proper) \emph{oriented $k$-colouring} of an oriented graph $G$ is a homomorphism to a tournament on $k$ vertices.
We therefore make the following definitions:
\begin{enumerate}
\item A \emph{proper ios-injective oriented $k$-colouring} of an oriented graph $G$ is an ios-injective homomorphism to an irreflexive tournament on $k$ vertices.
\item An \emph{improper ios-injective oriented $k$-colouring} of an oriented graph $G$ is an ios-injective homomorphism to a reflexive tournament on $k$ vertices.
\item An \emph{improper iot-injective oriented $k$-colouring} of an oriented graph $G$ is an iot-injective homomorphism to a reflexive tournament on $k$ vertices.
\end{enumerate}
A proper iot-injective oriented $k$-colouring of a graph $G$ would be an iot-injective homomorphism to
an irreflexive tournament on $k$ vertices. Since tournaments have no directed 2-cycles, these are the same as proper ios-injective oriented $k$-colourings.
For each fixed integer $k$ and each injective colouring problem defined above, we will determine the complexity of deciding whether a given oriented graph $G$ has an injective colouring with $k$ colours.
The approach to proving NP-completeness is similar to that for oriented colourings that are injective on in-neighbourhoods \cite{mrs,cobusthesis}: prove that it is NP-complete to decide the existence of an injective homomorphism of the given type to the tournament $U_m, \ m \geq 4$, that consists of a directed three cycle dominated by every vertex of a transitive tournament of size $m-3$, and then obtain the desired result as a corollary.
We consider the three situations in turn after establishing a useful lemma.
\begin{lem}
Let $G$ be an oriented graph such that $U_m$ is a subgraph of $G$. For $\mathcal{P} \in \{\mathrm{ios,\ iot}\}$, if $G$ has a $\mathcal{P}$-injective homomorphism to a tournament $T$ (respectively, reflexive tournament $T^r$), then $U_m$ (respectively, $U_m^r$) is a subgraph of $T$.
\label{ontoU_m}
\end{lem}
\noindent \textbf{Proof.}
The tournament $U_m$ has the property that every two different vertices have a common in-neighbour or a common out-neighbour. Hence no two of its vertices can be assigned the same image by a $\mathcal{P}$-injective homomorphism. Consequently, the image of $G$ must contain $U_m$.
\hfill $\square$
\subsection{Proper ios-injective colourings}
\begin{thm}
For each fixed $m \geq 4$, the problem of deciding if a given oriented graph has an ios-injective homomorphism to $U_m$ is NP-complete.
\end{thm}
\noindent \textbf{Proof.}
We first show that
the problem of deciding whether a given oriented graph $G$ has an ios-injective homomorphism to $U_4$ is NP-complete.
The transformation is from the problem of deciding if a given cubic graph is 3-edge-colourable~\cite{holyer}.
Let $G$ be a given cubic graph. Construct an oriented graph $G^\prime$ by replacing each edge $xy$ of $G$ by an oriented path $P_{xy}$ with vertices $x, v_1, v_2, v_3, v_4, y$ and arcs $xv_1, v_1v_2,$ $v_2v_3, v_3v_4,$ $yv_4$. The transformation can be accomplished in polynomial time. We claim that $G$ is $3$-edge-colourable if and only if there is an ios-injective homomorphism of $G^\prime$ to $U_4$.
Suppose that $G$ is $3$-edge colourable. Then, for each vertex $x$ of $G$, each of the colours $1, 2$, and $3$ appears on an edge incident with $x$. An ios-injective homomorphism of $G^\prime$ to $U_4$ is obtained by mapping all vertices of $G$ to the vertex of out-degree $3$ in $U_4$, assigning the colour of the edge $xy$ to the vertices $v_1$ and $v_4$ of $P_{xy}$, and extending this pre-colouring to the vertices $v_2$ and $v_3$ of $P_{xy}$.
Suppose that there is an ios-injective homomorphism of $G^\prime$ to $U_4$. Every vertex of $G$ has out-degree $3$ in $G^\prime$, so an ios-injective homomorphism of $G^\prime$ to $U_4$ must map it to the unique vertex of out-degree $3$ in $U_4$. Similarly, the vertices $v_1, v_2, v_3,$ and $v_4$ in each oriented path $P_{xy}$ have positive in-degree in $G^\prime$, so an ios-injective homomorphism of $G^\prime$ to $U_4$ must map each of them to a vertex of the directed 3-cycle. In any such mapping, $v_1$ and $v_4$ map to the same vertex, and the three out-neighbours of each vertex of $G$ (in $G^\prime$) map to different vertices of the 3-cycle. Assigning each edge $xy$ of $G$ the image of the vertex $v_1$ (and $v_4$) in $P_{xy}$ gives a 3-edge-colouring of $G$.
NP-completeness of ios-injective homomorphism to $U_m$ follows from NP-completeness of ios-injective homomorphism to $U_4$. Given an instance $G$ of ios-injective homomorphism to $U_4$, construct $G^\prime$ by adding the new vertices belonging to $V^\prime = \{x_i: x \in V(G), i = 1, 2, \ldots, (m-4)-\mathrm{d}^-(x)\}$ and the arcs $\{x_i x: x \in V(G), i = 1, 2, \ldots, (m-4)-\mathrm{d}^-(x)\}$. Since $m$ is a constant, the transformation can be accomplished in polynomial time. Each vertex of $G$ in $G^\prime$ has in-degree $m-4$ and therefore cannot map to the $m-4$ vertices of $U_m$ with in-degree less than $m-4$. An ios-injective homomorphism of $G$ to $U_4$ can be extended to an ios-injective homomorphism of $G^\prime$ to $U_m$.
\hfill $\square$
\begin{cor}
Let $k$ be a fixed positive integer. If $k \leq 3$, the problem of deciding if a given oriented graph $G$ has a proper ios-injective oriented $k$-colouring is Polynomial.
If $k \geq 4$, the problem of deciding if a given oriented graph $G$ has a proper ios-injective oriented $k$-colouring is NP-complete.
\end{cor}
\noindent \textbf{Proof.}
An oriented graph $G$ has a proper ios-injective oriented $k$-colouring if and only if $G \cup U_k$ has an ios-injective homomorphism to $U_k$.
\hfill $\square$
\subsection{Improper ios-injective colourings}
\begin{thm}
For each fixed $m \geq 4$, the problem of deciding if a given oriented graph has an ios-injective homomorphism to $U_m^r$ is NP-complete.
\label{ThmIOSU_k^r}
\end{thm}
\noindent \textbf{Proof.} The transformation is from the problem of deciding whether there exists an ios-injective homomorphism of a given oriented graph $G$ to $C_3^r$, which is NP-complete by
Theorem \ref{C_3^rNPc}.
Suppose the oriented graph $G$ is given.
We may assume that
$\Delta^+(G) \leq 2$ and $\Delta^-(G) \leq 2$, otherwise $G$ cannot have an ios-injective homomorphism to $C_3^r$.
Construct $G^\prime$ from $G$ as follows. For each $x \in V(G)$, if $x$ has in-degree at most one in $G$, add a set of $m-2$ new vertices and arcs joining each of them to $x$. If $x$ has in-degree two in $G$, do the same using a set of $m-3$ new vertices. The transformation can be accomplished in polynomial time. We claim that $G$ has an ios-injective homomorphism to $C_3^r$ if and only if $G^\prime$ has an ios-injective homomorphism to $U_m^r$.
An ios-injective homomorphism of $G$ to $C_3^r$ can clearly be extended to an ios-injective homomorphism of $G^\prime$ to $U_m^r$.
Suppose $f$ is an ios-injective homomorphism of $G^\prime$ to $U_m^r$. Since each vertex $x \in V(G)$ has in-degree at least $m-1$ in $G^\prime$ and every vertex of $U_m^r$ not belonging to the copy of $C_3^r$ has in-degree at most $m-3$, the vertex $x$ must map to a vertex of the directed 3-cycle in $U_m^r$. The restriction of $f$ to $V(G)$ is the desired mapping.
\hfill $\square$
\begin{cor}
Let $k$ be a fixed integer. If $k \leq 2$, the problem of deciding if a given oriented graph $G$ has an improper ios-injective oriented $k$-colouring is Polynomial.
If $k \geq 3$, the problem of deciding if a given oriented graph $G$ has an improper ios-injective oriented $k$-colouring is NP-complete.
\label{ImproperIOS}
\end{cor}
\noindent \textbf{Proof.}
When $k=3$ the transformation is from the problem of deciding whether there exists an ios-injective homomorphism of a given oriented graph $G$ to $C_3^r$, which is NP-complete
by Theorem \ref{C_3^rNPc}. Since there is no ios-injective homomorphism of $D_d$
(from Lemma \ref{LemmaD_d}) to $T_3^r$, an oriented graph $G$ has an improper ios-injective oriented $3$-colouring if and only if $G \cup D_6$ has an ios-injective homomorphism to $C_3^r$.
When $k \geq 4$, the transformation is from the problem of deciding whether there exists an ios-injective homomorphism of a given oriented graph $G$ to $U_k^r$. Given an oriented graph $G$, the transformed instance of improper ios-injective oriented $k$-colouring is the oriented graph $G \cup U_k$. The claim that this $G^\prime$ has an improper ios-injective oriented $k$-colouring if and only if $G$ has an ios-injective homomorphism to $U_k^r$ follows from Lemma \ref{ontoU_m}.
\hfill $\square$
\newpage
\subsection{Improper iot-injective colourings}
\begin{thm}
For each fixed $m \geq 4$, the problem of deciding if a given oriented graph has an improper iot-injective homomorphism to $U_m^r$ is NP-complete.
\end{thm}
\noindent \textbf{Proof.}
The transformation is from the problem of deciding whether there exists an iot-injective homomorphism of a given oriented graph $G$ to $C_3^r$, which is NP-complete by Theorem \ref{C_3^rNPc}. Given an oriented graph $G$, the transformed instance $G^\prime$ is constructed by starting with $G$ and proceeding as follows. For each vertex $x \in V(G)$, add a copy of $T_{m-3}$ and arcs from each of its vertices to $x$. Then for every vertex $t$ of each copy of $T_{m-3}$ that was added, add three vertices, $t_a,\ t_b,\ t_c$, and arcs from $t$ to each of them. The oriented graph $G$ has an iot-injective $C_3^r$-colouring if and only if $G'$ has an iot-injective $U_m^r$-colouring.
\hfill $\square$
\medskip
The proof of the following is identical to that of Corollary \ref{ImproperIOS}, except for replacing ``ios'' by ``iot''.
\begin{cor}
Let $k$ be a fixed integer.
If $k \leq 2$, the problem of deciding whether an oriented graph has an improper iot-injective $k$-colouring is Polynomial.
If $k \geq 3$, the problem of deciding whether an oriented graph has an improper iot-injective $k$-colouring is NP-complete.
\label{cor:np}
\end{cor}
For a given oriented graph $G$, we denote by $\chi_{\mathit{ios}}(G), \chi_{\mathit{ios}}^r(G)$ and $\chi_{\mathit{iot}}^r(G)$, the smallest number of colours in a proper ios-injective oriented colouring, an improper ios-injective oriented colouring, and an improper iot-injective oriented colouring of $G$, respectively. The superscript ``$r$'' is used to designate the improper colourings because the target graph being reflexive is what allows adjacent vertices to be assigned the same colour.
A project for future research is to find tight bounds for these parameters.
The upper bounds should be exponential in the in-degree and out-degree -- consider the disjoint union of all tournaments on a fixed number of vertices.
Weak upper bounds can be obtained using the methods in \cite{russell,mrs,mrs3,cobusthesis}.
Tight bounds and efficient algorithms for trees can be obtained as in \cite{russell,mrs,mrs3}.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,325 |
Convert HSL colors to RGB colors in hex format.
## API
```
require('hsl-to-hex') => Function
hsl(hue, saturation, luminosity)` => String
```
## Example
```js
var hsl = require('hsl-to-hex')
var hue = 133 hsl(133, 40, 60)
var saturation = 40
var luminosity = 60
var hex = hsl(hue, saturation, luminosity)
console.log(hex) // #70c282
```
## License
ISC | {
"redpajama_set_name": "RedPajamaGithub"
} | 6,309 |
\section{Introduction}
Deep generative models of all kinds have recently exhibited high quality samples in a wide variety of data modalities. Generative adversarial networks (GANs), autoregressive models, flows, and variational autoencoders (VAEs) have synthesized striking image and audio samples~\citep{goodfellow2014generative,karras2018progressive,brock2018large,oord2016pixel,menick2018generating,kalchbrenner2017video,dinh2016density,kingma2018glow,prenger2019waveglow,oord2016wavenet,kalchbrenner2018efficient,kingma2013auto,razavi2019generating}, and there have been remarkable advances in energy-based modeling and score matching that have produced images comparable to those of GANs~\citep{du2019implicit,song2019generative}.
This paper presents progress in diffusion probabilistic models~\citep{sohl2015deep}. A diffusion probabilistic model (which we will call a ``diffusion model'' for brevity) is a parameterized Markov chain trained using variational inference to produce samples matching the data after finite time. Transitions of this chain are learned to reverse a diffusion process, which is a Markov chain that gradually adds noise to the data in the opposite direction of sampling until signal is destroyed.
When the diffusion consists of small amounts of Gaussian noise, it is sufficient to set the sampling chain transitions to conditional Gaussians too, allowing for a particularly simple neural network parameterization.
Diffusion models are straightforward to define and efficient to train, but to the best of our knowledge, there has been no demonstration that they are capable of generating high quality samples. We show that diffusion models actually are capable of generating high quality samples, sometimes better than the published results on other types of generative models~(\cref{sec:experiments}).
In addition, we show that a certain parameterization of diffusion models reveals an equivalence with denoising score matching over multiple noise levels during training and with annealed Langevin dynamics during sampling~(\cref{sec:revproc_dsm_diffusion_connection})~\citep{song2019generative,vincent2011connection}.
We obtained our best sample quality results using this parameterization~(\cref{sec:loss_ablation}), so we consider this equivalence to be one of our primary contributions.
Despite their sample quality, our models do not have competitive log likelihoods compared to other likelihood-based models (our models do, however, have log likelihoods better than the large estimates annealed importance sampling has been reported to produce for energy based models and score matching~\citep{du2019implicit,song2019generative}).
We find that the majority of our models' lossless codelengths are consumed to describe imperceptible image details~(\cref{sec:coding}). We present a more refined analysis of this phenomenon in the language of lossy compression, and we show that the sampling procedure of diffusion models is a type of progressive decoding that resembles autoregressive decoding along a bit ordering that vastly generalizes what is normally possible with autoregressive models.
\section{Background}
Diffusion models~\citep{sohl2015deep} are latent variable models of the form $p_\theta(\mathbf{x}_0) \coloneqq \int p_\theta(\mathbf{x}_{0:T}) \,d\mathbf{x}_{1:T}$, where $\mathbf{x}_1, \dotsc, \mathbf{x}_T$ are latents of the same dimensionality as the data $\mathbf{x}_0 \sim q(\mathbf{x}_0)$. The joint distribution $p_\theta(\mathbf{x}_{0:T})$ is called the \emph{reverse process}, and it is defined as a Markov chain with learned Gaussian transitions starting at $p(\mathbf{x}_T)=\mathcal{N}(\mathbf{x}_T; \mathbf{0}, \mathbf{I})$:
\begin{align}
p_\theta(\mathbf{x}_{0:T}) &\coloneqq p(\mathbf{x}_T)\prod_{t=1}^T p_\theta(\mathbf{x}_{t-1}|\mathbf{x}_t), \qquad
p_\theta(\mathbf{x}_{t-1}|\mathbf{x}_t) \coloneqq \mathcal{N}(\mathbf{x}_{t-1}; {\boldsymbol{\mu}}_\theta(\mathbf{x}_t, t), {\boldsymbol{\Sigma}}_\theta(\mathbf{x}_t, t))
\end{align}
What distinguishes diffusion models from other types of latent variable models is that the approximate posterior $q(\mathbf{x}_{1:T}|\mathbf{x}_0)$, called the \emph{forward process} or \emph{diffusion process}, is fixed to a Markov chain that gradually adds Gaussian noise to the data according to a variance schedule $\beta_1, \dotsc, \beta_T$:
\begin{align}
q(\mathbf{x}_{1:T} | \mathbf{x}_0) &\coloneqq \prod_{t=1}^T q(\mathbf{x}_t | \mathbf{x}_{t-1} ), \qquad
q(\mathbf{x}_t|\mathbf{x}_{t-1}) \coloneqq \mathcal{N}(\mathbf{x}_t;\sqrt{1-\beta_t}\mathbf{x}_{t-1},\beta_t \mathbf{I}) \label{eq:forwardprocess}
\end{align}
\begin{figure}[t]
\centering
\includegraphics[width=0.75\textwidth]{\iftoggle{hqfigures}{images_high_quality/pgm_diagram_xarrow.pdf}{images/pgm_diagram_xarrow_small.pdf}}
\caption{\small{The directed graphical model considered in this work.}}
\label{fig:pgm}
\vspace{-1em}
\end{figure}
Training is performed by optimizing the usual variational bound on negative log likelihood:
\begin{align}
\Ea{-\log p_\theta(\mathbf{x}_0)} \leq \Eb{q}{ - \log \frac{p_\theta(\mathbf{x}_{0:T})}{q(\mathbf{x}_{1:T} | \mathbf{x}_0)}}
= \mathbb{E}_q\bigg[ -\log p(\mathbf{x}_T) - \sum_{t \geq 1} \log \frac{p_\theta(\mathbf{x}_{t-1} | \mathbf{x}_t)}{q(\mathbf{x}_t|\mathbf{x}_{t-1})} \bigg] \eqqcolon L \label{eq:vb_original}
\end{align}
The forward process variances $\beta_t$ can be learned by reparameterization~\citep{kingma2013auto} or held constant as hyperparameters, and
expressiveness of the reverse process is ensured in part by the choice of Gaussian conditionals in $p_\theta(\mathbf{x}_{t-1}|\mathbf{x}_t)$, because both processes have the same functional form when $\beta_t$ are small~\citep{sohl2015deep}.
A notable property of the forward process is that it admits sampling $\mathbf{x}_t$ at an arbitrary timestep $t$ in closed form: using the notation $\alpha_t \coloneqq 1-\beta_t$ and $\bar\alpha_t \coloneqq \prod_{s=1}^t \alpha_s$, we have
\begin{align}
q(\mathbf{x}_t|\mathbf{x}_0) = \mathcal{N}(\mathbf{x}_t; \sqrt{\bar\alpha_t}\mathbf{x}_0, (1-\bar\alpha_t)\mathbf{I}) \label{eq:q_marginal_arbitrary_t}
\end{align}
Efficient training is therefore possible by optimizing random terms of $L$ with stochastic gradient descent.
Further improvements come from variance reduction by rewriting $L$~\labelcref{eq:vb_original} as:
\begin{align}
\mathbb{E}_q \bigg[ \underbrace{\kl{q(\mathbf{x}_T|\mathbf{x}_0)}{p(\mathbf{x}_T)}}_{L_T} + \sum_{t > 1} \underbrace{\kl{q(\mathbf{x}_{t-1}|\mathbf{x}_t,\mathbf{x}_0)}{p_\theta(\mathbf{x}_{t-1}|\mathbf{x}_t)}}_{L_{t-1}} \underbrace{-\log p_\theta(\mathbf{x}_0|\mathbf{x}_1)}_{L_0} \bigg] \label{eq:vb}
\end{align}
(See~\cref{sec:extended_derivations} for details. The labels on the terms are used in \cref{sec:main}.) \Cref{eq:vb} uses KL divergence to directly compare $p_\theta(\mathbf{x}_{t-1}|\mathbf{x}_t)$ against forward process posteriors, which are tractable when conditioned on $\mathbf{x}_0$:
\begin{align}
q(\mathbf{x}_{t-1}|\mathbf{x}_t,\mathbf{x}_0) &= \mathcal{N}(\mathbf{x}_{t-1}; \tilde{\boldsymbol{\mu}}_t(\mathbf{x}_t, \mathbf{x}_0), \tilde\beta_t \mathbf{I}), \\
\text{where}\quad \tilde{\boldsymbol{\mu}}_t(\mathbf{x}_t, \mathbf{x}_0) &\coloneqq \frac{\sqrt{\bar\alpha_{t-1}}\beta_t }{1-\bar\alpha_t}\mathbf{x}_0 + \frac{\sqrt{\alpha_t}(1- \bar\alpha_{t-1})}{1-\bar\alpha_t} \mathbf{x}_t \quad \text{and} \quad
\tilde\beta_t \coloneqq \frac{1-\bar\alpha_{t-1}}{1-\bar\alpha_t}\beta_t \label{eq:q_posterior_mean_var}
\end{align}
Consequently, all KL divergences in \cref{eq:vb} are comparisons between Gaussians, so they can be calculated in a Rao-Blackwellized fashion with closed form expressions instead of high variance Monte Carlo estimates.
\section{Diffusion models and denoising autoencoders}
\label{sec:main}
Diffusion models might appear to be a restricted class of latent variable models, but they allow a large number of degrees of freedom in implementation. One must choose the variances $\beta_t$ of the forward process and the model architecture and Gaussian distribution parameterization of the reverse process.
To guide our choices, we establish a new explicit connection between diffusion models and denoising score matching~(\cref{sec:revproc_dsm_diffusion_connection}) that leads to a simplified, weighted variational bound objective for diffusion models~(\cref{sec:simplified_training_objective}). Ultimately, our model design is justified by simplicity and empirical results~(\cref{sec:experiments}). Our discussion is categorized by the terms of~\cref{eq:vb}.
\subsection[Forward process]{Forward process and $L_T$}
We ignore the fact that the forward process variances $\beta_t$ are learnable by reparameterization and instead fix them to constants (see~\cref{sec:experiments} for details).
Thus, in our implementation, the approximate posterior $q$ has no learnable parameters, so $L_T$ is a constant during training and can be ignored.
\subsection[Reverse process]{Reverse process and $L_{1:T-1}$} \label{sec:revproc_dsm_diffusion_connection}
Now we discuss our choices in $p_\theta(\mathbf{x}_{t-1}|\mathbf{x}_t) = \mathcal{N}(\mathbf{x}_{t-1}; {\boldsymbol{\mu}}_\theta(\mathbf{x}_t, t), {\boldsymbol{\Sigma}}_\theta(\mathbf{x}_t, t))$ for ${1 < t \leq T}$.
First, we set ${\boldsymbol{\Sigma}}_\theta(\mathbf{x}_t, t) = \sigma_t^2 \mathbf{I}$ to untrained time dependent constants. Experimentally, both $\sigma_t^2 = \beta_t$ and $\sigma_t^2 = \tilde\beta_t = \frac{1-\bar\alpha_{t-1}}{1-\bar\alpha_t}\beta_t$ had similar results. The first choice is optimal for $\mathbf{x}_0 \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$, and the second is optimal for $\mathbf{x}_0$ deterministically set to one point. These are the two extreme choices corresponding to upper and lower bounds on reverse process entropy for data with coordinatewise unit variance~\citep{sohl2015deep}.
Second, to represent the mean ${\boldsymbol{\mu}}_\theta(\mathbf{x}_t, t)$, we propose a specific parameterization motivated by the following analysis of $L_t$.
With $p_\theta(\mathbf{x}_{t-1} | \mathbf{x}_t) = \mathcal{N}(\mathbf{x}_{t-1}; {\boldsymbol{\mu}}_\theta(\mathbf{x}_t, t), \sigma_t^2\mathbf{I})$, we can write:
\begin{align}
L_{t-1}
= \Eb{q}{ \frac{1}{2\sigma_t^2} \|\tilde{\boldsymbol{\mu}}_t(\mathbf{x}_t,\mathbf{x}_0) - {\boldsymbol{\mu}}_\theta(\mathbf{x}_t, t)\|^2 } + C \label{eq:vb_term_orig}
\end{align}
where $C$ is a constant that does not depend on~$\theta$. So, we see that the most straightforward parameterization of ${\boldsymbol{\mu}}_\theta$ is a model that predicts $\tilde{\boldsymbol{\mu}}_t$, the forward process posterior mean.
However, we can expand \cref{eq:vb_term_orig} further by reparameterizing \cref{eq:q_marginal_arbitrary_t} as $\mathbf{x}_t(\mathbf{x}_0, {\boldsymbol{\epsilon}}) = \sqrt{\bar\alpha_t}\mathbf{x}_0 + \sqrt{1-\bar\alpha_t}{\boldsymbol{\epsilon}}$ for ${\boldsymbol{\epsilon}} \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$ and applying the forward process posterior formula~\labelcref{eq:q_posterior_mean_var}:
\begin{align}
L_{t-1} - C &= \Eb{\mathbf{x}_0, {\boldsymbol{\epsilon}}}{ \frac{1}{2\sigma_t^2} \left\| \tilde{\boldsymbol{\mu}}_t\!\left(\mathbf{x}_t(\mathbf{x}_0, {\boldsymbol{\epsilon}}), \frac{1}{\sqrt{\bar\alpha_t}} (\mathbf{x}_t(\mathbf{x}_0, {\boldsymbol{\epsilon}}) - \sqrt{1-\bar\alpha_t}{\boldsymbol{\epsilon}}) \right) - {\boldsymbol{\mu}}_\theta(\mathbf{x}_t(\mathbf{x}_0, {\boldsymbol{\epsilon}}), t) \right\|^2 } \\
&=\Eb{\mathbf{x}_0, {\boldsymbol{\epsilon}}}{ \frac{1}{2\sigma_t^2} \left\| \frac{1}{\sqrt{\alpha_t}}\left( \mathbf{x}_t(\mathbf{x}_0,{\boldsymbol{\epsilon}}) - \frac{\beta_t}{\sqrt{1-\bar\alpha_t}}{\boldsymbol{\epsilon}} \right) - {\boldsymbol{\mu}}_\theta(\mathbf{x}_t(\mathbf{x}_0,{\boldsymbol{\epsilon}}), t) \right\|^2 } \label{eq:vb_term_langevin_mu}
\end{align}
\Cref{eq:vb_term_langevin_mu} reveals that ${\boldsymbol{\mu}}_\theta$ must predict $\frac{1}{\sqrt{\alpha_t}}\left( \mathbf{x}_t - \frac{\beta_t}{\sqrt{1-\bar\alpha_t}}{\boldsymbol{\epsilon}} \right)$ given $\mathbf{x}_t$. Since $\mathbf{x}_t$ is available as input to the model, we may choose the parameterization
\begin{align}
{\boldsymbol{\mu}}_\theta(\mathbf{x}_t, t) = \tilde{\boldsymbol{\mu}}_t\!\left(\mathbf{x}_t, \frac{1}{\sqrt{\bar\alpha_t}} (\mathbf{x}_t - \sqrt{1-\bar\alpha_t}{\boldsymbol{\epsilon}}_\theta(\mathbf{x}_t)) \right) = \frac{1}{\sqrt{\alpha_t}}\left( \mathbf{x}_t - \frac{\beta_t}{\sqrt{1-\bar\alpha_t}} {\boldsymbol{\epsilon}}_\theta(\mathbf{x}_t, t) \right) \label{eq:mu_func_approx_langevin}
\end{align}
where ${\boldsymbol{\epsilon}}_\theta$ is a function approximator intended to predict ${\boldsymbol{\epsilon}}$ from $\mathbf{x}_t$. To sample $\mathbf{x}_{t-1} \sim p_\theta(\mathbf{x}_{t-1}|\mathbf{x}_t)$ is to compute
$
\mathbf{x}_{t-1} = \frac{1}{\sqrt{\alpha_t}}\left( \mathbf{x}_t - \frac{\beta_t}{\sqrt{1-\bar\alpha_t}} {\boldsymbol{\epsilon}}_\theta(\mathbf{x}_t, t) \right) + \sigma_t \mathbf{z}$, where $\mathbf{z} \sim \mathcal{N}(\mathbf{0}, \mathbf{I})
$.
The complete sampling procedure, \cref{alg:sampling}, resembles Langevin dynamics with ${\boldsymbol{\epsilon}}_\theta$ as a learned gradient of the data density.
Furthermore, with the parameterization~\labelcref{eq:mu_func_approx_langevin}, \cref{eq:vb_term_langevin_mu} simplifies to:
\begin{align}
\Eb{\mathbf{x}_0, {\boldsymbol{\epsilon}}}{ \frac{\beta_t^2}{2\sigma_t^2 \alpha_t (1-\bar\alpha_t)} \left\| {\boldsymbol{\epsilon}} - {\boldsymbol{\epsilon}}_\theta(\sqrt{\bar\alpha_t} \mathbf{x}_0 + \sqrt{1-\bar\alpha_t}{\boldsymbol{\epsilon}}, t) \right\|^2} \label{eq:vb_term_langevin_eps}
\end{align}
which resembles denoising score matching over multiple noise scales indexed by $t$~\citep{song2019generative}. As \cref{eq:vb_term_langevin_eps} is equal to (one term of) the variational bound for the Langevin-like reverse process~\labelcref{eq:mu_func_approx_langevin}, we see that optimizing an objective resembling denoising score matching is equivalent to using variational inference to fit the finite-time marginal of a sampling chain resembling Langevin dynamics.
To summarize, we can train the reverse process mean function approximator ${\boldsymbol{\mu}}_\theta$ to predict $\tilde{\boldsymbol{\mu}}_t$, or by modifying its parameterization, we can train it to predict ${\boldsymbol{\epsilon}}$. (There is also the possibility of predicting $\mathbf{x}_0$, but we found this to lead to worse sample quality early in our experiments.) We have shown that the ${\boldsymbol{\epsilon}}$-prediction parameterization both resembles Langevin dynamics and simplifies the diffusion model's variational bound to an objective that resembles denoising score matching.
Nonetheless, it is just another parameterization of $p_\theta(\mathbf{x}_{t-1}|\mathbf{x}_t)$, so we verify its effectiveness in \cref{sec:experiments} in an ablation where we compare predicting ${\boldsymbol{\epsilon}}$ against predicting $\tilde{\boldsymbol{\mu}}_t$.
\algrenewcommand\algorithmicindent{0.5em}%
\begin{figure}[t]
\begin{minipage}[t]{0.495\textwidth}
\begin{algorithm}[H]
\caption{Training} \label{alg:training}
\small
\begin{algorithmic}[1]
\Repeat
\State $\mathbf{x}_0 \sim q(\mathbf{x}_0)$
\State $t \sim \mathrm{Uniform}(\{1, \dotsc, T\})$
\State ${\boldsymbol{\epsilon}}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$
\State Take gradient descent step on
\Statex $\qquad \nabla_\theta \left\| {\boldsymbol{\epsilon}} - {\boldsymbol{\epsilon}}_\theta(\sqrt{\bar\alpha_t} \mathbf{x}_0 + \sqrt{1-\bar\alpha_t}{\boldsymbol{\epsilon}}, t) \right\|^2$
\Until{converged}
\end{algorithmic}
\end{algorithm}
\end{minipage}
\hfill
\begin{minipage}[t]{0.495\textwidth}
\begin{algorithm}[H]
\caption{Sampling} \label{alg:sampling}
\small
\begin{algorithmic}[1]
\vspace{.04in}
\State $\mathbf{x}_T \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$
\For{$t=T, \dotsc, 1$}
\State $\mathbf{z} \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$ if $t > 1$, else $\mathbf{z} = \mathbf{0}$
\State $\mathbf{x}_{t-1} = \frac{1}{\sqrt{\alpha_t}}\left( \mathbf{x}_t - \frac{1-\alpha_t}{\sqrt{1-\bar\alpha_t}} {\boldsymbol{\epsilon}}_\theta(\mathbf{x}_t, t) \right) + \sigma_t \mathbf{z}$
\EndFor
\State \textbf{return} $\mathbf{x}_0$
\vspace{.04in}
\end{algorithmic}
\end{algorithm}
\end{minipage}
\vspace{-1em}
\end{figure}
\subsection[Data scaling and reverse process decoder]{Data scaling, reverse process decoder, and $L_0$}
We assume that image data consists of integers in $\{ 0, 1, \dotsc, 255\}$ scaled linearly to $[-1, 1]$. This ensures that the neural network reverse process operates on consistently scaled inputs starting from the standard normal prior $p(\mathbf{x}_T)$.
To obtain discrete log likelihoods, we set the last term of the reverse process to an independent discrete decoder derived from the Gaussian $\mathcal{N}(\mathbf{x}_0 ; {\boldsymbol{\mu}}_\theta(\mathbf{x}_1, 1), \sigma_1^2 \mathbf{I})$:
\begin{align}\begin{split}
p_\theta(\mathbf{x}_0 | \mathbf{x}_1) &= \prod_{i=1}^D \int_{\delta_{-}(x_0^i)}^{\delta_{+}(x_0^i)} \mathcal{N}(x; \mu_\theta^i(\mathbf{x}_1, 1), \sigma_1^2) \, dx \label{eq:discrete_decoder} \\
\delta_{+}(x) &= \begin{cases}
\infty & \text{if}\ x=1 \\
x+\frac{1}{255} & \text{if}\ x < 1
\end{cases}
\qquad \delta_{-}(x) = \begin{cases}
-\infty & \text{if}\ x=-1 \\
x-\frac{1}{255} & \text{if}\ x > -1
\end{cases}
\end{split}\end{align}
where $D$ is the data dimensionality and the $i$ superscript indicates extraction of one coordinate.
(It would be straightforward to instead incorporate a more powerful decoder like a conditional autoregressive model, but we leave that to future work.) Similar to the discretized continuous distributions used in VAE decoders and autoregressive models~\citep{kingma2016improved,salimans2017pixelcnn++}, our choice here ensures that the variational bound is a lossless codelength of discrete data, without need of adding noise to the data or incorporating the Jacobian of the scaling operation into the log likelihood. At the end of sampling, we display ${\boldsymbol{\mu}}_\theta(\mathbf{x}_1,1)$ noiselessly.
\subsection{Simplified training objective}
\label{sec:simplified_training_objective}
With the reverse process and decoder defined above, the variational bound, consisting of terms derived from \cref{eq:vb_term_langevin_eps,eq:discrete_decoder}, is clearly differentiable with respect to $\theta$ and is ready to be employed for training. However, we found it beneficial to sample quality (and simpler to implement) to train on the following variant of the variational bound:
\begin{align}
L_\mathrm{simple}(\theta) \coloneqq \Eb{t, \mathbf{x}_0, {\boldsymbol{\epsilon}}}{ \left\| {\boldsymbol{\epsilon}} - {\boldsymbol{\epsilon}}_\theta(\sqrt{\bar\alpha_t} \mathbf{x}_0 + \sqrt{1-\bar\alpha_t}{\boldsymbol{\epsilon}}, t) \right\|^2} \label{eq:training_objective_simple}
\end{align}
where $t$ is uniform between $1$ and $T$. The $t=1$ case corresponds to $L_0$ with the integral in the discrete decoder definition \labelcref{eq:discrete_decoder} approximated by the Gaussian probability density function times the bin width, ignoring $\sigma_1^2$ and edge effects. The $t > 1$ cases correspond to an unweighted version of \cref{eq:vb_term_langevin_eps}, analogous to the loss weighting used by the NCSN denoising score matching model~\citep{song2019generative}. ($L_T$ does not appear because the forward process variances $\beta_t$ are fixed.)
\Cref{alg:training} displays the complete training procedure with this simplified objective.
Since our simplified objective~\labelcref{eq:training_objective_simple} discards the weighting in \cref{eq:vb_term_langevin_eps}, it is a weighted variational bound that emphasizes different aspects of reconstruction compared to the standard variational bound~\citep{gregor2016towards,higgins2017beta}.
In particular, our diffusion process setup in \cref{sec:experiments} causes the simplified objective to down-weight loss terms corresponding to small $t$. These terms train the network to denoise data with very small amounts of noise, so it is beneficial to down-weight them so that the network can focus on more difficult denoising tasks at larger $t$ terms. We will see in our experiments that this reweighting leads to better sample quality.
\section{Experiments}
\label{sec:experiments}
We set $T=1000$ for all experiments so that the number of neural network evaluations needed during sampling matches previous work~\citep{sohl2015deep,song2019generative}. We set the forward process variances to constants increasing linearly from $\beta_1 = 10^{-4}$ to $\beta_T = 0.02$. These constants were chosen to be small relative to data scaled to $[-1, 1]$, ensuring that reverse and forward processes have approximately the same functional form while keeping the signal-to-noise ratio at $\mathbf{x}_T$ as small as possible ($L_T = \kl{q(\mathbf{x}_T|\mathbf{x}_0)}{\mathcal{N}(\mathbf{0}, \mathbf{I})} \approx 10^{-5}$ bits per dimension in our experiments).
To represent the reverse process, we use a U-Net backbone similar to an unmasked \mbox{PixelCNN++}~\citep{salimans2017pixelcnn++,ronneberger2015unet} with group normalization throughout~\citep{wu2018group}. Parameters are shared across time, which is specified to the network using the Transformer sinusoidal position embedding~\citep{vaswani2017attention}. We use self-attention at the $16 \times 16$ feature map resolution~\citep{wang2018non,vaswani2017attention}. Details are in \cref{sec:experiment_details}.
\subsection{Sample quality}
\begin{figure}
\begin{minipage}{0.55\textwidth}
\captionof{table}{\small{CIFAR10 results. NLL measured in bits/dim.}} \label{table:cifar10_results}
\centering \scriptsize
\vspace{-1em}
\begin{tabular}{lccc}
\toprule
Model & IS & FID & NLL Test (Train) \\
\midrule
\textbf{Conditional} \\
\midrule
EBM~\citep{du2019implicit} & $8.30$ & $37.9$ \\
JEM~\citep{grathwohl2020your} & $8.76$ & $38.4$ \\
BigGAN~\citep{brock2018large} & $9.22$ & $14.73$ \\
StyleGAN2 + ADA (v1)~\cite{karras2020training} & $\mathbf{10.06}$ & $\mathbf{2.67}$ \\
\midrule
\textbf{Unconditional} \\
\midrule
Diffusion (original)~\citep{sohl2015deep} & & & $\leq 5.40$ \\
Gated PixelCNN~\citep{oord2016conditional} & $4.60$ & $65.93$ & $3.03$ $(2.90)$ \\
Sparse Transformer~\citep{child2019generating} & & & $\mathbf{2.80}$ \\
PixelIQN~\citep{ostrovski2018autoregressive} & $5.29$ & $49.46$ \\
EBM~\citep{du2019implicit} & $6.78$ & $38.2$ \\
NCSNv2~\citep{song2020improved} & & $31.75$ \\
NCSN~\citep{song2019generative} & $8.87\!\pm\!0.12$ & $25.32$ \\
SNGAN~\citep{miyato2018spectral} & $8.22\!\pm\!0.05$ & $21.7$ \\
SNGAN-DDLS~\citep{che2020your} & $9.09\!\pm\!0.10$ & $15.42$ \\
StyleGAN2 + ADA (v1)~\citep{karras2020training} & $\mathbf{9.74} \pm 0.05$ & $3.26$ \\
Ours ($L$, fixed isotropic ${\boldsymbol{\Sigma}}$) & $7.67\!\pm\!0.13$ & $13.51$ & $\leq 3.70 $ $(3.69)$ \\
\textbf{Ours ($L_\mathrm{simple}$)} & $9.46\!\pm\!0.11$ & $\mathbf{3.17}$ & $\leq 3.75$ $(3.72)$\\
\bottomrule
\end{tabular}
\end{minipage}\hfill
\begin{minipage}{0.4\textwidth}
\captionof{table}{\small{Unconditional CIFAR10 reverse process parameterization and training objective ablation. Blank entries were unstable to train and generated poor samples with out-of-range scores.}} \label{table:loss_ablation}
\centering \scriptsize
\vspace{-1em}
\begin{tabular}{lcc}
\toprule
Objective & IS & FID \\
\midrule
\textbf{$\tilde{\boldsymbol{\mu}}$ prediction (baseline)} \\
\midrule
$L$, learned diagonal ${\boldsymbol{\Sigma}}$ & $7.28\!\pm\!0.10$ & $23.69$ \\
$L$, fixed isotropic ${\boldsymbol{\Sigma}}$ & $8.06\!\pm\!0.09$ & $13.22$ \\
$\|\tilde{\boldsymbol{\mu}} - \tilde{\boldsymbol{\mu}}_\theta \|^2$ & -- & -- \\
\midrule
\textbf{${\boldsymbol{\epsilon}}$ prediction (ours)} \\
\midrule
$L$, learned diagonal ${\boldsymbol{\Sigma}}$ & -- & -- \\
$L$, fixed isotropic ${\boldsymbol{\Sigma}}$ & $7.67\!\pm\!0.13$ & $13.51$ \\
$\|\tilde{\boldsymbol{\epsilon}} - {\boldsymbol{\epsilon}}_\theta \|^2$ ($L_\mathrm{simple}$) & $\mathbf{9.46\!\pm\!0.11}$ & $\mathbf{3.17}$ \\
\bottomrule
\end{tabular}
\end{minipage}
\vspace{-1em}
\end{figure}
\Cref{table:cifar10_results} shows Inception scores, FID scores, and negative log likelihoods (lossless codelengths) on CIFAR10. With our FID score of 3.17, our unconditional model achieves better sample quality than most models in the literature, including class conditional models. Our FID score is computed with respect to the training set, as is standard practice; when we compute it with respect to the test set, the score is 5.24, which is still better than many of the training set FID scores in the literature.
We find that training our models on the true variational bound yields better codelengths than training on the simplified objective, as expected, but the latter yields the best sample quality. See \cref{fig:header_samples} for CIFAR10 and CelebA-HQ $256 \times 256$ samples, \cref{fig:samples_lsun_church} and \cref{fig:samples_lsun_bedroom} for LSUN $256 \times 256$ samples \cite{yu15lsun}, and \cref{sec:extended_samples} for more.
\begin{figure}
\begin{minipage}[t]{0.49\linewidth}
\centering
\includegraphics[width=\linewidth]{\iftoggle{hqfigures}{images_high_quality/lsun_church_layout.pdf}{images/lsun_church_layout.jpg}}
\caption{\small{LSUN Church samples. FID=$7.89$}}
\label{fig:samples_lsun_church}
\end{minipage}\hfill
\begin{minipage}[t]{0.49\linewidth}
\centering
\includegraphics[width=\linewidth]{\iftoggle{hqfigures}{images_high_quality/lsun_bedroom_layout_l192_step1556000.pdf}{images/lsun_bedroom_layout_l192_step155600.jpg}}
\caption{\small{LSUN Bedroom samples. FID=$4.90$}}
\label{fig:samples_lsun_bedroom}
\end{minipage}
\vspace{-1em}
\end{figure}
\subsection{Reverse process parameterization and training objective ablation}
\label{sec:loss_ablation}
In \cref{table:loss_ablation}, we show the sample quality effects of reverse process parameterizations and training objectives (\cref{sec:revproc_dsm_diffusion_connection}). We find that the baseline option of predicting $\tilde{\boldsymbol{\mu}}$ works well only when trained on the true variational bound instead of unweighted mean squared error, a simplified objective akin to~\cref{eq:training_objective_simple}. We also see that learning reverse process variances (by incorporating a parameterized diagonal ${\boldsymbol{\Sigma}}_\theta(\mathbf{x}_t)$ into the variational bound) leads to unstable training and poorer sample quality compared to fixed variances. Predicting ${\boldsymbol{\epsilon}}$, as we proposed, performs approximately as well as predicting $\tilde{\boldsymbol{\mu}}$ when trained on the variational bound with fixed variances, but much better when trained with our simplified objective.
\subsection{Progressive coding}
\label{sec:coding}
\Cref{table:cifar10_results} also shows the codelengths of our CIFAR10 models. The gap between train and test is at most 0.03 bits per dimension, which is comparable to the gaps reported with other likelihood-based models and indicates that our diffusion model is not overfitting (see~\cref{sec:extended_samples} for nearest neighbor visualizations).
Still, while our lossless codelengths are better than the large estimates reported for energy based models and score matching using annealed importance sampling~\citep{du2019implicit}, they are not competitive with other types of likelihood-based generative models~\citep{child2019generating}.
Since our samples are nonetheless of high quality, we conclude that diffusion models have an inductive bias that makes them excellent lossy compressors. Treating the variational bound terms $L_1 + \cdots + L_T$ as rate and $L_0$ as distortion, our CIFAR10 model with the highest quality samples has a rate of \textbf{1.78} bits/dim and a distortion of \textbf{1.97} bits/dim, which amounts to a root mean squared error of \text{0.95} on a scale from 0 to 255. More than half of the lossless codelength describes imperceptible distortions.
\paragraph{Progressive lossy compression} We can probe further into the rate-distortion behavior of our model by introducing a progressive lossy code that mirrors the form of \cref{eq:vb}: see \cref{alg:sending,alg:receiving}, which assume access to a procedure, such as minimal random coding~\citep{harsha2007communication,havasi2018minimal}, that can transmit a sample $\mathbf{x} \sim q(\mathbf{x})$ using approximately $\kl{q(\mathbf{x})}{p(\mathbf{x})}$ bits on average for any distributions $p$ and $q$, for which only $p$ is available to the receiver beforehand.
\begin{figure}[t]
\begin{minipage}[t]{0.55\textwidth}
\begin{algorithm}[H]
\caption{Sending $\mathbf{x}_0$} \label{alg:sending}
\small
\begin{algorithmic}[1]
\State Send $\mathbf{x}_T \sim q(\mathbf{x}_T|\mathbf{x}_0)$ using $p(\mathbf{x}_T)$
\For{$t=T-1, \dotsc, 2, 1$}
\State Send $\mathbf{x}_t \sim q(\mathbf{x}_t|\mathbf{x}_{t+1}, \mathbf{x}_0)$ using $p_\theta(\mathbf{x}_t | \mathbf{x}_{t+1})$
\EndFor
\State Send $\mathbf{x}_0$ using $p_\theta(\mathbf{x}_0|\mathbf{x}_1)$
\end{algorithmic}
\end{algorithm}
\end{minipage}
\hfill
\begin{minipage}[t]{0.44\textwidth}
\begin{algorithm}[H]
\caption{Receiving} \label{alg:receiving}
\small
\begin{algorithmic}[1]
\vspace{.01in}
\State Receive $\mathbf{x}_T$ using $p(\mathbf{x}_T)$
\For{$t=T-1, \dotsc, 1, 0$}
\State Receive $\mathbf{x}_t$ using $p_\theta(\mathbf{x}_t | \mathbf{x}_{t+1})$
\EndFor
\State \textbf{return} $\mathbf{x}_0$
\vspace{.01in}
\end{algorithmic}
\end{algorithm}
\end{minipage}
\vspace{-1em}
\end{figure}
When applied to $\mathbf{x}_0 \sim q(\mathbf{x}_0)$, \cref{alg:sending,alg:receiving} transmit $\mathbf{x}_T, \dotsc, \mathbf{x}_0$ in sequence using a total expected codelength equal to \cref{eq:vb}. The receiver, at any time~$t$, has the partial information $\mathbf{x}_t$ fully available and can progressively estimate:
\begin{align}
\mathbf{x}_0 \approx \hat\mathbf{x}_0 = \left( \mathbf{x}_t - \sqrt{1-\bar\alpha_t}{\boldsymbol{\epsilon}}_\theta(\mathbf{x}_t) \right) / \sqrt{\bar\alpha_t}
\end{align}
due to~\cref{eq:q_marginal_arbitrary_t}.
(A stochastic reconstruction $\mathbf{x}_0 \sim p_\theta(\mathbf{x}_0|\mathbf{x}_t)$ is also valid, but we do not consider it here because it makes distortion more difficult to evaluate.)
\Cref{fig:cifar10_rate_distortion} shows the resulting rate-distortion plot on the CIFAR10 test set. At each time $t$, the distortion is calculated as the root mean squared error $\sqrt{\|\mathbf{x}_0 - \hat\mathbf{x}_0\|^2 / D}$, and the rate is calculated as the cumulative number of bits received so far at time $t$. The distortion decreases steeply in the low-rate region of the rate-distortion plot, indicating that the majority of the bits are indeed allocated to imperceptible distortions.
\pgfplotsset{small,width=5.1cm,height=4cm,tick label style={font=\tiny},label style={font=\tiny}}
\begin{figure}[ht]
\begin{tikzpicture}
\begin{axis}[
grid=both,minor tick num=1,
xlabel=Reverse process steps ($T-t$),
ylabel=Distortion (RMSE)
]
\addplot+[only marks,mark size=0.9pt] table[x=timestep,y=rmse_scaled,col sep=comma] {plot_data/cifar10_eps-fixedlarge-mse_step790000.csv};
\end{axis}
\end{tikzpicture}
\begin{tikzpicture}
\begin{axis}[
grid=both,minor tick num=1,
xlabel=Reverse process steps ($T-t$),
ylabel=Rate (bits/dim)
]
\addplot+[only marks,mark size=0.9pt] table[x=timestep,y=rate,col sep=comma] {plot_data/cifar10_eps-fixedlarge-mse_step790000.csv};
\end{axis}
\end{tikzpicture}
\begin{tikzpicture}
\begin{axis}[
grid=both,minor tick num=1,
xlabel=Rate (bits/dim),
ylabel=Distortion (RMSE)
]
\addplot+[only marks,mark size=0.9pt] table[x=rate,y=rmse_scaled,col sep=comma] {plot_data/cifar10_eps-fixedlarge-mse_step790000.csv};
\end{axis}
\end{tikzpicture}
\caption{\small{Unconditional CIFAR10 test set rate-distortion vs. time. Distortion is measured in root mean squared error on a $[0,255]$ scale. See~\cref{table:cifar10_rate_distortion_table} for details.}}
\label{fig:cifar10_rate_distortion}
\vspace{-1em}
\end{figure}
\paragraph{Progressive generation} We also run a progressive unconditional generation process given by progressive decompression from random bits. In other words, we predict the result of the reverse process, $\hat\mathbf{x}_0$, while sampling from the reverse process using~\cref{alg:sampling}.
\Cref{fig:cropped_cifar10_progressive_samples,fig:cifar10_progressive_sample_quality} show the resulting sample quality of $\hat\mathbf{x}_0$ over the course of the reverse process.
Large scale image features appear first and details appear last.
\Cref{fig:ext_celeba_stochastic_decoding} shows stochastic predictions $\mathbf{x}_0 \sim p_\theta(\mathbf{x}_0|\mathbf{x}_t)$ with $\mathbf{x}_t$ frozen for various $t$. When $t$ is small, all but fine details are preserved, and when $t$ is large, only large scale features are preserved. Perhaps these are hints of conceptual compression~\citep{gregor2016towards}.
\begin{figure}[h]
\centering
\includegraphics[trim=0cm 19.15cm 0cm 0cm,clip,width=0.9\textwidth]{\iftoggle{hqfigures}{images_high_quality/cifar10_eps-fixedlarge-mse_20_progressive.png}{images/cifar10_eps-fixedlarge-mse_20_progressive.jpg}}
\caption{\small{Unconditional CIFAR10 progressive generation ($\hat\mathbf{x}_0$ over time, from left to right). Extended samples and sample quality metrics over time in the appendix~(\cref{fig:ext_cifar10_progressive_samples,fig:cifar10_progressive_sample_quality}).}}
\label{fig:cropped_cifar10_progressive_samples}
\vspace{-1em}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.8\textwidth]{\iftoggle{hqfigures}{images_high_quality/stochastic_decoding.pdf}{images/stochastic_decoding.jpg}}
\caption{\small{When conditioned on the same latent, CelebA-HQ $256 \times 256$ samples share high-level attributes. Bottom-right quadrants are $\mathbf{x}_t$, and other quadrants are samples from $p_\theta(\mathbf{x}_0 | \mathbf{x}_t)$.}}
\label{fig:ext_celeba_stochastic_decoding}
\vspace{-1em}
\end{figure}
\paragraph{Connection to autoregressive decoding} Note that the variational bound~\labelcref{eq:vb} can be rewritten as:
\begin{align}
L &= \kl{q(\mathbf{x}_T)}{p(\mathbf{x}_T)} + \mathbb{E}_{q}\Bigg[ \sum_{t \geq 1} \kl{q(\mathbf{x}_{t-1}|\mathbf{x}_t)}{p_\theta(\mathbf{x}_{t-1}|\mathbf{x}_t)} \Bigg] + H(\mathbf{x}_0)
\end{align}
(See \cref{sec:extended_derivations} for a derivation.) Now consider setting the diffusion process length $T$ to the dimensionality of the data, defining the forward process so that $q(\mathbf{x}_t|\mathbf{x}_0)$ places all probability mass on $\mathbf{x}_0$ with the first $t$ coordinates masked out (i.e. $q(\mathbf{x}_t|\mathbf{x}_{t-1})$ masks out the $t^\text{th}$ coordinate), setting $p(\mathbf{x}_T)$ to place all mass on a blank image, and, for the sake of argument, taking $p_\theta(\mathbf{x}_{t-1}|\mathbf{x}_t)$ to be a fully expressive conditional distribution. With these choices, $\kl{q(\mathbf{x}_T)}{p(\mathbf{x}_T)}=0$, and minimizing $\kl{q(\mathbf{x}_{t-1}|\mathbf{x}_t)}{p_\theta(\mathbf{x}_{t-1}|\mathbf{x}_t)}$ trains $p_\theta$ to copy coordinates $t+1, \dotsc, T$ unchanged and to predict the $t^{\text{th}}$ coordinate given $t+1, \dotsc, T$. Thus, training $p_\theta$ with this particular diffusion is training an autoregressive model.
We can therefore interpret the Gaussian diffusion model~\labelcref{eq:forwardprocess} as a kind of autoregressive model with a generalized bit ordering that cannot be expressed by reordering data coordinates. Prior work has shown that such reorderings introduce inductive biases that have an impact on sample quality~\citep{menick2018generating}, so we speculate that the Gaussian diffusion serves a similar purpose, perhaps to greater effect since Gaussian noise might be more natural to add to images compared to masking noise. Moreover, the Gaussian diffusion length is not restricted to equal the data dimension; for instance, we use $T=1000$, which is less than the dimension of the $32\times 32 \times 3$ or $256 \times 256 \times 3$ images in our experiments. Gaussian diffusions can be made shorter for fast sampling or longer for model expressiveness.
\subsection{Interpolation}
We can interpolate source images $\mathbf{x}_0, \mathbf{x}'_0 \sim q(\mathbf{x}_0)$ in latent space using $q$ as a stochastic encoder, $\mathbf{x}_t, \mathbf{x}'_t \sim q(\mathbf{x}_t | \mathbf{x}_0)$, then decoding the linearly interpolated latent $\bar{\mathbf{x}}_t = (1-\lambda) \mathbf{x}_0 + \lambda \mathbf{x}'_0$ into image space by the reverse process, $\bar{\mathbf{x}}_0 \sim p(\mathbf{x}_0 | \bar{\mathbf{x}}_t)$. In effect, we use the reverse process to remove artifacts from linearly interpolating corrupted versions of the source images, as depicted in \cref{fig:interpolations} (left). We fixed the noise for different values of $\lambda$ so $\mathbf{x}_t$ and $\mathbf{x}'_t$ remain the same. \cref{fig:interpolations} (right) shows interpolations and reconstructions of original CelebA-HQ $256 \times 256$ images ($t=500$). The reverse process produces high-quality reconstructions, and plausible interpolations that smoothly vary attributes such as pose, skin tone, hairstyle, expression and background, but not eyewear. Larger $t$ results in coarser and more varied interpolations, with novel samples at $t=1000$ (Appendix \cref{fig:interpolations_coarse_to_fine}).
\begin{figure}
\centering
\includegraphics[width=\linewidth]{\iftoggle{hqfigures}{images_high_quality/interp_with_diagram.pdf}{images/interp_with_diagram.jpg}}
\caption{\small{Interpolations of CelebA-HQ 256x256 images with 500 timesteps of diffusion.}}
\label{fig:interpolations}
\vspace{-1em}
\end{figure}
\section{Related Work}
While diffusion models might resemble flows~\citep{dinh2014nice,rezende2015variational,dinh2016density,kingma2018glow,chen2018neural,grathwohl2019ffjord,ho2019flow++} and VAEs~\citep{kingma2013auto,rezende2014stochastic,maaloe2019biva}, diffusion models are designed so that $q$ has no parameters and the top-level latent $\mathbf{x}_T$ has nearly zero mutual information with the data $\mathbf{x}_0$.
Our ${\boldsymbol{\epsilon}}$-prediction reverse process parameterization establishes a connection between diffusion models and denoising score matching over multiple noise levels with annealed Langevin dynamics for sampling~\citep{song2019generative,song2020improved}. Diffusion models, however, admit straightforward log likelihood evaluation, and the training procedure explicitly trains the Langevin dynamics sampler using variational inference (see \cref{sec:extended_related_work} for details).
The connection also has the reverse implication that a certain weighted form of denoising score matching is the same as variational inference to train a Langevin-like sampler. Other methods for learning transition operators of Markov chains include infusion training~\citep{bordes2016learning}, variational walkback~\citep{goyal2017variational}, generative stochastic networks~\citep{alain2016gsns}, and others~\citep{salimans2015markov,song2017nice,levy2018generalizing,nijkamp2019learning,lawson2019energy,wu2020stochastic}.
By the known connection between score matching and energy-based modeling, our work could have implications for other recent work on energy-based models~\citep{xie2016theory,xie2017synthesizing,xie2018learning,gao2018learning,xie2019learning,gao2020flow,du2019implicit,nijkamp2019anatomy,grathwohl2020your,deng2020residual}. Our rate-distortion curves are computed over time in one evaluation of the variational bound, reminiscent of how rate-distortion curves can be computed over distortion penalties in one run of annealed importance sampling~\citep{huang2020evaluating}. Our progressive decoding argument can be seen in convolutional DRAW and related models~\citep{gregor2016towards,nichol2020vq} and may also lead to more general designs for subscale orderings or sampling strategies for autoregressive models~\citep{menick2018generating,wiggers2020predictive}.
\section{Conclusion}
We have presented high quality image samples using diffusion models, and we have found connections among diffusion models and variational inference for training Markov chains, denoising score matching and annealed Langevin dynamics (and energy-based models by extension), autoregressive models, and progressive lossy compression. Since diffusion models seem to have excellent inductive biases for image data, we look forward to investigating their utility in other data modalities and as components in other types of generative models and machine learning systems.
\section*{Broader Impact}
Our work on diffusion models takes on a similar scope as existing work on other types of deep generative models, such as efforts to improve the sample quality of GANs, flows, autoregressive models, and so forth. Our paper represents progress in making diffusion models a generally useful tool in this family of techniques, so it may serve to amplify any impacts that generative models have had (and will have) on the broader world.
Unfortunately, there are numerous well-known malicious uses of generative models. Sample generation techniques can be employed to produce fake images and videos of high profile figures for political purposes. While fake images were manually created long before software tools were available, generative models such as ours make the process easier. Fortunately, CNN-generated images currently have subtle flaws that allow detection \cite{wang2019cnngenerated}, but improvements in generative models may make this more difficult.
Generative models also reflect the biases in the datasets on which they are trained. As many large datasets are collected from the internet by automated systems, it can be difficult to remove these biases, especially when the images are unlabeled. If samples from generative models trained on these datasets proliferate throughout the internet, then these biases will only be reinforced further.
On the other hand, diffusion models may be useful for data compression, which, as data becomes higher resolution and as global internet traffic increases, might be crucial to ensure accessibility of the internet to wide audiences. Our work might contribute to representation learning on unlabeled raw data for a large range of downstream tasks, from image classification to reinforcement learning, and diffusion models might also become viable for creative uses in art, photography, and music.
\begin{ack}
This work was supported by ONR PECASE and the NSF Graduate Research Fellowship under grant number DGE-1752814. Google's TensorFlow Research Cloud (TFRC) provided Cloud TPUs.
\end{ack}
\setlength{\bibsep}{5pt}
\bibliographystyle{plainnat}
{\small | {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,310 |
/*
* Axamit, gc.support@axamit.com
*/
package com.axamit.gc.core.services.impl;
import com.axamit.gc.api.GCContext;
import com.axamit.gc.api.dto.*;
import com.axamit.gc.api.services.GCContentApi;
import com.axamit.gc.api.services.GCContentNewApi;
import com.axamit.gc.core.exception.GCException;
import com.axamit.gc.core.pojo.FieldMappingProperties;
import com.axamit.gc.core.pojo.ImportItem;
import com.axamit.gc.core.pojo.ImportResultItem;
import com.axamit.gc.core.pojo.MappingType;
import com.axamit.gc.core.services.AbstractPageModifier;
import com.axamit.gc.core.services.GCPageModifier;
import com.axamit.gc.core.services.plugins.GCPluginManager;
import com.axamit.gc.core.services.plugins.GCPlugin;
import com.axamit.gc.core.sightly.models.MapperModel;
import com.axamit.gc.core.util.Constants;
import com.axamit.gc.core.util.GCStringUtil;
import com.axamit.gc.core.util.GCUtil;
import com.axamit.gc.core.util.ResourceResolverUtil;
import com.day.cq.wcm.api.Page;
import com.day.cq.wcm.api.PageManager;
import com.google.common.collect.ImmutableList;
import org.apache.commons.lang3.StringUtils;
import org.apache.felix.scr.annotations.Component;
import org.apache.felix.scr.annotations.Reference;
import org.apache.felix.scr.annotations.Service;
import org.apache.sling.api.resource.LoginException;
import org.apache.sling.api.resource.ModifiableValueMap;
import org.apache.sling.api.resource.PersistenceException;
import org.apache.sling.api.resource.Resource;
import org.apache.sling.api.resource.ResourceResolver;
import org.apache.sling.api.resource.ResourceResolverFactory;
import org.apache.sling.api.resource.ResourceUtil;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
import java.util.List;
import java.util.Map;
/**
* The <tt>GCItemCreator</tt> interface provides methods to create items in GatherContent.
*
* @author Axamit, gc.support@axamit.com
*/
@Service(value = GCPageModifier.class)
@Component(description = "GC Page Modifier Service", name = "GC Page Modifier", immediate = true, metatype = true)
public final class GCPageModifierImpl extends AbstractPageModifier implements GCPageModifier {
private static final Logger LOGGER = LoggerFactory.getLogger(GCPageModifierImpl.class);
@Reference
private ResourceResolverFactory resourceResolverFactory;
@Reference
private GCContentApi gContentApi;
@Reference
private GCContentNewApi gcContentNewApi;
@Reference
private GCPluginManager gcPluginManager;
private static GCItem createGCItem(final Integer projectId, final MapperModel mapperModel, final String folderUuid,
final String name) {
GCItem gcItem = new GCItem();
if (mapperModel != null && MappingType.TEMPLATE == mapperModel.getMappingType()) {
gcItem.setTemplateId(mapperModel.getTemplateId());
}
// if (mapperModel != null) {
// gcItem.setConfig(mapperModel.getGcGroups());
// }
gcItem.setProjectId(projectId);
gcItem.setFolderUuid(folderUuid);
gcItem.setName(name);
return gcItem;
}
private static String findOutFolderOrPage(PageManager pageManager, ImportItem importItem) {
return pageManager.getPage(importItem.getImportPath()) == null ? "folder" : "page";
}
private static MapperModel getMapperModel(ResourceResolver resourceResolver, ImportItem importItem) {
Resource mappingResource = resourceResolver.resolve(importItem.getMappingPath());
if (mappingResource == null || ResourceUtil.isNonExistingResource(mappingResource)) {
LOGGER.error("Mapping \"{}\" not found", importItem.getMappingPath());
return null;
}
MapperModel mapperModel = mappingResource.adaptTo(MapperModel.class);
if (mapperModel == null) {
LOGGER.error("Can not adapt mapping \"{}\" to model {}", importItem.getMappingPath(),
MapperModel.class.getName());
return null;
}
return mapperModel;
}
private static void updateGCSpecialPropertiesInAEMPage(ResourceResolver resourceResolver, PageManager pageManager,
Integer projectId, Integer itemId, ImportItem importItem) {
ModifiableValueMap modifiableValueMap = pageManager.getPage(importItem.getImportPath()) != null
? pageManager.getPage(importItem.getImportPath()).getContentResource().adaptTo(ModifiableValueMap.class)
: resourceResolver.getResource(importItem.getImportPath()).adaptTo(ModifiableValueMap.class);
try {
GCUtil.addGCExportProperties(resourceResolver, modifiableValueMap, projectId,
itemId, importItem.getMappingPath() == null ? StringUtils.EMPTY : importItem.getMappingPath());
} catch (PersistenceException | GCException e) {
LOGGER.error(e.getMessage(), e);
}
}
private void updateGCProperty(final GCItem gcItem, final Page page, final String key,
final FieldMappingProperties fieldMappingProperties,
final ResourceResolver resourceResolver, final String configurationPath, final GCContext gcContext) {
GCContent gcContent = findContentByKey(gcItem, key);
if (gcContent != null) {
if (!fieldMappingProperties.getPath().isEmpty()) {
for (String relativePropertyPath : fieldMappingProperties.getPath()) {
if (!ResourceUtil.isNonExistingResource(resourceResolver.resolve(
GCStringUtil.appendNewLevelToPath(page.getPath(), relativePropertyPath)))) {
String fullPath = GCStringUtil.appendNewLevelToPath(page.getPath(), relativePropertyPath);
String propertyPath = GCStringUtil.getRelativeNodePathFromPropertyPath(fullPath);
String propertyValue = GCStringUtil.getPropertyNameFromPropertyPath(fullPath);
try {
GCPlugin gcPlugin = gcPluginManager.getPlugin(resourceResolver, configurationPath,
gcContent.getType().getValue(), StringUtils.EMPTY, fieldMappingProperties.getPlugin());
if (isNewEditorMultifieldElement(gcContext, gcContent)) {
//TODO
// gcContent.setType(GCElementType.MULTIVALUE_NEW_EDITOR);
}
if (gcPlugin != null) {
gcPlugin.transformFromAEMtoGC(resourceResolver, page, gcContent, propertyPath, propertyValue);
}
} catch (Exception e) {
LOGGER.error(e.getMessage(), e);
}
break;
}
}
} else if (gcItem.getTemplateId() == null) {
if (GCElementType.TEXT.equals(gcContent.getType())) {
gcContent.setText(StringUtils.EMPTY);
} /*else if (GCElementType.SECTION.equals(gcContent.getType())) {*/
//TODO
// gcContent.setSubtitle(StringUtils.EMPTY);
// }
}
}
}
private boolean isNewEditorMultifieldElement(GCContext gcContext, GCContent gcContent) {
GCElementType elementType = gcContent.getType();
return (GCElementType.CHOICE_CHECKBOX.equals(elementType) || GCElementType.CHOICE_RADIO.equals(elementType));
}
/**
* @inheritDoc
*/
@Override
public List<ImportResultItem> createPage(final List<ImportItem> importItemsToMerge, final GCContext gcContext,
final List<ImportItem> childrenItems, final Integer projectId) {
ResourceResolver resourceResolver = null;
try {
resourceResolver = getPageCreatorResourceResolver();
PageManager pageManager = resourceResolver.adaptTo(PageManager.class);
GCItem gcItem = createMergedGCItem(importItemsToMerge, projectId, resourceResolver, pageManager, gcContext);
if (gcItem != null) { //! Else? I would log and early break/return
Integer createdItemId = gcContentNewApi.createItem(gcItem, gcContext);
if (createdItemId != null) { //! Else? I would log and early break/return
GCStatus statusData = new GCStatus();
if (!importItemsToMerge.isEmpty() && importItemsToMerge.get(0).getNewStatusData().getId() != null
&& gContentApi.updateItemStatus(gcContext, createdItemId,
importItemsToMerge.get(0).getNewStatusData().getId())) {
statusData = importItemsToMerge.get(0).getNewStatusData();
} else {
List<GCStatus> gcStatusList = gContentApi.statusesByProjectId(gcContext, projectId);
for (GCStatus gcStatus : gcStatusList) {
if (gcStatus.getIsDefault()) {
statusData = gcStatus;
break;
}
}
}
for (ImportItem childItem : childrenItems) {
childItem.setGcTargetItemId(createdItemId);
}
ImmutableList.Builder<ImportResultItem> importResultItemList = ImmutableList.builder();
for (ImportItem importItem : importItemsToMerge) {
updateGCSpecialPropertiesInAEMPage(resourceResolver, pageManager, gcItem.getProjectId(),
createdItemId, importItem);
final ImportResultItem importResultItem = new ImportResultItem(statusData.getName(),
gcItem.getName(),
importItem.getAemTitle(),
ImportResultItem.IMPORTED,
importItem.getTemplate(),
//! OSGI configurable mask like "https://{0}.gathercontent.com/item/" would be great.
"https://" + importItem.getSlug() + ".gathercontent.com/item/" + createdItemId,
importItem.getImportPath(),
statusData.getColor(),
gcItem.getPosition(),
createdItemId,
// gcItem.getParentId(),
"0",
importItem.getImportIndex(),
importItem.getMappingName());
importResultItemList.add(importResultItem.setType(findOutFolderOrPage(pageManager, importItem)));
}
return importResultItemList.build();
}
}
} catch (LoginException e) {
LOGGER.error("Failed to get ServiceResourceResolver {}", e.getMessage());
} catch (GCException e) {
LOGGER.error("Failed to get get data from GC {}", e.getMessage());
} catch (Exception e) {
LOGGER.error("Failed to export AEM page {}", e.getMessage());
} finally {
if (resourceResolver != null && resourceResolver.isLive()) {
resourceResolver.close();
}
}
ImmutableList.Builder<ImportResultItem> importResultItemList = ImmutableList.builder();
for (ImportItem importItem : importItemsToMerge) {
LOGGER.error("Couldn't export AEM page {}", importItem.getImportPath());
importResultItemList.add(new ImportResultItem().setImportStatus(ImportResultItem.NOT_IMPORTED)
.setGcTemplateName(importItem.getTemplate()).setAemLink(importItem.getImportPath())
.setAemTitle(importItem.getAemTitle()).setImportIndex(importItem.getImportIndex()));
}
return importResultItemList.build();
}
private GCItem createMergedGCItem(final Iterable<ImportItem> importItemsToMerge, final Integer projectId,
final ResourceResolver resourceResolver, final PageManager pageManager, final GCContext gcContext) {
GCItem gcItem = null;
for (ImportItem importItem : importItemsToMerge) {
if (importItem.getMappingPath() != null) {
Page page = pageManager.getPage(importItem.getImportPath());
MapperModel mapperModel = getMapperModel(resourceResolver, importItem);
if (mapperModel != null) {
Map<String, FieldMappingProperties> mapping = mapperModel.getMapper();
importItem.setMappingName(mapperModel.getMappingName());
String pluginConfigPath = mapperModel.getPluginConfigPath();
//! First-time create in loop?
if (gcItem == null) {
//TODO
// gcItem =
// createGCItem(projectId, mapperModel, importItem.getFolderUuid(), importItem.getTitle());
}
if (mapping != null && page != null) {
for (Map.Entry<String, FieldMappingProperties> mapEntry : mapping.entrySet()) {
updateGCProperty(gcItem, page, mapEntry.getKey(), mapEntry.getValue(), resourceResolver,
pluginConfigPath, gcContext);
}
} else {
LOGGER.error("No mapped properties in the mapping \"{}\"", importItem.getMappingPath());
//! Break/return?
}
}
} else {
//! Overwrite in loop?
//TODO
// gcItem = createGCItem(projectId, null, importItem.getFolderUuid(), importItem.getTitle());
}
}
return gcItem;
}
/**
* @inheritDoc
*/
@Override
public ImportResultItem updatePage(final GCContext gcContext, final ImportItem importItem) {
ResourceResolver resourceResolver = null;
String mappingName = null;
try {
resourceResolver = getPageCreatorResourceResolver();
PageManager pageManager = resourceResolver.adaptTo(PageManager.class);
Page page = pageManager.getPage(importItem.getImportPath());
MapperModel mapperModel = getMapperModel(resourceResolver, importItem);
if (mapperModel != null) { //! Else? I would log and early break/return
String pluginConfigPath = mapperModel.getPluginConfigPath();
Map<String, FieldMappingProperties> mapping = mapperModel.getMapper();
mappingName = mapperModel.getMappingName();
GCItem gcItem = gcContentNewApi.itemById(gcContext, importItem.getItemId());
if (gcItem != null) {
if (mapping == null) {
LOGGER.error("No mapped properties in the mapping \"{}\"", importItem.getMappingPath());
//! Break/return?
} else {
for (Map.Entry<String, FieldMappingProperties> mapEntry : mapping.entrySet()) {
updateGCProperty(gcItem, page, mapEntry.getKey(), mapEntry.getValue(),
resourceResolver, pluginConfigPath, gcContext);
}
}
//TODO
// Boolean isUpdatedSuccessfully = gcContentNewApi.updateItem(gcItem.getConfig(), gcItem.getId(), gcContext);
boolean isUpdatedSuccessfully = false;
if (isUpdatedSuccessfully) {
boolean updateItemStatus = false;
if (importItem.getNewStatusData().getId() != null) {
updateItemStatus = gContentApi
.updateItemStatus(gcContext, gcItem.getId(), importItem.getNewStatusData().getId());
}
//TODO
// final GCData statusData =
// updateItemStatus ? importItem.getNewStatusData() : gcItem.getStatus().getData();
// updateGCSpecialPropertiesInAEMPage(resourceResolver, pageManager, gcItem.getProjectId(),
// gcItem.getId(), importItem);
// return new ImportResultItem(statusData.getName(),
// gcItem.getName(),
// page.getTitle(),
// ImportResultItem.IMPORTED,
// importItem.getTemplate(),
// //! OSGI configurable mask like "https://{0}.gathercontent.com/item/" would be great.
// "https://" + importItem.getSlug() + ".gathercontent.com/item/" + gcItem.getId(),
// importItem.getImportPath(),
// statusData.getColor(),
// gcItem.getPosition(),
// gcItem.getId(),
// gcItem.getParentId(),
// importItem.getImportIndex(),
// mappingName);
return null;
} else {
LOGGER.error("Couldn't update item {}", importItem.getItemId());
//TODO
return null;
// return new ImportResultItem(null,
// null,
// null,
// ImportResultItem.NOT_IMPORTED,
// importItem.getTemplate(),
// "https://" + importItem.getSlug() + ".gathercontent.com/item/" + gcItem.getId(),
// importItem.getImportPath(),
// null,
// gcItem.getPosition(),
// gcItem.getId(),
// gcItem.getParentId(),
// importItem.getImportIndex(),
// mappingName);
}
}
}
} catch (LoginException e) {
LOGGER.error("Failed to get ServiceResourceResolver {}", e.getMessage());
} catch (GCException e) {
LOGGER.error("Failed to get get data from GC {}", e.getMessage());
} catch (Exception e) {
LOGGER.error("Failed to export update AEM page {}", e.getMessage());
} finally {
if (resourceResolver != null && resourceResolver.isLive()) {
resourceResolver.close();
}
}
LOGGER.error("Couldn't update item {}", importItem.getItemId());
return new ImportResultItem(null, null, null, ImportResultItem.NOT_IMPORTED, importItem.getTemplate(),
null, importItem.getImportPath(), null, null, null, null, importItem.getImportIndex(), mappingName);
}
private ResourceResolver getPageCreatorResourceResolver() throws LoginException {
return ResourceResolverUtil.getResourceResolver(resourceResolverFactory,
Constants.PAGE_CREATOR_SUBSERVICE_NAME);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,374 |
<?php
namespace Marvin\Pages\Controller;
use Silex\Application;
use Silex\ControllerProviderInterface;
use Symfony\Component\HttpFoundation\Request;
class AdminControllerProvider implements ControllerProviderInterface
{
public function connect(Application $app)
{
$controllers = $app['controllers_factory'];
$controllers->get('/', function () use ($app) {
$pages = $app['db']->fetchAll("SELECT * FROM page ORDER BY sort ASC");
return $app['twig']->render('admin/pages/list.twig', array(
'pages' => $pages,
));
})
->bind('admin_pages');
$controllers->match('/form/{id}', function (Request $request, $id) use ($app) {
$pageData = array();
if ($id > 0) {
$pageData = $app['db']->fetchAssoc("SELECT * FROM page WHERE id = ?", array($id));
}
$form = $app['form.factory']->createBuilder('form', $pageData)
->add('id', 'hidden')
->add('name', 'text')
->add('content', 'textarea', array(
'required' => false,
))
->getForm();
$form->handleRequest($request);
if ($form->isValid()) {
$data = $form->getData();
$slug = $originalSlug = $app['slugify']->slugify($data['name']);
$i = 2;
do {
$find = $app['db']->fetchAssoc("SELECT COUNT(*) AS count FROM page WHERE slug = ?". ($data['id'] > 0 ? " AND id != ". $data['id'] : ""), array($slug));
if ($find['count'] > 0) {
$slug = $originalSlug .'-'. $i;
$i++;
}
} while ($find['count'] > 0);
if ($data['id'] == 0) {
$maxSort = $app['db']->fetchAssoc("SELECT MAX(sort) AS sort FROM page");
$app['db']->executeUpdate("INSERT INTO page (name, slug, content, sort, created_at, updated_at) VALUES (?, ?, ?, ?, ?, ?)", array(
$data['name'],
$slug,
$data['content'],
$maxSort['sort']+1,
date('Y-m-d H:i:s'),
date('Y-m-d H:i:s'),
));
$data['id'] = $app['db']->lastInsertId();
$app['session']->getFlashBag()->add('message', $app['translator']->trans('The new page was added.'));
} else {
$app['db']->executeUpdate("UPDATE page SET name = ?, slug = ?, content = ?, updated_at = ? WHERE id = ?", array(
$data['name'],
$slug,
$data['content'],
date('Y-m-d H:i:s'),
$data['id'],
));
$app['session']->getFlashBag()->add('message', $app['translator']->trans('Changes were saved.'));
}
return $app->redirect('/admin/pages/form/'. $data['id']);
}
return $app['twig']->render('admin/pages/form.twig', array(
'form' => $form->createView(),
));
})
->value('id', 0)
->assert('id', '\d+');
$controllers->get('/delete/{id}', function ($id) use ($app) {
$page = $app['db']->fetchAssoc("SELECT sort FROM page WHERE id = ?", array($id));
$app['db']->executeUpdate("UPDATE page SET sort=sort-1 WHERE sort > ?", array($page['sort']));
$app['db']->delete('page', array('id' => $id));
$app['session']->getFlashBag()->add('message', $app['translator']->trans('The page was deleted'));
return $app->redirect('/admin/pages');
})
->assert('id', '\d+');
$controllers->match('/move/{id}/{type}', function ($id, $type) use ($app) {
$action = $type == 'down' ? 1 : -1;
$page = $app['db']->fetchAssoc("SELECT sort FROM page WHERE id = ?", array($id));
$app['db']->executeUpdate("UPDATE page SET sort=sort+? WHERE sort = ?", array(-$action, $page['sort']+$action));
$app['db']->executeUpdate("UPDATE page SET sort=sort+? WHERE id = ?", array($action, $id));
$app['session']->getFlashBag()->add('message', $app['translator']->trans('Order of pages was changed'));
return $app->redirect('/admin/pages');
})
->assert('id', '\d+')
->assert('type', '(up|down)');
$controllers->match('/file/upload', function (Request $request) use ($app) {
$file = $request->files->get('file');
if ($file) {
if (file_exists($app['config']['upload_dir']) === false) {
mkdir($app['config']['upload_dir']);
}
$file->move($app['config']['upload_dir'], $file->getClientOriginalName());
return $app['config']['public_upload_dir'] .'/'. $file->getClientOriginalName();
}
});
$controllers->match('/file/delete', function (Request $request) use ($app) {
$file = $request->get('file');
if (file_exists($app['config']['web_dir'] .'/'. $file) && dirname($file) == $app['config']['public_upload_dir']) {
unlink($app['config']['web_dir'] .'/'. $file);
}
});
return $controllers;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,500 |
As you might know Kivio, the KOffice diagramming and flowcharting application, hasn't been released with KOffice 2.0. Since the release a lot of people asked on IRC how long it will take until Kivio will be released and what's still missing for the release.
Of course the small code size doesn't mean that it doesn't have any features. Due to the high integration between the apps and the KOffice libs most of the features of KPresenter/Karbon are already available. For example load/save, inserting and arranging shapes, managing pages etc are done. Shapes can be rotated which was really missing in 1.6. Thanks to flake all shapes can now be interchanged seamlessy between Kivio and the other KOffice apps. At the moment there is also ongoing work to improve the connection tool.
What's missing in Kivio? Most of the work to make Kivio usefull has been done KOffice libs. What Kivio needs now are developers to fill the missing gaps. The most important missing features at the moment are stencils sets as we had them in 1.6 and support for text on shapes. Unfortunately we are all already busy with developing the other KOffice apps and day jobs, so nobody is working on this at the moment.
You are currently browsing the Sven's Blog blog archives for July, 2009. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,642 |
Produced by David Starner, Paul Marshall and the Online
Distributed Proofreading Team at http://www.pgdp.net (This
file was produced from images generously made available
by The Internet Archive)
[** Transcriber's Notes:
Superscripts have been represented using regular characters,
e.g. "ye 27th".
The [oe] ligature has been replaced with simply an oe. **]
BALLADS OF BOOKS
[Illustration]
[Illustration]
BALLADS OF BOOKS
CHOSEN BY
BRANDER MATTHEWS
[Illustration]
NEW YORK
DODD, MEAD AND COMPANY
1900
_Copyright, 1886_
BY GEORGE J. COOMBES
PRINTED BY
THE UNIVERSITY PRESS, CAMBRIDGE, U. S. A.
TO
FREDERICK LOCKER
POET AND LOVER OF BOOKS
_Come and take a choice of all my library_
Titus Andronicus, iv. 1
[Illustration]
[Illustration]
PREFATORY NOTE.
_______________
The poets have ever been lovers of books; indeed, one might ask how
should a man be a poet who did not admire a treasure as precious and as
beautiful as a book may be. With evident enjoyment, Keats describes
A viol, bowstrings torn, cross-wise upon
A glorious folio of Anacreon;
and it was a glorious folio of Beaumont and Fletcher which another
English poet (whose most poetic work was done in prose) "dragged home
late at night from Barker's in Covent Garden," and to pacify his
conscience for the purchase of which he kept to his overworn suit of
clothes for four or five weeks longer than he ought. Charles Lamb was a
true bibliophile, in the earlier and more exact sense of the term; he
loved his ragged volumes as he loved his fellow-men, and he was as
intolerant of books that are not books as he was of men who were not
manly. He conferred the dukedom of his library on Coleridge, who was no
respecter of books, though he could not but enrich them with his
marginal notes. Southey and Lord Houghton and Mr. Locker are English
poets with libraries of their own, more orderly and far richer than the
fortuitous congregation of printed atoms, a mere medley of unrelated
tomes, which often masquerades as The Library in the mansions of the
noble and the wealthy. Shelley said that he thought Southey had a
secret in every one of his books which he was afraid the stranger might
discover: but this was probably no more, and no other, than the secret
of comfort, consolation, refreshment, and happiness to be found in any
library by him who shall bring with him the golden key that unlocks its
silent door.
Mr. Lowell has recently dwelt on the difference between literature and
books: and, accepting this distinction, the editor desires to declare
at once that as a whole this collection is devoted rather to books than
to literature. The poems in the following pages celebrate the
bric-a-brac of the one rather than the masterpieces of the other. The
stanzas here garnered into one sheaf sing of books as books, of books
valuable and valued for their perfection of type and page and
printing,--for their beauty and for their rarity,--or for their
association with some famous man or woman of the storied past
Two centuries and a half ago Drummond of Hawthornden prefixed to the
'Varieties' of his friend Persons a braggart distich:--
This book a world is; here, if errors be,
The like, nay worse, in the great world we see.
The present collection of varieties in verse has little or naught to do
with the great world and its errors: it has to do chiefly, not to say
wholly, with the world of the Bookmen--the little world of the
Book-lover, the Bibliophile, the Bibliomaniac--a mad world, my masters,
in which there are to be found not a few poets who cherish old wine and
old wood, old friends and old books, and who believe that old books are
the best of old friends.
Books, books again, and books once more!
These are our theme, which some miscall
Mere madness, setting little store
By copies either short or tall,
But you, O slaves of shelf and stall!
We rather write for you that hold
Patched folios dear, and prize "the small
Rare volume, black with burnished gold."
as Mr. Austin Dobson sang on the threshold of Mr. Lang's delightfully
discursive little book about the 'Library.'
The editor has much pleasure in thanking the poets who have allowed him
to reprint their poems in these pages; and he acknowledges a double
debt of gratitude to the friends who have written poems expressly for
this collection. Encouraged by their support, and remembering that he
is not a contributor to his own pages, the editor ventures to conclude
his harmless necessary catalogue of the things contained and not
contained within these covers, by quoting Herrick's address to his
Book:--
Be bold, my Book, nor be abash'd, or fear,
The cutting thumb-nail, or the brow severe;
But by the muses swear, all here is good,
If but well read, or ill read, understood.
BRANDER MATTHEWS.
NEW YORK, _November_, 1886.
[Illustration]
[Illustration]
=Proem.=
_BALLADE OF THE BOOKWORM._
_Deep in the Past I peer, and see
A Child upon the Nursery floor,
A Child with book, upon his knee,
Who asks, like Oliver, for more!
The number of his years is IV,
And yet in Letters hath he skill,
How deep he dives in Fairy-lore!
The Books I loved, I love them still!_
_One gift the Fairies gave me: (Three
They commonly bestowed of yore)
The Love of Books, the Golden Key
That opens the Enchanted Door;
Behind it BLUEBEARD lurks and o'er
And o'er doth JACK his Giants kill,
And there is all ALADDIN'S store,--
The Books I loved, I love them still!_
_Take all, but leave my Books to me!
These heavy creels of old we love
We fill not now, nor wander free,
Nor wear the heart that once we wore;
Not now each River seems to pour
His waters from the Muse's hill;
Though something's gone from stream and shore,
The Books I love, I love them still!_
_ENVOY!_
_Fate, that art Queen by shore and sea,
We bow submissive to thy will,
Ah grant, by some benign decree,
The Books I loved--to love them still._
A. LANG.
[Illustration]
[Illustration]
=Contents.=
PAGE
_Prefatory Note_ v
PROEM. [1]_Ballade of the Bookworm_ (A. Lang) ix
EDWARD D. ANDERSON. _The Baby in the Library_ 17
FRANCIS BENNOCH. _My Books_ 19
LAMAN BLANCHARD. _The Art of Book-Keeping_ 20
ANNE C. L. BOTTA. _In the Library_ 26
H. C. BUNNER. [1]_My Shakspere_ 28
ROBERT BURNS. _The Bookworms_ 31
CATULLUS. [1]_To his Book_ (Translated by A. Lang) 32
BEVERLY CHEW. _Old Books are best_ 33
THOMAS S. COLLIER. [1]_The Forgotten Books_ 34
HELEN GRAY CONE. _An Invocation in a Library_ 36
SAMUEL DANIEL. _Concerning the Honor of Books_ 38
ISAAC D'ISRAELI. _Lines_ 39
AUSTIN DOBSON. _My Books_ 40
_To a Missal of the Thirteenth Century_ 42
_The Book-Plate's Petition_ 44
HENRY DRURY. _Over the Threshold of my Library_ 46
MAURICE F. EGAN. _The Chrysalis of a Bookworm_ 47
EVENUS. _Epigram_ (Translated by A. Lang) 48
JOHN FERRIAR. _The Bibliomania_ 49
F. FERTIAULT. _Triolet to her Husband_
(Translated by A. Lang) 57
WILLIAM FREELAND. _A Nook and a Book_ 58
EDMUND GOSSE. [1]_The Sultan of my Books_ 60
THOMAS GORDON HAKE. _Our Book-Shelves_ 64
ROBERT HERRICK. _To his Book_ 66
_To his Book_ 67
HORACE. [1]_To his Books_
(Translated by Austin Dobson) 68
LEIGH HUNT. _Sonnet_ 70
WILLIS FLETCHER JOHNSON. _My Books_ 71
BEN JONSON. _To my Bookseller_ 73
_To Sir Henry Goodyere_ 74
CHARLES LAMB. _In the Album of Lucy Barton_ 75
A. LANG. _Ballade of the Book-Hunter_ 77
_Ballade of True Wisdom_ 79
_Ballade of the Bookman's Paradise_ 81
_The Rowfant Books_ 83
_The Rowfant Library_ 85
_Ghosts in the Library_ 87
GEORGE PARSONS LATHROP. [1]_The Book Battalion_ 91
WALTER LEARNED. [1]_On the Fly-Leaf of a Book of
Old Plays_ 93
ROBERT LEIGHTON. _Too Many Books_ 95
FREDERICK LOCKER. [1]_From the Fly-Leaf of the
Rowfant Montaigne_ 97
HENRY WADSWORTH LONGFELLOW. _My Books_ 98
LORD LYTTON. _The Souls of Books_ 99
COSMO MONKHOUSE. [1]_De Libris_ 105
ARTHUR J. MUNBY. [1]_Ex Libris_ 107
[1]_On an Inscription_ 108
CAROLINE NORTON. _To my Books_ 110
F. M. P. _'Desultory Reading'_ 111
THOMAS PARNELL. _The Bookworm_ 112
SAMUEL MINTURN PECK. _Among my Books_ 116
WALTER HERRIES POLLOCK. [1]_A Ruined Library_ 117
BRYAN WALLER PROCTER (Barry Cornwall). _My Books_ 119
WILLIAM ROSCOE. _To my Books on Parting with Them_ 120
LORD ROSSLYN. _Among my Books_ 121
JOHN GODFREY SAXE. _The Library_ 123
CLINTON SCOLLARD. _In the Library_ 124
FRANK DEMPSTER SHERMAN. _The Book-Hunter_ 126
ROBERT SOUTHEY. _The Library_ 128
ROBERT LOUIS STEVENSON. _Picture-Books in Winter_ 130
RICHARD HENRY STODDARD. _Companions_ 131
RICHARD THOMSON. _The Book of Life_ 133
CHARLES TENNYSON TURNER. _On Certain Books_ 135
HENRY VAUGHAN. _To his Books_ 136
SAMUEL WADDINGTON. [1]_Literature and Nature_ 138
JOHN GREENLEAF WHITTIER. _The Library_ 139
TOMAS YRIARTE. _The Country Squire_ 141
ANONYMOUS. _Old Books_ 144
_APPENDIX._
GEORGE CRABBE. _The Library_ 149
A FINAL WORD. [1]_The Collector to his Library_
(Austin Dobson) 173
[Illustration]
[1] The poems thus marked were written or translated for the present
collection.
=Ballads of Books=
BALLADS OF BOOKS.
THE BABY IN THE LIBRARY.
EDWARD D. ANDERSON. _From 'Wide-Awake' for May, 1885._
Within these solemn, book-lined walls,
Did mortal ever see
A critic so unprejudiced,
So full of mirthful glee?
Just watch her at that lower shelf:
See, there she's thumped her nose
Against the place where Webster stands
In dignified repose.
Such heavy books she scorns; and she
Considers Vapereau,
And Beeton, too, though full of life,
Quite stupid, dull, and slow.
She wants to take a higher flight,
Aspiring little elf!
And on her mother's arm at length
She gains a higher shelf.
But, oh! what liberties she takes
With those grave, learned men;
Historians, and scientists,
And even "Rare old Ben!"
At times she takes a spiteful turn,
And pommels, with her fists,
De Quincey, Jeffrey, and Carlyle,
And other essayists.
And, when her wrath is fully roused,
And she's disposed for strife,
It almost looks as if she'd like
To take Macaulay's 'Life.'
Again, in sympathetic mood,
She gayly smiles at Gay,
And punches Punch, and frowns at Sterne
In quite a dreadful way.
In vain the Sermons shake their heads:
She does not care for these;
But catches, with intense delight,
At all the Tales she sees.
Where authors chance to meet her views,
Just praise they never lack;
To comfort and encourage them,
She pats them on the back.
MY BOOKS.
FRANCIS BENNOCH. _From the 'Storm and Other Poems.' 1878._
I love my books as drinkers love their wine;
The more I drink, the more they seem divine;
With joy elate my soul in love runs o'er,
And each fresh draught is sweeter than before.
Books bring me friends where'er on earth I be,--
Solace of solitude,--bonds of society!
I love my books! they are companions dear,
Sterling in worth, in friendship most sincere;
Here talk I with the wise in ages gone,
And with the nobly gifted of our own.
If love, joy, laughter, sorrow please my mind,
Love, joy, grief, laughter in my books I find.
THE ART OF BOOK-KEEPING.
LAMAN BLANCHARD. _From his 'Poetical Works.' 1876._
How hard, when those who do not wish
To lend, that's lose, their books,
Are snared by anglers--folks that fish
With literary hooks;
Who call and take some favorite tome,
But never read it through,--
They thus complete their set at home,
By making one at you.
Behold the bookshelf of a dunce
Who borrows--never lends:
Yon work, in twenty volumes, once
Belonged to twenty friends.
New tales and novels you may shut
From view--'tis all in vain;
They're gone--and though the leaves are "cut"
They never "come again."
For pamphlets lent I look around,
For tracts my tears are spilt;
But when they take a book that's bound,
'Tis surely extra-gilt.
A circulating library
Is mine--my birds are flown;
There's one odd volume left to be
Like all the rest, a-lone.
I, of my Spenser quite bereft,
Last winter sore was shaken;
Of Lamb I've but a quarter left,
Nor could I save my Bacon.
My Hall and Hill were levelled flat,
But Moore was still the cry;
And then, although I threw them Sprat,
They swallowed up my Pye.
O'er everything, however slight,
They seized some airy trammel;
They snatched my Hogg and Fox one night,
And pocketed my Campbell.
And then I saw my Crabbe at last,
Like Hamlet's, backward go;
And, as my tide was ebbing fast,
Of course I lost my Rowe.
I wondered into what balloon
My books their course had bent;
And yet, with all my marvelling, soon
I found my Marvell went.
My Mallet served to knock me down,
Which makes me thus a talker;
And once, while I was out of town,
My Johnson proved a Walker.
While studying o'er the fire one day
My Hobbes amidst the smoke,
They bore my Colman clean away,
And carried off my Coke.
They picked my Locke, to me far more
Than Bramah's patent's worth;
And now my losses I deplore
Without a Home on earth.
If once a book you let them lift,
Another they conceal;
For though I caught them stealing Swift,
As swiftly went my Steele.
Hope is not now upon my shelf,
Where late he stood elated;
But, what is strange, my Pope himself
Is excommunicated.
My little Suckling in the grave
Is sunk to swell the ravage;
And what 'twas Crusoe's fate to save
'Twas mine to lose--a Savage.
Even Glover's works I cannot put
My frozen hands upon;
Though ever since I lost my Foote
My Bunyan has been gone.
My Hoyle with Cotton went; oppressed,
My Taylor too must sail;
To save my Goldsmith from arrest,
In vain I offered Bayle.
I Prior sought, but could not see
The Hood so late in front;
And when I turned to hunt for Lee,
Oh! where was my Leigh Hunt.
I tried to laugh, old Care to tickle,
Yet could not Tickell touch;
And then, alas! I missed my Mickle,
And surely mickle's much.
'Tis quite enough my griefs to feed,
My sorrows to excuse,
To think I cannot read my Reid,
Nor even use my Hughes.
To West, to South, I turn my head,
Exposed alike to odd jeers;
For since my Roger Ascham's fled,
I ask 'em for my Rogers.
They took my Horne--and Horne Tooke, too,
And thus my treasures flit;
I feel, when I would Hazlitt view,
The flames that it has lit.
My word's worth little, Wordsworth gone,
If I survive its doom;
How many a bard I doated on
Was swept off--with my Broome.
My classics would not quiet lie,
A thing so fondly hoped;
Like Dr. Primrose, I may cry,
"My Livy has eloped!"
My life is wasting fast away--
I suffer from these shocks;
And though I've fixed a lock on Gray,
There's gray upon my locks.
I'm far from young--am growing pale--
I see my Butter fly;
And when they ask about my _ail_,
'Tis Burton! I reply.
They still have made me slight returns,
And thus my griefs divide;
For oh! they've cured me of my Burns,
And eased my Akenside.
But all I think I shall not say,
Nor let my anger burn;
For as they never found me Gay,
They have not left me Sterne.
IN THE LIBRARY.
ANNE C. L. BOTTA. _From her collected 'Poems.' 1882._
Speak low--tread softly through these halls;
Here genius lives enshrined,--
Here reign, in silent majesty,
The monarchs of the mind.
A mighty spirit-host, they come
From every age and clime;
Above the buried wrecks of years
They breast the tide of time.
And in their presence-chamber here
They hold their regal state,
And round them throng a noble train,
The gifted and the great.
O child of earth, when round thy path
The storms of life arise,
And when thy brothers pass thee by
With stern, unloving eyes,--
Here shall the Poets chant for thee
Their sweetest, loftiest lays;
And Prophets wait to guide thy steps
In wisdom's pleasant ways.
Come, with these God-anointed kings
Be thou companion here,
And in the mighty realm of mind
Thou shalt go forth a peer.
MY SHAKSPERE.
H. C. BUNNER. _Written expressly for this collection._
With bevelled binding, with uncut edge,
With broad white margin and gilded top,
Fit for my library's choicest ledge,
Fresh from the bindery, smelling of shop,
In tinted cloth, with a strange design--
Buskin and scroll-work and mask and crown,
And an arabesque legend tumbling down--
"The Works of Shakspere" were never so fine.
Fresh from the shop! I turn the page--
Its "ample margin" is wide and fair--
Its type is chosen with daintiest care;
There's a "New French Elzevir" strutting there
That would shame its prototypic age.
Fresh from the shop! O Shakspere mine,
I've half a notion you're much too fine!
There's an ancient volume that I recall,
In foxy leather much chafed and worn;
Its back is broken by many a fall,
The stitches are loose and the leaves are torn;
And gone is the bastard-title, next
To the title-page scribbled with owners' names,
That in straggling old-style type proclaims
That the work is from the corrected text
Left by the late Geo. Steevens, Esquire.
The broad sky burns like a great blue fire,
And the Lake shines blue as shimmering steel,
And it cuts the horizon like a blade--
But behind the poplar's a strip of shade--
The great tall Lombardy on the lawn.
And lying there in the grass, I feel
The wind that blows from the Canada shore,
And in cool, sweet puffs comes stealing o'er,
Fresh as any October dawn.
I lie on my breast in the grass, my feet
Lifted boy-fashion, and swinging free,
The old brown Shakspere in front of me.
And big are my eyes, and my heart's a-beat;
And my whole soul's lost--in what?--who knows?
Perdita's charms or Perdita's woes--
Perdita fairy-like, fair and sweet.
Is any one jealous, I wonder, now,
Of my love for Perdita? For I vow
I loved her well. And who can say
That life would be quite the same life to-day--
That Love would mean so much, if she
Had not taught me its A B C?
The Grandmother, thin and bent and old,
But her hair still dark and her eyes still bright,
Totters around among her flowers--
Old-fashioned flowers of pink and white;
And turns with a trowel the dark rich mould
That feeds the blooms of her heart's delight.
Ah me! for her and for me the hours
Go by, and for her the smell of earth--
And for me the breeze and a far love's birth,
And the sun and the sky and all the things
That a boy's heart hopes and a poet sings.
Fresh from the shop! O Shakspere mine,
It wasn't the binding made you divine!
I knew you first in a foxy brown,
In the old, old home, where I laid me down,
In the idle summer afternoons,
With you alone in the odorous grass,
And set your thoughts to the wind's low tunes,
And saw your children rise up and pass--
And dreamed and dreamed of the things to be,
Known only, I think, to you and me.
I've hardly a heart for you dressed so fine--
Fresh from the shop, O Shakspere mine!
THE BOOKWORMS.
_Burns saw a splendidly bound but sadly
neglected copy of Shakspere in the_
ROBERT BURNS. _library of a nobleman in Edinburgh,
and he wrote these lines on the ample
margin of one of its pages, where they
were found long after the poet's death._
Through and through the inspired leaves,
Ye maggots, make your windings;
But oh, respect his lordship's taste,
And spare the golden bindings.
CATULLUS TO HIS BOOK.
QVOI DONO LEPIDVM NOVVM LIBELLVM.
CAIUS VALERIUS CATULLUS. _Translated by A. Lang expressly
for this collection._
My little book, that's neat and new,
Fresh polished with dry pumice stone,
To whom, Cornelius, but to you,
Shall _this_ be sent, for you alone--
(Who used to praise my lines, my own)--
Have dared, in weighty volumes three,
(What labors, Jove, what learning thine!)
To tell the Tale of Italy,
And all the legend of our line.
So take, whate'er its worth may be,
My Book,--but Lady and Queen of Song,
This one kind gift I crave of thee,
That it may live for ages long!
OLD BOOKS ARE BEST.
TO J. H. P.
BEVERLY CHEW. _From the 'Critic' of March 13, 1886._
Old Books are best! With what delight
Does "Faithorne fecit" greet our sight
On frontispiece or title-page
Of that old time, when on the stage
"Sweet Nell" set "Rowley's" heart alight!
And you, O Friend, to whom I write,
Must not deny, e'en though you might,
Through fear of modern pirate's rage,
Old Books are best.
What though the prints be not so bright,
The paper dark, the binding slight?
Our author, be he dull or sage,
Returning from that distant age
So lives again, we say of right:
Old Books are best.
THE FORGOTTEN BOOKS.
THOMAS S. COLLIER. _Written expressly for this collection._
Hid by the garret's dust, and lost
Amid the cobwebs wreathed above,
They lie, these volumes that have cost
Such weeks of hope and waste of love.
The Theologian's garnered lore
Of Scripture text, and words divine;
And verse, that to some fair one bore
Thoughts that like fadeless stars would shine;
The grand wrought epics, that were born
From mighty throes of heart and brain,--
Here rest, their covers all unworn,
And all their pages free from stain.
Here lie the chronicles that told
Of man, and his heroic deeds--
Alas! the words once "writ in gold"
Are tarnished so that no one reads.
And tracts that smote each other hard,
While loud the friendly plaudits rang,
All animosities discard,
Where old, moth-eaten garments hang.
The heroes that were made to strut
In tinsel on "life's mimic stage"
Found, all too soon, the deepening rut
Which kept them silent in the page;
And heroines, whose loveless plight
Should wake the sympathetic tear,
In volumes sombre as the night
Sleep on through each succeeding year.
Here Phyllis languishes forlorn,
And Strephon waits beside his flocks,
And early huntsmen wind the horn,
Within the boundaries of a box.
Here, by the irony of fate,
Beside the "peasant's humble board,"
The monarch "flaunts his robes of state,"
And spendthrifts find the miser's hoard.
Days come and go, and still we write,
And hope for some far happier lot
Than that our work should meet this blight--
And yet--some books must be forgot.
AN INVOCATION IN A LIBRARY.
HELEN GRAY CONE. _From 'Oberon and Puck.' 1885._
O brotherhood, with bay-crowned brows undaunted,
Who passed serene along our crowded ways,
Speak with us still! For we, like Saul, are haunted:
Harp sullen spirits from these later days!
Whate'er high hope ye had for man your brother,
Breathe it, nor leave him, like a prisoned slave,
To stare through bars upon a sight no other
Than clouded skies that lighten on a grave.
In these still alcoves give us gentle meeting,
From dusky shelves kind arms about us fold,
Till the New Age shall feel her cold heart beating
Restfully on the warm heart of the Old:
Till we shall hear your voices, mild and winning
Steal through our doubt and discord, as outswells
At fiercest noon, above a city's dinning,
The chiming music of cathedral bells:
Music that lifts the thought from trodden places,
And coarse confusions that around us lie,
Up to the calm of high, cloud-silvered spaces,
Where the tall spire points through the soundless sky.
CONCERNING THE HONOR OF BOOKS.
_This sonnet, prefixed to the second edition
of Florio's Montaigne, 1613, is_
SAMUEL DANIEL. _generally attributed to the translator,
but the best critics now incline
to the belief that it is by his friend,
Daniel._
Since honor from the honorer proceeds,
How well do they deserve, that memorize
And leave in books for all posterity
The names of worthies and their virtuous deeds;
When all their glory else, like water-weeds
Without their element, presently dies,
And all their greatness quite forgotten lies,
And when and how they flourished no man heeds;
How poor remembrances are statues, tombs,
And other monuments that men erect
To princes, which remain in closed rooms,
Where but a few behold them, in respect
Of books, that to the universal eye
Show how they lived; the other where they lie!
LINES.
ISAAC D'ISRAELI. _Imitated from Rantzau, the founder
of the library at Copenhagen._
Golden volumes! richest treasures!
Objects of delicious pleasures!
You my eyes rejoicing please,
You my hands in rapture seize!
Brilliant wits, and musing sages,
Lights who beamed through many ages,
Left to your conscious leaves their story,
And dared to trust you with their glory;
And now their hope of fame achieved!
Dear volumes! you have not deceived!
MY BOOKS.
AUSTIN DOBSON. _From 'At the Sign of the Lyre.' 1885._
They dwell in the odor of camphor,
They stand in a Sheraton shrine,
They are "warranted early editions,"
These worshipful tomes of mine;--
In their creamy "Oxford vellum,"
In their redolent "crushed Levant,"
With their delicate watered linings,
They are jewels of price, I grant;--
Blind-tooled and morocco-jointed,
They have Bedford's daintiest dress,
They are graceful, attenuate, polished,
But they gather the dust, no less;--
For the row that I prize is yonder,
Away on the unglazed shelves,
The bulged and the bruised _octavos_,
The dear and the dumpy twelves,--
Montaigne with his sheepskin blistered,
And Howell the worse for wear,
And the worm-drilled Jesuits' Horace,
And the little old cropped Moliere,--
And the Burton I bought for a florin,
And the Rabelais foxed and flea'd,--
For the others I never have opened,
But those are the ones I read.
TO A MISSAL OF THE THIRTEENTH CENTURY.
AUSTIN DOBSON. _From 'At the Sign of the Lyre.' 1885._
Missal of the Gothic age,
Missal with the blazoned page,
Whence, O Missal, hither come,
From what dim scriptorium?
Whose the name that wrought thee thus,
Ambrose or Theophilus,
Bending, through the waning light,
O'er thy vellum scraped and white;
Weaving 'twixt thy rubric lines
Sprays and leaves and quaint designs:
Setting round thy border scrolled
Buds of purple and of gold?
Ah!--a wondering brotherhood,
Doubtless, round that artist stood,
Strewing o'er his careful ways
Little choruses of praise;
Glad when his deft hand would paint
Strife of Sathanas and Saint,
Or in secret coign entwist
Jest of cloister humorist.
Well the worker earned his wage,
Bending o'er the blazoned page!
Tired the hand and tired the wit
Ere the final _Explicit_!
Not as ours the books of old--
Things that steam can stamp and fold;
Not as ours the books of yore--
Rows of type, and nothing more.
Then a book was still a Book,
Where a wistful man might look,
Finding something through the whole,
Beating--like a human soul.
In that growth of day by day,
When to labor was to pray,
Surely something vital passed
To the patient page at last;
Something that one still perceives
Vaguely present in the leaves;
Something from the worker lent;
Something mute--but eloquent!
THE BOOK-PLATE'S PETITION.
BY A GENTLEMAN OF THE TEMPLE.
AUSTIN DOBSON. _Published originally in 'Notes and
Queries,' January 8, 1881._
While cynic CHARLES still trimm'd the vane
'Twixt _Querouaille_ and _Castlemaine_,
In days that shocked JOHN EVELYN,
My First Possessor fix'd me in.
In days of _Dutchmen_ and of frost,
The narrow sea with JAMES I crossed;
Returning when once more began
The Age of _Saturn_ and of ANNE.
I am a part of all the past;
I knew the GEORGES, first and last;
I have been oft where else was none
Save the great wig of ADDISON;
And seen on shelves beneath me grope
The little eager form of POPE.
I lost the Third that own'd me when
French NOAILLES fled at Dettingen;
The year JAMES WOLFE surpris'd Quebec,
The Fourth in hunting broke his neck;
The day that WILLIAM HOGARTH dy'd,
The Fifth one found me in Cheapside.
This was a _Scholar_, one of those
Whose _Greek_ is sounder than their _hose_;
He lov'd old books, and nappy ale,
So liv'd at Streatham, next to THRALE.
'Twas there this stain of grease I boast
Was made by DR. JOHNSON'S toast.
(He did it, as I think, for spite;
My Master called him _Jacobite_!)
And now that I so long to-day
Have rested _post discrimina_,
Safe in the brass-wir'd book-case where
I watched the Vicar's whit'ning hair
Must I these travell'd bones inter
In some _Collector's_ sepulchre!
Must I be torn from hence and thrown
With _frontispiece_ and _colophon_!
With vagrant _E_'s, and _I_'s and _O_'s,
The spoil of plunder'd _Folios_!
With scraps and snippets that to Me
Are naught but _kitchen company_!
Nay, rather, Friend, this favor grant me;
Tear me at once; _but don't transplant me_.
CHELTENHAM,
_Sept. 31, 1792._
OVER THE THRESHOLD OF MY LIBRARY.
_Quoted from the supplement of Dibdin's_
HENRY DRURY. _'Bibliomania,' where the original
Latin lines may be found._
From mouldering Abbey's dark Scriptorium brought,
See vellum tomes by Monkish labor wrought;
Nor yet the comma born, Papyri see,
And uncial letters' wizard grammary:
View my _fifteeners_ in their ragged line;
What ink! What linen! Only known long syne--
Entering where Aldus might have fixed his throne,
Or Harry Stephens coveted his own.
THE CHRYSALIS OF A BOOKWORM.
MAURICE F. EGAN. _From 'Songs and Sonnets.' 1885._
I read, O friend, no pages of old lore,
Which I loved well, and yet the flying days,
That softly passed as wind through green spring ways
And left a perfume, swift fly as of yore,
Though in clear Plato's stream I look no more,
Neither with Moschus sing Sicilian lays,
Nor with bold Dante wander in amaze,
Nor see our Will the Golden Age restore.
I read a book to which old books are new,
And new books old. A living book is mine--
In age, three years: in it I read no lies--
In it to myriad truths I find the clew--
A tender, little child: but I divine
Thoughts high as Dante's in its clear blue eyes.
EPIGRAM.
EVENUS (the grammarian). _Rendered into English by A. Lang
in the 'Library.' 1881._
Pest of the Muses, devourer of pages, in crannies hat lurkest,
Fruits of the Muses to taint, labor of learning to spoil;
Wherefore, O black-fleshed worm! wert thou born
for the evil thou workest?
Wherefore thine own foul form shap'st thou with envious toil?
THE BIBLIOMANIA.
Hic, inquis, veto quisquam fuit oletum.
Pinge duos angues.
Pers. _Sat._ i. l. 108.
JOHN FERRIAR. "_An Epistle to Richard Heber, Esq."
Manchester, April, 1809_.
What wild desires, what restless torments
seize The hapless man, who feels the book-disease,
If niggard Fortune cramp his gen'rous mind
And Prudence quench the Spark by heaven assign'd!
With wistful glance his aching eyes behold
The Princeps-copy, clad in blue and gold,
Where the tall Book-case, with partition thin,
Displays, yet guards the tempting charms within:
So great Facardin view'd, as sages[2] tell,
Fair Crystalline immur'd in lucid cell.
Not thus the few, by happier fortune grac'd,
And blest, like you, with talents, wealth, and taste,
Who gather nobly, with judicious hand,
The Muse's treasures from each letter'd strand.
For you the Monk illum'd his pictur'd page,
For you the press defies the Spoils of age;
FAUSTUS for you infernal tortures bore,
For you ERASMUS[3] starv'd on Adria's shore.
The FOLIO-ALDUS loads your happy Shelves,
And dapper ELZEVIRS, like fairy elves,
Shew their light forms amidst the well-gilt Twelves:
In slender type the GIOLITOS shine,
And bold BODONI stamps his Roman line.
For you the LOUVRE opes its regal doors,
And either DIDOT lends his brilliant stores:
With faultless types, and costly sculptures bright,
IBARRA'S Quixote charms your ravish'd sight:
LABORDE in splendid tablets shall explain
Thy beauties, glorious, tho' unhappy SPAIN!
O, hallowed name, the theme of future years,
Embalm'd in Patriot-blood, and England's tears,
Be thine fresh honors from the tuneful tongue,
By Isis' stream which mourning Zion sung!
But devious oft' from ev'ry classic Muse,
The keen Collector meaner paths will choose:
And first the Margin's breadth his soul employs,
Pure, snowy, broad, the type of nobler joys.
In vain might HOMER roll the tide of song,
Or HORACE smile, or TULLY charm the throng;
If crost by Pallas' ire, the trenchant blade
Or too oblique, or near, the edge invade,
The Bibliomane exclaims, with haggard eye,
"No Margin!" turns in haste, and scorns to buy.
He turns where PYBUS rears his Atlas-head,
Or MADOC'S mass conceals its veins of lead.
The glossy lines in polish'd order stand,
While the vast margin spreads on either hand,
Like Russian wastes, that edge the frozen deep,
Chill with pale glare, and lull to mortal sleep.[4]
Or English books, neglected and forgot,
Excite his wish in many a dusty lot:
Whatever trash _Midwinter_ gave to day,
Or _Harper's_ rhiming sons, in paper gray,
At ev'ry auction, bent on fresh supplies,
He cons his Catalogue with anxious eyes:
Where'er the slim Italics mark the page,
_Curious and rare_ his ardent mind engage.
Unlike the Swans, in Tuscan Song display'd,
He hovers eager o'er Oblivion's Shade,
To snatch obscurest names from endless night,
And give COKAIN or FLETCHER[5] back to light.
In red morocco drest he loves to boast
The bloody murder, or the yelling ghost;
Or dismal ballads, sung to crouds of old,
Now cheaply bought for thrice their weight in gold.
Yet to th' unhonor'd dead be Satire just;
Some flow'rs[6] "smell sweet and blossom in their dust."
'Tis thus ev'n SHIRLEY boasts a golden line,
And LOVELACE strikes, by fits, a note divine.
Th' unequal gleams like midnight-lightnings play,
And deepen'd gloom succeeds, in place of day.
But human bliss still meets some envious storm;
He droops to view his PAYNTERS' mangled form:
Presumptuous grief, while pensive Taste repines
O'er the frail relics of her Attic Shrines!
O for that power, for which Magicians vye.
To look through earth, and secret hoards descry!
I'd spurn such gems as Marinel[7] beheld,
And all the wealth Aladdin's cavern held,
Might I divine in what mysterious gloom
The rolls of sacred bards have found their tomb:
Beneath what mould'ring tower, or waste champain,
Is hid MENANDER, sweetest of the train:
Where rests ANTIMACHUS' forgotten lyre,
Where gentle SAPPHO'S still seductive fire;
Or he,[8] whom chief the laughing Muses own,
Yet skill'd with softest accents to bemoan
Sweet Philomel[9] in strains so like her own.
The menial train has prov'd the Scourge of wit,
Ev'n OMAR burnt less Science than the spit.
Earthquakes and wars remit their deadly rage,
But ev'ry feast demands some fated page.
Ye Towers of Julius,[10] ye alone remain
Of all the piles that saw our nation's stain,
When HARRY'S sway opprest the groaning realm,
And Lust and Rapine seiz'd the wav'ring helm.
Then ruffian-hands defaced the sacred fanes,
Their saintly statues and their storied panes;
Then from the chest, with ancient art embost,
The Penman's pious scrolls were rudely tost;
Then richest manuscripts, profusely spread,
The brawny Churls' devouring Oven fed:
And thence Collectors date the heav'nly ire
That wrapt Augusta's domes in sheets of fire.[11]
Taste, tho' misled, may yet some purpose gain,
But Fashion guides a book-compelling train.[12]
Once, far apart from Learning's moping crew,
The travell'd beau display'd his red-heel'd shoe,
Till ORFORD rose, and told of rhiming Peers,
Repeating _noble_ words to polish'd ears;[13]
Taught the gay croud to prize a fluttering name,
In trifling toil'd, nor "blush'd to find it fame."
The letter'd <DW2>, now takes a larger scope,
With classic furniture, design'd by HOPE,
(HOPE whom Upholst'rers eye with mute despair,
The doughty pedant of an elbow-chair;)
Now warm'd by ORFORD, and by GRANGER school'd,
In Paper-books, superbly gilt and tool'd,
He pastes, from injur'd volumes snipt away,
His _English Heads_, in chronicled array.
Torn from their destin'd page (unworthy meed
Of knightly counsel, and heroic deed)
Not FAITHORNE'S stroke, nor FIELD'S own types can save
[14] The gallant Veres, and one-eyed OGLE brave.
Indignant readers seek the image fled,
And curse the busy fool, who _wants a head_.
Proudly he shews, with many a smile elate,
The scrambling subjects of the _private plate_;
While Time their actions and their names bereaves,
They grin for ever in the guarded leaves.
Like Poets, born, in vain Collectors strive
To cross their Fate, and learn the art to thrive.
Like Cacus, bent to tame their struggling will,
The Tyrant-passion drags them backward still:
Ev'n I, debarr'd of ease, and studious hours,
Confess, mid' anxious toil, its lurking pow'rs.
How pure the joy, when first my hands unfold
The small, rare volume, black with tarnish'd gold!
The Eye skims restless, like the roving bee,
O'er flowers of wit, or song, or repartee,
While sweet as Springs, new-bubbling from the stone,
Glides through the breast some pleasing theme unknown.
Now dipt in ROSSI'S[15] terse and classic style,
His harmless tales awake a transient smile.
Now BOUCHET'S motley stores my thoughts arrest,
With wond'rous reading, and with learned jest.
Bouchet[16] whose tomes a grateful line demand,
The valued gift of STANLEY'S lib'ral hand.
Now sadly pleased, through faded Rome I stray,
And mix regrets with gentle DU BELLAY;[17]
Or turn, with keen delight, the curious page,
Where hardy Pasquin[18] braves the Pontiff's rage.
But D----n's strains should tell the sad reverse,
When Business calls, invet'rate foe to verse!
Tell how "the Demon claps his iron hands,"
"Waves his lank locks, and scours along the lands."
Through wintry blasts, or summer's fire I go,
To scenes of danger, and to sights of woe.
Ev'n when to Margate ev'ry Cockney roves,
And brainsick-poets long for shelt'ring groves,
Whose lofty shades exclude the noontide glow,
While Zephyrs breathe, and waters trill below,[19]
Me rigid Fate averts, by tasks like these,
From heav'nly musings, and from letter'd ease.
Such wholesome checks the better Genius sends,
From dire rehearsals to protect our friends:
Else when the social rites our joys renew,
The stuff'd Portfolio would alarm your view,
Whence volleying rhimes your patience would o'er-come,
And, spite of kindness, drive you early home.
So when the traveller's hasty footsteps glide
Near smoking lava on Vesuvio's side,
Hoarse-mutt'ring thunders from the depths proceed,
And spouting fires incite his eager speed.
Appall'd he flies, while rattling show'rs invade,
Invoking ev'ry Saint for instant aid:
Breathless, amaz'd, he seeks the distant shore,
And vows to tempt the dang'rous gulph no more.
[2] _Sages_, Count Hamilton, in the 'Quatre Facardins,' and Mr. M.
Lewis, in his 'Tales of Romance.'
[3] See the 'Opulentia Sordida,' in his 'Colloquies,' where he
complains feelingly of the spare Venetian diet.
[4] It may be said that Quintilian recommends margins; but it is with
a view to their being occasionally occupied: Debet vacare etiam
locus, in quo notentur quae scribentibus solent extra ordinem, id est
ex aliis quam qui sunt in manibus loci, occurrere. Irrumpunt enim
optimi nonnunquam Sensus, quos neque inserere oportet, neque differre
tutum est. 'Instit.' lib. x. c. 3.
He was therefore no _Margin-man_, in the modern sense.
[5] _Fletcher._ A translator of Martial. A very bad Poet, but
_exceedingly scarce_.
[6] Only the actions of the just
Smell sweet, and blossom in the dust.
SHIRLEY.
Perhaps Shirley had in view this passage of Persius,--
Nunc non e tumulo, fortunataque favilla
Nascentur Violae?
'Sat.' i. l. 37.
[7] 'Faerie Queene.'
[8] Aristophanes.
[9] See his exquisite hymn to the Nightingale in his =Ornithes=.
[10] Gray.
[11] The fire of London.
[12] Cloud-compelling Jove.--Pope's 'Iliad.'
[13] . . . gaudent praenomine molles
Auriculae.
JUVENAL.
[14] _The gallant Veres and one-eyed Ogle._ Three fine heads, for the
sake of which, the beautiful and interesting 'Commentaries' of Sir
Francis Veres have been mutilated by the Collectors of English
portraits.
[15] Generally known by the name of James Nicius Erythraeus. The
allusion is to his 'Pinacotheca.'
[16] 'Les Serees de Gillaume Bouchet,' a book of uncommon rarity. I
possess a handsome copy by the kindness of Colonel Stanley.
[17] 'Les Regrets,' by Joachim du Bellay, contain a most amusing and
instructive account of Rome in the sixteenth century.
[18] 'Pasquillorum Tomi duo.'
[19] Errare per lucos, aemaenae,
Quos et aquae subeunt et aurae.
HORAT.
TRIOLET TO HER HUSBAND.
F. FERTIAULT. _Rendered into English by A. Lang in
the 'Library.' 1881._
Books rule thy mind, so let it be!
Thy heart is mine, and mine alone.
What more can I require of thee?
Books rule thy mind, so let it be!
Contented when thy bliss I see,
I wish a world of books thine own.
Books rule thy mind, so let it be!
Thy heart is mine, and mine alone.
A NOOK AND A BOOK.
WILLIAM FREELAND. _From 'A Birth Song and other Poems.' 1882._
Give me a nook and a book,
And let the proud world spin round;
Let it scramble by hook or by crook
For wealth or a name with a sound.
You are welcome to amble your ways,
Aspirers to place or to glory;
May big bells jangle your praise,
And golden pens blazon your story!
For me, let me dwell in my nook,
Here by the curve of this brook,
That croons to the tune of my book,
Whose melody wafts me forever
On the waves of an unseen river.
Give me a book and a nook
Far away from the glitter and strife;
Give me a staff and a crook,
The calm and the sweetness of life;
Let me pause--let me brood as I list,
On the marvels of heaven's own spinning--
Sunlight and moonlight and mist,
Glorious without slaying or sinning.
Vain world, let me reign in my nook,
King of this kingdom, my book,
A region by fashion forsook;
Pass on, ye lean gamblers for glory,
Nor mar the sweet tune of my story!
THE SULTAN OF MY BOOKS.
There is many a true word spoken in doggerel.--_Czech Folk-Song._
EDMUND GOSSE. _Written for the present collection._
Come hither, my Wither,
My Suckling, my Dryden!
My Hudibras, hither!
My Heinsius from Leyden!
Dear Play-books in quarto,
Fat tomes in brown leather,
Stray never too far to
Come back here together!
Books writ on occult and
Heretical letters,
I, I am the Sultan
Of you and your betters.
I need you all round me;
When wits have grown muddy,
My best hours have found me
With you in my study.
I've varied departments
To give my books shelter;
Shelves, open apartments
For tomes helter-skelter;
There are artisans' flats, fit
For common editions,--
I find them, as that's fit,
Good wholesome positions.
But books that I cherish
Live under glass cases;
In the waste lest they perish
I build them oases;
Where gas cannot find them,
Where worms cannot grapple,
Those panes hold behind them,
My eye and its apple.
And here you see flirting
Fine folks of distinction:
Unique books just skirting
The verge of extinction;
Old texts with one error
And long notes upon it;
The 'Magistrates' Mirror'
(With Nottingham's sonnet);
Tooled Russias to gaze on,
Moroccos to fondle,
My Denham, in blazon,
My vellum-backed Vondel,
My Marvell,--a copy
Was never seen taller,--
My Jones's 'Love's Poppy,'
My dear little Waller;
My Sandys, a real jewel!
My exquisite, 'Adamo!'
My Dean Donne's 'Death's Duel!'
My Behn (naughty madam O!);
Ephelia's! Orinda's!
Ma'am Pix and Ma'am Barker!--
The rhymsters you find, as
The morals grow darker!
I never upbraid these
Old periwigged sinners,
Their songs and light ladies,
Their dances and dinners;
My book-shelf's a haven
From storms puritanic,--
We sure may be gay when
Of death we've no panic!
My parlor is little,
And poor are its treasures;
All pleasures are brittle,
And so are my pleasures;
But though I shall never
Be Beckford or Locker,
While Fate does not sever
The door from the knocker,
No book shall tap vainly
At latch or at lattice
(If costumed urbanely,
And worth our care, that is):
My poets from slumber
Shall rise in morocco,
To shield the new comer
From storm or sirocco.
* * * * *
I might prate thus for pages,
The theme is so pleasant;
But the gloom of the ages
Lies on me at present;
All business and fear to
The cold world I banish.
Hush! like the Ameer, to
My harem I vanish!
OUR BOOK-SHELVES.
THOMAS GORDON HAKE. _From the 'State' of April 17, 1886._
What solace would those books afford,
In gold and vellum cover,
Could men but say them word for word
Who never turn them over!
Books that must know themselves by heart
As by endowment vital,
Could they their truths to us impart
Not stopping with the title!
Line after line their wisdom flows,
Page after page repeating;
Yet never on our ears bestows
A single sound of greeting.
As thus they lie upon the shelves,
Such wisdom in their pages,
Do they rehearse it to themselves,
Or rest like silent sages?
One book we know such fun invokes,
As well were worth the telling:
Must it not chuckle o'er the jokes
That it is ever spelling?
And for the Holy Bible there,
It greets us with mild teaching;
Though no one its contents may hear,
Does it not go on preaching?
TO HIS BOOK.
ROBERT HERRICK. _Prefixed to 'Hesperides.' 1648._
While thou didst keep thy candor undefiled,
Dearly I loved thee, as my first-born child;
But when I sent thee wantonly to roam
From house to house, and never stay at home;
I brake my bonds of love, and bade thee go,
Regardless whether well thou sped'st or no,
On with thy fortunes then, whate'er they be;
If good I'll smile, if bad I'll sigh for thee.
TO HIS BOOK.
ROBERT HERRICK.
Make haste away, and let one be
A friendly patron unto thee;
Lest, rapt from hence, I see thee lie
Torn for the use of pastery;
Or see thy injured leaves serve well
To make loose gowns for mackerel;
Or see the grocers, in a trice,
Make hoods of thee to serve out spice.
TO HIS BOOKS.
_Imitated by Austin Dobson from the_
Q. HORATIUS FLACCUS. _'Epistles,' i. 20, for the present
collection._
For mart and street you seem to pine
With restless glances, Book of mine!
Still craving on some stall to stand,
Fresh pumiced from the binder's hand.
You chafe at locks, and burn to quit
Your modest haunt and audience fit,
For hearers less discriminate.
I reared you up for no such fate.
Still, if you _must_ be published, go;
But mind, you can't come back, you know!
"What have I done?"--I hear you cry,
And writhe beneath some critic's eye;
'What did I want?'--when, scarce polite,
They do but yawn, and roll you tight.
And yet, methinks, if I may guess
(Putting aside your heartlessness
In leaving me, and this your home),
You should find favor, too, at Rome.
That is, they'll like you while you're young.
When you are old, you'll pass among
The Great Unwashed,--then thumbed and sped,
Be fretted of slow moths, unread,
Or to Ilerda you'll be sent,
Or Utica, for banishment!
And I, whose counsel you disdain,
At that your lot shall laugh amain,
Wryly, as he who, like a fool,
Pushed o'er the cliff his restive mule.
Stay, there is worse behind. In age
They e'en may take your babbling page
In some remotest "slum" to teach
Mere boys the rudiments of speech!
But go. When on warm days you see
A chance of listeners, speak of me.
Tell them I soared from low estate,
A freedman's son, to higher fate
(That is, make up to me in worth
What you must take in point of birth);
Then tell them that I won renown
In peace and war, and pleased the Town;
Paint me as early gray, and one
Little of stature, fond of sun,
Quick-tempered, too,--but nothing more.
Add (if they ask) I'm forty-four,
Or was, the year that over us
Both Lollius ruled and Lepidus.
SONNET.
_Found by Mr. Alexander Ireland in_
LEIGH HUNT. _the London 'Examiner' of December 24, 1815,
and not anywhere included in the poet's
collected works._
Were I to name, out of the times gone by,
The poets dearest to me, I should say,
Pulci for spirits, and a fine, free way;
Chaucer for manners, and close, silent eye;
Milton for classic taste, and harp strung high;
Spenser for luxury, and sweet, sylvan play;
Horace for chatting with, from day to day;
Shakspere for all, but most society.
But which take with me, could I take but one?
Shakspere, as long as I was unoppressed
With the world's weight, making sad thoughts intenser;
But did I wish, out of the common sun,
To lay a wounded heart in leafy rest,
And dream of things far off and healing,--Spenser.
MY BOOKS.
WILLIS FLETCHER JOHNSON. _From the Boston 'Transcript.'_
On my study shelves they stand,
Well known all to eye and hand,
Bound in gorgeous cloth of gold,
In morocco rich and old.
Some in paper, plain and cheap,
Some in muslin, calf, and sheep;
Volumes great and volumes small,
Ranged along my study wall;
But their contents are past finding
By their size or by their binding.
There is one with gold agleam,
Like the Sangreal in a dream,
Back and boards in every part
Triumph of the binder's art;
Costing more, 'tis well believed,
Than the author e'er received.
But its contents? Idle tales,
Flappings of a shallop's sails!
In the treasury of learning
Scarcely worth a penny's turning.
Here's a tome in paper plain,
Soiled and torn and marred with stain,
Cowering from each statelier book
In the darkest, dustiest nook.
Take it down, and lo! each page
Breathes the wisdom of a sage:
Weighed a thousand times in gold,
Half its worth would not be told,
For all truth of ancient story
Crowns each line with deathless glory.
On my study shelves they stand;
But my study walls expand,
As thought's pinions are unfurled,
Till they compass all the world.
Endless files go marching by,
Men of lowly rank and high,
Some in broadcloth, gem-adorned,
Some in homespun, fortune-scorned;
But God's scales that all are weighed in
Heed not what each man's arrayed in!
TO MY BOOKSELLER.
_This is from the third of the poet's books_
BEN JONSON. _of epigrams. Bucklersbury was the
street most affected by grocers and
apothecaries._
Thou that mak'st gain thy end, and wisely well,
Call'st a book good, or bad, as it doth sell,
Use mine so too; I give thee leave; but crave,
For the luck's sake, it thus much favor have,
To lie upon thy stall, till it be sought;
Not offered, as it made suit to be bought;
Nor have my title-leaf on posts or walls,
Or in cleft-sticks, advanced to make calls
For termers, or some clerk-like serving-man,
Who scarce can spell thy hard names; whose knight less can.
If without these vile arts it will not sell,
Send it to Bucklersbury, there 't will well.
TO SIR HENRY GOODYERE.
_This is the eighty-sixth of the poet's first
book of epigrams, and, like its immediate_
BEN JONSON. _predecessor, it was addressed
to a gentleman bound in bonds of
friendship to many of the men of
genius of his time._
When I would know thee, Goodyere, my thought looks
Upon thy well-made choice of friends and books;
Then do I love thee, and behold thy ends
In making thy friends books, and thy books friends:
Now must I give thy life and deed the voice
Attending such a study, such a choice;
Where, though 't be love that to thy praise doth move,
It was a knowledge that begat that love.
IN THE ALBUM OF LUCY BARTON.
CHARLES LAMB. _Written in 1824 for the daughter of his
friend Bernard Barton._
Little Book, surnamed of _white_,
Clean as yet and fair to sight,
Keep thy attribution right.
Never disproportioned scrawl;
Ugly blot, that's worse than all;
On thy maiden clearness fall!
In each letter, here designed,
Let the reader emblemed find
Neatness of the owner's mind.
Gilded margins count a sin,
Let thy leaves attraction win
By the golden rules within;
Saying fetched from sages old;
Laws which Holy Writ unfold,
Worthy to be graved in gold:
Lighter fancies not excluding;
Blameless wit, with nothing rude in,
Sometimes mildly interluding,
Amid strains of graver measure:
Virtue's self hath oft her pleasure
In sweet Muses' groves of leisure.
Riddles dark, perplexing sense;
Darker meanings of offence;
What but _shades_--he banished hence.
Whitest thoughts in whitest dress,
Candid meanings, best express
Mind of quiet Quakeress.
BALLADE OF THE BOOK-HUNTER.
A. LANG. _From 'Ballades in Blue China.' 1880._
In torrid heats of late July,
In March, beneath the bitter _bise_,
He book-hunts while the loungers fly,--
He book-hunts, though December freeze;
In breeches baggy at the knees,
And heedless of the public jeers,
For these, for these, he hoards his fees,--
Aldines, Bodonis, Elzevirs.
No dismal stall escapes his eye,
He turns o'er tomes of low degrees,
There soiled Romanticists may lie,
Or Restoration comedies;
Each tract that flutters in the breeze
For him is charged with hopes and fears,
In mouldy novels fancy sees
Aldines, Bodonis, Elzevirs!
With restless eyes that peer and spy,
Sad eyes that heed not skies nor trees,
In dismal nooks he loves to pry,
Whose motto evermore is _Spes_!
But ah! the fabled treasure flees;
Grown rarer with the fleeting years,
In rich men's shelves they take their ease,
Aldines, Bodonis, Elzevirs!
ENVOY.
Prince, all the things that tease and please,
Fame, love, wealth, kisses, cheers, and tears,
What are they but such toys as these--
Aldines, Bodonis, Elzevirs?
BALLADE OF TRUE WISDOM.
A. LANG. _From 'Ballades in Blue China.' 1880._
While others are asking for beauty or fame,
Or praying to know that for which they should pray,
Or courting Queen Venus, that affable dame,
Or chasing the Muses the weary and gray,
The sage has found out a more excellent way,--
To Pan and to Pallas his incense he showers,
And his humble petition puts up day by day,
For a house full of books, and a garden of flowers.
Inventors may bow to the God that is lame,
And crave from the light of his stithy a ray;
Philosophers kneel to the God without name,
Like the people of Athens, agnostics are they;
The hunter a fawn to Diana will slay,
The maiden wild roses will wreathe for the Hours,--
But the wise man will ask, ere libation he pay,
For a house full of books, and a garden of flowers.
Oh grant me a life without pleasure or blame
(As mortals count pleasure who rush through their day
With a speed to which that of the tempest is tame).
Oh grant me a house by the beach of a bay,
Where the waves can be surly in winter, and play
With the sea-weed in summer, ye bountiful powers!
And I'd leave all the hurry, the noise, and the fray,
For a house full of books, and a garden of flowers.
ENVOY.
Gods, give or withhold it! Your "yea" and your "nay"
Are immutable, heedless of outcry of ours:
But life _is_ worth living, and here we would stay
For a house full of books, and a garden of flowers.
BALLADE OF THE BOOKMAN'S PARADISE.
A. LANG. _From 'Rhymes a la Mode.' 1885._
There _is_ a Heaven, or here, or there,--
A Heaven there is, for me and you,
Where bargains meet for purses spare,
Like ours, are not so far and few.
Thuanus' bees go humming through
The learned groves, 'neath rainless skies,
O'er volumes old and volumes new,
Within that Bookman's Paradise!
There treasures bound for Longepierre
Keep brilliant their morocco blue,
There Hookes' 'Amanda' is not rare,
Nor early tracts upon Peru!
Racine is common as Rotrou,
No Shakspere Quarto search defies,
And Caxtons grow as blossoms grew,
Within that Bookman's Paradise!
There's Eve,--not our first mother fair,--
But Clovis Eve, a binder true;
Thither does Bauzonnet repair,
Derome, Le Gascon, Padeloup!
But never come the cropping crew,
That dock a volume's honest size,
Nor they that "letter" backs askew,
Within that Bookman's Paradise!
ENVOY.
Friend, do not Heber and De Thou,
And Scott, and Southey, kind and wise,
_La chasse au bouquin_ still pursue
Within that Bookman's Paradise?
THE ROWFANT BOOKS.
_Ballade en guise de rondeau, written for_
A. LANG. _the catalogue of Mr. Frederick Locker's
books._
The Rowfant books, how fair they show,
The Quarto quaint, the Aldine tall,
Print, autograph, portfolio!
Back from the outer air they call,
The athletes from the Tennis ball,
This Rhymer from his rod and hooks,--
Would I could sing them, one and all,--
The Rowfant books!
The Rowfant books! In sun and snow
They're dear, but most when tempests fall;
The folio towers above the row
As once, o'er minor prophets,--Saul!
What jolly jest books, and what small
"Dear dumpy Twelves" to fill the nooks.
You do not find on every stall
The Rowfant books!
The Rowfant books! These long ago
Were chained within some College hall;
These manuscripts retain the glow
Of many a capital;
While yet the satires keep their gall,
While the Pastissier puzzles cooks,
Theirs is a joy that does not pall,--
The Rowfant books!
ENVOY.
The Rowfant books,--ah, magical
As famed Armida's golden looks,
They hold the Rhymer for their thrall,--
The Rowfant books!
THE ROWFANT LIBRARY.
A. LANG. _Written for the catalogue of Mr. Frederick
Locker's books._
I mind me of the Shepherd's saw,
For, when men spoke of Heaven, quoth he,
"It's everything that's bright and braw,
But _Bourhope's_ good enough for me."
Among the green deep bosomed hills
That guard St. Mary's Loch it lies,
The silence of the pastures fills
That yeoman's homely paradise!
Enough for him his mountain lake,
His glen the burn goes singing through;
And _Rowfant_, when the thrushes wake,
Might well seem Paradise to you!
For all is old, and tried, and dear,
And all is fair, and all about
The brook that murmurs from the mere
Is dimpled with the rising trout.
And when the skies of shorter days
Are dark, and all the paths are mire,
How kindly o'er your _Books_ the blaze
Sports from the cheerful study fire;
O'er Quartos, where our Fathers read
Entranced, the Book of Shakspere's play,
O'er all that Poe has dreamed of dread,
And all that Herrick sang of gay!
Rare First Editions, duly prized,
Among them dearest far I rate
The tome where _Walton's_ hand revised
His magical receipts for bait.
Happy, who rich in toys like these
Forgets a weary nation's ills,
Who, from his study window sees
The circle of the Sussex hills!
But back to town my Muse must fly,
And taste the smoke, and list to them
Who cry the News, and seem to cry
(With each Gladstonian victory),
_Woe, woe unto Jerusalem!_[20]
[20] During the General Election, November, 1885.
GHOSTS IN THE LIBRARY.
A. LANG. _From 'Longman's Magazine,' July, 1886._
Suppose, when now the house is dumb,
When lights are out, and ashes fall,--
Suppose their ancient owners come
To claim our spoils of shop and stall,
Ah me! within the narrow hall
How strange a mob would meet and go,
What famous folk would haunt them all,
Octavo, quarto, folio!
The great Napoleon lays his hand
Upon this eagle-headed N,
That marks for his a pamphlet banned
By all but scandal-loving men,--
A libel from some nameless den
Of Frankfort--_Arnaud, a la Sphere_,
Wherein one spilt, with venal pen,
Lies o'er the loves of Moliere.[21]
Another shade--he does not see
"Boney," the foeman of his race--
The great Sir Walter, this is he
With that grave homely Border face.
He claims his poem of the chase
That rang Benvoirlich's valley through;
And _this_, that doth the lineage trace
And fortunes of the bold Buccleuch;[22]
For these were his, and these he gave
To one who dwelt beside the Peel,
That murmurs with its tiny wave
To join the Tweed at Ashestiel.
Now thick as motes the shadows wheel,
And find their own, and claim a share
Of books wherein Ribou did deal,
Or Roulland sold to wise Colbert.[23]
What famous folk of old are here!
A royal duke comes down to us,
And greatly wants his Elzevir,
His Pagan tutor, Lucius.[24]
And Beckford claims an amorous
Old heathen in morocco blue;[25]
And who demands Eobanus
But stately Jacques Auguste de Thou![26]
They come, the wise, the great, the true,
They jostle on the narrow stair,
The frolic Countess de Verrue,
Lamoignon, ay, and Longepierre,
The new and elder dead are there--
The lords of speech, and song, and pen,
Gambetta,[27] Schlegel,[28] and the rare
Drummond of haunted Hawthornden.[29]
Ah, and with those, a hundred more,
Whose names, whose deeds, are quite forgot:
Brave 'Smiths' and 'Thompsons' by the score,
Scrawled upon many a shabby 'lot.'
This play-book was the joy of Pott[30]--
Pott, for whom now no mortal grieves.
Our names, like his, remembered not,
Like his, shall flutter on fly-leaves!
At least in pleasant company
We bookish ghosts, perchance, may flit;
A man may turn a page, and sigh,
Seeing one's name, to think of it.
Beauty, or Poet, Sage, or Wit,
May ope our book, and muse awhile,
And fall into a dreaming fit,
As now we dream, and wake, and smile!
[21] 'Histoire des Intrigues Amoureuses de Moliere et de celles de sa
femme. (A la Sphere.) A Francfort, chez Frederic Arnaud, MDCXCVII.'
This anonymous tract has actually been attributed, among others, to
Racine. The copy referred to is marked with a large N in red, with an
eagle's head.
[22] 'The Lady of the Lake,' 1810.
'The Lay of the Last Minstrel,' 1806.
"To Mrs. Robert Laidlaw. Peel. From the Author."
[23] 'Dictys Cretensis.' Apud Lambertum Roulland. Lut. Paris. 1680. In
red morocco, with the arms of Colbert.
[24] 'L. Annaei Senecae Opera Omnia.' Lug. Bat., apud Elzevirios. 1649.
With book-plate of the Duke of Sussex.
[25] 'Stratonis Epigrammata.' Altenburgi, 1764. Straton bound up in
one volume with Epictetus! From the Beckford library.
[26] 'Opera Helii Eobani Hessi.' Yellow morocco, with the first arms
of De Thou. Include a poem addressed "LANGE, _decus meum_." Quantity
of penultimate "Eobanus" taken for granted, _metri gratia_.
[27] 'La Journee du Chretien.' Coutances, 1831. With inscription,
"Leon Gambetta. Rue St. Honore. Janvier 1, 1848."
[28] Villoison's 'Homer.' Venice, 1788. With Tessier's ticket and
Schlegel's book-plate.
[29] 'Les Essais de Michel.' Seigneur de Montaigne. "Pour Francois le
Febvre de Lyon, 1695." With autograph of Gul. Drummond, and _cipresso
e palma_.
[30] "The little old foxed Moliere," once the property of William
Pott, unknown to fame.
THE BOOK BATTALION.
GEORGE PARSONS LATHROP. _Written for the present collection._
Wherever I go, there's a trusty battalion
That follows me faithfully, steady, and true;
Their force, when I falter, I safely may rally on,
Knowing their stoutness will carry me through:
Some fifteen hundred in order impartial,
So ranged that they tell what they mean by their looks.
Of all the armies the world can marshal
There are no better soldiers than well-tried books.
Dumb in their ranks on the shelves imprisoned,
They never retreat. Give the word, and they'll fire!
A few with scarlet and gold are bedizened,
But many muster in rough attire;
And some, with service and scars grown wizened,
Seem hardly the mates for their fellows in youth;
Yet they, and the troops armed only with quiz and
Light laughter, all battle alike for the truth.
Here are those who gave motive to sock and to buskin;
With critics, historians, poets galore;
A cheaply uniformed set of Ruskin,
Which Ruskin would hate from his heart's very core;
Moliere ('99), an old calf-bound edition,
"_De Pierre Didot l'aine, et de Firmin Didot_."
Which, meek and demure, with a sort of contrition,
Is masking its gun-lights, with fun all aglow;
And Smollett and Fielding, as veterans battered--
Cloth stripped from their backs, and their sides out of joint,
Their pictures of life all naked and tattered
Being thus applied to themselves with a point;
And six or eight books that I wrote myself,
To look at which, even, I'm half afraid;
They brought me more labor and pleasure than pelf,
And are clamoring still because they're not paid.
But these raw levies remain still faithful,
Because they know that volumes old
Stand by me, although their eyes dim and wraithful
Remind me they seldom at profit were sold.
So I say, be they splendid or tatterdemalion,
If only you know what they mean by their looks,
You will never find a better battalion
Of soldiers to serve you than well-tried books.
ON THE FLY-LEAF OF A BOOK OF OLD PLAYS.
WALTER LEARNED. _Written for the present collection._
At Cato's-Head in Russell Street
These leaves she sat a-stitching;
I fancy she was trim and neat,
Blue-eyed and quite bewitching.
Before her, in the street below,
All powder, ruffs, and laces,
There strutted idle London beaux
To ogle pretty faces;
While, filling many a Sedan chair
With hoop and monstrous feather,
In patch and powder London's fair
Went trooping past together.
Swift, Addison, and Pope, mayhap
They sauntered slowly past her,
Or printer's boy, with gown and cap
For Steele, went trotting faster.
For beau nor wit had she a look,
Nor lord nor lady minding;
She bent her head above this book,
Attentive to her binding.
And one stray thread of golden hair,
Caught on her nimble fingers,
Was stitched within this volume, where
Until to-day it lingers.
Past and forgotten, beaux and fair;
Wigs, powder, all out-dated;
A queer antique, the Sedan chair;
Pope, stiff and antiquated.
Yet as I turn these odd old plays,
This single stray lock finding,
I'm back in those forgotten days
And watch her at her binding.
TOO MANY BOOKS.
ROBERT LEIGHTON. _From 'Reuben, and Other Poems.' 1875_
I would that we were only readers now,
And wrote no more, or in rare heats of soul
Sweated out thoughts when the o'er-burden'd brow
Was powerless to control.
Then would all future books be small and few,
And, freed of dross, the soul's refined gold;
So should we have a chance to read the new,
Yet not forego the old.
But as it is, Lord help us, in this flood
Of daily papers, books, and magazines!
We scramble blind as reptiles in the mud,
And know not what it means.
Is it the myriad spawn of vagrant tides,
Whose growth would overwhelm both sea and shore,
Yet often necessary loss, provides
Sufficient and no more?
Is it the broadcast sowing of the seeds,
And from the stones, the thorns and fertile soil,
Only enough to serve the world's great needs
Rewards the sower's toil?
Is it all needed for the varied mind?
Gives not the teeming press a book too much--
Not one, but in its dense neglect shall find
Some needful heart to touch?
Ah, who can say that even this blade of grass
No mission has--superfluous as it looks?
Then wherefore feel oppressed and cry, Alas,
There are too many books!
FROM THE FLY-LEAF OF THE ROWFANT MONTAIGNE (FLORIO, 1603).
FREDERICK LOCKER. _Written for the present collection._
Of yore, when books were few and fine,
Will Shakspere cut these leaves of mine,
But when he passed I went astray
Till bought by Pope, a gift for Gay.
Then, later on, betwixt my pages
A nose was poked--the Bolt-Court Sage's.
But though the Fame began with Rawleigh,
And had not dwindled with Macaulay,
Though still I tincture many tomes
Like Lowell's pointed sense, and Holmes',
For me the halcyon days have past--
I'm here, and with a dunce at last.
MY BOOKS.
HENRY WADSWORTH LONGFELLOW. _Written in December, 1881._
Sadly as some old mediaeval knight
Gazed at the arms he could no longer wield,
The sword two-handed and the shining shield
Suspended in the hall, and full in sight,
While secret longings for the lost delight
Of tourney or adventure in the field
Came over him, and tears but half concealed
Trembled and fell upon his beard of white,
So I behold these books upon their shelf,
My ornaments and arms of other days;
Not wholly useless, though no longer used,
For they remind me of my other self,
Younger and stronger, and the pleasant ways,
In which I walked, now clouded and confused.
THE SOULS OF BOOKS.
EDWARD BULWER, LORD LYTTON. _From 'Earlier Poems.'_
I.
Sit here and muse!--it is an antique room--
High-roof'd, with casements, through whose purple pane
Unwilling Daylight steals amidst the gloom,
Shy as a fearful stranger.
There THEY reign
(In loftier pomp than waking life had known),
The Kings of Thought!--not crown'd until the grave.
When Agamemnon sinks into the tomb,
The beggar Homer mounts the Monarch's throne!
Ye ever-living and imperial Souls,
Who rule us from the page in which ye breathe,
All that divide us from the clod ye gave!--
Law--Order--Love--Intelligence--the Sense
Of Beauty--Music and the Minstrel's wreath!--
What were our wanderings if without your goals?
As air and light, the glory ye dispense
Becomes our being--who of us can tell
What he had been, had Cadmus never taught
The art that fixes into form the thought--
Had Plato never spoken from his cell,
Or his high harp blind Homer never strung?
Kinder all earth hath grown since genial Shakspere sung!
II.
Hark! while we muse, without the walls is heard
The various murmur of the laboring crowd,
How still, within those archive-cells interr'd,
The Calm Ones reign!--and yet they rouse the loud
Passions and tumults of the circling world!
From them, how many a youthful Tully caught
The zest and ardor of the eager Bar;
From them, how many a young Ambition sought
Gay meteors glancing o'er the sands afar--
By them each restless wing has been unfurl'd,
And their ghosts urge each rival's rushing car!
They made yon Preacher zealous for the truth;
They made yon Poet wistful for the star;
Gave Age its pastime--fired the cheek of Youth--
The unseen sires of all our beings are,--
III.
And now so still! This, Cicero, is thy heart;
I hear it beating through each purple line.
This is thyself, Anacreon--yet, thou art
Wreath'd, as in Athens, with the Cnidian vine.
I ope thy pages, Milton, and, behold,
Thy spirit meets me in the haunted ground!--
Sublime and eloquent, as while, of old,
"It flamed and sparkled in its crystal bound;"[31]
These _are_ yourselves--your life of life! The Wise,
(Minstrel or Sage) _out_ of their books are clay;
But _in_ their books, as from their graves, they rise,
Angels--that, side by side, upon our way,
Walk with and warn us!
Hark! the world so loud,
And they, the movers of the world, so still!
What gives this beauty to the grave? the shroud
Scarce wraps the Poet, than at once there cease
Envy and Hate! "Nine cities claim him dead,
Through which the living Homer begg'd his bread!"
And what the charm that can such health distil
From wither'd leaves--oft poisons in their bloom?
We call some books immoral! _Do they live?_
If so, believe me, TIME hath made them pure.
In Books, the veriest wicked rest in peace--
God wills that nothing evil shall endure;
The grosser parts fly off and leave the whole,
As the dust leaves the disembodied soul!
Come from thy niche, Lucretius! Thou didst give
Man the black creed of Nothing in the tomb!
Well, when we read thee, does the dogma taint?
No; with a listless eye we pass it o'er,
And linger only on the hues that paint
The Poet's spirit lovelier than his lore.
None learn from thee to cavil with their God;
None commune with thy genius to depart
Without a loftier instinct of the heart.
Thou mak'st no Atheist--thou but mak'st the mind
Richer in gifts which Atheists best confute--
FANCY AND THOUGHT! 'Tis these that from the sod
Lift us! The life which soars above the brute
Ever and mightiest, breathes from a great Poet's lute!
Lo! that grim Merriment of Hatred;[32]--born
Of him,--the Master-Mocker of Mankind,
Beside the grin of whose malignant spleen,
Voltaire's gay sarcasm seems a smile serene,--
Do we not place it in our children's hands,
Leading young Hope through Lemuel's fabled lands?--
God's and man's libel in that foul yahoo!--
Well, and what mischief can the libel do?
O impotence of Genius to belie
Its glorious task--its mission from the sky!
Swift wrote this book to wreak a ribald scorn
On aught the Man should love or Priest should mourn--
And lo! the book, from all its ends beguil'd,
A harmless wonder to some happy child!
[31] 'Comus.'
[32] 'Gulliver's Travels.'
IV.
All books grow homilies by time; they are
Temples, at once, and Landmarks. In them, we
Who _but_ for them, upon that inch of ground
We call "THE PRESENT," from the cell could see
No daylight trembling on the dungeon bar;
Turn, as we list, the globe's great axle round,
And feel the Near less household than the Far!
Traverse all space, and number every star,
There is no Past, so long as Books shall live!
A disinterr'd Pompeii wakes again
For him who seeks yon well; lost cities give
Up their untarnish'd wonders, and the reign
Of Jove revives and Saturn:--at our will
Rise dome and tower on Delphi's sacred hill;
Bloom Cimon's trees in Academe;[33]--along
Leucadia's headland, sighs the Lesbian's song;
With AEgypt's Queen once more we sail the Nile,
And learn how worlds are barter'd for a smile:--
Rise up, ye walls, with gardens blooming o'er,
Ope but that page--lo, Babylon once more!
[33] Plut. in 'Vit. Cim.'
V.
Ye make the Past our heritage and home:
And is this all? No; by each prophet-sage--
No; by the herald souls that Greece and Rome
Sent forth, like hymns, to greet the Morning Star
That rose on Bethlehem--by thy golden page,
Melodious Plato--by thy solemn dreams,
World-wearied Tully!--and, above ye all,
By THIS, the Everlasting Monument
Of God to mortals, on whose front the beams
Flash glory-breathing day--our lights ye are
To the dark Bourne beyond; in you are sent
The types of Truths whose life is THE TO-COME;
In you soars up the Adam from the fall;
In you the FUTURE as the PAST is given--
Ev'n in our death ye bid us hail our birth;--
Unfold these pages, and behold the Heaven,
Without one gravestone left upon the Earth?
DE LIBRIS.
COSMO MONKHOUSE. _Written for the present collection._
True--there are books and books. There's Gray,
For instance, and there's Bacon;
There's Longfellow, and Monstrelet,
And also Colton's 'Lacon,'
With 'Laws of Whist' and those of Libel,
And Euclid, and the Mormon Bible.
And some are dear as friends, and some
We keep because we need them;
And some we ward from worm and thumb,
And love too well to read them.
My own are poor, and mostly new,
But I've an Elzevir or two.
That as a gift is prized, the next
For trouble in the finding;
This Aldine for its early text,
That Plantin for the binding;
This sorry Herrick hides a flower,
The record of one perfect hour.
But whether it be worth or looks
We gently love or strongly,
Such virtue doth reside in books
We scarce can love them wrongly;
To sages an eternal school,
A hobby (harmless) to the fool.
Nor altogether fool is he
Who orders, free from doubt,
Those books which "no good library
Should ever be without,"
And blandly locks the well-glazed door
On tomes that issue never more.
Less may we scorn his cases grand,
Where safely, surely linger
Fair virgin fields of type, unscanned
And innocent of finger.
There rest, preserved from dust accurst,
The first editions--and the worst.
And least of all should we that write
With easy jest deride them,
Who hope to leave when "lost to sight"
The best of us inside them,
Dear shrines! where many a scribbler's name
Has lasted--longer than his fame.
EX LIBRIS.
ARTHUR J. MUNBY. _Written for the present collection._
Man that is born of woman finds a charm
In that which he is born of. She it is
Who moulds him with a frown or with a kiss
To good or ill, to welfare or to harm:
But, when he has attain'd her soft round arm
And drawn it through his own, and made her his,
He through her eyes beholds a wider bliss,
As sweet as that she gives him, and as warm.
What bliss? We dare not name it: her fond looks
Are jealous too; she hardly understands,
Girt by her children's laughter or their cries,
The stately smooth companionship of books:
And yet to her we owe it, to her hands
And to her heart, that books can make us wise.
ON AN INSCRIPTION.
"_Edward Danenhill: Book given him
by Joseph Wise, April ye 27th, 1741,"_
ARTHUR J. MUNBY. _was the inscription in a copy of Carew's
'Poems' (1651). Written for
the present collection._
A man unknown this volume gave,
So long since, to his unknown friend,
Ages ago, their lives had end,
And each in some obscurest grave
Lies mixt with earth: none now would care
To ask or who or what they were.
But, though these two are underground,
Their book is here, all safe and sound;
And he who wrote it (yea, and more
Than a whole hundred years before)
He, the trim courtier, old Carew,
And all the loves he feign'd or knew,
Have won from Aphrodite's eye
Some show of immortality.
'Tis ever thus; by Nature's will
The gift outlasts the giver still;
And Love itself lives not so long
As doth a lover's feeblest song.
But doubly hard is that man's case,
For whom and for his earnest rhymes
Neither his own nor after-times
Have any work, have any place:
Who through a hundred years shall find
No echoing voice, no answering mind;
And, when this tann'd and tawny page
Has one more century of age,
And others buy the book anew,
Because they care for old Carew,
Not one who reads shall care or know
What name was his, who owns it now:
But all he wrote and all he did
Shall be in such oblivion hid
As hides the blurr'd and broken stones
That cover his forgotten bones.
TO MY BOOKS.
CAROLINE NORTON. _From the 'Dream and other Poems.' 1840._
Silent companions of the lonely hour,
Friends, who can never alter or forsake,
Who for inconstant roving have no power,
And all neglect, perforce, must calmly take,
Let me return to YOU; this turmoil ending
Which worldly cares have in my spirit wrought,
And, o'er your old familiar pages bending,
Refresh my mind with many a tranquil thought;
Till, haply meeting there, from time to time,
Fancies, the audible echo of my own,
'T will be like hearing in a foreign clime
My native language spoke in friendly tone,
And with a sort of welcome I shall dwell
On these, my unripe musings, told so well.
'DESULTORY READING.'
F. M. P. _From the London 'Spectator' of January 16, 1886._
O finest essence of delicious rest!
To bid for some short space the busy mill
Of anxious, ever-grinding thought be still;
And let the weary brain and throbbing breast
Be by another's cooling hand caressed.
This volume in my hand, I hold a charm
Which lifts me out of reach of wrong or harm.
I sail away from trouble; and most blessed
Of every blessing, can myself forget:
Can rise above the instance low and poor
Into the mighty law that governs yet.
This hinged cover, like a well hung door,
Shuts out the noises of the jangling day,
These fair leaves fan unwelcome thoughts away.
THE BOOKWORM.
THOMAS PARNELL. _Translated from the Latin of Theodore Beza._
Come hither, boy, we'll hunt to-day
The bookworm, ravening beast of prey,
Produc'd by parent Earth, at odds,
As fame reports it, with the gods.
Him frantic hunger wildly drives
Against a thousand authors' lives:
Through all the fields of wit he flies;
Dreadful his head with clustering eyes,
With horns without, and tusks within,
And scales to serve him for a skin.
Observe him nearly, lest he climb
To wound the bards of ancient time,
Or down the vale of fancy go
To tear some modern wretch below.
On every corner fix thine eye,
Or ten to one he slips thee by.
See where his teeth a passage eat:
We'll rouse him from his deep retreat.
But who the shelter's forc'd to give?
'Tis sacred Virgil, as I live!
From leaf to leaf, from song to song
He draws the tadpole form along,
He mounts the gilded edge before,
He's up, he scuds the cover o'er,
He turns, he doubles, there he past,
And here we have him, caught at last.
Insatiate brute, whose teeth abuse
The sweetest servants of the Muse--
Nay, never offer to deny,
I took thee in the fact to fly.
His rose nipt in every page,
My poor Anacreon mourns thy rage;
By thee my Ovid wounded lies;
By thee my Lesbia's Sparrow dies;
Thy rabid teeth have half destroy'd
The work of love in Biddy Floyd;
They rent Belinda's locks away,
And spoil'd the Blouzelind of Gay.
For all, for every single deed,
Relentless justice bids thee bleed:
Then fall a victim to the Nine
Myself the priest, my desk the shrine.
Bring Homer, Virgil, Tasso near,
To pile a sacred altar here:
Hold, boy, thy hand outruns thy wit,
You reach'd the plays that Dennis writ;
You reach'd me Philips' rustic strain;
Pray take your mortal bards again.
Come, bind the victim,--there he lies,
And here between his numerous eyes
This venerable dust I lay
From manuscripts just swept away.
The goblet in my hand I take,
For the libation's yet to make:
A health to poets! all their days
May they have bread, as well as praise;
Sense may they seek, and less engage
In papers fill'd with party rage.
But if their riches spoil their vein,
Ye Muses, make them poor again.
Now bring the weapon, yonder blade
With which my tuneful pens are made.
I strike the scales that arm thee round,
And twice and thrice I print the wound;
The sacred altar floats with red,
And now he dies, and now he's dead.
How like the son of Jove I stand,
This Hydra stretch'd beneath the hand!
Lay bare the monster's entrails here,
And see what dangers threat the year:
Ye gods! what sonnet on a wench!
What lean translations out of French!
'Tis plain, this lobe is so unsound,
S--prints, before the months go round.
But hold, before I close the scene
The sacred altar should be clean.
O had I Shadwell's second bays,
Or, Tate, thy pert and humble lays!
(Ye pair, forgive me, when I vow
I never miss'd your works till now,)
I'd tear the leaves to wipe the shrine,
That only way you please the Nine:
But since I chance to want these two,
I'll make the songs of Durfey do.
Rent from the corps, on yonder pin,
I hang the scales that brac'd it in;
I hang my studious morning gown,
And write my own inscription down.
"This trophy from the Python won,
This robe, in which the deed was done,
These, Parnell, glorying in the feat
Hung on these shelves, the Muses seat.
Here Ignorance and Hunger found
Large realms of wit to ravage round;
Here Ignorance and Hunger fell
Two foes in one I sent to hell.
Ye poets who my labors see
Come share the triumph all with me!
Ye critics, born to vex the Muse,
Go mourn the grand ally you lose!"
AMONG MY BOOKS.
SAMUEL MINTURN PECK. _From 'Cap and Bells.' 1886._
Among my books--what rest is there
From wasting woes! what balm for care!
If ills appall or clouds hang low,
And drooping, dim the fleeting show,
I revel still in visions rare.
At will I breathe the classic air,
The wanderings of Ulysses share;
Or see the plume of Bayard flow
Among my books.
Whatever face the world may wear--
If Lillian has no smile to spare,
For others let her beauty blow,
Such favors I can well forego;
Perchance forget the frowning fair
Among my books.
A RUINED LIBRARY.
WALTER HERRIES POLLOCK. _Written for the present collection._
"Imperious Caesar dead and turn'd to clay
Might stop a hole to keep the wind away."
Here the live thought of buried Caesar's brain
Has served a lazy slut to lay the train
That lights a dunce's fire. Here Homer's seen
All torn or crumpled in the pettish spleen
Of some spoilt urchin. Here a leaf from Glanvil
Is reft to mark a place in 'On the Anvil.'
Here, too, a heavy-blotted Shakspere's page
Holds up an inky mirror to the age;
Here looking round you're but too sure to see a
Heart-breaking wreck from the 'Via Jacobaea;'
Here some rare pamphlet, long a-missing, lurks
In an odd volume of 'Lord Bacon's Works;'
Here may you find a Stillingfleet or Blair
Usurp the binding of a lost Voltaire;
And here a tattered Boyle doth gape ungently
Upon a damp-disfigured 'Life of Bentley.'
Here half a Rabelais jostles for position
The quarter of a 'Spanish Inquisition;'
Here Young's 'Night Thoughts' lie mixed with Swinburne's 'Ballads'
'Mid scraps of works on Poisons and on Salads;
And here a rent and gilt-edged Sterne doth lack a ray
Of sun that falls upon a bulging Thackeray;
Here--but the tale's too sad at length to tell
How a book-heaven's been turned to a book-hell.
MY BOOKS.
BRYAN WALLER PROCTER. _From 'An Autobiographical_
(BARRY CORNWALL.) _Fragment.' 1877._
All round the room my silent servants wait,--
My friends in every season, bright and dim;
Angels and seraphim
Come down and murmur to me, sweet and low,
And spirits of the skies all come and go
Early and late;
All from the old world's divine and distant date,
From the sublimer few,
Down to the poet who but yester-eve
Sang sweet and made us grieve,
All come, assembling here in order due.
And here I dwell with Poesy, my mate,
With Erato and all her vernal sighs,
Great Clio with her victories elate,
Or pale Urania's deep and starry eyes.
O friends, whom chance and change can never harm,
Whom Death the tyrant cannot doom to die,
Within whose folding soft eternal charm
I love to lie,
And meditate upon your verse that flows,
And fertilizes whereso'er it goes,
Whether....
TO MY BOOKS ON PARTING WITH THEM.
_The sale of the famous Roscoe library,
made necessary by reverses in business,_
WILLIAM ROSCOE. _took place in August and September, 1816._
As one who, destined from his friends to part,
Regrets his loss, yet hopes again erewhile,
To share their converse and enjoy their smile,
And tempers as he may affliction's dart,--
Thus, loved associates! chiefs of elder Art!
Teachers of wisdom! who could once beguile
My tedious hours, and lighten every toil,
I now resign you; nor with fainting heart;
For pass a few short years, or days, or hours.
And happier seasons may their dawn unfold,
And all your sacred fellowship restore;
When, freed from earth, unlimited its powers,
Mind shall with mind direct communion hold,
And kindred spirits meet to part no more.
AMONG MY BOOKS.
FRANCIS ST. CLAIR-ERSKINE, _From 'Sonnets.' 1883._
EARL OF ROSSLYN.
Alone, 'midst living works of mighty dead,
Poets and Scholars versed in history's lore,
With thoughts that reached beyond them and before,
I dream, and leave their glorious works unread;
Their greatness numbs me both in heart and head.
I cannot weep with Petrarch, and still more
I fail when I would delve the depths of yore,
And learn old Truths of modern lies instead;
The shelves frown on me blackly, with a life
That ne'er can die, and helpless to begin,
I can but own my weakness, and deplore
This waste, this barren brain, ah! once so rife
With hope and fancy. Pardon all my sin,
Great Ghosts that wander on the Eternal Shore.
THE LIBRARY.
_One of the excerpts from 'Occasional_
JOHN GODFREY SAXE. _Poems' included in his 'Complete Poems.'_
Here, e'en the sturdy democrat may find,
Nor scorn their rank, the nobles of the mind;
While kings may learn, nor blush at being shown,
How Learning's patents abrogate their own.
A goodly company and fair to see;
Royal plebeians; earls of low degree;
Beggars whose wealth enriches every clime;
Princes who scarce can boast a mental dime;
Crowd here together like the quaint array
Of jostling neighbors on a market day.
Homer and Milton,--can we call them blind?--
Of godlike sight, the vision of the mind;
Shakspere, who calmly looked creation through,
"Exhausted worlds, and then imagined new;"
Plato the sage, so thoughtful and serene,
He seems a prophet by his heavenly mien;
Shrewd Socrates, whose philosophic power
Xantippe proved in many a trying hour;
And Aristophanes, whose humor run
In vain endeavor to be-"cloud" the sun;
Majestic AEschylus, whose glowing page
Holds half the grandeur of the Athenian stage;
Pindar, whose odes, replete with heavenly fire,
Proclaim the master of the Grecian lyre;
Anacreon, famed for many a luscious line
Devote to Venus and the god of wine.
I love vast libraries; yet there is a doubt
If one be better with them or without,--
Unless he use them wisely, and indeed,
Knows the high art of what and how to read,
At learning's fountain it is sweet to drink,
But 'tis a nobler privilege to think;
And oft from books apart, the thirsting mind
May make the nectar which it cannot find,
'T is well to borrow from the good and great;
'T is wise to learn; 't is godlike to create!
IN THE LIBRARY.
CLINTON SCOLLARD. _From 'With Reed and Lyre.' 1886._
From the oriels one by one,
Slowly fades the setting sun;
On the marge of afternoon
Stands the new-born crescent moon.
In the twilight's crimson glow
Dim the quiet alcoves grow.
Drowsy-lidded Silence smiles
On the long deserted aisles;
Out of every shadowy nook
Spirit faces seem to look.
Some with smiling eyes, and some
With a sad entreaty dumb;
He who shepherded his sheep
On the wild Sicilian steep,
He above whose grave are set
Sprays of Roman violet;
Poets, sages--all who wrought
In the crucible of thought.
Day by day as seasons glide
On the great eternal tide,
Noiselessly they gather thus
In the twilight beauteous,
Hold communion each with each,
Closer than our earthly speech,
Till within the east are born
Premonitions of the morn!
THE BOOK-HUNTER.
FRANK DEMPSTER SHERMAN. _From the 'Century Magazine,'
November, 1885._
A cup of coffee, eggs, and rolls
Sustain him on his morning strolls:
Unconscious of the passers-by,
He trudges on with downcast eye;
He wears a queer old hat and coat,
Suggestive of a style remote;
His manner is preoccupied,--
A shambling gait, from side to side.
For him the sleek, bright-windowed shop
Is all in vain,--he does not stop.
His thoughts are fixed on dusty shelves
Where musty volumes hide themselves,--
Rare prints of poetry and prose,
And quaintly lettered folios,--
Perchance a parchment manuscript,
In some forgotten corner slipped,
Or monk-illumined missal bound
In vellum with brass clasps around;
These are the pictured things that throng
His mind the while he walks along.
A dingy street, a cellar dim,
With book-lined walls, suffices him.
The dust is white upon his sleeves;
He turns the yellow, dog-eared leaves
With just the same religious look
That priests give to the Holy Book.
He does not heed the stifling air
If so he find a treasure there.
He knows rare books, like precious wines,
Are hidden where the sun ne'er shines;
For him delicious flavors dwell
In books as in old Muscatel;
He finds in features of the type
A clew to prove the grape was ripe.
And when he leaves this dismal place,
Behold, a smile lights up his face!
Upon his cheeks a genial glow,--
Within his hand Boccaccio,
A first edition worn with age,
"Firenze" on the title-page.
THE LIBRARY.
ROBERT SOUTHEY. _Written at Keswick in 1818._
My days among the Dead are past;
Around me I behold,
Where'er these casual eyes are cast,
The mighty minds of old;
My never-failing friends are they,
With whom I converse day by day.
With them I take delight in weal,
And seek relief in woe;
And while I understand and feel
How much to them I owe,
My cheeks have often been dedew'd
With tears of thoughtful gratitude.
My thoughts are with the Dead, with them
I live in long-past years,
Their virtues love, their faults condemn;
Partake their hopes and fears,
And from their lessons seek and find
Instruction with an humble mind.
My hopes are with the Dead, anon
My place with them shall be,
And I with them shall travel on
Through all futurity;
Yet leaving here a name, I trust,
That will not perish in the dust.
PICTURE-BOOKS IN WINTER.
ROBERT LOUIS STEVENSON. _From 'A Child's Garden of Verses.' 1885._
Summer fading, winter comes--
Frosty mornings, tingling thumbs,
Window robins, winter rooks,
And the picture story-books.
Water now is turned to stone
Nurse and I can walk upon;
Still we find the flowing brooks
And the picture story-books.
All the pretty things put by,
Wait upon the children's eye
Sheep and shepherds, trees and crooks,
In the picture story-books.
We may see how all things are,
Seas and cities, near and far,
And the flying fairies' looks,
In the picture story-books.
How am I to sing your praise,
Happy chimney-corner days,
Sitting safe in nursery nooks,
Reading picture story-books?
COMPANIONS.
A French writer (whom I love well) speaks of three kinds of
companions, men, women, and books.
SIR JOHN DAVYS.
RICHARD HENRY STODDARD. _From the 'Atlantic Monthly,' June, 1877._
We have companions, comrade mine:
Jolly good fellows, tried and true,
Are filling their cups with the Rhenish wine,
And pledging each other, as I do you.
Never a man in all the land
But has, in his hour of need, a friend,
Who stretches to him a helping hand
And stands by him to the bitter end.
If not before, there is comfort then,
In the strong companionship of men.
But better than that, old friend of mine,
Is the love of woman, the life of life,
Whether in maiden's eyes it shine,
Or melts in the tender kiss of wife;
A heart contented to feel, not know,
That finds in the other its sole delight;
White hands that are loath to let us go,
The tenderness that is more than might!
On earth below, in heaven above,
Is there anything better than woman's love?
I do not say so, companion mine,
For what, without it, would I be here?
It lightens my troubles, like this good wine,
And, if I must weep, sheds tear for tear!
But books, old friends that are always new,
Of all good things that we know are best;
They never forsake us, as others do,
And never disturb our inward rest.
Here is truth in a world of lies,
And all that in man is great and wise!
Better than men and women, friend,
That are dust, though dear in our joy and pain,
Are the books their cunning hands have penned,
For they depart, but the books remain;
Through these they speak to us what was best
In the loving heart and the noble mind:
All their royal souls possessed
Belongs forever to all mankind!
When others fail him, the wise man looks
To the sure companionship of books.
THE BOOK OF LIFE.
_A Bibliographical Melody, printed in_
RICHARD THOMSON. _1820 at the press of John Johnson, as
a gift to the members of the Roxburghe
Club._
That Life is a Comedy oft hath been shown,
By all who Mortality's changes have known;
But more like a Volume its actions appear,
Where each Day is a Page and each Chapter a year.
'Tis a Manuscript Time shall full surely unfold,
Though with Black-Letter shaded, or shining with gold;
The Initial, like Youth, glitters bright on its Page,
But its Text is as dark--as the gloom of Old Age.
Then Life's Counsels of Wisdom engrave on thy breast,
And deep on thine Heart be her lessons imprest.
Though the Title stands first it can little declare
The Contents which the Pages ensuing shall bear;
As little the first day of Life can explain
The succeeding events which shall glide in its train,
The Book follows next, and, delighted, we trace
An Elzevir's beauty, a Guttemberg's grace;
Thus on pleasure we gaze with as raptured an eye,
Till, cut off like a Volume imperfect, we die!
Then Life's Counsels of Wisdom engrave on thy breast,
And deep on thine Heart be her lessons imprest.
Yet e'en thus imperfect, complete, or defaced,
The skill of the Printer is still to be traced;
And though death bend us early in life to his will,
The wise hand of our Author is visible still.
Like the Colophon lines is the Epitaph's lay,
Which tells of what age and what nation our day,
And, like the Device of the Printer, we bear
The form of the Founder, whose Image we wear.
Then Life's Counsels of Wisdom engrave on thy breast,
And deep on thine Heart be her lessons imprest.
The work thus completed its Boards shall inclose,
Till a Binding more bright and more beauteous it shows;
And who can deny, when Life's Vision hath past,
That the dark Boards of Death shall surround us at last.
Yet our Volume illumed with fresh splendors shall rise,
To be gazed at by Angels, and read to the skies,
Reviewed by its Author, revised by his Pen,
In a fair new Edition to flourish again.
Then Life's Counsels of Wisdom engrave on thy breast,
And deep on thine Heart be her lessons imprest.
ON CERTAIN BOOKS.
CHARLES TENNYSON TURNER. _From 'Sonnets.' 1864._
Faith and fixt hope these pages may peruse,
And still be faith and hope; but, O ye winds!
Blow them far off from all unstable minds,
And foolish grasping hands of youth! Ye dews
Of heaven! be pleased to rot them where they fall,
Lest loitering boys their fancies should abuse,
And they get harm by chance, that cannot choose;
So be they stain'd and sodden, each and all!
And if, perforce, on dry and gusty days,
Upon the breeze some truant leaf should rise,
Brittle with many weathers, to the skies,
Or flit and dodge about the public ways--
Man's choral shout, or organ's peal of praise
Shall shake it into dust, like older lies.
TO HIS BOOKS.
HENRY VAUGHAN. _From 'Silex Scintillans: Sacred Poems
and Pious Ejaculations.' 1678._
Bright books: perspectives on our weak sights,
The clear projections of discerning lights,
Burning in shining thoughts, man's posthume day,
The track of fled souls in their milkie way,
The dead alive and busy, the still voice
Of enlarged spirits, kind heaven's white decoys!
Who lives with you lives like those knowing flowers
Which in commerce with light spend all their hours;
Which shut to clouds, and shadows nicely shun,
But with glad haste unveil to kiss the sun.
Beneath you all is dark and a dead night,
Which whoso lives in wants both health and sight.
By sucking you, the wise, like bees, do grow
Healing and rich, though this they do most slow,
Because most choicely; for as great a store
Have we of books as bees, of herbs, or more;
And the great task to try, then know, the good,
To discern weeds, and judge of wholesome food,
Is a rare scant performance. For man dies
Oft ere 'tis done, while the bee feeds and flies.
But you were all choice flowers; all set and drest
By old sage florists, who well knew the best;
And I amidst you all am turned to weed!
Not wanting knowledge, but for want of heed.
Then thank thyself, wild fool, that would'st not be
Content to know what was too much for thee!
LITERATURE AND NATURE.
SAMUEL WADDINGTON. _Written for the present collection._
'Mid Cambrian heights around Dolgelly vale,
What time we scaled great Cader's rugged pile,
Or loitered idly where still meadows smile
Beside the Mawddach-stream, or far Cynfael--
Nor tome, nor rhythmic page, nor pastoral tale,
Our summer-sated senses would beguile;
Or lull our ears to melody, the while
The voiceful rill ran lilting down the dale.
In London town once more--behold, once more
The old delight returns! 'Mid heights how vast,
In Milton's verse, through what dim paths we wind;
How Keats's canvas glows, and Wordsworth's lore,
As tarn or torrent pure, by none surpass'd,
Sheds light and love--unfathomed, undefined.
THE LIBRARY.
JOHN GREENLEAF WHITTIER. _Sung at the opening of the
Library at Haverhill, Mass._
"Let there be Light!" God spake of old,
And over chaos dark and cold,
And through the dead and formless frame
Of nature, life and order came.
Faint was the light at first that shone
On giant fern and mastodon,
On half-formed plant and beast of prey,
And man as rude and wild as they.
Age after age, like waves o'erran
The earth, uplifting brute and man;
And mind, at length, in symbols dark
Its meanings traced on stone and bark.
On leaf of palm, on sedge-wrought roll,
On plastic clay and leathern scroll,
Man wrote his thoughts; the ages passed,
And lo! the Press was found at last!
Then dead souls woke; the thoughts of men
Whose bones were dust revived again;
The cloister's silence found a tongue,
Old prophets spake, old poets sung.
And here, to-day, the dead look down,
The kings of mind again we crown;
We hear the voices lost so long,
The sage's word, the sibyl's song.
Here Greek and Roman find themselves
Alive along these crowded shelves;
And Shakspere treads again his stage,
And Chaucer paints anew his age.
As if some Pantheon's marbles broke
Their stony trance, and lived and spoke,
Life thrills along the alcoved hall,
The lords of thought awake our call.
THE COUNTRY SQUIRE.
TOMAS YRIARTE. _An anonymous translation of one of the
'Literary Fables.'_
A country squire, of greater wealth than wit
(For fools are often blessed with fortune's smile),
Had built a splendid house, and furnished it
In splendid style.
"One thing is wanting," said a friend; "for, though
The rooms are fine, the furniture profuse,
You lack a library, dear sir, for show,
If not for use."
"'Tis true; but 'zounds!" replied the squire with glee,
"The lumber-room in yonder northern wing
(I wonder I ne'er thought of it) will be
The very thing.
"I'll have it fitted up without delay
With shelves and presses of the newest mode
And rarest wood, befitting every way
A squire's abode."
"And when the whole is ready, I'll dispatch
My coachman--a most knowing fellow--down
To buy me, by admeasurement, a batch
Of books in town."
But ere the library was half supplied
With all its pomps of cabinet and shelf,
The booby squire repented him, and cried
Unto himself:--
"This room is much more roomy than I thought;
Ten thousand volumes hardly would suffice
To fill it, and would cost, however bought,
A plaguy price."
"Now as I only want them for their looks,
It might, on second thoughts, be just as good,
And cost me next to nothing, if the books
Were made of wood."
"It shall be so, I'll give the shaven deal
A coat of paint--a colorable dress,
To look like calf or vellum, and conceal
Its nakedness."
"And, gilt and lettered with the author's name,
Whatever is most excellent and rare
Shall be, or seem to be ('tis all the same),
Assembled there."
The work was done; the simulated hoards
Of wit and wisdom round the chamber stood,
In binding some; and some, of course, in _boards_,
Where all were wood.
From bulky folios down to slender twelves
The choicest tomes, in many an even row
Displayed their lettered backs upon the shelves,
A goodly show.
With such a stock as seemingly surpassed
The best collection ever formed in Spain,
What wonder if the owner grew at last
Supremely vain?
What wonder, as he paced from shelf to shelf,
And conned their titles, that the squire began,
Despite his ignorance, to think himself
A learned man?
_Let every amateur, who merely looks
To backs and binding, take the hint, and sell
His costly library--for painted books
Would serve as well._
OLD BOOKS.
_From the appendix of 'How to Read_
ANON. _a Book in the Best Way.'
New York, n. d._
I must confess I love old books!
The dearest, too, perhaps most dearly;
Thick, clumpy tomes, of antique looks,
In pigskin covers fashioned queerly.
Clasped, chained, or thonged, stamped quaintly too,
With figures wondrous strange, or holy
Men and women, and cherubs, few
Might well from owls distinguish duly.
I love black-letter books that saw
The light of day at least three hundred
Long years ago; and look with awe
On works that live, so often plundered.
I love the sacred dust the more
It clings to ancient lore, enshrining
Thoughts of the dead, renowned of yore,
Embalmed in books, for age declining.
Fit solace, food, and friends more sure
To have around one, always handy,
When sinking spirits find no cure
In news, election brawls, or brandy.
In these old books, more soothing far
Than balm of Gilead or Nepenthe,
I seek an antidote for care--
Of which most men indeed have plenty.
"Five hundred times at least," I've said--
My wife assures me--"I would never
Buy more old books;" yet lists are made,
And shelves are lumbered more than ever.
Ah! that our wives could only see
How well the money is invested
In these old books, which seem to be
By them, alas! so much detested.
There's nothing hath enduring youth,
Eternal newness, strength unfailing,
Except old books, old friends, old truth,
That's ever battling--still prevailing.
'T is better in the past to live
Than grovel in the present vilely,
In clubs, and cliques, where placemen hive,
And faction hums, and dolts rank highly.
To be enlightened, counselled, led,
By master minds of former ages,
Come to old books--consult the dead--
Commune with silent saints and sages.
Leave me, ye gods! to my old books--
Polemics yield to sects that wrangle--
Vile "parish politics" to folks
Who love to squabble, scheme, and jangle.
Dearly beloved old pigskin tomes!
Of dingy hue--old bookish darlings!
Oh, cluster ever round my rooms,
And banish strifes, disputes, and snarlings.
=Appendix=
________________
THE LIBRARY
BY
GEORGE CRABBE
THE LIBRARY.
_In want and danger, the unknown
poet sent this poem to Edmund_
GEORGE CRABBE. _Burke, who saw its merit, befriended
its author, and procured its
publication._
When the sad soul, by care and grief oppressed,
Looks round the world, but looks in vain for rest,
When every object that appears in view
Partakes her gloom and seems dejected too;
Where shall affliction from itself retire?
Where fade away and placidly expire?
Alas! we fly to silent scenes in vain;
Care blasts the honors of the flowery plain;
Care veils in clouds the sun's meridian beam,
Sighs through the grove, and murmurs in the stream;
For when the soul is laboring in despair,
In vain the body breathes a purer air:
No storm-tost sailor sighs for slumbering seas--
He dreads the tempest, but invokes the breeze;
On the smooth mirror of the deep resides
Reflected woe, and o'er unruffled tides
The ghost of every former danger glides.
Thus, in the calms of life, we only see
A steadier image of our misery;
But lively gales and gently clouded skies
Disperse the sad reflections as they rise;
And busy thoughts and little cares avail
To ease the mind, when rest and reason fail.
When the dull thought, by no designs employed,
Dwells on the past, or suffered or enjoyed,
We bleed anew in every former grief,
And joys departed furnish no relief.
Not Hope herself, with all her flattering art,
Can cure this stubborn sickness of the heart:
The soul disdains each comfort she prepares,
And anxious searches for congenial cares;
Those lenient cares, which, with our own combined,
By mixed sensations ease th' afflicted mind,
And steal our grief away, and leave their own behind;
A lighter grief! which feeling hearts endure
Without regret, nor e'en demand a cure.
But what strange art, what magic can dispose
The troubled mind to change its native woes?
Or lead us, willing from ourselves, to see
Others more wretched, more undone than we?
This BOOKS can do;--nor this alone; they give
New views to life, and teach us how to live;
They soothe the grieved, the stubborn they chastise,
Fools they admonish and confirm the wise:
Their aid they yield to all: they never shun
The man of sorrow, nor the wretch undone:
Unlike the hard, the selfish, and the proud,
They fly not sullen from the suppliant crowd;
Nor tell to various people various things,
But show to subjects what they show to kings.
Come, Child of Care! to make thy soul serene,
Approach the treasures of this tranquil scene;
Survey the dome, and, as the doors unfold,
The soul's best cure, in all her cares behold!
Where mental wealth the poor in thought may find,
And mental physic the diseased in mind;
See here the balms that passion's wounds assuage;
See coolers here, that damp the fire of rage;
Here alteratives, by slow degrees control
The chronic habits of the sickly soul;
And round the heart, and o'er the aching head,
Mild opiates here their sober influence shed.
Now bid thy soul man's busy scenes exclude,
And view composed this silent multitude:--
Silent they are--but though deprived of sound,
Here all the living languages abound;
Here all that live no more; preserved they lie,
In tombs that open to the curious eye.
Blest be the gracious Power, who taught mankind
To stamp a lasting image of the mind!
Beasts may convey, and tuneful birds may sing,
Their mutual feelings, in the opening spring;
But Man alone has skill and power to send
The heart's warm dictates to the distant friend;
'Tis his alone to please, instruct, advise
Ages remote, and nations yet to rise.
In sweet repose, when Labor's children sleep,
When Joy forgets to smile and Care to weep,
When Passion slumbers in the lover's breast,
And Fear and Guilt partake the balm of rest,
Why then denies the studious man to share
Man's common good, who feels his common care?
Because the hope is his that bids him fly
Night's soft repose, and sleep's mild power defy,
That after-ages may repeat his praise,
And fame's fair meed be his, for length of days.
Delightful prospect! when we leave behind
A worthy offspring of the fruitful mind!
Which, born and nursed through many an anxious day,
Shall all our labor, all our care repay.
Yet all are not these births of noble kind,
Not all the children of a vigorous mind;
But where the wisest should alone preside,
The weak would rule us, and the blind would guide;
Nay, man's best efforts taste of man, and show
The poor and troubled source from which they flow;
Where most he triumphs we his wants perceive,
And for his weakness in his wisdom grieve.
But though imperfect all; yet wisdom loves
This seat serene, and virtue's self approves:--
Here come the grieved, a change of thought to find;
The curious here to feed a craving mind;
Here the devout their peaceful temple choose;
And here the poet meets his favoring Muse.
With awe, around these silent walks I tread;
These are the lasting mansions of the dead:--
"The dead!" methinks a thousand tongues reply;
"These are the tombs of such as cannot die!
Crowned with eternal fame, they sit sublime,
And laugh at all the little strife of time.
Hail, then, immortals! ye who shine above,
Each, in his sphere, the literary Jove;
And ye, the common people of these skies,
A humbler crowd of nameless deities;
Whether 't is yours to lead the willing mind
Through History's mazes, and the turnings find;
Or, whether led by Science, ye retire,
Lost and bewildered in the vast desire,
Whether the Muse invites you to her bowers,
And crowns your placid brows with living flowers!
Or godlike Wisdom teaches you to show
The noblest road to happiness below;
Or men and manners prompt the easy page
To mark the flying follies of the age;
Whatever good ye boast, that good impart;
Inform the head and rectify the heart.
Lo, all in silence, all in order stand,
And mighty folios, first a lordly band;
Then quartos their well-ordered ranks maintain,
And light octavos fill a spacious plain:
See yonder, ranged in more frequented rows,
A humbler band of duodecimos;
While undistinguish'd trifles swell the scene,
The last new play and frittered magazine.
Thus 't is in life, where first the proud, the great,
In leagued assembly keep their cumbrous state:
Heavy and huge, they fill the world with dread,
Are much admired, and are but little read:
The commons next, a middle rank, are found;
Professions fruitful pour their offspring round;
Reasoners and wits are next their place allowed,
And last, of vulgar tribes a countless crowd.
First, let us view the form, the size, the dress:
For these the manners, nay the mind, express:
That weight of wood, with leathern coat o'erlaid;
Those ample clasps of solid metal made;
The close-pressed leaves, unclosed for many an age;
The dull red edging of the well-filled page;
On the broad back the stubborn ridges rolled,
Where yet the title stands in tarnished gold;
These all a sage and labored work proclaim,
A painful candidate for lasting fame:
No idle wit, no trifling verse can lurk
In the deep bosom of that weighty work;
No playful thoughts degrade the solemn style,
Nor one light sentence claims a transient smile.
Hence, in these times, untouched the pages lie,
And slumber out their immortality:
They _had_ their day, when, after all his toil,
His morning study, and his midnight oil,
At length an author's ONE great work appeared,
By patient hope, and length of days endeared:
Expecting nations haled it from the press;
Poetic friends prefixed each kind address;
Princes and kings received the pond'rous gift,
And ladies read the work they could not lift.
Fashion, though Folly's child, and guide of fools,
Rules e'en the wisest, and in learning rules;
From crowds and courts to Wisdom's seat she goes,
And reigns triumphant o'er her mother's foes.
For lo! these favorites of the ancient mode
Lie all neglected like the Birthday Ode.
Ah! needless now this weight of massy chain,
Safe in themselves, the once-loved works remain;
No readers now invade their still retreat,
None try to steal them from their parent seat;
Like ancient beauties, they may now discard
Chains, bolts, and locks, and lie without a guard.
Our patient fathers trifling themes laid by,
And rolled, o'er labored works, th' attentive eye:
Page after page the much enduring men
Explored the deeps and shallows of the pen:
Till, every former note and comment known,
They marked the spacious margin with their own;
Minute corrections proved their studious care;
The little index, pointing, told us where;
And many an emendation showed the age
Looked far beyond the rubric title-page.
Our nicer palates lighter labors seek,
Cloyed with a folio-_Number_ once a week;
Bibles, with cuts and comments, thus go down:
E'en light Voltaire is _numbered_ through the town:
Thus physic flies abroad, and thus the law,
From men of study, and from men of straw;
Abstracts, abridgments, please the fickle times,
Pamphlets and plays, and politics and rhymes:
But though to write be now a task of ease,
The task is hard by manly arts to please,
When all our weakness is exposed to view,
And half our judges are our rivals too.
Amid these works, on which the eager eye
Delights to fix, or glides reluctant by,
When all combined, their decent pomp display,
Where shall we first our early offering pay?--
To thee, DIVINITY! to thee, the light
And guide of mortals, through their mental night;
By whom we learn our hopes and fears to guide;
To bear with pain, and to contend with pride;
When grieved, to pray; when injured, to forgive;
And with the world in charity to live.
Not truths like these inspired that numerous race,
Whose pious labors fill this ample space;
But questions nice, where doubt on doubt arose,
Awaked to war the long-contending foes.
For dubious meanings, learned polemics strove,
And wars on faith prevented works of love;
The brands of discord far around were hurled,
And holy wrath inflamed a sinful world:--
Dull though impatient, peevish though devout,
With wit, disgusting and despised without;
Saints in design, in execution men,
Peace in their looks, and vengeance in their pen.
Methinks I see, and sicken at the sight,
Spirits of spleen from yonder pile alight;
Spirits who prompted every damning page,
With pontiff pride, and still increasing rage:
Lo! how they stretch their gloomy wings around,
And lash with furious strokes the trembling ground!
They pray, they fight, they murder, and they weep,
Wolves in their vengeance, in their manners sheep;
Too well they act the prophet's fatal part,
Denouncing evil with a zealous heart;
And each, like Jonah, is displeased if God
Repent his anger, or withold his rod.
But here the dormant fury rests unsought,
And Zeal sleeps soundly by the foes she fought;
Here all the rage of controversy ends,
And rival zealots rest like bosom friends:
An Athanasian here, in deep repose,
Sleeps with the fiercest of his Arian foes;
Socinians here with Calvinists abide,
And thin partitions angry chiefs divide;
Here wily Jesuits simple Quakers meet,
And Bellarmine has rest at Luther's feet.
Great authors, for the church's glory fired,
Are for the church's peace to rest retired;
And close beside, a mystic, maudlin race,
Lie "Crumbs of Comfort for the Babes of Grace."
Against her foes Religion well defends
Her sacred truths, but often fears her friends;
If learned, their pride, if weak, their zeal she dreads,
And their hearts' weakness, who have soundest heads.
But most she fears the controversial pen,
The holy strife of disputatious men;
Who the blest Gospel's peaceful page explore,
Only to fight against its precepts more.
Near to these seats behold yon slender frames,
All closely filled and marked with modern names;
Where no fair science ever shows her face,
Few sparks of genius, and no spark of grace;
There sceptics rest, a still increasing throng,
And stretch their widening wings ten thousand strong;
Some in close fight their dubious claims maintain;
Some skirmish lightly, fly, and fight again;
Coldly profane, and impiously gay,
Their end the same, though various in their way.
When first Religion came to bless the land,
Her friends were then a firm believing band;
To doubt was then to plunge in guilt extreme,
And all was gospel that a monk could dream;
Insulted Reason fled the grov'lling soul,
For Fear to guide and visions to control:
But now, when Reason has assumed her throne,
She, in her turn demands to reign alone;
Rejecting all that lies beyond her view,
And, being judge, will be a witness too:
Insulted Faith then leaves the doubtful mind,
To seek for truth, without a power to find:
Ah! when will both in friendly beams unite,
And pour on erring man resistless light!
Next to the seats, well stored with works divine,
An ample space, PHILOSOPHY! is thine;
Our reason's guide, by whose assisting light
We trace the moral bounds of wrong and right;
Our guide through nature, from the sterile clay,
To the bright orbs of yon celestial way!
'T is thine, the great, the golden chain to trace,
Which runs through all, connecting race with race
Save where those puzzling, stubborn links remain,
Which thy inferior light pursues in vain:--
How vice and virtue in the soul contend;
How widely differ, yet how nearly blend;
What various passions war on either part,
And now confirm, now melt the yielding heart:
How Fancy loves around the world to stray,
While Judgment slowly picks his sober way;
The stores of memory and the flights sublime
Of genius, bound by neither space nor time;--
All these divine Philosophy explores,
Till, lost in awe, she wonders and adores.
From these, descending to the earth, she turns,
And matter, in its various forms, discerns;
She parts the beamy light with skill profound,
Metes the thin air, and weighs the flying sound;
'T is hers the lightning from the clouds to call,
And teach the fiery mischief where to fall.
Yet more her volumes teach--on these we look
Abstracts drawn from Nature's larger book;
Here, first described, the torpid earth appears,
And next, the vegetable robe it wears;
Where flowery tribes in valleys, fields, and groves,
Nurse the still flame, and feed the silent loves;
Loves where no grief, nor joy, nor bliss, nor pain,
Warm the glad heart or vex the laboring brain;
But as the green blood moves along the blade,
The bed of Flora on the branch is made;
Where, without passion, love instinctive lives,
And gives new life, unconscious that it gives.
Advancing still in Nature's maze, we trace,
In dens and burning plains, her savage race
With those tame tribes who on their lord attend,
And find in man a master and a friend;
Man crowns the scene, a world of wonders new,
A moral world, that well demands our view.
This world is here; for, of more lofty kind,
These neighboring volumes reason on the mind;
They paint the state of man ere yet endued
With knowledge;--man, poor, ignorant, and rude;
Then, as his state improves, their pages swell,
And all its cares, and all its comforts tell:
Here we behold how inexperience buys,
At little price, the wisdom of the wise;
Without the troubles of an active state,
Without the cares and dangers of the great,
Without the miseries of the poor, we know
What wisdom, wealth, and poverty bestow;
We see how reason calms the raging mind,
And how contending passions urge mankind:
Some, won by virtue, glow with sacred fire;
Some, lured by vice, indulge the low desire;
Whilst others, won by either, now pursue
The guilty chase, now keep the good in view;
Forever wretched, with themselves at strife,
They lead a puzzled, vexed, uncertain life;
For transient vice bequeaths a lingering pain,
Which transient virtue seeks to cure in vain.
Whilst thus engaged, high views enlarge the soul,
New interest draws, new principles control:
Nor thus the soul alone resigns her grief,
But here the tortured body finds relief;
For see where yonder sage Arachne shapes
Her subtle gin, that not a fly escapes!
There PHYSIC fills the space, and far around,
Pile above pile her learned works abound:
Glorious their aim--to ease the laboring heart;
To war with death, and stop his flying dart;
To trace the source whence the fierce contest grew;
And life's short lease on easier terms renew;
To calm the frenzy of the burning brain;
To heal the tortures of imploring pain;
Or, when more powerful ills all efforts brave,
To ease the victim no device can save,
And smooth the stormy passage to the grave.
But man, who knows no good unmixed and pure,
Oft finds a poison where he sought a cure;
For grave deceivers lodge their labors here,
And cloud the science they pretend to clear;
Scourges for sin, the solemn tribe are sent;
Like fire and storms, they call us to repent;
But storms subside, and fires forget to rage.
_These_ are eternal scourges of the age:
'T is not enough that each terrific hand
Spreads desolation round a guilty land;
But trained to ill, and hardened by its crimes,
Their pen relentless kills through future times,
Say, ye, who search these records of the dead--
Who read huge works, to boast what ye have read,
Can all the real knowledge ye possess,
Or those--if such there are--who more than guess,
Atone for each impostor's wild mistakes,
And mend the blunders pride or folly makes?
What thought so wild, what airy dream so light,
That will not prompt a theorist to write?
What art so prevalent, what proofs so strong,
That will convince him his attempt is wrong?
One in the solids finds each lurking ill,
Nor grants the passive fluids power to kill;
A learned friend some subtler reason brings,
Absolves the channels, but condemns their spring;
The subtile nerves, that shun the doctor's eye,
Escape no more his subtler theory;
The vital heat, that warms the laboring heart,
Lends a fair system to these sons of art;
The vital air, a pure and subtile stream,
Serves a foundation for an airy scheme,
Assists the doctor and supports his dream.
Some have their favorite ills, and each disease
Is but a younger branch that kills from these;
One to the gout contracts all human pain;
He views it raging in the frantic brain;
Finds it in fevers all his efforts mar,
And sees it lurking in the cold catarrh;
Bilious by some, by others nervous seen,
Rage the fantastic demons of the spleen;
And every symptom of the strange disease
With every system of the sage agrees.
Ye frigid tribe, on whom I wasted long
The tedious hours, and ne'er indulged in song;
Ye first seducers of my easy heart,
Who promised knowledge ye could not impart;
Ye dull deluders, truth's destructive foes;
Ye sons of fiction, clad in stupid prose;
Ye treacherous leaders, who, yourselves in doubt,
Light up false fires, and send us far about;--
Still may yon spider round your pages spin,
Subtile and slow, her emblematic gin!
Buried in dust and lost in silence, dwell,
Most potent, grave, and reverend friends--farewell!
Near these, and where the setting sun displays,
Through the dim window, his departing rays,
And gilds yon columns, there, on either side,
The huge Abridgments of the LAW abide;
Fruitful as vice, the dread correctors stand,
And spread their guardian terrors round the land;
Yet, as the best that human care can do
Is mixed with error, oft with evil too,
Skilled in deceit, and practised to evade,
Knaves stand secure, for whom these laws were made,
And justice vainly each expedient tries,
While art eludes it, or while power defies.
"Ah! happy age," the youthful poet sings,
"When the free nations knew not laws nor kings,
When all were blest to share a common store,
And none were proud of wealth, for none were poor,
No wars nor tumults vexed each still domain,
No thirst of empire, no desire of gain;
No proud great man, nor one who would be great,
Drove modest merit from its proper state;
Nor into distant climes would Avarice roam,
To fetch delights for Luxury at home:
Bound by no ties which kept the soul in awe,
They dwelt at liberty, and love was law!"
"Mistaken youth! each nation first was rude,
Each man a cheerless son of solitude,
To whom no joys of social life were known,
None felt a care that was not all his own;
Or in some languid clime his abject soul
Bowed to a little tyrant's stern control;
A slave, with slaves his monarch's throne he raised,
And in rude song his ruder idol praised;
The meaner cares of life were all he knew;
Bounded his pleasures, and his wishes few;
But when by slow degrees the Arts arose,
And Science wakened from her long repose;
When Commerce, rising from the bed of ease,
Ran round the land, and pointed to the seas;
When Emulation, born with jealous eye,
And Avarice, lent their spurs to industry;
Then one by one the numerous laws were made,
Those to control, and these to succor trade;
To curb the insolence of rude command,
To snatch the victim from the usurer's hand;
To awe the bold, to yield the wronged redress,
And feed the poor with Luxury's excess."
Like some vast flood, unbounded, fierce, and strong,
His nature leads ungoverned man along;
Like mighty bulwarks made to stem that tide,
The laws are formed and placed on every side;
Whene'er it breaks the bounds by these decreed,
New statutes rise, and stronger laws succeed;
More and more gentle grows the dying stream,
More and more strong the rising bulwarks seem;
Till, like a miner working sure and slow,
Luxury creeps on, and ruins all below;
The basis sinks, the ample piles decay;
The stately fabric shakes and falls away;
Primeval want and ignorance come on,
But Freedom, that exalts the savage state, is gone.
Next HISTORY ranks;--there full in front she lies,
And every nation her dread tale supplies;
Yet History has her doubts, and every age
With sceptic queries marks the passing page;
Records of old nor later date are clear,
Too distant those, and these are placed too near;
There time conceals the objects from our view,
Here our own passions and a writer's too:
Yet, in these volumes, see how states arose!
Guarded by virtue from surrounding foes;
Their virtue lost, and of their triumphs vain,
Lo! how they sunk to slavery again!
Satiate with power, of fame and wealth possessed,
A nation grows too glorious to be blest;
Conspicuous made, she stands the mark of all,
And foes join foes to triumph in her fall.
Thus speaks the page that paints ambition's race,
The monarch's pride, his glory, his disgrace;
The headlong course that maddening heroes run,
How soon triumphant, and how soon undone;
How slaves, turned tyrants, offer crowns to sale,
And each fallen nation's melancholy tale.
Lo! where of late the Book of Martyrs stood,
Old pious tracts, and Bibles bound in wood;
There, such the taste of our degenerate age,
Stand the profane delusions of the STAGE:
Yet virtue owns the TRAGIC MUSE a friend,
Fable her means, morality her end;
For this she rules all passions in their turns,
And now the bosom bleeds, and now it burns;
Pity with weeping eye surveys her bowl,
Her anger swells, her terror chills the soul;
She makes the vile to virtue yield applause,
And own her sceptre while they break her laws;
For vice in others is abhorred of all,
And villains triumph when the worthless fall.
Not thus her sister COMEDY prevails,
Who shoots at Folly, for her arrow fails;
Folly, by Dulness armed, eludes the wound,
And harmless sees the feathered shafts rebound;
Unhurt she stands, applauds the archer's skill,
Laughs at her malice, and is Folly still.
Yet well the Muse portrays, in fancied scenes,
What pride will stoop to, what profession means;
How formal fools the farce of state applaud;
How caution watches at the lips of fraud;
The wordy variance of domestic life;
The tyrant husband, the retorting wife;
The snares for innocence, the lie of trade,
And the smooth tongue's habitual masquerade.
With her the Virtues to obtain a place,
Each gentle passion, each becoming grace;
The social joy in life's securer road,
Its easy pleasure, its substantial good;
The happy thought that conscious virtue gives,
And all that ought to live, and all that lives.
But who are these? Methinks a noble mien
And awful grandeur in their form are seen,
Now in disgrace: what though by time is spread
Polluting dust o'er every reverend head;
What though beneath yon gilded tribe they lie,
And dull observers pass insulting by:
Forbid it shame, forbid it decent awe,
What seems so grave, should no attention draw!
Come, let us then with reverend step advance,
And greet--the ancient worthies of ROMANCE.
Hence, ye profane! I feel a former dread,
A thousand visions float around my head:
Hark! hollow blasts through empty courts resound,
And shadowy forms with staring eyes stalk round;
See! moats and bridges, walls and castles rise,
Ghosts, fairies, demons, dance before our eyes;
Lo! magic verse inscribed on golden gate;
And bloody hand that beckons on to fate:--
"And who art thou, thou little page, unfold?
Say, doth thy lord my Claribel withhold?
Go tell him straight, Sir Knight, thou must resign
The captive queen;--for Claribel is mine."
Away he flies; and now for bloody deeds,
Black suits of armor, masks, and foaming steeds;
The giant falls; his recreant throat I seize,
And from his corselet take the massy keys:--
Dukes, lords, and knights in long procession move,
Released from bondage with my virgin love:--
She comes! she comes! in all the charms of youth,
Unequalled love, and unsuspected truth!
Ah! happy he who thus, in magic themes,
O'er worlds bewitched, in early rapture dreams,
Where wild Enchantment waves her potent wand,
And Fancy's beauties fill her fairy land;
Where doubtful objects strange desires excite,
And Fear and Ignorance afford delight.
But lost, for ever lost, to me these joys,
Which Reason scatters, and which Time destroys;
Too dearly bought: maturer judgment calls
My busied mind from tales and madrigals;
My doughty giants all are slain or fled
And all my knights--blue, green, and yellow--dead!
No more the midnight fairy tribe I view,
All in the merry moonshine tippling dew;
E'en the last lingering fiction of the brain,
The churchyard ghost is now at rest again;
And all these wayward wanderings of my youth
Fly Reason's power, and shun the light of Truth.
With Fiction then does real joy reside,
And is our reason the delusive guide?
Is it then right to dream the sirens sing?
Or mount enraptured on the dragon's wing?
No; 't is the infant mind, to care unknown,
That makes th' imagined paradise its own;
Soon as reflections in the bosom rise,
Light slumbers vanish from the clouded eyes:
The tear and smile, that once together rose,
Are then divorced; the head and heart are foes:
Enchantment bows to Wisdom's serious plan,
And Pain and Prudence make and mar the man.
While thus, of power and fancied empire vain,
With various thoughts my mind I entertain;
While books, my slaves, with tyrant hand I seize,
Pleased with the pride that will not let them please,
Sudden I find terrific thoughts arise,
And sympathetic sorrow fills my eyes;
For, lo! while yet my heart admits the wound,
I see the CRITIC army ranged around.
Foes to our race! if ever ye have known
A father's fears for offspring of your own;
If ever, smiling o'er a lucky line,
Ye thought the sudden sentiment divine,
Then paused and doubted, and then, tired of doubt,
With rage as sudden dashed the stanza out;--
If, after fearing much and pausing long,
Ye ventured on the world your labored song,
And from the crusty critics of those days
Implored the feeble tribute of their praise;
Remember now the fears that moved you then,
And, spite of truth, let mercy guide your pen.
What vent'rous race are ours! what mighty foes
Lie waiting all around them to oppose!
What treacherous friends betray them to the fight!
What dangers threaten them:--yet still they write:
A hapless tribe! to every evil born,
Whom villains hate, and fools affect to scorn:
Strangers they come, amid a world of woe,
And taste the largest portion ere they go.
Pensive I spoke, and cast mine eyes around;
The roof, methought, returned a solemn sound;
Each column seemed to shake, and clouds, like smoke,
From dusty piles and ancient volumes broke;
Gathering above, like mists condensed they seem,
Exhaled in summer from the rushy stream;
Like flowing robes they now appear, and twine
Round the large members of a form divine;
His silver beard, that swept his aged breast,
His piercing eye, that inward light expressed,
Were seen--but clouds and darkness veiled the rest.
Fear chilled my heart: to one of mortal race,
How awful seemed the Genius of the place!
So in Cimmerian shores, Ulysses saw
His parent-shade, and shrunk in pious awe;
Like him I stood, and wrapped in thought profound,
When from the pitying power broke forth a solemn sound:--
"Care lives with all; no rules, no precepts save
The wise from woe, no fortitude the brave;
Grief is to man as certain as the grave:
Tempests and storms in life's whole progress rise,
And hope shines dimly through o'erclouded skies.
Some drops of comfort on the favored fall,
But showers of sorrow are the lot of _all_:
Partial to talents, then, shall Heaven withdraw
Th' afflicting rod, or break the general law?
Shall he who soars, inspired by loftier views,
Life's little cares and little pains refuse?
Shall he not rather feel a double share
Of mortal woe, when doubly armed to bear?
"Hard is his fate who builds his peace of mind
On the precarious mercy of mankind;
Who hopes for wild and visionary things,
And mounts o'er unknown seas with vent'rous wings;
But as, of various evils that befall
The human race, some portion goes to all;
To him perhaps the milder lot's assigned
Who feels his consolation in his mind.
And, locked within his bosom, bears about
A mental charm for every care without.
E'en in the pangs of each domestic grief,
Or health or vigorous hope affords relief;
And every wound the tortured bosom feels,
Or virtue bears, or some preserver heals;
Some generous friend of ample power possessed;
Some feeling heart, that bleeds for the distressed;
Some breast that glows with virtues all divine;
Some noble RUTLAND, misery's friend and thine.
"Nor say, the Muse's song, the Poet's pen,
Merit the scorn they meet from little men.
With cautious freedom if the numbers flow,
Not wildly high, nor pitifully low;
If vice alone their honest aims oppose,
Why so ashamed their friends, so loud their foes?
Happy for men in every age and clime,
If all the sons of vision dealt in rhyme.
Go on, then, Son of Vision! still pursue
Thy airy dreams; the world is dreaming too.
Ambition's lofty views, the pomp of state,
The pride of wealth, the splendor of the great,
Stripped of their mask, their cares and troubles known,
Are visions far less happy than thy own:
Go on! and, while the sons of care complain,
Be wisely gay and innocently vain;
While serious souls are by their fears undone,
Blow sportive bladders in the beamy sun,
And call them worlds! and bid the greatest show
More radiant colors in their worlds below:
Then, as they break, the slaves of care reprove,
And tell them, Such are all the toys they love.
[Illustration]
=A Final Word.=
_THE COLLECTOR TO HIS LIBRARY._
_Brown Books of mine, who never yet
Have caused me anguish or regret,--
Save when some fiend in human shape
Has set your tender sides agape,
Or soiled with some unmanly smear
The whiteness of your page sincere,
Or scored you with some phrase inane,
The bantling of his idle brain,--
I love you: and because must end
This commerce between friend and friend,
I do beseech each kindly fate--
To each and all I supplicate--
That you whom I have loved so long
May not be vended "for a song,"--
That you, my dear desire and care,
May 'scape the common thoroughfare,
The dust, the eating rain, and all
The shame and squalor of the stall.
Rather I trust your lot may touch
Some Croesus--if there should be such--
To buy you, and that you may so
From Croesus unto Croesus go
Till that inevitable day
When comes your moment of decay._
_This, more than other good, I pray._
AUSTIN DOBSON.
[Illustration]
End of the Project Gutenberg EBook of Ballads of Books, by Various
*** | {
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Edytor – program służący do wprowadzania zmian (edycji) w jakimś dokumencie. Zazwyczaj używane w znaczeniu edytor tekstu. Bardziej zaawansowane edytory tekstu bywają nazywane procesorami tekstu.
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10 Best Data Analysis & Management Tools To Eliminate Programming
30 October 2019 / 1 min read
RAPID MINER
2. WEKA
3. ORANGE
4. KNIME
5. TABLEAU PUBLIC
6. OPEN REFINE
7. TALEND
8. LOOKER
9. QLIK SENSE
10. NODEXL
By Rachael Chapman
Those who breathe the coding language invest in a profession popularly named as Data Science. They work on pulling out relevant data that is going to add a high value when used for numerous purposes. Such kinds are referred as Data Scientists. Their main focus lies on the management and analysis of large heaps of data.
On the other hand, not everyone in this world is born with a special talent such as the coders or developers . With high proficient coders using tools that we merely understand, there are tools which can help you analyse information even if you are not technically sound. In this article, we will discuss few non-technical tools that you would use in your day to day life that is prominently used in the world of Data analytics.
Before we jump into this, let's see how data analysis is conducted, It is a long but an effective procedure of extracting useful data from a data dump. This procedure includes a lot of qualitative and quantitative methods to extract the required data, of course, this is not done manually, there are a lot of tools that are used to process these data and get a clean output.
Let's have a look at some of these tools.
Rapidminer is a cross-platform software that was developed by the company RapidMiner in 2006. This tool is very easy to use and helps pre-processing, segregating and machine learning of huge dumps of data. Being a paid tool, there are certain free versions available but they have their own restrictions. The best free version tool, however, is the Rapidminer Studio Free edition.
This tool, however, came to limelight in 2016 at Gartner Quadrant of Advanced Analytics. With the capability of building machine learning models, this is not just a data mining or cleaning tool. This tool supports various high-end algorithms which are used in day to day lives. With the ability to help build models using Python or R, this tool is the best choice for people who use Data analysis with model building.
Weka is an abbreviation for Waikato Environment for Knowledge Analysis, it is a machine learning application developed on Java platform,where it got its name and was developed at the University of Waikato NZ.
Weka is a free licensed software which makes it a favorite to all the data analysts. It contains various list of algorithms and visualization tools that help in analyzing data and structuring them. This application can be used on Windows, Mac and Linux as of now.
The first version was designed mainly for analysis of data, however, the full version built on Java was initiated in 1997 which included all the features a present day data analyst needs.
Weka is widely used due to the following reasons:
Free under the GNU
Portability, since it is built on Java and thus can be used on almost any modern computing platform.
Collection of data preprocessing and modeling techniques.
Ease of use due to its GUI.
Orange is an open source software for data mining, visualization and machine learning developed at the University of Ljubljana in 1997.
The features of orange range from simple data visualization, data mining to complex evaluation of understanding algorithms and modelling structures.
At one end when data analysts use Orange as a data visualization tool, advanced users/developers can use it as a Python library for manipulating data and altering the widgets. Orange is also released under a free license allowing users to enjoy a great experience in data visualization.
Building a model on Orange will be done by creating a flowchart which sounds interesting as you would be able to understand the exact procedure of data mining. This application is available on Windows, OSX, and Linux, but can also be run on Python repository.
For advanced users wanting to use this on Python, 2 stable versions are available, one on Python 3.0 which is the most recent release and its predecessor which runs on Python 2.7.
KNIME is a free and open source application which is an abbreviation for Konstanz Information Miner and was introduced in 2006. Being a data pipeline concept, it has integrated data mining and machine learning. KNIME is built on Java which makes it easy to be installed on all the platforms with ease.
KNIME is mostly used in other areas like CRM's for analyzing custom data and in business intelligence as well.
The main feature of KNIME is that it allows its users to visually create data flows or flowcharts, select certain steps at a time to execute them or maybe all at a time and then form a collective analysis to discuss results, views on the same.
In the Gartner Quadrant 2018, KNIME was rated one among the top 4 products used for data visualization and analytics.
The most recent and stable release of version 3.6.1 was on Sept 9, 2018.
Tableau is a data visualization software developed by Tableau, a software company in Seattle, Washington, US which provides visualization on business intelligence. This application was introduced in 2003 by 3 developers whose idea was to simplify visualization techniques analyzing data structures and cubes.
According to a recent survey, Tableau and QlikView are recognised to be the most powerful software for business intelligence. The main advantage of this software is the ability to allow you to understand data structuring in real time, apart from allowing users to share the analysis with each other.
Having a series of change in names for the application, The purpose was still the same, Open Refine was previously called Google Refine and was called Freebase gridworks prior to it and is an open source application which is used for data cleaning and transformation to other formats which are called data wrangling, it works similar to excel and google spreadsheets,
The main function of this tool is that of its portable like feature allows actions used on one dataset to be implemented on another dataset.
Open Refine does not store formulae like spreadsheets in their cell, however, the transformation or structuring of data is done using the formula manually on each cell. All the expressions/formulae are written in General Refine Expression Language(GREL).
Open Refine is a web UI which runs on a web server but is also available to use on local machines.
Import files are available in the following formats for Open Refine:
TSV, CSV
RDF triples
Google Spreadsheets,
Export is supported in the following formats:
HTML table
Templating exporter
Talend is a high-speed enterprise data integration tool, founded by NASDAQ 13 years ago in 2005. Allowing data integration on all forms of cloud formats such as public, private and hybrid cloud, this tool is also available in on-premises environments.
Talend publishes its code and code modules in Apache License, however, all the applications are designed on Java platform. Basically, talend is a tool that can clean, transform and visualize data at one go.
With its capability to save and redo tasks from one data set to another makes it unique among so many applications in today's world. It also has the auto-discovery feature or auto-suggestion feature that suggests options to the user enhancing data analysis
Being a cloud-based platform, Looker specializes in 4 aspects, data management, visualizations, business intelligence, and analytics. It has a user-friendly interface as well as provides content in a graphical manner where it becomes easier for you to learn and understand the insights being provided. It has high data security and has a unique feature where customizations of Google and Facebook ad reports are conducted. View its extended visualization library and indulge in its assistance of creating small applications for your use. For their visualization options, they retrieve data directly from the source.
Qlik sense Offers multiple essential and easy to use features such as drop-down, smart search and access to real-time analytics, this could based application is a tool that can be viewed anywhere. Supporting databases such as IBM, Microsoft, Oracle, and many others, this tool is ideal for any size of business. Choose between a general or a business plan and view your analytic data on any device.
Another tool that gives you data analysis but has a dedicated platform for the online world. Functioning in the fields of social network and content analysis, you can improve your content marketing strategies with data that can help you enhance your social networks. Specialized for data-driven marketers, this tool has also included in its list social media analytics feature allowing you to serve your online users better. From giving you the option of importing data to various sources to provide you with better researching features, NodeXL is the finest social networking data analytics tool you require.
Analysis can be simple and easy if you understand how data analytics is done using different tools and techniques. Above are some important tools which will help all the analysis to' sort and structure data systematically in the long run.
With these tools, it is always recommended to use proxies to be safe and untraceable. With a tool like LimeProxies, you can always be sure that you will be safe and anonymous on the web.
Rachael Chapman
A Complete Gamer and a Tech Geek. Brings out all her thoughts and Love in Writing Techie Blogs.
How to become a Data Scientist? (The 2020 version)
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If you've shot an HDR, you know that minimizing movement is essential.
Tripod… check. Steady hands… check. Wind… DAMN!
One problem that plagues HDR is subtle movement in the scene. That could be clouds, walking people, or branches blowing in the wind. In this quick tip, Rich Harrington shows you how one checkbox can solve your ghosting problems.
For more information on tips, tricks, reviews, punditry, training videos, podcasts and anything else they can think of that might interest photographers using time lapse, HDR or panoramic photography check out Triple Exposure!
There are more creative tools for professional photographers today than at any time in the history of photography. The challenge is making sure you're learning from the very best instructors. It's not just about understanding the tools, but being able to tell the story with the greatest impact and efficient use of both your time and your client's.
The SCU video on the home page was done by Rich Harrington and RHED Pixel. They're an outstanding visual communications company based out of Washington D.C. The expertise Rich Harrington and his team have built into the company will be available to help SCU attendees raise the bar on the quality of their mixed media productions.
As you watch a few examples of RHED Pixel's work, think about how you'd tell your story. How would you tell the story of your company and combine your passion for photography and your skill set, getting the message out in minimal time, while still maintaining the quality of your image and brand? Better yet, do you have the skill set you need to tell the story of your clients as part of the services you offer?
Meet RHED Pixel and our new Dean of Video/Fusion, Rich Harrington.
National Foundation for Credit Counseling - Mortgage PSA from RHED Pixel on Vimeo.
Diabetes2025 Union Station Drawing - http://www.altfutures.org/diabetes2025 from RHED Pixel on Vimeo.
ARC Blood Donor from RHED Pixel on Vimeo. | {
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Memorial Day Is 'Sacred' In Virgin Islands
CaribbeanDays.com LLC thanks all that serve
Governor Kenneth Mapp, USVI encouraged the people of the Virgin Islands to remember the service men and women who have made the ultimate sacrifice to this great nation by attending the ceremonies and parade honoring the Military for Memorial Day, Monday, 5/28/2018 on St. Thomas and St. Croix. The parade route begins at Bassin Triangle in Christiansted and at Western Cemetery in Charlotte Amalie.
Governor Mapp, in his Memorial Day 2018 statement, Mapp noted the long relationship the territory has had with the U.S. military.
"As we honor the nation's dearly departed military service members, I join with my fellow Americans in remembering those, who since the Revolutionary War, continue to preserve and protect the vital interest of the United States at home and across the world," Governor Mapp said. "Let us seek God's merciful grace on those now serving and proudly honor the memory of the fallen, not only on this national holiday, but every day."
"There is documented evidence that men from the Danish West Indies fought in the battle of Gettysburg in 1863 more than 50 years before the transfer to the United States," Mapp wrote. "During the Second World War, Virgin Islanders petitioned to be included in the draft and two units of Virgin Islanders, including two future governors, departed these shores to be a part of the United States Army. Today, scores of Virgin Islanders are stationed throughout the globe on Freedom's watch."
The United States Coast Guard, Eagle pulls into port for the Memorial Day Events:
CaribbeanDays Villas on St. Thomas
Vintage Military pictures of Water Island, USVI | {
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Le gardian (du provençal gardian, en français « gardien ») est le gardien d'une manade camarguaise ou troupe de taureaux ou de chevaux élevée en semi-liberté et appartenant à un manadier. Pour le Code du travail, le gardian est un ouvrier agricole. Dans le monde rural de la France du , c'est le bouvier de Camargue.
Avant le , les gardians ont une tenue adaptée à leur travail et à la saison ; l'outil des gardians à pied est le bâton, celui des gardians montés est le trident ; leur habitation la cabane en roseaux. Ils sont souvent en sabots.
Origine et vogue du terme
La popularisation du terme provençal gardian est liée à la folklorisation qu'a connue ce métier au cours du . Certaines cartes postales des années 1900 emploient le terme français « gardien » dans leur légende : « Le Capitaine des Gardiens Raynaud » ou encore « Le gardien BICHETTE de la Manade Combet » entre autres exemples.
Le féminin de gardian, gardianeto (francisé en « gardianette »), désigne, dans certaines cartes postales anciennes, une cavalière en costume d'Arlésienne montant en amazone à l'arrière d'un gardian lors de fêtes provençales ou de spectacles taurins. Quant à la « gardianne », c'est un plat typiquement camarguais à base de viande de taureau marinée.
Le statut social
Au et au tout début du , le gardian ou bouvier est un ouvrier agricole occupant le bas de l'échelle sociale du monde rural camarguais. Il appartient à la masse des manouvriers qui louent leurs bras dans les campagnes françaises. Il est attaché à un domaine, le mas, et travaille sous les ordres d'un régisseur ou baile-gardian, sinon du maître lui-même (le pelot ou mèstre). Il n'est propriétaire ni de son cheval (quand on lui en confie un), ni de sa cabane (construite par un artisan sur les instructions du maître). Son rôle principal est de garder les bious ou taureaux lorsqu'ils se sustentent, ce qu'il fait à pied et avec un bâton court (le calos) généralement en frêne, ou de trier le bétail, ce qu'il fait à cheval et avec une pique terminée par un trident (ficheiroun). Pour cette raison, il est souvent appelé gardo-bèsti (« garde-bêtes »). Une besace (ou brasso) lui sert à emporter sur son lieu de travail repas, boisson ou divers objets dont il a besoin.
La tenue
Avant normalisation
Si la Confrérie arlésienne de Saint Georges, rénovée dans les années 1910-1920 sous l'appellation de « Confrérie des gardians » par le félibre Carle Naudot, a été fondée en 1512, l'habit des gardians est de création très récente. Au début du , « le gardian de Provence n'a pas de tenue spéciale particulière au métier, il a pantalon (braio) et gilet (courset) en peau de taupe, étoffe beige [...], ou bien en peau de diable, étoffe très solide, indéchirable, à petits carreaux noirs et blancs ou bien gris et blancs, [...] parfois [...] en veau mort-né avec le poil », note Carle Naudot en 1945, ajoutant que « le chapeau n'est pas porté couramment ». Des photos du début du montrent des gardians en sabots et casquette qui ressemblent davantage à des ouvriers agricoles qu'à des cow-boys de l'Ouest américain.
Après normalisation
« Nous nous vêtirons à la mode gardiane », Rul d'Elly, Chant de guerre.
C'est le marquis de Baroncelli, promoteur du folklore camarguais dans la première moitié du , qui impose aux gardians amateurs, jeunes gens tous fils de pelot (propriétaire ou fermier), un costume bien précis, afin de donner plus d'unité et d'allure à leur troupe (chourmo) lors de ses fêtes gardianes, où ils caracolent, une jeune Arlésienne coiffée en croupe. Ce costume comprend une veste de velours noir à soutaches, un pantalon en tissu « peau de taupe » avec liseré noir, une chemise voyante à grands carreaux, une ceinture (taiolo, ou taillole) (qui a évolué depuis), un grand chapeau de feutre noir (valergo, ou valergue)), une cravate (régate). Le cavalier porte des jambières ou gamaches en étoffe de laine à carreaux, servant à le protéger de la rosée ou mouillure du matin.
La monture
La race
Même le type de monture est spécifié : un cheval de race Camargue uniquement. En croupe, pour les fêtes, une jeune Arlésienne coiffée.
Cette folklorisation continue de nos jours : le 17 avril 2008, une charte a été signée au Parc naturel régional de Camargue sur la tenue des cavaliers et de leurs chevaux, prévoyant notamment de « privilégier au maximum les chevaux de type Camargue, au détriment de races nettement moins représentatives de Camargue. Les chevaux de couleur n'entrant pas dans le type devront être écartés, tout comme les chevaux ressemblant TROP à des chevaux de trait pour le devant d'abrivado ».
Le harnachement
L'outil de travail principal du gardian est le trident de fer ou ficheiroun, longue gaule de frêne ou de châtaignier armée d'une douille conique en fer terminée par trois pointes. C'est avec cet instrument que le bouvier se fait obéir du taureau, en particulier lors du triage du bétail.
Un autre outil est le seden, corde tressée avec du crin de jument et servant de lasso pour capturer le cheval dans la manade et de licol pour l'attacher.
La selle gardiane, différente de sa cousine anglaise, dérive de la selle à piquer française (encore en usage au Cadre Noir de Saumur) et comporte un troussequin élevé en forme de dossier et un pommeau également élevé.
Les étriers du gardian sont en forme de cage car lorsque le gardian n'était qu'un paysan il portait des sabots et la cage servait à ce que le sabot puisse rentrer à l'intérieur mais elle sert aussi pour que le cavalier, en cas de chute, ne reste pas accroché et se dégage facilement.
La cabane
Avant le
Héritières des premières cabanes d'habitation apparues au en Camargue et apparentées aux cabanes de roseaux qui parsemaient au le littoral languedocien et roussillonnais, les cabanes de gardians étaient construites à l'aide des matériaux végétaux disponibles localement, et ce uniquement pour des raisons de coût. Les matériaux nobles, acheminés depuis les régions limitrophes, étaient réservés à la construction des mas.
Des cabanes peu différentes servaient d'habitations permanentes ou saisonnières aux pêcheurs, bergers, agriculteurs, vanniers, sauniers qui travaillaient en Camargue. De celles-là, il ne reste que quelques clichés : le détail de leur architecture et leur nomenclature terminologique est perdu.
Au début du
La cabane de gardian du début du est un bâtiment à façade en pignon, à la toiture à deux versants inclinés de 45 %, dont la partie exposée au mistral est en abside et à croupe de façon à donner le moins de prise possible à celui-ci.
Elle possède une armature de piquets verticaux en bois d'ormeau supportant des pannes sablières (areniés). Sur ces dernières, s'appuient les chevrons (travetos, ou travettes), lesquels reposent en haut sur la panne faîtière (arenié mestre). Lorsque le pignon est en matériaux végétaux, la faîtière est soutenue par deux poteaux montant de fond, l'un à l'avant, l'autre à l'arrière; lorsque le pignon est en pierres maçonnées, elle repose à l'apex de celui-ci et sur un poteau de fond à l'arrière.
Le chevron central de la croupe dépasse systématiquement le faîte de la toiture pour se retrouver coiffé d'une corne (bano) ou barré transversalement en forme de croix. Par grand vent, pour éviter que la cabane ne se soulève, on attachait au bout saillant de ce chevron, des cordes fixées au sol.
La couverture est faite de rangées de javelles (manouns, ou manons) de roseau des marais (sagno ou sagne) posés sur des lattes (coundorsos, ou condorses). Pour obtenir une meilleure étanchéité, une rangée de tuiles canal scellées au mortier vient souvent coiffer le faîtage, et un enduit de mortier à la chaux (cacho-faio) est appliqué le long de ce dernier, formant une chemise (camiso) ou chape. Celle-ci a aussi comme avantages de réfléchir, par sa blancheur, les rayons du soleil, de protéger du vent le sommet de la toiture en le caparaçonnant et de réduire les risques d'incendie liés à la présence du conduit de cheminée.
Les ouvertures sont étroites et il n'y a pas de fenêtre au nord. L'entrée est toujours en pignon. La porte en est en bois. Une toile contre le soleil et les moustiques est suspendue au linteau en été.
La cabane occupe une surface de 80 à 120 m en moyenne.
Lorsque le pignon est en dur, une cheminée à hotte montante est adossée contre la paroi intérieure de celui-ci, et sous l'un des rampants, l'autre moitié du pignon étant prise par l'entrée. La souche de cheminée, généralement de section rectangulaire, dépasse toujours du rampant opposé à celui au-dessus de l'entrée.
La cabane s'allonge en fonction des besoins de l'habitant : soit pièce unique, l'occupant mangeant au mas, soit pièce à vivre et chambre. La chambre, séparée de la pièce principale, au mieux par un cloison, au pire par un simple rideau de tissu, occupe alors la partie arrondie ou culotte de l'édifice. Elle abrite un lit en forme de caisse, dit brèsso (fém.), sur lequel pose en guise de matelas un sac à sel de 50 kilos rempli d'herbes sèches, dites baunco. La pièce à vivre, pour sa part, est meublée simplement : une table, deux bancs, quelques étagères et coffres. En pignon, un auvent (laùpio, fém.), armature rudimentaire coiffée de sagne, complétée par une table et un banc en bois, sert aux tâches ménagères (préparation de la cuisine, vaisselle).
Le sol de la cabane est en terre battue ou en béton de terre (bétun), mélange de mortier de chaux et d'agrégats roulés.
Devant certaines cabanes, se dresse un poteau muni d'échelons, appelé escalassoun (échelier ou rancher), auquel le gardian est censé monter pour surveiller son troupeau. L'ethnologue camarguais Carle Naudot décrit semblable dispositif observé avant 1925 au lieu-dit Le Cardinal, terrain de l'Esquinau en basse Camargue : il s'agit du mât d'un navire naufragé sur la côte de Faraman, il est non pas planté en terre mais appuyé sur un peuplier blanc. On a donc fait de ce cas unique un équipement traditionnel de la cabane de gardian.
Aujourd'hui
Il n'existe plus aujourd'hui d'anciennes cabanes de gardians en dehors de celle qui a été remontée au Musée Arlaten à Arles (Bouches-du-Rhône). Aux Saintes-Maries-de-la-Mer, les cabanes des gardians du mas de l'Amarée, popularisées par les cartes postales de la première moitié du , ont été rasées. De même, celles du mas du Simbèu, construites vers 1930, ont été détruites une douzaine d'années plus tard par l'armée allemande.
Les cabanes visibles actuellement aux Saintes-Maries-de-la-Mer sont des variantes modernisées des cabanes à pignon en dur de la fin du XIX et du début du , en particulier la trentaine qui sont visibles entre le front de mer et l'étang des Launes, bâties par les derniers maîtres cabaniers dans les années 1950-1960 à l'initiative du maire de l'époque, Roger Delagnes.
Construites en matériaux modernes (à l'exception du matériau de couverture), les cabanes du milieu du sont assises sur des fondations et la chemise de leur faîtage est réalisée en ciment sur du grillage. La structure porteuse est constituée non plus par des poteaux de fond placés dans l'axe du bâtiment mais par des fermes triangulées reposant sur les murs gouttereaux. Si l'on ne peut plus parler d'« authenticité » ni de « respect des techniques anciennes » à leur sujet, ces cabanes modernes permettent toutefois à la forme et à l'image de la cabane de gardian de perdurer dans le paysage et les esprits.
Avec leurs murs en maçonnerie blanchis au lait de chaux, leur pignon aux rampants saillants, leurs larges baies (en pignon, sur les côtés et même parfois dans l'abside), et tout le confort moderne, certaines cabanes modernes servent de résidences secondaires, de gîtes, de chambres d'hôtel, de restaurants, etc., pour les touristes et les vacanciers.
Le métier
Il existe aujourd'hui des gardians professionnels et des gardians amateurs :
les premiers sont des salariés regroupés au sein de l'Association des gardians professionnels de taureaux et toros de Provence et de Languedoc ;
les seconds sont des bénévoles, regroupés pour la plupart dans l'Association des gardians non salariés.
La muselade
Introduction d'une muselière, petite pièce de bois plate et allongée, dans le cartilage du nez du veau d'un an, pour l'empêcher de téter et ainsi le sevrer.
La ferrade
Application au fer rouge de la marque de la manade sur la cuisse gauche du bouvillon d'un an.
Le triage
Le matin d'une course ou d'une ferrade, rassemblement et enfermement des bêtes de la manade dans un enclos pour les trier.
Le bistournage
Castration des jeunes taureaux par torsion du cordon testiculaire à l'aide d'une pince pour les rendre moins agressifs et plus aptes à la course camarguaise.
L'engasade
Immersion des troupeaux lors de la traversée du Rhône ou d'un marais.
L'abrivade
À l'origine, traversée au galop des taureaux de la course jusqu'aux arènes, encadrés par des gardians à cheval disposés en V. On disait aussi « la charge ».
L'escaumage
La tonte des juments pour obtenir des crins servant à confectionner les sedens pendant l'hiver. Les juments sont enfermées dans le bouvau (cour des taureaux ou toril) ou dans une bergerie (jasso) et saisies l'une après l'autre au cou par un nœud coulant.
Les jeux équestres
Il existe aussi des jeux de gardian comprenant, entre autres :
le saut de cheval à cheval : le gardian saute sur un autre cheval qui n'est pas sellé.
le saut de cheval à taureau : le gardian saute sur le taureau.
Notes et références
Bibliographie
Juliette Figuier, Le gardian de la Camargue - Mos de Lavêne, coll. « Auteurs célèbres », C. Marpon et E. Flammarion, Paris, 1889, 249 p.
Pierre Lanéry d'Arc, Les maisons-types de la Provence, chap. 35 de Enquête sur les conditions de l'habitation en France. Les maisons-types, t. 1, Ministère de l'instruction publique, Ernest Leroux, Paris, 1894, pp. 207-248.
de) Flandresy Jeanne, Charles-Roux Jules, Mellier Etienne, Le livre d'or de la Camargue, tome I, Le pays; les mas et les châteaux; le Rhône camarguais, Librairie A. Lemerre, Paris, 1916, 437 p.
Fernand Benoit, Les chaumières à abside de la Camargue : la cabane, origine, description, mode de construction, dans Revue du folklore français, t. 9, 1938, No 2, avril-juin, pp. 51-53, pl. h. t.
Fernand Benoit, Les coutumes, l'habitation et les fêtes [en Camargue], dans Le Chêne, numéro spécial, No 16, 1938, pp. 100-112.
D'Elly (Jean Rul D'Elly), La Camargue gardiane, Michel Delaveau, Paris, 1938, 165 p.
Carle Naudot, Ethnographie du pays d'Arles. Contribution au folklore de Camargue, Le Seden, 1947.
Henri Marc, Carle Naudot, Victor Quenin, Terre de camargue - Terro Camarguenco, Arthaud, Grenoble-Paris, 1948, 159 p.
Jean-Luc Massot, Maisons rurales et vie paysanne en Provence, Serg, 1975.
Georges Martin, La Camargue "gardianne" au temps passé, chez l'auteur, 1975, non paginé.
René Baranger, En Camargue avec Baroncelli, l'auteur, Clichy, 1983, 164 p.
Guy Châtel, La selle gardiane et le harnachement camarguais, dans Courrier du Parc, No 45-46, 1995.
.
.
Sophie Vignon, Les manadières et les gardianes dans la tauromachie camarguaise, Cahiers du Genre, 2019/1 (No 66), pp. 181-199.
Voir aussi
Gaucho
Liens externes
Cabanes entièrement en roseau des années 1900, Centre d'études et de recherches sur l'architecture vernaculaire
Texte publié à l'occasion de l'exposition « Cabanes de Camargue » réalisée par le Parc Naturel Régional de Camargue en 1983
Cavalier agricole
Éleveur
Manade | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,973 |
Bodyguard in Blue
Trust and Triumph
Hirofumi Fukuzawa
Yuichi Hachisuka
Yasuhiro Takeuchi
Actors, Deceased Power Rangers Actors
Richard Genelle
This article is about a/an actor in the Power Rangers franchise.
Richard Genelle as Ernie
December 30, 2008 (age 47)
Died in
Roles(s)
Appearance(s):
Mighty Morphin' Power Rangers (1993-1996)
Power Rangers Zeo (1996)
Turbo: A Power Rangers Movie (1997)
Richard Genelle (October 21, 1961 - December 30, 2008) was an actor who played Ernie, the owner of the Angel Grove Youth Center and frequent hang out of the Power Rangers, from the first season of Mighty Morphin' Power Rangers until the end of Power Rangers: Zeo. Genelle also made a brief appearence in Turbo: A Power Rangers Movie. Genelle's only other acting credit was as a guard in the 1991 film The Death Merchant.
Though Genelle played a popular supporting character on the show, he began having health problems due to his weight and was forced to leave the show because of his declining health (due to obesity and smoking problems). After leaving the show, Genelle successfully lost 40 pounds. He later founded a company called Retail Logistics Solutions, Inc. in Cerritos, California, providing transportation services.
Other appearances as Ernie
Mighty Morphin Power Rangers (Sega CD) (1995) (archive footage)
Mighty Morphin' Power Rangers: The Movie (1995) (scenes deleted)
Zeo Serial (1996)
The Good, the Bad, and the Stupid (1996) (archive footage)
The Lost Episode (1999) (archive footage)
Genelle passed away on December 30, 2008 due to a heart attack. He was forty-seven years old. His internment was in Riverside, California's Pierce Brothers Crestlawn Memorial Park.
Retrieved from "https://powerrangers.fandom.com/wiki/Richard_Genelle?oldid=432492"
Deceased Power Rangers Actors | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,872 |
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There's an Explanation for Those Aurora-Like Lights in Sulu
PAGASA gives an answer.
by Mia Rodriguez
(SPOT.ph) You might have seen pictures online of strange candle-shaped lights floating in the Sulu night sky, and like us, you likely have questions. No, unfortunately the strange shapes weren't aliens, but they are as interesting. The phenomenon was caused by ice crystals reflecting moonlight, said the Philippine Atmospheric, Geophysical and Astronomical Services Administration, in a report by GMA News Online.
PHOTO BY Amarkhan Jidara
"Ang nangyari kasi dyan, mayroon silang clear skies. 'Yong manipis na cirrus clouds, mayroong ice crystals. 'Yon 'yong nag-reflect ng ilaw galing sa buwan," Raymond Ordinario, Philippine Atmospheric, Geophysical and Astronomical Services Administration (PAGASA) weather specialist, told GMA News Online. This phenomenon, more commonly known as light pillars, usually happens in colder parts of the world.
The pictures were taken by Amarkhan Jidara on June 30 in the province of Sulu, but this isn't the first time the lights have been seen on our side of the world. In fact, the Tausug already have a local name for it: Lansuk-Lansuk, which translates to candles.
Science Pagasa Explainer | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,408 |
\section{Introduction}
To a real sequence $a_1, a_2, \ldots, a_m$ we can associate a permutation $\pi$ of $[m]:=\{1, \ldots, m\}$, which gives information about the shape of the histogram of the sequence, via
\begin{equation} \label{perm-assoc}
a_{\pi(1)} \leq a_{\pi(2)} \leq \cdots \leq a_{\pi(m)}.
\end{equation}
If there are some repetitions among the $a_i$ then $\pi$ is not unique. For example, the sequence $(5,10,10,5,1)$ has associated with it each of the sequences $51423$, $54123$, $51432$ and $54132$. (Here and elsewhere we present permutations in one-line notation, so for example $51423$ represents the permutation $\pi$ with $\pi(1)=5$, $\pi(2)=1$, et cetera.)
This association was introduced by Alavi, Malde, Schwenk and Erd\H{o}s in \cite{AMSE}, where they proposed using it to investigate sequences associated with graphs. For example, let ${\mathcal I}_m$ denote the set of (simple, finite) graphs $G$ with $\alpha(G)=m$, that is, whose largest independent set (set of mutually non-adjacent vertices) has size $m$. The {\em independent set sequence} of $G \in {\mathcal I}_m$ is the sequence $(i_k(G))_{k=1}^m$ where $i_k(G)$ is the number of independent sets of size $k$ in $G$. Say that $\pi$ is an {\em independent set permutation} of $G$ if $\pi$ is one of the permutations that can be associated to the independent set sequence of $G$ via (\ref{perm-assoc}). (We do not consider $i_0(G)$, as it equals $1$ for every $G$.)
The main theorem of \cite{AMSE} is that all $m!$ permutations of $[m]$ are independent set permutations.
\begin{thm} \label{thm-AMSE-main} \cite{AMSE}
Given $m \geq 1$ and a permutation $\pi$ of $[m]$, there is a graph $G$ with $\alpha(G)=m$ and with
\begin{equation} \label{AMSE-main-inq}
i_{\pi(1)} < i_{\pi(2)} < \cdots < i_{\pi(m)}.
\end{equation}
\end{thm}
In the language of \cite{AMSE} the independent set sequence of a graph is {\em unconstrained} --- it can exhibit arbitrary patterns of rises and falls.
For a permutation $\pi$ denote by $g(\pi)$ the minimum order (number of vertices) over all graphs $G$ for which $\pi$ is an independent set permutation of $G$, and for each $m$ denote by $f(m)$ the maximum, over all permutations $\pi$ of $[m]$, of $g(\pi)$. Alavi et al. showed that $f(m)$ is at most roughly $m^{2m+1}$ (they did not calculate their upper bound explicitly). They speculated that $f(m)\geq m^m$, and proposed the question of determining $f(m)$.
\begin{prob} \label{AMSE-ind-Q}
\cite[Problem 1]{AMSE} Determine the smallest order large enough to realize every permutation of order $m$ as the sorted indices of the vertex independent set sequence of some graph.
\end{prob}
Our first result settles this question exactly.
\begin{thm} \label{thm-ind-set-main}
(Part 1, $f(m)\leq m^m$) For each $m \geq 1$ there is a graph $G_m$ on $m^m$ vertices with $\alpha(G)=m$ and with
\begin{equation} \label{all-equal}
i_1(G_m)=i_2(G_m)=\cdots= i_m(G_m)=m^m
\end{equation}
(and so for every permutation $\pi$ of $[m]$, $\pi$ is an independent set permutation of $G_m$).
(Part 2, $f(m)\geq m^m$) On the other hand, if $\alpha(G)=m$ and $i_m(G) < m^m$ then $i_m(G) < i_{m-1}(G)$ (and so for a permutation of $[m]$ of the form $\cdots(m-1)\cdots m \cdots 1 \cdots$ to be an independent set permutation of some graph $G$, $G$ must have at least $m^m$ vertices).
\end{thm}
Our proof that $f(m)\geq m^m$ follows fairly quickly from results of Frankl, F\"uredi and Kalai \cite{FFK} and Frohmader \cite{Frohmader} on Kruskal-Katona type theorems for flag complexes. Our construction of $G_m$, to establish $f(m)\leq m^m$, follows the same general scheme introduced in \cite{AMSE}. There, it is shown how to construct a graph $G$ with $\alpha(G)=m$, with $i_k(G)$ being a sum. The first term of the sum is $\pi^{-1}(k)T$ (for some arbitrary constant $T$), and for $T$ sufficiently large the sum of the remaining terms can be bounded above by $T$. This puts $i_k(G)$ in the interval $[\pi^{-1}(k)T, \pi^{-1}(k)T + 1)$, and so $\pi$ is a (actually, the unique) independent set permutation of $G$. (We describe this construction in more detail in Section \ref{sec-isperm}). We obtain $f(m)\leq m^m$ by carefully carrying out the construction in a way that allows perfect control over the lower order terms in the sum.
This finer control allows us to extend Theorem \ref{thm-AMSE-main}. To a real sequence $a_1, a_2, \ldots, a_m$ we can associate a unique weak order (an ordered partition $(B_1,\ldots, B_\ell)$ of $[m]$ into non-empty blocks)
via $B_i=\{b_{i1}, b_{i2}, \ldots\}$, where
$$
a_{b_{11}} = a_{b_{12}} = \cdots < a_{b_{21}} = a_{b_{22}} = \cdots < \cdots < a_{b_{\ell1}} = a_{b_{\ell2}} = \cdots.
$$
For example the sequence $(4,6,4,1)$ (the independent set sequence of the edgeless graph on four vertices) induces the weak order $B_1=\{4\}$, $B_2=\{1,3\}$, $B_3=\{2\}$. Theorem \ref{thm-AMSE-main} says that every weak order in which all blocks are singletons is the weak order induced by some graph, while Part 1 of Theorem \ref{thm-ind-set-main} says the same for every weak order with a single block.
\begin{thm} \label{thm-ind-set-weak}
For $m \geq 1$, for every weak order $w$ on $[m]$ there is a graph $G$ with $\alpha(G)=m$, and with fewer than $m^{m+2}$ vertices, which induces $w$.
\end{thm}
Note that the number of weak orders on $[m]$ grows like $(1/2)m!(\log_2e)^{m+1}$ \cite{Barthelemy}, faster than the number of permutations. Note also that by Theorem \ref{thm-ind-set-main}, any weak order on $[m]$ that has $m-1$ and $m$ in the same block, and $1$ in a block with a higher index, cannot be induced by a graph with $m^m$ or fewer vertices. The analog of Problem \ref{AMSE-ind-Q} for weak orders --- where in the range $(m^m, m^{m+2})$ is the smallest order sufficient to realize every weak order on $[m]$? --- remains open.
\medskip
Alavi et al. also considered the {\em edge independent set sequence} or {\em matching sequence} of a graph. Let ${\mathcal M}_n$ denote the set of graphs with $\nu(G)=n$, that is, whose largest matching (set of edges no two sharing a vertex) has $n$ edges.
The matching sequence of $G \in {\mathcal M}_n$ is $(m_k(G))_{k=1}^n$ where $m_k(G)$ is the number of matchings in $G$ with $k$ edges. Say that $\pi$ is a {\em matching permutation} of $G$ if $\pi$ is one of the permutations that can be associated to the matching sequence of $G$ via (\ref{perm-assoc}).
In contrast to independent set permutations, there are permutations that are not the matching permutation of any graph. Indeed, Schwenk \cite{Schwenk} showed that the matching sequence of any graph $G \in {\mathcal M}_n$ is unimodal in the strong sense that for some $k$,
$$
m_1(G) < m_2(G) < \cdots < m_k(G) \geq m_{k+1}(G) > m_{k+2}(G) > \cdots > m_m(G).
$$
It follows that the permutations of $[n]$ that can be the matching permutations of a graph in ${\mathcal M}_n$ must have
\begin{equation}\label{unimodal}
\begin{array}{c}
\pi^{-1}(1)<\pi^{-1}(2)<\cdots < \pi^{-1}(k-1)\\~\mbox{and}\\
\pi^{-1}(n)< \pi^{-1}(n-1)< \cdots < \pi^{-1}(k+1),
\end{array}
\end{equation}
where $k:=\pi(n)$.
(This restriction on $\pi$ can also be deduced from the real-rootedness of the matching polynomial, first established by Heilmann and Lieb \cite{HL}.) Following Alavi et al., we refer to permutations satisfying (\ref{unimodal}) as {\em unimodal} permutations.
There are
$\sum_{k=0}^{n-1} \binom{n-1}{k} = 2^{n-1}$
unimodal permutations of $[n]$ and so, writing $M_n$ for the set of permutations $\pi$ that are the matching permutations of some graph in ${\mathcal M}_n$, we have
$M_n \leq 2^{n-1}$.
This bound was observed in \cite{AMSE}, where the following problem was posed.
\begin{prob} \label{AMSE-matching-q}
\cite[Problem 2]{AMSE}
Characterize the permutations realized by the edge independence sequence. In particular, can all $2^{n-1}$ unimodal permutations of $[n]$ be realized?
\end{prob}
We do not address the characterization problem, but our next result answers the particular question: a vanishing proportion of unimodal permutations are the matching permutations of some graph.
\begin{thm} \label{thm-ub-on-matching}
We have $M_n=o(2^n)$. More precisely, there is a constant $c$ such that for $n \geq 1$
\begin{equation} \label{eq-up-on-count}
M_n \leq \frac{c2^n}{\sqrt{n}}.
\end{equation}
\end{thm}
In the other direction, the perfect matching gives a lower bound on $M_n$ of $2^{\lfloor (n-1)/2 \rfloor}$. We can improve this by an additive term of $\Omega(n)$, but we do not give the details here.
\medskip
We give the proofs of our results concerning independent set permutations and weak orders in Section \ref{sec-isperm}, and address matching permutations in Section \ref{sec-matperm}. We end with some questions and comments in Section \ref{sec-questions}.
\section{Independent set permutations} \label{sec-isperm}
We begin with the proof of Part 2 of Theorem \ref{thm-ind-set-main}, $f(m)\geq m^m$.
The set ${\mathcal I}(G)$ of independent sets in a graph $G$ forms a flag complex, whose ground set is the vertex set $V(G)$ of $G$: it is closed under taking subsets, all (singleton) elements of $V(G)$ are independent sets (so ${\mathcal I}(G)$ is a simplicial complex), and, since a subset of vertices that is not an independent set spans at least one edge, minimal non-independent sets have size 2 (so ${\mathcal I}(G)$ is a flag complex).
The dimension of ${\mathcal I}(G)$ is $m-1$, where $m=\alpha(G)$, and its $1$-skeleton is $\overline{G}$, the complement of $G$. The face sequence of ${\mathcal I}(G)$ --- the sequence whose $k$th term is the number of elements of ${\mathcal I}(G)$ of size $k$ --- is exactly the independent set sequence of $G$.
A flag complex of dimension $m-1$ is said to be {\em balanced} if its $1$-skeleton has chromatic number $m$. The complex ${\mathcal I}(G)$ is not necessarily balanced; consider, for example, the graph $G=C_5$ (the cycle on five vertices), which has dimension $1$ but whose $1$-skeleton is $C_5$, which has chromatic number $3$. However, Frohmader \cite[Theorem 1.1]{Frohmader}, settling a conjecture of Eckhoff and (independently) Kalai, showed that
\begin{equation} \label{thm-Froh}
\begin{array}{c}
\mbox{for any flag complex ${\mathcal S}$ there is a balanced simplicial} \\
\mbox{complex ${\mathcal S}'$ with the same face sequence as ${\mathcal S}$.}
\end{array}
\end{equation}
(For example, the set of independent sets of the graph on vertex set $\{a,b,c,d,e\}$, with edges $ab$, $ac$, $bc$, $cd$ and $de$, forms a balanced simplicial complex, in fact a flag complex, whose face sequence agrees with that of ${\mathcal I}(C_5)$.)
We now need a result of Frankl, F\"uredi and Kalai \cite[Theorem 5.1]{FFK}, which addresses the question of how the bounds in the Kruskal-Katona theorem change in the presence of information about the chromatic number of the underlying set system. Fix natural numbers $1 \leq \ell \leq k \leq r$. Suppose ${\mathcal F}$ is a family of $k$-subsets of ${\mathbb N}$ such that for any member of ${\mathcal F}$, no two of its elements are congruent modulo $r$. The {\em $\ell$-shadow} of ${\mathcal F}$, denoted $\partial_\ell({\mathcal F})$, is the family of $\ell$-subsets of ${\mathbb N}$ that are subsets of some element of ${\mathcal F}$. Frankl, F\"uredi and Kalai show that if $|{\mathcal F}|=\binom{r}{k}x^k$ for some $x \geq 0$, then
\begin{equation} \label{thm-FFK}
|\partial_\ell({\mathcal F})| \geq \binom{r}{\ell}x^\ell.
\end{equation}
\begin{proof} (Theorem \ref{thm-ind-set-main}, Part 1)
Let $G$ be any graph with $\alpha(G)=m$, and with $i_m < m^m$.
By (\ref{thm-Froh}) there is a balanced simplicial complex ${\mathcal S}'$ whose face sequence is the independent set sequence of $G$. Because ${\mathcal S}'$ is balanced and has dimension $m-1$, it can be realized as a set of subsets of ${\mathbb N}$, each of which has the property that no two of its elements are congruent modulo $m$. Let ${\mathcal F}$ be the set of elements of ${\mathcal S}'$ in this realization, that have size $m$, and ${\mathcal F}'$ the set of elements of size $m-1$. We have
$$
|{\mathcal F}|=i_m(G)=x^m
$$
for some $0 \leq x < m$ (since $i_m(G)<m^m$), so by (\ref{thm-FFK}) (in the case $r=k=m$, $\ell=m-1$) we have
\begin{equation} \label{FFK2}
|\partial_{m-1}({\mathcal F})| \geq mx^{m-1} > x^m = i_m(G).
\end{equation}
But also, because ${\mathcal S}'$ is a simplicial complex, we have $\partial_{m-1}({\mathcal F})\subseteq {\mathcal F}'$ and so
\begin{equation} \label{FFK1}
|\partial_{m-1}({\mathcal F})| \leq |{\mathcal F}'|=i_{m-1}(G).
\end{equation}
Combining (\ref{FFK2}) and (\ref{FFK1}) we obtain $i_m(G) < i_{m-1}(G)$, as claimed.
\end{proof}
\medskip
We now move on to the proof of Part 1 of Theorem \ref{thm-ind-set-main}, $f(m)\geq m^m$. We begin with an outline of the construction, which is very similar to one described in \cite{AMSE}. Recall that our goal is to construct a graph $G_m$ with $\alpha(G)=m$ that has $m^m$ independent sets of size $k$ for each $k \in [m]$. A key idea that we use throughout is the effect of the join operation on independent set sequences. For a collection $\{G_j : j \in J\}$ of graphs, denote by $\oplus_{j \in J} G_j$ the graph consisting of a union of disjoint copies of the $G_j$, with every vertex in each $G_j$ adjacent to every vertex in $G_{j'}$ for each $j' \neq j$ --- the {\em mutual join} of the $G_j$. The effect of $\oplus$ on independent set sequences is additive: if $G=\oplus_{j \in J} G_j$ then for $k \geq 1$,
\begin{equation} \label{join}
i_k(G) = \sum_{j \in J} i_k(G_j),
\end{equation}
because no independent set in $G$ can have vertices in two different $G_j$'s.
Given a permutation $\pi$ of $[m]$, to construct a graph $G$ satisfying (\ref{AMSE-main-inq}) (i.e., $i_{\pi(1)}(G) < \cdots < i_{\pi(m)}(G)$) Alavi et al. \cite{AMSE} consider a graph of the form
$$
G_{\pi}:=\oplus_{k=1}^m kK_{n_k},
$$
where $n_k = (\pi^{-1}(k)T)^{1/k}$ for some large integer $T$, and where $kK_{n_k}$ denotes $k$ vertex disjoint copies of the complete graph $K_{n_k}$ on $n_k$ vertices. By (\ref{join}) we have
\begin{equation} \label{AMSE-sum}
i_k(G_{\pi}) = \pi^{-1}(k)T + \sum_{j=k+1}^m \binom{j}{k}(\pi^{-1}(j)T)^\frac{k}{j}.
\end{equation}
Here the term $\pi^{-1}(k)T$ is the count of independent sets of size $k$ in $kK_{n_k}$, and for $j>k$ the summand $\binom{j}{k}(\pi^{-1}(j)T)^\frac{k}{j}$ counts independent sets of size $k$ in $jK_{n_j}$; there are no independent sets of size $k$ in any $jK_{n_j}$ for $j < k$. For large enough $T=T(m)$ the sum in (\ref{AMSE-sum}) is strictly smaller than $T$, so that $\pi^{-1}(k)T \leq i_k(G_{\pi}) < (\pi^{-1}(k)+1)T$ and (\ref{AMSE-main-inq}) holds.
To more carefully control the sum in (\ref{AMSE-sum}), and allow us to construct a graph $G_m$ with $m^m$ independent sets of all sizes from $1$ to $m$, we modify this construction. Before doing so, we give some intuition.
The graph $G_0:=mK_m$ has $\alpha(G_0)=m$, $i_m(G_0)=i_{m-1}(G_0)=m^m$, and $i_k(G_0) = \binom{m}{k}m^k < m^m$ for $k < m-1$. We need to increase the count of independent sets of size $m-2$ by
$$
m^m-\binom{m}{2}m^{m-2}=m^{m-2}\left(m^2-\binom{m}{2}\right):=a_2m^{m-2},
$$
without changing the number of independent sets of sizes $m$ or $m-1$. By (\ref{join}), the graph $G_2:=\oplus_{i=1}^{a_2} (m-2)K_m$ (the mutual join of $a_2$ copies of $(m-2)K_m$) has $i_{m-2}(G_2)=a_2m^{m-2}$, and also has $i_m(G_2)=i_{m-1}(G_2)=0$. Hence, again by (\ref{join}), $\alpha(G_0 \oplus G_2)=m$, $i_m(G_0 \oplus G_2)=i_{m-1}(G_0 \oplus G_2)=i_{m-2}(G_0 \oplus G_2)=m^m$, and $i_{m-3}(G_0 \oplus G_2)=\binom{m}{3}m^{m-3} + a_2(m-2)m^{m-3}$. We need to add
$$
m^{m-3}\left(m^3 - \binom{m}{3} - a_2(m-2)\right) := a_3m^{m-3}
$$
independent sets of size $m-3$ (without adding any independent sets of sizes $m, m-1$ or $m-2$). We achieve this by setting
$$
G_3:= \oplus_{i=1}^{a_3} (m-3)K_m
$$
and considering $G_0\oplus G_2 \oplus G_3$. (Note that $a_3 \geq 0$, being a cubic in $m$ with non-negative coefficients.)
We continue in this manner until we reach a graph which satisfies (\ref{all-equal}), which we declare to be $G_m$. We have to check that at no point, while fixing the number of independent sets of size $k$ to be $m^m$, do we cause the number of independent sets of size $j$ to be greater than $m^m$, for some $1 \leq j < k$. This check is the main point of the formal proof of Theorem \ref{thm-ind-set-main}, Part 1.
\medskip
\begin{proof} (Theorem \ref{thm-ind-set-main}, Part 1)
For $m \geq 1$, define a sequence $(a_0, a_1, \ldots, a_{m-1})$ via
\begin{equation} \label{recurrence}
m^k = a_0\binom{m}{k} + a_1\binom{m-1}{k-1} + \cdots + a_{k-1}\binom{m-(k-1)}{1} + a_k\binom{m-k}{0}
\end{equation}
for $k=0, \ldots, m-1$ (so $a_0=1$, $a_1=0$, $a_2=m^2-\binom{m}{2}$, et cetera). The motivation behind this definition as follows: we will go through an iterative procedure (the one described above) to set the number of independent sets of each size to be $m^m$, starting with independent sets of size $m$, and working down. When we come to fix the number of independent sets of size $m-k$ to be $m^m$, it will turn out that we need to add $a_km^{m-k}$ such, which we will achieve by successively joining $a_k$ copies of $(m-k)K_m$ to what has thus far been constructed. Evidently each $a_i$ is an integer; but in fact $a_i \geq 0$, as we now show by induction.
For $m=1$ the sequence $(a_0, a_1, \ldots, a_{m-1})$ consists of the single term $a_0=1$, and for $m=2$ the sequence is $(1,0)$. So consider $m \geq 3$. We have $a_0=1$. Now suppose $a_0, \ldots, a_k$ are all non-negative, for some $k$, $0 \leq k \leq m-2$. We have
\begin{eqnarray*}
m^{k+1} & = & a_0m\binom{m}{k} + a_1m\binom{m-1}{k-1} + \cdots + a_{k-1}m\binom{m-(k-1)}{1} + a_km\binom{m-k}{0} \\
& \geq & a_0\binom{m}{k+1} + a_1\binom{m-1}{k} + \cdots + a_{k-1}\binom{m-(k-1)}{2} + a_k\binom{m-k}{1} \\
& = & m^{k+1} - a_{k+1},
\end{eqnarray*}
so $a_{k+1}\geq 0$.
The first equality here uses (\ref{recurrence}), the inequality uses
$$
m\binom{m-j}{k-j} \geq \binom{m-j}{k-j+1},
$$
valid for $m \geq 3$, $k \in \{0, \ldots, m-2\}$ and $j \in \{0, \ldots, k\}$, and the second equality uses (\ref{recurrence}) again (this time with $k$ replaced by $k+1$).
Now consider the graph
$G_m = \oplus_{k=0}^{m-1} G_k$ where $G_k = \oplus_{j=1}^{a_k} (m-k)K_m$.
We have $\alpha(G_m)=m$ and, for each $k \in \{0 \ldots, m-1\}$
\begin{eqnarray*}
i_{m-k}(G_m) & = & a_0\binom{m}{k}m^{m-k} + a_1\binom{m-1}{k-1}m^{m-k} + \cdots + a_k\binom{m-k}{0}m^{m-k} \\
& = & m^{m-k}\left(a_0\binom{m}{k} + a_1\binom{m-1}{k-1} + \cdots + a_k\binom{m-k}{0}\right) \\
& = & m^m,
\end{eqnarray*}
the last inequality by (\ref{recurrence}). The main points of the calculation above are that the only parts of $G_m$ that contribute to $i_{m-k}(G_m)$ are those of the form $aK_m$ for $a\geq m-k$, and that
$$
i_{m-k}(aK_m) = \binom{a}{m-k}m^{m-k} = \binom{m-(m-a)}{k-(m-a)}m^{m-k}.
$$
\end{proof}
\medskip
We now turn to the proof of Theorem \ref{thm-ind-set-weak}, concerning weak orders. The case $m=1$ is trivial, and $m=2$ is easy: the three weak orders on $[2]$ are achieved by $K_2$, $2K_2$ and $K_3 \cup K_2$. So from here on we assume $m \geq 3$.
We will construct
\begin{itemize}
\item a graph $H_1$ with $m^m+m^{m-1}$ vertices, with $m^m$ independent sets of each size in $\{2, \ldots, m\}$, $m^m+m^{m-1}$ independent sets of size $1$, and with $\alpha(H_1)=m$;
\item a graph $H_m$ with $2m^m-m^{m-1}$ vertices, with $2m^m-m^{m-1}$ independent sets of each size in $\{1, \ldots, m-1\}$, $2m^m$ independent sets of size $m$, and with $\alpha(H_m)=m$;
\item and for each $k \in \{2, \ldots, m-1\}$, we will construct a graph $H_k$ with $m^m$ vertices, with $m^m$ independent sets of each size in $\{1, \ldots, m\}\setminus\{k\}$, with $m^m+m^{m-1}$ independent sets of size $k$, and with $\alpha(H_m)=m$.
\end{itemize}
The main point here is that for each $k$ there is $s(k)$ such that $H_k$ has $s(k)$ independent sets of all sizes except $k$, and has $s(k)+m^{m-1}$ independent sets of size $k$ (specifically $s(k)=m^m$ for $k \neq m$ and $s(m)=2m^m-m^{m-1}$).
Let $w=(B_1, \ldots, B_\ell)$ be a weak order on $[m]$. Construct a graph $H(w)$ as follows: $H(w)$ is the mutual join of one copy of $G_m$ for each $k \in B_1$ (here and later, $G_m$ is the graph from Theorem \ref{thm-ind-set-main}, Part 1); one copy of $H_k$ for each $k \in B_2$, and in general $j-1$ copies of $H_k$ for each $k \in B_j$. For $k \in B_j$, for any $1 \leq j \leq \ell$, we have
$$
i_k(H_w) = \left(m^m|B_1| + \sum_{i \in B_2} s(i) + 2\sum_{i \in B_3} s(i) + \cdots + \sum_{i \in B_\ell} (\ell-1)s(i)\right) + (j-1)m^{m-1}.
$$
Noting that the term in parenthesis above is independent of $j$ and $k$, we see that the weak order induced by $H(w)$ is indeed $w$.
Among the $H_k$ none has more than $2m^m-m^{m-1}$ vertices, so the order of $H(w)$ is at most
\begin{equation} \label{verts-in-Hw}
m^m|B_1| + (|B_2|+2|B_3|+\cdots + (\ell-1)|B_\ell|)(2m^m-m^{m-1}).
\end{equation}
If any of the $B_i$'s has size at least $2$, then the quantity in (\ref{verts-in-Hw}) can be increased by replacing $|B_i|$ with $|B_i|-1$ and $|B_{i+1}|$ with $|B_{i+1}|+1$ (creating a new, $(\ell+1)$st, block if $i=\ell$). It follows that subject to the constraints $\sum_i |B_i| =m$ and $|B_i| \geq 1$, the quantity in (\ref{verts-in-Hw}) is maximized by
$$
m^m + (1+2+\ldots + (m-1))(2m^m+m^{m-1}) < m^{m+2}.
$$
This gives Theorem \ref{thm-ind-set-weak}; so our goal (which occupies the rest of the section) is to construct $H_k$, for $k \in\{1, \ldots,m\}$.
\medskip
In the proof of Theorem \ref{thm-ind-set-main}, we required $a_k \geq 0$. To construct $H_k$, we need a better bound.
\begin{lemma} \label{lem-a_k-large}
For $k \geq 2$ (and $m \geq 3$), $a_k \geq m^{k-1}$.
\end{lemma}
\begin{proof}
We will use an explicit expression for the $a_k$. It will be convenient in what follows to extend the sequence $(a_0, \ldots, a_{m-1})$ to $(a_0, \ldots, a_m)$, by using (\ref{recurrence}) to also define $a_m$.
Let $\vec{a}$ be the column vector with $a_j$ in the $j$th position (with the positions indexed from $0$ to $m$), and $\vec{m}$ the column vector with $m^j$ in the $j$th position. We have $M\vec{a}=\vec{m}$ where $M$ is the $(m+1)$ by $(m+1)$ matrix with $\binom{m-j}{i-j}$ in the $(i,j)$ position (rows and columns indexed from $0$). Here we understand $\binom{n}{c}$ to be $0$ for negative $c$. $M$ is lower triangular, with $1$'s down the diagonal, so invertible. We claim that $M^{-1}$ has $(-1)^{i-j}\binom{m-j}{i-j}$ in the $(i,j)$ position.
Indeed, consider the matrix $M\overline{M}$, where $\overline{M}$ has $(-1)^{i-j}\binom{m-j}{i-j}$ in the $(i,j)$ position. The $(k,\ell)$ entry of $M\overline{M}$ is clearly $0$ for $k<\ell$, and $1$ for $k=\ell$. For $\ell<k$ the $(k,\ell)$ entry is
$$
\sum_{t=\ell}^k (-1)^{t-\ell} \binom{m-t}{k-t}\binom{m-\ell}{t-\ell} = (-1)^{\ell-k} \binom{m-\ell}{m-k} \sum_{t=\ell}^k (-1)^{k-t}\binom{k-\ell}{k-t} = 0,
$$
the first inequality via some elementary rearrangements, and the second via the standard fact that the alternating sum of binomial coefficients is $0$. This shows that $M\overline{M}$ is the identity, and so the inverse of $M$ is as claimed.
Since $\vec{a}=M^{-1}\vec{m}$ we have
\begin{equation} \label{eq-a_k-exp}
a_k = m^k -m^{k-1}\binom{m-(k-1)}{1}+m^{k-2}\binom{m-(k-2)}{2} - \cdots + (-1)^k\binom{m}{k}.
\end{equation}
For $m \geq 3$ and $k \geq 2$, it is easily checked that the sequence
$$
m^k, ~m^{k-1}\binom{m-(k-1)}{1}, ~ m^{k-2}\binom{m-(k-2)}{2}, \ldots,~\binom{m}{k}
$$
is strictly decreasing.
Lower bounding $a_k$ by the sum of the first two terms of the decreasing alternating sum on the right-hand side of (\ref{eq-a_k-exp}) we get
$$
a_k > m^k - m^{k-1}\binom{m-(k-1)}{1} = (k-1)m^{k-1} \geq m^{k-1},
$$
as claimed.
\end{proof}
\medskip
Another tool we will need in the construction of the $H_k$ is the following easy observation.
\begin{lemma} \label{lem-obsv-dec}
If $k \leq n$ (with $k, n$ natural numbers), then the sequence
$$
n^k,~ \binom{k}{1}n^{k-1},~\binom{k}{2}n^{k-2}, \ldots, ~\binom{k}{k-1}n
$$
is non-increasing. In fact it is strictly decreasing, except that when $k=n$ the first two terms are equal.
\end{lemma}
\medskip
Lemma \ref{lem-obsv-dec} gives an alternate proof that the procedure described in the proof of Theorem \ref{thm-ind-set-main} (the construction of $G_m$) is valid. The construction starts with $mKm$, or $G_0 \oplus G_1$, which has $m^m$ independent sets of size $m$, and $m^m$ of size $m-1$. By Lemma \ref{lem-obsv-dec} the sequence $(i_{m-2}(mK_m), \ldots, i_1(mK_m))$ is strictly decreasing, with first term at most $m^m$, and with $i_{m-k}(mK_m)$ a multiple of $m^{m-k}$.
The construction continues by successively joining $a_2$ copies of $(m-2)K_m$ to what has currently been constructed, to obtain $G_0 \oplus G_1 \oplus G_2$ where $a_2 \geq 0$ is defined by $a_2m^{m-2}=m^m-\binom{m}{2}m^{m-k} ~(=i_{m-2}(mK_m))$. This brings the number of independent sets of sizes $m-2$ up to $m^m$, and by Lemma \ref{lem-obsv-dec} the sequence $(i_{m-3}(G_0 \oplus G_1 \oplus G_2), \ldots, i_1(G_0 \oplus G_1 \oplus G_2))$ is still strictly decreasing, with first term at most $m^m$, and with $i_{m-k}(G_0 \oplus G_1 \oplus G_2)$ a multiple of $m^{m-k}$.
Lemma \ref{lem-obsv-dec} now allows this construction to be inductively continued until $G_m$ is reached.
We modify things slightly to obtain $H_k$.
\medskip
\noindent {\bf Case 1}, $k=1$: Set $H_1=G_m \oplus K_{m^{m-1}}$. Note that this requires neither Lemma \ref{lem-a_k-large} nor Lemma \ref{lem-obsv-dec}.
\medskip
\noindent {\bf Case 2}, $k \neq m, 1$:
At the moment when the number of independent sets of size $k$ has reached $m^m$, there are $m^m$ independent sets of all sizes at least $k$, while the sequence $(i_{k-1}(G), \ldots, i_1(G))$ (where $G$ is the graph constructed so far) is strictly decreasing, with $i_{k-1}(G)=m^m - a_{m-(k-1)}m^{k-1} \leq m^m-m^{m-1}$ (the equality coming from the proof of Theorem \ref{thm-ind-set-main}, Part 1, and the inequality using Lemma \ref{lem-a_k-large}), and with $i_j(G)$ a multiple of $m^j$.
Successively join $m^{m-k-1}$ copies of $kK_m$ to $G$. This brings the number of independent sets of size $k$ up to $m^m + m^{m-1}$, and it adds
$$
km^{k-1}m^{m-k-1} \leq m^{m-1}
$$
independent sets of size $k-1$. The result is a graph $G'$ with $i_m(G')=\cdots=i_{k+1}(G')=m^m$, $i_k(G')=m^m+m^{m-1}$, with $(i_{k-1}(G'), \ldots, i_1(G'))$ strictly decreasing, with $i_{k-1}(G') \leq (m^m-m^{m-1})+m^{m-1} = m^m$, and with $i_j(G)$ a multiple of $m^j$. The inductive procedure described earlier (for the construction of $G_m$) can now be continued to obtain $H_k$.
\medskip
\noindent {\bf Case 3}, $k=m$: Instead of starting the construction with $mK_m$, we start with $K_{2m} \cup (m-1)K_m$. This has $2m^m$ independent sets of size $m$, and in general
$$
\binom{m-1}{k}m^k + 2m\binom{m-1}{k-1}m^{k-1}
$$
independent sets of size $k$. Two applications of Lemma \ref{lem-obsv-dec} give that the sequence $(i_{m-k}(K_{2m} \cup (m-1)K_m))_{k=1}^{m-1}$ is strictly decreasing, with $i_{m-k}(K_{2m} \cup (m-1)K_m)$ a multiple of $m^{m-k}$, and with $i_{m-1}(K_{2m} \cup (m-1)K_m)=2m^m-m^{m-1}$. The inductive procedure described earlier can now be implemented to obtain $H_m$.
\section{Matching permutations} \label{sec-matperm}
We begin by observing quickly that not all $2^{n-1}$ unimodal permutations of $\{1,\ldots,n\}$ are realizable as the permutation associated to a graph with largest matching $n$. Indeed, the following lemma shows that $m_1(G)$ cannot be the largest entry of a matching sequence of any graph whose largest matching has size at least $4$, so that for $n \geq 4$ the permutation $n(n-1)\cdots321$ is not realizable.
\begin{lemma} \label{lem-2beats1}
If $\nu(G)\geq 4$ then $m_2(G)>m_1(G)$.
\end{lemma}
\begin{proof}
We proceed by induction on $e(G)$, the number of edges of $G$. In the base case, $e(G)=4$, $G$ must consist of four vertex disjoint edges, and we have $m_2(G)=6>4=m_1(G)$. For the induction step, let $G$ be a graph on more than four edges with $\nu(G)\geq 4$ and let $uv$ be an arbitrary edge in $G$, joining vertices $u$ and $v$. Let $G_1$ be obtained from $G$ by deleting the edge $uv$, and $G_2$ by deleting the vertices $u$ and $v$. We have $m_2(G)=m_2(G_1)+m_1(G_2)$ (the set of matchings of size $2$ in $G$ partitions into those that do not include $uv$ --- $m_2(G_1)$ many --- and those that do --- $m_1(G_2)$ many). Also, $m_1(G)=m_1(G_1)+1$. Now by induction $m_2(G_1)>m_1(G_1)$, and also $m_1(G_2)\geq 2>1$, because on deleting $u$ and $v$ from $G$ at least two of the edges of any matching of size $4$ remain. Combining we get $m_2(G)=m_2(G_1)+m_1(G_2) > m_1(G_1)+1=m_1(G)$.
\end{proof}
\medskip
We make an incidental observation at this point. The matching polynomial of a graph with maximum matching size $n$ can be expressed in the form $(1+r_1x)(1+r_2x)\cdots (1+r_nx)$ where the $r_i$'s are real and non-negative; this is a consequence of a theorem of Heilmann and Lieb \cite{HL}. To a sequence that arises as the coefficient sequence of a polynomial of the form $(1+r_1x)(1+r_2x)\cdots (1+r_nx)$ with $r_i$ real and non-negative, we can associate permutations via (\ref{perm-assoc}). Because real-rooted polynomials have unimodal coefficient sequences, at most only the $2^{n-1}$ unimodal permutations of $[n]$ can arise in this context. The permutation $n(n-1)(n-2)\cdots 321$ can arise: let all $r_i$ be equal, say equal to $r$, so the polynomial becomes
$$
1 + \binom{n}{1}rx + \binom{n}{2}r^2x^2 + \cdots + \binom{n}{n-1}r^{n-1}x^{n-1} + r^nx^n.
$$
It's easy to check that if $r$ is sufficiently small,
$$
r^n < r^{n-1} \binom{n}{n-1} < \cdots < \binom{n}{2}r^2 < \binom{n}{1}r
$$
so that this polynomial has associated with it the unique permutation $n(n-1)(n-2)\cdots 321$. This shows that our observations about restrictions on the matching sequence are not just restrictions coming in disguise from the real-rooted property of the matching polynomial.
\medskip
The proof of Lemma \ref{lem-2beats1} generalizes considerably. We state and prove the generalization first, and then consider the consequences for matching permutations.
\begin{thm} \label{thm-main-res}
For each $n \geq 1$, and for each $k=0,\ldots, \lfloor n/2 \rfloor - 1$, if $\nu(G)\geq n$ then $m_k(G) < m_\ell(G)$ for each $\ell$ satisfying $k < \ell < n-k$.
\end{thm}
\begin{proof}
We begin by dealing with some easy boundary cases. The result is vacuously true for $n=1$. For $n\geq 2$ and $k=0$, the claim is that if $\nu(G)\geq n$ then $m_0(G) < m_\ell(G)$ for $1\leq \ell\leq n-1$. But $m_0(G)=1$, while $m_\ell(G) \geq \binom{n}{\ell}>1$ (just consider matchings of size $\ell$ that are subsets of any particular matching of size $n$ in $G$), so the claim is valid in this case.
This deals completely with the cases $n=2, 3$, as well as $n=4$, $k=0$. For $n=4$, $k=1$, the claim is that if $\nu(G)\geq 4$ then $m_1(G) < m_2(G)$, which is exactly Lemma \ref{lem-2beats1}. This completes the case $n=4$.
We now proceed by induction on $n$. For a particular $n>4$, assume that we already have the result for all $1 \leq n' < n$. Fix $k$, $1 \leq k \leq \lfloor n/2 \rfloor -1$. We will prove, by induction on number $e(G)$ of edges of $G$, that if $\nu(G) \geq n$ then $m_k(G)<m_\ell(G)$ for any $\ell$ strictly between $k$ and $n-k$.
In the base case, $e(G)=n$, $G$ must consist of $n$ vertex disjoint edges, and we have $m_\ell(G)=\binom{n}{\ell}>\binom{n}{k}=m_k(G)$.
For the induction step, let $G$ be a graph on more than $n$ edges, with $\nu(G)\geq n$, and let $uv$ be an arbitrary edge in $G$, joining vertices $u$ and $v$. As in the proof of Lemma \ref{lem-2beats1}, let $G_1$ be obtained from $G$ by deleting the edge $uv$, and $G_2$ by deleting the vertices $u$ and $v$. As in the proof of Lemma \ref{lem-2beats1} we have
\begin{equation} \label{e0}
m_\ell(G)=m_\ell(G_1)+m_{\ell-1}(G_2)~~\mbox{and}~~m_k(G)=m_k(G_1)+m_{k-1}(G_2).
\end{equation}
Now by the induction hypothesis on $e(G)$, we have
\begin{equation} \label{e1}
m_\ell(G_1) > m_k(G_1).
\end{equation}
But also,
\begin{equation} \label{e2}
m_{\ell-1}(G_2) > m_{k-1}(G_2).
\end{equation}
This follows from the $n-2$ case of the of the main induction. Indeed, $\nu(G_2)\geq n-2$ (removing $u, v$ can delete at most two of the edges from any matching of size $n$). Set $n'=n-2$, $k'=k-1$ and $\ell'=\ell-1$. We have $1 \leq k \leq \lfloor n/2\rfloor -1$ and $k < \ell < n-k$, so
$0 \leq k-1 \leq \lfloor n/2\rfloor -2$ and $k-1 < \ell-1 < n-k-1$, or
$0 \leq k' \leq \lfloor n'/2\rfloor -1$ and $k' < \ell' < n'-k'$,
and so the appeal to the earlier case of the main induction is valid.
Combining (\ref{e1}) and (\ref{e2}) with (\ref{e0}) yields $m_\ell(G) > m_k(G)$, as required.
\end{proof}
\medskip
An immediate consequence of Theorem \ref{thm-main-res} is that for any graph $G$ with $\nu(G)\geq n$ we have $m_{\lfloor n/2\rfloor -1}(G) < m_{\lfloor n/2\rfloor}(G)$, which says that the mode of the matching sequence must occur at $\lfloor n/2\rfloor$ or later. This means that $M_n$, the number of permutations of $[n]$ that can arise as the permutation associated with a graph with largest matching having size $n$, satisfies
$M_n \leq \sum_{k=\lfloor n/2\rfloor -1}^{n-1} \binom{n-1}{k}$.
This is asymptotically $2^{n-2}$ as $n$ goes to infinity; a factor of $2$ smaller than the upper bound observed in \cite{AMSE}.
A finer analysis of Theorem \ref{thm-main-res} yields the substantially smaller bound (\ref{eq-up-on-count}) on $M_n$. Let $(m_1, \ldots, m_n)$ be a matching sequence, with mode $m_t$ (perhaps obtained after breaking a tie). Any associated permutation (in one-line notation) puts $\{1, \ldots, t-1\}$ in increasing order and $\{t+1, \ldots, n\}$ in decreasing order in the first $n-1$ spots, and puts $t$ at the end.
This permutation can be encoded by an U-D sequence of length $n-1$ --- each time one sees a U, one enters the first as-yet-unused number from $\{1,\ldots, t-1\}$ (remembering that these numbers should be used in increasing order); each time one sees a D, one enters the first as-yet-unused number from $\{t+1,\ldots, n\}$ (remembering that these numbers should be used in decreasing order). For example,
$$
UUDDDUUDUDDUU
$$
would correspond to $n=14, t=8$, and would yield the permutation
$$
1~2~14~13~12~3~4~11~5~10~9~6~7~8.
$$
Notice that this is a bijective encoding --- a unique permutation can be read from a sequence. Notice also that in the U-D sequence one is never allowed to have an initial substring that has three more D's than U's, because the first time we see such an initial string, say after $j$ U's and $(j+3)$ D's, we would have seen $1$ through $j$, but not $j+1$, and we would have seen $n$ through $n-(j+2)$, in particular including $n-(j+2)$, so we would have
$m_{j+1}>m_{n-(j+2)}$, violating Theorem \ref{thm-main-res}. It follows that $M_n$ is bounded above by the number of U-D sequences of length $n-1$ having no initial substring with three more D's than U's.
This sequence begins $(1, 2, 4, 7, 14, 25, 50, \ldots)$ \cite[A001405]{Sloane}, and satisfies the formula
$$
a_n = \left\{
\begin{array}{cc}
\frac{3n/2+1}{2n+2}\binom{n+1}{n/2+1} & \mbox{for even $n$}, \\
2a_{n-1} & \mbox{for odd $n$},
\end{array}
\right.
$$
so that $a_n \sim c2^n/\sqrt{n}$ (with the constant $c$ depending on the parity of $n$). This verifies (\ref{eq-up-on-count}).
An alternate approach is to say that $M_n$ is bounded above by the number of U-D sequences of length $n+1$ that start with UU and have no initial substring with more D's than U's. This in turn is upper bounded by the number of U-D sequences of length $n+1$ having no initial substring with more D's than U's (with no restriction on how the strings start). These sequences are also known as {\em left factors} of Dyck words, and it is well-known (see, for example, \cite[A001405]{Sloane}) that there are $\binom{n+1}{\lfloor (n+1)/2 \rfloor}$ such. This is asymptotically $c2^n/\sqrt{n}$ (the constant $c$ again depending on the parity of $n$).
\section{Questions and problems} \label{sec-questions}
A number of interesting problems remain concerning the behavior of the independent set sequence of a graph. We begin with the natural refinement of our determination of $f(m)$.
\begin{prob}
For each permutation $\pi$, determine $g(\pi)$, the minimum order over all graphs $G$ for which $\pi$ is an independent set permutation of $G$.
\end{prob}
\medskip
We have shown that at most $m^m$ vertices is enough to induce the constant weak order on $[m]$ from an independent set sequence, but this is definitely not enough to realize all weak orders; for example, the weak order $m-1 < m < m-2 < m-3 < \cdots < 2 < 1$ requires at least $m^m + m - 1$ vertices. In the other direction, we have shown that fewer than $m^{m+2}$ vertices are sufficient to induce any weak order on $m$.
\begin{prob}
Determine the smallest order large enough to realize every weak order on $[m]$ as the weak order induced by the independent set sequence of some graph.
\end{prob}
\begin{prob} \label{prob-AMSE}
Do the same for weak orders consisting of singleton blocks; equivalently, answer Problem \ref{AMSE-ind-Q} with the additional constraint that the permutations associated with independent set sequences are required to be unique.
\end{prob}
In \cite{AMSE} the comment is made that Problem \ref{AMSE-ind-Q} ``is likely to remain exceeding difficult''. Given the surrounding discussion in \cite{AMSE}, it seems that the authors were implicitly thinking about Problem \ref{prob-AMSE} when they made this comment.
\medskip
A fascinating question is raised in \cite{AMSE}, that has attracted some attention, but has remained mostly open. Although the independent set sequence of a graph is unconstrained, if we restrict to special classes of graphs, then it can become constrained. For example the independent set sequence of a claw-free graph is unimodal \cite{Hamidoune}, and so at most only the $2^{m-1}$ unimodal permutations of $[m]$ can arise as the independent set permutation of a claw-free graph with largest independent set size $m$. Alavi et al. observed that the independent set sequences of stars and paths are both unimodal, and asked:
\begin{question}
\cite[Problem 3]{AMSE} Is the independent set sequence of every tree unimodal?
\end{question}
It is for all trees on 24 or fewer vertices \cite{Radcliffe}. See, for example, \cite{GalvinHilyard} for recent work and other references.
\medskip
It had been conjectured by Levit and Mandrescu \cite{LM} that every bipartite graph has unimodal independent sequence, and they obtained a partial result: if $G$ is a bipartite graph with $\alpha(G)=m \geq 1$, then the final third of the independent set sequence is weakly decreasing, i.e.,
$$
i_{\lceil (2m-1)/3 \rceil}(G) \geq \cdots \geq i_{m-1}(G) \geq i_m(G).
$$
The unimodality conjecture was, however, disproved by Bhattacharyya and Kahn \cite{BhattacharyyaKahn}.
\begin{prob} \label{bip-prob}
Characterize the permutations that can occur as the independent set permutations of a bipartite graph.
\end{prob}
There is an interesting parallel to the case of well covered graphs. A graph is {\em well covered} if all its maximal independent sets have the same size. It had been conjectured by Brown, Dilcher, and Nowakowski \cite{BDN} that every well covered graph has unimodal independent sequence, but this was disproved by Michael and Traves \cite{MT}, who also showed that the first half of the independent set sequence of a well covered graph is increasing, i.e.,
$$
i_1(G) < i_2(G) < \cdots < i_{\lceil m/2 \rceil}(G).
$$
They formulated the {\em roller-coaster} conjecture, that for any $m \geq 1$ and any permutation $\pi$ of $[\lceil m/2 \rceil, m]$ there is a well covered graph $G$ with $\alpha(G)=m$ and with
$$
i_{\pi([\lceil m/2 \rceil)}(G) < i_{\pi([\lceil m/2 \rceil)+1}(G) < \cdots < i_{\pi(m)}(G).
$$
This was subsequently proved by Cutler and Pebody \cite{CP}. The analog of the roller-coaster conjecture does not hold for Problem \ref{bip-prob}; for example, it is easy to see that for $n \geq 7$, any bipartite graph $G$ on $n$ vertices has $i_2(G) > i_1(G)$.
\medskip
Turning to matching permutations, the incidental observation made after the proof of Lemma \ref{lem-2beats1} raises the following (perhaps easy) question.
\begin{question}
Which unimodal permutations of $[n]$ can arises via (\ref{perm-assoc}) from the coefficient sequence of a polynomial of the form $(1+r_1x)(1+r_2x)\cdots (1+r_nx)$ with $r_i$ real and non-negative?
\end{question}
\medskip
Finally, the greater part of Problem \ref{AMSE-matching-q} remains open.
\begin{prob}
Characterize the permutations that can occur as the matching permutation of a graph,
and determine the growth rate of $M_n$, the number of permutations of $[n]$ that are matching permutations of some graph.
\end{prob}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,859 |
La chapelle Sainte-Marie-du-Ménez-Hom est une chapelle située à Sainte-Marie-du-Ménez-Hom, commune de Plomodiern en Bretagne, France.
La chapelle et l'enclos
La chapelle de Sainte-Marie-du-Ménez-Hom est une ancienne chapelle des Templiers ; sa sacristie porte l'étrange nom de « Chambre des moines rouges », allusion à la couleur de la croix des Templiers. Elle servait de lieu de réunion et de recueillement aux Templiers qui séjournaient dans la région. La légende dit qu'entre la chapelle, les calvaires et la fontaine se trouverait un trésor<ref>Le calvaire de Croas-Rhu porte la marque des Templiers, journal Le Télégramme de Brest et de l'Ouest, numéro du 19 mai 2004</ref>.
En , lors d'une violente tempête, le clocher de la chapelle fut frappé par la foudre et la toiture s'effondra partiellement
La chapelle de Sainte-Marie-du-Ménez-Hom est classée au titre des monuments historiques depuis le . En breton on l'appelle Ti ar Werc'hez (la Maison de la Vierge) comme le langage populaire dénommait les chapelles ayant la Vierge Marie pour patronne. La chapelle et son placître sont situés au pied des deux sommets du Ménez-Hom à une altitude de , au bord de la D 887, ancienne route nationale, important axe est-ouest qui fut longtemps l'axe principal d'accès à la presqu'île de Crozon.
L'existence jusqu'au début du de foires importantes à cet endroit explique les importantes collectes d'argent qui ont permis d'édifier la chapelle construite entre 1570 (une pierre du pignon ouest porte cette date) et 1773. Cette chapelle qui fut précédée d'une chapelle romane totalement disparue. Les dons affluaient, ce qui explique la beauté de l'édifice et la richesse du mobilier. En 1780 par exemple, au rôle des décimes, Sainte-Marie-du-Ménez-Hom figurait pour 26 livres, bien avant d'autres chapelles de la région pourtant fort bien pourvues.
l'enclos paroissial : on y pénètre par un arc de triomphe daté de 1739 qui porte une statue de saint Hervé aveugle représenté avec son petit compagnon-guide Guic'haran (saint Hervé est localement assimilé à saint Mahouarn, patron de la paroisse de Plomodiern). Une autre entrée de l'enclos montre des dalles de schiste ardoisier creusées de cupules qui sont probablement un réemploi d'un temple préchrétien, servant sans doute à recueillir de l'eau considérée alors comme sacrée.
le calvaire date de 1544 et est à trois fûts. À son sommet est représenté le Christ crucifié, l'étage en dessous porte deux cavaliers puis le second croisillon porte une pietà et deux statues géminées, l'une de saint Pierre et saint Jean, l'autre de sainte Marie-Madeleine et de saint Yves. Une autre statue au pied du calvaire représente Marie-Madeleine agenouillée et regardant le Christ crucifié. De part et d'autre du Christ sont aussi représentés le bon larron et le mauvais larron.
le clocher-porche avec ses trois étages de balustrades superposées, construites à des dates différentes échelonnées de 1668 à 1773, est surmonté d'un dôme à lanternon imité de ceux de Rome.
la chapelle : la façade est de la chapelle donnant sur l'enclos illustre bien les étapes successives de sa construction : un granit de Logonna pour la partie sud, la plus proche du clocher, un granit plus sombre d'une autre provenance pour la partie médiane (elle porte une inscription la datant : « Missire M. Cravec Recteur de Plomodiern, Guil Le Doaré Prêtre Vicaire C. Roignant. F. 1766 »), un simple appareillage de grès armoricain avec du granit uniquement en entourage des portes et vitres pour la dernière extension construite entre 1570 et 1591 et dénommée « la chambre des moines ».
L'intérieur de la chapelle : les trois retables baroques du maître-autel et des deux autels latéraux sud et nord, avec leurs boiseries, le tout datant du début du (1703 et 1710), ont été classés monuments historiques dès le , avant la chapelle elle-même et viennent d'être restaurés (achèvement des travaux en 2010). Le retable central a été fait par Jean Le Séven, maître menuisier et Jean Cévaër, maître sculpteur. Des colonnes torses, une abondante statuaire, de nombreux bas-reliefs encadrent la maîtresse-vitre. La partie centrale illustre la Sainte Famille et porte des statues de la Vierge Marie, de saint Joseph, de sainte Anne et de saint Joachim. Le retable sud est consacré aux saints fondateurs de l'Église : saint Pierre, saint Jacques, saint André, saint Paul. Le retable nord illustre saint Jean-Baptiste, saint Louis, saint Laurent et sainte Marie-Madeleine.
Le Christ en croix, en bois, datant lui aussi du début du , a été classé monument historique le . Une statue en pierre de Kersanton datée du se trouve dans la nef et représente saint Laurent avec le gril de son martyre. Des sablières remarquables sont encore en place dont celle évoquant la légende du riche laboureur écrasé par son attelage parce qu'il s'était moqué de la Sainte Famille juchée sur un âne lors de la fuite en Égypte.
En dehors de l'enclos, à quelques centaines de mètres de la chapelle sur la route de Plomodiern, la croix de Park ar Groaz Ru ("Champ de la Croix Rouge") évoque les Templiers, moines-soldats du Moyen Âge surnommés « moines rouges » en Bretagne.
Classement
La chapelle, ainsi que l'arc de triomphe et le calvaire font l'objet d'un classement au titre des monuments historiques depuis le .
Annexes
Notes et références
Bibliographie
A. H. Dizerbo, La chapelle de Sainte-Marie du Ménez-Hom '', Bulletin de la Société archéologique du Finistère, tome CXVI, 1987 (première partie) et tome CXVIII, 1989 ().
Monument historique dans le Finistère
Monument historique classé en 1916
Chapelle monument historique en Bretagne
Sainte-Marie-du-Ménez-Hom
Sainte-Marie-du-Ménez-Hom | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,828 |
Klučov kan verwijzen naar de volgende Tsjechische gemeenten:
Klučov (okres Třebíč)
Klučov (okres Kolín) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,049 |
\section{Introduction
\label{sec:introduction:background}}
Mechanical heating is required to maintain the energy balance in the solar chromosphere, as suggested by the temperature difference between the radiative equilibrium atmospheric model \citep{1989ApJ...346.1010A} and the observation-based semi-empirical model \citep{1981ApJS...45..635V}. Waves have been recognized as important contributors to chromospheric heating, although their heating mechanisms are elusive \citep[see][for a review]{2015SSRv..190..103J}. \rvr{While} the propagation of waves in the chromosphere has been well studied from both the observational and theoretical perspectives \rvr{\cite[e.g.,][]{2003ApJ...599..626B,2004A&A...422.1085H,2005ApJ...631.1270H,2008ApJ...680.1542H,2009A&A...508..951V,2011ApJ...743..142H,2012ApJ...755...18V,2013ApJ...764L..11D,2016A&A...585A.110K,2016A&A...590L...3S,2018MNRAS.479.5512K,2020ApJ...890...22A}}, a firm quantitative conclusion is still a distance away.
\rv{In the chromosphere, physical parameters change drastically, leading to difficulties in studying chromospheric dynamics.} The plasma beta varies in both the vertical and horizontal directions, and waves can change their modes when crossing the equipartition layer \citep{2006RSPTA.364..333C,2019ApJ...881L..21P} where the speed of sound is identical to the Alfv\'en speed. Density stratification also adds to the complexity by increasing the amplitude of the acoustic waves, leading to increased nonlinearity and formation of shocks.
In the high-beta regions of the chromosphere where the role of the magnetic field could be ignored, the propagation of acoustic waves has been well studied by hydrodynamic simulations with non-local thermodynamic equilibrium (non-LTE) radiative transfer \citep{1995ApJ...440L..29C,1997ApJ...481..500C}. In these studies, waves are generated by longitudinal piston motion. They succeed in reproducing the Ca II spectral profile which agrees with the observations.
The situation becomes even more complicated in the low-beta chromosphere with the participation of the magnetic field. \cite{2016ApJ...817...94A} and \cite{2016ApJ...829...80B} show that the shock heating rate in the chromosphere is larger than or consistent with the observation-based radiative cooling rate. A similar result is also obtained in \cite{2020ApJ...891..110W} with an improved treatment of the radiative loss term introduced by \cite{2012A&A...539A..39C}. In \cite{2016ApJ...817...94A}, \cite{2016ApJ...829...80B}, and \cite{2020ApJ...891..110W}, waves are generated by artificial transverse torque or transverse motion at the bottom of the flux tube. These studies do not include the effect of waves originating from outside the flux tube.
\rv{Theoretical studies could be divided into two categories, idealized models and realistic models. \cite{2016ApJ...817...94A}, \cite{2016ApJ...829...80B}, and \cite{2020ApJ...891..110W} are examples of idealized models. The physical process is clear in idealized models, but the results are affected by artificial settings in the model.} On the other hand, there are also realistic models \rvr{\cite[e.g.,][]{2011ApJ...730L..24K,2016A&A...585A...4C,2017ApJ...848...38I,2017Sci...356.1269M}} that aim to include complicated physical processes to approach reality. Realistic models are used to reproduce the synthesized images or spectral profiles for comparison with observations \cite[e.g.,][]{2013ApJ...772...90L,2009ApJ...694L.128L,2019MNRAS.486.4203Q}, but their complexity makes it difficult to understand the underlying elemental physical processes involved in heating. These studies do not focus on the physical processes that occur during wave propagation such as thermalization, nonlinear steepening, or mode conversions in the chromosphere.
\rv{The purpose of our study is to conduct a quantitative investigation on wave heating in the chromosphere. \rv{In particular, previous studies do not focus on the role of the fast magnetic wave in heating the low-beta chromosphere.} We perform a realistic two-dimensional radiative MHD simulation while conducting a detailed investigation on the propagation of waves to estimate the contribution to chromospheric heating by different modes of waves. To achieve this goal, we develop a novel method of automatically identifying the mode of waves and calculating the heating rate due to different modes of waves.}
\section{Numerical model}\label{sec:2}
We use RAMENS code \citep{2016PhDT.........5I,2015ApJ...812L..30I} which solves MHD equations with gravity, heat conduction, equation of state under local thermodynamic equilibrium (LTE) condition, radiative transfer in the photosphere, and approximated radiative loss term in the chromosphere and the corona. The basic equations of the simulation are the same as those in \cite{2017ApJ...848...38I}. One could refer to \cite{2016PhDT.........5I} for a detailed description of this code. We modified the original RAMENS code by replacing the treatment of the chromospheric radiative loss term with the improved recipe developed by \cite{2012A&A...539A..39C}.
The simulation domain is a 16 Mm $\times$ 16 Mm two-dimensional square extending from $-$ 2 Mm below the photosphere to 14 Mm in the corona with a uniform grid spacing of 8.5 km. The temperature of the corona is 1 MK which is maintained by the top boundary condition. The initial magnetic field is vertical and has a strength of 6 G. We start with a plane parallel atmosphere in the hydrostatic equilibrium state, though this setup does not strongly influence the later results obtained after the well-developed magneto-convections. The data analyzed cover 1000 s of the simulation which is approximately 10 times the transit time for acoustic waves in the chromosphere.
\section{Shock identification and heating rate calculation}\label{sec:21}
Our study focuses on wave heating in the low-beta chromosphere. Comparing this mechanism with other possible heating mechanisms (e.g., reconnection and turbulence with ambipolar diffusion), there is observational evidence showing that waves can carry enough energy for chromospheric heating \citep{2010ApJ...723L.134B}. Waves are generated by photospheric convection and steepen to shocks as they propagate upward in the chromosphere. To estimate the shock-heating rate, we identify the shock front in the chromosphere, determine the mode of each shock, and calculate the corresponding heating rate.
The positions of the shock fronts are identified by the local minimum of $\nabla \cdot \mathbf{V}$ with
\begin{equation}\label{eq:sel22}
-\nabla \cdot \mathbf{V} \ge c_{\text{th}} (\cs/\Delta x),
\end{equation}
where $c_{\text{th}}$ is a parameter indicating the threshold for identification, $\cs$ is the speed of sound, and $\Delta x$ is the grid size. The value of the parameter $c_{\text{th}}$ should depend on the shock-capturing quality of the numerical scheme and was taken to be $c_{\text{th}}=0.25$ in this study \citep[see Appendix in][]{2020ApJ...891..110W}.
\rv{The heating rate at the shock front is calculated using the following steps}. First, we extract the density, temperature, velocity, gas pressure, and magnetic pressure along the direction of propagation which is assumed to be identical to the direction of the gradient of the total pressure. The upstream and downstream quantities of the detected shock are determined as the first local maximum and minimum of $\partial^2 V_l/\partial l^2$ beside $l = l_{\text{c}}$ where $V_l$ is the velocity along the direction of propagation, $l$ is the distance along the direction of propagation, and $l_{\text{c}}$ is the position of the shock front. The upstream side is determined by the side with the lower density. We estimate the increment of the thermal energy flux at the shock front:
\begin{equation}\label{eq:ql}
\Delta F_{\text{th}} = U_1 \rho_1 T_1 (\smone-\smtwo),
\end{equation}
where $\Delta F_{\text{th}}$ is the increase in the thermal energy flux. Subscripts 1 and 2 denote the physical parameters that are sampled at the upstream and downstream region, respectively. $U$ is the shock-normal velocity in the shock rest frame, $T$ is the temperature, and $\sm$ is the entropy per unit mass. $U_1$ is calculated by mass conservation, $U_1 \rho_1 = U_2 \rho_2$, and the velocity relationship in different frames of reference, $v_1 - v_2 = U_1 - U_2$ where $\rho$ is the density and $v$ is the shock-normal velocity in the laboratory frame \rvr{(see Figure \ref{fig:sc} for a schematic plot)}. To estimate the heating rate per unit volume, we assume that the heating is evenly distributed in the volume of one grid point at the shock front. As a result, the heating rate per unit volume is calculated by
\begin{equation}\label{eq:cal}
Q_{\text{heat}}=\Delta F_{\text{th}}/w_{\text{shock}},
\end{equation}
where $w_{\text{shock}}$ is the width of the shock wave. Although the actual thickness in the real shocks should be given by the microscopic dissipation process, we here use the grid spacing $w_{\text{shock}} = \Delta x$ for convenience. \rvr{The heating rate is calculated each time step. We assume that the heating rate at a fixed position do not change within one time step}. Note that the spatially integrated amount of $Q_{\text{heat}}$ is independent of the choice of $w_{\text{shock}}$ and is used only for the later discussion.
\begin{figure}
\centering
\includegraphics[width=7cm]{s1.png}
\caption{\rvr{A schematic figure showing the calculation of thermal energy flux. $t$ is time. $A$ is the area on the shock front. $m=U_1 \rho_1\Delta t\Delta A=U_2 \rho_2\Delta t\Delta A$ is the mass of plasma that crosses the shock front. Color in the upstream and the downstream regions denotes the value of entropy per unit mass (red: higher value, blue: lower value). $\Delta Q_{\text{m}}=T\Delta\sm$ is the increment of thermal energy per unit mass. Thus, we can obtain the thermal flux by $\Delta F_{\text{th}}=m\Delta Q_{\text{m}}/(\Delta t\Delta A)$}.}
\label{fig:sc}
\end{figure}
\rv{Finally, we determine the mode of each shock wave \rvr{by checking whether the gas pressure and the magnetic pressure across the shock front change in the same direction}. The sign of $\int (\partial P_g / \partial l)(\partial P_m/ \partial l) \text{d} l$ across the shock front is used to determine whether it is a fast shock (positive value) or slow shock (negative value), where $P_g$ is the gas pressure and $P_m$ is the magnetic pressure. \rvr{We do not use phase speed to determine the mode of waves since it is difficult to obtain the local fast speed and slow speed in the dynamic chromosphere.}}
\section{Results} \label{sec:3}
Figure \ref{fig:fig1} shows the identified shock fronts in the dynamic simulation of solar chromosphere. Waves are generated by photospheric convection and they steepen to shocks in the chromosphere. Shocks dissipate their energy continuously in the chromosphere. A number of shocks gradually become undetectable during their propagation due to dissipation. When shocks impinge on the transition region, they drive the upward motion of the transition region that forms spicules.
\rv{We focus on the low-beta chromospheric plasma. Due to the large deformation of the transition region by the spicules, we cannot distinguish the chromosphere and the corona using a simple threshold on the geometrical height. The \rvr{low-beta} chromospheric plasma is defined by the following criteria}: (1) $\cmass> 10^{-5.5}$ $\cmassu$, (2) $T< 10^4$ K, and (3) Alfv\'en speed is larger than sound speed. The variable $\cmass$ is the column mass, and $\cmass(z)=\int_z^{z_{\text{top}}} \rho(z') \text{d} z'$, where $z_{\text{top}}$ is the height of the top of the simulation box. The temperature and column mass threshold are used to exclude coronal plasma. The values of the thresholds are chosen from the joint probability density distribution of the temperature and the column mass (Figure \ref{fig:fig2-1}).
\rv{The time and horizontal averaged radiative loss rate and heating rate of the low-beta chromospheric plasma are shown in Figure \ref{fig:fig2}}. It is shown that the shock heating is well balanced with radiative cooling below 2.5 Mm. At locations higher than 2.5 Mm, the energy balance is gradually disrupted due to the formation of spicules (in the presence of spicules, the energy balance at a fixed position is determined by the entropy flow carried by them).
\begin{figure*}
\centering
\includegraphics[width=15cm]{fig1c.png}
\caption{Snapshot of the simulation result with shock identification. The green line marks the position of the transition region (characterized by $T = 10^4$ K). The black solid lines are magnetic field lines. The gray shadow indicates the region where speed of sound is larger than the Alfv\'en speed. Identified shocks are plotted in blue (fast shock) and red (slow shock). Only a part of the simulation domain is shown in this figure.
(An animation of this figure is available.)}
\label{fig:fig1}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=5cm]{fig2-1.png}
\caption{Joint probability density distribution of the temperature and the column mass. The yellow line shows the average temperature at each column mass. The brown lines show the threshold values for chromospheric plasma at $\cmass=10^{-5.5}$ $\cmassu$ and $T = 10^4$ K.}
\label{fig:fig2-1}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=8cm]{fig2bbb.png}
\caption{Heating and radiative loss rate of the low-beta chromospheric plasma as a function of height. The black dashed line is the radiative loss rate in the simulation. The brown line is the sum of the heating rates due to fast and slow shocks. The blue solid line is the fast wave heating rate. The red solid line is the slow wave heating rate. For the blue and red lines, the thin lines with perturbation are the results that are directly calculated from the simulation; we also smooth the results with a Savitsky--Golay filter and plot them in thick lines. The green line represents the heating rate due to heat conduction. The average column mass at each height is shown in the secondary axis. \rvr{Only heating and cooling in the low-beta regions are included in this figure.}}
\label{fig:fig2}
\end{figure}
\rv{Where do these fast mode waves in the low-beta regions originate? }We find that low-beta fast magnetic waves originate from high-beta fast acoustic waves through mode conversion. An example of mode conversion is shown in Figure \ref{fig:fig3}. Mode conversion occurs when fast acoustic waves propagate from the high-beta region to the low-beta region and cross the equipartition layer. An attacking angle (the angle between the wavevector and the magnetic field) close to $90^{\circ}$ will result in a larger conversion rate \citep{2006RSPTA.364..333C,2019ApJ...881L..21P}.
\begin{figure*}
\centering
\includegraphics[width=13cm]{fig6.png}
\caption{Example of a fast acoustic wave to fast magnetic wave mode conversion. The upper panels show the time evolution (from left to right: $t_0 - 12$ s, $t_0$, and $t_0 + 6$ s, where $t_0$ is the time of the snapshot shown in Figure \ref{fig:fig1}). In the upper panels, gray lines represent magnetic field lines. The blue line shows the position of a fast shock. Shadows mark the region where the speed of sound is larger than the Alfv\'en speed. Orange lines mark the position of slices used in the lower panels. The lower panels show the distribution of the gas pressure (solid line) and magnetic pressure (dashed line) across the shock front. In each panel, the horizontal axis is the distance along the slice, in which zero corresponds to the location of the shock front.}
\label{fig:fig3}
\end{figure*}
\label{fig:fig3}
\section{Discussion
\label{sec:4}}
The propagation of waves in MHD simulation with an idealized setting is also \rvr{carried on in previous researches \citep{2005ApJ...631.1270H,2008ApJ...680.1542H,2009A&A...508..951V,2012ApJ...755...18V}. These studies mainly focus on waves that originate inside a flux tube. For these waves, as they propagate upwards, they propagate along the magnetic field lines thus the attacking angle is small and mode conversion is less efficient}. \cite{2008ApJ...680.1542H} do mention the waves that originate outside a flux tube could generate fast magnetic waves in the flux tube through mode conversion but they do not discuss the heating by the fast magnetic waves in detail. Our result shows that, with quantification of the heating rate, fast waves do play a role in heating the low-beta chromosphere.
\rvr{\cite{2006ApJ...653..739K} show that refraction could affect the propagation of fast waves and prevent their efficient energy transport to the chromosphere. They focus on waves inside a strong flux tube (sunspots). On the other hand, in our simulation, fast waves in the regions between two flux tubes are less affected by refraction since there is no substantial horizontal gradient of fast speed in these regions. In addition, the intensity of magnetic field in the flux tube is weaker in our simulation which also reduces the horizontal gradient of fast speed.}
Our simulation shows that shock heating is the dominant heating process in the chromosphere. This result is consistent with those from previous studies. However, the wave modes contributing to heating are different. In \cite{2016ApJ...817...94A}, \cite{2020ApJ...891..110W} and \rvr{\cite{2004A&A...422.1085H}}, transverse waves at the foot of a low-beta flux tube undergo nonlinear mode coupling and generate slow acoustic waves. They steepen to shocks which dissipate and contribute to chromospheric heating. In our simulation, Alfv\`en waves vanish because of the two-dimensional geometry. As a result, the nonlinear mode coupling is also absent.
As fast waves propagate like an expanding sphere, the strongest perturbation of the vertical velocity appears at the top of the sphere, whereas compression of the vertical magnetic field appears at the lateral sides. In our simulation, the background magnetic field is 6 G, mimicking the quiet sun region. The resultant intensity of the magnetic field perturbation in the chromosphere could be as large as 10--20 G. The combination of vertical velocity and vertical magnetic field perturbation can be used as a signal of the fast wave. Such a signal can hopefully be detected by next-generation solar telescopes such as Daniel K. Inouye Solar Telescope \citep[DKIST;][]{2020SoPh..295..172R} and Chinese Giant Solar Telescope \citep[CGST;][]{2011ASInC...2...31D}.
\rvr{In order to investigate effect of the topology of magnetic field line, we carry on another simulation with the same initial and boundary condition described in Section \ref{sec:2}. The only difference is that we increase the intensity of the initial background magnetic field from 6 G to 20 G. In this new setting, the magnetic field lines are less inclined which results in smaller attacking angle for waves that propagate upward. We find that the percentage of heating by slow wave increases, especially in the higher part of the chromosphere characterized by $\cmass<10^{-4.2}$ $\cmassu$. However, our main result that fast magnetic shock waves play a significant role in heating the low-beta chromosphere remains unchanged.}
\rvr{Our study is limited in the quiet region. In sunspots, observations show that wave energy is insufficient for chromospheric heating \citep{2011ApJ...735...65F}. In these regions, other effects related with magnetic field such as reconnection should be taken into consideration.}
In this study, the ambipolar diffusion and dynamic ionization of hydrogen are not considered. The dissipation of ambipolar diffusion could lead to substantial heating locally \rvr{\citep{2012ApJ...747...87K,2016ApJ...819L..11S,2017Sci...356.1269M,2019ApJ...871....3S}.} On the other hand, \cite{2016ApJ...817...94A} compare the time-averaged heating rate resulting from ambipolar diffusion and shock dissipation and find that shock heating is much stronger than the heating resulting from ambipolar diffusion. \cite{2007A&A...473..625L} compare simulations with the LTE assumption and dynamic ionization. It is shown that in the simulation with dynamic ionization, shock temperatures are higher and the intershock temperatures are lower than in the simulation with the LTE assumption. This effect could affect the measurement of the entropy jump. Moreover, dynamic ionization is important to determine the electron and the ion number density and will further affect the estimation of ambipolar diffusion, especially when the ionization degree is low. Further studies that compare shock heating, turbulence heating \citep[][]{2011ApJ...736....3V}, and ambipolar diffusion \rvr{\citep[][]{2005A&A...442.1091L,2018A&A...618A..87K,2020ApJ...889...95M,2020A&A...642A.220G}} in realistic simulations are expected to be conducted in the future.
\section{Conclusion
\label{chap:summary}}
We perform a two-dimensional MHD simulation to study the propagation of MHD waves in the chromosphere. We identify the mode of the shock waves in the chromosphere, calculate the heating rate from the entropy jump, and find that the heating rate balances with the radiative loss. Fast magnetic shock waves play a significant role in heating the low-beta chromosphere. These low-beta fast magnetic waves are generated by mode conversion.
\rvr{We acknowledge the referee for valuable comments.} The authors thank M. Carlsson for providing numerical tables for the recipe of the chromospheric radiative loss. \rvr{The authors thank B. Yu's assistance in making Figure \ref{fig:sc}.} Numerical computations were carried out on Cray XC50 at Center for Computational Astrophysics, National Astronomical Observatory of Japan. T.Y. is supported by JSPS KAKENHI grant No. 15H03640, No. 20KK0072, and No. 21H01124. H.I. is supported by JSPS KAKENHI grant No. 19K14756.
\clearpage
\bibliographystyle{apj}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,853 |
Visit two of the most iconic landscapes in the American Southwest! Travel in luxury and experience the unique landscapes of the region. The journey begins with a drive up the beautiful Oak Creek Canyon, past Flagstaff and then across the open plains of Navajo land. Antelope Canyon X is a narrow canyon with breathtaking shapes in its interior, created by natural forces. The only visible light shines down from a gap above, and at certain times of the day shafts of sunlight pierce the apex of the canyon and illuminate the vibrant colors of the sandstone chambers below. | {
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} | 9,813 |
Q: Android using two countdowntimers In my app, i'm using two different CountDownTimers that have same values. I have two buttons to control them but when i press the button twice, it starting from the beginning. I want to keep its last value.
Here is my code:
t1 = new CountDownTimer(white, 1000) {
@Override
public void onTick(long l) {
btnWhite.setText("seconds remaining: " + l / 1000);
white = l;
}
@Override
public void onFinish() {
}
};
t2 = new CountDownTimer(black, 1000) {
@Override
public void onTick(long l) {
btnBlack.setText("seconds remaining: " + l / 1000);
black = l;
}
@Override
public void onFinish() {
}
};
btnBlack.setOnClickListener(new View.OnClickListener() {
@Override
public void onClick(View view) {
t1.start();
t2.cancel();
}
});
btnWhite.setOnClickListener(new View.OnClickListener() {
@Override
public void onClick(View view) {
t2.start();
t1.cancel();
}
});
A: I have checked your code.
It is obvious because your timers initialised with default values. when you start again it won't take new values of white/black.
To achieve what you want you have to initialise timer with new values before starting it.
I have done some correction in your code. you can check that out.
Make Two methods
public void timerStart1(long timeLengthMilli) {
t1 = new CountDownTimer(timeLengthMilli, 1000) {
@Override
public void onTick(long l) {
isRunning1 = true;
tv1.setText("seconds remaining: " + l / 1000);
white = l;
}
@Override
public void onFinish() {
isRunning1 = false;
}
}.start();
}
public void timerStart2(long timeLengthMilli) {
t2 = new CountDownTimer(timeLengthMilli, 1000) {
@Override
public void onTick(long l) {
isRunning2 = true;
tv2.setText("seconds remaining: " + l / 1000);
black = l;
}
@Override
public void onFinish() {
isRunning2 = false;
}
}.start();
}
and set setOnClickListener like this
button1.setOnClickListener(new View.OnClickListener() {
@Override
public void onClick(View view) {
if (!isRunning1) {
isRunning2 = false;
timerStart1(white);
if (t2 != null)
t2.cancel();
}
}
});
button2.setOnClickListener(new View.OnClickListener() {
@Override
public void onClick(View view) {
if (!isRunning2) {
isRunning1 = false;
timerStart2(black);
if (t1 != null)
t1.cancel();
}
}
});
UPDATE :
Please check updated code and take these extra variables
boolean isRunning1 = false, isRunning2 = false;
Hope this will help you.
Happy Coding.
A: I have tested this and it works!
I have two TextViews and two Buttons. The black button is next to the black text view and the white button is next to the white text view.
First I declare the important constants.
//contains the elapsed time for each of the timers
long blackElapsed=0,whiteElapsed=0;
//contains the total time with which we start new timers
long totalWhite = 30000;
long totalBlack = 30000;
Next I initialise the CountDownTimers. Whatever you put in here doesn't matter. I only have this so that the timers will be initialised with some value.
The reason is that they have to be initialised in order to be able to .cancel() them later in the OnClickListeners.
black = new CountDownTimer(totalWhite, 1000){
@Override
public void onTick(long l) {
}
@Override
public void onFinish() {
}
};
white = new CountDownTimer(totalBlack, 1000){
@Override
public void onTick(long l) {
}
@Override
public void onFinish() {
}
};
Finally the OnClickListeners for the buttons. (W is white textView and B is black textView and b is black button and w is white button)
w.setOnClickListener(new View.OnClickListener() {
@Override
public void onClick(View view) {
black.cancel();
//using the elapsed time to start a new timer
totalBlack = totalBlack - blackElapsed;
//this preserves milliseconds by ticking every millisecond
white = new CountDownTimer(totalBlack, 1){
@Override
public void onTick(long l) {
B.setText(l+"");
blackElapsed=totalBlack-l; //updating the elapsed time
}
@Override
public void onFinish() {
}
}.start();
}
});
//we do a similar thing with the other player's button
b.setOnClickListener(new View.OnClickListener() {
@Override
public void onClick(View view) {
white.cancel();
totalWhite = totalWhite - whiteElapsed;
black = new CountDownTimer(totalWhite, 1){
@Override
public void onTick(long l) {
W.setText(l+"");
whiteElapsed=totalWhite-l;
}
@Override
public void onFinish() {
}
}.start();
}
});
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,684 |
Due to the extreme cold weather this winter, water customers in the City of Cheboygan are requested to run a constant stream of water – the size of a pencil in one sink – 24 hours per day until further notice. Credit for running your water will be calculated on the first bill which has an actual meter read.
If you do not run your water, you are likely going to experience frozen water lines and NO water service. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,884 |
\section{Introduction}
In order to explain the motivation and the purpose of this paper, we start with a well-known example proposed by Fourman--Scott \cite{Fourman1979}. Given a topological space $X$, let $\OX$ be the frame of open sets of $X$, and let
\begin{equation} \label{PCX-def}
\PCX=\{f\mid f\ \text{is a real-valued continuous map on an open subset}\ D(f):=U\subseteq X\}.
\end{equation}
For any $f,g\in\PCX$, the value
\begin{equation} \label{sim-PCX}
\al(f,g):=\Int\{x\in D(f)\cap D(g)\mid f(x)=g(x)\},
\end{equation}
i.e., the interior of the set
\[\{x\in D(f)\cap D(g)\mid f(x)=g(x)\}\]
in $X$, may be treated as the truth-value, computed in the frame $\OX$, of the statement that \emph{$f$ is equal to $g$}. The pair $(\PCX,\al)$ is a prototype of \emph{frame-valued sets} in the sense of Fourman--Scott \cite{Fourman1979} and Higgs \cite{Higgs1970,Higgs1984}. Explicitly, considering a frame $\Om$ as the table of truth-values, an \emph{$\Om$-set} is a set $A$ that comes equipped with a map
\[\al:A\times A\to\Om\]
such that
\begin{itemize}
\item (symmetry) $\al(x,y)=\al(y,x)$,
\item (transitivity) $\al(y,z)\wedge\al(x,y)\leq\al(x,z)$
\end{itemize}
for all $x,y,z\in A$, where $\al(x,y)$ is interpreted as the truth-value that $x$ is \emph{similar} (or \emph{equal}, or \emph{equivalent}) to $y$. It is well known that the category of $\Om$-sets is equivalent to the topos ${\sf Sh}(\Om)$ of sheaves over $\Om$ \cite{Fourman1979}.
As a dualization of \eqref{sim-PCX}, it is natural to consider the value
\begin{equation} \label{dissim-PCX}
\be(f,g):=\Int(X-\Int\{x\in D(f)\cap D(g)\mid f(x)=g(x)\})
\end{equation}
for any $f,g\in \PCX$, i.e., the interior of the complement of the interior of the set
\[\{x\in D(f)\cap D(g)\mid f(x)=g(x)\}\]
in $X$, as the truth-value of the statement that $f$ is \emph{dissimilar} (or \emph{unequal}, or \emph{inequivalent}) to $g$ (also computed in the frame $\OX$). In other words, $\be$ may be thought of as an $\OX$-valued \emph{dissimilarity} on the set $\PCX$.
In classical logic, with the law of double negation in our arsenal, a dissimilarity (or inequivalence) relation on a set may be postulated as the complement (or negation) of a similarity (or equivalence) relation; that is, similarity and dissimilarity are interdefinable in classical logic. However, in a non-classical logic, e.g., intuitionistic logic and many-valued logic, the law of double negation may fail, and thus similarity and dissimilarity may not be deduced from each other via negation. In the 1970s, Scott \cite{Scott1979} pointed out that an independent \emph{positive} theory of inequalities is required in intuitionistic logic. To achieve this, he established the theory of \emph{apartness relations}.
During the past decades, different approaches have been adopted in the search for a reasonable definition of dissimilarities in the many-valued setting; we refer to \cite{Cross2002,Dubois1980} for an overview. Some typical approaches are listed below, which are all defined through some variations of similarity:
\begin{itemize}
\item A dissimilarity relation is assumed to be the ``fuzzy complement'' (also called ``inverse'') of a similarity relation; see, e.g., \cite{Nakamura1982,Zwick1987}.
\item A dissimilarity between fuzzy sets is postulated as a similarity between their ``fuzzy complements'' \cite{Dubois1982c}.
\item A dissimilarity is defined as an analogue of the distance in a metric space \cite{Restle1959,Ruspini1973}.
\end{itemize}
The aim of this paper is to establish a \emph{positive} theory of dissimilarity valued in an \emph{involutive quantale} \cite{Mulvey1992}
\[\sQ=(\sQ,\with,k,{}^{\circ});\]
that is, our notion of dissimilarity will not be postulated as a ``negation'' or a ``complement'' of that of similarity. It should be noted that our notion of \emph{$\sQ$-valued dissimilarity}, in spite of being motivated by the theory of Scott \cite{Scott1979}, is conceptually different from his notion of apartness relation. In particular, the map $\be$ given by \eqref{dissim-PCX} is an $\OX$-valued dissimilarity on $\PCX$, but in general it is not an $\OX$-valued apartness relation on $\PCX$.
The paper is structured as follows. Section \ref{Quantales} reviews some basic notions about involutive quantales. Section \ref{Sim-Dissim-Def} presents the key notion of this paper, i.e., that of \emph{$\sQ$-valued dissimilarity}. $\sQ$-valued dissimilarity is a ``dualization'' of \emph{$\sQ$-valued similarity}, which originates from a series of works of H{\"o}hle and his collaborators \cite{Hoehle1992,Hoehle1995b,Hoehle1998,Hoehle2005,Hoehle1991}. Semantic meanings of the axioms of these two notions are analyzed in this section.
It is known from \cite{Hoehle2011a} that a set equipped with a $\sQ$-valued similarity is exactly a \emph{symmetric} category enriched in the quantaloid $\HsQ$, which is a subquantaloid of the quantaloid $\DsQ$ of \emph{diagonals} of $\sQ$ \cite{Hoehle2011a,Pu2012,Stubbe2014}. Section \ref{Sim-Dissim-Cat} shows that a dual conclusion holds for $\sQ$-valued similarities. Explicitly, it is demonstrated that a set equipped with a $\sQ$-valued dissimilarity is precisely a \emph{symmetric} category enriched in $\KsQ$, where $\KsQ$ is a subquantaloid of the quantaloid $\BsQ$ of \emph{back diagonals} of $\sQ$ \cite{Shen2016a}. Therefore, $\sQ$-valued similarity and $\sQ$-valued dissimilarity are both instances of the thesis of Lawvere \cite{Lawvere1973} that \emph{fundamental structures are themselves categories}.
Based on the categorical perspective, in Section \ref{Sim-vs-Dissim} we investigate the connections between $\sQ$-valued similarities and $\sQ$-valued dissimilarities by constructing \emph{lax functors} between the quantaloids $\HsQ$ and $\KsQ$, which are deeply affected by the structure of the quantale $\sQ$:
\begin{itemize}
\item If $\sQ$ is a divisible quantale with the bottom element being cyclic, then the negations of $\sQ$-valued dissimilarities are $\sQ$-valued similarities (Proposition \ref{divisible-neg-dissim}).
\item If $\sQ$ is a frame, then the negations of $\sQ$-valued dissimilarities are $\sQ$-valued similarities, and vice versa (Proposition \ref{frame-neg-sim-dissim}).
\end{itemize}
Furthermore, we confirm the intuition that similarity and dissimilarity are interdefinable when $\sQ$ satisfies the law of double negation. Explicitly, if $\sQ$ is a Girard quantale \cite{Rosenthal1990,Yetter1990}, then we have isomorphisms
\[\DsQ\cong\BsQ\quad\text{and}\quad\HsQ\cong\KsQ\]
of quantaloids (Theorem \ref{Girard-iso}); and moreover, if $\sQ$ is an involutive Girard quantale with a hermitian and cyclic dualizing element, then $\sQ$-valued similarity and $\sQ$-valued dissimilarity are fully decidable by each other (Theorem \ref{Girard-sim-dissim}). Conversely, for a commutative quantale $\sQ$, it is shown in Theorem \ref{DQ-BQ-Girard} that the existence of an isomorphism $\DsQ\cong\BsQ$ of quantaloids necessarily forces $\sQ$ to be a Girard quantale; hence, if $\sQ$ is commutative and integral, then
\[\HsQ\cong\KsQ\iff\sQ\ \text{is a Girard quantale},\]
which is recorded as Corollary \ref{DQ-BQ-Girard-integral}.
\section{Quantales} \label{Quantales}
A \emph{(unital) quantale} \cite{Mulvey1986,Rosenthal1990}
\[\sQ=(\sQ,\with,k)\]
is a monoid with $k$ being the unit, such that the underlying set $\sQ$ is a complete lattice (with a top element $\top$ and a bottom element $\bot$) and the multiplication $\with$ distributes over arbitrary suprema, i.e.,
\[p\with\Big(\bv_{i\in I}q_i\Big)=\bv_{i\in I}p\with q_i\quad\text{and}\quad\Big(\bv_{i\in I}p_i\Big)\with q=\bv_{i\in I}p_i\with q\]
for all $p,q,p_i,q_i\in\sQ$ $(i\in I)$. The induced right adjoints
\[(-\with q)\dv(-\ldd q):\ \sQ\to\sQ\quad\text{and}\quad(p\with -)\dv(p\rdd -):\ \sQ\to\sQ,\]
called \emph{left} and \emph{right implications} in $\sQ$, are given by
\begin{equation} \label{imp-def}
r\ldd q=\bv\{p'\in\sQ\mid p'\with q\leq r\}\quad\text{and}\quad p\rdd r=\bv\{q'\in\sQ\mid p\with q'\leq r\},
\end{equation}
respectively, which satisfy
\[p\with q\leq r\iff p\leq r\ldd q\iff q\leq p\rdd r\]
for all $p,q,r\in\sQ$. We say that
\begin{itemize}
\item $\sQ$ is \emph{commutative}, if $p\with q=q\with p$ for all $p,q\in\sQ$, in which case we write
\[p\ra q:=q\ldd p=p\rdd q\]
for all $p,q\in\sQ$;
\item $\sQ$ is \emph{integral}, if the unit $k=\top$, the top element of the complete lattice $\sQ$;
\item $\sQ$ is \emph{divisible}, if
\begin{equation} \label{divisible-def}
(u\ldd q)\with q=u=q\with(q\rdd u)
\end{equation}
whenever $u\leq q$ in $\sQ$, in which case $\sQ$ is necessarily integral.
\item $\sQ$ is a \emph{complete MV-algebra} \cite{Chang1958}, if $\sQ$ is commutative and
\begin{equation} \label{MV-def}
(p\ra q)\ra q=p\vee q
\end{equation}
for all $p,q\in\sQ$, in which case $\sQ$ is necessarily divisible (cf. \cite[Lemma 2.5]{Galatos2005}).
\item $\sQ$ is \emph{involutive} \cite{Mulvey1992}, if there exists an \emph{involution} on $\sQ$; that is, a map $(-)^{\circ}:\sQ\to\sQ$ such that
\[k^{\circ}=k,\quad q^{\circ\circ}=q,\quad (p\with q)^{\circ} = q^{\circ}\with p^{\circ}\quad\text{and}\quad \Big(\bv_{i\in I}q_i\Big)^{\circ}=\bv_{i\in I}q_i^{\circ}\]
for all $p,q,q_i\in\sQ$ $(i\in I)$. In this case,
\begin{itemize}
\item it is easy to verify that
\begin{equation} \label{invo-imp}
(p\ldd q)^{\circ}= q^{\circ}\rdd p^{\circ}
\end{equation}
for all $p,q\in\sQ$;
\item an element $q\in\sQ$ is called \emph{hermitian} (also \emph{self-adjoint}) if $q^{\circ}=q$, and $k$, $\top$, $\bot$ are clearly hermitian.
\end{itemize}
\end{itemize}
\begin{exmp} \label{quantale-exmp}
We list here some quantales that are of concern in this paper:
\begin{enumerate}[label=(\arabic*)]
\item \label{quantale-exmp:Lawvere} Lawvere's quantale $[0,\infty]=([0,\infty],+,0)$ \cite{Lawvere1973} is commutative and divisible, where $[0,\infty]$ is the extended non-negative real line equipped with the order ``$\geq$'' (so that $0$ becomes the top element and $\infty$ the bottom element), and ``$+$'' is the usual addition extended via
\[p+\infty=\infty+p=\infty\]
to $[0,\infty]$, with $0$ being the unit and making $[0,\infty]$ a commutative and integral quantale. The implication in $[0,\infty]$ is given by
\[p\ra q=\begin{cases}
q-p & \text{if}\ p<q,\\
0 & \text{else}
\end{cases}\]
for all $p,q\in[0,\infty]$, where the subtraction ``$-$'' is extended via
\[\infty-p=\begin{cases}
\infty & \text{if}\ p<\infty,\\
0 & \text{if}\ p=\infty
\end{cases}\]
to $[0,\infty]$.
\item \label{quantale-exmp:frame} Every \emph{frame} $\Om=(\Om,\wedge,\top)$ is a commutative, divisible and idempotent quantale, and vice versa. In particular, the two-element Boolean algebra, denoted by ${\bf 2}$, is a frame. Moreover, each topological space $X$ gives rise to the frame $\OX=(\OX,\cap,X)$ of open sets of $X$.
\item \label{quantale-exmp:t-norm} Every \emph{complete BL-algebra} \cite{Hajek1998} is a commutative and divisible quantale. In particular, the unit interval $[0,1]$ equipped with a \emph{continuous t-norm} \cite{Klement2000} is a commutative and divisible quantale.
\item \label{quantale-exmp:nil-min} The unit interval $[0,1]$ equipped with the \emph{nilpotent minimum t-norm} \cite{Klement2000} is a commutative, integral and non-divisible quantale.
\item \label{quantale-exmp:C3} The three-chain $C_3=\{\bot,k,\top\}$ is equipped with a commutative and non-integral quantale structure $(C_3,\with,k)$, with
\[\top\with\top=\top\ra\top=\top,\quad\top\ra\bot=\top\ra k=\bot\]
and the other multiplications\,/\,implications being trivial.
\item \label{quantale-exmp:Rel} Let $\Rel(X)$ denote the set of (binary) relations on a non-empty set $X$. Then $(\Rel(X),\circ,\id_X)$ is an involutive quantale, where $\circ$ refers to the composition of relations, and
\[\id_X=\{(x,x)\mid x\in X\}\]
is the identity relation on $X$. It is obvious that the \emph{opposite} $R^{\circ}$ of relations $R\in\Rel(X)$, i.e.,
\[R^{\circ}=\{(y,x)\in X\times X\mid(x,y)\in R\},\]
defines an involution on $\Rel(X)$. Note that $\Rel(X)$ is non-commutative and non-integral as long as $X$ contains at least two elements.
\item \label{quantale-exmp:Sup} Let $\Sup[0,1]$ denote the set of $\sup$-preserving maps on the unit interval $[0,1]$. Then $(\Sup[0,1],\circ,1_{[0,1]})$ is a non-commutative, non-integral and unital quantale, where $\circ$ refers to the composition of maps, and $1_{[0,1]}$ is the identity map on $[0,1]$. An involution on $\Sup[0,1]$ is given by
\[f^{\circ}:[0,1]\to[0,1],\quad f^{\circ}(x)=1-f^{\star}(1-x)\]
for all $f\in\Sup[0,1]$, where $f^{\star}:[0,1]\to[0,1]$ is the right adjoint of $f$.
\item \label{quantale-exmp:c-inv} Every commutative quantale $\sQ$ is involutive, with a trivial involution given by the identity map on $\sQ$. In particular, all the commutative quantales mentioned in \ref{quantale-exmp:Lawvere}--\ref{quantale-exmp:C3} are involutive.
\end{enumerate}
\end{exmp}
\begin{SA}
Throughout this paper, we fix an involutive quantale
\[\sQ=(\sQ,\with,k,{}^{\circ})\]
as the table of truth-values, unless otherwise specified.
\end{SA}
\section{Quantale-valued similarity and dissimilarity: Definitions and examples} \label{Sim-Dissim-Def}
In order to throw light on the postulation of dissimilarity, let us recall the notion of \emph{$\sQ$-valued similarity}\footnote{A map $\al:X\times X\to\sQ$ is a $\sQ$-valued similarity in the sense of Definition \ref{sim-def} if, and only if,
\[\overline{\alpha}(x,y):=\al(y,x)\]
defines a $\sQ$-valued equality in the sense of H\"{o}hle-Kubiak (see \cite[Definition 2.1]{Hoehle2011a}). So, $\sQ$-valued similarity and $\sQ$-valued equality are equivalent concepts.} :
\begin{defn} \label{sim-def} (cf. \cite[Definition 2.1 and Lemma 2.3]{Hoehle2011a}.)
A \emph{$\sQ$-valued similarity} on a set $X$ is a map
\[\al:X\times X\to\sQ\]
such that
\begin{enumerate}[label=(S\arabic*)]
\item \label{sim-def:str} (strictness) \ $\al(x,y)\leq\al(x,x)\wedge\al(y,y)$,
\item \label{sim-def:sym} (symmetry) \ $\al(x,y)=\al(y,x)^{\circ}$,
\item \label{sim-def:div} (divisibility) \ $\al(x,y)=(\al(x,y)\ldd\al(x,x))\with\al(x,x)$,
\item \label{sim-def:tran} (transitivity) \ $(\al(y,z)\ldd\al(y,y))\with\al(x,y)\leq\al(x,z)$
\end{enumerate}
for all $x,y,z\in X$.
\end{defn}
Note that $\al(x,x)$ is hermitian for all $x\in X$ by \ref{sim-def:sym}. Moreover, in the presence of \ref{sim-def:sym}, the axiom \ref{sim-def:div} of divisibility implies that $\al(x,y)=\al(y,y)\with(\al(y,y)\rdd\al(x,y))$ (see \cite[Lemma 2.3]{Hoehle2011a}), and the axiom \ref{sim-def:tran} implies that $\al(y,z)\with(\al(y,y)\rdd\al(x,y))\leq\al(x,z)$.
An easy analysis of the axioms in Definition \ref{sim-def} tells us that \ref{sim-def:str} is implied by \ref{sim-def:div} if $\sQ$ is integral, and \ref{sim-def:str} is equivalent to \ref{sim-def:div} if $\sQ$ is divisible:
\begin{prop} \label{sim-divisible} (See \cite{Hoehle2011a}.)
If $\sQ$ is a divisible quantale, then the axiom of strictness is equivalent to the axiom of divisibility. Hence, a map $\al:X\times X\to\sQ$ defines a $\sQ$-valued similarity on a set $X$ if, and only if,
\begin{enumerate}[label={\rm(\arabic*)}]
\item \label{sim-divisible:str} $\al(x,y)\leq\al(x,x)\wedge\al(y,y)$,
\item \label{sim-divisible:sym} $\al(x,y)=\al(y,x)^{\circ}$,
\item \label{sim-divisible:tran} $(\al(y,z)\ldd\al(y,y))\with\al(x,y)\leq\al(x,z)$
\end{enumerate}
for all $x,y,z\in X$.
\end{prop}
Moreover, both \ref{sim-def:str} and \ref{sim-def:div} are subsumed by \ref{sim-def:sym} and \ref{sim-def:tran} when $\sQ$ is a frame, in which case a set with a $\sQ$-valued similarity is exactly a frame-valued set in the sense of Fourman--Scott \cite{Fourman1979} and Higgs \cite{Higgs1970,Higgs1984}:
\begin{prop} \label{sim-frame} (See \cite{Fourman1979}.)
If $\sQ$ is a frame, then a map $\al:A\times A\to\sQ$ defines a $\sQ$-valued similarity on a set $A$ if, and only if,
\begin{enumerate}[label={\rm(\arabic*)}]
\item \label{sim-frame:sym} $\al(x,y)=\al(y,x)$,
\item \label{sim-frame:tran} $\al(y,z)\wedge\al(x,y)\leq\al(x,z)$
\end{enumerate}
for all $x,y,z\in A$.
\end{prop}
\begin{Discus}[of the axioms of $\sQ$-valued similarity]
Let $\al:X\times X\to\sQ$ be a $\sQ$-valued similarity.
\begin{itemize}
\item[\ref{sim-def:str}] The value
\[\al(x,y)\]
is understood as the truth-value of the statement that \emph{$x$ is similar to $y$}. Since each entity is supposed to be similar to itself as long as it exists (or, once it is defined), the value $\al(x,x)$ may be understood as the \emph{extent of existence} \cite{Fourman1979,Hoehle2011a} of $x$. The axiom of strictness then indicates that each entity is more similar to itself than to any other one.
\item[\ref{sim-def:sym}] Similarity is symmetric; that is, if $x$ is similar to $y$, then $y$ is similar to $x$.
\item[\ref{sim-def:div}] The value
\[\al(x,y)\ldd\al(x,x)\]
measures to what extent the existence of $x$ forces $x$ to be similar to $y$. The equation
\[(\al(x,y)\ldd\al(x,x))\with\al(x,x)=\al(x,y)\]
says that $x$ is similar to $y$ if, and only if, $x$ has been proved to exist and the existence of $x$ forces $x$ to be similar to $y$.
\item[\ref{sim-def:tran}] The inequality
\[(\al(y,z)\ldd\al(y,y))\with\al(x,y) \leq\al(x,z)\]
says that if $x$ is similar to $y$, and if the existence of $y$ forces $y$ to be similar to $z$, then $x$ is similar to $z$. So, this axiom refers to the transitivity of similarity.
\end{itemize}
\end{Discus}
\begin{rem} \label{Q-preorder}
A map $\al:X\times X\to\sQ$ satisfying \ref{sim-def:div} and \ref{sim-def:tran} actually defines a \emph{$\sQ$-preorder} on the \emph{$\sQ$-subset} $(X,\mu)$ with
\[\mu:X\to\sQ,\quad\mu(x)=\al(x,x);\]
see \cite{GutierrezGarcia2018,Pu2012}. Hence, $\sQ$-valued similarities are a special kind of $\sQ$-preordered $\sQ$-subsets. In particular, if
\[\al(x,x)=k\]
for all $x\in X$, then $(X,\al)$ reduces to a $\sQ$-preorder on the (crisp) set $X$; see, e.g., \cite{Bvelohlavek2004,Hoehle2015,Lai2006}.
\end{rem}
\begin{rem} \label{indistinguishability}
If $\sQ$ is an integral quantale, let $\al:X\times X\to\sQ$ be a map with
\[\al(x,x)=k=\top\]
for all $x\in X$. Then $\al$ is a $\sQ$-valued similarity on $X$ if, and only if,
\begin{enumerate}[label=(\arabic*)]
\item (symmetry) \ $\al(x,y)=\al(y,x)^{\circ}$,
\item (transitivity) \ $\al(y,z)\with\al(x,y)\leq\al(x,z)$
\end{enumerate}
for all $x,y,z\in X$. Hence, for an integral quantale $\sQ$, $\sQ$-valued similarities $\al$ with $\al(x,x)=k=\top$ for all $x\in X$ generalize \emph{probabilistic relations} in the sense of Menger \cite{Menger1951}, \emph{similarity relations} in the sense of Zadeh \cite{Zadeh1971}, \emph{likeness relations} in the sense of Ruspini \cite{Ruspini1982} and \emph{indistinguishability operators} in the sense of Trillas--Valverde \cite{Trillas1984,Valverde1985}.
\end{rem}
\begin{exmp} \label{sim-2}
For $\sQ={\bf 2}$, a ${\bf 2}$-valued similarity $\al$ on a set $X$ is just an equivalence relation on a subset of $X$. Explicitly,
\[\{(x,y)\in X\times X\mid \al(x,y)=1\}\]
is an equivalence relation on the subset $\{x\mid\al(x,x)=1\}$ consisting of elements that ``have been defined''.
\end{exmp}
\begin{exmp}[Guiding example] \label{sim-exmp-PCX}
Let $X$ be a topological space. Then it follows from Proposition \ref{sim-frame} that the map
\[\al: \PCX\times \PCX\to\OX\]
given by Equation \eqref{sim-PCX} is an $\OX$-valued similarity on the set $\PCX$ of partially defined real-valued continuous maps on $X$ (see Equation \eqref{PCX-def}).
\end{exmp}
\begin{exmp} \label{sim-exmp-real}
Every \emph{partial metric space} (cf. \cite[Definition 3.1]{Matthews1994} and \cite[Definition 2]{Bukatin2009}) is a $[0,\infty]$-valued similarity. A \emph{generalized} partial metric space (cf. \cite[Example 3.10]{Pu2012} and \cite[Example 2.14]{Stubbe2014}) becomes a $[0,\infty]$-valued similarity whenever it is symmetric. Explicitly, a (generalized) partial metric space is a set $X$ equipped with a map
\[\al:X\times X\to[0,\infty]\]
such that
\[\al(x,x)\vee \al(y,y)\leq \al(x,y)\quad\text{and}\quad\al(x,z)\leq\al(y,z)-\al(y,y)+\al(x,y)\]
for all $x,y,z\in X$. As a concrete instance of such examples, let
\[\CI=\{[a,b]\mid 0\leq a<b\leq\infty\}\]
be the set of closed intervals contained in $[0,\infty]$. Then
\[\al([a,b],[c,d])=b\vee d-a\wedge c\]
defines a $[0,\infty]$-valued similarity on $\CI$; that is, $(\CI,\al)$ is a symmetric (generalized) partial metric space.
\end{exmp}
Now we are ready to present the key notion of this paper:
\begin{defn} \label{dissim-def}
A \emph{$\sQ$-valued dissimilarity} on a set $X$ is a map
\[\be:X\times X\to\sQ\]
such that
\begin{enumerate}[label=(D\arabic*)]
\item \label{dissim-def:str} (strictness) \ $\be(x,y)\geq\be(x,x)\vee\be(y,y)$,
\item \label{dissim-def:sym} (symmetry) \ $\be(x,y)=\be(y,x)^{\circ}$,
\item \label{dissim-def:reg} (regularity) \ $\be(x,y)=\be(x,x)\ldd(\be(x,y)\rdd \be(x,x))$,
\item \label{dissim-def:tran} (contrapositive transitivity) \ $\be(x,z)\leq\be(x,y)\ldd(\be(y,z)\rdd\be(y,y))$
\end{enumerate}
for all $x,y,z\in X$.
\end{defn}
Note that $\be(x,x)$ is hermitian for all $x\in X$ by \ref{dissim-def:sym}. Moreover, in the presence of \ref{dissim-def:sym}, the axiom \ref{dissim-def:reg} implies that $\be(x,y)=(\be(y,y)\ldd\be(x,y))\rdd \be(y,y)$, and the axiom \ref{dissim-def:tran} implies that $\be(x,z)\leq(\be(y,y)\ldd\be(x,y))\rdd\be(y,z)$. With a direct computation it is easy to see that \ref{dissim-def:str} is implied by \ref{dissim-def:reg} if $\sQ$ is integral, and \ref{dissim-def:str} is equivalent to \ref{dissim-def:reg} if $\sQ$ is a complete MV-algebra.
\begin{Discus}[of the axioms of $\sQ$-valued dissimilarity]
Let $\be:X\times X\to\sQ$ be a $\sQ$-valued dissimilarity.
\begin{itemize}
\item[\ref{dissim-def:str}] The value
\[\be(x,y)\]
is understood as the truth-value of the statement that \emph{$x$ is dissimilar to $y$}. The axiom of strictness dictates that each entity is less dissimilar to itself than to any other one, which is parallel to the assertion that each entity is more similar to itself than to any other one.
Since each entity is supposed to be similar to itself unless it is still \emph{undefined}, the value $\be(x,x)$ may be understood as the \emph{extent of $x$ being undefined}, or the \emph{degree of non-existence} of $x$. Therefore, the underlying logical principle of the axiom of strictness is that \emph{non-existence implies dissimilarity}.
\item[\ref{dissim-def:sym}] Dissimilarity is symmetric; that is, if $x$ is dissimilar to $y$, then $y$ is dissimilar to $x$.
\item[\ref{dissim-def:reg}] The value
\[\be(x,y)\rdd \be(x,x)\]
measures the extent that the dissimilarity between $x$ and $y$ forces $x$ to be undefined; in other words, it is the truth-value of the \emph{contrapositive} of the assertion that ``if $x$ is defined, then $x$ similar to $y$". The equation
\begin{equation} \label{regularity}
\be(x,y)=\be(x,x)\ldd(\be(x,y)\rdd\be(x,x))
\end{equation}
then asserts that $x$ is dissimilar to $y$ if, and only if, $x$ being ``similar'' to $y$ would force $x$ to be undefined.
In order to explain the name ``regularity'' of this axiom, let us recall that in a frame $\sQ$, an element $q\in\sQ$ is \emph{regular} \cite{Johnstone1986} if
\begin{equation} \label{regular-def}
\bot\ldd(q\rdd\bot)=q=(\bot\ldd q)\rdd\bot.
\end{equation}
The term ``regular'' stems from the fact that regular open sets in a topological space $X$ are exactly regular elements in the frame $\OX$. Analogously, in a quantale $\sQ$ we may call an element $q\in\sQ$ \emph{regular} if
\[\bot\ldd(q\rdd\bot)=q=(\bot\ldd q)\rdd\bot.\]
If $\sQ$ is integral and $r\in\sQ$, then it is easy to verify that the operation
\[p\with_r q:=(p\with q)\vee r\]
defines a quantale structure on $\ua\!r:=\{q\in\sQ\mid r\leq q\}$, and regular elements in this quantale are precisely those $q\in\ua\!r$ satisfying
\[r\ldd(q\rdd r)=q=(r\ldd q)\rdd r.\]
Hence, with a slight abuse of language, it makes sense to read \eqref{regularity} as ``$\be(x,y)$ is \emph{regular} with respect to $\be(x,x)$''.
\item[\ref{dissim-def:tran}] The inequality
\[\be(x,z)\leq\be(x,y)\ldd(\be(y,z)\rdd\be(y,y))\]
is equivalent to
\[\be(x,z)\with(\be(y,z)\rdd\be(y,y))\leq\be(x,y),\]
which claims that if $x$ is dissimilar to $z$, and if the dissimilarity between $y$ and $z$ forces $y$ to be undefined, then $x$ is dissimilar to $y$; in other words, if $x$ dissimilar to $z$ and $y$ is ``similar'' to $z$, then $x$ is dissimilar to $y$. So, this axiom is actually the \emph{contrapositive transitivity} of dissimilarity.
\end{itemize}
\end{Discus}
\begin{exmp} \label{dissim-2}
For $\sQ={\bf 2}$, a ${\bf 2}$-valued dissimilarity $\be$ on a set $X$ is the complement of an equivalence relation on a subset of $X$. Explicitly,
\[\{(x,y)\in X\times X\mid \be(x,y)=1\}\]
is the complement (in $X\times X$) of an equivalence relation on the subset $\{x\mid\be(x,x)=0\}$ consisting of elements that ``have been defined''. So, as one expects, in this case each dissimilarity relation is the negation of a similarity relation, and vice versa.
\end{exmp}
\begin{exmp}[Guiding example] \label{dissim-exmp-PCX}
Let $\PCX$ be given as in Example \ref{sim-exmp-PCX}, and define
\[\be(f,g):=\Int(X-\Int\{x\in D(f)\cap D(g)\mid f(x)=g(x)\})\]
for all $f,g\in\PCX$. Then, one can check, via a straightforward but quite lengthy verification, that $\be$ is an $\OX$-valued dissimilarity on $\PCX$. The conclusion is also an immediate consequence of Example \ref{sim-exmp-PCX} and Proposition \ref{frame-neg-sim-dissim} that will be explained later. It is clear that $\be(f,f)$ is the largest open set on which $f$ is \emph{undefined}.
\end{exmp}
\begin{exmp} \label{dissim-exmp-real}
Let
\[\CI=\{[a,b]\mid 0\leq a<b\leq\infty\}\]
as in Example \ref{sim-exmp-real}, and define
\[\be([a,b],[c,d])=\begin{cases}
0 & \text{if}\ b\vee d=\infty,\\
\max\{0,b\wedge d-a\vee c\} & \text{else}.
\end{cases}\]
Then it is straightforward to check that $\be$ is a $[0,\infty]$-valued dissimilarity on $\CI$.
\end{exmp}
\begin{rem} \label{dissim-bot} (to be continued in Remark \ref{dissim-bot-2})
Let $\be:X\times X\to\sQ$ be a map with
\[\be(x,x)=\bot\]
for all $x\in X$. Then $\be$ is a $\sQ$-valued dissimilarity on $X$ if, and only if,
\begin{enumerate}[label=(\arabic*)]
\item (symmetry) \ $\be(x,y)=\be(y,x)^{\circ}$,
\item (regularity) \ $\be(x,y)$ is a regular element of $\sQ$ (see Equation \eqref{regular-def}),
\item (contrapositive transitivity) \ $\be(x,z)\leq\be(x,y)\ldd(\be(y,z)\rdd\bot)$
\end{enumerate}
for all $x,y,z\in X$. The semantic meaning of the inequality $\be(x,z)\leq\be(x,y)\ldd(\be(y,z)\rdd\bot)$ is that ``if $x$ is dissimilar to $z$ and if $y$ is not dissimilar to $z$, then $x$ is dissimilar to $y$.'' In particular, if $\sQ$ is a complete Boolean algebra, then, in the presence of the axiom of symmetry, the axiom of contrapositive transitivity is actually equivalent to \[\be(x,z)\leq\be(x,y)\vee\be(y,z),\]
which means that ``if $x$ is dissimilar to $z$, then for each $y$, either $x$ is dissimilar to $y$ or $y$ is dissimilar to $z$.''
In what follows, a $\sQ$-valued dissimilarity $\beta$ with
\[\beta(x,x)=\bot\]
for all $x$ will be called \emph{rigid}. In a rigid $\sQ$-valued dissimilarity, every entity is never dissimilar to itself no matter whether it has been ``fully defined''.
\end{rem}
\begin{rem} \label{dissim-vs-apart}
Although our notion of dissimilarity is inspired by that of \emph{apartness relation} of Scott (see \cite[Section 4]{Scott1979}), they are conceptually different. If $\sQ$ is a frame, then a $\sQ$-valued model of apartness relation consists of the following data:
\begin{itemize}
\item a set $X$;
\item a map $E:X\to\sQ$, where the value $E(x)$ is interpreted as the \emph{extent of existence} of $x$;
\item a map $\ga:X\times X\to\sQ$, where the value $\ga(x,y)$ is interpreted as the degree of $x$ being \emph{apart} from $y$.
\end{itemize}
These data are subject to the following requirements for all $x,y,z\in X$:
\begin{enumerate}[label=(\arabic*)]
\item $\ga(x,y)\leq E(x)\wedge E(y)$,
\item $\ga(x,x)=\bot$,
\item $\ga(x,y)=\ga(y,x)$,
\item $\ga(x,z)\wedge E(y)\leq \ga(x,y)\vee\ga(z,y)$.
\end{enumerate}
It is easy to see that the map $\be$ given in Example \ref{dissim-exmp-PCX} cannot be made into an $\OX$-valued apartness relation on $\PCX$, and thus $\sQ$-valued apartness relations are essentially different from $\sQ$-valued similarities. However, they are closely related if $\sQ$ is a complete Boolean algebra as we see below.
Let ${\sf B}$ be a complete Boolean algebra. Then a rigid ${\sf B}$-valued dissimilarity on a set $X$ is a map $\be:X\times X\to{\sf B}$ such that
\[\be(x,x)=\bot\quad\text{and}\quad\be(x,z)\leq\be(x,y)\vee\be(y,z)\]
for all $x,y,z\in X$. If $(X,E,\ga)$ is a $\sB$-valued apartness relation with $E(x)=\top$ for all $x\in X$, then $\ga$ is a rigid $\sB$-valued similarity on $X$.
Conversely, if $\be$ is a rigid $\sB$-valued dissimilarity on $X$, then $(X,E,\be)$ is a $\sB$-valued apartness relation with $E(x)=\top$ for all $x\in X$.
Therefore, for a complete Boolean algebra $\sB$, a $\sB$-valued apartness relation on a set whose elements have all been ``proved to exist'' is precisely a rigid $\sB$-valued dissimilarity relation.
For connections between Boolean-valued similarities Boolean-valued apartness relations, see Remark \ref{sim-vs-apart-Boolean}.
\end{rem}
\section{Similarities and dissimilarities as enriched categories} \label{Sim-Dissim-Cat}
It is already known from \cite{Hoehle2011a} that sets equipped with a $\sQ$-valued similarity are \emph{symmetric} categories enriched in a subquantaloid of the quantaloid $\DsQ$ of \emph{diagonals} of $\sQ$ \cite{Hoehle2011a,Pu2012,Stubbe2014}. The aim of this section is to show that there is an analogous categorical interpretation for $\sQ$-valued dissimilarities; that is, a set equipped with a $\sQ$-dissimilarity can be made into a \emph{symmetric} category enriched in a subquantaloid $\KsQ$ of the quantaloid $\BsQ$ of \emph{back diagonals} of $\sQ$ introduced in \cite{Shen2016a}. Therefore, $\sQ$-valued similarities and $\sQ$-valued dissimilarities are both instances of enriched categories.
\subsection{Quantaloid-enriched categories}
A \emph{quantaloid} \cite{Rosenthal1996} $\CQ$ is a category in which every hom-set is a complete lattice, and the composition $\circ$ of $\CQ$-arrows preserves suprema on both sides, i.e.,
\[v\circ\Big(\bv_{i\in I} u_i\Big)=\bv_{i\in I}v\circ u_i\quad\text{and}\quad\Big(\bv_{i\in I} v_i\Big)\circ u=\bv_{i\in I}v_i\circ u\]
for all $\CQ$-arrows $u,u_i:p\to q$, $v,v_i:q\to r$ $(i\in I)$. The corresponding right adjoints induced by the composition maps \[-\circ u\dv -\lda u:\ \CQ(p,r)\to\CQ(q,r)\quad\text{and}\quad v\circ -\dv v\rda -:\ \CQ(p,r)\to\CQ(p,q)\] satisfy \[v\circ u\leq w\iff v\leq w\lda u\iff u\leq v\rda w\] for all $\CQ$-arrows $u:p\to q$, $v:q\to r$, $w:p\to r$, where the operations $\lda$ and $\rda$ are called \emph{left} and \emph{right implications} in $\CQ$, respectively.
A (unital) quantale $\sQ=(\sQ,\with,k)$ is exactly a one-object quantaloid. As we will construct several quantaloids out of a quantale $\sQ$ later, in order to eliminate ambiguity we denote implications in a quantale $\sQ$ by $\ldd$ and $\rdd$ as in \eqref{imp-def}, and reserve the notations $\lda$ and $\rda$ for the quantaloids constructed from $\sQ$.
Given a \emph{small} quantaloid $\CQ$ (i.e., $\ob\CQ$ is a set), a \emph{$\CQ$-category} (also \emph{category enriched in $\CQ$}) \cite{Stubbe2005} consists of a \emph{$\CQ$-typed set} $X$ (i.e., a set $X$ equipped with a \emph{type} map $|\text{-}|:X\to\ob\CQ$) and a family of $\CQ$-arrows $\al(x,y)\in\CQ(|x|,|y|)$ $(x,y\in X)$, such that
\[1_{|x|}\leq\al(x,x)\quad\text{and}\quad\al(y,z)\circ\al(x,y)\leq\al(x,z)\]
for all $x,y,z\in X$.
A \emph{$\CQ$-functor} $f:(X,\al)\to(Y,\be)$ between $\CQ$-categories is a map $f:X\to Y$ such that
\[|x|=|fx|\quad\text{and}\quad\al(x,y)\leq\be(fx,fy)\]
for all $x,y\in X$. The category of $\CQ$-categories and $\CQ$-functors is denoted by
\[\QCat.\]
A \emph{homomorphism} $F:\CQ\to\CR$ of quantaloids is a functor of the underlying categories that preserves suprema of $\CQ$-arrows. By an \emph{involution} on a quantaloid $\CQ$ we mean a homomorphism
\[(-)^{\circ}:\CQ^{\op}\to\CQ\]
of quantaloids whose composition with itself outputs the identity homomorphism on $\CQ$. Explicitly, an involution on $\CQ$ is given by maps
\[(-)^{\circ}:\ob\CQ\to\ob\CQ\quad\text{and}\quad (-)^{\circ}:\CQ(p,q)\to\CQ(q^{\circ},p^{\circ})\]
for all $p,q\in\ob\CQ$, such that
\[q^{\circ\circ}=q,\quad (1_q)^{\circ}=1_{q^{\circ}},\quad u^{\circ\circ}=u,\quad (v\circ u)^{\circ}=u^{\circ}\circ v^{\circ}\quad\text{and}\quad\Big(\bv_{i\in I}u_i\Big)^{\circ}=\bv_{i\in I}u_i^{\circ}\]
for all $q\in\ob\CQ$ and $\CQ$-arrows $u,u_i:p\to q$ $(i\in I)$. Given a small \emph{involutive} quantaloid $\CQ$, i.e., a small quantaloid $\CQ$ equipped with an involution, we say that a $\CQ$-category $(X,\al)$ is \emph{symmetric} if
\begin{equation} \label{sym-Q-cat}
\al(x,y)=\al(y,x)^{\circ}
\end{equation}
for all $x,y\in X$.
\begin{rem} \label{sym-Q-cat-Stubbe}
Our definition of involutive quantaloids here slightly generalizes that of Rosenthal (see \cite[Definition 2.5.1]{Rosenthal1996}), which requires an involution to be the identity on objects. Indeed, as explained below, these definitions make no difference for the purpose of defining the symmetry of $\CQ$-categories.
Let $(X,\al)$ be a symmetric $\CQ$-category. Since $\al(x,y)\in\CQ(|x|,|y|)$ and $\al(y,x)^{\circ}\in\CQ(|x|^{\circ},|y|^{\circ})$, Equation \eqref{sym-Q-cat} actually forces
\[|x|=|x|^{\circ}\]
for all $x\in X$. Therefore, a symmetric $\CQ$-category is in fact a category enriched in the full subquantaloid $\CQ^{\circ}$ of $\CQ$ with
\[\ob\CQ^{\circ}=\{q\in\ob\CQ\mid q=q^{\circ}\},\]
which is equipped with the involution inherited from $\CQ$ that is clearly neutral on objects. Hence, symmetric $\CQ$-categories defined by \eqref{sym-Q-cat} are precisely symmetric $\CQ^{\circ}$-categories as postulated by \cite[Definition 2.3]{Heymans2011} and \cite[Definition 6.2]{Hoehle2011a}, whose prototype comes from \cite{Betti1982a}.
\end{rem}
\subsection{$\sQ$-valued similarities as enriched categories}
In this subsection we recall how $\sQ$-valued similarities are represented as symmetric categories enriched in a quantaloid $\HsQ$ (see \cite{Hoehle2011a}), which is a subquantaloid of the quantaloid $\DsQ$ of \emph{diagonals} of $\sQ$ \cite{Hoehle2011a,Stubbe2014}.
Let $p,q\in\sQ$. By a \emph{diagonal} \cite{Stubbe2014} from $p$ to $q$ we mean an element $d\in\sQ$ such that
\begin{equation} \label{diagonal-def}
(d\ldd p)\with p=d=q\with(q\rdd d).
\end{equation}
\begin{lem} \label{diagonal-property}
Let $p,q\in\sQ$.
\begin{enumerate}[label=\rm(\arabic*)]
\item \label{diagonal-property:bot} $\bot$ is a diagonal from $p$ to $q$.
\item \label{diagonal-property:id} $q$ is a diagonal from $q$ to $q$.
\item \label{diagonal-property:inf} $\bv\limits_{i\in I}d_i$ is a diagonal from $p$ to $q$ if so is each $d_i$ $(i\in I)$.
\item \label{diagonal-property:involution} If $d$ is a diagonal from $p$ to $q$, then $d^{\circ}$ is a diagonal from $q^{\circ}$ to $p^{\circ}$.
\end{enumerate}
\end{lem}
If $d$ is a diagonal from $p$ to $q$ and $e$ is a diagonal from $q$ to $r$, then it is not difficult to verify that
\[(e\ldd q)\with d=e\with(q\rdd d),\]
and thus we set
\begin{equation} \label{diagonal-comp-def}
e\diamond d:=(e\ldd q)\with d=e\with(q\rdd d).
\end{equation}
\begin{lem} \label{diagonal-comp}
Let $d$ be a diagonal from $p$ to $q$ and let $e$ be a diagonal from $q$ to $r$.
\begin{enumerate}[label=\rm(\arabic*)]
\item $e\diamond d$ is a diagonal from $p$ to $r$, called the \emph{composite} of $d$ and $e$.
\item $d\diamond p=d=q\diamond d$.
\item The composition of diagonals is associative.
\item The composition of diagonals preserves suprema on both sides, i.e.,
\[e\diamond\bv\limits_{i\in I}d_i=\bv_{i\in I}(e\diamond d_i)\quad\Big(\bv\limits_{i\in I} e_i\Big)\diamond d=\bv_{i\in I}(e_i\diamond d)\]
for all diagonals $d_i$ from $p$ to $q$ and $e_i$ from $q$ to $r$ $(i\in I)$.
\end{enumerate}
\end{lem}
Lemmas \ref{diagonal-property} and \ref{diagonal-comp} guarantee the existence of a quantaloid $\DsQ$ given by the following data, called the quantaloid of \emph{diagonals} of $\sQ$:
\begin{itemize}
\item objects of $\DsQ$ are elements $p,q,r,\dots$ of $\sQ$;
\item for $p,q\in\sQ$, morphisms from $p$ to $q$ in $\DsQ$ are diagonals from $p$ to $q$;
\item the composition of diagonals $d\in\BsQ(p,q)$ and $e\in\BsQ(q,r)$ is given by $e\diamond d$;
\item the identity diagonal on $q\in\sQ$ is $q$ itself;
\item each hom-set $\DsQ(p,q)$ is equipped with the order inherited from $\sQ$.
\end{itemize}
As pointed out in \cite[Example 2.14]{Stubbe2014}, the above construction makes sense not only for a general quantaloid $\CQ$, but also for a general category $\CC$ \cite{Grandis2000,Grandis2002}. It is easily seen that
\[\HsQ(p,q):=\{d\in\DsQ(p,q)\mid d\leq p\wedge q\}\]
for all $p,q\in\sQ$ defines a subquantaloid $\HsQ$ of $\DsQ$ (see \cite[Remark 4.4]{Hoehle2011a}), and we denote by
\[d:p\rqa q\]
a morphism $d\in\HsQ(p,q)$; that is,
\[d:p\rqa q\iff d\leq p\wedge q\quad\text{and}\quad(d\ldd p)\with p=d=q\with(q\rdd d).\]
In the case that $\sQ$ is integral, we have
\[\HsQ=\DsQ\]
since $d\leq p\wedge q$ would be a consequence of $(d\ldd p)\with p=d=q\with(q\rdd d)$. Moreover,
\[\HsQ(p,q)=\DsQ(p,q)=\{d\in\sQ\mid d\leq p\wedge q\}\]
if $\sQ$ is divisible.
\begin{exmp}
Since Lawvere's quantale $[0,\infty]$ (see Example \ref{quantale-exmp}\ref{quantale-exmp:Lawvere}) is divisible, it holds that
\[\BD_*[0,\infty](p,q)=[p\vee q,\infty]\]
for all $p,q\in[0,\infty]$, and
\[e\diamond d=e-q+d \]
for all $d:p\rqa q$ and $e:q\rqa r$.
\end{exmp}
Note that by Lemma \ref{diagonal-property}\ref{diagonal-property:involution}, $\HsQ$ is also an involutive quantaloid with the involution lifted from $\sQ$. From the definition we see that a $\HsQ$-category consists of a set $X$, a map $|\text{-}|:X\to\sQ$ and a map $\al:X\times X\to\sQ$ such that
\begin{enumerate}[label=(\arabic*)]
\item \label{HsQ-cat:str} $\al(x,y)\leq|x|\wedge|y|$,
\item \label{HsQ-cat:div} $(\al(x,y)\ldd |x|)\with |x|=\al(x,y)=|y|\with(|y|\rdd\al(x,y))$,
\item \label{HsQ-cat:ref} $|x|\leq\al(x,x)$,
\item \label{HsQ-cat:tran} $(\al(y,z)\ldd |y|)\with\al(x,y)=\al(y,z)\with(|y|\rdd\al(x,y))\leq\al(x,z)$
\end{enumerate}
for all $x,y,z\in X$, where \ref{HsQ-cat:str} and \ref{HsQ-cat:div} follows from $\al(x,y)\in\HsQ(|x|,|y|)$. Then, \ref{KsQ-cat:str} in conjunction with \ref{KsQ-cat:ref} leads to
\[\al(x,x)=|x|\]
for all $x\in X$, and thus a $\HsQ$-category is exactly given by a map $\al:X\times X\to\sQ$ such that (cf. Definition \ref{sim-def})
\begin{itemize}
\item $\al(x,y)\leq\al(x,x)\wedge\al(y,y)$,
\item $(\al(x,y)\ldd\al(x,x))\with\al(x,x)=\al(x,y)=\al(y,y)\with(\al(y,y)\rdd\al(x,y))$,
\item $(\al(y,z)\ldd\al(y,y))\with\al(x,y)=\al(y,z)\with(\al(y,y)\rdd\al(x,y))\leq\al(x,z)$
\end{itemize}
for all $x,y,z\in X$. Therefore, a $\sQ$-valued similarity $\al$ on a set $X$ is exactly a $\HsQ$-category satisfying
\begin{itemize}
\item $\al(x,y)=\al(y,x)^{\circ}$
\end{itemize}
for all $x,y\in X$; that is, a symmetric $\HsQ$-category:
\begin{thm} \label{sim-DsQ-Cat} (See \cite{Hoehle2011a}.)
A set equipped with a $\sQ$-valued similarity is precisely a symmetric $\HsQ$-category.
\end{thm}
\begin{rem} \label{sym-DQ-cat}
As elaborated in Remark \ref{sym-Q-cat-Stubbe}, a symmetric $\HsQ$-category is exactly a symmetric $\HssQ$-category, where $\HssQ$ is the involutive quantaloid constructed from $\sQ$ by H{\"o}hle--Kubiak in \cite[Proposition 6.3]{Hoehle2011a}.
\end{rem}
\subsection{$\sQ$-valued dissimilarities as enriched categories}
In this subsection, we construct a quantaloid $\KsQ$ for each quantale $\sQ$ and reveal that sets equipped with a $\sQ$-valued dissimilarity are precisely \emph{symmetric} categories enriched in $\KsQ$.
Let $p,q\in\sQ$. By a \emph{back diagonal} \cite{Shen2016a} from $p$ to $q$ we mean an element $b\in\sQ$ such that
\begin{equation} \label{back-diagonal-def}
p\ldd(b\rdd p)=b=(q\ldd b)\rdd q.
\end{equation}
The verification of the following lemma is straightforward:
\begin{lem} \label{back-diagonal-property}
Let $p,q\in\sQ$.
\begin{enumerate}[label=\rm(\arabic*)]
\item \label{back-diagonal-property:top} $\top$ is a back diagonal from $p$ to $q$.
\item \label{back-diagonal-property:id} $q$ is a back diagonal from $q$ to $q$.
\item \label{back-diagonal-property:inf} $\bw\limits_{i\in I}b_i$ is a back diagonal from $p$ to $q$ if so is each $b_i$ $(i\in I)$.
\item \label{back-diagonal-property:involution} If $b$ is a back diagonal from $p$ to $q$, then $b^{\circ}$ is a back diagonal from $q^{\circ}$ to $p^{\circ}$.
\end{enumerate}
\end{lem}
If $b$ is a back diagonal from $p$ to $q$ and $c$ is a back diagonal from $q$ to $r$, then
\[b\ldd(c\rdd q)\leq(q\ldd b)\rdd c\]
since
\[(q\ldd b)\with(b\ldd(c\rdd q))\leq q\ldd(c\rdd q)=c,\]
and similarly
\[(q\ldd b)\rdd c\leq b\ldd(c\rdd q).\]
Thus it makes sense to define
\begin{equation} \label{back-diagonal-comp-def}
c\bullet b:=b\ldd(c\rdd q)=(q\ldd b)\rdd c,
\end{equation}
which turns out to be a back diagonal from $p$ to $r$:
\begin{lem} \label{back-diagonal-comp}
Let $b$ be a back diagonal from $p$ to $q$ and let $c$ be a back diagonal from $q$ to $r$.
\begin{enumerate}[label=\rm(\arabic*)]
\item $c\bullet b$ is a back diagonal from $p$ to $r$, called the \emph{composite} of $b$ and $c$.
\item $b\bullet p=b=q\bullet b$.
\item The composition of back diagonals is associative.
\item The composition of back diagonals preserves infima on both sides, i.e.,
\[c\bullet\bw_{i\in I}b_i=\bw_{i\in I}(c\bullet b_i)\quad\Big(\bw_{i\in I} c_i\Big)\bullet b=\bw_{i\in I}(c_i\bullet b)\]
for all back diagonals $b_i$ from $p$ to $q$ and $c_i$ from $q$ to $r$ $(i\in I)$.
\end{enumerate}
\end{lem}
From Lemmas \ref{back-diagonal-property} and \ref{back-diagonal-comp} we actually obtain a quantaloid $\BsQ$ from each quantale $\sQ$, called the quantaloid of \emph{back diagonals} of $\sQ$:
\begin{itemize}
\item objects of $\BsQ$ are elements $p,q,r,\dots$ of $\sQ$;
\item for $p,q\in\sQ$, morphisms from $p$ to $q$ in $\BsQ$ are back diagonals from $p$ to $q$;
\item the composition of back diagonals $b\in\BsQ(p,q)$ and $c\in\BsQ(q,r)$ is given by $c\bullet b$;
\item the identity back diagonal on $q\in\sQ$ is $q$ itself;
\item each hom-set $\BsQ(p,q)$ is equipped with the \emph{reversed} order inherited from $\sQ$.
\end{itemize}
It should be noted that the construction of $\BsQ$ makes sense not only for a quantale $\sQ$, but also for a general quantaloid $\CQ$; see \cite{Shen2016a}.
For each $p,q\in\sQ$, let
\[\KsQ(p,q):=\{b\in\BsQ\mid p\vee q\leq b\};\]
that is, $\KsQ(p,q)$ consists of back diagonals from $p$ to $q$ that are above both $p$ and $q$. Then, it is easy to see that $\KsQ$ is a subquantaloid of $\BsQ$, and we write
\[b:p\lar q\]
for a morphism $b\in\KsQ(p,q)$; that is,
\[b:p\lar q\iff p\vee q\leq b\quad\text{and}\quad p\ldd(b\rdd p)=b=(q\ldd b)\rdd q.\]
Note that if $\sQ$ is integral, then $p\vee q\leq b$ is implied by $p\ldd(b\rdd p)=b=(q\ldd b)\rdd q$, and thus
\[\KsQ=\BsQ\]
in this case.
\begin{exmp}
For Lawvere's quantale $[0,\infty]$ given in Example \ref{quantale-exmp}\ref{quantale-exmp:Lawvere},
\[\BB_*[0,\infty](p,q)=\begin{cases}
[0,p\wedge q] & \text{if}\ p,q<\infty,\\
\{0,\infty\} & \text{if}\ p=q=\infty,\\
\{0\} & \text{else}
\end{cases}\]
for all $p,q\in[0,\infty]$, and
\[ c\bullet b=\begin{cases}
0 & \text{if}\ c\wedge b<\infty\ \text{and}\ q=\infty,\\
\max\{0,c-q+b\} & \text{else}.
\end{cases}\]
for all $b:p\lar q$ and $c:q\lar r$.
\end{exmp}
Now let us look at categories enriched in the quantaloid $\KsQ$. As a direct consequence of Lemma \ref{back-diagonal-property}\ref{back-diagonal-property:involution}, $\KsQ$ is an involutive quantaloid with the involution lifted from $\sQ$; that is, the involution $(-)^{\circ}$ on $\sQ$ actually gives rise to an involution on $\KsQ$. By definition, a $\KsQ$-category consists of a set $X$, a map $|\text{-}|:X\to\sQ$ and a map $\be:X\times X\to\sQ$ such that
\begin{enumerate}[label=(\arabic*)]
\item \label{KsQ-cat:str} $|x|\vee |y|\leq\be(x,y)$,
\item \label{KsQ-cat:reg} $|x|\ldd(\be(x,y)\rdd|x|)=\be(x,y)=(|y|\ldd\be(x,y))\rdd |y|$,
\item \label{KsQ-cat:ref} $\be(x,x)\leq|x|$,
\item \label{KsQ-cat:tran} $\be(x,z)\leq\be(x,y)\ldd(\be(y,z)\rdd|y|)=(|y|\ldd\be(x,y))\rdd\be(y,z)$
\end{enumerate}
for all $x,y,z\in X$, where \ref{KsQ-cat:str} and \ref{KsQ-cat:reg} follows from $\be(x,y)\in\KsQ(|x|,|y|)$. Note that the combination of \ref{KsQ-cat:str} and \ref{KsQ-cat:ref} forces
\begin{equation} \label{bexx=x}
\be(x,x)=|x|
\end{equation}
for all $x\in X$, and thus a $\KsQ$-category is precisely given by a map $\be:X\times X\to\sQ$ such that (cf. Definition \ref{dissim-def})
\begin{itemize}
\item $\be(x,y)\geq\be(x,x)\vee\be(y,y)$,
\item $\be(x,x)\ldd(\be(x,y)\rdd \be(x,x))=\be(x,y)=(\be(y,y)\ldd\be(x,y))\rdd\be(y,y)$,
\item $\be(x,z)\leq\be(x,y)\ldd(\be(y,z)\rdd\be(y,y))=(\be(y,y)\ldd\be(x,y))\rdd\be(y,z)$
\end{itemize}
for all $x,y,z\in X$. Therefore, a $\sQ$-valued dissimilarity $\be$ on a set $X$ is exactly a $\KsQ$-category satisfying
\begin{itemize}
\item $\be(x,y)=\be(y,x)^{\circ}$
\end{itemize}
for all $x,y\in X$; that is, a symmetric $\KsQ$-category:
\begin{thm} \label{dissim-BsQ-Cat}
A set equipped with a $\sQ$-valued dissimilarity is precisely a symmetric $\KsQ$-category.
\end{thm}
\begin{rem} \label{dissim-bot-2}
It is clear that $\KsQ(\bot,\bot)=\BsQ(\bot,\bot)$ for every quantale $\sQ$ and an element of $\KsQ(\bot,\bot)$ is exactly a regular element of $\sQ$ (see Equation \eqref{regular-def}). Furthermore, the quantale $\KsQ(\bot,\bot)$ is integral and a rigid $\sQ$-valued dissimilarity (see Remark \ref{dissim-bot}) on a set $X$ is precisely a symmetric category structure enriched in the involutive quantale $\KsQ(\bot,\bot)$.
\end{rem}
\section{Similarity vs. dissimilarity} \label{Sim-vs-Dissim}
In classical logic, the negation of a similarity relation is a dissimilarity relation, and vice versa. (cf. Examples \ref{sim-2} and \ref{dissim-2}). It is natural to ask whether it still holds in the quantale-valued setting; that is, whether the negation of a $\sQ$-valued dissimilarity is a $\sQ$-valued similarity, and vice versa. With the help of \emph{lax functors} between the quantaloids $\HsQ$ and $\KsQ$, in this section we provide some partial answers to this question in the case that $\sQ$ is a divisible quantale, a frame or a Girard quantale.
Before proceeding on, we would like to remind the readers of the fact that although being of unequivocal importance, inquiring what is really meant by \emph{negation} remains a sensitive question in fuzzy set theory, and it will not be discussed here. In what follows we just focus on two kinds of negations in a quantale, one of which is determined by the bottom element of the quantale, and the other is the \emph{linear negation} in a Girard quantale. Both of the negations under concern are of residuation-type; that is, they are determined by the operator $\with$ via adjoint property.
Recall that a \emph{lax functor} \cite{Hofmann2014,Street1972a} $F:\CQ\to\CR$ of quantaloids is given by maps
\[F:\ob\CQ\to\ob\CR\quad\text{and}\quad F_{p,q}:\CQ(p,q)\to\CR(Fp,Fq)\]
for all $p,q\in\ob\CQ$ (with $F_{p,q}$ usually written as $F$ for short), such that
\begin{enumerate}[label=(\arabic*)]
\item \label{lax-functor:mono} $F_{p,q}$ is monotone,
\item \label{lax-functor:mor} $Fv\circ Fu\leq F(v\circ u)$,
\item \label{lax-functor:unit} $1_{Fq}\leq F1_q$
\end{enumerate}
for all $p,q,r\in\ob\CQ$ and $\CQ$-arrows $u:p\to q$, $v:q\to r$. A lax functor $F:\CQ\to\CR$ becomes a homomorphism of quantaloids if it preserves suprema of $\CQ$-arrows and the inequalities ``$\leq$'' in \ref{lax-functor:mor} and \ref{lax-functor:unit} are replaced by ``$=$'', and a homomorphism of quantaloids becomes an \emph{isomorphism} of quantaloids if so is the underlying functor.
Every lax functor $F:\CQ\to\CR$ of quantaloids induces a functor
\[\QCat\to\RCat,\]
which assigns to each $\CQ$-category $(X,|\text{-}|,\al)$ an $\CR$-category $(X,F|\text{-}|,F\al)$, and each $\CQ$-functor $f:(X,|\text{-}|,\al)\to(Y,|\text{-}|,\be)$ will be mapped to an $\CR$-functor $f:(X,F|\text{-}|,F\al)\to(Y,F|\text{-}|,F\be)$. Therefore, the existence of a lax functor
\[\HsQ\to\KsQ\]
would allow us to construct a $\KsQ$-category from a $\HsQ$-category, and vice versa. Furthermore, if a lax functor
\[F:\HsQ\to\KsQ\]
preserves the involution of $\sQ$ in the sense that
\[F(d:p\rqa q)^{\circ}=(Fd:Fp\lar Fq)^{\circ},\quad\text{i.e.,}\quad Fq^{\circ}=(Fq)^{\circ}\quad\text{and}\quad Fd^{\circ}=(Fd)^{\circ}\]
for all $p,q\in\sQ$, $d\in\HsQ(p,q)$, then each $\sQ$-valued similarity would generate a $\sQ$-valued dissimilarity, and vice versa.
\subsection{When $\sQ$ is a divisible quantale}
In each quantale $\sQ$, we may define
\begin{equation} \label{negation-def}
\neg_l q:=\bot\ldd q\quad\text{and}\quad \neg_r q:=q\rdd\bot
\end{equation}
as the \emph{left} and \emph{right negations} of $q$, respectively, which can be unified to
\begin{equation} \label{negation-commutative-def}
\neg q:=\neg_l q=\neg_r q=q\ra\bot
\end{equation}
if the bottom element $\bot$ is \emph{cyclic} in the sense that
\[\bot\ldd q=q\rdd\bot\]
for all $q\in\sQ$. It is clear that the negation operators on $\sQ$ admit pointwise extensions to maps $X\times X\to\sQ$. The main result of this subsection is:
\begin{prop} \label{divisible-neg-dissim}
If $\sQ$ is a divisible quantale with the bottom element $\bot$ being cyclic, then the negation $\neg\be$ of each $\sQ$-valued dissimilarity $\be$ is a $\sQ$-valued similarity.
\end{prop}
Proposition \ref{divisible-neg-dissim} follows immediately from Equation \eqref{invo-imp} and the following lemma:
\begin{lem} \label{neg-BQ-divisible}
If $\sQ$ is a divisible quantale, then both the assignments
\[(b:p\lar q)\mapsto(\neg_l b:\neg_l p\rqa\neg_l q)\quad\text{and}\quad(b:p\lar q)\mapsto(\neg_r b:\neg_r p\rqa\neg_r q)\]
define lax functors $\neg_l,\neg_r:\KsQ\to\HsQ$.
\end{lem}
\begin{proof}
We only verify that $\neg_l:\KsQ\to\HsQ$ is a lax functor, and the lax functoriality of $\neg_r$ can be obtained dually.
If $b\in\KsQ(p,q)$, then it follows from $b\geq p\vee q$ that
\[\neg_l b\leq\neg_l p\wedge\neg_l q.\]
Thus $\neg_l b\in\HsQ(\neg_l p,\neg_l q)$ by the divisibility of $\sQ$.
Since
$\neg_l$ is clearly monotone on hom-sets and preserves identities, it remains to prove that
\[\neg_l c\diamond\neg_l b\leq\neg_l(c\bullet b)\]
for all morphisms $b:p\lar q$, $c:q\lar r$ in $\KsQ$. Indeed, since $q\leq c$ and $\neg_l(c\rdd q)\leq q\ldd(c\rdd q)=c$, we have
\begin{align*}
\neg_l q\with c&=\neg_l(c\with(c\rdd q))\with c&(q\leq c)\\
&=(\neg_l(c\rdd q)\ldd c)\with c\\
&=\neg_l(c\rdd q),&(\neg_l(c\rdd q)\leq c)
\end{align*}
and consequently
\begin{align}
\neg_l c\diamond\neg_l b&=(\neg_l c\ldd\neg_l q)\with\neg_l b \nonumber\\
&=\neg_l(\neg_l q\with c)\with\neg_l b \nonumber\\
&=\neg_l\neg_l(c\rdd q)\with\neg_l b \label{neg-c-diamond-neg-b}
\end{align}
by Equation \eqref{diagonal-comp-def}. It follows that
\[(\neg_l c\diamond\neg_l b)\with(b\ldd(c\rdd q))=\neg_l\neg_l(c\rdd q)\with\neg_l b\with(b\ldd(c\rdd q))\leq\bot,\]
and therefore
\[\neg_l c\diamond\neg_l b\leq\neg_l(b\ldd(c\rdd q))=\neg_l(c\bullet b)\]
by Equation \eqref{back-diagonal-comp-def}, as desired.
\end{proof}
\begin{cor} \label{neg-BQCat-divisible}
If $\sQ$ is a divisible quantale, then both the assignments
\[(X,\be)\mapsto(X,\neg_l\be)\quad\text{and}\quad(X,\be)\mapsto(X,\neg_r\be)\]
define functors $\KsQ\text{-}\Cat\to\HsQ\text{-}\Cat$.
\end{cor}
\subsection{When $\sQ$ is a frame}
In the case that $\sQ$ is a frame, the negation of a $\sQ$-valued dissimilarity is a $\sQ$-valued similarity, and vice versa:
\begin{prop} \label{frame-neg-sim-dissim}
If $\sQ$ is a frame, then
\begin{enumerate}[label={\rm(\arabic*)}]
\item \label{frame-neg-sim-dissim:neg-dissim} the negation $\neg\be$ of each $\sQ$-valued dissimilarity $\be$ is a $\sQ$-valued similarity, and
\item \label{frame-neg-sim-dissim:neg-sim} the negation $\neg\al$ of each $\sQ$-valued similarity $\al$ is a $\sQ$-valued dissimilarity.
\end{enumerate}
\end{prop}
Since each frame is a commutative and divisible quantale, Proposition \ref{divisible-neg-dissim} guarantees the validity of Proposition \ref{frame-neg-sim-dissim}\ref{frame-neg-sim-dissim:neg-dissim}. In fact, in this case Lemma \ref{neg-BQ-divisible} can be strengthened to the following:
\begin{lem} \label{frame-neg-BQ-DQ}
If $\sQ$ is a frame, then the assignment
\[(b:p\lar q)\mapsto(\neg b:\neg p \rqa\neg q)\]
defines a quantaloid homomorphism $\neg:\KsQ\to\HsQ$.
\end{lem}
\begin{proof}
It is clear that the assignment
\[(b:p\lar q)\mapsto(\neg b:\neg p \rqa\neg q)\]
preserves identities and local suprema. With Lemma \ref{neg-BQ-divisible} in hand, it remains to show that
\[\neg(c\bullet b)\leq\neg c\diamond\neg b=\neg\neg(c\ra q)\wedge\neg b\]
by Equation \eqref{neg-c-diamond-neg-b}; that is,
\[\neg((c\ra q)\ra b)\leq\neg\neg(c\ra q)\wedge\neg b=\neg(\neg(c\ra q)\vee b).\]
This is easy since
\[\neg(c\ra q)\leq(c\ra q)\ra b\quad\text{and}\quad b\leq(c\ra q)\ra b\]
are both obvious.
\end{proof}
Moreover, Proposition \ref{frame-neg-sim-dissim}\ref{frame-neg-sim-dissim:neg-sim} is a direct consequence of Lemma \ref{frame-neg-DQ-BQ} below. Before proceeding to prove this lemma, we point out that the open set $\be(f,g)$ given by Example \ref{dissim-exmp-PCX} is precisely the negation of the open set $\al(f,g)$ given by Example \ref{sim-exmp-PCX} in the frame $\OX$, i.e.,
\[\be(f,g)=\neg\al(f,g)=\al(f,g)\ra\varnothing.\]
So, by applying Proposition \ref{frame-neg-sim-dissim} to the $\OX$-valued similarity $\al$, the $\OX$-valued dissimilarity $\be$ on $\PCX$ is soon obtained.
\begin{lem}\label{frame-neg-DQ-BQ}
If $\sQ$ is a frame, then the assignment
\[(d:p\rqa q)\mapsto(\neg d:\neg p \lar\neg q)\]
defines a quantaloid homomorphism $\neg:\HsQ\to\KsQ$.
\end{lem}
\begin{proof}
First, if $d\in\HsQ(p,q)$, then $d=p\wedge(p\ra d)=q\wedge(q\ra d)$ by Equation \eqref{diagonal-def}. It follows that
\[\neg d=\neg(p\wedge(p\ra d))=(p\ra d)\ra\neg p=(((p\ra d)\ra\neg p)\ra\neg p)\ra\neg p=(\neg d\ra\neg p)\ra\neg p,\]
and similarly $\neg d=(\neg d\ra\neg q)\ra\neg q$. Hence $\neg d\in\BsQ(\neg p,\neg q)=\KsQ(\neg p,\neg q)$ by Equation \eqref{back-diagonal-def}.
Second, since $\neg:\HsQ\to\KsQ$ obviously preserves identities and local suprema, it remains to verify that
\[\neg e\bullet\neg d=\neg(e\diamond d)\]
for all $d:p\rqa q$ and $e:q\rqa r$.
Since frames are divisible, it follows that
\[\HsQ(p,q)=\DsQ(p,q)=\{d\in\sQ\mid d\leq p\wedge q\}\]
for all $p,q\in\sQ$, and the composite of $d:p\rqa q$ and $e:q\rqa r$ is given by
\[e\diamond d=e\wedge(q\ra d)= e\wedge q\wedge(q\ra d)=e\wedge d.\]
Thus we only need to show that
\[(d\leq p\wedge q\ \text{and}\ e\leq q\wedge r)\implies\neg(e\wedge d)=(\neg d\ra\neg q)\ra\neg e\]
because, by definition, $\neg e\bullet\neg d=(\neg d\ra\neg q)\ra\neg e$. On one hand, $d\leq\neg d\ra\neg q$ implies that
\[\neg(e\wedge d)=d\ra\neg e\geq(\neg d\ra\neg q)\ra\neg e.\]
On the other hand,
\[\neg e\wedge(\neg d\ra\neg q)\leq\neg e\quad\text{and}\quad\neg d\wedge(\neg d\ra\neg q)\leq\neg q\leq\neg e\]
implies that
\begin{equation} \label{neg-e-neg-d-leq}
\neg e\vee\neg d\leq(\neg d\ra\neg q)\ra\neg e,
\end{equation}
and consequently
\begin{align*}
\neg(e\wedge d)&=\neg\neg\neg(e\wedge d)\\
&=\neg(\neg\neg e\wedge\neg\neg d) \\
&=\neg\neg(\neg e\vee\neg d)\\
&\leq\neg\neg((\neg d\ra\neg q)\ra\neg e)&(\text{Inequality \eqref{neg-e-neg-d-leq}})\\
&=\neg\neg\neg(e\wedge(\neg d\ra\neg q))\\
&=\neg(e\wedge(\neg d\ra\neg q))\\
&=(\neg d\ra\neg q)\ra\neg e,
\end{align*}
where the second equality holds since
\[\neg\neg(p\wedge q)=\neg\neg p\wedge\neg\neg q\]
for all elements $p,q$ in a frame (see, e.g., \cite[Exercise I.1.11(ii)]{Johnstone1986}).
\end{proof}
\begin{exmp} \label{C3-neg}
The assumptions on $\sQ$ in Propositions \ref{divisible-neg-dissim} and \ref{frame-neg-sim-dissim} are not indispensable. Note that the commutative quantale $C_3$ (see Example \ref{quantale-exmp}\ref{quantale-exmp:C3}) is not integral, and thus not divisible, and it holds that
\begin{align*}
&\neg\bot=\top,\quad\neg k=\neg\top=\bot;\\
&\HC(\bot,q)=\HC(q,\bot)=\HC(k,\top)=\HC(\top,k)=\{\bot\}\quad(q\in C_3),\\
&\HC(\top,\top)=\{\bot,\top\},\quad\HC(k,k)=\{\bot,k\};\\
&\KC(\top,q)=\KC(q,\top)=\KC(k,\bot)=\KC(\bot,k)=\{\top\}\quad (q\in C_3),\\
&\KC(\bot,\bot)=\{\bot,\top\},\quad\KC(k,k)=\{k,\top\}.
\end{align*}
With a direct computation we deduce that $\neg$ yields homomorphisms of quantaloids
\[\neg:\HC\to\KC\quad\text{and}\quad\neg:\KC\to\HC.\]
Therefore, the negation $\neg\al$ of each $C_3$-valued similarity $\al$ is also a $C_3$-valued dissimilarity, and vice versa.
\end{exmp}
\begin{cor} \label{frame-neg-BQCat-DQCat}
If $\sQ$ is a frame, then the assignments
\[(X,\be)\mapsto(X,\neg\be)\quad\text{and}\quad(X,\al)\mapsto(X,\neg\al)\]
define functors $\KsQ\text{-}\Cat\to\HsQ\text{-}\Cat$ and $\HsQ\text{-}\Cat\to\KsQ\text{-}\Cat$, respectively.
\end{cor}
\subsection{When $\sQ$ is a Girard quantale}
Let $m\in\sQ$. We say that
\begin{itemize}
\item $m$ is \emph{cyclic}, if $m\ldd q=q\rdd m$ for all $q\in\sQ$;
\item $m$ is \emph{dualizing}, if $(m\ldd q)\rdd m=q=m\ldd(q\rdd m)$ for all $q\in\sQ$.
\end{itemize}
It is easy to observe the following facts:
\begin{itemize}
\item If $\sQ$ is commutative, then every element of $\sQ$ is cyclic.
\item If $\sQ$ is integral, then a dualizing element of $\sQ$, whenever it exists, has to be the bottom element $\bot$ of $\sQ$.
\end{itemize}
$\sQ$ is said to be a \emph{Girard quantale} \cite{Rosenthal1990,Yetter1990} if it has a cyclic dualizing element.
\begin{exmp} \label{Girard-quantale-exmp}
For the quantales listed in Example \ref{quantale-exmp}:
\begin{enumerate}[label={\rm(\arabic*)}]
\item \label{Girard-quantale-exmp:Lawvere} Lawvere's quantale $[0,\infty]$ is not Girard.
\item \label{Girard-quantale-exmp:frame} A frame is Girard if, and only if, it is a complete Boolean algebra.
\item \label{Girard-quantale-exmp:BL} A complete BL-algebra is Girard if, and only if, it is a complete MV-algebra. In particular, the unit interval $[0,1]$ equipped with a continuous t-norm becomes a Girard quantale if, and only if, it is isomorphic to $[0,1]$ equipped with the {\L}ukasiewicz t-norm.
\item \label{Girard-quantale-exmp:nil-min} The unit interval $[0,1]$ equipped with the nilpotent minimum t-norm is Girard, in which the bottom $0$ is the only cyclic dualizing element.
\item \label{Girard-quantale-exmp:C3} $C_3$ is Girard, in which the unit $k$ is the only cyclic dualizing element (see \cite[Exercise 2.6.1]{Eklund2018}).
\item \label{Girard-quantale-exmp:Rel} The involutive quantale $\Rel(X)$ is Girard, with a cyclic dualizing element given by $X\times X-\id_X$.
\item \label{Girard-quantale-exmp:Sup} The involutive quantale $\Sup[0,1]$ is Girard, with a cyclic dualizing element given by its unit $1_{[0,1]}$, i.e., the identity map on $[0,1]$ (see \cite[Example 2.6.17(a)]{Eklund2018}).
\end{enumerate}
\end{exmp}
In a Girard quantale $\sQ$ with a cyclic dualizing element $m$, following the notation of \cite{Rosenthal1990}, we define the \emph{linear negation} of $q\in\sQ$ as
\begin{equation} \label{linear-negation-def}
q^{\perp}:=m\ldd q=q\rdd m,
\end{equation}
which clearly satisfies
\begin{equation} \label{q-bot-bot=q}
q^{\perp\perp}=q.
\end{equation}
Hence, a Girard quantale may be considered as a table of truth-values in which the law of double negation is satisfied.
\begin{rem} \label{linear-negation-vs-negation}
If a Girard quantale $\sQ$ is integral, then the linear negation coincides with the negation, i.e.,
\begin{equation} \label{integral-bot=neg}
q^{\perp}=\neg q
\end{equation}
for all $q\in\sQ$. However, Equation \eqref{integral-bot=neg} may fail in a Girard quantale whose bottom $\bot$ fails to be a cyclic dualizing element, e.g., the Girard quantales $C_3$, $\Rel(X)$ and $\Sup[0,1]$ listed in Example \ref{Girard-quantale-exmp}.
\end{rem}
\begin{lem} \label{neg-BQ-Girard}
If $\sQ$ is a Girard quantale, then the assignment
\[(b:p\lar q)\mapsto(b^{\perp}:p^{\perp}\rqa q^{\perp})\]
defines a homomorphism of quantaloids $(-)^{\perp}:\KsQ\to\HsQ$.
\end{lem}
\begin{proof}
First, $b^{\perp}\in\HsQ(p^{\perp},q^{\perp})$ if $b\in\KsQ(p,q)$. Since $\sQ$ is Girard,
\begin{align*}
b^{\perp}&=(p\ldd(b\rdd p))^{\perp} & (\text{Equation \eqref{back-diagonal-def}})\\
&=(p^{\perp\perp}\ldd(b^{\perp}\ldd p^{\perp}))^{\perp} & (b\rdd p=b^{\perp}\ldd p^{\perp})\\
&=((b^{\perp}\ldd p^{\perp})\with p^{\perp})^{\perp\perp}\\
&=(b^{\perp}\ldd p^{\perp})\with p^{\perp},
\end{align*}
and similarly $b^{\perp}=q^{\perp}\with(q^{\perp}\rdd b^{\perp})$. Hence, $b^{\perp}:p^{\perp}\rqa q^{\perp}$ is a morphism in $\HsQ$ as $b^{\perp}\leq p^{\perp}\wedge q^{\perp}$ is obvious.
Second, since $(-)^{\perp}$ preserves identities and local suprema, it remains to show that
\[c^{\perp}\diamond b^{\perp}=(c\bullet b)^{\perp}\]
for all $b:p\lar q$ and $c:q\lar r$. Indeed,
\begin{align*}
c^{\perp}\diamond b^{\perp}&=(c^{\perp}\ldd q^{\perp})\with b^{\perp} & (\text{Equation \eqref{diagonal-comp-def}})\\
&=(c\rdd q)\with b^{\perp} & (c\rdd q=c^{\perp}\ldd q^{\perp})\\
&=((c\rdd q)\with b^{\perp})^{\perp\perp}\\
&=(b^{\perp\perp}\ldd(c\rdd q))^{\perp}\\
&=(b\ldd(c\rdd q))^{\perp}\\
&=(c\bullet b)^{\perp}, & (\text{Equation \eqref{back-diagonal-comp-def}})
\end{align*}
which completes the proof.
\end{proof}
\begin{lem} \label{neg-DQ-Girard}
If $\sQ$ is a Girard quantale, then the assignment
\[(d:p\rqa q)\mapsto(d^{\perp}:p^{\perp}\lar q^{\perp})\]
defines a homomorphism of quantaloids $(-)^{\perp}:\HsQ\to\KsQ$.
\end{lem}
\begin{proof}
First, if $d\in\HsQ(p,q)$, then $d\leq p\wedge q$ and $d=(d\ldd p)\with p=q\with(q\rdd d)$ by Equation \eqref{diagonal-def}. It follows that $d^{\perp}\geq p^{\perp}\vee q^{\perp}$ and
\[d^{\perp}=((d\ldd p)\with p)^{\perp}=p^{\perp}\ldd(d\ldd p)=p^{\perp}\ldd((p^{\perp}\ldd(d\ldd p))\rdd p^{\perp})=p^{\perp}\ldd(d^{\perp}\rdd p^{\perp}),\]
and similarly $d^{\perp}=(q^{\perp}\ldd d^{\perp})\rdd q^{\perp}$. Hence $d^{\perp}\in\KsQ(p^{\perp},q^{\perp})$ by Equation \eqref{back-diagonal-def}.
Second, since $(-)^{\perp}:\HsQ\to\KsQ$ obviously preserves identities and local suprema, it remains to check that
\[e^{\perp}\bullet d^{\perp}=(e\diamond d)^{\perp}\]
for all $d:p\rqa q$ and $e:q\rqa r$. Indeed,
\begin{align*}
e^{\perp}\bullet d^{\perp}&=d^{\perp}\ldd(e^{\perp}\rdd q^{\perp})&(\text{Equation \eqref{back-diagonal-comp-def}})\\
&=d^{\perp}\ldd((q\with e^{\perp})^{\perp})\\
&=d^{\perp}\ldd(e^{\perp\perp}\ldd q)\\
&=d^{\perp}\ldd(e\ldd q)\\
&=((e\ldd q)\with d)^{\perp}\\
&=(e\diamond d)^{\perp}, &(\text{Equation \eqref{diagonal-comp-def}})
\end{align*}
which completes the proof.
\end{proof}
The homomorphisms of quantaloids given in Lemmas \ref{neg-BQ-Girard} and \ref{neg-DQ-Girard} are obviously inverse to each other by Equation \eqref{q-bot-bot=q}, and thus they are both isomorphisms between the quantaloids $\HsQ$ and $\KsQ$. Moreover, it is clear that both of them can be extended to isomorphisms of quantaloids between $\DsQ$ and $\BsQ$, and therefore:
\begin{thm} \label{Girard-iso}
If $\sQ$ is a Girard quantale, then there are isomorphisms
\[\DsQ\cong\BsQ\quad\text{and}\quad\HsQ\cong\KsQ\]
of quantaloids, and consequently, the assignment $(X,\al)\mapsto(X,\al^{\perp})$ defines an isomorphism
\[\HsQ\text{-}\Cat\cong\KsQ\text{-}\Cat\]
of categories.
\end{thm}
Note that for each of the involutive Girard quantales listed in Example \ref{Girard-quantale-exmp}, the cyclic dualizing element given there is hermitian. Actually, whenever $\sQ$ is an involutive Girard quantale with a hermitian and cyclic dualizing element, it is easy to verify that the homomorphisms of quantaloids given in Lemmas \ref{neg-BQ-Girard} and \ref{neg-DQ-Girard} both preserve the involution of $\sQ$, and in this case:
\begin{thm} \label{Girard-sim-dissim} \footnote{The authors are grateful to an anonymous referee for helpful remarks on this theorem.}
If $\sQ$ is an involutive Girard quantale with a hermitian and cyclic dualizing element, then $\sQ$-valued similarities and $\sQ$-valued dissimilarities are interdefinable by the aid of linear negation; that is, the linear negation $\al^{\perp}$ of each $\sQ$-valued similarity $\al$ is a $\sQ$-valued dissimilarity, and conversely, the linear negation $\be^{\perp}$ of each $\sQ$-valued dissimilarity $\be$ is a $\sQ$-valued similarity.
\end{thm}
\begin{rem} \label{sim-vs-apart-Boolean}
If $\sB$ is a complete Boolean algebra, from Theorem \ref{Girard-iso} we know that $\sB$-valued similarities and $\sB$-valued dissimilarities are interdefinable by passing to complements. Note that
\[\al(p,q):=p\wedge q\]
for all $p,q\in\sB$ defines a $\sB$-valued similarity $\al$ on $\sB$ itself, but its negation cannot be made into a $\sB$-valued apartness relation on $\sB$ (see Remark \ref{dissim-vs-apart}), because
\[\neg\al(q,q)=\neg q\neq\bot\]
as long as $q\neq\top$. So, even in the Boolean-valued case, similarities and apartness relations are not interdefinable by passing to complements. However, as pointed out to us by an anonymous referee, there is also a natural way to switch between similarities and apartness relations in the Boolean-valued case.
If $(X,E,\gamma)$ is a $\sB$-valued apartness relation, then
\[\al(x,y):=E(x)\wedge E(y)\wedge \neg\gamma(x,y)\]
defines a $\sB$-valued similarity on $X$. Conversely, if $\al$ is a $\sB$-valued similarity on $X$, then $(X,E,\gamma)$ is a $\sB$-valued apartness relation, where $E(x):=\al(x,x)$ and \[\gamma(x,y):=E(x)\wedge E(y)\wedge \neg\al(x,y)\]
for all $x,y\in X$. We note en passe that the principal lower set $\da\!(E(x)\wedge E(y))$ of $\sB$ is itself a complete Boolean algebra and $\al(x,y)$ is the complement of $\gamma(x,y)$ in this Boolean algebra ($\gamma(x,y)$ belongs to the Boolean algebra because $\gamma(x,y)\leq E(x)\wedge E(y)$).
\end{rem}
\begin{exmp} \label{nil-min-neg}
As an immediate consequence of Theorem \ref{Girard-iso}, we may find another example for the non-necessity of the assumptions on $\sQ$ in Propositions \ref{divisible-neg-dissim} and \ref{frame-neg-sim-dissim}. Let $\sQ$ be the unit interval $[0,1]$ equipped with the nilpotent minimum t-norm (see Example \ref{quantale-exmp}\ref{quantale-exmp:nil-min}). Then $\sQ$ is not a divisible quantale, hence not a frame, but the negation operator
\[\neg:\HsQ\to\KsQ\]
is an isomorphism of quantaloids since $\sQ$ is a Girard quantale with the bottom $0$ being the cyclic dualizing element (see Example \ref{Girard-quantale-exmp}\ref{Girard-quantale-exmp:nil-min}).
\end{exmp}
\begin{rem}
Since the quantale $C_3$ is Girard (see Example \ref{Girard-quantale-exmp}\ref{Girard-quantale-exmp:C3}), by applying the linear negation \eqref{linear-negation-def} we are able to switch between $C_3$-valued similarities and $C_3$-valued dissimilarities. It is interesting that for this quantale, as Example \ref{C3-neg} shows, the negation \eqref{negation-commutative-def} also makes sense while considering the interactions between similarities and dissimilarities.
\end{rem}
In Theorem \ref{Girard-iso}, the interdefinability of $\sQ$-valued similarities and $\sQ$-valued dissimilarities follows from the isomorphism
\[\HsQ\cong\KsQ\]
when $\sQ$ is Girard. It is now natural to ask whether $\sQ$ being Girard is essential for establishing the isomorphism $\HsQ\cong\KsQ$. In what follows we are able to provide an affirmative answer for a commutative and integral quantale $\sQ$ (see Corollary \ref{DQ-BQ-Girard-integral}). Actually, we have the following:
\begin{thm} \label{DQ-BQ-Girard}
Let $\sQ$ be a commutative quantale. Then there is an isomorphism
\[\DsQ\cong\BsQ\]
of quantaloids if, and only if, $\sQ$ is a Girard quantale.
\end{thm}
As a preparation, let us investigate properties of the quantale
\[\BsQ(q,q)\]
for a given quantale $\sQ=(\sQ,\with,k)$ and a \emph{cyclic} element $q\in\sQ$. Since $\BsQ(q,q)$ is equipped with the \emph{reverse} order inherited from $\sQ$, in order to eliminate ambiguity we use they symbol ``$\preceq$'' for the order in $\BsQ(q,q)$; that is,
\[b\preceq b'\ \text{in}\ \BsQ(q,q)\iff b'\leq b\ \text{in}\ \sQ.\]
Moreover, we denote by $\lda$, $\rda$ the implications in $\BsQ$, and reserve $\ldd$, $\rdd$ for implications in $\sQ$.
\begin{lem} \label{BQq-cyclic}
If $q\in\sQ$ is a cyclic element, then
\[b'\lda b=q\ldd(b'\rdd b)\quad\text{and}\quad b\rda b'=(b\ldd b')\rdd q\]
for all $b,b'\in\BsQ(q,q)$.
\end{lem}
\begin{proof}
It is clear that $q\ldd(b'\rdd b)\in\BsQ(q,q)$ by Equation \eqref{back-diagonal-def}, and
\begin{align*}
b''\bullet b\preceq b'&\iff b'\leq b''\bullet b=(q\ldd b)\rdd b''&(\text{Equation \eqref{back-diagonal-comp-def}})\\
&\iff (q\ldd b)\with b'\leq b''\\
&\iff q\ldd(((q\ldd b)\with b')\rdd q)\leq b''&(b''=q\ldd(b''\rdd q))\\
&\iff q\ldd(b'\rdd((q\ldd b)\rdd q))\leq b''\\
&\iff q\ldd(b'\rdd b)\leq b''&(b=(q\ldd b)\rdd q)\\
&\iff b''\preceq q\ldd(b'\rdd b)
\end{align*}
for all $b''\in\BsQ(q,q)$. Thus $b'\lda b=q\ldd(b'\rdd b)$. Similarly we obtain $b\rda b'=(b\ldd b')\rdd q$.
\end{proof}
\begin{prop} \label{BQq-Girard}
If $q\in\sQ$ is a cyclic element, then $\BsQ(q,q)$ is a Girard quantale. In particular, if $\sQ$ is a commutative quantale, then $\BsQ(q,q)$ is a Girard quantale for all $q\in\sQ$.
\end{prop}
\begin{proof}
We show that $m_q:=q\ldd q=q\rdd q$ is a cyclic dualizing element of the quantale $\BsQ(q,q)$.
First, $m_q$ is cyclic. For any $b\in\BsQ(q,q)$, the conjunction of
\begin{align*}
b\ldd m_q&=b\ldd(q\rdd q)\\
&=((q\ldd b)\rdd q)\ldd(q\rdd q)&(b=(q\ldd b)\rdd q) \\
&=(q\ldd b)\rdd(q\ldd(q\rdd q))\\
&=(q\ldd b)\rdd q\\
&=b \\
&=q\ldd(b\rdd q) \\
&=((q\ldd q)\rdd q)\ldd(b\rdd q)\\
&=(q\ldd q)\rdd(q\ldd(b\rdd q))\\
&=(q\ldd q)\rdd b &(b=q\ldd(b\rdd q))\\
&=m_q\rdd b
\end{align*}
and Lemma \ref{BQq-cyclic} yields that
\begin{equation} \label{b-imp-mq}
m_q\lda b=q\ldd(m_q\rdd b)=q\ldd b=b\rdd q=(b\ldd m_q)\rdd q=b\rda m_q.
\end{equation}
Second, $m_q$ is dualizing. Since for any $b\in\BsQ(q,q)$,
\begin{align*}
m_q\lda(b\rda m_q)&=q\ldd(b\rdd q)&(\text{Equation \eqref{b-imp-mq}})\\
&=(q\ldd b)\rdd q \\
&=(m_q\lda b)\rda m_q,&(\text{Equation \eqref{b-imp-mq}})
\end{align*}
the conclusion thus follows.
\end{proof}
\begin{rem}
For a cyclic element $q$ of a quantale $\sQ$, it is shown in \cite{Rosenthal1990a} that
\[\sj=((-)\rdd q)\rdd q:\sQ\to\sQ\]
is a \emph{nucleus} \cite{Rosenthal1990} on $\sQ$, and the resulting quotient quantale
\[\sQ_{\sj}=(\sQ_{\sj},\with_{\sj},q\rdd q)\]
is a Girard quantale (see \cite[Theorem 3.1.1]{Rosenthal1990} for the construction of $\sQ_{\sj}$). As we will see below, the Girard quantale $\BsQ(q,q)$ obtained in Proposition \ref{BQq-Girard} is isomorphic to $\sQ_{\sj}$.
Let $\sR=(\sR,\with,k)$ be a Girard quantale with a cyclic dualizing element $m$. Then from $\sR$ we may construct another Girard quantale $\sR^d=(\sR^d,\with^d,m)$, which is isomorphic to $\sR$ with the correspondence $q\mapsto q^{\perp}$ being an isomorphism of quantales:
\begin{itemize}
\item elements of $\sR^d$ are the same as those of $\sR$, and $\sR^d$ is equipped with the \emph{reversed} order of $\sR$;
\item the multiplication on $\sR^d$ is defined by
\begin{equation} \label{Girard-dual-comp}
p\with^d q=(p^{\perp}\with q^{\perp})^{\perp}=q\ldd p^{\perp}=q^{\perp}\rdd p
\end{equation}
for all $p,q\in\sR^d$;
\item the unit of $\sR^d$ is the cyclic dualizing element $m$ of $\sR$, and the unit $k$ of $\sR$ is a cyclic dualizing element of $\sR^d$.
\end{itemize}
Now we show that $\sQ_{\sj}$ is isomorphic to $(\BsQ(q,q))^d$, and hence to $\BsQ(q,q)$. To see this, just note that the underlying sets of $\sQ_{\sj}$ and $\BsQ(q,q)$ are the same, and
\begin{align*}
b\with_{\sj}c&=\sj(b\with c)&\text{(see \cite[Theorem 3.1.1]{Rosenthal1990})}\\
&=((b\with c)\rdd q)\rdd q\\
&=(c\rdd(b\rdd q))\rdd q\\
&=q\ldd(c\rdd(b\rdd q))&(q\ \text{is cyclic})\\
&=c\lda(b\rdd q)&(\text{Lemma \ref{BQq-cyclic}})\\
&=c\lda(m_q\lda b)&(\text{Equation \eqref{b-imp-mq}})\\
&=b\bullet^d c&(\text{Equation \eqref{Girard-dual-comp}})
\end{align*}
for all $b,c\in\sQ_{\sj}$.
\end{rem}
Note that every $q\in\sQ$ satisfies
\[(q\ldd k)\with k=q=k\with(k\rdd q);\]
that is, $q\in\DsQ(k,k)$ for all $q\in\sQ$. Moreover,
\[p\diamond q=(p\ldd k)\with q=p\with(k\rdd q)=p\with q\]
for all $p,q\in\DsQ(k,k)$. Hence, $\sQ$ and $\DsQ(k,k)$ are the same quantales, upon which the proof of Theorem \ref{DQ-BQ-Girard} is obtained:
\begin{proof}[Proof of Theorem \ref{DQ-BQ-Girard}]
The ``if'' part is already obtained in Theorem \ref{Girard-iso}. For the ``only if'' part, note that
\[\sQ=\DsQ(k,k).\]
Hence, the isomorphism $\BsQ\cong\DsQ$ guarantees that $\sQ\cong\BsQ(q,q)$ for some $q\in\sQ$. Since $\sQ$ is commutative, Proposition \ref{BQq-Girard} ensures that $\BsQ(q,q)$ is a Girard quantale, and therefore so is $\sQ$.
\end{proof}
Since $\DsQ\cong\HsQ$ and $\BsQ\cong\KsQ$ when $\sQ$ is integral, the following corollary is an immediate consequence of Theorem \ref{DQ-BQ-Girard}:
\begin{cor} \label{DQ-BQ-Girard-integral}
Let $\sQ$ be a commutative and integral quantale. Then there is an isomorphism
\[\HsQ\cong\KsQ\]
of quantaloids if, and only if, $\sQ$ is a Girard quantale.
\end{cor}
\section*{Acknowledgement}
The first, the second and the fourth named authors acknowledge the support of National Natural Science Foundation of China (No. 11771310, No. 11701396 and No. 11871358).
The authors thank the referees gratefully for their valuable comments and suggestions which help improve the paper significantly.
| {
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