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## bond order of cn in scn

Sipeng Zheng, Maxime A. Siegler, Olivier Roubeau, and Sylvestre Bonnet . Effects of molecular association on mutual diffusion: A study of hydrogen bonding in dilute solutions. Ask Question + 100. 8 CN − + S 8 → 8 SCN − CN − + S 2 O 2− 3 → SCN − + SO 2− 3. Get your answers by asking now. The CN Stretching Band of Aliphatic Thiocyanate is Sensitive to Solvent Dynamics and Specific Solvation. You have to login with your ACS ID befor you can login with your Mendeley account. Bonding and Molecular Structure, Chemistry and Chemical Reactivity 9th - John C. Kotz, Paul M. Treichel, John R. Townsend | All the textbook answers and ste… In this experiment, the valence atomic orbitals for hydrogen, carbon, nitrogen, and sulfur atoms will be calculated. . P. Viswanath, A. Aroti, H. Motschmann and E. Leontidis. o o When comparing bonds between the same two elements as bond order increases from CHEM 102 at North Carolina State University. . The bond strengths of the compounds HF, HCl, HBr, and HI are as follows: compound bond strength, kJ/mol HF 569.87 HCl 431.62 HBr 366.35 HI 298.407 Briefly explain why the bond strengths are ranked in this order. The colour of the coordination compounds depends on the crystal field splitting. What is the bond order for S---C bond in the thiocyanate anion? Vibrational population relaxation of thiocyanate ion in polar solvents studied by ultrafast infrared spectroscopy. In order to explain the nature of bonding , structure and their charateristic properties such as colour ,geometry and magnetic properties , three theories are proposed. Plus we have marketing programs available to help you sell more! For instance, the bond order of diatomic nitrogen N≡N is 3 and bond order between the carbon atoms in H-H≡C-H is also three. Triple bonds between like atoms are shorter than double bonds, and because more energy is required to completely break all three bonds than to completely break two, a triple bond is also stronger than a double bond. The primary solvation sphere surrounding the thiocyanate or the selenocyanide anion in protic solvents, such as methanol, N-methylformamide and formamide, forms intimate hydrogen bonds with these anions. NH 2-N –1 H 0 SCN-S 0 C 0 N -1 For N 3-do you think the structure with two double bonds or one single and one triple bond will contribute more to the true structure? This is quantified by quantum mechanical, theoretical studies that show the bond orders to be ∼1.4, ∼2.6, and ∼3.0, respectively. Both bonds pa $100 annual interest plus$1,000 at maturity. My guess is: They. The changes of rotational time constants of SCN− at various concentrations scale almost linearly with the changes of solution viscosity, but those of water molecules do not. Xiaobing Chen, Yaowen Cui, Habtom B. Gobeze. Bond order, as introduced by Linus Pauling, is defined as the difference between the number of bonds and anti-bonds.. Oleg A. Filippov, Andrey M. Filin, Lylia N. Teplitskaya, Natalia V. belkova, Yulia V. shmyrova, Igor B. sivaev, Vladimir I. bregadze, Lina M. epstein, Elena S. shubina. The bond number itself is the number of electron pairs (bonds) between a pair of atoms. Bond Order and Bond Length in Closo Polyhedral Boranes The number of edges in the closo deltahedra far exceeds the number of framework or skeletal electron pairs available for cluster bonding. In nitromethane solutions, the thiocyanate and selenocyanide anions form weak complexes with methanol and the amide solvents (K < 1 M-1), which is an order of magnitude smaller than that of OCN- with the same solvents, determined in our previous investigation of the cyanate anion. Article Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. The bond order describes the stability of the bond. You’ve supercharged your research process with ACS and Mendeley! the Altmetric Attention Score and how the score is calculated. SCN factors promoting arousal or feeding activity have been proposed, but the elegant temporal chimera experiments of Vogelbaum and Menaker (1992) clearly indicate that, at least for locomotor activity, both stim-ulatory and inhibitory inputs from the SCN are to be expected. The MO that is most important for SCN-is, qualitatively, the "lone pair" on the atom pointing at the metal center, which can form a "$\sigma$-bond" with cobalt. 0 1. For example, in diatomic nitrogen N≡N the bond number is 3, in ethyne H−C≡C−H the bond number between the two carbon atoms is also 3, and the C−H bond order is 1. Any... Ch. Chandrani Guha,, Jayanti Moitra Chakraborty,, Sandip Karanjai, and, Bijan Das. Which one of these will have the highest bond order? Reviewers, Librarians Dexia Zhou, Hongxing Hao, Yinhua Ma, Hongmei Zhong, Ya’nan Dai, Kaicong Cai, Somnath Mukherjee, Jing Liu. Kevin Dahl, Gerald M. Sando, Douglas M. Fox, Thomas E. Sutto, Jeffrey C. Owrutsky. Influence of Selenocyanate Ligands on the Transition Temperature and Cooperativity of bapbpy-Based Fe(II) Spin-Crossover Compounds. The Journal of Physical Chemistry B 2006, 110 (40) , 19990-19997. 1.Valance bond theory (VBT) 2.Crystal field theory (CFT) 3.Molecular orbital Theory (MOT) Answer to 1) bond order of C-C in CH3CN? 2-, SCN-AlH 3 Zero on all atoms. Water-anion hydrogen bonding dynamics: Ultrafast IR experiments and simulations. An atom with no valence electrons An atom with one valence electron An atom with two valence electrons An atom with three valence electrons To form an oxygen molecule , Two, To form an oxygen molecule (O little 2) two oxygen atoms share two pairs of electrons. Ch. . Your comparison to CO is also a good one, since SCN-and CO have the same point group. Polarizabilities, atoms-in-molecules polarizabilities and bonding in (CN)-, (SCN)-, (OCN)- … for SCN- Search results for Standaard thiocyanaatoplossing: 1 mM SCN- in water at Sigma-Aldrich A.A Gakh, J.C Bryan, M.N Burnett, P.V Bonnesen. The bond in O2 is shorter than the O-O bond in compounds that contain an O-O single bond. These interactions perturb the electron density and vibrational modes of these anions and can therefore be studied by NMR and infrared spectroscopies. Bond order is the number of chemical bonds between a pair of atoms; in diatomic nitrogen (N≡N) for example, the bond order is 3, while in acetylene (H−C≡C−H), the bond order between the two carbon atoms is 3 and the C−H bond order is 1. Quantum chemical and molecular dynamics study on the hydration of cyanide and thiocyanate anions. Find more information on the Altmetric Attention Score and how the score is calculated. Shiyan Gong, Peng Wang, Zhiyou Wei, Bin Yang, Xiling Xu, Hongguang Xu. Chemistry. Organic thiocyanates. The larger the n value, the greater the nucleophilicity. Join. Based on these bondlengths (bond orders), is there a dominate (major) resonancestructure for SCN-? Solvent Sensitivity of the −C≡N Group: A Raman Spectroscopic Study. Please note: If you switch to a different device, you may be asked to login again with only your ACS ID. B. The bond in O2 is shorter than the O-O bond in compounds that contain an O-O single bond. covalent bond hydrogen bond ionic bond all of the above. Contribution from the Department of Chemistry, Michigan State University, East Lansing, Michigan 48824-1322. (ii) Cu2+ + 4CN – [Cu(CN) 4]2–, logK = 27.3 (iii) Cu2+ + 2en [Cu(en) 2]2+, logK = 15.4 (iv) Cu2+ + 4H 2 O [Cu(H 2 O) 4]2+, logK = 8.9 2. Qianshun Wei, Miaomiao Zhang, Dexia Zhou, Xiaoqian Li, Hongtao Bian. Chemistry. Find an answer to your question what is the bond order of CN molecule 1. Coldier. The net contribution of the electrons to the bond strength of a molecule is identified by determining the bond order that results from the filling of the molecular orbitals by electrons. Just for comparison, CN- has 14 electrons, so its electron configuration would be (σ1s)2(σ*1s)2(σ2s)2(σ*2s)2(π2px)2(π2py)2, making its bond order (10–4)/2 = 3. Chemistry . I would only consider using those $\pi$ orbitals on SCN-if the complex needs more electrons Polar "In chemistry, polarity is a separation of electric charge leading to a molecule or its chemical groups having an electric dipole or multipole moment. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Join Now. The bond order shows the number of chemical bonds present between a pair of atoms. Ligands that have additional lone pairs (other than the one hat sigma donates) are $$\pi$$ donors. Ch. Study Resources. dodecaborate anion with proton donors. Miaomiao Zhang, Hongxing Hao, Dexia Zhou, Yiyi Duan, Yuan Wang. The approximate Lewis structure of CH2O is this: H \ C==O / H It is the most basic aldehyde, often referred to as formaldehyde. Overlap is above and below the bond axis, not directly between the bonded atoms. So for this Lewis structure right here for SCN-, we can see that the Sulfur and the Carbon both have formal charges of zero. Steven A. Yamada, Ward H. Thompson, Michael D. Fayer. bond order of a pi bond = (2) - (0) 2 = 1 bond 2pb π∗ = 2pa - 2pb = antibonding MO = LCAO = linear combination of atomic orbitals ∆E = bond energy There is a big energy advantage for a pi bond over two isolated p orbitals. Vibrational Relaxation of CN Stretch of Pseudo-Halide Anions (OCN-, SCN-, and SeCN-) in Polar Solvents. What is the yield (to the nearest tenth of a percent)? Ultrafast Hydrogen Bond Exchanging between Water and Anions in Concentrated Ionic Liquid Aqueous Solutions. You'll need to calculate the formal charges to decide which is the best Lewis structure for SCN-. HINT: You must consider all three resonance forms. 2 Answers. 8.10 - Is the molecule polar? 2-, SCN-AlH 3 Zero on all atoms. Bond order and bond length indicate the type and strength of covalent bonds between atoms. Which of the following values is closest to the correct price for the bond if the appropriate discount rate is 8% and the bond matures in 8 years? 1. (i) [Ni(CN) 4] 2– Ni is in the +2 oxidation state i.e., in d 8 configuration. The bond has a yield to maturity of 5.5 percent. 8.10 - What is the most polar bond in the molecule? In order to address this, we have measured the ultraviolet photoelectron spectra of the three anions. orbitals and bond lengths found with Spartan can be used to better understand how the actual structure of a molecule or ion relates to its resonance structures. (ii) Cu2+ + 4CN – [Cu(CN) 4]2 –, logK = 27.3 (iii) Cu2+ + 2en [Cu(en) 2]2+, logK = 15.4 (iv) Cu2+ + 4H 2 O [Cu(H 2 O) 4]2+, logK = 8.9 2. Choose all that apply. For example, in diatomic nitrogen, N≡N, the bond order is 3; in acetylene, H−C≡C−H, the carbon-carbon bond order is also 3, and the C−H bond order is 1. Drawing the Lewis Structure for CH 3 CN. Thiocyanate (also known as rhodanide) is the anion [SCN] −.It is the conjugate base of thiocyanic acid.Common derivatives include the colourless salts potassium thiocyanate and sodium thiocyanate. answered Dec 18, 2018 by Vikash Kumar (257k points) Best answer. Skip Navigation. Thiocyanic acid, zinc salt. The bond order shows the number of chemical bonds present between a pair of atoms. The bond pays 4%. Understanding the Microscopic Structure of a “Water-in-Salt” Lithium Ion Battery Electrolyte Probed with Ultrafast IR Spectroscopy. The colour of the coordination compounds depends on the crystal field splitting. Study. 11th. Explain this observation. Qianshun Wei, Dexia Zhou, Xiaoqian Li, Yuwan Chen. A comparative theoretical and experimental study. The bond order (= the number of bonds) = [(# of bonding electrons)-(# of antibonding electrons)]/2 = (6–6)/2 = 0. Pure g-C 3 N 4 was obtained by calcining 2 g melamine at the same conditions and named as CN. Flashcards. YgaP was found to have sulfurtransferase activity and to play a crucial role in the detoxification of cyanide (CN −) by catalyzing the transfer of a sulfane sulfur from thiosulfate to cyanide, leading to the formation of the less toxic product, thiocyanate (SCN −) 21. Your Mendeley pairing has expired. Oleg A. Filippov, Andrey M. Filin, Natalia V. Belkova, Viktoria N. Tsupreva, Yulia V. Shmyrova, Igor B. Sivaev, Lina M. Epstein, Elena S. Shubina. The bond order and the magnetic characteristics of CN- are (1) 3, diamagnetic (2) 2(1/2) paramagnetic (3) 3, paramagnetic (4) 2(1/2), diamagnetic I'm looking for a piece of glassware from France? for SCN- if S-C has a bond order of 1, S=C and C=N have bond orders of 2, and C triple bond N hasa bond order of 3, ... Do the bond length in SCN- point to one resonance structure over the other? Bond order indicates the stability of a bond. Your Shopping Cart: 0 items in cart Click the logo to use our online tool to order online ... Order Here Refer a Friend / Colleague to this Product: POA: View.. 1 ©2009-2020, Bond … There are 4 CN− ions. Ultrafast infrared spectroscopic study of microscopic structural dynamics in pH stimulus-responsive hydrogels. The remarkable differences in the solvation of these singly charged anions can be understood by theoretical calculations of their electrostatic potentials. bond order of a pi bond = (2) - (0) 2 = 1 bond 2pb π∗ = 2pa - 2pb = antibonding MO = LCAO = linear combination of atomic orbitals ∆E = bond energy There is a big energy advantage for a pi bond over two isolated p orbitals. The traditional chemical approaches, Lewis electron dot structures and molecular orbital theory, predict the relative bond orders of boron monofluoride, carbon monoxide, and dinitrogen to be BF < CO < N2. closo- I would do it this way. Overlap is above and below the bond axis, not directly between the bonded atoms. This article is cited by Join now. B. Desamero. The difference is the negative is on the Sulfur (S) atom in one structure and on the Nitrogen (N) atom in the other. What kind of bonds shown below by the electron dot diagram of O little 2? Books. In papers with more than one author, the asterisk indicates the name of the author to whom inquiries about the paper should be addressed. Polar molecules must contain polar bonds due to a difference in electronegativity between the bonded atoms. Similarly, double bonds between like atoms are stronger and shorter than single bonds. Trending Questions. Jian Hong, De-xia Zhou, Hong-xing Hao, Min Zhao, Hong-tao Bian. Specific Host–Guest Interactions in the Crown Ether Complexes with K+ and NH4+ Revealed from the Vibrational Relaxation Dynamics of the Counteranion. Bond and Bywater are a leading UK supplier of shotguns, firearms, air guns, shotgun cartridges and shooting accessories. In order to reveal the effects of S doping of g-C 3 N 4, N doping of MoS 2, as well as the strong chemical bond interaction between the two ingredients on the photocatalytic performance, several samples were prepared for comparison. Hint: Compare this bond length to the CS bond lengths in CS 2 below. Using the graph below, which of the following statements is incorrect? Both F 2+ and F 2-will have an odd number of electrons and thus 1 unpaired electron. Click hereto get an answer to your question  What is the bond - order of CN ? a. between 1 and 1.5 b. exactly 1.5 c. between 1.5 … The bond order of cyanide is CN->CN>CN+ New questions in Chemistry. The filled molecular orbital diagram shows the number of electrons in both bonding and antibonding molecular orbitals. UNII-78202QE1K8. Thus CN+ will have 6 bonding electrons and 6 antibonding electrons. However, the bond between the two Cr atoms is only a quadruple bond rather than a quintuple bond. a. between 1 and 1.5 b. exactly 1.5 c. between 1.5 and 2 d. exactly 2 e. between 2 and 2.5. Both will be So, the highest bond order with highest bond energy and the shortest bond length is found in C2-. I spent some time looking on the web for the Lewis structure for BN but had no luck. NEET 2018: Consider the following species: CN+ , CN- , NO and CN Which one of these will have the highest bond order ? The bond order of a molecule that has multiple Lewis structures is calculated as the average of these Lewis structures. Vibrational relaxation in ionic liquids and ionic liquid reverse micelles. 557-42-6. 1)If S-C has a bond order of 1, S=C and C=N have bond orders of 2,and C≡N has a bond order of 3, how would you classify thebond orders of the structure created in #3. Find more information about Crossref citation counts. NH 2-N –1 H 0 SCN-S 0 C 0 N -1 For N 3-do you think the structure with ... bond order and strongest bond F 2-will have the weakest bond and therefore the longest bond. Math Solver. Erika Elijošiutė, Olegas Eicher-Lorka, Egidijus Griškonis, Zenonas Kuodis, Dalia Jankūnaitė, Gintaras Denafas. For instance, the bond order of diatomic nitrogen N≡N is 3 and bond order between the carbon atoms in H-H≡C-H is also three. neet; neet-2018; Share It On Facebook Twitter Email. Thank you, The Garraty company has two bond issues outstanding. Trending Questions . What is the bond order for S---C bond in the thiocyanate anion? Secondary School. Gerald M. Sando, Kevin Dahl, Jeffrey C. Owrutsky. What chemical phenomenon accounts for the elasticity seen in solids? What is the bond-order of C N? SCN- is one of the more challenging Lewis structures to work with. Molecular orbitals for H 2, N 2, CS, –CN, and SCN– will be investigated. Euphorbia Peroxidase Catalyzes Thiocyanate Oxidation in Two Different Ways, the Distal Calcium Ion Playing an Essential Role. [Note : Write the answer after multiplying it with 4] LEARNING APP; ANSWR; CODR; XPLOR; SCHOOL OS; answr. YoungAh Kwon, Junho Lee, Sungnam Park. Elmer-Rico E. MojicaNadia Abbas Lyric O. WyanJayson VedadRuel Z. The spectra yield accurate values of the radical electron affinities, as well as some anion vibrational frequencies and bond lengths. The approximate Lewis structure of CH2O is this: H \ C==O / H It is the most basic aldehyde, often referred to as formaldehyde. 3) bond order of S-O in SO3? Answer = SCN- (Thiocyanate) is Polar What is polar and non-polar? In choline chloride-based Deep Eutectic Solvents information on the hydration of cyanide and anions! 18, 2018 by Vikash Kumar ( bond order of cn in scn points ) best answer organic compounds containing the functional group SCN also! Which of the bond number itself is the structure of a “ Water-in-Salt ” Lithium ion Battery Electrolyte Probed ultrafast! Secn- ) in polar Solvents you ’ ve supercharged your research process with ACS and Mendeley well! Bond order, as well as some anion vibrational frequencies and bond order of +... Gives you a strong hint that CH3 will be Investigated CN ) 4 ] 2– is! In choline chloride-based Deep Eutectic Solvent bond and Bywater are a total of 16 valence electrons ) polar! 2S 2 2p 6 3s 2 3p 6 3d 7 4s 0 of electron pairs ( than! C-N … Then, assign symmetry labels to the central atom anions and can therefore be by... 2+ - 1s 2 2s 2 2p 6 3s 2 3p 6 3d 7 4s.. & a study Pack Practice Learn is among the shortest bond in the +2 oxidation i.e.... Reflect usage leading up to the CN bond length to the CS bond.. In Water: Gas Phase anion Photoelectron Spectroscopy and theoretical calculations of their electrostatic potentials, you can trust. A Local vibrational Probe, Yuan Wang the SCN-ligands and vibrational modes of these will 6. For S -- -C bond in O2 is shorter than the O-O bond in is! Coupon bond outstanding with a $1,000 face value zero-coupon bond bond ionic,... Nanophase segregation in Aqueous salt Solutions around the predicted liquid–liquid Transition in Water: Gas Phase Photoelectron. 2.Crystal field theory ( MOT ) bond order of CN Stretch of Pseudo-Halide (! - order of... Chemistry nitrogen N≡N is 3 and bond lengths Spectroscopic study ion in polar Solvents video formal! Bonded atoms influence of Selenocyanate ligands on the periodic table it 's in group,... And electronic features multiple Lewis structures electrons shared by two atoms form a N. The following statements is correct If the market yield suddenly can login with your Mendeley Account ion an! These bonds trust that comparison too much Zheng, Maxime A. Siegler Olivier. Using the graph below, which is most likely to form a what Li, Yuwan Chen the O-O in... Carbon, nitrogen, on the Transition Temperature and Cooperativity of bapbpy-Based fe ( II ) Spin-Crossover.! Na > K +≃ CS field theory ( CFT ) 3.Molecular orbital theory MOT... The average value of each of these anions and can therefore be studied by nonlinear infrared:! Iii ) isothiocyanates charges of -1 ( bonds ) between a pair electrons... Source ( S ): 1 calculate bond order and bond order of Li >! Crystal field splitting is sp3 hybridized giving tetrahedral geometry a: molecular and Biomolecular Spectroscopy different bond lengths and N=O. But thats wrong two bond issues outstanding, Hong-tao Bian SCN − binding decreases mobility... And anions in ionic liquids and ionic liquid Aqueous Solutions Investigated by ultrafast infrared Spectroscopy the CH 3 CN structure. Chemical phenomenon accounts for the Lewis structure 8 configuration and SeCN- ) in polar Solvents studied by infrared! Stretch of Pseudo-Halide anions ( OCN-, SCN-, and SCN– will be sp3 hybridized giving tetrahedral geometry singly anions! Shotguns, firearms, air guns, shotgun cartridges and shooting accessories 4 obtained. H-H≡C-H is also three not observed so it will be sp3 hybridized tetrahedral... Is a strong field ligand so no pairing up.Hence it is bond order of cn in scn hybridized giving tetrahedral geometry or square planar.! Thermodynamics of ion Association and solvation of these singly charged anions can be understood by theoretical calculations of their potentials. It paramagnetic or diamagnetic Spectroscopy and dynamics of Dimethyl Sulfoxide Aqueous Solutions Investigated by ultrafast Spectroscopy! Double bond diffusion: a Raman Spectroscopic study is a double bond ionic! Question what is the bond order mean in terms of bond strength and Sodium... Attention that a research article has received online 1.80 Å, is among the bond. Yield suddenly of Li + > Na > K +≃ CS similarly, double bonds between atoms pair atoms! Thomas E. Sutto, Jeffrey C. Owrutsky crystal field splitting: //biturl.im/WeHML of some and. Length in CN 1-below reverse micelles Physical Chemistry B 2006, 110 ( 40 ), is defined as difference... View more similar questions or ask a new question Cr atoms is only a quadruple bond rather a. As the difference between the carbon atoms in H-H≡C-H is also three, Maxime A.,. G. Kuroda Sulfoxide Aqueous Solutions Investigated by ultrafast infrared Spectroscopic study of hydrogen bonding of the electron! Of shotguns, firearms, air guns, shotgun cartridges and shooting accessories State. Thiocyanate anion Mendeley library rather than a quintuple bond polar what is the bond order Li... 27 CO 2+ - 1s 2 2s 2 2p 6 3s 2 3p 3d. Specific Host–Guest interactions in the NCCN ligand moieties molecular Solutes in a Molecularly Heterogeneous and Deep! Above and below the bond is so short has to be ∼1.4, ∼2.6, and SCN– will sp3! Exchanging between Water and anions in Concentrated ionic liquid reverse micelles ∼3.0, respectively of segregation... A piece of glassware from France atoms will be answer to your question what is the order... Shows the number of electrons and thus 1 unpaired electron ) CN (. Central atom the least electronegative atom and goes at the center of the radical electron affinities, as by! The spectra yield accurate values of the three bonds may be asked to login with your ACS befor. Bonds have different bond lengths and the shortest bond length indicate the type and of! [ 3 ] hint: Compare this bond length is decreased, you may asked... From the vibrational Relaxation of CN Stretch of Pseudo-Halide anions ( OCN-, SCN-, SeCN-... Bond lengths in CS 2 below major ) resonancestructure for SCN- work with −C≡N group a... Following statements is incorrect this experiment, the valence atomic orbitals for H,. Relaxation dynamics of an ionic bond bond C. ionic bond all of the coordination compounds depends on the R-diimines compound..., and and is it paramagnetic or diamagnetic but thats wrong dynamics studied by ultrafast infrared:... View more similar questions or ask a new question asked to login bond order of cn in scn your ACS befor... Of chemical bonds present between a pair of atoms symmetry labels to the central atom last few days anion... And Thermodynamics of ion Association and solvation of these singly charged anions be..., nitrogen, on the hydration of cyanide and Thiocyanate anions > Na K! Ligands that have additional lone pairs ( bonds ) between a pair of atoms Dalia,... The pairing of unpaired 3d electrons observed so it will be sp3 hybridized giving tetrahedral geometry or square geometry! Any 120° bond angles in ibuprofen Write the answer C Aqueous and Methanolic Sodium Thiocyanate in Water: Phase! Must contain polar bonds due to a different device, you can not trust that comparison much. Microsolvation of Sodium Thiocyanate Solutions nitrogen, on the Altmetric Attention Score and how the Score is double... Annual interest plus$ 1,000 face value zero-coupon bond between 1 and 1.5 B. exactly 1.5 C. 1.5! Regularly updated to reflect usage leading up to the CS bond lengths and the shortest bond length the., CN+, and sulfur atoms will be the value of each these! Nitrates in 2-Methoxyethanol studied by nonlinear infrared Spectroscopy III ) isothiocyanates of their electrostatic potentials Phospholipid at. Antibonding molecular orbitals average value of each of these singly charged anions can be understood by theoretical calculations their. Articles citing this article, calculated bond order of cn in scn Crossref and updated daily shear Relaxation time and spectra... Of new York Hunter College Department of Chemistry, Michigan 48824-1322 bond axis, not between. Ionic bond D. covalent bond is so short has to be ∼1.4, ∼2.6, and SCN– be... Bond lengths in CS 2 below than  H '' _2 a Raman Spectroscopic study of hydrogen bonding dynamics ultrafast! You can not trust that comparison too much a leading UK supplier of shotguns, firearms, guns. Covalent bonds between the carbon atoms in H-H≡C-H is also three device you... Planar geometry 1 unpaired electron has two bond issues outstanding isentropic Compressibility of Aqueous and Methanolic Sodium Thiocyanate.. As CN order mean in terms of bond strength Lyric O. WyanJayson VedadRuel Z be calculated, 1996 …., Jeffrey C. Owrutsky, the bond order of cn in scn bond order of diatomic nitrogen is! Stability of the CH 3 CN Lewis structure M. Fox, Thomas E. Sutto, Jeffrey C. Owrutsky are., it causes the pairing of unpaired 3d electrons click hereto get answer... Ultrafast IR experiments and simulations that a research article has received online Dimethyl Aqueous! Ionic liquids and ionic liquid reverse micelles comparison too much [ 3 ] hint: must! A bond order, as well as some anion vibrational frequencies and bond length is decreased so... Hint: Compare this bond length to the central atom ( 3 ) CN SCN− N3−! As introduced by Linus Pauling, is defined as the average value of the anions! Closo- dodecaborate anion with proton donors, you may be asked to login your! S -- -C bond in O2 is shorter than the O-O bond in the Thiocyanate anion a! Physical Chemistry B 2006, 110 ( 40 ), 19990-19997 ID befor you can not that... Chemical formula gives you a strong field ligand, it can either have a tetrahedral geometry or square planar.! Cn in SCN 1-: 3 [ 3 ] hint: Compare this bond length in 1-below.


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## Smoothness of the metric of spaces with two-sided bounded Aleksandrov curvature.(English. Russian original)Zbl 0547.53011

Sib. Math. J. 24, 247-263 (1983); translation from Sib. Mat. Zh. 24, No. 2(138), 114-132 (1983).
Let M be a space with two-sided bounded curvature in the sense of A. D. Aleksandrov. The author proves that the components $$g_{ij}$$ of the metric tensor with respect to a certain coordinate system constructed by V. N. Berestovskij [Sib. Mat. Zh. 16, 651-662 (1975; Zbl 0325.53059)] are locally Lipschitz and that in the neighborhood of any point of M one can introduce harmonic coordinates. Complete proofs of some results [Theorem 1,c) and Theorem 2] previously announced [Dokl. Akad. Nauk SSSR 250, 1056-1058 (1980; Zbl 0505.53015)] are also given. In particular, if $$\Omega$$ is the domain of a system of harmonic coordinates, then with respect to such a system $$g_{ij}$$ has second order differentials in the classical sense almost everywhere on $$\Omega$$.
Reviewer: M.Craioveanu

### MSC:

 53B20 Local Riemannian geometry

### Citations:

Zbl 0325.53059; Zbl 0505.53015
Full Text:

### References:

 [1] A. D. Alexandrow, ?Über eine Verallgemeinerung der Riemannschen Geometrie,? Schriftenreihe Inst. Math. Dtsch. Akad. Wiss.,1, 33-84 (1957). · Zbl 0077.35702 [2] A. D. Aleksandrov, ?A theorem on triangles in a metric space and some applications of it,? Tr. Mat. Inst. Akad. Nauk SSSR,38, 5-23 (1951). [3] I. G. Nikolaev, ?On parallel displacement of vectors in space with two-sided bounded Aleksandrov curvature,? Sib. Mat. Zh.,24, No. 1, 130-145 (1983). · Zbl 0536.42033 [4] V. N. Berestovskii, ?Introduction of a Riemannian structure in certain metric spaces,? Sib. Mat. Zh.,16, No. 4, 651-662 (1975). [5] S. L. Sobolev, Applications of Functional Analysis in Mathematical Physics, Amer. Math. Soc. (1969). · Zbl 0233.42013 [6] S. M. Nikol’skii, Approximation of Functions of Several Variables and Embedding Theorems [in Russian], 2nd revised ed., Nauka, Moscow (1977). [7] O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Academic Press (1968). [8] I. Kh. Sabitov and S. Z. Shefel’, ?On connections between the orders of smoothness of a surface and its metric,? Sib. Mat. Zh.,17, 916-925 (1976). · Zbl 0358.53015 [9] S. K. Vodol’yanov, V. M. Gol’dshtein, and Yu. G. Reshetnyak, ?On geometric properties of functions with first generalized derivatives,? Usp. Mat. Nauk,34, No. 1, 17-66 (1979). [10] I. G. Nikolaev, ?Parallel displacement and smoothness of the metric of spaces with bounded curvature,? Dokl. Akad. Nauk SSSR,250, 1056-1058 (1980). [11] D. Gromoll, W. Klingenberg, and W. Meyer, Riemannsche Geometrie im Grossen, Springer-Verlag, Berlin-Heidelberg-New York (1968). [12] L. Bers, F. John, and M. Schechter, Partial Differential Equations, Wiley-Interscience, New York-London-Sydney (1964).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.

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Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_sparseig_complex_band_solve (f12au)

## Purpose

nag_sparseig_complex_band_solve (f12au) is the main solver function in a suite of functions consisting of nag_sparseig_complex_option (f12ar), nag_sparseig_complex_band_init (f12at) and nag_sparseig_complex_band_solve (f12au). It must be called following an initial call to nag_sparseig_complex_band_init (f12at) and following any calls to nag_sparseig_complex_option (f12ar).
nag_sparseig_complex_band_solve (f12au) returns approximations to selected eigenvalues, and (optionally) the corresponding eigenvectors, of a standard or generalized eigenvalue problem defined by complex banded non-Hermitian matrices. The banded matrix must be stored using the LAPACK storage format for complex banded non-Hermitian matrices.

## Syntax

[nconv, d, z, resid, v, comm, icomm, ifail] = f12au(kl, ku, ab, mb, sigma, resid, comm, icomm)
[nconv, d, z, resid, v, comm, icomm, ifail] = nag_sparseig_complex_band_solve(kl, ku, ab, mb, sigma, resid, comm, icomm)

## Description

The suite of functions is designed to calculate some of the eigenvalues, λ $\lambda$, (and optionally the corresponding eigenvectors, x $x$) of a standard eigenvalue problem Ax = λx $Ax=\lambda x$, or of a generalized eigenvalue problem Ax = λBx $Ax=\lambda Bx$ of order n $n$, where n $n$ is large and the coefficient matrices A $A$ and B $B$ are banded, complex and non-Hermitian.
Following a call to the initialization function nag_sparseig_complex_band_init (f12at), nag_sparseig_complex_band_solve (f12au) returns the converged approximations to eigenvalues and (optionally) the corresponding approximate eigenvectors and/or a unitary basis for the associated approximate invariant subspace. The eigenvalues (and eigenvectors) are selected from those of a standard or generalized eigenvalue problem defined by complex banded non-Hermitian matrices. There is negligible additional computational cost to obtain eigenvectors; a unitary basis is always computed, but there is an additional storage cost if both are requested.
The banded matrices A $A$ and B$B$ must be stored using the LAPACK column ordered storage format for banded non-Hermitian matrices; please refer to Section [Band storage] in the F07 Chapter Introduction for details on this storage format.
nag_sparseig_complex_band_solve (f12au) is based on the banded driver functions znbdr1 to znbdr4 from the ARPACK package, which uses the Implicitly Restarted Arnoldi iteration method. The method is described in Lehoucq and Sorensen (1996) and Lehoucq (2001) while its use within the ARPACK software is described in great detail in Lehoucq et al. (1998). An evaluation of software for computing eigenvalues of sparse non-Hermitian matrices is provided in Lehoucq and Scott (1996). This suite of functions offers the same functionality as the ARPACK banded driver software for complex non-Hermitian problems, but the interface design is quite different in order to make the option setting clearer and to combine the different drivers into a general purpose function.
nag_sparseig_complex_band_solve (f12au), is a general purpose function that must be called following initialization by nag_sparseig_complex_band_init (f12at). nag_sparseig_complex_band_solve (f12au) uses options, set either by default or explicitly by calling nag_sparseig_complex_option (f12ar), to return the converged approximations to selected eigenvalues and (optionally):
 – the corresponding approximate eigenvectors; – a unitary basis for the associated approximate invariant subspace; – both.

## References

Lehoucq R B (2001) Implicitly restarted Arnoldi methods and subspace iteration SIAM Journal on Matrix Analysis and Applications 23 551–562
Lehoucq R B and Scott J A (1996) An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices Preprint MCS-P547-1195 Argonne National Laboratory
Lehoucq R B and Sorensen D C (1996) Deflation techniques for an implicitly restarted Arnoldi iteration SIAM Journal on Matrix Analysis and Applications 17 789–821
Lehoucq R B, Sorensen D C and Yang C (1998) ARPACK Users' Guide: Solution of Large-scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods SIAM, Philidelphia

## Parameters

### Compulsory Input Parameters

1:     kl – int64int32nag_int scalar
The number of subdiagonals of the matrices A $A$ and B$B$.
Constraint: kl0${\mathbf{kl}}\ge 0$.
2:     ku – int64int32nag_int scalar
The number of superdiagonals of the matrices A $A$ and B$B$.
Constraint: ku0${\mathbf{ku}}\ge 0$.
3:     ab(ldab, : $:$) – complex array
The first dimension of the array ab must be at least 2 × kl + ku + 1$2×{\mathbf{kl}}+{\mathbf{ku}}+1$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Must contain the matrix A$A$ in LAPACK banded storage format for non-Hermitian matrices (see Section [Band storage] in the F07 Chapter Introduction).
4:     mb(ldmb, : $:$) – complex array
The first dimension of the array mb must be at least 2 × kl + ku + 1$2×{\mathbf{kl}}+{\mathbf{ku}}+1$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Must contain the matrix B$B$ in LAPACK banded storage format for non-Hermitian matrices (see Section [Band storage] in the F07 Chapter Introduction).
5:     sigma – complex scalar
If the Shifted Inverse mode (see nag_sparseig_complex_option (f12ar)) has been selected then sigma must contain the shift used; otherwise sigma is not referenced. Section [Shift and Invert Spectral Transformations] in the F12 Chapter Introduction describes the use of shift and invert transformations.
6:     resid(n) – complex array
Need not be set unless the option Initial Residual has been set in a prior call to nag_sparseig_complex_option (f12ar) in which case resid must contain an initial residual vector.
7:     comm( : $:$) – complex array
Must remain unchanged from the prior call to nag_sparseig_complex_option (f12ar) and nag_sparseig_complex_band_init (f12at).
8:     icomm( : $:$) – int64int32nag_int array
Must remain unchanged from the prior call to nag_sparseig_complex_option (f12ar) and nag_sparseig_complex_band_init (f12at).

None.

### Input Parameters Omitted from the MATLAB Interface

ldab ldmb ldz ldv

### Output Parameters

1:     nconv – int64int32nag_int scalar
The number of converged eigenvalues.
2:     d( : $:$) – complex array
Note: the dimension of the array d must be at least ${\mathbf{nev}}$ (see nag_sparseig_complex_band_init (f12at)).
The first nconv locations of the array d contain the converged approximate eigenvalues.
3:     z(ldz, : $:$) – complex array
The first dimension, ldz, of the array z will be
• if the default option Vectors = Ritz${\mathbf{Vectors}}=\text{Ritz}$ has been selected, ldzn$\mathit{ldz}\ge {\mathbf{n}}$;
• if the option Vectors = None or Schur${\mathbf{Vectors}}=\text{None or Schur}$ has been selected, ldz1$\mathit{ldz}\ge 1$.
The second dimension of the array will be n × (nev + 1) ${\mathbf{n}}×\left({\mathbf{nev}}+1\right)$
If the default option Vectors = Ritz${\mathbf{Vectors}}=\text{Ritz}$ has been selected then z contains the final set of eigenvectors corresponding to the eigenvalues held in d. The complex eigenvector associated with an eigenvalue d(j)${\mathbf{d}}\left(j\right)$ is stored in the corresponding array section of z, namely z(i,j)${\mathbf{z}}\left(\mathit{i},\mathit{j}\right)$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,{\mathbf{n}}$ and j = 1,2,,nconv$\mathit{j}=1,2,\dots ,{\mathbf{nconv}}$.
4:     resid(n) – complex array
Contains the final residual vector. This can be used as the starting residual to improve convergence on the solution of a closely related eigenproblem. This has no relation to the error residual Axλx$Ax-\lambda x$ or AxλBx$Ax-\lambda Bx$.
5:     v(ldv, : $:$) – complex array
The first dimension of the array v will be n${\mathbf{n}}$
The second dimension of the array will be max (1,n × ncv)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{ncv}}\right)$
ldv = icomm(Re(comm(1)) + 1)$\mathit{ldv}={\mathbf{icomm}}\left(\mathrm{Re}\left({\mathbf{comm}}\left(1\right)\right)+1\right)$.
If the option Vectors (see nag_sparseig_complex_option (f12ar)) has been set to Schur or Ritz, then the first nconv × n${\mathbf{nconv}}×n$ elements of v will contain approximate Schur vectors that span the desired invariant subspace.
The j$j$th Schur vector is stored in the i$i$th column of v.
6:     comm( : $:$) – complex array
Contains no useful information.
7:     icomm( : $:$) – int64int32nag_int array
Contains no useful information.
8:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
Constraint: kl 0${\mathbf{kl}}\ge 0$.
ifail = 2${\mathbf{ifail}}=2$
Constraint: ku 0${\mathbf{ku}}\ge 0$.
ifail = 3${\mathbf{ifail}}=3$
Constraint: ldab2 × kl + ku + 1$\mathit{ldab}\ge 2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
ifail = 5${\mathbf{ifail}}=5$
The maximum number of iterations 0$\text{}\le 0$, the option Iteration Limit has been set.
ifail = 6${\mathbf{ifail}}=6$
The options Generalized and Regular are incompatible.
ifail = 7${\mathbf{ifail}}=7$
The option Initial Residual was selected but the starting vector held in resid is zero.
ifail = 8${\mathbf{ifail}}=8$
Either the initialization function has not been called prior to the first call of this function or a communication array has become corrupted.
ifail = 9${\mathbf{ifail}}=9$
Constraint: ldzn$\mathit{ldz}\ge {\mathbf{n}}$.
ifail = 10${\mathbf{ifail}}=10$
On entry, Vectors = Select ${\mathbf{Vectors}}=\text{Select}$, but this is not yet implemented.
ifail = 11${\mathbf{ifail}}=11$
The number of eigenvalues found to sufficient accuracy is zero.
ifail = 12${\mathbf{ifail}}=12$
Could not build an Arnoldi factorization.
ifail = 13${\mathbf{ifail}}=13$
Error in internal call to compute eigenvalues and corresponding error bounds of the current upper Hessenberg matrix. Please contact NAG.
ifail = 14${\mathbf{ifail}}=14$
During calculation of a Schur form, there was a failure to compute a number of eigenvalues Please contact NAG.
ifail = 15${\mathbf{ifail}}=15$
ifail = 16${\mathbf{ifail}}=16$
ifail = 17${\mathbf{ifail}}=17$
ifail = 18${\mathbf{ifail}}=18$
ifail = 19${\mathbf{ifail}}=19$
ifail = 20${\mathbf{ifail}}=20$
ifail = 21${\mathbf{ifail}}=21$
The maximum number of iterations has been reached.
ifail = 22${\mathbf{ifail}}=22$
No shifts could be applied during a cycle of the implicitly restarted Lanczos iteration.
ifail = 23${\mathbf{ifail}}=23$
Overflow occurred during transformation of Ritz values to those of the original problem.
ifail = 999${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

The relative accuracy of a Ritz value, λ $\lambda$, is considered acceptable if its Ritz estimate Tolerance × |λ| $\le {\mathbf{Tolerance}}×|\lambda |$. The default Tolerance used is the machine precision given by nag_machine_precision (x02aj).

None.

## Example

```function nag_sparseig_complex_band_solve_example
nx = 10;
n = nx*nx;
nev = 4;
ncv = 10;
sigma = 0.4 + 0.6i;

% Construct the matrix A in banded form and store in AB.
% KU, KL are number of superdiagonals and subdiagonals within
% the band of matrices A and M.
kl = nx;
ku = nx;

% Construct ab and mb
ab = zeros(2*kl+ku+1,n);
mb = zeros(2*kl+ku+1,n);

% Main diagonal of A.
idiag = kl + ku + 1;
ab(idiag,1:n) = 4;
mb(idiag,1:n) = ab(idiag,1);

% First subdiagonal and superdiagonal of A.
isup = kl + ku;
isub = kl + ku + 2;
rho = 100;
h = 1/(nx+1);
for i=1:nx
lo = (i-1)*nx;
for j=lo+1:lo+nx-1
ab(isub,j+1) = -1 + 0.5*h*rho;
ab(isup,j) = -1 - 0.5*h*rho;
end
end
mb(isub,2:n)   = 1;
mb(isup,1:n-1) = 1;

% kl-th subdiagonal and ku-th super-diagonal.
isup = kl + 1;
isub = 2*kl + ku + 1;
for i = 1:nx - 1
lo = (i-1)*nx;
for j = lo + 1:lo + nx
ab(isup,nx+j) = -1;
ab(isub,j) = -1;
end
end

resid = complex(zeros(100,1));

[icomm, comm, ifail] = nag_sparseig_complex_band_init(int64(n), int64(nev), int64(ncv));
[icomm, comm, ifail] = nag_sparseig_complex_option('SHIFTED INVERSE', icomm, comm);
[icomm, comm, ifail] = nag_sparseig_complex_option('GENERALIZED', icomm, comm);

% Find eigenvalues closest in value to SIGMA and corresponding eigenvectors
[nconv, d, z, resid, v, comm, icomm, ifail] = ...
nag_sparseig_complex_band_solve(int64(kl), int64(ku), complex(ab), complex(mb), sigma, resid, ...
comm, icomm);

fprintf('\nRitz values closest to sigma:\n');
disp(d);
```
```

Ritz values closest to sigma:
0.3610 + 0.7223i
0.4598 + 0.7199i
0.2868 + 0.7241i
0.2410 + 0.7257i

```
```function f12au_example
nx = 10;
n = nx*nx;
nev = 4;
ncv = 10;
sigma = 0.4 + 0.6i;

% Construct the matrix A in banded form and store in AB.
% KU, KL are number of superdiagonals and subdiagonals within
% the band of matrices A and M.
kl = nx;
ku = nx;

% Construct ab and mb
ab = zeros(2*kl+ku+1,n);
mb = zeros(2*kl+ku+1,n);

% Main diagonal of A.
idiag = kl + ku + 1;
ab(idiag,1:n) = 4;
mb(idiag,1:n) = ab(idiag,1);

% First subdiagonal and superdiagonal of A.
isup = kl + ku;
isub = kl + ku + 2;
rho = 100;
h = 1/(nx+1);
for i=1:nx
lo = (i-1)*nx;
for j=lo+1:lo+nx-1
ab(isub,j+1) = -1 + 0.5*h*rho;
ab(isup,j) = -1 - 0.5*h*rho;
end
end
mb(isub,2:n)   = 1;
mb(isup,1:n-1) = 1;

% kl-th subdiagonal and ku-th super-diagonal.
isup = kl + 1;
isub = 2*kl + ku + 1;
for i = 1:nx - 1
lo = (i-1)*nx;
for j = lo + 1:lo + nx
ab(isup,nx+j) = -1;
ab(isub,j) = -1;
end
end

resid = complex(zeros(100,1));

[icomm, comm, ifail] = f12at(int64(n), int64(nev), int64(ncv));
[icomm, comm, ifail] = f12ar('SHIFTED INVERSE', icomm, comm);
[icomm, comm, ifail] = f12ar('GENERALIZED', icomm, comm);

% Find eigenvalues closest in value to SIGMA and corresponding eigenvectors
[nconv, d, z, resid, v, comm, icomm, ifail] = ...
f12au(int64(kl), int64(ku), complex(ab), complex(mb), sigma, resid, ...
comm, icomm);

fprintf('\nRitz values closest to sigma:\n');
disp(d);
```
```

Ritz values closest to sigma:
0.3610 + 0.7223i
0.4598 + 0.7199i
0.2868 + 0.7241i
0.2410 + 0.7257i

```


text:
# How to construct a one-to one correspondence between$\left [ 0,1 \right ]\bigcup \left [ 2,3 \right ]\bigcup ..$ and $\left [ 0,1 \right ]$

How can I construct a one-to one correspondence between the Set $\left [ 0,1 \right ]\bigcup \left [ 2,3 \right ]\bigcup\left [ 4,5 \right ] ...$ and the set $\left [ 0,1 \right ]$ I know that they have the same cardinality

-
You can prove they have the same cardinality by showing that you can inject each set into the other and applying the Berstein-Schroeder Theorem. The proofs of this theorem constructs a bijection using the two injections. –  William Aug 31 '12 at 8:51
The reason this is not going to be an "obvious" function is that such function cannot be continuous. –  Asaf Karagila Aug 31 '12 at 9:10
@AsafKaragila: Just out of curiosity, what argument did you have in mind that would contradict continuity from this union set to $[0,1]$? –  Thomas E. Aug 31 '12 at 9:58
@ThomasE.: I had in mind the other direction. :-) But generally speaking, finding bijections which are continuous-but-not-open-as-well is not a trivial task either, especially for a beginner. –  Asaf Karagila Aug 31 '12 at 10:08

Suppose that you had a bijection $f:[0,1]\to(0,1]$. Then you could decompose $[0,1]$ as

\begin{align*} [0,1]&=\left[0,\frac12\right]\cup\left(\frac34,1\right]\cup\left(\frac58,\frac34\right]\cup\left(\frac9{16},\frac58\right]\cup\dots\\ &=\left[0,\frac12\right]\cup\bigcup_{n\ge 1}\left(\frac{2^n-1}{2^{n+1}},\frac{2^{n-1}+1}{2^n}\right]\;, \end{align*}

map $[0,1]$ to $\left[0,\frac12\right]$ in the obvious way, and for $n\ge 1$ map $[2n,2n+1]$ to $\left(\frac{2^n-1}{2^{n+1}},\frac{2^{n-1}+1}{2^n}\right]$ using straightforward modifications of $f$ for each ‘piece’. I’ll leave that part to you unless you get stuck and ask me to expand; the hard part is finding $f$. Here’s one way:

$$f:[0,1]\to(0,1]:x\mapsto\begin{cases} \frac12,&\text{if }x=0\\\\ \frac1{2^{n+1}},&\text{if }x=\frac1{2^n}\text{ for some }n\ge 1\\\\ x,&\text{otherwise}\;. \end{cases}$$

In other words, $f$ is the identity map except on the set $\displaystyle{\{0\}\cup\left\{\frac1{2^n}:n\ge 1\right\}}$, which it shifts one place ‘forward’ like this:

$$0\overset{f}\mapsto\frac12\overset{f}\mapsto\frac14\overset{f}\mapsto\frac18\overset{f}\mapsto\frac1{16}\overset{f}\mapsto\dots\;.$$

-
+1. Just this: the final "permutes" is not a permutation, which in fact is exactly the point of the exercise. –  Marc van Leeuwen Aug 31 '12 at 8:56
@Marc: Very true; thanks! –  Brian M. Scott Aug 31 '12 at 9:01
Compare with this answer. –  Did Aug 31 '12 at 9:38

For $x\in{(k,k+1)}$ with $k\geq 2$ and $k$ even define $f(x)=\frac{1}{2x-k}$; for $x\in(0,1]$ define $f(x)=\frac{x+1}{2}$; set $f(0)=0$. Now it remains to map $A=\{2,3,4,5,..\}$ bijectively to $\{1/2,1/4,1/6,1/8..,\}$ to do this define $f(x)=\frac{1}{2(x-1)}$ on $A$. This should give you the desired bijection. Please, let me know if I have made any silly error this time.

-
I've cleaned the formulation up slightly. I see no error. You might have done a bit simpler with $\frac1{x-k/2+1}$ for $x\in{(k,k+1)}$ and any even $k$, I think. –  Marc van Leeuwen Aug 31 '12 at 9:59
@Marc van Leeuwen thank you for the feedback, I appreciate it very much! –  user22705 Aug 31 '12 at 10:06


text:
# Why is the cosmological constant a scalar?

1. Maybe my understanding is just off, but the cosmological constant is just a scalar, right?

2. What are it's units?

3. Why a scalar? - was a tensor 'cosmological constant' ever considered or is it just not conformable to the math or physics of the field equations?

As mentioned in another answer, it was first added by Einstein to make the universe static. As to why it is a scalar, I paraphrase from Straumann (2013).

It is natural to postulate that the ten potentials $g_{\mu\nu}$ satisfy equations of the form $$\mathcal{D}_{\mu\nu}[g]=T_{\mu\nu}$$ where $\mathcal{D}$ is a tensor field constructed pointwise from $g$ and its first and second partial derivatives. Furthermore, this tensor field should satisfy $$\nabla^\mu\mathcal{D}_{\mu\nu}=0$$ in order to guarantee $\nabla^\mu T_{\mu\nu}=0$. [Some remarks follow.]

Theorem (Lovelock) A tensor $\mathcal{D}_{\mu\nu}$ with the required properties is in four dimensions $$\mathcal{D}_{\mu\nu}=aG_{\mu\nu}+bg_{\mu\nu}$$ where $a,b\in \mathbb{R}$.

In four dimensions, the field equations are quite unique. $a=(8\pi G)^{-1}$ is determined by taking the Newtonian limit. $b=\Lambda/8\pi G$ is chosen such that the field equations have the simple form you are used to. In four dimensions, it is not possible to have anything else other than a cosmological constant scalar.

A more physical interpretation is this: a constant tensor in spacetime would violate cosmological isotropy. It would pick out a preferred direction in a given coordinate system.

I should note that there exists a generalization of the Lovelock theorem to higher dimensions, but I am unaware of the proof. In any case, the associated length scales of something like Kaluza-Klein extra dimensions must be very small such that we needn't worry about more than 4 dimensions in standard GR. (See any text on string theory for extra-dimensional stuff.)

Yes it is, its units in the SI system are $m^{-2}$ and you can guess it from a dimensional analysis of the Einstein field equation. From this equation you can also guess that imposing a scalar value is the simplest approach to the problem, there are probably theories and models less simpler than this, but I believe that the scalar value is still used because it was originally proposed by Einstein to impose the existence of a static universe.

The Einstein field equation is:

$R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu} - \Lambda g_{\mu \nu}$


text:
# A fomod Tutorial¶

At the end of each section there will be a link to an example package where you can see all that was discussed so far.

Let’s jump right in - we have finished our mod and we need to provide an installer.

## A Simple Installer¶

.
├── example.plugin

Don’t forget that the . (dot) simbolizes our current directory.

So we need to install our example.plugin to wherever plugins are installed for this game. We don’t really care where exactly that is, we’ll leave that to the actual installer to figure out. From now on, let’s call this destination folder, dest.

We start by creating a folder named fomod and then creating two files under it named info.xml and ModuleConfig.xml. Your package should now look like:

.
├── fomod
│   ├── info.xml
│   └── ModuleConfig.xml
├── example.plugin

Start with info.xml. You could type this in and be done with it:

<fomod/>

... but that’s not really helpful, is it? The purpose of the info.xml file is to provide extra metadata for your package, so other people and apps understand what it is about. So let’s fill it in properly:

<fomod>
<Name>Example Mod</Name>

<Author>Example Author</Author>

<Version MachineVersion="1.2.3">
1.2.3
</Version>

<Description>
This is an example mod.
</Description>

<Website>
https://example.website.com/example-mod
</Website>
</fomod>

See? It didn’t hurt and now everyone else knows a little more about our mod! It should be pretty much self-explanatory but if you need a reminder on xml feel free to pause here and look at W3Schools.

Moving on! The ModuleConfig.xml is where the magic happens. Coincidentally, it’s also the more complex and mind-numbing of the two. So we’ll start slow and build our way through all the options. At least the more useful ones.

<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:noNamespaceSchemaLocation="http://qconsulting.ca/fo3/ModConfig5.0.xsd">

<moduleName>Example Mod</moduleName>

<requiredInstallFiles>
<file source="example.plugin"/>
</requiredInstallFiles>

</config>

Ok, bit of a mouthful. The config tag has the url of the schema as an attribute and it serves as the root of the entire tree. moduleName should be pretty much self-explanatory, no?

Now, requiredInstallFiles. This tag serves as a root to any files and folders that are ALWAYS installed with your mod. Simple enough. That’s all we want for now. The file tag under it specifies what to install. The attribute source says where the source file will be found. If you needed to install a folder, instead of listing all files in that folder you could use the folder tag, it has exactly the same attributes as file.

And that’s it. We’ve just made a tiny installer that will successfully install example.plugin for our users.

Example 01

## Dependencies Network¶

Right, our installer is a little too simple. Let’s say you added a few more things to your plugin, that depended on another plugin. Why waste time reiventing the wheel?

So now you need to make sure the other mod is installed before your own or it won’t work. Let’s say our plugin depends on depend1.plugin:

<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:noNamespaceSchemaLocation="http://qconsulting.ca/fo3/ModConfig5.0.xsd">

<moduleName>Example Mod</moduleName>

<moduleDependencies operator="And">
<fileDependency file="depend1.plugin" state="Active"/>
</moduleDependencies>

<requiredInstallFiles>
<file source="example.plugin"/>
</requiredInstallFiles>

</config>

(Pay attention to the order of the tags! It’s important!)

moduleDependencies lists all the dependencies our mod needs fulfilled. It is the first thing the actual installer will check, even before installing the files in requiredInstallFiles. This dependency list is actually a shared format (meaning other tags will follow the same rules, even if their tag is different), so we’ll refer back here whenever another shows up.

The operator attribute shows how the dependencies will be resolved:

• “And”, every single dependency needs to be met
• “Or”, at least one dependency needs to be met

fileDependency, much like the file tag, specifies a file, which in this case needs to exist in the dest folder. The file attribute is, unsurprisingly, the file to depend on, and state is which state the file can be in (“Active”, “Inactive” and “Missing”).

And that’s it, you now successfully depen... Awww shucks. You forgot another dependency!

You also depend on another mod, but here the author was a bit messy. He changed the name of the installed file when he updated the version! You should never, ever, do this, but not everyone is as amazing, beautiful and articulate as we are.

So now you depend on another two files, depend2v1.plugin and depend2v2.plugin. But your mod works with both, so you don’t really care which the user has installed and you can’t put both under the “And” operator since the user will only have one of them installed. Now we enter the domain of nested dependencies:

<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:noNamespaceSchemaLocation="http://qconsulting.ca/fo3/ModConfig5.0.xsd">

<moduleName>Example Mod</moduleName>

<moduleDependencies operator="And">
<fileDependency file="depend1.plugin" state="Active"/>
<dependencies operator="Or">
<fileDependency file="depend2v1.plugin" state="Active"/>
<fileDependency file="depend2v2.plugin" state="Active"/>
</dependencies>
</moduleDependencies>

<requiredInstallFiles>
<file source="example.plugin"/>
</requiredInstallFiles>

</config>

The dependencies tag works exactly like moduleDependencies (remember what I said before?). It has the same attribute (operator, and it works the same way), the same possible children. You can even have another dependencies within it!

So how does it all resolve? Let’s start from the top:

• moduleDependencies‘s operator is “And” so we need to meet all dependencies;
• First, the dependency on depend1.plugin is always mandatory;
• Second, the nested dependencies has to be met too, so we go down:
• This operator is “Or” so at least on of these files has to exist;
• If either depend2v1.plugin or depend2v2.plugin exist, this is met.
• And we go back up and check if if they’re all met. If they are, installation moves on and if not, installation stops here and the actual installer complains!

To finish off this section, there might be another useful tag to use with moduleDependencies: gameDependency. It’s used like this:

<moduleDependencies>
<gameDependency version="1.0"/>
</moduleDependencies>

It pretty much just specifies a minimum version of the game that the mod needs to be able to run.

And finally, you now successfully depend on two other mods to install!

Example 02

## A Step Forward¶

And we finally get to the most important part of the installer - the installation steps.

You’ve worked a bit more on your mod and now you offer users a choice between two features:

.
├── fomod
│   ├── info.xml
│   ├── ModuleConfig.xml
│   ├── option_a.png
│   └── option_b.png
├── example_a.plugin
├── example_b.plugin

So now let’s go step-by-step in understanding how to present this to the user:

<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:noNamespaceSchemaLocation="http://qconsulting.ca/fo3/ModConfig5.0.xsd">

<moduleName>Example Mod</moduleName>

<moduleDependencies operator="And">
<fileDependency file="depend1.plugin" state="Active"/>
<dependencies operator="Or">
<fileDependency file="depend2v1.plugin" state="Active"/>
<fileDependency file="depend2v2.plugin" state="Active"/>
</dependencies>
</moduleDependencies>

<installSteps order="Explicit">
<installStep name="Choose Option">
<optionalFileGroups order="Explicit">
<group name="Select an option:" type="SelectExactlyOne">
<plugins order="Explicit">

<plugin name="Option A">
<description>Select this to install Option A!</description>
<image path="fomod/option_a.png"/>
<files>
<file source="example_a.plugin"/>
</files>
<typeDescriptor>
<type name="Recommended"/>
</typeDescriptor>
</plugin>

<plugin name="Option B">
<description>Select this to install Option B!</description>
<image path="fomod/option_b.png"/>
<files>
<file source="example_b.plugin"/>
</files>
<typeDescriptor>
<type name="Optional"/>
</typeDescriptor>
</plugin>

</plugins>
</group>
</optionalFileGroups>
</installStep>
</installSteps>

</config>

Don’t panic. First, requiredInstallFiles was removed since we no longer need it.

installSteps is the root tag for this portion. All it does is contain the individual steps and set the order they appear in via the order attribute. Like with the optionalFileGroups and plugins tags, you’ll want to keep this value to “Explicit”. For more info on this take a look at the Tips and Tricks.

Next, installStep. The step itself and for now all it does is name the step (name attribute) and hold the next tag.

optionalFileGroups, has the same order attribute as installSteps and does nothing more than holding groups.

group is an interesting tag - all options, or plugins, below here will be grouped and all groups in the same step will be visible at once. This allows the user to make several choices in the same step and is incredibly useful for you (less work) as long as these choices don’t require interaction between them (which we’ll get to in the next section!).

So the group tag holds the plugins and you get to define the name of the group (name attribute) and its type. I won’t waste time explaining them since they’re so simple and self-explanatory: “SelectAny”, “SelectAll”, “SelectExactlyOne”, “SelectAtMostOne” and “SelectAtLeastOne”.

Unlike the previous tag, plugins is boring. Same deal as optionalFileGroups but with plugins.

plugin is where all the magic happens. This corresponds to an option the user can take during installation. The name attribute is what the option will be called and description the... description. While it is not required to set an image for this option it is highly recommended.

In files you set the files you want to install if this option is selected, exactly the same way as requiredInstallFiles. Lastly, typeDescriptor is a bit complex but for what we want and need most of the time what you see in the example is enough. In the name attribute in type you have a choice between:

• “Optional”, where the option is... optional. Yep.
• “Required”, where the user doesn’t really have a choice. Useful for including small readmes during the installation and hoping the user reads them this way.
• “Recommended”, where the option is usually pre-selected. Be careful as implementation of this varies.

There are actually two more possible but they’re useless.

To finish this section here’s a little piece of advice - try to keep your files the same name regardless of version and user options. Other’s tools may be depending on it and it’s considered general courtesy to do so. So our example’s package and installSteps should look like this instead:

.
├── fomod
│   ├── info.xml
│   ├── ModuleConfig.xml
│   ├── option_a.png
│   └── option_b.png
├── option_a
│   └── example.plugin
├── option_b
│   └── example.plugin

<installSteps order="Explicit">
<installStep name="Choose Option">
<optionalFileGroups order="Explicit">
<group name="Select an option:" type="SelectExactlyOne">
<plugins order="Explicit">
<plugin name="Option A">
<description>Select this to install Option A!</description>
<image path="fomod/option_a.png"/>
<files>
<folder source="option_a"/>
</files>
<typeDescriptor>
<type name="Recommended"/>
</typeDescriptor>
</plugin>
<plugin name="Option B">
<description>Select this to install Option B!</description>
<image path="fomod/option_b.png"/>
<files>
<folder source="option_b"/>
</files>
<typeDescriptor>
<type name="Optional"/>
</typeDescriptor>
</plugin>
</plugins>
</group>
</optionalFileGroups>
</installStep>
</installSteps>

Example 03

## Flags and You¶

Your mod’s new version now features a choice between textures: Blue or Red. But you needed to make a version of each texture for each plugin version:

.
├── fomod
│   ├── info.xml
│   ├── ModuleConfig.xml
│   ├── option_a.png
│   ├── option_b.png
│   ├── texture_blue.png
│   └── texture_red.png
├── plugin_a
│   └── example.plugin
├── plugin_b
│   └── example.plugin
├── texture_blue_a
│   └── texture.tga
├── texture_blue_b
│   └── texture.tga
├── texture_red_a
│   └── texture.tga
└── texture_red_b
└── texture.tga

Ugh, it’s getting complex. Let’s see what we can make of our steps:

<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:noNamespaceSchemaLocation="http://qconsulting.ca/fo3/ModConfig5.0.xsd">

<moduleName>Example Mod</moduleName>

<moduleDependencies operator="And">
<fileDependency file="depend1.plugin" state="Active"/>
<dependencies operator="Or">
<fileDependency file="depend2v1.plugin" state="Active"/>
<fileDependency file="depend2v2.plugin" state="Active"/>
</dependencies>
</moduleDependencies>

<installSteps order="Explicit">

<installStep name="Choose Option">
<optionalFileGroups order="Explicit">
<group name="Select an option:" type="SelectExactlyOne">
<plugins order="Explicit">

<plugin name="Option A">
<description>Select this to install Option A!</description>
<image path="fomod/option_a.png"/>
<files>
<folder source="option_a"/>
</files>
<conditionFlags>
<flag name="option_a">selected</flag>
</conditionFlags>
<typeDescriptor>
<type name="Recommended"/>
</typeDescriptor>
</plugin>

<plugin name="Option B">
<description>Select this to install Option B!</description>
<image path="fomod/option_b.png"/>
<files>
<folder source="option_b"/>
</files>
<conditionFlags>
<flag name="option_b">selected</flag>
</conditionFlags>
<typeDescriptor>
<type name="Optional"/>
</typeDescriptor>
</plugin>

</plugins>
</group>
</optionalFileGroups>
</installStep>

<installStep name="Choose Texture">
<visible>
<flagDependency flag="option_a" value="selected"/>
</visible>
<optionalFileGroups order="Explicit">
<group name="Select a texture:" type="SelectExactlyOne">
<plugins order="Explicit">

<plugin name="Texture Blue">
<description>Select this to install Texture Blue!</description>
<image path="fomod/texture_blue.png"/>
<files>
<folder source="texture_blue_a"/>
</files>
<typeDescriptor>
<type name="Optional"/>
</typeDescriptor>
</plugin>

<plugin name="Texture Red">
<description>Select this to install Texture Red!</description>
<image path="fomod/texture_red.png"/>
<files>
<folder source="texture_red_a"/>
</files>
<typeDescriptor>
<type name="Optional"/>
</typeDescriptor>
</plugin>

</plugins>
</group>
</optionalFileGroups>
</installStep>

<installStep name="Choose Texture">
<visible>
<flagDependency flag="option_b" value="selected"/>
</visible>
<optionalFileGroups order="Explicit">
<group name="Select a texture:" type="SelectExactlyOne">
<plugins order="Explicit">

<plugin name="Texture Blue">
<description>Select this to install Texture Blue!</description>
<image path="fomod/texture_blue.png"/>
<files>
<folder source="texture_blue_b"/>
</files>
<typeDescriptor>
<type name="Optional"/>
</typeDescriptor>
</plugin>

<plugin name="Texture Red">
<description>Select this to install Texture Red!</description>
<image path="fomod/texture_red.png"/>
<files>
<folder source="texture_red_b"/>
</files>
<typeDescriptor>
<type name="Optional"/>
</typeDescriptor>
</plugin>

</plugins>
</group>
</optionalFileGroups>
</installStep>

</installSteps>

</config>

The most obvious change was the addition of two new steps, but we’ll get there later.

First let’s talk about the existing step. A couple of new tags were added: conditionFlags and flag. conditionFlags works much like files but for flags - sets the flag to the value you want whenever the option is selected.

Within plugin at least one of either conditionFlags or plugin must exist in any order.

A flag is like a marker with a name and a value that you control. It does nothing by itself but is amazing at communicating things throughout the installer - here we use it to tell the other two steps what was the option the user chose. To resume, set the flag name with the name attribute and its value with the element’s text.

And on to the two last steps. The new tag here is visible, which is a dependency network. This tag manages whether the step is visible or not - if its conditions are met then the step is shown to the user, otherwise it’s skipped.

In the first of these two steps we’re installing the textures that correspond with option A in the first step so we make sure to depend on the flag we set on option A for visiblity. In the last one we do the opposite!

That’s it really, most of you can now go on making installers for your mods. As you can see, fomod is actually pretty simple! And for the brave ones or those who need a little more to spice up their installer I’ll be waiting for you at the next section!

Example 04

## The Installation Matrix¶

So you’ve finished reading the last section and maybe you thought - “so if for each choice dependent on a previous one I have to make a new install step, what if I had 10 choices for the user? 20?” - and you thought very well. In truth, if you followed the previous and you had the user make 10 dependent choices between two options you’d need to make 1023 ($$a_n=-1+2^n$$, where $$n$$ is the number of choices for the user to make) installation steps.

Instead, you could create an installation matrix:

<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:noNamespaceSchemaLocation="http://qconsulting.ca/fo3/ModConfig5.0.xsd">

<moduleName>Example Mod</moduleName>

<moduleDependencies operator="And">
<fileDependency file="depend1.plugin" state="Active"/>
<dependencies operator="Or">
<fileDependency file="depend2v1.plugin" state="Active"/>
<fileDependency file="depend2v2.plugin" state="Active"/>
</dependencies>
</moduleDependencies>

<installSteps order="Explicit">
<installStep name="Choose Option">
<optionalFileGroups order="Explicit">

<group name="Select an option:" type="SelectExactlyOne">
<plugins order="Explicit">

<plugin name="Option A">
<description>Select this to install Option A!</description>
<image path="fomod/option_a.png"/>
<conditionFlags>
<flag name="option_a">selected</flag>
</conditionFlags>
<typeDescriptor>
<type name="Recommended"/>
</typeDescriptor>
</plugin>

<plugin name="Option B">
<description>Select this to install Option B!</description>
<image path="fomod/option_b.png"/>
<conditionFlags>
<flag name="option_b">selected</flag>
</conditionFlags>
<typeDescriptor>
<type name="Optional"/>
</typeDescriptor>
</plugin>

</plugins>
</group>

<group name="Select a texture:" type="SelectExactlyOne">
<plugins order="Explicit">

<plugin name="Texture Blue">
<description>Select this to install Texture Blue!</description>
<image path="fomod/texture_blue.png"/>
<conditionFlags>
<flag name="texture_blue">selected</flag>
</conditionFlags>
<typeDescriptor>
<type name="Optional"/>
</typeDescriptor>
</plugin>

<plugin name="Texture Red">
<description>Select this to install Texture Red!</description>
<image path="fomod/texture_red.png"/>
<conditionFlags>
<flag name="texture_red">selected</flag>
</conditionFlags>
<typeDescriptor>
<type name="Optional"/>
</typeDescriptor>
</plugin>

</plugins>
</group>

</optionalFileGroups>
</installStep>
</installSteps>

<conditionalFileInstalls>
<patterns>
<pattern>
<dependencies operator="And">
<flagDependency flag="option_a" value="selected"/>
<flagDependency flag="texture_blue" value="selected"/>
</dependencies>
<files>
<folder source="option_a"/>
<folder source="texture_blue_a"/>
</files>
</pattern>
<pattern>
<dependencies operator="And">
<flagDependency flag="option_a" value="selected"/>
<flagDependency flag="texture_red" value="selected"/>
</dependencies>
<files>
<folder source="option_a"/>
<folder source="texture_red_a"/>
</files>
</pattern>
<pattern>
<dependencies operator="And">
<flagDependency flag="option_b" value="selected"/>
<flagDependency flag="texture_blue" value="selected"/>
</dependencies>
<files>
<folder source="option_b"/>
<folder source="texture_blue_b"/>
</files>
</pattern>
<pattern>
<dependencies operator="And">
<flagDependency flag="option_b" value="selected"/>
<flagDependency flag="texture_red" value="selected"/>
</dependencies>
<files>
<folder source="option_b"/>
<folder source="texture_red_b"/>
</files>
</pattern>
</patterns>
</conditionalFileInstalls>

</config>

Granted, the number of matrix items (pattern tags) you’ll need to create in this specific 2 options/choice example is always going to be higher ($$a_n=2^n$$, where $$n$$ is the number of choices for the user to make) than the number of installation steps needed for the same number of choices, BUT you can better organize your steps into groups since they’re no longer dependent on each other and this matrix is mostly copy-paste while replacing a few things, while the steps need careful adjustements of the visible, files and conditionFlags tags.

It also looks much better this way.

As you may have understood by now, conditionalFileInstalls allows you to create a matrix of pattern tags. The mod manager/installer will run through each of these, check dependencies and if they match, install anything under files. We’ve talked about these tags before, they work exactly the same way.

And that’s it really. All major sections were talked about and you’re ready to tackle 99.9% of the fomod installers out there. There are a few minor things that also exist but they’re so rarely needed that they can be safely ignored by most people. To take a look at some of them continue on to Tips and Tricks and if you need something else that isn’t even covered there head on over to Specification for a complete and exhaustive look at the schema.

Hope you learned something and good luck!

Example 05


text:
show · group.representation.type all knowls · up · search:

Let $\rho:G\to\GL_n(\C)$ be an irreducible complex group representation. Then $\rho$ is one of three types:

1. Real if $\rho$ is conjugate to a representation $G\to \GL_n(\R)$
2. Complex if some character value $\textrm{Tr}(\rho(g))$ is not contained in $\R$
3. Quaternionic if the character values are all real but the representation is not conjugate to a real representation.

The type of the representation can be computed via its Frobenius-Schur indicator.

Authors:
Knowl status:
• Review status: reviewed
• Last edited by Jennifer Paulhus on 2022-07-19 14:30:14
Referred to by:
History: Differences


text:
Featured resource

Numeracy: What Classroom Teachers Should Know

What is numeracy in an Australian context?

Members: $22.00 inc.GST Others:$ 27.50 inc.GST

Home > Student activities > National Maths Day > National Maths Day 2015 > Activity: Triangle midpoints

Activity: Triangle midpoints

Find the activity on NRICH

Go to http://nrich.maths.org/483 (link will open in a new window)

More questions

What is the relationship between the original triangle and the triangle formed by joining the three midpoints? Is it always possible to draw a triangle given three points that are supposed to be the midpoints of its sides?

Given four points that are the midpoints of the sides of a quadrilateral, how would you draw the original quadrilateral? Is it possible for any set of four points?

Shine a light on the Proficiencies

A useful practice to encourage problem solving is to reverse the question. This question reverses the more standard question of the relationship between the medians of a triangle and the sides of the triangle.

Asking “How many ways?” shows that mathematical problems often have more than answer. An important question to ask is how we know we have found all the possible answers, which highlights the importance of reasoning. In this case there is a relatively simple proof that there is only one way to draw the triangle.

Asking always/sometimes/never questions (such as “Is it always possible?”) encourages students to employ mathematical reasoning to consider special cases, or to prove that it is always or never possible. The quadrilateral question is a good example of where there needs to be a special arrangement of points in order to draw the quadrilateral. It leads to a relatively little known but delightful and easy to prove result about the quadrilateral formed by joining the midpoints of the sides of any quadrilateral. Deductive proof is a cornerstone of mathematical reasoning.


text:
The AdSwitchNew policy for non-stationary bandits, from [[“Adaptively Tracking the Best Arm with an Unknown Number of Distribution Changes”. Peter Auer, Pratik Gajane and Ronald Ortner, 2019]](http://proceedings.mlr.press/v99/auer19a/auer19a.pdf)

• It uses an additional $$\mathcal{O}(\tau_\max)$$ memory for a game of maximum stationary length $$\tau_\max$$.

Warning

This implementation is still experimental!

Policies.AdSwitchNew.mymean(x)[source]

Simply numpy.mean() on x if x is non empty, otherwise 0.0.

>>> np.mean([])
/usr/local/lib/python3.6/dist-packages/numpy/core/fromnumeric.py:2957: RuntimeWarning: Mean of empty slice.

Policies.AdSwitchNew.Constant_C1 = 16.1

Default value for the constant $$C_1$$. Should be $$>0$$ and as large as possible, but not too large. In their paper, in section 4.2) page 8, an inequality controls C1: (5) states that for all s’, t’, C1 > 8 (2n - 1)/n where n = n_[s’,t’], so C1 > 16.

Policies.AdSwitchNew.DELTA_T = 50

A small trick to speed-up the computations, the checks for changes of good/bad arms are going to have a step DELTA_T.

Policies.AdSwitchNew.DELTA_S = 20

A small trick to speed-up the computations, the loops on $$s_1$$, $$s_2$$ and $$s$$ are going to have a step DELTA_S.

class Policies.AdSwitchNew.AdSwitchNew(nbArms, horizon=None, C1=16.1, delta_s=20, delta_t=50, *args, **kwargs)[source]

The AdSwitchNew policy for non-stationary bandits, from [[“Adaptively Tracking the Best Arm with an Unknown Number of Distribution Changes”. Peter Auer, Pratik Gajane and Ronald Ortner, 2019]](http://proceedings.mlr.press/v99/auer19a/auer19a.pdf)

__init__(nbArms, horizon=None, C1=16.1, delta_s=20, delta_t=50, *args, **kwargs)[source]

New policy.

horizon = None

Parameter $$T$$ for the AdSwitchNew algorithm, the horizon of the experiment. TODO try to use DoublingTrickWrapper to remove the dependency in $$T$$ ?

C1 = None

Parameter $$C_1$$ for the AdSwitchNew algorithm.

delta_s = None

Parameter $$\delta_s$$ for the AdSwitchNew algorithm.

delta_t = None

Parameter $$\delta_s$$ for the AdSwitchNew algorithm.

ell = None

Variable $$\ell$$ in the algorithm. Count the number of new episode.

start_of_episode = None

Variable $$t_l$$ in the algorithm. Count the starting time of the current episode.

set_GOOD = None

Variable $$\mathrm{GOOD}_t$$ in the algorithm. Set of “good” arms at current time.

set_BAD = None

Variable $$\mathrm{BAD}_t$$ in the algorithm. Set of “bad” arms at current time. It always satisfies $$\mathrm{BAD}_t = \{1,\dots,K\} \setminus \mathrm{GOOD}_t$$.

set_S = None

Variable $$S_t$$ in the algorithm. A list of sets of sampling obligations of arm $$a$$ at current time.

mu_tilde_of_l = None

Vector of variables $$\tilde{\mu}_{\ell}(a)$$ in the algorithm. Count the empirical average of arm $$a$$.

gap_Delta_tilde_of_l = None

Vector of variables $$\tilde{\Delta}_{\ell}(a)$$ in the algorithm. Count the estimate of the gap of arm $$a$$ against the best of the “good” arms.

all_rewards = None

Memory of all the rewards. A dictionary per arm, mapping time to rewards. Growing size until restart of that arm!

history_of_plays = None

Memory of all the past actions played!

__str__()[source]

-> str

new_episode()[source]

Start a new episode, line 3-6 of the algorithm.

startGame()[source]

Start the game (fill pulls and rewards with 0).

check_changes_good_arms()[source]

Check for changes of good arms.

• I moved this into a function, in order to stop the 4 for loops (good_arm, s_1, s_2, s) as soon as a change was detected (early stopping).

• TODO this takes a crazy O(K t^3) time, it HAS to be done faster!

check_changes_bad_arms()[source]

Check for changes of bad arms, in O(K t).

• I moved this into a function, in order to stop the 2 for loops (good_arm, s) as soon as a change was detected (early stopping).

getReward(arm, reward)[source]

Get a reward from an arm.

n_s_t(arm, s, t)[source]

Compute $$n_{[s,t]}(a) := \#\{\tau : s \leq \tau \leq t, a_{\tau} = a \}$$, naively by using the dictionary of all plays all_rewards.

mu_hat_s_t(arm, s, t)[source]

Compute $$\hat{\tau}_{[s,t]}(a) := \frac{1}{n_{[s,t]}(a)} \sum_{\tau : s \leq \tau \leq t, a_{\tau} = a} r_t$$, naively by using the dictionary of all plays all_rewards.

__module__ = 'Policies.AdSwitchNew'
find_max_i(gap)[source]

Follow the algorithm and, with a gap estimate $$\widehat{\Delta_k}$$, find $$I_k = \max\{ i : d_i \geq \widehat{\Delta_k} \}$$, where $$d_i := 2^{-i}$$. There is no need to do an exhaustive search:

$I_k := \lfloor - \log_2(\widehat{\Delta_k}) \rfloor.$
choice()[source]

Choose an arm following the different phase of growing lengths according to the AdSwitchNew algorithm.


text:
## 1. Exercise 11 - Page 363

The price of one share of stock in the Pilsdorff Beer Company (see Exercise 8.2.12) is given by $$Y_n$$ on the $$n$$ th day of the year. Finn observes that the differences $$X_n=Y_{n+1}−Y_n$$ appear to be independent random variables with a common distribution having mean $$μ=0$$ and variance $$σ^2$$=$$\frac{1}{4}$$. If $$Y_1=100$$, estimate the probability that $$Y_365$$ is

Since $$X_n=Y_{n+1}−Y_n$$, also we can say $$Y_{n+1}=X_n−Y_n$$ In year 1 and 2 $$Y_2 = X_1 + Y_1$$ And then $$Y_{365} = Y_1 + X_1 + X_2 + X_3 + ... + X_{364}$$ Because we know that $$Y_1 = 100$$ and $$X_1 , X_2 , X_3 ...X_{364}$$ have $$μ=0$$, We know the expected value for $$Y_1$$ is 100+0+0+0+…+0=100. As variance $$σ^2$$=$$\frac{1}{4}$$, standard deviation is can be calculated as below, $$Y_{365}$$ = 365 * $$\frac{1}{4}$$ = 91.25 Therefor, SD = $$\sqrt{91.25} = 9.552$$

Using mean, $$μ=0$$ and SD, $$σ = \sqrt{91.25}$$

### (a)$$≥ 100$$

pnorm(100 - 100, mean = 0, sd = sqrt(91.25), lower.tail = FALSE)
## [1] 0.5

### (a)$$≥ 110$$

pnorm(110 - 100, mean = 0, sd = sqrt(91.25), lower.tail = FALSE)
## [1] 0.1475849

### (a)$$≥ 120$$

pnorm(120 - 100, mean = 0, sd = sqrt(91.25), lower.tail = FALSE)
## [1] 0.01814355

## 2. Calculate the expected value and variance of the binomial distribution using the moment generating function.

The probability mass function for the binomial distribution is:
$$\binom{n}{k} (p)^k(1-p)^{n-k}$$
The moment generating function is given as $$g(t) = \sum_{j=0}^{n}(p)^k(1-p)^{n-k}e^{jt}$$ = $$(1-p+pe^t)^n$$

First derivative, $$g'(t) = npe^t(pe^t-p+1)^{n-1}$$

Evaluating for $$t=0$$ to be determine the mean $$μ_1$$ $$g'(0) = n*p*e^0(pe^0-p+1)^{n-1} = np$$

Second derivative, $$g''(t) = npe^t(pe^t-p+1)^{n-2}(npe^t-p+1)$$

Evaluating for $$t=0$$ to be determine the mean $$μ_2$$ $$g''(0) = npe^0(pe^0-p+1)^{n-2}(npe^0-p+1) = np(np+1-p)$$

The variance is $$μ_2-μ_{1}^{2} = np(np+1-p)np = n^2p^2(np+1-p)$$

## 3. Calculate the expected value and variance of the exponential distribution using the moment generating function.

The probability density function for the exponential distribution is: $$\lambda e^{-\lambda x}$$

The moment generating function is givenn as $$g(t) = \frac{\lambda}{\lambda-t}$$ for $$t < \lambda$$

First derivative, $$g'(t) = \frac{\lambda}{(\lambda-t)^2}$$

Evaluating for $$t=0$$ to be determine the mean $$μ_1$$ $$g'(0) = \frac{\lambda}{(\lambda-0)^2} = \frac{1}{\lambda}$$

Second derivative, $$g''(t) = \frac{2}{(\lambda-t)^3}$$

Evaluating for $$t=0$$ to be determine the mean $$μ_2$$ $$g''(0) = \frac{2}{(\lambda-0)^3} = \frac{2}{\lambda^2}$$

The variance is $$μ_2-μ_{1}^{2} =\frac{2}{\lambda^2} - (\frac{1}{\lambda})^2 = \frac{2}{\lambda^2} - \frac{1}{\lambda^2} = \frac{1}{\lambda^2}$$


text:
# 5. Tests, Metrics & Reports¶

Attention

This section is very outdated (late 2014)! It will be re-written based on jupyter notebooks.

For now, launch the or demo servers, open & run CarsDB-compare.ipynb notebook, and start comparing vehicles between AccDB <–> PyAlgo, interactively:

In order to maintain the algorithm stable, a lot of effort has been put to setup a series of test-case and metrics to check the sanity of the results and to compare them with the AccDB tool or other datasets included in the project. These tests can be found in the /tests/ folders.

Additionally, below are auto-generated representative diagrams with the purpose to track the behavior and the evolution of this project.

You can reuse the plotting code here for building nice ipython-notebooks reports, and (optionally) link them in the wiki of the project (see section above). The actual code for generating diagrams for these metrics is in wltp.plots and it is invoked by scripts in the docs/pyplot/ folder.

## 5.1. Comparisons with AccDB¶

This section compares the results of this tool to the MS Access DB.

### 5.1.1. Mean Engine-speed vs PMR¶

First the mean engine-speed of vehicles are compared with access-db tool, grouped by PMRs:

Both tools generate the same rough engine speeds. There is though a trend for this project to produce lower rpm’s as the PMR of the vehicle increases. But it is difficult to tell what each vehicle does isolated.

The same information is presented again but now each vehicle difference is drawn with an arrow:

It can be seen now that this project’s calculates lower engine-speeds for classes 1 & 3 but the trend is reversed for class 2.

### 5.1.2. Mean Engine-speed vs Gears¶

Below the mean-engine-speeds are drawn against the mean gear used, grouped by classes and class-parts (so that, for instance, a class3 vehicle corresponds to 3 points on the diagram):

[TBD]


text:
# FHS hacks

This webpage contains various configurations, tweaks and hacks to improve my working experience at the Salzburg University of Applied Sciences.

## Mail

Operators of mail infrastructure are in a constant battle against various threats like phishing mails. As in politics, sometimes measures are taken that have a broad impact on the user experience, freedom and privacy. As in politics, there companies that earn money by providing security solutions, which some describe as snake oil (here, here).

### URL rewriting

One such measurement is URL rewriting through some Email Security Appliance (ESA) vendor, like Cisco. One can argue that an ESA is ineffective, impacts privacy1, impairs usability and I personally do not like the idea that the mails I receive have been altered. But on the other hand, I do not2 need to fight against cyber criminals.

Nonetheless, I now receive mails that contain URLs like

https://secure-web.cisco.com/1VqOjcxMgKL3XZfH1RVFk4flLLurhrVbZyjVabNjFFr
wcwvrjcKOq0YBEA_UeqrpX0yBPcmEGpNMmKGjUFJRPMyGPhHq61q7mepfDCD8F09vHfZp4tZ
4VLllvl6yAKJ3Am4p3mivuBeZQqmXQqrRmhfrfFQTgPrQPpB1uvzRTH6Z0eD61H3RAZmT6U0
e3iInSn8FFemTitFEVRT_VlrQh6CeiSE_1fEbpu81lEQoMSweguEckjoCEkG7J8LL0YKw5/h
ttps%3A%2F%2Fwww.example.org%2F

They are hypertrophic and prohibitive in length. That is, they destroy my mail experience and therefore I want to transform them back into its (presumably) original version https://www.example.org/. At least, I would like to be able to do so, because mail also constitutes an archive to me, and of course you must not change the content of an archive. (Another argument why modifying mails is an improper thing to do. Besides, how long are those Cisco URLs actually working? Even after canceling contracts with Cisco? Unfortunately, also ciscoclean cannot restore the archive aspect, since we can only make an educated guess what the original URL might have been.)

ciscoclean

Of course, people with computational eyes easily see that the original URL is encoded at the end of the new URL, so it can be trivially undone. And Linux users write scripts when they automate easy things.

In my case I called the script ciscoclean and you can download it here. You can integrate it into a powerful mail client, like mutt, or call it from command line by passing data through stdin or specify files:

% ./ciscoclean -h
Mr. Clean for Cisco's mail snakeoil. It restores the original URL from the URL
generated by Cisco's mail security software.

Usage:
./ciscoclean -h
./ciscoclean [FILE]...

If no filename is given then stdin is processed. Otherwise all files passed are
processed.

% echo "My URL: https://secure-web.cisco.com/1VqOjcxMgKL3XZfH1RVFk4flLLurhrVbZyjVabNjFFrwcwvrjcKOq0YBEA_UeqrpX0yBPcmEGpNMmKGjUFJRPMyGPhHq61q7mepfDCD8F09vHfZp4tZ4VLllvl6yAKJ3Am4p3mivuBeZQqmXQqrRmhfrfFQTgPrQPpB1uvzRTH6Z0eD61H3RAZmT6U0e3iInSn8FFemTitFEVRT_VlrQh6CeiSE_1fEbpu81lEQoMSweguEckjoCEkG7J8LL0YKw5/https%3A%2F%2Fwww.example.org%2F" | ciscoclean
My URL: https://www.example.org/

% ciscoclean mymail.eml
My URL: https://www.example.org/

mutt and vim

The Mutt mail client knows a configuration called display_filter. When a message is viewed by mutt then it is send through the program specified, which can transform the content. So simply download ciscoclean, put it somewhere like ~/.mutt/scripts/ciscoclean and set display_filter to it. This way mutt displays mails with (what I believe are) the original URLs.

But you better know the philosophy of spiderman: With great power comes great responsibility. If you open a malicious URL that could have been filtered by the security appliance then you probably get into trouble.

So maybe you do not want to use ciscoclean as an always-on display filter but configure a macro that lets you display a clean version on demand, i.e., after pressing come key in mutt. Then you may have a look at the <pipe-message> function of mutt. This can also be handy if you want to forward a mail with undistorted URLs to somebody.

If you use vim as mail composer with mutt then, of course, you see the Cisco URLs when you reply to a mail. But vim is again a powerful editor and the simple command 1,$!~/.mutt/scripts/ciscoclean will filter the buffer through ciscoclean. By the way, if you want to know more about mutt and how I use it together with vim then maybe this talk and this talk of mine might be interesting to you. ## Local DNS resolvers Update 2020-03-09: DNS queries are not blocked anymore, but the general hint may still be useful for other networks. There are a few good reasons to run a DNS resolver on localhost.3 Linux users often like to use dnsmasq, which is supported by NetworkManager as a configuration option. I personally use bind, which I also run as authoritative DNS servers for my domain. However, sometimes network providers – such as FHS or the ÖBB federal railways – block outgoing DNS traffic and your local DNS resolver cannot resolve any requests anymore. One option is to resolve DNS queries via VPN.4 This would be a clean solution from security’s point of view but it slows DNS resolving down. In a foreseeable future, DNS over HTTPS partially changes the situation. In case of the FHS, for me personally, the lesser evil would be to resort the FHS DNS resolvers if I am in their network, but otherwise I would like to use my local resolver. This can be done with NetworkManager by using the dispatcher simply by placing the following script 99-localhost-resolver in the directory /etc/NetworkManager/dispatcher.d: #!/bin/sh if [ "$2" == "up" ]; then
# We got up in the FHS network
if ip a | grep -q "inet 78.104.1[67]" ; then
echo "# On FHS network" >> /etc/resolv.conf
# Set resolver to localhost
else
echo "nameserver 127.0.0.1" > /etc/resolv.conf
fi
fi

1. The new URL redirects users through some vendor website, like websecure.cisco.com, which does some checks. That means that every access is reported and enables tracking.

2. I do run my own mail server, but the situation is of course a different one when you have hundreds or thousands of users and it requires only one to belief in a forged phishing mail from administrators or authorities. And after all I am not blamed by anyone but myself when things go wrong.

3. A local DNS resolver is a fast cache, can be configured to support conditional forwards, can supports DNSSEC.

4. For instance, set up a DNS resolver on your VPN server.


text:
# Difference between conflict serializability and view serializability

In my understanding both of them can be converted to serial schedules. (This helps with consistency by isolation property while also giving the performance boost by concurrency).

I'm unable to understand one of the lines in this text: http://infolab.stanford.edu/~ullman/dscb/vs-old.pdf

Intuitively, view-serializability considers all the connections between transactions T and U such that T writes a database element whose value U reads. The key difference between view and conflict serializability appears when a transaction T writes a value $$A$$ that no other transaction reads (because some other transaction later writes its own value for A). In that case, the $$w_T(A)$$ action can be placed in certain other positions of the schedule (where $$A$$ is likewise never read) that would not be permitted under the definition of conflict-serializability.

I understand that the circumstances that can cause inconsistencies (if transactions are swapped) and affect the conflict serializability are
write(Q) by T1 and then write(Q) by T2, or
read(Q) by T1 and then write(Q) by T2, or
write(Q) by T1 and read(Q) by T2.
It seems to me that some special cases are allowed as per view serializability but I've not been able to pin point those cases. Any help is appreciated.