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• Volume 57, Issue 4

October 2001,   pages  683-837

• Statistical constraints on state preparation for a quantum computer

Quantum computing algorithms require that the quantum register be initially present in a superposition state. To achieve this, we consider the practical problem of creating a coherent superposition state of several qubits. We show that the constraints of quantum statistics require that the entropy of the system be brought down when several independent qubits are assembled together. In particular, we have: (i) not all initial states are realizable as pure states; (ii) the temperature of the system must be reduced. These factors, in addition to decoherence and sensitivity to errors, must be considered in the implementation of quantum computers.

• Measuring the top quark mass in the channel: A study

We describe a simple method to measure the top quark mass in the $$t\overline t \to WbW\overline b \to e\mu$$ channel that may be useful in Run II of DØ detector. The method is validated by applying it to the Run Ib data.

• Investigation of halo structure of 6He by hyperspherical three-body method

Hyperspherical harmonics expansion method is applied to a three-body model of two neutron halo nuclei. Convergence of the expansion has been ensured. A repulsive part is introduced in the interaction between the core and the extra-core neutron, to simulate Pauli principle. Two neutron separation energy, r.m.s. radii, correlation factor and probability density distributions have been calculated for 6He. It is found that the convergence of the two neutron separation energy is relatively slow, while other quantities reach convergence quickly.

• A simple coordinate space approach to three-body problems — Examples: Halo nucleus and double-λ hypernucleus

We show how to treat the dynamics of an asymmetric three-body system consisting of one heavy and two identical light particles in simple coordinate space variational approach. The method is constructive and gives an efficient way of resolving a three-body system to an effective two-body system. It is illustrated by explaining the structural properties of some nuclei of current interest, namely halo nuclei and double-λ hypernuclei. The ansatz used here may be of value in a number of three particle problems of similar nature.

• Rovibrational matrix elements of the multipole moments and of the polarizability of the H2 molecule in the solid phase: Effect of intermolecular potential

Rovibrational matrix elements of the multiple moments Ql up to rank 10 and of the linear polarizability of the H2 molecule in the condensed phase have been computed taking into account the effect of the intermolecular potential. Comparison with gas phase matrix elements shows that the effect of solid state interactions is marginal.

• Scattering and absorption differential cross sections for double photon Compton scattering

The scattering and absorption differential cross sections for nonlinear QED process such as double photon Compton scattering have been measured as a function of independent final photon energy. The incident gamma photons are of 0.662 MeV in energy as produced by an 8 Ci137Cs radioactive source and thin aluminum foils are used as scatterer. The two simultaneously emitted photons in this process are detected in coincidence using two Nal(T1) scintillation detectors and a slow-fast coincidence set-up of 30 nsec resolving time. The measured values of scattering and absorption differential cross sections agree with theory within experimental estimated error.

• Polarization modulational instability in a birefringent optical fiber with fourth order dispersion

We obtain conditions for the occurrence of polarization modulational instability in the anomalous and normal dispersion regimes for the coupled nonlinear Schrödinger equation modelling fourth order dispersion effects when the linearly polarized pump is oriented at arbitrary angles with respect to the slow and fast axes of the birefringent fiber.

• Neutrino beam plasma instability

We derive relativistic fluid set of equations for neutrinos and electrons from relativistic Vlasov equations with Fermi weak interaction force. Using these fluid equations, we obtain a dispersion relation describing neutrino beam plasma instability, which is little different from normal dispersion relation of streaming instability. It contains new, nonelectromagnetic, neutrino-plasma (or electroweak) stable and unstable modes also. The growth of the instability is weak for the highly relativistic neutrino flux, but becomes stronger for weakly relativistic neutrino flux in the case of parameters appropriate to the early universe and supernova explosions. However, this mode is dominant only for the beam velocity greater than 0.25c and in the other limit electroweak unstable mode takes over.

• Magnetization curves for general cylindrical samples in a transverse field

Using the recent results for the surface current density on cylindrical surfaces of arbitrary cross-section producing uniform interior magnetic field we propose a method for obtaining solutions of Bean’s critical state model for general cylindrical samples. The method uses the technique of conformal mapping to express the sample surface and the flux-fronts in terms of a set of coefficients that depend on a parameter. The flux-fronts are to be determined by solving a system of nonlinear ordinary differential equations for the coefficients. Retaining only a certain finite number of leading coefficients we get an approximate solution. The procedure is illustrated by considering two cyclindrical samples — one with an elliptical cross-section and the other with a non-elliptical cross-section. The virgin curve and small and large magnetization hysteresis loops for the two samples are obtained.

• Dielectric relaxation phenomena of rigid polar liquid molecules under giga hertz electric field

The dielectric relaxation phenomena of rigid polar liquid molecules chloral and ethyltrichloroacetate (j) in benzene, n-hexane and n-heptane (i) under 4.2, 9.8 and 24.6 GHz electric fields at 30°C are studied to show the possible existence of double relaxation times τ2 and τ1 for rotations of the whole and the flexible parts of molecules. The probability of showing double relaxation is more in aliphatic solvents indicating their nonrigidity. The symmetric and asymmetric distribution parameters γ and δ are obtained from Xij/X0ij and Xij/X0ij and wj→0 where Xij and Xij are real and imaginary parts of the complex orientational susceptibility Xi* and X0ij is the low frequency susceptibility which is real. Xij’s are involved with the measured dielectric relative permittivities εij, εin, ε0ij and ε∞ij of solutions. The theoretical weighted contributions c1 and c2 towards dielectric dispersions by Fröhlich’s method are compared with the experimental ones obtained from the graphical variation of Xij/X0ij and Xij/X0ij with weight fractions wj’s at wj → 0. The measured dipole moments μ2 and μ1 of the whole and the flexible part of a polar molecule in terms of the linear coefficients β’s of Xij’s with wj’s and the estimated τ2 and τ1 reveal their associations with aliphatic solvents. The theoretical dipole moments μtheo’s from the available bond angles and bond moments of the substituent polar groups of the molecules with the estimated μ’s suggest the mesomeric, inductive and electromeric effects in them under GHz electric field.

• Theory of anisotropic diamagnetism, local moment magnetization and carrier spin-polarization in Pb1−xEuxTe

We present theoretical analyses of anisotropic lattice diamagnetism, magnetization due to magnetic ions and carrier spin-polarization in the diluted magnetic semiconductor, Pb1-xEuxTe. The lattice diamagnetism results from orbital susceptibility due to inter band effects and spin-orbit contributions. The spin-orbit contribution is found to be dominant. However, both the contributions show pronounced anisotropy. With increase in x, the diamagnetism decreases. We consider contributions from randomly distributed isolated magnetic ions and clusters of pairs and triads for the local moment magnetization. The isolated magnetic-ion contribution is the dominant one.We calculate the magnetization for two typical magnetic ion concentrations: x=0.03 and x=0.06. Temperature dependence of the magnetization is also considered. Apart from lattice and localized magnetic ions, the carrier contribution to the spin-density is also calculated for a carrier density of p=1018 cm−3. the relative spin-density of carriers increases with increase in the magnetic field strength and magnetic ion concentration. The agreement with experiment where available is reasonably good.

• Near surface composition of some alloys by X-ray photoelectron spectroscopy

Chemical compositions of the alloys of CuNi (Cu0.10Ni0.90, Cu0.30Ni0.70, Cu0.70Ni0.30) and BiSb (Bi0.80Sb0.20, Bi0.64Sb0.34, Bi0.55Sb0.45) are determined by X-ray photoelectron spectroscopy. The stoichiometries are determined and are compared with the bulk compositions. Possible sources of systematic errors contributing to the results are discussed. Errors arising out of preferential etching in these alloys have been investigated. It has been inferred from such studies that the preferential etching does not enrich the surface composition with a particular component for the two systems reported here. Quantitative results of CuNi system indicate that the surface regions of the Cu0.70Ni0.30 alloy is Cu-rich, although no such evidence is observed in case of BiSb system.

• Masses of S and P wave mesons and pseudoscalar decay constants using a confinement scheme

In the framework of relativistic harmonic confinement model for quarks and antiquarks, the masses of S- and P-wave mesons and pseudoscalar decay constants from light flavour to heavy flavour sectors are computed. The residual two-body Coulomb interaction and the spin-dependent interaction of the confined one gluon exchange effects (COGEP) such as spin-spin and spin-orbit interactions are perturbatively incorporated with the confinement energy to get the respective vector-pseudoscalar meson mass differences. Here we employ the same parametrization and model parameters as used in a recent study of low-lying hadron masses and leptonic decay widths. The results are being compared with the values obtained from other theoretical models and the experimental values.

• Lattice variation and thermal parameters of gel grown KDP crystals added with some ammonium compounds

Pure and impurity added (with NH4Cl, NH4NO3, NH4H2PO4, and (NH4)2SO4) KDP single crystals were grown by the gel method using silica gels. X-ray diffraction data were collected for powder samples and used for the estimation of lattice variation and thermal parameters like Debye-Waller factor, mean-square amplitude of vibration, Debye temperature and Debye frequency. The thermal parameters do not vary in a particular order with respect to impurity concentration. The results obtained are reported and discussed.

• Rejoinder

• # Pramana – Journal of Physics

Current Issue
Volume 93 | Issue 6
December 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019


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# Chemical Bonding Class 11 Questions with Answers-Important for Exams

Get Chemical Bonding important questions for class 11 exams. View the Important Question bank for Class 11& 12 Chemistry syllabus. These important questions will play significant role in clearing concepts of Chemistry. These questions are designed keeping NCERT class 11 chemical bonding topic in mind for various exams. The questions are updated with respect to upcoming Board exams. You will get here all the important questions for class 11 chemistry chapters. Learn all the concepts of Chemical Bonding class 11 chemistry with these questions and answers. Also, you can get notes of eact topic of Chemistry class 11 & 12 o our website. Click Here for Detailed Chapter-wise Notes of Chemistry for Class 11th, JEE & NEET.  You can access free study material for all three subject’s Physics, Chemistry and Mathematics. Click Here for Detailed Notes of any chapter.  eSaral provides you complete edge to prepare for Board and Competitive Exams like JEE, NEET, BITSAT, etc. We have transformed classroom in such a way that a student can study anytime anywhere. With the help of AI we have made the learning Personalized, adaptive and accessible for each and every one. Visit eSaral Website to download or view free study material for JEE & NEET. Also get to know about the strategies to Crack Exam in limited time period.
Q. Arrange the following molecules in increasing order of ionic character of their bonds $L i F, K_{2} O, N_{2}, S O_{2}, C I F_{3}$
Ans. $\mathrm{N}_{2}<\mathrm{SO}_{2}<\mathrm{ClF}_{3}<\mathrm{K}_{2} \mathrm{O}<\mathrm{LiF}$
Q. Arrange the following bonds in increasing order of ionic character $C-H, F-H, B r-H, N a-I, K-F, L i-C l$
Ans. $\mathrm{C}-\mathrm{H}<\mathrm{Br}-\mathrm{H}<\mathrm{Li}-\mathrm{Cl}<\mathrm{Na}-\mathrm{I}<\mathrm{K}-\mathrm{F}$
Q. Write Lewis dot symbols for the following elements: $M g, N a, B, O, N, B r, B e$ [NCERT]
Ans.
Q. Write Lewis symbol for the following atoms and ions : $S$ and $S^{2-}, \quad A l$ and $A I^{3+}, \quad H$ and $H^{-}$ [NCERT]
Ans.
Q. Write Lewis dot symbols for the following ions: $L i^{+}, C l^{-}, O^{2-}, M g^{2+}$ and $N^{3-} . \quad$ [NCERT]
Ans.
Q. Why is $\sigma$ -bond stronger than $\pi$ -bond?
Ans. It is because extent of overlapping is more in $\sigma$ -bond than $\pi$ bond.
Q. What is the total number of sigma and $p i$ bonds in the following molecules: (i) $\mathrm{C}_{2} \mathrm{H}_{2}$ (ii) $C_{2} H_{4}$ [NCERT]
Ans.
Q. Considering Z-axis as the inter nuclear axis, which out of the following will form a sigma bond : (i) $1 \mathrm{s}$ and $1 \mathrm{s}$ (ii) $1 \mathrm{s}$ and $2 \mathrm{p}_{\mathrm{z}}$ (iii) $2 p_{x}$ and $2 p_{x}$ (iv) $2 p_{x}$ and $2 p_{y} (\mathrm{v}) 2 p_{y}$ and $2 p_{y}$ [NCERT]
Ans. (i) and (ii)
Q. The skeletal form of structure of $C H_{3} C O O H$ as shown below is correct, but some of the bonds are wrongly shown. Write the correct Lewis structure for acetic acid,
Ans.
Q. Correct Lewis structure of $C H_{3} C O O H$ is
Ans.
Q. Which type of hybridisation explain the trigonal bipyramidal shape of $S F_{4} ?$ [NCERT]
Ans. $s p^{3} d$ hybridisation.
Q. Explain why $B e H_{2}$ molecule has a zero dipole moment although the $B e-H$ bonds are polar. $\quad$
Ans. BeH_ $_{2}$ molecule is linear due to presence of sp-hybridization in $B e .$ Therefore, dipole moments due to the polar $B e-H$ bond get cancel out. Therefore, molecule has zero dipole moment.
Q. Predict which out of the following molecule will have higer dipole moment $\mathrm{CS}_{2}$ and $\mathrm{OCS}$.
Ans. $-O C S$ will have higher dipole moment because bond moment of $O=$ $-C$ and that of $C=S$ do not cancel each other
Q. $H_{3} P O_{3}$ can be represented by structure $I$ and $I I$ as shown below. Can these two structures be taken as the canonical forms of the resonance hybrid of $H_{3} P O_{3} .$ If not, state reason for the same.
Ans. No, these cannot be taken as canonical forms because the position of atoms have changed.
Q. Use molecular orbital theory to explain why $B e_{2}$ molecule does not exist.
Ans. Molecular orbital configuration of $B e_{2}$ is $\left(\sigma_{1 s}\right)^{2},\left(\sigma^{*}_{1 s}\right)^{2}$ $\left(\sigma_{2 s}\right)^{2},\left(\sigma_{2 s}\right)^{2} .$ Bond order of $B e_{2}=\frac{1}{2}\left(N_{b}-N_{a}\right)=\frac{1}{2}(4-4)=0$
Q. Compare the relative stability of the following species and indicate their magnetic (diamagnetic or paramagnetic) properties : $C_{2}^{2-}$ and $O_{2}^{-}$
Ans. $C_{2}^{-2}(B . O=3)$ Diamagnetic is more stable than $\quad O_{2}^{-}\left(B . O=1 \frac{1}{2}\right)$ paramagnetic
Q. Why is it that in the $S F_{4}$ molecule, the lone pair of electrons occupies an equatorial position in the overall trigonal bi-pyramidal arrangement in preference to an axial position. [NCERT]
Ans. In $S F_{4}$ molecule, lone pair of electron occupy an equatorial position because $l . p .-b . p .$ repulsions are less in that case. As a result, energy is less which leads to more stability.
Q. Define hydrogen bond. Is it weaker or stronger than the vander Waals forces ?
Ans. Hydrogen bond is attractive force which binds H-atom of one molecule with electronegative atom of another molecule. It is stronger than vander waals force.
Q. How can one non-polar molecule induce a dipole in a nearby non-polar molecule ? [NCERT]
Ans. In a non-polar molecule, there may be instantaneous dipole created by specific position of electrons. This instantaneous dipole can induce a dipole in a nearby non-polar molecules.
Q. Define H-bond.
Ans. H-bond is force of attraction between hydrogen and most electronegative element like F, O and N. It is a weak bond.
Q. List two main conditions for forming hydrogen bonds.
Ans. (i) High electronegativity of atom bonded to hydrogen. (ii) Small size of electronegative atom.
Q. Arrange the following as indicated : $O, F, S, C l, N$ in the order of increasing strength of hydrogen bonding (X……H-X).
Ans. S < Cl < N < O < F..
Q. Give electron dot formula of (i) $\mathrm{CN}^{-},$ (ii) $\mathrm{SO}_{3}^{2-},$ (iii) $\mathrm{ClO}_{2}^{-}$ (iv) $\mathrm{ClO}_{4}^{-}$
Ans.
Q. Write Lewis dot symbols for the following atoms and ions : $S \quad$ and $\quad S^{2-} ; \quad P$ and $P^{3-} ; \quad N a$ and $N a^{+} ; \quad A l$ and $A l^{3+}$; $H$ and $H^{-}$ [NCERT]
Ans.
Q. Write the Lewis dot structure of (i) $\mathrm{Cl}_{4},$ (ii) $P H_{3}$ and (iii) $B C l_{3} .$ Is the octet rule obeyed in these structures? [NCERT]
Ans. In $B C l_{3},$ Octet rule is not obeyed because boron contains only six electrons around it.
Q. What is the formal charge on Nitrite ion ?
Ans. Nitrite : The Lewis structure of nitrite ion is
Q. Enlist four major differences between $\sigma$ (sigma) and $\pi$ (pie) bond.
Ans.
Q. Describe the change in hybridisation of aluminium atom (if any) during the reaction $A l C l_{3}+C l^{-} \longrightarrow A l C l_{4}^{-}$
Ans. In $A I C I_{3}$ the hybrid state of aluminium is $s p^{2}$ but in $A I C I_{4}^{-}$ the hybrid state of aluminium is $s p^{3} .$ The $A l C l_{4}^{-}$ adopts tetrahedral arrangement. Thus hybrid state of $A l$ atom changes from $s p^{2}$ to $s p^{3}$
Q. Draw the shapes of the following hybrid orbitals :
Ans. All the hybrid orbitals have same shape, $i . e .,$ one lobe larger than the other, however, their sizes are in the order $s p Q. Is there change in the hybridization of the B and N atoms as a result of the following reaction :$B F_{3}+N H_{3} \longrightarrow F_{3} B \leftarrow N H_{3} ?$Ans. In$B F_{3}, B$is$s p^{2}$hybridised and in$N H_{3}, N$is$s p^{3}$hybridised. After the reaction, hybridisation of$B$changes to$s p^{3}$but that of$N$remains unchanged. Hybridisation of Boron changes to involve a vacant orbital to accommodate electron pair coming from$N H_{3} .$Q. Apart from tetrahedral geometry, another possible geometry for$C H_{4}$is square planar with the four$H$atoms at the corners of the square and the$C$atom at its centre. Explain why$C H_{4}$is not square-planar? [NCERT] Ans. In$C H_{4},$central carbon atom is surrounded by four bond pairs of electrons. These bond pairs arrange themselves in tetrahedral manner with maximum possible bond angle of$109^{\circ} 28^{\prime} .$In square planar geometry, the bond angles will be$90^{\circ}$with larger bond pair-bond pair repulsions. Therefore, square planar geometry is not possible. Q. On the basis of VSEPR theory discuss the geometry of (i)$N H_{3}$molecule (ii)$C H_{4}$molecule. Ans. (i) 5 valence electron in nitrogen. 3 bond pairs and one lone pair of electron. Therefore it's shape is pyramidal. since the Bond pairBond pair repulsion (ii)$C H_{4}$is tetrahedral because it has four bonded pair of electrons. Q. Explain why the bond order of$N_{2}$is greater than that of$N_{2}^{+}$but the bond order of$O_{2}$is less than that of$O_{2}^{+}$Explain why$N_{2}$has greater bond dissociation energy than$N_{2}^{+}$whereas$O_{2}^{+}$has greater bond dissociation energy than$O_{2}$Ans. Q. During formation of$N_{2}^{+}$from$N_{2}$the electron is to be removed from bonding molecular orbital$\left(\sigma_{2 p z}\right)$while during formation of$\mathrm{O}_{2}^{+}$from$\mathrm{O}_{2}$the electron is to be removed from anti bonding molecular orbital$\pi_{2 p x}^{*}$or$\pi_{2 p y}^{*} .$Thus, the bond order decreases during formation of$N_{2}^{+}$from 3 to 2.5 while it increases during the formation of$\mathrm{O}_{2}^{+}$from 2 to 2.5 Ans. Q. Sketch the bond moments and resultant dipole moments in the following molecules :$H_{2} \mathrm{O}, P C l_{3}, N H_{3}, N F_{3}$Ans. Q. Out of the following four resonance structures for the$C O_{2}$molecule, which are important for describing the bonding in the molecule and why? Ans. Out of the given structure$1,2$and 3 are important for describing the bonding in$C O_{2}$molecule. These structure are isovalent (same number of bonds) and thus have higher contribution. Structure 4 has unsaturated carbon with only 4 electrons and thus has negligible stability. Q. Write resonance structures for$S O_{3}, N O_{2}$and$N O_{3}^{-}$Ans. Q. How is the bonding molecular orbital in a molecule of hydrogen different from its antibonding molecular orbital? Ans. (i) Bonding molecular orbital has a single lobe and no node whereas antibonding molecular orbital has two lobes which are separated by a node. (ii) Bonding molecular orbital has less energy whereas antibonding has higher energy than atomic orbitals. Q. Explain why the bond order of$N_{2}$is greater than$N_{2}^{+}$but the bond order of$O_{2}$is less than that of Ans. When$N_{2}$change to$N_{2}^{+}$the electron is removed from bonding molecular orbital while when$O_{2}$changes to$O_{2}^{+},$the electron is removed from antibonding molecular orbital. Q. Explain the significance of bond order. Can bond order be used for quantitative comparisons of the strengths of chemical bond ? Calculate the bond order of$N_{2}, O_{2}, O_{2}^{+}$and$O_{2}^{-}$Ans. Bond order cannot be used for quantitative comparison of the strengths of chemical bonds. It can give us only approximate idea about the strength of the bond. Greater the bond order, greater is the strength of the bond.$N_{2}(3), O_{2}(2), O_{2}^{+}(2.5) . O_{2}^{-}(1.5)$Q. Although geometries of$\mathrm{NH}_{3}$and$\mathrm{H}_{2} \mathrm{O}$are distorted tetrahedral, bond angle in water is less than that of ammonia. Explain. Ans.$N H_{3}$molecule has one lone pair while$H_{2} O$has two lone pairs of electrons. Due to the presence of lone pairs the geometries of and are distorted. Due to the presence of stronger$l p-b p$repulsion than$b p-b p$repulsion the bond angle in is reduced from normal tetrahedral bond angle$\left(109.5^{\circ}\right)$to$107^{\circ} .$In case of, two$l p$of electrons force the$O-H$bonds more closely than$N-H$bonds in. So bond angle decreases to a larger extent i.e. to$104.5^{\circ}$Q. Although both$\mathrm{CO}_{2}$and$\mathrm{H}_{2} \mathrm{O}$are triatomic molecules, the shape of$H_{2} O$molecule in bent while that of$C O_{2}$is linear. Explain this on the basis of dipole moment. Ans.$\mathrm{CO}_{2}$has zero dipole moment. This means that molecules is linear so that the two$C=O$bond moments get cancelled giving zero resultant dipole moment. However, molecule has resultant dipole moment showing that it cannot be linear. The two$O-H$bonds are arranged in angular shape and the bond moment of two$O-H$bonds give resultant dipole moment. Q. How do the bonding and anti-bonding MOs formed from a givenpair of AOs compare to each other with respect to: (i) energy (ii) presence of nodes (iii) internuclear electron density? Ans. Q. What is meant by the term bond order in Lewis concept? Calculate the bond order of$N_{2}, O_{2}$and$\mathrm{CO}$. Ans. According to Lewis concept, bond order is number of bonds present between two atoms of a molecule or ion. Bond order in$N \quad N\left(N_{2}\right)$is 3 Bond order in$\mathrm{O} \quad \mathrm{O}\left(\mathrm{O}_{2}\right)$is 2 Bond order in$C \quad O(C O)$is 3 Q. Give the structure of$\left(\mathrm{CH}_{3}\right)_{3} \mathrm{N}$and$\left[\left(\mathrm{CH}_{3}\right)_{3} \mathrm{Si}\right]_{3} \mathrm{N}$. Are they isostructural? Ans.$\left(\mathrm{CH}_{3}\right)_{3} \mathrm{N}$is trimethyl amine. It has a pyramidal geometry as shown in structure I.$\left[\left(C H_{3}\right)_{3} \mathrm{Si}\right]_{3} N$is planar as shown is structure II. Thus, the two species are not isostructural. In$\left(C H_{3}\right)_{3} N . N$eatom assumes$s p^{3}$hybrid state whereas in , the$N$atom assumes Inybrid state. Q. Draw Lewis structures for$H_{2} \mathrm{CO}_{3}, \mathrm{SF}_{6}, \mathrm{PF}_{5}, \mathrm{IF}_{7}$and$\mathrm{CS}_{2} .$Is the octet rule obeyed in these cases? Ans. Q. Which hybrid orbitals are used by carbon atoms in the following molecules ? (i)$H_{3} C-C H_{3}$(ii)$H_{3} C-C H=C H_{2}$(iii)$\mathrm{CH}_{3} \mathrm{CHO}$(iv)$\mathrm{CH}_{3} \mathrm{COOH}$(v)$\mathrm{CH}_{3}-\mathrm{CH}_{2}-\mathrm{OH}$Ans. Q. Discuss the shapes of following molecules using VSEPR model Ans. Q. (i) Explain why$B e C l_{2}$has a zero dipole moment although$B e-C l$bonds are polar. (ii) Explain important aspects of resonance with reference to$\mathrm{CO}_{3}^{2-}$ion. (iii) Why$H_{2} S$is gas and$H_{2} O$is liquid? Ans. (i)$\quad B e C l_{2}$has zero dipole moment because it has linear shape both dipole are equal and opposite such that net dipole moment is zero. (ii) In resonating structure, the position of atom is same. The bond lengths are equal due to resonance. Larger the number of canonical structures, greater will be the stability of molecule / ion. (iii)$\quad H_{2} O$is liquid because water molecule are associated with inter molecular$H$-bonding where as$H_{2} S$is not associated with inter molecular$H$-bonding that is why it is a gas. Q. Write down the$M . O .$configuration of$\mathrm{O}_{2}^{+}, \mathrm{O}_{2}, \mathrm{O}_{2}^{-}$and$\mathrm{O}_{2}^{-2} .$Also compare their stability. Ans. Q. Explain the following: (i)$\mathrm{O}_{2}^{-}$is paramagnetic but$\mathrm{O}_{2}^{2-}$is not. (ii)$N_{2}$has higher bond order than$N O .$Ans. Q. Write down the$M . O .$configuration of$N_{2}, N_{2}^{+}$and$N_{2}^{-}\$ and compare their stability.
Ans. But since, the population of antibonding orbitals is more in case of and thus, is slightly less stable than.
Q. Which of the following substances exhibit hydrogen bonding ? Draw the hydrogen bonds between two molecules of the substance where appropriate :
Ans.

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# Geometry Seminar

#### Event Details

Date: April 9, 2021
Time: 12h-13h
Author:Marilena Moruz (Al.I. Cuza University of Iasi)
Title: Ruled real hypersurfaces in $$\mathbb CP^n_p$$
Summary: H. Anciaux and K. Panagiotidou [1] initiated the study of non-degenerate real hypersurfaces in non-flat indefinite complex space forms in 2015. Next, in 2019 M. Kimura and M. Ortega [2] further developed their ideas, with a focus on Hopf real hypersurfaces in the indefinite complex projective space $$\mathbb CP^n_p$$. In this work we are interested in the study of non-degenerate ruled real hypersurfaces in $$\mathbb CP^n_p$$. We first define such hypersurfaces, then give basic characterizations. We also construct their parameterization. They are described as follows. Given a regular curve $$\alpha$$ in $$\mathbb CP^n_p$$, then the family of the complete, connected, complex $$(n - 1)$$-dimensional totally geodesic submanifolds orthogonal to $$\alpha'$$ and $$J\alpha'$$, where $$J$$ is the complex structure, generates a ruled real hypersurface. This representation agrees with the one given by M. Lohnherr and H. Reckziegel in the Riemannian case [3]. Further insights are given into the cases when the ruled real hypersurfaces are minimal or have constant sectional curvatures. The present results are part of a joint work together with prof. M. Ortega and prof. J.D. Pérez.
[1] H. Anciaux, K. Panagiotidou, Hopf Hypersurfaces in pseudo-Riemannian complex and para-complex space forms, Diff. Geom. Appl. 42 (2015) 1-14.
[2] M. Kimura, M. Ortega, Hopf Real Hypersurfaces in Indefinite Complex Projective, Mediterr. J. Math. (2019) 16:27.
[3] M. Lohnherr, H. Reckziegel, On ruled real hypersurfaces in complex space forms. Geom. Dedicata 74 (1999), no. 3, 267–286.
Where: https://oficinavirtual.ugr.es/redes/SOR/SALVEUGR/accesosala.jsp?IDSALA=22960561


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# Grassmann elements and geometric algebra Λ(V)

The Grassmann SubManifold elements $v_k\in\Lambda^1V$ and $w^k\in\Lambda^1V'$ are linearly independent vector and covector elements of $V$, while the Leibniz Operator elements $\partial_k\in L^1V$ are partial tangent derivations and $\epsilon_k\in L^1V'$ are dependent functions of the tangent manifold. Let $V\in\text{Vect}_{\mathbb k}$ be a TensorBundle with dual space $V'$ and the basis elements $w_k:V\rightarrow\mathbb K$, then for all $x\in V,c\in\mathbb K$ it holds: $(w^i+w^j)(x) = w^i(x)+w^j(x)$ and $(cw^k)(x) = cw^k(x)$ hold. An element of a mixed-symmetry TensorAlgebra{V} is a multilinear mapping that is formally constructed by taking the tensor products of linear and multilinear maps, $(\bigotimes_k \omega_k)(v_1,\dots,v_{\sum_k p_k}) = \prod_k \omega_k(v_1,\dots,v_{p_k})$. Higher grade elements correspond to SubManifold subspaces, while higher order function elements become homogenous polynomials and Taylor series.

julia> Λ(ℝ^3)
DirectSum.Basis{⟨+++⟩,8}(v, v₁, v₂, v₃, v₁₂, v₁₃, v₂₃, v₁₂₃)

julia> Λ(tangent(ℝ^2))
DirectSum.Basis{T¹⟨++₁⟩,8}(v, v₁, v₂, ∂₁, v₁₂, ∂₁v₁, ∂₁v₂, ∂₁v₁₂)

julia> Λ(tangent((ℝ^0)',3,3))
DirectSum.Basis{T³⟨¹²³⟩',8}(w, ϵ₁, ϵ₂, ϵ₃, ϵ₁₂, ϵ₁₃, ϵ₂₃, ϵ₁₂₃)

Combining the linear basis generating elements with each other using the multilinear tensor product yields a graded (decomposable) tensor SubManifold $\langle v_{i_1}\otimes\cdots\otimes v_{i_k}\rangle_k : V'^k\rightarrow\mathbb K$, where rank is determined by the sum of basis index multiplicities in the tensor product decomposition. The Grassmann anti-symmetric exterior basis is denoted by $v_{i_1\dots i_g}\in\Lambda^gV$ having the dual elements $w^{i_1\cdots i_g}\in\Lambda^gV'$, while the Leibniz symmetric basis will be denoted by $\partial_{i_1}^{\mu_1}\dots\partial_{i_g}^{\mu_g}\in L^gV$ with corresponding $\epsilon_{i_1}^{\mu_1}\dots\epsilon_{i_g}^{\mu_g}\in L^gV'$ adjoint elements. Combined, this space produces the full Leibniz tangent algebra $T^\mu V=V\oplus (\bigoplus_{g=1}^\mu L^g V)$ and the Grassmann exterior algebra $\Lambda V = \bigoplus_{g=1}^n\Lambda^g V$ with $2^n$ elements. The mixed index algebra $\Lambda(T^\mu V) = (\bigoplus_{g=1}^n\Lambda^g V)\oplus(\bigoplus_{g=1}^\mu L^g V)$ is partitioned into both symmetric and anti-symmetric tensor equivalence classes. Any mixed tensor SubManifold pair $\omega,\eta$ satisfies either

$\underbrace{\omega\otimes\eta = -\eta\otimes\omega}_{\text{anti-symmetric}} \qquad \text{or} \qquad \underbrace{\omega\otimes\eta = \eta\otimes\omega}_{\text{symmetric}}.$

For the oriented sets of the Grassmann exterior algebra, the parity of $(-1)^\Pi$ is factored into transposition compositions when interchanging ordering of the tensor product argument permutations. The symmetrical algebra does not need to track this parity, but has higher multiplicities in its indices. Symmetric differential function algebra of Leibniz trivializes the orientation into a single class of index multi-sets, while Grassmann's exterior algebra is partitioned into two oriented equivalence classes by anti-symmetry. Full tensor algebra can be sub-partitioned into equivalence classes in multiple ways based on the element symmetry, grade, and metric signature composite properties. Both symmetry classes can be characterized by the same geometric product.

julia> indices(Λ(3).v12)
2-element Array{Int64,1}:
1
2

A higher-order composite tensor element is an oriented-multi-set $X$ such that $v_X = \bigotimes_k v_{i_k}^{\otimes\mu_k}$ with the indices $X = \left((i_1,\mu_1),\dots,(i_g,\mu_g)\right)$ and $|X|=\sum_k\mu_k$ is tensor rank. Anti-symmetric indices $\Lambda X\subseteq\Lambda V$ have two orientations and higher multiplicities of them result in zero values, so the only interesting multiplicity is $\mu_k\equiv1$. The Leibniz-Taylor algebra is a quotient polynomial ring $LV\cong R[x_1,\dots,x_n]/\{\prod_{k=1}^{\mu+1} x_{p_k}\}$ so that $\partial_k^{\mu+1}$ is zero. Typically the $k$ in a product $\left(\partial_{p_1}\otimes\cdots\otimes\partial_{p_k}\right)^{(k)}$ is referred to as the order of the element if it is fully symmetric, which is overall tracked separately from the grade such that $\partial_k\langle v_j\rangle_r = \langle\partial_kv_j\rangle_r$ and $(\partial_k)^{(r)}\omega_j = (\partial_kv_j)^{(r)}$. There is a partitioning into even grade components $\omega_+$ and odd grade components $\omega_-$ such that $\omega_++\omega_-=\omega$.

Grassmann's exterior algebra doesn't invoke the properties of multi-sets, as it is related to the algebra of oriented sets; while the Leibniz symmetric algebra is that of unoriented multi-sets. Combined, the mixed-symmetry algebra yield a multi-linear propositional lattice. The formal sum of equal grade elements is an oriented Chain and with mixed grade it is a MultiVector simplicial complex. Thus, various standard operations on the oriented multi-sets are possible including $\cup,\cap,\oplus$ and the index operation $\ominus$, which is symmetric difference operation.

By virtue of Julia's multiple dispatch on the field type $\mathbb K$, methods can specialize on the dimension $n$ and grade $G$ with a TensorBundle{n} via the TensorAlgebra{V} subtypes, such as SubManifold{V,G}, Simplex{V,G,B,𝕂}, Chain{V,G,𝕂}, SparseChain{V,G,𝕂}, MultiVector{V,𝕂}, and MultiGrade{V,G} types.

The elements of the Basis can be generated in many ways using the SubManifold elements created by the @basis macro,

julia> using Grassmann; @basis ℝ'⊕ℝ^3 # equivalent to basis"-+++"
(⟨-+++⟩, v, v₁, v₂, v₃, v₄, v₁₂, v₁₃, v₁₄, v₂₃, v₂₄, v₃₄, v₁₂₃, v₁₂₄, v₁₃₄, v₂₃₄, v₁₂₃₄)

As a result of this macro, all of the SubManifold{V,G} elements generated by that TensorBundle become available in the local workspace with the specified naming. The first argument provides signature specifications, the second argument is the variable name for the TensorBundle, and the third and fourth argument are prefixes of the SubManifold vector names (and covector basis names). By default, $V$ is assigned the TensorBundle and $v$ is the prefix for the SubManifold elements.

julia> V # Minkowski spacetime
⟨-+++⟩

julia> typeof(V) # dispatch by vector space
SubManifold{⟨-+++⟩,4,0x000000000000000f}

julia> typeof(v13) # extensive type info
SubManifold{⟨-+++⟩,2,0x0000000000000005}

julia> 2v1 + v3 # vector Chain{V,1} element
2v₁ + 0v₂ + 1v₃ + 0v₄

julia> 5 + v2 + v234 # MultiVector{V} element
5 + 1v₂ + 1v₂₃₄

It is entirely possible to assign multiple different bases with different signatures without any problems. In the following command, the @basis macro arguments are used to assign the vector space name to $S$ instead of $V$ and basis elements to $b$ instead of $v$, so that their local names do not interfere:

julia> @basis "++++" S b;

julia> let k = (b1 + b2) - b3
for j ∈ 1:9
k = k * (b234 + b134)
println(k)
end end
0 + 1v₁₄ + 1v₂₄ + 2v₃₄
0 - 2v₁ - 2v₂ + 2v₃
0 - 2v₁₄ - 2v₂₄ - 4v₃₄
0 + 4v₁ + 4v₂ - 4v₃
0 + 4v₁₄ + 4v₂₄ + 8v₃₄
0 - 8v₁ - 8v₂ + 8v₃
0 - 8v₁₄ - 8v₂₄ - 16v₃₄
0 + 16v₁ + 16v₂ - 16v₃
0 + 16v₁₄ + 16v₂₄ + 32v₃₄

Alternatively, if you do not wish to assign these variables to your local workspace, the versatile constructors of DirectSum.Basis{V} can be used to contain them, which is exported to the user as the method Λ(V),

julia> G3 = Λ(3) # equivalent to Λ(V"+++"), Λ(ℝ^3), Λ.V3
DirectSum.Basis{⟨+++⟩,8}(v, v₁, v₂, v₃, v₁₂, v₁₃, v₂₃, v₁₂₃)

julia> G3.v13 ⊖ G3.v12
1v₂₃

The multiplication product used: $*$ or $\ominus$ is the geometric algebraic product.

Definition. The geometric algebraic product is the oriented symmetric difference operator $\ominus$ (weighted by the bilinear form $g$) and multi-set sum $\oplus$ applied to multilinear tensor products $\otimes$ in a single operation.

$\omega_X\ominus \eta_Y = \underbrace{\overbrace{(-1)^{\Pi(X,Y)}}^{\text{orient parity}}\overbrace{\det\left[g_{\Lambda(X\cap Y)}\right]}^{\text{intersect metric}} (\overbrace{\bigotimes_{k\in \Lambda(X\ominus Y)} v^{i_k}}^{(X\cup Y)\backslash(X\cap Y)}}_{\Lambda^1-anti-symmetric,\, \Lambda^g-mixed-symmetry})\otimes (\underbrace{\overbrace{\bigotimes_{k\in L(X\oplus Y)} \partial_{i_k}^{\otimes\mu_k}}^{\text{multi-set sum}}}_{L^g-symmetric})$

Remark: The product symbol $\ominus$ will be used to denote explicitly usage of the geometric algebraic product, although the standard number product $*$ notation could also be used. The $\ominus$ choice helps emphasize that the geometric algebraic product is characterized by symmetric differencing of anti-symmetric indices.

julia> (1 + 2v34) ⊖ (3 + 4v34), (1 + 2v34) * (3 + 4v34), (1 + 2im) * (3 + 4im)
(-5 + 10v₃₄, -5 + 10v₃₄, -5 + 10im)

Symmetry properties of the tensor algebra can be characterized in terms of the geometric product by two averaging operations, which are the symmetrization $\odot$ and anti-symmetrization $\boxtimes$ operators. These products satisfy various MultiVector properties, including the associative and distributive laws.

Definition (Exterior product): Let $w_k\in\Lambda^{p_k}V$, then for all $\sigma\in S_{\sum p_k}$ define an equivalence relation $\sim$ such that

$\bigwedge_k \omega_k(v_{1},\dots,v_{p_k}) \sim (-1)^{\Pi(\sigma)}(\bigotimes_k \omega_k)(v_{\sigma(1)},\dots,v_{\sigma(\sum p_k)})$

if and only if $\ominus_k\omega_k = \boxtimes_k\omega_k$ holds. It has become typical to use the $\wedge$ product symbol to denote products of such elements as $\bigwedge\Lambda V \equiv \bigotimes\Lambda V/\sim$ modulo anti-symmetrization.

julia> v3 ∧ v4, v4 ∧ v3, v3 ∧ v3
(v₃₄, -1v₃₄, 0v)

Remark. Observe that the anti-symmetric property implies that $\omega\otimes\omega=0$, while the symmetric property neither implies nor denies such a property. Grassmann remarked in 1862 that the symmetric algebra of functions is by far more complicated than his anti-symmetric exterior algebra. The first part of the book focused on anti-symmetric exterior algebra, while the more complex symmetric function algebra of Leibniz was subject of the second multivariable part of the book. Elements $\omega_k$ in the space $\Lambda V$ of anti-symmetric algebra are often studied as unit quantum state vectors in a unitary probability space, where $\sum_k\omega_k\neq\bigotimes_k\omega_k$ is entanglement.

Definition (Reverse, involute, conjugate). The reverse of $\langle\omega\rangle_r$ is defined as $\langle\tilde\omega\rangle_r = (-1)^{(r-1)r/2}\langle\omega\rangle_r$, while the involute is $\langle\omega\rangle_r^\times=(-1)^r\langle\omega\rangle_r$ and clifford $\langle\omega\rangle_r^\ddagger$ is the composition of involute and reverse.

julia> clifford(v234) == involute(~v234)
true

Definition (Reversed product). Define the index reversed product $\ast$ which yields a Hilbert space structure:

$\omega\ast\eta = \tilde\omega\ominus\eta, \quad \omega\ast'\eta = \omega\ominus\tilde\eta, \qquad |\omega|^2 = \omega\ast\omega, \quad |\omega| = \sqrt{\omega\ast\omega}, \quad ||\omega|| = \text{Euclidean }|\omega|.$

Remark. Observe that $\ast$ and $\ast'$ could both be exchanged in abs, abs2, and norm; however, these are different products. The scalar product $\circledast$ is the scalar part, so $\eta\circledast\omega = \langle\eta\ast\omega\rangle$.

julia> 2v34 ⊖ 2v34, 2v34 * 2v34, 2v34 ∗ 2v34, 2v34 ⊛ 2v34 # (gp, gp, rp, sp)
(-4v, -4v, 4v, 4v)

julia> abs2(2v34), abs(2v34), norm(2v34) # application of reverse product
(4v, 2.0v, 2.0)

Definition (Inverse). $\omega^{-1} = \omega\ast(\omega\ast\omega)^{-1} = \tilde\omega/|\omega|^2$, with $\eta/\omega = \eta\ominus\omega^{-1}$ and $\eta\backslash\omega = \eta^{-1}\ominus\omega$.

julia> 1/v34, inv(v34) == ~v34/abs2(v34)
(-1.0v₃₄, true)

Definition (Sandwich product). This product can be defined as $\eta\oslash\omega = \omega\backslash\eta\ominus\omega^\times$. Alternatively, the reversed definition is $\eta^\times\ominus\omega/\eta$ or in Julia η>>>ω, which is often found in literature.

julia> (2v3+5v4) ⊘ v3 == inv(v3)*(2v3+5v4)*involute(v3)
true

Remark. Observe that it is overall more simple and consistent to use $\{\ast,\oslash\}$ operations instead of the reversed.

The real part $\Re\omega = (\omega+\tilde\omega)/2$ is defined by $|\Re\omega|^2 = (\Re\omega)^{\ominus2}$ and the imag part $\Im\omega = (\omega-\tilde\omega)/2$ by $|\Im\omega|^2 = -(\Im\omega)^{\ominus2}$, such that $\omega = \Re\omega+\Im\omega$ has real and imaginary partitioned by

$\langle\tilde\omega\rangle_r/\left|\langle\omega\rangle_r\right| = \sqrt{\langle\tilde\omega\rangle_r^2/\big|\langle\omega\rangle_r\big|^2} = \sqrt{\langle\omega\rangle_r\ast\langle\omega\rangle_r^{-1}} = \sqrt{\langle\tilde\omega\rangle_r/\langle\omega\rangle_r}=\sqrt{(-1)^{(r-1)r/2}} \in\{1,\sqrt{-1}\},$

which is a unique partitioning completely independent of the metric space and manifold of the algebra.

$\omega\ast\omega = |\omega|^2 = |\Re\omega+\Im\omega|^2 = |\Re\omega|^2+|\Im\omega|^2 + 2\Re(\Re\omega\ast\Im\omega)$

The radial and angular components in a multivector exponential are partitioned by the parity of their metric.

It is possible to assign the quaternion generators $i,j,k$ with

julia> i,j,k = hyperplanes(ℝ^3)
3-element Array{Simplex{⟨+++⟩,2,B,Int64} where B,1}:
-1v₂₃
1v₁₃
-1v₁₂

julia> i^2, j^2, k^2, i*j*k
(-1v, -1v, -1v, -1v)

julia> -(j+k) * (j+k)
2v⃖

julia> -(j+k) * i
0 - 1v₁₂ - 1v₁₃

Alternatively, another representation of the quaternions is

julia> basis"--"
(⟨--⟩, v, v₁, v₂, v₁₂)

julia> v1^2, v2^2, v12^2, v1*v2*v12
(-1v, -1v, -1v, -1v)

The parametric type formalism in Grassmann is highly expressive to enable the pre-allocation of geometric algebra computations for specific sparse-subalgebras, including the representation of rotational groups, Lie bivector algebras, and affine projective geometry. All of this is enabled by the psuedoscalar complement duality.

Definition (Poincare-Hodge dual complement): Let $\star\langle\omega\rangle_p = \langle\omega\rangle_p\ast I = \langle\tilde\omega\rangle_p\ominus I$, then $\star : \Lambda^pV\rightarrow\Lambda^{n-p}V$.

Remark. While $\star\omega$ is complementrighthodge of $\omega$, the complementlefthodge would be $I\ast'\omega$. The $\star$ symbol was added to the Julia language as unary operator for ease of use with Grassmann on Julia's v1.2 release.

With LightGraphs.jl, GraphPlot.jl, Cairo.jl, Compose.jl it is possible to convert Grassmann numbers into graphs.

using Grassmann, Compose # environment: LightGraphs, GraphPlot
x = Λ(ℝ^7).v123
Grassmann.graph(x+!x)
draw(PDF("simplex.pdf",16cm,16cm),x+!x)

Figure. Triangle with its tetrahedron complement $v_{123} + \star v_{123}$ in $\mathbb R^7$.

John Browne has discussed Grassmann duality principle in book, stating that every theorem (involving either of the exterior and regressive products) can be translated into its dual theorem by replacing the $\wedge$ and $\vee$ operations and applying Poincare duality (homology). Applying this Grassmann duality principle to the $\wedge$ product, let $P=\sum_kp_k$, $\{\omega_k\}_k\in\Lambda^{p_k}V$, then it is possible to obtain the co-product $\bigvee :\Lambda^{p_1}V_1\times\dots\times\Lambda^{p_g}V_g \rightarrow \Lambda^{P-(g-1)\#V}\bigcup_k V_k$. Grassmann's original notation implicitly combined $\wedge,\vee,\star$. The join $\wedge$ product is analogous to union $\cup$, the meet $\vee$ product is analogous to intersection $\cap$, and the orthogonal complement $\star\mapsto^\perp$ is negation. Together, $(\wedge,\vee,\star)$ yield an orthocomplementary propositional lattice (quantum logic):

$(\star\bigvee_k \omega_k)(v_1,\dots,v_P) = (\bigwedge_k\star\omega_k)(v_1,\dots,v_P) \quad DeMorgan's\,Law,$

where DeMorgan's law is used to derive tensor contractions.

Definition. Symmetrically define skew left $\lrcorner$ and right $\llcorner$ contractions $\langle\omega\rangle_r\cdot\langle\eta\rangle_s = \begin{cases} \omega\llcorner\eta=\omega\vee\star\eta & r\geq s \\ \omega\lrcorner\eta=\eta\vee\star\omega & r\leq s \end{cases}$. Note for $\omega,\eta$ of equal grade, $\omega\circledast\eta = \omega\odot\eta = \omega\cdot\eta = \omega\llcorner\eta = \omega\lrcorner\eta$ are all symmetric. In Julia, $\lrcorner$ is $<$ and $\llcorner$ is $>$.

julia> (G3.v1 + G3.v2) ⋅ (1.5G3.v2 + G3.v3)
1.5v

Definition. Let $\nabla = \sum_k\partial_kv_k$ be a vector field and $\epsilon = \sum_k\epsilon_k(x)w_k \in \Omega^1V$ be unit sums of the mixed-symmetry basis. Elements of $\Omega^pV$ are known as differential $p$-forms and both $\nabla$ and $\epsilon$ are tensor fields dependent on $x\in W$. Another notation for a differential form is $dx_k = \epsilon_k(x)w_k$, such that $\epsilon_k = dx_k/w_k$ and $\partial_k\omega(x) = \omega'(x)$.

julia> tangent(ℝ^3)(∇)
0v₁₂ + 0v₁₃ + 1∂₁v₁ + 0v₂₃ + 1∂₁v₂ + 1∂₁v₃

julia> (ℝ^3)(∇)
1v₁ + 1v₂ + 1v₃

Remark. The space $W$ does not have to equal $V\in\text{Vect}_{\mathbb K}$ above, as $\Omega^pV$ could have coefficients from $\mathbb K = LW$.

Definition. Define differential $d:\Omega^p V\rightarrow\Omega^{p+1}V$ and co-differential $\delta:\Omega^pV\rightarrow\Omega^{p-1}V$ such that

$\star d\omega = \star(\nabla\wedge\omega) = \nabla\times\omega, \qquad \omega\cdot\nabla = \omega\vee\star\nabla = \partial\omega =-\delta\omega.$

Vorticity curl of vector-field: $\star d(dx_1+dx_2+dx_3) = (∂_2 -∂_3)dx_1 + (∂_3 -∂_1)dx_2 + (∂_1 -∂_2)dx_3$.

julia> @basis tangent(ℝ^3,2,3); ⋆d(v1+v2+v3)
0 - 1∂₂v₁ + 1∂₃v₁ + 1∂₁v₂ - 1∂₃v₂ - 1∂₁v₃ + 1∂₂v₃

Boundary of 3-simplex, faces of simplex (oriented): $\partial(v_{1234}) = -\partial_4v_{123}+\partial_3v_{124}-\partial_2v_{134}+\partial_1v_{234}$.

julia> ∂(Λ(tangent(ℝ^4,2,4)).v1234)
0.0 - 1∂₄v₁₂₃ + 1∂₃v₁₂₄ - 1∂₂v₁₃₄ + 1∂₁v₂₃₄

These two maps have the special properties $d\circ d=0$ and $\partial\circ\partial = 0$ for any form $\omega$ and vector field $\nabla$. In topology there is boundary operator $\partial$ defined by $\partial\epsilon = \epsilon\cdot\nabla = \sum_k\partial_k\epsilon_k$ and is commonly discussed in terms the limit $\epsilon(x)\cdot\nabla\omega(x) = \lim_{h\rightarrow0} \frac{\omega(x+h\epsilon)-\omega(x)}{h}$, which is the directional derivative.

Theorem (Integration by parts & Stokes). Let $\nabla \in\Omega_1 V$ be a Leibnizian vector field operator, then $d,-\partial$ are Hilbert adjoint Hodge-DeRahm operators with

$\int_M d\omega\wedge\star\eta +\int_M \omega\wedge\star\partial\eta = 0, \qquad \langle d\omega\ast\eta\rangle =\langle\omega\ast-\partial\eta\rangle.$

Proof. Recall, $\partial\omega = \omega\cdot\nabla = \star^{-1}(\star\omega\wedge\star^2\nabla) = (-1)^n(-1)^{nk}\star d\star\omega$. Then substitute this into the integral $\int_M \omega\wedge(-1)^{mk+m+1}\star\star d\star\eta = (-1)^{km+m+1}(-1)^{(m-k+1)(k-1)}\int_M\omega\wedge d\star\eta$, and apply the identity $(-1)^{km+m+1}(-1)^{(m-k+1)(k-1)}=(-1)^k$ and $(-1)^k\int_M\omega\wedge d\star\eta = \int_M d(\omega\wedge\star\eta) - (-1)^{k-1}\omega\wedge d\star\eta = \int_M d\omega\wedge\star\eta$. Stokes identity can be proved by relying on a variant of the common factor theorem by Browne.

Theorem (Clifford-Dirac-Laplacian) Dirac operator is $(\nabla^2)^\frac12\omega = \pm\nabla\ominus\omega = \pm\nabla\wedge\omega \pm \nabla\cdot\omega = \pm d\omega\pm\partial\omega$.

$\nabla^2\omega = \nabla\wedge(\omega\cdot\nabla) + (\nabla\wedge\omega)\cdot\nabla) = \mp(\mp\omega\ominus\nabla)\ominus\nabla).$

Elements $\omega\in\mathcal H^p M = \{\nabla\omega = 0\mid\omega\in \Omega^pM\}$ are harmonic forms if $\nabla\omega = 0$ and hence both closed $d\omega=0$ and coclosed $\delta\omega=0$. Hodge decomposition: $\Omega^pM=\mathcal H^pM\oplus\text{im}(d\Omega^{p-1}M)\oplus\text{im}(\partial\Omega^{p+1}M)$.

julia> ω = 4.5v12 + 7.4v13
4.5v₁₂ + 7.4v₁₃ + 0.0∂₁v₁ + 0.0∂₂v₁ + 0.0∂₃v₁ + 0.0v₂₃ + 0.0∂₁v₂ + 0.0∂₂v₂ + 0.0∂₃v₂ + 0.0∂₁v₃ + 0.0∂₂v₃ + 0.0∂₃v₃ + 0.0∂₁₂ + 0.0∂₁₃ + 0.0∂₂₃

julia> V(∇^2)*ω == V(∇)*V(∇)*ω == d(∂(ω)) + ∂(d(ω))
true

Let $\nabla\in\Lambda^1V$, then $\omega = (\nabla\backslash\nabla)\ominus\omega = \nabla\backslash(d\omega + \partial\omega)$ where $\nabla\parallel\partial\omega$ and $\nabla\perp d\omega$. Let's reflect across the hyperplane $\star\nabla$, then $\nabla\backslash (d\omega-\partial\omega) = \nabla\backslash(d\omega-\partial\omega)\ominus(\nabla\backslash\nabla) = -\nabla^2\backslash(d\omega+\partial\omega)\ominus\nabla = -\nabla\backslash\omega\ominus\nabla$. Hence, reflection by hyperplane $\star\nabla$ has isometry $\omega\oslash\nabla$ which is a versor outermorphism.

Theorem (Cartan-Dieudonne). Every isometry of $V\rightarrow V$ is the composite of at most $k$ reflections across non-singular hyperplanes. Hence there exist vectors $\nabla_j$ such that

$(((\omega\oslash\nabla_1)\oslash\nabla_2)\oslash\cdots)\oslash\nabla_k = \omega\oslash(\nabla_1\ominus\nabla_2\ominus\dots\ominus\nabla_k)$

for any isometry element of the orthogonal group $O(p,q)$. Note that elements under transformations of this group preserve inner product relations. The even grade operators make up the rotational group, where each bivector isometry is a composition of two reflections.

Consider the differential equation $\partial_i\epsilon_j = \epsilon_j\oslash\omega$ with the solution $\epsilon_j(x) = \epsilon_j(0)\oslash e^{x_i\omega}$ where $\theta =2 x_i$ is the parameter of the Lie group. Then for a normalized $\omega$,

$e^{\theta\omega} = \sum_k \frac{(\theta\omega)^{\ominus k}}{k!} = \begin{cases} \cosh\theta+\omega\sinh\theta, & \text{if } \omega^2 = 1, \\ \cos\theta + \omega\sin\theta, & \text{if } \omega^2=-1, \\ 1+\theta\omega, & \text{if } \omega^2=0. \end{cases}$

Note that $\nabla\oslash e^{\theta\omega/2} = \nabla \ominus e^{\theta\omega}$ is a double covering when using the complex numbers in the Euclidean plane.

Due to GeometryTypes.jl Point interoperability, plotting and visualizing with Makie.jl is easily possible. For example, the vectorfield method creates an anonymous Point function that applies a versor outermorphism:

using Grassmann, Makie
basis"2" # Euclidean
streamplot(vectorfield(exp(π*v12/2)),-1.5..1.5,-1.5..1.5)
streamplot(vectorfield(exp((π/2)*v12/2)),-1.5..1.5,-1.5..1.5)
streamplot(vectorfield(exp((π/4)*v12/2)),-1.5..1.5,-1.5..1.5)
streamplot(vectorfield(v1*exp((π/4)*v12/2)),-1.5..1.5,-1.5..1.5)
@basis S"+-" # Hyperbolic
streamplot(vectorfield(exp((π/8)*v12/2)),-1.5..1.5,-1.5..1.5)
streamplot(vectorfield(v1*exp((π/4)*v12/2)),-1.5..1.5,-1.5..1.5)

using Grassmann, Makie
@basis S"∞+++"
f(t) = (↓(exp(π*t*((3/7)*v12+v∞3))>>>↑(v1+v2+v3)))
lines(V(2,3,4).(points(f)))
@basis S"∞∅+++"
f(t) = (↓(exp(π*t*((3/7)*v12+v∞3))>>>↑(v1+v2+v3)))
lines(V(3,4,5).(points(f)))

using Grassmann, Makie; @basis S"∞+++"
streamplot(vectorfield(exp((π/4)*(v12+v∞3)),V(2,3,4)),-1.5..1.5,-1.5..1.5,-1.5..1.5,gridsize=(10,10))

using Grassmann, Makie; @basis S"∞+++"
streamplot(vectorfield(exp((π/4)*(v12+v∞3)),V(2,3,4),V(1,2,3)),-1.5..1.5,-1.5..1.5,-1.5..1.5,gridsize=(10,10))

using Grassmann, Makie; @basis S"∞+++"
f(t) = ↓(exp(t*v∞*(sin(3t)*3v1+cos(2t)*7v2-sin(5t)*4v3)/2)>>>↑(v1+v2-v3))
lines(V(2,3,4).(points(f)))

using Grassmann, Makie; @basis S"∞+++"
f(t) = ↓(exp(t*(v12+0.07v∞*(sin(3t)*3v1+cos(2t)*7v2-sin(5t)*4v3)/2))>>>↑(v1+v2-v3))
lines(V(2,3,4).(points(f)))

As a result of Grassmann's exterior & interior products, the Hodge-DeRahm chain complex from cohomology theory has dimensional equivalence brought by the Grassmann-Poincare-Hodge complement duality,

$\mathcal H^{n-p}M \cong \frac{\text{ker}(d\Omega^{n-p}M)}{\text{im}(d\Omega^{n-p+1}M)}, \qquad \dim\mathcal H^pM = \dim\frac{\text{ker}(\partial\Omega^pM)}{\text{im}(\partial\Omega^{p+1}M)}.$

The rank of the grade $p$ boundary incidence operator is

$\text{rank}\langle\partial\langle M\rangle_{p+1}\rangle_p = \min\{\dim\langle\partial\langle M\rangle_{p+1}\rangle_p,\dim\langle M\rangle_{p+1}\}.$

Invariant topological information can be computed using the rank of homology groups, where $b_p(M)=\dim\mathcal H^pM$

$b_p(M) = \dim\langle M\rangle_{p+1} - \text{rank}\langle\partial\langle M\rangle_{p+1}\rangle_p - \text{rank}\langle\partial\langle M\rangle_{p+2}\rangle_{p+1}$

are the Betti numbers with Euler characteristic $\chi(M) = \sum_p (-1)^pb_p$.

Let's obtain the full skeleton of a simplical complex $\Delta(\omega)=\mathcal P(\omega)\backslash\Lambda^0(V)$ from the power set $\mathcal P(\omega)$ of all vertices with each subcomplex $\Delta(\partial(\omega))$ contained in the edge graph:

$\Delta(\omega) = \sum_{g=1}^n\sum_{k=1}^{n\choose g}\left(\text{abs}\langle\omega\rangle_{g,k} + \Delta\left(\text{abs}\,\partial\langle\omega\rangle_{g,k}\right)\right).$

Compute the value $\chi(\Delta(\omega))=1$ and $\chi(\Delta(\partial(\omega))) = \, ?$ for any Simplex $\omega$. As an exercise, also compute the corresponding betti numbers..

julia> [(χ(Δ(ω)),χ(Δ(∂(ω)))) for ω ∈ (Λ(5).v12,Λ(5).v123,Λ(5).v1234,Λ(5).v12345)]
4-element Array{Tuple{Int64,Int64},1}:
(1, 2)
(1, 0)
(1, 2)
(1, 0)

These methods can be applied to any MultiVector simplicial complex.

## Null-basis of the projective split

Let $v_\pm^2 = \pm1$ be a basis with $v_\infty = v_++v_-$ and $v_\emptyset = (v_--v_+)/2$ An embedding space $\mathbb R^{p+1,q+1}$ carrying the action from the group $O(p+1,q+1)$ then has $v_\infty^2 =0$, $v_\emptyset^2 =0$, $v_\infty \cdot v_\emptyset = -1$, and $v_{\infty\emptyset}^2 = 1$ with Minkowski plane $v_{\infty\emptyset}$ having the Hestenes-Dirac-Clifford product properties,

julia> using Grassmann; @basis S"∞∅++"
(⟨∞∅++⟩, v, v∞, v∅, v₁, v₂, v∞∅, v∞₁, v∞₂, v∅₁, v∅₂, v₁₂, v∞∅₁, v∞∅₂, v∞₁₂, v∅₁₂, v∞∅₁₂)

julia> v∞^2, v∅^2, v1^2, v2^2
(0v, 0v, 1v, 1v)

julia> v∞ ⋅ v∅, v∞∅^2
(-1v, 1v)

julia> v∞∅ * v∞, v∞∅ * v∅
(-1v∞, 1v∅)

julia> v∞ * v∅, v∅ * v∞
(-1 + 1v∞∅, -1 - 1v∞∅)

For the null-basis, complement operations are different:

$\star v_\infty = \star(v_++v_-) = (v_- + v_+)v_{1...n} = v_{\infty1...n}$
$\star 2v_\emptyset = \star(v_--v_+) = (v_+ - v_-)v_{1...n} = -2v_{\emptyset1...n}$

The Hodge complement satisfies $\langle\omega\ast\omega\rangle I=\omega\wedge\star\omega$. This property is naturally a result of using the geometric product in the definition. An additional metric independent version of the complement operation is available with the ! operator,

$!v_\infty = !(v_++v_-) = (v_- - v_+)v_{1...n} = 2v_{\emptyset1...n}$
$!2v_\emptyset = !(v_--v_+) = (v_+ + v_-)v_{1...n} = -v_{\infty1...n}$

For that variation of complement, $||\omega||^2 I = \omega\,\wedge\,!\omega$ holds.

julia> ⋆v∞, !v∞, ⋆v∅, !v∅
(1v∞₁₂, 2v∅₁₂, -1v∅₁₂, -0.5v∞₁₂)

julia> !v∞ * v12 == -2v∅, !v∅ * v12 == v∞/2
(true, true)

julia> ⋆v∞ * v12 == -v∞, ⋆v∅ * v12 == v∅
(true, true)

julia> v∞ * !v∞, v∅ * !v∅
(0 - 2v₁₂ + 2v∞∅₁₂, -0.0 + 0.5v₁₂ + 0.5v∞∅₁₂)

In this example, the null-basis properties from the projective split are shown.

julia> tangent(S"∞∅++",2,4)(∇^2)
0 + 1∂₃∂₃ + 1∂₄∂₄ - 2∂₁₂

## Differential forms and tangent algebra

Definition (Symmetric Leibniz differentials): Let $\partial_k = \frac\partial{\partial x_k}\in L_gV\,$ be Leibnizian symmetric tensors, then there is an equivalence relation $\asymp$ which holds for each $\sigma\in S_p$

$(\partial_p \circ \dots\circ \partial_1)\omega \asymp(\bigotimes_k \partial_{\sigma(k)})\omega \iff \ominus_k\partial_k = \bigodot_k\partial_k,$

along with each derivation $\partial_k(\omega\eta) = \partial_k(\omega)\eta + \omega\partial_k(\eta)$.

The product rule is encoded into Grassmann algebra when a tangent bundle is used, demonstrated here symbolically with Reduce by using the dual number definition:

julia> using Grassmann, Reduce
Reduce (Free CSL version, revision 4590), 11-May-18 ...

julia> @mixedbasis tangent(ℝ^1)
(⟨+-₁¹⟩*, v, v₁, w¹, ϵ₁, ∂¹, v₁w¹, v₁ϵ₁, v₁∂¹, w¹ϵ₁, w¹∂¹, ϵ₁∂¹, v₁w¹ϵ₁, v₁w¹∂¹, v₁ϵ₁∂¹, w¹ϵ₁∂¹, v₁w¹ϵ₁∂¹)

julia> a,b = :x*v1 + :dx*ϵ1, :y*v1 + :dy*ϵ1
(xv₁ + dxϵ₁, yv₁ + dyϵ₁)

julia> a * b
x * y + (dy * x + dx * y)v₁ϵ₁

Higher order and multivariable Taylor numbers are also supported.

julia> @basis tangent(ℝ,2,2) # 1D Grade, 2nd Order, 2 Variables
(T²⟨+₁₂⟩, v, v₁, ∂₁, ∂₂, ∂₁v₁, ∂₂v₁, ∂₁₂, ∂₁₂v₁)

julia> ∂1 * ∂1v1
∂₁∂₁v₁

julia> ∂1 * ∂2
∂₁₂

julia> v1*∂12
∂₁₂v₁

julia> ∂12*∂2 # 3rd order is zero
0v

julia> @mixedbasis tangent(ℝ^2,2,2); # 2D Grade, 2nd Order, 2 Variables

julia> V(∇) # vector field
0v₁₂ + 1∂₁v₁ + 0∂₂v₁ + 0∂₁v₂ + 1∂₂v₂ + 0∂₁₂

julia> V(∇) ⋅ V(∇) # Laplacian
0 + 1∂₁∂₁ + 1∂₂∂₂

julia> ans*∂1 # 3rd order is zero
0.0v⃖

Multiplication with an $\epsilon_i$ element is used help signify tensor fields so that differential operators are automatically applied in the SubManifold algebra as ∂ⱼ⊖(ω⊗ϵᵢ) = ∂ⱼ(ωϵᵢ) ≠ (∂ⱼ⊗ω)⊖ϵᵢ.

julia> using Reduce, Grassmann; @mixedbasis tangent(ℝ^2,3,2);

julia> (∂1+∂12) * (:(x1^2*x2^2)*ϵ1 + :(sin(x1))*ϵ2)
0.0 + (2 * x1 * x2 ^ 2)∂₁ϵ¹ + (cos(x1))∂₁ϵ² + (4 * x1 * x2)∂₁₂ϵ¹

Although fully generalized, the implementation in this release is still experimental.

## Symbolic coefficients by declaring algebra

Due to the abstract generality of the code generation of the Grassmann product algebra, it is easily possible to extend the entire set of operations to other kinds of scalar coefficient types.

julia> using GaloisFields, Grassmann

julia> const F = GaloisField(7)
𝔽₇

julia> basis"2"
(⟨++⟩, v, v₁, v₂, v₁₂)

julia> F(3)*v1
3v₁

julia> inv(ans)
5v₁

By default, the coefficients are required to be <:Number. However, if this does not suit your needs, alternative scalar product algebras can be specified with

Grassmann.generate_algebra(:AbstractAlgebra,:SetElem)

where :SetElem is the desired scalar field and :AbstractAlgebra is the scope which contains the scalar field.

With the usage of Requires, symbolic scalar computation with Reduce.jl and other packages is automatically enabled,

julia> using Reduce, Grassmann
Reduce (Free CSL version, revision 4590), 11-May-18 ...

julia> basis"2"
(⟨++⟩, v, v₁, v₂, v₁₂)

julia> (:a*v1 + :b*v2) ⋅ (:c*v1 + :d*v2)
(a * c + b * d)v

julia> (:a*v1 + :b*v2) ∧ (:c*v1 + :d*v2)
0.0 + (a * d - b * c)v₁₂

julia> (:a*v1 + :b*v2) * (:c*v1 + :d*v2)
a * c + b * d + (a * d - b * c)v₁₂

If these compatibility steps are followed, then Grassmann will automatically declare the product algebra to use the Reduce.Algebra symbolic field operation scope.

julia> using Reduce,Grassmann; basis"4"
Reduce (Free CSL version, revision 4590), 11-May-18 ...
(⟨++++⟩, v, v₁, v₂, v₃, v₄, v₁₂, v₁₃, v₁₄, v₂₃, v₂₄, v₃₄, v₁₂₃, v₁₂₄, v₁₃₄, v₂₃₄, v₁₂₃₄)

julia> P,Q = :px*v1 + :py*v2 + :pz* v3 + v4, :qx*v1 + :qy*v2 + :qz*v3 + v4
(pxv₁ + pyv₂ + pzv₃ + 1.0v₄, qxv₁ + qyv₂ + qzv₃ + 1.0v₄)

julia> P∧Q
0.0 + (px * qy - py * qx)v₁₂ + (px * qz - pz * qx)v₁₃ + (px - qx)v₁₄ + (py * qz - pz * qy)v₂₃ + (py - qy)v₂₄ + (pz - qz)v₃₄

julia> R = :rx*v1 + :ry*v2 + :rz*v3 + v4
rxv₁ + ryv₂ + rzv₃ + 1.0v₄

julia> P∧Q∧R
0.0 + ((px * qy - py * qx) * rz - ((px * qz - pz * qx) * ry - (py * qz - pz * qy) * rx))v₁₂₃ + (((px * qy - py * qx) + (py - qy) * rx) - (px - qx) * ry)v₁₂₄ + (((px * qz - pz * qx) + (pz - qz) * rx) - (px - qx) * rz)v₁₃₄ + (((py * qz - pz * qy) + (pz - qz) * ry) - (py - qy) * rz)v₂₃₄

It should be straight-forward to easily substitute any other extended algebraic operations and fields; issues with questions or pull-requests to that end are welcome.


text:
### Generalized Channels from Limited Blockchain Scripts and Adaptor Signatures

Lukas Aumayr, Oguzhan Ersoy, Andreas Erwig, Sebastian Faust, Kristina Hostakova, Matteo Maffei, Pedro Moreno-Sanchez, and Siavash Riahi

##### Abstract

Decentralized and permissionless ledgers offer an inherently low transaction rate, as a result of their consensus protocol demanding the storage of each transaction on-chain. A prominent proposal to tackle this scalability issue is to utilize off-chain protocols, where parties only need to post a limited number of transactions on-chain. Existing solutions can roughly be categorized into: (i) application-specific channels (e.g., payment channels), offering strictly weaker functionality than the underlying blockchain; and (ii) state channels, supporting arbitrary smart contracts at the cost of being compatible only with the few blockchains having Turing-complete scripting languages (e.g., Ethereum). In this work, we introduce and formalize the notion of generalized channels allowing users to perform any operation supported by the underlying blockchain in an off-chain manner. Generalized channels thus extend the functionality of payment channels and relax the definition of state channels. We present a concrete construction compatible with any blockchain supporting transaction authorization, time-locks and constant number of Boolean $\land$ and $\lor$ operations -- requirements fulfilled by many (non-Turing-complete) blockchains including the popular Bitcoin. To this end, we leverage adaptor signatures -- a cryptographic primitive already used in the cryptocurrency literature but formalized as a standalone primitive in this work for the first time. We formally prove the security of our generalized channel construction in the Universal Composability framework. As an important practical contribution, our generalized channel construction outperforms the state-of-the-art payment channel construction, the Lightning Network, in efficiency. Concretely, it halves the off-chain communication complexity and reduces the on-chain footprint in case of disputes from linear to constant in the number of off-chain applications funded by the channel. Finally, we evaluate the practicality of our construction via a prototype implementation and discuss various applications including financially secured fair two-party computation.

Available format(s)
Category
Cryptographic protocols
Publication info
A major revision of an IACR publication in ASIACRYPT 2021
Keywords
Contact author(s)
o ersoy @ tudelft nl
History
2022-05-18: last of 2 revisions
See all versions
Short URL
https://ia.cr/2020/476

CC BY

BibTeX

@misc{cryptoeprint:2020/476,
author = {Lukas Aumayr and Oguzhan Ersoy and Andreas Erwig and Sebastian Faust and Kristina Hostakova and Matteo Maffei and Pedro Moreno-Sanchez and Siavash Riahi},
title = {Generalized Channels from Limited Blockchain Scripts and Adaptor Signatures},
howpublished = {Cryptology ePrint Archive, Paper 2020/476},
year = {2020},
note = {\url{https://eprint.iacr.org/2020/476}},
url = {https://eprint.iacr.org/2020/476}
}

Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.


text:
Collaborate
This blog is managed and hosted on GitHub. If you wish to update the contents of this post or if you have found an inaccuracy and wish to make corrections, we recommend that you please submit a pull request to this repository.

# Overview

When CAS is configured to act as an OAuth identity provider, it begins to issue access tokens that are by default opaque identifiers. There is also the option to generate JWTs as access tokens on a per-application basis. Using JWTs, CAS can create JSON documents to encode all relevant parts of an access token into the token itself. The main benefit of this is that API servers can verify access tokens without doing a token lookup on every API request, making the API much more easily scalable. Also, this means that applications don’t need to be aware of how CAS implements access tokens which makes it possible to change the implementation later without affecting clients.

Our starting position is based on:

# Configuration

First, let’s create a few mock attributes that ought to be released to our sample yet-to-be-registered OAuth application:

cas.authn.attributeRepository.stub.attributes.cn=Misagh
cas.authn.attributeRepository.stub.attributes.sn=Moayyed
cas.authn.attributeRepository.stub.attributes.mail=mm1844@gmail.com

Once the OAuth module is included in the WAR Overlay, we can begin to register a simple OAuth application with CAS using the following JSON service definition:

{
"@class" : "org.apereo.cas.support.oauth.services.OAuthRegisteredService",
"clientId": "client",
"clientSecret": "secret",
"serviceId" : "https://example.net/dashboard",
"name" : "OAUTH",
"id" : 1,
"attributeReleasePolicy" : {
"@class" : "org.apereo.cas.services.ReturnAllowedAttributeReleasePolicy",
"allowedAttributes" : [ "java.util.ArrayList", [ "cn", "mail", "sn" ] ]
},
"supportedGrantTypes": [ "java.util.HashSet", [ "password" ] ]
}

A few things to note:

• Our application has the usual clientId, clientSecret and redirectUri (i.e. serviceId) defined.
• The cn, mail, and sn attributes are selectively defined to be released to the application.
• CAS will only interact with the application using the password grant, which we will use to request access tokens either in plain or JWT format.

Let’s start simple, by using the password grant to request an access token without any extra configurations:

\$ curl https://sso.example.org/cas/oauth2.0/token?grant_type=password'&'\

The above request first authenticates the request using the provided username and password. Once the application policy is located and verified by CAS, an access token can be provided in the response:

{
"access_token": "AT-1-wiNsTgaHzXLUIyaaoFoip-znohWPihea",
"token_type": "bearer",
"expires_in": 28800,
"scope": ""
}

We can, of course, use the access token in exchange for user profile information:

curl -k --user client:secret https://sso.example.org/cas/oauth2.0/profile?\
access_token=AT-1-wiNsTgaHzXLUIyaaoFoip-znohWPihea

{
"cn": "Misagh",
"mail": "mm1844@gmail.com",
"sn": "Moayyed",
"service": "client",
"id": "casuser",
"client_id": "client"
}

{
"@class" : "org.apereo.cas.support.oauth.services.OAuthRegisteredService",
"clientId": "client",
"clientSecret": "secret",
"serviceId" : "https://example.net/dashboard",
"name" : "OAUTH",
"jwtAccessToken": true,
"id" : 1,
"attributeReleasePolicy" : {
"@class" : "org.apereo.cas.services.ReturnAllowedAttributeReleasePolicy",
"allowedAttributes" : [ "java.util.ArrayList", [ "cn", "mail", "sn" ] ]
},
"supportedGrantTypes": [ "java.util.HashSet", [ "password" ] ]
}

With the addition of the jwtAccessToken field, CAS will render access tokens as JWTs that are by default signed and encrypted using (pre-generated, if undefined) keys. So, let’s start simple and force CAS to disable signing and encryption of such tokens so we can unpack them easier later for verification:

# Force keys to be blank
cas.authn.oauth.access-token.crypto.encryption.key=
cas.authn.oauth.access-token.crypto.signing.key=

cas.authn.oauth.access-token.crypto.enabled=false
cas.authn.oauth.access-token.crypto.signing-enabled=false
cas.authn.oauth.access-token.crypto.encryption-enabled=false

{
"access_token": "eyJhbGciOi...",
"token_type": "bearer",
"expires_in": 28800,
"scope": ""
}

Since the JWT is plain this time around, we can easily unpack it using a service like jwt.io to verify the embedded JSON:

{
"sub": "casuser",
"mail": "mm1844@gmail.com",
"roles": [],
"iss": "https://sso.example.org/cas",
"cn": "Misagh",
"nonce": "",
"client_id": "client",
"aud": "client",
"permissions": [],
"scope": [],
"claims": [],
"scopes": [],
"state": "",
"sn": "Moayyed",
"exp": 1572837100,
"iat": 1572808300,
"jti": "AT-1-ibYxeSXhcU1N-0sF1JQXdgX4YAmBgCXY"
}

Of course, we can exchange the very same JWT for user profile information just as we did with a plain access token:

{
"cn": "Misagh",
"mail": "mm1844@gmail.com",
"sn": "Moayyed",
"service": "client",
"id": "casuser",
"client_id": "client"
}

## Signing & Encryption

If we wanted, we could turn on signing and encryption of our JWT access tokens:

cas.authn.oauth.accessToken.crypto.encryption.key=4fdqpa_mlx1XMtQR...
cas.authn.oauth.accessToken.crypto.signing.key=FXdUERkUNGqmai8oociQOyrHCQVYSW...
cas.authn.oauth.accessToken.crypto.enabled=true
cas.authn.oauth.accessToken.crypto.signing-enabled=true
cas.authn.oauth.accessToken.crypto.encryption-enabled=true

The same exercise can be repeated to make sure an encrypted/signed JWT can be decoded back to produce user profile information.

Of course, keys can always belong to a specific service definition, overriding the global default. If we wanted to, we could modify our sample service definition as such:

{
"@class" : "org.apereo.cas.support.oauth.services.OAuthRegisteredService",
"clientId": "client",
"clientSecret": "secret",
"serviceId" : "https://example.net/dashboard",
"name" : "OAUTH",
"jwtAccessToken": true,
"id" : 1,
"attributeReleasePolicy" : {
"@class" : "org.apereo.cas.services.ReturnAllowedAttributeReleasePolicy",
"allowedAttributes" : [ "java.util.ArrayList", [ "cn", "mail", "sn" ] ]
},
"supportedGrantTypes": [ "java.util.HashSet", [ "password" ] ],
"properties" : {
"@class" : "java.util.HashMap",
"accessTokenAsJwtSigningKey" : {
"@class" : "org.apereo.cas.services.DefaultRegisteredServiceProperty",
"values" : [ "java.util.HashSet", [ "..." ] ]
},
"accessTokenAsJwtEncryptionKey" : {
"@class" : "org.apereo.cas.services.DefaultRegisteredServiceProperty",
"values" : [ "java.util.HashSet", [ "..." ] ]
},
"accessTokenAsJwtSigningEnabled" : {
"@class" : "org.apereo.cas.services.DefaultRegisteredServiceProperty",
"values" : [ "java.util.HashSet", [ "true" ] ]
},
"accessTokenAsJwtEncryptionEnabled" : {
"@class" : "org.apereo.cas.services.DefaultRegisteredServiceProperty",
"values" : [ "java.util.HashSet", [ "true" ] ]
}
}
}

All properties should be optional; You may only specify that which you intend to override.

While it’s nice to allow JWT access tokens on a per-service basis, you may want to extend that behavior to all applications and make JWT access tokens the global default. To do, you would need to turn on the following setting:

cas.authn.oauth.accessToken.createAsJwt=true

When ciphers are turned on, JWT access tokens are by default (whether it’s global or for a specific service) are always encrypted first and then signed. You can certainly change the strategy type to reverse this behavior either globally or for a specific relying party:

# cas.authn.oauth.accessToken.crypto.strategy-type=ENCRYPT_AND_SIGN
cas.authn.oauth.accessToken.crypto.strategy-type=SIGN_AND_ENCRYPT

## Bonus

You may have noticed that our JSON service definition contains a client secret in plain text. However, client secrets can also be kept as encrypted secrets; To be clear, authorized relying parties always have access to and submit the client secret in plain text and CAS will auto-reverse the encryption of the secret found in the service definition file for verification and matching.

Skipping other details for brevity, our service file could take on the following form:

{
"@class" : "org.apereo.cas.support.oauth.services.OAuthRegisteredService",
"clientId": "client",
"clientSecret": "{cas-cipher}eyJhbGciOiJIUzUxMiIs...",
"serviceId" : "https://example.net/dashboard",
"name" : "OAUTH",
"jwtAccessToken": true,
"id" : 1
...
}

All you’d have to do is to take a plain secret and use the CAS Command-line Shell to transform it into encrypted form. The encryption and signing keys for client secrets may be defined via the following settings:

cas.authn.oauth.crypto.encryption.key=...
cas.authn.oauth.crypto.signing.key=...
cas.authn.oauth.crypto.enabled=true
cas.authn.oauth.crypto.signing-enabled=true
cas.authn.oauth.crypto.encryption-enabled=true

Configuration Namespaces
Note the similarities of the above configuration block and that of access tokens when it comes to the crypto namespace. This is not by chance, as configuration namespaces in CAS are internally reused everywhere to streamline the specification and validation process as much as possible for maximum code re-use. In most cases, such namespaces in CAS configuration settings are transferable to other areas that declare support for the same feature or namespace.

# So…

I hope this review was of some help to you and I am sure that both this post as well as the functionality it attempts to explain can be improved in any number of ways. Please know that all other use cases, scenarios, features, and theories certainly are possible as well. Feel free to engage and contribute as best as you can.

Finally, if you benefit from Apereo CAS as free and open-source software, we invite you to join the Apereo Foundation and financially support the project at a capacity that best suits your deployment. If you consider your CAS deployment to be a critical part of the identity and access management ecosystem and care about its long-term success and sustainability, this is a viable option to consider.

Happy Coding,

Misagh Moayyed