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Cryptology ePrint Archive: Report 2011/560

Randomized Secure Two-Party Computation for Modular Conversion, Zero Test, Comparison, MOD and Exponentiation

Ching-Hua Yu and Bo-Yin Yang

Abstract: When secure arithmetic is required, computation based on secure multiplication ($\MULT$) is much more efficient than computation based on secure boolean circuits. However, a typical application can also require other building blocks, such as comparison, exponentiation and the modulo (MOD) operation. Secure solutions for these functions proposed in the literature rely on bit-decomposition or other bit-oriented methods, which require $O(\ell)$ $\MULT$s for $\ell$-bit inputs. In the absence of a known bit-length independent solution, the complexity of the whole computation is often dominated by these non-arithmetic functions.

To resolve the above problem, we start with a general modular conversion, which converts secret shares over distinct moduli. For this, we proposed a probabilistically correct protocol for this with a complexity that is independent of $\ell$. Then, we show that when these non-arithmetic functions are based on secure modular conversions, they can be computed in constant rounds and $O(k)$ $\MULT$s, where $k$ is a parameter for an error rate of $2^{-\Omega(k)}$. To promote our protocols to be actively secure, we apply $O(k)$ basic zero-knowledge proofs, which cost at most $O(k)$ exponentiation computation, $O(1)$ rounds and $O(k(\ell+\kappa))$ communication bits, where $\kappa$ is the security parameter used in the commitment scheme.

Category / Keywords: applications / secure two-party computation, randomized algorithm, efficiency

Date: received 14 Oct 2011, last revised 15 Nov 2011

Contact author: chinghua yu at gmail com

Available format(s): PDF | BibTeX Citation

Short URL: ia.cr/2011/560

[ Cryptology ePrint archive ]


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# Dirichlet-to-Neumann Maps: Spectral Theory, Inverse Problems and Applications (16w5083)

Arriving in Oaxaca, Mexico Sunday, May 29 and departing Friday June 3, 2016

## Organizers

(University College London)

(Université de Montréal)

The DtN map is a first order elliptic pseudodifferential operator defined on the boundary of a Riemannian manifold. It has a discrete spectrum, and its eigenvalues coincide with the eigenvalues of the Steklov boundary value problem. For this reason, the spectrum of the DtN map is often referred to as the Steklov spectrum. The Steklov problem appears in various applications to physics, notably to hydrodynamics, vibration theory and the study of heat diffusion. Recently, there has been a lot interest in the properties of the Steklov eigenvalues and eigenfunctions from the viewpoint of geometric spectral theory. In particular, significant progress was achieved in the study of eigenvalue inequalities, spectral invariants and nodal geometry. Isoperimetric inequalities for eigenvalues is an important topic in spectral geometry, going back to the classical Faber-Krahn and Szeg\H{o}--Weinberger inequalities for the first Dirichlet and Neumann eigenvalues. In 1954, R. Weinstock proved that the disk is a unique maximizer of the first nonzero Steklov eigenvalue among all simply-connected planar domains of given perimeter. In 2010, Girouard and Polterovich extended this result to higher eigenvalues, showing that under similar conditions, the $k$-th Steklov eigenvalue achieves its maximum in the limit on a disjoint union of $k$ identical disks. For non-simply connected domains and surfaces, the problem is much more difficult. It was recently proved by Fraser and Schoen that among all surfaces of genus zero with $\ell \ge 1$ boundary components of fixed total length, there exists a smooth maximizer of the first nonzero Steklov eigenvalue. Moreover, Fraser and Schoen exhibited a link between the extremal surfaces for Steklov eigenvalues and the theory of free boundary minimal surfaces in Euclidean balls, and described explicitly the "critical catenoid" which is the maximizer for the case $\ell=2$. Other important inequalities for Steklov eigenvalues on Riemannian manifolds were recently obtained by Colbois, El Soufi, Girouard, Kokarev and Hassannezhad. In particular, the importance of the isoperimetric ratio for the control on the Steklov spectrum was emphasized. Inverse spectral problems for the Dirichlet-to-Neumann operator is a topic that unifies the two major themes of the workshop. Rephrasing the celebrated question of Marc Kac "Can one hear the shape of a drum?", one may ask which geometric quantities of a Riemannian manifold with boundary can be recovered from the Steklov spectrum. It follows from the standard eigenvalue asymptotics for pseudodifferential operators that the dimension and the volume of the boundary are Steklov spectral invariants. In a recent paper by Sher and Polterovich, further spectral invariants were computed using the heat trace expansion for the Dirichlet-to-Neumann operator. As a corollary, it was shown that the three-dimensional ball is uniquely determined by its Steklov spectrum among domains in $R^3$ with connected boundary. It is conjectured that a similar result should hold in arbitrary dimension. The same authors, together with Girouard and Parnovski, have also shown that the Steklov spectrum of a surface uniquely determines the number of its boundary components, as well as their lengths. This result does not generalize in a straightforward way to higher dimensions. In particular, it is an open question whether in higher dimensions the number of boundary components is a Steklov spectral invariant. Some other interesting results and conjectures on inverse problems for the Steklov spectrum on surfaces were stated by Jollivet and Sharafutdinov. The study of nodal domains and nodal lines of eigenfunctions has fascinated mathematicians and physicists for more than two centuries, starting with the famous experiments of Chladni with vibrating plates. For Laplace eigenfunctions, it was shown by Courant that the $k$-th eigenfunction has at most $k$ nodal domains. The same is true for the eigenfunctions of the Steklov problem, considered as functions in the "interior". However, the nodal count for eigenfunctions of the Dirichlet-to-Neumann operator, defined on the boundary of a manifold, is far more difficult, and essentially no general results are known when the boundary is of dimension two or higher. Very recently, Zelditch proved a sharp upper bound (conjectured earlier by Bellova-Lin) on the size of the nodal set of DtN eigenfunctions on real-analytic Riemannian manifolds. A similar result for the Laplace eigenfunctions was earlier proved by Donnelly-Fefferman, as a partial solution of Yau's conjecture. Unlike the case of Laplace eigenfunctions, lower bounds on the volume of the nodal sets of DtN eigenfunctions remain unknown even in the analytic category. Apart from the usual DtN operator acting on functions, one may also study its counterparts acting on differential forms. Different constructions of such operators were introduced by Belishev-Sharafutdinov, Joshi-Lionheart and, more recently, by Raulot-Savo. In particular, the latter authors proved interesting geometric inequalities for eigenvalues of such operators involving the isoperimetric ratio as well as the principal curvatures. The DtN map can be interpreted to encode information on measurements on a boundary of an object, and this motivates the study of inverse problems of the form: given the DtN map, determine the coefficients of the direct problem, that is, the governing PDE. These inverse problems appear in a wide variety of applications such as Electrical Impedance Tomography, where the direct problem is elliptic, and Reflection Seismology, with a hyperbolic direct problem. Inverse scattering problems can also be reduced to recovery from the Dirichlet-Neumann map when the coefficients are known outside a bounded domain. It would be interesting to relate the Steklov spectrum to spectral properties of the Scattering Operator. A canonical inverse boundary value problem is to determine a compact Riemannian or Lorentzian manifold with boundary from the Dirichlet-to-Neumann map of the Laplace operator on the manifold. The Riemannian or elliptic version of the inverse boundary value problem is called Calderon's problem. In the two dimensional case the problem was solved by Nachman, Sylvester,Lassas and Uhlmann, and this workshop will focus on the three and higher dimensional cases. Recently, Dos Santos Ferreira, Kurylev, Lassas and Salo solved Calderon's problem on a class of product type manifolds, which they call transversally anisotropic, via a reduction to the hyperbolic version using Wick rotation.The Lorentzian or hyperbolic version of the problem was solved by Eskin in the time independent case by using ideas from the Boundary Control method. This method was introduced by Belishev, and he and Kurylev used it to solve Gelfand's inverse boundary spectral problem.
A variant of the above inverse boundary value problem is to determine the manifold in a known conformal class. This problem has been solved on a class of manifolds that are transversally anisotropic modulo conformal scaling. In the hyperbolic case, the proof uses geometrical optics solutions to reduce the inverse boundary value problem to the problem of inverting the geodesic ray transform. In the elliptic case, Dos Santos Ferreira, Kenig, Salo and Uhlmann performed a similar reduction using complex geometrical optics solutions, that originate from the now classical work of Sylvester and Uhlmann. Concerning inversion of the geodesic ray transform, the recent result by Uhlmann and Vasy is a breakthrough but still poses restrictions on the geometry. On the other hand, Eskin has solved the hyperbolic conformally transversally anisotropic case supposing that the dependence on time is analytic and under the mild assumption that the geometry is non-trapping. In a setting related to partial data, that is, when the Dirichlet-to-Neumann map is known only partially, Lassas and Oksanen replaced the non-trapping assumption by a strictly weaker assumption on the asymptotics of the Neumann traces of the Dirichlet eigenfunctions of the manifold. In all the solved cases discussed above, there is a non-trivial conformal Killing field on the manifold. The problem is open in the case of general geometry, and under active research. In the Lorentzian case, black holes are natural counterexamples to unique solvability of the inverse boundary value problem. In the Riemannian case, non-smooth counterexamples can be obtained by using transformation optics as first realized by Greenleaf, Lassas and Uhlmann. The study of transformation optics has become a highly active field motivated by the possibility of building actual cloaking devices using metamaterials. Partly motivated by the existence of non-smooth counter examples, the inverse boundary value problem has been widely studied when the coefficients of the direct problem are less smooth. In the two dimensional case, Astala and Päivärinta solved the elliptic problem with $L^\infty$ coefficients. In higher dimensions, the recent exciting result by Haberman and Tataru assumes $C^1$ coefficients (and conformally Euclidean geometry). There is ongoing research to generalize their result for Lipschitz coefficients.

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The NH3 ligand is neutral, but the chloro ligand has a charge of 1â. For example, the CN ligand can bind through the carbon atom (cyano) or through the nitrogen atom (isocyano). Plastics are made with the aid of transition metal catalysts, along with detergents, fertilizers, paints, and more (see Figure 15). Metals in the f-block (the lanthanoids and actinoids) can accommodate higher coordination number due to their greater ionic radii and availability of more orbitals for bonding. In this case, the number of ligands and the coordination number are equal. Very complicated pharmaceuticals are manufactured with catalysts that are selective, reacting with one specific bond out of a large number of possibilities. Therefore, coordination number of Fe in Fe[Fe(CN) 6] 4- ,[Fe(CN) 6] 3- and [FeCl 4] - are 6 , 6 , 4 respectively. Seven chloride ligands have Pb-Cl distances of 280–309 pm. Fluoride anions have 4 neighbors of opposite charge arranged at vertices of an tetrahedron. Coordination number is the term proposed by Werner to denote the total number of bonds from the ligands to the metal atom. The ability to bind ligands and change oxidation states makes transition metal catalysts well suited for catalytic applications. In accordance with the predictions of the Kepert model these give linear complexes. Though they are normally used for transition metal atoms in a compound, they can also refer to nonmetallic substances as well. b)Does not give quantitative interpretation of magnetic data. Summary â Coordination Number vs Oxidation Number The oxide ions are 3-coordinate in a trigonal planar configuration.[17]. Many of these compounds are highly colored (Figure 1). [1][3] The concept is most commonly applied to coordination complexes. Coordination number should be â¦ In contrast, once oxidized, lithium, magnesium, aluminum, and titanium cannot be reduced because they have elevated oxidation energies. The name is potassium trisoxalatoferrate(III) (note that tris is used instead of tri because the ligand name starts with a vowel). As we shall see, the coordination number depends on the relative size of the atoms or ions. Brackets in a formula enclose the coordination sphere; species outside the brackets are not part of the coordination sphere. Check Your Learning Geometric Isomers In the complex [Co(en) 2 Cl 2] Br, the co-ordination number and oxidation state of cobalt are. The common geometries found in complexes are tetrahedral and square planar (both with a coordination number of four) and octahedral (with a coordination number of six). Isomers are different chemical species that have the same chemical formula. One of the most important applications of transition metals is as industrial catalysts. The other form is shown in Figure 11. Click hereðto get an answer to your question ï¸ A coordination compound of iron is P that has the coordination number of iron 6 and oxidation number of iron is + 3. (iv) Number â¦ 3 and +3 . For example, DNA helices are optical isomers, and the form that occurs in nature (right-handed DNA) will bind to only one isomer of [M(en)3]n+ and not the other. Composition of Substances and Solutions, 3.2 Determining Empirical and Molecular Formulas, 3.4 Other Units for Solution Concentrations, Chapter 4. α-Aluminium has a regular cubic close packed structure, fcc, where each aluminium atom has 12 nearest neighbors, 6 in the same plane and 3 above and below and the coordination polyhedron is a cuboctahedron. A simple example of two ionization isomers are [CoCl6][Br] and [CoCl5Br][Cl]. Similarly, SCNâ can be bound through the sulfur or nitrogen atom, affording two distinct compounds ([Co(NH3)5SCN]2+ or [Co(NH3)5NCS]2+). The water ligands are neutral, and the chloride ligands are anionic with a charge of 1â each. Indicate any, Name each of the compounds or ions given in. Favorite Answer. 8. Coordination complexes are useful as preservatives. Iron(0) mostly coordinates ï¬ve or six ligands with trigonal bipyramidal and octahedral geometry. In chemistry, crystallography, and materials science, the coordination number, also called ligancy, of a central atom in a molecule or crystal is the number of atoms, molecules or ions bonded to it. The compound P may be? Related techniques include neutron or electron diffraction. Name the ligands first, followed by the central metal. Martyn V. Twigg. Most often, this involves a donor atom with a lone pair of electrons that can form a coordinate bond to the metal. Complexing agents often are used for water softening because they tie up such ions as Ca2+, Mg2+, and Fe2+, which make water hard. The oxidation states of iron, nickel, and copper change readily according to the oxygen atmosphere because the transition energies between different oxidation states are small for these metal oxides. Therefore, this complex is trans-diaminedichloroplatinum(II). The influence of the coordination number of the iron on the isomer shift is well demonstrated by comparing salts of the [FeCl6]3âtrianion with â¦ One of those definition are as follows: Denoting the position of the first peak as rp. Alpha phase. Figure 4 showed one example of a chelate. You can look over more examples and test yourself with online quizzes at the University of Sydneyâs site. The nomenclature of the complexes is patterned after a system suggested by Alfred Werner, a Swiss chemist and Nobel laureate, whose outstanding work more than 100 years ago laid the foundation for a clearer understanding of these compounds. [6], Coordination numbers become ambiguous when dealing with polyhapto ligands. Over 90% of all manufactured products are made with the aid of one or more catalysts. The coordination number is the number of ions that immediately surround an ion of the opposite charge within a crystal lattice. Chemical Bonding and Molecular Geometry, 7.5 Strengths of Ionic and Covalent Bonds, Chapter 8. The coordination number of systems with disorder cannot be precisely defined. The oxidation states of iron, nickel, and copper change readily according to the oxygen atmosphere because the transition energies between different oxidation states are small for these metal oxides. If a coordination compound is ionic, name the cation first and the anion second, in accordance with the usual nomenclature. Check Your Learning They exhibit different properties, even though they have the same formula. C. 6 and 3. For surfaces, the number of neighbors is more limited, so the surface coordination number is smaller than the bulk coordination number. Often the color of a compound is affected by the specific materials coordinated to that central ion. Cations form the cubic close packing while anions fill all tetrahedral holes. CHEM1902 Coordination Chemistry The total number of points of attachment to the central element is termed the coordination numberand this can vary from 2 to as many as 16, but is usually 6. In diamond, each carbon atom is at the centre of a regular tetrahedron formed by four other carbon atoms, the coordination number is four, as for methane. [12], Two other examples of commonly-encountered chemicals are Fe2O3 and TiO2. [12][16] In zinc there are only 6 nearest neighbours at 266 pm in the same close packed plane with six other, next-nearest neighbours, equidistant, three in each of the close packed planes above and below at 291 pm. Overall charge is -3. The coordination sphere consists of the central metal ion or atom plus its attached ligands. B. 12 and 6. John Burgess. The inhibition of cell division indicated that this square planar compound could be an anticancer agent. Optical isomers differ only in the way they affect polarized light and how they react with other optical isomers. Many other ligands coordinate to the metal in more complex fashions. en can only bind in a cis fashion i.e. Transition metals often form geometric isomers, in which the same atoms are connected through the same types of bonds but with differences in their orientation in space. The ion/molecule/atom surrounding the central ion/molecule/atom is called a ligand. Coordination Numbers and Geometry Lecture 2. This is an example of a main group element in a coordination complex. In chemistry, coordination number (C.N. The coordination sphere consists of the central metal ion or atom plus its attached ligands. The structure is quite complex, the oxygen atoms are coordinated to four iron atoms and the iron atoms in turn share vertices, edges and faces of the distorted octahedra. For transition metal complexes, coordination numbers range from 2 (e.g., AuI in Ph3PAuCl) to 9 (e.g., ReVII in [ReH9]2−). Determine the name of the following complexes and give the coordination number of the central metal atom. We can obtain this value by removing all the ligands (CN ligands) with their charge. Zigya App. In one isomer, cis chloride ligands cause more electron density on one side of the molecule than on the other, making it polar. Ionization isomers (or coordination isomers) occur when one anionic ligand in the inner coordination sphere is replaced with the counter ion from the outer coordination sphere. This coordination number is rare outside of d 10 complexes of the group 11 metals and mercury, specifically, Cu +, Ag +, Au +, and Hg 2 +. 6 and +3. The CN does not distinguish the geometry of such complexes, i.e. If Ca is two plus, Na one plus, and each CN minus one, what is the charge on Fe? For the copper(II) ion in [CuCl 4] 2â, the coordination number is four, whereas for the cobalt(II) ion in [Co(H 2 O) 6] 2+ the coordination number is six. octahedral vs trigonal prismatic. The bluish-gray Fe 3+ ions are surrounded by green Cl â ions. Coordination numbers of 8 to 12 are commonly observed for f-block elements. For example, in [Cr(H2O)4Cl2]Br, the coordination sphere (in brackets) has a charge of 1+ to balance the bromide ion. The chloride and nitrate anions in [Co(H2O)6]Cl2 and [Cr(en)3](NO3)3, and the potassium cations in K2[PtCl6], are outside the brackets and are not bonded to the metal ion. 3 and +3 . Iron is a notably versatile transition metal for preparing coordination compounds [4]. Table 5 compares coordination numbers to the molecular geometry: Unlike main group atoms in which both the bonding and nonbonding electrons determine the molecular shape, the nonbonding d-electrons do not change the arrangement of the ligands. Give the formula and coordination number. Among these, Cu + more commonly gives tetrahedral complexes but can be coaxed to give linear ones. Solution (a) There are two Na + ions, so the coordination sphere has a negative two charge: [PtCl 6] 2â. Solution: In 1st compound CN is an unidentate ligand and hence it will surround Fe atom from 6 different sides. In all cases, these are the cis isomers and never the trans isomers. Lv 4. Vanadium oxide is used to produce 230,000,000 tons of sulfuric acid worldwide each year, which in turn is used to make everything from fertilizers to cans for food. The coordination number is the the number of attachments that all of the ligands make to the transition metal. When the hemoglobin loses its oxygen, the color changes to a bluish red. The coordination number of the central metal ion or atom is the number of donor atoms bonded to it. Johnson Matthey plc, Royston, UK. So, we must first discuss their sizes. There are various ways of assigning the contribution made to the coordination number of the central iron atom by each cyclopentadienide ligand. Enterobactin forms a water-soluble complex with excess iron, and the body can safely eliminate this complex. (a) There are two Na+ ions, so the coordination sphere has a negative two charge: [PtCl6]2â. [4] Among the Frank–Kasper phases, the packing of metallic atoms can give coordination numbers of up to 16. Many metal ions are also undesirable in food products because these ions can catalyze reactions that change the color of food. Coordination number: 4: Bond angle(s) â109.5° Î¼ (Polarity) 0: In a tetrahedral molecular geometry, a central atom is located at the center with four substituents that are located at the corners of a tetrahedron. The coordination number of sodium is 6. The transition elements and main group elements can form coordination compounds, or complexes, in which a central metal atom or ion is bonded to one or more ligands by coordinate covalent bonds. Identify which geometric isomer of [Pt(NH3)2Cl2] is shown in Figure 9. For example, the ligand EDTA, (HO2CCH2)2NCH2CH2N(CH2CO2H)2, coordinates to metal ions through six donor atoms and prevents the metals from reacting (Figure 19). Hemoglobin will only transport oxygen if the iron is Fe2+; oxidation of the iron to Fe3+ prevents oxygen transport. The term was originally defined in 1893 by Swiss chemist Alfred Werner (1866â1919). As an example of how the arrangement in space can influence the molecular properties, consider the polarity of the two [Co(NH3)4Cl2]NO3 isomers. For molecules and polyatomic ions the coordination number of an atom is determined by simply counting the other atoms to which it is bonded (by either single or multiple bonds). It is ionically bonded with the Ag+ A g + center of the coordination complex shown. Answers of In the fluorite structure, the coordination number of Ca2+ ion is : [1993]a)4b)6c)8d)3Correct answer is option 'C'. Formula of the unit cell will be 4 N a 2 O = N a 8 O 4 The coordination number of N a + = 4 It has antifluorite-type structures having 4: 8 co-ordination number and N a + ions are in all tetrahedral voids. Electronic configuration of Fe2+is 4s0 3d6.As CNâ is a strong field ligand, it causes the pairing of the unpaired 3d electrons.Since there are six ligands around the central metal ion, the most feasible hybridization is d2sp3. The Lewis base donors, called ligands, can be a wide variety of chemicalsâatoms, molecules, or ions. [8], The coordination numbers are well defined for atoms in the interior of a crystal lattice: one counts the nearest neighbors in all directions. Two chloride ligands are more distant, with a Pb-Cl distances of 370 pm. Another type of crystal is illustrated by titanium(IV) The iron is surrounded by the 6 cyanide ligands, so 6. 153. tetrahedral and square planar) geometries are well known, as are a variety of mixed oxidation state compounds. Another important type of isomers are optical isomers, or enantiomers, in which two objects are exact mirror images of each other but cannot be lined up so that all parts match. Electronic Structure and Periodic Properties of Elements, 6.4 Electronic Structure of Atoms (Electron Configurations), 6.5 Periodic Variations in Element Properties, Chapter 7. Ethylenediamine, H2NCH2CH2NH2, and the anion of the acid glycine, $\text{NH}_2\text{CH}_2\text{CO}_2^{\;\;-}$ (Figure 6) are examples of bidentate ligands. Identify which atoms in this molecule could act as donor atoms. In the complex [Co(en) 2 Cl 2] Br, the co-ordination number and oxidation state of cobalt are. In chemistry, coordination number (c.n. [1] Identify at least two biologically important metals that could be disrupted by chelation therapy. 6 and +1. For example, the octahedral [Co(NH3)4Cl2]+ ion has two isomers. For the trans isomer, each ligand is directly across from an identical ligand, so the bond dipoles cancel out, and the molecule is nonpolar. 3. Representative Metals, Metalloids, and Nonmetals, 18.2 Occurrence and Preparation of the Representative Metals, 18.3 Structure and General Properties of the Metalloids, 18.4 Structure and General Properties of the Nonmetals, 18.5 Occurrence, Preparation, and Compounds of Hydrogen, 18.6 Occurrence, Preparation, and Properties of Carbonates, 18.7 Occurrence, Preparation, and Properties of Nitrogen, 18.8 Occurrence, Preparation, and Properties of Phosphorus, 18.9 Occurrence, Preparation, and Compounds of Oxygen, 18.10 Occurrence, Preparation, and Properties of Sulfur, 18.11 Occurrence, Preparation, and Properties of Halogens, 18.12 Occurrence, Preparation, and Properties of the Noble Gases, Chapter 19. The energy resulting from the absorption of light is used in photosynthesis. Atomic and Ionic Radii . Likewise, six sodium ions immediately surround each chloride ions, making the coordination number of chloride also equal to 6. Advanced Theories of Covalent Bonding, 9.2 Relating Pressure, Volume, Amount, and Temperature: The Ideal Gas Law, 9.3 Stoichiometry of Gaseous Substances, Mixtures, and Reactions, 10.6 Lattice Structures in Crystalline Solids, Chapter 13. There are many definitions of the term coordination number, but there is no one simple unambiguous definition that works in all cases.For simple monodentate and chelating ligands, the coordination number can be defined as the number of atoms or ligands directly bonded to the metal atom.For example, [Fe(NH 3) 6] 3+ and [Fe(en) 3] 3+ are both â¦ For the complex [Fe(en) 2 Cl 2] Cl, identify the following: (i) Oxidation number of iron. The bond angles are cos â1 (â 1 â 3) = 109.4712206...° â 109.5° when all four substituents are the same, as in methane (CH 4) as well as its heavier analogues. The Lewis acid in coordination complexes, often called a central metal ion (or atom), is often a transition metal or inner transition metal, although main group elements can also form coordination compounds. the two NH2groups must always be 90 degrees apart. Zigya App. [9] The surface coordination number is also dependent on the Miller indices of the surface. 1 mol of compound P form 4 mole of ions when dissolved in water. If you examine the figure below, you will see that there are six chloride ions immediately surrounding a single sodium ion. The diamine (NH3)2 portion is retained with other groups, replacing the dichloro [(Cl)2] portion. [Fe(CN)4] = coordination number 4. Although not a complex, self-darkening sunglasses are an example of a photoactive substance. Equilibria of Other Reaction Classes, 16.3 The Second and Third Laws of Thermodynamics, 17.1 Balancing Oxidation-Reduction Reactions, Chapter 18. Bidentate ligands are those in which two atoms coordinate to the metal center. Therefore, there are 2 different measures for ligancy in the case of crystals, namely the bulk coordination number and the surface coordination number. Chelation Therapy Iron is bounded with three atoms of oxalato ion which is a bidentate ligand, Thus, coordination number is 6. Number of possible isomers for the complex [Co(en)2Cl2]Cl will be (en = ethylenediamine) asked Dec 25, 2018 in Chemistry by sonuk ( 44.5k points) coordination compounds The latter numbers are in close agreement with those reported more recently by Weigel et al. Since that time, many similar platinum compounds have been developed for the treatment of cancer. Coordination numbers range from two to 16, with four and six being the most common. Figure $$\PageIndex{9}$$: Transition metals with a coordination number of four can adopt a tetrahedral geometry (a) as in K 2 [Zn(CN) 4] or a square planar geometry (b) as shown in [Pt(NH 3) 2 Cl 2]. All of the oxygen and sulfur atoms have lone pairs of electrons that can be used to coordinate to a metal center, so there are six possible donor atoms. In sodium chloride each sodium ion has 6 chloride ions as nearest neighbours (at 276 pm) at the corners of an octahedron and each chloride ion has 6 sodium atoms (also at 276 pm) at the corners of an octahedron. Predict whether the carbonate ligand ${\text{CO}_3}^{2-}$ will coordinate to a metal center as a monodentate, bidentate, or tridentate ligand. Last Modified Date: November 19, 2020 Defined by Alfred Werner in 1893, a coordination number is a term used in chemistry to denote the number of bonds a central metal ionhas in a coordination compound. Many polydentate ligands are chelating ligands, and a complex consisting of one or more of these ligands and a central metal is a chelate. Name the ligands alphabetically. [15] Similar considerations can be applied to the regular body centred cube structure where in addition to the 8 nearest neighbors there 6 more, approximately 15% more distant,[12] and in this case the coordination number is often considered to be 14. University of Leicester, Leicester, UK. 1 decade ago. The newer drugs include carboplatin, oxaliplatin, and satraplatin. So, we must first discuss their sizes. PREPARATION OF AN IRON COORDINATION COMPOUND A. For example, one type of compound she is developing captures carbon dioxide waste from power plants and catalytically converts it into useful products (see Figure 16). The coordination sphere consists of the central metal ion or atom plus its attached ligands. The nickel ions are 6-coordinate with a distorted octahedral coordination polyhedron where columns of octahedra share opposite faces. It has been found that many metals plate out as a smoother, more uniform, better-looking, and more adherent surface when plated from a bath containing the metal as a complex ion. The first coordination number can be defined using the radial distribution function g(r):[18][19]. A chelating ligand holds the metal ion rather like a crabâs claw would hold a marble. If compound P shows geometrical as well as optical isomerism. [7] In ferrocene the hapticity, η, of each cyclopentadienide anion is five, Fe(η5-C5H5)2. The most common structures of the complexes in coordination compounds are octahedral, tetrahedral, and square planar (see Figure 7). are used to designate the number of each type of ligand in the complex ion. octahedral vs trigonal prismatic. For transition metals with a coordination number of four, two different geometries are possible: tetrahedral or square planar. Some alternative medicine practitioners recommend chelation treatments for ailments that are not clearly related to heavy metals, such as cancer and autism, although the practice is discouraged by many scientific organizations. Complexing agents that tie up metal ions are also used as drugs. The most common binding mode involves the coordination of one sulfur atom and one oxygen atom, forming a five-member ring with the metal. B. When metal ions are reduced to produce thin metal coatings, metals can clump together to form clusters and nanoparticles. Negative ligands (anions) have names formed by adding –, If more than one ligand of a given type is present, the number is indicated by the prefixes. Calculate the lattice parameter of the iron. Isomerism in Complexes. (a) tricarbonatocobaltate(III) ion; (b) tetraaminecopper(II) ion; (c) tetraaminedibromocobalt(III) sulfate; (d) tetraamineplatinum(II) tetrachloroplatinate(II); (e) tris-(ethylenediamine)chromium(III) nitrate; (f) diaminedibromopalladium(II); (g) potassium pentachlorocuprate(II); (h) diaminedichlorozinc(II). Coordination number should be distinguished from oxidation number (defined in the previous paragraph). Oxygen molecules are transported by hemoglobin in the blood by being bound to the iron center. For the copper(II) ion in [CuCl 4] 2â, the coordination number is four, whereas for the cobalt(II) ion in [Co(H 2 O) 6] 2+ the coordination number is six.

## coordination number of iron

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CodeGeneration[LanguageDefinition] - Maple Help

Home : Support : Online Help : Programming : Code Generation Package : Language Definition : CodeGeneration/LanguageDefinition/IsDefined

CodeGeneration[LanguageDefinition]

 IsDefined
 test if target language is defined

 Calling Sequence CodeGeneration[LanguageDefinition][IsDefined](lang)

Parameters

 lang - string; name of target language

Description

 • The IsDefined command returns true if the language lang has been defined for use by CodeGeneration, and false otherwise.
 • A language has been defined for use by CodeGeneration if it is one of the built-in languages provided with the package, or it has been added with Add or Define.

Examples

Test that the language C is ready for use by CodeGeneration.

 > $\mathrm{CodeGeneration}[\mathrm{LanguageDefinition}][\mathrm{IsDefined}]\left("C"\right)$
 ${\mathrm{true}}$ (1)


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# OHMSR-71-R 1.2.1 (for Orbiter 2016)

#### OrbitHangar

Author: blake

A fictional space plane based on the SR71 Blackbird featuring a detailed and functional virtual cockpit.

Changes:

- New cockpit graphics.

- Reworked autopilot integrated with flight computer.  This is done to enable further work on the flight computer procedures.

- Now fully open sourced:  https://github.com/BMCDad/orbiter-sr71r

- (1.2.1) fix animation issues, gauge issues with D3D9 client.

#### 4throck

##### Enthusiast !
Great add-on. The working VC and system simulation is simple but effective.

Flight is perhaps a bit too stable at low speeds (~150knots) and controls are too precise for a big and heavy plane.
But that's a positive thing if you just want to do rough reentry and then fly to a distant runway.

So congratulations on a great add-on!

#### marcogavazzeni

The VC is wonderful :thumbup:

#### Kyle_E

##### Begining Double Bassist

just does not work with D3D9 client
Still very nice!

#### Blake

just does not work with D3D9 client
Still very nice!
I did test it with the D3D9 client so it should work. What error are you getting?

#### Kyle_E

##### Begining Double Bassist
On startup it will finish loading but does not show anything though I can hear it.

I won't post the whole log file because it is 300kb of the same error, but here is an abbreviated version.View attachment AbbrevOrbiter.log
To shut it down I had to use alt-F4.
I can get you the full log but as I said it's BIG.

#### Kyle_E

##### Begining Double Bassist
Thanks!:thankyou:

#### Tex

Tutorial Publisher
Great work!! I even used a shot from your addon in my upcoming movie. Thanks for your efforts!

#### Blake

Great work!! I even used a shot from your addon in my upcoming movie. Thanks for your efforts!
Cool. Can't wait to see it.

#### AnakinSkyobiliviator

##### Member
Very nice! I'm such a noob but I'll have fun hitting every toggle the VC has to offer!

#### reicou

##### New member
I really like it, is great.

thanks, dude :thumbup:

#### Kyle_E

##### Begining Double Bassist
Here is a quick XRSound config file I made for this vessel using the config for the DeltaGlider as a template.
Code:
#--------------------------------------------------------------------------
# XRSound Configuration File -- overrides for SR71r vessels.
#
# For XRSound Version 1.00.
#
# Any values altered will not take effect until Orbiter is relaunched via the launchpad.
#
# D. Beachy, 24-Jan-2018
#--------------------------------------------------------------------------

#==========================================================================
# This section defines vessel-specific settings that apply only to this
# vessel class; i.e., the SR71r in this case.
#==========================================================================
[VESSEL]

#==========================================================================
# This section defines sounds that override the default sounds in
# XRSound.cfg.  These overrides only apply to this vessel class; i.e.,
# the SR71r in this case.
#==========================================================================
[SOUND_FILES]

#--------------------------------------------------------------------------
# Callout played when the ship's landing gear touches down.
# Sound ID = 10021
# Default (for DG) = XRSound\Default\Wheels Down.wav
#--------------------------------------------------------------------------
Touchdown = XRSound\Default\Wheels Down.wav

#--------------------------------------------------------------------------
# Callout played when the ship's landing gear lifts off.
# Sound ID = 10022
# Default (for DG) = XRSound\Default\Wheels Up.wav
#--------------------------------------------------------------------------
Liftoff = XRSound\Default\Wheels Up.wav

#==========================================================================
# This section defines which sounds should play when a given animation
# event occurs.  Orbiter assigns animation IDs to a vessel as the vessel
# creates them.  To determine what animation ID Orbiter assigns to a given
# animation for a given vessel, set 'LogVesselAnimations = 1' in XRVessel.cfg.
# Then launch the vessel in Orbiter and activate the animation (e.g., landing gear)
# for which you want to assign sounds.  Then use the ID listed in XRSound.log for
# that animation to define which sounds to use in the this file.
#
# Also, you may add/define *additional* animation sounds however you wish for any vessel
# you wish.  You may also manipluate these sounds via their sound IDs from your
# C++ vessel code via the XRSound API (see $ORBITER_ROOT\Orbitersdk\XRSound.h). # # Each [animation_#] section applies to an animation with the supplied ID. # For example, section [animation_52] applies to animation ID 52 for this vessel, the SR71r # in Orbiter. # # Each setting is optional. However, if OpenCloseSoundID is missing, then the # Opening / Open / Closing / Closed sounds cannot play. Similarly, if MovingSoundID # is missing, then the Moving sound cannot play. This is not an error, however: it # simply disables those sounds. # # OpenCloseSoundID = <vessel-unique sound ID> # defines the sound ID for this vessel this animation's Opening/Open/Closing/Closed sounds are tied to. # MovingSoundID = <vessel-unique sound ID> # defines the sound ID for this vessel this animation's Moving sound is tied to. # Opening = <file> # Plays when the animation starts opening. # Open = <file> # Plays when the animation finishes opening. # Moving = <file> # Plays in a loop whenever the animation is moving. # Closing = <file> # Plays when the animation starts closing. # Closed = <file> # Plays when the animation finishes closing. # IsLandingGear = 0 (false, the default) or 1 (true): used by XRSound to treat this animation as the vessel's gear for playing gear-related sounds. #-------------------------------------------------------------------------- # PlaybackType = BothViewClose # Determines how and where the sounds for this animation are played (see valid values below); if not set, the default is BothViewClose. # # Valid PlaybackType values are: # InternalOnly = In cockpit view only, always full volume. (Implies the vessel has focus, as well). # BothViewFar = Full volume in internal view, faded (less) by distance and pressure in external view (this will be silent in vacuum). Vessel does not need focus. # BothViewMedium = Full volume in internal view, faded (middle) by distance and pressure in external view (this will be silent in vacuum). Vessel does not need focus. # BothViewClose = Full volume in internal view, faded (more) by distance and pressure in external view (this will be silent in vacuum). Vessel does not need focus. # Radio = Full volume in both internal and external view (vessel must have focus). # Wind = External view only, faded by static pressure but not distance. # # Note that all <file> paths above are relative to$ORBITER_ROOT.
# To disable a custom sound, simply delete or comment out the line.
#==========================================================================

# Landing Gear
[animation_52]
OpenCloseSoundID = 12002
MovingSoundID = 12003
PlaybackType = BothViewClose
Opening = XRSound\Default\Gear Down.wav
Open = XRSound\Default\Gear Down and Locked.wav
Moving = XRSound\Default\Gear Whine.wav
Closing = XRSound\Default\Gear Up.wav
Closed = XRSound\Default\Gear Up and Locked.wav
IsLandingGear = 1

# Hover
[animation_82]
OpenCloseSoundID = 12006
MovingSoundID = 12007
Moving = XRSound\Default\Hydraulics1.wav
Closed = XRSound\Default\Thump.wav

# Hover2
[animation_83]
OpenCloseSoundID = 12006
MovingSoundID = 12007
Moving = XRSound\Default\Hydraulics1.wav
Closed = XRSound\Default\Thump.wav

# Retro Doors
[animation_85]
OpenCloseSoundID = 12008
MovingSoundID = 12009
Moving = XRSound\Default\Hydraulics1.wav
Closed = XRSound\Default\Thump.wav

# Canopy
[animation_48]
OpenCloseSoundID = 12010
MovingSoundID = 12011
Open = XRSound\Default\Hatch Open.wav
Moving = XRSound\Default\Hydraulics1.wav
Closed = XRSound\Default\Hatch Closed.wav

# Cargo
[animation_43]
OpenCloseSoundID = 12012
MovingSoundID = 12013
Moving = XRSound\Default\Hydraulics1.wav
Closed = XRSound\Default\Thump.wav

# Cargo2
[animation_45]
OpenCloseSoundID = 12014
MovingSoundID = 12015
Moving = XRSound\Default\Hydraulics1.wav
Closed = XRSound\Default\Thump.wav
To use, copy the information to a file named "XRSound-SR71r.cfg" and paste that into the XRSound directory.

Do not use this with version 1.20; this only works with version 1.0

Last edited:

#### Blake

New version:
Minor changes to code. I have updated the VC and changed how the autopilot works with regard to the flight computer, which is still a work in progress.

Also, I have put the source code, GIMP files, and Blender files on github: https://github.com/BMCDad/orbiter-sr71r

Thanks so much for making such a great plane.

I've noticed the indicators are hidden and in an incorrect position when reaching a certain level. Here are some photos:

Also, I would like to know some specs about it. How much time can it spend in space (Fuel cell)? How to re-entry (Target periapses, re-entry angle, etc)? I didn't understand the manual re-entry way. The manual doesn't contain these informations. It needs to be updated for the current cockpit, as there is no explanation for the computer, plus some outdated images.

Otherwise, it's a good plane. It deserves 10 stars :thumbup:

Here is a quick XRSound config file I made for this vessel using the config for the DeltaGlider as a template.

To use, copy the information to a file named "XRSound-SR71r.cfg" and paste that into the XRSound directory.
It doesn't work well. When I open the cargo door, the sound is still running even when the door is fully opened.

---------- Post added at 01:46 PM ---------- Previous post was at 06:40 AM ----------

I've successfully integrated XRSound into the source code. All components now have sound (Hover and retro doors, switches, knobs, push buttons, cargo and canopy, etc). Where can I send it?

I've noticed some problems in animations, pictured below:

Airbrake animation is broken.

Hover doors animation is broken.

Touchdown points aren't updated without landing gear.

#### Blake

Thanks for the feed back Abullah, looks like I missed some tweaks on the animations.

What client are you using?

I am not sure what you mean by 'client'. I am using Orbiter 2016, XRSound 1.1 and D3D9 Client r996m.

I'll PM you the modified source code. Just added sound, nothing else. It's based on the latest GitHub repository revision. I can send it in another way if you want.

#### Blake

I am not sure what you mean by 'client'. I am using Orbiter 2016, XRSound 1.1 and D3D9 Client r996m.

I'll PM you the modified source code. Just added sound, nothing else. It's based on the latest GitHub repository revision. I can send it in another way if you want.
By client I mean D3D9 vs inline. The mesh tolerance seems to be different between the two, so what works inline does not always behave the same D3D9. I do all development inline, and then run through a test in D3D9, but I don't always catch everything.

By sound, do you mean changes to the XRSound config Kyle_E did? There is no sound in the SR71 code itself.


text:
# A corollary of Mermin-Wagner Theorem

The picture above shows Mermin-Wagner Theorem and its corollary. How can the corollary be derived from Mermin-Wagner Theorem?

-


text:
# Why don't most physics programs study the primary sources?

Why don't most physics programs study the primary sources? In other words: Why don't they include Newton's Principia, Lagrange's Analytical Mechanics, etc., in the curricula?

-

## 4 Answers

Much of Physics is now far better understood than when it was first formulated. For example, if you wanted to learn Maxwell's equations you are far better off using a modern textbook than going back to Maxwell's original work. Likewise Einstein's General Relativity. You mention Newton, but he used a notation that would flummox most modern students.

There are exceptions, for example Dirac's book on quantum mechanics is widely regarded as a classic and valuable even for modern students. However, apart from these exceptions the original documents are of most interest to historians of science.

-

The reason is that this is too time-consuming. The old papers are very difficult to read, because they have suboptimal presentation. Newton's Principia is not written well for the purposes of teaching mechanics. It is designed to emulate Aristotle in scope and completeness (but with the advantage of being correct physics), and it is optimal for the goal it served historically, getting Aristotle out of physics. It's absurd latin and old-fasioned geometrical phrasing with Greek-style geometry is suboptimal for pedagogy and clarity. One needs to use coordinate geometry and vector components in a modern presentation, there is no advantage to the old style.

In order to assign the Principia, one needs a modern translation of all the ideas in it. This is not provided by any source. Such a translation would keep all the arguments, but rearrange the order and the phrasing to be in common language and using modern calculus and vector component notation. This is very simple to do for someone who knows any physics, and Chandrashekhar did something like this in his annotated Principia (this is a very nice and respectful presentation). But Chandrashekhar wasn't writing an introductory physics book, rather he was trying to place the Principia's monumental achievement in historical context, to let modern readers know exactly what Newton was doing.

But I think it is possible to assign the Principia as an elementary physics book, if the translation is authentic and modernized. It requires rewriting the whole thing from scratch, but without throwing away any of Newton's insights. These insights are mostly contained in the special problems he solves.

I tried to provide an example of how to do this for less difficult things. The oldest is Archimedes "The Method of Mechanical Theorems", and you can read the gloss on Wikipedia, under Archimedes' "The Method". Using modern calculus, you can explain what Archimedes is doing in two minutes, and you can see exactly why his paper is so laborious--- he didn't have an algebraic notation or a standardized way to talk about infinitesimals. One should do a similar thing for "on floating bodies", "the equilibrium of planes" (this is simple to do, since it is all center-of-mass results which are trivial with modern calculus), "on the sphere and cylinder" (also simple, but it requires summing the squares, which is simple with modern calculus of finite differences), and the other works. All physics comes from Archimedes originally, especially the concept of force and torque.

One of the ideas in "the method" is the observation that certain integrals involving boundaries which are curvilinear algebraic varieties end up being polynomials in certain plane slicings, and so give rational number integrals. The simplest example he gives is of two cylinders intersecting at right angles. The question of which figures enclose rational volumes has never been investigated again, as far as I know, although it is interesting. It is similar to the question of periods, but it is different because of the slicing freedom. If people are familiar with the method, maybe there is interesting mathematics left undiscovered here.

With Newton, people keep the statement of the three laws of motion (the least important part of the Principia) and throw away all the special problems Newton analyzes. This is exactly the opposite of what they should be doing. One can state Newton's laws today as follows:

• There is a conserved momentum, equal to the total mass times the velocity for an isolated body, and when it flows from one body to another, this is a force.

That's it. That's Newton's three laws. There is a further result

• There is a conserved angular momentum, equal to the momentum cross the position for a point particle, and it is conserved under pairwise attraction repulsion events.

If you add the assumption that all matter is composed of points with pairwise forces which are attractions and repulsions, you get Newton's model of the world. The division of the laws into "1st law" "2nd law" "3rd law" is arbitrary and unnecessary, and only of historical interest. If you start with these two assumptions, you can state the three laws and show why and how they work, and the angular momentum law, and how and why it works, and then you can go to the special problems.

Newton's argument using infinitesimal triangles that the angular momentum is conserved during a radial attraction/repulsion event is essential, and it needs to be preserved unchanged. Newton's argument about the parabolic shape of spinning water needs to be preserved, although one should use the notion of potential energy in the argument. Newton's derivation of the speed of sound needs to be transmitted with both a derivation of the wave equation and an explanation of the adiabatic/isothermal distinction. Newton's analysis of the orbits of the planets is a little difficult to translate--- Feynman attempted to do this in "The Lost Lecture", but this is a place where improvements are possible, because one wants to use the modern insights about the central force, while keeping Newton's insights about the inverse-square special case. I don't know how to best present this.

All these things are usually skipped in elementary books, and instead you get an endless stream of unimaginative problems that are copied from one book to the next. These can be summarized in about a dozen or two orthogonal mechanical problems.

Regarding Laplace, the issue here is that the method of secular perturbations is greatly clarified by using action angle variables, and some special transcendental functions which flip from angle to time. The presentation in Laplace (which I haven't read) is not going to be optimal.

Translating Maxwell is going to be a pain, because you need to keep the modern insight that E and B (and A and $\phi$) are fundamental, not E and H. Maxwell is mislead by the fact that it wasn't appreciated that electric fields come from point sources, while B fields come from currents, so he wants the continuity conditions across material boundaries to be the same for the electric and magnetic field, and this picks out E and H as analogous and fundamental. This disease infects the literature right up to the 1950s, it is only fixed in the 1960s (and Feynman and Landau and Lifschitz help here).

The insights in Maxwell are so enormous that it is impossible to describe what is lost. These are the material dielectric and magnetic properties that are described phenomenologically. In my opinion, a good translation of Maxwell is essential to keep the 19th century E&M thinking alive.

For thermodynamics the problem is more severe, because the original authors did not understand that entropy was statistical, and energy was fundamental. They thought that the energy was the fundamental thermodynamic function, not the entropy. Here, one can start with stat mech, explain the thermodynamic potentials, and then go back and translate all the old papers into modern notation. These are all enormous projects, however. Thermodynamics has not been transmitted well--- most of the old thinking is lost. Statistical mechanics has been transmitted well, and includes the old thinking in principle, but the intuitions are different. If you don't believe me, look at any 19th century physics journals where they are busy measuring the heat capacities of materials at different temperatures, nd there are dozens of annoying thermodynamic identities you have to internalize. There is no way that studying this nonsense is productive, but it needs to be translated to modern language. The chemistry departments are responsible for this, and I don't think they are doing a great job of this.

Regarding 20th century physics, it is essential to modernize the treatment of Bohr, so that old-quantum theory is preserved. This is done in a wonderful way by Ter-Haar in the 1950s in a little book called "Old quantum theory". The Wikipedia pages on Adiabatic Invariant, Correspondence principle, and Old quantum theory try to present the results in a modern translation, without fussing over the ambiguities that are now known to be caused by working to leading order in h-bar.

Some things missing from all modern books:

• The Lee model: This was historically essential for understanding renormalization, and this needs a modern translation. It doesn't appear in any modern field theory book.
• The Bethe-Salpeter equation: This is important for bound states, although a full bound-state formalism is lacking today.
• The pion-nucleon models: these don't work, but they are an important source of intuition. The magical pion decay rate calculations, the false theorem on identity of the pseudovector vs. vector coupling, etc, these are all forgotten.
• Regge theory: This is unforgivable. There is a short intro in Landau and Lifschitz quantum mechanics, but the only good source is Gribov's book "The theory of Complex Angular Momentum". You need to know which hadrons are on which trajectory, and you won't find this presented in modern language or notation.
• S-matrix theory: This is also unforgivable. I don't know the best book here, there are a few. This is coming back in style, so it might get fixed soon. This includes what Chew was doing with pions and nucleons, that I still can't figure out. But Weinberg eventually concludes it's all equivalent to effective field theory (it should be, but then you need to show the equivalence in some way).

There are a bunch of other things that can be dispensed of quickly today, I can't think of them at this moment.

So the main reason is that the historical documents need to be modernized to be pedagogical, and nobody does this. The only person who (secretly) did this is Feynman! He would rework the results with incredible fidelity to the historical literature, but without saying he is doing history. Almost all his elementary textbook arguments are reworked forgotten results from the old literature, done in his own way. This is why they are such classics, in my opinion, they are extremely historically faithful.

-
I think even Maxwell isn't worth reading today because he predates special relativity. Far better to read the works of talented teachers who teach as a profession and get feedback such as Griffiths with his Classical Electrodynamics book. Prof Lewin is another fantastic teacher who hasn't written a book, but has lectured on the web and maybe this is his medium. Maybe you're starting to fall asleep in the past and not keeping upto date with modern books. – Physiks lover Jun 22 '12 at 22:14
@Physikslover: I think the modern books on everything elementary are a total scandal, with the exception of Feynman's lectures and Landau and Lifschitz. The issue is that you just haven't read Maxwell. Maxwell's concerns do not involve special relativity, but analyzing systems of materials in interaction with electricity, magnetism and light. The reason it is annoying to read is the lack of vector notation and the annoying multiplicity of fields (E,H,B,D), plus the lack of computation. But it would be worthwhile to read a condensed modern version which updates the thing, but keeps the results. – Ron Maimon Jun 23 '12 at 2:13

Because it's in Latin.

That is not the only reason though, just read for yourself a relative modern translation:

SCHOLIUM. If the ellipsis, by having its centre removed to an infinite distance, de generates into a parabola, the body will move in this parabola; and the force, now tending to a centre infinitely remote, will become equable. Which is Galileo's theorem. And if the parabolic section of the cone (by changing the inclination of the cutting plane to the cone) degenerates into an hyperbola, the body will move in the perimeter of this hyperbola, having its centripetal force changed into a centrifugal force. And in like manner as in the circle, or in the ellipsis, if the forces are directed to the centre of the figure placed in the abscissa, those forces by increasing or diminishing the ordinates in any given ratio; or even by changing the angle of the inclination of the ordinates to the abscissa, are always augmented or diminished in the ratio of the distances from the centre; provided the periodic times remain equal; so also in all figures whatsoever, if the ordinates are augmented or diminished in any given ratio, or their inclination is any way changed, the periodic time remaining the same, the forces directed to any centre placed in the abscissa are in the several ordinates augmented or diminished in the ratio of the distances from the centre.

I personally think this is not a straightforward way to express a physical law. As an undergraduate I would have really disliked to read such a text, even if it is written by Newton himself. As our knowledge advances we can connect different areas of physics, these connections might have not been obvious at all at the time of their discovery. Compare this with the lecture from Feynman (can be found online) about the same law.

-
Yes, but this passage has some ideas that can be rephrased in modern language in two or three sentences: the first part says that if the gravitating mass is infinitely far, the gravitational field becomes constant, and the motion parabolic a-la Galileo. The second part is simply deriving the scaling law for the force by dimensional analysis from the speed with which you traverse the orbit. Both need to be expressed in any modern textbook, and neither are, because the textbooks don't read the Principia and translate as I did in the comment. – Ron Maimon Jun 22 '12 at 19:47
@RonMaimon: I agree that it can be rephrased in a few sentences. But isn't this exactly the reason that we don't read the original because it is in a language that is hard to understand? Your idea to create a better translation is tempting and I don't know any publisher that started a project in that direction. – Alexander Jun 22 '12 at 19:53
This is something I try to do online when I can. Chandrashekhar did something like this. The issue for me is that nobody translates the ideas, so that all of them are present. They always pick the least interesting points and transmit these. So you end up having to go back to the original to get all the ideas. – Ron Maimon Jun 22 '12 at 19:54

It's because knowledge is being continually updated and errors corrected. Newton's Principia Mathematica doesn't even contain his second law where force is proportional to the rate of change of momentum. Lagranges's Analytical Mechanics is very difficult to follow where he used concepts such as "the living force" to represent $mv^2$ and levers and pulleys to justify his use of the principle of virtual work. Both these books can be summarised in a few pages without losing the basic principles of what they were doing.

Productive young people at the cutting edge of research are lead by their peers and modern textbooks. Please, please don't look at the old text books until you understand the modern ones, otherwise you'll end up on a merry go around in the past, convinced you've discovered something new.

-
-1: sorry, this answer is both factually incorrect and ridiculous. Newton's Principia contains the second law, and the concept of "vis-viva" is not "living force", but Leibnitz's term for what we now call energy. The levers and pullies justification for virtual work is essential, and must be presented to students in order to explain the concept of work. Feynman does this in vol 1, and this is one of the highlights of his presentation. There is not a single physicist who does original research who doesn't know the history, and the modern textbook treatments are woefully inadequate in every way. – Ron Maimon Jun 23 '12 at 11:21
yeah, his second law as in "A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed" which isn't expressed as proportional to the rate of change of momentum. And Lagranges Analytical Mechanics is very long winded when it comes to explaining virtual work, whereas Feynman does it in a paragraph - why? because we've moved on from 250 years ago as teachers and researchers refine and improve the interpretation and techniques. Please remove the -1 – Physiks lover Jun 25 '12 at 17:05
Newton defines "quantity of motion" two paragraphs previously as mass times velocity! So that "a change in motion" is a change in momentum, except he doesn't use the word "momentum", and calls it "motion". Please read the book, I would -1 the comment too if I could. Newton doesn't mean "proportional", by the way, but "proportional with a universal coefficient", i.e. what we would today call "equal" (up to units). One reason you know the coefficient of proportionality between F and $\dot{p}$ is universal is the third law, but Newton assumes you have some other F notion to fix units. – Ron Maimon Jun 25 '12 at 18:37
@RonMaimon goddamnit you're right on Newton. – Physiks lover Jun 25 '12 at 22:24
@Physicslover: I read the book. As far as "moving on", if you don't read the book, it is hard to determine if we've moved on or not. We have, but at the cost of leaving behind some stuff which is not obsolete. – Ron Maimon Jun 26 '12 at 3:20


text:
# What phenomenon causes the increase of drag with rain?

I know that rain generally worsens the aerodynamic propeties of an aircraft.

I'm wondering what processes cause the increase in drag.

I'm mostly interested in General Aviation, thus relatively low speed and low altitudes.

Several things I can imagine, although the effects might be small:

• The water in the air makes it heavier, increasing the effective value of the air density, $\rho$
• Rain might lower the temperature, again having an effect on $\rho$
• Rain drops on the wing might cause earlier transition on the wing, increasing drag.

I hope somebody can clarify what magnitude these effects have, and if there are any more effects that cause an increase in drag when flying in the rain.

• On the glider rain drops can reduce glide ratio substantially. It depends on the wing profile but it can drop from 45 to 30. – Andrius Feb 18 '16 at 13:37
• @Andrius, I know rain can change the glide ratio significantly, but I'm mostly interested in what causes these changes. I'll reword my question to make it a bit clearer. I'm also focusing more on the increase in drag, to keep a narrow focus. – ROIMaison Feb 18 '16 at 13:45
• For us physics challenged readers, what does $\rho$ stand for? – TomMcW Feb 18 '16 at 19:43
• Sorry, $\rho$ is the density of the air – ROIMaison Feb 18 '16 at 19:45

While Peter Kämpf already commented the effects I'll take a look at the numbers. Don't expect this calculations to be exact predictions, rather worst cases to know the magnitude of following effects:

• Mass increase due to rain drops
• Forces due to drops hitting the wing
• Early tripping of the boundary layer

After deducing some equations I'll feed them with the data of two different airplanes.

Mass increase due to rain drops

The dew layer you find on your airplane in the morning can reach a maximum altitude of 0.8 mm. Raindrops, which do not fill the complete wing surface, don't usually exceed 2 mm diameter.Therefore taking an average distributed height of $h_W=1mm$ for the water on the wing should be still a worst-case prediction.

$$m_{Drops}=\rho_{W} \cdot A\cdot h_W$$

$A$ is the projected area of the aircraft on the horizontal plane and $\rho_W$ water density.

Forces due to drops hitting the wing

Normal rainfall intensities are about 5 mm/hour and we already speak about violent rain with precipitation rates beyond 50 mm/hour. But let us take the heaviest rate ever measured: $I_R=38mm/min$.

Drops fall normally with $v_{Drop}=10m/s$.

To simplify the calculation I'll assume vertical rain and that once a drop hits the airplane it sticks to it. In reality you don't accelerate every drop to the airplane speed nor slow its vertical speed down to 0.

i) Drag increase

$$\Delta D= \rho_{W} \cdot I_R \cdot A \cdot v_{airplane}$$

ii) Lift decrease

$$\Delta L= \rho_{W} \cdot I_R \cdot A \cdot v_{Drop}$$

To calculate the total drag ($D=C_D\cdot \frac{\rho_{Air}}{2} V^2 \cdot A$) I've used $C_D=0.035$. $V$ is the cruise speed of the aircraft.

Early tripping of the boundary layer

While it's possible to find airfoil polars in rain searching the Internet, I'll first look at a "bugs influence" diagram where you can see the effect of small disturbances at the nose without other effects overlayed. Bugs and raindrops are similar in size, therefore the comparison should be valid.

Source

The polar above shows how the drag can double (100% increase)just because of some small disturbances at the leading edge. Keep in mind that in modern profiles the effect will probably be smaller. The next polar is from a transport aircraft airfoil.

Source

While not so important as in the glider airfoil, the drag coefficient increase (experimental results) is everywhere at least a two-digit percentage.

Airplanes

I've taken two airplanes with really different wing loadings: A1 would be something between the two Solar Impulse models and A2 a single engine four-seater like a Cessna 172. The last three rows give the drag, lift and mass difference as described above as a percentage of its total value.

|                       |            A1 |     A2 |
|-----------------------+---------------+--------|
| Mass m [kg]           |          1600 |   1000 |
| Area A [m^2]          |           210 |     20 |
| Cruise speed V [m/s]  |            20 |     58 |
| Lift L [N]            |         15696 |   9810 |
| Drag D [N]            |          1800 |   1323 |
| Drops mass mW [kg]    |           210 |     20 |
| Drag increase dD [N]  |          2660 |    703 |
| Lift decrease dL [N]  |          1330 |    127 |
|-----------------------+---------------+--------|
| dD/D [%]              |           147 |     53 |
| dL/L [%]              |           8.5 |    1.3 |
| mW/m [%]              |            13 |      2 |
|-----------------------+---------------+--------|

SUMMARY

For a normal GA airplane the water mass on the wings and the lift decrease influence due to raindrops hitting the wing even considering the worst possible conditions are low single digit percentages.

While drag increase due to drops hitting the wing is relevant in the extreme case I took here, under normal conditions it should be below single digit percentages too.

On the other hand the drag increase due to the early tripping of the boundary layer is at least a two digit percentage, at high lift coefficients even more.


text:
:: Kleene Algebra of Partial Predicates
:: by Artur Korni{\l}owicz , Ievgen Ivanov and Mykola Nikitchenko
::
:: Copyright (c) 2018 Association of Mizar Users

definition
let D be set ;
func Pr D -> set equals :: PARTPR_1:def 1
PFuncs (D,BOOLEAN);
coherence
PFuncs (D,BOOLEAN) is set
;
end;

:: deftheorem defines Pr PARTPR_1:def 1 :
for D being set holds Pr D = PFuncs (D,BOOLEAN);

registration
let D be set ;
cluster Pr D -> functional non empty ;
coherence
( not Pr D is empty & Pr D is functional )
;
end;

definition
let D be set ;
mode PartialPredicate of D is PartFunc of D,BOOLEAN;
end;

theorem :: PARTPR_1:1
for x being object
for D being set st x in Pr D holds
x is PartialPredicate of D by PARTFUN1:46;

theorem :: PARTPR_1:2
for D being set
for p being PartialPredicate of D holds p in Pr D by PARTFUN1:45;

theorem Th3: :: PARTPR_1:3
for x being object
for D being set
for p being PartialPredicate of D holds
( p . x = TRUE or p . x = FALSE )
proof end;

definition
let D be set ;
defpred S1[ object , Function, object ] means ( $1 in dom$2 & $2 .$1 = $3 ); func PPnegation D -> Function of (Pr D),(Pr D) means :Def2: :: PARTPR_1:def 2 for p being PartialPredicate of D holds ( dom (it . p) = dom p & ( for d being object holds ( ( d in dom p & p . d = TRUE implies (it . p) . d = FALSE ) & ( d in dom p & p . d = FALSE implies (it . p) . d = TRUE ) ) ) ); existence ex b1 being Function of (Pr D),(Pr D) st for p being PartialPredicate of D holds ( dom (b1 . p) = dom p & ( for d being object holds ( ( d in dom p & p . d = TRUE implies (b1 . p) . d = FALSE ) & ( d in dom p & p . d = FALSE implies (b1 . p) . d = TRUE ) ) ) ) proof end; uniqueness for b1, b2 being Function of (Pr D),(Pr D) st ( for p being PartialPredicate of D holds ( dom (b1 . p) = dom p & ( for d being object holds ( ( d in dom p & p . d = TRUE implies (b1 . p) . d = FALSE ) & ( d in dom p & p . d = FALSE implies (b1 . p) . d = TRUE ) ) ) ) ) & ( for p being PartialPredicate of D holds ( dom (b2 . p) = dom p & ( for d being object holds ( ( d in dom p & p . d = TRUE implies (b2 . p) . d = FALSE ) & ( d in dom p & p . d = FALSE implies (b2 . p) . d = TRUE ) ) ) ) ) holds b1 = b2 proof end; end; :: deftheorem Def2 defines PPnegation PARTPR_1:def 2 : for D being set for b2 being Function of (Pr D),(Pr D) holds ( b2 = PPnegation D iff for p being PartialPredicate of D holds ( dom (b2 . p) = dom p & ( for d being object holds ( ( d in dom p & p . d = TRUE implies (b2 . p) . d = FALSE ) & ( d in dom p & p . d = FALSE implies (b2 . p) . d = TRUE ) ) ) ) ); definition let D be set ; let p be PartialPredicate of D; func PP_not p -> PartialPredicate of D equals :: PARTPR_1:def 3 () . p; coherence () . p is PartialPredicate of D proof end; involutiveness for b1, b2 being PartialPredicate of D st b1 = () . b2 holds b2 = () . b1 proof end; end; :: deftheorem defines PP_not PARTPR_1:def 3 : for D being set for p being PartialPredicate of D holds PP_not p = () . p; theorem Th4: :: PARTPR_1:4 for x being object for D being set for p being PartialPredicate of D st x in dom p & () . x = FALSE holds p . x = TRUE proof end; theorem Th5: :: PARTPR_1:5 for x being object for D being set for p being PartialPredicate of D st x in dom p & () . x = TRUE holds p . x = FALSE proof end; theorem :: PARTPR_1:6 for x being object for D being set for p being PartialPredicate of D st x in dom () & () . x = FALSE holds ( x in dom p & p . x = TRUE ) proof end; theorem :: PARTPR_1:7 for x being object for D being set for p being PartialPredicate of D st x in dom () & () . x = TRUE holds ( x in dom p & p . x = FALSE ) proof end; definition let D be non empty set ; defpred S1[ object , Function] means ($1 in dom $2 &$2 . $1 = TRUE ); defpred S2[ object , Function, Function] means ($1 in dom $2 &$2 . $1 = FALSE &$1 in dom $3 &$3 . $1 = FALSE ); deffunc H1( Function, Function) -> set = { d where d is Element of D : ( S1[d,$1] or S1[d,$2] or S2[d,$1,\$2] ) } ;
func PPdisjunction D -> Function of [:(Pr D),(Pr D):],(Pr D) means :Def4: :: PARTPR_1:def 4
for p, q being PartialPredicate of D holds
( dom (it . (p,q)) = { d where d is Element of D : ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) or ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE ) ) } & ( for d being object holds
( ( ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) ) implies (it . (p,q)) . d = TRUE ) & ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE implies (it . (p,q)) . d = FALSE ) ) ) );
existence
ex b1 being Function of [:(Pr D),(Pr D):],(Pr D) st
for p, q being PartialPredicate of D holds
( dom (b1 . (p,q)) = { d where d is Element of D : ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) or ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE ) ) } & ( for d being object holds
( ( ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) ) implies (b1 . (p,q)) . d = TRUE ) & ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE implies (b1 . (p,q)) . d = FALSE ) ) ) )
proof end;
uniqueness
for b1, b2 being Function of [:(Pr D),(Pr D):],(Pr D) st ( for p, q being PartialPredicate of D holds
( dom (b1 . (p,q)) = { d where d is Element of D : ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) or ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE ) ) } & ( for d being object holds
( ( ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) ) implies (b1 . (p,q)) . d = TRUE ) & ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE implies (b1 . (p,q)) . d = FALSE ) ) ) ) ) & ( for p, q being PartialPredicate of D holds
( dom (b2 . (p,q)) = { d where d is Element of D : ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) or ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE ) ) } & ( for d being object holds
( ( ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) ) implies (b2 . (p,q)) . d = TRUE ) & ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE implies (b2 . (p,q)) . d = FALSE ) ) ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def4 defines PPdisjunction PARTPR_1:def 4 :
for D being non empty set
for b2 being Function of [:(Pr D),(Pr D):],(Pr D) holds
( b2 = PPdisjunction D iff for p, q being PartialPredicate of D holds
( dom (b2 . (p,q)) = { d where d is Element of D : ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) or ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE ) ) } & ( for d being object holds
( ( ( ( d in dom p & p . d = TRUE ) or ( d in dom q & q . d = TRUE ) ) implies (b2 . (p,q)) . d = TRUE ) & ( d in dom p & p . d = FALSE & d in dom q & q . d = FALSE implies (b2 . (p,q)) . d = FALSE ) ) ) ) );

definition
let D be non empty set ;
let p, q be PartialPredicate of D;
func PP_or (p,q) -> PartialPredicate of D equals :: PARTPR_1:def 5
() . (p,q);
coherence
() . (p,q) is PartialPredicate of D
proof end;
commutativity
for b1, p, q being PartialPredicate of D st b1 = () . (p,q) holds
b1 = () . (q,p)
proof end;
idempotence
for p being PartialPredicate of D holds p = () . (p,p)
proof end;
end;

:: deftheorem defines PP_or PARTPR_1:def 5 :
for D being non empty set
for p, q being PartialPredicate of D holds PP_or (p,q) = () . (p,q);

theorem Th8: :: PARTPR_1:8
for x being object
for D being non empty set
for p, q being PartialPredicate of D holds
( not x in dom (PP_or (p,q)) or ( x in dom p & p . x = TRUE ) or ( x in dom q & q . x = TRUE ) or ( x in dom p & p . x = FALSE & x in dom q & q . x = FALSE ) )
proof end;

theorem Th9: :: PARTPR_1:9
for x being object
for D being non empty set
for p, q being PartialPredicate of D st x in dom p & x in dom q & (PP_or (p,q)) . x = TRUE & not p . x = TRUE holds
q . x = TRUE
proof end;

theorem Th10: :: PARTPR_1:10
for x being object
for D being non empty set
for p, q being PartialPredicate of D st x in dom (PP_or (p,q)) & (PP_or (p,q)) . x = TRUE & not ( x in dom p & p . x = TRUE ) holds
( x in dom q & q . x = TRUE )
proof end;

theorem Th11: :: PARTPR_1:11
for x being object
for D being non empty set
for p, q being PartialPredicate of D st x in dom p & (PP_or (p,q)) . x = FALSE holds
p . x = FALSE
proof end;

theorem Th12: :: PARTPR_1:12
for x being object
for D being non empty set
for p, q being PartialPredicate of D st x in dom q & (PP_or (p,q)) . x = FALSE holds
q . x = FALSE
proof end;

theorem Th13: :: PARTPR_1:13
for x being object
for D being non empty set
for p, q being PartialPredicate of D st x in dom (PP_or (p,q)) & (PP_or (p,q)) . x = FALSE holds
( x in dom p & p . x = FALSE & x in dom q & q . x = FALSE )
proof end;

:: Associativity law
theorem Th14: :: PARTPR_1:14
for D being non empty set
for p, q, r being PartialPredicate of D holds PP_or (p,(PP_or (q,r))) = PP_or ((PP_or (p,q)),r)
proof end;

theorem Th15: :: PARTPR_1:15
for D being non empty set
for p, q, r being PartialPredicate of D holds PP_or ((PP_or (p,q)),(PP_or (p,r))) = PP_or ((PP_or (p,q)),r)
proof end;

definition
let D be non empty set ;
let p, q be PartialPredicate of D;
func PP_and (p,q) -> PartialPredicate of D equals :: PARTPR_1:def 6
PP_not (PP_or ((),()));
coherence
PP_not (PP_or ((),())) is PartialPredicate of D
;
commutativity
for b1, p, q being PartialPredicate of D st b1 = PP_not (PP_or ((),())) holds
b1 = PP_not (PP_or ((),()))
;
idempotence
for p being PartialPredicate of D holds p = PP_not (PP_or ((),()))
;
func PP_imp (p,q) -> PartialPredicate of D equals :: PARTPR_1:def 7
PP_or ((),q);
coherence
PP_or ((),q) is PartialPredicate of D
;
end;

:: deftheorem defines PP_and PARTPR_1:def 6 :
for D being non empty set
for p, q being PartialPredicate of D holds PP_and (p,q) = PP_not (PP_or ((),()));

:: deftheorem defines PP_imp PARTPR_1:def 7 :
for D being non empty set
for p, q being PartialPredicate of D holds PP_imp (p,q) = PP_or ((),q);

theorem Th16: :: PARTPR_1:16
for D being non empty set
for p, q being PartialPredicate of D holds dom (PP_and (p,q)) = { d where d is Element of D : ( ( d in dom p & p . d = FALSE ) or ( d in dom q & q . d = FALSE ) or ( d in dom p & p . d = TRUE & d in dom q & q . d = TRUE ) ) }
proof end;

theorem Th17: :: PARTPR_1:17
for x being object
for D being non empty set
for p, q being PartialPredicate of D holds
( not x in dom (PP_and (p,q)) or ( x in dom p & p . x = FALSE ) or ( x in dom q & q . x = FALSE ) or ( x in dom p & p . x = TRUE & x in dom q & q . x = TRUE ) )
proof end;

theorem Th18: :: PARTPR_1:18
for x being object
for D being non empty set
for p, q being PartialPredicate of D st x in dom p & p . x = TRUE & x in dom q & q . x = TRUE holds
(PP_and (p,q)) . x = TRUE
proof end;

theorem Th19: :: PARTPR_1:19
for x being object
for D being non empty set
for p, q being PartialPredicate of D st x in dom p & p . x = FALSE holds
(PP_and (p,q)) . x = FALSE
proof end;

theorem :: PARTPR_1:20
for x being object
for D being non empty set
for p, q being PartialPredicate of D st x in dom q & q . x = FALSE holds
(PP_and (p,q)) . x = FALSE by Th19;

theorem :: PARTPR_1:21
for x being object
for D being non empty set
for p, q being PartialPredicate of D st x in dom p & (PP_and (p,q)) . x = TRUE holds
p . x = TRUE by ;

theorem :: PARTPR_1:22
for x being object
for D being non empty set
for p, q being PartialPredicate of D st x in dom q & (PP_and (p,q)) . x = TRUE holds
q . x = TRUE by ;

theorem Th23: :: PARTPR_1:23
for x being object
for D being non empty set
for p, q being PartialPredicate of D st x in dom (PP_and (p,q)) & (PP_and (p,q)) . x = TRUE holds
( x in dom p & p . x = TRUE & x in dom q & q . x = TRUE )
proof end;

theorem Th24: :: PARTPR_1:24
for x being object
for D being non empty set
for p, q being PartialPredicate of D st x in dom p & x in dom q & (PP_and (p,q)) . x = FALSE & not p . x = FALSE holds
q . x = FALSE
proof end;

theorem Th25: :: PARTPR_1:25
for x being object
for D being non empty set
for p, q being PartialPredicate of D st x in dom (PP_and (p,q)) & (PP_and (p,q)) . x = FALSE & not ( x in dom p & p . x = FALSE ) holds
( x in dom q & q . x = FALSE )
proof end;

:: Associativity law
theorem :: PARTPR_1:26
for D being non empty set
for p, q, r being PartialPredicate of D holds PP_and (p,(PP_and (q,r))) = PP_and ((PP_and (p,q)),r) by Th14;

theorem :: PARTPR_1:27
for D being non empty set
for p, q, r being PartialPredicate of D holds PP_and ((PP_and (p,q)),(PP_and (p,r))) = PP_and ((PP_and (p,q)),r) by Th15;

:: Meet-absorbing law
theorem Th28: :: PARTPR_1:28
for D being non empty set
for p, q being PartialPredicate of D holds PP_or ((PP_and (p,q)),q) = q
proof end;

:: Join-absorbing law
theorem Th29: :: PARTPR_1:29
for D being non empty set
for p, q being PartialPredicate of D holds PP_and (p,(PP_or (p,q))) = p
proof end;

:: Distributivity law
theorem Th30: :: PARTPR_1:30
for D being non empty set
for p, q, r being PartialPredicate of D holds PP_and (p,(PP_or (q,r))) = PP_or ((PP_and (p,q)),(PP_and (p,r)))
proof end;

theorem Th31: :: PARTPR_1:31
for D being non empty set
for p, q being PartialPredicate of D holds dom (PP_imp (p,q)) = { d where d is Element of D : ( ( d in dom p & p . d = FALSE ) or ( d in dom q & q . d = TRUE ) or ( d in dom p & p . d = TRUE & d in dom q & q . d = FALSE ) ) }
proof end;

theorem Th32: :: PARTPR_1:32
for x being object
for D being non empty set
for p, q being PartialPredicate of D holds
( not x in dom (PP_imp (p,q)) or ( x in dom p & p . x = FALSE ) or ( x in dom q & q . x = TRUE ) or ( x in dom p & p . x = TRUE & x in dom q & q . x = FALSE ) )
proof end;

theorem Th33: :: PARTPR_1:33
for x being object
for D being non empty set
for p, q being PartialPredicate of D st x in dom p & p . x = FALSE holds
(PP_imp (p,q)) . x = TRUE
proof end;

theorem Th34: :: PARTPR_1:34
for x being object
for D being non empty set
for p, q being PartialPredicate of D st x in dom q & q . x = TRUE holds
(PP_imp (p,q)) . x = TRUE by Def4;

theorem Th35: :: PARTPR_1:35
for x being object
for D being non empty set
for p, q being PartialPredicate of D st x in dom p & p . x = TRUE & x in dom q & q . x = FALSE holds
(PP_imp (p,q)) . x = FALSE
proof end;

theorem :: PARTPR_1:36
for x being object
for D being non empty set
for p, q being PartialPredicate of D st x in dom p & x in dom q & (PP_imp (p,q)) . x = TRUE & not p . x = FALSE holds
q . x = TRUE
proof end;

theorem :: PARTPR_1:37
for x being object
for D being non empty set
for p, q being PartialPredicate of D st x in dom p & (PP_imp (p,q)) . x = FALSE holds
p . x = TRUE by ;

theorem :: PARTPR_1:38
for x being object
for D being non empty set
for p, q being PartialPredicate of D st x in dom q & (PP_imp (p,q)) . x = FALSE holds
q . x = FALSE by ;

theorem :: PARTPR_1:39
for x being object
for D being non empty set
for p, q being PartialPredicate of D st x in dom (PP_imp (p,q)) & (PP_imp (p,q)) . x = FALSE holds
( x in dom p & p . x = TRUE & x in dom q & q . x = FALSE )
proof end;

theorem :: PARTPR_1:40
for x being object
for D being non empty set
for p, q being PartialPredicate of D st x in dom (PP_imp (p,q)) & (PP_imp (p,q)) . x = TRUE & not ( x in dom p & p . x = FALSE ) holds
( x in dom q & q . x = TRUE )
proof end;

theorem :: PARTPR_1:41
for D being non empty set
for p, q, r being PartialPredicate of D holds PP_and ((PP_imp (p,r)),(PP_imp (q,r))) = PP_imp ((PP_or (p,q)),r)
proof end;

theorem :: PARTPR_1:42
for D being non empty set
for p, q, r being PartialPredicate of D holds PP_or ((PP_imp (p,r)),(PP_imp (q,r))) = PP_imp ((PP_and (p,q)),r)
proof end;

:: True constant predicate
definition
let D be set ;
func PP_True D -> PartialPredicate of D equals :: PARTPR_1:def 8
D --> TRUE;
coherence
D --> TRUE is PartialPredicate of D
proof end;
end;

:: deftheorem defines PP_True PARTPR_1:def 8 :
for D being set holds PP_True D = D --> TRUE;

:: False constant predicate
definition
let D be set ;
func PP_False D -> PartialPredicate of D equals :: PARTPR_1:def 9
D --> FALSE;
coherence
D --> FALSE is PartialPredicate of D
proof end;
end;

:: deftheorem defines PP_False PARTPR_1:def 9 :
for D being set holds PP_False D = D --> FALSE;

theorem Th43: :: PARTPR_1:43
for D being set holds PP_not () = PP_True D
proof end;

theorem :: PARTPR_1:44
for D being set holds PP_not () = PP_False D
proof end;

theorem Th45: :: PARTPR_1:45
for D being non empty set
for p being PartialPredicate of D holds PP_or (p,()) = PP_True D
proof end;

theorem :: PARTPR_1:46
for D being non empty set
for p being PartialPredicate of D holds PP_or ((),p) = PP_True D by Th45;

theorem Th47: :: PARTPR_1:47
for D being non empty set
for p being PartialPredicate of D holds PP_and (p,()) = PP_False D
proof end;

theorem :: PARTPR_1:48
for D being non empty set
for p being PartialPredicate of D holds PP_and ((),p) = PP_False D by Th47;

theorem Th49: :: PARTPR_1:49
for D being non empty set
for p being PartialPredicate of D holds PP_or (p,()) = () | (dom p)
proof end;

theorem :: PARTPR_1:50
for D being non empty set
for p being PartialPredicate of D holds PP_or ((),p) = () | (dom p) by Th49;

theorem Th51: :: PARTPR_1:51
for D being non empty set
for p being PartialPredicate of D holds PP_and (p,()) = () | (dom p)
proof end;

theorem :: PARTPR_1:52
for D being non empty set
for p being PartialPredicate of D holds PP_and ((),p) = () | (dom p) by Th51;

theorem :: PARTPR_1:53
for D being non empty set
for p being PartialPredicate of D holds PP_imp ((),p) = PP_True D
proof end;

theorem :: PARTPR_1:54
for D being non empty set
for p being PartialPredicate of D holds PP_imp (p,()) = PP_True D by Th45;

theorem Th55: :: PARTPR_1:55
for D being non empty set
for p, q being PartialPredicate of D holds PP_or ((() | (dom p)),(() | (dom q))) = () | (dom q)
proof end;

:: Empty predicate
definition
let D be set ;
coherence
{} is PartialPredicate of D
by RELSET_1:12;
end;

:: deftheorem defines PP_BottomPred PARTPR_1:def 10 :
for D being set holds PP_BottomPred D = {} ;

theorem Th56: :: PARTPR_1:56
for D being set holds PP_not () = PP_BottomPred D
proof end;

theorem Th57: :: PARTPR_1:57
for D being non empty set holds PP_or ((),()) = PP_True D
proof end;

theorem :: PARTPR_1:58
for D being non empty set holds PP_and ((),()) = PP_False D
proof end;

theorem :: PARTPR_1:59
for D being non empty set holds PP_imp ((),()) = PP_True D
proof end;

definition
let D be non empty set ;
func PartialPredConnectivesMeet D -> BinOp of (Pr D) means :Def11: :: PARTPR_1:def 11
for p, q being PartialPredicate of D holds it . (p,q) = PP_and (p,q);
existence
ex b1 being BinOp of (Pr D) st
for p, q being PartialPredicate of D holds b1 . (p,q) = PP_and (p,q)
proof end;
uniqueness
for b1, b2 being BinOp of (Pr D) st ( for p, q being PartialPredicate of D holds b1 . (p,q) = PP_and (p,q) ) & ( for p, q being PartialPredicate of D holds b2 . (p,q) = PP_and (p,q) ) holds
b1 = b2
proof end;
func PartialPredConnectivesJoin D -> BinOp of (Pr D) means :Def12: :: PARTPR_1:def 12
for p, q being PartialPredicate of D holds it . (p,q) = PP_or (p,q);
existence
ex b1 being BinOp of (Pr D) st
for p, q being PartialPredicate of D holds b1 . (p,q) = PP_or (p,q)
proof end;
uniqueness
for b1, b2 being BinOp of (Pr D) st ( for p, q being PartialPredicate of D holds b1 . (p,q) = PP_or (p,q) ) & ( for p, q being PartialPredicate of D holds b2 . (p,q) = PP_or (p,q) ) holds
b1 = b2
proof end;
func PartialPredConnectivesCompl D -> UnOp of (Pr D) means :Def13: :: PARTPR_1:def 13
for p being PartialPredicate of D holds it . p = PP_not p;
existence
ex b1 being UnOp of (Pr D) st
for p being PartialPredicate of D holds b1 . p = PP_not p
proof end;
uniqueness
for b1, b2 being UnOp of (Pr D) st ( for p being PartialPredicate of D holds b1 . p = PP_not p ) & ( for p being PartialPredicate of D holds b2 . p = PP_not p ) holds
b1 = b2
proof end;
end;

:: deftheorem Def11 defines PartialPredConnectivesMeet PARTPR_1:def 11 :
for D being non empty set
for b2 being BinOp of (Pr D) holds
( b2 = PartialPredConnectivesMeet D iff for p, q being PartialPredicate of D holds b2 . (p,q) = PP_and (p,q) );

:: deftheorem Def12 defines PartialPredConnectivesJoin PARTPR_1:def 12 :
for D being non empty set
for b2 being BinOp of (Pr D) holds
( b2 = PartialPredConnectivesJoin D iff for p, q being PartialPredicate of D holds b2 . (p,q) = PP_or (p,q) );

:: deftheorem Def13 defines PartialPredConnectivesCompl PARTPR_1:def 13 :
for D being non empty set
for b2 being UnOp of (Pr D) holds
( b2 = PartialPredConnectivesCompl D iff for p being PartialPredicate of D holds b2 . p = PP_not p );

definition
let D be non empty set ;
coherence ;
end;

:: deftheorem defines PartialPredConnectivesLatt PARTPR_1:def 14 :
for D being non empty set holds PartialPredConnectivesLatt D = OrthoLattStr(# (Pr D),,, #);

registration
let D be non empty set ;
let f, g be BinOp of D;
let h be UnOp of D;
cluster OrthoLattStr(# D,f,g,h #) -> non empty ;
coherence
not OrthoLattStr(# D,f,g,h #) is empty
proof end;
end;

registration
let D be non empty set ;
coherence ;
end;

registration
existence
ex b1 being LattStr st
( not b1 is empty & b1 is constituted-Functions )
proof end;
end;

registration
existence
ex b1 being OrthoLattStr st
( b1 is strict & not b1 is empty & b1 is constituted-Functions )
proof end;
end;

registration
let D be non empty set ;
coherence
proof end;
coherence
proof end;
coherence
proof end;
end;

theorem Th60: :: PARTPR_1:60
for D being non empty set holds Top = PP_True D
proof end;

theorem Th61: :: PARTPR_1:61
for D being non empty set holds Bottom = PP_False D
proof end;

definition
let L be non empty constituted-Functions LattStr ;
let a, b be Element of L;
pred a is_a_partial_complement_of b means :: PARTPR_1:def 15
( a "\/" b = (Top L) | (dom b) & b "\/" a = (Top L) | (dom b) & a "/\" b = () | (dom b) & b "/\" a = () | (dom b) );
end;

:: deftheorem defines is_a_partial_complement_of PARTPR_1:def 15 :
for L being non empty constituted-Functions LattStr
for a, b being Element of L holds
( a is_a_partial_complement_of b iff ( a "\/" b = (Top L) | (dom b) & b "\/" a = (Top L) | (dom b) & a "/\" b = () | (dom b) & b "/\" a = () | (dom b) ) );

definition
let L be non empty constituted-Functions LattStr ;
attr L is partially_complemented means :: PARTPR_1:def 16
for b being Element of L ex a being Element of L st a is_a_partial_complement_of b;
end;

:: deftheorem defines partially_complemented PARTPR_1:def 16 :
for L being non empty constituted-Functions LattStr holds
( L is partially_complemented iff for b being Element of L ex a being Element of L st a is_a_partial_complement_of b );

definition
let IT be non empty constituted-Functions LattStr ;
attr IT is partially_Boolean means :: PARTPR_1:def 17
( IT is bounded & IT is partially_complemented & IT is distributive );
end;

:: deftheorem defines partially_Boolean PARTPR_1:def 17 :
for IT being non empty constituted-Functions LattStr holds
( IT is partially_Boolean iff ( IT is bounded & IT is partially_complemented & IT is distributive ) );

registration
coherence
for b1 being non empty constituted-Functions LattStr st b1 is partially_Boolean holds
( b1 is bounded & b1 is partially_complemented & b1 is distributive )
;
coherence
for b1 being non empty constituted-Functions LattStr st b1 is bounded & b1 is partially_complemented & b1 is distributive holds
b1 is partially_Boolean
;
end;

theorem Th62: :: PARTPR_1:62
for D being non empty set
for p being PartialPredicate of D
for a, b being Element of st a = p & b = PP_not p holds
b is_a_partial_complement_of a
proof end;

registration
let D be non empty set ;
coherence
proof end;
end;

:: Distributivity law
theorem :: PARTPR_1:63
for D being non empty set
for p, q, r being PartialPredicate of D holds PP_or (p,(PP_and (q,r))) = PP_and ((PP_or (p,q)),(PP_or (p,r)))
proof end;

definition
let L be non empty OrthoLattStr ;
attr L is Kleene means :: PARTPR_1:def 18
for x, y being Element of L holds x "/\" (x ) [= y "\/" (y );
end;

:: deftheorem defines Kleene PARTPR_1:def 18 :
for L being non empty OrthoLattStr holds
( L is Kleene iff for x, y being Element of L holds x "/\" (x ) [= y "\/" (y ) );

registration
coherence
for b1 being non empty meet-commutative meet-absorbing join-absorbing OrthoLattStr st b1 is Boolean & b1 is well-complemented holds
b1 is Kleene
proof end;
end;

registration
let D be non empty set ;
coherence
proof end;
end;

registration
existence
ex b1 being non empty constituted-Functions LattStr st
( b1 is partially_Boolean & b1 is join-idempotent & b1 is Lattice-like )
proof end;
end;

registration
existence
ex b1 being non empty OrthoLattStr st
( b1 is Kleene & b1 is de_Morgan & b1 is join-idempotent & b1 is with_idempotent_element & b1 is Lattice-like & b1 is strict )
proof end;
end;

registration
existence
ex b1 being non empty constituted-Functions OrthoLattStr st
( b1 is partially_Boolean & b1 is Kleene & b1 is de_Morgan & b1 is join-idempotent & b1 is with_idempotent_element & b1 is Lattice-like & b1 is strict )
proof end;
end;


text:
# zbMATH — the first resource for mathematics

New multiple harmonic sum identities. (English) Zbl 1308.11022
As one of the authors did also elsewhere (H. Prodinger [Appl. Anal. Discrete Math. 2, No. 1, 65–68 (2008; Zbl 1273.11034); Integers 8, No. 1, Article A10, 8 p. (2008; Zbl 1162.05004); Util. Math. 83, 291–299 (2010; Zbl 1242.05023)]), this paper uses the partial fraction decomposition, a technique derived from the seminal study on Rice’s integrals by P. Flajolet and R. Sedgewick [Theor. Comput. Sci. 144, No. 1–2, 101–124 (1995; Zbl 0869.68056)], in order to establish three new identities about a class of binomial sums involving the harmonic numbers.
Two applications are supplied: the first, to infinite series like the exotic sums, recalls the evaluations for Euler sums in terms of zeta values given, e.g., by P. Flajolet and B. Salvy [Exp. Math. 7, No. 1, 15–35 (1998; Zbl 0920.11061)]; the second application, to congruences, employs the Wolstenholme’s theorem and results provided by J. Zhao [Int. J. Number Theory 4, No. 1, 73-106 (2008; Zbl 1218.11005)] and by Kh. Hessami Pilehrood et al. [Int. J. Number Theory 8, No. 7, 1789–1811 (2012; Zbl 1261.11001)], [Trans. Am. Math. Soc. 366, No. 6, 3131–3159 (2014; Zbl 1308.11018)]

##### MSC:
 11B65 Binomial coefficients; factorials; $$q$$-identities 05A19 Combinatorial identities, bijective combinatorics 11A07 Congruences; primitive roots; residue systems 11M32 Multiple Dirichlet series and zeta functions and multizeta values
Full Text: