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## Über dieses Buch

Within the tradition of meetings devoted to potential theory, a conference on potential theory took place in Prague on 19-24, July 1987. The Conference was organized by the Faculty of Mathematics and Physics, Charles University, with the collaboration of the Institute of Mathematics, Czechoslovak Academy of Sciences, the Department of Mathematics, Czech University of Technology, the Union of Czechoslovak Mathematicians and Physicists, the Czechoslovak Scientific and Technical Society, and supported by IMU. During the Conference, 69 scientific communications from different branches of potential theory were presented; the majority of them are in cluded in the present volume. (Papers based on survey lectures delivered at the Conference, its program as well as a collection of problems from potential theory will appear in a special volume of the Lecture Notes Series published by Springer-Verlag). Topics of these communications truly reflect the vast scope of contemporary potential theory. Some contributions deal with applications in physics and engineering, other concern potential theoretic aspects of function theory and complex analysis. Numerous papers are devoted to the theory of partial differential equations. Included are also many articles on axiomatic and abstract potential theory with its relations to probability theory. The present volume may thus be of intrest to mathematicians speciali zing in the above-mentioned fields and also to everybody interested in the present state of potential theory as a whole.

## Inhaltsverzeichnis

### On Weighted Beppo Levi Functions

Let 1 < p < ∞ and let w be a weight satisfying the Muckenhoupt AP condition. We write $$\left\| f \right\|_{L^p ,w} = (\int\limits_{R^n } {\left| {f(x)} \right|} ^p w(x)dx^{1/p} ,L^p (R^n ,w) = \{ f;\left\| f \right\|_{L^p ,w} < \infty$$.

Hiroaki Aikawa

### Generalized Cauchy-Riemann Equations and the Positive -Subharmonic Functions

In this paper we will give a statement about the positive solutions of the generalized Cauchy-Riemann system in R2n. It will also be shown that the positive solutions are related with the “$$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{k}$$-subharmonicity”.

Ömer Akin, A. Okay Çelebi

### Iteration Methods for Potential Problems

Two iteration methods are considered as a means of constructing solutions of various integral equation formulations of the exterior Dirichlet and Neumann potential problems. These integral equations may be of the second kind, in which case the integral operators will in general be non self-adjoint, or of the first kind, which in the Dirichlet case leads to self-adjoint operators.

Thomas S. Angell, Ralph E. Kleinman, Gary F. Roach

### On the Poisson Equation on the Infinite Dimensional Torus

Let {μt}t>0 be the Wiener semigroup on the infinite dimensional torus T∞ (cf. Heyer,5 Ch. 5). Let us consider the Green measure $$G(dy) = \int\limits_0^\infty {e^{ - t} \mu _t (dy)dt}$$.

Alexander D. Bendikov, Igor V. Pavlov

### A Characterization of Elementary Potential Theory

In the following (Vλ) will be a Ray resolvent on a compact metric space E with set D of non-branchpoints, E denotes the class of positive Borel functions on E, and S is the cone of functions on D, which are excessive for (Vλ). For simplification we assume V1:= V01 to be continuous.

### Wiener Estimates for Solutions of Elliptic Equations in Nondivergence Form

In this paper we deal with a Wiener estimate of the modulus of continuity at the boundary for the solutions of elliptic problems in nondivergence form. It is well known that the Wiener criterion for the continuity of the Laplace equation at a point of the boundary was given at first by Wiener (1923) and was extended to general linear elliptic (second order) equations in divergence form by Littman, Stampacchia, Weinberger (1963) (for the intermediate stages see the references of this last paper).

Marco Biroli

### Approximation by Continuous Potentials

In this note we improve theorems in [1] and [2] dealing with approximation of (super)harmonic functions by continuous potentials. That is, we intend to show that for every finely open set G of a balayage space (X, W) there exists a continuous potential q ε P such that $$S(G) = \overline {P + \mathbb{R}q} ,H(G) = \overline {H(q)}$$.

Jürgen Bliedtner, Wolfhard Hansen

### The Schrödinger Equation Δu = µu in a Neighbourhood of an Isolated Singularity of µ

In this paper we present new results about the Picard principle and the behaviour of the continuous solutions of the Schrödinger equation Δu = µu at an isolated singularity of the measure µ. Using the axiomatic potential theory, a criterion for the integrability of µ is given.

Abderrahman Boukricha

### A Characterization of the Fine Sheaf Property

Throughout this article S will be a standard H-cone of functions on a set X. For definitions and notations see Springer Lecture Notes in Mathematics 853.

Aurel Cornea

### Harmonic Morphisms and Ray Processes

In the following we will let (X, U) and (Y, V) denote P-harmonic spaces in the sense of Constantinescu and Cornea [3]. (X, Y are locally compact Hausdorff spaces with countable bases and U, V are hyperharmonic sheaves on respectively.) We will assume that the constant function 1 is hyperharmonic. It is now well known that there exist Hunt processes Xt,Yt on X, V with continuous paths (i.e. diffusions) such that the family of Xt-excessive, resp. Yt-excessive, functions coincide with the family of non-negative U-hyperharmonic, resp. V-hyperharmonic, functions.

Laszlo Csink, Bernt Øksendal

### On Dirichlet’s Boundary Value Problem for Certain Anisotropic Differential and Pseudo-Differential Operators

In our previous paper [1] a calculus for pseudo-differential operators was used to find periodic solutions for a large class of not necessarily elliptic pseudo-differential equations with constant coefficients. In this paper we consider a generalized homogeneous Dirichlet-problem for pseudo — differential operators with constant coefficients and prove Fredholm’s alternative theorem. We give two examples how to apply this result. The first one is a differential operator arising in stochastics. In the second example we deal with an anisotropic pseudo-differential operator and using the theory of Dirichlet forms (see [4]) it follows that this operator generates a symmetric Hunt process. Some of our results had been already used in [6].

Karl Doppel, Niels Jacob

### Hyperharmonic Cones

The theory of H-cones ([3]) covers the superharmonic case in potential theory. This work continues the algebraic axiomatization of the hyperharmonic case. Our basic structure is a hyperharmonic structure defined by M. Arsove and H. Leutwiler in [1]. The cancellation law does not hold in hyperharmonic structures. In order to characterize cancellable elements we assume that a hyperharmonic structure is a convex cone and the greatest lower bound of (u/n) nεℕ exists for all u.

Sirkka-Liisa Eriksson-Bique

### Jump Relations for Surface Potentials in Case of Continuous Densities

It is well-known that the single layer surface potential with uniform HÖLDER continuous density is continuously differentiable with respect to each side of the surface and fulfills jump relations for the first order derivatives. In this paper it is shown that the jump relations still hold in the CAUCHY sense if the density is supposed to be only continuous.

Alexander Hornberg

### On a Unified Characterization of Capacity

The notion of capacity as a fundamental quantity associated with a smooth surface was treated exhaustively by Polya and Szego in a series of papers over more than twenty years following earlier work by Fekete (1923) and many others who considered the problem of determining the charge accumulated on an electrostatic condenser maintained at a constant potential. See Polya and Szego (1951) and Payne (1967) for extensive references.

George C. Hsiao, Ralph E. Kleinman

### A Note on the Computation of Green Functions for Sublaplacians on Lie Algebras of Type H

In [4] A. Kaplan introduced a class of step-2 nilpotent Lie groups for which the fundamental solution corresponding to the standard sublaplacian may be written down in an explicit elementary form. He also says that heuristic evidence suggests that these groups are the only ones which admit such nice explicit formulas. When I tried to convince myself of this evidence I found some modifications of Kaplan’s proof, which give additional insight into the problem and may be — at least for people working in potential theory — regarded as a simplification.

H. Hueber

### On Thermoelastic Potential

S.Klainerman, in [1] proved that for suficiently small nonlinear perturbations of linear dissipative equations and systems like: the wave equation, heat equation, linear isotropic elasticity equations etc., the corresponding solution of the initial-value problem for perturbed equations and system behave asymptotically like the solutions of the initial-value problem for linear unperturbed equations and system.

Hung Do Duc, Jerzy Gawinecki, Adam Piskorek

### The Dirichlet Problem on Ends

In the recent investigations of the Dirichlet problem in an axiomatic framework we focus our interest on the following two problems: (1) the boundary behavior of the normalized PWB-solutions, originally studied by O. Frostman[3] and generalized by Constantinescu-Cornea[2] and Lukeš-Malý[11] and (2) the unicity of Keldych operators formulated by J. Lukeš[10], given by Bliedtner-Hansen[1] its essential contribution. The normalized PWB-solutions HfU,X on an open set U is formed by the Perron-Wiener-Brelot’s method taking the closure of U in the one-point compactified space and putting f to be 0 at the infinity, and the effect of the ideal boundary is neglected. In this article we consider above problem under the full influence of the ideal boundary and in the conclusing result we reflect that our result is a generalized version of that obtained formerly.

Teruo Ikegami

### Diffusion Kernels of Logarithmic Type

Let X be a locally compact, non-compact Hausdorff space with countable basis. We denote by: CK(X) the usual topological vector space of all finite continuous functions with compact support;C(X) the usual Fréchet space of all finite continuous functions on X;MK(X) the usual topological vector space of all real Radon measures with compact support;M(X) the topological vector space of real Radon measures on X with the weak topology.

Masayuki ItÔ

### A Remark on Capacitary Measures for Diffusion Processes

In this note we shall give a characterization of capacitary measures of semipolar sets for diffusion processes, which is closely related to Dellacherie, Feyel and Mokobodzki2. In this study, the process X = (Xt) is assumed to be a standard process on (E, E), where E is a locally compact separable Hausdorff space and E is the Borel algebra on E. For X we suppose the following: i) X satisfies the strong duality hypotheses relative to a σ-finite measure ξ(dx) = dx; ii) U(x, K) and Û(x, K) are bounded on E for each compact subset K of E; iii) the kernel density u(x, y) of U(x, dy) relative to ξ is continuous off the diagonal and x → u(x, y), y → u(x, y) are lower semicontinuous. The brief explanation of the notations above will be given in the subsequent section. But we assume that the reader is familiar with terminologies in Blumenthal and Getoor1. A process is called a diffusion process on E if it is a standard process on (E, E) with continuous paths. For a subset B of E, TB denotes the hitting time of B, that is, TB = inf(t>0, XtεB).

Mamoru Kanda

### On the Convexity of Level Sets for Elliptic and Parabolic Exterior Boundary Value Problems

The talk is divided into two parts. First I study an elliptic free boundary problem and show that sometimes it is useful to work with curvilinear coordinates. Then I present some results on parabolic problems, which I obtained in cooperation with I. Diaz.

Bernhard Kawohl

### Harnack Inequality and Some Characteristics of Non-Regular Points of Simplexes

Let S be a compact convex set, E=E(S) — the set of all extreme points of S, Shs=Ē\E — the set of all non-regular points of S; C(Ē) — the vector space of all real continuous functions on Ē.

D. G. Keselman

### Polar Sets in a Nonlinear Potential Theory

In this lecture we discuss nonlinear potential theory based on “A-super-harmonic functions”; the theory can be viewed as a (nonlinear) extension of the classical study of superharmonic functions in ℝn.

T. Kilpeläinen

### A Sharper Form of a Theorem of Kolmogorov

1°. Let H+ be the class of functions f(z) holomorphic in $$\mathbb{D} = \left\{ {z \in \mathbb{C}:\left| z \right| < 1} \right\}$$ and such that Ref(z)>0. Functions in H+ have the representation (1)$$\int_{D} {f(w)dudv = L(f)} (w = u + iv)$$

Boris Korenblum

### Extremal Problems for the Distributed Moment Problem

The present work continues the investigation of the socalled distributed moment problem (DMP) (see [13] and references there). We use the last name instead of its alternative one, inverse potential problem [19], to stress the analogy with the one-dimensional moment problem [1,8,15].

Ognyan Kounchev

### On Mathematical Modelling of Non-Newtonian Flow by Perturbation Methods

Isothermal incompressible flow in consideration of friction influences can generally be described by the non-linear system of partial differential equations (1)$$\text{Re}\,\frac{{\text{dv}}}{{\text{dt}}}\, = \, - \operatorname{Re} \,.\,\text{grad}\,\text{p}\,\text{ + }\,\text{DIV}\,\text{T}\,\text{ + }\,\text{f}\,\text{,}\,\text{div v}\,\text{ = }\,\text{0}\text{.}$$

R. Kreul, H. Kretzschmar

### On Diffusion Semigroups Generated by Semi-Elliptic Differential Operators in Infinite Dimensions

We want to study some continuity properties of operator semigroups, generated by a semi-elliptic differential operator on a real separable Hilbert space ℍ. To this end, let us begin by writing the finite-dimensional semi-elliptic differential operator (1)$$\text{Lu(x) = }\frac{\text{1}}{\text{2}}\sum\limits_{i,j = 1}^n {a_{ij} } (x)\frac{{\partial ^2 u}}{{\partial x_i \partial x_j }}(x) + \sum\limits_{i = 1}^n {b_i } (x)\frac{{\partial u}}{{\partial x_i }}(x)$$ on ℍ = ℝn in coordinate-free form as (1a)$$Lu(x) = \frac{1}{2} tr u''(x)(a(x)\cdot ,\cdot ) + u'(x)(b(x))$$

Gottlieb Leha

### The Obstacle Problem in a Non-Linear Potential Theory

M. Brelot gave rise to the concept harmonic space when he extended classical potential theory on ℝn to an axiomatic system on a locally compact space. I have recently constructed1 a non-linear harmonic space by dropping the assumption that the sum of two harmonic functions is harmonic and considering some other axioms instead. This approach has its origin in the work of O. Martio, P. Lindqvist and S. Granlund2,3,4, who have developed a non-linear potential theory on ℝn connected with variational integrals of the type ∫ F(x,∇u(x)) dm(x), where F(x, h) ≈ |h|p.

P. Lehtola

### On the Appell Transformation

The main purpose of this paper is to show that the well known Appell transformation [1] is essentially — i.e. up to compositions with the obvious transformations of the form (1.2) mentioned below — the only transformation, mapping one solution of the n-dimensional heat equation $$\Delta u\, = \,\frac{{\partial u}}{{\partial t}}$$ into another (see Corollary 2.2 for a precise statement).

Heinz Leutwiler

### Perturbation and Excessive Functions

Originally, N.Boboc and Gh.Bucur [2] studied perturbations of a resolvent or a semigroup on a measurable space, by bounded kernels. They gave a characterization of the perturbed excessive structures.

Habib Maagli

### Capacities on Harmonic Spaces with Adjoint Structure

In the classical potential theory, the capacities defined in terms of Green potentials coincide with the capacity defined by Dirichlet integrals; more precisely, for a compact set K in a Greenian domain Ω in ℝd, $$Sup\{ \mu (\Omega )|G\mu \leqq 1,\,Supp\,\mu \subset K\} = \inf \{ \smallint G\mu \,d\mu |G\mu \geqq 1\,on\,K\} = \,\inf \{ D[f]\,|\,f:\,potential\,on\,\Omega \,with\,f \geqq 1\,on\,K\}$$, where Gμ is the Green potential of μ ≧ 0 on Ω and D[f] is the Dirichlet integral of f.

Fumi-Yuki Maeda

### Balayage Spaces on Topological Sums

Several authors, e.g. N. Bouleau /6/, N. Boboc and Gh. Bucur /4/, H. Ben Saad and K. Janßen /2/, A. Boukricha /5/, studied potentialtheoretic structures on topological sums and their connection uith the biharmonic spaces of E.P. Smyrnélis /10/.

Michael Meyer

### Subordination for Balayage Spaces

Let X be a Iccb space, Uand W two sub-Markov resolvents on X possessing proper potential kernels U and V such that W is subordinated to U (i. e. Vλ ≤ Uλ Vλ]0, ∞[). Denote by S U , S W respectively E U E` W the cone of super-median respectively excessive functions associated with U and W. The purpose of this article is: (i) to study the situation where (X, E U ) and (X, E W ) are both balayage spaces (in the sense of [3]) and P is the sub-Markov kernel on X satisfying U=V+PU, and (ii) to give conditions on U, V and P under which (X, E W ) respectively (X, E U ) form a balayage space, provided (X, E U ) respectively (X, E W ) is a balayage space. The notations are the notations of [3].

Hans-Helge Müller

### On the New Problem in the Potential Gravity Theory

The results presented in this paper were obtained in the course of an investigation of a simple model problem of the time-dependent Earth s gravity field, but they are equally applicable to other problems governed by thermo-magnetohydro-dynamics and geophysics. The author has studied the problem of the existence, uniqueness and regularity of the solution of the variational problem of the coupled gravity field, velocity field in thermo-Bingham’s fluid (by which the Earth s interior is simulated) and a quasi-stationary magnetic field.

Jiři Nedoma

### Stochastic Differentiation Formula and Convergence Theorem for Finely Hypoharmonic Functions

As a consequence of a convergence theorem for finely hyperharmonic functions given by us in [4e], we give here an elementary proof of the following fact: Let K ⊂ C be a compact set in the complex plane and let (fn) be a Cauchy sequence in (Cc(K)), sup | |) of functions holomorphic in the neighborhoods of K. Then for any point z in the fine interior K’ of k there is a compact fine neighborhood V of Z in K’ of K there is a compact fine neighborhood V of Z in K’ such that: The process (∂zfn(Zτ)) is locally a cauchy sequence in L2([0,∞[xΩ, μz,) for any Z’ in K’, where μz, denotes the Doléans measure on[0,+∞[xΩ associated with the complex standard Brownian motion started at z’ and stopped at the fine boundary of K’.

Nguyen Xuan-Loc

### Morphisms of Stonian Cones

The main result of this paper puts forward a class of morphisms of H-cones, possessing image: namely those commuting with arbitrary infimum. Applying this result to the canonical embedding in the bidual, one gets a decomposition of any H-cone In the particular case when the cone has the specific order, this decomposition gives a well-known theorem of Dixmier [1], on the structure of stonian compact spaces.

Eugen Popa

### On the Transition Function of the Infinite Dimensional Ornstein-Uhlenbeck Process Given by the Free Quantum Field

In this paper we want to investigate a particular semigroup (πt)t>0 of probability kernels defined on some infinite dimensional Banach space Bα contained in Y’(ℝd−1) (i.e. the tempered distributions on ℝd-1), d ≥ 2, $$\alpha \in ]\frac{{d - 2}}{2},\infty$$. This semigroup (πt)t>0 is the transition function of the infinite dimensional Ornstein-Uhlenbeck process given by the free field of Euclidean quantum field theory on ℝd (cf. [1], [2]). We will establish that (πt)t>0 has the Feller property, determine its generator (on a suitable domain) and its Dirichlet form. Furthermore, we will characterise the associated entrance space in the sense of Dynkin (cf. [3], [4], [5]).

Michael Röckner

### Finiteness of the Family of Simply Connected Quadrature Domains

A quadrature domain D of a functional L is a domain with finite area satisfying (1.1)$$\begin{array}{*{20}c}{\smallint _D f(w)dudv = L(f)} {(w = u + iv)} \\\end{array}$$ for every f in the class AL1(D) of functions analytic and integrable in D. A simple example is a disk: The unit disk is a quadrature domain of f ↦ πf(0). In this case, the quadrature domain is determined uniquely. But, in general, it is not easy to determine all quadrature domains of a given functional.

Makoto Sakai

### On an Inverse Problem for the Newtonian Potential

In a joint work with C. Maderna and C. Pagani,1 we have considered the following classical inverse problem of potential theory: to determine the figure of a Homogeneous material body G whose shape is unknown, from measurements of the Newtonian potential created by G, taken on the surface of a sphere containing G in its interior.

Sandro Salsa

### On the Integration of Elastostatic Displacement Equation

The Cauchy-Navier equation of motion for the linear elastic isotropic continuum, for the vanishing body forces, may be written in the form

Abdullah Shidfar

### Non-Linear Potential Theory in Lebesgue Spaces with Mixed Norm

The classical potential theory for the Riesz kernel in ℝd has been generalized in many different ways. In this paper we are concerned with the Lp-potential theory which appeared around 1970. See N.G. Meyers [Me] and V.G. Maz’ya and V.P. Havin [MH] for an introduction to this theory. Applications and further developments of the theory are found in the books V.G. Maz’ya [Ma], V.G. Maz’ya, T.O. Shaposhnikova [MS], the lecture notes D.R. Adams [A] and the survey article L.I. Hedberg [H]. See also [S1] and [S2].

Tord Sjödin

### Schroedinger Equations with Discontinuous Solutions

In this paper we study the Schrödinger equation $$( - \frac{1}{2}\Delta + \mu )h = 0$$ on a (possibly unbounded) domain U of Rd(d ≥ 1). µ will be a signed measure with µ+ ε Dloc and µ−ε D1. By definition, a measure v ≥ 0 is in the class Dloc if it is associated with an increasing continuous additive functional A v of standard Brownian motion (Px,X t : x ε Rd, t ≥ 0) such that for some t > 0 and every compact set C ⊂ Rd$$\mathop {\sup }\limits_{x \in R^d } E^x [A_v (t).1_c (X_t )] < \infty$$.

Theo Sturm

### Polynomial Hulls and Envelopes of Holomorphy

This is a report on recent joint work with Herbert Alexander.

John Wermer

### Green Function and Uniform Harnack Inequality

We show that any harmonic space which possesses a Green function with certain properties satisfies the uniform inequality. Also a lower bound for the harmonic measure of balls is given.

Rainer Wittmann

### On the Asymptotic Behavior of Solutions of a System of Integral Equations of Mixed Boundary Value Problem of Plane Elasticity in a Neighborhood of Corner Points of the Contour

In the papers [1–2] a method for investigation of asymptotics of solutions near singularities of the boundary of boundary integral equations, arising in problems of potential theory was proposed. This method is based on the fact that solutions of integral equations can be expressed in terms of solutions of some exterior and interior boundary value problems. In the author’s papers [3–4] asymptotics of solutions of boundary integral equations near corner points of the contour in plane problems of elasticity of the first two boundary value problems for the Lame’s system was obtained.

Stepan Zargaryan

### Backmatter

Weitere Informationen


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Is there a list of ways R scripts (and/or Shiny data products) can be put into productionisation? – General

It would be really nice if they can be compiled into some .app or .exe files, but I’m guessing this is not possible at the moment, with some attempts in the past (I found an abandoned Electron project, for example).

I’m guessing R Studio Server may let a client/user load the script via a web browser, and web browsers are widely used. But this requires an R Studio Server to be installed and cannot be provided headless.

An HTML file can be generated with R Markdowns (and Bookdowns as well, right?) where client/user can just open the HTML file on their phone also, but is this really industry standard?

What about using R with Jenkins? I saw some tutorials, but I’m missing few puzzle pieces to really understand how they are setup (e.g. RStudio Connect) to execute the script. What are some examples Jenkins taking input from the client/user to create an output? How would Jenkins get R scripts from Github and not just sit there with the R script files, but actually execute them?

format: plain


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#### Vol. 1, No. 1, 2016

 Recent Issues Volume 4, Issue 4 Volume 3, Issue 4 Volume 3, Issue 3 Volume 3, Issue 2 Volume 3, Issue 1 Volume 2, Issue 4 Volume 2, Issue 3 Volume 2, Issue 2 Volume 2, Issue 1
 The Journal About the Journal Editorial Board Subscriptions Ethics Statement Submission Guidelines Submission Form Editorial Login Ethics Statement ISSN: 2379-1691 (e-only) ISSN: 2379-1683 (print) Author Index To Appear Other MSP Journals
A plethora of inertial products

### Dan Edidin, Tyler J. Jarvis and Takashi Kimura

Vol. 1 (2016), No. 1, 85–108
DOI: 10.2140/akt.2016.1.85
##### Abstract

For a smooth Deligne–Mumford stack $\mathsc{X}$, we describe a large number of inertial products on $K\left(I\mathsc{X}\right)$ and ${A}^{\ast }\left(I\mathsc{X}\right)$ and inertial Chern characters. We do this by developing a theory of inertial pairs. Each inertial pair determines an inertial product on $K\left(I\mathsc{X}\right)$ and an inertial product on ${A}^{\ast }\left(I\mathsc{X}\right)$ and Chern character ring homomorphisms between them. We show that there are many inertial pairs; indeed, every vector bundle $V$ on $\mathsc{X}$ defines two new inertial pairs. We recover, as special cases, the orbifold products considered by Chen and Ruan (2004), Abramovich, Graber and Vistoli (2002), Fantechi and Göttsche (2003), Jarvis, Kaufmann and Kimura (2007) and by the authors (2010), and the virtual product of González, Lupercio, Segovia, Uribe and Xicoténcatl (2007).

We also introduce an entirely new product we call the localized orbifold product, which is defined on $K\left(I\mathsc{X}\right)\otimes ℂ$.

The inertial products developed in this paper are used in a subsequent paper to describe a theory of inertial Chern classes and power operations in inertial $K$-theory. These constructions provide new manifestations of mirror symmetry, in the spirit of the hyper-Kähler resolution conjecture.

##### Keywords
quantum $K$-theory, orbifold, product, orbifold cohomology, Gromov–Witten, equivariant, stringy, inertia, Deligne–Mumford, stack
##### Mathematical Subject Classification 2010
Primary: 55N32, 55N15
Secondary: 53D45, 57R18, 14N35, 19L10, 19L47, 14H10


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# How do you draw the Lewis structure for the hypobromite ion, BrO^-?

Then teach the underlying concepts
Don't copy without citing sources
preview
?

#### Explanation

Explain in detail...

#### Explanation:

I want someone to double check my answer

2
Jan 19, 2018

Well, if two atoms are in space, how do they connect?

Given that $\text{Br}$ has $7$ valence electrons and ${\text{O}}^{-}$ has $7$ valence electrons (i.e. including the $-$ charge), we have $14$ valence electrons to distribute.

Halogens usually make one bond and have three lone pairs of electrons. With that, it is enough to come up with:

$: \ddot{\text{Br"-ddot"O}} {:}^{\left(-\right)}$
""color(white)(a)^(.." "" "..)

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Using Prime factorization, find the HCF and LCM of $24,36,40$
| 231 views

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Prime factors of $24,36,40$
\begin{aligned} &24=2 \times 2 \times 2 \times 3=2^{3} \times 3 \\ &36=2 \times 2 \times 3 \times=2^{2} \times 3^{2} \\ &40=2 \times 2 \times 2 \times 2 \times 5=2^{3} \times 5 \end{aligned}
\begin{aligned} &\operatorname{HCF}(24,36,40)=2^{2}=4 \\ &\operatorname{LCM}(24,36,40)=2^{2} \times 3^{2} \times 5=360 \end{aligned}
by Platinum (164,290 points)

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# coons

Creates a coons patch through a selected loop of 2, 3 or 4 curves.

## Inline mode usage

coons -h
Displays the help file that summarizes the parameters for this command.
coons –s all
Selects all curves of the scene.
coons –s <name>
Selects <name> curve.

## Interactive mode usage

Invocation
coons
Parameters
• Two, three or four curves used for the creation of the coons patch. The set of selected curves must form a closed loop.

Example:

We are going to create a coons patch from a set of four curves, so the first step is to create the curves. For this, we will use the curveInterpolation command:

>curveInterpolation
Degree [3] 3
Select curve points [Press Enter when done]
Point 0 [x y z] 0 0 0
Point 1 [x y z] 1 0 2
Point 2 [x y z] 2 0 1
Point 3 [x y z] 4 0 0
Point 4 [x y z]:
>curveInterpolation
Degree [3] 3
Select curve points [Press Enter when done]
Point 0 [x y z] 4 0 0
Point 1 [x y z] 4 1 0.5
Point 2 [x y z] 4 3 1
Point 3 [x y z] 4 4 0
Point 4 [x y z]:
>curveInterpolation
Degree [3] 3
Select curve points [Press Enter when done]
Point 0 [x y z] 4 4 0
Point 1 [x y z] 3 4 0.5
Point 2 [x y z] 1 4 0
Point 3 [x y z] 0 4 2
Point 4 [x y z]:
>curveInterpolation
Degree [3] 3
Select curve points [Press Enter when done]
Point 0 [x y z] 0 4 2
Point 1 [x y z] 0 2 1
Point 2 [x y z] 0 1 0.25
Point 3 [x y z] 0 0 0
Point 4 [x y z]:

Remember that you need to press Enter when all the points of a curve have been introduced. The result of these curves are shown in the following figure:

Now we are going to create the coons surface. To do this, use the coons command:

>coons
Select a closed set of 2, 3 or 4 curves (press Enter when done)
>

The result is shown in the following figure:


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## FANDOM

10,264 Pages

The Fibonatetris, coined by SuperJedi224, is equal to the trillionth term of the fibonacci sequence (tetris being a modified form of the SI Tera-, for one trillion), starting with the first one.

It has exactly 208,987,640,250 digits.

Its prime factorization starts with 3 × 512 × 7 × 11 × 41 × 47 × 101 × 127 × 151 × 251 × 401 × 641 × 1087 × 1601 × 2161 × 2207 × 3001 × 3041 × 4001 × 4481 × 9601 × …

## Approximations in other notationsEdit

Notation Lower bound Upper bound
Scientific notation $$4\times10^{208\,987\,640\,249}$$ $$5\times10^{208\,987\,640\,249}$$
Arrow notation $$26\uparrow147\,697\,227\,336$$ $$416\uparrow79\,793\,888\,139$$
Down-arrow notation $$526\downarrow\downarrow5$$ $$157\downarrow\downarrow6$$
Steinhaus-Moser Notation 10[3][3] 11[3][3]
Copy notation 1[1[12]] 2[2[12]]
Chained arrow notation $$26→147\,697\,227\,336$$ $$416→79\,793\,888\,139$$
H* function H(69H(2)) H(70H(2))
Taro's multivariable Ackermann function A(3,A(3,36)) A(3,A(3,37))
Pound-Star Notation #*((1))*(36)*3 #*((1))*(37)*3
BEAF {26,147697227336} {416,79793888139}
Hyper-E notation 4E208,987,640,249 5E208,987,640,249
Bashicu matrix system (0)(1)[35] (0)(1)[36]
Hyperfactorial array notation (13!)! (14!)!
Fast-growing hierarchy $$f_2(f_2(34))$$ $$f_2(f_2(35))$$
Hardy hierarchy $$H_{\omega^22}(34)$$ $$H_{\omega^22}(35)$$
Slow-growing hierarchy $$g_{\omega^{\omega^{\omega+1}2}}(10)$$ $$g_{\omega^{\omega^{\omega+1}2+\omega^\omega}}(10)$$

Source: [1]


text:
What is the homotopy from the empty set to a non-empty set?

It is said that the empty set is the initial object of the category of sets, and the empty space is the initial object in the category of topological spaces. So there exists a unique arrow from the initial object to any object in those categories. But I wonder what would be that map?

It seems that at $t = 0$ the map would give an empty set or space, while at $t = 1$ the map would give a non-empty set or space. How is that possible with a continuous map?

• What is $t=0, t=1$ ? What does a simple continuous map have to do with homotopies ? – Maxime Ramzi Sep 5 '18 at 9:12
• A homotopy $X\to Y$ can be looked at as a continuous function $F:X\times\mathbb I\to Y$. In special case $X=\varnothing$ then $X\times\mathbb I=\varnothing$ and the empty function is the only homotopy $\varnothing\to Y$. – drhab Sep 5 '18 at 9:13
• The homotopy does not go from X to Y, rather from one such map to another such map. – Leon Hendrian Sep 5 '18 at 12:47
• Yes, @LeonHendrian , as long as X is empty and Y is non-empty, is what I need. – Yan King Yin Sep 6 '18 at 13:14
• @YanKingYin A function $f:A\to B$ is a set of ordered pairs with the special property that for every $a\in A$ there is a unique $b\in B$ with $(a,b)\in f$. The set $\varnothing$ is subset of $\varnothing\times B$. Further vacuously all elements of $\varnothing$ are ordered pairs (no element can be found that is not) and also the mentioned condition that makes a relation a function is satisfied vacuously. Conclusion: $\varnothing$ is a function $\varnothing\to B$. No matter whether $B$ is empty or not. An empty function has an empty range/image, but that is not the same thing as co-domain. – Vera Sep 6 '18 at 15:29

The statement "the empty space is the initial object in the category of topological spaces" has nothing to do with homotopies. It simply means there is a unique continuous map $\emptyset\mapsto Y$ for any space $Y$, namely the empty map. It is continuous because the preimage of every open set is open.

• But a continuous function is a homotopy, no? – Yan King Yin Sep 6 '18 at 6:35
• I'm not sure what you mean. How is an arbitrary continuous function a homotopy? – Michal Adamaszek Sep 6 '18 at 6:38
• as per @drhab 's comment above, there exists a homotopy $\varnothing \rightarrow Y$ which is the empty function. – Yan King Yin Sep 6 '18 at 13:10
• @YanKingYin You can also define a homotopy $X\to Y$ to be a family of continuous functions $f_t:X\to Y$ where $t$ ranges over $[0,1]$. This in such a way that the function $F:X\times[0,1]\to Y$ prescribed by $(x,t)\mapsto f_t(x)$ is continuous. Then if $X=\varnothing$ for every $t\in[0,1]$ the function $f_t$ is the empty function with domain $\varnothing$ and codomain $Y$. It is a co-called constant homotopy (i.e. $f_t:\varnothing\to Y$ does not depend on $t$). See the comment of Vera on your question where it concerns the fact that $Y$ as codomain is not necessarily empty. – drhab Sep 6 '18 at 18:28


text: The standard deviation of the sample and population is represented as σ ͞x and σ. Here, The mean of the sample and population are represented by µ͞x and µ. 3. Standard Deviation Formula for Discrete Frequency Distribution. N = size of the sample data set. The mean is (1 + 2 + 4 + 5 + 8) / 5 = 20/5 =4. Usually, we are interested in the standard deviation of a population. For a sample size of more than 30, the sampling distribution formula is given below –. 2 - 4 = -2. This is called the variance. Practice: Sample standard deviation. So the full original data Set is an array of numbers 5,7,8,3,10,21,4,13,1,0,0,9,17. In the example shown, the formulas in F6 and F7 are: = STDEV.P( C5:C14) // F6 = STDEV.S( C5:C14) // F7. EX: μ = (1+3+4+7+8) / 5 = 4.6. σ = √ [ (1 - 4.6)2 + (3 - 4.6)2 + ... + (8 - 4.6)2)]/5. More on standard deviation (optional) s = sample standard deviation. 1. Population SD formula is S = √∑ (X - M) 2 / n. The standard deviation is a measure of the spread of scores within a set of data. Mean and standard deviation versus median and IQR. To calculate the standard deviation of a data set, you can use the STEDV.S or STEDV.P function, depending on whether the data set is a sample, or represents the entire population. µ͞x =µ and σ͞x =σ / √n. σ = √ (12.96 + 2.56 + 0.36 + 5.76 + 11.56)/5 = 2.577. Hence the summation notation simply means to perform the operation of (xi - μ2) on each value through N, which in this case is 5 since there are 5 values in this data set. Sample Standard Deviation - s = $\sqrt{s^{2}}$ Here in the above variance and std deviation formula, σ 2 is the population variance, s 2 is the sample variance, m is the midpoint of a class. y : … Divide the sum by n-1. Standard deviation (σ) is the measure of spread of numbers from the mean value in a given set of data. 2. Sample standard deviation and bias. So, for an assignment for a Python class at college I have to demonstrate that the Sample Standard Deviation formula is more accurate than the population standard population formula on a sample data Set. 4. The sample size of more than 30 represents as n. Sample SD formula is S = √∑ (X - M) 2 / n - 1. Practice: Visually assessing standard deviation. The deviations are found by subtracting the mean from each value: 1 - 4 = -3. Add those values up. Following this out calculations will diverge from one another and we will distinguish between the population and sample standard deviations. Next lesson. x̄ = mean value of the sample data set. In case you are not given the entire population and only have a sample (Let’s say X is the sample data set of the population), then the formula for sample standard deviation is given by: Sample Standard Deviation = √ [Σ (X i – X m ) 2 / (n – 1)] x 1, ..., x N = the sample data set. This is the currently selected item. 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text: Find the oxidation number of carbon in the following compounds : CH3OH, CH2O, HCOOH, C2H2 . The oxidation number of fluorine is always –1. Any free element has an oxidation number equal to zero. c.The region inside the nucleus where the electrons are located. b.theory Therefore, x + 5 (0) + 1 = 2 x = + 1 Thus, oxidation number of F e in [F e (H 2 O) 5 N O] is + 1 due to charge transfer from N O to F e 2 + (N O becomes N O +). Click hereto get an answer to your question  BALANCING EQUATIONS BY OXIDATION NUMBER METHOD 3. An oxidation number can be assigned to a given element or compound by following the following rules. November 10, 2020 By Leave a Comment. Favorite Answer. They are +1 and -2. Which of the following words/phrases can be used to describe table salt (NaCl)? Record to the correct number of significant figures. {eq}HNO_3 + H_2S \to S + NO + H_2O {/eq} For the redox reaction: a. state the oxidation number for each ion in the reaction. The usual oxidation number of hydrogen is +1. The water molecule is neutral; therefore, the oxygen must have an oxidation number of to balance the charge. Oxidation number of oxygen is -2. MnO2 + H2C2O4 + H2SO4 --> MnSO4 + CO2 + H2O? now balance by ooc method. Learn about oxidation-reduction (redox) reactions. Oxidation corresponds to increasing the oxidation number of some atom. I'm a little confused here. a) 7.95 g Water, or H2O is a free-standing neutral compound, so it's oxidation number is 0. There are a few exceptions to this rule: When oxygen is in its elemental state (O 2), its oxidation number is 0, as is the case for all elemental atoms. Chemistry. The oxidation number of an atom is a number that represents the total number of electrons lost or gained by it. How long will the footprints on the moon last? An oxidation number of -1 is possible only for metal hydride. Hydrogen's oxidation number in water is +1, and oxygen's is -2. State which element is the oxidizing agent and which element is the reducing agent, as well as how you know. A. a solid Oxygen almost always has an oxidation number of -2, except in peroxides (H 2 O 2) where it is -1 and in compounds with fluorine (OF 2) where it is +2. What is the oxidation number of hydrogen? Hydrogen –usually +1, except when bonded to Group I or Group II, when it forms hydrides, -1. Let x be the oxidation number of iron in [F e (H 2 O) 5 N O] 2 +. Here is the full equation: SO4-2 + 2 CH2O + 2H > H2S + 2CO2 + 2H2O Oxidation Number: The number that is assigned to an element to indicate the loss or gain of electrons by an atom of that element is called as the oxidation number. - the O is O atom is (-2). Select one: Therefore, x + 5 (0) + 1 = 2 x = + 1 Thus, oxidation number of F e in [F e (H 2 O) 5 N O] is + 1 due to charge transfer from N O to F e 2 + (N O becomes N O +). first find oxidation number . How to find the Oxidation Numbers for H2O (Water) - YouTube How much money do you start with in monopoly revolution? For example, the oxidation number of Na + is +1; the oxidation number of N 3- is -3. What did women and children do at San Jose? The oxidation state, sometimes referred to as oxidation number, describes the degree of oxidation (loss of electrons) of an atom in a chemical compound.Conceptually, the oxidation state, which may be positive, negative or zero, is the hypothetical charge that an atom would have if all bonds to atoms of different elements were 100% ionic, with no covalent component. Sulfide ion is oxidized to sulfate ion. Oxidation number of sulfur is increased from -2 to +6. The oxidation number of "Cr" is +3. An oxidation number can be assigned to a given element or compound by following the following rules. value. Select one or more: boyerariel79 boyerariel79 Answer: 2 ^0. Its atoms have oxidation number though. The sum of the oxidation numbers in a monatomic ion is equal to the overall charge of that ion. [Assume both get equal amounts of sunlight] *, Questions1. The oxidation number of the sulfur atom in the SO 4 2-ion must be +6, for example, because the sum of the oxidation numbers of the atoms in this ion must equal -2. The formula for water is . No, the oxidation number for O is always -2, so the oxidation number for H in H2O has to be +1. In O2, the oxidation number is 0 on either oxygen atom. Water is a molecule!!!! Worksheet 25 - Oxidation/Reduction Reactions Oxidation number rules: Elements have an oxidation number of 0 Group I and II – In addition to the elemental oxidation state of 0, Group I has an oxidation state of +1 and Group II has an oxidation state of +2. State which element is being oxidized and which element is being reduced, as well as how you know. The oxidation number is 0 only if you … Add the half-reactions together. Assign an oxidation number of -2 to oxygen (with exceptions). For example, in CO₂, the more electronegative O atoms get all of the shared electrons. What is the oxidation number of CH2O? How many grams of water were produced? Let x be the oxidation number of iron in [F e (H 2 O) 5 N O] 2 +. H 2 O 2 reduction to H 2 O and SO 42- Oxygen atom is at -1 oxidation state in H 2 O 2. Use the chart below to determine which element has the larger Zeff …, At room temperature, the greatest attractive forces exist between particles of dastranjania dastranjania 6 minutes ago Chemistry High School H2O oxidation number See answer dastranjania is waiting for your help. …, snow begins to melt. B. a gas The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. They are positive and negative numbers used for balancing the redox reaction. Hydrogen has an oxidation number of +1 when combined with non-metals, but it has an oxidation number of -1 when combined with metals. …. The skeleton equation is: H 2 S + KMnO 4 + H 2 SO 4 → S+ MnSO 4 + K H SO 4 + H 2 O . 7.94 g + 0.0132 g = ? All Rights Reserved. There is now 1 inch of snow on your car and 1 inch of snow on your wooden deck. Select one: Applying the oxidation number rules to the following equation, we have. Since the overall charge on the complex is 2 +, the sum of oxidation states of all elements in it should be equal to 2 +. 11th - 12th grade. Oxidation corresponds to increasing the oxidation number of some atom. Add your answer and earn points. Which melts first? I know O is -2 and H is +1 unless there is a metal which there isnt so anything you put for C would unbalance the compound no? Since there are two of them, the hydrogen atoms contribute to a charge of +2. The oxidation number of an atom is a number that represents the total number of electrons lost or gained by it. a.Compound Applying the oxidation number rules to the following equation, we have. Heterogeneous Water Oxidation ... Where n is the number of electrons per mole products and F is the Faraday constant. Their partially negative end is pulled by aluminum's + … …, What is the result of the following calculation? For the following REDOX reaction, state the oxidation number for each element in the reaction. The oxidation number of a free element is always 0. What is the oxidation number of #"NaH"_2"PO"_4#? There are only three oxidation numbers possible for hydrogen: +1, -1, and 0. Since the oxidation number of copper increased from 0 to +2, we say that copper was oxidized and lost two negatively charged electrons. …. (i) Fe2+ + H+ + Cr2O72- → Cr3+ + Fe3+ + H2O For compounds containing hydrogen, the oxidation number of hydrogen is always +1. 15 grams of hydrogen and 7.5 grams of oxygen were placed in a beaker. The beaker was never h2o oxidation number. Who is the longest reigning WWE Champion of all time? In almost all cases, oxygen atoms have oxidation numbers of -2. now balance br. d.The region outside the nucleus where the proton is likely to be moving. The oxidation state of O atom is not always - 2 but it can be - 2 to +2. br2 + naoh = nabro3 + 5nabr . Oxidation number (also called oxidation state) is a measure of the degree of oxidation of an atom in a substance (see: Rules for assigning oxidation numbers). 1. What was the weather in Pretoria on 14 February 2013? This site is using cookies under cookie policy. when balancing any reactions water should be removed. Oxidation Number: The number that is assigned to an element to indicate the loss or gain of electrons by an atom of that element is called as the oxidation number. opened. When did sir Edmund barton get the title sir and how? Find out how to identify redox reactions using oxidation numbers. Water, or H2O is a free-standing neutral compound, so it's oxidation number is 0. Hydrogen's oxidation number in water is +1, and oxygen's is -2. br2+6naoh =nabro3 + 5nabr. The oxidation number of diatomic and uncombined elements is zero. The sum of oxidation numbers in a neutral compound is 0. Tube worms that survive near geothermal vents of hydrogen sulfide rely on bacteria living inside them to obtain energy by the oxidation of H2S to SO4^2-. (b) … i was wondering why the oxidation number of h2o is -2. i always think its 0 because Hydrogen is +1 multiply by 2 and oxygen is -2 so +2-2 would obviously equal 0. but the answer is always -2. why is it that h2o's oxidation number is -2. e.g the question to check whether or not a redox reaction has occured: CaO + H2O -> Ca(OH)2 CaO = +2 H2O =-2 Ca(OH)2 = +2-2 so a redox reaction has not … b.Element 11. C. a liquid. Hydrogen's oxidation number in water is +1, and oxygen's is -2. The algebraic sum of the oxidation numbers of elements in a compound is zero. d.Pure substance, What is an explanation of a natural phenomenon supported by large amounts of data called? Explanation: Mg + 4HNO3 → Mg (NO3)2 + 2NO2 + H2O Step – 4 To balance the number of … Its atoms have oxidation number though. The sun comes out and all the > We can use the rules for oxidation numbers. H2O has actually oxidation number 0. c.Mixture asked Oct 7, 2018 in Chemistry by Afreen ( 30.7k points) oxidation-reduction balance na first. Select one: In H2O, H is +1 and O is -2, no matter how many H2O molecules you have. The oxidation number of a monatomic ion equals the charge of the ion. When did organ music become associated with baseball? The oxidation numbers of hydrogen and oxygen in H2O. Similarly oxidation state of H atom is from - 1 to +1(+1 in a compound and - 1 in metal hydride but it can also be 0 as the range is - 1,0,+1) . Calculating Oxidation Numbers. like I was stating, THE H IS H atoms are in the (+1) each. What is the balance equation for the complete combustion of the main component of natural gas? H2O oxidation number Get the answers you need, now! The sum of the oxidation numbers of all atoms in an ion equals charge on the ion. Assign an oxidation number of -2 to oxygen (with exceptions). To balance the number of oxygen atoms and hydrogen atoms 2HNO3 is multiplied by 2. You can specify conditions of storing and accessing cookies in your browser, So do u even know what oxidation number is, Water doesn't really have an oxidation number. Its oxidation number of oxygen that is -2, also remember that to see whether there is a redox reaction, u have to calculate the oxidation number of only the atom with variable oxidation number not of whole compound (the oxidation number of whole compound is always constant, usually 0). It is traditional to place the oxidation number of each element above the symbol for the element, as shown here. Since the overall charge on the complex is 2 +, the sum of oxidation states of all elements in it should be equal to 2 +. They are positive and negative numbers used for balancing the redox reaction. (11.... An oxidation number of 0 is possible for hydrogen gas only because it is purely made of hydrogen atoms. For the hydrogen atoms the oxidation state is 1+, for the oxygen why is Net cash provided from investing activities is preferred to net cash used? The oxidation number of a monatomic ion equals the charge of the ion. The oxidation number of hydrogen is -1 in compounds containing elements that are less  electronegative than hydrogen, as in CaH 2. It is neutral so what can you do with something that is Neutral? ; When oxygen is part of a peroxide, its oxidation number is -1. d.hypothe ; When oxygen is part of a peroxide, its oxidation number is -1. The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. Step 2 (a) The oxidation number of various atoms involved in the reaction. b.The region outside the nucleus where The oxidation number of any atom in its elemental form is 0. An explosion happened inside the baker that produced water. Any free element has an oxidation number equal to zero. SO 4 2-: (+6) + 4(-2) = -2. …, It snows one inch of snow. The oxidation number of hydrogen is +1. The sum of the oxidation numbers of all atoms in a neutral compound is 0. The oxidation number of diatomic and uncombined elements is zero. oxidation number of br in nabro3 = 5. oxidation number of br in nabr = 1. now inter change oxidation numbers with other atom. Copyright © 2021 Multiply Media, LLC. a) Li or K b) C or O c) C or P d) C or How would you describe life and why?, What is an energy level? The atoms in He and N 2, for example, have oxidation numbers of 0. Performance & security by Cloudflare, Please complete the security check to access. Why don't libraries smell like bookstores? There are a few exceptions to this rule: When oxygen is in its elemental state (O 2), its oxidation number is 0, as is the case for all elemental atoms. a.scientific law Calculating Oxidation Numbers. c.conclusion Circle an answer for a-e: it is 2-. Balance the following chemical equation using the oxidation number method: K2Cr2O7 + SnCl2 + HCl → CrCl3 + SnCl4 + H2O + KCl . Snow is ice. Since the oxidation number of copper increased from 0 to +2, we say that copper was oxidized and lost two negatively charged electrons. snswer is 6 In almost all cases, oxygen atoms have oxidation numbers of -2. Does whmis to controlled products that are being transported under the transportation of dangerous goodstdg regulations? And this will be the case in all O2 molecules, no matter how many you have. There are a few exceptions to this rule: When oxygen is in its elemental state (O 2), its oxidation number is 0, as is the case for all elemental atoms. The essential rules for this problem are: The oxidation number of a halogen in a compound is -1 unless it is with a more electronegative element. a.The region outside the nucleus where the neutron is likely to be moving. The usual oxidation number of hydrogen is +1. 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# Need Help

Can someone Explain Newton's Sums to me?

The explanation,Some examples and a few exercise Problems would be really appreciated.

Note by Mehul Arora
3 years, 1 month ago

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*italics* or _italics_ italics
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MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

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Do you still need help ?

- 3 years, 1 month ago

Yeah... Newton sums are still going over my head, although I understood like 20% of it XD

- 3 years, 1 month ago

Ok. Where do you have a doubt ?

- 3 years, 1 month ago

Like I'm trying but kinda not understanding the terminology basically.. Like ini the Brilliant wiki, What is P1? etc... Maybe if I can understand the terminology Ill be able to comprehend it.

- 3 years, 1 month ago

$$P_n$$ means (For a quadratic polynomial) that $$P_n = a^n + b^n$$ where $$(a,b)$$ are roots of Polynomial $$f(x)$$
Any other Problem ?

- 3 years, 1 month ago

Start solving questions :)

- 3 years, 1 month ago

Yeah.. I am trying. Just learnt Vieta's. Trying to perfect it. :)

- 3 years, 1 month ago

Best of luck :)

- 3 years, 1 month ago

Thanks!!

- 3 years, 1 month ago

Sorry, not at home. Answering from mobile (difficult to type). Will be home in 2-3 days. Will surely help you then.

- 3 years, 1 month ago

When's your vacation going to be over ?

- 3 years, 1 month ago

I 'll be at home on 25th. And my School.starts on 1st April.

- 3 years, 1 month ago

Ok. Enjoy :)

- 3 years, 1 month ago

Check out the Brilliant wiki - Newton's Identities, which is in the Vieta formula chapter in algebra.

Staff - 3 years, 1 month ago

Sir, I did go through to the wiki but it went over my head. Can someone please explain it in an easier fashion?

- 3 years, 1 month ago

Have you tried the Brilliant wiki on it ?

- 3 years, 1 month ago

Yeah.. I didn't learn much. The wiki must be good though it went over my head.

- 3 years, 1 month ago

Then try the AoPS wiki on Newton's Sums . I had learnt it from there since when I had learnt it , the wiki on Brilliant wasn't filled out . Or you can always download some pdfs on the topic . Now that always works (for me atleast!)

- 3 years, 1 month ago

Can you give me some links (apart from AoPS)

- 3 years, 1 month ago

Hmm ,strange . I couldn't find any good pdf for this topic . But try it out yourself and if you have any doubt in it , ask here and I'll only be glad to help you out :)

- 3 years, 1 month ago

Do you have Hall and Knight ? If so , please check pg 468 .

- 3 years, 1 month ago

Nope.. Which class is it aimed at?

- 3 years, 1 month ago

It's just a ref. book for Algebra .

- 3 years, 1 month ago

Well, it'd be better if you could point out the parts you don't understand in the Brilliant wiki on Newton's sums.

Also, I'd recommend the Wikipedia article on Newton's Identities. I learnt it from there. :D

There are two points to note when learning Newton's Identities. It is a method that helps you to form a recurrence for power sums of roots, so it'd be better if you have a good understanding of recurrence relations. Also, applications of Newton's Identities mostly use Vieta's formulas because you need to find the elementary symmetric polynomials first before calculating the power sums. If you get confused with "too many formulas" there, just remember the concise form stated in the Brilliant wiki under the section "General Form".

- 3 years, 1 month ago

I have learnt all the Vietas formulae but Ill start working on the Recurrence relations right away...

Wah wah!! XD

- 3 years, 1 month ago


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news

# Biopharmaceutical bioseparation systems market to value $20bn 2 SHARES Share via With demand for biopharmaceuticals rising and advances being made in bioseparation systems, the global market it expected to witness significant growth. According to recent market research, the global market for biopharmaceutical bioseparation systems, estimated to value$13 billion in 2022, is projected to grow to \$19.6 billion by 2026, achieving a compound annual growth rate (CAGR) of 13.2 percent.

The analysts suggest the rising demand for biopharmaceuticals, equipment advancements and the pharma industry’s increasing focus on production efficiency are propelling growth in the global bioseparation systems market, as is the growing use of single-use/disposable instruments. Single-use systems, such as disposable continuous chromatography, are designed to be more precise and enable more accurate analysis of data.

Downstream bioprocessing represents an expensive, complex and time-consuming step in the production of biopharmaceutical drugs. There is a growing emphasis on developing technologies to enhance efficiency and reduce costs, driving innovation both in existing and novel solutions. Advances in upstream cell culture techniques are also creating enormous bottlenecks in downstream separation and purification processes.

While various techniques or systems are available for bioseparation, the market for biopharmaceutical bioseparation systems is primarily chromatography systems, centrifuges and membranes/filters, which are the most commonly used products. Of these, chromatography represents the major technique used in downstream processing of biopharmaceutical, remaining the gold standard technique for downstream purification processes, with limited alternatives that can match the yield and purity levels of the technique.

Chromatography can be used throughout the drug development process, from the identification od potential drug candidates (with HPLC enabling high throughput screening) to ensuring drug formulation purity, analysing the bioavailability and release time of novel drug formulations to determine dosages to devising manufacturing techniques.

Advancements are underway in terms of product developments and innovations, with significant emphasis being placed on optimising and streamlining existing methods.

The chromatography segment of the market is projected to grow at a 12.4 percent CAGR between 2022 and 2026.

In downstream bioprocessing, the development of single-use/disposable technologies represents a major advancement, conferring significant benefits to various stages of bioprocessing, including eliminating cleaning steps; minimising cross-infection risks; improving flexibility and convenience; lowering time to bring facility to operations; and expediting turnaround times.

Technologies/techniques themselves are also undergoing tremendous advances. For instance, alternatives to Protein A based purification in affinity chromatography are emerging, Protein A based purification is the preferred technique but significantly expensive. Advancements have also been made in HPLC, gas chromatography, supercritical fluid-chromatography, electrophoresis and centrifugation, for separating sensitive and large molecules.

The rise of advanced therapeutics, such as cell-based therapies and personalised medicines, is also promoting growth in the market.

According to the report, while high demand is expected, growth in the market is restrained by the stringent regulations governing bioseparation systems approval, high initial capital costs of bioseparation systems, and the presence of alternative technologies, such as magnetic separation for chromatography.

North America is overall the largest market for biopharmaceutical bioseparation systems. In this market, growth is anticipated in automated bioseparation systems capable of handling multiple samples at a time, with greater levels of accuracy.

Europe is also a lucrative market, with the region being at the forefront of drug research and the base of several reputed biopharmaceuticals companies. Growth in the biopharmaceutical bioseparation systems market in this area is promoted by the higher adoption of advanced treatment options, funding from governments and the extensive presence of high-quality research infrastructure.

The Asia-Pacific market is expected to grow the most in years to come, as people’s income rises, healthcare expenditure grows and the population expands. Several biopharmaceutical companies based in the West planning to set up their production in the Asia-Pacific countries owing to low labour cost, which will also contribute to market growth in the region.

Key players in the global market for biopharmaceutical bioseparation systems include 3M Company, 3M Purification, Agilent Technologies, Alfa Wassermann Separation Technologies LLC; Flottweg Separation Technology; GEA Westfalia Separator Group GmbH; GE Healthcare Life Sciences; KNAUER Wissenschaftliche GerÃ¤te GmbH; Konik Nanobiotech SL; Merck KGaA; Pall Corporation; Liminal BioSciences Inc; Sartorius Stedim Biotech; Repligen Corporation; Thermo Fisher Scientific; Tosoh Bioscience Inc.; YMC Europe GmbH.

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Answer
Comment
Share
Q)

# The correction factor 'a' to the ideal gas equation corresponds to

( A ) Volume of the gas molecules
( B ) Density of the gas molecules
( C ) Electric field present between the gas
( D ) Forces of attraction between the gas molecules

## 1 Answer

Comment
A)
Volume of the gas molecules


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# Angular velocity formula for a particle?

I know that when the motion of a particle is circular about the origin then: $$\vec v=\vec \omega \times \vec r$$ But that this does not hold for any motion with a radial as well as tangential component. So how can we define angular velocity [mathematically] for general motion?

When you fix a reference point (take it to be the origin of your reference frame) you can write the position as

$$\vec{r} = r \hat{r}$$

where $\hat{r}$ is the unit vector pointing toward the particle. Deriving you obtain

$$\vec{v} = \frac{dr}{dt} \hat{r}+ r \frac{d \hat{r}}{dt}$$

The first term is the radial component of the velocity, the second one is proportional to the derivative of a vector with fixed modulus, so it can be represented as

$$\frac{d \hat{r}}{dt} = \vec{\omega} \times \hat{r}$$

and inserting this in the previous equation you get

$$\vec{v} = \frac{dr}{dt} \hat{r}+ \vec{\omega} \times \vec{r}$$

You can extract the second term from the velocity by projecting in the plane orthogonal to the radial vector,

$$\vec{v}-(\vec{v}\cdot \hat{r})\hat{r} = \vec{\omega} \times \vec{r}$$ which generalize your equation. The angular velocity is perpendicular to the plane defined by $\vec{r}$ and $\vec{v}$, and depends on the reference point. A preferred reference point is the center of the osculator circle of the trajectory, at a given time. In this case the radial component of the velocity is zero, but unless the trajectory is circular this preferred point is not fixed.


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## Trigonometry (10th Edition)

$\cot$ 30$^{\circ}$ = $\frac{\sqrt3}{3}$ $\cot$ 30$^{\circ}$ = $\sqrt3$
$\cot$ 30$^{\circ}$ = $\frac{1}{\sqrt3}$ = $\frac{\sqrt3}{3}$ $\cot$ 30$^{\circ}$ = $\frac{\sqrt3}{1}$ = $\sqrt3$


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Next: Rational numbers Up: The classical sets of Previous: The Euclidean algorithm   Contents

# Applications of the Euclidean algorithm

Take , and . Suppose we want to find all the solutions , of . A necessary condition for a solution to exist is that , so assume this.

Lemma 6.3.1   If then has solutions in .

Proof. Take and such that . Then if then if and , .

Lemma 6.3.2   Any other solution is of the form , for .

Proof. These certainly work as solutions. Now suppose and is also a solution. Then . Since and are coprime we have and . Say that , . Then .

root 2002-06-10


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# Thread: Equations with fractional exponents.

1. ## Equations with fractional exponents.

I have to solve a few equations that are similar to radial equations. I know how fractional exponents relate to square roots but I have something majorly wrong in my operations. I can't solve any of these. Here's one I could use help with. If I see enough steps I can easily apply it to the rest.

$5x^5/3 +160=0$

It's supposed to read 5x to the 5/3rd power.

I have a test tomorrow and I'm trying to get a grip on some of this stuff that I'm not understanding. Oddly enough I can handle quadratics with fractional exponents already, I just can't do this simpler version.

2. Originally Posted by Turtle13
I have to solve a few equations that are similar to radial equations. I know how fractional exponents relate to square roots but I have something majorly wrong in my operations. I can't solve any of these. Here's one I could use help with. If I see enough steps I can easily apply it to the rest.

$5x^5/3 +160=0$

It's supposed to read 5x to the 5/3rd power.

I have a test tomorrow and I'm trying to get a grip on some of this stuff that I'm not understanding. Oddly enough I can handle quadratics with fractional exponents already, I just can't do this simpler version.
you mean $5x^{5/3} + 160 = 0$ ?

Note that this means $x^{5/3} = -32$

raise both sides to the 3/5 power (do you see why?) you get

$x = (-32)^{3/5}$

which i think you can finish off

3. Yes, that helps immensely. I was able to solve them all with no problems.

BTW, how do I properly type that fractional exponent with the math brackets?

4. Originally Posted by Turtle13
Yes, that helps immensely. I was able to solve them all with no problems.

BTW, how do I properly type that fractional exponent with the math brackets?
if an argument or power has more than one elements in it, you must include it is {} brackets.

$$x^2/3$$ gives $x^2/3$

while

$$x^{2/3}$$ gives $x^{2/3}$

5. Originally Posted by Jhevon
if an argument or power has more than one elements in it, you must include it is {} brackets.

$$x^2/3$$ gives $x^2/3$

while

$$x^{2/3}$$ gives $x^{2/3}$
Or....

$$x^{\tfrac{2}{3}}$$ gives you $x^{\tfrac{2}{3}}$, but I kinda like yours better. It's easier to read.

6. Originally Posted by masters
Or....

$$x^{\frac{2}{3}}$$ gives you $x^{\frac{2}{3}}$, but I kinda like yours better. It's easier to read.
indeed, i usually use \frac in a power only if the base is huge, like $\left( \frac {x^3 + 4x + 1}{2x + 3} \right)^{\frac 23}$

in that case, the power might look too small, so you can use \tfrac: $\left( \frac {x^3 + 4x + 1}{2x + 3} \right)^{\tfrac 23}$

7. Originally Posted by Jhevon

in that case, the power might look too small, so you can use \tfrac:
Yeah, \tfrac looks better.

\frac: $x^{\frac{2}{3}}$

\tfrac: $x^{\tfrac{2}{3}}$