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Published by Springer-Verlag [2216 journals]
• Isentropic Gas Flow for the Compressible Euler Equation in a Nozzle
• Abstract: Abstract We study the motion of isentropic gas in a nozzle. Nozzles are used to increase the thrust of engines or to accelerate a flow from subsonic to supersonic. Nozzles are essential parts for jet engines, rocket engines and supersonicwind tunnels. In the present paper, we consider unsteady flow, which is governed by the compressible Euler equation, and prove the existence of global solutions for the Cauchy problem. For this problem, the existence theorem has already been obtained for initial data away from the sonic state, (Liu in Commun Math Phys 68:141–172, 1979). Here, we are interested in the transonic flow, which is essential for engineering and physics. Although the transonic flow has recently been studied (Tsuge in J Math Kyoto Univ 46:457–524, 2006; Lu in Nonlinear Anal Real World Appl 12:2802–2810, 2011), these papers assume monotonicity of the cross section area. Here, we consider the transonic flow in a nozzle with a general cross section area. When we prove global existence, the most difficult point is obtaining a bounded estimate for approximate solutions. To overcome this, we employ a new invariant region that depends on the space variable. Moreover, we introduce a modified Godunov scheme. The corresponding approximate solutions consist of piecewise steady-state solutions of an auxiliary equation, which yield a desired bounded estimate. In order to prove their convergence, we use the compensated compactness framework.
PubDate: 2013-08-01
• Asymptotic Behaviour of a Pile-Up of Infinite Walls of Edge Dislocations
• Abstract: Abstract We consider a system of parallel straight edge dislocations and we analyse its asymptotic behaviour in the limit of many dislocations. The dislocations are represented by points in a plane, and they are arranged in vertical walls; each wall is free to move in the horizontal direction. The system is described by a discrete energy depending on the one-dimensional horizontal positions x i > 0 of the n walls; the energy contains contributions from repulsive pairwise interactions between all walls, a global shear stress forcing the walls to the left, and a pinned wall at x = 0 that prevents the walls from leaving through the left boundary. We study the behaviour of the energy as the number of walls, n, tends to infinity, and characterise this behaviour in terms of Γ-convergence. There are five different cases, depending on the asymptotic behaviour of the single dimensionless parameter β n , corresponding to ${\beta_n \ll 1/n, 1/n \ll \beta_n \ll 1}$ , and ${\beta_n \gg 1}$ , and the two critical regimes β n ~ 1/n and β n ~ 1. As a consequence we obtain characterisations of the limiting behaviour of stationary states in each of these five regimes. The results shed new light on the open problem of upscaling large numbers of dislocations. We show how various existing upscaled models arise as special cases of the theorems of this paper. The wide variety of behaviour suggests that upscaled models should incorporate more information than just dislocation densities. This additional information is encoded in the limit of the dimensionless parameter β n .
PubDate: 2013-08-01
• On the Stability in Weak Topology of the Set of Global Solutions to the Navier–Stokes Equations
• Abstract: Abstract Let X be a suitable function space and let ${\mathcal{G} \subset X}$ be the set of divergence free vector fields generating a global, smooth solution to the incompressible, homogeneous three-dimensional Navier–Stokes equations. We prove that a sequence of divergence free vector fields converging in the sense of distributions to an element of ${\mathcal{G}}$ belongs to ${\mathcal{G}}$ if n is large enough, provided the convergence holds “anisotropically” in frequency space. Typically, this excludes self-similar type convergence. Anisotropy appears as an important qualitative feature in the analysis of the Navier–Stokes equations; it is also shown that initial data which do not belong to ${\mathcal{G}}$ (hence which produce a solution blowing up in finite time) cannot have a strong anisotropy in their frequency support.
PubDate: 2013-08-01
• Mappings of Least Dirichlet Energy and their Hopf Differentials
• Abstract: Abstract The paper is concerned with mappings ${h \colon \mathbb{X}}$ ${{\begin{array}{ll} {\rm onto} \\ \longrightarrow \end{array}}}$ ${\mathbb{Y}}$ between planar domains having least Dirichlet energy. The existence and uniqueness (up to a conformal change of variables in ${\mathbb{X}}$ ) of the energy-minimal mappings is established within the class ${\overline{\fancyscript{H}}_2(\mathbb{X}, \mathbb{Y})}$ of strong limits of homeomorphisms in the Sobolev space ${\fancyscript{W}^{1,2}(\mathbb{X}, \mathbb{Y})}$ , a result of considerable interest in the mathematical models of nonlinear elasticity. The inner variation of the independent variable in ${\mathbb{X}}$ leads to the Hopf differential ${h_{z} \overline{h_{\bar{z}}} {\rm d}z \otimes {\rm d}z}$ and its trajectories. For a pair of doubly connected domains, in which ${\mathbb{X}}$ has finite conformal modulus, we establish the following principle: A mapping ${h \in \overline{\fancyscript{H}}_{2} (\mathbb{X}, \mathbb{Y})}$ is energy-minimal if and only if its Hopf-differential is analytic in ${\mathbb{X}}$ and real along ${\partial \mathbb{X}}$ . In general, the energy-minimal mappings may not be injective, in which case one observes the occurrence of slits in ${\mathbb{X}}$ (cognate with cracks). Slits are triggered by points of concavity of ${\mathbb{Y}}$ . They originate from ${\partial \mathbb{X}}$ and advance along vertical trajectories of the Hopf differential toward ${\mathbb{X}}$ where they eventually terminate, so no crosscuts are created.
PubDate: 2013-08-01
• Global Solutions of Nonlinear Wave Equations in Time Dependent Inhomogeneous Media
• Abstract: Abstract We consider the problem of small data global existence for a class of semilinear wave equations with null condition on a Lorentzian background ${(\mathbb{R}^{3 + 1}, g)}$ with a time dependent metric g coinciding with the Minkowski metric outside the cylinder ${\{(t, x) x \leq R\}}$ . We show that the small data global existence result can be reduced to two integrated local energy estimates and demonstrate that these estimates work in the particular case when g is merely C 1 close to the Minkowski metric. One of the novel aspects of this work is that it applies to equations on backgrounds which do not settle to any particular stationary metric.
PubDate: 2013-08-01
• Global Well-posedness of Incompressible Inhomogeneous Fluid Systems with Bounded Density or Non-Lipschitz Velocity
• Abstract: Abstract In this paper, we first prove the global existence of weak solutions to the d-dimensional incompressible inhomogeneous Navier–Stokes equations with initial data ${a_0 \in L^\infty (\mathbb{R}^d), u_0 = (u_0^h, u_0^d) \in \dot{B}^{-1+\frac{d}{p}}_{p, r} (\mathbb{R}^d)}$ , which satisfy ${(\mu \ a_0 \ _{L^\infty} + \ u_0^h\ _{\dot{B}^{-1+\frac{d}{p}}_{p, r}}) {\rm exp}(C_r{\mu^{-2r}}\ u_0^d\ _{\dot{B}^{-1+\frac{d}{p}}_{p,r}}^{2r}) \leqq c_0\mu}$ for some positive constants c 0, C r and 1 < p < d, 1 < r < ∞. The regularity of the initial velocity is critical to the scaling of this system and is general enough to generate non-Lipschitz velocity fields. Furthermore, with additional regularity assumptions on the initial velocity or on the initial density, we can also prove the uniqueness of such a solution. We should mention that the classical maximal L p (L q ) regularity theorem for the heat kernel plays an essential role in this context.
PubDate: 2013-08-01
• Asymptotics of the Solutions of the Stochastic Lattice Wave Equation
• Abstract: Abstract We consider the long time limit for the solutions of a discrete wave equation with weak stochastic forcing. The multiplicative noise conserves energy, and in the unpinned case also conserves momentum. We obtain a time-inhomogeneous Ornstein-Uhlenbeck equation for the limit wave function that holds for both square integrable and statistically homogeneous initial data. The limit is understood in the point-wise sense in the former case, and in the weak sense in the latter. On the other hand, the weak limit for square integrable initial data is deterministic.
PubDate: 2013-08-01
• On the Spectrum of the Poincaré Variational Problem for Two Close-to-Touching Inclusions in 2D
• Abstract: Abstract We study the spectrum of the Poincaré variational problem for two close to touching inclusions in R 2. We derive the asymptotics of its eigenvalues as the distance between the inclusions tends to zero.
PubDate: 2013-08-01
• Wigner Measure Propagation and Conical Singularity for General Initial Data
• Abstract: Abstract We study the evolution of Wigner measures of a family of solutions of a Schrödinger equation with a scalar potential displaying a conical singularity. Under a genericity assumption, classical trajectories exist and are unique, thus the question of the propagation of Wigner measures along these trajectories becomes relevant. We prove the propagation for general initial data.
PubDate: 2013-07-01
• Boundary Regularity of Rotating Vortex Patches
• Abstract: Abstract We show that the boundary of a rotating vortex patch (or V-state, in the terminology of Deem and Zabusky) is C ∞, provided the patch is close to the bifurcation circle in the Lipschitz norm. The rotating patch is also convex if it is close to the bifurcation circle in the C 2 norm. Our proof is based on Burbea’s approach to V-states.
PubDate: 2013-07-01
• Multiple Blow-Up Phenomena for the Sinh-Poisson Equation
• Abstract: Abstract We consider the sinh-Poisson equation $$(P) _ \lambda - \Delta{u} = \lambda \, {\rm sinh} \, u \quad {\rm in} \, \Omega, \quad u = 0 \quad {\rm on} \, \partial\Omega$$ , where Ω is a smooth bounded domain in ${\mathbb{R}^2}$ and λ is a small positive parameter. If ${0 \in \Omega}$ and Ω is symmetric with respect to the origin, for any integer k if λ is small enough, we construct a family of solutions to (P) λ , which blows up at the origin, whose positive mass is 4πk(k−1) and negative mass is 4πk(k + 1). This gives a complete answer to an open problem formulated by Jost et al. (Calc Var PDE 31(2):263–276, 2008).
PubDate: 2013-07-01
• An Estimate for the Morse Index of a Stokes Wave
• Abstract: Abstract Stokes waves are steady periodic water waves on the free surface of an infinitely deep irrotational two-dimensional flow under gravity without surface tension. They can be described in terms of solutions of the Euler–Lagrange equation of a certain functional; this allows one to define the Morse index of a Stokes wave. It is well known that if the Morse indices of the elements of a set of non-singular Stokes waves are bounded, then none of them is close to a singular one. The paper presents a quantitative variant of this result.
PubDate: 2013-07-01
• Nonlocal Nonlinear Schrödinger Equations in R 3
• Abstract: Abstract This paper studies a class of nonlocal nonlinear Schrödinger equations in R 3, which occurs in the infinite ion acoustic speed limit of the Zakharov system with magnetic fields in a cold plasma. The magnetic fields induce some nonlocal effects in these nonlinear Schrödinger systems, and the main goal of this paper is to understand these effects. The key is to establish some a priori estimates on the nonlocal terms generated by the magnetic field, through which we obtain various conclusions including finite time blow-ups, sharp threshold of global existence and instability of standing waves for these equations.
PubDate: 2013-07-01
• Existence of Quasipattern Solutions of the Swift–Hohenberg Equation
• Abstract: Abstract We consider the steady Swift–Hohenberg partial differential equation, a one-parameter family of PDEs on the plane that models, for example, Rayleigh–Bénard convection. For values of the parameter near its critical value, we look for small solutions, quasiperiodic in all directions of the plane, and which are invariant under rotations of angle ${\pi/q, q \geqq 4}$ . We solve an unusual small divisor problem and prove the existence of solutions for small parameter values, then address their stability with respect to quasi-periodic perturbations.
PubDate: 2013-07-01
• The Two-Dimensional Euler Equations on Singular Domains
• Abstract: Abstract We establish the existence of global weak solutions of the two-dimensional incompressible Euler equations for a large class of non-smooth open sets. Loosely, these open sets are the complements (in a simply connected domain) of a finite number of obstacles with positive Sobolev capacity. Existence of weak solutions with L p vorticity is deduced from a property of domain continuity for the Euler equations that relates to the so-called γ-convergence of open sets. Our results complete those obtained for convex domains in Taylor (Progress in Nonlinear Differential Equations and their Applications, Vol. 42, 2000), or for domains with asymptotically small holes (Iftimie et al. in Commun Partial Differ Equ 28(1–2), 349–379, 2003; Lopes Filho in SIAM J Math Anal 39(2), 422–436, 2007).
PubDate: 2013-07-01
• Compatibility Equations of Nonlinear Elasticity for Non-Simply-Connected Bodies
• Abstract: Abstract Compatibility equations of elasticity are almost 150 years old. Interestingly, they do not seem to have been rigorously studied, to date, for non-simply-connected bodies. In this paper we derive necessary and sufficient compatibility equations of nonlinear elasticity for arbitrary non-simply-connected bodies when the ambient space is Euclidean. For a non-simply-connected body, a measure of strain may not be compatible, even if the standard compatibility equations (“bulk” compatibility equations) are satisfied. It turns out that there may be topological obstructions to compatibility; this paper aims to understand them for both deformation gradient F and the right Cauchy-Green strain C = F T F. We show that the necessary and sufficient conditions for compatibility of deformation gradient F are the vanishing of its exterior derivative and all its periods, that is, its integral over generators of the first homology group of the material manifold. We will show that not every non-null-homotopic path requires supplementary compatibility equations for F and linearized strain e. We then find both necessary and sufficient compatibility conditions for the right Cauchy-Green strain tensor C for arbitrary non-simply-connected bodies when the material and ambient space manifolds have the same dimensions. We discuss the well-known necessary compatibility equations in the linearized setting and the Cesàro-Volterra path integral. We then obtain the sufficient conditions of compatibility for the linearized strain when the body is not simply-connected. To summarize, the question of compatibility reduces to two issues: i) an integrability condition, which is d(F dX) = 0 for the deformation gradient and a curvature vanishing condition for C, and ii) a topological condition. For F dx this is a homological condition because the equation one is trying to solve takes the form dφ = F dX. For C, however, parallel transport is involved, which means that one needs to solve an equation of the form dR/ ds = RK, where R takes values in the orthogonal group. This is, therefore, a question about an orthogonal representation of the fundamental group, which, as the orthogonal group is not commutative, cannot, in general, be reduced to a homological question.
PubDate: 2013-07-01
• Classical Limit for a System of Non-Linear Random Schrödinger Equations
• Abstract: Abstract This work is concerned with the semi-classical analysis of mixed state solutions to a Schrödinger–Position equation perturbed by a random potential with weak amplitude and fast oscillations in time and space. We show that the Wigner transform of the density matrix converges weakly and in probability to solutions of a Vlasov–Poisson–Boltzmann equation with a linear collision kernel.Aconsequence of this result is that a smooth non-linearity such as the Poisson potential (repulsive or attractive) does not change the statistical stability property of the Wigner transform observed in linear problems.We obtain, in addition, that the local density and current are self-averaging, which is of importance for some imaging problems in random media. The proof brings together the martingale method for stochastic equations with compactness techniques for non-linear PDEs in a semi-classical regime. It relies partly on the derivation of an energy estimate that is straightforward in a deterministic setting but requires the use of a martingale formulation and well-chosen perturbed test functions in the random context.
PubDate: 2013-07-01
• Hadamard Variational Formula for the Green’s Function of the Boundary Value Problem on the Stokes Equations
• Abstract: Abstract For every ${\varepsilon > 0}$ , we consider the Green’s matrix ${G_{\varepsilon}(x, y)}$ of the Stokes equations describing the motion of incompressible fluids in a bounded domain ${\Omega_{\varepsilon} \subset \mathbb{R}^d}$ , which is a family of perturbation of domains from ${\Omega\equiv \Omega_0}$ with the smooth boundary ${\partial\Omega}$ . Assuming the volume preserving property, that is, ${\mbox{vol.}\Omega_{\varepsilon} = \mbox{vol.}\Omega}$ for all ${\varepsilon > 0}$ , we give an explicit representation formula for ${\delta G(x, y) \equiv \lim_{\varepsilon\to +0}\varepsilon^{-1}(G_{\varepsilon}(x, y) - G_0(x, y))}$ in terms of the boundary integral on ${\partial \Omega}$ of ${G_0(x, y)}$ . Our result may be regarded as a classical Hadamard variational formula for the Green’s functions of the elliptic boundary value problems.
PubDate: 2013-06-01
• Nondispersive solutions to the L 2-critical Half-Wave Equation
• Abstract: Abstract We consider the focusing L 2-critical half-wave equation in one space dimension, $$i \partial_t u = D u - u ^2 u$$ , where D denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold ${M_{*} > 0}$ such that all H 1/2 solutions with ${\ u\ _{L^2} < M_*}$ extend globally in time, while solutions with ${\ u\ _{L^2} \geq M_*}$ may develop singularities in finite time. In this paper, we first prove the existence of a family of traveling waves with subcritical arbitrarily small mass. We then give a second example of nondispersive dynamics and show the existence of finite-time blowup solutions with minimal mass ${\ u_0\ _{L^2} = M_*}$ . More precisely, we construct a family of minimal mass blowup solutions that are parametrized by the energy E 0 > 0 and the linear momentum ${P_0 \in \mathbb{R}}$ . In particular, our main result (and its proof) can be seen as a model scenario of minimal mass blowup for L 2-critical nonlinear PDEs with nonlocal dispersion.
PubDate: 2013-05-03
• Breakdown of Smoothness for the Muskat Problem
• Abstract: Abstract In this paper we show that there exists analytic initial data in the stable regime for the Muskat problem such that the solution turns to the unstable regime and later breaks down, that is, no longer belongs to C 4.
PubDate: 2013-04-06
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SUNCAT is the largest freely available source of information about serials holdings in the UK. Researchers are able to locate serials held in 85 UK research libraries. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9102621674537659, "perplexity": 1257.8066702806502}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368704664826/warc/CC-MAIN-20130516114424-00025-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://www.gradesaver.com/textbooks/math/precalculus/precalculus-6th-edition-blitzer/chapter-11-test-page-1180/16 | ## Precalculus (6th Edition) Blitzer
$-24$ feet /second in the downward direction
$s(t)=-16t^2+72t$ Thus, we have $v(t)=s′(t)=-16 \dfrac{dt^2}{dt}+72 \times \dfrac{dt}{dt}$ So, $v(t)=-32t +72$ Set $t=3$ Then $v(3)=-32(3) +72=-24$ feet /second in the downward direction | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8758205771446228, "perplexity": 3063.3527714948527}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986670928.29/warc/CC-MAIN-20191016213112-20191017000612-00107.warc.gz"} |
http://clay6.com/qa/9127/if-a-and-b-are-two-sets-such-that-a-b-cup-b-a-then-what-can-you-say-about-a | # If $A\:and\:B$ are two sets such that $(A-B)\cup B=A$, then what can you say about $A\:and\:B$?
$\begin{array}{1 1} A\subset B \\ B\subset A \\ A=\phi \\ B=\phi \end{array}$
Ans- (B)
We know that $(A-B)\cup B=A\cup B$
But given that $(A-B)\cup B=A$
$\Rightarrow\:A\cup B=A$
$\Rightarrow B\subset A$
edited May 17, 2014 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9814920425415039, "perplexity": 270.6372501373605}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824570.79/warc/CC-MAIN-20171021043111-20171021063111-00246.warc.gz"} |
https://proofwiki.org/wiki/Elements_of_Ordered_Pair_do_not_Commute | Elements of Ordered Pair do not Commute
Theorem
Let $\set {a, b}$ be a doubleton, so that $a$ and $b$ are distinct objects.
Let $\tuple {a, b}$ denote the ordered pair such that the first coordinate is $a$ and the second coordinate is $b$.
Then:
$\tuple {a, b} \ne \tuple {b, a}$
Proof
$\tuple {a, b} = \set {\set a, \set {a, b} }$
and by Equality of Ordered Pairs:
$\tuple {a, b} = \tuple {b, a} \iff a = b$
But $a \ne b$ and so:
$\tuple {a, b} \ne \tuple {b, a}$
$\blacksquare$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9892860651016235, "perplexity": 307.56487815914033}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655886178.40/warc/CC-MAIN-20200704135515-20200704165515-00451.warc.gz"} |
https://www.scec.org/meetings/2021/am/poster/089 | ## Poster #089, San Andreas Fault System (SAFS)
### A unified perspective of seismicity and fault coupling along the San Andreas Fault
Poster Image:
#### Poster Presentation
2021 SCEC Annual Meeting, Poster #089, SCEC Contribution #11496
The San Andreas Fault (SAF) showcases the breadth of possible earthquake sizes and occurrence behavior, from repeating earthquakes to total quiescence, to large damaging earthquakes. In particular, the central SAF is a microcosm of such diversity. This section also exhibits the spectrum of fault coupling from locked to creeping. Here, we show that these varied observations are in fact tightly connected. Specifically, the creep rate along the central SAF is shown to be directly proportional to the fraction of non-clustered earthquakes for the period 1984–2020. This relationship provides a unified perspective of earthquake phenomenology along the SAF, where lower coupling manifests in weaker t...emporal clustering, with repeating earthquakes as an end-member.
We compute a metric called the "fraction of background events", to describe relative dominance between aseismic and seismic processes. We show that this quantity is highly correlated (at a 93% level) with the rate of creep as measured from geodesy. The degree of fault coupling thus has a first-order effect on the long-term seismicity dynamics of the entire central San Andreas. Regions that exhibit a lower fraction of non-clustered background seismicity are interpreted as having a higher likelihood of triggering large mainshock-aftershock sequences by taking up an increasingly larger area of the fault surface via seismic slip.
Under this unified paradigm, the northwest ~75 km of the creeping segment may be more aptly described as a transition zone, with an increased likelihood of a rupture propagating through the entire ~150 km-long creeping segment. Our findings would be directly relevant to other fault systems globally. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8901774883270264, "perplexity": 4608.231041154965}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572304.13/warc/CC-MAIN-20220816120802-20220816150802-00307.warc.gz"} |
https://www.varsitytutors.com/ap_physics_1-help/tension | # AP Physics 1 : Tension
## Example Questions
← Previous 1 3
### Example Question #1 : Tension
A 10kg block is suspended by two ropes. Each rope makes an angle of 45 degrees to the horizontal.
What is the magnitude of the tension force in each rope?
Explanation:
Luckily enough, the angles of the two ropes are the same. Therefore, the tension in each will be the same. This immediately eliminates two of the five answers. Now we just need to calculate what that force is.
We know that together, the vertical components of the tension must equal the weight of the block. Therefore we can write:
Since we know that the two tension forces are equal, we can rewrite:
Rearranging for T, we get:
### Example Question #1 : Tension
Consider the following system:
If the mass is and , what is the tension, ? Assume no frictional forces.
Explanation:
Since there is no friction between the mass and slope, there are only two relevant forces acting on the mass: gravity and tension. Furthermore, since the block is not in motion, we know that these forces are equal to each other. Therefore:
Substituting in an expression for the force of gravity, we get:
We know all of these values, allowing us to solve for the tension:
### Example Question #1 : Tension
Consider the following system:
If the force of tension is , the force of static friction is , the block has a mass of , and the block is motionless, what is the angle ?
Explanation:
There are three relevant forces acting on the block in this scenario: friction, tension, and gravity. We are given two of these values, so we simply need to develop an expression for the force of gravity in the direction of the slope. Since the block is motionless, we can write:
Substituting in an expression for the force of gravity, we get:
Rearrange to solve for the angle:
We know all of these values, allowing us to solve:
### Example Question #1 : Tension
Consider the following system:
If the coefficient of static friction is , the angle measures , the force of tension is , and the block is motionless, what is the mass of the block?
Explanation:
There are three relevant forces acting on the block in this scenario: tension, friction, and gravity. We are given tension, so we will need to develop expressions for friction and gravity. Since the block is motionless, we can say:
Plugging in expressions for the force of gravity and friction, we get:
Rearranging for the mass, we get:
We know all of these values, allowing us to solve:
### Example Question #1 : Tension
Consider the following system:
If the block has a mass of and the angle measures , what is the minimum value of the coefficient of static friction that will result in a tension of ?
Explanation:
Since there is no tension, there are only two relevant forces acting on the block: friction and gravity. Since the block is motionless, we can also write:
Substitute the expressions for these two forces:
Canceling out mass and gravitational acceleration, and rearranging for the coefficient of static friction, we get:
### Example Question #1 : Tension
Consider the following system:
If the mass is accelerating at a rate of , the angle measures , the mass of the block is , and the coefficient of kinetic friction is , what is the tension ?
Explanation:
There are three relevant forces acting on the block in this situtation: friction, gravity, and tension. We can use Newton's second law to express the system:
Substituting expressions in for the forces, we get:
Canceling out mass and rearranging to solve for tension, we get:
We have values for each variable, allowing us to solve:
### Example Question #2 : Tension
A 12kg block is sliding down a incline with an acceleration of as shown in the diagram. If the coefficient of kinetic friction of block 1 on the ramp is 0.18, what is the mass of block 2?
Explanation:
In order to find the mass of block 2, we're going to need to calculate a few other things, such as the tension in the rope.
To begin with, we'll need to identify the various forces on our free-body diagram. To do this, we will begin with block 1 and use a rotated coordinate system to simplify things. In such a system, the x-axis will run parallel to the surface of the ramp, while the y-axis will be perpendicular to the ramp's surface, as shown below:
Now we can identify the forces acting on block 1. Along the rotated y-axis, the force of gravity acting on the block is equal to , and the force of the ramp on the block is just the normal force, . Since block 1 is not moving in the y direction, we can set these two forces equal to each other.
Now, considering the forces acting along the rotated x-axis, we have a force pointing downwards equal to . Pointing upwards, we have the tension force and we also have the frictional force, .
The formula for calculating the force due to kinetic friction is:
Since we have already determined what the normal force is, we can substitute that expression into the above equation to obtain:
Now, we can write an expression for the net force acting upon block 1 in the x direction:
Rearrange the above expression to solve for tension.
So far, we have only been looking at block 1. Now let's turn our attention to block 2 and see what forces are acting on it. In the downward direction we have the weight of the block due to gravity, which is equal to . In the upward direction, as we can see in the diagram, we have the tension of the rope, . We need to write an expression that tells us the net force acting upon block 2.
Since we calculated the expression for tension from the information regarding block 1, we can plug that expression into the above equation in order to obtain:
Now rearrange to solve for the mass of block 2.
Then plugging in values, we can finally calculate block 2's mass:
### Example Question #1 : Tension
What is the tension force on a wire holding a 10kg ball 20ft above the ground, if the ball is not moving at that height?
Explanation:
Since the gravitational force must be cancelled by the tension force, as the ball is experiencing no acceleration, and no other forces are being applied to it:
### Example Question #1 : Tension
A block weighing is hanging from a string. Bruce begins applying a force up on the block. What is the force of tension in the string?
Explanation:
The block has three forces on it: the force of tension, the force of gravity, and the force from Bruce. The force of gravity is:
The force from Bruce plus the force of tension has to equal gravity (since Bruce's force and tension are up while gravity is down) so the block is in equilibrium.
### Example Question #1 : Tension
A helicopter is lifting a box of mass with a rope. The helicopter and box are accelerating upward at . Determine the tension in the rope. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9479341506958008, "perplexity": 397.5739847378098}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583514314.87/warc/CC-MAIN-20181021181851-20181021203351-00228.warc.gz"} |
http://publish.illinois.edu/ymb/2017/03/28/exterior-differential-forms/ | # Exterior Differential forms
$$\def\pd{\partial} \def\Real{\mathbb{R}}$$
• We define a differential $$k$$-form as
$$\omega(x) = \sum_{I=\{i_1<i_2<\ldots<i_k\}} c_I(x) dx_{i_1}\wedge \cdots\wedge dx_{i_k},$$
an exterior form with coefficients depending on the positions.
(We will be using sometimes simplifying notation $$\xi_i = dx_i$$ or $$\eta_i=dy_i$$.)
• Pullback maps: Let $$\Omega_1,\Omega_2$$ be two Euclidean domains
(not necessarily embedded in the same space) such that
$$\psi:\Omega_1 \to \Omega_2$$ is a smooth (continuously differentiable enough times) mapping between them.Let $$f$$ be a real function on $$\Omega_2$$. Then one can pull it back, by defining
$(\psi^*f)(x):=f(\psi(x)).$There is no mystery here – this is just a way to substitute.
Example: If $$\psi:\Real^2\to\Real^3$$ is given by
$\psi(x_1,x_2)=(x_1^2, x_1x_2, x_2^2),$
and $$f:\Real^3\to\Real$$ is a linear function $$ay_1+by_2+cy_3+d$$, then
$\psi^*f(x_1,x_2)=ax_1^2+bx_1x_2+cx_2^2+d.$
• Similarly one can define the pullback of a differential form. Again, the pullback of a differential form is just a substitution.
If $$\omega$$ is a $$k$$-form on $$\Omega_2$$, given by
$$\omega(x) = \sum_{I=\{i_1<i_2<\ldots<i_k\}} c_I(x) dx_{i_1}\wedge \cdots\wedge dx_{i_k},$$
and $$\psi:\Omega_1\to\Omega_2$$ is given, in coordinates, as
$x_i=\psi_i(y_1,\ldots,y_m),$
then
$(\psi^*\omega)(y)=\sum_{I=\{i_1<i_2<\ldots<i_k\}} c_I(\psi(y)) dx_{i_1}(y)\wedge \cdots\wedge dx_{i_k}(y)= \sum_{I=\{i_1<i_2<\ldots<i_k\}} c_I(\psi(y))(\sum_l\frac{\pd x_{i_1}}{\pd y_l}dy_l)(y)\wedge \cdots\wedge (\sum_l\frac{\pd x_{i_1}}{\pd y_l}(y)dy_l),$
Example: if $$\psi(r,\phi)=(r\cos(\phi),r\sin(\phi))=(x,y)$$,
then
$\psi^*(dx\wedge dy)=(\cos(\phi)dr-r\sin(\phi)d\phi)\wedge(\sin(\phi)dr+r\cos(\phi)d\phi)=r dr\wedge d\phi.$
• If $$\gamma$$ is a curve in
$$\Omega_1$$, it is mapped by $$\psi$$ into a curve on $$\Omega_2$$, again, by composition.
• One can verify immediately that for 1-forms
the composed path integrals of $$\psi^*\omega(\gamma)$$ and its image in $$\Omega_2$$
are the same:
$$\underbrace{\int_{\psi(\gamma)}df}_{\text{Integration in } \Omega_2} = \underbrace{\int_\gamma d \psi^*(f)}_{\text{Integration in } \Omega_1}.$$
• Examples
1. Define the 1-form $$\omega_1 = xdx + ydy = rdr$$ – this is, clearly, just the differential $$dr^2/2$$.
2. Another 1-form $$\omega_2 = xdy-ydx$$. Applied on a vector $$f=(s,t)$$ at a point $$r=(x,y)$$, it evaluates to
$$\omega_2(v) = \frac{xt – ys}.$$
The physical meaning is therefore the angular momentum of the force $$v$$ applied at $$r$$.
3. One more form: $$\omega_3=\omega_2/|r|^2$$ – defined everywhere outside of the origin.
In the polar coordinates $$d\phi := \frac{\mathbf{r}\times v}{|\mathbf{r}|^2}$$,
and we get the well-known identity:
$$\omega_1 \wedge \omega_2 = (rdr)\wedge d\phi = \frac{1}{2} (xdx+ydy)\wedge \frac{xdy+ydx}{x^2+y^2} = \frac{x^2dx\wedge dy – y^2 dy\wedge dx}{x^2+y^2} = dx\wedge dy$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9935276508331299, "perplexity": 364.25688042902794}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187825147.83/warc/CC-MAIN-20171022060353-20171022080353-00070.warc.gz"} |
https://developers.google.com/machine-learning/crash-course/multi-class-neural-networks/softmax | # Multi-Class Neural Networks: Softmax
Recall that logistic regression produces a decimal between 0 and 1.0. For example, a logistic regression output of 0.8 from an email classifier suggests an 80% chance of an email being spam and a 20% chance of it being not spam. Clearly, the sum of the probabilities of an email being either spam or not spam is 1.0.
Softmax extends this idea into a multi-class world. That is, Softmax assigns decimal probabilities to each class in a multi-class problem. Those decimal probabilities must add up to 1.0. This additional constraint helps training converge more quickly than it otherwise would.
For example, returning to the image analysis we saw in Figure 1, Softmax might produce the following likelihoods of an image belonging to a particular class:
Class Probability
apple 0.001
bear 0.04
candy 0.008
dog 0.95
egg 0.001
Softmax is implemented through a neural network layer just before the output layer. The Softmax layer must have the same number of nodes as the output layer.
Figure 2. A Softmax layer within a neural network.
## Softmax Options
Consider the following variants of Softmax:
• Full Softmax is the Softmax we've been discussing; that is, Softmax calculates a probability for every possible class.
• Candidate sampling means that Softmax calculates a probability for all the positive labels but only for a random sample of negative labels. For example, if we are interested in determining whether an input image is a beagle or a bloodhound, we don't have to provide probabilities for every non-doggy example.
Full Softmax is fairly cheap when the number of classes is small but becomes prohibitively expensive when the number of classes climbs. Candidate sampling can improve efficiency in problems having a large number of classes.
## One Label vs. Many Labels
Softmax assumes that each example is a member of exactly one class. Some examples, however, can simultaneously be a member of multiple classes. For such examples:
• You may not use Softmax.
• You must rely on multiple logistic regressions.
For example, suppose your examples are images containing exactly one item—a piece of fruit. Softmax can determine the likelihood of that one item being a pear, an orange, an apple, and so on. If your examples are images containing all sorts of things—bowls of different kinds of fruit—then you'll have to use multiple logistic regressions instead. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8527825474739075, "perplexity": 929.2189729680183}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247504790.66/warc/CC-MAIN-20190221132217-20190221154217-00292.warc.gz"} |
https://eventuallyalmosteverywhere.wordpress.com/tag/total-population-size/ | # Generating uniform trees
A long time ago, I wrote quite a few a things about uniform trees. That is, a uniform choice from the $n^{n-2}$ unrooted trees with vertex set [n]. This enumeration, normally called Cayley’s formula, has several elegant arguments, including the classical Prufer bijection. But making a uniform choice from a large set is awkward, and so we seek more probabilistic methods to sample such a tree, which might also give insight into the structure of a ‘typical’ uniform tree.
In another historic post, I talked about the Aldous-Broder algorithm. Here’s a quick summary. We run a random walk on the complete graph $K_n$ started from a uniformly-chosen vertex. Every time we arrive at a vertex we haven’t visited before, we record the edge just traversed. Eventually we have visited all n vertices, so have recorded n-1 edges. It’s easy enough to convince yourself that these n-1 edges form a tree (how could there be a cycle?) and a bit more complicated to decide that the distribution of this tree is uniform.
It’s worth noting that this algorithm works to construct a uniform spanning tree on any connected base graph.
This post is about a few alternative constructions and interpretations of the uniform random tree. The first construction uses a Galton-Watson process. We take a Galton-Watson process where the offspring distribution is Poisson(1), and condition that the total population size is n. The resulting random tree has a root but no labels, however if we assign labels in [n] uniformly at random, the resulting rooted tree has the uniform distribution among rooted trees on [n].
Proof
This is all about moving from ordered trees to non-ordered trees. That is, when setting up a Galton-Watson tree, we distinguish between the following two trees, drawn extremely roughly in Paint:
That is, it matters which of the first-generation vertices have three children. Anyway, for such a (rooted) ordered tree T with n vertices, the probability that the Galton-Watson process ends up equal to T is
$\mathbb{P}(GW = T) = \prod_{v\in T} \frac{e^{-1}}{C(v)!} = e^{-n} \prod_{v\in T}\frac{1}{C(v)!},$
where $C(v)$ is the number of children of a vertex $v\in T$. Then, since $\mathbb{P}( |GW|=n )$ is a function of n, we find
$\mathbb{P}(GW=T \,\big|\, |GW|=n) = f(n)\prod_{v\in T} \frac{1}{C(v)!},$
where f(n) is a function of n alone (ie depends on T only through its size n).
But given an unordered rooted tree t, labelled by [n], there are $\prod_{v \in t} C(v)!$ ordered trees associated to t in the natural way. Furthermore, if we take the Poisson Galton-Watson tree conditioned to have total population size n, and label uniformly at random with [n], we obtain any one of these ordered trees with probability $\frac{f(n)}{n!} \prod_{v\in t} \frac{1}{C(v)!}$. So the probability that we have t after we forget about the ordering is $\frac{f(n)}{n!}$, which is a function of n alone, and so the distribution is uniform among the set of rooted unordered trees labelled by [n], exactly as required.
Heuristic for Poisson offspring distribution
In this proof, the fact that $\mathbb{P}(C(v)=k)\propto \frac{1}{k!}$ exactly balances the number of orderings of the k children explains why Poisson(1) works out. Indeed, you can see in the proof that Poisson(c) works equally well, though when $c\ne 1$, the event we are conditioning on (namely that the total population size is n) has probability decaying exponentially in n, whereas for c=1, the branching process is critical, and the probability decays polynomially.
We can provide independent motivation though, from the Aldous-Broder construction. Both the conditioned Galton-Watson construction and the A-B algorithm supply the tree with a root, so we’ll keep that, and look at the distribution of the degree of the root as constructed by A-B. Let $\rho=v_1,v_2,v_3,\ldots$ be the vertices [n], ordered by their discovery during the construction. Then $\rho$ is definitely connected by an edge to $v_2$, but thereafter it follows by an elementary check that the probability $\rho$ is connected to $v_m$ is $\frac{1}{n-1}$, independently across all m. In other words, the distribution of the degree of $\rho$ in the tree as constructed by A-B is
$1+ \mathrm{Bin}\left(n-2,\frac{1}{n-1}\right) \approx 1+\mathrm{Poisson}(1).$
Now, in the Galton-Watson process, conditioning the tree to have fixed, large size changes the offspring distribution of the root. Conveniently though, in a limiting sense it’s the same change as conditioning the tree to have size at least n. Since these events are monotone in n, it’s possible to take a limit of the conditioning events, and interpret the result as the Galton-Watson tree conditioned to survive. It’s a beautiful result that this interpretation can be formalised as a local limit. The limiting spine decomposition consists of an infinite spine, where the offspring distribution is a size-biased version of the original offspring distribution (and so in particular, always has at least one child) and where non-spine vertices have the original distribution.
In particular, the number of the offspring of the root is size-biased, and it is well-known and not hard to check that size-biasing Poisson(c) gives 1+Poisson(c) ! So in fact we have, in an appropriate limiting sense in both objects, a match between the degree distribution of the root in the uniform tree, and in the conditioned Galton-Watson tree.
This isn’t supposed to justify why a conditioned Galton-Watson tree is relevant a priori (especially the unconditional independence of degrees), but it does explain why Poisson offspring distributions are relevant.
Construction via G(N,p) and the random cluster model
The main reason uniform trees were important to my thesis was their appearance in the Erdos-Renyi random graph G(N,p). The probability that vertices {1, …, n} form a tree component in G(N,p) with some particular structure is
$p^{n-1} (1-p)^{\binom{n}{2}-(n-1)} \times (1-p)^{n(N-m)}.$
Here, the first two terms give the probability that the graph structure on {1, …, n} is correct, and the the final term gives the probability of the (independent) event that these vertices are not connected to anything else in the graph. In particular, this has no dependence on the tree structure chosen on [n] (for example, whether it should be a path or a star – both examples of trees). So the conditional distribution is uniform among all trees.
If we work in some limiting regime, where $pn\rightarrow 0$ (for example if n is fixed and $p=\frac{1}{N}\rightarrow 0$), then we can get away asymptotically with less strong conditioning. Suppose we condition instead just that [n] form a component. Now, there are more ways to form a connected graph with one cycle on [n] than there are trees on [n], but the former all require an extra edge, and so the probability that a given one such tree-with-extra-edge appears as the restriction to [n] in G(N,p) is asymptotically negligible compared to the probability that the restriction to [n] of G(N,p) is a tree. Naturally, the local limit of components in G(N,c/N) is a Poisson(c) Galton-Watson branching process, and so this is all consistent with the original construction.
One slightly unsatisfying aspect to this construction is that we have to embed the tree of size [n] within a much larger graph on [N] to see uniform trees. We can’t choose a scaling p=p(n) such that G(n,p) itself concentrates on trees. To guarantee connectivity with high probability, we need to take $p> \frac{\log n}{n}$, but by this threshold, the graph has (many) cycles with high probability.
At this PIMS summer school in Vancouver, one of the courses is focusing on lattice spin models, including the random cluster model, which we now briefly define. We start with some underlying graph G. From a physical motivation, we might take G to be $\mathbb{Z}^d$ or some finite subset of it, or a d-ary tree, or the complete graph $K_N$. As in classical bond percolation (note G(N,p) is bond percolation on $K_N$), a random subset of the edges of G are included, or declared open. The probability of a given configuration w, with e open edges is proportional to
$p^e (1-p)^{|E(G)| - e} q^{k(w)},$ (*)
where the edge-weight $p\in(0,1)$ as usual, and cluster weight $q\in (0,\infty)$, and $k(w)$ counts the number of connected components in configuration w. When q=1, we recover classical bond percolation (including G(N,p) ), while for q>1, this cluster-reweighting favours having more components, and q<1 favours fewer components. Note that in the case $q\ne 1$, the normalising constant (or partition function) of (*) is generally intractable to calculate explicitly.
As in the Erdos-Renyi graph, consider fixing the underlying graph G, and taking $p\rightarrow 0$, but also taking $\frac{q}{p}\rightarrow 0$. So the resulting graph asymptotically ‘wants to have as few edges as possible, but really wants to have as few components as possible’. In particular, 1) all spanning trees of G are equally likely; 2) any configuration with more than one component has asymptotically negligible probability relative to any tree; 3) any graph with a cycle has #components + #edges greater than that of a tree, and so is asymptotically negligible probability relative to any tree.
In other words, the limit of the distribution is the uniform spanning tree of G, and so this (like Aldous-Broder) is a substantial generalisation, which constructs the uniform random tree in the special case where $G=K_n$.
# Analytic vs Probabilistic Arguments for a Supercritical BP
This follows on directly from the previous post. I was originally going to talk only about what follows, but I got rather carried away with the branching process account. I was stuck on a particular exercise, and we ended up coming up with two arguments: one analytic and one probabilistic. Since the typical flavour of this blog is to present problems which show the advantage of the probabilistic approach, it seems only fair to remark on this case, where the analytic method was less interesting, but much simpler.
Recall that we have a supercritical random graph $G(n,\frac{\lambda}{n}), \lambda>1$, and we are considering the rescaled exploration process $S_{nt}$, which has asymptotic mean $\mu_t=1-t-e^{-\lambda t}$. We can calculate similarly an expression for the asymptotic variance
$\frac{\text{Var}(S_{nt})}{n}\rightarrow v_t=e^{-\lambda t}(1-e^{-\lambda t}).$
To use this to verify the result about the size of the giant component, we verify that $\mu_{\zeta_\lambda+x/\sqrt{n}}$ is negative, and has small variance, which would confirm that the giant component has size bounded above by $\zeta_\lambda$ almost surely. A similar argument is required for the lower bound. The variance is a separate matter, but it is therefore necessary that $\mu_t$ should be decreasing at $t=\zeta_\lambda$, that is $\mu_t'=\lambda e^{-\lambda \zeta_\lambda}<0$. This is what we try to prove in the remainder of this post. Recall that in the previous post we have checked that it is equal to zero here.
Heuristic Explanation
$\mu_t$ has been rescaled from the original definition of the exploration process in both size and time-scale so some care is needed to see why this should hold in the limit. Remember that all components apart from the giant component are of size O(log n). So immediately after exhausting the giant component, you are likely to be visiting components of size roughly log n. A time interval of dt for $\mu$ corresponds to ndt for S, during which S will visit some components of size log n and some of O(1) and some in between. In particular, some fixed proportion of vertices are isolated, that is, in a component of size 1.
There is then a complicated size-biasing train of thought. A component of size log n is more likely to come up than an isolated vertex, but there are not as many of them. The log n components push the derivative $\mu_t'$ towards zero, because S_t decreases by 1 over a time-interval of length log n, which gives a gradient of zero in the limit. However, the isolated vertices give a gradient of -1, because S_t decreases by 1 over a time interval of 1. Despite the fact that log n intervals are likely to appear earlier, it still remains the case that after exhausting a component (in particular, at time $t=\zeta_\lambda$, after exhausting the giant component), with some bounded below positive probability you will choose an isolated vertex next. The component size only affects that time-scale if it is O(n), which none of the remaining components are, so the derivative $\mu_{\zeta_\lambda}'$ consists of some complicated weighted mean of 0 and -1. In particular, it is negative.
Analytic solution
Obviously, that won’t do in practice. Suppressing lambdas for ease of notation, the key fact is: $e^{-\lambda \zeta}=1-\zeta$. We want to show that $\lambda e^{-\lambda \zeta}<1$. Substituting
$\lambda=-\frac{\log(1-\zeta)}{\zeta},$
means that it is required to show:
$-\frac{1-\zeta}{\zeta}\log(1-\zeta)<1.$
Differentiating the left hand side gives:
$\frac{\log(1-\zeta)+\zeta}{\zeta^2}<0,$
since of course $\log(1-\zeta)=\zeta+\frac{\zeta^2}{2}+\frac{\zeta^3}{3}+\dots$. So it suffice to check the result for small $\zeta$. But, again using a Taylor series:
$-\frac{1-\zeta}{\zeta}\log(1-\zeta)=1-\frac12\zeta+O(\zeta^2)<1,$
for small $\zeta$. This gives the required result.
Probabilistic Interpretation and Solution
First, we observe that $\lambda e^{-\lambda\zeta}=\lambda(1-\zeta)$ is the expected number of vertices in the first generation of a $\text{Po}(\lambda)$ whose progeny become extinct. This motivates considering the canonical decomposition of a supercritical branching process Z into the skeleton process and the dual process. The skeleton $Z^+$ consists of all vertices which have infinitely many successors. It is relatively easy to show that this is a branching process with offspring distribution $\text{Po}(\lambda\zeta)$ conditioned on being positive. The dual process $Z^*$ is a G-W branching process with offspring distribution $\text{Po}(\lambda)$ conditioned on dying. This is the same as a branching process with offspring distribution $\text{Po}(\lambda(1-\zeta)$, by a sprinkling argument, which says that if we begin with a Poisson number of things, then remove each one independently with some fixed probability, the remaining number of things is Poisson also.
We can construct the original branching process by
• With probability $\zeta$, take the skeleton, and affixe independent copies of $Z^*$ at every vertex in the skeleton.
• With probability $1-\zeta$, just take a copy of $Z^*$.
It is immediately clear that $\lambda(1-\zeta)\leq 1$. After all, the dual process is almost surely finite, so the offspring distribution cannot have expectation greater than 1. Checking that this is strong is more fiddly. The best way I have come up with is to examine the tail of the distribution of total population size of the original branching process.
The total population size T of a branching process has an exponential tail if the offspring distribution is subcritical. It isn’t hugely surprising that this behaves like a large deviation for iid RVs, since in the limit such an event requires a lot of the offspring counts to deviate substantially from the mean. The same holds in the supercritical case, with the additional complication that though the finite tail decays exponential, there is positive probability that the total size will be infinite. In the critical case, however, there is a power-law decay. This is not hugely surprising as it marks the threshhold for the appearance of the infinite population, just as in a multiplicative coalescent at time 1, we have a load of very large components just about to form a giant component. The tool for all of these results is Dwass’s Theorem, which says:
$\mathbb{P}(T=n)=\frac{1}{n}\mathbb{P}(X_1+\ldots+X_n=n-1),$
where $X_1$ are iid with the offspring distribution. When $\mathbb{E}X_1\neq 1$, this is a large deviation event, for which Cramer’s theorem applies (assuming, as is the case for the Poisson distribution, that the offspring distribution has finite variance). When, $\mathbb{E}X=1$, the Central Limit Theorem says that with high probability,
$X_1+\ldots+X_n\in [n-n^{3/4},n+n^{3/4}],$
so, skating over the details of whether everything is exactly uniform within this CLT scaling window,
$\mathbb{P}(T=n)\geq \frac{1}{n}\cdot\frac{1}{2n^{3/4}}.$
The true exponent of the power law decay is substantially slower than this, but the above argument works as a back-of-the-envelope bound.
In particular, if the dual process has mean 1, then the population size of the original branching process is given by taking a distribution with exponential tail with some probability and a distribution with power-law tail with some probability. Obviously the power-law will dominate, which contradicts the assumption that the original branching process was supercritical, and so has an exponential tail.
# Exploring the Supercritical Random Graph
I’ve spent a bit of time this week reading and doing all the exercises from some excellent notes by van der Hofstad about random graphs. I think they are absolutely excellent and would not be surprised if they become the standard text for an introduction to probabilistic combinatorics. You can find them hosted on the author’s website. I’ve been reading chapters 4 and 5, which approaches the properties of phase transitions in G(n,p) by formalising the analogy between component sizes and population sizes in a binomial branching process. When I met this sort of material for the first time during Part III, the proofs generally relied on careful first and second moment bounds, which is fine in many ways, but I enjoyed vdH’s (perhaps more modern?) approach, as it seems to give a more accurate picture of what is actually going on. In this post, I am going to talk about using the branching process picture to explain why the giant component emerges when it does, and how to get a grip on how large it is at any time after it has emerged.
Background
A quick tour through the background, and in particular the notation will be required. At some point I will write a post about this topic in a more digestible format, but for now I want to move on as quickly as possible.
We are looking at the sparse random graph $G(n,\frac{\lambda}{n})$, in the super-critical phase $\lambda>1$. With high probability (that is, with probability tending to 1 as n grows), we have a so-called giant component, with O(n) vertices.
Because all the edges in the configuration are independent, we can view the component containing a fixed vertex as a branching process. Given vertex v(1), the number of neighbours is distributed like $\text{Bi}(n-1,\frac{\lambda}{n})$. The number of neighbours of each of these which we haven’t already considered is then $\text{Bi}(n-k,\frac{\lambda}{n})$, conditional on k, the number of vertices we have already discounted. After any finite number of steps, k=o(n), and so it is fairly reasonable to approximate this just by $\text{Bi}(n,\frac{\lambda}{n})$. Furthermore, as n grows, this distribution converges to $\text{Po}(\lambda)$, and so it is natural to expect that the probability that the fixed vertex lies in a giant component is equal to the survival probability $\zeta_\lambda$ (that is, the probability that it is infinite) of a branching process with $\text{Po}(\lambda)$ offspring distribution. Note that given a graph, the probability of a fixed vertex lying in a giant component is equal to the fraction of the vertex in the giant component. At this point it is clear why the emergence of the giant component must happen at $\lambda=1$, because we require $\mathbb{E}\text{Po}(\lambda)>1$ for the survival probability to be non-zero. Obviously, all of this needs to be made precise and rigorous, and this is treated in sections 4.3 and 4.4 of the notes.
Exploration Process
A common functional of a rooted branching process to consider is the following. This is called in various places an exploration process, a depth-first process or a Lukasiewicz path. We take a depth-first labelling of the tree v(0), v(1), v(2),… , and define c(k) to be the number of children of vertex v(k). We then define the exploration process by:
$S(0)=0,\quad S(k+1)=S(k)+c(k)-1.$
By far the best way to think of this is to imagine we are making the depth-first walk on the tree. S(k) records how many vertices we have seen (because they are connected by an edge to a vertex we have visited) but have not yet visited. To clarify understanding of the definition, note that when you arrive at a vertex with no children, this should decrease by one, as you can see no new vertices, but have visited an extra one.
This exploration process is useful to consider for a couple of reasons. Firstly, you can reconstruct the branching process directly from it. Secondly, while other functionals (eg the height, or contour process) look like random walks, the exploration process genuinely is a random walk. The distribution of the number of children of the next vertex we arrive at is independent of everything we have previously seen in the tree, and is the same for every vertex. If we were looking at branching processes in a different context, we might observe that this gives some information in a suitably-rescaled limit, as rescaled random walks converge to Brownian motion if the variance of the (offspring) distribution is finite. (This is Donsker’s result, which I should write something about soon…)
The most important property is that the exploration process returns to 0 precisely when we have exhausted all the vertices in a component. At that point, we have seen exactly the vertices which we have explored. There is no reason not to extend the definition to forests, that is a union of trees. The depth-first exploration is the same – but when we have exhausted one component, we move onto another component, chosen according to some labelling property. Then, running minima of the exploration process (ie times when it is smaller than it has been before) correspond to jumping between components, and thus excursions above the minimum to components themselves. The running minimum will be non-positive, with absolute value equal to the number of components already exhausted.
Although the exploration process was defined with reference to and in the language of trees, the result of a branching process, this is not necessary. With some vertex denoted as the root, we can construct a depth-first labelling of a general graph, and the exploration process follows exactly as before. Note that we end up ignoring all edges except a set that forms a forest. This is what we will apply to G(n,p).
Exploring G(n,p)
When we jump between components in the exploration process on a supercritical (that is $\lambda>1$) random graph, we move to a component chosen randomly with size-biased distribution. If there is a giant component, as we know there is in the supercritical case, then this will dominate the size-biased distribution. Precisely, if the giant component takes up a fraction H of the vertices, then the number of components to be explored before we get to the giant component is geometrically distributed with parameter H. All other components have size O(log n), so the expected number of vertices explored before we get to the giant component is O(log n)/H = o(n), and so in the limit, we explore the giant component immediately.
The exploration process therefore gives good control on the giant component in the limit, as roughly speaking the first time it returns to 0 is the size of the giant component. Fortunately, we can also control the distribution of S_t, the exploration process at time t. We have that:
$S_t+(t-1)\sim \text{Bi}(n-1,1-(1-p)^t).$
This is not too hard to see. $S_t+(t-1)$ is number of vertices we have explored or seen, ie are connected to a vertex we have explored. Suppose the remaining vertices are called unseen, and we began the exploration at vertex 1. Then any vertex with label in {2,…,n} is unseen if it successively avoids being in the neighbourhood of v(1), v(2), … v(t). This happens with probability $(1-p)^t$, and so the probability of being an explored or seen vertex is the complement of this.
In the supercritical case, we are taking $p=\frac{\lambda}{n}$ with $\lambda>1$, and we also want to speed up S, so that all the exploration processes are defined on [0,1], and rescale the sizes by n, so that it records the fraction of the graph rather than the number of vertices. So we set consider the rescaling $\frac{1}{n}S_{nt}$.
It is straightforward to use the distribution of S_t we deduce that the asymptotic mean $\mathbb{E}\frac{1}{n}S_{nt}=\mu_t = 1-t-e^{-\lambda t}$.
Now we are in a position to provide more concrete motivation for the claim that the proportion of vertices in the giant component is $\zeta_\lambda$, the survival probability of a branching process with $\text{Po}(\lambda)$ offspring distribution. It helps to consider instead the extinction probability $1-\zeta_\lambda$. We have:
$1-\zeta_\lambda=\sum_{k\geq 0}\mathbb{P}(\text{Po}(\lambda)=k)(1-\zeta_\lambda)^k=e^{-\lambda\zeta_\lambda},$
where the second equality is a consequence of the simple form for the moment generating function of the Poisson distribution.
As a result, we have that $\mu_{\zeta_\lambda}=0$. In fact we also have a central limit theorem for S_t, which enables us to deduce that $\frac{1}{n}S_{n\zeta_\lambda}=0$ with high probability, as well as in expectation, which is precisely what is required to prove that the giant component of $G(n,\frac{\lambda}{n})$ has size $n(\zeta_\lambda+o(1))$.
# Branching Processes and Dwass’s Theorem
This is something I had to think about when writing my Part III essay, and it turns out to be relevant to some of the literature I’ve been reading this week. The main result is hugely helpful for reducing a potentially complicated combinatorial object to a finite sum of i.i.d. random variables, which in general we do know quite a lot about. I was very pleased with the proof I came up with while writing the essay, even if in the end it turned out to have appeared elsewhere before. (Citation at end)
Galton-Watson processes
A Galton-Watson process is a stochastic process describing a simple model for evolution of a population. At each stage of the evolution, a new generation is created as every member of the current generation produces some number of `offspring’ with identical and independent (both across all generations and within generations) distributions. Such processes were introduced by Galton and Watson to examine the evolution of surnames through history.
More precisely, we specify an offspring distribution, a probability distribution supported on $\mathbb{N}_0$. Then define a sequence of random variables $(Z_n,n\in\mathbb{N})$ by:
$Z_{n+1}=Y_1^n+\ldots+Y_{Z_n}^n,$
where $(Y_k^n,k\geq 1,n\geq 0)$ is a family of i.i.d. random variables with the offspring distribution $Y$. We say $Z_n$ is the size of the $n$th generation. From now on, assume $Z_0=1$ and then we call $(Z_n,n\geq 0)$ a Galton-Watson process. We also define the total population size to be
$X:=Z_0+Z_1+Z_2+\ldots,$
noting that this might be infinite. We refer to the situation where $X<\infty$ finite as extinction, and can show that extinction occurs almost surely when $\mathbb{E}Y\leq 1$, excepting the trivial case $Y=\delta_1$. The strict inequality parts are as you would expect. We say the process is critical if $\mathbb{E}Y=1$, and this is less obvious to visualise, but works equally well in the proof, which is usually driven using generating functions.
Total Population Size and Dwass’s Theorem
Of particular interest is $X$, the total population size, and its distribution. The following result gives us a precise and useful result linking the probability of the population having size $n$ and the distribution of the sum of $n$ RVs with the relevant offspring distribution. Among the consequences are that we can conclude immediately, by CLT and Cramer’s Large Deviations Theorem, that the total population size distribution has power-law decay in the critical case, and exponential decay otherwise.
Theorem (Dwass (1)): For a general branching process with a single time-0 ancestor and offspring distribution $Y$ and total population size $X$:
$\mathbb{P}(X=k)=\frac{1}{k}\mathbb{P}(Y^1+\ldots+ Y^k=k-1),\quad k\geq 1$
where $Y^1,\ldots,Y^k$ are independent copies of $Y$.
We now give a proof via a combinatorial argument. The approach is similar to that given in (2). Much of the literature gives a proof using generating functions.
Proof: For motivation, consider the following. It is natural to consider a branching process as a tree, with the time-0 ancestor as the root. Suppose the event $\{X=k\}$ in holds, which means that the tree has $k$ vertices. Now consider the numbers of offspring of each vertex in the tree. Since every vertex except the root has exactly one parent, and there are no vertices outside the tree, we must have $Y^1+\ldots+Y^k=k-1$ where $Y^1,\ldots,Y^k$ are the offspring numbers in some order. However, observe that this is not sufficient. For example, if $Y^1$ is the number of offspring of the root, and $k\geq 2$, then we must have $Y^1\geq 1$. Continue reading | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 130, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.923873245716095, "perplexity": 260.9392042215778}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676596204.93/warc/CC-MAIN-20180723090751-20180723110751-00192.warc.gz"} |
https://southgalwayfrs.ie/project-info/environmental-assessment/ | # Environmental Assessment
The Environmental Consultant, Mott MacDonald will produce the necessary Environmental Assessments for the scheme as outlined below.
Provided an environmentally acceptable and economically feasible scheme is identified, an Environmental Impact Assessment Report (EIAR) will be prepared for the scheme. The EIAR shall be prepared to meet the requirements set out by Directive 2014/52/EU and the Environmental Protection Agency (EPA) in the ‘Draft Guidelines on the Information to be contained in Environmental Impact Assessment Reports’ (EPA, 2017).
The purpose of the Environmental Impact Assessment Report (EIAR) will be to document the environment in the vicinity of the proposed scheme in an effort to quantify the possible impacts, if any on the environment. The assessment process will serve to highlight areas where mitigation measures may be necessary in order to protect the surrounding environment from any negative impacts of the proposed scheme. The objective is to facilitate the most efficient and positive design of the proposed scheme insofar as possible and that measures are in place to ensure that any adverse impacts are avoided, reduced or remedied as appropriate.
The Environmental Consultant will also prepare an Appropriate Assessment Screening Report. The AA Screening will consider the likely impacts of the preferred Scheme on relevant Natura Sites. Should significant effects be identified, a Natura Impact Statement will be produced. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8350549340248108, "perplexity": 1684.4210089708326}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943589.10/warc/CC-MAIN-20230321002050-20230321032050-00707.warc.gz"} |
https://www.gradesaver.com/textbooks/math/calculus/thomas-calculus-13th-edition/chapter-14-partial-derivatives-section-14-2-limits-and-continuity-in-higher-dimensions-exercises-14-2-page-796/21 | ## Thomas' Calculus 13th Edition
$1$
Here, we have $P(x,y) \to O(0,0)$ This implies that $D(P,O) \to 0$ Therefore, $D^2=x^2+y^2; D \to 0$ As we can see that $\lim\limits_{D \to 0}\dfrac{\sin D^2}{D^2}=\dfrac{0}{0}$. This shows that the limit of Indeterminate form thus, we will have to apply L-Hospital's rule Plug the limits, then we have $\lim\limits_{D \to 0}\dfrac{2 D\cos D^2}{2D}=\cos (0)=1$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.993532657623291, "perplexity": 101.08661978146186}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540519149.79/warc/CC-MAIN-20191209145254-20191209173254-00160.warc.gz"} |
https://www.physicsforums.com/threads/1st-order-ivp.584802/ | 1st order ivp
1. Mar 7, 2012
Ry122
y' = x
x' = -5y-4x
y(0) = 1
x(0) = 0
after finding the general solution as shown here
http://www.wolframalpha.com/input/?i=y'+=+x,+x'+=+-5y-4x
how do you go about applying the initial values and finding the complete solution?
2. Mar 7, 2012
HallsofIvy
Staff Emeritus
Your general solution should have two undetermined coefficients. Substitute 0 for t, set x= 0, y= 1 and you will have two equations to solve for the two coefficients.
3. Mar 7, 2012
Ry122
actually I don't think wolfram alpha has done the correct thing in making x and y a function of t as there's no mention of another variable in the original equations. What would you do when they aren't functions of t?
4. Mar 8, 2012
HallsofIvy
Staff Emeritus
You can call the independent variable whatever you want! What did you mean by x' and y'? I assumed the primes were derivatives. With respect to what variable?
Similar Discussions: 1st order ivp | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8383994698524475, "perplexity": 1144.484903514322}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823482.25/warc/CC-MAIN-20171019231858-20171020011858-00325.warc.gz"} |
https://conference.ippp.dur.ac.uk/event/470/contributions/2532/ | # The 34th International Symposium on Lattice Field Theory (Lattice 2016)
24-30 July 2016
Highfield Campus, University of Southampton
Europe/London timezone
## Spontaneous symmetry breaking induced by complex fermion determinant --- yet another success of the complex Langevin method
26 Jul 2016, 15:40
20m
Building 32 Room 1015 (Highfield Campus, University of Southampton)
### Building 32 Room 1015
#### Highfield Campus, University of Southampton
Talk Nonzero Temperature and Density
### Speaker
Dr Yuta Ito (KEK)
### Description
In many interesting systems, the fermion determinant becomes complex and its phase plays a crucial role in the determination of the vacuum. For instance, in finite density QCD at low temperature and high density, exotic fermion condensates are conjectured to form due to such effects. When one applies the complex Langevin method to such a complex action system naively, one cannot obtain the correct results because of the singular-drift problem associated with the appearance of small eigenvalues of the Dirac operator. Here we propose to add a fermion bilinear term to the action to avoid this problem and extrapolate its coefficient to zero. We test this idea in an SO(4)-invariant matrix model with a Gaussian action and a complex fermion determinant, whose phase is expected to induce the spontaneous breaking of the SO(4) symmetry. Our results agree well with the previous results obtained by the Gaussian expansion method.
### Primary author
Dr Yuta Ito (KEK)
### Co-author
Prof. Jun Nishimura (KEK,SOKENDAI)
Slides | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9176946878433228, "perplexity": 1760.5194174264711}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370496669.0/warc/CC-MAIN-20200330054217-20200330084217-00059.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/214972-eigenvalues-eigenvectors.html | # Math Help - Eigenvalues and Eigenvectors
1. ## Eigenvalues and Eigenvectors
Let L be the line through the origin of R2 that makes an angle of ∏/4 with the positive x-axis, and let A be the standard matrix for the reflection of R2 about that line. Make a conjecture
about the eigenvalues and eigenvectors of A and confirm your conjecture by computing them in the usual way.
2. ## Re: Eigenvalues and Eigenvectors
Seeing that I know what is going to happen, it's best for you to make the conjecture yourself. Draw a picture and see what happens to arbitrary vectors. Are any vectors invariant under the reflection (i.e. do they stay the same)? Are any vectors scaled, but otherwise lie on the same line spanned by itself?
3. ## Re: Eigenvalues and Eigenvectors
I think the eigenvalues of A are λ=1 and λ=-1 with corresponding eigenspaces?
4. ## Re: Eigenvalues and Eigenvectors
Yes, sounds about right. Nevertheless, you should check it as suggested by your question. Do you know what the matrix for A is? Hint: the matrix depends entirely on where you take the basis vectors (0,1) and (1,0). | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9552090764045715, "perplexity": 596.4825518170201}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413507443438.42/warc/CC-MAIN-20141017005723-00053-ip-10-16-133-185.ec2.internal.warc.gz"} |
http://mathhelpforum.com/differential-geometry/114740-solved-limit-function-exists-but-limit-its-derivative-doesn-t-2.html | # Math Help - [SOLVED] limit of function exists, but limit of its derivative doesn't.
1. Originally Posted by redsoxfan325
$\lim_{x\to\infty}\frac{-1}{x}\leq\lim_{x\to\infty}\frac{\sin(x^2)}{x}\leq\ lim_{x\to\infty}\frac{1}{x}$
So the Squeeze Theorem says $\lim_{x\to\infty}\frac{\sin(x^2)}{x}=0$
The derivative of $\frac{\sin x}{x^2}$ is $\frac{\cos x}{x^2}-\frac{2\sin x}{x^3}$, which clearly goes to $0$ as $x\to\infty$.
This is a mess and everyone understands what she/he wants : I meant that in my example $\frac{sin x}{x}$ I forgot to square the denominator when differentiating and thus got a wrong derivative.
Tonio
2. Originally Posted by tonio
This is a mess and everyone understands what she/he wants : I meant that in my example $\frac{sin x}{x}$ I forgot to square the denominator when differentiating and thus got a wrong derivative.
Tonio
Oh, ok. no prob.
Page 2 of 2 First 12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.995052695274353, "perplexity": 634.5726593556806}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500830323.35/warc/CC-MAIN-20140820021350-00109-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/orbital-path-of-the-sun.847736/ | # Orbital path of the sun
1. Dec 11, 2015
### whatisreality
I've written a java program to model the solar system. All my planets move in very well defined, stable orbits. The sun, on the other hand, is doing some very weird stuff. I've attached a plot of its path, a very zoomed in and a very zoomed out one. It isn't spiralling gradually inwards or outwards, as far as I can tell it's going in weird, wobbly loops. How do I explain that?! Especially as the rest of the planets work so well??
In my model, all bodies are point masses and have circular orbits, and they orbit around their common centre of mass.
#### Attached Files:
File size:
72.7 KB
Views:
59
• ###### sun orbit.jpg
File size:
43.9 KB
Views:
54
2. Dec 11, 2015
### whatisreality
All good! Adding jupiter fixed it.
Similar Discussions: Orbital path of the sun | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8052768707275391, "perplexity": 2528.1200850428836}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084886939.10/warc/CC-MAIN-20180117122304-20180117142304-00796.warc.gz"} |
https://www.physicsforums.com/threads/trig-substituion-integral.218154/ | # Trig Substituion Integral
1. Feb 26, 2008
### kuahji
Evaluate
$$\int1/sqrt(4x^2-49)$$ for x>7/2
Where I get lost really, is why do I set x = 7/2 sec u? The textbook just shows a generic formula where you always set x=a sec u. The only thing I could see is that anything less than 7/2 yiels a negative under the square root. But then again, this goes against the little formula which isn't really a problem, but take this other integral for example
$$\int8dx/(4x^2+1)^2$$ here it shows setting x=1/2 tan u. But here I'm not really understanding the reason why. I'm guessing its because I'm not really sure why I set x equal to say tanget, sine, or whatever else.
Last edited: Feb 26, 2008
2. Feb 26, 2008
### rock.freak667
If you take 2x=7sec$\theta$ then:
$$2 dx=7 sec\theta tan\theta \theta$$
$$\int \frac{1}{\sqrt{4x^2-49}} dx$$
$$\equiv \int \frac{\frac{7}{2}sec \theta tan \theta}{\sqrt{49sec^2 \theta-49}$$
then use $sec^2\theta-1=tan^2 \theta$
for the 2nd integral. they use tan because when you substitute x=1/2 tanu then the denomination becomes
(4(1/2 tanu)^2 +1)^2 = (tan^2u+1)^2 and well tan^u+1=sec^2 u
so it becomes sec^4u.
But basically what you want to do is make a substitution where the denominator (after substitution) can be used as another trig identity. e.g sin^2u+cos^2=1 etc.
(EDIT: Not sure if my LaTex is showing up correctly[My browser is showing LaTex from questions I typed out many days ago!] so I don't know if you will understand what I wanted to say)
Last edited: Feb 26, 2008
3. Feb 26, 2008
### kuahji
Yes, it makes sense now. Its usually the little stuff that gets me all confused ^_^. If you take 2x=7sec is what I wasn't getting.
Similar Discussions: Trig Substituion Integral | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9886487722396851, "perplexity": 2074.1368257380013}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886133032.51/warc/CC-MAIN-20170824043524-20170824063524-00069.warc.gz"} |
https://tug.org/pipermail/pstricks/2008/005117.html | Jonny Smith joesm77 at gmail.com
Fri Jan 4 14:18:58 CET 2008
```Hi,
I'm wondering if one can put on several single omni spotlights as in
That is, I'm thninking of photoshop where they have these omni spotlights,
see
and other fancy things.
I had an idea how to get this done. I thought one could overlap a few shapes
flowing from transparent into say white
but I'm not sure if this is going to work.
Or is there a way to tell pst-grad to do a bunch of gradients points at
once? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9128990769386292, "perplexity": 3236.447304099017}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514572517.50/warc/CC-MAIN-20190916100041-20190916122041-00323.warc.gz"} |
https://www.physicsforums.com/threads/proof-of-the-riemann-hypothesis.238947/ | # Proof of the Riemann Hypothesis
1. Jun 5, 2008
2. Jun 5, 2008
### Hurkyl
Staff Emeritus
... we may have proved the Riemann Hypothesis...
I confess if they aren't even convinced, I find it hard to be optimistic.
3. Jun 5, 2008
### Dragonfall
Crackpots don't doubt themselves, so this guy passes the crackpot test.
4. Jun 6, 2008
### elfboy
I wish I could understand it better even with many years of college math and wikipedia
5. Jun 6, 2008
### mhill
perhaps he tries to be humble... the positivity 'test' of the double integral given by Polya seems correct , perhaps an expert of number theory should take a look at this paper
6. Jun 6, 2008
### Count Iblis
Did Wiles also pass the Crackpot test when he announced he found the proof of Fermat's theorem, only to discover later that there was a fatal flaw in it [which he was able to fix later at the very moment when he was taking a final look to understand better why he had faled and why he would not be able to succeed (making it easier to put the matter to rest in his mind)].
Or does the Crackpot test itself pass the Crackpot test :rofl:
7. Jun 6, 2008
### ice109
you pass the crackpot test but fail the reading comprehension test. to pass the crackpot test is to not be a crackpot. you passed since you have doubts.
8. Jun 6, 2008
### Hurkyl
Staff Emeritus
I suppose this is a catch-22, for if they were convinced, I'd generally be even less optimistic.
9. Jun 6, 2008
### shalayka
When asked "what if" his general theory of relativity had been disproven by experiment, Albert Einstein replied: "Then I would feel sorry for the good Lord. The theory is correct."
Sounds pretty self-assured to me, but he was no crackpot.
10. Jul 2, 2008
### neutrino
11. Jul 2, 2008
### kts123
Good lord, that's a hell'a'va proof he has there. I wish I understood the half of it (I barely understand the zeta function by itself...) What do you fella's make of it?
12. Jul 3, 2008
### Gib Z
Just a gut feeling that none of them are even close. Of course I know nothing either.
13. Jul 3, 2008
### neutrino
14. Jul 3, 2008
### Dragonfall
Well, far be it for us to question a Fields medalist.
EDIT: Ok I applaud the effort of this auto-keyword-link thing, but this has gone too far!
15. Jul 3, 2008
### CRGreathouse
Heh, yeah. Maybe we can have a list of stop-phrases?
16. Jul 6, 2008 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.804819643497467, "perplexity": 3917.0715104047463}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501171418.79/warc/CC-MAIN-20170219104611-00308-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://www.varsitytutors.com/algebra_1-help/how-to-find-the-solution-to-an-inequality-with-division?page=1 | # Algebra 1 : How to find the solution to an inequality with division
## Example Questions
← Previous 1 3 4
### Example Question #1 : How To Find The Solution To An Inequality With Division
Solve for :
Explanation:
To solve for , separate the integers and 's by adding 1 and subtracting from both sides to get . Then, divide both sides by 2 to get . Since you didn't divide by a negative number, the sign does not need to be reversed.
### Example Question #1 : How To Find The Solution To An Inequality With Division
Solve the following:
Explanation:
Don't forget to change the direction of the inequality sign when dividing by a negative number!
### Example Question #1 : How To Find The Solution To An Inequality With Division
Give the solution set of the inequality:
The set of all real numbers
Explanation:
Note change in direction of the inequality symbol when the expressions are divided by a negative number.
or, in interval form,
### Example Question #1 : How To Find The Solution To An Inequality With Division
Give the solution set of the inequality:
The inequality has no solution.
Explanation:
Note change in direction of the inequality symbol when the expressions are divided by a negative number.
or, in interval form,
### Example Question #1 : How To Find The Solution To An Inequality With Division
Give the solution set of the inequality:
The inequality has no solution.
Explanation:
Note change in direction of the inequality symbol when the expressions are divided by a negative number.
or, in interval form,
### Example Question #1 : How To Find The Solution To An Inequality With Division
Give the solution set of the inequality:
The set of all real numbers
Explanation:
Note change in direction of the inequality symbol when the expressions are divided by a negative number.
or, in interval form,
### Example Question #1 : How To Find The Solution To An Inequality With Division
Give the solution set of the inequality:
The set of all real numbers
Explanation:
Note change in direction of the inequality symbol when the expressions are divided by a negative number.
or, in interval form,
### Example Question #2 : How To Find The Solution To An Inequality With Division
Solve for :
Explanation:
First, add and subtract from both sides of the inequality to get .
Then, divide both sides by and reverse the sign since you are dividing by a negative number.
This gives you .
### Example Question #2 : How To Find The Solution To An Inequality With Division
Find the solution set to the following compound inequality statement:
Explanation:
Solve each of these two inequalities separately:
, or, in interval form,
, or, in interval form,
The two inequalities are connected with an "and", so we take the intersection of the two intervals.
Solve for : | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8992888927459717, "perplexity": 485.05123748567166}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875147116.85/warc/CC-MAIN-20200228073640-20200228103640-00380.warc.gz"} |
http://projecteuclid.org/euclid.rae/1256835201 | ## Real Analysis Exchange
### Divergence in Measure of Rearranged Multiple Orthononal Fourier Series
#### Abstract
Let $\{\varphi_n(x)$, $n=1,2,\dots\}$ be an arbitrary complete orthonormal system (ONS) on the interval $I:=[0,1)$ that consists of a.e. bounded functions. Then there exists a rearrangement $\{ \varphi_{\sigma_1(n)}$, $n=1,2, \dots\}$ of the system $\{\varphi_n(x)$, $n=1,2,\dots\}$ that has the following property: for arbitrary nonnegative, continuous and nondecreasing on $[0,\infty)$ function $\phi(u)$ such that $u\phi (u)$ is a convex function on $[0,\infty)$ and $\phi (u) = o(\ln u)$, $u \to \infty$, there exists a function $f \in L(I^2)$ such that $\int_{I^2} | f(x,y) |$ $\phi( | f(x,y) | )\;dx\; dy \infty$ and the sequence of the square partial sums of the Fourier series of $f$ with respect to the double system $\{ \varphi_{\sigma_1 (m)}(x)\varphi_{\sigma_1 (n)}(y)$, $m,n \in\N \}$ on $I^2$ is essentially unbounded in measure on $I^2$.
#### Article information
Source
Real Anal. Exchange Volume 34, Number 2 (2008), 501-520.
Dates
First available: 29 October 2009
http://projecteuclid.org/euclid.rae/1256835201
Mathematical Reviews number (MathSciNet)
MR2569201
#### Citation
Getsadze, Rostom. Divergence in Measure of Rearranged Multiple Orthononal Fourier Series. Real Analysis Exchange 34 (2008), no. 2, 501--520. http://projecteuclid.org/euclid.rae/1256835201.
#### References
• A. M. Garsia, Topics in Almost Everywhere Convergence, Lectures in Advanced Mathematics, 4, Markham Publishing Co., Chicago, 1970.
• G. A. Karagulyan, Divergence of double Fourier series in complete orthonormal systems, (Russian) Izv. Akad. Nauk Armyan. SSR Ser. Mat., 24(2) (1989), 147–159, 200; translation in Soviet J. Contemp. Math. Anal., 24(2) (1989), no. 2, 44–56.
• B. S. Kashin, A. A. Saakyan, Orthogonal Series, Translated from the Russian by Ralph P. Boas, Translation edited by Ben Silver, Translations of Mathematical Monographs, 75, American Mathematical Society, Providence, RI, 1989.
• A. M. Olevskiĭ, Fourier Series with Respect to General Orthogonal Systems, Translated from the Russian by B. P. Marshall and H. J. Christoffers, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band, 86, Springer-Verlag, New York-Heidelberg, 1975.
• A. M. Olevskiĭ, Divergent series for complete systems in $L\sp{2}$, (Russian) Dokl. Akad. Nauk SSSR, 138 (1961), 545–548. English translation: Soviet Math. Dokl., 2(6) (1961), 669–672.
• A. M. Olevskiĭ, Divergent Fourier series, Izv. Akad. Nauk SSSR Ser. Mat., 27 (1963), 343–366.
• G. E. Tkebuchava, Divergence of multiple Fourier series with respect to bases, (English translation) Soviet Math. Dokl., 40(2) (1990), 346–348. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9174163937568665, "perplexity": 854.5417402694646}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997877693.48/warc/CC-MAIN-20140722025757-00209-ip-10-33-131-23.ec2.internal.warc.gz"} |
http://www.physicsforums.com/showthread.php?t=241326 | # electric field and magnetic field - proton deflection
by scholio
Tags: deflection, electric, field, magnetic, proton
P: 160 1. The problem statement, all variables and given/known data when protons travelling north in a horizontal plane enter a region of uniform magnetic field of 0.8Teslas in the downward direction, they are deflected into a horizontal circle of radius 0.2 meters. what is the magnitude and direction of a uniform electric field applied over the same region of space that will allow the protons to pass through the region undelflected 2. Relevant equations radius r = mv/qB where m is mass, v is velocity, q is charge, B is magnetic field electromagnetic force F = qE + qv X B where X indicates cross product electric field force F_E = qE where E is electric field magnetic force F_B = qv X B where B is magnetic field charge of electron/proton = 1.6*10^-19 coulombs mass proton m = 1.67*10^-27 kg 3. The attempt at a solution i used : since the proton must not be deflected, i assumed electric field force must equal magnetic force so: qE = qv X B i then used radius eq, r = mv/qB and solved for v --> v = qBr/m and subbed it in for v to get: qE = q(qBr/m) X B and solved for E --> E = (qBr/m)Bsin(theta), i have q = 1.6*10^-19 coulombs, B = 0.8 Teslas, r = 0.2 meters, mass m = 1.67*10^-27 kg but what is theta? is my approach correct? cheers
HW Helper P: 2,688 Your approach for finding the magnitude of the E field is correct. The direction of the B-field is given in the problem, though it is worded in a somewhat confusing way. They tell you that the protons are moving north and that the B field is pointing in the "downward" direction. I believe this means "into the plane of the page." Considering this to be the case, what is your angle?
P: 160 which page? the page the problem is written on? in that case, the angle would be 90 degrees. correct?
P: 1,133
## electric field and magnetic field - proton deflection
If you were to take a path northward over the Earth's surface, southward would be the direction opposite of northward (which is antiparallel to the motion and so the proton would be unaffected), and downward would be the direction into the Earth (which is perpendicular to the motion and is what the question is probably referring to---so you're correct, angle is 90 degrees)...heh, question was worded kind of confusingly. There is a trick to determining the angle (if I'm right about this): the particle will move in a circle if the field is completely 90 degrees perpendicular to the motion...otherwise the path is helical.
HW Helper
P: 2,688
Quote by scholio which page? the page the problem is written on? in that case, the angle would be 90 degrees. correct?
Yup. You got it.
P: 160 thanks, so using theta = 90 degrees and E = (qBr/m)Bsin(theta), i have q = 1.6*10^-19 coulombs, B = 0.8 Teslas, r = 0.2 meters, mass m = 1.67*10^-27 kg i got electric field E = 1.23*10^7 coulombs/meter
HW Helper P: 2,688 Looks good. Good job!
Related Discussions Introductory Physics Homework 9 Introductory Physics Homework 0 Advanced Physics Homework 5 Introductory Physics Homework 5 Introductory Physics Homework 9 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9290519952774048, "perplexity": 956.6761705962945}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00060-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://www.ms.u-tokyo.ac.jp/seminar/topology/future.html | トポロジー火曜セミナー
開催情報 火曜日 17:00～18:30 数理科学研究科棟(駒場) 056号室 河野 俊丈, 河澄 響矢, 北山 貴裕, 逆井卓也 http://faculty.ms.u-tokyo.ac.jp/~topology/index.html Tea: 16:30 - 17:00 コモンルーム
2018年10月23日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
François Fillastre 氏 (Université de Cergy-Pontoise)
Co-Minkowski space and hyperbolic surfaces (ENGLISH)
[ 講演概要 ]
There are many ways to parametrize two copies of Teichmueller space by constant curvature -1 Riemannian or Lorentzian 3d manifolds (for example the Bers double uniformization theorem). We present the co-Minkowski space (or half-pipe space), which is a constant curvature -1 degenerated 3d space, and which is related to the tangent space of Teichmueller space. As an illustration, we give a new proof of a theorem of Thurston saying that, once the space of measured geodesic laminations on a compact hyperbolic surface is identified with the tangent space of Teichmueller space via infinitesimal earthquake, then the length of laminations is an asymmetric norm. Joint work with Thierry Barbot (Avignon).
2018年10月30日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
The quasiconformal equivalence of Riemann surfaces and a universality of Schottky spaces (JAPANESE)
[ 講演概要 ]
In the theory of Teichmüller space of Riemann surfaces, we consider the set of Riemann surfaces which are quasiconformally equivalent. For topologically finite Riemann surfaces, it is quite easy to examine if they are quasiconformally equivalent or not. On the other hand, for Riemann surfaces of topologically infinite type, the situation is rather complicated.
In this talk, after constructing an example which shows the complexity of the problem, we give some geometric conditions for Riemann surfaces to be quasiconformally equivalent. Our argument enables us to see a universality of Schottky spaces.
2018年11月06日(火)
17:30-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 17:00-17:30
Coarsely convex spaces and a coarse Cartan-Hadamard theorem (JAPANESE)
[ 講演概要 ]
A coarse version of negatively-curved spaces have been very well studied as Gromov hyperbolic spaces. Recently we introduced a coarse version of non-positively curved spaces, named them coarsely convex spaces and showed a coarse version of the Cartan-Hadamard theorem for such spaces in a joint-work with Tomohiro Fukaya (arXiv:1705.05588). Based on the work, I introduce coarsely convex spaces and explain a coarse Cartan-Hadamard theorem, ideas for proof and its applications to differential topology.
2018年11月08日(木)
10:30-12:00 数理科学研究科棟(駒場) 056号室
Michael Heusener 氏 (Université Clermont Auvergne)
Deformations of diagonal representations of knot groups into $\mathrm{SL}(n,\mathbb{C})$ (ENGLISH)
[ 講演概要 ]
This is joint work with Leila Ben Abdelghani, Monastir (Tunisia).
Given a manifold $M$, the variety of representations of $\pi_1(M)$ into $\mathrm{SL}(2,\mathbb{C})$ and the variety of characters of such representations both contain information of the topology of $M$. Since the foundational work of W.P. Thurston and Culler & Shalen, the varieties of $\mathrm{SL}(2,\mathbb{C})$-characters have been extensively studied. This is specially interesting for $3$-dimensional manifolds, where the fundamental group and the geometrical properties of the manifold are strongly related.
However, much less is known of the character varieties for other groups, notably for $\mathrm{SL}(n,\mathbb{C})$ with $n\geq 3$. The $\mathrm{SL}(n,\mathbb{C})$-character varieties for free groups have been studied by S. Lawton and P. Will, and the $\mathrm{SL}(3,\mathbb{C})$-character variety of torus knot groups has been determined by V. Munoz and J. Porti.
In this talk I will present some results concerning the deformations of diagonal representations of knot groups in basic notations and some recent results concerning the representation and character varieties of $3$-manifold groups and in particular knot groups. In particular, we are interested in the local structure of the $\mathrm{SL}(n,\mathbb{C})$-representation variety at the diagonal representation. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9011102318763733, "perplexity": 1011.2948215357885}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583515375.86/warc/CC-MAIN-20181022180558-20181022202058-00081.warc.gz"} |
http://math.stackexchange.com/questions/38592/give-an-example-of-a-sequence-of-uniformly-continuous-functions-on-mathbbr | # Give an example of a sequence of uniformly continuous functions on $\mathbb{R}$ that converge pointwise to a non-uniformly continuous function
Give an example of a sequence of uniformly continuous functions on $\mathbb{R}$ that converge pointwise to a non-uniformly continuous function.
My thoughts:
I'm trying to work backwards: by choosing a non-uniformly continuous function, but I can't find anything that works. Any help would be appreciated.
Thanks
-
If you don't care about the limit function being continuous, one of the simplest examples is the sequence of functions $f_{n}(x) = x^{n}$ on $[0,1]$. This sequence converges pointwise to the function which is zero for $x \lt 1$ and $1$ for $x = 1$. This is of course an example on $[0,1]$, but you get an example on $\mathbb{R}$ by extending all the functions by zero for $x \leq 0$ and by $1$ for $x \geq 1$.
Recall that a continuous function on a compact interval is automatically uniformly continuous (prove this, in case you don't know that statement!). To get an example of a non-uniformly continuous function, we need to look for a function on an unbounded interval.
A very simple example of a continuous but not uniformly continuous function is $f(x) = x^{k}$ on $[0,\infty)$ for $k \neq 0,1$. Now simply define $f_{n}(x) = f(x)$ if $0 \leq x \leq n$ and $f_{n}(x) = f(n)$ if $x \geq n$. You should be able to check yourself that each $f_{n}$ is uniformly continuous and that the sequence $f_{n}$ converges pointwise to $f$.
To extend this example to all of $\mathbb{R}$, simply extend the functions by zero to the left.
A very similar idea works for $e^{x}$. If you want a slightly more interesting example, you can try to tackle $f(x) = \sin{(e^x)}$ with the same idea of truncating and extending constantly to the left and right.
Added: In fact, the procedure I outlined is one that always works (there are other ways but this probably is the most straightforward one). More precisely, if $f: \mathbb{R} \to \mathbb{R}$ is continuous, put $$f_{n}(x) = \begin{cases} f(x), &\text{if } |x| \leq n,\\f(n), &\text{if }x \geq n,\\f(-n), &\text{if } x \leq -n\end{cases}$$ and check that $f_{n}$ is uniformly continuous. This is because $f_{n}$ is uniformly continuous inside the compact interval $[-(n+1),n+1]$ due to the fact I mentioned above and constant outside. It is easy to see that $f_{n}(x) \to f(x)$ for all $x$, so $f_{n} \to f$ pointwise.
So to get an example of the kind you're asking about, the only thing you really need to think about is how to find a continuous but not uniformly continuous function, and I've given a few examples that should illustrate the kinds of functions you should be looking at.
-
I know it's totally useless (and it obfuscates the obvious), but if $g$ is the characteristic function of $[0,1]$, then you can write $f_n(x) = \left(\int_0^x g \right)^n$ in your first example. For those who just love formulas... – Joel Cohen May 12 '11 at 17:27
@Joel: I had to chuckle at that. I don't know exactly how useless it is, though (think of distributions :)). But probably for the present exercise it's really really really point- and useless... Thanks! – t.b. May 12 '11 at 17:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9214590787887573, "perplexity": 72.72199057157408}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440644064019.39/warc/CC-MAIN-20150827025424-00096-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://www.groundai.com/project/efficient-formulation-of-full-configuration-interaction-quantum-monte-carlo-in-a-spin-eigenbasis-via-the-graphical-unitary-group-approach/1 | Efficient Formulation of Full Configuration Interaction Quantum Monte Carlo in a Spin Eigenbasis via the Graphical Unitary Group Approach
# Efficient Formulation of Full Configuration Interaction Quantum Monte Carlo in a Spin Eigenbasis via the Graphical Unitary Group Approach
Werner Dobrautz Max Planck Institute for Solid State Research, Heisenbergstr. 1, 70569 Stuttgart, Germany Simon Smart European Centre for Medium-Range Weather Forecasts, Shinfield Rd, Reading RG2 9AX, United Kingdom Ali Alavi Max Planck Institute for Solid State Research, Heisenbergstr. 1, 70569 Stuttgart, Germany Dept of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
July 23, 2019
###### Abstract
We provide a spin-adapted formulation of the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) algorithm, based on the Graphical Unitary Group Approach (GUGA), which enables the exploitation of SU(2) symmetry within this stochastic framework. Random excitation generation and matrix element calculation on the Shavitt graph of GUGA can be efficiently implemented via a biasing procedure on the branching diagram. The use of a spin-pure basis explicitly resolves the different spin-sectors and ensures that the stochastically sampled wavefunction is an eigenfunction of the total spin operator . The method allows for the calculation of states with low or intermediate spin in systems dominated by Hund’s first rule, which are otherwise generally inaccessible. Furthermore, in systems with small spin gaps, the new methodology enables much more rapid convergence with respect to walker number and simulation time. Some illustrative applications of the GUGA-FCIQMC method are provided: computation of the spin gap of the cobalt atom in large basis sets, achieving chemical accuracy to experiment, and the , , , spin-gaps of the stretched N molecule, an archetypal strongly correlated system.
Quantum Monte Carlo, SU(2) symmetry
###### pacs:
02.70.Ss, 31.10.+z, 31.15.xh, 31.25.âv
## I Introduction
The concept of symmetry is of paramount importance in physics and chemistry. The exploitation of the inherent symmetries and corresponding conservation laws in electronic structure calculations not only reduces the degrees of freedom by block-diagonalization of the Hamiltonian into different symmetry sectors, but also ensures the conservation of “good” quantum numbers and thus the physical correctness of calculated quantities. It also allows to target a specific many-body subspace of the problem at hand. Commonly utilized symmetries in electronic structure calculations are discrete translational and point group symmetries, angular momentum and projected spin conservation.
Due to a non-straight-forward implementation and accompanying increased computational cost, one often ignored symmetry is the global spin-rotation symmetry of spin-preserving, nonrelativistic Hamiltonians, common to many molecular systems studied. This symmetry arises from the vanishing commutator
[^H,^S2]=0, (1)
and leads to a conservation of the total spin quantum number .
In addition to the above-mentioned Hilbert space size reduction and conservation of the total spin , solving for the eigenstates of in a simultaneous spin-eigenbasis of allows targeting distinct—even (near-)degenerate—spin eigenstates, which allows the calculation of spin gaps between states inaccessible otherwise, and facilitates a correct physical interpretation of calculations and description of chemical processes governed by the intricate interplay between them. Moreover, by working in a specific spin sector, convergence of projective techniques which rely on the repeated application of a propagator to an evolving wavefunction is greatly improved, especially where there are near spin-degeneracies in the exact spectrum.
The Full Configuration Interaction Quantum Monte Carlo (FCIQMC) approach Booth, Thom, and Alavi (2009); Cleland, Booth, and Alavi (2010) is one such methodology which can be expected to benefit from working in a spin-pure many-body basis. Formulated in Slater determinant (SD) Hilbert spaces, at the heart of the FCIQMC algorithm is excitation generation, in which from a given Slater determinant, another Slater determinant (a single or double excitation thereof) is randomly selected to be spawned on, with probability and sign determined by the corresponding Hamiltonian matrix element. Such individual determinant-to-determinant moves cannot, in general, preserve the total spin, which instead would require a collective move involving several SDs. Therefore, although the FCI wavefunction is a spin eigenvector, this global property of the wavefunction needs to emerge from the random sampling of the wavefunction, and is not guaranteed from step to step. Especially in systems in which the wavefunctions consist of determinants with many open-shell orbitals, this poses a very difficult challenge. If, instead, excitation generation between spin-pure entities could be ensured, this would immensely help in achieving convergence, especially in the aforementioned problems.
To benefit from the above mentioned advantages of a spin-eigenbasis, we present in this work the theoretical framework to efficiently formulate FCIQMC in a spin-adapted basis, via the mathematically elegant unitary group approach (UGA) and its graphical (GUGA) extension, and discuss the actual computational implementation in depth.
There are several other schemes to construct a basis of eigenfunctions, such as the Half-Projected Hartree-Fock (HPHF) functions Smeyers and Doreste-Suarez (1973); Helgaker, Jørgensen, and Olsen (2000), Rumer spin-paired spin eigenfunctions Rumer (1932); Weyl, Rumer, and Teller (1932); Simonetta, Gianinetti, and Vandoni (1968); Smart (2013); Reeves (1966), Kotani-Yamanouchi (KY) genealogical spin eigenfunctions Kotani and Amemiya (1955); Van Vleck and Sherman (1935); Pauncz (1979), Serber-type spin eigenfunctions, Serber (1934); Pauncz (1979); Salmon and Ruedenberg (1972), Löwdin spin-projected Slater determinants Löwdin (1955) and the Symmetric Group Approach Duch and Karwowski (1982); Ruedenberg (1971); Pauncz (1995)—closely related to the UGA—, which are widely used in electronic structure calculations. Some of these have partially been previously implemented in FCIQMC (HPHF, Rumer, KY and Serber)—but with severe computational limitations. Booth and Alavi et. al. (2013); Booth et al. (2011); Booth, Smart, and Alavi (2014). The GUGA approach turns out to be quite well suited to the FCIQMC algorithm, and is able to alleviate many of the problems previously encountered.
Concerning other computational approaches in electronic structure theory, there is a spin-adapted version of the Density Matrix Renormalization Group algorithm McCulloch and Gulácsi (2002); Tatsuaki (2000); Zgid and Nooijen (2008); Sharma and Chan (2012); Li and Chan (2017), a symmetry-adapted cluster (SAC) approach in the coupled cluster (CC) theory Ohtsuka et al. (2007); Nakatsuji and Hirao (1978, 1977), where is conserved due to fully spin- and symmetry-adapted cluster operators and the projected CC method Qiu et al. (2017); Tsuchimochi and Ten-no (2019); He and Cremer (2000); Tsuchimochi and Ten-no (2018), where the spin-symmetry of a broken symmetry reference state is restored by a projection, similar to the Löwdin spin-projected Slater determinants Löwdin (1955).
The use of spin-eigenfunctions in the Columbus Lischka et al. (2001, 2011, 2017), Molcas Aquilante et al. (2016) and GAMESS software package Schmidt et al. (1993); Gordon and Schmidt (2005) packages rely on the graphical unitary group approach (GUGA), where the CI method in GAMESS is based on the loop-driven GUGA implementation of Brooks and Schaefer Brooks and Schaefer (1979); Brooks et al. (1980).
Based on the GUGA introduced by Shavitt Shavitt (1977, 1978), Shepard et al. Shepard and Simons (1980); Lischka et al. (1981) made extensive use of the graphical representation of spin eigenfunctions in form of Shavitt’s distinct row table (DRT). In the multifacet graphically contracted method Shepard (2005, 2006); Gidofalvi and Shepard (2009); Öhrn et al. (2010); Shepard, Gidofalvi, and Brozell (2014a, b); Gidofalvi, Brozell, and Shepard (2014) the ground state wavefunction is formulated nonlinearly based on the DRT, conserving the total spin .
In this paper, we begin by reviewing the GUGA approach, concentrating on those aspects of the formalism that are especially relevant to the FCIQMC method, including the concept of branching diagrams in excitation generation. We then present a brief overview of the FCIQMC algorithm in the context of the GUGA method, including a discussion of optimal excitation generation and control of the time step. Next we provide application of this methodology to spin-gaps of the N atom, the N molecule and the cobalt atom, which illustrate several aspects of the GUGA formulation. In Sec. IX we conclude our findings and give an outlook to future applications and possible extensions or our implementation.
## Ii The (Graphical) Unitary Group Approach
In this section we discuss the use of the Unitary Group Approach (UGA) Paldus (1974) to formulate the FCIQMC method in spin eigenfunctions. The UGA is used to construct a spin-adapted basis—also known as configuration state functions (CSFs)—, which allows to preserve the total spin quantum number in FCIQMC calculations. With the help of the Graphical Unitary Group Approach (GUGA), introduced by Shavitt Shavitt (1977), an efficient calculation of matrix elements entirely in the space of CSFs is possible, without the necessity to transform to a Slater determinant (SD) basis. The GUGA additionally allows effective excitation generation, the cornerstone of the FCIQMC method, without reference to a non spin-pure basis and the need of storage of auxiliary information.
In this work we concern ourselves exclusively with spin-preserving, nonrelativistic Hamiltonians in the Born-Oppenheimer approximation Born and Oppenheimer (1927) in a finite basis set. The basis of the unitary group approach (UGA), which goes back to Moshinsky Moshinsky (1968), is the spin-free formulation of the spin-independent, non-relativistic, electronic Hamiltonian in the Born-Oppenheimer approximation, given as
^H=n∑ijtij∑σ=↑,↓a†iσajσ+12n∑ijkl⟨ik|r−112|jl⟩∑σ,τ=↑,↓a†iσa†kτalτajσ, (2)
where .With the reformulation
a†iσa†kτalτajσ=a†iσajσa†kτalτ−δjkδστa†iσalσ,
we can define
∑σa†iσajσ=^Eij (3)
and
∑στa†iσa†kτalτajσ=^Eij^Ekl−δjk^Eil=^eij,kl. (4)
as the singlet one- and two-body excitation operators Helgaker, Jørgensen, and Olsen (2000), which do not change and upon acting on a state, , with definite total and z-projection value of the spin. With Eqs. (3) and (4) the Hamiltonian (2) can be expressed in terms of these spin-free excitation operators as Matsen (1964)
^H=∑ijtij^Eij+12∑ij,klVij,kl^eij,kl. (5)
where . An elegant and efficient method to create a spin-adapted basis and calculate the Hamiltonian matrix elements in this basis is based on the important observation that the spin-free excitation operators (3) and (4) in the non-relativistic Hamiltonian (5) obey the same commutation relations as the generators of the Unitary Group Paldus (1974, 1975, 1976), being the number of spatial orbitals. The commutator of the spin-preserving excitation operators can be calculated as
[^Eij,^Ekl]= ∑στa†iσajσa†kτalτ−a†kτalτa†iσajσ = ∑στa†iσajσa†kτalτ−a†iσa†kτalτajσ−δila†kτajσ = ∑στa†iσajσa†kτalτ−a†iσajσa†kτalτ+δjka†iσalτ−δila†kτajσ [^Eij,^Ekl]= δjk^Eil−δil^Ekj, (6)
which is the same as for the basic matrix units and the generators of the unitary group .
The Unitary Group Approach (UGA) was pioneered by Moshinsky Moshinsky (1968), Paldus Paldus (1974) and Shavitt Shavitt (1977, 1978), who introduced the graphical-UGA (GUGA) for practical calculation of matrix elements. With the observation that the spin-free, nonrelativistic Hamiltonian (5) is expressed in terms of the generators of the unitary group, the use of a basis that is invariant and irreducible under the action of these generators is desirable. This approach to use dynamic symmetry to block-diagonalize the Hamiltonian is different to the case where the Hamiltonian commutes with a symmetry operator. In the UGA does not commute with the generators of , but rather is expressed in terms of them. Block diagonalization occurs, due to the use of an invariant and irreducible basis under the action of these generators. Hence, the UGA is an example of a spectrum generating algebra with dynamic symmetry Iachello (1993); Sonnad et al. (2016).
### ii.1 The Gel’fand-Tsetlin Basis
The Gel’fand-Tsetlin (GT) Gel’fand and Cetlin (1950a, b); Gel’fand (1950) basis is invariant and irreducible under the action of the generators of . The group has generators, , and a total of Casimir operators, commuting with all generators of the group, and the GT basis is based on the group chain
U(n)⊃U(n−1)⊃⋯⊃U(2)⊃U(1), (7)
where is Abelian and has one-dimensional irreducible representations (irreps). Each subgroup
has Casimir operators, resulting in a total of commuting operators, named Gel’fand invariants Gel’fand (1950). The simultaneous eigenfunctions of these invariants form the GT basis and are uniquely labeled by a set of integers related to the eigenvalues of the invariants. Thus, based on the branching law of Weyl Weyl (1931), a general -electron CSF can be represented by a Gel’fand pattern Gel’fand and Cetlin (1950a)
(8)
The integers in the top row (and all subsequent rows) of (8) are nonincreasing, , and the integers in the subsequent rows fulfill the condition
mi,j+1≥mij≥mi+1,j+1, (9)
called the “in-between” condition Louck (1970).
The non-increasing integers of the top row of Eq. (8), , are called the highest weight or weight vector of the representation and specify the chosen irrep of ; the following rows uniquely label the states belonging to the chosen irrep.
Let be the irreducible representation of , uniquely specified by the weight vector . Any representation of a group G yields a representation of any of its subgroups , , subduced by , . of subduced by is simply reducible Paldus (2006), due to the branching law of the unitary group Weyl (1946),
Γ{mn}↓U(n−1)=∑⊕Γ{mn−1}, (10)
where the direct sum extends over all irreps of for which the ”in-between” condition (9) holds and each irrep is contained once at most Louck (1970); Weyl, Rumer, and Teller (1932).
This fact and since is Abelian with one-dimensional irreps led Gel’fand and Tsetlin to the realization that the permissible highest weights of the subgroups in the chain (7) can be used to uniquely label the basis vectors of a general irrep space.
In CI calculations one usually employs a one-particle basis of spin-orbitals with creation and annihilation operators of electrons in spatial orbital with spin . The operators
^Aiσ,jτ=^a†iσ^ajτ;i,j=1,…,n;σ,τ=↑,↓ (11)
can be associated with the generators of with the commutation relation
[^Aiσ,jτ,^Ai′σ′,j′τ′]=δji′δτσ′^Aiσ,j′τ′−δij′δστ′^Ai′σ′,jτ. (12)
The partial sums over spin or orbital indices of these operators
^Eij=∑σ=↑,↓^Aiσ,jσand^Eστ=n∑i=1^Aiσ,iτ (13)
are related to the orbital and spin generators. (Superscript denotes the pure orbital space and the pure spin space.) Since we deal with fermions we have to restrict ourselves to the totally antisymmetric representations of , denoted as . Since the molecular Hamiltonian (5) is spin independent, we can consider the proper subgroup of the direct product of the spin-free orbital space , with generators , and the pure spin space with the four generators Paldus (1974), given as
U(2n)⊃Uo(n)⊗Us(2). (14)
With the total antisymmetric representation of , and and as the highest weights representing the irreps of and respectively, the subduced representation of contains only those representations of for which and are mutually conjugate Paldus (2006); Matsen (1974, 1964); Moshinsky (1968).
Plainly spoken, this means the irreps of and are related in a specific manner to obtain physically plausible states satisfying the Pauli exclusion principle and antisymmetry of fermionic wavefunctions. This is another aspect of the fact that in the totally antisymmetric wavefunction, the totally antisymmetric orbital part must be combined with the totally symmetric spin part and vice versa. E.g. an antisymmetric spin function forces a symmetric spatial function ( or ), yielding an antisymmetric singlet state. On the other hand, a symmetric spin function () is combined with an antisymmetric spatial function to yield the antisymmetric triplet states.
Moreover, since the Hamiltonian (5) is spin-independent, does not contribute to the matrix element evaluation, so we only have to concern ourselves with the irreps of the orbital subgroup, following Matsen’s spin-free approach Matsen (1974, 1964).
The consequence of the mutually conjugate relationship between and irreps for electronic structure calculations is that the integers in a Gel’fand pattern (8) for are related to occupation numbers of spatial orbitals. This means they are restricted to , due to the Pauli exclusion principle. The highest weight, , indicates the chosen electronic state with the conditions
n∑i=1min=Nand12n∑i=1δ1,min=S, (15)
with being the total number of electrons and the number of singly occupied orbitals, is equal to twice the total spin value .
### ii.2 The Paldus table
The restriction of in electronic Gel’fand patterns led Paldus Paldus (1974) to the more compact formulation by a table of integers. It is sufficient to count the appearances , and in each row of a Gel’fand pattern and store this information, denoted by and in a table, named a Paldus table.
The first column, , contains the number of doubly occupied orbitals, the second column, , the number of singly occupied and the last one, , the number of empty orbitals, as shown by the example of an state:
\footnotesize⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣221100002211000211000211002100110101⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦≡⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣aibici224223123122112021011010⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦→⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣ΔaiΔbiΔci0−011−000−010−101−110−100−010−10⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦, (16)
where the differences , with , of subsequent rows are also indicated. For each row the condition
ai+bi+ci=i,(i=1,…,n) (17)
holds, thus any two columns are sufficient to uniquely determine the state. The top row satisfies the following properties
a=an=12N−S,b=bn=2S,c=cn=n−a−b=n−12N−S, (18)
completely specifying the chosen electronic state as an irrep of .
The total number of CSFs for a given number of orbitals , electrons and total spin is given by the Weyl-Paldus Paldus (1974); Weyl (1946) dimension formula
NCSF=b+1n+1(n+1a)(n+1c)=2S+1n+1(n+1N2−S)(n+1n−N2−S). (19)
As it can be seen from Eq. (19), the number of possible CSFs—of course—still scales combinatorially with the number of electrons and orbitals, as seen in Fig. 1 with a comparison to the total number of possible SDs (without any symmetry restriction). The ratio of the total number of SDs and CSFs for can be estimated by Stirling’s formula (for sufficiently large and ) as
NSDNCSF≈√πnn2(2S+1), (20)
which shows orbital dependent, , decrease of the efficient Hilbert space size for a spin-adapted basis. The Paldus table also emphasizes the cumulative aspects of the coupling between electrons, with the i-th row providing information on number of electrons, (up to i-th level) and the spin, , by
Ni=ai+bi,Si=12bi. (21)
As can be seen in Eq. 16, there are four permissible difference vectors (, with ) between consecutive rows of a Paldus table, which corresponds to the possible ways of coupling a spatial orbital based on the group chain (7). This information can be condensed in the four-valued step value, shown in Table 1.
All possible CSFs of a chosen irrep can then be encoded by the collection of the step values in a step-vector, where starting from the “vacuum” ’th row , an empty spatial orbital is indicated by , a “positively spin-coupled” orbital, , by , a “negatively spin-coupled”, , by and a doubly occupied spatial orbital by . To retain physically allowed states the condition applies. (As a side note: Another common notation —e.g. in Molcas—is to indicate positive spin-coupling as , negative spin-coupling by and a doubly occupied orbital by .)
The step-value in Tab. 1 is given by and the collection of all into the step-vector representation is the most compact form of representing a CSF, with the same storage cost as a Slater determinant, with 2 bits per spatial orbital. One can create all basis function of a chosen irrep of by constructing all possible distinct step-vectors which lead to the same top-row of the Paldus table (18), specifying the chosen irrep with definite spin and number of electrons, with the restriction .
## Iii The Graphical Unitary Group Approach (GUGA)
The graphical unitary group approach (GUGA) of Shavitt Shavitt (1977, 1981) is based on this step-vector representation and the observation that there is a lot of repetition of possible rows in the Paldus tables specifying the CSFs of a chosen irrep of . Instead of all possible Paldus tables, Shavitt suggested to just list the possible sets of distinct rows in a table, called the distinct row table (DRT). The number of possible elements of this table is given by Shavitt (1977)
NDRT= (a+1)(c+1)(b+1+d2)−d(d+1)(d+2)2 = (N2−S+1)(n−N2−S+1)(2S+1+d2)−d(d+1)(d+2)2, (22)
with , which is drastically smaller than the total number of possible CSFs (19) or Slater determinants (without any symmetry restrictions) as seen in Fig. 1. Each row in the DRT is identified by a pair of indices , with being the level index, related to the orbital index and being the lexical row index such that if or if and .
A simple example of the DRT of a system with , and is shown in Table 2.
Relations between elements of the DRT belonging to two neighboring levels and are indicated by the so called downward, , and upward, , chaining indices, with . These indices indicate the connection to a lexical row index in a neighboring level by a step-value , where a zero entry indicates an invalid connection associated with this step-value. Given a DRT table any of the possible CSFs can be generated by connecting distinct rows linked by the chaining indices.
This DRT table can be represented as a graph, see Fig. 2, where each distinct row is represented by a vertex (node) and nonzero chaining indices are indicated by an arc (directed edge). The vertices are labeled according to the lexical row index , starting at the unique head node at the top, which corresponds to the highest row . It ends at the second unique null row , which is called the tail of the graph. Vertices with the same -value of Table 2 are at the same level on this grid. The highest -value is on top and the lowest at the bottom. Vertices also have left-right order with respect to their value and vertices that share the same value are further ordered—still horizontally—with respect to their value. With the above mentioned ordering of the vertices according to their and values, the slope of each arc is in direct correspondence to the step-value , connecting two vertices. corresponds to vertical lines, and the tilt of the other arcs increases with the step-value .
Each CSFs in the chosen irrep of , is represented by a directed walk through the graph starting from the tail and ending at the head, e.g. the green and orange lines in Fig. 2 (color online), representing the states and in step-vector representation. Such a walk spans arcs (number of orbitals) and visits one node at each level . There is a direct correspondence between the Paldus table, Gel’fand patterns and directed walks on Shavitt graphs for representing all possible CSFs in a chosen irrep of .
### iii.1 Evaluation of Nonvanishing Hamiltonian Matrix Elements
Given the expression of the nonrelativistic spin-free Hamiltonian in (5) a matrix element between two CSFs, and , is given by:
⟨m′|^H|m⟩=∑ijtij⟨m′|^Eij|m⟩+12∑ij,klVij,kl⟨m′|^eij,kl|m⟩. (23)
The matrix elements, and , provide the coupling coefficients between two given CSFs and and are the integral contributions. The coupling coefficients are independent of the orbital shape and only depend on the involved CSFs, and - Therefore, for a given set of integrals the problem of computing Hamiltonian matrix elements in the GT basis is reduced to the evaluation of these coupling coefficients. The graphical representation of CSFs has been proven a powerful tool to evaluate these coupling coefficients thanks to the formidable contribution of Paldus, Boyle, Shavitt and others Paldus and Boyle (1980); Shavitt (1978); Downward and Robb (1977).
The great strength of the graphical approach is the identification and evaluation of nonvanishing matrix elements of the excitation operators (generators) , between two GT states (CSFs), . The generators are classified according to their indices, with being diagonal weight (W) and with being raising (R) and lowering (L) operators (or generators). In contrast to Slater determinants, applied to yields a linear combination of CSFs ,
^Eij|m⟩=∑m′|m′⟩⟨m′|^Eij|m⟩, (24)
with an electron moved from spatial orbital to orbital without changing the spin of the resulting states . They are called raising (lowering) operators since the resulting will have a higher (lower) lexical order than the starting CSF .
The distance, , from to , is an important quantity and is called the range of the generator . For the one-body term in (5) Shavitt Shavitt (1977) was able to show that the walks on the graph, representing the CSFs and , must coincide outside of this range to yield a non-zero matrix element. The two vertices in the DRT graph, related to orbital and (with ) represent the points of separation of the walks and they are named loop head and loop tail. And the matrix element only depends on the shape of the loop formed by the two graphs in the range , shown in Fig. 3.
Shavitt Shavitt (1978) showed that the relations
N′k=Nk±1,b′k=bk±1andS′k=Sk±12fork∈S0, (25)
between and must be fulfilled to yield a nonzero matrix element ( for a raising and for a lowering generator).
This allows two possible relations between the vertices at each level in terms of Paldus array quantities depending on the type of generator (R,L). For raising generators R:
a′k=ak,b′k=bk+1,c′k=ck−1, →Δbk=−1–––––––––––, (26) a′k=ak+1,b′k=bk−1,c′k=ck →Δbk=+1–––––––––––, (27)
where and for lowering generators L:
a′k=ak−1,b′k=bk+1,c′k=ck →Δbk=−1–––––––––––, (28) a′k=ak,b′k=bk−1,c′k=ck+1 →Δbk=+1–––––––––––. (29)
At each vertex of the loop in range one of the relations (26-29) must be fulfilled for the one-body matrix element to be non-zero.
Based on the graphical approach, Shavitt Shavitt (1978) showed that the matrix elements of the generators can be factorized in a product, where each term corresponds to a segment of the loop in the range and is given by
⟨m′|^Eij|m⟩=j∏k=iW(Qk;d′k,dk,Δbk,bk), (30)
where is the value of state at level . additionally depends on the segment shape of the loop at level , determined by the type of the generator , the step values and and . The nonzero segment shapes for a raising (R) generator are shown in Fig. 4. In Table 3 the nonzero matrix elements of the one-electron operator —an over/under-bar indicates the loop head/tail—depending on the segment shape symbol, the step-values and the -value are given in terms of the auxiliary functions
A(b,x,y)=√b+xb+y,C(b,x)=√(b+x−1)(b+x+1)b+x. (31)
### iii.2 Two-Body Matrix Elements
The matrix elements of the two-body operators are more involved than the one-body operators, especially the product of singlet excitation generators, . Similar to the one-electron operators, the GT states and must coincide outside the total range to for to be nonzero. The form of the matrix element depends on the overlap range of the two ranges
S1=(i,j)∩(k,l). (32)
One possibility to calculate the matrix element would be to sum over all possible intermediate states, ,
⟨m′|^Eij^Ekl|m⟩=∑m′′⟨m′|^Eij|m′′⟩⟨m′′|^Ekl|m⟩, (33)
but in practice this is very inefficient. For non-overlapping ranges the matrix element just reduces to the product
⟨m′|^eij,kl|m⟩=⟨m′|^Eij^Ekl|m⟩=⟨m′|^Eij|m′′⟩⟨m′′|^Ekl|m⟩, (34)
where must coincide with in the range and with in range . The same rules and matrix elements as for one-body operators apply in this case. An example of this is shown in the left panel of Fig. 5.
For , we define the non-overlap range
S2=(i,j)∪(k,l)−S1, (35)
where the same restrictions and matrix elements as for one-body operators apply. In the overlap range, , different restrictions for the visited Paldus table vertices apply for the matrix element to be nonzero. This depends on the type of the two generators involved and were worked out by Shavitt Shavitt (1981). For two raising generators (RR) the following conditions apply
a′p=ap,b′p=bp+2,c′p=cp−2 →Δbp=−2––––––––––– (36) a′p=ap+2,b′p=bp+2,c′p=cp →Δbp=+2––––––––––– (37) a′p=ap+1,b′p=bp,c′p=cp−1 →Δbp=0–––––––––. (38)
For two lowering generators (LL):
a′p=ap+2,b′p=bp+2,c′p=cp, →Δbp=−2––––––––––– (39) a′p=ap,b′p=bp−2,c′p=cp+2 →Δbp=+2––––––––––– (40) a′p=ap−1,b′p=bp,c′p=cp+2 →Δbp=0–––––––––. (41)
And for a mixed combination of raising and lowering generators (RL)
a′p=ap−1,b′p=bp+2,c′p=cp−1, →Δbp=−2––––––––––– (42) a′p=ap+1,b′p=bp−2,c′p=cp+1 →Δbp=+2––––––––––– (43) a′p=ap,b′p=bp,c′p=cp →Δbp=0–––––––––. (44)
Drake and Schlesinger Drake and Schlesinger (1977), Paldus and Boyle Paldus and Boyle (1980), Payne Payne (1982) and Shavitt and Paldus Shavitt (1981) were able to derive a scheme, where the two-body matrix elements can be computed as a product of segment values similar to the one-body case (30)
⟨m′|^eij,kl|m⟩=∏p∈S2W(Qp;d′p,dp,Δbp,bp)×∑x=0,1∏p∈S1Wx(Qp;d′p,dp,Δbp,bp), (45)
where and are the overlap (32) and non-overlap (35) ranges defined above.
are the already defined single operator segment values, listed in Table 3, and are new segment values of the overlap range (their listing is omitted for brevity here, but can be found in Refs. [dccclxxx(69; 74)]. The sum over two products in corresponds to the singlet coupled intermediate states (), with a nonzero contribution if and the triplet intermediate coupling ().
This product formulation of the two-body matrix elements in a spin-adapted basis is the great strength of the graphical unitary group approach, which allows an efficient implementation of the GT basis in the FCIQMC algorithm. The details of the matrix element calculation in this basis are, however, tedious and will be omitted here for brevity and clarity. More details on the matrix element calculation, especially the contributions of the two-body term to diagonal and one-body matrix elements can be found in Appendix B or in Refs. [dccclxxx(69; 74)].
## Iv Spin-Adapted Full Configuration Interaction Quantum Monte Carlo
The Full Configuration Interaction Quantum Monte Carlo (FCIQMC) method Booth, Thom, and Alavi (2009); Cleland, Booth, and Alavi (2010) attempts to obtain the exact solution of a quantum mechanical problem in a given single-particle basis set by an efficient sampling of a stochastic representation of the wavefunction—originally expanded in a discrete antisymmetrised basis of Slater determinants (SDs)—through the random walk of walkers, governed by imaginary-time the Schrödinger equation. For brevity of this manuscript we refer the interested reader to Refs. [dccclxxx(1; 2)] and [dccclxxx(21)] for an in-depth explanation of the FCIQMC method.
Having introduced the theoretical basis of the unitary group approach (UGA) and its graphical extension (GUGA) to permit a mathematically elegant and computationally efficient incorporation of the total spin symmetry in form of the Gel’fand-Tsetlin basis, here we will present the actual implementation of these ideas in the FCIQMC framework, termed GUGA-FCIQMC.
Fundamentally, the three necessary ingredients for an efficient spin-adapted formulation of FCIQMC are:
1. Efficient storage of the spin-adapted basis
2. Efficient excitation identification and matrix element computation
3. Symmetry adapted excitation generation with manageable computational cost
The first point is guaranteed with the UGA, since storing the information content of a CSF and a SD amounts to the same memory requirement, with CSFs represented in the step-vector representation. Efficient identification of valid excitations is rather technical and explained in Appendix A and in Ref. [dccclxxx(74)]. For the present discussion we simply need to know, although it is more involved to determine if two CSFs are connected by a single application of than for SDs, it is possible to do so efficiently. Matrix element computation is based on the product structure of the one- (30) and two-body (45) matrix elements derived by Shavitt Shavitt (1978) explained above and presented in more detail in App. B and in Ref. [dccclxxx(74)]. Concerning point (iii): symmetry adaptation in FCIQMC is most efficiently implemented at the excitation generation step, by creating only symmetry-allowed excitations. For the continuous spin symmetry this is based on Shavitt’s DRT and the restriction for nonzero matrix elements in the GUGA. This, in addition to the formulation in a spin-pure GT basis, ensures that the total spin quantum number is conserved in a FCIQMC calculation.
### iv.1 Excitation Generation: Singles
The concept of efficient excitation generation in the spin-adapted GT basis via the GUGA will be explained in detail by the example of single excitations. Although more complex, the same concepts apply for generation of double excitation, which are discussed below.
In contrast to excitation generation for SDs, there are now two steps involved for a CSF basis. The first, being the same as in a formulation of FCIQMC in Slater determinants, is the choice of the two spatial orbitals and , with probability . This should be done in a way to ensure the generation probability to be proportional to the Hamiltonian matrix element involved. However, here comes the first difference of a CSF-based implementation compared to a SD-based one. For Slater determinants, the choice of an electron in spin-orbital and an empty spin-orbital is sufficient to uniquely specify the excitation , and to calculate the involved matrix element . However, in a CSF basis, the choice of an occupied spatial orbital , and empty or singly occupied spatial orbital , only determines the type of excitation generator acting on an CSF basis state as well as the involved integral contributions and of the matrix element . To ensure , the occupied orbital and (partially) empty are picked in the same way as for SDs, but with an additional restriction to ensure . However, the choice of does not uniquely determine the excited CSF as there are multiple possible ones, as explained above.
As a consequence, the choice of spatial orbitals and does not determine the coupling coefficient of the matrix element . Optimally, for a given and generator , the connected CSF has to be created with a probability proportional to the coupling coefficient . By ensuring is proportional to the integral contributions and to the coupling coefficients, the total spawning probability
ps(m′|m)=p(i)p(j|i)p(m′|m) (46)
will be proportional to the magnitude of Hamiltonian matrix element . The efficiency of the FCIQMC algorithm depends on the ratio of the Hamiltonian matrix element between two connected states and the probability to choose the excitation , as the imaginary timestep of the simulation is adapted to faithfully account for all excitations
Δτ−1∝|Hm′m|ps(m′|m). (47)
In a primitive implementation, is determined by the “worst-case” ratio during a simulation. A less strict approach to this problem is discussed below. By choosing nonzero and ensuring is achieved by a branching tree approach, we obtain one of the different possible walks on the Shavitt graph with nonzero loop contributions with the starting CSF .
### iv.2 The Branching Tree
In the spin-adapted excitation generation, after a certain generator is picked with a probability based on the integral contributions of the Hamiltonian matrix element, the type of generator is determined, raising (R) if and lowering (L) if . One connecting single excitation is then chosen by looping from starting orbital to and stochastically choosing a valid nonzero Shavitt graph, based on the restrictions (26-29), mentioned in the GUGA section above. As an example, let us have a closer look at a chosen raising generator. As can be seen in the single segment value Table 3 there are 4 possible nonzero starting matrix elements. These starting segments are associated with a relative difference of the total spin | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8686475157737732, "perplexity": 1478.883109086679}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496667262.54/warc/CC-MAIN-20191113140725-20191113164725-00557.warc.gz"} |
http://www.boundaryvalueproblems.com/content/2011/1/29/ | Research
# Blow-up for an evolution p-laplace system with nonlocal sources and inner absorptions
Yan Zhang1, Dengming Liu2*, Chunlai Mu2 and Pan Zheng2
Author Affiliations
1 School of Mathematics and Computer Engineering, Xihua University, Chengdu, Sichuan 610039, PR China
2 College of Mathematics and Statistics, Chongqing University, Chongqing 410031, PR China
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Boundary Value Problems 2011, 2011:29 doi:10.1186/1687-2770-2011-29
Received: 10 June 2011 Accepted: 6 October 2011 Published: 6 October 2011
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
### Abstract
This paper investigates the blow-up properties of positive solutions to the following system of evolution p-Laplace equations with nonlocal sources and inner absorptions
with homogeneous Dirichlet boundary conditions in a smooth bounded domain Ω ∈ RN(N ≥ 1), where p, q > 2, m, n, r, s ≥ 1, α, β > 0. Under appropriate hypotheses, the authors discuss the global existence and blow-up of positive weak solutions by using a comparison principle.
2010 Mathematics Subject Classification: 35B35; 35K60; 35K65; 35K57.
##### Keywords:
evolution p-Laplace system; global existence; blow-up; nonlocal sources; absorptions
### 1 Introduction
In this paper, we deal with the blow-up properties of positive solutions to an evolution p-Laplace system of the form
(1.1)
where p, q > 2, m, n, r, s ≥ 1, α, β > 0, Ω is a bounded domain in RN(N ≥ 1) with a smooth boundary ∂Ω, the initial data , and , , where v denotes the unit outer normal vector on ∂Ω.
System (1.1) is the classical reaction-diffusion system of Fujita-type for p = q = 2. If p ≠ 2, q ≠ 2, (1.1) appears in the theory of non-Newtonian fluids [1,2] and in nonlinear filtration theory [3]. In the non-Newtonian fluids theory, the pair (p, q) is a characteristic quantity of the medium. Media with (p, q) > (2, 2) are called dilatant fluids and those with (p, q) < (2, 2) are called pseudoplastics. If (p, q) = (2, 2), they are Newtonian fluids.
System (1.1) has been studied by many authors. For p = q = 2, Escobedo and Herrero [4] considered the following problem
(1.2)
where p, q > 0. Their main results read as follows. (i) If pq ≤ 1, every solution of (1.2) is global in time. (ii) If pq > 1, some solutions are global while some others blow up in finite time.
In the last three decades, many authors studied the following degenerate parabolic problem
(1.3)
under different conditions (see [5,6] for nonlinear boundary conditions; see [7-10] for local nonlinear reaction terms; see [11] for nonlocal nonlinear reaction terms). In [12], the existence, uniqueness, and regularity of solutions were obtained. When f(u) = -uq, q > 0 or f(u) ≡ 0 extinction phenomenon of the solution may appear [13-15]; However, if f(u) = uq, q > 1 the solution may blow up in finite time [7-10,14].
Especially, in [11], Li and Xie dealt with the following p-Laplace equation
(1.4)
Under appropriate hypotheses, they established the local existence and uniqueness of its solution. Furthermore, they obtained that the solution u exists globally if q < p - 1; u blows up in finite time if q > p - 1 and u0(x) is large enough.
Recently, in [16], Li generalized (1.4) to system and studied the following problem
(1.5)
Similar to [11], he proved that whether the solution blows up in finite time depends on the initial data, constants α, β, and the relations between mn and (p - 1)(q - 1).
For other works on parabolic system like (1.1), we refer readers to [17-30] and the references therein.
When p = q, m = n, r = s, α = β, u0(x) = v0(x), system (1.1) is then reduced to a single p-Laplace equation
(1.6)
However, to the authors' best knowledge, there is little literature on the study of the global existence and blow-up properties for problems (1.1) and (1.6). Motivated by the above works, in this paper, we investigate the blow-up properties of solutions of the problem (1.1) and extend the results of [4,11,16,19] to more generalized cases.
In order to state our results, we introduce some useful symbols. Throughout this paper, we let φ(x), ψ(x) be the unique solution of the following elliptic problem
(1.7)
and
(1.8)
respectively. For convenience, we denote
Before starting the main results, we introduce a pair of parameters (μ, γ) solving the following characteristic algebraic system
namely,
with
It is obvious that 1/τ and 1/θ share the same signs. We claim that the critical exponent of problem (1.1) should be (1/τ, 1/θ) = (0, 0), described by the following theorems.
Theorem 1.1. Assume that (1/τ, 1/θ) < (0, 0), then there exist solutions of (1.1) being globally bounded.
Theorem 1.2. Assume that (1/τ, 1/θ) > (0, 0), then the nonnegative solution of (1.1) blows up in finite time for sufficiently large initial values and exists globally for sufficiently small initial values.
Theorem 1.3. Assume that (1/τ, 1/θ) = (0, 0), φ(x) and ψ(x) are defined in (1.7) and (1.8), respectively.
(i) Suppose that r > p - 1 and s > q - 1. If αnβr ≥ |Ω|n+r, then the solutions are globally bounded for small initial data; if , , then the solutions blow up in finite time for large data.
(ii) Suppose that p - 1 > r and q - 1 > s. If , then the solutions are globally bounded for small initial data; if , then the solutions blow up in finite time for large data.
(iii) Suppose that p - 1 > r and s > q - 1. If , then the solutions are globally bounded for small initial data; if , , then the solutions blow up in finite time for large data.
(iv) Suppose that r > p - 1 and q - 1 > s. If , then the solutions are globally bounded for small initial data; if , , then the solutions blow up in finite time for sufficiently large data.
The rest of this paper is organized as follows. In Section 2, we shall establish the comparison principle and local existence theorem for problem (1.1). Theorems 1.1 and 1.2 will be proved in Section 3 and Section 4, respectively. Finally, we will give the proof of Theorem 1.3 in Section 5.
### 2 Preliminaries
Since the equations in (1.1) are degenerate at points where ∇u = 0 or ∇v = 0, there is no classical solution in general, and we therefore consider its weak solutions. Let ΩT = Ω × (0, T), ST = ∂Ω × (0, T) and . We begin with the precise definition of a weak solution of problem (1.1).
Definition 2.1 A pair of functions (u(x, t), v(x, t)) is called a weak solution of problem (1.1) in if and only if
(i) (u, v) is in the space and (ut, vt) ∈ L2(0, T; L2(Ω)) × L2(0, T; L2(Ω)).
(ii) the following equalities
and
hold for all ϕ1, ϕ2, which belong to the class of test functions
(iii) u(x, t)|t = 0 = u0(x), v(x, t)|t = 0 = v0(x) for all .
In a natural way, the notion of a weak subsolution for (1.1) is given as follows.
Definition 2.2 A pair of functions (u(x, t), v(x, t)) is called a weak subsolution of problem (1.1) in if and only if
(i) (u, v) is in the space and (ut, vt) ∈ L2(0, T; L2(Ω)) × L2(0, T; L2(Ω)).
(ii) the following inequalities
and
hold for any ϕ1, ϕ2, which belong to the class of test functions
(iii) u(x, t)|t = 0 u0(x), v(x, t)|t = 0 v0(x) for all .
Similarly, a pair of functions is a weak supersolution of (1.1) if the reversed inequalities hold in Definition 2.2. A weak solution of (1.1) is both a weak subsolution and a weak supersolution of (1.1).
We shall use the following comparison principle to prove our global and nonglobal existence results.
Proposition 2.3 Let (u, v) and be a nonnegative subsolution and supersolution of (1.1), respectively, with for all . Then, a.e. in .
Proof. From the definitions of weak subsolution and supersolution, for any ϕ1, ϕ2 ∈ Θ2, we could obtain that
(2.1)
and
(2.2)
In addition, inequalities (2.1) and (2.2) remain true for any subcylinder of the form Ωτ = Ω × (0, τ) ⊂ ΩT and corresponding lateral boundary Sτ = ∂Ω × (0, τ) ⊂ ST. Taking a special test function in (2.1), where χ[0, τ] is the characteristic function defined on [0, τ] and s+ = max{s, 0}, we find that
(2.3)
where |Ω| denotes the Lebesgue measure of Ω and
Next, our task is to estimate the first term on the right-side of (2.3). In view of Cauchy's inequality, we see that
(2.4)
Furthermore, by Lemma 1.4.4 in [12], we know that there exists δ > 0 such that
(2.5)
Combining now (2.3)-(2.5), we deduce that
(2.6)
here , .
Likewise, taking test function in (2.2), we have that
(2.7)
where C3, C4 denote some positive constants. Moreover, there exists a large enough constant C, such that
(2.8)
Now, we write
then, (2.8) implies that
(2.9)
By Gronwall's inequality, we know that y(τ) = 0, for any τ ∈ [0, T]. Thus, , this means that , in as desired. The proof of Proposition 2.3 is complete. □
With the above established comparison principle in hand, we are able to show the basic existence theorem of weak solutions. Here, we only state the local existence theorem, and its proof is standard [12, 16, for more details].
Theorem 2.1 Given , there is some T0 > 0 such that the problem (1.1) admits a nonnegative unique weak solution (u, v) for each t < T0, and . Furthermore, either T0 = ∞ or
### 3 Proof of Theorem 1.1
Proof of Theorem 1.1. Notice that (1/τ, 1/θ) < (0, 0) implies
We will prove Theorem 1.1 in four subcases.
(a) For μ = r, γ = s, we then have mn < rs. Let , where , will be determined later. After a simple computation, we have
and
So, is a time-independent supersolution of problem (1.1) if
i.e.,
(3.1)
(b) For μ = p - 1, γ = q - 1, we then have mn < (p - 1)(q - 1). Let
where φ, ψ satisfying (1.7) and (1.8), respectively. Taking
and
then it is easy to verify that is a global supersolution for system (1.1).
(c) For μ = r, γ = q - 1, we then have mn < r(q - 1). Choose and satisfy
Let with ψ defined by (1.8). By direct Computation, we arrive at
(3.2)
and
(3.3)
(d) For μ = p - 1, γ = s, we then have mn < r(q - 1). Let with φ defined by (1.7), where and . Then, (3.2) and (3.3) hold if
The proof of Theorem 1.1 is complete. □
### 4 Proof of Theorem 1.2
Proof of Theorem 1.2. Observe that 1/τ, 1/θ > 0 implies
For μ = r, γ = s. Choosing
then is a global supersolution for problem (1.1) provided that and .
For μ = p - 1, γ = q - 1. Let , where φ and ψ satisfying (1.7) and (1.8), respectively. Choosing
and
therefore, is a global supersolution for system (1.1) if and .
For other cases, the solutions of (1.1) should be global due to the above discussion.
Next, we begin to prove our blow-up conclusion under large enough initial data. Due to the requirement of the comparison principle, we will construct blow-up subsolutions in some subdomain of Ω in which u, v > 0. We use an idea from Souplet [31] and apply it to degenerate equations. Since problem (1.1) does not make sense for negative values of (u, v), we actually consider the following problem
(4.1)
where u+ = max{0, u}, v+ = max{0, v}. Let ϖ(x) be a nontrivial nonnegative continuous function and vanish on ∂Ω. Without loss of generality, we may assume that 0 ∈ Ω and ϖ(0) > 0. We shall construct a self-similar blow-up subsolution to complete our proof.
Set
(4.2)
here
and li, σi > 0(i = 1, 2), 0 < T < 1 are to be determined later. Notice the fact that
(4.3)
for sufficiently small T > 0.
Calculating directly, we obtain
and
where
On the other hand, we know
(4.4)
(4.5)
here Hx(u), Hx(v) denotes the Hessian matrix of u(x, t), v(x, t) respect to x, respectively. Use the notation d(Ω) = diam(Ω), then from (4.4) and (4.5), it follows that
Further, we have
(4.6)
and
(4.7)
Since 1/τ, 1/θ < 0, we see that μγ < mn. In addition, it is clear that
(4.8)
For , we choose l1 and l2 such that
(4.9)
Recall that μ = max{p - 1, r} and γ = max{q - 1, s}, then (4.9) implies
and
Next, we can choose positive constants σ1, σ2 sufficiently small such that
consequently, we have
(4.10)
For , we fix l1 and l2 to satisfy
(4.11)
then we can also select σ1, σ2 small enough such that (4.10) holds.
From (4.6), (4.7) and (4.10), for sufficiently small T > 0, it follows that
(4.12)
Since ϖ(0) > 0 and ϖ(x) are continuous, there exist two positive constants ρ and ε such that ϖ(x) ≥ ε for all x B(0, ρ) ⊂ Ω. Choose T small enough to insure , hence u≤ 0, v≤ 0 on ST. From (4.1) and (4.2), it follows that , for sufficiently large . By comparison principle, we have (u, v) ≤ (u, v) provided that and . It shows that (u, v) blows up in finite time. The proof of Theorem 1.2 is complete. □
### 5 Proof of Theorem 1.3
Proof of Theorem 1.3. In the critical case of (1/τ, 1/θ) = (0, 0), we have mn = μγ.
(i) For r > p - 1, s > q - 1, we know mn = rs. Thanks to αnβr ≥ |Ω|n+r, we can choose A and B sufficiently large such that , and
Clearly, is a supersolution of problem (1.1), then by comparison principle, the solution of (1.1) should be global.
Next, we begin to prove our blow-up conclusion. Since mn = rs, we can choose constants l1, l2 > 1 such that
(5.1)
According to Proposition 2.3, we only need to construct a suitable blow-up subsolution of problem (1.1) on . Let y(t) be the solution of the following ordinary differential equation
where
Since and , we have c1 > 0. On the other hand, by virtue of (5.1), it is easy to see that δ1 > δ2. Then, it is obvious that there exists a constant 0 < T' < +∞ such that
Construct
where φ, ψ satisfying (1.7) and (1.8), respectively. Moreover, by the assumptions on initial data, we can take small enough constant y0 such that
(5.2)
Now, we begin to verify that (u(x, t), v(x, t)) is a blow-up subsolution of the problem (1.1) on , T < T'. In fact, ∀(x, t) ∈ ΩT × (0, T), a series of computations show
(5.3)
Similarly, we also have
(5.4)
On the other hand, ∀t ∈ [0, T], we have
(5.5)
and
(5.6)
Combining now (5.2)-(5.6), we see that (u, v) is a subsolution of (1.1) and (u, v) < (u, v) on by comparison principle, thus (u, v) must blow up in finite time since (u, v) does.
(ii) For p - 1 > r, q - 1 > s, we know mn = (p - 1)(q - 1). Under the assumption , we can choose A, B such that
Then, is a global supersolution of (1.1).
Since mn = (p - 1)(q - 1), we can choose constants l1, l2 > 1 such that
(5.7)
Next, we consider the following ordinary differential equation
where
Since , , we have c1 > 0. On the other hand, in light of (5.7), it is easy to show that δ1 > δ2. Then, it is clear that y(t) will become infinite in a finite time T' < +∞.
Let
where φ(x), ψ(x) satisfies (1.7) and (1.8), respectively. Similar to the arguments for the case r > p - 1, s > q - 1, we can prove that (u(x, t), v(x, t)) is a blow-up subsolution of the problem (1.1) on , T < T'. Then, the solution (u, v) of (1.1) blows up in finite time.
(iii) For p - 1> r, s > q - 1, we know mn = s(p - 1). Since , we can choose A, B such that
We can check is a global supersolution of (1.1).
Thanks to mn = s(p - 1), we can choose constants l1, l2 > 1 such that
(5.8)
Let
where φ(x), ψ(x) are defined in (1.7) and (1.8), respectively, and y(t) satisfies the following Cauchy problem
where
Then, the left arguments are the same as those for the case r > p - 1, s > q - 1, so we omit them.
(iv) The proof of this case is parallel to (iii). The proof of Theorem 1.3 is complete. □
### Competing interests
The authors declare that they have no competing interests.
### Authors' contributions
The authors declare that the work was realized in collaboration with same responsibility. All authors read and approved the final manuscript.
### Acknowledgements
The authors are very grateful to the anonymous referees and the editor for their careful reading and useful suggestions, which greatly improved the presentation of the paper. Dengming Liu is supported by the Fundamental Research Funds for the Central Universities (Project No. CDJXS 11 10 00 19). Chunlai Mu is supported in part by NSF of China (Project No. 10771226) and in part by Natural Science Foundation Project of CQ CSTC (Project No. 2007BB0124).
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31. Souplet, P: Blow-up in nonlocal reaction-diffusion equations. SIAM J Math Anal. 29, 1301–1334 (1998). Publisher Full Text | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9640193581581116, "perplexity": 1284.4713490928584}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368698411148/warc/CC-MAIN-20130516100011-00008-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://scholarlyrepository.miami.edu/dissertations/1659/ | ## Dissertations from ProQuest
#### Title
Short Term Variability Of Hydrogen-Peroxide In Surface Oceans
1987
Article
#### Degree Name
Doctor of Philosophy (Ph.D.)
#### Department
Marine and Atmospheric Chemistry
#### Abstract
This study reports on several of the factors affecting the short term variability of hydrogen peroxide in surface oceans. A major pathway in the formation of H\$\sb{2}\$O\$\sb{2}\$ in surface waters is thought to be the in situ photochemical formation of H\$\sb{2}\$O\$\sb{2}\$ via humic substances. The photochemical rate of formation of H\$\sb{2}\$O\$\sb{2}\$ was studied in several waters with varying concentrations of natural organic carbon, 0.2 to 17 mg C L\$\sp{-1}\$. The results of this study indicated that the rate of formation of hydrogen peroxide is related to the concentration of humic substances as measured by dissolved organic carbon and absorbance at wavelengths \$>295\$ nm.To quantitatively describe the photochemical formation of hydrogen peroxide, it was also necessary to obtain the quantum yield at various wavelengths. The apparent quantum yield was determined at individual wavelengths and over spectral regions. All of the waters studied showed the same general trend, i.e. a gradually decreasing apparent quantum yield with increasing wavelength.The impact of marine rain on surface ocean H\$\sb{2}\$O\$\sb{2}\$ concentration was studied in several oceanic environments. It was found that the concentration of H\$\sb{2}\$O\$\sb{2}\$ is higher in marine rain than rain collected over land. It was also found that the concentration of H\$\sb{2}\$O\$\sb{2}\$ in marine rain was about two orders of magnitude higher, approximately 10-80 uM, than surface ocean concentrations, 0.05-0.1 uM. Therefore, rain may have a substantial impact on the surface ocean H\$\sb{2}\$O\$\sb{2}\$ concentration, the magnitude of the affect largely dependent on the extent of mixing in the area of the rain.A continuous analyzer for determining H\$\sb{2}\$O\$\sb{2}\$, fluorescence and chlorophyll was developed and used, and the three variables measure in coastal, shelf-break and open ocean environments. Wide fluctuations occurred in all three variables in the coastal waters studied. As expected, open ocean environments did not show the wide variations in any of the variables. Decomposition rates of H\$\sb{2}\$O\$\sb{2}\$ were measured in several coastal environments and were very fast when compared to open ocean environments. Although the processes leading to the decomposition of H\$\sb{2}\$O\$\sb{2}\$ were not studied, it is clear that these processes control the concentration of H\$\sb{2}\$O\$\sb{2}\$ in coastal environments. (Abstract shortened with permission of author.)
#### Keywords
Environmental Sciences | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9208295345306396, "perplexity": 3526.068054546249}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670635.48/warc/CC-MAIN-20191120213017-20191121001017-00320.warc.gz"} |
https://zbmath.org/?q=an%3A0967.32007 | ×
zbMATH — the first resource for mathematics
Composition operators on the Bloch space of several complex variables. (English) Zbl 0967.32007
Let $$\Omega$$ be a homogeneous bounded domain in $${\mathbb C}^n$$. For any $$\phi\in Hol(\Omega, \Omega)$$ and $$f\in Hol(\Omega)$$, denote $$C_\phi f:=f\circ \phi$$ and call $$C_\phi$$ the composition operator induced by $$\phi$$.
The authors obtain the necessary and sufficient conditions for $$C_\phi$$ to be compact on the Bloch space $$\beta(B_n)$$ or the little Bloch space $$\beta_0(B_n)$$.
MSC:
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) 32A30 Other generalizations of function theory of one complex variable (should also be assigned at least one classification number from Section 30-XX) 47B38 Linear operators on function spaces (general) 30D45 Normal functions of one complex variable, normal families
Keywords:
Bloch space; composition operator
Full Text:
References:
[1] K Madigan, A Matheson. Compact Composition operators on the Bloch Space. Trans Amer Math Soc, 1995, 347: 2679-2687 · Zbl 0826.47023 · doi:10.2307/2154848 [2] R Timoney. Bloch Function in Several Complex Variables I. Bull London Math Soc, 1980, 12, 241-267 · Zbl 0428.32018 · doi:10.1112/blms/12.4.241 [3] R Timoney. Bloch Function in Several Complex Variables II. J Reine Angew Math, 1980, 319, 1-22 · Zbl 0425.32008 · doi:10.1515/crll.1980.319.1
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8603862524032593, "perplexity": 1220.6619441892913}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991659.54/warc/CC-MAIN-20210516013713-20210516043713-00136.warc.gz"} |
https://brilliant.org/problems/box-in-the-air/?group=UuznI0l2it5G | Box in the Air
Geometry Level 3
In the $xyz$ coordinates, the centroids of the 3 faces of a cuboid are located at points $(2 , 2 , 2), (7 , 5 , 2)$, and $(7 , 2 , 4)$, where each side is parallel to one of the axes.
What is the volume of this cuboid?
× | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 3, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8282347321510315, "perplexity": 357.5329479738362}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987751039.81/warc/CC-MAIN-20191021020335-20191021043835-00445.warc.gz"} |
https://reference.wolfram.com/language/ref/GreenFunction.html | # GreenFunction
GreenFunction[{[u[x]],[u[x]]},u,{x,xmin,xmax},y]
gives a Green's function for the linear differential operator with boundary conditions in the range xmin to xmax.
GreenFunction[{[u[x1,x2,]],[u[x1,x2,]]},u,{x1,x2,}Ω,{y1,y2,}]
gives a Green's function for the linear partial differential operator over the region Ω.
GreenFunction[{[u[x,t]],[u[x,t]]},u,{x,xmin,xmax},t,{y,τ}]
gives a Green's function for the linear time-dependent operator in the range xmin to xmax.
GreenFunction[{[u[x1,,t]],[u[x1,,t]]},u,{x1,}Ω,t,{y1,,τ}]
gives a Green's function for the linear time-dependent operator over the region Ω.
# Details and Options
• GreenFunction represents the response of a system to an impulsive DiracDelta driving function.
• GreenFunction for a differential operator is defined to be a solution of that satisfies the given homogeneous boundary conditions .
• A particular solution of with homogeneous boundary conditions can be obtained by performing a convolution integral .
• GreenFunction for a time-dependent differential operator is defined to be a solution of that satisfies the given homogeneous boundary conditions .
• A particular solution of with homogeneous boundary conditions can be obtained by performing a convolution integral .
• The Green's functions for classical PDEs have characteristic geometrical properties:
• is given as an expression in and if the dependent variable is of the form , and as a pure function with formal parameters and if the dependent variable is of the form instead of . »
• The region Ω can be anything for which RegionQ[Ω] is True.
• All the necessary initial and boundary conditions for ODEs must be specified in .
• Boundary conditions for PDEs must be specified using DirichletCondition or NeumannValue in .
• Assumptions on parameters may be specified using the Assumptions option.
# Examples
open allclose all
## Basic Examples(2)
Green's function for a boundary value problem:
Green's function for the heat operator on the real line:
## Scope(22)
### Basic Uses(2)
Compute the Green's function for an ordinary differential operator:
Obtain a pure function in the result by using u instead of u[x] in the second argument:
Compute the Green's function for a partial differential operator:
Obtain a pure function in the result by using u instead of u[x,t] in the second argument:
### Ordinary Differential Equations(4)
Compute the Green's function for an initial value problem:
Compute the Green's function for a Dirichlet problem:
Compute the Green's function for a Neumann problem:
Compute the Green's function for a Robin problem:
### Wave Equation(4)
Green's function for the wave operator on the real line:
Green's function for the wave operator with a Dirichlet condition on a half-line:
Green's function for the wave operator with a Neumann condition on a half-line:
Green's function for the wave operator with a Dirichlet condition on an interval:
### Heat Equation(5)
Green's function for the heat operator on the real line:
Green's function for the heat operator with a Dirichlet condition on a half-line:
Green's function for the heat operator with a Dirichlet condition on an interval:
Green's function for the heat operator with a Neumann condition on an interval:
Green's function for the heat operator in the plane:
### Laplace Equation(4)
Green's function for the Laplacian in two dimensions:
Dirichlet problem for the Laplacian in a quadrant of the plane:
Dirichlet problem for the Laplacian in a rectangle:
Green's function for the Laplacian in three dimensions:
### Helmholtz Equation(3)
Green's function for the Helmholtz operator in two dimensions:
Dirichlet problem for the Helmholtz operator in the upper half-plane:
Dirichlet problem for the Helmholtz operator in a rectangle:
## Options(1)
### Assumptions(1)
Specify Assumptions on parameters in GreenFunction:
Obtain a simpler result under the assumption that t>s:
## Applications(9)
### Ordinary Differential Equations(4)
Solve an initial value problem for an inhomogeneous differential equation using GreenFunction:
Define a forcing function:
Perform a convolution of the Green's function with the forcing function:
Compare with the result given by DSolveValue:
Solve a Dirichlet problem for an inhomogeneous differential equation using GreenFunction:
Define a forcing function:
Perform a convolution of the Green's function with the forcing function:
Compare with the result given by DSolveValue:
Solve a Neumann problem for an inhomogeneous differential equation using GreenFunction:
Define a forcing function:
Perform a convolution of the Green's function with the forcing function:
Compare with the result given by DSolveValue:
Solve a Robin problem for an inhomogeneous differential equation using GreenFunction:
Define a forcing function:
Perform a convolution of the Green's function with the forcing function:
Compare with the result given by DSolveValue:
### Partial Differential Equations(2)
Solve the inhomogeneous wave equation using GreenFunction:
Define the inhomogeneous term:
Solve the inhomogeneous equation using :
Compare with the solution given by DSolveValue:
Solve an initial value problem for the heat equation using GreenFunction:
Specify an initial value:
Solve the initial value problem using :
Compare with the solution given by DSolveValue:
### Physics and Engineering(3)
Compute the current i[t] in a circuit with a voltage source v[t] that is connected to a resistor R and an inductor L. The operator for this circuit is given by:
Diagram for the circuit:
Compute the Green's function:
Find the current for a given voltage source:
Compute the displacement u[x] for a string of length p and tension T that is fixed at the two ends and is subjected to a force per unit length of f[x]. The operator for the displacement is given by:
Force diagram:
Compute the Green's function:
Find the displacement for a given force:
The impulse response of a continuous linear time-invariant system can be found by using the Green's function for the system with homogeneous initial conditions. Compute the impulse response for the system defined by:
Green's function for the system with homogeneous initial conditions:
Obtain the impulse response by setting s=0:
Plot the impulse response:
## Properties & Relations(2)
Compute a Green's function for a differential equation:
Obtain the same result using DSolve:
GreenFunction is related to OutputResponse and TransferFunctionModel:
Wolfram Research (2016), GreenFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/GreenFunction.html.
#### Text
Wolfram Research (2016), GreenFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/GreenFunction.html.
#### CMS
Wolfram Language. 2016. "GreenFunction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GreenFunction.html.
#### APA
Wolfram Language. (2016). GreenFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GreenFunction.html
#### BibTeX
@misc{reference.wolfram_2022_greenfunction, author="Wolfram Research", title="{GreenFunction}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/GreenFunction.html}", note=[Accessed: 29-September-2022 ]}
#### BibLaTeX
@online{reference.wolfram_2022_greenfunction, organization={Wolfram Research}, title={GreenFunction}, year={2016}, url={https://reference.wolfram.com/language/ref/GreenFunction.html}, note=[Accessed: 29-September-2022 ]} | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.934064507484436, "perplexity": 1179.2545612549145}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335424.32/warc/CC-MAIN-20220930020521-20220930050521-00110.warc.gz"} |
https://www.ias.ac.in/listing/articles/pmsc/088/01 | • Volume 88, Issue 1
January 1979, pages 1-92
• On the covering of syntax-directed translations for context-free grammars
A necessary and sufficient set of conditions is obtained that relates any two context-free grammarsG1 andG2 with the property that wheneverG2 left—or right—coversG1, the syntax-directed translations (SDT’s) with underlying grammarG1 is a subset of those with underlying grammarG2. Also the case thatG2 left—or right—coversG1 but the SDT’s with underlying grammarG1 is not a subset of the SDT’s with underlying grammarG2 is considered; in this case an algorithm is described to obtain the syntax-directed translation schema (SDTS) with underlying grammarG2 to the given SDTS with underlying grammarG1, if it exists.
• Convergence of a numerical procedure for the solution of a fourth order boundary value problem
The convergence of a proposed second order finite difference method for the determination of an approximate solution of the fourth order differential equationy(4)+fy=g is proved. The matrix associated with the system of linear equations that arises is not even assumed to be monotone, as is often the case in practice. The only requirement is that the functionf(x) be nonnegative. In a typical numerical illustration, the observed maximum errors in absolute value are compared with the respective theoretical error bound for a series of the values of the step size.
• On a theorem of Wigner on products of positive matrices
A simple algebraic proof of a theorem due to Wigner on the product of three positive matrices is given. It is shown that the theorem holds for four matrices under an additional condition. The proofs are valid in the more general case of operators in a Hilbert space.
• On pulsatile flow of a viscous fluid in a rotating channel
A study is made on the pulsatile flow superposed on a steady laminar flow of a viscous fluid in a parallel plate channel rotating with an angular velocity Ω about an axis perpendicular to the plates. An exact solution of the governing equations of motion is obtained. The solution in dimensionless form contain two parametersK2L2/v which is reciprocal of Ekmann Number and frequency parameter σ=αL2/v. The effects of these parameters on the principal flow characters such as mean sectional velocity and shear stresses at the plates have been examined. For large σ andK2 the flow near the plates has a multiple boundary layer character.
• Temperature distribution in a laminar circular jet
The effect of frictional heat on the temperature distribution in a laminar circular jet has been studied. It is found from the analysis and the graphs that as the Prandtl number decreases from unity the overall temperature difference near the axis of the jet increases but as we move away from the axis it goes on decreasing. The reverse phenomenon happens in the case of increasing Prandtl number.
• Rayleigh-Taylor instability of compressible finitely conducting rotating fluid in the presence of a magnetic field
The character of the equilibrium of a non-viscous, compressible finitely conducting rotating fluid in the presence of a vertical magnetic field along the direction of gravitational field has been investigated. It is shown that the solution is characterised by a variational principle. Based on the existence of variational principle, an approximate solution has been derived for the case of a fluid having exponentially varying density in the vertical direction. Due to finite resistivity of the medium it is found that potentially stable or unstable configuration retains its character. Further the growth rate of disturbance has been obtained corresponding to short and long wavelengths and it is found that electrical resistivity suppresses the growth rate for large wavelengths but it increases the same for small wavelengths. It is further shown that magnetic field has a destabilizing influence for large wavelengths and a stabilizing influence for small wavelengths.
• Descriptions of operators in quantum mechanics
The problem of expressing a general dynamical variable in quantum mechanics as a function of a primitive set of operators is studied from several points of view. In the context of the Heisenberg commutation relation, the Weyl representation for operators and a new Fourier-Mellin representation are related to the Heisenberg group and the groupSL(2,R) respectively. The description of unitary transformations via generating functions is analysed in detail. The relation between functions and ordered functions of noncommuting operators is discussed, and results closely paralleling classical results are obtained.
• # Proceedings – Mathematical Sciences
Volume 130, 2020
All articles
Continuous Article Publishing mode
• # Editorial Note on Continuous Article Publication
Posted on July 25, 2019 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9222893118858337, "perplexity": 537.6841208680121}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590348523564.99/warc/CC-MAIN-20200607044626-20200607074626-00227.warc.gz"} |
https://dl.asminternational.org/handbooks/book/46/chapter-abstract/544229/Stray-Current-Corrosion | ## Abstract
Stray-current corrosion is an accelerated form of corrosion caused by externally induced electric current. It can occur in unprotected pipelines and submerged metal structures located near electric power sources or anywhere voltage differences exist. This article describes common scenarios and sources of stray current along with ways to detect it and prevent the type of corrosion it can cause. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9312113523483276, "perplexity": 2703.054216668788}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986669057.0/warc/CC-MAIN-20191016163146-20191016190646-00327.warc.gz"} |
https://resonaances.blogspot.com/2013/08/a-kingdom-for-scale.html?showComment=1376886088073 | ## Tuesday, 13 August 2013
### A kingdom for a scale
The recent hiatus, so far the longest in the history of Résonaances, was caused by a unique combination of work, travel, frustration, depression, and sloth. Sorry :-| A day may come when this blog will fall silent forever; but it is not this day ;)
After the first run of the LHC particle physics finds itself in an unprecedented situation. During most of the history of the discipline we had a high energy scale that allowed us to organize our theoretical and experimental efforts. It first appeared back in the 1930s when Fermi wrote down his theory of weak interactions which contained a 4-fermion operator mediating the beta decay of the neutron. For dimensional reasons, 4-fermion operators appear in the Lagrangian divided by an energy scale squared, and in the case of the Fermi operator this scale is what we now know as the electroweak scale v=174 GeV. This scale come with well defined physical consequences. Scattering amplitudes in the Fermi theory misbehave at energies above v, and some new physics must appear to regulate them. Later several details of this picture were modified. In particular, it was found that the Fermi 4-fermion operator is a low energy effective description of the exchange of a W boson between pairs of fermions. However the argument for new physics near the electroweak scale remained in place, this time to regulate the W and Z bosons scattering amplitudes. That's why even before the LHC kicked off we could give an almost risk-free promise that it was going to discover something.
Now things have changed dramatically. The LHC has explored the energy range up to about 1 TeV and definitively crossed the electroweak scale. The promised new physics phenomenon was found: a spin-0 boson coupled to mass, as predicted in the Standard Model. This little addition miraculously cures all the woes of the theory. Ignoring gravity, the Standard Model with the 125 GeV Higgs boson can be extended to arbitrarily high scales. Only the coupling of matter to gravity guarantees some new phenomena, like maybe strong gravitational effects and production of black holes. But that should happen at an immensely high scale of 10^19 GeV that we may never be able to reach in collider experiments. We are not sure if there is any other physical scale between the electroweak and Planck scale. There's no well defined energy frontier we can head toward. Particle physics no longer has a firm reference point. An artist's view of the current situation is this:
There's actually one important practical consequence. Regardless how high energy collider we build next: 30 TeV, 100 TeV, or 1000 TeV, we cannot be sure it will discover any new phenomena rather than just confirm the old theory in the new energy range.
This is not to say that the Standard Model must be valid all the way to the Planck scale. On the contrary we have strong hints it is otherwise. The existence of dark matter, the observations of neutrino oscillations, the matter-antimatter asymmetry in the Universe, and the cosmological inflation, they all require some physics beyond the Standard Model. However none of the above points to a concrete scale where new phenomena must show up. The answers may be just behind the corner and be revealed by the run-II of the LHC. Or the answer may be due to Planck-scale physics and will never be directly explored; or else it may be due to very light and very weakly coupled degrees of freedom that should be probed by other means than colliders. For example, for dark matter particles we know theoretically motivated models with the mass ranging from sub-eV (axions) to the GUT scale (wimpzillas), and there is no mass between these two extremes that is clearly favored from the theory point of view. The case of the neutrino oscillations is a bit different because, as soon as we prove experimentally that neutrinos are Majorana particles, we will confirm the existence of a set of dimension-5 operators beyond the Standard Model, the so-called Weinberg operators of the form (H L)^2/Λ. Then the scale Λ is the maximum energy scale where new physics (singlet Majorana neutrinos or something more complicated) has to show up. This is however little consolation given the scale emerging from neutrino experiments is Λ∼10^15 GeV, obviously beyond the direct reach of accelerators in a foreseeable future.
So, while pushing up the energy frontier in accelerators will continue, I think that currently searching high and low for a new scale is the top priority. Indeed, increasing the collision energy has become an expensive and time consuming endeavor; we will achieve an almost factor of 2 increase in 2 years, and, optimistically, we can hope for another factor of 2 at the time scale of ∼25 years. On the other hand, indirect sensitivity to high scales via searches for higher dimensional operators beyond the standard model can often be improved by orders of magnitude in the near future. The hope is that Fermi's trick will work again and we may discover the new scale indirectly, by means of experiments at much lower energies. There are literally hundreds of dimension-6 operators beyond the standard model that can be searched for in experiments. For example, operators involving the Higgs fields would affect the Higgs couplings measured, and in this case the LHC and later the ILC can probe the operators suppressed by up to ∼10 TeV. Flavor and CP violating processes offer an even more sensitive probe, with the typical sensitivity between 10 and 10^5 TeV. Who knows, maybe the recent anomaly in B→K*μμ decays is not yet another false alarm, but an effect of the flavor violating dimension-6 operators of the form
with Λ of order 30 TeV. And if not, there are hundreds other doors to knock on. Demonstrating the presence of a nearby new physics scale would surely bring back momentum to the particle physics program. At least, we would know where we stand, and how big a collider we must build to be guaranteed new physics. So yes, a kingdom and on my part I'm adding the hand of a princess too..
wolfgang said...
>> A day may come when this blog will fall silent forever; but it is not this day...
Thank you!
Ap said...
Hi Jester,
I was getting worried that something serious had occurred so very glad to have you back. Yours is by far the best BSM blog & the field needs you!
Concerning physics, I totally agree about the need to identify a scale of new physics, but I'm not quite so pessimistic about the chance of LHC13 discoveries as a factor of 2 at this particular energy scale is still a big deal. (LHCb is a great expt but I don't at all believe the LHCb B->K*\mu\mu anomaly is new physics. Angular distributions have a long history of giving false results, and the `true' look-elsewhere correction is huge, taking it well below the 2.8 \sigma quoted.)
The silver lining of the present headless chicken situation is that there are rich opportunities for completely new non-collider experiments with associated discoveries. Given the collider timescale/cost problem that's been hanging over traditional HEP since the 1970's a movement away from big colliders might not be bad thing.
fiat lux said...
8.6 GeV FTW!
Cop Shoot Cop said...
Keep up the writing! Enjoying your blog!
Anonymous said...
Please produce a picture of the princess.
Anonymous said...
Why do you need to see a picture? You only get the hand.
Jester said...
but you can choose the left or the right hand.
Anonymous said...
I wish I could get a bunch of those headless chickens in front of a whiteboard for an hour, because I think there’s just loads of new physics lying around like low-hanging fruit. But it’s in the orchard next door, in things like classical electromagnetism and TQFT and little papers that HEP guys never read. And all this new physics is within the standard model, showing why some of the beyond-the-standard-model ideas are ill-founded. I try to tell people about this “top down” stuff, but they’re just not hearing it. But there again, I suppose headless chickens aren’t too good at listening.
Robert L. Oldershaw said...
You can lead a theoretical physicist to knowledge, but you cannot make him think.
Anonymous said...
You note 4 important and promising subjects. Some are more equal than others. For those considering which to pursue, I offer some hard-earned, strongly biased opinions:
neutrino oscillations: guaranteed to be interesting
matter-antimatter asymmetry: a profound but approachable if challenging problem
cosmic inflation: a morass of theoretical nonsense
dark matter: guaranteed to frustrate and disappoint.
andrew said...
"The case of the neutrino oscillations is a bit different because, as soon as we prove experimentally that neutrinos are Majorana particles, we will confirm the existence of a set of dimension-5 operators beyond the Standard Model, the so-called Weinberg operators of the form (H L)^2/Λ. Then the scale Λ is the maximum energy scale where new physics (singlet Majorana neutrinos or something more complicated) has to show up. This is however little consolation given the scale emerging from neutrino experiments is Λ∼10^15 GeV, obviously beyond the direct reach of accelerators in a foreseeable future."
I very much doubt that neutrinos will be found to be Majorana and don't really understand why this, rather than a Dirac hypothesis, would be so attractive to theorists.
Mario said...
Hey! I guess you mean LHCb's latest paper http://arxiv.org/abs/1308.1707, not the one you mentioned. There is a deviation from form-factor invariant values. The other (you posted) is consistent with the SM.
Anonymous said...
Anon at 16.16
I also offer some hard-earned and biased opinions:
1) New results from neutrino oscillations may be experimentally interesting but they are almost entirely useless theoretically. Despite the *enormous amount of data we already have* about both lepton and quark flavor mixing and masses (for decades now) there is no good theory of why the flavor structure is what it is.
The small amount of extra data from upcoming neutrino osc expts will not help us at all. The only really useful thing will be to find out if neutrinos masses are Majorana or Dirac (as Jester correctly says). All else in the experimental direction is essentially hype about its importance. What is needed in this field is *new theoretical ideas*.
2) Dark matter (and possibly connections with matter-antimatter asymmetry) are guaranteed beyond-the-SM physics with an enormously exciting and varied experimental program that could have big impact on our understanding of the world. Saying this field is "guaranteed to frustrate and disappoint" is unthinking cynicism.
3) Given the huge amount of primordial fluctuation data which supports the basic idea of (sub-Planckian, single field) inflation this is much more likely to be true than not. Given this is one of the most profound changes in our understanding of the world that has occurred since QM it is very worth while to invest a great deal of effort in a) trying to integrate the inflationary sector into the known structure and facts of the SM, and b) collecting much more data on all length scales, eg from the future large scale structure surveys. Saying that inflation is a "morass of theoretical nonsense" shows that you fail to understand both the theory and the importance of the what we are learning from data.
Anonymous said...
You must enter the gates of Mordor to find new Physics.
Jester said...
Anonymous said...
Andrew -- Without changing the standard model we can provide the neutrino a mass via Majorana terms using an effective operator LHLH/M where M is a large cut-off scale. Dirac mass requires the addition of right handed neutrinos as well as a symmetry to be imposed (lepton number symmetry). If we don't impose this additional symmetry then once again the neutrinos will have Majorana masses.
Michel Beekveld said...
Welcome back Jester.
You were missed.
Unknown said...
"But there again, I suppose headless chickens aren’t too good at listening."
People who get new buildings written into their contracts can't really complain about feeling like headless chickens, especially when they can't admit when they're wrong.
Loren said...
Dark matter has a chance of being identified if various in-the-works experiments and observations succeed.
If direct detection of WIMP's succeeds, we may get an interesting harvest of physics, especially if several different detector teams detect them. They use or propose to use a variety of different materials, covering the periodic table and spinning vs. spinless.
I've thought about this in detail, and one may get proton vs. neutron, spin-independent vs. spin-dependent, and even the mass of the WIMP if it's less than a few hundred GeV.
If annihilation detection succeeds, then we may get a precise WIMP mass and annihilation cross sections into both photons and neutrinos. Not quite as good, but still good.
Anonymous said...
I notice that B -> K* mu mu paper speaks only from the 1/fb worth of 7 TeV data. Any guesses on how long it will be before the 2/fb worth of 8 TeV data is crunched to shed more light on this specific subject?
Anonymous said...
Might there be some new stuff coming out of Daya Bay?
Michel Beekveld
Anonymous said...
So, I understand your point about it not being clear what scale we need to be at for direct production, but what are you experimentally suggesting? Intensity frontier experiments? Precision collider work? Something else...?
Jester said...
Suggesting? I'm just lamenting ;-)
But seriously, I think precision physics (collider or non-collider) is now the key, as we cannot start building a new high-energy machine until we're sure there's something out there to find.
Jester said...
Sure, it is possible that neutrino experiments (Daya Bay or another) will find something interesting one day.
LHCb takes its time. My guess the full statistics in B->K*mumu will be out for winter conferences.
cb said...
Instead of looking for the magic effective operator in a vast landscape randomly (Nature/God will select/choose the good one), may be courageous model-building physicists (dépités mais pas décapités/dismayed but not beheaded ;-) could use and test the hypothetical educated guess from algebraic constraints of the non-commutative geometric framework for instance? Anyway it could be that the naturalness paradigm in "standard" quantum field theory living on a Minkowski space-time is heuristically obsolete and it is required to understand better the compass role of the spin to blend somehow 4D-spacetime and internal gauge degrees of freedom before contemplating any super-space and extra-dimensions...
Anonymous said...
" Any guesses on how long it will be before the 2/fb worth of 8 TeV data is crunched to shed more light on this specific subject?"
I guess 3-8 months, depending on the interest in the LHCb group.
Chris Austin said...
What do you think of the arguments of Dicus and He, who argue in hep-ph/0409131 that by considering the onset of tree level unitarity violation in f \bar{f} -> n longitudinal W's, where f is any SM fermion, and n is typically around 10, (see the graph on page 30), the SM itself, with no assumptions about new physics, leads to the conclusion that the scale of mass generation for the SM fermions cannot be more than about 120 TeV for the charged fermions, and about 170 TeV for the neutrinos?
Stefan said...
"Ignoring gravity, the Standard Model with the 125 GeV Higgs boson can be extended to arbitrarily high scales."
<-- Do you mean with that that the current model is renormalizable? If not could you clarify that a little bit?
Jester said...
I meant just renormalizability + unitarity + vacuum stability.
Jester said...
Chris, i think this paper deal with the SM without a Higgs boson. With the 125 GeV Higgs there's no problems with unitarity of fermion scattering | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8044846653938293, "perplexity": 1377.9267783248786}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305288.57/warc/CC-MAIN-20220127193303-20220127223303-00380.warc.gz"} |
http://math.stackexchange.com/questions/280335/another-series-convergence-question | # Another series convergence question
Does this series converge?
$\displaystyle\sum\limits_{n=2}^\infty \frac{\sqrt{n+1}}{(2n^2-3n+1) (\ln n +(\ln n)^2)}$
-
What do you think? – Ron Gordon Jan 16 '13 at 21:58
I tried using the Cauchy Condensation Test, but the resultant expression doesn't seem helpful. – Ryan Jan 16 '13 at 22:07
The terms of the series are all positive; all you need do is use the Comparison Test, after proving convergence of the series of terms $\sum_{n=2}^\infty \frac{\sqrt{n+1}}{2n^2 - 3n + 1}$ – Chris Jan 16 '13 at 22:08
Ah, thank you, user_blahblah. – Ryan Jan 16 '13 at 22:31
@Ryan: is this a problem created by you? It seems unusual for a book. (subjectively speaking) – Chris's sis Jan 16 '13 at 22:40
Informally, for large $n$, $\sqrt{n+1}$ behaves like $n^{1/2}$, the quadratic in the denominator behaves like $2n^2$, giving a combined behaviour of $\frac{1}{2}\cdot\frac{1}{n^{3/2}}$, plenty good enough for convergence. And the $\log$ stuff at the bottom gives our series a minor (and unnecessary) boost towards convergence.
More formally, we can note that for $n\ge 3$, $$0 \lt \frac{\sqrt{n+1}}{(2n^2-3n+1)(\ln n+\ln^2 n)}\lt \frac{\sqrt{n+1}}{n^2-3n+1}$$ So if we can prove that $\sum_2^\infty \frac{\sqrt{n+1}}{n^2-3n+1}$ converges, it will follow by Comparison that our series converges.
Now note that $\sqrt{n+1}\le 2\sqrt{n}$, and that if $n \ge 6$, then $n^2-3n+1\ge \frac{1}{2}n^2$. It follows that for $n\ge 6$, we have $$\frac{\sqrt{n+1}}{n^2-3n+1}\lt \frac{4}{n^{3/2}}.$$ Since $\sum_2^\infty \frac{1}{n^{3/2}}$ converges, the series $\sum_2^\infty \frac{\sqrt{n+1}}{n^2-3n+1}$ converges.
-
Very nice pedagogical answer, thank you for the insights. – Ryan Jan 16 '13 at 22:34
Yes.${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$
-
This one requires no tricks - look at the ratio of the numerator and denominator in the limit $n \to \infty.$ In fact, you can even throw out the $\ln n$ term.
There is $n_0$ such that $$\displaystyle\sum\limits_{n=2}^\infty \frac{\sqrt{n+1}}{(2n^2-3n+1) (\ln n +(\ln n)^2)}<\displaystyle\sum\limits_{n=n_0}^\infty \frac{\sqrt{n+1}}{(n+1)\cdot(\sqrt{n+1})^2}=\sum\limits_{n=n_0+1}^\infty \frac{1}{n\cdot\sqrt{n}}$$ Thus the series clearly converges. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9888924956321716, "perplexity": 419.62757054201546}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802765002.8/warc/CC-MAIN-20141217075245-00098-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/2926959/suppose-that-x-is-a-finite-set-and-is-hausdorff-as-a-topological-space-show/2927257 | # Suppose that $X$ is a finite set, and is Hausdorff as a topological space. Show that $X$ is discrete (as a topological space).
Problem : Suppose that $$X$$ is a finite set, and is Hausdorff as a topological space. Show that $$X$$ is discrete (as a topological space).
Thoughts: I'm not even quite sure what the question is asking. I know the definition of a discrete topology is that a set is open in $$X$$ if it is a subset of $$\mathcal P(X)$$, so then can't the discrete topology be applied to any set and so any $$X$$ is discrete? Any help appreciated.
• It is enough to prove that every singleton is open. Fix $a$ and for every $b\neq a$ find a neighborhood $U_b$ of $a$ not containing $b$. Then consider $\bigcap _{b\neq a}U_b$. – SMM Sep 22 '18 at 21:07
• The discrete topology can be applied to any set, but that's not what's happening here. Here all we know is that the space is Hausdorff, and you are supposed to show that this means that the space is discrete. In general, Hausdorff is a weaker condition than discrete (e.g. $\Bbb R$ with standard topology is Hausdorff but not discrete). Your task is to show that when the space is finite, the notions of Hausdorff and discrete coincides. – Arthur Sep 22 '18 at 21:08
• @Arthur I think I am confused what a discrete topology means. – IntegrateThis Sep 22 '18 at 21:09
• It means that all subsets are open. – Arthur Sep 22 '18 at 21:09
Since $$X$$ is Hausdorff, we know that singletons are closed. Therefore, all finite subsets of $$X$$ are closed (finite unions of closed sets are closed). Since all subsets of $$X$$ are finite, it follows that every subset of $$X$$ is closed. Using this it is not hard to see that every subset of $$X$$ is open. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8725103139877319, "perplexity": 84.51774459093778}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514572516.46/warc/CC-MAIN-20190916080044-20190916102044-00137.warc.gz"} |
https://aviation.stackexchange.com/questions/63800/is-goce-a-satellite-or-aircraft | # Is GOCE a satellite or aircraft?
This vessel (GOCE) is in orbit and maneuvers by using air similar to a plane to create lift and turn. The solar powered ion thrusters powers it continuously inside the atmosphere for years. Is this technically a plane, satellite, or a missile?
• I took the liberty of editing out the off-topic side question (not only because it's off-topic but also because you should only have one question per post). Feel free to roll back if you disagree. – Sanchises Apr 28 at 20:42
• By definition it's a spacecraft instead of an aircraft, because it relies on orbital velocity to stay up, not air in any form. It's a low drag satellite. – user3528438 Apr 28 at 21:40
A satellite is an object in orbit. A satellite keeps a constant altitude if it is traveling at the correct orbital speed for their altitude. Otherwise it will be in an elliptical orbit where the altitude is constantly changing. In any case, satellites are (almost) in free-fall
An airplane travels at much, much lower speed than orbital speed. Airplanes keep their altitude because their wings produce lift.
Satellites in lower orbited are subject to atmospheric drag, albeit to much smaller extentd than planes. GOCE was orbiting Earth at an - for satellites - very low altitude (224 km). There is hardly any atmosphere at this altitude, but at an orbital velocity of 7.8 m/s (51337 kts), atmospheric drag is still a problem.
Therefore the structure of GOCE was particularly designed for low drag. The wings were flat without camber. They did not produce any lift. GOCE used ion thrusters to compensate the atmospheric drag and maintain orbital velocity.
Photo: ESA
Photo: ESA
Missiles are self-propelled guided weapons. I can assure you that GOCE was not a weapon.
[TL;DR]: GOCE was a satellite, because it orbited an astronomical body. It was not a plane, because its wings did not produce lift.
GOCE´s 20 month mission ended on November 11, 2013 with the planned destructive re-enty into the atmosphere. It measured Earth´s gravity field and acquired a precise model of the Earth geoid:
Image: ESA
• "The wings were flat without camber. They did not produce any lift." - What were the wings for, then? I would have thought that if a surface doesn't produce lift, then that surface is not considered a wing. – Terran Swett Apr 29 at 17:14
• @TannerSwett, They provide area for solar panels, and they stabilize the space craft. Indeed, it would be more correct to call them fins instead of wings – bogl Apr 29 at 18:49 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8095813989639282, "perplexity": 1780.1982471225933}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496672313.95/warc/CC-MAIN-20191123005913-20191123034913-00350.warc.gz"} |
http://mathhelpforum.com/calculus/280050-function-function-ln-x.html | # Thread: Function of a function with ln(x)
1. ## Function of a function with ln(x)
Consider the equation: $f(x) = \ln (f(x))$
Does there exist a function that satisfies this equation? It is obvious that $f(x)$, if it exists, is not differentiable. Taking the derivative of both sides would yield $f'(x) = \dfrac{f'(x)}{f(x)}$. But, then you could cancel out the $f'(x)$ terms and you find that $f(x)=1$, which is a contradiction. Therefore, $f(x)$ could not be differentiable (if it exists at all).
It could also be written as the limit to the sequence of functions $f_0(x) = x$ and $f_n(x) = \ln \left( f_{n-1}(x) \right)$ should the limit function exist.
Never mind. I figured it out. No, the function does not exist. Proof:
The domain of $f_n(x)$ would be $^ne$ where the superscript before the number refers to tetration. So, $^2e = e^e$, $^3e = e^{e^e}$. But then the limit domain would not exist.
2. ## Re: Function of a function with ln(x)
Originally Posted by SlipEternal
Proof:
The domain of $f_n(x)$ would be $^ne$ where the superscript before the number refers to tetration. So, $^2e = e^e$, $^3e = e^{e^e}$. But then the limit domain would not exist.
The editing timed out before I could finish updating it. I meant the domain for $f_n(x)$ with $n\ge 2$ would be $x \in (^{n-2}e,\infty)$.
3. ## Re: Function of a function with ln(x)
What about the complex logarithm instead of the real logarithm? It has two fixed points. I think the limit function would tend to converge solutions of the form $x=e^{e^{\cdot^{\cdot^{\cdot^x}}}}$
4. ## Re: Function of a function with ln(x)
Originally Posted by SlipEternal
What about the complex logarithm instead of the real logarithm? It has two fixed points. I think the limit function would tend to converge solutions of the form $x=e^{e^{\cdot^{\cdot^{\cdot^x}}}}$
$x = -W(-1) \approx 0.318132 -1.33724 i$
where $W(x)$ is the Lambert W function is one of these fixed points.
It is a very unstable fixed point. The slightest variation sends $e^{e^{e^{\vdots^x}}}$ off to infinity.
5. ## Re: Function of a function with ln(x)
Originally Posted by romsek
$x = -W(-1) \approx 0.318132 -1.33724 i$
where $W(x)$ is the Lambert W function is one of these fixed points.
It is a very unstable fixed point. The slightest variation sends $e^{e^{e^{\vdots^x}}}$ off to infinity.
Very true. But, check out what Wolframalpha gives for $x=e^{e^{e^x}}$: Wolfram|Alpha: Computational Knowledge Engine
That has four solutions.
Then $x=e^{e^{e^{e^x}}}$ also has four solutions according to wolframalpha, but obviously not the same as the solutions for $e^{e^{e^x}}$.
So, it is unclear how many solutions you might find as you take increasing heights to the tower of $e$'s. I don't have any intuition to determine how many solutions there might be at each step (other than multiples would preserve roots). So, if I had six levels of $e$, it would include all of the solutions to three levels. I'm not sure the correct terminology here, but I hope it is clear what I mean by "levels".
Perhaps after a finite number of steps, there are an infinite number of solutions, and perhaps those would become more stable. But, I really have no intuition to know why the number of solutions changes as we add levels to the tower. Perhaps it could be an interesting topic to consider one day, but not right now. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.94879150390625, "perplexity": 264.74056419323927}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257650685.77/warc/CC-MAIN-20180324132337-20180324152337-00112.warc.gz"} |
http://www.nag.com/numeric/cl/nagdoc_cl23/html/F08/f08msc.html | f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual
# NAG Library Function Documentnag_zbdsqr (f08msc)
## 1 Purpose
nag_zbdsqr (f08msc) computes the singular value decomposition of a complex general matrix which has been reduced to bidiagonal form.
## 2 Specification
#include #include
void nag_zbdsqr (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer ncvt, Integer nru, Integer ncc, double d[], double e[], Complex vt[], Integer pdvt, Complex u[], Integer pdu, Complex c[], Integer pdc, NagError *fail)
## 3 Description
nag_zbdsqr (f08msc) computes the singular values and, optionally, the left or right singular vectors of a real upper or lower bidiagonal matrix $B$. In other words, it can compute the singular value decomposition (SVD) of $B$ as
$B = U Σ VT .$
Here $\Sigma$ is a diagonal matrix with real diagonal elements ${\sigma }_{i}$ (the singular values of $B$), such that
$σ1 ≥ σ2 ≥ ⋯ ≥ σn ≥ 0 ;$
$U$ is an orthogonal matrix whose columns are the left singular vectors ${u}_{i}$; $V$ is an orthogonal matrix whose rows are the right singular vectors ${v}_{i}$. Thus
$Bui = σi vi and BT vi = σi ui , i = 1,2,…,n .$
To compute $U$ and/or ${V}^{\mathrm{T}}$, the arrays u and/or vt must be initialized to the unit matrix before nag_zbdsqr (f08msc) is called.
The function stores the real orthogonal matrices $U$ and ${V}^{\mathrm{T}}$ in complex arrays u and vt, so that it may also be used to compute the SVD of a complex general matrix $A$ which has been reduced to bidiagonal form by a unitary transformation: $A=QB{P}^{\mathrm{H}}$. If $A$ is $m$ by $n$ with $m\ge n$, then $Q$ is $m$ by $n$ and ${P}^{\mathrm{H}}$ is $n$ by $n$; if $A$ is $n$ by $p$ with $n, then $Q$ is $n$ by $n$ and ${P}^{\mathrm{H}}$ is $n$ by $p$. In this case, the matrices $Q$ and/or ${P}^{\mathrm{H}}$ must be formed explicitly by nag_zungbr (f08ktc) and passed to nag_zbdsqr (f08msc) in the arrays u and/or vt respectively.
nag_zbdsqr (f08msc) also has the capability of forming ${U}^{\mathrm{H}}C$, where $C$ is an arbitrary complex matrix; this is needed when using the SVD to solve linear least squares problems.
nag_zbdsqr (f08msc) uses two different algorithms. If any singular vectors are required (i.e., if ${\mathbf{ncvt}}>0$ or ${\mathbf{nru}}>0$ or ${\mathbf{ncc}}>0$), the bidiagonal $QR$ algorithm is used, switching between zero-shift and implicitly shifted forms to preserve the accuracy of small singular values, and switching between $QR$ and $QL$ variants in order to handle graded matrices effectively (see Demmel and Kahan (1990)). If only singular values are required (i.e., if ${\mathbf{ncvt}}={\mathbf{nru}}={\mathbf{ncc}}=0$), they are computed by the differential qd algorithm (see Fernando and Parlett (1994)), which is faster and can achieve even greater accuracy.
The singular vectors are normalized so that $‖{u}_{i}‖=‖{v}_{i}‖=1$, but are determined only to within a complex factor of absolute value $1$.
## 4 References
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912
Fernando K V and Parlett B N (1994) Accurate singular values and differential qd algorithms Numer. Math. 67 191–229
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
## 5 Arguments
1: orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or Nag_ColMajor.
2: uploNag_UploTypeInput
On entry: indicates whether $B$ is an upper or lower bidiagonal matrix.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
$B$ is an upper bidiagonal matrix.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
$B$ is a lower bidiagonal matrix.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3: nIntegerInput
On entry: $n$, the order of the matrix $B$.
Constraint: ${\mathbf{n}}\ge 0$.
4: ncvtIntegerInput
On entry: $\mathit{ncvt}$, the number of columns of the matrix ${V}^{\mathrm{H}}$${\mathbf{ncvt}}=0$ of right singular vectors. Set ${\mathbf{ncvt}}=0$ if no right singular vectors are required.
Constraint: ${\mathbf{ncvt}}\ge 0$.
5: nruIntegerInput
On entry: $\mathit{nru}$, the number of rows of the matrix $U$ of left singular vectors. Set ${\mathbf{nru}}=0$ if no left singular vectors are required.
Constraint: ${\mathbf{nru}}\ge 0$.
6: nccIntegerInput
On entry: $\mathit{ncc}$, the number of columns of the matrix $C$. Set ${\mathbf{ncc}}=0$ if no matrix $C$ is supplied.
Constraint: ${\mathbf{ncc}}\ge 0$.
7: d[$\mathit{dim}$]doubleInput/Output
Note: the dimension, dim, of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the diagonal elements of the bidiagonal matrix $B$.
On exit: the singular values in decreasing order of magnitude, unless NE_CONVERGENCE (in which case see Section 6).
8: e[$\mathit{dim}$]doubleInput/Output
Note: the dimension, dim, of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: the off-diagonal elements of the bidiagonal matrix $B$.
On exit: e is overwritten, but if NE_CONVERGENCE see Section 6.
9: vt[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array vt must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdvt}}×{\mathbf{ncvt}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdvt}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{vt}}\left[\left(j-1\right)×{\mathbf{pdvt}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{vt}}\left[\left(i-1\right)×{\mathbf{pdvt}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: if ${\mathbf{ncvt}}>0$, vt must contain an $n$ by $\mathit{ncvt}$ matrix. If the right singular vectors of $B$ are required, $\mathit{ncvt}=n$ and vt must contain the unit matrix; if the right singular vectors of $A$ are required, vt must contain the unitary matrix ${P}^{\mathrm{H}}$ returned by nag_zungbr (f08ktc) with ${\mathbf{vect}}=\mathrm{Nag_FormP}$ .
On exit: the $n$ by $\mathit{ncvt}$ matrix ${V}^{\mathrm{H}}$ or ${V}^{\mathrm{H}}$ of right singular vectors, stored by rows.
If ${\mathbf{ncvt}}=0$, vt is not referenced.
10: pdvtIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array vt.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$,
• if ${\mathbf{ncvt}}>0$, ${\mathbf{pdvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdvt}}\ge 1$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{ncvt}}>0$, ${\mathbf{pdvt}}\ge {\mathbf{ncvt}}$;
• otherwise ${\mathbf{pdvt}}\ge 1$.
11: u[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array u must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdu}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nru}}×{\mathbf{pdu}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $U$ is stored in
• ${\mathbf{u}}\left[\left(j-1\right)×{\mathbf{pdu}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{u}}\left[\left(i-1\right)×{\mathbf{pdu}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: if ${\mathbf{nru}}>0$, u must contain an $\mathit{nru}$ by $n$ matrix. If the left singular vectors of $B$ are required, $\mathit{nru}=n$ and u must contain the unit matrix; if the left singular vectors of $A$ are required, u must contain the unitary matrix $Q$ returned by nag_zungbr (f08ktc) with ${\mathbf{vect}}=\mathrm{Nag_FormQ}$ .
On exit: the $\mathit{nru}$ by $n$ matrix $U$ or $QU$ of left singular vectors, stored as columns of the matrix.
If ${\mathbf{nru}}=0$, u is not referenced.
12: pduIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array u.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nru}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
13: c[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array c must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdc}}×{\mathbf{ncc}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdc}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $C$ is stored in
• ${\mathbf{c}}\left[\left(j-1\right)×{\mathbf{pdc}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{c}}\left[\left(i-1\right)×{\mathbf{pdc}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $n$ by $\mathit{ncc}$ matrix $C$ if ${\mathbf{ncc}}>0$.
On exit: c is overwritten by the matrix ${U}^{\mathrm{H}}C$. If ${\mathbf{ncc}}=0$, c is not referenced.
14: pdcIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$,
• if ${\mathbf{ncc}}>0$, ${\mathbf{pdc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdc}}\ge 1$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{ncc}}\right)$.
15: failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).
## 6 Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_CONVERGENCE
$〈\mathit{\text{value}}〉$ off-diagonals did not converge. The arrays d and e contain the diagonal and off-diagonal elements, respectively, of a bidiagonal matrix orthogonally equivalent to $B$.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{ncc}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ncc}}\ge 0$.
On entry, ${\mathbf{ncvt}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ncvt}}\ge 0$.
On entry, ${\mathbf{nru}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nru}}\ge 0$.
On entry, ${\mathbf{pdc}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdc}}>0$.
On entry, ${\mathbf{pdu}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdu}}>0$.
On entry, ${\mathbf{pdvt}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdvt}}>0$.
NE_INT_2
On entry, ${\mathbf{ncvt}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ncvt}}>0$.
On entry, ${\mathbf{pdc}}=〈\mathit{\text{value}}〉$ and ${\mathbf{ncc}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{ncc}}\right)$.
On entry, ${\mathbf{pdu}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdu}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nru}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nru}}\right)$.
NE_INT_3
On entry, ${\mathbf{ncc}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdc}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{ncc}}>0$, ${\mathbf{pdc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
otherwise ${\mathbf{pdc}}\ge 1$.
On entry, ${\mathbf{pdvt}}=〈\mathit{\text{value}}〉$, ${\mathbf{ncvt}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{ncvt}}>0$, ${\mathbf{pdvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
otherwise ${\mathbf{pdvt}}\ge 1$.
On entry, ${\mathbf{pdvt}}=〈\mathit{\text{value}}〉$, ${\mathbf{ncvt}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{ncvt}}>0$, ${\mathbf{pdvt}}\ge {\mathbf{ncvt}}$;
otherwise ${\mathbf{pdvt}}\ge 1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
## 7 Accuracy
Each singular value and singular vector is computed to high relative accuracy. However, the reduction to bidiagonal form (prior to calling the function) may exclude the possibility of obtaining high relative accuracy in the small singular values of the original matrix if its singular values vary widely in magnitude.
If ${\sigma }_{i}$ is an exact singular value of $B$ and ${\stackrel{~}{\sigma }}_{i}$ is the corresponding computed value, then
$σ~i - σi ≤ p m,n ε σi$
where $p\left(m,n\right)$ is a modestly increasing function of $m$ and $n$, and $\epsilon$ is the machine precision. If only singular values are computed, they are computed more accurately (i.e., the function $p\left(m,n\right)$ is smaller), than when some singular vectors are also computed.
If ${u}_{i}$ is an exact left singular vector of $B$, and ${\stackrel{~}{u}}_{i}$ is the corresponding computed left singular vector, then the angle $\theta \left({\stackrel{~}{u}}_{i},{u}_{i}\right)$ between them is bounded as follows:
$θ u~i,ui ≤ p m,n ε relgapi$
where ${\mathit{relgap}}_{i}$ is the relative gap between ${\sigma }_{i}$ and the other singular values, defined by
$relgapi = min i≠j σi - σj σi + σj .$
A similar error bound holds for the right singular vectors.
The total number of real floating point operations is roughly proportional to ${n}^{2}$ if only the singular values are computed. About $12{n}^{2}×\mathit{nru}$ additional operations are required to compute the left singular vectors and about $12{n}^{2}×\mathit{ncvt}$ to compute the right singular vectors. The operations to compute the singular values must all be performed in scalar mode; the additional operations to compute the singular vectors can be vectorized and on some machines may be performed much faster. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 232, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9987316727638245, "perplexity": 1921.2185029477234}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246643283.35/warc/CC-MAIN-20150417045723-00185-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/subset-of-a-group.266197/ | # Subset of a group
1. Oct 22, 2008
### the_fox
Let G be a group and A a subset of G with n elements such that if x is in A then x^(-1) is not in A. Let X={(a,b) a in A, b in A, ab in A}. Prove that X contains at most n(n-1)/2 elements.
2. Oct 22, 2008
### morphism
Hint: X is largest when a,b in A => ab in A.
3. Oct 23, 2008
### the_fox
That's how X is defined. What exactly do you mean?
4. Oct 31, 2008
### BryanP
Let's see what we are given. We have G as a group and A as a subset of G. Thus, since G is a group, we have the following giveaways:
1) G is closed under the binary operation
2) G is associative under the binary operation
3) G has an identity element
4) G has an inverse for each element
A is a subset of G with n elements, such that each element has it's inverse NOT contained in the set.
Since A is just a subset of G, we cannot assume that it will have an identity (this is only a giveaway if it were a subgroup). Thus, we will have some sort of a max here, meaning, there is a version of A that has the identity, and a version without the identity.
This matters because the inverse of an identity is itself. Thus if we have A with an identity, we'd have to remove it completely from the set since a = a^-1 and a^-1 is not permitted in A. This gives us (n-1) elements.
Now, what happens when A does NOT have the identity? This means you have n elements since each a in A is not a self-inverse.
Thus you have to account for a set that factors in self-inverses, and a set that doesn't. Getting the number of non-inverses just amounts to taking half of the given number of elements of the particular set (since we know through G that each element of A does have an inverse).
Try applying this to X.
Similar Discussions: Subset of a group | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8792172074317932, "perplexity": 493.9890291071888}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917121893.62/warc/CC-MAIN-20170423031201-00413-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://www.fact-archive.com/encyclopedia/Speed_of_sound | ## Online Encylopedia and Dictionary Research Site
Online Encyclopedia Search Online Encyclopedia Browse
# Speed of sound
The speed of sound varies depending on the medium through which the sound waves pass. It is usually quoted in describing properties of substances (e.g. see the article on sodium).
More commonly the term refers to the speed of sound in air. The speed varies depending on atmospheric conditions; the most important factor is the temperature. The humidity has very little effect on the speed of sound, while the static sound pressure (air pressure) has none. Sound travels slower with an increased altitude (elevation if you are on solid earth), primarily as a result of temperature and humidity changes. An approximate speed (in metres/second) can be calculated from: (The proposal to take the letter v for speed of sound instead of c for speed of light is not generally accepted.)
$c_{\mathrm{air}} = (331{.}5 + 0{.}6 \cdot \vartheta) \ \mathrm{m/s}$
where $\vartheta$ (theta) is the temperature in degrees Celsius.
A more accurate expression is
$c = \sqrt {\kappa \cdot R\cdot T}$
where R (287.05 J/kgK for air) is the universal gas constant R divided by the molar mass of air, κ (kappa) is the adiabatic index (1.402 for air), sometimes called γ, and T is the absolute temperature in kelvins. In the standard atmosphere:
T0 is 273.15 K (= 0 °C = 32 °F), giving a value of 331.5 m/s (= 1193 km/h = 741.5 mph = 643.9 knots).
T20 is 293.15 K (= 20 °C = 68 °F), giving a value of 343.4 m/s (= 1236 km/h = 768.2 mph = 667.1 knots).
T25 is 298.15 K (= 25 °C = 77 °F), giving a value of 346.3 m/s (= 1246 km/h = 774.7 mph = 672.7 knots).
In fact, assuming a perfect gas the speed of sound depends on temperature only, not on the pressure. Air is almost a perfect gas. The temperature of the air varies with altitude, giving the following variations in the speed of sound using the standard atmosphere (actual conditions may vary).
Altitude Temperature m/s km/h mph knots Sea level 15 °C (59 °F) 340 1225 761 661 11000m-20000m (Cruising altitude of commercial jets, and first supersonic flight) -57 °C (-70 °F) 295 1062 660 573 29000m (Flight of X-43A) -48 °C (-53 °F) 301 1083 673 585
In a Non-Dispersive Medium – Sound speed is independent of frequency, therefore the speed of energy transport and sound propagation are the same. Air is a non-dispersive medium.
In a Dispersive Medium – Sound speed is a function of frequency. The spatial and temporal distribution of a propagating disturbance will continually change. Each frequency component propagates at each its own phase speed, while the energy of the disturbance propagates at the group velocity. Water is an example of a dispersive medium.
In general, the speed of sound is given by
$c=\sqrt{\frac{C}{\rho}}$
where C is a coefficient of stiffness and ρ is the density.
Thus the speed of sound increases with the stiffness of the material, and decreases with the density.
In a fluid the only non-zero stiffness is to volumetric deformation ( a fluid does not sustain shear forces).
Hence the speed of sound in a fluid is given by
$c=\sqrt{\frac{K}{\rho}}$
where K is the adiabatic bulk modulus For a gas, K is approximately given by
$K=\kappa\, p$
where
κ is the adiabatic index, sometimes called γ.
p is the pressure.
Thus, for a gas the speed of sound can be calculated using:
$c = \sqrt{{\kappa \cdot p}\over\rho}$
which using the ideal gas law is identical to:
$c = \sqrt {\kappa \cdot R\cdot T}$
(Newton famously used isothermal calculations and omitted the κ from the numerator.)
In a solid, there is a non-zero stiffness both for volumetric and shear deformations. Hence, in a solid it is possible to generate sound waves with different velocities dependent on the deformation mode.
In a solid rod (with thickness much smaller than the wavelength) the speed of sound is given by:
$c = \sqrt{\frac{E}{\rho}}$
where
ρ (rho) is density
Thus in steel the speed of sound is approximately 5100 m/s.
In a solid with lateral dimensions much larger than the wavelength, the sound velocity is higher. It is found be replacing Young's modulus with the plane wave modulus , which can be expressed in terms of the Young's modulus and Poisson's ratio as:
$M=E \frac{1-\nu}{1-\nu-2\nu^2}$
For air, see density of air.
The speed of sound in water is of interest to those mapping the ocean floor. In saltwater, sound travels at about 1500 m/s and in freshwater 1435 m/s. These speeds vary due to pressure, depth, temperature, salinity and other factors.
For general equations of state, if classical mechanics is used, the speed of sound c is given by
$c^2=\frac{\partial p}{\partial\rho}$
where differentiation is taken with respect to adiabatic change. If relativistic effects are important, the speed of sound S is given by:
$S^2=c^2 \left. \frac{\partial p}{\partial e} \right|_{\rm adiabatic}$
(note that $e=\rho (c^2+e^C) \,$ is the relativisic internal energy density; see relativistic Euler equations). This formula differs from the classical case in that ρ has been replaced by $e/c^2 \,$.
## Table - Speed of sound in air c, density of air ρ and acoustic impedance Z vs. temperature °C
Impact of temperature °C c in m/s ρ in kg/m³ Z in N·s/m³ -10 325.4 1.341 436.5 -5 328.5 1.316 432.4 0 331.5 1.293 428.3 +5 334.5 1.269 424.5 +10 337.5 1.247 420.7 +15 340.5 1.225 417.0 +20 343.4 1.204 413.5 +25 346.3 1.184 410.0 +30 349.2 1.164 406.6
Mach number is the ratio of the object's speed to the speed of sound in air (medium). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 14, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.923498272895813, "perplexity": 1090.1683120619282}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358560.75/warc/CC-MAIN-20211128134516-20211128164516-00250.warc.gz"} |
https://www.freemathhelp.com/forum/threads/converting-a-cartesian-plane-to-spherical-coordinates.115446/ | # Converting a Cartesian plane to spherical coordinates?
#### seralin
##### New member
Joined
Apr 9, 2019
Messages
1
If I have the equation of a plane like z = 9 or y = 3, how can I rewrite them in spherical coordinates?
I know that with a point in 3D you would find ρ,θ,φ - for a plane like z = 9 how would I write ρ?
I'm guessing that θ might be 2π, but I'm lost as to how to find ρ and φ for a plane instead of a point.
Thanks!
#### Romsek
##### Full Member
Joined
Nov 16, 2013
Messages
336
If you have some subset of $$\displaystyle \mathbb{R}^3$$ specified by an equation in cartesian coordinates you can always
just substitute in the appropriate expressions in spherical coordinates.
$$\displaystyle x=\rho \sin(\theta)\cos(\phi),~y=\rho \sin(\theta)\sin(\phi),~z=\rho \sin(\theta)$$
for example $$\displaystyle z=9 \Rightarrow \rho \sin(\theta) = 9,~\rho = 9\csc(\theta)$$
#### Dr.Peterson
##### Elite Member
Joined
Nov 12, 2017
Messages
4,110
If I have the equation of a plane like z = 9 or y = 3, how can I rewrite them in spherical coordinates?
I know that with a point in 3D you would find ρ,θ,φ - for a plane like z = 9 how would I write ρ?
I'm guessing that θ might be 2π, but I'm lost as to how to find ρ and φ for a plane instead of a point.
Thanks!
I hope you understand that what you will get is not specific values of the coordinates (as for a point), but an equation in the three coordinates. Romsek gave you the full answer in the case of the plane z=9. In particular, θ does not have one value; it can have any value, and then ρ is a function of that. Moreover, φ is not present in his equation at all; it can have any value independent of the other coordinates.
You can do something similar for y=3, but then all three coordinates are involved.
But beware: not all sources define the coordinates in the same way, so it's possible that your θ and φ are swapped or otherwise modified from Romsek's version. Check for the conversion formulas as given in your own textbook!
#### Romsek
##### Full Member
Joined
Nov 16, 2013
Messages
336
If you have some subset of $$\displaystyle \mathbb{R}^3$$ specified by an equation in cartesian coordinates you can always
just substitute in the appropriate expressions in spherical coordinates.
$$\displaystyle x=\rho \sin(\theta)\cos(\phi),~y=\rho \sin(\theta)\sin(\phi),~z=\rho \sin(\theta)$$
for example $$\displaystyle z=9 \Rightarrow \rho \sin(\theta) = 9,~\rho = 9\csc(\theta)$$
There is an error in above.
$$\displaystyle z = \rho \cos(\theta)$$ and thus $$\displaystyle z=9 \Rightarrow \rho = 9 \sec(\theta)$$
my apologies | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8492541909217834, "perplexity": 574.7259230146941}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195525187.9/warc/CC-MAIN-20190717121559-20190717143559-00421.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-and-trigonometry-10th-edition/chapter-7-7-5-multiple-angle-and-product-to-sum-formulas-7-5-exercises-page-548/45 | ## Algebra and Trigonometry 10th Edition
$x=\frac{5\pi}{3}+2n\pi$ $x=\pi+2n\pi$ where $n$ is an integer.
$sin\frac{x}{2}+cos~x=0$ $sin\frac{x}{2}=-cos~x$ $\pm\sqrt {\frac{1-cos~x}{2}}=-cos~x~~$ (square both sides) $\frac{1-cos~x}{2}=cos^2x$ $2~cos^2x+cos~x-1=0$ $2~cos^2x+2~cos~x-cos~x-1=0$ $2~cos~x(cos~x+1)-(cos~x+1)=0$ $(2~cos~x-1)(cos~x+1)=0$ $cos~x=\frac{1}{2}$ or $cos~x=-1$ $cos~x=\frac{1}{2}$: $x=\frac{\pi}{3}$ or $x=\frac{5\pi}{3}$ Testing the solutions: $sin\frac{\frac{\pi}{3}}{2}+cos~\frac{\pi}{3}=\frac{1}{2}+\frac{1}{2}\ne0$ $sin\frac{\frac{5\pi}{3}}{2}+cos~\frac{5\pi}{3}=-\frac{1}{2}+\frac{1}{2}=0$ $cos~x=-1$: $x=\pi$ Testing the solution: $sin\frac{\pi}{2}+cos~\pi=1-1=0$ The period of $cos~x$ is $2\pi$. The general solutions are: $x=\frac{5\pi}{3}+2n\pi$ $x=\pi+2n\pi$ where $b$ is an integer. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9883429408073425, "perplexity": 157.64425472441496}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711069.79/warc/CC-MAIN-20221206024911-20221206054911-00031.warc.gz"} |
https://economics.stackexchange.com/questions/19520/calculating-the-real-interest-rate-why-would-my-thesis-supervisor-want-me-to-su/19887 | # Calculating the real interest rate, why would my thesis supervisor want me to subtract inflation instead of dividing by inflation?
I'm writing my thesis and for the data I'm using I had calculated the real interest rate by dividing the nominal interest rate by the inflation rate. My supervisor recommended me to change it to instead just subtracting the inflation but I didn't want to ask him why since I've already asked him too many stupid questions.
I always thought dividing is the "real" way and IIRC one of my professors called subtracting the inflation rate a mere "approximation". So why would my supervisor want me to subtract it?
Two things to note here. First, subtracting inflation from the nominal interest rate is an approximation to the real interest rate, but only in discrete time. Furthermore, the "true" relationship it's approximating isn't division of one rate by the other--you have to add 1 to all three of your quantities (inflation, real interest rate, and nominal interest rate) first to get the true relationship.
Here's a brief overview. Consider the Fisher equation of
$r = i - \pi$
where $r$ is the real interest rate, $i$ is the nominal interest rate, and $\pi$ is the inflation rate.
This equation is often introduced as a linear approximation to the true real rate of interest, given by the equation
$\frac{1 + i}{1 + \pi} = 1 + r$
Let's see how this holds in a discrete time model. Denote your nominal income as $Y$ and the price level as $P$. Your real income is $Y/P$. If all this income is invested in some interest-bearing asset in a discrete time model, your real income in the next time period becomes
$\frac{Y (1 + i)}{P (1 + \pi)}$
and if we want to find a real interest rate that summarizes this change in real income, we would need to write your real income in the next time period as
$\frac{Y}{P} (1 + r)$
which gives us the identity $\frac{1 + i}{1 + \pi} = 1 + r$ that the Fisher equation approximates. However, if we work in continuous time, this breaks down. First, the units don't work out--inflation rates and interest rates are measured in percentage change per year (or some other unit of time), so they cannot be added to the dimension-less number $1$. Second, it turns out that Fisher's approximation is actually completely correct in continuous time. Using derivatives, we define our quantities as follows:
$i = \frac{dY}{dt} \frac{1}{Y}$
$\pi = \frac{dP}{dt} \frac{1}{P}$
$r = \frac{d(Y/P)}{dt} \frac{1}{(Y/P)}$.
Using the quotient rule, we can rewrite $r$ as
$r = \frac{\frac{dY}{dt}P - \frac{P}{dt}Y}{P^{2}} \frac{1}{(Y/P)}$
which simplifies to
$r = (\frac{dY}{dt} \frac{1}{P} - \frac{dP}{dt} \frac{Y}{P^{2}}) \frac{P}{Y} = \frac{dY}{dt} \frac{1}{Y} - \frac{dP}{dt} \frac{1}{P} = i - \pi$
which gives us the Fisher equation, no approximations about it!
Note that this assumes continuous compounding from your nominal interest rate. If you instead have a compounding rate of $\tau$, we would define $i$ differently, and our equation becomes
$r = \tau \ln(1 + \frac{i}{\tau}) - \pi$.
However, for the purposes of the data you're working with, I am 90% sure that this modification is completely superfluous. If you want to use something more precise than the Fisher equation, you need to know exactly how your data was computed. What price index was used to calculate the inflation rate? Under what assumptions was the nominal interest rate calculated?
In short, while I can't speak to the reasons for your supervisor's recommendation, they're definitely correct that you should subtract inflation rather than divide by it. There's a reason why we use the Fisher equation.
• Can you explain how the assumption of continuous compounding plays into your definition of i in continuous time? – tjnel Dec 30 '17 at 2:45
• @tjnel Sure! Without continuous compounding, an initial nominal investment of Y becomes Y(1+i/n)^tn at time t, with n as the compounding rate. If we treat the price level and the real investment as growing continuously (with instant compounding), we have (Y(1+i/n)^tn)/(Pe^t(pi)) = (Y/P)e^rt. Dividing both sides by Y/P and taking the natural logarithm of both sides, we have (n)ln(1+i/n) - pi = r. If this is the case, we can't define the nominal interest rate as (dy/dt)/Y, because i is not equal to (n)ln(1+i/n). – SilasLock Dec 30 '17 at 9:36
Rules to keep in mind when dealing with nominal variables.
• To convert a nominal quantity (i.e. nominal GDP) to real value (i.e. real GDP, you divide the nominal value by a measure of price level (i.e. GDP deflator or CPI).
• To convert a nominal interest rate to real interest rate, you subtract inflation rate from nominal interest rate. Read the post about fisher equation @Herr K. suggested above.
Your supervisor wants you to do this because its based on the fisher equation.
$$i =r+\pi$$
where $i$ is nominal interest rate, $r$ is real interest rate and $\pi$ is inflation rate.
rearranging this equation we get the basis for your supervisors recommendation.
$$r=i-\pi$$
Hope this helps. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8480541706085205, "perplexity": 557.2011930320893}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875146665.7/warc/CC-MAIN-20200227063824-20200227093824-00464.warc.gz"} |
https://www.physicsforums.com/threads/how-can-i-diagonalize-this-symmetric-matrix.693505/ | # How can I diagonalize this symmetric matrix?
1. May 24, 2013
### Denisse
\begin{array}{ccc}
2 & 1 & 0 \\
1 & 2 & 0 \\
0 & 0 & 3 \end{array}
2. May 24, 2013
### rock.freak667
If A is your matrix, then first you will need to get your eigenvalues λ by solving
| A-λI| = 0 where I is the identity matrix.
Then you will need to get your eigenvectors. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9196139574050903, "perplexity": 554.5473415908831}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084887067.27/warc/CC-MAIN-20180118051833-20180118071833-00385.warc.gz"} |
https://www.neetprep.com/video-class/7107---Reynolds-Number?subjectId=55&chapterId=685 | #31 | Reynold's Number
(Physics) > Mechanical Properties of Fluids
Related Practice Questions :
In which one of the following cases will the liquid flow in a pipe be most streamlined ?
(a) Liquid of high viscosity and high density flowing through a pipe of small radius
(b) Liquid of high viscosity and low density flowing through a pipe of small radius
(c) Liquid of low viscosity and low density flowing through a pipe of large radius
(d) Liquid of low viscosity and high density flowing through a pipe of large radius
Complete Question Bank + Test Series
Complete Question Bank
Difficulty Level:
The Reynolds number of a flow is the ratio of
(a) Gravity to viscous force
(b) Gravity force to pressure force
(c) Inertia forces to viscous force
(d) Viscous forces to pressure forces | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.939943253993988, "perplexity": 2214.515137219704}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875143373.18/warc/CC-MAIN-20200217205657-20200217235657-00508.warc.gz"} |
https://www.careerstoday.in/maths?conid=11&start=600 | # Maths
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The whole numbers are the part of the number system in which it includes all the positive integers f ... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.88153076171875, "perplexity": 1213.4984910650903}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964362297.22/warc/CC-MAIN-20211202205828-20211202235828-00280.warc.gz"} |
https://brg.a2hosted.com/?page_id=1468 | # Sunday Times Teaser 2653 – Tough Guys
by H Bradley and C Higgins
A ship has two identical vertical masts on the centre line of its deck. These masts are a whole number of feet tall and are seven feet apart horizontally.
The tops of these masts mast are each attached by straight guy ropes to the same anchor point on the centre line of the deck.
One rope is two feet longer than the other and their combined length is a whole number of feet.
What is the height of the masts?
This teaser can be solved analytically as follows. First let the height of the masts be $$h$$, the length of the guy ropes be $$l_1$$ and $$l_2$$ with $$l$$ and $$s$$ being the sum and difference of these lengths respectively. Then let the distance of the anchor point from the closest mast be $$a$$ and the distance between the masts be $$d$$. We hence have $$l_1=(l+s)/2$$ and $$l_2=(l-s)/2$$. Now using the two pythagorean triangles formed by the deck, the masts and the guy ropes we have:$l_1^2=h^2+(a+d)^2=(l+s)^2/4$ $l_2^2=h^2+a^2=(l-s)^2/4$ Taking the difference of these two equations gives $$2ad+d^2=ls$$ and hence: $a=(ls-d^2)/(2d)$ We can now substitute for $$a$$ in the above equations, which gives: $(2dl_1)^2=(2dh)^2+(ls+d^2)^2=d^2(l+s)^2$ $(2dl_2)^2=(2dh)^2+(ls-d^2)^2=d^2(l-s)^2$ Finally by adding these two equations and simplifying we obtain:$(2dh)^2=(d^2-s^2)(l^2-d^2)$
Substituting the given values $$d=7$$ and $$s=2$$ and simplifing the result gives the quadratic diophantine equation $x^2 – 5y^2=-5$ where $$x=2h/3$$ and $$y=l/7$$. This is a variant of Pell’s equation and has multiple solutions, the first being the trivial one $$x=0,y=1$$, the n’th solution being given by expanding$x_n+\sqrt{5}y_n=(9+4\sqrt{5})^n$ and equating the terms on either side. These solutions can also be generated recursively using: $x_{n+1}=9x_n+20y_n$ $y_{n+1}=4x_n+9y_n$ which provides the basis for this Python solution:
with the output:
including the intended solution of 30 feet masts.
Yes, you are right – thanks – i’ll modify my comment.
My empirical solution for this problem is as follows.
Let $$x$$ be the height of the masts, $$y$$ and $$y+2$$ be the lengths of the guy ropes with the anchor point on the left side of the mast holding the shorter rope. Using Pythagorean Theorem we can get: $\sqrt{(y+2)^2-x^2}-\sqrt{y^2-x^2}=7$ giving us $x=(3/14)\sqrt{5(2y-5)(2y+9)}$ whereby by trial and error and using the fact that $$y>7$$ and $$y=a/2$$ where $$a$$ is an integer, we get $$y=(61/2)$$ ft and $$x=30$$ ft.
But your general solution is really very neat and explicit! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9223034977912903, "perplexity": 291.45337703142053}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250594662.6/warc/CC-MAIN-20200119151736-20200119175736-00201.warc.gz"} |
http://mathoverflow.net/questions/64286/dual-borel-conjecture-in-lavers-model | Dual Borel conjecture in Laver's model
A set $X\subseteq 2^\omega$ of reals is of strong measure zero (smz) if $X+M\not=2^\omega$ for every meager set $M$. (This is a theorem of Galvin-Mycielski-Solovay, but for the question I am going to ask we may as well take it as a definition.)
A set $Y$ is strongly meager (sm) if $Y+N\not=2^\omega$ for every Lebesgue null set $N$.
The Borel conjecture (BC) says that every smz set is countable; the dual Borel conjecture (dBC) says that every sm set is countable.
In Laver's model (obtained by a countable support iteration of Laver reals of length $\aleph_2$) the BC holds. Same for the Mathias model.
In a paper that I (with Kellner+Shelah+Wohofsky) just sent to arxiv.org, we claim that it is not clear if Laver's model satisfies the dBC.
QUESTION: Is that correct? Or is it perhaps known that Laver's model has uncountable sm sets?
Additional remark 1: Bartoszynski and Shelah (MR 2020043) proved in 2003 that in Laver's model there are no sm sets of size continuum ($\aleph_2$). (The MR review states that the paper proves that the sm sets are exactly $[\mathbb R] ^{\le \aleph_0}$. This is obviously a typo in the review.)
Additional remark 2: If many random reals are added to Laver's model (either during the iteration, or afterwards), then BC still holds, but there will be sm sets of size continuum, so dBC fails in a strong sense.
-
The paper mentioned in the question is arxiv.org/abs/1105.0823 – Andres Caicedo May 8 '11 at 13:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9779952764511108, "perplexity": 950.1320387811893}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394010749774/warc/CC-MAIN-20140305091229-00012-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://tex.stackexchange.com/questions/473706/7-0pt-%E2%89%A0-x%E2%8B%8510-0pt-for-all-x | # 7.0pt ≠ x⋅10.0pt for all x
Consider the following input:
\documentclass{article}
\begin{document}
\newlength{\smallertopskip}
\setlength{\smallertopskip}{.700004577636718749999999999999999999999999999999\topskip}
\newlength{\largertopskip}
\setlength{\largertopskip}{.700004577636718750000000000000000000000000000000\topskip}
Topskip: \the\topskip
Smaller: \the\smallertopskip
Larger: \the\largertopskip
\end{document}
Running pdflatex on it results in
Topskip: 10.0pt
Smaller: 6.99997pt
Larger: 7.00012pt
As you see above, I tried hard to get exactly 7pt as a result of multiplication of some fixed-point constant with \topskip, but failed. Sure, it's very well-known that fixed-point computations are really inaccurate in TeX, but I'm wondering whether some external package could provide us with a rather general-purpose multiplication function (say, \mult) that is more precise than the built-in multiplication such that
\newlength{\myLength}
\setlength{\myLength}{\mult{x}{\topskip}}
\the\myLength
(or similar code) would result in 7.0pt for some verbatim fixed-point constant x assuming that \topskip is 10.0pt? Notice that both 7.0 and 10.0 are representable in binary, and the mathematically correct answer x=7/10=0.7 is not representable in binary, but the fixed-point arithmetics is different anyway...
Aside: As some answers and comments notice, the difference between the under- and overapproximation is invisible to the naked eye. But the difference has an effect on potential further computations with \myLength, including choosing, e.g., a font size (which is a non-continuous operation).
• To achieve exact results on any system, TeX does not use floating-point at all for lengths. It does everything with integer units of sp, where 1 pt equals 65536 sp. (Another way of saying it is that it uses fixed-point arithmetic.) Your 10 pt corresponds to 655360 sp, your \smallertopskip to 458750 sp, and your \largertopskip to 458760 sp. (And your desired “target” 7 pt corresponds to 458752 sp.) Knuth regrets using binary instead of decimal here, probably because it leads to confusion like this. – ShreevatsaR Feb 7 at 4:55
• You wrote, "Sure, it's very well-known that floating-point computations are really inaccurate in TeX." Actually, TeX does not use floating-point arithmetic, for reasons that have been noted many years ago -- by Knuth himself, as well as by others. Instead, as @ShreevatsaR has already point out in a comment, TeX uses fixed-point arithmetic. A separate comment: Are you concerned that there might be a meaningful, i.e., visually observable, difference between 6.9997pt and 7.00012pt? Please clarify. – Mico Feb 7 at 5:39
• (Since I realized an ambiguity in my first comment) Using decimal instead of binary won't change the mathematics (with bounded memory you can have only limited precision), but just makes the human experience easier: e.g. instead of having fractions that are multiples of 1/65536 if you had say 1/10000 or 1/100000, then instead of 0.70000457763671875 being an interesting bound (it's halfway between two representable numbers 45875/65536 = 0.6999969482421875 and 45876/65536 = 0.70001220703125), some consecutive representable numbers may be (say) 0.69999, 0.70000, 0.70001 -- easier to understand. – ShreevatsaR Feb 7 at 18:38
It's not really that TeX floating point calculations are inaccurate, it simply isn't doing floating point at all, so you either need a macro implementation such as the xfp package in Werner's answer, or suitably scale the calculation so that it is accurate in the range of fixed point arithmetic being used by TeX and using integer quantities that can be stored exactly (as opposed to 0.7) also helps.
\documentclass{article}
\begin{document}
\newlength{\smallertopskip}
\setlength{\smallertopskip}{\dimexpr 7\topskip/10\relax}
Topskip: \the\topskip
Smaller: \the\smallertopskip
\end{document}
Although it only really makes a difference in this special case that the value is a known integer. In general, if you are going to need decimal places in the answer just rounding to 1 or two decimal places will ensure a consistent result and not make any real difference to the output, .000001pt isn't very big.
• That's probably better than loading thousands of lines of expl3 macros. – Henri Menke Feb 7 at 8:07
• @HenriMenke yes but if you know you want a zero decimal places answer it's easy to arrange an integer answer, if you want 75% of 11pt, then perhaps less so, it depends why you are worrying about a +/-.00001 pt difference I suppose. – David Carlisle Feb 7 at 11:42
• @user0 you need an awful lot of rounding to make that visible, it really isn't ever going to happen. How many further calculations are you ever going to do with topskip? and dimexpr and gluexpr are the same here as the form <factor><glue> discards the stretch and shrink components anyway. – David Carlisle Feb 7 at 20:11
• but given that you can not store .7 exactly in binary, you really can't worry about a difference in the 5th decimal place or you'd never be able to do any calculations at all. @user0 – David Carlisle Feb 7 at 20:24
• @user0 yes that's why I constructed the arithmetic as I did in this answer, keeping all intermediate steps exactly representable *7 divide 10 rather than multiply by 0.7 – David Carlisle Feb 7 at 20:48
Use xfp:
7.0pt
\documentclass{article}
\usepackage{xfp}
\begin{document}
\newlength{\myLength}%
\setlength{\myLength}{\fpeval{round(0.7 * \topskip, 0)}pt}%
\the\myLength
\end{document}
Dimensions in \fpeval are converted to pt and stripped of the dimension part in order to perform calculations (hence the addition of a "closing pt").
If you want you can define
\newcommand{\mult}[2]{\fpeval{round(#1 * #2, 0)}pt}
\setlength{\myLength}{\mult{0.7}{\topskip}}
• @user0: It may counter the errors that could result from floating point computations. – Werner Feb 7 at 5:28
• @user0 Of course, even with xfp (or any approach) once you assign to a length, you'll only get integer multiples of 1 sp (= 1/65536 pt), so this only affects the rounding that happens to 0.7 (or whatever) before the multiplication; it doesn't affect the range of possible values for the length at the end. – ShreevatsaR Feb 7 at 6:25
• This looks like a quite complicated way to say \setlength{\mylenth}{7pt}. – Ulrike Fischer Feb 7 at 7:50
• @user0 that was a joke. But I would use \setlength{\myLength}{\fpeval{\dim_to_decimal:n {\topskip} * 0.7}pt} instead of rounding the result. – Ulrike Fischer Feb 7 at 19:12
There is no way to get a length of 458752sp from <factor>\topskip if \topskip has the value 10pt, that is, 655360sp, because TeX don't do floating-point computations, but fixed-point base two arithmetic.1
The binary representation of 7/10 is 0.10(1100), parentheses denote the period. and the multiplication rules of TeX can only provide either 458750sp or 458760sp, represented respectively as 6.99997pt and 7.00012pt.
The difference between the upper and lower best representations is 10sp, which is less than 0.000435 millimeters or 0.00016 points.
Since the usual value of \vfuzz is 0.1pt (less than 0.03mm), there should be no concern about getting an “exact” value: you'd need to cumulate more than 680 such errors in order to exceed the \vfuzz.
\documentclass{article}
\newlength{\multipletopskip}
\begin{document}
Topskip: \the\topskip (\number\topskip sp)
70\% topskip: \the\dimexpr 7\topskip/10\relax (\number\dimexpr 7\topskip/10\relax sp)
\setlength{\multipletopskip}{0.7\topskip}
0.7 topskip: \the\multipletopskip (\number\multipletopskip sp)
\setlength{\multipletopskip}{0.70001\topskip}
0.70001 topskip: \the\multipletopskip (\number\multipletopskip sp)
\end{document}
As you see, 0.7\topskip is accurate up to 2sp, less than 0.00009mm.2
Unless you completely override TeX's computation by using a different model such as IEEE754 (decimal32) as is done in Werner's answer, you can't get “exact” values.
Footnotes
1 When TeX was written, there was no agreed upon standard for floating-point computations and Knuth's aim was to obtain the same output on every machine TeX was implemented on. Using 64 bits instead of 32 could have achieved “better” accuracy, but at the expense of speed and need for memory: PC's of that time might have even less than 640 kiB of RAM.
2 Being a skip, it would be more sensible to use \glueexpr rather than \dimexpr, as noted by GuM in comments. Note that <factor><skip register> will discard the plus and minus components, whereas \multiply and \divide don't. So
\setlength{\multipletopskip}{\glueexpr\topskip*7/10\relax}
could be better.
• In fact, how tiny these differences are becomes clearer when we switch to small units: a difference of 10 sp is about 53 nanometres. As The TeXbook says “Since the wavelength of visible light is approximately 100 sp, [DEK adds comment in texbook.tex: in fact, violet=75sp, red=135sp] rounding errors of a few sp make no difference to the eye”. – ShreevatsaR Feb 7 at 12:49
• Don't you mean 0.1pt is approx. 0.03mm, not 0.3mm? 0.3mm is about 1/3 of a mm, which is about 1/75 of an inch, which is about 1pt not 0.1pt. – alephzero Feb 7 at 16:09
• This answer contains the implicit remark that eTeX’s \dimexpr (or \glueexpr) provides better precision: why not to make this remark explicit? – GuM Feb 7 at 18:59
• @user0: I don’t think it is a good idea to allow the \topskip glue to stretch or shrink; nonetheless, Knuth decided to make \topskip a <glue parameter>, not a <dimen parameter>, so, in the principle, using \glueexpr should be safer (and more elegant). It depends, however, on what you are trying to achieve: you may well want to kill \tpskip’s shrinkability/stretchability intentionally. – GuM Feb 7 at 19:15
• @user0 7\topskip discards the stretch, if you don't want to do that use \setlength{\myLengthG}{\glueexpr\topskip*7/10\relax} – David Carlisle Feb 7 at 20:15
This should not be an answer, but rather a comment both to @egreg’s answer and to David Carlisle’s; unfortunately, I need to include some sample code, and this can only be done (in an intelligible form) in an answer. I willingly concede that it doesn’t add anything substantial to those two answers—except, perhaps, a bit of clarity. I’m ready to remove this answer if either of the abovementioned authors clarifies his.
As already repeatedly remarked, Knuth’s TeX does (or rather, did) its calculations with dimensions using 32-bit fixed points arithmetics; all modern typesetting engines, however, incorporate the so-called “e-TeX” (for Extended, or Enhanced, TeX) extensions, among which is the ability to perform dimension scaling, that is, multiplication of a dimension for a fraction, with 64-bit precision. More precisely, e-TeX extensions introduce a new type of syntax by means of which dimensions can be specified, that enables the use of “expressions”, in the customary sense of the term; in particular, you are allowed to multiply a certain dimension for a fraction, as in
\setlength\someotherdimen{\dimexpr \somedimen * 7/10}
(the primitive \dimexpr marks the beginning of a “dimen-valued” expression; the end is implicitly marked by the first token that cannot be interpreted as part of the expression itself—you can assume it is the closing brace, in the above example). When this is done, a 64-bit temporary register is used to hold the intermediate result of the multiplication of the dimension for the numerator of the fraction (\somedimen * 7, in the above example); thank to this expedient, when the subsequent division for the denominator (10, in our case) is performed, all bits of the final (32-bit) result are significant.
As already said, \dimexpr starts a “dimen-valued” expression; there exists a similar primitive for “glue-valued” expressions, named \glueexpr. Note, however, the difference, that others have already explained, between
\setlength\someotherskip{\glueexpr 7\someskip /10}
and
\setlength\someotherskip{\glueexpr \someskip * 7/10}
In the former, 7\someskip is converted to a dimen value, destroying (that is, zeroing) its stretch and shrink components, if any; this doesn’t happen with the latter.
The following code proves that
\setlength{\someotherskip}{\glueexpr \topskip * 7/10}
attains the same precision as explicitly setting \someotherskip to 7pt (and similarly for the stretch and shrink components, if present):
% My standard header for TeX.SX answers:
\documentclass[a4paper]{article} % To avoid confusion, let us explicitly
% declare the paper format.
\usepackage[T1]{fontenc} % Not always necessary, but recommended.
% End of standard header. What follows pertains to the problem at hand.
\newcommand*{\ReportDimen}[2]{%
#1:\>\texttt{\the #2}\\\>\texttt{(\number #2sp)}\\%
}
\newcommand*{\ReportGlue}[2]{%
#1:\>\texttt{\the #2}\\\>%
\texttt{(%
\number #2sp
plus \number\expandafter\dimexpr\the\gluestretch #2\relax sp
minus \number\expandafter\dimexpr\the\glueshrink #2\relax sp%
)}\\%
}
\setlength{\topskip}{10.0pt plus 1pt minus 1 pt}
\newlength{\smallertopskip}
\setlength{\smallertopskip}{.700004577636718749999999999999999999999999999999\topskip}
\newlength{\largertopskip}
\setlength{\largertopskip}{.700004577636718750000000000000000000000000000000\topskip}
\newlength{\eTeXTopskip}
\setlength{\eTeXTopskip}{\glueexpr \topskip * 7/10}
\newlength{\sevenpoints}
\setlength{\sevenpoints}{7.0pt plus .7pt minus .7pt}
\begin{document}
\begin{tabbing}
\ReportGlue {Topskip}{\topskip}
\ReportDimen{Smaller}{\smallertopskip}
\ReportDimen{Larger}{\largertopskip}
\ReportGlue {eTeX's}{\eTeXTopskip}
\ReportGlue {Exact}{\sevenpoints}
\end{tabbing}
\end{document}
(values in scaled points are shown as well).
• Why the \expandafter after \number and before \dimexpr in \ReportGlue? Those shouldn't be necessary, as well as \the in that expression and the \dimexpr. \number\gluestretch #2sp should suffice. – Skillmon Apr 1 at 22:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9305142164230347, "perplexity": 2749.152237469804}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526536.46/warc/CC-MAIN-20190720153215-20190720175215-00274.warc.gz"} |
http://math.stackexchange.com/questions/288918/about-the-use-of-stirling-approximation/288979 | # About the use of Stirling approximation
How to prove this inequality: $$\ln \Gamma \left( x \right)-2\ln \Gamma \left( \frac{x+1}{2} \right)>\frac{2x}{3}$$ Sry I forgot to mention that $x>300$
-
Try plugging in $x=1$ to see what happens. – Byron Schmuland Jan 28 '13 at 13:41
@gauss115 - please revise the question, as Byron pointed out the statement as it stands is false. – nbubis Jan 28 '13 at 13:54
@nbubis sry! The condition is $x>300$ – gauss115 Jan 28 '13 at 14:00
How do you know this is true? What did you try to show this is true? – Did Jan 28 '13 at 14:06
Another approach would be to use approximations. There is a quickly convergent version of Stirling's formula which goes like this:$$\ln{\Gamma(x)}=\left(x-\frac{1}{2}\right)\ln{x}-x+\frac{\ln{2\pi}}{2}+\frac{1}{12(x+1)}+O(x^{-2})$$ (see http://goo.gl/9hsnO). Derive upper and lower bound from this and plug back into your inequality. You'll end up with a relatively straightforward logarithmic inequality.
Simply take the derivative of your function: $$f'(x) = \psi(x)-\psi\left(\frac{x+1}{2}\right)$$ And then show that $f''(x) >0$ and that $f'(x)>2/3$ for say $x=20$. This shows that the derivative is larger for all $x>20$.
Now calculate $f(300) > 200$ to prove that $f(x) > 2x/3$ for all $x>300$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9113307595252991, "perplexity": 244.57890121482424}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398445219.14/warc/CC-MAIN-20151124205405-00002-ip-10-71-132-137.ec2.internal.warc.gz"} |
http://math.stackexchange.com/users/59019/yanpeng?tab=activity | # Yanpeng
less info
reputation
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# 11 Actions
Jan21 awarded Scholar Jan21 accepted How should I prove a set is convex? Jan21 awarded Supporter Jan21 comment How should I prove a set is convex? thanks for your answer @hardmath Jan21 comment How should I prove a set is convex? @cardinal, thank you too. Jan21 comment How should I prove a set is convex? @Mercy, thank you very much. Jan21 comment How should I prove a set is convex? @JuliánAguirre, I'm sure there is equality in the definition of $\mathbf{S}$. Jan21 comment How should I prove a set is convex? @JaivirBaweja, I don't understand how could this $x^Tx = 1$ condition be obtained from $\mathbf{S}$? Jan21 comment How should I prove a set is convex? @Mercy, it will be pleasure if any hint could be given. Jan21 awarded Student Jan21 asked How should I prove a set is convex? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9376379251480103, "perplexity": 2410.817465466381}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413507444385.33/warc/CC-MAIN-20141017005724-00288-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://en.wikipedia.org/wiki/Weil_restriction | Weil restriction
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In mathematics, restriction of scalars (also known as "Weil restriction") is a functor which, for any finite extension of fields L/k and any algebraic variety X over L, produces another variety ResL/kX, defined over k. It is useful for reducing questions about varieties over large fields to questions about more complicated varieties over smaller fields.
Definition
Let L/k be a finite extension of fields, and X a variety defined over L. The functor ${\displaystyle \mathrm {Res} _{L/k}X}$ from k-schemesop to sets is defined by
${\displaystyle \mathrm {Res} _{L/k}X(S)=X(S\times _{k}L)}$
(In particular, the k-rational points of ${\displaystyle \mathrm {Res} _{L/k}X}$ are the L-rational points of X.) The variety that represents this functor is called the restriction of scalars, and is unique up to unique isomorphism if it exists.
From the standpoint of sheaves of sets, restriction of scalars is just a pushforward along the morphism Spec L ${\displaystyle \to }$ Spec k and is right adjoint to fiber product of schemes, so the above definition can be rephrased in much more generality. In particular, one can replace the extension of fields by any morphism of ringed topoi, and the hypotheses on X can be weakened to e.g. stacks. This comes at the cost of having less control over the behavior of the restriction of scalars.
Properties
For any finite extension of fields, the restriction of scalars takes quasiprojective varieties to quasiprojective varieties. The dimension of the resulting variety is multiplied by the degree of the extension.
Under appropriate hypotheses (e.g., flat, proper, finitely presented), any morphism ${\displaystyle T\to S}$ of algebraic spaces yields a restriction of scalars functor that takes algebraic stacks to algebraic stacks, preserving properties such as Artin, Deligne-Mumford, and representability.
Examples and applications
1) Let L be a finite extension of k of degree s. Then ${\displaystyle \mathrm {Res} _{L/k}}$(Spec L) = Spec(k) and ${\displaystyle Res_{L/k}\mathbb {A} ^{1}}$ is an s-dimensional affine space ${\displaystyle \mathbb {A} ^{s}}$ over Spec k.
2) If X is an affine L-variety, defined by
${\displaystyle X={\text{Spec}}L[x_{1},\dots ,x_{n}]/(f_{1},\dots ,f_{m})}$
we can write ${\displaystyle \mathrm {Res} _{L/k}X}$ as Spec ${\displaystyle k[y_{i,j}]/(g_{l,r})}$, where yi,j (${\displaystyle 1\leq i\leq n,1\leq j\leq s}$) are new variables, and gl,r (${\displaystyle 1\leq l\leq m,1\leq r\leq s}$) are polynomials in ${\displaystyle y_{i,j}}$ given by taking a k-basis ${\displaystyle e_{1},\dots ,e_{s}}$ of L and setting ${\displaystyle x_{i}=y_{i,1}e_{1}+\dots +y_{i,s}e_{s}}$ and ${\displaystyle f_{t}=g_{t,1}e_{1}+\dots +g_{t,s}e_{s}}$.
3) Restriction of scalars over a finite extension of fields takes group schemes to group schemes.
In particular:
4) The torus
${\displaystyle \mathbb {S} :=\mathrm {Res} _{\mathbb {C} /\mathbb {R} }\mathbb {G} _{m}}$
where Gm denotes the multiplicative group, plays a significant role in Hodge theory, since the Tannakian category of real Hodge structures is equivalent to the category of representations of S. The real points have a Lie group structure isomorphic to ${\displaystyle \mathbb {C} ^{\times }}$. See Mumford–Tate group.
5) The Weil restriction ${\displaystyle \mathrm {Res} _{L/k}\mathbb {G} }$ of a (commutative) group variety ${\displaystyle \mathbb {G} }$ is again a (commutative) group variety of dimension ${\displaystyle [L:k]\dim \mathbb {G} }$, if L is separable over k. Aleksander Momot applied Weil restrictions of commutative group varieties with ${\displaystyle k=\mathbb {R} }$ and ${\displaystyle L=\mathbb {C} }$ in order to derive new results in transcendence theory which were based on the increase in algebraic dimension.
6) Restriction of scalars on abelian varieties (e.g. elliptic curves) yields abelian varieties, if L is separable over k. James Milne used this to reduce the Birch and Swinnerton-Dyer conjecture for abelian varieties over all number fields to the same conjecture over the rationals.
7) In elliptic curve cryptography, the Weil descent attack uses the Weil restriction to transform a discrete logarithm problem on an elliptic curve over a finite extension field L/K, into a discrete log problem on the Jacobian variety of a hyperelliptic curve over the base field K, that is potentially easier to solve because of K's smaller size.
Weil restrictions vs. Greenberg transforms
Restriction of scalars is similar to the Greenberg transform, but does not generalize it, since the ring of Witt vectors on a commutative algebra A is not in general an A-algebra.
References
The original reference is Section 1.3 of Weil's 1959-1960 Lectures, published as:
• Andre Weil. "Adeles and Algebraic Groups", Progress in Math. 23, Birkhäuser 1982. Notes of Lectures given 1959-1960.
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http://mathhelpforum.com/calculus/212835-finding-slope-tangent-line.html | # Math Help - Finding a slope of tangent line
1. ## Finding a slope of tangent line
Hey guys
So what I really need help with is not finding the slope it's how do I combine the square root into the F(4+H) part of the derivative formula ?
Let
The slope of the tangent line to the graph of at the point is .
The equation of the tangent line to the graph of at is for
and
.
Hint: the slope is given by the derivative at , ie.
2. ## Re: Finding a slope of tangent line
Just substitute "4 + h" for "x" under the radical.
3. ## Re: Finding a slope of tangent line
This question just doesn't seem to be working out... I calculated my derivative as 0 which means a slope of 0 .
?
4. ## Re: Finding a slope of tangent line
See the attachment. Try the conjugate in the Newton ratio. Remember x^2 - y^2 = (x - y)(x + y).
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http://math.stackexchange.com/questions/170827/finite-non-abelian-p-group-cannot-split-over-its-center | Finite non-abelian $p$-group cannot split over its center
Show that a finite non-abelian $p$-group cannot split over its center.
I'd be happy for some clues.
-
Write $P = Q \times Z(P)$. How are $Z(Q)$ and $Z(P)$ related? How big is $Z(Q)$? – Jack Schmidt Jul 14 '12 at 19:13
@JackSchmidt: I know that splitting gives a direct sum in abelian categories, but I've never seen a corresponding result in the category of groups. Are you asserting that if we have maps $H\to G \to H$ with composite the identity map, then $G=H\times K$ for some group $K$? If so, is there an easy proof? Or is that just what it means to spilt in group theory? – Aaron Jul 14 '12 at 23:22
@Aaron: I think Jack just misspoke. "Splitting over $K$" means that $G$ is a semidirect product with $K$ normal. – Arturo Magidin Jul 14 '12 at 23:48
@ArturoMagidin: Ahh, thanks. – Aaron Jul 15 '12 at 1:36
@Aaron: Splitting over a central subgroup is always a direct product. In general it is only semi. – Jack Schmidt Jul 15 '12 at 2:12
This definition is not very common, so it may be worth mentioning here:
Definition. Let $G$ be a group. A subgroup $K$ of $G$ is said to be co-central in $G$ if $G=Z(G)K$.
Theorem. Let $G$ be a group. If $K$ is co-central in $G$, then $Z(K)\subseteq Z(G)$.
Proof. Let $z\in Z(K)$, and $x\in G$. We need to show that $zx=xz$. Since $G=Z(G)K$, there exists $a\in Z(G)$ and $k\in K$ such that $x=ak$. Then \begin{align*} zx &= zak\\ &= azk &&\text{(since }a\in Z(G)\text{)}\\ &= akz &&\text{(since }z\in Z(K)\text{ and }k\in K\text{)}\\ &= xz. \end{align*} Thus, $z\in Z(G)$, as claimed. $\Box$
Now assume that we can write a $p$-group $P$ as $P=Z(P)H$ with $H$ a subgroup. Then by the lemma, $Z(H)\subseteq Z(P)$. If $P$ were split over $Z(P)$, then we would have $Z(P)\cap H = \{e\}$, hence $Z(H)=\{e\}$.
Why is that a very big problem for $H$, which can only be solved if $H=\{e\}$?
-
+1 Nice explanation and even nicer theorem! – DonAntonio Jul 15 '12 at 2:05
Suppose $P=Z(P)H$ with $1<H<P$ nontrivial and $H\cap Z(P)=1$. Two questions:
• Can $Z(H)\subseteq H$ and $Z(P)$ intersect nontrivially?
• Can $Z(H)=1$? Observe that $H$ is a $p$-group.
Show that every element of $Z(P)Z(H)$ commutes with every element of $Z(P)H=P$, and therefore derive the containment $Z(P)Z(H)\subseteq Z(P)\implies Z(H)\subseteq Z(P)$, contra trivial intersection.
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https://solarenergyengineering.asmedigitalcollection.asme.org/POWER/proceedings-abstract/POWER2011/44595/727/351428 | The spacer of Boiling Water Reactor (BWR) becomes the flow obstacle. Moreover the clearance between fuel rods of the innovative Water Reactor for Flexible fuel cycle (FLWR) becomes very narrow. Thus understanding of the thermal-fluid characteristics in the boiling channel equipped with the flow obstacle becomes more important. In this study, to clarify the flow obstacle effect, the experimental investigation was conducted with the forced convective boiling system. The test section was 8 mm in inner diameter, and the rod-type flow obstacle, which had 3.6 mm in diameter and 20 mm in length, was installed. The blockage ratio β = 20.3% which was the similar value of the present BWR reactor. The heating length LT was taken three different lengths, LT = 810, 840 and 900 mm. As the experimental results, the CHF was increased by the installing of the flow obstacle, and it was strongly influenced by the axial position of the flow obstacle. In the most of the case, the liquid film dryout was detected at the exit of the test section, while the CHF was observed at the upstream of the flow obstacle in the case of the LT = 810 mm. The calculation results of the liquid film flow model have shown a good agreement with these experimental results by using the presented influence length of the flow obstacle.
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https://chem.libretexts.org/Bookshelves/General_Chemistry/Map%3A_Chemistry_-_The_Central_Science_(Brown_et_al.)/11%3A_Liquids_and_Intermolecular_Forces | # 11: Liquids and Intermolecular Forces
The physical properties of a substance depends upon its physical state. Water vapor, liquid water and ice all have the same chemical properties, but their physical properties are considerably different. In general Covalent bonds determine: molecular shape, bond energies, chemical properties, while intermolecular forces (non-covalent bonds) influence the physical properties of liquids and solids. The kinetic molecular theory of gases described in Chapter 10 gives a reasonably accurate description of the behavior of gases. A similar model can be applied to liquids, but it must take into account the nonzero volumes of particles and the presence of strong intermolecular attractive forces.
• 11.1: A Molecular Comparison of Gases, Liquids, and Solids
The state of a substance depends on the balance between the kinetic energy of the individual particles (molecules or atoms) and the intermolecular forces. The kinetic energy keeps the molecules apart and moving around, and is a function of the temperature of the substance and the intermolecular forces try to draw the particles together.
• 11.2: Intermolecular Forces
Molecules in liquids are held to other molecules by intermolecular interactions, which are weaker than the intramolecular interactions that hold molecules and polyatomic ions together. The three major types of intermolecular interactions are dipole–dipole interactions, London dispersion forces (these two are often referred to collectively as van der Waals forces), and hydrogen bonds.
• 11.3: Some Properties of Liquids
Surface tension, capillary action, and viscosity are unique properties of liquids that depend on the nature of intermolecular interactions. Surface tension is the energy required to increase the surface area of a liquid. Surfactants are molecules that reduce the surface tension of polar liquids like water. Capillary action is the phenomenon in which liquids rise up into a narrow tube called a capillary. The viscosity of a liquid is its resistance to flow.
• 11.4: Phase Changes
Fusion, vaporization, and sublimation are endothermic processes, whereas freezing, condensation, and deposition are exothermic processes. Changes of state are examples of phase changes, or phase transitions. All phase changes are accompanied by changes in the energy of a system. Changes from a more-ordered state to a less-ordered state (such as a liquid to a gas) are endothermic. Changes from a less-ordered state to a more-ordered state (such as a liquid to a solid) are always exothermic.
• 11.5: Vapor Pressure
Because the molecules of a liquid are in constant motion and possess a wide range of kinetic energies, at any moment some fraction of them has enough energy to escape from the surface of the liquid to enter the gas or vapor phase. This process, called vaporization or evaporation, generates a vapor pressure above the liquid. Molecules in the gas phase can collide with the liquid surface and reenter the liquid via condensation. Eventually, a steady state or dynamic equilibrium is reached.
• 11.6: Phase Diagrams
The states of matter exhibited by a substance under different temperatures and pressures can be summarized graphically in a phase diagram, which is a plot of pressure versus temperature. Phase diagrams contain discrete regions corresponding to the solid, liquid, and gas phases. The solid and liquid regions are separated by the melting curve of the substance, and the liquid and gas regions are separated by its vapor pressure curve, which ends at the critical point.
• 11.7: Structure of Solids
A crystalline solid can be represented by its unit cell, which is the smallest identical unit that when stacked together produces the characteristic three-dimensional structure. Solids are characterized by an extended three-dimensional arrangement of atoms, ions, or molecules in which the components are generally locked into their positions. The components can be arranged in a regular repeating three-dimensional array. The smallest repeating unit of a crystal lattice is the unit cell.
• 11.8: Bonding in Solids
The major types of solids are ionic, molecular, covalent, and metallic. Ionic solids consist of positively and negatively charged ions held together by electrostatic forces; the strength of the bonding is reflected in the lattice energy. Ionic solids tend to have high melting points and are rather hard. Molecular solids are held together by relatively weak forces, such as dipole–dipole interactions, hydrogen bonds, and London dispersion forces. Metallic solids have unusual properties.
• 11.E: Liquids and Intermolecular Forces (Exercises)
These are homework exercises to accompany the Textmap created for "Chemistry: The Central Science" by Brown et al.
• 11.S: Liquids and Intermolecular Forces (Summary)
This is the summary Module for the chapter "Liquids and Intermolecular Forces" in the Brown et al. General Chemistry Textmap.
Thumbnail: A water drop. (CC BY 2.0; José Manuel Suárez). | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8170715570449829, "perplexity": 1017.2340568956412}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320300010.26/warc/CC-MAIN-20220116180715-20220116210715-00681.warc.gz"} |
https://mattbaker.blog/tag/diophantine-approximation/ | # The Stern-Brocot tree, Hurwitz’s theorem, and the Markoff uniqueness conjecture
My goal in this post is to describe a surprising and beautiful method (the Stern-Brocot tree) for generating all positive reduced fractions. I’ll then discuss how properties of the tree yield a simple, direct proof of a famous result in Diophantine approximation due to Hurwitz. Finally, I’ll discuss how improvements to Hurwitz’s theorem led Markoff to define another tree with some mysterious (and partly conjectural) similarities to the Stern-Brocot tree.
# Real Numbers and Infinite Games, Part II
In my last post, I wrote about two infinite games whose analysis leads to interesting questions about subsets of the real numbers. In this post, I will talk about two more infinite games, one related to the Baire Category Theorem and one to Diophantine approximation. I’ll then talk about the role which such Diophantine approximation questions play in the theory of dynamical systems.
The Choquet game and the Baire Category Theorem
The Cantor game from Part I of this post can be used to prove that every perfect subset of ${\mathbf R}$ is uncountable. There is a similar game which can be used to prove the Baire Category Theorem. Let $X$ be a metric space. In Choquet’s game, Alice moves first by choosing a non-empty open set $U_1$ in $X$. Then Bob moves by choosing a non-empty open set $V_1 \subseteq U_1$. Alice then chooses a non-empty open set $U_2 \subseteq V_1$, and so on, yielding two decreasing sequences $U_n$ and $V_n$ of non-empty open sets with $U_n \supseteq V_n \supseteq U_{n+1}$ for all $n$. Note that $\bigcap U_n = \bigcap V_n$; we denote this set by $U$. Alice wins if $U$ is empty, and Bob wins if $U$ is non-empty. Continue reading | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 14, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9452060461044312, "perplexity": 395.2042288754994}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662627464.60/warc/CC-MAIN-20220526224902-20220527014902-00371.warc.gz"} |
http://umj.imath.kiev.ua/article/?lang=en&article=6887 | 2017
Том 69
№ 9
# A Mixed Problem for One Pseudoparabolic System in an Unbounded Domain
Abstract
We prove the existence and uniqueness of a solution of a mixed problem for a system of pseudoparabolic equations in an unbounded (with respect to space variables) domain.
English version (Springer): Ukrainian Mathematical Journal 53 (2001), no. 1, pp 141-148.
Citation Example: Domans'ka G. P., Lavrenyuk S. P. A Mixed Problem for One Pseudoparabolic System in an Unbounded Domain // Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 123-129.
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https://physics.stackexchange.com/questions/640382/what-is-the-inverse-map-of-r-jku-frac12-rm-tr-sigma-j-u-sigma-ku | # What is the inverse map of $R_{jk}(U)=\frac{1}{2}{\rm Tr}(\sigma_j U \sigma_kU^\dagger)$?
Given a $$2\times 2$$ unitary, unimodular matrix $$U\in {\rm SU}(2)$$, the (elements of the) corresponding $$3\times 3$$ rotation matrix $$R\in {\rm SO}(3)$$ can be obtained from the map $$R_{jk}(U)=\frac{1}{2}{\rm Tr}(\sigma_j U \sigma_kU^\dagger)\tag{1}$$ where $$\sigma_j,\sigma_k$$ represent the Pauli matrices. What is the inverse map that determines (the elements of) $$U$$ from $$R$$?
• See (3.23) to (3.25) here, and (4.59) to (4.61) for the Lorentz analog. May 31 at 22:18
1. OP's formula (1) follows from the relation [1] $$U \sigma_k U^{-1}~=~ \sum_{j=1}^3\sigma_j R^j{}_k.\tag{A}$$ See also e.g. this & this related Phys.SE posts.
2. The map
$$SU(2)~\ni~ U~\mapsto~ R~\in~ SO(3)\tag{B}$$ is 2:1, so the inverse map does strictly speaking not exist. However, there exists a double-valued map, cf. e.g. my Phys.SE answer here.
3. There is an injective map from rotation vectors $$\vec{\alpha}~\in~B(\vec{0},\pi)~:=~\{\vec{\alpha} \in\mathbb{R}^3 \mid |\vec{\alpha}|< \pi\} \tag{C}$$ (belonging to an open neighborhood) to $$3\times 3$$ rotation matrices $$R(\vec{\alpha})~=~\exp(i\vec{\alpha}\cdot \vec{L})~\in~ SO(3)~ \subseteq ~{\rm Mat}_{3\times 3}(\mathbb{R}). \tag{D}$$ Then we can pick a continuous branch $$U(\vec{\alpha})~=~\exp(\frac{i}{2}\vec{\alpha}\cdot \vec{\sigma})~\in~ SU(2)~ \subseteq ~{\rm Mat}_{2\times 2}(\mathbb{C}), \tag{E}$$ of the double-valued map in an open neighborhood. This establishes a continuous inverse map in an open neighborhood.
References:
1. G 't Hooft, Introduction to Lie Groups in Physics, lecture notes; chapters 3 + 6. The pdf file is available here.
• I was looking for an inverse map i.e. finding (the elements of) $U$ knowing the elements of $R$, maybe up to a sign. With this, can we make an inversion?
– SRS
May 31 at 16:22
I am addressing the comment of the OP to the complete answer of @Qmechanic. Given the real orthogonal 3×3 matrix (D), $$R(\vec \alpha)= e^{i\vec \alpha\cdot \vec L} \equiv e^{\theta \mathbb L},\\ \vec \alpha\equiv \theta \hat n, ~~~|\hat n|=1, ~~~{\mathbb L} =\begin{pmatrix} 0&-n_z& n_y\\ n_z&0 &-n_x\\-n_y& n_x&0\end{pmatrix} ,$$ real antisymmetric, where the three real coefficients $$\vec \alpha$$ of the Lie algebra element are broken down to an angle $$\theta$$ and a unit axis of rotation $$\hat n$$, so that $${\mathbb L}^3 =-{\mathbb L}~~\leadsto$$, $$R(\theta \hat n) = {\mathbb 1}+ \sin\theta ~{\mathbb L}+ (1-\cos \theta)~{\mathbb L}^2.$$
This is the celebrated matrix form of Rodrigues' rotation formula, with three terms, the middle one being real antisymmetric, and the extremal ones being real symmetric. So, manifestly, $$(\operatorname{Tr}~R-1)/2= \cos{\theta},$$ to solve for θ, whence $$R-R^T= 2\sin \theta ~{\mathbb L}~~\leadsto \\ \hat n= (R_{32}-R_{23}, R_{13}-R_{31}, R_{21}-R_{12})^T/2\sin\theta,$$ so you have determined $$\vec \alpha$$, which specifies $$U=\cos\theta/2 +i\sin\theta/2 ~~~\hat n\cdot \vec\sigma$$, as per the answer commented on. Choose your quadrants and half-angle values to your convenience.
Let $$g_{rs}$$ be a matrix in $$\text{SO(3)}$$, while $$U$$ be a matrix in $$\text{SU(2)}$$. We can find as in the OP:
$$g_{rs} = \frac{1}{2}\text{Tr}\left(\sigma^r U \sigma^{s} U^{\dagger}\right) \tag{1}$$
The inverse of this $$2:1$$ mapping is
$$U = \mp \frac{1+\sigma^r \sigma^s g_{rs}}{2\left(1+\text{Tr}~g\right)^{\frac{1}{2}}} \tag{2}$$
Bibliography: Carmeli, M., Malin, S. "Theory of spinors. An introduction", WS, 2000, page 7, eqns. 1.14 and 1.15.
For this problem the completness relation of the pauli matrices is here very useful:
$$\sigma^j_{\alpha \beta}\sigma^j_{\gamma \delta}=2\delta_{\alpha \delta}\delta_{\beta \gamma}-\delta_{\alpha\beta}\delta_{\gamma \delta},$$ where the superindices denote the typ of the pauli matrix and the subindices denotes the matrix index and we sum over repeated indices.
Given a matrix $$R_{jk}(U)$$, we can get the product of the matrix elements of $$U$$ and $$U^\dagger$$ back (so the map can not be uniquely inverted as mentioned above.) as shown in the following:
We take $$R_{jk}(U)$$ and multiply it with $$\sigma^j_{\alpha \beta}$$ and $$\sigma^k_{\gamma \delta}$$ where we in the following assume summation over all repeated indices. We get after writing the trace in $$R_{jk}(U)$$ also in index notation (using latin indices for the trace and greek letters for the new free indices):
$$\sigma^{j}_{\alpha\beta}R_{jk}(U)\sigma^k_{\gamma \delta}=\frac{1}{2} \sigma^{j}_{\alpha\beta}\sigma^j_{ab} U_{bc} \sigma^k_{cd}U^\dagger_{da}\sigma^k_{\gamma \delta}$$
Using the above completeness relation twice for the summations over j and k, then resolving all appearing kronecker deltas and using that $$UU^\dagger=1$$, we get in the end:
$$\sigma^{j}_{\alpha\beta}R_{jk}(U)\sigma^k_{\gamma \delta}=U_{\alpha\delta}U^\dagger_{\gamma \delta}-\frac{1}{2}\delta_{\alpha \beta}\delta_{\gamma \delta}$$
(if I do not messed something up....)
From that formula, one can extra the matrix elements of $$U$$ up to the an factor of $$-1$$ by probing different index combinations of $$\alpha,\beta,\gamma,\delta$$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 45, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9739528894424438, "perplexity": 384.59896985070253}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964361253.38/warc/CC-MAIN-20211202084644-20211202114644-00340.warc.gz"} |
https://www.physicsforums.com/threads/spring-with-weight-question-potential-energy-of-the-spring.809448/ | # Spring with weight question / Potential energy of the spring
1. Apr 19, 2015
### masterchiefo
1. The problem statement, all variables and given/known data
The spring of a dynamo-meter grows 20cm when one suspends a 4N on weight. What is the potential energy of the spring when one suspends a weight of 12N?
2. Relevant equations
Ue = 1/2*k*r2
Ug= m*g*h
3. The attempt at a solution
so to find k I do this step.
1/2*k*0.22 = 4*0.2
k=40N/M
Now that I have k I can find the potential energy of the spring with 12N
1/2*40*r2 =12*r
I find r to be 0.6
now I do 12*0.6 = 7.2J to fnd the potential energy of the spring.
The real answer is 3.6J but I cant figure out how to get this.
2. Apr 19, 2015
### PhanthomJay
Looks like you tried to set the PE of the spring equal to the work done by the weight force, but you calculated the work wrong when doing it this way. Try Hookes law to calculate k.
3. Apr 19, 2015
### masterchiefo
I dont understand, I am already using Hookes law to calculate K.
How am I supposed to find K with only this formula W= 1/2*k*r2
I dont have K nor W and only r
4. Apr 19, 2015
### PhanthomJay
Write down Hookes law. Solve for k knowing that the spring extends 20 cm in the equilibrium position when the weight is 4 N.
5. Apr 19, 2015
### masterchiefo
Thats Hookes law... W= 1/2*k*r2
4*0.2= 1/2*k*0.22
6. Apr 19, 2015
### PhanthomJay
No that is not Hookes law. The work done by a spring is the negative of its change in PE which somewhat resembles your equation, but what you want is Hookes law to make life easier, please look it up.
7. Apr 20, 2015
### collinsmark
As PhanthomJay said, W = (1/2)kr2 is not Hooke's law. You don't want to use W = (1/2)kr2 when calculating the spring constant. (Although you will use it later when calculating the spring's potential energy.)
If you are puzzled as to why this conservation of energy approach does not work for calculating the spring constant k, I might be of help. Using the equation
mgh = (1/2)kh2
Implies the following: Initially, the weight is put on the relaxed spring, then released such that weight falls on its own accord. Eventually, the weight's velocity will momentarily drop back to zero when it falls a distance h below its initial position. At this point in time the initial energy of mgh has been transferred to the potential energy of the spring, (1/2)kh2.
But it won't stop there! The mass will then shoot back up to its initial position and oscillate back and forth like that until frictional forces slowly dampen the oscillation.
So that h that I describe above is not the h that you are looking for.
In this problem, instead of letting the mass fall on its own accord, the mass is gently lowered until it reaches a distance of 20 cm, at which point the system stays in a state of equilibrium (no oscillations). That's where you want to use Hooke's law to find the spring constant. Hooke's law describes force; not potential energy (at least not directly). | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8479426503181458, "perplexity": 873.1709297882398}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257650993.91/warc/CC-MAIN-20180324190917-20180324210917-00284.warc.gz"} |
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[1] Hari Bercovici and Dan Timotin. Truncated Toeplitz operators and complex symmetries. Proc. Amer. Math. Soc. 146 (2018) 261-266. Abstract, references, and article information View Article: PDF [2] György Pál Gehér. Asymptotic limits of operators similar to normal operators. Proc. Amer. Math. Soc. 143 (2015) 4823-4834. MR 3391040. Abstract, references, and article information View Article: PDF [3] Kunyu Guo, Youqing Ji and Sen Zhu. A $C^*$-algebra approach to complex symmetric operators. Trans. Amer. Math. Soc. 367 (2015) 6903-6942. Abstract, references, and article information View Article: PDF [4] Dan Timotin. Note on a Julia operator related to model spaces. Contemporary Mathematics 638 (2015) 247-254. Book volume table of contents View Article: PDF [5] Dan Timotin. A short introduction to de Branges--Rovnyak spaces. Contemporary Mathematics 638 (2015) 21-38. Book volume table of contents View Article: PDF [6] Yueshi Qin and Rongwei Yang. 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Linearly similar Szökefalvi-Nagy--Foias model in a domain. St. Petersburg Math. J. 15 (2004) 289-321. MR 2052133. Abstract, references, and article information View Article: PDF This article is available free of charge [15] Jonathan Arazy and Miroslav Englis. Analytic models for commuting operator tuples on bounded symmetric domains. Trans. Amer. Math. Soc. 355 (2003) 837-864. MR 1932728. Abstract, references, and article information View Article: PDF This article is available free of charge [16] Dale R. Buske. Hilbert modules over a class of semicrossed products. Proc. Amer. Math. Soc. 129 (2001) 1721-1726. MR 1814102. Abstract, references, and article information View Article: PDF This article is available free of charge [17] Kehe Zhu. A sharp estimate for extremal functions. Proc. Amer. Math. Soc. 128 (2000) 2577-2583. MR 1662242. Abstract, references, and article information View Article: PDF This article is available free of charge [18] Zhidong Pan. Triangular extension spectrum of weighted shifts. Proc. Amer. Math. Soc. 126 (1998) 3293-3298. MR 1476383. Abstract, references, and article information View Article: PDF This article is available free of charge [19] Ralph deLaubenfels. Similarity to a contraction, for power-bounded operators with finite peripheral spectrum. Trans. Amer. Math. Soc. 350 (1998) 3169-3191. MR 1603894. Abstract, references, and article information View Article: PDF This article is available free of charge [20] Michael A. Dritschel and Hugo J. Woerdeman. Model theory and linear extreme points in the numerical radius unit ball. Memoirs of the AMS 129 (1997) MR 1401492. Book volume table of contents [21] Adele Zucchi. Operators of class $C_0$ with spectra in multiply connected regions. Memoirs of the AMS 127 (1997) MR 1388899. Book volume table of contents [22] Hari Bercovici and Adele Zucchi. Generalized interpolation in a multiply connected region. Proc. Amer. Math. Soc. 124 (1996) 2109-2113. MR 1322912. Abstract, references, and article information View Article: PDF This article is available free of charge [23] James Guyker. The de Branges-Rovnyak model with finite-dimensional coefficients . Trans. Amer. Math. Soc. 347 (1995) 1383-1389. MR 1257108. Abstract, references, and article information View Article: PDF This article is available free of charge [24] Hari Bercovici and Srdjan Petrović. Generalized scalar operators as dilations . Proc. Amer. Math. Soc. 123 (1995) 2173-2180. MR 1246516. Abstract, references, and article information View Article: PDF This article is available free of charge [25] A. A. Jafarian, H. Radjavi, P. Rosenthal and A. R. Sourour. Simultaneous triangularizability, near commutativity and Rota's theorem . Trans. Amer. Math. Soc. 347 (1995) 2191-2199. MR 1257112. Abstract, references, and article information View Article: PDF This article is available free of charge [26] Srdjan Petrović. On the similarity of centered operators to contractions . Proc. Amer. Math. Soc. 121 (1994) 533-541. MR 1182705. Abstract, references, and article information View Article: PDF This article is available free of charge [27] Joel Pincus and Shao Jie Zhou. Principal currents for a pair of unitary operators. Memoirs of the AMS 109 (1994) MR 1243584. Book volume table of contents [28] Cheng Zu Zou. The unicellularity of contractions of class $C\sb 0$ . Proc. Amer. Math. Soc. 119 (1993) 775-782. MR 1163329. Abstract, references, and article information View Article: PDF This article is available free of charge [29] V. Müller and F.-H. Vasilescu. Standard models for some commuting multioperators . Proc. Amer. Math. Soc. 117 (1993) 979-989. MR 1112498. Abstract, references, and article information View Article: PDF This article is available free of charge [30] M. A. Kaashoek and S. M. Verduyn Lunel. Characteristic matrices and spectral properties of evolutionary systems . Trans. Amer. Math. Soc. 334 (1992) 479-517. MR 1155350. 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Results: 1 to 30 of 75 found Go to page: 1 2 3 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9544675350189209, "perplexity": 2396.229192626019}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812306.16/warc/CC-MAIN-20180219012716-20180219032716-00289.warc.gz"} |
http://rspa.royalsocietypublishing.org/content/462/2076/3575 | # Chirality and domain growth in the gyroid mesophase
Jonathan Chin, Peter V Coveney
## Abstract
We describe the first dynamical simulations of domain growth during the self-assembly of the gyroid mesophase from a ternary amphiphilic mixture, using the lattice Boltzmann method. The gyroid is a chiral structure; we demonstrate that, for a symmetric amphiphile with no innate preference for left- or right-handed morphologies, the self-assembly process may give rise to a racemic mixture of domains. We use measurements of the averaged mean curvature to analyse the behaviour of domain walls, and suggest that diffusive domain growth may be present in this system.
Keywords:
## 1. Introduction
Emergent phenomena occur when many elements, each with very simple properties, display unexpectedly complex behaviour when gathered together in large numbers (Skår & Coveney 2003; Coveney & Fowler 2005). Such behaviour gives rise to a vast number of patterns and structures in nature. A good example of this is provided by amphiphile molecules, which, despite having a very simple structure in general, can spontaneously assemble into a wide variety of ordered structures; in particular, the membranes of living cells are constructed from these molecules.
One of these structures is called the gyroid mesophase, whose properties, once assembled, have been studied experimentally and theoretically. However, little is understood about the process of self-assembly. Using a lattice Boltzmann model which can capture both surfactant dynamics and fluid flow effects, we are able to simulate the self-assembly of such a phase. In particular, we demonstrate below that the self-assembly process of the gyroid mesophase can give rise to imperfections or defects in its structure.
## 2. Amphiphile mesophases
The word amphiphile comes from the Greek words amfi, for ‘both’, and filein, ‘to like’, and means a kind of molecule that is composed of two parts, each attracted to a different species. Soap is a commonly encountered amphiphile: soap molecules are composed of a polar head group which is hydrophilic, or attracted to water, and a long hydrophobic or lipophilic tail group which is repelled by water and attracted to oil. In a mixture of oil, water and soap, the soap molecules are strongly attracted to the interface between the oil and the water, allowing them to sit in the minimal-energy configuration with the head group facing towards water and the tail group facing into the oil. This tendency to migrate towards interfaces is why such molecules are commonly termed surfactants, or surface-active agents. Surfactants have immense industrial importance, mainly due to their tendency to sit at interfaces and thereby reduce surface tension; this is the principle by which all detergents work.
While a solution of surfactant in water is easily prepared in the kitchen sink, such solutions show very rich and complicated behaviour, summarized by, for example, Gompper & Schick (1994) and Langevin (1999), or the reviews of Seddon & Templer (1993, 1995). In a very dilute solution, amphiphile molecules exist as monomers, but beyond a certain critical concentration, they may assemble into a variety of morphologies, such as spherical, wormlike, or branching micellar clusters, or bicontinuous structures, comprised of interpenetrating networks of oil and water channels separated by a monolayer of amphiphile. The networks may be isotropic sponges with no long-range order, but can also form liquid crystalline structures with translational symmetry. This behaviour depends on many variables, such as temperature, surfactant volume fraction, non-polar chain length, polar head-group strength (which, in turn, depends on factors such as pH and ionic concentration) and the form of the interaction between tail groups, between head and tail groups and between head groups.
### (a) The Canham–Helfrich Hamiltonian
A surfactant molecule in a monolayer will have a variety of lateral forces acting on it. Repulsion between tail groups and between head groups acts to push molecules apart; surface tension effects act to pull them together. The effects of these forces can often be difficult to separate, and the nature of the individual forces is still poorly understood (Seddon & Templer 1995).
A useful idealization of this system is to think of it as being composed of an oil-filled region, a separate water-filled region and a surfactant membrane separating the two. The oil and water regions may have a highly tortuous morphology, giving rise to a correspondingly complicated shape for the surfactant membrane. However, the complicated interactions between molecules can now be described in terms of their net effect on the surface, and, in particular, their effect on its curvature.
Consider a hypothetical symmetric amphiphile, whose head and tail groups are of the same size and exert the same magnitude of forces. At an interface, the ideal configuration of amphiphile molecules is planar; by symmetry, there is no innate preference for the interface to curve in one direction or another. If one bends the surface, then head groups will be pushed together and tail groups pulled apart, or vice versa; this incurs an energy cost. At any point on a surface, maximum and minimum curvatures can be found, corresponding to the minimum and maximum radii of curvature, respectively; the sign of the curvature indicates its direction. These are called the principal curvatures and ; the mean and Gaussian curvatures are defined as and , respectively. The free-energy cost of curvature at a point on an interface was characterized by Canham and Helfrich (Canham 1970),(2.1)where κ is called the splay curvature modulus and is the saddle splay modulus. This is integrated over the interfacial surface to find the Hamiltonian for the complete system. The Helfrich Hamiltonian, which describes the energetics of a fictional surface of zero thickness (Schröder et al. 2004), is still far from the full story for surfactants; packing frustration and hydrocarbon chain stretching are important, for example, in the inverse hexagonal wormlike micellar phases (Seddon & Templer 1993). However, it is sufficient to predict many of the possible mesophase morphologies.
### (b) Bicontinuous triply periodic minimal surface phases
The existence of bicontinuous phases was first suggested by Scriven (1976). In these phases, the amphiphile surface must both extend throughout space and minimize its curvature energy. These constraints are satisfied by a class of structures known as triply periodic minimal surfaces (TPMS). A ‘minimal surface’ has zero mean curvature and negative Gaussian curvature at all points; a TPMS is a minimal surface with cubic symmetry, repeating in the X, Y and Z directions. A brief history of the TPMS is given by Schwarz & Gompper (2002); until 1970, only five TPMS were known, discovered by H. A. Schwarz and his students in the late nineteenth century. Schoen (1970) discovered 12 more, describing them in a NASA technical report. Little attention was paid to the new surfaces, until their existence was rigorously proven by Karcher, who discovered the existence of others in addition (Karcher & Polthier 1996). Further theoretical details of various aspects of periodic minimal surfaces are described in the literature (Mackay 1985; Góźdź & Hołyst 1996; Karcher & Polthier 1996; Klinowski et al. 1996; Hyde & Ramsden 2000; Hyde & Schroeder 2003; Lord & Mackay 2003; Enlow et al. 2004).
Three surfaces of particular interest in surfactant morphologies are called the P, D and G surfaces (figure 1). P and D were discovered by H. A. Schwarz; G, the ‘gyroid’, was discovered by Schoen. Fogden & Hyde (1999) give a detailed analysis of the P, D and G surfaces. An analytical form for these three surfaces (among others) is known in terms of the Enneper–Weierstrass representation (Fogden & Hyde 1999; Gandy & Klinowski 2000), which maps the fundamental patch of each surface into the complex plane. It can be shown (Gandy & Klinowski 2000) that the P, D and G surfaces all have the same Weierstrass representation, except for a single parameter θ, called the Bonnet angle. Variation of θ maps the surfaces on to one another, and is called a Bonnet transformation. The Enneper–Weierstrass representation of these surfaces is unfortunately a little cumbersome to work with, since it involves several elliptic integrals. Nonetheless, it has been used to derive many properties of the surfaces, such as the area, Euler characteristic and Gaussian curvature, per unit cell. These have also been derived numerically for the gyroid and related surfaces of constant mean curvature (Große-Brauckmann 1997), by discretizing a surface and permitting it to evolve, under volume constraints, to H=0.
Figure 1
Three common cubic triply periodic minimal surfaces. (a) The P, or ‘plumber's nightmare’ surface. (b) The D, or ‘diamond’ surface, so called because each labyrinth has the same structure as the carbon atoms in a diamond crystal (Hyde 1989). (c) The G, or ‘gyroid’ surface.
## 3. The gyroid
Of these three surfaces, the ‘gyroid’ G is perhaps the most interesting. It is the only known TPMS, which is balanced (i.e. the two labyrinths can be mapped onto each other through a Euclidean transformation) while containing no straight lines; it is also the only known TPMS composed entirely from triple junctions. The gyroid has space group symmetry ; the unit cell consists of 96 copies of a fundamental surface patch, related through the symmetry operations (Gandy & Klinowski 2000) of this space group. Channels run through the gyroid labyrinths in the (100) and (111) directions; passages emerge perpendicular to any given channel as it is traversed, the direction at which they do so gyrating down the channel, giving rise to the ‘gyroid’ name (Große-Brauckmann 1997). The labyrinths are chiral, so that the channels of one labyrinth gyrate in the opposite sense to the channels of the other. Looking down the (111) direction of a gyroid shows a distinctive ‘wagon wheel’ pattern (Avgeropoulos et al. 1997), which has been observed experimentally in transmission electron micrographs (TEM) of gyroid phases (Shefelbine et al. 1999). Gyroids have been observed in many experimental systems, and are usually regarded as the most commonly occurring of the cubic TPMS geometries. They have been seen in triblock copolymers (Shefelbine et al. 1999) and lipid–water mixtures (Seddon & Templer 1993, 1995; Mariani et al. 1996; Czeslik & Winter 2002). Attempts have been made (Chan et al. 1999) to construct a ceramic nanostructured film with the gyroid morphology through the use of a self-assembling polymer template, raising the interesting possibility of fabricating chirally selective porous media. The possibility of using TPMS morphologies for photonic crystals is under active investigation (Urbas et al. 2002).
There have been hints that lyotropic liquid crystals may play a part in biological processes such as lipid digestion (Patton & Carey 1979; Rigler et al. 1986), and offer insights into cell membrane properties and dynamics (Stier et al. 1978; Rand 1981; Prestegard & O'Brien 1987). Donnay & Pawson (1969) suggested that periodic minimal surfaces could be found in nature, and pointed at the microscopic structure of sea urchin skeletons (Nissen 1969) as a possible example; micrographic evidence has since been emerged to suggest that cubic TPMS phases may be present in certain plant cell organelles, and Landh (1995) suggests, with micrographic evidence, that gyroid structures may exist in the endoplasmic reticulum. Schwarz & Gompper (1999, 2000) examined several TPMS morphologies, and suggested that, for oil–water symmetric systems giving rise to zero spontaneous mean curvature, the gyroid was the most stable structure.
## 4. Dynamical simulations
Despite the significant progress (Schwarz & Gompper 1999) made in free-energy models, which through analysis and Monte Carlo simulation give a good understanding of equilibrium properties and phase stability, comparatively little is known about the dynamics of the cubic phases, or indeed mesophases in general; free-energy functionals will not give information about the dynamics of a system far from equilibrium. The limitation to equilibrium states is a severe one, for several reasons. First, real-world systems may contain boundaries or imperfections, which prevent ‘perfect’ equilibrium phases from forming. Second, rheological or otherwise dynamical properties of a mesophase are non-equilibrium by definition, and so cannot be captured by such treatments. Many of the mechanical properties of metals and electrical properties of semiconductors are determined primarily by the nature of defects and impurities, so an understanding of the non-equilibrium behaviour is essential for the investigations of material properties. Mesoscale techniques, such as dissipative particle dynamics or lattice Boltzmann, since they can handle kinetic descriptions, offer a non-equilibrium alternative to the free-energy descriptions. Groot & Madden (1998) performed dissipative particle dynamics simulations of diblock copolymer melts, which allowed reproduction of several well-known morphologies, and also showed the dynamical pathways through which they were reached. In particular, they showed the existence of a ‘gyroid-like’ phase with many triple junctions. However, the phase was not exactly of cubic symmetry, possibly because of finite size effects and frustration. Various course-grained molecular dynamics based mesoscale simulation methods have been developed in the past few years, with applications inter alia to surfactant self-assembly (Marrink et al. 2004).
Using an early version of the LB3D code, Nekovee & Coveney (2001) showed that it was possible to use the lattice Boltzmann model of Chen et al. (2000) to simulate the assembly dynamics of lamellar and bicontinuous phases of binary water–surfactant systems; indeed, they showed the existence of a P-surface cubic phase of the model. Then, while investigating the effect of the presence of surfactant on spinodal decomposition, González-Segredo & Coveney (2004a,b) discovered the existence of a gyroid cubic phase. Importantly, use of the lattice Boltzmann method allowed modelling of the dynamics of the self-assembly of the gyroid. These lattice Boltzmann simulations of the gyroid phase demonstrated several important features. First, it had been suggested (Prinsen et al. 2002) that simulation of cubic phases might require a model with more complicated amphiphiles than symmetric dimers, and also that long-range interactions might be required. The lattice Boltzmann simulations, which used symmetric amphiphiles with short-range interactions, refuted this. Second, analysis of the X-ray structure factor of the simulation data showed the existence of oscillating modes at long times; visualization of the simulation output strongly suggested that these were due to Marangoni effects (Scriven & Sternling 1960; Grotberg & Gaver 1996).
Theissen et al. (1998) describe a lattice Boltzmann model of amphiphilic systems which is based around a Ginzburg–Landau free-energy functional (Gompper & Schick 1990; Schwarz & Gompper 2000); however, it should be noted that this model lacks an explicit surfactant density, making the assumption that the surfactant is all adsorbed onto the oil–water interface. It is not expected, therefore, that this model would reproduce Marangoni effects at the interface. Lamura et al. (1999) describe a Ginzburg–Landau model with explicit surfactant density, but their model, in turn, lacks explicit surfactant orientation. The model used in LB3D has both explicit density and orientation for surfactant particles.
The gyroid self-assembly simulations used initial conditions of a randomized mixture of oil, water and surfactant, with the oil and water in 1 : 1 ratio. Phase separation would occur rapidly, within the first thousand time-steps or so. After phase separation, the system would form a ‘molten gyroid’ phase, consisting of many oil or water rods surrounded by surfactant; these rods would join up to form triple junctions, giving rise to a gyroid morphology. Towards the end of these simulations, after around 30 000 time-steps, the Marangoni oscillations were observed. These simulations were originally performed to investigate the dynamics of surfactant-limited phase separation; discovery of the gyroid phase was an interesting side effect. Owing to this, the simulations of González and Coveney were not optimally suited to examining the gyroid phase, in several ways. First, they were limited in size: most of the simulation work was done with 643 systems. These are perfectly adequate for phase-separation studies; however, it was observed that while at early times there might be several different gyroid regions with different orientations, at long times these would join up to form a single gyroid grain spanning the entire simulation grid, a so-called ‘perfect gyroid’. In the real world, cubic mesophases are far from perfect: despite careful experimental procedures (Hajduk et al. 1994), artificially created gyroid materials may not consist of a perfect gyroid repeating throughout space, but rather of many gyroid grains with different orientations. In addition, there may exist dislocations and defects within these grains, analogous to the defects found in ordinary solid crystals (Feynman et al. 1964; Kittel 1996). Any investigation of the gyroid domains or the nature of their interactions would require a simulation of sufficient physical size to accommodate several domains for the majority of the simulation.
Second, they were limited in time: the time-scales probed (maximum of 35 000 time-steps) offered few hints as to how transient the effects observed were. Phase separation was observed to happen extremely quickly, over a 1000 time-steps or so, and formation of gyroid morphology on a slower but comparable time-scale. The behaviour of gyroid defects, on the other hand, takes place on a much slower scale, at least tens of thousands of time-steps, and these time-scales were not probed. With these limitations in mind, a new set of simulations was performed specifically to look at the nature of defect dynamics in the gyroid liquid crystal phase.
## 5. The TeraGyroid simulations
It was clear that larger simulations were required; how much larger was not so clear, since defects had not been observed in the unambiguous absence of finite size effects. Defect dynamics simulations were performed as part of a larger project (Blake et al. 2005) in which a transatlantic supercomputing grid was constructed. As part of this project, a very substantial amount of computing power became available for use. The simulations of most interest used the parameter sets (González-Segredo & Coveney 2004a) called numbers 8 and 9, which are shown in table 1.
View this table:
Table 1
Parameter sets used in TeraGyroid simulations.
Simulations were run at sizes of, among others, 643, 1283 and 2563, for durations listed in table 2. If a stable gyroid state was reached, then a simulation was terminated; the systems which did not reach this state ran for as long as possible under the resource constraints at the time. During each simulation, the order parameter field was stored to disk, as well as the surfactant density field . This was performed every time-steps, for values of also shown in table 2. The output rate was chosen according to several constraints; it had to be sufficiently rapid to allow observation of the gyroid dynamics, but limits were imposed by local disk quotas, which would often shift due to data from other concurrent jobs.
View this table:
Table 2
Simulation durations and end states.
## 6. Analysis techniques
### (a) The structure function
A commonly examined property of mesoscale systems is the structure function , defined as the spherically averaged Fourier transform of the order parameter ,(6.1)This is very closely related to the ‘structure factor’ measured in small angle X-ray scattering (SAXS) experiments, which are commonly used in experimental examination of mesophases (Avgeropoulos et al. 1997; Laurer et al. 1997; Sakurai et al. 1998; Shefelbine et al. 1999), and has been an important tool in the analysis of many mesoscale simulations (Velasco & Toxvaerd 1993; Rybka et al. 1995; Wagner & Yeomans 1998; González-Segredo & Coveney 2004b). To measure the dominant length-scale of a system, the reciprocal space first moment of is usually calculated,(6.2)The corresponding length-scale is then .
### (b) Polygonal approximation of the interface
Since the interfacial properties are so crucial to the behaviour of the gyroid phase, it is important to be able to extract the location of the two-dimensional interfacial surface from the three-dimensional simulation data. In a continuum system, the interface between the oil and water phases can be defined as the implicit surface , for . In contrast, the simulation output data consists of the values of ϕ at discrete sites on the lattice; it is not quite so straightforward to define an interfacial surface because, in general, while can be positive or negative, corresponding to the lattice site containing a majority of red or blue particles, respectively, it is almost never exactly zero. However, a continuum order parameter field can still be estimated from the discrete simulation output, by linear interpolation; this interpolated function is single-valued and continuous across the simulated region, and can be regarded as the simulated approximation to the order parameter field; the isosurface is then well defined.
Generation of a polygon mesh approximation to this surface can be achieved through several means. The most popular method is known as marching cubes, this technique was developed by Lorensen & Cline (1987). In marching cubes, each point on the lattice is assigned a value of 1 if is ‘negative’, or unambiguously less than zero, and 0 if is ‘positive’, i.e. greater than zero or within floating-point error of zero. If a positive site is adjacent to a negative site, then changes sign somewhere along the link between the two sites, and therefore the link between the sites intersects the contour surface; the position of these intersections can be found easily through interpolation. The marching cubes algorithm proceeds by considering each cube of eight sites in turn, calculating a number between 0 and 255, representing the state of these sites, and then using this number to find a set of polygons approximating the interface shape inside the cube. The polygon sets can be stored in a lookup table, making the algorithm extremely fast.
Unfortunately, the algorithm is not guaranteed to produce a surface with the same topology as the implicit interpolated surface . In fact, it is possible for marching cubes to produce ‘non-manifold’ surfaces containing holes; this arises because some of the 256 possible states of a cube have more than one way of being tiled with polygons, and the tilings are topologically different.
Several approaches have been suggested to correct this problem. One such technique is to tile the space with tetrahedra rather than cubes (Carneiro et al. 1996; Treece et al. 1999), which produces a topologically consistent surface without holes, but at the expense of speed, memory and geometrical accuracy (Lewiner et al. 2003). Another approach is the recursive dividing cubes algorithm (Cline et al. 1988), which has similar performance penalties, particularly for large datasets.
Chernyaev (1995) observed that it was possible to extend the marching cubes approach to produce polygon surface meshes which are not only topologically consistent (i.e. do not have any holes), but also topologically correct (i.e. homeomorphic to the surface ). An implementation of this technique was described by Lewiner et al. (2003), who used lookup tables to determine which tests to perform, resulting in a very efficient algorithm to produce a triangle mesh which is both topologically consistent and topologically correct.
### (c) Fluid properties at the interface
By applying this algorithm to the simulation output datasets, a polygonal approximation to the interface between the two fluid components can be found. This is useful not only for direct visualization of the system, but also for calculating many interesting properties.
First, the interfacial area of the system can be calculated, simply by adding up the areas of all of the triangles in the surface. Without knowing the location of the interface, any space-varying property , such as surfactant density, can only be averaged over the bulk of the three-dimensional system. However, once the interface has been polygonized, the variation of α over the interface can now be found, by interpolating α to the surface vertices and taking an area-weighted average. Consider a single triangle , area , vertices . The value of α averaged over the triangle is(6.3)This relationship is exact if α varies linearly over the surface of the triangle. Integrating α over the entire closed polygon mesh M is equivalent to summing the values integrated over each individual triangle,(6.4)Hence, the average of α over the surface is(6.5)
### (d) Interfacial curvature
Any point on the surface has a unit normal , defined as . The shape operator is a rank-2 tensor defined as .
The Gaussian and mean curvatures can then be found from the determinant and half the trace of the shape operator, respectively. More usefully, the surface-averaged value of the squared mean curvature, , can also be calculated; this quantity is physically interesting because it appears as a term in the Helfrich Hamiltonian. The contribution to the surface-averaged curvature from each lattice site can also be calculated. This curvature field has low values where there is no interface, or where the interface has a minimal surface structure, and it has a high value where the interface is highly curved; consequently, it can be used to distinguish non-minimal-surface defect regions from the bulk gyroid phase (figure 2).
Figure 2
Orthographic-projection volume rendering highlighting domain walls for a 1283 simulation using parameter set 9, showing the late-time collapse of the last domain into a pair of line dislocations. Periodic boundaries are enforced; one of the dislocations straddles an edge of the simulation box.
### (e) Interfacial Euler characteristic
Consider a two-dimensional closed surface tiled with a polygon mesh; each polygon is joined to its neighbours by a straight edge, and the edges meet at point vertices. Suppose that the surface has V vertices, E edges and F polygon faces; the Euler characteristic or Euler number of the surface is defined as . A surprising and very powerful result of elementary topology (Kinsey 1993; Nakahara 1993; Armstrong 1997) is that the Euler characteristic depends only on the topology of the surface, and not on its shape or the manner in which it is tiled with polygons. More precisely (Kinsey 1993), if and are compact, connected surfaces without boundary, then is topologically equivalent to if and only if , and either both are orientable or both are non-orientable. A surface is non-orientable if it contains Möbius bands; all of the surfaces under consideration here are orientable, so determination of topological equivalence reduces to comparison of the Euler characteristic.
The Euler number is additive; a system with n surfaces each with Euler number has total Euler number . The surface meshes produced by the marching cubes algorithm are composed entirely of triangles; this property simplifies calculation of the Euler characteristic. Each triangle on the mesh has three edges; each edge on the mesh joins exactly two triangles. Hence, the number of edges is 3/2 times the number of triangles, .
The Euler number of a surface is directly related to another topological quantity called the genus g, which is often regarded as simply the number of holes in the surface. For an orientable surface (Kinsey 1993), . The Euler number is an extremely useful concept when trying to understand the output of mesoscale simulations. Imagine a lattice Boltzmann simulation of a spherical-droplet phase, where the droplets are defined as regions for which the order parameter . Since the Euler number of a sphere is 2, and since it is additive, the number of droplets in the system is simply half the Euler number of the ϕ=0 surface. Moreover, the Euler number, being a topological invariant, is insensitive to fluctuations in the droplet shape or size, provided that each droplet remains simply connected.
Recently, the Euler number was used by Hołyst & Oswald (1998) to characterize fluctuating wormhole-like passages (Goos & Gompper 1993) appearing in Monte Carlo simulations of lamellar phases; similar wormholes have been observed in a lattice gas dynamical model (Boghosian et al. 1996). In a system with periodic boundary conditions, a membrane spanning both the x and y directions is topologically equivalent to the surface of a torus, and therefore has Euler number zero; thus, a lamellar stack of membranes will also have Euler number zero. However, if a wormhole forms between two lamellae, the Euler number will suddenly jump to −2, hence the Euler number is for n such wormholes. The Euler number has also been used in examinations of TPMS surfactant morphologies (Schwarz & Gompper 2000) and experimental MRI data (Gerig et al. 1999). A gyroid surface has Euler characteristic −8 per unit cell.
## 7. Analysis of the simulation data
So far, both the techniques used to simulate assembly of the gyroid mesophase and the techniques used to analyse the resulting data have been presented. The following sections present the actual analysis of the generated data, and a discussion of the physics involved.
### (a) Initial phase separation
The simulated system is initialized to a very highly mixed and disordered state, often called a ‘quench’ in the literature. The system undergoes spinodal decomposition (Bray 1994; Kendon et al. 1999; Chin & Coveney 2002; González-Segredo et al. 2003), and rapidly divides into interpenetrating regions of bulk water and bulk oil, separated by a layer of surfactant. If the system contained only water and oil, each component would eventually form a single large domain; however, this situation would leave insufficient surface area to allow all of the surfactant to sit at the oil–water interface. Therefore, the surfactant has the effect of limiting the domain growth (Love et al. 2001).
Figure 3 shows the dominant length-scale for the first 5000 time-steps of three simulations using parameter set 8. This shows that the length-scale rapidly increases as the oil and water components separate, but stays relatively constant after around 1000 time-steps. Figure 4 shows renderings of the oil–water interface at ϕ=0 over the first 500 time-steps, indicating that the rapid increase in structure factor corresponds to the formation of a mesh of interconnecting channels, all of roughly the same width. Visual inspection of the interface shows that no gyroid structure is present during the channel formation.
Figure 3
Typical length-scale L1(t) as a function of time, for three simulations on parameter set 8.
Figure 4
The oil–water interface (ϕ=0) during early time evolution of a parameter set 8 system running on a 1283 lattice.
### (b) Gyroid formation: parameter set 9
Figure 5 shows the mean curvature H of the ϕ=0 interface, averaged over the whole surface. The most obvious feature of this graph is the magnitude of the curvature fluctuations in each system. The 2563 system, being the largest, had the most interfacial area over which to average; therefore had the lowest amount of fluctuation in this system, with a standard deviation of 8.41×10−5 inverse lattice units. The 1283 system had fluctuations roughly twice as large, with a standard deviation of 1.70×10−4, and the 643 a little over twice as large again, at 3.86×10−4 inverse lattice units.
Figure 5
Mean curvature H averaged over the ϕ=0 surface for parameter set 9, for three different simulation lattice sizes.
The next obvious feature of figure 5 is that, while the 1283 and 2563 systems fluctuated about zero for the entire simulation duration, the 643 simulation dropped to a negative-mean-curvature geometry after around 105 time-steps. Visual inspection showed that this corresponded to the system collapsing into a stable state, forming a regular structure which filled the entire simulation lattice, apart from two point defects.
Looking along the y-axis, the structure reconnected with itself through the periodic boundary conditions, moving half a unit cell in each of the x and z directions after traversing the lattice. There were seven unit cells in the x and z directions, and six and a half in the y direction; a topologically perfect gyroid with this many unit cells would have an Euler characteristic of −2548. The simulated structure had due to the presence of the point defects. The skewing of the ‘gyroid’ in the x and z directions and consequent rhombohedral (rather than cubic) liquid crystal structure meant that the oil–water interface did not form a zero-mean-curvature gyroid structure, but a skewed gyroid with, on average, negative mean curvature. It should be noted that the production of a gyroid with this orientation is a reassuring sign that the production of a gyroid mesophase is not a simulation artefact due to some anisotropy induced by the lattice, in which case such misalignment would not be expected.
Figure 6 shows the ϕ=0 isosurface, coloured red on the oil-majority side and blue on the water-majority side, at time-step 2.5×105. The region shown lies at the interface between the two domains; the domain wall is shown running vertically through the centre of the diagram. Careful examination of the channels running through the diagram shows that, while the two domains contained gyroids with roughly the same orientation, the domains had opposite chirality: red channels spiral clockwise away from the viewer in the right-hand side domain, but anticlockwise away from the viewer in the left-hand side domain. Since the model used is oil–water symmetric, there should be no preference for, say, oil channels to be left-handed, so the existence of chiral domains is to be expected.
Figure 6
The oil–water interface at time-step 2.5×105 for the 1283 simulation of parameter set 9. Note that the domains have opposite chirality.
Figure 2 shows volume renderings of the curvature contribution, highlighting regions with high interfacial curvatures. At time-step 10 000, the gyroid structure had not emerged, but by time-step 40 000, multiple gyroid domains were visible. These domains were observed to merge, until only two domains were left in the system, around time-step 200 000. One of the domains then collapsed over the next hundred thousand time-steps, leaving behind two columnar defect regions which persisted for as long as the simulation continued.
Figure 7 shows a volume rendering of the order parameter field at time-step 5×105, with the transfer function chosen to highlight regions of high oil density and regions of high water density in red and blue, respectively. By this time, the domain walls had collapsed into a pair of dislocation lines running right through the system. The green lines show a Burgers loop (Kittel 1996) around each dislocation; it can be seen that the Burgers vectors for each dislocation are equal in magnitude and opposite in direction.
Figure 7
Orthographic-projection volume rendering of the 1283 simulation of parameter set 9 at time-step 5×105. Oil-bearing regions are highlighted in red, water-bearing regions in blue and the interfacial regions are suppressed. The green lines show Burgers circuits around the two dislocations.
### (c) Gyroid formation: parameter set 8
Figures 8 and 9 show the surface-averaged mean curvature and surface-averaged squared mean curvature, respectively, for simulations using parameter set 8. The fluctuations in the mean curvature again dropped in magnitude with increased system size, just as observed with parameter set 9. On closer examination, the fluctuations for the 1283 simulation dropped in magnitude around time-step 2.5×105, with a similar halt in the evolution of the other bulk and interface properties. The domain wall visualizations (figure 10) show that the system produced two domains at late times, and one of these disappeared around time-step 2.5×105, leaving no dislocations or other major defects in the system. It was verified by direct inspection that the last two domains were again of opposite chirality. At the end of the simulation, time-step 378 500, the system was found to contain a gyroid with 15 unit cells in each direction, with an Euler characteristic of −26 828, corresponding to 3353.5 gyroid unit cells. A 153 periodic gyroid would be expected to contain 3375 unit cells; the discrepancy is attributed to persistent point defects, which appeared as the domains formed, up to time-step 105, and remained unchanged for the rest of the simulation.
Figure 8
Mean curvature H averaged over the ϕ=0 surface for parameter set 8, for three different simulation sizes, showing that the fluctuation in mean curvature is reduced with system size.
Figure 9
Squared mean curvature H2 averaged over the ϕ=0 surface for parameter set 8, showing smaller fluctuations in larger systems, and perfect minimal surface formation in the 1283 simulation.
Figure 10
Images from a gyroid self-assembly simulation. The images in the top row show slices through the order parameter field, with oil-majority regions in black and water-majority regions in white. Images in the middle row show the corresponding interfacial curvature per lattice site, with high curvature white and low curvature black. Images in the bottom row show an orthographic-projection volume rendering of the order parameter, looking down the (111) direction to show the characteristic ‘wagon wheel’ pattern seen in electron micrographs (Thomas et al. 1988). The simulation was performed on a 1283 lattice using parameter set 8.
The 2563 simulation, limited only by available computer time, ran until time-step 57 500, at which point it was still composed of multiple chiral gyroid domains. The domains were observed to interpenetrate in a manner not unlike the morphology of single-component fluid domains during spinodal decomposition.
Figure 11 shows the averaged squared mean curvature plotted against time for 1283 and 2563 simulations of parameter sets 8 and 9. Both of the systems which used parameter set 8 appeared to take on fairly close power-law behaviour (seen as straight lines on the log–log graphs) after time-step 104, when the system recognizably contained gyroid regions (see figure 12). The parameter set 9 systems also did this, but took longer to do so; the value of for the set 9 systems was somewhat larger than that of the set 8 systems, until around time-step 5×104, at which point they appeared to show the same behaviour. The 1283 systems eventually deviated from the power-law behaviour as finite-size effects set in, around time-step 2×105 for set 8 and 3×105 for set 9.
Figure 11
Averaged squared curvature against time for four simulations of different sizes and parameter sets. Late-time finite size effect behaviour is clear in the 1283 simulations, where the curvature drops close to zero as a perfect minimal surface is formed.
Figure 12
Averaged squared curvature in the power-law scaling régime for two parameter sets and sizes, with a t−1/2 power law plotted as a guide to the eye.
Figure 13 shows renderings of the domain walls and defects of the parameter set 9 system at points before and just after it reached power-law behaviour, with the defects for the parameter set 8 system plotted for comparison. During the period when the set 8 system showed power-law scaling but the set 9 system did not, the latter appeared to have a rather higher density of defective regions, shown as dark regions in the volume renderings. After time-step 50 000, both systems appeared clearly separated into domains.
Figure 13
Volume rendering of domain walls in two 2563 systems. At earlier times, the parameter set 9 system has a larger density of defective, non-gyroidal regions. By time-step 55 000, both systems are much more clearly separated into domains.
The datasets for for each simulation were trimmed to only include the time-steps after 10 000, when gyroid structures form, for the parameter set 8 simulations, and the time-steps after 50 000, after clear domain formation, for the parameter set 9 simulations; they were also restricted to the time-steps before the sudden drop in and stabilization of the squared mean curvature for the 1283 simulations, to exclude the corresponding finite size effects.
A relation of the form was then fitted to each trimmed dataset; the fitted values of the exponent n are given, along with the relevant time periods, in table 3. The fitted exponents together had a mean of −0.481 and standard deviation of 0.037. This is quite close to an exponent of ; the data points to which these fits were made are plotted together in figure 11, along with a power law for comparison. A possible reason for power-law scaling and for why the exponent should be −1/2 is given below.
View this table:
Table 3
Exponents n for the relation 〈H2〉=atn fitted to the squared mean curvature of four simulations between the times tmin, when clear domains have formed, and tmax, when the finite size of either the system or the available computer resources limits the simulation.
Consider a large system of volume V, containing many gyroid domains. Let the length-scale of a typical single domain be , and suppose that the gyroid length-scale grows as a power law of time, , just as domains often do, e.g. in spinodal decomposition. The surface area of a single domain will then scale as , and the volume of a single domain as .
The total number N of domains in the system is , so . The total surface area A of domains is proportional to both the number of domains and the typical surface area of a single domain, so therefore . If the domain walls all have roughly the same width λ, then the total volume of the domain walls scales as . Therefore, the contribution of the domain walls to any extensive property of the system would also be expected to scale as .
A gyroid is a minimal surface, so one would expect the contribution to the average squared curvature of the system from the well-formed gyroid regions inside domains to be small. On the other hand, the domain walls are non-gyroidal, and have non-zero mean curvature; if there is a roughly constant contribution to per unit volume of domain wall, then could also be expected to scale as . If this is the case, then that would further suggest that the length-scale of the gyroid domains indeed scales as , indicative of random-walk or diffusive behaviour.
## 8. Conclusions
Simulations of the self-assembly of a surfactant gyroid mesophase were performed; for the first time, the simulations were sufficiently large to clearly show the formation and evolution of multiple gyroid domains.
All systems underwent a rapid phase separation, during which oil and water phases formed a network of channels, separated by surfactant. After the phase separation, the systems began to form a gyroid mesophase. At late times, when clear domain walls had formed, power-law scaling of the average squared mean curvature of the system was observed, with exponent close to −1/2. This may indicate diffusive scaling of the length-scale of the domains themselves.
Analysis of simulated mesophases is non-trivial, due to the difficulty of defining a gyroid order parameter. In a ferromagnetic system, one can use the magnetization as an order parameter; all points within a ferromagnetic domain will have the same value of . In an ordinary fluid phase transition such as spinodal decomposition, the density of one fluid at a point can be subtracted from the density of the other, and normalized to give a scalar order parameter which varies from +1 in one fluid to −1 in the other, and which is zero in disordered, mixed regions. This kind of localized order parameter is sufficient to characterize the rapid separation of components before a mesophase is formed; however, after the initial demixing, further ordering of the system is morphological in nature, as gyroid regions are formed. Such a localized order parameter cannot describe this sort of ordering, since it is defined on too small a length-scale, and is not morphological in nature.
A set of tools was developed for calculating a polygonal approximation to the ϕ=0 surface of a simulated mesophase, in a topologically consistent manner; calculations of useful quantities such as the topologically invariant Euler characteristic, total interfacial area and interfacially averaged squared mean curvature were then possible. The average squared mean curvature was observed to vary as a power law with exponent approximately −1/2 over the régime where domains are clearly formed but finite size effects have not yet set in; this may indicate random-walk motion of the domain walls. Direct visualization of the interface shows the existence of chiral domains, as should perhaps be expected for a symmetric amphiphile which has no intrinsic chirality. Once domains effectively disappeared due to reaching the simulation system size, the formation of line dislocations analogous to those in solid crystals was observed.
These simulations have opened up several avenues for further work. Even disregarding the physics involved, the techniques for characterization of lyotropic liquid crystal phases could still be improved, by looking at, for example, localized contributions to the interfacial curvature or Euler characteristic. Gyroid domain walls were observed to have a high density of points where both the local order parameter ϕ and the local order parameter gradient were small; while this may be due to the structure of the ϕ=0 surface inside domain walls, an improvement of this technique beyond a simple visual tool could lead to an algorithm for spatial localization of domain walls, and, consequently, the ability to make more quantitative analyses of the domain behaviour.
The simulations raise a number of physical questions. Those performed so far have all had ‘oil’/‘water’ symmetry and a symmetric amphiphile, a situation which may be hard to realize exactly in experiment. The simulations produced chiral domains; it would be interesting to see if these could ever be produced experimentally, since the existence of such domains would affect the feasibility of producing gyroidal chirally selective filters. Observations of gyroid chirality would be difficult to make using TEM or SAXS techniques, but might be possible using, for example, transmission electron microtomography (Laurer et al. 1997).
Finally, it is interesting to note that the lattice Boltzmann method allowed the bridging of two different length-scales in these simulations: the scale (approx. five lattice sites) of the single-fluid-component domains, and the scale (greater than or approx. 50 lattice sites) of gyroid domains.
## Acknowledgements
We thank EPSRC for funding this research under RealityGrid grant GR/R67699 and GR/R94916/01. Most of the scientific data on which this paper is based were obtained during the TeraGyroid project (http://www.realitygrid.org/TeraGyroid.html), jointly funded by NSF and EPSRC, in which we were able to use over 6000 processors from a federated computational grid formed from the US TeraGrid and the UK national supercomputing centres at HPCx and CSAR, to generate and visualize the results over a very short period of time. We are grateful to Thomas Lewiner for helpful discussions and triangulation code. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8752601146697998, "perplexity": 1186.0336371672563}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802768397.103/warc/CC-MAIN-20141217075248-00019-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/388530/proving-an-identity-with-a-combinatorial-proof | # Proving an identity with a combinatorial proof
For any integers $n$, $k$, $r$ where $n\geq k\geq r \geq 0$, give a combinatorial proof of the following identity:
$$\binom{n}{k}\binom{k}{r}=\binom{n}{r}\binom{n-r}{k-r}$$
The problem is that I can't come up with a good counting argument of what exactly the two sides are doing. The left hand side is quite mysterious, and the right hand side is apparently choosing $r$ items and then choosing $k-r$ items from the remaining, which should be equivalent to $\binom{n}{k}$ but somehow isn't. How should I approach this problem?
-
I think the reason the RHS is not $\binom{n}{k}$ is that in choosing $r$ items and then $k-r$ items, you can distinguish between the items chosen in the first batch and those chosen in the second. – Alex Becker May 11 '13 at 14:46
Think of the problem of how to choose a sports team team consisting of $k$ people, and then to choose $r$ people from that team to play in a particular match (leaving $k-r$ people on the sidelines). How many ways are there to do this if you have $n$ total people from which to choose?
In more general terms, we you can think of this identity as saying that if we make a selection from a set of $n$ things so that $k$ of them have a property $1$ and $r$ of them have properties $1$ and $2$, it doesn't matter the order in which we assaign the properties.
Good way of explaining this. That is, let $A$ be a set with $n$ elements. You are counting the pairs $(B, C)$, where $B \subseteq C \subseteq A$, with $\lvert B \rvert = r$, and $\lvert C \rvert = k$, in two ways, either $C$ first (LHS), or $B$ first (RHS). – Andreas Caranti May 11 '13 at 14:51
I'm not failing to get the identity. How can we formalize this? How can the left hand side be formulated into the size of a set $S$? – user54609 May 11 '13 at 15:00
@EricDong, look at my comment, it is the set of all pairs $(B, C)$ etc. – Andreas Caranti May 11 '13 at 15:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8935447931289673, "perplexity": 164.11976084238603}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375098071.98/warc/CC-MAIN-20150627031818-00257-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://pritschet.me/wiki/python/fipy/fipy-solving-pdes-python/ | # Solving a PDE
In this first example we want to solve the Laplace Equation (2) a special case of the Poisson Equation (1) for the absence of any charges.
(1)
(2)
Prior to actually solving the PDE we have to define a mesh (or grid), on which the equation shall be solved, and a couple of boundary conditions. Lines 6-9 define some support variables and a 2D mesh. Lines 10-12 define further support variables, which are used later to define boundary conditions and save the result in NumPy arrays.
Lines 14-21 define the physical quantitiy (potential in the code, \phi in the above equaitons), the equation with this quantity, that needs to be solved, and the boundary conditions. In this example the boundary conditions represent three capacitor plates thus that the mesh shows the space between the plates. One plate is located at the top and extends along the entire x-range. The two other plates are at the left and right hand side, respectively, cover only the y<y0 range.
As shown in this example, boundary conditions (BCs) are objects of type FixedValue or FixedFlux (not shown). If more than one BC need to be defined, this can be done in a Python list or tuple. But notice: FiPy takes the term boundary condition very seriously. Boundary conditions can only be defined on so called exterior faces. Thus, if boundary conditions for a given problem cannot be applied onto the surfaces of a "common" mesh (line, square or cube), one has to construct a mesh suitable for the problem using a tool called gmsh [2].
Line 23 starts the actual PDE solving algorthm.
Lines 26-27 store the solution in a file. The specified file extensions determines the format of the output. Omitting arguments to viewer.plot() starts an interactive display module (e.g. Matplotlib, Mayavi2, etc.).
In case of "common" meshes it is rather easy to convert the CellVariable into a NumPy array as is shown in lines 31-39. In case of a uncommon mesh one would have to call the __call__() method of the CellVariable and evaluate each point for a given coordinate (will be shown in another example).
# References
1. http://www.ctcms.nist.gov/fipy/
2. http://geuz.org/gmsh/
3. R. Pavelka, "Numerical solving of anisotropic elliptic equation
on disconnected mesh using FiPy and Gmsh", 2012 (Mirror)
4. J. E. Guyer, D. Wheeler & J. A. Warren, "FiPy: Partial Differential Equations with Python", Computing in Science & Engineering 11(3) pp. 6—15 (2009)
5. C. Geuzaine and J.-F. Remacle. "Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities". International Journal for Numerical Methods in Engineering, Volume 79, Issue 11, pages 1309-1331, 2009 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8015623688697815, "perplexity": 1385.9806248990137}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247512461.73/warc/CC-MAIN-20190222013546-20190222035546-00190.warc.gz"} |
http://mathoverflow.net/questions/118931/stabilization-in-banach-algebras?sort=votes | # Stabilization in Banach algebras
In $C^\ast$-algebras we use $K(H)$, the algebra of compact operators on a separable Hilbert space, for stabilization of a $C^\ast$-algebra, i.e. $S(A):=A\otimes K(H)$. Is there any similar stabilization functor in Banach algebras? What is the substitute of $K(H)$?
-
@Z254R, are you pointing out deficiencies in a question? ;-) Well in any case I agree completely with you. @Vahid, what do you want a stabilization functor to do? Do you have in mind K-theory? – Captain Oates Jan 14 at 23:58
After being able to define (an appropriate notion for) the stabilization of a Banach algebra, say $A$, I'd like to see if it is Morita equivalent with $A$. Of course, the equality of $K$-groups is the next. And so on. Stabilization of $C^\ast$-algebras is a elementary notion, so I thought, maybe there is a similar notion for Banach algebras too. That's why I asked this question. – Vahid Shirbisheh Jan 15 at 0:14
Is it clear which tensor product should be used for Banach algebras? (For $C^*$-algebras, one secretly enjoys nuclearity of $K(H)$...) – Alain Valette Jan 15 at 17:58
@Yemon: I list some instances that clarify the importance of stabilization in $C^*$-algebras: 1. Both K-theory and KK-theory are stable functors meaning $K(A)\simeq K(A\otimes K(H))$. 2. Two separable $C^*$-algebras $A$ and $B$ are Morita equivalent if and only if they are stably isomorphic, i.e. $A\otimes K(H)\simeq B\otimes K(H)$. 3. Tensoring by $K(L^2(G))$ also appears in some theorems too, for example see Takai-Takesaki duality. So, it is nice to have a similar notion in Banach algebras, for instance proving item 2 for Banach algebras would be a good start. – Vahid Shirbisheh Jan 17 at 5:58
@Yemon: You are welcome. Following Vincent Lafforgue's works, Walter Paravicini has studied Morita equivalence of Banach algebras too. – Vahid Shirbisheh Jan 25 at 20:58
This is not an answer, but too long for a comment: Even for $C^*$-algebras there is more than one method of stabilization. The critical feature of the compact operators is that $K(H) \otimes K(H) \cong K(H)$ for an infinite dimensional Hilbert space and that $K_*(K(H)) \cong K_*(\mathbb{C})$. But this also holds true for other $C^*$-algebras like for example the infinite Cuntz algebra $\mathcal{O}_{\infty}$ or the Jiang-Su algebra $\mathcal{Z}$. In general - again for the $C^*$-case - you might want to search for strongly self-absorbing $C^*$-algebras.
There is the following theorem:
If $A$ is a simple, separable and nuclear $C^*$-algebra, then $A \cong A \otimes \mathcal{O}_{\infty}$ if and only if $A$ is purely infinite.
And there is a famous theorem by Kirchberg, which you may read like this:
Assume $A$ and $B$ are simple, separable, nuclear and $\mathcal{O}_{\infty} \otimes K(H)$-stable $C^*$-algebras (i.e. $A \otimes \mathcal{O}_{\infty} \otimes K(H) \cong A$ and likewise for $B$), then $A$ and $B$ are isomorphic if and only if they are $KK$-equivalent.
If you are looking for a setup like this, you may want to search for Banach algebras, which have the $K$-theory of $\mathbb{C}$ and are isomorphic to the tensor product of two copies of themselves. Or one might be able to generalize the notion of strongly self-absorbing to Banach algebras.
That was a lot about $C^*$-algebras and not much about the Banach case. Sorry about that.
-
Thanks Ulrich. I guess one can choose is $K(H)$ for stabilization of Banach algebras too. But I am not sure if this gives rise to similar theorems about Morita equivalence and $K$-theory of Banach algebras. So, it seems there is no obvious choose "known yet" and it is open for more investigations. – Vahid Shirbisheh Jan 15 at 12:44
Out of curiosity, Ulrich: *what is it that one gains by tensoring a given $C^\ast$-algebra with $K(H)$? (not a rhetorical question; I am genuinely ignorant of, or have forgotten, the reason one does this). – Captain Oates Jan 16 at 19:38
Yemon, one gains, for example idempotents. The group $K_0(A)$ can be constructed looking at the projections of $A\otimes K$ (looking at $\bigcup_n M_n(A)$ also works here, but passing to the completion may be more appealing at times). One also looses after tensoring with $K$ (which can be a good thing). E.g., Brown's theorem says that if $a,b\in A$ are positive then thr C*-algebras $\overline{aAa}$ and $\overline{bAb}$ become isomorphic after tensoring with $K$ if and only if $a$ and $b$ generate the same closed two-sided ideal. – Leonel Robert Jan 17 at 17:05
Stabilization with K is also interesting from the point of view of Hilbert bimodules, since the Brown-Green-Rieffel theorem says that two C*-algebras with countable approximate identities are strongly Morita equivalent if and only if they are stably isomorphic. Apart from that we have an exact sequence $0 \to Inn(A \otimes \mathbb{K}) \to Aut(A \otimes \mathbb{K}) \to Pic(A \otimes \mathbb{K}) \to 0$. – Ulrich Pennig Jan 17 at 18:22
Oops, I just realized my last comment was mentioned above already. – Ulrich Pennig Jan 17 at 18:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8460084795951843, "perplexity": 390.81640510873586}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386163047523/warc/CC-MAIN-20131204131727-00069-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://journal.gpps.global/Generation-Mechanism-of-Diffuser-Stall-in-a-Centrifugal-Compressor-with-Vaneless,128974,0,2.html | ## Introduction
A centrifugal compressor is mostly used in a wide range of applications due to a high pressure ratio and small size. Centrifugal compressors with a vaneless diffuser are often employed due to their wider stability range, compared to those with a vaned diffuser. However, the operating range is limited by the onset of rotating stall and surge. Prior to the onset of the surge, the rotating stall was generally occurred. Therefore, to enhance the operating range and suppress the surge onset, one should must focus on understanding the rotating stall generation mechanism.
Numerous researchers have reported rotating stall behavior in centrifugal compressors with vaneless diffuser. The tornado-type separation vortex caused by full blade leading-edge separation was observed under the flow field of a developed rotating condition (Iwakiri et al., 2009). Zhang et al. investigated the volute influence on stall inception onset and development. Under the volute influence, the spillage vortices at a leading edge of impeller blades were developed by a static pressure peak near the tongue (Zhang et al., 2019). Therefore, the centrifugal compressor impeller stall was strongly related to a suction side separation and tip-leakage vortex. Additionally, many reports discuss a diffuser stall behavior in a vaneless diffuser. Senoo et al. were the first to investigate the relationship between reverse flow and the initiation of a vaneless diffuser rotating stall in a centrifugal blower. They proposed initiation criteria for a vaneless diffuser rotating stall (Senoo et al., 1977; Senoo and Kinoshita, 1978). Ohuchida et al. suggested both a boundary layer separation pattern and a stall propagation mechanism in a vaneless diffuser by PIV experiments (Ohuchida et al., 2013). They indicated that a rotating reverse flow disturbance in the diffuser was related to developing of a rotating stall cell, blocking interior diffuser flow at near-surge. However, the detailed diffuser stall generation mechanism in a vaneless diffuser was not completely understood.
Previous publications revealed the mechanism of rotating stall in the centrifugal compressor with vaned diffuser (Fujisawa and Ohta, 2017; Fujisawa et al., 2019a,b). The current study is focused on the generation mechanism of a diffuser stall in vaneless diffuser. Both experimental and numerical analysis were conducted on stall inception of centrifugal compressor with vaneless diffuser. The unsteady velocity was measured in circumferential and meridional direction within vaneless diffuser. Furthermore, Detached Eddy Simulation (DES) analysis over the entire compressor region, including the volute, was performed to reveal more detailed diffuser stall cell behavior.
## Experimental apparatus and procedure
The tested centrifugal compressor schematic view for this study is shown in Figure 1. The test rig at Waseda University was used for a shipboard turbocharger, consisting of an impeller, vaneless diffuser, and volute. The impeller is equipped with seven full blades and seven splitter blades. The geometry details, as well as key parameters, of the tested compressor are summarized in Table 1. The impeller is an open-type impeller with a tip clearance that is 1.0% of the impeller blade width at B1 = 105.6 mm. The impeller rotational speed was set to 6,000 min−1.
#### Dimensions of tested compressor.
Tested Centrifugal CompressorDiffuser
Rotational SpeedN6,000 min−1Inlet DiameterD3328 mm
Mass Flow Rate (design point)G1.64 kg/sOutlet DiameterD4559 mm
Pressure Ratio (design point)p5/p11.1Diffuser WidthB419.55 mm
Impeller
(Main + Splitter)(7 + 7)
Inlet DiameterD1248 mm
Outlet DiameterD2328 mm
The steady pressure was measured at the compressor exit by a differential pressure transducer (Yamatake JTD920A). The orifice flow meter was set at the outlet duct to measure the steady mass flow rate. Velocity fluctuations, which defined as the difference between the time averaged value and the peak one, were measured on the test rig using a hot-wire anemometer (DANTEC, 55R57). The circumferential measurement points were placed at five positions (0°, 90°, 150°, 180° and 240°), respectively as shown in Figure 1. The meridional velocity measurement points were at the diffuser inlet (D.I.), diffuser midsection (D.M.), and diffuser exit (D.E.). The hot-wire probes traversed in the spanwise direction in 5% increments of the diffuser width. The probe body is a 2.3 mm diameter, which is about 10% of the diffuser passage width. Although this probe size slightly influenced on the flow blockage, the scale of stall behavior would be greatly larger compared with the velocity fluctuation induced by the probe effect.
### CFD methodology
A detached eddy simulation (DES) code was developed for investigating the internal flow. The Computational Fluid Dynamics (CFD) code was constructed to solve the three-dimensional compressible Navier-Stokes equations. The convective flux was evaluated using the flux difference splitting (FDS) method and extended to the third-order MUSCL interpolation. The viscous flux was evaluated as a second-order central difference. The Matrix Free Gauss-Seidel (MFGS) implicit algorithm was employed for time integration. The shear stress transport (SST) k-ω turbulence model was used in this code and applied to DES. This model depends on the local turbulent length scale and grid spacing size. The dissipation term in the k-equation of the SST k-ω turbulence model is modified as follows:
DDESk=ρk3/2/l~
##### (2)
l~=min(lkω,CDESΔ)
where l is the turbulent length scale, Δ is the local grid spacing, and the model’s constant coefficient CDES was set to the recommended value by the report (Strelets, 2001). The several studies reported that DES analysis is more suitable than URANS for detailed flow field investigations near stall in both axial and centrifugal compressor (Im et al., 2012; Yamada et al., 2017). Our In-house code was successfully validated for unsteady flow fields of various turbomachinery flows (Fujisawa et al., 2019b; Zhang et al., 2020).
The computational domains applied in the numerical simulations are illustrated in Figure 2. The grid system included 14 impeller passages, vaneless diffuser, and a volute. The impeller (including clearance region) and vaneless diffuser systems had 32.0 and 11.0 million cells, respectively. The volute region had 22.0 million cells. In total, the computational grid had 65.0 million cells. The wall cell width was 0.1 µm, corresponding to a y+ parameter, equal to approximately one along all solid surfaces. At the inflow boundary, the total pressure and temperature were fixed, while mass flow rates were fixed at the outflow boundary. Across a sliding boundary separating the moving impeller and stationary diffuser frames, the most recent data on one side were interpolated to obtain opposite side data by using an unsteady simulation sliding mesh. Nonslip and adiabatic conditions were adopted for the wall conditions. The main CFD setup was summarized in Table 2.
### Computational setup.
Numerical Scheme Convective Term FDS (MUSCL 3rd-order Interpolation) Viscous Term 2nd-order Central Differential Method Time Integral MFGS Turbulence Model SST k-ω Turbulence Model (DES) Boundary Conditions Inlet Boundary Total Pressure and Temperature Fixed Outlet Boundary Mass Flow Rate Fixed RS Boundary Sliding Mesh Time Steps 5,000 step/rev Mesh Impeller Region 32.0 million cells Diffuser Region 11.0 million cells Volute Region 22.0 million cells Wall y+ <1
### Compressor performance and stall characteristics
The experimental and numerical compressor performance results are shown in Figure 3. The numerical total pressure-rise characteristics were obtained from the time-averaged results of unsteady DES analysis and steady RANS analyses. The flow and total pressure-rise coefficients are defined as follows:
### Compressor Performance.
ϕ=Qπ2D22B2(N/60)
##### (4)
Ψt=Δptρπ2D22(N/60)2/2
The unsteady simulations were conducted at the operational design point (ϕ = 0.24), and two off-design points (ϕ = 0.10 and 0.08). The steady simulations were conducted at three operational points from ϕ = 0.24 to ϕ = 0.14. The steady RANS analysis results were in good agreement with measured results. The unsteady CFD results were overestimated particularly at the off-design points as compared with the test results. The mixing loss especially within scroll passages was assumed to be underpredicted at near stall point, but the main factor of overprediction of pressure rise coefficient still was not confirmed. However, the relative error between experimental and unsteady CFD results at off design points was within 5%, as shown in Figure 3. Therefore, it was no problem to investigate the characteristics of a diffuser stall because the relative error was small.
To investigate the unsteady rotating stall characteristics, the radial velocity fluctuation of impeller discharge flow was measured with a split-film anemometer at ϕ = 0.08. The radial velocity was measured at D.I.0, 90, 180 and 240, which were indicated in Figure 1. The fast Fourier transformation (FFT) spatial distribution results at each measurement point in the spanwise direction are shown in Figure 4. The vertical axis represents the radial velocity fluctuation magnitude. From Figure 4, two types of velocity fluctuations were observed. One occurred at around 58 Hz and the other occurred at around 25–30 Hz. Based on previous report (Fujisawa et al., 2016), the unsteady phenomena at around 58 Hz were caused by a rotating stall within the impeller passages. On the other hand, the lower frequency fluctuation at around 25–30 Hz, was equivalent to 25–30% of the impeller rotating speed, and was caused by a rotating stall within the vaneless diffuser. Particularly, focusing on the diffuser stall, its characteristics varied depending on the circumferential position. From Figure 4(i), the small velocity fluctuation magnitudes caused by diffuser stall were observed near the shroud wall at D.I.0. Whereas, from Figure 4(ii), the large velocity fluctuation magnitude around 28 Hz was observed near the hub wall at D.I.90. Furthermore, Figure 4(iii) and (iv) show that the largest velocity fluctuation magnitude caused by the diffuser stall was observed near both shroud and hub walls at D.I.180 and 240, located near the cutoff. The mass flow fluctuations at the diffuser exit obtained from CFD analysis are shown in Figure 5. The red colored line indicated low mass flow regions and this rotational speed was about 25% of the impeller rotation speed. Therefore, the red colored low mass flow regions was assumed to be caused by the rotating diffuser stall, which rotated at 25% of the impeller rotation speed. This low mass flow region appeared at 90° and disappeared after passing the cutoff. Both experiment and CFD analysis indicated that the diffuser rotating stall initiated at 90° and developed further as the diffuser stall cell approached the cutoff. After passing the cutoff, the diffuser rotating stall was attenuated. The detailed unsteady diffuser stall behavior is discussed later.
### Generation mechanism of diffuser stall
To investigate the detailed unsteady diffuser stall behavior, an unsteady DES analysis was conducted. First, to understand the flow field, Figure 6 shows the distributions of radial velocity fluctuation within the diffuser passage obtained by CFD analysis. The radial velocity fluctuation was defined as follows:
### Distribution of Radial Velocity Fluctuations (CFD:ϕ = 0.08).
##### (5)
ΔVr(t)=Vr(t)Vr_ave
Vr_ave indicates the time-averaged radial velocity. The red color indicates the large magnitudes of radial velocity fluctuations. The region indicated by the white dashed circle is the diffuser stall cell that induced the largest mass flow fluctuations in Figure 5. Furthermore, the time-averaged mass flow rate of impeller discharge flow obtained from CFD analysis at ϕ = 0.08 is shown in Figure 7. The impeller exit plane mass flow rate was calculated on sections obtained by dividing the diffuser inlet into 32 circumferential sections. At t* = 2.2, the region with the large velocity fluctuations, as shown in Figure 6, was first generated at around 45°, where a strong circumferential adverse pressure gradient is observed, as depicted in Figure 8. Then, the large fluctuations region was formed at the diffuser exit area by the accumulation of several small backflow regions at t* = 2.9. After that, the diffuser stall cell expanded both radially and circumferentially as it approached the cutoff at t* = 3.7 and 4.2. Finally, the stall cell size reduced after passing the cutoff at t* = 5.4 due to increased impeller discharge mass flow rate at near the cutoff, as shown in Figure 7.
### Distribution of Static Pressure in Circumferential Direction (CFD:ϕ = 0.24 and 0.08).
The circumferential static pressure distributions at ϕ = 0.24 and ϕ = 0.08 obtained from the CFD analysis are shown in Figure 8. The static pressure was measured at shroud wall D.M. From this figure, at around 300°, the static pressure at off-design point ϕ = 0.08 was the same level as that at design point ϕ = 0.24. However, from 0° to the cutoff, the static pressure at ϕ = 0.08 was higher than that of ϕ = 0.24. The magnitude of circumferential adverse pressure gradient was strong near 0° for ϕ = 0.08. Therefore, the diffuser stall was initiated at around 45°. Furthermore, the time-averaged meridional distribution of radial velocity and streamlines at 0°, 45°, and 240° are shown in Figure 9. The boundary layer separation occurred at hub and shroud wall by turns at 240deg, with numerous papers reporting the same phenomena (Van den Braembussche, 2019). From this paper, the spanwise pressure gradient induced the flow concentration on the side with the highest pressure. In addition, the spanwise pressure gradient inversion was generated once separation occurred, and this pressure gradient change continued until flow reversal took place on the other side. Therefore, the separations generated at both hub and shroud by turns. First, the flow concentrated on the shroud side where the total pressure was highest and boundary layer separation occurred on the hub side at 240deg (point (a)), as shown in Figure 9. At point (b), the pressure gradient inversion occurred and the flow concentrated on the hub side. Therefore, the next boundary layer separation occurred at the opposite wall. Particularly, at 45deg, the separation region was observed near the diffuser inlet shroud wall. Accordingly, the hub side boundary layer separations at around 45°, triggering the diffuser stall cell generation, were caused by the shroud side separation region at the diffuser inlet and a large circumferential adverse pressure gradient. Figure 10 shows the time-averaged limiting streamlines on hub wall obtained by measurement and CFD analysis. In the experiment, the limiting streamlines were visualized by using an oil film method. The limiting hub wall streamlines were bent at around 45° where a small backflow region generated.
### Limiting streamlines on hub wall (EXP and CFD:ϕ = 0.08).
For improved understanding of the diffuser stall cell generation mechanism, the unsteady flow field was visualized. The instantaneous flow field at two different times are shown in Figure 11. The distributions of radial velocity fluctuation within the diffuser passage at each time are shown. The developing behavior of diffuser stall cell was investigated from Figures 12 and 13. Figures 12(i) and 13(i) show the meridional radial velocity distribution around the region A and B at t* = 2.13 and 3.00. Figures 12(ii) and 13(ii) show the iso-surface of Vr = −10 m/s colored by the distance from the hub wall. From Figure 12, the hub wall boundary layer separation initiated at around 45°. Then, this hub wall separation induced the next shroud wall separation because of a spanwise pressure gradient, as shown in Figure 13. The boundary layer separations on the shroud wall developed, as it can be seen by the low velocity region at diffuser exit. The hub side Backflow Region I was the impeller wake. From numerical results, the impeller wake mainly expanded on the hub side from 90° to the cutoff. The shroud side Backflow Region II consisted of the boundary layer separations. The region indicated by the black circle was the low velocity region. Backflow Region I rotated faster than Backflow Region II, as shown in Figure 13a and b. Therefore, Region I caught up with Region II and induced the boundary layer separation enlargement on the shroud wall (Region II). Furthermore, the shroud wall boundary layer separations (Region II) induced the next hub wall boundary layer separations (Region III) because of a spanwise pressure gradient. The low velocity region within vaneless diffuser developed due to the boundary layer separations occurring on the shroud and hub wall by turns. Finally, the low velocity region formed the entire diffuser passage span blockage.
### Development of Diffuser Stall Cell at t* = 2.88 and 3.00 (CFD:ϕ = 0.08).
In order to validate the numerical results, the radial velocity fluctuation of impeller discharge flow was measured at the meridional direction (D.I.150, D.M.150 and D.E.150), as shown in Figure 14. The diffuser stall fluctuations were observed near hub side at D.I.150. On the other hand, the region with large magnitudes of the diffuser stall was shifted to the shroud side at D.M.150 and D.E.150. It is because the diffuser stall was consisted of the boundary layer separations occurring on the shroud and hub wall by turns. Therefore, the boundary layer separations occurring on the shroud and hub wall were considered as the key factors contributing to the diffuser stall generation by both experimental and CFD results. In our future work, the general generation mechanism of the boundary layer separations, which occurred at the hub and then the shroud end wall, will be revealed by several CFD investigations.
## Conclusions
The generation mechanism of a diffuser rotating stall in a centrifugal compressor with a vaneless diffuser was investigated by experiments and CFD analysis. The results can be summarized as follows:
1. In the test compressor, fluctuations occurred at around 25–30 Hz, were caused by diffuser stall at off design point ϕ = 0.08. The fluctuation magnitude varied depending on circumferential position with the largest fluctuations observed near the cutoff.
2. The diffuser stall behavior varied depending on circumferential position. First, the boundary layer separations initiated on the hub wall at around 45° where a large circumferential adverse pressure gradient magnitude was observed. Next, the hub side separations induced the next shroud wall boundary layer separations. The shroud side boundary layer separations developed and induced a low velocity region located at the diffuser stall cell center.
3. The diffuser stall cell developed further as it approached the cutoff. The boundary layer separations developed on both shroud and hub walls by turns due to the impeller wake interaction that developed on the hub side. The boundary layer separations occurring on the shroud and hub wall were considered as the key factors contributing to the diffuser stall generation by both experimental and CFD results.
## Nomenclature
B
diffuser passage height (m)
D
diameter (m)
G
mass flow rate (kg/s)
N
rotational speed (min−1)
Q
volume flow rate (m3/s)
Vr
p
static pressure (Pa)
pt
total pressure (Pa)
t
time (s)
trev
one impeller revolution time (s)
t*
non-dimensional time (=t/trev)
Z
ρ
air density (kg/m3)
ϕ
flow coefficient
ψt
total pressure rise coefficient
1
impeller inlet
2
impeller outlet
3
diffuser inlet
4
diffuser outlet | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8809607028961182, "perplexity": 2838.7274138895027}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296944452.97/warc/CC-MAIN-20230322211955-20230323001955-00303.warc.gz"} |
http://math.stackexchange.com/questions/45720/amount-of-moving-balls-on-chessboard | # Amount of moving balls on chessboard
With given chessboard $N\times M$ we have to put moving balls (they are moving in any direction, they can move from one square to any adjacent square).
If any ball touches border, it bounces from the border with angle 90 degrees.
How many balls can we put on the chessboard such that two different balls won't touch each other?
-
This needs thought to carefully specify the problem. How big are the balls? Do you mean the balls reflect from the border (as you say they can move in any direction) or after one reflection they are moving orthogonally (as you say they bounce from the border at 90 degrees)? Presumably the duration is "infinite", but it would be good to say that. – Ross Millikan Jun 16 '11 at 12:30
In addition to Ross's questions: are all ball speeds constant and identical? Are they supposed to be reflecting from the border in the way an unspun snooker/pool ball would do? That's only a 90 degree angle if the angle of incidence is 45 degrees: so is that angle of 45 degrees a constraint? – EnergyNumbers Jun 16 '11 at 12:57
I assume that one ball fills one square on the chess board and that two balls touch iff they enter the same square at the same timestep. But what are the rules of reflection? And what are the rules of motion, can balls move like bishops along diagonals or only like rocks? – Tim van Beek Jun 16 '11 at 12:57
Oh sorry, yes, speed of balls are constant, and time is infinite. @Tim, yes they can move along diagonals. If balls hits the border the move direction is changed by 90*, obviously if ball hits the corner the move direction is changed by 180*. Sorry for lack of informations. – Chris Jun 16 '11 at 13:26
State more clearly in which direction can the balls move? Horizontally, vertically and along a diagonal(like the bishops do in chess)? If a ball touches the border, I understand that it reflects like in all reflection laws, that is if a ball moves horizontally and then hits the border, it continues to move horizontally. Please state more clearly what you want, so that you can get an answer. – Beni Bogosel Jun 16 '11 at 15:05
This is an incomplete solution, but it's a starting point and I would be open to any further suggestions to get the final solution
I hope my assumptions are correct, but I'm going to assume that you get to choose the initial direction of the balls, and that the balls are the size of 1 square each.
Using this, notice that the obvious upper bound are $N\cdot M$ balls. Just as obvious, the lower bound are $max \left\{ N,M \right\}$ balls. We get the lower bound by taking the side of the board with the longest length, and filling every square of it with a ball. Now we choose the initial direction to go straight, so it only bounces off 2 walls forever, going back and forth.
It's also important to note that with this setup, we can NEVER have 2 balls in the same column/row, since they are only going straight and their paths would eventually intersect. So in that sense, we have a potentially better upper bound of $N + M$ balls.
Now, WLOG, suppose that the bottom half of the board is the longest(w/ length $N$), and thus we use that bottom side to have $N$ balls going straight up and down. This is where I get stuck, is there a way for you to have a ball travel horizontally on the board while not ever hitting one of the vertically traveling balls? My initial thought is yes, if we have the horizontal balls move like a sine wave function, and just have the horizontal ball go through the flow of the balls, but I still think they would eventually hit... any suggestions guys/gals? O_o
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If $M = N$ then there's a lower bound of $2M - 2$ by having a single loop going diagonally, and in general you can set up a loop of length $2M - 2$ starting anywhere other than on a diagonal. For $M = 8$ you can set up two such loops, each alterating ball, non-ball; such that at $t=0$ one has a ball one cell to the right of the top-left corner, and the other one cell to the right of the bottom-left corner. Then parity arguments show that you can get a horizontal-moving ball in the top and bottom rows, for a total of 16. This is, of course, only a lower bound. – Peter Taylor Jun 16 '11 at 15:23
@Peter That sounds right to me... it would look like a diamond inside of a square if I'm imagining it correctly, right? but I'm not sure it would be $2M$ balls if the paths of the balls do make a diamond shaped loop on the inside, since the board is discrete in shape. Another interesting thought is that if we can find the longest unique single loop on the board with these rules, then we can just place balls for each square the paths takes up. – Nicolas Villanueva Jun 16 '11 at 15:29
it's $2M-2$. The extra 2 to get up to 16 are horizontal balls which manage to use parity to dodge past the corners of the diamonds. It's probably possible to squeeze in two vertical ones too for 18, now that I think about it. – Peter Taylor Jun 16 '11 at 15:32
In fact for a square board you can set up one loop of length $2M-2$ on white squares and another on black squares, for a lower bound of $4(M-1)$ – Peter Taylor Jun 16 '11 at 16:55
Incomplete answer; too long for a comment;
The speed and movemend can be considered in two ways: in one second, the ball "teleports" to a neighbor square (in one of the three directions), or the ball is considered to move like in the real life, continuously, with constant speed. Both problems are interesting, and maybe the first one is easy
Suppose the balls move continuously
My first thought is that if there is a ball moving horizontally and one diagonally, then they must meet. First, we can find the "times" the diagonal ball is in the strip containing the horizontal ball. Since these moments of time would be some irrational multiple of the period of the horizontal ball (the ratio is a multiple of $\sqrt{2}$), by a density argument, the balls must meet.
As in the previous answer, if all the balls have horizontal or vertical direction, the situation is easy. Suppose now that all the balls move diagonally, and each one moves in a rectangular strip with cut corners if the dimensions are equal, but when the dimensions are not equal, the trajectory can cover a big portion of the board. Each two such regions intersect, as can easily be seen, visualising the problem. The problem is the following: can we position the balls such that even if the trajectories meet, they are never simultaneously in the intersection? Even in a $8\times 8$ chessboard this problem seems hard, although the trajectories have the same length (when not moving on the great diagonal.
I will not continue with this case, since maybe this was not intended by the OP.
In the case where balls move "discretely" from a square to another, again, I feel that if the balls move diagonally, fewer balls can be positioned on the board.
[edit:] Ok, here's what came to my mind. I'll take for example the $8\times 8$ board. And take a rectangular diagonal loop and fill it with $14$ balls all moving in a "snake" style. We can make another loop on the other color and get another $14$ balls. I think this is an example $28$ balls. Don't know if the previous comments refer to this kind of structure, at least I didn't understand that.
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in the discrete case allowing diagonal movement lets you fit more balls in. For 8x8 with horizontal and vertical movement only there's an easy upper bound of 16. Allowing diagonal movement there's a lower bound of 28 - see comments on Nicolas' reply. – Peter Taylor Jun 16 '11 at 16:57
I'm no seasoned expert when it comes to solving problems like these, but I would've thought the discrete case might be harder than the continuous one... – Josh Chen Jun 17 '11 at 10:34
In any case I do rather get the feeling that the OP's intention was the discrete case. – Josh Chen Jun 17 '11 at 10:34
what you describe is precisely what I was describing as a 28-ball solution. – Peter Taylor Jun 17 '11 at 11:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8614559173583984, "perplexity": 425.215296803981}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394010484313/warc/CC-MAIN-20140305090804-00000-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://mathhelpforum.com/discrete-math/189158-bit-theory-functions-one-one-onto-invertible.html | # Thread: bit theory, functions (one-one,onto, invertible)
1. ## bit theory, functions (one-one,onto, invertible)
Hey guys
Now I understand perfectly fine what one to one, onto and invertible functions are, my troubles are when bit theory is involved with this question giving you only the length, they all seem like none are well defined.
So question is, Let Q be the set of bit strings, q, of length 64. Consider the following functions, and for those that are well defined, determine the range, and whether they are one-to-one, onto, invertible.
a) f:Q->Q, f(q) sets the left most bit of s to zero
I was told you could assume the set could be anything you want it to be, so I would just think its not onto or one to one and not well defined.
But now im not sure if thats right, and how exactly I would explain or word the answer.
b) g:Q->Q, g(q) removes the left most bit of s, I would just say its not well defined as now the length changes to 63.
c) h: Q->{0,1,2,3 .... 32}, h(q) counts the number of 1s in the leftmost 32bits of s.
This one flew over my head a bit as they seem to be giving u a set up to 32 now, and then I'm not sure what i'm comparing it against.
d) o:Q->Q, o(q) shifts the bits of s 4 places to the right discarding the 4 right most bits and setting the 4 left most bits to 0.
this seems similar to b), since its like your changing the length of the set again
e) a:Q->Q, a(q) = complement of q,
f) b:Q->Q, b(q) = q AND t where t is a fixed 64 bit string
g) y:Q->Q, y(q) = q OR t, where t is a fixed 64 bit string
Since my understanding when bit theory is involved just messes things up, doing the subsequent questions becomes impossible.
If you guys could help me do the question or point me in the right direction or atleast give me an example like this that involves bit theory with some kind of working solution.
I'd greatly appreciate any help.
Thanx
2. ## Re: bit theory, functions (one-one,onto, invertible)
Originally Posted by KavX
Hey guys
Now I understand perfectly fine what one to one, onto and invertible functions are, my troubles are when bit theory is involved with this question giving you only the length, they all seem like none are well defined.
So question is, Let Q be the set of bit strings, q, of length 64. Consider the following functions, and for those that are well defined, determine the range, and whether they are one-to-one, onto, invertible.
a) f:Q->Q, f(q) sets the left most bit of s to zero
I was told you could assume the set could be anything you want it to be, so I would just think its not onto or one to one and not well defined.
How can you assume anything about the set Q, you are told it is the set of bit strings of length 64. f maps Q to the subset of Q of 64 bit strings with a 0 in the left most bit. Every element in the range is the image of two elements in the domain.
But now im not sure if thats right, and how exactly I would explain or word the answer.
b) g:Q->Q, g(q) removes the left most bit of s, I would just say its not well defined as now the length changes to 63.
g does not take Q into Q, so is ill defined.
c) h: Q->{0,1,2,3 .... 32}, h(q) counts the number of 1s in the leftmost 32bits of s.
This one flew over my head a bit as they seem to be giving u a set up to 32 now, and then I'm not sure what i'm comparing it against.
The range is the set of integers 0, ... , 32, that is h maps Q to a integer between 0 and 32
d) o:Q->Q, o(q) shifts the bits of s 4 places to the right discarding the 4 right most bits and setting the 4 left most bits to 0.
this seems similar to b), since its like your changing the length of the set again
e) a:Q->Q, a(q) = complement of q,
f) b:Q->Q, b(q) = q AND t where t is a fixed 64 bit string
g) y:Q->Q, y(q) = q OR t, where t is a fixed 64 bit string
Since my understanding when bit theory is involved just messes things up, doing the subsequent questions becomes impossible.
If you guys could help me do the question or point me in the right direction or atleast give me an example like this that involves bit theory with some kind of working solution.
I'd greatly appreciate any help.
Thanx | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8736286163330078, "perplexity": 696.8500459427054}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471983578600.94/warc/CC-MAIN-20160823201938-00193-ip-10-153-172-175.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/moving-body-on-a-diagonal-path.679219/ | # Homework Help: Moving body on a diagonal path
1. Mar 18, 2013
### emutudeng
Just a question about simple moving body exercise. With no acceleration.
Body moves with speed 15 m/s, time 12 seconds, mass 0.8 kg, and the force have a side effect 5 Newtons.
Firstly, F=mg, so 0,8*0,98.Then I calculated the actual moving direction's force by pythagoros, thats 0,95N.
Should i use cosinus or what ?
2. Mar 18, 2013
### tiny-tim
hi emutudeng!
i don't understand
you say there's no acceleration, but then you use g,
and what do you mean by "side effect"?
and what angle is there (for the cosine)?
what exactly is the body doing?
3. Mar 18, 2013
### emutudeng
Lets say there is a body (0,8kg) moving in direct line for 12 seconds with balanced speed 15 m/s, but there is a wind blowing from a side with a force 0,5 N.
So the body is moving diagonally and I should get the diagonal with pytagoros. The thought that the angle is between the actual trajector and the first direct line, but maybe i dont even need it.
Edit: i need the distance
4. Mar 18, 2013
### tiny-tim
ah!
ok, then this is wrong …
F is the force, and that's already given, as 0.5 N
you need to find the acceleration, so use F = ma
then find the sideways distance moved, then use pythagoras | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8751033544540405, "perplexity": 2740.1611929878463}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589757.30/warc/CC-MAIN-20180717164437-20180717184437-00382.warc.gz"} |
https://cstheory.stackexchange.com/tags/dependent-type/hot?filter=year | # Tag Info
12
It turns out that $W$ types plus identity types (eq/= in Coq) allow you to construct pretty much all the general inductive types you want, with the expected computation rules, and even canonicity. This is a very recent result of mine, you can read a preprint at Why not W?, which has been accepted for publication in the TYPES 2020 post-proceedings. The idea ...
7
Definitional equality is the same as equality in the metatheory. It works exactly the same way as in 1-category theory. If I have a category $\mathbb{C}$ and some morphisms $f,g : \mathbb{C}(A, B)$, I write $f = g$ for their equality, where $=$ is a metatheoretical relation. I can assume a family structure on $\mathbb{C}$ to get a CwF, plus assume some type ...
6
The basic idea is clearest when you think about things in terms of Tarski-style universes. There, you have a data type of codes, and an interpretation function which maps codes to types. In this case, it is obvious that you can discriminate on type codes, even though this is a much more questionable operation on types. For example, $\mathbb{N} \to 0$ and $\... 6 The simplest one that I know about is the$\text{Set}$-based polynomial model ("container" model). Here, every context is interpreted as a family of sets, i.e. a$Q : \text{Set}$together with an$A : Q \to \text{Set}$. We can view this as a set of questions together with sets of possible answers for each question, or a request-response protocol ... 6 No: strong elimination of large inductive types (SELIT) is itself inconsistent because it breaks the layering of universes by trivially allowing you to smuggle a large value in a Prop box and take it back out unscathed. In Is Impredicativity Implicitly Implicit? I proposed a restriction on SELIT which is a bit more permissive than Coq's while still enjoying ... 4 Take any small symmetric monoidal category$V$. Then the category of$V$-valued presheaves will (a) have closed monoidal structure (via Day convolution), and (b) have enough stuff (inherited from$\mathrm{Set}$) to interpret dependent types. This gives you enough structure to interpret something like our LNL calculus pretty easily, because there is a nice ... 4 Use an auxiliary type of positive natural numbers. data positive : Set where one : positive s0 : positive → positive -- multiply by 2 s1 : positive → positive -- multiply by 2 and add 1 data N : Set where zero : N pos : positive → N Supplemental: Another option, which I found on my whiteboard today (probably put there by Egbert Rijke months ago)... 4 U : U is inconsistent in a wide variety of settings. It is safe to say that it's inconsistent in any type theory. Deriving False from it is feasible by hand. The simplest version of this is called Hurkens' paradox: Original source Coq implementation. Agda implementation. The Coq source additionally describes the sufficient conditions for getting False. In ... 4 May I have a reference to why η expansion is invalid for CoC? It's not invalid. It's up to choice whether$\eta$-conversion for functions (or other types) is included. The original CoC paper seems to omit it, but as you see ATAPL includes it. I'm not certain, but$\eta$may have been omitted from the original source because it was difficult to handle in the ... 4$\text{absurd} : (A : \text{U}) \to 0 \to A$and$\text{elim} : (A : 0 \to \text{U}) \to (x : 0) \to A\,x$are equivalent. To go right, use$\text{absurd}\,(A\,x)\,x$. To go left, use$\text{elim}\,(\lambda\,x.\,A)\,x$. Also, both types are propositions because of the$0$-s in domains. There's not much reason to assume or use$\text{elim}$instead of$\text{...
4
Definitional equality is essentially a syntactic notion of equality, not witnessed by a term in the type theory: when two types or terms are definitionally equal, we are saying that they are precisely the same. Therefore, definitional equality of types is interpreted as equality of objects, and definitional equality of terms is interpreted as equality of ...
4
To me, type theory bridges programming with models and proof theories. In particular, I can use category theory to think about programming languages when the underlying type theory has a categorical model (e.g. the intuitionistic type theory by Martin Löf). On the other hand, type theory to programming is like (point-set) topology to analysis -- it gives you ...
4
Disclaimer: I've never actually implemented any variation of cubical type theories and I am not an expert in cubical type theory. I write this answer according to my own intuition. I welcome corrections and complementaries. How does this actually compute a path from a to d? The comp operator creates a term that has the following properties (in your case, ...
3
We certainly do not need very many $W$-types. If we also have universes, we only need one $W$-type, namely the natural numbers. For example, the UniMath library uses just the natural numbers and no other inductive types (if we discount the fact that standard types constructors, such as products and sum, are defined inductively in Coq).
3
Here is one article that discussed induction-recursion. Here's their code: data Lang : Set ⟦_⟧ : Lang → Set data Lang where Zero One Two : Lang Pair Fun Tree : (A : Lang) (B : ⟦ A ⟧ → Lang) → Lang ⟦ Zero ⟧ = ⊥ ⟦ One ⟧ = ⊤ ⟦ Two ⟧ = Bool ⟦ Pair A B ⟧ = Σ ⟦ A ⟧ λ a → ⟦ B a ⟧ ⟦ Fun A B ⟧ = (a : ⟦ A ⟧) → ⟦ B a ⟧ ⟦ Tree A B ⟧ = W ⟦ A ⟧ ...
3
"Types are the leaven of computer programming; they make it digestible." Robin Milner
3
I think I just came up with one. The following code block is written in a syntax similar to Agda. test : (a : _) (B : Set) (b : B) -> a ≡ b test a B b = refl Assuming ≡ to be the homogeneous equality type and refl to be its constructor, the solution to the underscore is B, which is not defined there yet. Type checking the above code (with open import ...
3
They are different things. SClos is a case-split function, which is a function that pattern match on its parameter. Think of it as a lambda (if you know Haskell, you can think of it as a LambdaCase \case, which is a sugared lambda). This is definitely a canonical value in Mini-TT. The neutral value, OTOH, is an application on a case-split function, where the ...
3
This is impossible. Suppose that we have such a type $T$, with an implementation of addition $\mathit{add} : T \to T \to T$, which is judgementally commutative. Because MLTT is strongly normalising, we know that we can put $\mathit{add}$ in $\beta$-normal, $\eta$-long form. Now suppose that we have two variables $x, y$ of type $T$. Now consider the terms $\... 2 As an inhabited proposition$(W_{a:A}B(a)\to 0)\to 0$has only one term in normal form. First, without function extensionality, this type cannot be proven to be propositional. Second, propositional types do not necessarily have unique normal forms. Normal forms are up to$\beta\eta$rules, not propositional equality.$0 \to 0\$ has infinitely many normal ...
2
It is false that the only well-typed occurrence of Prf has to be of the form Prf(all ...). For example, in the context with a variable p : Prop we can form the type Prf(p) which is not of the stated form. Another possibility is that we have a Prf(t) for some closed term t : Prop which is not of the form all ... but it normalizes to it. The purpose of Prf is ...
2
You asked several questions. You asked about a type indexed by a list, so you can do this. data DataType (A : Type) (F : A -> Type) : List A -> Type where empty : DataType A F [] _bla_ : forall {xs} {x} -> DataType A F xs -> F x -> DataType A F (x ∷ xs) Where F is a type family that is indexed by the elements of the list. Apart from that, ...
2
I don't have an answer for that question, I'm afraid. But I'll just point out that the rule you cite doesn't say that we should use Γ ⊢ e₁ e₂ ⇒ τ just because function application is an eliminator. Instead, it says that within the typing rule for function application, the part that checks the actual function should synthesize (i.e. should be Γ ⊢ e₁ ⇒ τ₁ → ...
2
IMHO, it's not like constructors are checked, but introduction rules (including tuples, lambdas, data constructors, etc.) are checked (while elimination rules are synthesized). The kind constructors you mentioned are called formation rules, which are not introduction rules. You can also think this way: if a constructor is checked, then it has a corresponding ...
2
No, but only under a very specific condition (neither MLTT, or CoC, or LF) -- that your type theory has certain resource control mechanism (linear or affine). See Affine logic Wikipedia entry. Affine logic predated linear logic. V. N. Grishin used this logic in 1974, after observing that Russell's paradox cannot be derived in a set theory without ...
1
IMHO universe hierarchy is particularly complicated to implement while the benefit is very small. It complicates programming in the language and the implementation of the language. The only benefit is to ensure the logical consistency. I guess that's why everybody is not writing about it. Here are some answers to your questions. I don't know much type theory ...
1
I have a shorter answer: normalization is usually used in conversion check of terms (aka definitional equality), and CoC has untyped conversion check. In conversion check, we normalize terms and compare 'em syntactically (think of it as comparing the ASTs). This probably means the normalization process doesn't have access to the types of terms, so it may not ...
1
Ali Asaf worked out a hierachy of universes with explicit coercions (lifting) in A calculus of constructions with explicit subtyping and established a relationship with cummulative universes.
1
CoqMT (Coq Modulo Theory) was an extension of the Coq proof assistant that allows one to parametrize a development with a decidable first-order theory T. Since equality on natural number expressions with addition and multiplication is decidable, this would be a valid application of CoqMT. Unfortunately, the implementation has not been updated in over 10 ...
1
Here is an example exploiting positivity of an index to prove false: module Whatever where open import Level using (Level) open import Relation.Binary.PropositionalEquality open import Data.Empty variable ℓ : Level A B : Set ℓ data _≅_ (A : Set ℓ) : Set ℓ → Set ℓ where trefl : A ≅ A Subst : (P : Set ℓ → Set ℓ) → A ≅ B → P A → P B Subst P trefl PA = ...
Only top voted, non community-wiki answers of a minimum length are eligible | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8231363296508789, "perplexity": 657.1158191179677}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057225.38/warc/CC-MAIN-20210921131252-20210921161252-00437.warc.gz"} |
https://www.cuemath.com/ncert-solutions/q-1-exercise-2-3-polynomials-class-10-maths/ | # Ex.2.3 Q1 Polynomials Solution - NCERT Maths Class 10
Go back to 'Ex.2.3'
## Question
Divide the polynomial $$p(x)$$ by the polynomial $$g(x)$$ and find the quotient and remainder in each of the following:
(i)
\begin{align}\,\,\,p(x) &= {x^3} - 3{x^2} + 5x - 3, \\ \quad\;\; g(x) &= {x^2} - 2\end{align}
(ii)
\begin{align}\,\,\,p(x)& = {x^4} - 3{x^2} + 4x + 5, \\ \quad\; g(x) &= {x^2} + 1 - x\end{align}
(iii)
\begin{align}\,\,\,p(x) &= {x^4} - 5x + 6, \qquad \\ \quad g(x) &= 2 - {x^2}\end{align}
Video Solution
Polynomials
Ex 2.3 | Question 1
## Text Solution
What is unknown?
The quotient and remainder of the given polynomials.
Reasoning:
You can solve this question by following the steps given below:
First, arrange the divisor as well as dividend individually in decreasing order of their degree of terms.
In case of division, we seek to find the quotient. To find the very first term of the quotient, divide the first term of the dividend by the highest degree term in the divisor.
Now write the quotient.
Multiply the divisor by the quotient obtained. Put the product underneath the dividend.
Subtract the product obtained as happens in case of a division operation.
Write the result obtained after drawing another bar to separate it from prior operations performed.
Bring down the remaining terms of the dividend.
Again, divide the dividend by the highest degree term of the remaining divisor.
Repeat the previous three steps on the interim quotient.
Steps:
(i)\begin{align}\,\,\,p(x) &= {x^3} - 3{x^2} + 5x - 3, \\ \quad\;\; g(x) &= {x^2} - 2\end{align}
Quotient $$= x - 3,$$ Remainder $$= 7x - 9$$
(ii) \begin{align}\,\,\,p(x)& = {x^4} - 3{x^2} + 4x + 5, \\ \quad\; g(x) &= {x^2} + 1 - x\end{align}
$$={x}^{4}+0.{x}^{3}-3{{x}^{2}}+4x+5$$
Quotient = $${x^2} + x - 3$$ Remainder $$= 8$$
(iii) \begin{align}\,\,\,p(x) &= {x^4} - 5x + 6, \qquad \\ \quad g(x) &= 2 - {x^2}\end{align}
\begin{align}= {x^4} + 0.{x^2}-5x + {\text{ }}6\end{align}
Quotient $$= - {x^2} - 2,$$ Remainder $$= - 5x + 10$$
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• Personalized curriculum to keep up with school | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 14, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.999993085861206, "perplexity": 2277.581155181031}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251778272.69/warc/CC-MAIN-20200128122813-20200128152813-00325.warc.gz"} |
http://mathhelpforum.com/algebra/22379-stuff-again-print.html | # Stuff... again... :)
• November 9th 2007, 08:07 PM
Rocher
Stuff... again... :)
1. Frank and Ernest start jogging on a 110m circular track. They begin at the same time and from the same point but jog in opposite directions, one at $\frac{8}{3}m$ per second and the other at $\frac{7}{3}m$ per second. How many times will they meet during the first 15 minutes of jogging?
2. Two candles of the same height are lit at the same time.. The first is consumed in four hours, the second in three hours. Assuming that each candle burns at a constant rate, how many hours after being lit was the first candle twice the height of the second?
3. If a:b = 3:4 and a: (b+c)= 2:5, find the value of a:c.
4. $\frac{1}{2}a = \frac{4}{3}b = \frac{5}{6}c$
5. The volume of two boxes are in the ratio 3:8. If the volume of the larger box is $375cm^3$ more than the volume of the smaller box, find the volume of each box.
Express each of the following with denominator 1:
6. $(x^3)^\frac{1}{3}$
7. $((a-b)^4)^\frac{1}{4}$
Simplify / Evaluate
8. $(\frac{x^4y^6}{a^8b^10})^\frac{3}{2}$
Simplify
9. $\frac{(xy^2)^4}{2x^3y^3}$
Solve
10. $9x=27$
Thank you guys.
• November 9th 2007, 09:02 PM
earboth
Quote:
Originally Posted by Rocher
...
Express each of the following with denominator 1:
6. $(x^3)^\frac{1}{3}$
7. $((a-b)^4)^\frac{1}{4}$
Simplify / Evaluate
8. $(\frac{x^4y^6}{a^8b^{10}})^\frac{3}{2}$
Simplify
9. $\frac{(xy^2)^4}{2x^3y^3}$
Solve
10. $9x=27$
Thank you guys.
Hello,
to #6, #7:
$(x^3)^\frac{1}{3}=x=\frac{x}{1}$
$((a-b)^4)^\frac{1}{4}=|a-b|=\frac{|a-b|}{1}$
to #8:
$(\frac{x^4y^6}{a^8b^{10}})^\frac{3}{2}=x^{4\cdot \frac32} \cdot y^{6\cdot \frac32} \cdot a^{-8 \cdot \frac32} \cdot b^{-10 \cdot \frac32}$ = $x^6 \cdot y^9 \cdot a^{-12} \cdot b^{-15}$
to#9:
$\frac{(xy^2)^4}{2x^3y^3}=\frac12 \cdot x^{4-3} \cdot y^{2 \cdot 4 - 3}=\frac12 \cdot x \cdot y^5$
to #10:
Are you sure that you can't solve this equation? (Divide both sides by 9 [you do this to get 1 as the coefficient of x because you want to know the value of one x]. For confirmation only: I've got x = 3)
• November 9th 2007, 09:32 PM
earboth
Quote:
Originally Posted by Rocher
1. Frank and Ernest start jogging on a 110m circular track. They begin at the same time and from the same point but jog in opposite directions, one at $\frac{8}{3}m$ per second and the other at $\frac{7}{3}m$ per second. How many times will they meet during the first 15 minutes of jogging?
...
3. If a:b = 3:4 and a: (b+c)= 2:5, find the value of a:c.
4. $\frac{1}{2}a = \frac{4}{3}b = \frac{5}{6}c$
5. The volume of two boxes are in the ratio 3:8. If the volume of the larger box is $375cm^3$ more than the volume of the smaller box, find the volume of each box.
...
Hi,
to #1.: The distance between the two joggers increases by 5 m/s. Together they need 22 s to run the full distance of 110 m.
15 min = 900 s. Therefore they meet $\frac{900}{22} \text{ times} \approx 40\text{ times}$
#3.: From the first equation you get: $b=\frac43 a$ . Plug in this term into the 2nd equation:
$\frac{a}{\frac43 a + c}=\frac25$ . After a few steps of simplification you get:
$\frac{7}{15}a = \frac25 c~\iff~\frac ac=\frac25 \cdot \frac{15}{7} = \frac67$
to #4.: Split this "chain of equalities" into 3 equations and solv for a, b and c:
$\left \{\begin{array}{l}\frac{1}{2}a = \frac{4}{3}b \\ \frac{1}{2}a = \frac{5}{6}c \\ \frac{4}{3}b = \frac{5}{6}c\end{array} \right.$ (Remark: There doesn't exist an unique solution! I've got $[a,b,c]=[40k,15k,24k],~k\in\mathbb{Z}$
to #5.:
Let V be the volume of the smaller box. Then you have:
$\frac{V}{V+375}=\frac38$ Solve for V. You should get V = 225 cm³
• November 10th 2007, 06:57 AM
Rocher
to #10:
Are you sure that you can't solve this equation? (Divide both sides by 9 [you do this to get 1 as the coefficient of x because you want to know the value of one x]. For confirmation only: I've got x = 3)[/QUOTE]
Ah dammit, stupid Latex. It was 9^x. So x is an exponent.
• November 10th 2007, 07:25 AM
Soroban
Hello, Rocher!
Quote:
2. Two candles of the same height are lit at the same time.
The first is consumed in four hours, the second in three hours.
Assuming that each candle burns at a constant rate, how many hours
after being lit was the first candle twice the height of the second?
The first candle is consumed in 4 hours.
. . In one hour, $\frac{1}{4}$ iof the candle is gone.
. . In $x$ hours, $\frac{x}{4}$ of the candle is gone.
In $x$ hours, there will be: $\left(1 - \frac{x}{4}\right)$ of the candle left.
The second candle is consumed in 3 hours.
. . In one hour, $\frac{1}{3}$ of the candle is gone.
. . In $x$ hours, $\frac{x}{3}$ of the candle is gone.
In $x$ hours, there will be $\left(1 - \frac{x}{3}\right)$ of the candle left.
If the first candle is twice the height of the second candle,
. . we have: . $1 - \frac{x}{4} \;=\;2\left(1 - \frac{x}{3}\right)$
Now solve for $x.$
Quote:
3. If $a:b \:= \:3:4$ and $a: (b+c)\:= \:2:5$,
find the value of $a:c$
We have: . $\frac{a}{b} \:=\:\frac{3}{4}\quad\Rightarrow\quad b \:=\:\frac{4}{3}a$ .[1]
And: . $\frac{a}{b+c} \:=\:\frac{2}{5}\quad\Rightarrow\quad 5a \:=\:2b + 2c$ .[2]
Substitute [1] into [2]: . $5a \:=\:2\left(\frac{4}{3}a\right) + 2c\quad\Rightarrow\quad \frac{7}{3}a \:=\:2c\quad\Rightarrow\quad \frac{a}{c} \:=\:\frac{6}{7}$
Therefore: . $\boxed{a:c\:=\:6:7}$
Quote:
8. Simplify: . $\left(\frac{x^4y^6}{a^8b^{10}}\right)^\frac{3}{2}$
We have: . $\frac{(x^4)^{\frac{3}{2}}(y^6)^{\frac{3}{2}}} {(a^8)^{\frac{3}{2}} (b^{10})^{\frac{3}{2}} } \;=\;\frac{x^6y^9}{a^{12}b^{15}}$
Quote:
10. Solve: . $9x=27$ . ??
Since the problem is way too simple, I'll assume that it says: . $9^x \:=\:27$
We have: . $(3^2)^x \:=\:3^3\quad\Rightarrow\quad 3^{2x} \:=\:3^3$
Therefore: . $2x \:=\:3\quad\Rightarrow\quad\boxed{ x \:=\:\frac{3}{2}}$
• November 10th 2007, 07:33 AM
Rocher
Quote:
Originally Posted by Soroban
Hello, Rocher!
The first candle is consumed in 4 hours.
. . In one hour, $\frac{1}{4}$ iof the candle is gone.
. . In $x$ hours, $\frac{x}{4}$ of the candle is gone.
In $x$ hours, there will be: $\left(1 - \frac{x}{4}\right)$ of the candle left.
The second candle is consumed in 3 hours.
. . In one hour, $\frac{1}{3}$ of the candle is gone.
. . In $x$ hours, $\frac{x}{3}$ of the candle is gone.
In $x$ hours, there will be $\left(1 - \frac{x}{3}\right)$ of the candle left.
If the first candle is twice the height of the second candle,
. . we have: . $1 - \frac{x}{4} \;=\;2\left(1 - \frac{x}{3}\right)$
Now solve for $x.$
Is x -4??
Also, I have another question xD
$2^x-1=8$ >_< Latex doesn't work... It's 2 to the exponent x-1 = 8. x-1 is the exponent, the base is 2.
• November 10th 2007, 09:54 AM
earboth
Quote:
Originally Posted by Rocher
Is x -4??
Also, I have another question xD
$2^{x-1}=8$ ...
Hello,
Since $8 = 2^3$ your equation becomes:
$2^{x-1}=2^3$
Two powers with equal bases are equal if the exponents are equal too:
Therefore: $x-1 = 3~\iff~x = 4$
• November 10th 2007, 12:13 PM
topsquark
Quote:
Originally Posted by Rocher
$2^x-1=8$ >_< Latex doesn't work... It's 2 to the exponent x-1 = 8. x-1 is the exponent, the base is 2.
When you are coding the LaTeX, put what you want in the exponent inside a pair of { }:
2^{x - 1} generates $2^{x - 1}$
-Dan | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 74, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9013283848762512, "perplexity": 1498.8712858639399}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657136545.14/warc/CC-MAIN-20140914011216-00332-ip-10-234-18-248.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/333052/interpretaton-of-confidence-interval/333097 | # Interpretaton of confidence interval
I have read two books that explicitly state that the $(1-\alpha)$% confidence interval should be interpreted as:
If you construct 100 such confidence intervals, $\alpha$ of them are expected to not contain the true population statistic and $(1-\alpha)$ of them are expected to contain the true population statistic.
and not as
There is a $(1-\alpha)$% probability that the true population statistic is contained in the confidence interval.
In my view, they both equivalent: If you make the first statement, you implicitly make the second statement. You are looking at any one arbitrary confidence interval, which in itself is a random variable, the generic confidence interval should, therefore, be subject to the second statement. Do things change when this random variable is actually realized?
-
I struggled with this concept for quite a while, but an intuitive explanation for it I found on Wikipedia (as background context a $95\%$ confidence interval of the number of voters voting for a particular party was found to be $[36\%,44\%])$:
I think confidence intervals are better understood in the following way. There is an infinite set of constant open intervals. Some of them cover the population statistic $\theta$ and some of them don't. Now the (random) confidence interval is just a mechanism to draw an interval from this set. So the $(1-\alpha)\%$ is the chance of picking a constant interval that covers $\theta$. But whatever interval is drawn (or realized), since each of these interval is constant, the probability that it contains $\theta$ can only be $0$ or $1$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8799468874931335, "perplexity": 202.28181265889344}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645338295.91/warc/CC-MAIN-20150827031538-00202-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://www.vosesoftware.com/riskwiki/TheJacobiantransformation.php | The Jacobian transformation | Vose Software
# The Jacobian transformation
The Jacobian transformation is an algebraic method for determining the probability distribution of a variable y that is a function of just one other variable x (i.e. y is a transformation of x) when we know the probability distribution for x.
• Let x be a variable with probability density function f(x) and cumulative distribution function F(x);
• Let y be another variable with probability density function f(y) and cumulative distribution function F(y);
• Let y be related to x by some function such that x and y increase monotonically, then we can equate changes dF(y) and dF(x) together, i.e.:
|f(y)dy| = |f(x)dx|
Rearranging a little, we get:
is known as the Jacobian.
Example
If x = Uniform(0,c) and y = 1/x:
so
so the Jacobian is
which gives the distribution for y: | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9971397519111633, "perplexity": 783.6337987248337}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488517048.78/warc/CC-MAIN-20210622093910-20210622123910-00430.warc.gz"} |
https://thatsmaths.com/2013/08/22/ternary-variations/ | Ternary Variations
Georg Cantor (1845-1918) was led, through his study of trigonometric series, to distinguish between denumerably infinite sets like the rationals and uncountable sets like the reals. He introduced a set that is an abstract form of what we now call Cantor’s Ternary Set. In fact, the ternary set had been studied some ten years earlier by the Dublin-born mathematician Henry Smith and, independently, by the Italian Vito Volterra. General sets of this form are now called Smith-Volterra-Cantor sets (SVC sets).
Construction of the Ternary Set
To construct the ternary set we proceed iteratively. Starting with the unit interval ${C_0 = [0,1]}$, we remove the open interval ${I_1 = (\frac{1}{3},\frac{2}{3})}$ corresponding to the “middle third” to get ${C_1 = [0,\frac{1}{3}]\cup[\frac{2}{3},1]}$. Next, we remove the open middle third of each remaining interval, ${I_2 = (\frac{1}{9},\frac{2}{9})\cup(\frac{7}{9},\frac{8}{9})}$ to get ${C_2}$, a union of four closed intervals. Continuing this process, we arrive ultimately at the ternary set
$\displaystyle C = \bigcap_{n=0}^{\infty} C_n = [0,1] - \bigcup_{n=1}^{\infty} I_n \,.$
The initial stages of the construction are shown in the figure below.
The first seven stages in constructing the Cantor ternary set.
Properties of ${C}$
1. The set ${C}$ is “large“: it is uncountable, making it large compared to the rational numbers.
2. The set ${C}$ is “small“: it cannot have a positive length: at each stage, the length is decreased by a factor ${\frac{2}{3}}$ and the limit of ${(2/3)^n}$ is zero.
3. ${C}$ is self-similar: If we scale it up by a factor of 3, we obtain two pieces, each identical to ${C}$ itself.
4. ${C}$ is fractal, with a fractal dimension of ${(\log 2 /\log 3 )}$
5. ${C}$ is a perfect set: For every point in ${C}$, there are other points arbitrarily close to it; there are no isolated points.
6. ${C}$ is nowhere dense: The interior of the closure is empty.
7. ${C}$ is totally disconnected.
We will not prove these properties, which are demonstrated in many standard texts on point set topology. But a few remarks are apposite.
(1) Length: The length removed at stage ${n}$ is ${2^{n-1}/3^n}$. Summing these, the total length rermoved is 1. This implies that the remaining length is 0. Technically, the set ${C}$ is of Lebesgue measure zero.
(2) Size: for any point ${x}$ in ${C}$, we construct a binary number as follows: if ${x}$ is to the left of a middle third removed at stage ${n}$ the ${n}$th digit is 0. If to the right, the ${n}$th digit is 1. Clearly this gives a one-to-one correspondence between ${C}$ and all binary numbers in ${[0,1]}$ so ${C}$ must be uncountable.
(3) Self-similar: Scaling by 3 maps all numbers in the interval ${[0,\frac{1}{3}]}$ to ${[0,1]}$ in such a way that the original set ${C}$ is reproduced. The elements of ${C}$ in ${[\frac{2}{3},1]}$ give a copy of ${C}$ shifted to the interval ${[2,3]}$.
(4) Fractal: In coordinate geometry, if all axes are scaled by a factor ${S}$, the length of a line segment is increased by ${S^1}$, the area of a square by ${S^2}$, the volume of a cube by ${S^3}$, etc. In general, a ${D}$-dimensional set scales as ${S^D}$. But we have seen that scaling ${C}$ by ${S = 3}$ doubles the set. So ${3^D = 2}$ or ${D = (\log 2 /\log 3 ) \approx 0.631}$.
For discussion of (5), (6) and (7) see topology texts.
Generalization
Suppose ${0 < r < 1}$. Starting with ${[0,1]}$, we remove from the centre of each component an open interval that is a fraction ${r}$ of its length,leaving two closed intervals of length ${s=(1-r)/2}$. Iterating this process, the length remaining at stage ${n}$ is ${(1-r)^n}$. This tends to zero. The result is a ternary set that, expanded by a factor ${1/s = 2/(1-r)}$ yields two copies of itself. So
$\displaystyle \left( \frac{1}{s} \right)^{\!\!\!^D} = 2 \qquad\mbox{or}\qquad D = \left( \frac{\log 2}{\log(1/s)} \right) \,.$
By choosing ${s}$ correctly, we can obtain a set of any fractal dimension between 0 and 1.
The choice ${r = 4/5}$ or ${s = 1/10}$ is interesting: the first stage removes all numbers except those whose decimal expansion begins with 0 or 9. (A technicality: numbers with terminating decimals are assumed to be represented by infinite expansions, e.g. 0.1 = 0.0999…). At the second stage, only numbers beginning with .00, .09, .90 or .99 remain. Ultimately, we obtain all numbers with decimal expansions containing only zeros and nines. The fractal dimension of this set is ${(\log 2 /\log 10 ) =\log_{10}2=0.3010}$.
Fat Fractal. The SVC set.
So far, all the ternary sets were of measure zero. It is possible, and quite easy, to modify the construction procedure so that the resulting set ${S}$ is of positive measure. For example, at stage 1 we remove 1/4 from the middle. At stage 2 we remove 1/16 (rather than 1/8). At stage ${n}$ we remove (${1/2^{2n}}$) from each interval. The total length removed is
$\displaystyle \frac{1}{4} + \frac{2}{16} + \frac{4}{64} + \dots = \frac{1}{4}\left( 1 + \frac{1}{2} + \frac{1}{4} + \dots\right) = \frac{1}{2}$
and the length remaining is also 1/2. Such a set, with positive measure, is called a fat fractal. The set ${S}$ is a particular example of a Smith-Volterra-Cantor or SVC set.
The first six stages in constructing the Smith-Volterra-Cantor set.
Volterra used such a set to define a function ${V(x)}$ with a remarkable property: ${V(x)}$ is differentialbe everywhere on ${[0,1]}$ but its derivative, although bounded, is not (Riemann) integrable. This defies the Fundamental Theorem of Calculus. As a consequence of results like this, a completely new method of integration, Lebesgue integration, was developed.
But that is another story, to which we shall return. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 68, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9231820106506348, "perplexity": 280.1567737417814}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256494.24/warc/CC-MAIN-20190521162634-20190521184634-00275.warc.gz"} |
https://brilliant.org/discussions/thread/ekene-franklins-messageboard/ | # Ekene Franklin's messageboard.
Feel free to discuss anything here with me friends! We're here to help each other!
Note by Ekene Franklin
9 months ago
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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}$
Sort by:
My country's national olympiads 1st round is to be conducted tomorrow. I will most likely post all of them here for our consumption! I expect them to be pretty easy.
- 9 months ago
Ok. Thanks and also are you writing that exam.
- 9 months ago
It is actually meant for students two years older than I am, but I will give it a shot.
- 9 months ago
Oh !!!
- 9 months ago
It there a minimum age requirement to be eligible to write your country's olympiads?
- 9 months ago
Yes, I think it is 13 or 14. People from class 8 or 9 onwards can write the exam.
- 9 months ago
Are you also writing the olympiads soon?
- 9 months ago
No. At present I am not interested. The main goal of student student studying in class 11 and 12 in India is to get into IIT (Indian Institute Of Technology)
- 9 months ago
Oh, so that is what JEE and IIT are all about. I hope you get into it seamlessly. But say, do you hope to write olympiads in the future?
- 9 months ago
Yes. I may.
- 9 months ago
This is a marvelous secret and I benefited a lot from this. If you want to write any question for the problems of the week the better option is to write a physics problem rather than maths. Out of 5 problems 3 are allotted for maths and 2 for science. In the maths category there will be a much competition from many users. But in the science category there are only 4 competitors namely Rohit Gupta, Pranshu Gaba, and two staff members Blake Farrow, Danielle Scarano. So, I posted some good physics problems and 3 of them are featured now. But I won't force you to post only physics problems then there will be much competition for science. If you have good problems in maths you can post them (as 3 of my maths problems are also featured although I say that I am basically a physics man).
- 8 months, 3 weeks ago
Where is the show full site option??(from ur mt. Everest Q discussion)
- 8 months, 3 weeks ago
It will be in the main option bar. It is the place where you can find links like "profile", "stats", "search", etc. The "view" full site option will be present there.
- 8 months, 3 weeks ago
Main option bar : Those 3 lines on left corner?? Or in the stats section of taskbar?
- 8 months, 3 weeks ago
The three line aside the option "Premium"
- 8 months, 3 weeks ago
Oh so this is on website. Can this be done in app?
- 8 months, 3 weeks ago
Ohh !! You are doing in the app. Sorry, I don't know whether this is possible in app or not. But can't you find any option like this. in the screenshot.
- 8 months, 3 weeks ago
No :-( how can I send screenshot on this chat?
- 8 months, 3 weeks ago
You have to copy the link of the picture and paste it here.
- 8 months, 3 weeks ago
K thnx a lot for helping, and sorry for disturbing..
- 8 months, 3 weeks ago
It's OK. No problem. I went to brilliant only once in the mobile app so I am not much aware of the options present in it.
- 8 months, 3 weeks ago
Mauj Rajpal, from which part of India are you from.
- 8 months, 3 weeks ago
Delhi ;-) and u??
- 8 months, 3 weeks ago
- 8 months, 3 weeks ago
Oh nice. From where do you put your problems in sets(source) and how do you have sets of other people in yours??
- 8 months, 3 weeks ago
In every set there will be a option at the bottom called "save". If you click that, that particular set will be saved into your sets. Some of my problems are original whereas some other are completely taken from books or the problems and the remaining are the problems which I have taken from books and modified them.
- 8 months, 3 weeks ago
Can you suggest me some of your book(especially maths)
- 8 months, 3 weeks ago
It's our college material. It is a collection of all good problems from various books.
- 8 months, 3 weeks ago
Here is the screenshot of where is it present. Can you find it now ?
- 8 months, 3 weeks ago
Is this screenshot from app?
- 8 months, 3 weeks ago
No. It is from from the main website.
- 8 months, 3 weeks ago
How do you send questions to them to select for weekly problem??Do they themselves select from your sets?
- 8 months, 3 weeks ago
No. Only staff will select the problems of the week. In general, the notification will be as if I have written the problem of the week.
- 8 months, 3 weeks ago
They themselves select it by seeing your sets or you send them via some forum and then they choose one of the all sent questions??
- 8 months, 3 weeks ago
The community members write problems and post them to community and the staff selects the best 15 questions among all the problems and they will feature them as problems of the week. There is no forum involved in this process.
- 8 months, 3 weeks ago
I personally think Brilliant should include a biology and biochemistry section. What do you think?
- 7 months ago
Yes.
- 7 months ago
Before brilliant used to have biology. But recently when the staff updated brilliant they removed biology topic due to lack of problems in that category.
- 7 months ago
Actually I asked them to remove it, because the biology problems could not be seen in the community page. You can see Calvin Lin's messageboard.
- 7 months ago
@X X Calvin Lin's message board is currently unavailable. I searched for the website for a long time but in vain. Maybe he have locked it.
- 7 months ago
Really? Here is the link. It's still available.
- 7 months ago
@X X But last time when I went it was locked
- 7 months ago
yes
- 6 months, 3 weeks ago
Here is a link of a biology problem.
- 6 months, 3 weeks ago
Yes, but they cannot be found in the community page.
- 6 months, 3 weeks ago
@X X I know. And there is no biology when you're making a problem.
- 6 months, 3 weeks ago
@B D Previously there was, but they removed it.
- 6 months, 3 weeks ago
@X X OK
- 6 months, 3 weeks ago
- 6 months, 3 weeks ago
Hello Everyone !!! As this note has got some immense response from young students I would like to continue this further. But the problem is that this note is getting more and more comments day by day due to which the process of loading and opening any notifications is taking more time. So, on the request of @EKENE FRANKLIN any further discussion can be continued in the new note Brilliant young students messageboard
- 6 months, 3 weeks ago
$\text{Doubt : 1}$
If the mole fraction of a solute is changed from $\dfrac14$ to $\dfrac12$ by adding some solute in the $800~g$ of $H_2O$ solvent, then what will be the ratio of molarity of the two solutions ?
- 9 months ago
$\text{Doubt : 2}$
In the following reaction : $\large aNO_3^- + bCl^- + cH^+ \rightarrow dNO + eCl_2 + fH_2O$ find the value of $\dfrac{a^2(b + c)}{d \times f}$
- 9 months ago
Simply balance the redox equation and substitute the stoichiometric coefficients of the balanced reaction into your expression.( I would have posted a solution if only my device could type LaTeX.)
- 9 months ago
Ok. Try now after keeping it in \ (....) without spaces which will give you $...$
- 9 months ago
Did you tried it. What is the answer you got.
- 9 months ago
Yeah, I solved it. My answer is 7.
- 9 months ago
That's correct answer. Can you atleast keep the values of a,b,cd,e,f.
- 9 months ago
a=2, b=6, c=8, d=2, e=3 and f=4
- 9 months ago
Ok. Thanks i will check with mine.
- 9 months ago
$\text{Doubt : 3}$
Uncertainty in position and momentum are equal. Then what will be the Uncertainty in velocity ?
- 9 months ago
I don't think uncertainty in position and momentum are relativistically equal. However, if a non-relativistic assumption is to be made, then from Heisenberg's, and taking p=mv, then uncertainty in velocity>=(h/2pi)X(1/(uncertainty in position X mass)). Sorry for the math ambiguity. My device doesn't type LaTeX.
- 9 months ago
Yes you are correct. You got the correct answer. Regarding the latex you can keep it in \ (...\ ).
- 9 months ago
The LaTeX is not still working. It prevents me from posting some reeeeeeallly cool problems.;(
- 9 months ago
$\text{Doubt : 4}$
In $(Si_4O_{12})^{8-}$, how many $Si - O - Si$ linkages are there ?
- 9 months ago
I have an answer, but I am not totally sure it is correct. Coordination structure chemistry of silicates and silicones is my weakest point in chemistry. Perhaps I need to practise more!:)
- 9 months ago
- 9 months ago
It is 4.
- 9 months ago
I was actually sure of 4 linkages, but I was not sure of one more. Thanks anyways.
- 9 months ago
It's OK.
- 9 months ago
You are a great guy, Ram.
- 9 months ago
That's my quality quantitatively speaking.
- 9 months ago
My goal in many interactions is to make them frictionless. Trivially speaking, I resist every enmity and conduct friendliness. I have even successfully transmuted people from an angry ground state, to a happy excited state!
- 9 months ago
Oh !!! That's good.
- 9 months ago
Thanks. Ask if you want more!
- 9 months ago
Thanks guys. Keep sending problems and lets discuss!
- 9 months ago
Hi guys! For sure we all have noticed that the derivative of kinetic energy is momentum. Is there a physical explanation for this?
- 9 months ago
Quantitatively speaking, momentum is the product of mass of a body and its velocity. It is denoted by $\overline{P}$. $\large \overline{P} = m \overline{v}$ But the above definition is valid only when velocity of the body is constant. If the velocity of the body is variable we cannot use this formula and we should go by calculus. Physically speaking, momentum is the rate of change of kinetic energy with respect to its velocity. So, you ca say it as kinetic energy at that instance. Here is how we get $\overline{P} = m \overline{v}$ from kinetic energy. $K.E = \dfrac12 m (\overline{v})^2$ Now, differentiate both side with respect to velocity. We get : $\dfrac{d}{dx} K.E = \dfrac{d}{dx} \left ( \dfrac12 m (\overline{v})^2 \right )$ As, we know $\dfrac{d}{dx} K.E = \overline{P}$, we get : $\overline{P} = \dfrac12 \cdot m \cdot 2 \overline{v}$ So, we get : $\large \boxed{\color{blue} \overline{P} = m \overline{v}}$
- 9 months ago
Perfect bro!
- 9 months ago
Thanks. But what I should say you, brother or sister
- 9 months ago
I am a young boy:)
- 9 months ago
Ok
- 9 months ago
Explain the Bohr theory of atoms, combining both mathematics and physics in your explanations.
- 9 months ago
Does anyone know how to post a solution of a problem that doesn't have a posted solution yet?
- 9 months ago
If you got it correct you click the button "No solution" under Continue and write. And if you got it wrong you can't.
- 9 months ago
Thanks a lot, bro!
- 9 months ago
You are welcome!
- 9 months ago
@Mohmmad Farhan why can't you come here. I think this is a best place for a student like you. You can post your quires here.
- 9 months ago
Sorry! I am not really active these days because of exams
- 8 months, 4 weeks ago
Ok. No problem !!!
- 8 months, 4 weeks ago
How do I post numerical answer problems on Brilliant?
- 9 months ago
I think you have posted many problems in brilliant. However, when you are writng a question there will be two options, either Number or Multiple choice. You can select whatever you wish.
- 9 months ago
Gearing up for today's 1st round olympiads!
- 9 months ago
Good luck!
- 9 months ago
Thanks. The results will be uploaded online in about 1.5 hours. Meanwhile I am posting them here for our consumption. Solve some, post solutions and invite others to do same!
- 9 months ago
Maybe you can post them like problems.
- 9 months ago
@B D Ok.
- 9 months ago
Problem 1: A password consists of $4$ distinct digits such that their sum is $19$ and exactly two of its digits are prime, e.g. $0397$. How many possible such passwords exist?
- 9 months ago
2,3: (2,3,6,8)
2,5: (2,5,4,8)
2,7: (2,7,4,6),(2,7,1,9)
3,5: x
3,7: (3,7,1,8),(3,7,0,9)
5,7: (5,7,1,6)
There are $7\times4!=168$ possibilities
- 8 months, 3 weeks ago
How did you get all possibilities?
- 8 months, 3 weeks ago
Find the 7 possibilities one by one, then times 24
- 8 months, 3 weeks ago
Problem 2: What is the maximum possible area of a quadrilateral of sides $1, 4, 7, 8$?
- 9 months ago
I think we should use the concept of maxima and minima.
- 9 months ago
Perhaps you should solve it. I used the maximality condition of Bretschneider's formula. My answer was $18$.
- 9 months ago
Yes I too got like that.
- 9 months ago
What is that?
- 8 months, 2 weeks ago
Looks similar to:
https://brilliant.org/problems/hexagon-octagon-in-a-rectangle/?ref_id=1538063
Maybe a harder version perhaps! :D
- 8 months, 2 weeks ago
Problem 3: $20$ lines are drawn in a plane. What is the maximum number of parts into which the plane can be divided by the $20$ straight lines?
- 9 months ago
I think it is $211$
- 8 months, 4 weeks ago
There is a trick. First, take the 20th triangular number. $\displaystyle\frac{20(20+1)}{2} = 210$ Add 1. $210+1=211$ These numbers are known as pizza numbers and are always one more than a triangular number.
P.S. @EKENE FRANKLIN did you use this method?
- 8 months, 4 weeks ago
Great! Just what I wanted, and what I used during the examination!
- 8 months, 4 weeks ago
Thumbs Up!
- 8 months, 3 weeks ago
What is the remainder when $1^{241} + 2^{241} + 3^{241} + 4^{241}$ is divided by $5$?
- 9 months ago
$0$
- 8 months, 4 weeks ago
Of course that is the answer, but prove it.
- 8 months, 4 weeks ago
Since the divisibility rule of 5 only concerns the last digit, we have to find the last digits of the values. $1^{124}$ is obviously 1. Since the repetition in the last digit is (for 2): 2, 4, 8, 6 then $241 \div 4 = k (\text{Remainder}1$) the first number is 2 (in this set). Apply this to the other powers. Summing the last digits, we get 10 and $10 \div 5 = k (\text{Remainder}0)$
- 8 months, 4 weeks ago
Excellent
- 8 months, 4 weeks ago
Great job. I did modular arithmetic during the examination.
- 8 months, 4 weeks ago
Me too
- 8 months, 3 weeks ago
Problem 5: Find the general term of the sequence described by $a_n = 4a_{n - 1} - 3a_{n - 2}$ given that $a_1 = 1$ and $a_2 = 9$.
- 9 months ago
$a_n=4\times3^{n-1}-3$
- 8 months, 3 weeks ago
Great. This was what I got. But do you mind showing your solution so I can clarify mine?
- 8 months, 3 weeks ago
Solve the equation $x^2=4x-3$ and get $x=3$ or $1$. Let $b_n=3^n-1^n$.
The first few terms of $b_n$ is $2,8,26,80...$ and $b_n$ also satisfies the recursion $b_n=4b_{n-1}-3b_{n-2}$
Compare the two sequences: $a_n=\frac43b_n-\frac53=4\times3^{n-1}-3$
- 8 months, 3 weeks ago
@X X Not exactly what I did, but something similar. Thanks a lot. If I may ask, have you ever participated in an olympiad before, or do you hope to participate in one?
- 8 months, 3 weeks ago
No, I haven't. I knew few things about it, so I'm not sure about participating in one.
- 8 months, 3 weeks ago
@X X Ok. Thanks. By the way, I really admire your problems. They are very cool. One of my major problems here is creating problems. My original problems are too easy(you can check them out). But solving others problems(of course not those extremely hard ones) is not a big problem for me.
- 8 months, 3 weeks ago
@X X I think this is your first problem of the week in easy catogery.
- 8 months, 3 weeks ago
You're right! (Ha, ha! I'm a little jealous of you because your problems are often the first problem in the easy category, and in your contributions page, the number of people are banging up:p)
- 8 months, 3 weeks ago
@X X But I am not fully done. I expected that Number + Reciprocal will take me to 1 lakh (100 K) solvers. But it fell short of 7 K. But luckily this week too my question came in one of the problems of week (the first one) so it is confirmed that I will reach the milestone in one or two days.
- 8 months, 3 weeks ago
@X X But some of your problems came in Intermediate and Advanced catogery.
- 8 months, 3 weeks ago
I know. (Fewer people solves those, that is why I was so happy to see my problem as the second of the basic category)
- 8 months, 3 weeks ago
@X X OH !!! Your problem is featured as the second one in the problems of the week. But my first problem (Thundering Question) is the third one.
- 8 months, 3 weeks ago
Oh, I remember that one! (I loved that week because a lot of my solutions appeared in that week. )
- 8 months, 3 weeks ago
@X X Yes, Vamprie Fan Club. It is were I had my first individual century. After that, the next week I had got 150 upvotes for my question Rising Balloons and the next week is a memorable for me, as I had my first double century for the question Counting 0's.
- 8 months, 3 weeks ago
@X X Ram Mohith and X X, both of you are excellent problem contributors, some of the best here. I hope I can reach your level!
- 8 months, 3 weeks ago
@X X It's a Fact :
I have made a hatrick in June 9 - June 29 (3 weeks) as 3 of my problem continuously came in easy part of the POTW. Now again the chance has come for me. Last week and this week too two of my problems are featured. So, if luck favors next week to I can have a problem in POTW. I am thinking very hard on which question should I post the next.
- 8 months, 3 weeks ago
The sequel to the Mt Everest problem about pressure cookers?
- 8 months, 3 weeks ago
- 8 months, 3 weeks ago
Is it released yet?
- 8 months, 3 weeks ago
No. Still under construction. Before that one I have two more questions, Raising Balloons - 2 and Popping balloons which are yet to be released. After these two the sequel for My. Everest vs Mt K 2 will be released.
- 8 months, 3 weeks ago
OH!
- 8 months, 3 weeks ago
How is my new problem, check it here. It is posed by B.D
- 8 months, 3 weeks ago
Cool video!
- 8 months, 3 weeks ago
@X X Yes. I liked that so much that I framed the question based on that.
- 8 months, 3 weeks ago
@X X Don't worry mister !!! You have the skill to post and solve tough questions (level 4 and 5).
- 8 months, 3 weeks ago
I made the cut for the 2nd round! I scored $76.70$ out of a maximum possible 110. Hurray!
- 9 months ago
Oh !!! Congrats.
- 8 months, 4 weeks ago
Thanks a lot. But no one apart from us is solving the problems. I want to check my solutions and my faulty areas. Please direct the problem solving masters of Brilliant here to help me out.
- 8 months, 4 weeks ago
If you want to direct anyone to this place just keep the symbol @ and type the name of the person. Immediately a notification will be sent to that particular user.
- 8 months, 4 weeks ago
Hi guys !!! I want to tell you some secret. I think next week would be a week of physics. Because this week I have found six problems which have equal chances in coming to problems of the week. There are Rohit Gupta's Trick Spring Balance and Spring Balance, Pranshu Gaba's Three blocks Sliding and Relative Stuntmen Motion, Danielle Scarano's Thermographic Dog and my Mt. Everest vs Mt. Godwin Austin. I think these question will be featured definitely this week if not possible then next week.
- 8 months, 4 weeks ago
@Calvin Lin
- 8 months, 4 weeks ago
I think you there is some error in spelling or you have kept a gap between the symbol and name. Once check again. If it is correct then you should get it in blue color link and not just normal text.
For example you should type it as @ _ Ekene Franklin ( _ symbol means no gap) then you will get it as @EKENE FRANKLIN
- 8 months, 4 weeks ago
It is not still working. Do I need to add something like a code or anything, or should I just type it?
- 8 months, 4 weeks ago
You should select from the list which will appear when you type the name. Also the name should be exactly matching with that particular user even capital letters should be taken into consideration.
- 8 months, 4 weeks ago
Which list?
- 8 months, 4 weeks ago
Ok. If you want I will do it on behalf of you.
- 8 months, 4 weeks ago
Of course I want! Thanks!
- 8 months, 4 weeks ago
Tell the names of whom do you want to be called to this note.
- 8 months, 4 weeks ago
@Calvin Lin, @Harsh Shrivastava, @Chew Seong Cheong, @Michael Mendrin, @X X, @Aniket Sanghi.
- 8 months, 4 weeks ago
- 8 months, 4 weeks ago
- 6 months, 3 weeks ago
@X X, @Michael Mendrin, @Chew-Seong Cheong, @Mohmmad Farhan,and the rest of all, Can you please participate in this message-board (as requested by Ekene Franklin)
- 8 months, 4 weeks ago
What courses would you recommend for a 4th grader like me?
- 8 months, 4 weeks ago
You are 4 the grade ????!!!!!
- 8 months, 4 weeks ago
10 years old. I told you
- 8 months, 3 weeks ago
But, I didn't think you are 4 th grade.
- 8 months, 3 weeks ago
I am! REALLY
- 8 months, 3 weeks ago
I am going to be 5th grade
- 8 months, 1 week ago
Oh !!! Good. All the best
- 8 months, 1 week ago
Actually I am 1 year away from board exam (in 2020)
- 8 months, 1 week ago
My first trophy was in Grade 1(third)[english], Second in grade 2 (1st)[first scrabble competition] and third in grade 2(Best school player award)[first scrabble competiton]
- 8 months, 1 week ago
# BrilliantForLife
Dare you to toggle $\LaTeX$
- 8 months, 4 weeks ago
I have a proof for the existence of infinite dimensions.
On a line (1 Dimension), a number can be represented by a point. On a grid (2 Dimensions), an intersection is represented for the data in a cell. In a cube (3 Dimensions), data can still be represented on a point (where straight lines intersect). Following this pattern, there are infinite dimensions (because there are $\aleph_0$ numbers).
- 8 months, 4 weeks ago
Problem 6: Find all primes which are the sum of two primes and difference of two primes.
- 8 months, 4 weeks ago
5=7-2=3+2,this is the only possibility.
- 8 months, 3 weeks ago
Exactly. Just what I did during the exam, using the logic that if a prime satisfies those conditions, then the only even prime, 2 must be in the sum and difference.
- 8 months, 3 weeks ago
Problem 7: Simplify $S = (cot 10^\circ - 3\sqrt{3})(cosec 20^\circ + 2cot 20^\circ)$
- 8 months, 4 weeks ago
Why are you keeping square brackets. For square root you just need to keep flower brackets.
- 8 months, 4 weeks ago
For trigonometric functions, logarithms, limts, etc you must keep a slash before them. Like \cot will be as $\cot$ and \log will be as $\log$.
- 8 months, 3 weeks ago
Problem 8: A unit square is completely covered by $3$ identical circles. Find the smallest possible diameter of the circles.
- 8 months, 3 weeks ago
Problem 9: How many $6$ digit numbers can be formed using only odd digits?
- 8 months, 3 weeks ago
Is it 5^6?
- 8 months, 3 weeks ago
Correct. Nice stuff. What did you do?
- 8 months, 3 weeks ago
There are 1,3,5,7,9 five odd digits. There are six digits, each digit has 5 possibilities, so there are 5^6 possibilities
- 8 months, 3 weeks ago
@X X Bro, u are amazing
- 8 months, 3 weeks ago
DO YOU SERIOUSLY HAVE TO GIVE ME 46 NOTIFICATIONS OVERNIGHT!
- 8 months, 3 weeks ago
Is that bad, Mohmmad? I am usually on Brilliant throughout the night.
- 8 months, 3 weeks ago
I HAVE EXAMS
- 8 months, 3 weeks ago
I see. But you can use Brilliant to prepare.
- 8 months, 3 weeks ago
English EXAMS
- 8 months, 3 weeks ago
Farhat, from which part of India are were from ?
- 8 months, 3 weeks ago
- 8 months, 3 weeks ago
I am also from Andhra Pradesh. From Visakhapatnam.
- 8 months, 3 weeks ago
I am Telugu. You are Telugu right because you are from Andhra Pradesh right?
- 8 months, 3 weeks ago
Yes. I am born in Visakhapatnam.
- 8 months, 3 weeks ago
- 8 months, 3 weeks ago
Sorry, but why?
- 8 months, 3 weeks ago
See, My brother's name is Farhan but I am Farhat. And Mohmmad is his surname and Mohammad is my surname. (Stupid Passport errors)
- 8 months, 3 weeks ago
- 8 months, 3 weeks ago
YES! FINALLY!
- 8 months, 3 weeks ago
You can change your name in the account settings
- 8 months, 3 weeks ago
@X X I don't bother
- 8 months, 3 weeks ago
Did you know that :
$1 + \omega = \omega \ but \ \omega + 1 \neq \omega$
Freakin' cool right?
P.S. $\displaystyle \omega \ is \ the \ second \ lowest \ level \ of \ \infty \ after \ \aleph_0$
- 8 months, 3 weeks ago
Finally I reached it. I have reached 1 lakh (100 k) solvers in 91 problems.
- 8 months, 3 weeks ago
Congratulations!
- 8 months, 3 weeks ago
Do you know the minimum people to reach?
- 8 months, 2 weeks ago
I didn't get your point !!! Are you asking about minimum people required to get contribution page.
- 8 months, 2 weeks ago
Yes he is.
- 8 months, 2 weeks ago
@X X I think you must minimum have 5K people reached. But the main parameter I think is that the contribution page is mostly obtained either by up votes or by solvers to the question (more than 1K). When I got my contribution page I have around 120 upvotes, around 300 solvers, around 30 views for my notes.
- 8 months, 2 weeks ago
OK. Thanks!
- 8 months, 2 weeks ago
@X X @X X, I want your help. I am trying to write a new problem but I got struck up at one point. Here is my doubt :
Suppose lets say a man jumps from a building with initial velocity $u$. After $x$ seconds another man freely falls from the same building and from the same height. Now, we can say that the displacement of both the man's are equal as they both reach the ground simultaneously after they jump from same point, so we can equate their distances. Here is what I did : $ut + \dfrac12 gt^2 = \dfrac12 g(t - x)^2$ where $t$ is the time taken by the first person to reach the ground. The main part of my doubt is that whether we should write the time taken by second person as $(t - x)$ or $(t + x)$.
- 8 months, 2 weeks ago
I'm not good at physics, though. I think the time taken by the second person should be as same as the first one, maybe?
- 8 months, 2 weeks ago
@X X What I think is that as the second person jumps after $x$ seconds his time period will be $(t - x)$ seconds. Let's ask it to staff.
- 8 months, 2 weeks ago
So, are you caculating " when the first man fell to the ground, what is the distance between the two man"?
- 8 months, 2 weeks ago
@X X No. Both man reach the ground simultaneously so can we write the equation as in my before comment in which we equate displacements.
- 8 months, 2 weeks ago
Oh! Sorry, I misread the problem.
- 8 months, 2 weeks ago
Yes, I too think the time taken by the second person is $t-x$
- 8 months, 2 weeks ago
It should be t+x as he is free falling and the previous one jumped with velocity u
- 8 months, 2 weeks ago
DO or DIE
You are lost in a forest on a cold midnight day. The temperature is quite low and it is bitter cold. Now you have only two chances :
• You should lie on the ground and should shrink your body like a tight pack.
• Go and stay in the water in a pond nearby.
There not much time left. Take your decision before cold kills you. So, what wold you choose ?
- 8 months, 3 weeks ago
First option.
- 8 months, 3 weeks ago
But the second one is the correct choice you have to make.
- 8 months, 3 weeks ago
Why? Because of anomaly in expansion and contraction of water.?
- 8 months, 3 weeks ago
And also due to the high specific heat capacity of water. This question is a slight modification of air currents (due to which it is cool during the day and warm during night near coastal areas) and I am going to post this very soon.
- 8 months, 3 weeks ago | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 81, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8005084991455078, "perplexity": 2500.82878283053}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998844.16/warc/CC-MAIN-20190618223541-20190619005541-00177.warc.gz"} |
https://statkat.com/stattest.php?t=9&t2=18&t3=9 | # Two sample t test - equal variances not assumed - overview
This page offers structured overviews of one or more selected methods. Add additional methods for comparisons by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table
Two sample $t$ test - equal variances not assumed
Spearman's rho
Two sample $t$ test - equal variances not assumed
Independent/grouping variableVariable 1Independent/grouping variable
One categorical with 2 independent groupsOne of ordinal levelOne categorical with 2 independent groups
Dependent variableVariable 2Dependent variable
One quantitative of interval or ratio levelOne of ordinal levelOne quantitative of interval or ratio level
Null hypothesisNull hypothesisNull hypothesis
H0: $\mu_1 = \mu_2$
Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2.
H0: $\rho_s = 0$
Here $\rho_s$ is the Spearman correlation in the population. The Spearman correlation is a measure for the strength and direction of the monotonic relationship between two variables of at least ordinal measurement level.
In words, the null hypothesis would be:
H0: there is no monotonic relationship between the two variables in the population.
H0: $\mu_1 = \mu_2$
Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2.
Alternative hypothesisAlternative hypothesisAlternative hypothesis
H1 two sided: $\mu_1 \neq \mu_2$
H1 right sided: $\mu_1 > \mu_2$
H1 left sided: $\mu_1 < \mu_2$
H1 two sided: $\rho_s \neq 0$
H1 right sided: $\rho_s > 0$
H1 left sided: $\rho_s < 0$
H1 two sided: $\mu_1 \neq \mu_2$
H1 right sided: $\mu_1 > \mu_2$
H1 left sided: $\mu_1 < \mu_2$
AssumptionsAssumptionsAssumptions
• Within each population, the scores on the dependent variable are normally distributed
• Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2. That is, within and between groups, observations are independent of one another
• Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
Note: this assumption is only important for the significance test, not for the correlation coefficient itself. The correlation coefficient itself just measures the strength of the monotonic relationship between two variables.
• Within each population, the scores on the dependent variable are normally distributed
• Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2. That is, within and between groups, observations are independent of one another
Test statisticTest statisticTest statistic
$t = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}}$
Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $s^2_1$ is the sample variance in group 1, $s^2_2$ is the sample variance in group 2, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to the null hypothesis.
The denominator $\sqrt{\frac{s^2_1}{n_1} + \frac{s^2_2}{n_2}}$ is the standard error of the sampling distribution of $\bar{y}_1 - \bar{y}_2$. The $t$ value indicates how many standard errors $\bar{y}_1 - \bar{y}_2$ is removed from 0.
Note: we could just as well compute $\bar{y}_2 - \bar{y}_1$ in the numerator, but then the left sided alternative becomes $\mu_2 < \mu_1$, and the right sided alternative becomes $\mu_2 > \mu_1$.
$t = \dfrac{r_s \times \sqrt{N - 2}}{\sqrt{1 - r_s^2}}$
Here $r_s$ is the sample Spearman correlation and $N$ is the sample size. The sample Spearman correlation $r_s$ is equal to the Pearson correlation applied to the rank scores.
$t = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}}$
Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $s^2_1$ is the sample variance in group 1, $s^2_2$ is the sample variance in group 2, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to the null hypothesis.
The denominator $\sqrt{\frac{s^2_1}{n_1} + \frac{s^2_2}{n_2}}$ is the standard error of the sampling distribution of $\bar{y}_1 - \bar{y}_2$. The $t$ value indicates how many standard errors $\bar{y}_1 - \bar{y}_2$ is removed from 0.
Note: we could just as well compute $\bar{y}_2 - \bar{y}_1$ in the numerator, but then the left sided alternative becomes $\mu_2 < \mu_1$, and the right sided alternative becomes $\mu_2 > \mu_1$.
Sampling distribution of $t$ if H0 were trueSampling distribution of $t$ if H0 were trueSampling distribution of $t$ if H0 were true
Approximately the $t$ distribution with $k$ degrees of freedom, with $k$ equal to
$k = \dfrac{\Bigg(\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}\Bigg)^2}{\dfrac{1}{n_1 - 1} \Bigg(\dfrac{s^2_1}{n_1}\Bigg)^2 + \dfrac{1}{n_2 - 1} \Bigg(\dfrac{s^2_2}{n_2}\Bigg)^2}$
or
$k$ = the smaller of $n_1$ - 1 and $n_2$ - 1
First definition of $k$ is used by computer programs, second definition is often used for hand calculations.
Approximately the $t$ distribution with $N - 2$ degrees of freedomApproximately the $t$ distribution with $k$ degrees of freedom, with $k$ equal to
$k = \dfrac{\Bigg(\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}\Bigg)^2}{\dfrac{1}{n_1 - 1} \Bigg(\dfrac{s^2_1}{n_1}\Bigg)^2 + \dfrac{1}{n_2 - 1} \Bigg(\dfrac{s^2_2}{n_2}\Bigg)^2}$
or
$k$ = the smaller of $n_1$ - 1 and $n_2$ - 1
First definition of $k$ is used by computer programs, second definition is often used for hand calculations.
Significant?Significant?Significant?
Two sided:
Right sided:
Left sided:
Two sided:
Right sided:
Left sided:
Two sided:
Right sided:
Left sided:
Approximate $C\%$ confidence interval for $\mu_1 - \mu_2$n.a.Approximate $C\%$ confidence interval for $\mu_1 - \mu_2$
$(\bar{y}_1 - \bar{y}_2) \pm t^* \times \sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}$
where the critical value $t^*$ is the value under the $t_{k}$ distribution with the area $C / 100$ between $-t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20).
The confidence interval for $\mu_1 - \mu_2$ can also be used as significance test.
-$(\bar{y}_1 - \bar{y}_2) \pm t^* \times \sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}$
where the critical value $t^*$ is the value under the $t_{k}$ distribution with the area $C / 100$ between $-t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20).
The confidence interval for $\mu_1 - \mu_2$ can also be used as significance test.
Visual representationn.a.Visual representation
-
Example contextExample contextExample context
Is the average mental health score different between men and women?Is there a monotonic relationship between physical health and mental health?Is the average mental health score different between men and women?
SPSSSPSSSPSS
Analyze > Compare Means > Independent-Samples T Test...
• Put your dependent (quantitative) variable in the box below Test Variable(s) and your independent (grouping) variable in the box below Grouping Variable
• Click on the Define Groups... button. If you can't click on it, first click on the grouping variable so its background turns yellow
• Fill in the value you have used to indicate your first group in the box next to Group 1, and the value you have used to indicate your second group in the box next to Group 2
• Continue and click OK
Analyze > Correlate > Bivariate...
• Put your two variables in the box below Variables
• Under Correlation Coefficients, select Spearman
Analyze > Compare Means > Independent-Samples T Test...
• Put your dependent (quantitative) variable in the box below Test Variable(s) and your independent (grouping) variable in the box below Grouping Variable
• Click on the Define Groups... button. If you can't click on it, first click on the grouping variable so its background turns yellow
• Fill in the value you have used to indicate your first group in the box next to Group 1, and the value you have used to indicate your second group in the box next to Group 2
• Continue and click OK
JamoviJamoviJamovi
T-Tests > Independent Samples T-Test
• Put your dependent (quantitative) variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
• Under Tests, select Welch's
• Under Hypothesis, select your alternative hypothesis
Regression > Correlation Matrix
• Put your two variables in the white box at the right
• Under Correlation Coefficients, select Spearman
• Under Hypothesis, select your alternative hypothesis
T-Tests > Independent Samples T-Test
• Put your dependent (quantitative) variable in the box below Dependent Variables and your independent (grouping) variable in the box below Grouping Variable
• Under Tests, select Welch's
• Under Hypothesis, select your alternative hypothesis
Practice questionsPractice questionsPractice questions | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8666266202926636, "perplexity": 1170.8434331043209}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320306335.77/warc/CC-MAIN-20220128182552-20220128212552-00302.warc.gz"} |
https://openlearning.aalto.fi/mod/book/view.php?id=10324&chapterid=593 | Table of Content
### Basics of sequences
This section contains the most important definitions about sequences. Through these definitions the general notion of sequences will be explained, but then restricted to real number sequences.
##### Definition: Sequence
Let $$M$$ be a non-empty set. A sequence is a function:
$f:\mathbb{N}\rightarrow M.$
Occasionally we speak about a sequence in $$M$$.
Note. Characteristics of the set $$\mathbb{N}$$ give certain characteristics to the sequence. Because $$\mathbb{N}$$ is ordered, the terms of the sequence are ordered.
##### Definition: Terms and Indices
A sequence can be denoted denoted as
$$(a_{1}, a_{2}, a_{3}, \ldots) = (a_{n})_{n\in\mathbb{N}} = (a_{n})_{n=1}^{\infty} = (a_{n})_{n}$$
instead of $$f(n).$$ The numbers $$a_{1},a_{2},a_{3},\ldots\in M$$ are called the terms of the sequence.
Because of the mapping \begin{aligned} f:\mathbb{N} \rightarrow & M \\ n \mapsto & a_{n}\end{aligned} we can assign a unique number $$n\in\mathbb{N}$$ to each term. We write this number as a subscript and define it as the index; it follows that we can identify any term of the sequence by its index.
n 1 2 3 4 5 6 7 8 9 $$\ldots$$
$$\downarrow$$ $$\downarrow$$ $$\downarrow$$ $$\downarrow$$ $$\downarrow$$ $$\downarrow$$ $$\downarrow$$ $$\downarrow$$ $$\downarrow$$
$$a_{n}$$
$$a_{1}$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$\ldots$$
#### A few easy examples
##### Example 1: The sequence of natural numbers
The sequence $$(a_{n})_{n}$$ defined by $$a_{n}:=n,\,n\in \mathbb{N}$$ is called the sequence of natural numbers. Its first few terms are: $a_1=1,\, a_2=2,\, a_3=3, \ldots$ This special sequence has the property that every term is the same as its index.
##### Example 2: The sequence of triangular numbers
Triangular numbers get their name due to the following geometric visualization: Stacking coins to form a triangular shape gives the following diagram:
To the first coin in the first layer we add two coins in a second layer to form the second picture $$a_2$$. In turn, adding three coins to $$a_2$$ forms $$a_3$$. From a mathematical point of view, this sequence is the result of summing natural numbers. To calculate the 10th triangular number we need to add the first 10 natural numbers: $D_{10} = 1+2+3+\ldots+9+10$ In general form the sequence is defined as: $$D_{n} = 1+2+3+\ldots+(n-1)+n.$$
This motivates the following definition:
##### Notation and Definition: Sum sequence
Let $$(a_n)_n, a_n: \mathbb{N}\to M$$ be a sequence with terms $$a_n$$, the sum is written: $a_1 + a_2 + a_3 + \ldots + a_{n-1} + a_n =: \sum_{k=1}^n a_k$ The sign $$\sum$$ is called sigma. Here, the index $$k$$ increases from 1 to $$n$$.
Sum sequences are sequences whose terms are formed by summation of previous terms.
Thus the nth triangular number can be written as: $D_n = \sum_{k=1}^n k$
##### Example 3: Sequence of square numbers
The sequence of square numbers $$(q_n)_n$$ is defined by: $$q_n=n^2$$. The terms of this sequence can also be illustrated by the addition of coins.
Interestingly, the sum of two consecutive triangular numbers is a square number. So, for example, we have: $$3+1=4$$ and $$6+3=9$$. In general this gives the relationship:
$q_n=D_n + D_{n-1}$
##### Example 4: Sequence of cube numbers
Analogously to the sequence of square number, we give the definition of cube numbers as $a_n := n^3.$ The first terms of the sequence are: $$(1,8,27,64,125,\ldots)$$.
##### Example 5.
Let $$(q_n)_n$$ with $$q_n := n^2$$ be the sequence of square numbers \begin{aligned}(1,4,9,16,25,36,49,64,81,100 \ldots)\end{aligned} and define the function $$\varphi(n) = 2n$$. The composition $$(q_{2n})_n$$ yields: \begin{aligned}(q_{2n})_n &= (q_2,q_4,q_6,q_8,q_{10},\ldots) \\ &= (4,16,36,64,100,\ldots).\end{aligned}
##### Definition: Sequence of differences
Given a sequence $$(a_{n})_{n}=a_{1},\, a_{2},\, a_{3},\ldots,\, a_{n},\ldots$$; then $(a_{n+1}-a_{n})_{n}:=a_{2}-a_{1}, a_{3}-a_{2},\dots$ is called the 1st difference sequence of $$(a_{n})_{n}$$
The 1st difference sequence of the 1st difference sequence is called the 2nd difference sequence. Analogously the $$n$$th difference< sequence is defined.
##### Example 6.
Given the sequence $$(a_n)_n$$ with $$a_n := \frac{n^2+n}{2}$$, i.e. \begin{aligned}(a_n)_n &= (1,3,6,10,15,21,28,36,\ldots)\end{aligned} Let $$(b_n)_n$$ be its 1st difference sequence. Then it follows that \begin{aligned}(b_n)_n &= (a_2-a_1, a_3-a_2, a_4-a_3,\ldots) \\ &= (2,3,4,5,6,7,8,9)\end{aligned} A term of $$(b_n)_n$$ has the general form \begin{aligned}b_n &= a_{n+1}-a_{n} \\ &= \frac{(n+1)^2+(n+1)}{2} - \frac{n^2+n)}{2} \\ &= \frac{(n+1)^2+(n+1)-n^2 - n }{2} \\ &= \frac{(n^2+2n+1)+1-n^2}{2} \\ &= \frac{2n+2}{2} \\ &= n + 1.\end{aligned}
### Some important sequences
There are a number of sequences that can be regarded as the basis of many ideas in mathematics, but also can be used in other areas (e.g. physics, biology, or financial calculations) to model real situations. We will consider three of these sequences: the arithmetic sequence, the geometric sequence, and Fibonacci sequence, i.e. the sequence of Fibonacci numbers.
#### The arithmetic sequence
There are many definitions of the arithmetic sequence:
##### Definition A: Arithmetic sequence
A sequence $$(a_{n})_{n}$$ is called the arithmetic sequence, when the difference $$d \in \mathbb{R}$$ between two consecutive terms is constant, thus: $a_{n+1}-a_{n}=d \text{ with } d=const.$
Note: The explicit rule of formation follows directly from definition A: $a_{n}=a_{1}+(n-1)\cdot d$ For the $$n$$th term of an arithmetic sequence we also have the recursive formation rule: $a_{n+1}=a_n + d.$
##### Definition B: Arithmetic sequence
A non-constant sequence $$(a_{n})_{n}$$ is called an arithmetic sequence (1st order) when its 1st difference sequence is a sequence of constant value.
This rule of formation gives the arithmetic sequence its name: The middle term of any three consecutive terms is the arithmetic mean of the other two, for example:
$a_2 = \frac{a_1+a_3}{2}.$
##### Example 1.
The sequence of natural numbers $(a_n)_n = (1,2,3,4,5,6,7,8,9,\ldots)$ is an arithmetic sequence, because the difference, $$d$$, between two consecutive terms is always given as $$d=1$$.
#### The geometric sequence
The geometric sequence has multiple definitions:
##### Definition: Geometric sequence
A sequence $$(a_{n})_{n}$$ is called a geometric sequence when the ratio of any two consecutive terms is always constant $$q\in\mathbb{R}$$, thus $\frac{a_{n+1}}{a_{n}}=q \text{ for all } n\in\mathbb{N}.$
Note.The recursive relationship $$a_{n+1} = q\cdot a_n$$ of the terms of the geometric sequence and the explicit formula for the calculation of the n th term of a geometric sequence $a_n=a_1\cdot q^{n-1}$ follows directly from the definition.
Again the name and the rule of formation of this sequence are connected: Here, the middle term of three consecutive terms is the geometric mean of the other two, e.g.: $a_2 = \sqrt{a_1\cdot a_3}.$
##### Example 2.
Let $$a$$ and $$q$$ be fixed positive numbers. The sequence $$(a_n)_n$$ with $$a_n := aq^{n-1}$$, i.e. $\left( a_1, a_2, a_3, a_4,\ldots \right) = \left( a, aq, aq^2, aq^3,\ldots \right)$ is a geometric sequence. If $$q\geq1$$ the sequence is monotonically increasing. If $$q<1$$ it is strictly decreasing. The corresponding range $${a,aq,aq^2, aq^3}$$ is finite in the case $$q=1$$ (namely, a singleton), otherwise it is infinite.
#### The Fibonacci sequence
The Fibonacci sequence is famous because it plays a role in many biological processes, for instance in plant growth, and is frequently found in nature. The recursive definition is:
##### Definition: Fibonacci sequence
Let $$a_0 = a_1 = 1$$ and let $a_n := a_{n-2}+a_{n-1}$ for $$n\geq2$$. The sequence $$(a_n)_n$$ is then called the Fibonacci sequence. The terms of the sequence are called the Fibonacci numbers.
The sequence is named after the Italian mathematician Leonardo of Pisa (ca. 1200 AD), also known as Fibonacci (son of Bonacci). He considered the size of a rabbit population and discovered the number sequence: $(1,1,2,3,5,8,13,21,34,55,\ldots),$
##### Example 3.
The structure of sunflower heads can be described by a system of two spirals, which radiate out symmetrically but contra rotating from the centre; there are 55 spirals which run clockwise and 34 which run counter-clockwise.
Pineapples behave very similarly. There we have 21 spirals running in one direction and 34 running in the other. Cauliflower, cacti, and fir cones are also constructed in this manner.
### Convergence, divergence and limits
The following chapter deals with the convergence of sequences. We will first introduce the idea of zero sequences. After that we will define the concept of general convergence.
### Preliminary remark: Absolute value in $$\mathbb{R}$$
The absolute value function $$x \mapsto |x|$$ is fundamental in the study of convergence of real number sequences. Therefore we should summarise again some of the main characteristics of the absolute value function:
##### Definition: Absolute Value
For any given number $$x\in\mathbb{R}$$ its absolute value $$|x|$$ is defined by \begin{aligned}|x|:=\begin{cases}x & \text{for }x\geq0,\\ -x & \text{for }x<0.\end{cases}\end{aligned}
##### Theorem: Calculation Rule for the Absolute Value
For $$x,y\in\mathbb{R}$$ the following is always true:
1. $$|x|\geq0,$$
2. $$|x|=0$$ if and only if $$x=0.$$
3. $$|x\cdot y|=|x|\cdot|y|$$ (Multiplicativity)
4. $$|x+y|\leq|x|+|y|$$ (Triangle Inequality)
Proof.
Parts 1.-3. Results follow directly from the definition and by dividing it up into separate cases of the different signs of $$x$$ and $$y$$
Part 4. Here we divide the triangle inequality into different cases.
Case 1.
First let $$x,y \geq 0$$. Then it follows that \begin{aligned}|x+y|=x+y=|x|+|y|\end{aligned} and the desired inequality is shown.
Case 2.
Next let $$x,y < 0$$. Then: \begin{aligned}|x+y|=-(x+y)=(-x)+ (-y)=|x|+|y|\end{aligned}
Case 3.
Finally we consider the case $$x\geq 0$$ and $$y<0$$. Here we have two subcases:
• For $$x \geq -y$$ we have $$x+y\geq 0$$ and thus $$|x+y|=x+y$$ from the definition of absolute value. Because $$y<0$$ then $$y<-y$$ and therefore also $$x+y < x-y$$. Overall we have: \begin{aligned}|x+y| = x+y < x-y = |x|+|y|\end{aligned}
• For $$x < -y$$ then $$x+y<0$$. We have $$|x+y|=-(x+y)=-x-y$$. Because $$x\geq0$$, we have $$-x < x$$ and thus $$-x-y\leq x-y$$. Overall we have: \begin{aligned}|x+y| = -x-y \leq x-y = |x|+|y|\end{aligned}
Case 4.
The case $$x<0$$ and $$y\geq0$$ we prove it analogously to the case 3, in which $$x$$ and $$y$$ are exchanged.
$$\square$$
### Zero sequences
##### Definition: Zero sequence
A sequence $$(a_{n})_{n}$$ s called a zero sequence, if for every $$\varepsilon>0,$$ there exists an index $$n_{0}\in\mathbb{N}$$ such that $|a_{n}| < \varepsilon$ for every $$n\geq n_{0},\, n\in\mathbb{N}$$. In this case we also say that the sequence converges to zero.
Informally: We have a zero sequence, if the terms of the sequence with high enough indices are arbitrarily close to zero.
##### Example 1.
The sequence $$(a_n)_n$$ defined by $$a_{n}:=\frac{1}{n}$$, i.e. $\left(a_{1},a_{2},a_{3},a_{4},\ldots\right):=\left(\frac{1}{1},\frac{1}{2},\frac{1}{3},\frac{1}{4},\ldots\right)$ is called the harmonic sequence. Clearly, it is positive for all $$n\in\mathbb{N}$$, however as $$n$$ increases the absolute value of each term decreases getting closer and closer to zero.
Take for example $$\varepsilon := \frac{1}{5000}$$, then choosing the index $$n_0 = 5000$$, it follows that $$a_n<\frac{1}{5000}=\varepsilon$$, for all $$n\geq n_0$$.
##### Example 2.
Consider the sequence $(a_n)_n \text{ where } a_n:=\frac{1}{\sqrt{n}}.$ Let $$\varepsilon := \frac{1}{1000}$$.We then obtain the index $$n_0=1000000$$ in this manner that for all terms $$a_n$$ where $$n\geq n_0$$ $$a_n < \frac{1}{1000}=\varepsilon$$.
Note. To check whether a sequence is a zero sequence, you must choose an (arbitrary) $$\varepsilon \in \mathbb{R}$$ where $$\varepsilon > 0$$. Then search for a index $$n_0$$, after which all terms $$n$$ are smaller then said $$\varepsilon$$.
##### Example 3.
We consider the sequence $$(a_n)_n$$, defined by $a_n := \left( -1 \right)^n \cdot \frac{1}{n^2}.$
Because of the factors $$(-1)^n$$ two consecutive terms have different signs; we call a sequence whose signs change in this way an alternating sequence.
We want to show that this sequence is a zero sequence. According to the definition we have to show that for every $$\varepsilon > 0$$ there exist $$n_0 \in \mathbb{N}$$, such that we have the inequality: $|a_n|< \varepsilon$ for every term $$a_n$$ where $$n\geq n_0$$.
Proof.
Firstly we let $$\varepsilon > 0$$ be an arbitrary constant. Because the inequality $$|a_n|< \varepsilon$$ must hold true for an arbitrary $$\varepsilon$$ we must find the index $$n_0$$ which depends on each $$\varepsilon$$. More exactly: The inequality $|a_{n_0}|=\left| \frac{1}{{n_0}^2} \right|= \frac{1}{{n_0}^2}<\varepsilon$ must be true for the index $$n_0$$. Solve for $$n_0$$: $n_0 > \frac{1}{\sqrt{\varepsilon}},$ this index $$n_0$$ gives our desired characteristic for every $$\varepsilon$$.
##### Negative examples
The following are examples of non-convergent alternating sequences:
• $$a_n = (-1)^n$$
• $$a_n = (-1)^n \cdot n$$
##### Theorem: Characteristics of Zero sequences
Let $$(a_n)_n$$ and $$(b_n)_n$$ be two sequences. Then:
1. Let $$(a_n)_n$$ be a zero sequence, if $$b_n = a_n$$ or $$b_n = -a_n$$ for all $$n\in\mathbb{N}$$ then $$(b_n)_n$$ is also a zero sequence.
2. Let $$(a_n)_n$$ be a zero sequence, if $$-a_n\leq b_n \leq a_n$$ for all $$n\in\mathbb{N}$$ then $$(b_n)_n$$ is also a zero sequence.
3. Let $$(a_n)_n$$ be a zero sequence, then $$(c\cdot a_n)_n$$ where $$c \in \mathbb{R}$$ is also a zero sequence.
4. If $$(a_n)_n$$ and $$(b_n)_n$$ are zero sequences, then $$(a_n + b_n)_n$$ is also a zero sequence.
Proof.
Parts 1 and 2. If $$(a_n)_n$$ is a zero sequence, then according to the definition there is an index $$n_0 \in \mathbb{N}$$, such that $$|a_n|<\varepsilon$$ for every $$n\geq n_0$$ and an arbitrary $$\varepsilon\in\mathbb{R}$$. But then we have $$|b_n|\leq|a_n|<\varepsilon$$; this proves parts 1 and 2 are correct.
Part 3. If $$c=0$$, then the result is trivial. Let $$c\neq0$$ and choose $$\varepsilon > 0$$ such that \begin{aligned}|a_n|<\frac{\varepsilon}{|c|}\end{aligned} for all $$n\geq n_0$$. Rearranging we get: \begin{aligned} |c|\cdot|a_n|=|c\cdot a_n|<\varepsilon\end{aligned}
Part 4.
Because $$(a_n)_n$$ is a zero sequence, by the definition we have $$|a_n|<\frac{\varepsilon}{2}$$ for all $$n\geq n_0$$. Analogously, for the zero sequence $$(b_n)_n$$ there is a $$m_0 \in \mathbb{N}$$ with $$|b_n|<\frac{\varepsilon}{2}$$ for all $$n\geq m_0$$.
Then for all $$n > \max(n_0,m_0)$$ it follows (using the triangle inequality) that: \begin{aligned}|a_n + b_n|\leq|a_n|+|b_n|<\frac{\varepsilon}{2}+\frac{\varepsilon}{2} = \varepsilon\end{aligned}
$$\square$$
### Convergence, divergence
The concept of zero sequences can be expanded to give us the convergence of general sequences:
##### Definition: Convergence and Divergence
A sequence $$(a_{n})_{n}$$ is called convergent to $$a\in\mathbb{R}$$, if for every $$\varepsilon>0$$ there exists a $$n_{0}$$ such that: $|a_{n}-a| \lt \varepsilon \text{ for all }n\in\mathbb{N}_{0},\text{ where }n\geq n_{0}$
An equivalent definition can be defined by:
A sequence $$(a_{n})_{n}$$ is called convergent to $$a\in\mathbb{R}$$, if $$(a_{n}-a)_{n}$$ is a zero sequence.
##### Example 4.
We consider the sequence $$(a_n)_n$$ where $a_n=\frac{2n^2+1}{n^2+1}.$ By plugging in large values of $$n$$, we can see that for $$n\to\infty$$ $$a_n \to 2$$ and therefore we can postulate that the limit is $$a=2$$.
Proof.
For a vigorous proof, we show that for every $$\varepsilon > 0$$ there exists an index $$n_0\in\mathbb{N}$$, such that for every term $$a_n$$ with $$n>n_0$$ the following relationship holds: $\left| \frac{2n^2+1}{n^2+1} - 2\right| < \varepsilon.$
Firstly we estimate the inequality: \begin{aligned}\left|\frac{2n^2+1}{n^2+1}-2\right| =&\left|\frac{2n^2+1-2\cdot\left(n^2+1\right)}{n^2+1}\right| \\ =&\left|\frac{2n^2+1-2n^2-2}{n^2+1}\right| \\ =&\left|-\frac{1}{n^2+1}\right| \\ =&\left|\frac{1}{n^2+1}\right| \\ <&\frac{1}{n}.\end{aligned}
Now, let $$\varepsilon > 0$$ be an arbitrary constant. We then choose the index $$n_0\in\mathbb{N}$$, such that $n_0 > \frac{1}{\varepsilon} \text{, or equivalently, } \frac{1}{n_0} < \varepsilon.$ Finally from the above inequality we have: $\left|\frac{2n^2+1}{n^2+1}-2\right| < \frac{1}{n} < \frac{1}{n_0} < \varepsilon,$ Thus we have proven the claim and so by definition $$a=2$$ is the limit of the sequence.
$$\square$$
If a sequence is convergent, then there is exactly one number which is the limit. This characteristic is called the uniqueness of convergence.
##### Theorem: Uniqueness of Convergence
Let $$(a_{n})_{n}$$ be a sequence that converges to $$a\in\mathbb{R}$$ and to $$b\in\mathbb{R}$$. This implies $$a=b$$.
Proof.
Assume $$a\ne b$$; choose $$\varepsilon\in\mathbb{R}$$ with $$\varepsilon:=\frac{1}{3}|a-b|.$$ Then in particular $$[a-\varepsilon,a+\varepsilon]\cap[b-\varepsilon,b+\varepsilon]=\emptyset.$$
Because $$(a_{n})_{n}$$ converges to $$a$$, there is, according to the definition of convergence, a index $$n_{0}\in\mathbb{N}$$ with $$|a_{n}-a|< \varepsilon$$ for $$n\geq n_{0}.$$ Furthermore, because $$(a_{n})_{n}$$ converges to $$b$$ there is also a $$\widetilde{n_{0}}\in\mathbb{N}$$ with $$|a_{n}-b|< \varepsilon$$ for $$n\geq\widetilde{n_{0}}.$$ For $$n\geq\max\{n_{0},\widetilde{n_{0}}\}$$ we have: \begin{aligned}\varepsilon\ = &\ \frac{1}{3}|a-b| \Rightarrow\\ 3\varepsilon\ = &\ |a-b|\\ = &\ |(a-a_{n})+(a_{n}-b)|\\ \leq &\ |a_{n}-a|+|a_{b}-b|\\ < &\ \varepsilon+\varepsilon=2\varepsilon,\end{aligned} Consequently we have obtained $$3\varepsilon\leq2\varepsilon$$, which is a contradiction as $$\varepsilon>0$$. Therefore the assumption must be wrong, so $$a=b$$.
$$\square$$
##### Definition: Divergent, Limit
If provided that a sequence $$(a_{n})_{n}$$ and an $$a\in\mathbb{R}$$ exist, to which the sequence converges, then the sequence is called convergent and $$a$$ is called the limit of the sequence, otherwise it is called divergent.
Notation. $$(a_{n})_{n}$$ is convergent to $$a$$ is also written: $a_{n}\rightarrow a,\text{ or }\lim_{n\rightarrow\infty}a_{n}=a.$ Such notation is allowed, as the limit of a sequence is always unique by the above Theorem (provided it exists).
##### Theorem: Bounded Sequences
A convergent sequence $$(a_n)_n$$ is bounded i.e. there exists a constant $$r\in\mathbb{R}$$ such that: $|a_n| \lt r$ for all $$n\in\mathbb{N}$$.
Proof.
We assume that the sequence $$(a_n)_n$$ has the limit $$a$$. By the definition of convergence, we have that $$|a_n - a|<\varepsilon$$ for all $$\varepsilon \in \mathbb{R}$$ and $$n\geq n_0$$. Choosing $$\varepsilon = 1$$ gives:
\begin{aligned}|a_n|-|a|&\ \leq |a_n -a| \\ &\ < 1,\end{aligned} And therefore also $$|a_n|\leq |a|+1$$.
Thus for all $$n\in \mathbb{N}$$: $|a_n|\leq \max \left\{ |a_1|,|a_2|,\ldots,|a_{n_0}|,|a|+1 \right\}=:r$
$$\square$$
### Rules for convergent sequences
##### Theorem: Subsequences
Let $$(a_{n})_{n}$$ be a sequence such that $$a_{n}\rightarrow a$$ and let $$(a_{\varphi(n)})_{n}$$ be a subsequence of $$(a_{n})_{n}$$. Then it follows that $$(a_{\varphi(n)})_{n}\rightarrow a$$.
Informally: If a sequence is convergent then all of its subsequences are also convergent and in fact converge to the same limit as the original.
Proof.
By the definition of a subsequence $$\varphi(n)\geq n$$. Because $$a_{n}\rightarrow a$$ it is implicated that $$|a_{n}-a|<\varepsilon$$ for $$n\geq n_{0}$$, therefore $$|a_{\varphi(n)}-a|<\varepsilon$$ for these indices $$n$$.
$$\square$$
##### Theorem: Rules
Let $$(a_{n})_{n}$$ and $$(b_{n})_{n}$$ be sequences with $$a_{n}\rightarrow a$$ and $$b_{n}\rightarrow b$$. Then for $$\lambda, \mu \in \mathbb{R}$$ it follows that:
1. $$\lambda \cdot (a_n)+\mu \cdot (b_n) \to \lambda \cdot a + \mu \cdot b$$
2. $$(a_n)\cdot (b_n) \to a\cdot b$$
Informally: Sums, differences and products of convergent sequences are convergent.
Proof.
Part 1. Let $$\varepsilon > 0$$. We must show, that for all $$n \geq n_0$$ it follows that: $|\lambda \cdot a_n + \mu \cdot b_n - \lambda \cdot a - \mu \cdot b| < \varepsilon.$ The left hand side we estimate using: $|\lambda (a_n-a)+\mu (b_n - b)| \leq |\lambda|\cdot|a_n-a|+|\mu|\cdot|b_n-b|.$
Because $$(a_n)_n$$ and $$(b_n)_n$$ converge, for each given $$\varepsilon > 0$$ it holds true that: \begin{aligned}|a_n - a| <\ \varepsilon_1 := &\ \textstyle \frac{\varepsilon}{2|\lambda|} \text{ for all }n\geq n_0\\ |b_n - b| <\ \varepsilon_2 := &\ \textstyle \frac{\varepsilon}{2|\mu|} \text{ for all }n\geq n_1\end{aligned}
Therefore \begin{aligned}|\lambda|\cdot|a_n-a|+|\mu|\cdot|b_n-b| < &\ |\lambda|\varepsilon_1 + |\mu|\varepsilon_2 \\ = &\ \textstyle{ \frac{\varepsilon}{2} + \frac{\varepsilon}{2} } = \varepsilon\end{aligned} for all numbers $$n \geq \max \{n_0,n_1\}$$. Therefore the sequence $\left( \lambda \left( a_n - a \right) + \mu \left( b_n - b \right) \right)_n$ is a zero sequence and the desired inequality is shown.
Part 2. Let $$\varepsilon > 0$$. We have to show, that for all $$n > n_0$$ $|a_n b_n - a b| < \varepsilon.$ Furthermore an estimation of the left hand side follows: \begin{aligned} |a_n b_n - a b| =&\ |a_n b_n - a b_n + a b_n - ab| \\ \leq &\ |b_n|\cdot|a_n-a| + |a|\cdot|b_n - b|.\end{aligned} We choose a number $$B$$, such that $$|b_n| \lt b$$ for all $$n$$ and $$|a| \lt b$$. Such a value of $$B$$ exists by the Theorem of convergent sequences being bounded. We can then use the estimation: \begin{aligned}|b_n|\cdot|a_n-a| + |a|\cdot|b_n - b| <&\ B \cdot \left(|a_n - a| + |b_n - b| \right).\end{aligned} For all $$n>n_0$$ we have $$|a_n - a|<\frac{\varepsilon}{2\cdot B}$$ and $$|b_n - b|<\frac{\varepsilon}{2\cdot B}$$, and - putting everything together - the desired inequality it shown.
$$\square$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9991962313652039, "perplexity": 296.6495768944159}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949689.58/warc/CC-MAIN-20230331210803-20230401000803-00014.warc.gz"} |
http://en.wikipedia.org/wiki/Probabilistic_classifier | # Probabilistic classification
(Redirected from Probabilistic classifier)
In machine learning, a probabilistic classifier is a classifier that is able to predict, given a sample input, a probability distribution over a set of classes, rather than only predicting a class for the sample. Probabilistic classifiers provide classification with a degree of certainty, which can be useful in its own right,[1] or when combining classifiers into ensembles.
Formally, a probabilistic classifier is a conditional distribution $\operatorname{P}(Y \vert X)$ over a finite set of classes Y, given inputs X. Deciding on the best class label $\hat{y}$ for X can then be done using the optimal decision rule[2]:39–40
$\hat{y} = \operatorname{\arg\max}_{y} \operatorname{P}(Y=y \vert X)$
Binary probabilistic classifiers are also called binomial regression models in statistics. In econometrics, probabilistic classification in general is called discrete choice.
Some classification models, such as naive Bayes, logistic regression and multilayer perceptrons (when trained under an appropriate loss function) are naturally probabilistic. Other models such as support vector machines are not, but methods exist to turn them into probabilistic classifiers.
## Generative and conditional training
Some models, such as logistic regression, are conditionally trained: they optimize the conditional probability $\operatorname{P}(Y \vert X)$ directly on a training set (see empirical risk minimization). Other classifiers, such as naive Bayes, are trained generatively: at training time, the class-conditional distribution $\operatorname{P}(X \vert Y)$ and the class prior $P(Y)$ are found, and the conditional distribution $\operatorname{P}(Y \vert X)$ is derived using Bayes' rule.[2]:43
## Probability calibration
Not all classification models are naturally probabilistic, and some that are, notably naive Bayes classifiers and boosting methods, produce distorted class probability distributions.[3] However, for classification models that produce some kind of "score" on their outputs (such as a distorted probability distribution or the "signed distance to the hyperplane" in a support vector machine), there are several methods that turn these scores into properly calibrated class membership probabilities.
For the binary case, a common approach is to apply Platt scaling, which learns a logistic regression model on the scores.[4] An alternative method using isotonic regression[5] is generally superior to Platt's method when sufficient training data is available.[3]
In the multiclass case, one can use a reduction to binary tasks, followed by univariate calibration with an algorithm as described above and further application of the pairwise coupling algorithm by Hastie and Tibshirani.[6] An alternative one-step method, the Dirichlet calibration, is introduced by Gebel and Weihs.[7]
## Evaluating probabilistic classification
Commonly used loss functions for probabilistic classification include log loss and the mean squared error between the predicted and the true probability distributions. The former of these is commonly used to train logistic models.
## References
1. ^ Hastie, Trevor; Tibshirani, Robert; Friedman, Jerome (2009). The Elements of Statistical Learning. p. 348. "[I]n data mining applications the interest is often more in the class probabilities $p_\ell(x), \ell = 1, \dots, K$ themselves, rather than in performing a class assignment."
2. ^ a b Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer.
3. ^ a b Niculescu-Mizil, Alexandru; Caruana, Rich (2005). "Predicting good probabilities with supervised learning". ICML. doi:10.1145/1102351.1102430. edit
4. ^ Platt, John (1999). "Probabilistic outputs for support vector machines and comparisons to regularized likelihood methods". Advances in large margin classifiers 10 (3): 61–74.
5. ^ Zadrozny, Bianca; Elkan, Charles (2002). "Transforming classifier scores into accurate multiclass probability estimates". "Proceedings of the eighth ACM SIGKDD international conference on Knowledge discovery and data mining - KDD '02". pp. 694–699. doi:10.1145/775047.775151. ISBN 1-58113-567-X. CiteSeerX: 10.1.1.13.7457. edit
6. ^ Hastie, Trevor; Tibshirani, Robert (1998). "Classification by pairwise coupling". The Annals of Statistics 26 (2): 451–471. doi:10.1214/aos/1028144844. Zbl 0932.62071. CiteSeerX: 10.1.1.46.6032. edit
7. ^ Gebel, Martin; Weihs, Claus (2008). "Calibrating Margin-Based Classifier Scores into Polychotomous Probabilities". In Preisach, Christine; Burkhardt, Hans; Schmidt-Thieme, Lars; Decker, Reinhold. Data Analysis, Machine Learning and Applications. Studies in Classification, Data Analysis, and Knowledge Organization. pp. 29–36. doi:10.1007/978-3-540-78246-9_4. ISBN 978-3-540-78239-1. edit | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 8, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8406145572662354, "perplexity": 3537.5436184525747}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931007324.75/warc/CC-MAIN-20141125155647-00154-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://mathoverflow.net/questions/333480/proving-the-representability-of-a-functor-that-is-covered-by-open-subfunctors | # Proving the representability of a functor that is covered by open subfunctors
I want to prove Theorem 8.9 from Algebraic Geometry I ( U.Görtz, T.Wedhorn), which reads as follows:
Let $$S$$ be a scheme $$F: Sch/S°\rightarrow Set$$ a functor such that:
1. F is a sheaf for the Zariski topology
2. F has a cover by open subfunctors $$\alpha_i:F_i\rightarrow F$$, such that every $$F_i$$ is representable by a scheme $$X_i$$
Then F is representable.
A cover by open subfunctors means, that for every scheme $$T$$ and for every morphism $$h_T\rightarrow F$$, the pullback $$F_i\times_F h_T$$ is representable, say by $$Y_i$$ and the morphism of schemes $$Y_i\rightarrow T$$ corresponding to the projection $$F_i\times_F h_T\rightarrow h_T$$ is an open immersion. In addition the images of $$Y_i\rightarrow T$$ form an open covering of $$T$$
Let me explain what I have done so far and where I am stuck:
The $$X_i$$ can be glued to a scheme $$X$$. The morphisms $$\tilde{\alpha_i}:h_{X_i}\cong F_i\rightarrow F$$ correspond via the yoneda lemma to elements $$f_i\in F(X_i)$$. Using the sheaf property of F, the $$f_i$$ glue together to an element $$f\in F(X)$$ which gives us a natural transformation $$\alpha: h_X\rightarrow F$$. For a scheme $$T$$ and a morphism $$g\in\mathrm{Hom}(T,X)$$ this is given by $$\alpha(T)(g)=F(g)(f)$$. The last step is to show that this assignment is bijective. I managed to show the surjectivity, but can't find a proof for the injectivity. It would suffice to show that the following diagram is a pullback $$\require{AMScd}$$ $$\begin{CD} h_{X_i} @>>> h_X\\@VidVV @VV\alpha V\\ h_{X_i} @>>\tilde{\alpha_i}> F \end{CD}$$ where the morphism $$h_{X_i}\rightarrow h_X$$ is induced from the open immersion $$X_i\rightarrow X$$. The commutativity of this diagram is clear to me. To proof that this is a pullback we could try the following: Since $$h_{X_i}$$ is an open subfunctor, there is an open subscheme $$U_i$$ of $$X$$, such that the following square is cartesian: $$\begin{CD} h_{U_i} @>>> h_X\\@VVV @VV\alpha V\\ h_{X_i} @>>\tilde{\alpha_i}> F \end{CD}$$ By the commutativity of the first square the open immersion $$X_i\rightarrow X$$ factors through $$U_i\rightarrow X$$, so that $$X_i$$ is an open subscheme of $$U_i$$. But I couldn't find a way to show $$X=U_i$$. Another way would simply be checking that, $$h_{X_i}$$ satisfies the universal property. However, when trying to do so one needs that $$\alpha(T)$$ is injective for every scheme $$T$$.
I also looked at the proof in EGA I (Springer edition 1971), where this is Proposition 4.5.4 in chapter 0. There Grothendieck uses (without comment) that the fiber product $$F_i\times_F h_X$$ is represented by $$X_i$$, which I think is equivalent to saying that the square from above (the one with $$X_i$$) is cartesian.
I am thankful for any thoughts on this.
I found another way to prove this Theorem, which then also shows that this map is injective. Consider two Zariski sheaves $$F,G:\mathrm{Sch}^{\circ}\rightarrow\mathrm{Set}$$ and for each sheaf a cover by open subfunctors $$\alpha_i:F_i\rightarrow F$$ and $$\beta_i:G_i\rightarrow G$$. Suppose we have isomorphisms $$\varphi_i: F_i\rightarrow G_i$$, such that the diagram $$\require{AMScd}$$ $$\begin{CD} F_i\times_FF_j @>>> F_i @>\varphi_i>> G_i\\@VVV @. @VV\beta_iV \\ F_j @>\varphi_j>> G_j @>\beta_j>>G \end{CD}$$ commutes and the induced morphism $$F_i\times_F F_j\rightarrow F_i\times_G F_j$$ is an isomorphism. Then $$F\cong G$$. The proof of this statement is similiar to the situation, where $$F$$ and $$G$$ are sheaves on a topological space and $$\varphi_i:F_{|U_i}\rightarrow G_{|U_i}$$ are isomorphisms of sheaves agreeing on overlaps. We can apply this to the functor $$h_X$$ with open cover $$h_{U_i}\rightarrow h_X$$ and the functor $$F$$ with open cover $$\alpha_i:F_i\rightarrow F$$. One can check that the isomorphisms $$h_{U_i}\cong F_i$$ satisfy the conditions above. In this way we obtain an isomorphism $$h_X\rightarrow F$$. The yoneda lemma implies, that this morphsim is actually the same as the morphism defined in my question. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 59, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9897832274436951, "perplexity": 62.8026002974381}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178378043.81/warc/CC-MAIN-20210307170119-20210307200119-00112.warc.gz"} |
https://www.cut-the-knot.org/arithmetic/algebra/CyclingInequality2.shtml | # A Cycling Inequality with Integrals II
### Proof
Introduce $u=x^a,\;$ $v=x^b,\;$ $w=x^c.\;$ For $x\in [0,1],\;$ $u,v,w\in [0,1],\;$ such that also $1-u,1-v,1-w\in [0,1].\;$ From here, say, $(1-u)(1-v)\ge 0,\;$ implying $1+uv\ge u+v\;$ and, subsequently,
$1+w+uv\ge u+v+w.$
Therefore,
$\displaystyle \frac{1}{1+w+uv}\le \frac{1}{u+v+w}$
and
$\displaystyle \frac{w}{1+w+uv}\le \frac{w}{u+v+w}$
Similarly, $\displaystyle\frac{u}{1+u+vw}\le \displaystyle\frac{u}{u+v+w}\;$ and $\displaystyle\frac{v}{1+v+wu}\le \displaystyle\frac{v}{u+v+w}.$ Adding the three up we obtain
$\displaystyle\sum_{cycl}\frac{u}{1+u+vw}\le\sum_{cycl}\frac{u}{u+v+w}=1.$
Taking integral from $0\;$ to $1\;$ we obtain the required inequality. Equality is only possible for $a=b=c=0.$
### Acknowledgment
Dan Sitaru has kindly posted the above problem from his book Math Accent, at the CutTheKnotMath facebook page. He later communicated privately the solution above. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 4, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9831763505935669, "perplexity": 1303.873260048156}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103989282.58/warc/CC-MAIN-20220702071223-20220702101223-00649.warc.gz"} |
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# Basic Properties of Parabolas
In this section we’re going to discuss the most important properties of parabola, also we’ll show how to find its vertex, roots and intercepts. This information is needed to sketch the graph and also allows to analyze the behavior of given parabola. We’ve already shown how to sketch graphs of quadratic function, and now we’re going through most essential details. Let’s start with considering the equation of parabola. We already know how parabola looks like. Recall that graph of the parabola is U-shaped:
Equation of parabola is often written in the following way:
$$y(x)=ax^2+bx+c$$
This equation can represent any particular parabola, provided that we choose appropriate values of coefficients $a,b,c$. Say, to obtain the simplest parabola $y=x^2$ we should set $a=1, b=c=0$.
Parabola is a quadratic function. This means that the highest exponent of x must be two in our equation. You see coefficients $a,b,c$ , which are certain numbers, they can be either natural or rational, or even irrational. They define the shape of our parabola. Let’s take a closer look at the equation and coefficients $a,b,c$ and find out how the parabola depends on them.
Here’s video version of this tutorial:
First, consider coefficient $a$ beside the $x$ squared in the equation. The sign of this parameter defines whether branches of given parabola are directed upwards or downwards. Parabola marked red on the picture below has $a=1$, while for green one it makes $a=-1$:
Remember that if $a$ is positive then the parabola opens up. If $a$ is negative branches of parabola are directed downwards. Here we should also say that $a$ is supposed to be non zero because otherwise term with $x^2$ would disappear and we would obtain linear function $y(x)=bx+c$ or simply a straight line. Thus, when doing homework with parabolas keep in mind that $a \neq 1$.
The coefficient $a$ controls the speed of increase (or decrease) of the quadratic function from the vertex, greater absolute value of $a$ makes the function change faster and the graph appears narrower. Smaller values of $a$ lead to wider open graph. Consider parabolas $y=x^2$ (red curve), $y=4x^2$ (green) and $y=0.5x^2$ (blue):
Now let’s talk about coefficient $c$, or so called free term (it doesn’t contain $x$). It defines $y$-intercept of given parabola graph. Indeed, if we substitute $x=0$ into the equation of parabola we’ll get:
$$y(0)=a\cdot 0+b \cdot 0+c=c$$
and it means that graph intercepts $y$-axis at the point $x=0, y=c$. Consider parabola $y=x^2+c$ for several values of $c$ coefficient:
So, roughly speaking, the $c$ coefficient or, in other words, $y$-intercept defines how “high” parabola vertex is.
Let’s move on and discuss $x$-intercepts of the parabola. As you can see on the graph parabola can cross the $x$-axis in different ways. Basically, there are three possibilities: parabola either crosses the horizontal axis in two different points, or it touches the $x$-axis at one single point with ordinate $y=0$, or parabola is entirely above (or below) the $x$-axis.
To define exactly at what points given parabola crosses the $x$-axis we need to solve the equation $y(x)= ax^2+bx+c=0$ for $x$.
As you probably know, quadratic equation can have either two, or one, or no roots. These roots are the points of $x$-intercepts of parabola. In general, the roots of quadratic equation are defined by the following formula:
$$x_{1,2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$
where $a, b, c$ are coefficients of the equation $y(x)= ax^2+bx+c=0$.
Expression $D=b^2-4ac$ (staying under the square root in formula above) is called discriminant of quadratic equation. You can easily find out how many roots particular quadratic equation has by analyzing the value of the discriminant.
If $D>0$ there are two different roots because $\sqrt D >0$:
$$x_{1,2}=\frac{-b\pm \sqrt{D}}{2a}$$
If $D=0$ there is only one root because square root $\sqrt D=0$:
$$x=\frac{-b}{2a}$$
If $D<0$, given equation has no real roots because square root of negative number does not exist.
As we said before, graph of parabola is symmetric with respect to vertical $y$-axis. Recall $y=x^2$:
If we consider values $x>0$ and $–x$, corresponding values of $y$ will be the same: $$y(x)=x^2, y(-x)=(-x)^2=x^2$$
In general case parabola’s axis of symmetry is parallel to the $y$-axis.
Also every parabola has a vertex , i.e. a point where it turns, hence it’s also called the turning point (shown by arrows at the picture below):
$x$-coordinate of vertex is defined as following:
$$x_0=-\frac{b}{2a}$$
At this point parabola achieves minimum if $a>0$ (the parabola opens upwards) and maximum if $a<0$ (it opens downwards). This becomes clear after sketching the graph. If your parabola looks like a cup the vertex is the point of minimum, otherwise parabola has maximum. So doing the sketch is helpful for better understanding.
To obtain the formula for vertex let’s modify our equation a bit. We’re trying to get a complete square:
$$\begin{split}y(x)&=ax^2+bx+c=a(x^2+\frac{b}{a}x+\frac{c}{a})=a(x^2+\frac{b}{a}x+(\frac{b}{2a})^2-(\frac{b}{2a})^2+\frac{c}{a})\\ &=a(x+\frac{b}{2a})^2+a(-(\frac{b}{2a})^2+\frac{c}{a})= a(x+\frac{b}{2a})^2-\frac{b^2-4ac}{4a}\end{split}$$
The first term disappears if $x=-\frac{b}{2a}$. Suppose $a>0$. Then parabola achieves its minimum at $x_0=-\frac{b}{2a}$ because we have constant term $-\frac{b^2-4ac}{4a}$ and non-negative term $a(x+\frac{b}{2a})^2$. The latter is zero at the vertex and positive at all other points. Hence value of $y$ is minimal at the vertex. Similarly, if $a<0$, parabola achieves its maximum at the vertex.
Axis of symmetry of the parabola is defined by the equation $x=-\frac{b}{2a}$. It’s a vertical straight line any point of which has $x$-coordinate defined by the formula $x=-\frac{b}{2a}$.
Example.
We’re given the following equation:
$$y=3x^2+4x-5$$
This equation is quadratic, therefore it represents a parabola.
First, we need to determine coefficients $a, b, c$: in our case they are $a=3, b=4, c= – 5$.
$a=3>0$, therefore the given parabola opens up.
$c=-5$ is the $y$-intercept of the given parabola.
Ok, now we need to find the roots of our equation. We start by calculating the determinant:
$$D=b^2-4ac=16-4\cdot 3 \cdot (-5)=76>0$$
This means that our parabola crosses the $x$-axis at two points. Let’s find their coordinates. Recall the formula defining roots of qudratic equation:
$$x_{1,2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$
Substituting our coefficients we obtain:
$$x_{1,2}=\frac{-4\pm \sqrt{76}}{2\cdot 3}=\frac{-4\pm \sqrt{76}}{6}$$
$$x_1=\frac{-4+\sqrt{76}}{6}\approx 0.78$$
$$x_2=\frac{-4-\sqrt{76}}{6}\approx -2.12$$
Find the vertex of the parabola (as our parabola opens up, it will be the point of minimum):
$$x_0=-\frac{b}{2a}=-\frac{4}{6}=-\frac{2}{3}$$
To obtain $y$-coordinate of the vertex we need to substitute $x_0$ instead of $x$ into the initial equation:
$$\begin{split}y_0 &=3x_0^2+4x_0-5=3(-\frac{2}{3})^2+4(-\frac{2}{3})-5=\frac{12}{9}-\frac{8}{3}-5=\frac{12-24-45}{9}\\&=-\frac{57}{9}=-\frac{19}{3} \end{split}$$
Now we have enough information to sketch the graph:
Summing up, we’ve discussed general equation of parabola and how its coefficients $a, c$ influence the graph. The third coefficient $b$ will be considered in one of the next sections. Also we’ve talked about $y$- and $x$-intercepts and how to find them and we have obtained the formula for vertex of the parabola which is the point of maximum (or minimum, depending on the coefficient $a$). All this information can be useful for your algebra homework so it’s useful to memorize the formulas.
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http://mathhelpforum.com/advanced-algebra/160178-general-polynomial-solved.html | Thread: Is the general polynomial solved?
1. Is the general polynomial solved?
My science fair project is concerned with solving a general polynomial of arbitrary degree. Naturally, I've wanted it to be completely original, but I've found several citations regarding its already having been done:
Polynomial - Wikipedia, the free encyclopedia
Solving the Quintic (the section titled "solution based on series")
Abel's Impossibility [email protected] (mentioned at the very bottom of the article)
You can see that the wikipedia source is without citation for the part that mentions hypergeometric functions.
My questions is: really?! Part of my project is considering the implications of solving the problem, and they seem to be momentously profound. I'm talking prime number, Riemann hypothesis profound, so why have I been able to find only three sources and a research paper that I have to buy (click here to see what I'm talking about--it's the article that is mentioned in the 4th note of the above wikipedia article) on the subject? I'm currently studying linear algebra so I don't really know what an eigenvalue is, but I've looked up the motivation for the hypergeometric function and its application to finding zeros has at least held intuitively for me, and it may have some clear application. But apparently it's been shown to have a direct application! Gahh!!
Dfrtbx
I guess that there's actually two questions in there... I actually do want both to be answered =)
2. Originally Posted by Dfrtbx
My science fair project is concerned with solving a general polynomial of arbitrary degree. Naturally, I've wanted it to be completely original, but I've found several citations regarding its already having been done:
Polynomial - Wikipedia, the free encyclopedia
Solving the Quintic (the section above the colorful picture)
Abel's Impossibility [email protected] (mentioned at the very bottom of the article)
You can see that the wikipedia source is without citation for the part that mentions hypergeometric functions.
My questions is: really?! Part of my project is considering the implications of solving the problem, and they seem to be momentously profound. I'm talking prime number, Riemann hypothesis profound, so why have I been able to find only three sources and a research paper that I have to buy (click here to see what I'm talking about--it's the article that is mentioned in the 4th note of the above wikipedia article) on the subject? I'm currently studying linear algebra so I don't really know what an eigenvalue is, but I've looked up the motivation for the hypergeometric function and its application to finding zeros has at least held intuitively for me, and it may have some clear application. But apparently it's been shown to have a direct application! Gahh!!
Dfrtbx
I guess that there's actually two questions in there... I actually do want both to be answered =)
I can't even see one single, poor question...
Tonio
3. Originally Posted by Dfrtbx
My science fair project is concerned with solving a general polynomial of arbitrary degree. Naturally, I've wanted it to be completely original, but I've found several citations regarding its already having been done:
Polynomial - Wikipedia, the free encyclopedia
Solving the Quintic (the section above the colorful picture)
Abel's Impossibility [email protected] (mentioned at the very bottom of the article)
You can see that the wikipedia source is without citation for the part that mentions hypergeometric functions.
My questions is: really?! Part of my project is considering the implications of solving the problem, and they seem to be momentously profound. I'm talking prime number, Riemann hypothesis profound, so why have I been able to find only three sources and a research paper that I have to buy (click here to see what I'm talking about--it's the article that is mentioned in the 4th note of the above wikipedia article) on the subject? I'm currently studying linear algebra so I don't really know what an eigenvalue is, but I've looked up the motivation for the hypergeometric function and its application to finding zeros has at least held intuitively for me, and it may have some clear application. But apparently it's been shown to have a direct application! Gahh!!
Dfrtbx
I guess that there's actually two questions in there... I actually do want both to be answered =)
It has been proved that it's impossible to solve the general polynomial of degree 5 or greater algebraically. Contrary to your research, there is no scarcity of the proof. The proof will be found in any textbook that covers Field Theory. There is also an accessible account by Mario Livio (The Equation That Couldn't be Solved).
4. Originally Posted by mr fantastic
It has been proved that it's impossible to solve the general polynomial of degree 5 or greater algebraically. Contrary to your research, there is no scarcity of the proof. The proof will be found in any textbook that covers Field Theory. There is also an accessible account by Mario Livio (The Equation That Couldn't be Solved).
Yes: Abel. I think we may be on different pages here... I know about the proof about the impossibility of an algebraic solution. I'm concerned with a non-algebraic solution, and that is what is the citations are about--a non-algebraic solution. The article at everything2 says: "Felix Klein developed in 1877 a particularly elegant method for solving quintics based on the symmetries of the icosahedron." It says that he found a solution to every (I assume, it doesn't say so explicitly) quintic. It says in the very last paragraph that "Poincare among others" have extended Klein's approach to every polynomial of arbitrary degree.
Let me give you my motivation for a solution in terms of non-algebraic functions (this wasn't the initial motivation, but it's the one that makes the most sense): By the fundamental theorem of algebra, there must be n roots of an nth degree polynomial, and given integer coefficients, these roots must be algebraic. The Abel-Ruffini theorem concerns the impossibility of using algebraic functions in doing so--but there must be roots and these roots must be algebraic. I've taken this as a tell that non-algebraic functions can be used to find the roots.
Thanks for the book suggestion!
I can't even see one single, poor question...
Tonio
Haha, that made me laugh... My english teachers have never liked the way I write, and probably with good reason =P
"I'm talking prime number, Riemann hypothesis profound, so why have I been able to find only three sources and a research paper that I have to buy ... on the subject?"
To be more explicit: has it really been done? And if it really has been done, why is the evidence of its having been done so scarce? It doesn't seem like it should be difficult to find information...
Dfrtbx
5. Originally Posted by Dfrtbx
Yes: Abel. I think we may be on different pages here... I know about the proof about the impossibility of an algebraic solution. I'm concerned with a non-algebraic solution, and that is what is the citations are about--a non-algebraic solution. The article at everything2 says: "Felix Klein developed in 1877 a particularly elegant method for solving quintics based on the symmetries of the icosahedron." It says that he found a solution to every (I assume, it doesn't say so explicitly) quintic. It says in the very last paragraph that "Poincare among others" have extended Klein's approach to every polynomial of arbitrary degree.
Let me give you my motivation for a solution in terms of non-algebraic functions (this wasn't the initial motivation, but it's the one that makes the most sense): By the fundamental theorem of algebra, there must be n roots of an nth degree polynomial, and given integer coefficients, these roots must be algebraic. The Abel-Ruffini theorem concerns the impossibility of using algebraic functions in doing so--but there must be roots and these roots must be algebraic. I've taken this as a tell that non-algebraic functions can be used to find the roots.
[snip]
No.
Sorry, but I think you're quite out of your depth here.
6. Originally Posted by mr fantastic
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http://www.math.fu-berlin.de/users/lei/Etale%20%20Cohomology.html | # Étale cohomology
## Lectures at FU Berlin, Winter Semester 2016-2017
### Introduction
In this course we are going to follow closely SGA1 and SGA4 to develop an abstract framwork of fundamental groups and cohomology theory. To do this we first need a generalization of a topological space, and this would be the Grothendieck topology. The notion of sheaves on a topological space would be generalized to the notion of topos. The sheaf cohomology will be replaced by the derived category of a ringed topos. This general framwork serves like a machine: whenever one puts in a concrete Grothendieck topology one gets the corresponding cohomology theory out, and after some further work one may also get the corresponding fundamental group. In this course we are going to put in the étale topology in, and study the output, namely the étale cohomology and the étale fundamental group, which are also the most important output of this machine.
### Prerequisites
The prerequests for this course is a first course in algebraic geometry. A certain familiarity with the language of schemes and commutative algebra is prefered. But these will not be used to develop the general machine of cohomology theory and fundamental groups.
### Course outline
You can find the course outline here.
### Exercises
Every Tursday there will be a new exercise sheet. You can try to solve them, and if you have any questions please contact me at [email protected].
### Other Information
Place (course): SR 130/A3 (Arnimallee 3)
Place (exercise): SR 130/A3 (Arnimallee 3)
Date: Wednesday 10:00-12:00 (course) and 14:00-16:00 (exercise)
First Appointment: 19.10.2016
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https://brilliant.org/problems/gravity-is-back-again/ | # Gravity is back again!
The acceleration due to gravity on Moon($$g_m$$) is $$\frac16$$ that of Earth($$g_e$$) and the radius of the Moon($$R_m$$)is $$\frac14$$ that of Earth($$R_e$$). Find the escape velocity of Moon in terms of escape velocity of Earth.
The answer is in the from $$\dfrac{v_e}{\sqrt{\overline{ab}}}$$ where $$\overline{ab}$$ is a two digit number, find $$a^b$$.
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http://aas.org/archives/BAAS/v36n2/aas204/283.htm | AAS 204th Meeting, June 2004
Session 54 Solar Flares
SPD Poster, Wednesday, June 2, 2004, 10:00am-7:00pm, Ballroom
## [54.10] Observations of Doppler Shift Oscillations with the Bragg Crystal Spectrometer on Yohkoh
J. T. Mariska (Naval Research Laboratory)
Oscillations in solar coronal loops appear to be a common phenomenon. Transverse and longitudinal oscillations have been observed with both the TRACE and EIT imaging experiments. Damped Doppler shift oscillations have been observed in emission lines from ions formed at flare temperatures with the SUMER experiment on SOHO. These observations provide valuable diagnostic information on coronal conditions and may help refine our understanding of coronal heating mechanisms. We have initiated a study of the time dependence of Doppler shifts measured during flares with the Bragg Crystal Spectrometer on Yohkoh. In this presentation, we report some initial results on Doppler shifts as a function of time measured in the emission lines of \ion{S}{15} and \ion{Ca}{19}. For some flares, both lines exhibit damped Doppler shift oscillations with amplitudes of a few km~s-1 and periods and decay times of a few minutes.
Bulletin of the American Astronomical Society, 36 #2
© YEAR. The American Astronomical Soceity. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9448450207710266, "perplexity": 4137.619801978708}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207929561.98/warc/CC-MAIN-20150521113209-00239-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/1372771/kinetic-energy-approximation | # kinetic energy approximation
If a pitcher throws a pitch at a velocity $v_0$, then the kinetic energy is $E_0=\frac 12mv_0^2$. If the pitcher releases the pitch from x feet higher, then we will suppose that he can readjust his delivery to hit the same spot over home plate with the same initial speed. But this time, there is added energy from the drop in height x, so the total kinetic energy is
$$\frac 12mv^2=E=E_0+E_1$$. Find the formula for $v$ in terms of $v_0$ and $x$. Then use a linear approximation to estimate the difference $v−v_0$, which is the gain in velocity due to the extra height.
could anyone provide hints? very confused.
We have
$$E=\frac12mv^2=\frac12mv_0^2+mgx$$
from which we can write
$$v=v_0\left(1+\frac{2gx}{v_0^2}\right)^{1/2} \tag 1$$
Assuming that $\frac{2gx}{v_0^2}<<1$, we approximate the square root in $(1)$ as
$$\left(1+\frac{2gx}{v_0^2}\right)^{1/2}\approx. 1+\frac{gx}{v_0^2} \tag 2$$
where the approximation error is of order $\left(\frac{2gx}{v_0^2}\right)^2$. Using $(2)$ in $(1)$ reveals that
$$\bbox[5px,border:2px solid #C0A000]{v-v_0\approx. \frac{gx}{v_0}}$$
which is the linear approximation of the uplift in velocity from the increased elevation $x$.
A second way to arrive at the same result is to write
$$\frac12m(v^2-v_0^2)=mgx\implies(v-v_0)=\frac{2gx}{v+v_0} \tag 3$$
Assuming that $v\approx. v_0$ when $x$ is "small," we replace $v$ in the denominator of the right-hand side of $(3)$ with $v_0$ and obtain
$$v-v_0\approx.\frac{gx}{v_0}$$
as expected!
Notice, due to drop through a height $x$, with initial velocity $v_0$, the velocity $v$ is increased constantly under earth's gravitational acceleration $g$.
Now, using third equation of the motion $$v^2=v_o^2+2gx$$ $$\color{blue}{v=\sqrt{v_0^2+2gx}}$$ Again notice, $$v=\sqrt{v_o^2+2gx}$$$$=\left(v_0^2+2gx\right)^{1/2}$$ $$=v_0\left(1+\frac{2gx}{v_0^2}\right)^{1/2}$$ Now, assume that initial velocity $v_0$ is large enough then $\left|\frac{2gx}{v_0^2}\right|<1$ thus using binomial expansion of $\left(1+\frac{2gx}{v_0^2}\right)^{1/2}$ as follows $$v=v_0\left(1+\frac{\frac{1}{2}}{1!}\left(\frac{2gx}{v_0^2}\right)+\frac{\frac{1}{2}\left(\frac{1}{2}-1 \right)}{2!}\left(\frac{2gx}{v_0^2}\right)^2+\ldots\right)$$ Neglecting higher power terms, we get approximate value of velocity gain $(v-v_0)$ as follows $$v\approx v_0\left(1+\frac{1}{2}\frac{2gx}{v_0^2}\right)$$ $$v\approx v_0+\frac{gx}{v_0}$$ $$\color{blue}{v-v_0\approx\frac{gx}{v_0} }$$
Your velocity $v$ has two components, the $x-$ and $y-$direction. Assuming no friction, the $x-$component is constant, while there will be a uniform acceleration $g$ down due to gravity. You can then calculate the $y-$velocity at the moment the pitch reaches the home plate.
You now have two velocity components, but that doesn't mean that $E = E_x + E_y$. Can you see why? This is where the linear approximation comes into play. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9967615604400635, "perplexity": 220.60532334361764}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540490972.13/warc/CC-MAIN-20191206200121-20191206224121-00474.warc.gz"} |
https://math.stackexchange.com/questions/2416821/continuous-image-of-connected-set-is-connected-proof/2416997 | # Continuous image of connected set is connected: Proof
In De La Fuente's Mathematical Methods and Models for Economists, the following is said:
Let $f:X\to Y$ be a continuous mapping between two metric spaces. If C is a connected subset of $X$, then $f(C)$ is connected.
The proof goes as in Rudin's Principles, and I cannot understand exactly what Rudin also does not explain:
Suppose $f(C)$ is not connected. Then $f(C)=P\cup Q$, where $P$ and $Q$ are nonempty, separated subsets of $Y$, that is,
$clP\cap Q = \emptyset$ and $P\cap clQ = \emptyset$
Let $$A = C\cap f^{-1}(P) \\ B = C\cap f^{-1}(Q)$$ and notice that then $$C = A\cup B$$ where neither $A$ nor $B$ is empty, and $$f(A)=P \\ f(B) = Q$$
The proof goes on, but this is where De La Fuente loses me. I can clearly see that $f(A)\subseteq P$, but not that $P\subseteq f(A)$. Any thoughts?
Thanks!
• I have added (proof-explanation) tag - see the tag-info - since your question seems to be about this specific proof (rather than asking for any proof of the fact mentioned in the title). – Martin Sleziak Sep 4 '17 at 22:15
Let $p\in P$. Then as $P\subset f(C)$, we know that $f(c)=p$ for some $c\in C$. Can you go from there?
• Yes! Thanks a lot! – Miguel Santana Sep 5 '17 at 8:42
The proof has to be properly aligned with the OP's definition of connectedness:
Definition: A subspace $C$ of a topological space $X$ is connected if and only if for any non-trivial partition $A \cup B$ of $C$,
$\qquad \overline{A} \cap B \ne \emptyset\; \text{ OR }\; A \cap \overline{B} \ne \emptyset$.
Proposition 1: A function $f: X \to Y$ is continuous $\text{ iff for every } A \subset X \text{, } f(\overline{A})\subseteq \overline{f(A)}$.
The above proposition and some general properties about sets and functions should be enough to construct the desired proof. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.985593318939209, "perplexity": 122.09725434479807}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315811.47/warc/CC-MAIN-20190821065413-20190821091413-00555.warc.gz"} |
https://math.stackexchange.com/questions/3042059/what-are-the-foundations-of-probability-and-how-are-they-dependent-upon-a-sigm | # What are the foundations of probability and how are they dependent upon a $\sigma$-field?
I am reading Christopher D. Manning's Foundations of Statistical Natural Language Processing which gives an introduction on Probability Theory where it talks about $$\sigma$$-fields. It says,
The foundations of probability theory depend on the set of events $$\mathscr{F}$$ forming a $$\sigma$$-field".
I understand the definition of a $$\sigma$$-field, but what are these foundations of probability theory, and how are these foundations dependent upon a $$\sigma$$-field?
• But this is not a textbook on probability theory. This is a book on Natural Language Processing which includes some preliminaries of probability theory -- this is not the same thing. If you want to understand the foundations of probability, consult a textbook on that, not on something that merely uses probabilities. Dec 16 '18 at 0:48
• @eddard Some people are concerned with the scope of your question being too broad ("foundations of probability theory" is a topic of considerable size). Would you mind editing the post to be only about the second bullet point? Dec 16 '18 at 22:00
• @Lord_Farin someone tried to focus it and improve the title. Then someone didn't like it and rolled it back. Not sure why, since that would make this a good question. Dec 18 '18 at 8:09
Probability when there are only finitely many outcomes is a matter of counting. There are $$36$$ possible results from a roll of two dice and $$6$$ of them sum to $$7$$ so the probability of a sum of $$7$$ is $$6/36$$. You've measured the size of the set of outcomes that you are interested in.
It's harder to make rigorous sense of things when the set of possible results is infinite. What does it mean to choose two numbers at random in the interval $$[1,6]$$ and ask for their sum? Any particular pair, like $$(1.3, \pi)$$, will have probability $$0$$.
You deal with this problem by replacing counting with integration. Unfortunately, the integration you learn in first year calculus ("Riemann integration") isn't powerful enough to derive all you need about probability. (It is enough to determine the probability that your two rolls total exactly $$7$$ is $$0$$, and to find the probability that it's at least $$7$$.)
For the definitions and theorems of rigorous probability theory (those are the "foundations" you ask about) you need "Lebesgue integration". That requires first carefully specifying the sets that you are going to ask for the probabilities of - and not every set is allowed, for technical reasons without which you can't make the mathematics work the way you want. It turns out that the set of sets whose probability you are going to ask about carries the name "$$\sigma$$-field" or "sigma algebra". (It's not a field in the arithmetic sense.) The essential point is that it's closed under countable set operations. That's what the "$$\sigma$$" says. Your text may not provide a formal definition - you may not need it for NLP applications.
• I am reading it a couple of times. I think it is going to take me some time to understand all of it. Thanks for the answer anyways. Dec 16 '18 at 1:43
• I find the following answer related and useful Why do we need sigma-algebras to define probability spaces Dec 24 '18 at 15:18
• "Probability when there are only finitely many outcomes is a matter of counting." This is a rather dangerous simplification since it assumes a uniform distribution on those outcomes. Cue the old joke, "I have a 50% chance to win the lottery, since I'll either win or I'll lose." Dec 27 '18 at 17:02
• @user7530 Fair enough. I suppose I could have said "simple algebra" or "weighted averages" instead of "counting". But as an introduction to an answer to the OP's question I think what I wrote is OK. Dec 27 '18 at 17:07
To add more to the answer by Ethan Bolker and flesh this out, probability functions are defined on sets, representing events, i.e. some set of outcomes of which we're interested in the probability of whatever we are querying or observing to happen as falling into or not, e.g. the probability that the temperature at noon tomorrow will be in the range $$[25, 30]$$.
Every probability function, which assigns to each event set $$E$$, itself a subset of the total set of possible outcomes, or sample space, $$S$$, is required satisfy the following rules, called the Kolmogorov axioms. The reason for this is they capture the most basic rules of how we expect probabilities to behave intuitively:
1. Rule 1: There are no negative probabilities. That is, for every event $$E$$ we have $$P(E) \ge 0$$. Since probabilities are meant to formalize the idea of "how many chances in..." is there for something to happen, it makes no sense to talk of a negative number of chances for the same reason that it makes no sense to talk of a negative number of apples. What does it mean to have -3 occurrences of something, or -6 apples held in my hand right now at this very moment in time?
2. Rule 2: The probability of the entire sample space is 1. i.e. $$P(S) = 1$$. This should be intuitive, because at least some outcome must occur, and the set $$S$$ is the set of all possible outcomes, so whatever outcome occurs has to be within it. Thus the event $$S$$ will always occur no matter what.
3. Rule 3: Probabilities of mutually exclusive events add. If we have an up-to-countable sequence of mutually exclusive events $$E_1, E_2, E_3, \cdots$$, i.e. that $$E_i \cap E_j = \emptyset$$ for all possible pairs with $$i \ne j$$, then we should have
$$P(E_1 \cup E_2 \cup E_3 ...) = \sum_{i=1}^{\infty} P(E_i)$$
Now as mentioned, we may not be able to assign every event a probability. For the case of a discrete sample space, i.e. where $$S$$ is a finite or at most countably infinite set, this may be doable. But for continuous sample spaces (e.g. $$\mathbb{R}$$), there are subtleties that make it difficult to define a useful probability function in most cases for most sets using methods that are convenient to use such as integration, and thus we must restrict the domain of $$P$$ to not all subsets of $$S$$, but only some selected amount, which we call the $$\sigma$$-field, usually denoted $$\Sigma$$. That is, $$\mathrm{dom}(P) = \Sigma \subseteq 2^S$$, and we are not thus allowed to consider events $$E \notin \Sigma$$. The definition of a $$\sigma$$-field is just whatever is required to ensure that with regard to the above definition, all the sets involved in it make sense. Which basically means we must have
1. Because of rule 2, in order for us to have $$P(S) = 1$$ we need $$S$$ to be in the domain of $$P$$ in the first place, so we must have $$S \in \Sigma$$.
2. While this second stipulation is not strictly speaking required simply to make the above definition valid, we typically take that the complement $$\bar{E} = S \backslash E$$ of any event should be in $$\Sigma$$. This is because very often we are interested in the probability of something NOT happening (e.g. the probability that a given number of people do NOT get better with some sort of medical treatment we are testing), and we want that question to make sense and thus must be able to have the event corresponding to this as an available input to our probability function $$P$$.
3. Finally, so that rule 3 can make sense, given any countable sequence of members $$E_1, E_2, E_3, \cdots \in \Sigma$$ we must have $$(E_1 \cup E_2 \cup E_3 \cup \cdots) \in \Sigma$$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 48, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9161557555198669, "perplexity": 189.62549764458663}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585181.6/warc/CC-MAIN-20211017175237-20211017205237-00566.warc.gz"} |
http://www.ipm.ac.ir/ViewPaperInfo.jsp?PTID=6775&school=Physics | ## “School of Physics”
Back to Papers Home
Back to Papers of School of Physics
Paper IPM / P / 6775
School of Physics
Title: On the Relation of Weyl Geometry and Bohmian Quantum Gravity
Author(s):
1 A. Shojai 2 F. Shojai
Status: Published
Journal: Gravitation and Cosmology
Vol.: 9
Year: 2003
Pages: 163-168
Supported by: IPM
Abstract:
It is shown that the recently geometric formulation of quantum mechanics [,,,,,] implies the use of Weyl geometry. It is discussed that the natural framework for both gravity and quantum is Weyl geometry. At the end a Weyl invariant theory is built, and it is shown that both gravity and quantum are present at the level of equations of motion. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9280447363853455, "perplexity": 3294.31440397286}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662587158.57/warc/CC-MAIN-20220525120449-20220525150449-00514.warc.gz"} |
https://labs.tib.eu/arxiv/?author=M.%20Ichikawa | • ### One loop effects of natural SUSY in indirect searches for SUSY particles at the ILC(1703.07671)
May 24, 2017 hep-ph
We have found the possible region of parameters of the minimal supersymmetric standard model (MSSM) within the bounds from the experimental results of the Higgs mass, the rare decay mode of $b$-quark, the muon $g-2$, the dark matter abundance, and the direct searches for the lighter stop (i.e., one of the supersymmetric partners of top quark) at the LHC. We present numerical results of calculations for the one loop effects of supersymmetric particles in the processes of $\tau^+ \tau^-$, $b \overline{b}$, $t \overline{t}$, and $Z h$ production at the ILC by using benchmark points within the possible region of the MSSM parameters.
• ### Formation of In-plane Skyrmions in Epitaxial MnSi Thin Films as Revealed by Planar Hall Effect(1506.04821)
June 16, 2015 cond-mat.str-el
We investigate skyrmion formation in both a single crystalline bulk and epitaxial thin films of MnSi by measurements of planar Hall effect. A prominent stepwise field profile of planar Hall effect is observed in the well-established skyrmion phase region in the bulk sample, which is assigned to anisotropic magnetoresistance effect with respect to the magnetic modulation direction. We also detect the characteristic planar Hall anomalies in the thin films under the in-plane magnetic field at low temperatures, which indicates the formation of skyrmion strings lying in the film plane. Uniaxial magnetic anisotropy plays an important role in stabilizing the in-plane skyrmions in the MnSi thin film.
• ### Molecular behavior of DNA in a cell-sized compartment coated by lipids(1504.03143)
The behavior of long DNA molecules in a cell-sized confined space was investigated. We prepared water-in-oil droplets covered by phospholipids, which mimic the inner space of a cell, following the encapsulation of DNA molecules with unfolded coil and folded globule conformations. Microscopic observation revealed that the adsorption of coiled DNA onto the membrane surface depended on the size of the vesicular space. Globular DNA showed a cell-size-dependent unfolding transition after adsorption on the membrane. Furthermore, when DNA interacted with a two-phase membrane surface, DNA selectively adsorbed on the membrane phase, such as an ordered or disordered phase, depending on its conformation. We discuss the mechanism of these trends by considering the free energy of DNA together with a polyamine in the solution. The free energy of our model was consistent with the present experimental data. The cooperative interaction of DNA and polyamines with a membrane surface leads to the size-dependent behavior of molecular systems in a small space. These findings may contribute to a better understanding of the physical mechanism of molecular events and reactions inside a cell.
• ### Discretized Topological Hall Effect Emerging from Skyrmions in Constricted Geometry(1501.03290)
We investigate the skyrmion formation process in nano-structured FeGe Hall-bar devices by measurements of topological Hall effect, which extracts the winding number of a spin texture as an emergent magnetic field. Step-wise profiles of topological Hall resistivity are observed in the course of varying the applied magnetic field, which arise from instantaneous changes in the magnetic nano-structure such as creation, annihilation, and jittering motion of skyrmions. The discrete changes in topological Hall resistivity demonstrate the quantized nature of emergent magnetic flux inherent in each skyrmion, which had been indistinguishable in many-skyrmion systems on a macroscopic scale.
• ### Robust formation of skyrmions and topological Hall effect in epitaxial thin films of MnSi(1209.4480)
Magneto-transport properties have been investigated for epitaxial thin films of B20-type MnSi grown on Si(111) substrates. Both Lorentz transmission electron microscopy (TEM) images and topological Hall effect (THE) clearly point to the robust formation of skyrmions over a wide temperature-magnetic field region. New features distinct from those of bulk MnSi are observed for epitaxial MnSi films: a shorter (nearly half) period of the spin helix and skyrmions, and an opposite sign of THE. These observations suggest versatile features of skyrmion-induced THE beyond the current understanding.
• ### Analyzing power in elastic scattering of 6He from polarized proton target at 71 MeV/nucleon(1106.3903)
June 14, 2011 nucl-ex, nucl-th
The vector analyzing power has been measured for the elastic scattering of neutron-rich 6He from polarized protons at 71 MeV/nucleon making use of a newly constructed solid polarized proton target operated in a low magnetic field and at high temperature. Two approaches based on local one-body potentials were applied to investigate the spin-orbit interaction between a proton and a 6He nucleus. An optical model analysis revealed that the spin-orbit potential for 6He is characterized by a shallow and long-ranged shape compared with the global systematics of stable nuclei. A semimicroscopic analysis with a alpha+n+n cluster folding model suggests that the interaction between a proton and the alpha core is essentially important in describing the p+6He elastic scattering. The data are also compared with fully microscopic analyses using non-local optical potentials based on nucleon-nucleon g-matrices.
• ### Analyzing power for the proton elastic scattering from neutron-rich 6He nucleus(1007.3775)
July 22, 2010 nucl-ex, nucl-th
Vector analyzing power for the proton-6He elastic scattering at 71 MeV/nucleon has been measured for the first time, with a newly developed polarized proton solid target working at low magnetic field of 0.09 T. The results are found to be incompatible with a t-matrix folding model prediction. Comparisons of the data with g-matrix folding analyses clearly show that the vector analyzing power is sensitive to the nuclear structure model used in the reaction analysis. The alpha-core distribution in 6He is suggested to be a possible key to understand the nuclear structure sensitivity.
• ### The Giant Monopole Resonance in the $^{112-124}$Sn Isotopes and the Symmetry Energy Term in Nuclear Incompressibility(0709.3132)
Sept. 20, 2007 nucl-ex
We have investigated the isoscalar giant monopole resonance (GMR) in the Sn isotopes, using inelastic scattering of 400-MeV $\alpha$-particles at extremely forward angles, including 0 deg. A value of -550 \pm 100 MeV has been obtained for the asymmetry term, $K_\tau$, in the nuclear incompressibility.
• ### Isotopic dependence of the giant monopole resonance in the even-A ^{112-124}Sn isotopes and the asymmetry term in nuclear incompressibility(0709.0567)
Sept. 5, 2007 nucl-ex
The strength distributions of the giant monopole resonance (GMR) have been measured in the even-A Sn isotopes (A=112--124) with inelastic scattering of 400-MeV $\alpha$ particles in the angular range $0^\circ$--$8.5^\circ$. We find that the experimentally-observed GMR energies of the Sn isotopes are lower than the values predicted by theoretical calculations that reproduce the GMR energies in $^{208}$Pb and $^{90}$Zr very well. From the GMR data, a value of $K_{\tau} = -550 \pm 100$ MeV is obtained for the asymmetry-term in the nuclear incompressibility. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8487188220024109, "perplexity": 2363.3535806323275}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986677230.18/warc/CC-MAIN-20191017222820-20191018010320-00457.warc.gz"} |
http://mathhelpforum.com/calculus/59792-limit.html | # Math Help - limit
1. ## limit
I need to show that lim[cos(x^2)/x] as x goes to infinity is zero and the limit of its derivative does not exist.
Thx
I need to show that lim[cos(x^2)/x] as x goes to infinity is zero and the limit of its derivative does not exist.
Thx
$-1 \leq \cos x^2 \leq 1$...
As for proving that "the limit of its derivative does not exist", what is the derivative of $$cos(x^2)/x$$? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9866504073143005, "perplexity": 292.52091999443604}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131296603.6/warc/CC-MAIN-20150323172136-00247-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://answerdata.org/what-is-a-genetic-advance/ | # What is a genetic advance?
Thank you really much!
The increase in the level of a quantitative variable that results from recurrent mass selection. For example, after one screening generation, there might be a 5% increase in the yield, or in the level of horizontal resistance to a particular species of parasite."
http://www.opbf.org/open-plant-breeding/glossary/g
Progress in shifting the genotype mean and gene frequencies of a trait in the population toward the desired direction as a result of selection.
• Genetic advance is the measure of genetic gain under selection which depends on three factors such as genetic variability, heritability and selection intensity.
• Genetic advance is the difference between the mean of the selected
plants in the original population and the mean of the progeny raised from the selected plants
in the next generation. It can be predicted by the following formula.
Genetic advance(GA) = s P * H * K
K = selection intensity
P = phenotypic standard deviation of the character in the population | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9126116633415222, "perplexity": 1093.0241082405473}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945333.53/warc/CC-MAIN-20230325130029-20230325160029-00648.warc.gz"} |
https://forum.allaboutcircuits.com/threads/need-help-building-voltage-follower-and-amplifier-for-high-bandwidth.62393/ | # Need help building voltage follower and amplifier for high bandwidth.
#### MrLeitexxx
Joined Nov 16, 2011
2
I need help designing a circuit which will act as a buffer (or voltage follower) and also an other circuit which will act as an amplifier ( lets say X10 linear gain). Both must work well for a frequency between 30Hz and 10kHz. they must be single supply, (+9V/GND). | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8072006702423096, "perplexity": 3657.421552376605}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987835748.66/warc/CC-MAIN-20191023173708-20191023201208-00517.warc.gz"} |
http://forum.math.toronto.edu/index.php?topic=1485.0 | ### Author Topic: What is wrong with using the FTOC II in complex variable ? (Read 574 times)
#### Ende Jin
• Sr. Member
• Posts: 35
• Karma: 11
##### What is wrong with using the FTOC II in complex variable ?
« on: November 09, 2018, 06:48:49 PM »
In a tutorial, the TA showed us why FTOC II can be used in the complex context, the proof is simply just making the derivative of F into the imaginary part $v'$ and real part $u'$. Because that is a line integral, thus ultimately, replace the $\Gamma$ with the parameterization of $\gamma : [a, b] \rightarrow \mathbb{C}$, we are integrating the derivative of $u \circ \gamma: [a,b] \rightarrow \mathbb{R}$ which is just a integral in the real line. Now after using the real-function version of FTOC II on the derivative of a real-valued function, we get $u,v$ back and end up with $F$ again.
This is the outline of the proof. However, I don't see that F needs to be analytic on a simply-connected domain in this proof. That F can be analytic on only an annulus (i.e. differentiable on the range of parameterization), even F is undefined outside the annulus, we can still get that the integration is zero when integrating on a closed curve. But it is absolutely wrong because it contradicts the chapter of singularities.
Which part is wrong?
« Last Edit: November 09, 2018, 06:51:07 PM by Ende Jin »
#### Victor Ivrii
Assume that you need to calculate $I=\int_\gamma (P\,dx +Q\,dy)$ and $P=U_x$, $Q=U_y$. Then $P\,dx +Q\,dy= dU$ and $I= U(x_1,y_1)-U(x_0,y_0)$ where $\gamma$ goes from $(x_0,y_0)$ to $(x_1,y_1)$. So far there is no analytic functions or simple connectivity of the domain.
However $P=U_x$, $Q=U_y$ implies $P_y=Q_x$. If we integrate $f\,dz$ then $P=f$, $Q=if$ and this is equivalent to $f_y=if_x$ which is a Cauchy-Riemann condition (plug $f=u+iv$).
Further, if domain is not simply connected, then even "$f$ is analytic" does not imply that there exists an analytic single-valued function $F$ such that $F'=f$ (for non-analytic functions the "derivative": is not defined). F.e. in $\mathbb{C}\setminus 0$ let $f=z^{-1}$; then $F=\log (z)$ is a multivalued function and $I$ is an increment of $F$ along $\gamma$, depending on $\gamma$ (more precisely, it depends on the equivalency class $[\gamma]$ of $\gamma$: $\gamma_1 \sim \gamma_2$ if one could be continuously morphed into another without leaving domain. Only in simple connected domains all curves with the same start and end points are equivalent. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.973000705242157, "perplexity": 325.54166322880087}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600402132335.99/warc/CC-MAIN-20201001210429-20201002000429-00721.warc.gz"} |
http://mathhelpforum.com/trigonometry/124530-expressing.html | # Math Help - Expressing
1. ## Expressing
Given that $cotA=K$ and that $A$ is acute, express $sec2A$ in terms of $K$
2. Hello Punch
Originally Posted by Punch
Given that $cotA=K$ and that $A$ is acute, express $sec2A$ in terms of $K$
Here are the formulae you'll need:
If $A$ is acute, $\cot A = \frac{\text{adjacent}}{\text{opposite}}=\frac{K}{1 }$, so you can find $\cos A$ using Pythagoras' Theorem.
$\sec 2A = \frac{1}{\cos 2A}$
$\cos2A = 2 \cos^2A -1$
Can you complete it now? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 13, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8317621350288391, "perplexity": 755.8192847358894}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701161718.0/warc/CC-MAIN-20160205193921-00151-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://www.opposingviews.com/category/water-vapor-may-be-slowing-global-warming-rate | Water Vapor May be Slowing Global Warming Rate - Opposing Views
Water Vapor May be Slowing Global Warming Rate
Author:
Publish date:
A 10 percent drop in water vapor ten miles above Earth’s surface has had a big impact on global warming, say researchers in a study published online January 28 in the journal Science. The findings might help explain why global surface temperatures have not risen as fast in the last ten years as they did in the 1980s and 1990s.
Observations from satellites and balloons show that stratospheric water vapor has had its ups and downs lately, increasing in the 1980s and 1990s, and then dropping after 2000. The authors show that these changes occurred precisely in a narrow altitude region of the stratosphere where they would have the biggest effects on climate.
Water vapor is a highly variable gas and has long been recognized as an important player in the cocktail of greenhouse gases—carbon dioxide, methane, halocarbons, nitrous oxide, and others—that affect climate.
“Current climate models do a remarkable job on water vapor near the surface. But this is different — it’s a thin wedge of the upper atmosphere that packs a wallop from one decade to the next in a way we didn’t expect,” says Susan Solomon, NOAA senior scientist and first author of the study.
Since 2000, water vapor in the stratosphere decreased by about 10 percent. The reason for the recent decline in water vapor is unknown. The new study used calculations and models to show that the cooling from this change caused surface temperatures to increase about 25 percent more slowly than they would have otherwise, due only to the increases in carbon dioxide and other greenhouse gases.
An increase in stratospheric water vapor in the 1990s likely had the opposite effect of increasing the rate of warming observed during that time by about 30 percent, the authors found.
The stratosphere is a region of the atmosphere from about eight to 30 miles above the Earth’s surface. Water vapor enters the stratosphere mainly as air rises in the tropics. Previous studies suggested that stratospheric water vapor might contribute significantly to climate change. The new study is the first to relate water vapor in the stratosphere to the specific variations in warming of the past few decades.
Authors of the study are Susan Solomon, Karen Rosenlof, Robert Portmann, and John Daniel, all of the NOAA Earth System Research Laboratory (ESRL) in Boulder, Colo.; Sean Davis and Todd Sanford, NOAA/ESRL and the Cooperative Institute for Research in Environmental Sciences, University of Colorado; and Gian-Kasper Plattner, University of Bern, Switzerland.
NOAA understands and predicts changes in the Earth's environment, from the depths of the ocean to the surface of the sun, and conserves and manages our coastal and marine resources.
Image credit
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https://stke.sciencemag.org/content/4/175/ra35?ijkey=71d2e6a3a046314f7307ef4f58a3886ef2d767ba&keytype2=tf_ipsecsha | Research ArticleCOMPUTATIONAL BIOLOGY
# Reduction of Complex Signaling Networks to a Representative Kernel
See allHide authors and affiliations
Science Signaling 31 May 2011:
Vol. 4, Issue 175, pp. ra35
DOI: 10.1126/scisignal.2001390
## Abstract
The network of biomolecular interactions that occurs within cells is large and complex. When such a network is analyzed, it can be helpful to reduce the complexity of the network to a “kernel” that maintains the essential regulatory functions for the output under consideration. We developed an algorithm to identify such a kernel and showed that the resultant kernel preserves the network dynamics. Using an integrated network of all of the human signaling pathways retrieved from the KEGG (Kyoto Encyclopedia of Genes and Genomes) database, we identified this network’s kernel and compared the properties of the kernel to those of the original network. We found that the percentage of essential genes to the genes encoding nodes outside of the kernel was about 10%, whereas ~32% of the genes encoding nodes within the kernel were essential. In addition, we found that 95% of the kernel nodes corresponded to Mendelian disease genes and that 93% of synthetic lethal pairs associated with the network were contained in the kernel. Genes corresponding to nodes in the kernel had low evolutionary rates, were ubiquitously expressed in various tissues, and were well conserved between species. Furthermore, kernel genes included many drug targets, suggesting that other kernel nodes may be potential drug targets. Owing to the simplification of the entire network, the efficient modeling of a large-scale signaling network and an understanding of the core structure within a complex framework become possible.
## Introduction
Cellular systems have evolved molecular interaction networks to maintain their complex regulatory functions, which allow cells to perform processes such as differentiation and to respond to the environment. We speculated that such interaction networks were built around certain core structures or “kernels,” which would be simpler to analyze without losing essential information. An individual kernel can be defined broadly as a simplified framework of a given complex interaction network that preserves the dynamics and the output of the original network. Identification of such kernels would enable insights into the organization and evolution of biomolecular interaction networks, allow the generation of representative but simplified representations of complete networks that can be modeled, and, eventually, facilitate the exploration of interventions to manipulate cellular systems to perform desired responses (1).
When discussing biological networks, we refer to the proteins or genes as “nodes” and the relationships between the proteins or genes as “edges.” Although the networks are typically constructed with the names of the encoding genes, the functions are assumed to be performed by the encoded proteins or RNAs. Two general approaches to the study of biological networks are (i) component-wise analysis of individual components in the networks, as in studies of “minimal gene sets,” and (ii) computational analysis of simplified networks. Several studies have investigated the minimal gene sets (2) required for survival, using computational approaches (35) or experiments with bacterial mutants (2, 6, 7). One limitation of these component-wise approaches is that they cannot take into account regulatory interactions among the genes. The methods that involve simplifying complex networks generally strive to preserve “static” topological properties, such as the small-world property, scale-freeness, fractality, or modularity (821), and can largely be classified into two categories (17), coarse graining and filtering or pruning. Coarse graining refers to the grouping of nodes with respect to various topological properties and replacing each group of nodes with a single node called a coarse-graining unit (CGU), thereby achieving a simpler network representation (12, 18). The filtering or pruning approach deletes nodes classified as less important from scores assigned to the nodes on the basis of the network’s topological characteristics. One limitation of these simplified network approaches is that, by primarily focusing on preserving static topological properties of general complex networks, they fail to preserve the dynamical properties of cellular signaling networks. Cellular signaling networks exhibit properties, such as feedback loops, that make preservation of dynamical properties challenging.
The spanning tree network reduction approach of Kim et al. (13) reduces only the number of edges while preserving all the nodes of the original network. Because the resulting simplified network is a tree, it cannot preserve the dynamics of the original network if the original contains feedback loops (2225) or feedforward loops (26, 27). The approach taken by Itzkovitz et al. (12) replaces network motifs with CGUs, which in principle can preserve the dynamics of a network only if the intrinsic dynamics of each network motif are identically implemented in the CGU of the reduced network. However, it remains unclear how to implement such identical dynamics at each CGU. Song et al. (18) proposed a reduction scheme that tiles a network with boxes such that the shortest path length of any two nodes in a box is less than a given number called a box size, where the size of the box is 1 + m, with m the maximum of the shortest paths between two nodes in the box. However, the resulting network does not contain any information on the direction or interaction type (activation or inhibition) of the edges; thus, preservation of dynamic properties is not possible. Using network symmetry, Xiao et al. (19) proposed a network reduction scheme in which a set of nodes is grouped as one node if the rearrangement of their position within the set does not change the network topology. This approach can be effectively applied to a gene network containing many functionally redundant genes, but it is not effectively applicable to cell signaling networks that usually contain many long cascades.
Here, we describe the “kernel identification algorithm,” which is an algorithm that identifies a kernel systematically by considering the relationship between a network’s structure and its dynamics. Because of the enormous complexity of biomolecular interaction networks, it is not computationally feasible to find a representative kernel by simultaneously taking into account the dynamics of all possible subnetwork cases. The kernel identification algorithm overcomes this difficulty by recursive sequential replacement of the neighborhood subnetwork of each node with a smaller one that preserved the same dynamics. The neighborhood subnetwork of a node is the network composed of the nodes directly connected to the given node. We show that our algorithm can be applicable to large-scale cell signaling networks to produce smaller, simpler networks that retain the original network’s dynamics. Although some coarse-graining methods, such as fractal analysis (11, 18), also perform repetitive substitutions of subnetworks with smaller ones and are as efficient as our method in terms of computational complexity, they generally fail to preserve the dynamical properties of a network.
By applying the kernel identification algorithm, we identified kernels for various signaling networks ranging from bacterium (Escherichia coli) and yeast (Saccharomyces cerevisiae) to human, and we verified that the identified kernels preserved the input-output dynamics of the original networks. We found that a large proportion of the nodes within the kernels (kernel nodes) corresponded to essential genes, disease-associated genes, genes encoding drug targets, or genes that are part of synthetic lethal gene pairs. Moreover, we found that kernel nodes were encoded by genes conserved in multiple species, suggesting low evolutionary rates, and encoded proteins present in various tissues, suggesting that these kernel-associated genes may serve core cellular functions. The kernel identification algorithm can provide a reduced form of a given network, and this smaller network may provide insight into the design principles of complex biomolecular interaction networks, as well as suggest effective ways to perturb or manipulate the network.
## Results
### Kernel identification algorithm
We wanted to develop an algorithm that preserved the input and output nodes of a biomolecular interaction network and the input-output dynamics of the original network while reducing the complexity of the network (Fig. 1A). An input node in a network denotes a node without any regulatory inputs (indegree is zero). Likewise, an output node denotes any node that lacks any relationships with downstream nodes (outdegree is zero). For example, in some signaling networks, ligands or receptors (when ligands are not specified) may correspond to input nodes, and transcription factors (when their target genes are not specified) may correspond to output nodes. The remaining nodes in a network are intermediate nodes. We developed an algorithm that minimized the number of intermediate nodes by replacing certain subnetworks within a large network with smaller subnetworks.
To overcome the computational burden that would result from analyzing simultaneously all possible dynamics of biological networks and their subnetworks, the kernel identification network recursively replaces the neighborhood subnetwork of each node with a smaller network, either with fewer nodes or fewer edges or both, with the same dynamics until no further replacement is possible.
To determine the rules for subnetwork replacement, we developed and simulated the mathematical models of all two- and three-node networks with ordinary differential equations (see Materials and Methods and Supplementary Model Descriptions), and then clustered the two- and three-node networks according to the similarity in their dynamics (Fig. 1B). We verified that the clustering assignments were similar between linear and Hill-type mathematical models and among the parameter values used (see fig. S1 and Supplementary Model Descriptions). On the basis of the clustered networks, the algorithm attempts to replace the neighborhood subnetwork of each node with a smaller network (see Fig. 1C for examples of subnetwork replacement and Materials and Methods and fig. S2 for details). The algorithm cannot replace subnetworks either (i) when one node in a three-node subnetwork is also a component node of a self-feedback loop, a two-node feedback loop, or an intermediate node of an incoherent feedforward loop, or (ii) when both the indegree and the outdegree of the node are >1 (Fig. 1D). When a network cannot be reduced any further by the above reduction process, the algorithm reduces the network by replacing the neighborhood subnetwork of a set of edges, taking into account consistency of the types of regulation among the neighboring edges (see Fig. 1E for an example and Materials and Methods and fig. S2 for details). We defined the “node reduction percentage” as [(the number of intermediate nodes removed during reduction)/(the number of intermediate nodes in the original network)] × 100. For the sample network shown in Fig. 1A, the node reduction percentage equals 73% [(8/11) × 100].
We applied the algorithm to the signaling networks of E. coli, S. cerevisiae, and H. sapiens, where we define the signaling network as an integrated network of all the signaling pathways obtained from the KEGG (Kyoto Encyclopedia of Genes and Genomes) database (28) for each species, and identified the kernels of those networks (data S3 to S5). We refer to these three networks as the E. coli, yeast, and human signaling networks. Through Boolean simulations (29, 30), we verified that the kernels preserved the dynamical properties of the input-output response profiles of the original networks (see Supplementary Model Descriptions and table S1). Because it is not feasible to construct and simulate large-scale networks, such as the human signaling network, by ordinary differential equations, we used Boolean models to verify the preservation of network dynamics.
### Structural characteristics of networks and kernels
Application of our algorithm to relatively small-sized networks, the circadian regulation network (31, 32) in mammals (data S1) and a generalized integrin signaling pathway (unknown regulations were assumed to be activations) representing data from multiple species (33) (data S2), resulted in a node reduction percentage of 67% and 94%, respectively (Fig. 2A). The circadian kernel consisted of two negative feedback loops and one positive feedback loop, a structure consistent with the known core of circadian regulation (34). The considerable reduction that we achieved for the integrin network resulted from the following characteristics of the integrin pathway: It had only 8 negative edges (inhibitory regulations) out of 101 edges, and hence most feedforward loops in the pathway were of coherent type, and the pathway consisted of many long signaling cascades (the network diameter of the pathway, the maximum of the shortest path lengths between node pairs, was 14).
A signaling network with a high node reduction percentage contains numerous redundant nodes in terms that are not required to preserve input-output dynamics; thus, the node reduction rate can be considered as a measure of redundancy in signaling networks. To explore the amount of redundancy in signaling networks of three species, E. coli, S. cerevisiae, and H. sapiens (Fig. 2B), we compared the node reduction percentages for the kernels of the E. coli, yeast, and human signaling networks. The node reduction percentage for each network was ~80% (Fig. 2C), suggesting that these three signaling networks have a similar proportion of redundant intermediate signaling proteins. The amount of reduction in the number of nodes and edges increased as the proportion of intermediate nodes in the original network increased (Fig. 2C and fig. S3). For example, the human network with 1953 total nodes had the largest proportion of intermediate nodes (44%) and exhibited the greatest reduction in nodes and edges when the kernel was compared to the original network (fig. S3).
We examined the global topological properties of network density, clustering coefficient, network diameter, and characteristic path length between the original networks and their kernels (Fig. 3). The network densities and average clustering coefficients of the kernels were greater than those of the original networks (Fig. 3, A and B), which means that the nodes of the kernels were more densely connected, and neighborhood nodes of each node were more densely connected to each other. From the comparison of the network diameters and the characteristic path lengths (Fig. 3, C and D), we found that the small-world property of the kernels was stronger; that is, every node in the kernel was on average a smaller number of steps away from any other node in the kernel. For example, in the human network, the kernel nodes were 3.3 steps away from each other, whereas the average number of steps was 6.3 in the original network.
We also analyzed the local properties of the networks, such as the subnetwork structure and the properties of the most highly connected node, the “giant component.” We compared the distribution ratios of three-node subnetworks (numbered 1 through 13) between the human network and its kernel (Fig. 3E). The subnetwork structure, which plays the role of a signal splitter (subnetwork ID1), was the most frequently occurring (50%) in the original network, whereas a signal integrator (subnetwork ID3) was dominant (47%) in the kernel. This implies that the human network includes a large number of signal splitters and many signaling pathways are connected by signal integrators. Indeed, the giant component of the original human signaling network included 85% of the total number of nodes (fig. S4).
The local properties of the E. coli and yeast networks were different from those of the human network. In these two networks, the signal splitter subnetwork (ID1) was the most frequently occurring three-node subnetwork structure in both the original networks and the kernels (fig. S5), suggesting that these two networks have signal-splitting subnetworks but that most pathways in the networks are isolated. Indeed, most of the pathways in the E. coli and yeast networks were short in length (Figs. 2B and 3D), and the networks did not contain extensively connected components (fig. S4). The different structural features related to subnetwork occurrence and node interconnectedness between the human network and the networks of E. coli and yeast may relate to the multifunctionality of kinases, which is reflected in the number of connections that they make. We found that the average indegree and outdegree of human kinases were significantly higher than the average indegree and outdegree of total nodes in both the original and the kernel networks (Fig. 4, A to D). In contrast, indegree and outdegree of the kinases relative to the total nodes in the original and kernel networks of the two single-cell species were not significantly different (Fig. 4). Moreover, this tendency is enforced in the kernels (Fig. 4, B and D, and fig. S6). These results imply that human kinases function to connect multiple signaling pathways.
### Enrichment of essential genes, disease genes, and synthetic lethal genes in the kernel
Kernel nodes can be defined as those not deleted during the reduction process, which results in these nodes having similar or increased connectivity in the reduced network relative to the same nodes in the original network. It is possible that the kernel nodes play pivotal roles and that the non–kernel nodes have auxiliary roles in terms of biological processes. We investigated the enrichment of essential genes, disease genes, and synthetic lethal gene pairs in the sets of the kernel nodes and the non–kernel nodes for the human network. If the kernel nodes represented proteins with critical functions, then we would expect that the kernels would be enriched for nodes in each of these classes. We observed that 10% of the non–kernel nodes were essential genes, whereas 32% of the kernel nodes were essential genes (Fig. 5A) [essential genes were defined from Zhang and Lin (35); see Materials and Methods], and the difference is statistically significant (P = 1.38 × 10−21). In addition, we observed that most of the essential genes present in the original human network were included in the kernel (fig. S7). Essential genes also tended to be enriched in kernel networks of E. coli and yeast (fig. S8). Highly connected proteins in protein-protein interaction networks have a higher probability of being encoded by essential genes (36), and the original human network exhibited this property (fig. S9). To determine whether enrichment of essential genes in kernel nodes depended on node degree (the number of inputs and outputs), we considered only the nodes of small degree (<4) for the calculation of the ratio of essential genes (see fig. S10 for the degree distribution in the human network, and note that the number of nodes of degree <4 is about half of the total number of nodes). Even when only nodes with relatively few connections were considered, the kernel nodes were still enriched for essential genes (Fig. 5A). Thus, the kernel-identifying algorithm identified both essential genes represented by those in the kernel network and nonessential genes represented by nodes that were deleted by the network reduction.
In the human network, we found a similar enrichment for disease-associated genes, which were defined on the basis of the Online Mendelian Inheritance in Man (OMIM) database (37) in the National Center for Biotechnology Information (NCBI). Most kernel nodes (95%) corresponded to Mendelian disease genes (Fig. 5B), and their enrichment in the kernel compared to the non–kernel nodes was statistically significant. As with the essential gene enrichment, we found that the enrichment in disease-associated genes was also not dependent on degree and that even limiting the analysis to nodes with a degree <4 showed a significant enrichment in disease-associated genes in the kernel nodes compared to the non–kernel nodes (Fig. 5B). Many genes are both essential and disease-related. When the classes are taken together, the results suggest that kernel nodes represent biologically important points in the network and often correspond to critical genes.
Synthetic lethality is considered to be closely related to network structure (38, 39). We expected that the kernel nodes would be enriched in synthetic lethal gene pairs. Two genes are called a synthetic lethal gene pair if mutation of either alone is not lethal, but mutation of both leads to death or a significant decrease in the organism’s fitness (40). We analyzed how many synthetic lethal gene pairs were included in the kernel of the human network. Synthetic lethal pairs of human genes were based on Conde-Pueyo et al. (40). As expected, most synthetic lethal pairs (93%) occurred between two kernel nodes, and we did not identify any synthetic lethal pairs in the set of non–kernel nodes (Fig. 5C). Our finding that kernels contained not only most essential genes and disease genes but also most synthetic lethal gene pairs suggests that kernel nodes are critical in terms of individual components and, because the synthetic lethality is closely related to network structure (38, 39) and the kernel was obtained in consideration of network structure, in terms of network structure.
Because kernels are representative of biological networks, we speculate that kernel nodes may be ubiquitously expressed in various tissues and be conserved among diverse species. We investigated “tissue broadness” and “species broadness” for both kernel nodes and non–kernel nodes of the human network. The tissue broadness (41) of a gene is defined as the number of human tissues in which the gene is expressed (42), and species broadness as the number of species in which homologs of the gene exist (43) (see Materials and Methods). We found that both the tissue broadness (Fig. 5D) and the species broadness (Fig. 5E) of the kernel nodes were significantly larger than those of the non–kernel nodes.
A high value for tissue broadness suggests that the gene is expressed ubiquitously and that the gene plays a common basic cellular function of various types of cells. We found that many kernel nodes were related to metabolic and developmental processes (table S2). Similarly, a high value for species broadness of a gene implies that the gene is evolutionarily conserved; hence, the kernel nodes may represent a conserved core of the network.
### Evolutionary rates of kernel nodes and the relationship with function
Large values for tissue and species broadness of the kernel nodes imply that the gene sequences of the kernel nodes might have changed little during evolution. We explored the evolutionary rate (see Materials and Methods) of the kernel nodes with the hypothesis that conserved nodes would have lower evolutionary rates than non–kernel nodes and found that the kernel nodes had significantly lower evolutionary rates than did the non–kernel nodes (Fig. 5F), implying that the gene sequences of the kernel nodes are conserved during evolution.
From a Gene Ontology (GO) analysis, we observed that the functions assigned to the kernel nodes were different from those of the non–kernel nodes. The kernel nodes were mainly related to metabolic processes (48%, table S2) or to developmental processes (49%, table S2), whereas many of the non–kernel nodes were related to sensory perception (62%, table S3). We also observed a relationship between gene functions and evolutionary rates. The genes with relatively higher evolutionary rates were mainly related to immune processes and sensory perception, whereas those with lower evolutionary rates were related to developmental and metabolic processes (Fig. 5G). Because the genes related to metabolic processes play a pivotal role for cell survival (64.4% of the essential genes are related to metabolic processes with the enrichment of P = 4.9 × 10−29), these genes might have been evolutionarily stable. We noticed that the kernel nodes with high evolutionary rates were related to immune processes, whereas the non–kernel nodes with high evolutionary rates were related to sensory perception (Fig. 5G and tables S4 to S9). Because the kernel was determined from the network structure, this type of network reduction process can provide insight into gene functions.
Although genes associated with immune processes and sensory perception had relatively high evolutionary rates (Fig. 5G), these genes were associated with different parts of the network: Immune process–associated genes were enriched in the kernel nodes, whereas sensory perception–associated genes were enriched in the non–kernel nodes. We hypothesized that the genes related to immune processes were represented by nodes within elaborately and tightly regulated network substructures, such as feedback loops, and thus were not eliminated during network reduction. In comparison, we predicted that those genes related to sensory perception would not be represented by nodes in network substructures such as feedback loops. Indeed, the nodes representing the immune response genes were more enriched in feedback loops compared to the nodes representing the sensory perception genes (Fig. 5H). Genes with low evolutionary rate have been negatively selected (44) and genes with high evolutionary rate have been positively selected (45) during evolution. Hence, these results suggest that the functions of the kernel nodes might have been conserved by negative selection or the deleterious effects of mutations on organisms, whereas those of the non–kernel nodes might have evolved by positive selection or by beneficial effects of mutations.
### Network kernel and drug targets
Because most kernel nodes in the human signaling network can be mapped to diseases (fig. S7), we speculated that the kernel nodes might be related to drug targets. Hase et al. (46) showed that drug targets are enriched in the backbone network composed of middle-degree nodes (6 to 38 connections) in a protein-protein interaction network from Rual et al. (47). We compared the ratios of drug targets in the kernel and non–kernel nodes and found that drug targets were enriched in the kernel (Fig. 6, A and B), which is consistent with the previous work (46). Drug targets were identified on the basis of DrugBank (48). In addition, we examined the relationship between drug targets and node degree, and we found that nodes that were drug targets had middle degrees (Fig. 6C), which is also consistent with the previous work (see fig. S11 for the degree correlation between the protein-protein interaction network and the human network). We observed that the neighborhood nodes of drug targets had low degrees (Fig. 6D), and drug targets had low closeness centrality (Fig. 6E), which is a measure of how a given node is close to all other nodes (49). These characteristics suggest that the nodes that are drug targets are middle-degree hubs in the kernel but are peripheral nodes such that their perturbation would locally affect the network. On the basis of these results, we suggest that analysis of the topological properties of the kernel may enable discovery of drug targets.
## Discussion
Genomic and other experimental techniques have enabled the discovery and study of large-scale biomolecular interaction networks, such as the map of human cancer signaling (50). However, the scale of biological networks has typically required the study and modeling of either small subnetworks for performing detailed parameterization or larger sets of nodes with limited opportunity to parameterize the interactions. One means of solving this problem is to condense a biological network into a smaller one that is equivalent in terms of its dynamical and topological aspects. We considered whether interaction networks have evolved from certain core structures and if such reduced networks could represent the dynamics of the source network. To address these questions, we introduced the concept of a kernel of a biological network, which we defined as the minimal essential network that preserves the input-output dynamics of the original network. We created an algorithm by which we systematically identified a kernel by considering the relationship between the network structure and its dynamics (Fig. 1). Because the proposed algorithm reduces signaling networks by considering the locally equivalent subnetworks instead of global equivalents, it is fast and can be applied to large-scale networks containing tens of thousands of nodes and millions of edges. Several studies have simplified complex networks by considering the static properties of the network topology (819). Compared to our approach, these alternatives do not preserve the dynamical properties of the input-output response profiles of the original network (fig. S12). Among the coarse-graining methods, hierarchical modularization methods (20, 51) are most effectively applicable to various networks because they do not require any a priori information on the number or size of modules (that is, groups of clustered nodes) (51). However, coarse-graining methods still cannot reduce networks while preserving their dynamics because a module represented as a single node in the reduced network can actually contain many feedback or feedforward loops that entail complex dynamics.
For more detailed modeling of the behavior of individual genes, the parameters for each reduced node can be expanded while maintaining a simplified but fully representative network away from the area under study. With our algorithm, we identified the kernels of several networks ranging from E. coli and yeast to human and verified that the identified kernels preserved the fundamental input-output dynamics of the original networks. We found that the kernels comprise nodes representing essential genes, disease genes, drug targets, and synthetic lethal gene pairs (Figs. 5 and 6). Moreover, the kernels contained a high proportion of nodes representing genes with low evolutionary rates and genes that are ubiquitously expressed in various tissues and are present in many species. These results suggest that the kernels might be the backbones of biomolecular interaction networks, and interaction networks might have evolved on the basis of their kernels. We conclude that the analysis of a network kernel can provide new insights into the design principles of complex biomolecular interaction networks, identify potential drug targets, and facilitate modeling and parameterization of the resulting smaller-scale networks. This kernel identification network algorithm should only be applied to networks where neither stochasticity nor a time delay effect is dominant in determining the dynamical properties of input-output response profiles.
Genes corresponding to nodes in the reduced network may be most informative for phylogenetics or evolutionary studies, which may have implications for understanding the domestication of animals. As the number of organisms with well-defined signaling networks increases, it will be possible to investigate the effects of agricultural domestication of animals on both these genes represented by kernel nodes and non–kernel nodes. For example, reduced sensory perception (less flightiness and less need for detection of predators) and increased immune response (living in proximity and unnaturally large, genetically homogeneous groups) are signatures of domestication that are both intensively selected and probable preconditions for introduction to farms. Thus, one could ask if domesticated animals and their wild relatives (such as cows and aurochs) have the kernels representing gene networks that would produce these phenotypes compatible with domestication, whereas animals that have not been domesticated successfully (deer or all sub-Saharan large mammals) would have different kernels. One could also evaluate if the genes represented by the kernels between the domesticated and nondomesticated have these genes (and the kernel nodes) in domesticated animals changed under intense selection in a relatively short evolutionary period. Kernel nodes may also suggest points of genetic modification in agriculture because they lie on the critical path to generation of products and hence may show strong signatures of domestication or be future targets for modification.
The whole kernel can be modeled and then the behavior of a node in the original network can be elucidated from more detailed analyses of the kernel, and the kernel will show the modular organization of the original network as well as the critical input and output edges, which must be included in subnetwork models. We expect that the proposed kernel identification method can also be applied to facilitate the modeling and analysis of middle-scale signaling pathways, such as the epidermal growth factor receptor pathway (5254), which already comprises a large number of signaling proteins under various states. Borisov et al. proposed a model reduction scheme in which unfeasible protein states are eliminated on the basis of a domain analysis (55). Our method reduced the number of signaling proteins to be modeled, as shown in the examples of the circadian network and the integrin pathway (Fig. 2A). The kernel identification method can be used to screen the key signaling components that dominate the dynamics of a given signaling pathway. This is particularly useful in the study of complex signaling networks, because the identification of such key signaling components is a fundamental step for any further analysis (56, 57).
Although we considered a pulsatile stimulus in this study to examine the dynamical properties of the input-output response profiles of a cellular system because many cell signaling inputs can actually be approximated with this form of signal, other types of “biologically relevant” input stimulations, such as long-term constant inputs or oscillatory inputs, can also be approximated by controlling the parameter of the pulse signal (that is, the duration) or the combinations of pulse signals. Future studies will extend this method to determine its effectiveness in reproducing network dynamics in response to the other (even biologically irrelevant) input stimulation patterns.
## Materials and Methods
### Classification of two- and three-node networks with respect to dynamics
We constructed mathematical models of two- and three-node networks with ordinary differential equations (see Supplementary Model Descriptions for details) and simulated them 1000 times with random parameter values in the interval [0, 1], where the stimulus was given by a pulse type (fig. S13). Next, we classified 1000 response curves into six types (fig. S14) for each model and represented the numbers of response curves in each of the six classes by a six-dimensional vector (tables S10 and S11). If a stimulus, such as a pulse—most cell signaling inputs can actually be approximated with this form of signal—is given to a cellular system, it can produce one of the following response profiles: a pulsatile response, a monotonic increasing or decreasing response, a sustained oscillatory response, or a damped oscillatory response. We considered these types of response profiles in our examination of the dynamical properties. We defined the dynamical distance D(X,Y) between two models represented by X = (x1, x2, …, x6) and Y = (y1, y2, …, y6) byD(X,Y)=|Y/|Y|X/|X||/2where || denotes the Euclidean norm|X|=x12+x22+xn2 with X=(x1,x2,xn)On the basis of this distance, we applied the multidimensional scaling approach (58) to classify the two- or three-node networks with respect to their dynamics (Fig. 1B). We used a distance criterion D < 0.001 for the determination of the same network dynamics.
### Kernel identification algorithm
Our algorithm can be applied to directed networks, such as signaling networks, gene transcription networks, and metabolic networks. For simplicity, we assume that each edge in the networks has only one of two regulation types (activation or inhibition). Hence, a biological network can be represented by a signed graph G = (V,E), where V is a set of nodes and E is a set of edges with signs. Each edge can be represented by eij = (vi, vj, σij), where vi is a start node, vj is an end node, and σij is a sign (+1, 0, or −1) of the edge (referred to as the “signature”). σij = 0 denotes that two nodes vi and vj are not connected by an edge. For each node vj, we define the set of start nodes of the edges whose end node is vj, by VE(vj) = {vi|(vi, vj, σij) ∈ E}. Likewise, we define the set of end nodes of edges whose start node is vj by VS(vj) = {vk|(vj, vk, σjk) ∈ E}. Let Σ be the set of signed graphs. Our kernel identification algorithm is represented by a map F: Σ→Σ satisfying the following four conditions [for a given G = (V,E) ∈ Σ, let F(G) = (F(V),F(E))]:
(i) F preserves node-edge relations. That is, for a given edge, eij = (vi, vj, σij), F(eij) = (F(vi), F(vj), F(σij)). Here, F(σij) = 0 denotes an edge deletion.
(ii) F preserves input and output nodes.
(iii) Given vj satisfying either indegree(vj) = 1 or outdegree(vj) = 1, consider the case that vj is not involved in a self-feedback loop, an incoherent feed-forward loop, or a two-node feedback loop. For any viVE(vj) and vkVS(vj), if (vi, vk, σijσjk) ∈ E, then F(vj) = F(vk) and F(σij) = F(σjk) = 0. Otherwise, F(vj) = F(vk), F(σjk) = 0, and F(σik) = σijσjk. In this case, a new edge (F(vi), F(vk), F(σik)) is contained in F(E).
(iv) Consider the case that there is no node satisfying the conditions for the node-based reduction [step (iii)]. For each edge eij = (vi, vj, σij) ∈ E, if eij is not contained in any incoherent feedforward loop, then for any vkVS(vj) [vkVE(vj), respectively], F(vj) = F(vi), F(σij) = F(σjk) = 0, and F(σik) = σijσjk [F(σki) = σijσkj, respectively].
Our algorithm minimizes the size of F(G) by reordering the nodes in G. We repeated steps (iii) and (iv) for 1000 times reordering of nodes and selected the minimal network for each G.
### Essential genes, disease genes, synthetic lethal gene pairs, and drug targets
The essential gene lists for three species, E. coli, S. cerevisiae, and H. sapiens, were obtained from the database DEG (Database of Essential Genes, version 5.4) (35). The disease gene list was obtained from OMIM database (37) in the NCBI. This list contains 14,388 disease genes, 1536 of which are contained in the original human network. The list of synthetic lethal gene pairs for human was obtained from iHSLN (inferred human SL genes) (40). The drug target list was obtained from the DrugBank database (48). This list contains 1330 proteins that are drug targets, 275 of which are contained in the original human network.
The tissue broadness (41) of a gene is defined as the number of tissues in which the gene is the upper outlier, meaning that the mRNA abundance is higher than the sum of the upper quartile and 1.5 times the interquartile range (59). We calculated the tissue broadness information using mRNA expression data in 79 human tissues (42).
We defined the species broadness of a gene as the number of species in which homologs of the gene exist. The homolog information of 20 species (table S12) was extracted from the HomoloGene database (43) in the NCBI.
### Evolutionary rate
The evolutionary rates were defined by the ratios of the nonsynonymous substitution rates (dN) and the synonymous substitution rates (dS) for homologous gene pairs in human and mouse and they were obtained from the Human PAML Browser (60).
### GO analysis
GO (61) analysis was performed with the functional annotation tool in DAVID (62). We first divided the 1493 human kernel genes into four groups and the 460 deleted genes into three groups on the basis of the ranges of evolutionary rate (see Fig. 5G for the ranges). We then retrieved the GO terms significantly related with each gene group [Benjamini score (63) <0.05], using the functional annotation tool applied to the 1953 genes of the human network as a background set. We selected the child GO terms related to parent terms (metabolic process, developmental process, sensory perception, immune process, and signal transduction) in the GO hierarchy.
### Motif enrichment analysis
We identified network motifs using MAVisto (64) without considering both edge labels and vertex labels. For reliable statistics, 1000 random networks were generated, and the three-node subgraphs with P values <0.05 were considered as network motifs. We defined the enrichment of a motif for each gene class (Fig. 5H) as the ratio of the number of the genes related to the motif contained in the gene class to the expected number of genes related to the motif for the size of the gene class.
### Statistical analysis
We performed one-sided two-sample χ2 tests to evaluate the statistical abundance of the essential genes (Fig. 5A), disease genes (Fig. 5B), and drug targets (Fig. 6B) in the kernels. For the tissue broadness (Fig. 5D), species broadness (Fig. 5E), evolutionary rates (Fig. 5F), degree (Fig. 6C), neighborhood connectivity (Fig. 6D), and closeness centrality (Fig. 6E), the one-sided two-sample t test was applied.
### Availability of the software
We have implemented the proposed kernel identification algorithm as software. It is available from http://sbie.kaist.ac.kr/software and as part of the Supplementary Materials.
## Supplementary Materials
www.sciencesignaling.org/cgi/content/full/4/175/ra35/DC1
Model Descriptions
Fig. S1. The multidimensional scaling map for classification of responses of the nonlinear (Hill-type) models of two- and three-node networks.
Fig. S2. The flow diagram illustrating the kernel identification algorithm.
Fig. S3. Ratios of kernel to original in terms of nodes and edges for the signaling networks of E. coli, yeast, and human.
Fig. S4. Relative size of the giant component, which is the component with the most connections, in the original three networks.
Fig. S5. The frequency distributions of three-node subnetworks in the signaling networks of E. coli and yeast compared with the distributions of these subnetworks in their kernels.
Fig. S6. Average indegrees and outdegrees of kinases in the original networks and kernels for the networks of E. coli, yeast, and human.
Fig. S7. The frequency of essential genes and disease genes in the kernel nodes and non–kernel nodes.
Fig. S8. The ratio of essential genes contained in the kernel and non–kernel nodes of the networks of E. coli and yeast.
Fig. S9. The ratio of essential genes represented in the set of nodes of each degree in the human network.
Fig. S10. Degree distribution and cumulative frequency distribution of degrees in the human network.
Fig. S11. The degree in the human protein-protein interaction network versus the degree in the human signaling network.
Fig. S12. Comparison of the input-output dynamics of the original network and the reduced network after applying five different network-reduction approaches.
Fig. S13. The stimulus pattern used for the simulation of two- and three-node network models.
Fig. S14. Six representative response patterns used for classification of two- and three-node networks.
Table S1. Response coherency between the original signaling network and the corresponding kernel.
Table S2. GO terms related to genes represented by nodes in the kernels (kernel genes).
Table S3. GO terms related to genes that were excluded from the kernel, but were represented by nodes in the original network (non-kernel).
Table S4. GO terms related to the kernel genes that had evolutionary rates larger than 0.25.
Table S5. GO terms related to the kernel genes that had evolutionary rates between 0.125 and 0.25.
Table S6. GO terms related to the kernel genes that had evolutionary rates between 0.0625 and 0.125.
Table S7. GO terms related to the kernel genes that had evolutionary rates less than 0.0625.
Table S8. GO terms related to non–kernel genes that had evolutionary rates less than 0.25.
Table S9. GO terms related to the non–kernel genes that had evolutionary rates between 0.125 and 0.25.
Table S10. Simulation results for linear models of 18 network structures.
Table S11. Simulation results for Hill-type models of 18 network structures.
Table S12. The list of 20 species examined in the HomoloGene database.
Data S1. The circadian regulatory network data where the first, second, and third columns denote regulator, relation, and target, respectively. [Filename: Circadian_regulatory_network.txt]
Data S2. The integrin signaling pathway data where the first, second, and third columns denote regulator, relation, and target, respectively. [Filename: Integrin_signaling_pathway.txt]
Data S3. The E. coli signaling network data where the first, second, and third columns denote regulator, relation, and target, respectively. [Filename: Ecoli_network.txt]
Data S4. The yeast signaling network data where the first, second, and third columns denote regulator, relation, and target, respectively. [Filename: Yeast_network.txt]
Data S5. The human signaling network data where the first, second, and third columns denote regulator, relation, and target, respectively. [Filename: Human_network.txt]
Software. The software of the proposed kernel identification algorithm for MS-DOS–type operating systems. [Filename: kernelfinder.exe]
References
## References and Notes
1. Acknowledgments: We thank S. Baek, D. Kim, D. Lee, H. Byrne, and A. Fletcher for valuable comments on the manuscript. Funding: This work was supported by the National Research Foundation of Korea (NRF) grants funded by the Korea Government, the Ministry of Education, Science and Technology (MEST) (2009-0086964, 2010-0017662, and 2011-0002145). It was also supported by World Class University program through the NRF of Korea funded by the MEST (R32-2008-000-10218-0). Author contributions: K.-H.C. designed the research; J.-R.K., J.K., and H.-Y.L. performed simulations; J.-R.K., J.K., H.-Y.L., and K.-H.C. analyzed the data; Y.-K.K., J.-R.K., and K.-H.C. implemented the algorithm; and J.-R.K., J.K., P.H.-H., and K.-H.C. wrote the manuscript. Competing interests: P.H.-H. is the director of BioAstral Limited. The other authors declare that they have no competing interests.
View Abstract | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8266822695732117, "perplexity": 1655.7842823063368}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347390755.1/warc/CC-MAIN-20200526081547-20200526111547-00135.warc.gz"} |
https://www.physicsforums.com/threads/whats-the-microwave-s-parameter.702989/ | # What's the microwave S-parameter?
1. ### iScience
414
i haven't been able to obtain a simple answer
4
747
4. ### HallsofIvy
40,372
Staff Emeritus
So its worth is "many times" 0?
5. ### jasonRF
747
yes. I was being a bit silly. sorry!
jason
555
7. ### yungman
It is very similar to other port parameters like ABCD, Z, Y, etc. S-parameter deals with incidence and reflected power waves and the 2 port parameters are ##S_{11},\; S_{21},\;S_{12},\;S_{22}## and with certain condition like either the reflected or incident wave is zero etc. Keep that in mind, it's just another parameters.
Most RF device are spec with S-parameters. They are presented either as polar coordinates form where they give you the amplitude and the phase angle, or presented as complex numbers. You plot the parameters on Smith Chart and you can read out Z and Y parameters also. That will make it very easy to design circuits and matching networks.
Last edited: Aug 4, 2013 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8380830883979797, "perplexity": 2349.9064248603836}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246644083.49/warc/CC-MAIN-20150417045724-00286-ip-10-235-10-82.ec2.internal.warc.gz"} |
http://www.engineeringintro.com/mechanics-of-structures/sfd-bmd/how-to-find-point-of-contraflexure-example/ | How to Find Point of Contraflexure
In an overhanging beam, point of contraflexure is a point where bending moment changes it sign. Example: A point, where bending moment goes from positive to negative or from negative to positive. It can better explained by solving a numerical.
Point of Contraflexure Example
Find point of contraflexure of beam shown below.
Solution
In order to find point of contraflexure, first find the reactions at supports.
Consider,
∑M(at point D) = 0
R1 x 7 + (15 x2) = (30×2) + (20×5)
R1 = 18.571 KN
For Reaction R2, consider;
∑M(at point A) = 0
R2x7 = (20×2) + (30×5) (15×9)
R2 = 46.429 KN
Shear Force Diagram
Next step is to draw a shear force diagram. For shear force diagram, first we have to find shear force at point A, B, C, D and E.
Shear force at point A = S.FA = 18.571 KN
Shear force at point B(left) = S.FB(L) = 18.571 KN
Shear force at point B (right) = S.FB(R) = 18.571 – 20 = 1.429 KN
Shear force at point C (left) = S.FC(L) = – 1.429 KN
Shear force at point C (right) = S.FC(R) = 18.571 – 20 – 30 = 31.429 KN
Shear force at point D (left) = S.FD(L) = -31.429 KN
Shear force at point D (right) = S.FD(R) = 18.571 – 20 – 30 + 46.429 = 15 KN
Shear force at point E = S.FE = 15 KN
Bending Moment Diagram
Bending Moment at A = 18.571 x 0 = 0
Bending Moment at B = 18.571 x 2 = 37.142 KN.m
Bending Moment at C = (18.571 x 5) – (20 x 3) = 32.855 KN.m
Bending Moment at D = -(15 x 2) = -30KN.m
Bending Moment at E = 0
Position of Point of Contraflexure
From bending moment diagram, one can see that bending moment is going from positive to negative after some distance from point C. Let consider this point as P.
Given “x” distance from point C.
point of contraflexure can be calculated as;
18.571(x + 5) – 20(30*x) = 0
18.571x + 92.855 -20x -60 -30x = 0
31.429x = 32.855
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