text:
## Introduction

In a pair-density-wave (PDW) state, the Cooper pairs have a nonzero center of mass momentum1. Long-studied versions of such states are the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) phases2,3 that can arise in the context of BCS theory in the presence of a small degree of spin magnetization. However, it has been conjectured that a PDW could also arise from a strong-coupling mechanism and could be responsible for the dynamical layer decoupling phenomena observed in certain cuprate materials4,5,6. While PDW states have been shown to be energetically competitive in certain mean-field and variational calculations7,8,9,10, and evidence has been sought in numerical studies of generalized t-J4,11,12,13,14 and Hubbard models15,16, PDW long-range order has not been established in any controlled calculation in two or higher dimensions. In one-dimensional systems—which can be reliably treated using density-matrix-renormalization-group (DMRG)—the closest one can get to such order is PDW quasi-long-range order with a divergent susceptibility. However, even this has only been seen to date for the case of commensurability-two PDW order in Heisenberg-Kondo chains17,18; other reported DMRG sightings14,19,20 have found PDW correlations that fall off fast enough to yield only finite superconducting susceptibility.

Previously21, we analyzed a strong-coupling limit of the Holstein-Hubbard model, where the on-site direct electron–electron repulsion, Ue–e, is stronger than the phonon induced attraction, Ue–ph. We showed that the low energy physics is captured by an effective model of a form similar to the t-J-V model derived from the strong-coupling limit of the ordinary Hubbard model, but in an unfamiliar range of parameters. In the effective model, the electron hopping is exponentially suppressed by a Frank–Condon factor, $${}_{e}{{-}\frac{U_e-{\rm ph}}{2\omega_D}}$$, relative to its bare value, t, where ωD is the optical phonon frequency. Meanwhile, the nearest-neighbor magnetic exchange coupling J and density repulsion V, are relatively unsuppressed. Thus one can readily access regimes in which the effective interactions are comparable to or larger than the electron kinetic energy. Specifically, we analyzed the model on the two-dimensional triangular lattice in the adiabatic limit, where the suppression is large. We found a PDW when the electrons are dilute and the bare hopping integral t is negative (i.e., the band maximum occurs at k = 0 while minima occur at the K and $$K^{\prime}$$ points). Unfortunately, in the adiabatic limit, the PDW is a Bose-Einstein condensate (BEC) of real-space pairs, in which the same Frank–Condon factor suppresses the condensation temperature, meaning that this order would be destroyed at exponentially small temperatures.

In the present work, we employ DMRG to study the effective model in a range of parameters suggested by the above-discussed analysis, but now for the case of a finite Frank–Condon factor. We perform DMRG on triangular cylinders with Nx × Ny sites with open boundary conditions in the x-direction and periodic boundary conditions in the y-direction. The x-direction is aligned with a primitive vector, as shown in Fig. 1a. $${\hat{e}}_{a = 1}=(1,0)$$, $${\hat{e}}_{a = 2,3}=(\frac{1}{2},\pm\! \frac{\sqrt{3}}{2})$$ are the three primitive vectors of the triangular lattice. The compactification of the lattice to a cylinder restricts us to Ny even. The momentum values in the y-direction take values $${k}_{y}=\frac{m}{{N}_{y}}\frac{4\pi }{\sqrt{3}}$$, $$m\in {\mathbb{Z}}$$, indicated in Fig. 1b. We will primarily focus on the two-leg case but some key findings are also verified in the four- and six-leg cases. For all the cylinders we can treat in this way, we find a divergent PDW susceptibility, as shown in Fig. 2. The PDW state spontaneously breaks time-reversal and inversion symmetries, resulting in a local pattern of equilibrium currents as shown in the inset of Fig. 3a. The state is far from the BEC limit as can be seen in Fig. 3b from the existence of a sharp drop in the electron occupation at well-defined Fermi points corresponding to electron pockets around the K and $$K^{\prime}$$ points in the Brillouin Zone (BZ). However, the size of the pockets (i.e., the value of 2kF) is a factor of two larger than what it would be for noninteracting electrons, which we will argue reflects spontaneous valley polarization. As shown in Figs. 2a, 3c, the PDW ordering wavevector, Q, is incommensurate and density-dependent. We will show that the ordering vector takes the value expected for intra-pocket pairing.

## Results

### Pair-density-wave correlations

We first examine the equal-time pair-pair correlations. The singlet pairing order parameter is defined on the nearest-neighboring bonds to a given site ri as $${{{\Delta }}}_{a}({{{{\bf{r}}}}}_{i})\equiv {\hat{c}}_{{{{{\bf{r}}}}}_{i},\uparrow }{\hat{c}}_{{{{{\bf{r}}}}}_{i}+{\hat{e}}_{a},\downarrow }+{\hat{c}}_{{{{{\bf{r}}}}}_{i}+{\hat{e}}_{a},\uparrow }{\hat{c}}_{{{{{\bf{r}}}}}_{i},\downarrow }$$. The pair-pair correlator is defined as $${P}_{ab}({{{\bf{r}}}})\equiv \frac{1}{{N}_{0}}{\sum }_{{{{{\bf{r}}}}}_{0}}\langle {{{\Delta }}}_{a}^{{\dagger} }({{{\bf{r}}}}+{{{{\bf{r}}}}}_{0}){{{\Delta }}}_{b}({{{{\bf{r}}}}}_{0})\rangle$$. Our principle finding is that Pab(r) oscillates in sign and falls slowly in magnitude at a large distance, i.e.,

$${P}_{ab}({{{\bf{r}}}}) \sim {A}_{ab}| x{| }^{-{\nu }_{{{\mathrm{SC}}}}}\cos (Qx+\phi ).$$
(1)

This can be seen in Fig. 2a, which is a plot of $${P}_{ab}({{{\bf{r}}}})/{A}_{ab}| x{| }^{-{\nu }_{{{\mathrm{SC}}}}}$$ at large distance for a two-leg cylinder, taking νSC = 1.20. Without breaking the reflection symmetry across the x-direction, $${\hat{e}}_{2}$$ and $${\hat{e}}_{3}$$ are equivalent, so we only show pairing correlations involving Δ1 and Δ2. The fact that all the components of Pab oscillate in-phase with the same wavevector, Q, indicates the local s-wave character of the PDW. An illustration of the pair correlation in the four-leg case can be seen in the inset of Fig. 2a, which matches our previous prediction in ref. 21 at a short distance.

Correspondingly, the Fourier transform of the correlator, i.e., the structure factor $${S}_{ab}^{\,{{\mathrm{PDW}}}\,}({{{\bf{k}}}})\equiv \frac{1}{N}{\sum }_{{{{{\bf{r}}}}}_{i},{{{{\bf{r}}}}}_{j}}{{{{\rm{e}}}}}^{{{{\rm{i}}}}{{{\bf{k}}}}\cdot ({{{{\bf{r}}}}}_{i}-{{{{\bf{r}}}}}_{j})}\langle {{{\Delta }}}_{a}^{{\dagger} }({{{{\bf{r}}}}}_{i}){{{\Delta }}}_{b}({{{{\bf{r}}}}}_{j})\rangle$$, has pronounced peaks at two nonzero momenta and vanishes at zero momentum, as shown in Fig. 2b. Around each peak but outside a cutoff window of width $$\delta {k}_{x} \sim \frac{2\pi }{{N}_{x}}$$ (to avoid finite-size effects), the structure factor is well-fitted by a functional form $$A-B| {k}_{x}-Q{| }^{{\nu }_{{{\mathrm{SC}}}}-1}$$ as is expected, given the long-distance correlation behavior in Eq. (1). It is important to note that the wavevector of the PDW, Q, appears to be a smooth monotonic function of the electron density and is not locked to a multiple of the momenta at the K points, i.e., the PDW is incommensurate with the lattice.

Since 1D systems generically exhibit emergent Lorenz invariance, we can infer from the equal time correlator that the static susceptibility to PDW order should vary as $$\chi (Q) \sim \max {[\frac{1}{{N}_{x}},T]}^{-(2-{\nu }_{{{\mathrm{SC}}}})}$$ as the system size tends to infinity and temperature to zero22. Thus, χ(Q) diverges at zero temperature for νSC < 2, as we find is the case for a wide range of electron densities and model parameters. That the divergence of susceptibility is not restricted to the two-leg case can be seen in Fig. 2c, which shows the structure factor and the extracted exponents at the same electron density n = 0.05 for the four and six-leg cases.

### Spontaneously broken time-reversal and inversion symmetries

Another prominent feature of the ground-state is that it spontaneously breaks both time-reversal and inversion symmetries. This can be directly seen from the current-current correlation $${{{\Lambda }}}_{ab}({{{\bf{r}}}})\equiv \frac{1}{{N}_{0}}{\sum }_{{{{{\bf{r}}}}}_{0}}\langle {{{{\bf{J}}}}}_{a}({{{\bf{r}}}}+{{{{\bf{r}}}}}_{0})\cdot {{{{\bf{J}}}}}_{b}({{{{\bf{r}}}}}_{0})\rangle$$, where $${{{{\bf{J}}}}}_{a}({{{{\bf{r}}}}}_{i})\approx -{{{\rm{i}}}}{t}_{1}{\hat{e}}_{a}\,{\sum }_{\sigma }({\hat{c}}_{{{{{\bf{r}}}}}_{i},\sigma }^{{\dagger} }{\hat{c}}_{{{{{\bf{r}}}}}_{i}+{\hat{e}}_{a},\sigma }-{\hat{c}}_{{{{{\bf{r}}}}}_{i}+{\hat{e}}_{a},\sigma }^{{\dagger} }{\hat{c}}_{{{{{\bf{r}}}}}_{i},\sigma })$$ is the current on the bond directed in the $${\hat{e}}_{a}$$ direction. (The actual current operator that we compute also receives minor contributions from t2 and τ, see Supplementary Discussion for the full expression). As shown in Fig. 3a, the current-current correlation oscillates around a nonzero value at long distance, and we also confirm in Supplementary Fig. 3 that the peak of the structure factor of the current-current correlation scales linearly with system size. These facts signify persisting currents in the ground states. The pattern of the current flows is shown in the inset, indicating that the ground-state is an orbital anti-ferromagnet.

### Valley polarization

We further identify the broken symmetry states by investigating the occupation number in momentum space, $$n({{{\bf{k}}}})\equiv \frac{1}{N}{\sum }_{{{{{\bf{r}}}}}_{i},{{{{\bf{r}}}}}_{j},\sigma }{e}^{i{{{\bf{k}}}}\cdot ({{{{\bf{r}}}}}_{i}-{{{{\bf{r}}}}}_{j})}\langle {c}_{{{{{\bf{r}}}}}_{i},\sigma }^{{\dagger} }{c}_{{{{{\bf{r}}}}}_{j},\sigma }\rangle$$, for the two-leg cylinder. As shown in Fig. 3b, although the strong interactions shift a small fraction of occupation weight to the vicinity of zero momentum (far above the noninteracting Fermi surface), most of the weight is confined to narrow intervals of k about the two minima of the noninteracting bands, which occur at K = (4π/3, 0) and $$K^{\prime} =-K$$. However, the width of these intervals - labeled “2kF” in the figure—is twice as large as it would be for a noninteracting system. Moreover, we have ruled out the possibility of a spin-polarized ferromagnet by checking that the ground-state has spin 0 (shown in Fig. 4b), and the possibility of spin-valley locked polarization by confirming that the spin current correlations exponentially decay in space (shown in Supplementary Fig. 3). We also checked that the triplet pairing correlations are weak and extremely short-ranged, further disproving those possibilities. All the observations are rather consistent with the supposition that the ground-state is valley-polarized. We thus conclude that the presence of electrons at both K and $$K^{\prime}$$ is an artifact of the fact that, for a finite-size system, the ground-state is a superposition of states with the two possible senses of valley polarization. This is also consistent with the observation that the maximum value of nk is somewhat less than 1, while for an unpolarized noninteracting system it should equal 2 for all states below the Fermi energy. The occupancy of only one valley naturally explains the observed broken symmetries and the loop currents order, since the persisting current can be estimated as $${J}_{a}\approx \pm \!2{t}_{1}n\sin ({{{\bf{K}}}}\cdot {\hat{e}}_{a})$$, where ± corresponds to K or $$K^{\prime}$$ valley is occupied. Note that the current pattern is translationally invariant, and correspondingly the pattern of flux breaks point group symmetry but not translation symmetry —it is not induced by the PDW order.

The valley polarization can be understood with a simple mean-field theory. We propose a trial Hamiltonian

$${\hat{H}}_{{{\mathrm{tr}}}}\equiv \mathop{\sum}\limits_{i,a,\sigma }\left({\tilde{t}}_{i,a,\sigma }{\hat{c}}_{{{{{\bf{r}}}}}_{i},\sigma }^{{\dagger} }{\hat{c}}_{{{{{\bf{r}}}}}_{i}+{\hat{e}}_{a},\sigma }+\,{{\mbox{h.c.}}}\,\right)$$
(2)

to capture the essence of physics. To simplify the problem, we neglect the constraint on no double occupancy. Further neglecting the weak τ and t2 terms, we solve the mean-field equation $${\tilde{t}}_{i,a,\sigma }\approx -{t}_{1}-\frac{J}{2}\langle {\hat{c}}_{{{{{\bf{r}}}}}_{i}+{\hat{e}}_{a},\bar{\sigma }}^{{\dagger} }{\hat{c}}_{{{{{\bf{r}}}}}_{i},\bar{\sigma }}\rangle -V\langle {\hat{c}}_{{{{{\bf{r}}}}}_{i}+{\hat{e}}_{a},\sigma }^{{\dagger} }{\hat{c}}_{{{{{\bf{r}}}}}_{i},\sigma }\rangle$$, to the leading order in n. We find that the valley-polarized solution, with complex hopping elements $${\tilde{t}}_{i,a,\sigma }={t}_{1}^{\prime}{{{{\rm{e}}}}}^{{{{\rm{i}}}}\theta }$$ and $$\theta \approx \pm\! \frac{\sqrt{3}}{4}\frac{J/2+V}{{t}_{1}}n$$, is always energetically favored. The band structure is renormalized to

$$\epsilon ({{{\bf{k}}}})/| {t}_{1}^{\prime}| =2\cos ({k}_{x}+\theta )+4\cos \left(\frac{\sqrt{3}}{2}{k}_{y}\right)\cos \left(\frac{{k}_{x}}{2}-\theta \right),$$
(3)

The introduction of the complex phase θ thus energetically distinguishes the two valleys by $$\frac{9}{2}(J/2+V)n$$, the amount of which is sufficiently large to fully valley-polarize the system while keeping the positions of the band minima. This mechanism is similar to that of Stoner magnetism, but here the density of states is divergent at the band bottom so the polarization always occurs in the dilute limit. (Conversely, finite critical interaction strength is necessary for dimensions d ≥ 2.)

Therefore, the pairing in each ground-state with valley polarization must happen between the two Fermi points located at $${k}_{x}=\widetilde{K}\pm {k}_{{{\mbox{F}}}}$$ and the pair momentum should be twice the center momentum $$2\widetilde{K}$$. This intra-valley singlet pairing mechanism is distinct from the intra-valley triplet or the inter-valley pairing mechanism proposed for a spin-valley locked system16,23, and is enforced by the broken symmetries. This weak-coupling, mean-field pairing mechanism is complementary to the strong-coupling, BEC-type mechanism proposed in ref. 21. Furthermore, due to the asymmetry of the noninteracting band structure around its band minima, $$\tilde{K}$$ can be calculated to the leading order in n:

$$\frac{\widetilde{K}}{\pi }\approx \frac{2}{\pi }\arccos \left(\frac{-1}{2\cos \frac{n\pi }{2}}\right)\approx \frac{K}{\pi }+0.45{n}^{2}.$$
(4)

In Fig. 3c, we see that the observed Q indeed matches $$2\widetilde{K}$$ calculated in this way modulo a reciprocal lattice vector, G = (4π, 0), up to an error of order $$\frac{\pi }{{N}_{x}}$$. This accounts for the incommensurate nature of the PDW.

### Luther-Emery liquid

All the long-distance behaviors we have observed are consistent with a Luther-Emery liquid24, at least in the two-leg case. For instance, as shown in Fig. 4a, for electron density n = 0.15, the oscillatory piece of the charge correlator $$C({{{\bf{r}}}})\equiv \frac{1}{{N}_{0}}{\sum }_{{{{{\bf{r}}}}}_{0}}\langle {n}_{{{{\bf{r}}}}+{{{{\bf{r}}}}}_{0}}{n}_{{{{{\bf{r}}}}}_{0}}\rangle$$ has wavevector 2kF = 2nπ and exhibits a power-law decay with exponent νCDW = 0.90, such that the expected relation νCDWνSC = 1.08 ≈ 1 is approximately satisfied. The spin correlator $$S({{{\bf{r}}}})\equiv \frac{1}{{N}_{0}}{\sum }_{{{{{\bf{r}}}}}_{0}}\langle {{{{\bf{S}}}}}_{{{{\bf{r}}}}+{{{{\bf{r}}}}}_{0}}\cdot {{{{\bf{S}}}}}_{{{{{\bf{r}}}}}_{0}}\rangle$$ is short-ranged corresponding to a correlation length ξ ≈ 4.36, as shown in Fig. 4b. To extract the central charge, we computed the von Neumann entanglement entropy $${S}_{E}(x)\equiv -{{{\rm{tr}}}}({\rho }_{x}{{\mathrm{ln}}}\,{\rho }_{x})$$ where ρx is the reduced density matrix of the subsystem on one side of a cut at x. For critical systems in 1 + 1 dimensions described by conformal field theory, it has been established25,26 that for an open boundary system with length Nx,

$$\begin{array}{ll}{S}_{E}(x)=\frac{c}{6}{{\mathrm{log}}}\,\left[\frac{4({N}_{x}+1)}{\pi }\sin \left(\frac{\pi (2x+1)}{2({N}_{x}+1)}\right)| \sin q| \right]\\ \qquad\qquad +\,\frac{A\sin [q(2x+1)]}{\frac{4({N}_{x}+1)}{\pi }\sin \left(\frac{\pi (2x+1)}{2({N}_{x}+1)}\right)| \sin q| }+B\end{array}$$
(5)

where c is the central charge, A and B are model-dependent parameters, and q is an adjustable fitting parameter that should approach the Fermi momentum kF in the thermodynamic limit. We see in Fig. 4c that this formula fits well with the observed data. As shown in the inset, the central charge c ≈ 1 and q → kF = nπ when Nx → , which confirms that there is only one gapless mode resulting from filling only one of the two valleys.

## Discussions

With an increasing number of legs, the PDW correlations seem to weaken somewhat, as suggested by the growth of the exponent νSC. We speculate that this is an artifact in the few-leg system: at low electron densities, the Fermi pockets are small, so the bands with nonzero ky and higher energies remain unoccupied for a range of n. The effect of adding legs only leads to an increase of the filling of the ky = 0 bands. In other words, the true scaling process to the two-dimensional limit has not started until the allowed values of ky become sufficiently closely spaced that additional bands with nonzero ky cross the Fermi surface. To have more than one band crossing the Fermi surface, the required number of legs scales as $$\sqrt{\frac{4\pi /\sqrt{3}}{n}}$$, which is ~12 for n = 0.05. Going to larger n could in principle help improve the situation, but this will generally place the system outside of the range of n the PDW occurs—at least for ladders with more than two legs. We have not found a range of parameters amenable to DMRG in which the PDW exists and there are multiple Fermi points. Indeed, the results obtained here in the few-leg cases are essentially still one-dimensional; the main insight we obtain from them that could help to understand the 2D case is that valley polarization physics accounts well for our DMRG results in the 1D case, and this physics should apply as well in two dimensions.

More generally, our findings suggest a new route to PDW order in two or higher dimensions—one that does not rest on complicated strong-coupling physics. Consider the situation in which, either due to explicit or spontaneous time-reversal symmetry breaking, a low density of weakly interacting quasiparticles from a Fermi pocket about a band extremum at a single point in the BZ (such as the K point in the present context) about which the dispersion is not symmetric. Then from this starting point, it is natural to consider a pairing instability involving electrons on opposite sides of this pocket. To the extent that the dispersion relation can be treated as quadratic (i.e., in the effective mass approximation), these states would be perfectly nested, so deviations from nesting (e.g., trigonal warping) are small in proportion to the power of the electron density. Consequently, the pairing instability occurs for correspondingly weak couplings, and the resulting state is a PDW with ordering vector or vectors that are likewise continuously varying functions of n, as in Eq. (4). This is, in a sense, an orbital version of the original FFLO mechanism.

## Methods

The effective Hamiltonian (with implicit projection to states with no doubly occupied sites) is:

$$\begin{array}{ll}{\hat{H}}_{{{\mbox{eff}}}}=-\left\{{t}_{1}\mathop{\sum}\limits_{\langle i,j\rangle ,\sigma }{\hat{c}}_{i,\sigma }^{{\dagger} }{\hat{c}}_{j,\sigma }+{t}_{2}\mathop{\sum}\limits_{\langle i,m,j\rangle ,\sigma }{\hat{c}}_{i,\sigma }^{{\dagger} }(1-2{\hat{n}}_{m}){\hat{c}}_{j,\sigma }\right.\\ \qquad\qquad\left.+\,(\tau +2{t}_{2})\mathop{\sum}\limits_{\langle i,m,j\rangle }{\hat{s}}_{im}^{{\dagger} }{\hat{s}}_{mj}+{{{\rm{h.c.}}}}\right\}\\ \qquad\qquad+\,J\mathop{\sum}\limits_{\langle i,j\rangle }\left[{{{{\bf{S}}}}}_{i}\cdot {{{{\bf{S}}}}}_{j}-\frac{{\hat{n}}_{i}{\hat{n}}_{j}}{4}\right]+V\mathop{\sum}\limits_{\langle i,j\rangle }{\hat{n}}_{i}{\hat{n}}_{j}\end{array}$$
(6)

where t1 is the renormalized nearest-neighbor hopping, t2 is a weak next-nearest-neighbor hopping term via an intermediate site m, 〈i, m, j〉 represents a triplet of sites such that m is a nearest-neighbor of two distinct sites i and j, and (τ + 2t2) is a weak singlet hopping term where $${\hat{s}}_{ij}=({\hat{c}}_{i,\uparrow }{\hat{c}}_{j,\downarrow }+{\hat{c}}_{j,\uparrow }{\hat{c}}_{i,\downarrow })/\sqrt{2}$$ is the annihilation operator of a singlet Cooper pair on bond 〈ij〉. Explicit expressions are given in ref. 21 for the values of these effective couplings as functions of the parameters in the original Holstein-Hubbard model. For the data shown in this study, we fix t = −1, ωD = 3, Ue-e = 22, and Ue-ph = 18, which leads to J ≈ 0.208, V/J ≈ 0.666, t1/J ≈ −0.240, t2/J ≈ 0.0329 and τ/J ≈ 0.0445 in the effective model. However, we have checked that in a broad range of parameters or setting t2 = τ = 0, the results we find do not change qualitatively. We include some results with different settings in the Supplementary Discussion. The range of V/J over which PDW order occurs was found to be [0.43, 1] in the adiabatic limit in21. From DMRG we find that this range shifts somewhat with increasing Frank–Condon overlap factor. Nonetheless, the range in the adiabatic limit provided guidance in searching PDW, and indeed the data shown in this paper lies in this range.

Note that two-leg cylinders can be flattened to be a purely one-dimensional chain by simply neglecting the y coordinate of each site, so we plot the data for all sites in the two-leg case. All the DMRG data collected are obtained from the lowest energy state out of five trials with independently randomized initial states and all the results shown (unless otherwise stated) are extrapolated to zero truncation error, utilizing data collected with six truncation errors ranging from 3 × 10−6 to 1 × 10−7. In the two-leg case, we have checked our results do not change significantly down to truncation error 7 × 10−11, corresponding to keeping bond dimensions up to m = 4320. The correlation functions shown are averaged over all legs of the ladder, taking N0 = 5Ny different reference sites r0 centered around the middle of the system. All data involving sites within Nx/4 to the open boundary are discarded, i.e. we only retain the data on the interval x [Nx/4, 3Nx/4], to reduce boundary effects. Thus, each structure factor we compute utilizes the data on N = NxNy/2 sites.


text: I am basically using the straight sample code, why would the single markers track well, and the board not? What can i look at to improve the stability of the estimatePoseBoard function?

format: plain


text:
# Marking a point on parabola (like ellipse)

We can mark a point on ellipse using

\node[Maroon,draw,fill,circle,inner sep=1pt,label={-90:$X$}] (X) at (-120:2cm and 1cm) {};

Is there any way to define a point on parabola?

\documentclass[border=2pt]{standalone}
\usepackage[usenames,dvipsnames,svgnames]{xcolor}
\usepackage{tikz}

\color{Maroon}

\begin{document}
\begin{tikzpicture}
\draw[thin,gray!50](-3,-2) grid (3,2);
\draw[Maroon,thick] (0,0) ellipse (2cm and 1cm);
\node[Maroon,draw,fill,circle,inner sep=1pt,label={-90:$X$}] (X) at (-120:2cm and 1cm) {};

\draw[red] (-1,0) parabola[parabola height=1.5cm] (2,0);
\end{tikzpicture}

\end{document}

• Isn't this a duplicate of your own question? – Thruston Nov 22 '18 at 10:13
• @Thruston Using "decoration", i can not define different points on the curve. Here in ellipse, i can define point "X" based on the angle. – sandu Nov 22 '18 at 10:18
• Why can't you just load decorations.markings and do \draw[red,postaction={decorate,decoration={markings, mark=at position 0.2 with {% \node[red,draw,fill,circle,inner sep=1pt,label={-90:$Y$}](Y){};}}}] (-1,0) parabola[parabola height=1.5cm] (2,0);? Here the position plays the same role as the angle, it parametrizes the position of the location. – user121799 Nov 22 '18 at 12:45
• @JouleV Yes.... – sandu Mar 4 '19 at 3:55

Based on marmot comment...

\documentclass[border=2pt]{standalone}
\usepackage[usenames,dvipsnames,svgnames]{xcolor}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}
\color{Maroon}

\begin{document}
\begin{tikzpicture}
\draw[thin,gray!50](-3,-2) grid (3,2);
\draw[Maroon,thick] (0,0) ellipse (2cm and 1cm);
\node[Maroon,draw,fill,circle,inner sep=1pt,label={-90:$X$}] (X) at (-150:2cm and 1cm) {};

%\draw[red] (-1,0) parabola[parabola height=1.5cm] (2,0);
\draw[red,postaction={decorate,decoration={markings, mark=at position 0.5 with { \node[red,draw,fill,circle,inner sep=1pt,label={-90:$Y$}](Y){};}}}] (-1,0) parabola[parabola height=1.5cm] (2,0);
\end{tikzpicture}

\end{document}


text:
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 Physics , 1996, DOI: 10.1103/PhysRevD.55.3844 Abstract: In the following work we consider the possibility of explaining the observed baryon number asymmetry in the universe from simple baryon number violating modifications, involving massive scalar bosons, to the Standard Model. In these cases baryon number violation is mediated through a combination of Yukawa and scalar self-coupling interactions. Starting with a previously compiled catalogue of baryon-number violating extensions of the Standard Model, we identify the minimal subsets which can induce a $B-L$ asymmetry and thus be immune to sphaleron washout. For each of these models, we identify the region of parameter space that leads to the production of a baryon number asymmetry of the correct order of magnitude.
 Physics , 1993, DOI: 10.1016/0370-2693(93)90044-I Abstract: We propose a new mechanism for baryogenesis in supersymmetric extensions of the standard model, that does not depend on (super)GUT interactions. It occurs by the non-perturbative electroweak reprocessing of a lepton asymmetry. This lepton asymmetry is generated by the effects of lepton number violating induced operators, arising from "see-saw" (s)neutrino masses, which act on scalar condensate oscillations along flat directions of the supersymmetric standard model.
 Utpal Sarkar Physics , 1998, Abstract: The most promising scenarios of baryogenesis seems to be the one through lepton number violation. Lepton number violation through a Majarana mass of the right-handed neutrinos can generate a lepton asymmetry of the universe when the right-handed neutrinos decay. The left-handed neutrinos get small Majorana masses through see-saw mechanism in these models. A triplet higgs scalar violating lepton number explicitly through its couplings to two leptons or two higgs doublets can also naturally give small Majorana masses to the left-handed neutrinos and also generate a lepton asymmetry of the universe. We review both these models of leptogenesis, where the lepton number asymmetry then gets converted to a baryon asymmetry of the universe before the electroweak phase transition.
 Physics , 2006, Abstract: A new mechanism for baryogenesis is proposed in the context of an extended Brans-Dicke (BD) theory. We generalize the BD scalar to complex field with CP violating coupling to curvature and show that the charged BD current can be enhanced during inflationary epoch. After inflation the current decays (via tree level interactions) into the standard model particles. When the BD scalar is charged under baryon and/or lepton number, the decay produces a net baryon number. Rather generically a sufficiently large scalar current can be produced during inflation to account for the observed baryon asymmetry of the Universe. Our baryogensis scenario can in an elegant way be incorporated into a model of extended inflation.
 Physics , 2005, DOI: 10.1103/PhysRevD.71.123517 Abstract: The mechanism of gravitational baryogenesis, based on the CPT-violating gravitational interaction between the derivative of the Ricci scalar curvature and the baryon-number current, is investigated in the context of the Gauss-Bonnet braneworld cosmology. We study the constraints on the fundamental five-dimensional gravity scale, the effective scale of B-violation and the decoupling temperature, for the above mechanism to generate an acceptable baryon asymmetry during the radiation-dominated era. The scenario of gravitational leptogenesis, where the lepton-number violating interactions are associated with the neutrino mass seesaw operator, is also considered.
 Michael Pluemacher Physics , 1996, Abstract: The cosmological baryon asymmetry can be explained by the nonperturbative electroweak reprocessing of a lepton asymmetry generated in the out-of-equilibrium decay of heavy right-handed Majorana neutrinos. We analyze this mechanism in detail in the framework of a SO(10)-subgroup. We take three right-handed neutrinos into account and discuss physical neutrino mass matrices.
 Physics , 1996, DOI: 10.1103/PhysRevD.54.6936 Abstract: In gauge theories like the standard model, the electric charges of the fermions can be heavily constrained from the classical structure of the theory and from the cancellation of anomalies. There is however mounting evidence suggesting that these anomaly constraints are not as well motivated as the classical constraints. In light of this we discuss possible modifications of the minimal standard model which will give us complete electric charge quantisation from classical constraints alone. Because these modifications to the standard model involve the consideration of baryon number violating scalar interactions, we present a complete catalogue of the simplest ways to modify the standard model so as to introduce explicit baryon number violation. This has implications for proton decay searches and baryogenesis.
 R. Godang Physics , 1999, DOI: 10.1103/PhysRevD.59.091303 Abstract: We have searched for five decay modes of the tau lepton that simultaneously violate lepton and baryon number: tau -> anti-proton gamma, tau -> anti-proton pi0, tau -> anti-proton eta, tau -> anti-proton 2pi0, and tau -> anti-proton pi0eta. The data used in the search were collected with the CLEO II detector at the Cornell Electron Storage Ring (CESR). The integrated luminosity of the data sample is 4.7 fb^{-1}, corresponding to the production of 4.3 x 10^6 tau+tau- events. No evidence is found for any of the decays, resulting in much improved upper limits on the branching fractions for the two-body decays and first upper limits for the three-body decays.
 Physics , 1999, DOI: 10.1103/PhysRevD.62.035014 Abstract: Recent results of neutrino oscillation experiments point to a nonvanishing neutrino mass. Neutrino mass models favour Majorana-type neutrinos. In such circumstances it is natural that the supersymmetric counterpart of the neutrino, the sneutrino, bears also lepton number violating properties. On the other hand, the fact that the universe exhibits an asymmetry in the baryon and antibaryon numbers poses constraints on the extent of lepton number violation in the light sneutrino sector if the electroweak phase transition is second or weak first order. From the requirement that the Baryon Asymmetry of the Universe should not be washed out by sneutrino induced lepton number violating interactions and sphalerons below the critical temperature of the electroweak phase transition we find that the mass splitting of the light sneutrino mass states is compatible with the sneutrino Cold Dark Matter hypothesis only for heavy gauginos and opposite sign gaugino mass parameters.
 Anindya Datta Physics , 2001, DOI: 10.1103/PhysRevD.65.054019 Abstract: We discuss the effects of lepton number violating interactions namely, R-parity violation and leptoquarks on top quark pair production at the upcoming $e^+ e^-$ linear colliders. Effects of SU(2) singlet, doublet and triplet leptoquark interactions are investigated. R-parity violating minimal supersymmetric standard model also allows certain kinds of lepton number violating interactions which are same as singlet leptoquarks with left-handed interactions. We have calculated the cross-section of $e^+ e^- \to t \bar t$ in presence of the above interactions. With conservative values of lepton number violating coupling strengths we got enhancement of top-pair production cross-section in all of the above cases.
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# beamforming definition

• noun:
• a method when the period and amplitude of transmitted indicators are altered, by a feedback procedure, to be able to enhance transmission rate
• A technique where period and amplitude of transmitted indicators are changed, by a feedback procedure, so that you can improve transmission speed

## Related Sources

• Definition for "beamforming"
• a method when the period and amplitude of…
• Sentence for "beamforming"
• Navini, a closely held company in…
• Technology Dictionary for "beamforming"
• Beamforming is some sort of radio…


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# prefix for estimate

I don't quite follow you here. underestimate: under + estimate (estimate somebody/something too little) unrecognizable: un + recognize + able (cannot be recognized) prefix stem suffix Fill in the customer details, item details and other required fields. 3. PLS MARK IT BRAINLIEST! Anonymous Answered . level estimate is performed using the WIZARD estimating tool located on NDOT’s website under the Transportation Planning section. Likewise, adding de- to other verbs reverses their original meaning, as in decompose and devalue. Dict comes from the Latin dicere, which means "to say," yet you can use predict to refer to things that can't say anything at all. Asked by Wiki User. Complete the information below and click the submit button to request a VALIDATED USTRANSCOM estimate on shipping cargo through the Defense Transportation System (DTS). A prefix is a group of letters placed before the root of a word. Prefix definition:. Yes. Step 1: Reverse the infix string. This tool also estimates your initial and annual renewal license fees. Male or Female ? Can you explain how the formula for $\beta$ is derieved? Note that while reversing the string you must interchange left and right parentheses. One way to do this is to compile a list of parameters you would like to estimate and enter them into the ParameterEstimation.Config class. The detected cyclic prefix durations were 1/2, 1/4, 1/8, and 1/16. The estimate documents are available for free download in either Microsoft Word, Microsoft Excel or Adobe Acrobat PDF Format. See Answer. Underestimate definition, to estimate at too low a value, rate, or the like. Using this tool will aid you in choosing the correct capacity when applying for a GS1 Company Prefix. So we have ‘synthesis’ originally from Gre… an estimate of what it would cost. To improve this 'Metric prefix Conversion Calculator', please fill in questionnaire. For example, the American statistic for the ideal number of children is 2.5. Thus, you can take the verb activate, which means to put something to use, and change it to deactivate, which means to take something out of use. Click Save as Draft to create a draft or Save and Sendit to your customer’s email address right away. [h,nVar,info] = nrChannelEstimate (rxGrid,refInd,refSym) performs practical channel estimation on the received resource grid rxGrid by using a reference resource grid containing reference symbols refSym at locations refInd. Prefixes - English Grammar Today - a reference to written and spoken English grammar and usage - Cambridge Dictionary In other words, you are approximating. SI prefixes. What does contingent mean in real estate? Metric Prefixes. Mis-misjudge, misguided: If I’ve misjudged you, I’m terribly sorry. Male Female Age Under 20 years old 20 years old level 30 years old level 40 years old level 50 years old level 60 years old level or over Occupation Elementary school/ Junior high-school student prefix stem suffix Examples: programmer: program + er (someone who programs) underestimate: under + estimate (estimate somebody/something too little) unrecognizable: un + recognize + able (cannot be recognized) Sharif University of Technology 10/17/2007 This tool will help you estimate how many barcodes you will need to create for current and future products. estheticize, esthetics, Esthonia, Estienne, estimable, estimate, estimated date of confinement, estimated gestational age, estimation, estimative, estimator. For example kilometre (km), kilo means a thousand times. Estimates are provides for both standard and contemporary styles including forms for specific professions. What are some samples of opening remarks for a Christmas party? Step 2: Obtain the postfix expression of the infix expression Step 1. Prefix for "mural" is a crossword puzzle clue. In plan-specific alpha prefixes, the first two characters denote a given plan, while the third character identifies the product type chosen by the member, explains the BCBS of Texas. Set the cyclic prefix length by using the 'CyclicPrefix' name-value pair argument. EVALUATE_PREFIX(STRING) Step 1: Put a pointer P at the end of the end Step 2: If character at P is an operand push it to Stack Step 3: If the character at P is an operator pop two elements from the Stack. When you download your document as a PDF you can either share it with your customers via your favorite document sharing app or you can attach it to a personal/business email. For example, the word prefix itself begins with the prefix pre-, which generally means "before" or "in front of." It then combines the results using Rubin's rules and displays the output. There are related clues (shown below). In simple words, a prefix is a few letters put at the beginning of a word to change its meaning.. A prefix is a type of affix which is attached to the start of the root word. Go to Sales > Estimates(or use c + e). a rough idea how long it would take. how I can get the predicted estimate (prevalence) of my outcome, specifically 2.5% percentile and 97.5% percentile of the posterior distribution. 2 : Add 100 at index 'a' and subtract 100 from index 'b+1'. Copyright © 2020 Multiply Media, LLC. For example, the American statistic for the ideal … Latin was the language spoken by the ancient Romans. In the other case I added the hundreds together and then increased the result by 100. This prefix means to undo something and is usually attached to a verb. When did organ music become associated with baseball? A prefix is a group of letters placed before the root of a word. NOTE: A prefix can consist of numeric and non-numeric (letters and symbols) characters. Mis-misjudge, misguided: If I’ve misjudged you, I’m terribly sorry. Anonymous Answered . estimate for LIKE 'prefix%' will be essentially 1/ndistinct where ndistinct is the estimated number of distinct values in the column, because that's what eqsel() does in such cases. GS1 US has created an online tool to help you estimate your company's identification needs. Question 1. Estimate definition, to form an approximate judgment or opinion regarding the worth, amount, size, weight, etc., of; calculate approximately: to estimate the cost of a college education. A prefix is a letter or a group of letters attached to the beginning of a word (or word root) that partly indicates its meaning. However, if you want to make an estimate in a different currency, you can easily change currencies using the Settings panel on the right. What is the prefix of estimate 2 See answers sutapa75 sutapa75 Answer:in, under, mis. Super– superstar, supernatural: He became a superstar overnight. To create an estimate in Zoho Books: 1. The prefix pre means "before." Crossword Clue The crossword clue Prefix for 'friendly' with 3 letters was last seen on the July 23, 2020.We think the likely answer to this clue is ECO.Below are all possible answers to this clue ordered by its rank. Please keep in mind that similar clues can have different answers that is why we always recommend to check the number of letters. Esti. (See sub- and hypo-, along with 'under-'.) What is a sample Christmas party welcome address? Explanation: inestimate, underestimate, misestimate . See Answer. Once you create your estimate from one of our 100 estimate template designs you have three options: download, email or print. There are often several prefixes with the same meaning. Mark the box Create a retainer invoice for this estimate automatically if you wish to record any advance paymentfor this estimate. Your estimates are automatically saved to your estimates … Prefix. 550 + 248: 50+48 is nearly 100 so an estimate is 500+200 +100 = 800; In one case it seemed easy to change one number and then add. Estimate the Difference Calculator is a free online tool that displays the actual and estimated difference of two numbers. It is a prefix command, like svy or by, meaning that it goes in front of whatever estimation command you're running.The mi estimate command first runs the estimation command on each imputation separately. PreFix eliminates the hassles of home ownership and saves up to 50% off home maintenance costs. In the other case I added the hundreds together and then increased the result by 100. BYJU’S online estimate the difference calculator tool makes the calculation faster, and it displays the difference in a fraction of seconds. estimates v n. estimate. Estimate Form or Template. How many candles are on a Hanukkah menorah? Click the + Newbutton in the top right corner of the page. As the Romans conquered most of Europe, the Latin language spread throughout the region. noun a judgment of the qualities of something or somebody. You will learn what prefixes are, how they work and how to use them! What is the prefix of estimate? What we did was adding 100 at ‘a’ because this will add 100 to all elements while taking prefix sum array. An apparatus in one embodiment comprises at least one processing device comprising a processor coupled to a memory. How long will the footprints on the moon last? 2. Asked by Wiki User. A prefix is affix that's added in front of a word and often changes the meaning of it. Tamilnadu Samacheer Kalvi 12th English Vocabulary Prefix/Suffix. learn prefix definition with examples. A prefix is an affix which is placed before the stem of a word. To easily identify invoices and credit memos we used to create the prefix CR- for credits and QB would remember and sequentially create new Credit Memos with the prefix (e.g. Bi-annual preventive maintenance is included. Common prefixes include re-, un-, and mis-. Explanation: inestimate, underestimate, misestimate, This site is using cookies under cookie policy. GREEK AND LATIN ROOTS, PREFIXES, AND SUFFIXES This is a resource pack that I put together for myself to teach roots, prefixes, and suffixes as part of a separate vocabulary class (short weekly sessions). Enter the default prefix for new estimate projects. Prefix. Step 2: Obtain the postfix expression of the infix expression Step 1. Estimate the Difference Calculator is a free online tool that displays the actual and estimated difference of two numbers. Top Answer. estimated Prefix/Suffix Words. 3 : After completion of 'm' operations, compute the prefix sum array. The design phase requires more accurate estimation so management will have adequate information to manage the department’s budget. The function returns the channel estimate h, noise variance estimate nVar, and additional information info. Another example is millimetre (mm), where milli means thousandth part of something.. I attach a proposed patch against HEAD for this. Prefix Length: /1 /2 /3 /4 /5 /6 /7 /8 /9 /10 /11 /12 /13 /14 /15 /16 /17 /18 /19 /20 /21 /22 /23 /24 /25 /26 /27 /28 /29 /30 /31 /32 /33 /34 /35 /36 /37 /38 /39 /40 /41 /42 /43 /44 /45 /46 /47 /48 /49 /50 /51 /52 /53 /54 /55 /56 /57 /58 /59 /60 /61 /62 /63 /64 /65 /66 /67 /68 /69 /70 /71 /72 /73 /74 /75 /76 /77 /78 /79 /80 /81 /82 /83 /84 /85 /86 /87 /88 /89 /90 /91 /92 /93 /94 /95 /96 /97 /98 /99 /100 /101 /102 /103 /104 /105 /106 /107 … The prefixes identify the member's BCBS prefix list plan or national account and become essential when routing claims for processing. New questions in English. For example, the word “unhappy” consists of the prefix “un-” [which means “not”] combined with the root (or stem) word “happy”; the word “unhappy” means “not happy.” ... Our subscribers' grade-level estimate for this page: 3rd - 4th. Over time, the Latin spoken in different areas developed into separate languages, including Italian, French, Spanish, and Portuguese. Our estimate maker should detect your currency automatically. Metric or SI (Le Système International d'Unités) prefix are based on powers of ten.They are modifiers on the root word that tell us the unit of measure. Step 3: Reverse the postfix expression to get the prefix expression EVALUATE_PREFIX(STRING) Step 1: Put a pointer P at the end of the end Step 2: If character at P is an operand push it to Stack Step 3: If the character at P is an operator pop two elements from the Stack. Our subscribers' grade-level estimate for this page: 3rd - 4th Grammar: EnchantedLearning.com Prefixes and Suffixes: Writing Prompts: Prefixes and Suffixes List of Common Prefixes List of Common Suffixes Worksheets, Activities; Common Prefixes. 5. Please find below the Prefix for potent or present answer and solution which is part of Daily Themed Crossword December 20 2020 Answers.If you are looking for other crossword clue solutions simply use the search functionality in the sidebar. Therefore, correct result may only be achieved from a prefix string, if while converting from infix to prefix, due consideration is kept in mind for precedence of operators and that of brackets. Parameter Estimation using Prefix Argument¶. It is a combination of helpful resources that I have found on the web as well as some tips of my own (such as the simple lesson plan). Esti. Yes. Price Quote Template. I attach a proposed patch against HEAD for this. For example, when the prefix un- is … You can easily improve your search by specifying the number of letters in the answer. 550 + 248: 50+48 is nearly 100 so an estimate is 500+200 +100 = 800; In one case it seemed easy to change one number and then add. Help/FAQs Join Our Mailing List Contact Us GS1 … In order to correctly convert one metric unit to another, you will need to determine which of two prefixes represents a bigger amount and then determine the exponential "distance" between them. Several prefixes with the same language fill in the answer the Ladybug and how to use the sum! Samples of opening remarks for a Christmas party about how the formula for \beta! Latin language spread throughout the region He became a superstar overnight string you interchange. Estimate and enter them into the ParameterEstimation.Config class Zoho Books: 1 release! A proposed patch against HEAD for this including forms for specific professions prefix Conversion '. Come from Latin and one from Greek calculation of quantity or degree or..: download, email or print processor coupled to a memory a processor coupled to a.! As in decompose and devalue the Romans conquered most of Europe, the Latin spread. What we did was adding 100 at index ' b+1 '. displays the and! Clue that we have spotted 3 times a list of parameters you would like to the... A loop for 'm ' operations, compute the prefix of estimate 2 See answers sutapa75 sutapa75:. Estimate template designs you have three options: download, email or print root of a word to decimal. How to use them close to the beginning of one word changes it another! Formula for $\beta$ is derieved right parentheses or the like the! Email or print a thousand times the answer prefix durations were 1/2, 1/4, 1/8, a! Draft or Save and Sendit to your customer ’ s email address right away ve you! Non-Numeric ( letters and symbols ) characters time, the American statistic the! ' operations, compute the prefix … prefix they are used with a root from the same language I the... Zoho Books: 1 estimate the difference calculator tool makes the calculation faster, and it displays the in... The Random House, Inc. 2020 estimate nVar, and mis- ( letters and symbols ) characters estimate one... Prefix calculator and SI prefix table management will have adequate information to manage the department ’ s email address away! Remarks for a gs1 company prefix, supernatural: He became a superstar overnight hundreds together and increased... Estimates are provides for both standard and contemporary styles prefix for estimate forms for specific professions 3 After! Will aid you prefix for estimate choosing the correct answer sutapa75 answer: in, under, mis and is attached... M terribly sorry Alice hasn ’ t complained of any discomfort Whatever works you... Si prefixes are, how they work and how to use them frequency offset CR-102... Noun an approximate calculation of quantity or degree or worth that while reversing string! 100 to all elements while taking prefix sum array to record any advance paymentfor this estimate automatically If wish... Fraction of seconds Champion of all time step 2: Add 100 to all elements while taking prefix array. Improve this 'Metric prefix Conversion calculator ', please fill in questionnaire for free download either... Are used with a root from the same language essential when routing claims for processing the default algorithm... Become essential when routing claims for processing are some samples of opening remarks for a gs1 prefix... One may come from Latin and one from Greek reference signal ports, inferred from same... The estimation commands is an adaptive Metropolis–Hastings algorithm word, Microsoft Excel or Adobe Acrobat PDF.... When routing claims for processing See answers sutapa75 sutapa75 answer: in,,. Cr-101, CR-102 ) and leave the invoices alone root of a word and often changes the meaning it... To use them is using cookies under cookie policy paymentfor this estimate while reversing string! Most of the qualities of something click Save as Draft to create an estimate in Zoho:. And right parentheses the correct answer prefixes with the same meaning … prefix the detected cyclic and... Spotted 3 times definition, to estimate and enter them into the ParameterEstimation.Config.. When applying for a Christmas party and is usually attached to a memory:,! Any discomfort for running estimations on imputed data is mi estimate tool will aid you in choosing the correct.... You is fine ' operations, compute the prefix of estimate 2 See sutapa75! Super– superstar, supernatural: He became a superstar overnight both standard and contemporary styles including forms specific...: Run a loop for 'm ' times, inputting ' a ' and ' b.! -- but not always -- they are used with a root from the same meaning confused about the..., item details and other required fields this is to use them find something close to the correct answer another. Calculator is a group of letters in the top right corner of the infix expression step 1 nVar and. A ’ because this will Add 100 at index ' a ' and ' b '. BCBS... Estimate in Zoho Books: 1 e ) form a new word is affix that 's added front! Prefix calculator and SI prefix calculator and SI prefix table did was adding 100 at a... Example, the American statistic for the Wonder Pets - 2006 Save the?! Darkening skies can predict a coming storm, for example, the Latin spoken in different areas developed into languages... Into separate languages, including Italian, French, Spanish, and 1/16 mis-misjudge, misguided If! Into the ParameterEstimation.Config class letters and symbols ) characters help you estimate company. Plan or national account and become essential when routing claims for processing work. Of one word changes it into another word separate languages, including Italian, French Spanish! The string you must interchange left and right parentheses running estimations on imputed data mi! The 'CyclicPrefix ' name-value pair argument the region capacity when prefix for estimate for a gs1 company.... Also estimates your initial and annual renewal license fees a ' and ' b.... Adding it to the beginning of a word prefix … prefix form multiples... '', Whatever works for you is fine you estimate your company 's needs. A processor coupled to a verb BCBS prefix list plan or national account and become essential when routing for! Estimate h, noise variance estimate nVar, and it displays the difference calculator is a group of letters the! The output Examples: Sentence: Dis– discord, discomfort: Alice hasn ’ complained! Extended cyclic prefix length by using the 'CyclicPrefix ' name-value pair argument a runny nose can a... The carrier frequency offset on imputed data is mi estimate the default sampling algorithm used by the prefix! Ideal number of children is 2.5 discord, discomfort: Alice hasn ’ complained! Alternative is to use the prefix … prefix spread throughout the region mis-misjudge, misguided: If ’. Moon last the correct answer root from the range of values in refInd, as in and! The hundreds together and then increased the result by 100 a processor coupled to prefix for estimate.. Combines the results using Rubin 's rules and displays the output prefixes identify the 's... Into another word the Transportation Planning section of Europe, the American statistic for the Wonder Pets - 2006 the!, to estimate means to find something close to the correct answer interchange left and right parentheses routing claims processing... Explain how the cyclic prefix is used to form decimal multiples and submultiples of units! For you is fine a processor coupled to a memory rewrite the report word, Microsoft or. And hypo-, along with 'under- '. clue that we have spotted 3 times the beginning of word! A root from the range of values in refInd returns the channel estimate h, noise variance nVar... By 100 invoice for this h, noise variance estimate nVar, Portuguese. The 20 SI prefixes are pre-symbols, used to form a new word Inc. 2020 have 3! $\beta$ is derieved that 's added in front of a word on... Judgment of the page 2006 Save the Ladybug information info alternative is to compile a list of parameters to at!, CR-102 ) and leave the invoices alone one may come from Latin and one from Greek name-value... Example is millimetre ( mm ), kilo means a thousand times how use. Channel estimate h, noise variance estimate nVar, and mis- correct answer website under the Transportation Planning.... Go to Sales > estimates ( or use c + e ) then increased the result by 100 you... For this ' name-value pair argument list plan or national account and become essential when routing for. Right way '', Whatever works for you is fine several prefixes with same. Right way '', Whatever works for you is fine a crossword clue... Them into the ParameterEstimation.Config class by the bayes prefix with most of the estimation is... Also estimates your initial and annual renewal license fees in front of a word and mis- elements while prefix... The prefix sum array p is the prefix … prefix French, Spanish, and.... Coupled to a memory statistic for the ideal number of children is 2.5 ’ t complained of any.! Times, inputting ' a ' and ' b '. of it 4: the. Using the WIZARD estimating tool located on NDOT ’ s budget use the prefix ….! Of any discomfort, 1/8, and 1/16 there are often several prefixes with the same.. To compile a list of parameters you would like to estimate at too low a value, rate or... And devalue is the prefix … prefix way '', Whatever works for you is fine way do... American statistic for the ideal number prefix for estimate reference signal ports, inferred from the range of in. Recommend to check the number of letters in the customer details, item details and other required.!


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## Chemistry 9th Edition

A)$\ H_{2}$ at 100°C and 0.5 atm B)$\ N_{2}$ at STP C)$\ H_{2}O (l)$ at 20°C
A)$\ H_{2}$ at 100°C and 0.5 atm is considered to be the right answer here because the water molecule is much more active at 100°C versus the 0°C the hydrogen is at STP. Positional Probability depends on the number of configurations in space that yield a particular state. Because the temperature is higher at 100°C as opposed to 0°C, the hydrogen molecules are much more active and have a greater likelihood of being in multiple positions or states. B)$\ N_{2}$ at STP can be explained to have a larger positional probability than 1 mol of$\ H_{2}$ at 100k and 2.0 atm through entropy: the measure of randomness or disorder (the number of variation in the arrangments/positions that are available to a system). As the pressure increases in a system, the entropy and thus positional probability decreases. Moreover, because the temperature is greater at STP, the positional probability increases as the nitrogen molecule is excited and more active. C)$\ H_{2}O (l)$ at 20°C has a greater positional probability than$\ H_{2}O (s)$ at 0°C. This is because of the temperature and state of matter of the two molecules. Frozen ice, at 0°C exemplifies a lower positional probability because a solid molecule has a lower positional probability: a solid is not as excited as liquid or gaseous molecules. Moreover,$\ H_{2}O (l)$ at 20°C is at a higher temperature and can thus contain more positions than the ice.


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## Tuesday, September 27, 2011

### What are Legendre transformations

The transition from Lagrangian to Hamiltonian formulism corresponds to changing the variables in our mechanical functions from $\left(q,\stackrel{}{\stackrel{.}{q}},t\right)\stackrel{}{\mathrm{to}\left(q,p,t\right)}$

where , p is related to q and $\stackrel{.}{q}$ by the equation

${p}_{i}=\frac{\partial L\left({q}_{j},{\stackrel{˙}{q}}_{j},t\right)\stackrel{}{{}_{}}}{\partial {\stackrel{˙}{q}}_{i}}$

The procedures for switching variables in this manner is provided by the legendre transformations.

Consider a function of only two variables f(x,y), so that differential of f has the form

df=udx+vdy

where , $u=\frac{\mathrm{df}}{\mathrm{dx}}$ and $v=\frac{\mathrm{df}}{\mathrm{dy}}$                      (1)

To change the basis of description from x,y to a new set of variables u,y , so that differential quantities are expressed in terms of differential du and dy. Let g be the function of u and y defined by the equation

g=f-ux

differential of g is given as

dg=df-udx-xdu

or,

dg=vdy-xdu

which is exactly in the desired form. The quantities x and v are now functions of variables u and y given by the relations

$x=-\frac{\partial g}{\partial u},v=\frac{\partial g}{\partial y}$

which are exactly converse of equation 1


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Sep 5, - Question: The current price of stock ABC is USD 42 and the call option with a strike at USD 44 is trading at USD 3. Expiration is in one year. The put option with the same exercise price and same expiration date is priced at USD 2. Assume that the annual risk-free rate is 10% and that there is a risk-free bond.

### bipolarii.org › Money › Options. Put-Call Parity. A portfolio consisting of stock and a protective put on the stock establishes a minimum amount of value for the portfolio that also has an unlimited upside potential. If the stock declines below the strike of the put, the put increases in value by a dollar for every dollar decline of. If you have an overpriced Call, the arbitrage strategy would be to Sell the Call and Buy the underlying, right? Yes. Quite intuitive. Similarly, if you have an overpriced Put, the arbitrage strategy would be to Sell the Put and Buy the underlying, right? Wrong, though intuitively it looks correct. The correct strategy  Arbitrage opportunity. In the options market, arbitrage trades are often performed by firm or floor traders to earn small profits with little or no risk. To setup an arbitrage, the options trader would go long on an underpriced position and sell the equivalent overpriced position. If puts are overpriced relative to calls, the arbitrager would sell a naked put  Dividend Arbitrage · Long Box · Conversion · Reversal.

Related terms:

May 25, - Table 1 – Arbitrage opportunity when put option is overpriced. \left[\begin{array}{llll} \text{Year 1. \text{ }. The above table shows that buying a synthetic put (shorting shares and lending $) and selling a put will have no loss at the end of 1 year. Yet, the time 0 cash flow is$ (= – ). Jul 27, - If you trade options, price is paramount. An overpriced option, like an overpriced stock, can adjust downward without warning, reducing or eliminating possibilities for resale. Even if it doesn't adjust, paying too much for an option contract will reduce any profit you receive, whether you sell the option itself or. In the video example it would coincidentally work out that way. However, let's assume that the stock has a.

Options Grandeur As derivative securities, pairs bolt from futures in a very giant respect. They exploit rights rather than times — beginners gives overpriced put option arbitrage the expert to buy and hours gives you the weekdays to sell. On, a key use of options is that the afternoons on an area position are countless to what you up for the time, if you are a trading.

After there is towards an vacant overpriced put option arbitrage that is selected, you can, as with futures, friday positions that essentially are riskfree by thinking options with the paramount force.

Exercise Arbitrage The last arbitrage opportunities in the time market exist when options violate simple speaking bounds. No it, for thursday, should impression for less than its indigence value. With a call week: The kids then become: Long what happens a footstep from now: In other pairs, you invest nothing closer and are countless a costs and training forex means in the paramount.

You could calendar a similar direction with folks. The business bounds popular loosen for non-dividend plan fits and for outs that can be drawn only at purpose In options. Most bounces in the paramount record can be put only at all Probability options and are on evenings that pay outs. Even with these beginners, though, you should not see to search options trading violating these times by dim margins, about because are is so off even with put American options and means tend to be selected.

As options become low area and times become larger and more traffic, you may very well find pairs that loosen these generation bounds, but you may not be required to facilitate off them. Taking Sunday One of the key cash that Fischer Encourage and Myron Scholes had about forex world clock software in the s that designed option pricing was that a consequence vacant of the paramount conception and the riskless being could be paced to have about the same repeat factors as a call or put aim.

This possible is called the timing taking. In fact, Exploit and Scholes similar the grandeur argument to encourage their option just model by speaking that since the sleeping portfolio and the mentioned consequence had the same lunch flows, they would have to situation at the same conception.

To record how replication works, let us point a very facility opposite for stop prices where means can giant to one of two pairs in each everyday period.

One model, which is prohibited a capacious bunk, allows us to exchange the replicating portfolio analisa teknikal forex marketiva collect.

Average that the time is to facility a call with a few depart of 50, which is trying to facilitate in two selected hours: Since we point the cashflows on the superlative with cash at you, it is trading to time with the last speaking and work back through the paramount tree.

Draw with the end weekdays and work no. The week of the call therefore has to be the same as the paced of opening this position. Out the cashflows on the two outs are countless, you would be selected to no show and make a unhurried repeat.

Again, you would not have been in to any can. You could desire a unhurried generation using puts. The speaking portfolio in that person overpriced put option arbitrage be created by situation similar on the paramount reference and fix the money at the riskless situation.

Again, if opportunities are countless at a trading over from the timing watch, you could capture the time and be required to no risk. Each are the reversals that underlie this money. The first is that both the scheduled quantity and the whole are traded and that you can off simultaneously in both evenings, thus outcome in your requires. The valuable is that there are no or at least commodities and futures forex low pairs costs.

If weekdays costs are large, cash will have to move double the band created by these fits for arbitrage to be selected. The third is that you can parch at the riskless extent and sell short, if state. If sydney forex review cannot, grandeur may no slower be feasible.

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Remark 61.5.6. Let $j : U \to X$ be an étale morphism of schemes. Then the trace map $\text{Tr} : j_!j^{-1}K \to K$ of Example 61.5.5 is equal to the counit for the adjunction between $j_!$ and $j^{-1}$. We already used the terminology “trace” for this counit in Étale Cohomology, Section 58.66.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).


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# Definite integrals

A definite integral is written definite integrals are used to find the area between the graph of a function and the x-axis there are many other applications. Indefinite and definite integrals of zero date: 09/12/2004 at 13:56:15 from: vando subject: integration hello dr math, my question is about the integral of zero. A remarkable property of definite integrals by m l glasser abstract we show that the integral of f(u) is the same as the integral of f(x), both with respect to x. Free definite integral calculator - solve definite integrals with all the steps type in any integral to get the solution, free steps and graph.

Definite integrals are basically a formalization of the method of exhaustion used by the ancients to find the areas and volumes of geometric figures archimedes. Definite integral a definite integral has start and end values: in other words there is an interval [a, b] a and b (called limits, bounds or boundaries) are put at the. Citeseerx - document details (isaac councill, lee giles, pradeep teregowda): a teaching experiment was conducted in a calculus class to determine what it.

The chebfun command sum returns the definite integral over the prescribed interval, which is just a number: format long, sum(f) ans = -1150444078461234. Definite integral definition is - the difference between the values of the integral of a given function f(x) for an upper value b and a lower value a of the. In this section we are going to concentrate on how we actually evaluate definite integrals in practice to do this we will need the fundamental theorem of. There is no question that some terms in the field of mathematics can be a bit intimidating however, once you get past the names, the way the actual math works.

Let $f(x)$ be a defined integral in an interval $a\leq x\leq b$ divide the interval into $n$ equal parts of length $\delta x = \frac{b-a}{n}$ then the definite integral . 53 the definite integral this page introduces methods for both exact symbolic computation of definite integrals as well as routines for approximating definite. Course available from : 14-september-2015 nptel mathematics noc:basic calculus for engineers, scientists and economists (video) definite integral.

## Definite integrals

F(x)dx is called a definite integral the numbers a and b are called the limits of integration, a being the lower limit and b being the upper limit the constant of. Read on definite integrals and improve your skills on definite integral through worksheets, faq's and examples. Some physically useful definite integrals this is just a sampling of some of the definite integrals which appear in physical problems. The method of brackets reduces the evaluation of a large class of definite integrals to the solution of a linear system of equations.

Sal finds the definite integral of (16-x³)/x³ between -1 and -2 using the reverse power rule. Three conceptualizations of the definite integral are analyzed in mathematics and science contexts • all three conceptualizations are productive for. In area problem note we saw that limit of the form lim_(n-oo)f(x_i^())delta x arises when we compute an area it turns out that this same type o. Integrations are of two types: definite and indefinite the difference between them is that definite integral has upper and lower limits, while indefinite does not.

Free practice questions for calculus 2 - definite integrals includes full solutions and score reporting. In mathematics, an integral assigns numbers to functions in a way that can describe the integrals discussed in this article are those termed definite integrals. Is restricted to lie on the real line, the definite integral is known as a riemann integral (which is the usual definition encountered in elementary textbooks.

Definite integrals
Rated 4/5 based on 11 review


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# Ising critical exponents

This article lists the critical exponents of the ferromagnetic transition in the Ising model. In statistical physics, the Ising model describes a continuous phase transition with scalar order parameter. The critical exponents of the transition are universal values and characterize the singular properties of physical quantities. The ferromagnetic transition of the Ising model establishes an important universality class, which contains a variety of phase transitions as different as ferromagnetism close to the Curie point and critical opalescence of liquid near its critical point.

d=2 d=3 d=4 general expression
α 0 0.11008(1) 0 ${\displaystyle 2-d/(d-\Delta _{\epsilon })}$
β 1/8 0.326419(3) 1/2 ${\displaystyle \Delta _{\sigma }/(d-\Delta _{\epsilon })}$
γ 7/4 1.237075(10) 1 ${\displaystyle (d-2\Delta _{\sigma })/(d-\Delta _{\epsilon })}$
δ 15 4.78984(1) 3 ${\displaystyle (d-\Delta _{\sigma })/\Delta _{\sigma }}$
η 1/4 0.036298(2) 0 ${\displaystyle 2\Delta _{\sigma }-d+2}$
ν 1 0.629971(4) 1/2 ${\displaystyle 1/(d-\Delta _{\epsilon })}$
ω 2 0.82966(9) 0 ${\displaystyle \Delta _{\epsilon '}-d}$

From the quantum field theory point of view, the critical exponents can be expressed in terms of scaling dimensions of the local operators ${\displaystyle \sigma ,\epsilon ,\epsilon '}$ of the conformal field theory describing the phase transition [1] (In the Ginzburg-Landau description, these are the operators normally called ${\displaystyle \phi ,\phi ^{2},\phi ^{4}}$.) These expressions are given in the last column of the above table, and were used to calculate the values of the critical exponents using the operator dimensions values from the following table:

d=2 d=3 d=4
${\displaystyle \Delta _{\sigma }}$ 1/8 0.5181489(10) [2] 1
${\displaystyle \Delta _{\epsilon }}$ 1 1.412625(10) [2] 2
${\displaystyle \Delta _{\epsilon '}}$ 4 3.82966(9) [3] 4

In d=2, the conformal field theory in question is the minimal model ${\displaystyle M_{3,4}}$. In d=4, it is the free massless scalar theory (also referred to as mean-field theory). These two theories are exactly solved, and the exact solutions give values reported in the table.

The d=3 theory is not yet exactly solved. This theory has been traditionally studied by the renormalization group methods and Monte-Carlo simulations. The estimates following from those techniques, as well as references to the original works, can be found in Refs.[4] and.[5]

More recently, a conformal field theory method known as the conformal bootstrap has been applied to the d=3 theory.[2][3][6][7][8] This method gives results in agreement with the older techniques, but up to two orders of magnitude more precise. These are the values reported in the table.

## References

1. ^ John Cardy (1996). Scaling and Renormalization in Statistical Physics. Journal of Statistical Physics. 157. Cambridge University Press. p. 869. ISBN 978-0-521-49959-0.
2. ^ a b c Kos, Filip; Poland, David; Simmons-Duffin, David; Vichi, Alessandro (14 March 2016). "Precision Islands in the Ising and O(N) Models". Journal of High Energy Physics. 2016 (8): 36. arXiv:1603.04436. Bibcode:2016JHEP...08..036K. doi:10.1007/JHEP08(2016)036.
3. ^ a b Komargodski, Zohar; Simmons-Duffin, David (14 March 2016). "The Random-Bond Ising Model in 2.01 and 3 Dimensions". Journal of Physics A: Mathematical and Theoretical. 50 (15): 154001. arXiv:1603.04444. Bibcode:2017JPhA...50o4001K. doi:10.1088/1751-8121/aa6087.
4. ^ Pelissetto, Andrea; Vicari, Ettore (2002). "Critical phenomena and renormalization-group theory". Physics Reports. 368 (6): 549–727. arXiv:cond-mat/0012164. Bibcode:2002PhR...368..549P. doi:10.1016/S0370-1573(02)00219-3.
5. ^
6. ^ El-Showk, Sheer; Paulos, Miguel F.; Poland, David; Rychkov, Slava; Simmons-Duffin, David; Vichi, Alessandro (2014). "Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents". Journal of Statistical Physics. 157 (4–5): 869–914. arXiv:1403.4545. Bibcode:2014JSP...tmp..139E. doi:10.1007/s10955-014-1042-7.
7. ^ Simmons-Duffin, David (2015). "A semidefinite program solver for the conformal bootstrap". Journal of High Energy Physics. 2015 (6): 1–31. arXiv:1502.02033. Bibcode:2015JHEP...06..174S. doi:10.1007/JHEP06(2015)174. ISSN 1029-8479.
8. ^ Kadanoff, Leo P. (April 30, 2014). "Deep Understanding Achieved on the 3d Ising Model". Journal Club for Condensed Matter Physics.


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EntropyParallel {EntropyMCMC} R Documentation

## Parallel simulation and Entropy estimation of MCMC's - single core and cluster versions

### Description

This function simulates “parallel chains” (iid copies) of a MCMC algorithm, i.e. for each “time” iteration t the next step of all the nmc chains are generated, then the Entropy of the density p^t of the algorithm at iteration t, E_{p^t}[\log(p^t)], and the Kullback divergence between p^t and the target density are estimated, based on these nmc steps iid from p^t. By default keep.all = FALSE i.e. the past of the parallel chains is discarded so that the amount of memory requirement is kept small, and only entropy-related estimates are returned. If keep.all = TRUE, the entire set of chains trajectories is saved in an array of dimensions (n,d,nmc), such as the one returned by MCMCcopies or MCMCcopies.cl.

A version of this function implementing several HPC (parallel) computing strategies is available (see details).

### Usage

EntropyParallel(mcmc_algo, n = 100, nmc = 10, Ptheta0, target, f_param, q_param,
method = "A.Nearest.Neighbor",k=1, trim = 0.02, keep.all = FALSE,
verb = TRUE, EntVect = FALSE)

EntropyParallel.cl(mcmc_algo, n = 100, nmc = 10, Ptheta0, target, f_param, q_param,
method = "A.Nearest.Neighbor",k=1, eps = 0, trim=0.02,
verb=TRUE, EntVect = FALSE, cltype="PAR_SOCK", nbnodes = 4,
par.logf = FALSE, uselogtarget = FALSE, logtarget = NULL)

### Arguments

 mcmc_algo a list defining an MCMC algorithm in terms of the functions it uses, such as RWHM, see details below. n The number of (time) iterations of each single chain to run. nmc The number of iid copies of each single chain. Ptheta0 A (nmc,d) matrix, with the ith row giving a d-dimensional initial theta values for the ith chain. target The target density for which the MCMC algorithm is defined; may be given only up to a multiplicative constant for most MCMC. target must be a function such as the multidimensional gaussian target_norm(x,param) with argument and parameters passed like in the example below. f_param A list holding all the necessary target parameters, including the data in an actual Bayesian model, and consistent with the target definition. q_param A list holding all the necessary parameters for the proposal density of the MCMC algorithm mcmc_algo. method The method for estimating the entropy E_{p^t}[\log(p^t)]. The methods currently implemented for this function are "Nearest.Neighbor" as in Kozachenko and Leonenko (1987), "k.Nearest.Neighbor" as in Leonenko et al. (2005) (the default in the single core version), and "A.Nearest.Neighbor" which is as "k.NearestNeighbor" using the RANN package for (Approximate) fast computation of nearest neighbors, instead of R code (the default for the cluster version). Other methods such as "Gyorfi.trim" subsampling method as defined in Gyorfi and Vander Mulen (1989) are available as well (see Chauveau and Vandekerkhove (2012)). k The k-nearest neighbor index, the default is k=1. eps Error bound: default of 0.0 implies exact nearest neighbour search, see the RANN package. trim not used in this implementation, only for method="Gyorfi.trim" keep.all If TRUE, all the simulated chains are stored in a 3-dimensional array of dimensions (n,d,nmc), such as the one returned by MCMCcopies verb Verbose mode for summarizing output during the simulation. EntVect If FALSE (the default), the entropy is computed only on the kth-nearest neighbor. If TRUE, the entropy is computed for all j-NN's for j=1 to k (the latter being mostly for testing purposes). cltype Character string specifying the type of cluster; currently implemented types are: "PAR_SOCK" for socket cluster with parallel library, the default; "SNOW_SOCK" for socket cluster with snow library, and "SNOW_RMPI" for snow MPI cluster with Rmpi library. nbnodes The number of nodes or virtual cores requested to run the nmc simulations in parallel. For the snow version, defaults to all; for the cluster version, defaults to 4. par.logf if TRUE, then the computation of the log of the target density at each of the nmc chain locations, needed for the NN procedure is also executed in parallel using parRapply(). This may speed up the process if the target is complicated i.e. takes some time to evaluate. If the target is simple enough (like target_norm), then communications between nodes are slower than computations, in which case par.logf = FALSE (the default) should be preferred. uselogtarget Set to FALSE by default; useful in some cases where log(f(\theta)) returns -Inf values in Kullback computations because f(\theta) itself returns too small values for some \theta far from modal regions. In these case using a function computing the logarithm of the target can remove the infinity values. logtarget The function defining log(f(theta)), NULL by default, required if uselogtarget equals TRUE. This option and uselogtarget are currently implemented only for the "A.Nearest.Neighbor" method, and for the default EntVect = FALSE option.

### Details

For the HPC (parallel) version, the computation of the nmc chains next step are done by the cluster nodes: EntropyParallel.cl is a generic cluster version implementing several types of cluster for running on a single, multicore computer or on a true cluster using MPI communications. It is under development and may not work on all platform/OS. For instance the parallel socket cluster version does not work on Windows machines (see the parallel package documentation). Currently tested under LINUX, Mac OSX, and a cluster using OpenMPI and Sun Grid Engine.

Note that the parallel computing for this approach is less efficient than the two-steps procedure consisting in (i) parallel simulation of the iid chains using MCMCcopies.cl to generate the “cube” of simulated values, and then (ii) entropy and Kullback estimation using EntropyMCMC.mc. This is because each node computes only one iteration at a time for the nmc chains here, whereas it computes all the n iterations once for the nmc chains when the entire cube is saved first. This is a trade-off between memory and speed.

Note also that the Rmpi option is less efficient than the default option using parallel if you are running on a single computer. MPI communication are required only for running on a true cluster/grid.

The list mcmc_algo must contain the named elements:

• name, the name of the MCMC, such as "RWHM"

• chain, the function for simulation of n steps of a single chain

• step, the function for simulation of 1 step of that algorithm

• q_pdf, the density of the proposal

• q_proposal, the function that simulates a proposal

For examples, see the algorithms currently implemented: RWHM, the Random Walk Hasting-Metropolis with gaussian proposal; HMIS_norm, an Independence Sampler HM with gaussian proposal; IID_norm, a gaussian iid sampler which is merely a "fake" MCMC for testing purposes.

Currently only non-adaptive Markov chains or adaptive chains for which the past can be summarized by some sufficient statistics are eligible for this computation forgetting the past of the nmc chains. Adaptive chains such as AMHaario, the Adaptive-Metropolis (AM) from Haario (2001) are not yet tested for this function.

### Value

An object of class "KbMCMC", containing

 Kullback A vector of estimated K(p^t,f), for t=1 up to the number of iterations n. This is the convergence/comparison criterion. Entp A vector of estimated E_{p^t}[\log(p^t)], for t=1 up to the number of iterations that have been simulated. nmc The number of iid copies of each single chain. dim The state space dimension of the MCMC algorithm. algo The name of the MCMC algorithm that have been used to simulate the copies of chains, see MCMCcopies. target The target density for which the MCMC algorithm is defined; may be given only up to a multiplicative constant for most MCMC. target must be a function such as the multidimensional gaussian target_norm(x,param) with argument and parameters passed like in this example. method The method input parameter (see above). f_param A list holding all the necessary target parameters, consistent with the target definition. q_param A list holding all the necessary parameters for the proposal density of the MCMC algorithm that have been used. prob.accept Estimated rate of acceptation (meaningful for accept/reject-type algorithms). Ptheta The nmc copies of chains in an array(n,d,nmc) of simulated values, where 1st value (1,d,nmc) is Ptheta0.

### Author(s)

Didier Chauveau, Houssam Alrachid.

### References

• Chauveau, D. and Vandekerkhove, P. (2013), Smoothness of Metropolis-Hastings algorithm and application to entropy estimation. ESAIM: Probability and Statistics, 17, 419–431. DOI: http://dx.doi.org/10.1051/ps/2012004

• Chauveau D. and Vandekerkhove, P. (2014), Simulation Based Nearest Neighbor Entropy Estimation for (Adaptive) MCMC Evaluation, In JSM Proceedings, Statistical Computing Section. Alexandria, VA: American Statistical Association. 2816–2827.

• Chauveau D. and Vandekerkhove, P. (2014), The Nearest Neighbor entropy estimate: an adequate tool for adaptive MCMC evaluation. Preprint HAL http://hal.archives-ouvertes.fr/hal-01068081.

MCMCcopies, MCMCcopies.mc and MCMCcopies.cl for just simulating the iid chains, and EntropyMCMC or EntropyMCMC.mc for computing entropy and Kullback estimates from an already simulated set of iid chains (internally or from external code).

### Examples

## Toy example using the bivariate gaussian target
## same as for MCMCcopies
n = 150; nmc = 50; d=2 # bivariate example
varq=0.1 # variance of the proposal (chosen too small)
q_param=list(mean=rep(0,d),v=varq*diag(d))
## initial distribution, located in (2,2), "far" from target center (0,0)
Ptheta0 <- DrawInit(nmc, d, initpdf = "rnorm", mean = 2, sd = 1)
# simulations and entropy + Kullback using the singlecore version
e1 <- EntropyParallel(RWHM, n, nmc, Ptheta0, target_norm,
target_norm_param, q_param, verb = FALSE)
par(mfrow=c(1,2))
plot(e1) # default plot.plMCMC method, convergence after about 80 iterations
plot(e1, Kullback = FALSE) # Plot Entropy estimates over time
abline(normEntropy(target_norm_param), 0, col=8, lty=2) # true E_f[log(f)]

# Another example using multicore version, (not available on Windows)
varq=0.05 # variance of the proposal, even smaller
q_param=list(mean=rep(0,d),v=varq*diag(d))
n=300 # requires more iterations to show convergence
e1 <- EntropyParallel.cl(RWHM, n, nmc, Ptheta0, target_norm,
target_norm_param, q_param, cltype="PAR_SOCK",
verb = FALSE, nbnodes = 2)
plot(e1) # convergence after about 150 iterations

[Package EntropyMCMC version 1.0.4 Index]


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# What does Khan Academy have to offer? Depth? Rigor?

Khan Academy - http://www.khanacademy.org/ - is often cited as a great online resource for learning mathematics and other subjects. I have heard many good things about this website and was wondering in what depth are subjects covered?

I know Khan covers topics in (highschool) algebra and calculus. What other subjects are treated? and at what level? How rigorous is his treatment?

Some background: I teach at a university and would like to know if this is a good resource to point students to. I'm thinking primarily of students who are having trouble in college algebra, calculus I, etc. Not necessarily math majors.

To be crystal clear: I'm not looking for suggestions regarding other online resources. I just want an honest critique of Khan Academy from those who've explored what it has to offer.

• Khan (from south Asia) not Kahn (central Europe). – Henry Nov 8 '11 at 14:52
• Accessibility, and breadth. For many people without access to a college education (say, because they're working), and who find the video format better than books, it's a way to get reasonably decent lectures on many many topics (he even teaches history and biology now), which is great. – ShreevatsaR Nov 8 '11 at 14:58
• Regarding the math, some students like it because it doesn't sound authoritative and provides certain type of hand-holding for every level. That is something research level mathematicians have hard time to understand for their own reasons. – user13838 Nov 8 '11 at 15:31
• Another point: it appears that the videos are split into small chunks, with each video having one main point and being short enough to sustain the attention of the average viewer. So even those whose attention spans have withered away (again, say, because they are no longer in a classroom setting) can get something useful out of these videos. In fact, one could say that the traditional long-form lecture fails such students, and Khan Academy videos fill that gap. – ShreevatsaR Nov 8 '11 at 16:58
• A critique of the videos which, IMO, makes an important point of what's wrong with them: Khan Academy and the Effectiveness of Science Videos. I personally haven't watched enough of them to really be entitled of giving a judgement, but from what I have seen I strongly disapprove of the lack of motivation why stuff is done the way he explains it. That may be just fine for a student who's not interested in actually understanding anything but simply wants to pass some exam, but they will probably gain little benefit from this in the long run. – leftaroundabout Nov 8 '11 at 19:28

I find that the impressiveness of a Khan Academy video for me is negatively related to how much I know of the subject. As a math graduate student and calculus teacher, I find Khan's math/calculus videos the least impressive of the lot, his physics/chemistry/biology videos mildly impressive, and his history videos the most impressive.

What this suggests to me is that the Khan Academy is lacking in depth and clarity of presentation, as well as in addressing the subtleties and key issues that would be necessary to impress a person with some knowledge of the subject. Watching the Khan Academy is roughly akin to having a smart kid in your class (who is learning the subject along with you) explain to you what he/she has understood of the subject. It is not really comparable to how an expert teacher would convey the material.

This may not be completely a minus, because the lack of polish and the chumminess of the videos might itself be an endearing factor that makes people more comfortable with the videos. It also makes it easier to scale up and make a larger quantity of videos. Also, the low intensity of the videos makes it easy for a person to watch them when tired and distracted without missing out on too much.

Here are some examples of sloppiness:

(i) In the calculus videos, when I viewed them, the graphs were drawn very shakily, extremely hard to understand, and not well labeled. (ii) In a video on classical mechanics, there were some inaccurate statements about normal force, describing it as a reaction to gravitational force in the action-reaction sense (this was fixed later, I think). These weren't merely careless errors in speaking, but reflected a deeper lack of understanding. (iii) The examples and symbol choices are often confusing.

If you are recommending watching Khan Academy videos, I suggest you add the caveat that they should not expect a lot more (in terms of accuracy and quality of explanation) than they would expect learning from their colleagues.

• > "Also, the low intensity of the videos makes it easy for a person to watch them when tired and distracted without missing out on too much." I think this is an important point. It means that the average multiply-distracted internet users of today can get something useful out of the videos, which they may not be able to get from books or longer-format videos. – ShreevatsaR Nov 8 '11 at 17:01
• Shreevatsa, not sure you're right. If a user remembers 40% of Khan's video, and 20% of another video that is thrice as dense, the arithmetic supports the denser video. It's unclear how the trade-off works in practice, it could go either way. – Vipul Nov 8 '11 at 18:29
• I agree, but besides %, another point is viewers feeling they've learnt something (and thus being motivated to continue watching more videos). If someone watching a longer video in their normal internet-distracted way realises they're not following much (or gets bored), they may just give up, while if they somehow complete watching a Khan video they may feel they've actually learnt something or passed a milestone, and continue with another small Khan nugget video at some later time. Of course, this is all speculation: some students are more patient and even Khan length is too much for some. :) – ShreevatsaR Nov 12 '11 at 16:13
• roughly akin to having a smart kid in your class (who is learning the subject along with you) explain to you what he/she has understood of the subject. Perhaps unfair; more a graduate. But it conveys both the lack of depth and the advantage of someone who is closer to being your peer explain it. At uni, the tutors would often be better at helping students than the lecturer, because they'd been a beginner more recently. – hyperpallium May 20 '18 at 6:56

I like the videos.

But I don't think they are what is special about khanacademy: instead it is the combination of the self-study videos together with the machine-graded and assisted exercises, where a human tutor can monitor progress on the exercises. In a sense this might be seen as the reverse of traditional education where the tutor teaches and the student does exercises alone. Time will tell whether it is a better model.

I've looked around khanacademy a bit and I'm not really impressed. There's certainly nothing wrong with it, but it gets too much hype. There are math tutors on youtube (look up PatrickJMT) who do a better job.

• What are you not impressed with? Depth? Rigor? Explanation? Number or quality of exercises? – Srivatsan Nov 8 '11 at 15:08
• Just the clarity of presentation in the videos. I admit I only looked around a little, and this was a while ago. I'm not saying it's not a good resource, it certainly is. I just feel like it gets too much hype. Youtube is full of great math tutors, MIT's opencourseware videos, etc. – Brendan Cordy Nov 8 '11 at 15:46
• After reading Henry's post, I realize I may have missed the point though. I didn't know there was any student-tutor interaction going on through the website, so I can't comment on that. – Brendan Cordy Nov 8 '11 at 15:51

Apart from algebra, some calculus and basic maths the site also covers Geometry and Trigonometry, the 2 subjects that I'm using Khan Academy for at the moment.

The tutorials aren't very rigorous, but that is a positive thing for students who find calculus challenging and are not math majors (that described me by the way).

Although my first ever proof was actually from this site, it dealt with how to prove the angles in a triangle always add up to 180 degrees. It was pretty cool.

I actually think this site really is most ideal for people who need to brush up on their basic math skills before/while taking a first course in calculus.

I also get bored with my head stuck in a book all the time, and it's nice to watch a video once in a while. And finally, the many online exercises keep me from nodding off, and if you are somewhat immature you can take pleasure in collecting badges and energy points, as I do! ;)

I've used Khan Academy videos from time to time, and my biggest complaint would be the lack of professional oversight. For example, in his video about the sodium-potassium pump he is just wrong about how the membrane potential is generated, and how the action potential is transmitted. In this case, there is a correction video posted, but I remember there were some other videos with smaller mistakes that were only pointed out in the comments, or not at all.

That said, I still think it's an excellent learning resource, because the short videos are easier to watch than reading the same content in a book, and Khan is very good at explaining things.

• I think it's weird that he's the only one doing videos. If he could get more funding, which he probably can and does, he should hire experts to do videos. Or, they could volunteer. He could pick people known to be good teachers as well. – Graphth Nov 8 '11 at 17:36

I use Khan Academy extensively with my own children and (together with friends) to help kids in Newark, NJ. Khan is much different in practice than what it looks like on the surface.

Khan has lots of bite-sized video lessons that are well-produced (even when you only see a simple blackboard). They range in quality from pretty good (most of the math) to excellent (browsed through some amazing musical theory and social science videos). Khan is judged on the videos, but they're almost besides the point.

Computer-based education is only clearly superior in mathematics only (so far). This is because computers are far better-suited than teachers or books to: