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# Seiberg-Witten Theory 2: Clifford Structures and Spinors

Here is the second post on setting up the Seiberg-Witten equations on a 4-manifold, based on our learning seminar at UT Austin. The first post is here.

Clifford Algebras and Structures

For a vector space V with inner product g, its Clifford Algebra is defined as the tensor algebra of V modded out by all relations generated by setting $v\otimes v=-g(v,v)1$,

$Cl(V)= \otimes V/\langle v\otimes v +g(v,v)1: v\in V\rangle.$

For a vector bundle $E\to M$, any map $c: T^*M\to End(E)$ satisfying $c(u)c(v)+c(v)c(u)=-2g(u,v)I_E$ for all $u,v\in \Gamma(T^*M)$ (equivalently satisfying $c(v)^2=-|v|I_E$ for all v) extends to a representation
$c:Cl(T^*M)\to End(E)$
Such a map is called a Clifford structure.

There are two reasons we are interested in Clifford algebras and Clifford structures for Seiberg-Witten theory. The first is their relation to Spin and Spinc structures. The second is their relation to Dirac operators. In this post we will focus on their relation to Spin and Spinc structures, and discuss Dirac operators next.

Clifford algebras and Spin

Let $Cl(n)$ denote the Clifford algebra of $\mathbb{R}^n$ with its standard inner product. Let $(e_1,\cdots, e_n)$ denote the standard orthonormal basis for $\mathbb{R}^n$. Consider the multiplicative subgroup of $Cl(n)$ generated by unit vectors of $\mathbb{R}^n$. This is called $Pin(n)$.

There is a natural $\mathbb{Z}/2$ grading on $Cl(n)$ induced by a bijection $Cl(n)\leftrightarrow \bigwedge^* \mathbb{R}^n$ identifying $e_{i_1}\cdots e_{i_k} \leftrightarrow e_{i_1}\wedge \cdots \wedge e_{i_k}$. The integer grading on the exterior power reduces to a $\mathbb{Z}/2$ grading (even/odd) on the Clifford algebra. This yields a splitting $Cl(n)= Cl^+(n)\oplus Cl^-(n)$. Define $Spin(n)$ to be the intersection of $Pin(n)$ with the even summand $Cl^+(n)$.

Before, we defined $Spin(n)$ to be the universal double cover of $SO(n)$. We can show this new definition of Spin agrees with the old definition, by explicitly constructing a double cover map from this subset of $Cl(n)$ to $SO(n)$.

There is an action of $Cl(n)$ on $\mathbb{R}^n$ given by signed conjugation (using the multiplicative structure of the Clifford algebra). If $v\in \mathbb{R}^n$ is a unit vector (i.e. a generator of $Pin(n)$) then for any $x\in \mathbb{R}^n$
$-vxv^{-1} = vxv = x-2\langle x,v \rangle v$
Here we have used the fact that for unit vectors $-vv=1$ so $v^{-1}=-v$, and the relation $vx+xv=-2g(v,x)$ in the Clifford algebra. This can be interpreted geometrically: the action $(v,x)\mapsto -vxv^{-1}$ is the reflection of x over the hyperplane orthogonal to v.

The group of orthogonal transformations is generated by reflections over hyperplanes, so we have a representation called the twisted adjoint representation:
$\rho: Pin(k)\to O(k)$
defined by $\rho(y)x = yx\varepsilon(y^{-1})$ where $\varepsilon(Cl^\pm(n))=\pm 1$ (extend linearly). Restricting this to $Spin(k)$ this is just usual conjugation, which corresponds to an even number of reflections so the image lies in $SO(k)$:
$\rho: Spin(k)\to SO(k)$
This map is a surjective group homomorphism, and by studying the elements of $Spin(k)$ which lie in the center of $Cl(k)$, we see that the kernel of $\rho: Spin(k)\to SO(k)$ is two elements $\{\pm 1\}$. Because these are nice smooth compact Lie groups, this implies that $\rho$ is a covering map. To check it is not the trivial double cover, we can find a path in $Spin(k)$ between -1 and 1 given by
$\gamma(t)=\cos(t)+e_1e_2\sin(t)=-(e_1\cos(t/2)+e_2\sin(t/2))(e_1\cos(t/2)-e_2\sin(t/2))$
for $t\in [-\pi,\pi]$ [observe this path is a product of two unit vectors at each t and is thus in $Spin(n)$].
Therefore this definition of $Spin(n)$ agrees with the previous one.

The spinor representation

We have already seen that $Spin(4)\cong SU(2)\times SU(2)$ so $Spin(4)$ and $Spin^c(4)$ naturally admit two complex rank two representations coming from the projections onto the two factors of $SU(2)$. However, it is useful to understand these representations from the Clifford algebra perspective so that the representations carry the additional information of a Clifford structure. In fact, there is a complex representation of the entire (complexified) Clifford algebra $Cl(4)$ which splits into a direct sum of two complex rank two representations, which behave nicely with respect to the $\mathbb{Z}/2$ grading on the Clifford algebra. More specifically:

Theorem: There is a complex vector space $\mathbb{S}=\mathbb{S}^+\oplus \mathbb{S}^-$ with $\dim_{\mathbb{C}}\mathbb{S}^+=\dim_{\mathbb{C}}\mathbb{S}^-=2$, and an $\mathbb{C}$-linear isomorphism
$c: Cl(4)\otimes \mathbb{C}\to End(\mathbb{S})$
such that $c(Cl^+(4))\cong End(\mathbb{S}^+)\oplus End(\mathbb{S}^-)$ and $c(Cl^-(4)\cong Hom(\mathbb{S}^+,\mathbb{S}^-)\oplus Hom(\mathbb{S}^-,\mathbb{S}^+)$.

To prove this, we have to define $\mathbb{S}$, $\mathbb{S}^\pm$, and the map c, and then verify that c is an algebra isomorphism satisfying the specified properties. There are a lot of things to check so I will define everything, and say a few things about how the map c works which hopefully make it more believable that c is an algebra isomorphism.

Let $V=\mathbb{R}^4$ with standard coordinates and standard almost complex structure J. This almost complex structure gives rise to a splitting of $V\otimes \mathbb{C} = V^{1,0}\oplus V^{0,1}$, where $V^{1,0}$ is the i-eigenspace of J and $V^{0,1}$ is the -i-eigenspace of J. We have orthonormal bases for these pieces given by:
$V^{1,0}=span\left(\varepsilon_1 := \frac{1}{\sqrt{2}}(e_1-if_1), \varepsilon_2 := \frac{1}{\sqrt{2}}(e_2-if_2)\right)$
$V^{0,1}=span\left(\overline{\varepsilon}_1 := \frac{1}{\sqrt{2}}(e_1+if_1), \overline{\varepsilon}_2 := \frac{1}{\sqrt{2}}(e_2+if_2)\right)$

Define $\mathbb{S}:= \bigwedge^* V^{1,0}$, and its splitting by $\mathbb{S}^+ := \bigwedge^{even}V^{1,0}$ and $\mathbb{S}^- := \bigwedge^{odd}V^{1,0}$.

Now we need to define $c: Cl(V)\otimes \mathbb{C} \to End (\mathbb{S})$ with the properties specified in the theorem. We will define c on elements of $V\otimes \mathbb{C}$ and then extend this to a map on the Clifford algebra by setting $c(e_{i_1}\cdots e_{i_k})=c(e_{i_1})\cdot \cdots \cdot c(e_{i_k})$ and extending complex linearly. To specify c on $V\otimes \mathbb{C}$, it suffices to say what c does to vectors in $V^{1,0}$ and $V^{0,1}$.

For $v\in V^{1,0}$, $c(v)$ is the endomorphism of $\mathbb{S}$ obtained by wedging with v:
$c(v)(u_1\wedge \cdots u_k)=\sqrt{2}v\wedge u_1\wedge \cdots \wedge u_k$

For $\overline{v}\in V^{0,1}$ $c(\overline{v})$ is contraction with $\overline{v}$:
$c(\overline{v})(u_1\wedge \cdots u_k) = \sqrt{2}\sum_{j=1}^k (-1)^j g(v,u_j)u_1\wedge \cdots \wedge \widehat{u_j} \wedge \cdots u_k$

One needs to check that this respects the Clifford algebra structure, and is an isomorphism. Initially, this may look wrong because for example when $v\in V^{1,0}$
$c(v)^2(u_1\wedge \cdots \wedge u_k) = v\wedge v\wedge u_1\wedge \cdots \wedge u_k=0$
and it seems like we should have $c(v)^2=-|v|^2I$. However, the algebra structure we want to preserve is complex linear on $Cl(V)\otimes \mathbb{C}$ and has the Clifford structure only on the $Cl(V)$ piece. Therefore, for example when $v=e_j-if_j\in V^{1,0}$,
$0=c(e_j-if_j)^2 = (c(e_j)-ic(f_j))^2 = (c(e_j))^2-ic(e_j)c(f_j)-ic(f_j)c(e_j)-(c(f_j))^2 = |e_j|^2-i2g(e_j,f_j)-|f_j|^2$

For basis elements, the map c is a sum of the exterior and interior products. To compute for example, $c(e_j)$ we split this into the $V^{1,0}$ and $V^{0,1}$ parts, so
$c(e_j)=c\left(\frac{1}{2}(e_j-if_j)+\frac{1}{2}(e_j+if_j)\right)=\frac{\sqrt{2}}{2}\left((e_j-if_j)\wedge\cdot +\iota_{e_j-if_j} \right)$
If you want to be slightly more convinced without completing the proof that $c(v)^2=-|v|^2I$ for real elements of $Cl(V)$ it is fairly easy at this point to check that $c(e_j)^2=-I$ at least on the $\bigwedge^0V^{1,0}$ part of $\mathbb{S}=\bigwedge V^{1,0}$ (since any map that starts with contraction vanishes and $\iota_x(y\wedge f)=-fg(x,y)$ for $f\in \bigwedge^0V^{1,0}$ a complex number, and $x,y\in V\subset Cl(V)\otimes 1$).

We get the last property in the theorem easily from the definition of c. For $v\in V\otimes \mathbb{C}$, $c(v)$ either raises or lowers by 1, wedge power of an element of $\mathbb{S}=\bigwedge V^{1,0}$. Therefore $c(v)$ sends $\mathbb{S}^+$ to $\mathbb{S}^-$ and vice versa. Extending this over the entire Clifford algebra, we see that the endomorphisms in $c(Cl^+(4))$ preserve $\mathbb{S}^+$ and $\mathbb{S}^-$ (since they switch between $\mathbb{S}^\pm$ an even number of times) and $c(Cl^-(4))$ sends $\mathbb{S}^\pm$ to $\mathbb{S}^\mp$.

Note: We can rewrite the isomorphism $c: Cl(4)\otimes \mathbb{C}\to End(\mathbb{S})$ as a map
$c: Cl(4)\otimes \mathbb{C}\otimes \mathbb{S}\to \mathbb{S}$.
This will be useful when we use this representation to form associated bundles and consider sections of those bundles and maps between the spaces of sections.

This theorem generalizes for $Cl(2n)$, producing a complex vector space $\bigwedge V^{1,0}$ which splits where $dim(V)=2n$, whose endomorphisms are isomorphic to $Cl(2n)\otimes \mathbb{C}$, where $Cl^+$ preserves the splitting and $Cl^-$ switches the components. In the odd dimensional case, the situation is slightly different, but reduces to the even case by showing that $Cl(2n-1)\cong Cl^+(2n)$. For the purposes of Seiberg-Witten Floer homology, it will be useful to know $Cl(3)\cong Cl^+(4)$ which implies $Cl(3)\otimes \mathbb{C}\cong End(\mathbb{S}^+)\oplus End(\mathbb{S}^-)$.

Spinor bundles

Now that we have this representation of the complexification of the Clifford algebra, we can restrict to get a representation of Spin. Because $Spin(4)\subset Cl^+(4)$, and $c(Cl^+(4))$ preserves the splitting $\mathbb{S}=\mathbb{S}^+\oplus \mathbb{S}^-$, we get two representations
$\rho_{\pm}: Spin(4)\to Aut(\mathbb{S}^\pm)$
Note the image of $Spin(4)$ lands in automorphisms instead of only endomorphisms because elements of $Spin(4)$ are invertible in $Cl(4)$. These two representations correspond to the same ones we obtain by identifying $Spin(4)\cong SU(2)\times SU(2)$ and projecting onto one component.

We can extend these maps to $Spin^c$ by defining
$\rho^c_{\pm}: Spin^c(4)\to Aut(\mathbb{S}^\pm)$
by $\rho^c_\pm((g,z))=z\rho_{\pm}(g)$ for $g\in Spin(4)$, $z\in U(1)$.

Note this is well defined since $\rho^c_\pm((-g,-z))=\rho^c_\pm((g,z))$.

Given a Spin or Spinc structure on a manifold, these representations give rise to associated bundles $S^\pm \to M$. These bundles show up in the set-up for the Seiberg-Witten configuration space, which I will get to in another post.


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## Introduction

Tropical cyclones (TCs) are one of the most costly natural hazards impacting on coastal regions across the world1,2,3 and there is an increasing trend of TC related billion-dollar disasters in the US4. Over 70% of total tropical cyclone damage in the North Atlantic is caused by major TCs, category 3, 4 or 5 on the Saffir–Simpson scale, which make landfall5.

In 2017, the North Atlantic hurricane season involved 6 landfalls and is the most costly season to date, with estimates of damage at over \$360bn6,7. The season has broken several records. September had the largest Atlantic Accumulated Cyclone Energy (ACE) on record at 175, whilst the season ACE of 226 is the 7th highest. ACE (units, 10Kn2) is defined as the sum of the maximum sustained surface wind speed squared at six‐hourly intervals for all periods when the TC is at least of tropical storm strength (>=34 knots)8. The duration of hurricanes in 2017 was also record-breaking, setting a new September record of 41 hurricane days, principally due to Irma, Jose and Maria, each of which lasted for over 9 days.

The active season which unfolded was not, however, well predicted in the early season forecasts9. This suggests factors developed later in the season, leading to a higher level of activity than initially predicted. Here we compare the 2017 Atlantic hurricane season to the active seasons of 2005 and 2010 (as indicated in Fig. 1a). We explore the similarities and differences in the precursors to those seasons, predominantly from an ocean perspective, using a range of observational datasets. The seasons of 2005 and 2010 were chosen for their similarity in activity and intensity to 2017. The 2005 season has the 2nd highest Atlantic ACE on record at 250 with 15 hurricanes (>=64 knots) and 7 major hurricanes (>=96 knots). The 2010 season had an ACE of 165, 12 hurricanes (H) and 5 major hurricanes (MH) and was characterised by hurricanes with a significant duration, similar to 2017 (10 H, 6 MH), which formed further east. In addition, these seasons had the highest SSTAs in the MDR, between July and September for the period 1980–2017 (Fig. 1b).

As hurricanes intensify by extracting energy from the warm ocean surface, the underlying SSTs and ocean thermal structure are critical for their development10 with local SSTs greater than 26.5 °C usually considered to be a necessary condition for tropical cyclone development, although it can vary slightly by ocean basin. Over 95% of TC in the Atlantic form in waters warmer than 25.7 °C11,12,13. Favourable (warm) ocean thermal structure in the MDR, together with a low vertical wind shear, and atmospheric low-pressure disturbances such as African Easterly Waves (AEW), are together conducive to intense and sustained hurricane development5,9,11,14,15. The size of the Atlantic Warm Pool (AWP), where water is warmer than 28.5 °C, has also been shown to influence the TC track, with more TC genesis further east for a large AWP. Its size also affects the position and strength of the Azores high16. When a large AWP is present, the Azores high weakens and shifts north eastwards, enabling TCs to track poleward and recurve towards the east.

Observed Atlantic hurricane frequency has been found to correlate with long-term variability of sea surface temperatures particularly as measured by the Atlantic Multi-decadal Variability (AMV) index17,18,19,20,21,22,23, which is the North Atlantic area-averaged (0–60°N, 0–80°W) Sea Surface Temperature Anomaly (SSTA). In addition, positive Ocean Heat Content Anomalies (OHCA) have been found to increase the hurricane intensity and track length24,25,26. Variations in SST are influenced by the heat balance in the mixed layer of the ocean which is governed by air-sea fluxes, together with horizontal advection and vertical advection/mixing processes, and may be written as13,27:

$$\frac{{\partial T}}{{\partial t}} = \frac{{Q_{{\mathrm{net}}}}}{{(\rho C_{\mathrm{p}}H)}} + \left( {{\mathbf{U}}_{\mathrm{g}} + {\mathbf{U}}_{{\mathrm{ek}}}} \right) \cdot \nabla {\mathbf{T}} + \frac{{(w_{\mathrm{e}} + w_{{\mathrm{ek}}})(T - T_{\mathrm{b}})}}{H}$$
(1)

where T is the mixed layer temperature (equivalent to the SST), Qnet is the net surface heat flux i.e., the sum of the turbulent (sensible and latent) and radiative (solar and longwave) heat fluxes, ρ is the density of seawater, Cp is the specific heat of seawater, H is the mixed layer depth, Ug is the geostrophic current velocity, Uek is the Ekman current velocity, we is the vertical entrainment rate, wek is the Ekman pumping velocity and Tb is the temperature of the water just below the mixed layer. The first term on the right hand side is determined by net surface heat flux, the second by horizontal advection and the third relates to vertical heat exchanges. Our analysis focuses on the respective contributions of surface heat fluxes and ocean circulation to the SST anomalies that can develop prior to hurricane seasons.

As an index of ocean heat transport, and convergence thereof, the observed Atlantic meridional overturning circulation (AMOC) strength at 26°N is used, as measured by the RAPID Array28,29. Previous studies30,31,32,33 have identified links between ocean advection/AMOC strength and hurricane frequency, and also that the AMOC leads a SSTA dipole in the North Atlantic (with poles at 10–15°N and 45–60°N) that has maximum correlation strength at a 5-month lag34. Anomalies of the surface net heat flux (SFX) and wind stress curl (WSC) have also been analysed for their impact on SST although they have not previously been identified as precursors to hurricane season strength. A reduction in the WSC in the tropical Atlantic can lead to maximum positive SSTAs around 2 months later35, while SFX anomalies directly warm or cool the surface layer34,36,37,38.

Here we analyse SSTA alongside OHCA (using GODAS ocean re-analysis data39), and compare these to related indices and data. Our results reveal that the positive SSTAs prior to the 2017 active hurricane season were generated by processes not previously recognised to be important indicators of hurricane season strength. Furthermore, in contrast to other recent strong seasons, in which the SSTAs were evident in March, the anomalously warm ocean surface in 2017 developed later between April and July, making prediction of the Atlantic hurricane season severity even more difficult.

## Results

### Atlantic hurricane season activity

The time series of Atlantic tropical cyclones from 1980 shows an increase in activity over the period (Fig. 1a). The number of landfalls has fluctuated with 6 or more recorded in 1985, 1996, 2005, 2008, 2010 and 2017. To better understand hurricane season activity, Fig. 1b highlights the ACE index for the years 1980–2017 and the associated SSTA and vertical wind shear anomaly in the MDR. Active seasons occur when there is a positive SSTA and a negative shear anomaly (i.e., weaker than average vertical wind shear) in the MDR (40% of years considered). Less active seasons occur when there is a negative SSTA and positive shear anomaly (43%) and when there is a positive SSTA and positive shear anomaly (14%). The seasonally averaged negative shear anomaly is greater in 2005 and 2010 compared to 2017. In both 2005 and 2010, negative anomalies persisted throughout the season, whereas in 2017 negative shear anomalies developed later in August and September. Previous work has linked ENSO variability / La Niña conditions with negative shear anomalies40,41,42,43, SSTA44,45,46 and a potential relationship between shear and SSTA47. We note that in 2005, 2010 and 2017 La Niña persisted during August and September. There is also a significant correlation between SSTA and shear anomaly of –0.58 (p < 0.01) indicating that positive (negative) SSTA are often associated with negative (positive) shear anomalies.

To composite the disparate influences on individual TC tracks (as opposed to season activity), Fig. 1c highlights conditions at 6 hourly intervals along the TC tracks for August and September of 2005, 2010 and 2017, incorporating SSTA, vertical shear anomaly and wind speed along the track. Again, the most favourable conditions for TCs are positive SSTAs and a negative vertical shear anomaly as the majority of the 6 hourly values fall within the corresponding quadrant of Fig. 1c. A substantial number of values coincide with positive SSTAs and low positive vertical shear anomalies. Far less favourable for TC tracks are negative SSTAs and even when the wind shear anomalies are clearly negative we only find few hurricane track points in the corresponding quadrant. The relationships between SSTA and vertical shear, and number of tropical cyclones are significant, with correlations of 0.75 (p < 0.01) with SSTA and −0.67 (p < 0.01) with vertical shear between July and September, for the period 1980–2017. Figure 1c also highlights that wind speeds over 100 knots are only observed when there are positive SSTAs present and usually a negative vertical shear anomaly, indicating that both positive SSTA and weak vertical shear are critical to the full intensification of hurricanes. Previous studies have shown that 70% of major storms in the North Atlantic undergo rapid intensification48 and the interplay between SSTA and vertical shear in such intensification remains a matter for further research.

Of interest is the comparison between Fig. 1b and c, where the latter indicates TC tracks can exist where there are low positive values of vertical wind shear anomaly. The average vertical shear in the MDR in August and September is 6.14 ms−1 (for the period 1948–2017). Wind shear magnitudes of less than 10 ms−1 are generally considered favourable for TC genesis9,14,49; accordingly, positive shear anomaly values up to 4 ms−1 are still likely to be conducive to TC development, consistent with a significant number of tracks observed in the top right quadrant (Fig. 1c).

Figure 2 shows SSTA overlaid with sea level pressure (SLP) and tropical cyclone tracks for 2005, 2010 and 2017. Positive SSTA in the MDR is evident in each year, although strongest in 2010. The 2005 season featured hurricane activity in all months, whereas in 2010 and 2017, September was the most active month. The 2010 and 2017 seasons were characterized by large SST anomalies, persisting across the MDR from 10–20°N and 15–90°W, from July to September. Accordingly, some tropical cyclone (TC) genesis was located further east than is usual in both years, and as far east as 20°W, in contrast to 2005. In 2005, the largest SSTAs were further west at 40°W. In 2005 and 2010, positive temperature anomalies persisted to a depth of over 250 m during the season, whereas in 2017 the anomalies were generally only to a depth of 100 m between 10 and 60°W, although a positive anomaly to a depth of 250 m did exist during September 2017 between 60 and 80°W (Supplementary Fig. 1).

TC tracks are influenced by both SLP and SSTA. The positive SSTA across the MDR enabled an eastward shift in the genesis location in 2010 and 2017. In 2010, the Azores high was weaker and located further northeast in the Atlantic (especially in August), related to the large AWP16, resulting in a lower pressure anomaly off the east coast of America (Supplementary Fig. 2) and enabling TC tracks to curve northwards. Accordingly, there were no US landfalls despite a very active season, and the 6 tracks which did make landfall in central America predominantly originated in the Gulf of Mexico or Caribbean Sea. In 2005 and September 2017, the lower pressure anomaly off the east coast of America during July to September also existed, but not to the same extent as 2010. TC tracks did curve northwards but there were 9 and 6 landfalls in the respective years.

### Ocean and heat flux time series

Turning to the ocean variables and to highlight the connection between SSTAs, air-sea fluxes and the ocean circulation we focus on the period from 2000 to 2017 (Fig. 3b–d). Ocean temperature variability is characterised by SSTAs and OHCA105 (temperature anomalies averaged over the top 105 m). The impact of air-sea fluxes on SSTA and OHCA105 is dominated by latent heat fluxes (LHFX). To understand the timing of the development of SSTAs in different parts of the MDR, LHFX anomalies (LHFXA) are either averaged over the entire MDR (Fig. 3b), the southern part of the MDR (10–15°N, 40–60°W) or the north-eastern (NE) MDR (15–21°N, 24–36°W) (Fig. 3c). The location of each region is shown in Fig. 3a. The north-eastern and southern regions of the MDR were chosen because the air-sea fluxes reached a maximum in these areas in 2017 during April and May–August, respectively. The Meridional Overturning Circulation (MOC) is the main contributor to ocean heat transport in the North Atlantic and here we use observations of the MOC and of its Ekman component at 26°N for the period from 2004 to 2017 (Fig. 3d).

SSTA and OHCA105m are closely aligned (Fig. 3b), although the strength of the anomaly is greater at the surface. Clear peaks in SSTA and OHCA105m are seen in 2005 and 2010, and also positive anomalies have been seen since 2015. LHFXA in the MDR (green line) is significantly correlated with SSTA and OHCA105m; correlation coefficients are respectively 0.52 and 0.38, both significant at the 0.01 level (p < 0.01). The LHFXA variability is analysed further in Fig. 3c for the two areas (southern MDR and NE MDR) that experienced the strongest anomalies in 2017. The most extreme latent heat flux anomaly values occurred in 2017, with latent heat loss weaker by 33 Wm–2 sustained from May to August in the southern MDR region and weakening of over 60 Wm−2 in the NE MDR during April (unsmoothed data). In 2005 and 2010, the LHFXA values in the southern MDR region were noticeably smaller in magnitude (May–Aug mean: 7 Wm−2 in 2005; 13 Wm−2 in 2010).

A more detailed analysis of the April and May–Aug net surface heat flux anomalies (SFXA) for the MDRs from 1980 to 2017 is shown in Supplementary Fig. 3. In the NE MDR, the 2017 April SFXA of 93 Wm−2 was 3.4 standard deviations (SD) from the mean. In, the southern MDR, the 2017 May–August SFXA was 45 Wm−2, 2.5 SD from the mean. For each MDR, the 2017 SFXA was the most extreme in the period considered. Furthermore, for the NE MDR the April 2017 SFXA was nearly twice that for the next largest anomaly (55 Wm−2 in 2014) further emphasising the particularly unusual conditions prior to the 2017 season.

MOC values (Fig. 3d) are significantly below the seasonal mean values in February/March 2005 and 2010 at 11/13 Sv (1 Sv = 10m3s1) and 9/10 Sv, respectively. Average values for February and March are 15 Sv. Above average values are observed in February 2017 at 17 Sv (the latest available MOC data). Ekman transports are also below average in Feb/Mar 2005 and 2010 but are close to average in Feb/March 2017 and for the remainder of 2017. The February/ March values are important as Fig. 3b, d indicate that the observed MOC transport co-varies with the observed SSTA and OHCA105m in the MDR over the period from 2004 to 2017. The correlation between them is −0.35 for MOC-SSTA and −0.27 for MOC-OHCA105, when the MOC leads by 5 months, which is statistically significant at the 0.01 level (p < 0.01). This is in line with the results of Duchez et al.34 who found the strongest correlation between the SSTA and MOC occurred when the MOC leads by 5 months. In addition, Supplementary Fig. 4 shows how the MOC transport anomaly in February and March co-varies statistically with SSTA and OHCA105m in July, August and September. Correlations over 0.5 are statistically significant at the 0.05 level (p < 0.05). Anti-correlations over 0.6 are seen over large areas of the MDR between 10–20°N and 30–70°W during July–September. Importantly, the correlations are higher for OHCA105m than SSTA, highlighting that the variability of the MOC transport at 26°N influences the upper ocean layer in the MDR, which is of consequence for hurricane intensification as it potentially provides a significant heat source extending over the top 100 m of the ocean to power hurricane development.

### Hurricane season precursors in 2005, 2010 and 2017

Figure 4a shows the 2017 March–August monthly surface heat flux anomalies (SFXA) overlaid with the surface wind anomaly. In April, a strong positive SFXA of 100–150 Wm−2 developed (i.e., 300–350 Wm−2 heat gain compared to the climatological mean of typically about 200 Wm−2) with a maximum at over 150 Wm−2 between 24–36°W and 16–18°N. This is associated with a reduction in the strength of the north easterly (NE) trade winds (30°W, 18–27°N) revealed by the wind vector anomalies. The weaker NE winds meant there was less cold, dry air over the region. As a consequence, the humidity gradient (between the sea surface and the overlying air) was lower than normal, and so the associated latent heat loss was reduced. In addition, the more humid air over the region enabled an increase in the downwards LongWave Radiative (LWR) flux. Anomalies in these two flux components largely account for the positive SFXA seen. Additionally, April 2017 is also characterised by a negative wind stress curl anomaly (WSCA) in the region associated with the anomalous winds (Fig. 4b). This results in a downward anomaly in Ekman pumping effectively reducing the upwelling of cold water at the eastern boundary and further assisting development of the positive SSTAs. Similarly, weaker than average NE/E winds persisted in June, July and August between 40–60°W, 9–15°N, along the southern boundary of the MDR. Again, the latent heat loss was lower and the LWR flux into the ocean was higher, largely explaining the positive SFXA and associated positive SSTA and OHCA in the southern MDR at this time.

The extent to which the 2017 SFXA generated the observed SSTA, and how this compares to 2005 and 2010, is also explored (Fig. 5). In 2005 and 2010, positive SSTAs already existed in the MDR in March (Fig. 5a, b). The patterns of these SSTAs are consistent with those related to the below-average MOC and Ekman transport in February and March generating the dipole SSTA pattern observed, with negative anomalies in the northern part of the Atlantic and positive anomalies in the south32,34,50. In 2017, in contrast, the MOC and Ekman transport was close to the 2004–16 mean, and SSTs were close to average (Fig. 5c). The SFXA forcing over April–July generates the SSTAs shown in Fig. 5d–f. In the MDR vicinity, SFXA-generated SSTAs are most substantial in 2017 (Fig. 5f), concentrated in the southern area of the MDR between 9–15°N and 30–60°W with a maximum around 1.4 °C.

Adding SFXA-generated temperature anomalies to the initial March SSTA patterns, we obtain estimates of the August SSTAs (Fig. 5g–i). When comparing the estimated to the observed SSTAs in August (Fig. 5j–l), the spatial SSTA patterns and sign in the MDR look similar, however the amplitude in 2005 and 2010 is higher than observed. Discrepancies (Fig. 5m−o) between surface flux based estimates and observations, can be attributed to the ocean circulation (advection) and mixing. The MOC recovered from its below-average strength in February and March during both 2005 and 2010 (Fig. 3d). This stronger MOC means that in spring and early summer more heat was transported northwards by the ocean, partly compensating for the MDR temperature changes linked to SFXA and assisting the development of average SST conditions in the north, whilst reducing the amplitude of the positive SSTAs in the south, seen during August in these years. Hence, the observed August SSTA in 2005 and 2010 are weaker than those estimated from surface heat flux alone. In 2017, the weaker NE/E trade winds observed from June to August in the MDR will have reduced the northward Ekman transport into the region. As less heat was transported north during those months it is likely to have led to the lower observed SSTA in August than predicted from surface fluxes alone. Additionally, along the eastern boundary between 9–18°N the weaker trade winds will have led to a reduced upwelling and the warmer observed SSTA in the area. From these results, we conclude that in 2005 and 2010, the initial ocean condition in March (MOC-related) was the main influence on SSTAs in August (Fig. 5j–l), whereas in 2017, the SFXAs that developed from April to July were the dominant factor. These results are not specific to the GODAS ocean temperature reanalysis dataset used here and we found similar results with NCEP SST data (Supplementary Fig. 5).

Our analysis of a range of observations reveals the important role that surface heat fluxes played in positive SSTA development in the MDR in 2017, which was critical for subsequent hurricane activity in September. Figure 6 summarizes the mechanisms which were important during the season.

The reduction in the NE trade winds between April and July enabled positive latent heat flux and LWR anomalies to develop. The negative wind stress curl generated downward Ekman pumping anomalies, suppressing the upwelling at the eastern boundary. These factors generated positive SSTAs. The reduction in the NE trade winds also reduced the Ekman transport of warm water into the MDR region, explaining why SFXA-predicted temperature anomalies are higher than observed, by August. Reduced vertical wind shear was also evident in late August and September, which together with the positive SSTAs, played a key role in the active hurricane season of 2017. In contrast, in 2005 and 2010, it was the reduced AMOC in February-March of those years which was the predominant cause of the positive SSTA in the MDR, which again combined with a weak vertical shear to favour very active hurricane seasons. These findings reveal for the first time that different precursors can generate positive SSTA in the MDR region, conducive to an active hurricane season. In 2017, surface fluxes were the dominant factor, whereas in 2005 and 2010 the AMOC played a key role.

In terms of hurricane prediction, the 2017 season was more difficult to forecast, as the surface flux anomalies developed between April and July shortly before the main season (August–September). For the other strong hurricane years considered, the reduction in the MOC / Ekman transport occurred earlier in February–March potentially enabling a longer lead time forecast given a sufficient observing system in place and noting also the importance of atmospheric conditions (vertical wind shear). In conclusion, our results have revealed that drivers of recent active hurricane seasons involving the ocean can take two forms: late winter changes in the ocean circulation and late spring/early summer changes in the air-sea heat flux. Developing forecast systems that adequately represent these processes will potentially aid preparedness and mitigation for the financial and societal consequences of hurricanes.

## Methods

### Hurricanes and ocean properties

The observed Atlantic tropical cyclone and hurricane track data for the years 1980–2016 were obtained from HURDAT2, the revised Atlantic hurricane database51. Hurricane track data for 2017 was obtained from Unisys Weather (http://weather.unisys.com/hurricanes/). The NCEP Global Ocean Data Assimilation System39 (GODAS) was used for the ocean temperatures at the surface and depth. All anomalies were based on the reference period 1980 to 2017 unless stated otherwise. The ocean heat content anomalies are based on the average temperature anomaly over the depth indicated. The observed AMOC strength and Ekman transport at 26°N, for the period 2004–2017, were obtained from the RAPID MOC monitoring project52,53.

### Surface meteorology and air-sea exchanges

The NCEP/NCAR reanalysis54 was employed for sea level pressure (SLP), wind speeds and the air-sea heat flux. Wind speeds and associated anomalies were determined from the 1000 mb zonal and meridional components. The absolute vertical wind shear was calculated as the absolute difference between the 250-mb and 850-mb zonal wind using the reference period 1948 to 2017. The net surface heat flux (SFX) was determined as the sum of the net shortwave (Qsw), net longwave (Qlw), latent (Qlat) and sensible (Qsen) heat fluxes.

$${\mathrm{SFX}} = Q_{{\mathrm{sw}}} + Q_{{\mathrm{lw}}} + Q_{{\mathrm{lat}}} + Q_{{\mathrm{sen}}}$$
(2)

The estimated ocean temperature anomalies (ΔT) used in Fig. 5b and c are based on the surface flux anomalies (SFXA) and calculated as follows:

$$\Delta T\left( {x,y,t} \right) = \Delta T\left( {x,y,t_0} \right) + \frac{1}{{\rho DC_{\mathrm{p}}}}\mathop {\int }\nolimits_{\hskip -5pt t_0}^t {\mathrm{SFXA}}\left( {x,y,t} \right){\mathrm{d}}t$$
(3)

where D = mixed layer depth (100 m), ρ = density (1025 kg m−3), Cp = specific heat capacity 4182 JK−1 kg−1, t0: April, t: July. A mixed layer depth of 100 m was chosen for consistency with the findings of Cayan38, who showed the MLD in the North Atlantic to be between 150 m and 75 m between April and July. The spatial temperature pattern does not vary with depth chosen. A more detailed calculation would require a spatially and temporally varying MLD but the aim here is to show the magnitude of the heat flux related signal in line with the approach adopted by Duchez et al.55 The wind stress curl was calculated from NCEP/NCAR reanalysis54 monthly surface wind stress.

format: plain


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# Owen Shepherd

Physicist. Software Developer. Many things. (CV)

I develop the Public Domain C Library and Impeller (a client for pump.io) among other things, as well as contributing to projects such as Ogre3d.org.

Replied to a post on werd.io:

@benwerd SIL OFL is pretty standard for fonts. Very liberal, do what you want, just mandatory renaming if you make a derivative. Only slightly tricky case is if you're building your own WOFF or such, in which case your WOFF needs to bear a different name (derivatives clause)

Replied to a post on werd.io:

@benwerd I did that too last time my MacBook needed service. I was somewhat disappointed at the lack of looks of disapproval...

Replied to a post on werd.io:

@benwerd Nowhere, I expect. They discontinued production and discounted them a few months before N5 release, so I imagine all you'll be able to find are second hand ones.

Replied to a post on werd.io:

DPD in the UK do that. One hour delivery window, live tracking of the van your package is on, etc.

## Hashcash, Pingbacks, Webmention, Proof of Work and the #indieweb

Spam, spam, ubiquitous spam. Who likes it? Nobody. Everyone is aware of the issues of E-Mail spam; anybody who has a blog is aware of the problem of pingback spam

Pingback is an old technology, based upon XML-RPC (which should date it somewhat!) . Designed in simpler times, saying that the percentage of pingbacks which are spam today is 100% is accurate to a large number of significant figures.

About a year ago, the indieweb community looked at Pingback and asked the question "Why is this based on XML-RPC?" They had a point; Pingback is a single function, with two URL parameters, which hardly needs a complex RPC system, and so they developed WebMention, which takes Pingback and distills it to a simple HTTP POST of your standard URL-encoded-form-data.

While WebMention solves the problem of complexity, it doesn't solve the problem of Spam (in a long term manner at least - at present, the small deployment means it isn't a spam target). However, today I provide a method for combating spam in WebMention: hashcash

## Hashcash?

Hashcash is a mechanism originally proposed for combating spam in E-Mail. The idea is relatively simple: the sender must provide a value which, when combined with certain values from the E-Mail, evaluates to a hash where the last $$n$$ digits are zero. Its' called a "proof of work" system because that's what it provides: proof that the sender has done work. Specifically, on average it will take $$2^{n-1}$$ attempts to find a hash in which the last $$n$$ bits are zero.

This is useful: It lets us tune the work over time. Its' asymmetric: it puts all the effort on the sender; rate limiting them by their computational power (because verifying that the hash ends with the correct number of bits only takes one hash calculation). Every time we increase $$n$$ by one, we double the computational power that the spammer needs to send out the same quantity of spam

Hashcash isn't the perfect protocol, of course; it doesn't (and nothing will) completely eradicate spam. It does, however, provide a mechanism to reduce it; placing less load on more intensive spam filtering systems (such as the traditional Bayesian filters)

## Hashcash for WebMention

The proposed mechanism of implementing Hashcash for WebMention is as follows. For the sender side:

1. The sender builds the WebMention request body, in URL encoded format.
2. The sender computes the SHA-256 hash of the request body
3. If the hash contains the required number of trailing zero bits, then the mention is ready. Otherwise, a "nonce" value is added to the request and the hash is retried.
4. The client then submits the request to the destination's WebMention endpoint
5. If the hash meets the recipients' criteria (or the recipient implements WebMention 0.1/0.2, and therefore does not perform hashcash processing), then the recipient should return a success result code. Otherwise, the recipient should return a HTTP 403 error, with properties error=bad_hashcash&required_length=N, where N is the number of trailing zero bits required
6. If the recipient requires more zero bits, then the sender may choose to retry with more zero bits (but should verify that the number is reasonable - a required length of 32 is unlikely to complete in reasonable time and could comprise a DOS attempt)

For the receiver side, the process is much simpler:

1. Take the SHA-256 hash of the request body
2. If the hash does not meet the required number of trailing zero bits, return a HTTP 403 error with properties error=bad_hashcash&required_length=N. The endpoint must, at a minimum, support returning errors in application/x-www-url-form-encoded format.
3. If the hash does meet the required number of trailing zero bits, continue with the rest of the WebMention validation.

## Considerations

The hash length should be chosen to be reasonable; a good starting point might be 22 bits. The maximum permitted length (for a sender) should also be chosen to be reasonable; 28 bits seems like a reasonable limitation for the time being. 20 bits is a cost of 2MHash on average; recent CPUs are capable of 20Mhash/s. Information about the present cost of a hash computation can be approximated by looking at the statistics for BitCoin; BitCoin uses SHA-256 so is comparable

Of course, implementers may wish to be more lenient as a HashCash-type system is phased in

@benwerd I noticed that on your idno install you've set the homepage to be your profile page. How did you do that?

Replied to a post on werd.io:

@benwerd It's certainly a very useful desire from the perspective of propagating religion. Which way the association originated is an interesting question.

Replied to a post on twitter.com:

@stevestreeting The lack of hubs and KVMs (and where they exist, the expense) are painful too.

## We have met the enemy – and he is us

The conventional assumption in computer security was that our primary adversaries were criminals, miscreants, and the security services of our “political foe”. Attacks were liable to be active and involve the exploitation of vulnerabilities in our systems, because such foe were unlikely to be able to access . On the basis of these assumptions and allegations, there were comments from people such as the US government proposing bans upon the use of communications equipment by Chinese companies Huawei and ZTE in the US.

In the light of the PRISM and TEMPORA revelations, the hypocrisy was deafening. In the light of the latest revelations, its’ hard to find a source of humor at all.

We have met the enemy, and he is us; our governments have invaded our communications infrastructure, have access to the data behind our services, and have installed backdoors in our software.

If we are to preserve privacy on the web, the need for change has never been greater.

I think it can be said that, at an infrastructure level, our immediate priorities should be:

• Deploying TLS v1.2 with Perfect Forward Secrecy
Older versions of TLS have less effective PFS cipher-suites, and often require undesirable tradeoffs. When dealing with TLS 1.0, we are often required to judge between the weaknesses of RC4 (no longer considered secure when used as in TLS/SSL) and potential vulnerability to the BEAST attack. It’s safe to say that TLS 1.0, as deployed, is fundamentally broken; while the protocol itself is not completely so, the cipher-suites deployed in the wild all have issues. Deploying a patch to TLS 1.0 would be as difficult as updating to TLS 1.2; an action we should be doing anyway
• Deploying DNSSEC, DANE and Certificate Pinning
Given the known extensive partnerships of the secret services, I think it is fair to say that the CA model has outlived its’ usefulness. Browsers ship hundreds of security certificates; it is sheer naivety to assume that none of them have been compromised. I don’t see this as the complete end for CAs; they can continue to provide utility in the form of extended validation services, but their numbers will be reduced and, importantly, it will no longer be able to trust every CA to ensure the trust in a domain.DNSSEC itself poses issues; it has a single root of trust (the IANA), and it is a requirement that we trust our domain registrar as part of the chain of trust to ourselves. However, it vastly reduces the number of moving parts and trusted authorities, and makes validating that trust significantly easier. Attacks against DNSSEC need to be narrowly targetted to be effective; comparison of DNSSEC signed zones across multiple machines provides a simple method of watching for suspicious behavior.
The other result of DNSSEC is that it makes deploying encrypted services easier. Anything we can do to increase the proportion of encrypted traffic on the internet can only be a good thing.
• Development of TLS v2?
The existing TLS specification is a gradual evolution of SSL. It is therefore old, battle tested and, as a protocol, a well known quantity. As a side effect, it is also complex; it contains many misfeatures, and our evolving understanding of cryptography points out many parts of TLS which are at the very least suboptimal, or often highly problematic. Mitigating its’ many design flaws has resulted in huge increases in the complexity of the codebases implementing TLS; large portions now need to run in constant time to avoid timing-based side channel attacks. NSS and OpenSSL are monstrous, often convoluted code bases. Validating them is troublesome; performing constant time cryptography on modern processors is increasingly difficult, identifying potential ways in which this can become non-constant-time is nigh impossible. Given all we know today, I think it’s time to take a step back from TLS, take a hard look, and for every known issue, misfeature or design problem, fix it. TLS 2.0 need not have more in common with TLS 1.2 than an the initial hello packets; it should be built on modern, known good primitives; PCKS-#1 v1.5 padding wholly unsupported, replaced by OAEP; authenticated encryption used where possible, encrypt-then-MAC where not.The chipersuite list should be slimmed down; current “good practice” ciphers like RSA and AES and authenticated encryption modes like AES-GCM can stay, but state of the art ciphersystems like Salsa/20 and Curve 25519 should also be included and recommended. While it is likely that the NSA has cryptanalysis knowledge we do not, we do not know of any near-realistic attacks in the current good practice ciphers, and the nature of cryptanalysis says that their capabilities are unlikely to bring them close to breaking either. Even cryptanalytic attacks considered “groundbreaking” rarely result in practically exploitable flaws.

Instead, we should be changing because the NIST-sanctioned cipher-suites are not designed with us in mind. Constant time implementations of AES in software are notoriously difficult, and this goes double for systems like AES-GCM. SHA-3 contains many primitives which are efficient to implement in hardware, but difficult in software; the NIST elliptic curves use constants which result in inefficient software implementations (neither mind not providing rationale for the choice of said constants).

Cryptographic algorithms designed with software in mind reduce the cost of implementing it, increasing security, while also reducing the number of corner cases and attacks which can result in information leakage on common hardware. They additionally reduce the performance delta against a well equipped adversary (such as a government security agency).

The prime objective of a TLS revision should be that a TLS2 implementation be relatively concise and easy to validate, without the inscrutable complexity of TLS1. The TLS2 paths in OpenSSL and NSS should not be as convoluted and twisty as those for TLS1, and the preferred algorithms should be those which do not require large tables or similar constructions liable to suffer from side channel attacks.

A good rule of thumb is that the maths is hard to subvert; the code is easy. To that end, we should push towards simpler protocols which are easier to analyze. We should also push towards better defaults from the libraries we use; it is ridiculous that OpenSSL, for example, doesn’t come secure by default.

## Looking to the future – Buffers and Textures in Ogre 2.0

Being as they have an integrated resource system, the 1.x versions of Ogre have had a somewhat opaque buffer system. Anyone who has worked closely with the system, doing dynamic uploads for example, will have discovered that uploads are done by creating the object and then later uploading the actual data into it.

For its’ era, this was a perfectly adequate design. When it was conceived, this method largely reflected the way that graphics APIs worked – or at the very least permitted you to work. In OpenGL, for example, you create the buffer object then upload data into it using the glBufferData command.

Everything changed with Direct3D 10

## The old ways – A look at Direct3D 9

Direct3D9 divided buffer types using two properties – the “usage” which described how the application intended to use the buffer, and the “pool” which described how the memory was to be allocated. For our purposes, only one usage flag matters:

“D3DUSAGE_DYNAMIC – Set to indicate that the vertex buffer requires dynamic memory use. This is useful for drivers because it enables them to decide where to place the buffer. In general, static vertex buffers are placed in video memory and dynamic vertex buffers are placed in AGP memory. Note that there is no separate static use. If you do not specify D3DUSAGE_DYNAMIC, the vertex buffer is made static. D3DUSAGE_DYNAMIC is strictly enforced through the D3DLOCK_DISCARD and D3DLOCK_NOOVERWRITE locking flags. As a result, D3DLOCK_DISCARD and D3DLOCK_NOOVERWRITE are valid only on vertex buffers created with D3DUSAGE_DYNAMIC- MSDN

D3DUSAGE_DYNAMIC provides the driver with a hint that the application will be modifying the buffer frequently, and omitting it informs it that the application will be updating it infrequently, if at all.

This is a useful abstraction, but we can do better: if the buffer is entirely off limits to the GPU, then the driver can make further optimizations with the aim of increasing performance.

## The changes of Direct3D11

Direct3D11 takes the existing “dynamic/static” disjunction, renaming “static” to “default”, and adds two new modes, bringing us to a total of four:

• D3D11_USAGE_IMMUTABLE – Immutable buffers must be initialised with data when they are uploaded, and from then on are read-only for both the CPU and GPU
• D3D11_USAGE_DEFAULT – Essentially Direct3D 9′s default mode. This is optimized for infrequent updates
• D3D11_USAGE_DYNAMIC – Essentially Direct3D 9′s D3DUSAGE_DYNAMIC. Optimized for frequent updates and for use as a GPU render target
• D3D11_USAGE_STAGING – A specialized buffer usage that cannot be directly read or written by the GPU. D3D11_USAGE_STAGING is used to create DMA capable buffers for quickly copying data from the GPU.

The most important of these is D3D11_USAGE_IMMUTABLE. While Ogre does sometimes copy back data from the GPU, staging buffers are a much smaller performance win than immutable buffers, and have a higher logistical complexity for the engine.

## The changes in Ogre 2.0

The most obvious change is the addition of the enumerants HBU_IMMUTABLE and TU_IMMUTABLE for hardware buffers and textures respectively. These signal to Ogre, and therefore to the driver (where possible) that the contents of the buffer will never change.

Deeper changes have been required to the Texture and HardwareBuffer objects.

### Textures

The Texture API has changed significantly in Ogre 2.0 as the resource system has been changed. In older versions, the Texture object tracked the “source”, “desired” and “internal” versions of various properties – from width and height to pixel format. As Ogre 2.0 is separating the “resident” representation of the resources from their sources, all but the internal versions of the properties have been removed.

Instead, the loader sets the properties to those it desires for the texture object before calling the new _determineInternalFormat function. This function will then use its’ knowledge of the render system’s capabilities in order to determine the actual format to be used.

Once the internal format has been determined, the loader may use the _uploadBuffers function in order to actually fill the texture with data. For all the existing texture types, this is optional; the loader may later get the buffer by using the getBuffer method and then use the blit method of the returned buffer object to do the upload, as was the standard method in previous versions of Ogre. However, when using an immutable texture, the _uploadBuffers method must be used, in order to accommodate the contract associated with immutable textures.

### Other Buffers

Other buffer types are managed separately through the HardwareBufferManager. For each of these types, an additional pointer is being added to their constructor function in order to enable immediate upload.


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# Inverse Functions help

• April 11th 2010, 03:02 PM
baconfoot
Inverse Functions help
Hey guys i'm trying to figure these Inverse problems out. Any and all help would be greatly appreciated.

arcTan(- √3) . Give Answer in Exact Radians

and
is this possible?

Sin[arcCos(-1/3)+arcSin(2/5)] . Give Answer in Exact Radians
• April 11th 2010, 03:24 PM
stapel
Quote:

Originally Posted by baconfoot
arcTan(- √3) . Give Answer in Exact Radians

What angle measure $\theta$ gives you a tangent value of $\sqrt{3}$? Where (that is, in which quadrants) is tangent negative?

Quote:

Originally Posted by baconfoot
Sin[arcCos(-1/3)+arcSin(2/5)] . Give Answer in Exact Radians

If you've learned about angle-sum identities, this might be a good place to apply one. (Wink)


text:
# Which character to change (Robbers)

Robbers, your task is to crack a cops answer. Reveal the character that should be changed and what it should be changed to too. Please notify the cop if you cracked their challenge.

The robber winner is the user with the most cracks.
I will declare the winner cop and robber a week after this challenge is posted.

### Example Submission

# Cracked [math's answer](http://example.com/17290/Change-a-character)

1[a|b -> a

0[a|b -> b

$$$$

# R, cracks Robin Ryder's 5th answer

PO=0
TA=TO =min(0 * 0 * 0, FALSE, 0 * 0 * 0)
if(T |TO&TA&TO)PO=18
LETTERS[PO]

Try it online!

Maybe another unintended crack...

I know nothing about R. I just know TRUE is truthy.

import operator as o
import inspect as i
a,d = 37,lambda n:n if len(str(n))==1else d(sum(map(int,str(n))))
for k, v in{d(sum(map(ord, n))): f for n, f in i.getmembers(o,i.isbuiltin)[::11]}.items():
a+= int(v(a,k**2))
print(a)

Changed a space to a plus sign on line 5.

Try it online!

# brainfuck, cracks Zachary Cotton's answer

++++++++++[>+++++++++++<-]>+?.-------.+++++.++++++.

Try it online!

Simple, really. We just have to print each byte decremented by one. An obvious way to do this is to replace a + with a no-op, right before the printing begins.

13024e0 -> 13024

130b4e0 -> 69420

This is $$\(4,14,0)_{130}\$$.

Try it online!

{(⍵ ⍵)(⍵ ⌶)(⍵ ⍵)},8

Try it online!

Because of how APL's parsing works, it is possible to get a meaningful result only when the first or second (...) is changed into a function, and it is only possible in the form of either (⍵X⍵) (using a dyadic operator X) or (⍵ X) (using a monadic operator X).

So I went through the built-in operators in turn until I hit ⌶ (a monadic op that becomes an experimental function based on an integer code).

8⌶ is Inverted table index of. It takes "inverted tables" on its two sides, and computes "index of" rows of right arg in rows of left arg. ⍵ ⍵ is treated as a single-row, two-column table for this purpose, and the row to search is trivially found at index 1. Therefore the result is 1.

99>
!!@

9, right, 9, down, !, print 99, up, 9, right, >, pop 9 and rotate the bottom row to @!!, down, !, print 0, left, !, print 0, left, @, end.

Try it Online!

‛⟑»:K:Ẋf∑$βS2Ẏ  Try it Online! Probably not the intended solution, but this simply changes the compressed string from iraq to hours. The string itself doesn't matter, what matters is that it is a string made of 5 different characters. In fact, if you could change 2 characters, you could change it to Vyxal (₴ŀ) and it would still work. # Vyxals, 13 bytes, cracks A username’s answer khǍ⇩k•$Fk¹FsU

Try it Online!

@TwXµY *!ZøX ªX+YÑ}gB Bs

Try it here.

The trick here is to replace the unicode shortcut Ì (equal to gJ) with B (equals 11), breaking the chain and resulting in a comma operator instead. A nice brain teaser for sure.

# Yggdrasil, 3 bytes, cracks Aaron Miller's answer

?.:

Try it online!

Honestly just tried random commands until I found something that worked. I realised the . had to be there, and the : printed it all, so the first character needed modifying.

# brainfuck, cracks emanresu A's answer

.+[.[->+>+<<]+[>--<+++++]>++]

Try it online!

The unmodified program prints a single byte per iteration of the main loop. This means there are two ways to make it print more than the original two bytes:

• Make one of the inner loops print the entire output in one run. This seems unlikely as the loops are quite simple.
• Change the >+< in the end to get away from the zero cell. There are only a few changes that make some sense, so I got to >++ in a few attempts of trial and error.

 \   ÇÇÇÇÇÇÇÇ
/!\; ÇÇÇÇÇÇÇÇ
\n/!<ÇÇÇÇÇÇÇÇ
!   ÇÇÇÇÇÇÇÇ
>  ^ÇÇÇÇÇÇÇÇ
\\2   \! \/! ~! \
ÇÇÇ\ v! ~      /
ÇÇÇÇÇ\'"a"\ÇÇÇÇÇ
ÇÇÇÇÇÇÇÇ\\/ÇÇÇÇÇ

Try it online!

The changed character is the second backslash on the last line (previously a space), which sends the instruction pointer upwards to collect the string  /ÇÇÇÇÇ\ before continuing along a path that removes the last two values from the stack and prints 199 (the character code of Ç).

# Javascript (REPL?), cracks Nicolas B's answer

a=!![],a++,a+!![]

Replacing the += with +!.

# Python 3, cracks grandBagel's answer

a=6
b=++a++a++++a++a
a=5*5/5**2
print(b*1-b)

Try it online!

The thing that makes the output a float is multiplying b by a, which has a value of 1.0. If we multiply b by 1` instead, the result is an int.


text:
MathSciNet bibliographic data MR1091779 58F23 Gill, John Complex dynamics of the limit periodic system $F_n(z)=F_{n-1}(f_n(z)),\;f_n\to f$$F_n(z)=F_{n-1}(f_n(z)),\;f_n\to f$. Extrapolation and rational approximation (Luminy, 1989). J. Comput. Appl. Math. 32 (1990), no. 1-2, 89–96. Article

For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.


text:
# Difference between revisions of "1997 JBMO Problems/Problem 1"

## Problem

(Bulgaria) Show that given any 9 points inside a square of side length 1 we can always find 3 that form a triangle with area less than $\frac{1}{8}$


text:
# Force due to pressure on side of a container

1. ### loto

17
Hi gang,

I'm having an issue with one my homework problems. There is an L shaped tank, 3d in height, 1d in width. The verticle part of the L is 2d in height, and the horizontal part of the L is a 1d box. d = 8m
_
| |
| |_
|_ _|<--Force due to pressure needed for this side.

I need to find the pressure on the right wall of the horizontal part of the L. What I have managed to get so far is the equation:
F=(rho)(g)(width)(height^2)

Not very far, I know. I'm assuming I have to integrate this somehow, but I am unsure as to how to set it up. I'm not really looking for a solution, just a push in the right direction with regards to the integration, or in the correct direction if I am very, very wrong.

Thanks, in advance, for the help.

2. ### Fermat

876
Is this a Plan view (a view from on top) or a Side view?

3. ### loto

17
From the side, sorry.

4. ### Fermat

876
Ok, hang on a min.

5. ### LeonhardEuler

864
OK, I can explain this in general but I didn't completely follow all the dimentions you gave. From equilibrium considerations, the pressure at some depth, h is $P_0+\rho gh$ where $P_0$ is the pressure at the surface. The force at this hieght is PA where A is the area. Along the surface of the wall, the pressure is changing with the depth. At a given depth, h, the force is $(P_0+\rho gh)ldh$ where l is the length of the wall. Now it is a matter of integrating the infinitesimal forces at each height to get the total force.

6. ### loto

17
We are actually using gauge pressure, so it simplifies the formula you gave a bit. I did get the correct answer using your method, though, so thank you very much.

7. ### Fermat

876
Arrgh, too late!!

Oh, well. What LeonhardEuler said.

I enclose a small piece of work explaining how to do these types of integrals for pressure over a submerged area.

It may be useful for yourself or anyone else reading these posts.

File size:
84.6 KB
Views:
127


text:
## Electronic Communications in Probability

### On the infinite sums of deflated Gaussian products

#### Abstract

In this paper we derive the exact tail asymptotic behaviour of $S_\infty=\sum_{i=1}^\infty \lambda_i X_iY_i$, where $\lambda_i, i\ge 1,$ are non-negative square summable deflators (weights) and $X_i,Y_i, i\ge1,$ are independent standard Gaussian random variables. Further, we consider the tail asymptotics of $S_{\infty;p}=\sum_{i=1}^\infty\lambda_i X_i|Y_i|^p, p> 1$, and also discuss the influence on the asymptotic results when $\lambda_i$'s are independent random variables.

#### Article information

Source
Electron. Commun. Probab., Volume 17 (2012), paper no. 31, 8 pp.

Dates
Accepted: 23 July 2012
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465263164

Digital Object Identifier
doi:10.1214/ECP.v17-1921

Mathematical Reviews number (MathSciNet)
MR2955496

Zentralblatt MATH identifier
1266.60069

Subjects
Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G15: Gaussian processes

Rights

#### Citation

Hashorva, Enkelejd; Ji, Lanpeng; Tan, Zhongquan. On the infinite sums of deflated Gaussian products. Electron. Commun. Probab. 17 (2012), paper no. 31, 8 pp. doi:10.1214/ECP.v17-1921. https://projecteuclid.org/euclid.ecp/1465263164

#### References

• Arendarczyk, Marek; DÈ©bicki, Krzysztof. Asymptotics of supremum distribution of a Gaussian process over a Weibullian time. Bernoulli 17 (2011), no. 1, 194–210.
• Bovier, A.: Extreme values of random processes. Lecture Notes Technische Universität Berlin, 2005.
• DobriÄ‡, Vladimir; Marcus, Michael B.; Weber, Michel. The distribution of large values of the supremum of a Gaussian process. Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987). Astérisque No. 157-158 (1988), 95–127.
• Embrechts, Paul; KlÃ¼ppelberg, Claudia; Mikosch, Thomas. Modelling extremal events. For insurance and finance. Applications of Mathematics (New York), 33. Springer-Verlag, Berlin, 1997. xvi+645 pp. ISBN: 3-540-60931-8
• Hashorva, E., and Ji., L.: Asymptotics of the finite-time ruin probability for the Sparre Andersen risk model perturbed by inflated chi-process. (2012) Preprint.
• Hashorva, Enkelejd; Pakes, Anthony G.; Tang, Qihe. Asymptotics of random contractions. Insurance Math. Econom. 47 (2010), no. 3, 405–414.
• Hoeffding, W.: On a theorem of V.N. Zolotarev. Theory of Probab. Appl. IX 1, (1964), 89–91. (English translation)
• Ivanoff, B. Gail; Weber, N. C. Tail probabilities for weighted sums of products of normal random variables. Bull. Austral. Math. Soc. 58 (1998), no. 2, 239–244.
• Kallenberg, Olav. Some new representations in bivariate exchangeability. Probab. Theory Related Fields 77 (1988), no. 3, 415–455.
• Lifshits, M. A. Tail probabilities of Gaussian suprema and Laplace transform. Ann. Inst. H. Poincaré Probab. Statist. 30 (1994), no. 2, 163–179.
• Lifshits, M. A. Gaussian random functions. Mathematics and its Applications, 322. Kluwer Academic Publishers, Dordrecht, 1995. xii+333 pp. ISBN: 0-7923-3385-3
• Lifshits, M.A. Lectures on Gaussian Processes. Springer Briefs in Mathematics, Springer, 2012.
• Liu, Yan; Tang, Qihe. The subexponential product convolution of two Weibull-type distributions. J. Aust. Math. Soc. 89 (2010), no. 2, 277–288.
• Pakes, Anthony G. Convolution equivalence and infinite divisibility. J. Appl. Probab. 41 (2004), no. 2, 407–424.
• Resnick, Sidney I. Extreme values, regular variation and point processes. Reprint of the 1987 original. Springer Series in Operations Research and Financial Engineering. Springer, New York, 2008. xiv+320 pp. ISBN: 978-0-387-75952-4
• Zolotarev, V.M.: Concerning a certain probability problem. Theory of Probab. Appl. IX1, (1961), 201–204. (English translation)


text:
# Immune networks: multitasking capabilities near saturation

E. Agliari, A. Annibale, A. Barra, A. Coolen, D. Tantari

Journal of Physics A 46, 415003 (2013)

The immune system must simultaneously recall multiple defense strategies because many antigens can attack the host at the same time.

Pattern-diluted associative networks were introduced recently as models for the immune system, with nodes representing T-lymphocytes and stored patterns representing signalling protocols between T- and B-lymphocytes. It was shown earlier that in the regime of extreme pattern dilution, a system with NT T-lymphocytes can manage a number $N_B!=!\order(N_T^\delta)$ of B-lymphocytes simultaneously, with δ!<!1. Here we study this model in the extensive load regime NB!=!αNT, with also a high degree of pattern dilution, in agreement with immunological findings. We use graph theory and statistical mechanical analysis based on replica methods to show that in the finite-connectivity regime, where each T-lymphocyte interacts with a finite number of B-lymphocytes as NT→∞, the T-lymphocytes can coordinate effective immune responses to an extensive number of distinct antigen invasions in parallel. As α increases, the system eventually undergoes a second order transition to a phase with clonal cross-talk interference, where the system's performance degrades gracefully. Mathematically, the model is equivalent to a spin system on a finitely connected graph with many short loops, so one would expect the available analytical methods, which all assume locally tree-like graphs, to fail. Yet it turns out to be solvable. Our results are supported by numerical simulations.

#### Scale of non-locality for a system of n particles

S. Talaganis, I. Teimouri

Sub. to Physical Review D

#### A phase transition creates the geometry of the continuum from discrete space

R. Farr, T. Fink

Sub. to Physical Review E

#### How much can we influence the rate of innovation?

T. Fink, M. Reeves

#### Memristive networks: from graph theory to statistical physics

A. Zegarac, F. Caravelli

EPL

#### The statistical physics of real-world networks

G. Cimini, T. Squartini, F. Saracco, D. Garlaschelli, A. Gabrielli, G. Caldarelli

Nature Reviews Physics

#### On defining the Hamiltonian beyond quantum theory

D. Branford, O. Dahlsten, A. Garner

Foundations of Physics

#### Reconstructing grain-shape statistics from electron back-scatter diffraction microscopy

R. Farr, Z. Vukmanovic, M. Holness, E. Griffiths

Physical Review Materials

#### Tackling information asymmetry in networks: a new entropy-based ranking index

P. Barucca, G. Caldarelli, T. Squartini

Journal of Statistical Physics

#### Eigenvalues of subgraphs of the cube

B. Bollobás, J. Lee, S. Letzter

European Journal of Combinatorics

#### Maximum one-shot dissipated work from Rényi divergences

N. Halpern, A. Garner, O. Dahlsten, V. Vedral

Physical Review E

123 / 123 papers


text:
# Cohomological dimension of $\mathcal{B}_n$

What is the cohomological dimension of the braid group $\mathcal{B}_n$ on n-strands ? A reference would be appreciated.

-
An obvious upper bound would be $2n$ for $B_n$, by looking at the classifying space. I think you could even take an upper bound of $n-1$. –  Steve D Jun 29 '11 at 15:26
There is a tag already for cohomological-dimension, so I figured we may as well use it –  David White Jun 29 '11 at 16:30

The other answers are correct, but I wanted to point out a quick way to see that $B_n$ has cohomological dimension $n-1$.

One obtains a lower bound of $n-1$ since $\mathbb{Z}^{n-1}$ is a subgroup of $B_n$. Take $n-1$ disjoint non-isotopic loops forming a pants decomposition of the $n$ punctured plane, then Dehn twists about these give a subgroup isomorphic to $\mathbb{Z}^{n-1}$.

For an upper bound, one may use the fact that the moduli space of $n$ points in $\mathbb{C}$ (normalized to have sum $=0$) is a Stein manifold of complex dimension $n-1$, and therefore has a spine of dimension $n-1$. The fundamental group of this space is $B_n$. This space is equivalent to the space of monic polynomials of degree $n$ with zero trace (coefficient of degree $n-1=0$) and non-zero discriminant, which is how one may see that it is Stein (actually, an affine variety). The fact that the moduli space is a $K(B_n,1)$ follows from Teichmuller theory, or one may pass to the finite-sheeted cover of $n$ marked points, and see that this is an iterated surface bundle, and therefore its universal cover is contractible.

-

It's $n-1$.

See Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Inventiones Mathematicae, Volume 84, Number 1, 157-176.

-

There is a more geometric(?) method to obtain an upper bound of the cohomological dimension. Recall that the braid group $B_n$ is the fundamental group of the complement of the complexification of an essential hyperplane arrangement $\mathcal{A}_{n-1}$ in an $(n-1)$-dimensional vector space $\mathfrak{h}_{n-1}$ divided out by the action of the symmetric group $\Sigma_n$ of $n$ letters. Namely $$B_n = \pi_1((\mathfrak{h}_{n-1}\otimes\mathbb{C} - \cup_{H\in \mathcal{A}_{n-1}} H\otimes\mathbb{C})/\Sigma_n).$$ Since the complement is known to be $K(\pi,1)$, it serves as the classifying space of $B_n$.

In general, for any real hyperplane arrangement $\mathcal{A}$, Salvetti constructed a cell complex $\mathrm{Sal}(\mathcal{A})$ which is homotopy equivalent to the complement of the complexification. Furthermore if $\mathcal{A}$ is essential in an $n$ dimensional vector space, then $\dim\mathrm{Sal}(\mathcal{A}) = n$.

Thus the cohomological dimension of $B_n$ is bounded by $n-1$.

-

I also recommend M. Korkmaz' survey on low-dimensional homology of Mapping Class Groups:

Korkmaz, Mustafa(TR-MET) Low-dimensional homology groups of mapping class groups: a survey. (English summary) Turkish J. Math. 26 (2002), no. 1, 101–114. 57M05 (20J05 57M07 57N05)

It is in fact quite easy to find a subgroup of $P_n$ isomorphic to $\mathbb Z^{n-1}$. It is generated by the full twist of the first $k$ strands, where $k= 2,3,\dots, n$.


text:
Under the auspices of the Computational Complexity Foundation (CCF)

REPORTS > KEYWORD > DNF:
Reports tagged with DNF:
TR01-006 | 18th October 2000
Rocco Servedio

#### On Learning Monotone DNF under Product Distributions

We show that the class of monotone $2^{O(\sqrt{\log n})}$-term DNF
formulae can be PAC learned in polynomial time under the uniform
distribution. This is an exponential improvement over previous
algorithms in this model, which could learn monotone
$o(\log^2 n)$-term DNF, and is the first efficient algorithm
for ... more >>>

TR02-019 | 20th March 2002
Nader Bshouty, Lynn Burroughs

#### On the proper learning of axis parallel concepts

We study the proper learnability of axis parallel concept classes
in the PAC learning model and in the exact learning model with
membership and equivalence queries. These classes include union of boxes,
DNF, decision trees and multivariate polynomials.

For the {\it constant} dimensional axis parallel concepts $C$
we ... more >>>

TR02-069 | 14th November 2002
Luca Trevisan

#### A Note on Deterministic Approximate Counting for k-DNF

Revisions: 1

We describe a deterministic algorithm that, for constant k,
given a k-DNF or k-CNF formula f and a parameter e, runs in time
linear in the size of f and polynomial in 1/e and returns an
estimate of the fraction of satisfying assignments for f up to ... more >>>

TR06-066 | 5th May 2006
Vitaly Feldman

#### On Attribute Efficient and Non-adaptive Learning of Parities and DNF Expressions

Revisions: 1

We consider the problems of attribute-efficient PAC learning of two well-studied concept classes: parity functions and DNF expressions over $\{0,1\}^n$. We show that attribute-efficient learning of parities with respect to the uniform distribution is equivalent to decoding high-rate random linear codes from low number of errors, a long-standing open problem ... more >>>

TR09-141 | 19th December 2009
Anindya De, Omid Etesami, Luca Trevisan, Madhur Tulsiani

#### Improved Pseudorandom Generators for Depth 2 Circuits

We prove the existence of a $poly(n,m)$-time computable
pseudorandom generator which $1/poly(n,m)$-fools'' DNFs with $n$ variables
and $m$ terms, and has seed length $O(\log^2 nm \cdot \log\log nm)$.
Previously, the best pseudorandom generator for depth-2 circuits had seed
length $O(\log^3 nm)$, and was due to Bazzi (FOCS 2007).

It ... more >>>

TR14-099 | 7th August 2014
Gil Cohen, Igor Shinkar

#### The Complexity of DNF of Parities

We study depth 3 circuits of the form $\mathrm{OR} \circ \mathrm{AND} \circ \mathrm{XOR}$, or equivalently -- DNF of parities. This model was first explicitly studied by Jukna (CPC'06) who obtained a $2^{\Omega(n)}$ lower bound for explicit functions. Several related models have gained attention in the last few years, such as ... more >>>

TR15-100 | 16th June 2015
Bireswar Das, Patrick Scharpfenecker, Jacobo Toran

#### Succinct Encodings of Graph Isomorphism

It is well known that problems encoded with circuits or formulas generally gain an exponential complexity blow-up compared to their original complexity.

We introduce a new way for encoding graph problems, based on $\textrm{CNF}$ or $\textrm{DNF}$ formulas. We show that contrary to the other existing succinct models, there are ... more >>>

ISSN 1433-8092 | Imprint


text:
# Why use conda

The following quote is from Conda’s github page [1]

Conda is a cross-platform, Python-agnostic binary package manager. It is the package manager used by Anaconda installations, but it may be used for other systems as well. Conda makes environments first-class citizens, making it easy to create independent environments even for C libraries. Conda is written entirely in Python, and is BSD licensed open source.

The main advantage of using conda to manage your packages and environment is that it will work across platforms

conda also uses hard linking, so it is inexpensive to create multiple copies of the same package

# How to

One simple way to start is to first specify a environment.yml file

# environment.yml
name: psst-env
dependencies:
- python
- nose
- numpy
- pandas
- pip:
- pyomo

The name of the environment can be changed. Activate the environment by using the following.

source activate psst-env

Then you can create the environment by

conda env create

You can update the environment after adding a package to environment.yml by using the following

conda env update

Alternatively, you can create a new empty environment by using either one of the following

conda create -n pelican-env python=2
conda create --name pelican-env python=2

In this case, pelican-env is the name of the environment. You can follow the name of the environment with all the packages you want separated by spaces. You must have atleast one package to create a environment.

After the environment is created, you can source the environment :

source activate pelican-env

You can install packages here using one of the following :

conda install <PACKAGE-NAME>

When you have set up the environment and would like to share it, you can run the following to generate a .yml file

conda env export

I like to update by environment.yml by running the following

conda env export > environment.yml

# References

[1] “Conda/conda,” GitHub.

[2] “Conda - A Cross-Platform Package Manager for Any Binary Distribution.”.


text:
# Solve for x

• Aug 3rd 2010, 06:00 AM
saha.subham
Solve for x
21x^2 - 83x + 34 = 0

i know i will get two values of ex by factorization. but i am not being able to middle term factorize it.
• Aug 3rd 2010, 06:22 AM
Wilmer
• Aug 3rd 2010, 06:27 AM
eumyang
Quote:

Originally Posted by saha.subham
21x^2 - 83x + 34 = 0

i know i will get two values of ex by factorization. but i am not being able to middle term factorize it.

It's not factorable. For it to be factorable over the integers the discriminant would need to be a perfect square.
\begin{aligned}
b^2 - 4ac &= (-83)^2 - 4(21)34 \\
&= 6889 - 2856 \\
&= 4033
\end{aligned}

... and 4033 is not a perfect square.
• Aug 4th 2010, 03:03 AM
HallsofIvy
Relatively few quadratics can be factored with integer coefficients. "Complete the square" or use the quadratic formula instead.
• Aug 4th 2010, 09:38 AM
pencil09
if $ax^2+bx+c=0$
then $x=\frac{-b\pm \sqrt{(b^2-4ac)}}{2a}$


text:
lilypond-devel
[Top][All Lists]

## Re: Where to put new Scheme engravers?

 From: Trevor Daniels Subject: Re: Where to put new Scheme engravers? Date: Sun, 28 Sep 2014 11:21:06 +0100

```Urs, you wrote Sunday, September 28, 2014 10:38 AM

> Am 28.09.2014 11:36, schrieb Trevor Daniels:
>>
>> Most of the engravers are written in C++ and each is a separate
>> file in the lily/ directory.  There's a little above engravers here:
>>
>> http://www.lilypond.org/doc/v2.19/Documentation/contributor/engraver-tutorial
>
> Yes, I've seen that, and as you say this only gives information about
> C++ engravers.
>
> But I do have an engraver written in Scheme (not by me) that I find very
> useful. I could make it available in openlilylib, but I think it's a
> very common notation that should be available in LilyPond itself
> (printing double barlines before time signature changes).
>
> Do you want to say that it's inappropriate to add Scheme engravers to
> LilyPond and that it should be rewritten in C++ for that (which I couldn't?

Ah, sorry, I missed the keyword "Scheme" in your original post.

Currently there is just one Scheme engraver in the LilyPond codebase.
It is in scm/scheme-engravers.scm, and was added as part of issue 2445.
You may find it useful to read through that issue:


text:
×

# I took many hours to solve this problem given by my professor. I solve it manually like listing all possible answers but same as my earlier trials i didn't get the answer. Could someone help me with this. So much appreciated.

Okay so this is the problem...

The digits of a three-digit number are in geometric progression. If 596 is subtracted from this number ,the resulting three-digit number has its digits in arithmetic progression with a common difference equal to the reciprocal of the ratio of the geometric progression. Find the number.

Note by Crischell Baylon
3 years, 8 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

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

Sort by:

It is clear that the only possible cases are $$111, 124, 421, 248, 842, 139$$ and $$931$$. Further as the number is greater than $$596$$ we are left with only two cases: $$842$$ and $$931$$. Now just subtracting $$596$$ from each of these numbers it is clear that $$842$$ satisfies the given condition.

- 3 years, 8 months ago

You can also use the fact that the arithmetic progression is the reciprocal of the geometric progression ratio; that must mean that the geometric progression is decreasing, eliminating all but $$3$$ possibilities.

- 3 years, 8 months ago

Thank you very much for your response. I also found the number 842 but i felt uncertain of answering this number because of the rule of geometric progression. Therefore i tried other numbers..

- 3 years, 8 months ago


text:
# Homework Help: Permutation Group Proof

1. Mar 16, 2006

### thom

Sorry I am not used to using the tex code, but will learn and explain in words for now!

I am trying to prove that for all distinct x and y in a finite set X, there exists a function f in P(X) (the permutation group) such that f(x)=x' and f(y)=y'. Note: x' and y' are also distinct.

I have started by letting P(X) = the symmetry group and have shown that there exists a cycle (x y) which takes two distinct values and returns 2 distinct values. I am pretty sure x can equal y'. Now I am stuck! I have been told that a case-by-case analysis will probably be required. Can somebody point me in the right direction?

Last edited: Mar 16, 2006
2. Mar 16, 2006

### AKG

What are x' and y'? Do you mean to say that for all x, y, x' and y' in a finite set X, where x and y are distinct from one another, and x' and y' are distinct from one another, there exists a permutation f of X such that f(x) = x' and f(y) = y'? What do you mean "a cycle (x y) which takes two distinct values and returns 2 distinct values"? You should regard your cycle as a permutation of X, that is, it's domain is all of X, and it's range is also all of X. I can't see where you'd get stuck on this problem, but anyways, what's the first f that comes to mind? In which cases does this not work? What can you make work in these cases?

3. Mar 16, 2006

### thom

The cycle (x y) refers to a permutation which maps x to y and y to x. Here of course, f(x) = y and f(y) = x. This is the simplest bijection I can think of that holds the required conditions.
Basically I am thinking that the only bijections that don't 'work' are those that fix ANY points in the set X, the most trivial of these being the identity function. Hence, in the case of the symmetry group of the set X={1,...,n}, any n-cycle would satisfy f?
Even so, the trouble lies in extending this argument from a symmetry group to any permutation group of a nonempty finite set.

I can kind of see why you think this may seem a bit trivial, but this question is worth almost 25% of the weighting and my argument doesn't seem strong enough.

Last edited: Mar 17, 2006
4. Mar 17, 2006

### thom

Ah, no worries. I'm pretty sure I can prove this without using the symmetry group after all. I have acquired a paper on derangements which seems to suffice!

5. Mar 17, 2006

### matt grime

How can you prove it 'without using the symmetry group' when it is a question about the symmetry group (but not a very hard one)? I am assuming that 'symmetry group' of a finite set is 'the permutation' group. Why on earth would you want to use derangements, since it is trivial to find solutions fixing all but x,y,x',y' which are definitely not derangements. Indeed, If I were to ask this as a question about a 3 element set asking you to send 1 to 2 and 2 to 1 then it is impossible to do this with a derangement since 3 is necessarily fixed (and the element you require is (12)).

6. Mar 17, 2006

### AKG

I know what a I cycle is, my point was that (x y) does more than just map x to y and y to x. If z is different from both x and y, then it also maps z to z.
You mean EVERY point? There is only one function which fixes every point. Or do you mean the only bijections that won't work are those which fix SOME point in X? No, most bijections that work fix many points in X.
No, not any n cycle would satisfy f.
Do you mean to say that you have trouble extending the argument from the group of bijection of the set {1,...,n} to an arbitary set with n elements? This is not difficult, in fact it's so trivial, one hardly ever bothers to distinguish arbitrary sets of n elements from the set {1,...,n} in this context. Of course, it can be proved, and it is very, very simple.

Okay, first suppose x, y, x', and y' are all distinct. What's the most basic function that will satisfy the desired criteria? Will this work if x, y, x' and y' are not all distinct? In particular, what if x' $\neq$ y, but y' = x (of course, we still have x $\neq$ y and x' $\neq$ y'). Why won't your previous solution work? What changes can you make to get a function that does work? Now, what other possible cases are there?


text:
# Coulomb potential in Kohn-Sham DFT

1. Nov 20, 2012

### molkee

In Kohn-Sham DFT, the Coulomb potential, which is a component of the Kohn-Sham potential, is given by:

$v_H(\mathbf{r}) = \int \frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|}d\mathbf{r'}$

where $\rho(\mathbf{r'})$ is the electron density.

For molecular systems with exponential densities, this potential is known to be finite at any $\textbf{r}$.

How to prove it analytically just based on its definition?

Would a potential

$v(\mathbf{r}) = \int \frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|^n}d\mathbf{r'}$

(where $n$ is some nonnegative integer)

be also finite at any $\mathbf{r}$?

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted


text:
My watch list
my.chemeurope.com

# Nickel-iron battery

Battery specifications
Energy/weight 50 Wh/kg[1]
Energy/size
Power/weight 100 W/kg[1]
Charge/discharge efficiency 65%[2]
Energy/consumer-price 6-7 Wh/US\$[1]
Self-discharge rate 20%-40%/month[1]
Time durability 30[2] - 50 years[3][4]
Cycle durability Repeated deep discharge

does not reduce life significantly.[2]

Nominal Cell Voltage 1.2 V[1]
Charge temperature interval min.?-max.46°C[5]

The nickel-iron battery is a storage battery having a nickel(III) oxide-hydroxide cathode and an iron anode, with an electrolyte of potassium hydroxide. The active materials are held in nickel-plated steel tubes or perforated pockets. The nominal cell voltage is 1.2V. It is a very robust battery which is tolerant of abuse, (overcharge, overdischarge, short-circuiting and thermal shock) and can have very long life even if so treated. It is often used in backup situations where it can be continuously charged and can last for more than 20 years[citation needed]. Its limitations, namely, low specific energy, poor charge retention, and poor low-temperature performance, and its high cost of manufacture compared with the lead-acid battery led to a decline in usage. [1]

The ability of these batteries to survive frequent cycling is due to the low solubility of the reactants in the electrolyte. The formation of metallic iron during charge is slow because of the low solubility of the Fe3O4, which is good and bad. It is good because the slow formation of iron crystals preserves the electrodes, bad because it limits the high rate performance: these cells charge slowly, and are only able to discharge slowly.

Nickel-iron batteries have long been used in European mining operations because of their ability to withstand vibrations, high temperatures and other physical stress. They are being examined again for use in wind and solar power systems and for modern electric vehicle applications.

In many respects the Nickel/Iron battery was almost "too good". A battery that lasts for decades in many cases can outlast the equipment that it was originally designed to power. So from an economic standpoint lead acid, NiCd and other technologies have been deemed "good enough" and are the predominant technologies in use today even though they do not last as long as a Nickel/Iron counterpart.

## Electrochemistry

The half reactions at anode and cathode are given by:

$2 Ni O OH \; + \; 2 H_2O \; + \; 2e^- \quad \rightleftharpoons \quad 2 Ni(OH)_2 + 2 OH^-$

and

$Fe + 2 OH^- \quad \rightleftharpoons \quad Fe(OH)_2 + 2e^-.$

(Discharging is read left to right, charging is from right to left.) [2]

## History

Swedish inventor Waldemar Jungner had invented the Nickel-Cadmium battery in 1899. Junger experimented with substituting iron for the cadmium in varying proportions, including 100% iron. Jungner had already discovered that the main advantage over the nickel-cadmium chemistry was cost, but due to the poorer efficiency of the charging reaction and more pronounced formation of hydrogen (gassing), the nickel-iron technology was wanting and was abandoned. Jungner never patented the iron version of his battery. The battery was developed by Thomas Edison in 1901, and used as the energy source for electric vehicles, such as the Detroit Electric. Edison claimed the nickel-iron design to be, "... far superior to batteries using lead plates and acid"(lead-acid battery). Jungner's work was largely unknown in the US until the 1940's when Nickel Cadmium batteries went into production there. A 50 Volt Nickel-Iron battery was the main power supply in the WWII German V2 Rocket (together with two 16 Volt accumulators which powered the four gyroscopes), with a smaller version used in the V1 flying bomb. (viz. 1946 Operation Backfire blueprints.)

Edison's batteries were made from about 1903 to 1972 by the Edison Battery Storage Company located in East Orange, NJ. They were quite profitable for the company. In 1972 the battery company was sold to the Exide Battery Corporation which discontinued making the battery in 1975. Edison was disappointed that his battery was not adopted for starting internal combustion engines and that electric vehicles went out of production only a few years after his battery was introduced. He actually developed the battery to be the battery of choice for electric vehicles which he thought would be the preferred transportation mode in the early 1900's. The battery enjoyed wide use for railroad signaling, fork lift, and standby power applications. There are no Nickel Iron batteries manufactured in the Western world at this time (2007), but they are still manufactured in China.

## References

1. ^ a b c d e mpoweruk.com: Accumulator and battery comparisons (pdf)
2. ^ a b c Mpower: Nickel Iron Batteries, Axeonpower: Nickel Iron Batteries
3. ^ "Nickel Iron Battery Frequently Asked Questions" BeUtilityFree
4. ^ a description of the Chinese nickel-iron battery from BeUtilityFree
5. ^ Web archive backup: Edison Battery Booklet original instruction book for the Edison battery


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# Does Regularized Logistic Regression Produce Calibrated Results?

It has been asked and addressed here that logistic regression modelling is calibrated already and there is no need for calibration of it. To me it seems the argument provided there does not follow when there is a regularization involved. Can somebody please tell me if indeed it is the case that a regularized logistic regression model is not calibrated?

• This answer is perhaps more relevant: stats.stackexchange.com/a/16541. In a lot of regularised logistic regression I have seen (e.g. glmnet), the regularisation is not applied to the intercept. So, I do not see why the calibration should be affected in the case of regularisation. We should be able to check this by repeating cardinal's calculation using the penalised log-likelihood instead of the log-likelihood. – Alex Aug 23 '18 at 4:57
• @Alex, thanks for your input. Cardinl's reasoning in my humble opinion does not follows since after taking derivates there will be an extra term of $+\lambda \beta_i$ in the equations instead of zero on the other side. In other words the regularization factor of. $\lambda \|\beta\|^2$ will not cancel when taking derivatives. – Cupitor Aug 24 '18 at 0:02
• Maybe I am being dense here, but I don't see why, for the constant term, i.e. $\beta_0$, there would be an "extra term of $+\lambda\beta_0$"? The intercept is not usually included in the penalty term so shouldn't the penalty terms be treated as a constant with respect to the partial derivative? – Alex Aug 24 '18 at 1:39
• See for example, stats.stackexchange.com/a/161689, which deals with the linear regression case. – Alex Aug 24 '18 at 1:40
• Oh I am totally sorry. I get it now. Fair enough! That solves the issue :) – Cupitor Aug 24 '18 at 7:27

The answer depends entirely on the amount of penalization used. If too little, the model will be seen to be overfitted when evaluated on an independent sample. If too much penality, it will be found to be underfitted. The goal is to solve for the penalty that gets it "just right". Two ways to do this are cross-validation (e.g., 100 repeats of 10-fold cross-validation) or computing the effective AIC and solving for the penalty that optimizes it.

The calibration curve is the plot of x=predicted probability of the event vs. y=actual probability of the event. The actual probability is obtained (in an independent sample, or using the bootstrap to correct the "apparent" calibration) by running a smoother on $(\hat{P}, Y)$, where $Y$ is the vector of binary outcomes. If the calibration curve is linear, it can be summarized by its intercept and slope. When the slope is greater than 1 there is underfitting, and when the slope is less than 1 there is overfitting (regression to the mean; low $\hat{P}$ are too low and high ones are too high).

The Bayesian approach to this has major advantages over what is described above, including:

1. the penalty is automatically estimated if you do a reasonable job of specifying a prior distribution for the penalty parameters (usually stated in terms of the reciprocal, i.e., the variance of random effects)
2. once finished, the Bayesian posterior distribution works exactly as it should, whereas the frequentist approach does not give us confidence intervals or hypothesis tests once penalization is used
• Thank you. But I am not sure if I understand the connection of your answer to calibration. No matter how the regularization coefficient is chosen eventually we end up with a learned model based on optimization. The question is whether that model is calibrated meaning that $\hat{p}(y=1)\approx p(y=1)$. – Cupitor Aug 20 '18 at 22:39
• Me too. I don't see how that answer relates to calibration (maybe it's implicit in the answer). I know, empirically, that constrained regressions (e.g. using an offset term) are not necessarily calibrated, and I just performed a penalised regression that is close to, but not exactly, calibrated on the training sample (let alone a hold-out sample). – Ross Gayler Aug 21 '18 at 0:15
• @RossGayler thank you for sharing the sentiment. I was confused. I can see that the proof does not follow so it is not surprising. But it is surprising that there is nothing written about it out there. – Cupitor Aug 21 '18 at 20:11
• In light of these comments, Frank, could you update this answer to make any implicit connections to calibration easier for readers to perceive? – gung - Reinstate Monica Aug 21 '18 at 20:27
• I agree with @frank-harrell. Perhaps you are working from a different definition of calibration. Your (Cupitor) definition in the first comment after Frank's answer only requires that the global mean predicted probability be correct. Whereas the definition that Frank (and I) use is that the actual probability conditional on the predicted probability should equal the predicted probability. Hence Frank's reference to "the calibration curve". Anyway, empirically I have in front of me right now two calibration curves for the same regression: unpenalised (slope=1), penalised (slope=1.06). – Ross Gayler Aug 24 '18 at 22:18


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# Charge of polyatomic ions doesn't equal the charge of ions it consists of?

I'm taking entry level collage chem. class, and we got to this topic that I don't really understand. I think I understood what an ion is (a atom whose amount of electrons is not equal to amount of protons, so it has positive or negative charge). But then polyatomic ions... Lets say peroxide; consists of two Oxygens, whose ions have charge of -2. Consequently, two of these anions are suppose to have charge of -4, but peroxide has charge of -2. How does this work? If polyatomic ions don't just sum the charge of ions they consist of, how else can I predict and understand their charge?

H-O-O-H. The bond in the middle is covalent. Deprotonate to $\ce{{HOO}^{-}}$ (hydroperoxide) then $\ce^{^{-}{OO}^{-}}$, peroxide dianion, $\ce{O2^{2-}}$. You learn it by doing it, having examples, learning what to look for, and gaining understanding. Both oxygens are in the -1 oxidation state.