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• ### The white dwarf binary pathways survey - II. Radial velocities of 1453 FGK stars with white dwarf companions from LAMOST DR4(1708.09480)

Aug. 30, 2017 astro-ph.SR
We present the second paper of a series of publications aiming at obtaining a better understanding regarding the nature of type Ia supernovae (SNIa) progenitors by studying a large sample of detached F, G and K main sequence stars in close orbits with white dwarf companions (i.e. WD+FGK binaries). We employ the LAMOST (Large Sky Area Multi-Object Fibre Spectroscopic Telescope) data release 4 spectroscopic data base together with GALEX (Galaxy Evolution Explorer) ultraviolet fluxes to identify 1,549 WD+FGK binary candidates (1,057 of which are new), thus doubling the number of known sources. We measure the radial velocities of 1,453 of these binaries from the available LAMOST spectra and/or from spectra obtained by us at a wide variety of different telescopes around the globe. The analysis of the radial velocity data allows us to identify 24 systems displaying more than 3sigma radial velocity variation that we classify as close binaries. We also discuss the fraction of close binaries among WD+FGK systems, which we find to be ~10 per cent, and demonstrate that high-resolution spectroscopy is required to efficiently identify double-degenerate SNIa progenitor candidates.
• ### Variation Of The Tully-Fisher Relation As A Function Of The Magnitude Interval Of A Sample Of Galaxies(1609.08534)

Sept. 27, 2016 astro-ph.GA
In this paper we carry out a preliminary study of the dependence of the Tully-Fisher Relation (TFR) with the width and intensity level of the absolute magnitude interval of a limited sample of 2411 galaxies taken from Mathewson \& Ford (1996). The galaxies in this sample do not differ significantly in morphological type, and are distributed over an $\sim11$-magnitude interval ($-24.4 < I < -13.0$). We take as directives the papers by Nigoche-Netro et al. (2008, 2009, 2010) in which they study the dependence of the Kormendy (KR), the Fundamental Plane (FPR) and the Faber-Jackson Relations (FJR) with the magnitude interval within which the observed galaxies used to derive these relations are contained. We were able to characterise the behaviour of the TFR coefficients $(\alpha, \beta)$ with respect to the width of the magnitude interval as well as with the brightness of the galaxies within this magnitude interval. We concluded that the TFR for this specific sample of galaxies depends on observational biases caused by arbitrary magnitude cuts, which in turn depend on the width and intensity of the chosen brightness levels.
• ### Doppler Tomography and Photometry of the Cataclysmic Variable 1RXS J064434.5+334451(1609.03154)

Sept. 11, 2016 astro-ph.SR
We have obtained simultaneous photometric and spectroscopic observations of the cataclysmic variable 1RXS J064434.5+334451. We have calibrated the spectra for slit losses using the simultaneous photometry allowing to construct reliable Doppler images from H$\alpha$ and HeII 4686 emission lines. We have improved the ephemeris of the object based on new photometric eclipse timings, obtaining $HJD = 2453403.759533 + 0.26937446E$. Some eclipses present a clear internal structure which we attribute to a central HeII emission region surrounding the white dwarf, a finding supported by the Doppler tomography. This indicates that the system has a large inclination angle $i=78 \pm 2^{\circ}$. We have also analysed the radial velocity curve from the emission lines to measure its semi--amplitude, $K_1$, from H$\alpha$ and HeII 4686 and derive the masses of the components: $M_1=0.82\pm0.06$ M$_{\odot}$, $M_2=0.78\pm0.04$ M$_{\odot}$ and their separation $a=2.01\pm0.06$ $R_{\odot}$. The Doppler tomography and other observed features in this nova-like system strongly suggests that this is a SW Sex-type system.
• ### CCD photometry of the Globular Cluster NGC 6093(1606.04526)

June 14, 2016 astro-ph.GA, astro-ph.SR
In this paper we present photometric CCD observations of the globular cluster NGC 6093 (M80) in filters B, V , R and I. We produce the colour-magnitude diagrams for this object and obtain values for its metallicity [Fe/H], reddening E(V-B), E(V-I) and distance modulus (m - M)_0.
• ### A Radial Velocity Study of the Intermediate Polar EX Hydrae(1606.03734)

June 12, 2016 astro-ph.SR
A study on the intermediate polar EX Hya is presented, based on simultaneous photometry and high dispersion spectroscopic observations, during four consecutive nights. The strong photometric modulation related to with the 67-min spin period of the primary star is clearly present, as well as the narrow eclipses associated to the orbital modulation. Since our eclipse timings have been obtained almost 91,000 cycles since the last reported observations, we present new linear ephemeris, although we cannot rule out a sinusoidal variation suggested by previous authors. The system mainly shows double-peaked H$\alpha$, H$\beta$ and HeI $\lambda$5876 \AA emission lines. From the profile of the H$\alpha$ line, we find two components; one with a steep rise and velocities not larger than $\sim$1000 km s$^{-1}$ and another broader component extending up to $\sim$2000 km s$^{-1}$, which we interpret as coming mainly from the inner disc. A strong and variable hotspot is found and a stream-like structure is seen at times. We show that the best solution correspond to $K_1 = 58 \pm 5$ km s$^{-1}$ from H$\alpha$, from the two emission components, which are both in phase with the orbital modulation. We remark on a peculiar effect in the radial velocity curve around phase zero, which could be interpreted as a Rositter-MacLaughlin-like effect, which has been taken into account before deriving $K_1$. This value is compatible with the values found in high-resolution both in the ultraviolet and X-ray. We find: $M_{1} = 0.78 \pm 0.03$ M$_{\odot}$, $M_{2} = 0.10 \pm 0.02$ M$_{\odot}$ and $a = 0.67 \pm 0.01$ R$_{\odot}$. Doppler Tomography has been applied, to construct six Doppler tomograms for single orbital cycles spanning the four days of observations to support our conclusions. Our results indicate that EX Hya has a well formed disc and that the magnetosphere should extend only to about $3.75\,R_{\rm{WD}}$.
• ### ROTSE1 J164341.65+251748.1: a new W UMa-type eclipsing binary(1606.03464)

June 10, 2016 astro-ph.SR, astro-ph.IM
ROTSE1 J164341.65+251748.1 was photometrically observed in the V band during three epochs with the 0.84-m telescope of the San Pedro M\'artir Observatory in Mexico. Based on additional BVR photometry, we find that the primary star has a spectral type around G0V. The light curve of the system is typical of a W~UMa type binary stars and has an orbital period of $\sim$ 0.323 days. In an effort to gain a better understanding of the binary system and determine its physical properties, we analyzed the light curve with the Wilson and Devinney method. We found that ROTSE1 J164341.65+251748.1 has a mass ratio of $\sim$ 0.34 and that the less massive component is over 230 K hotter than the primary star. The inclination of the system is $\sim$ 84.6 degrees, and the {\bf degree} of over-contact is 11\%. The analysis shows the presence of variable bright spots on the primary star.
• ### Photometry of the eclipsing cataclysmic variable SDSS J152419.33+220920.0(1304.7977)

April 30, 2013 astro-ph.SR, astro-ph.HE
Aims. We present new photometry of the faint and poorly studied cataclysmic variable SDSS J152419.33+220920.0, analyze its light curve and provide an accurate ephemeris for this system. Methods. Time-resolved CCD differential photometry was carried out using the 1.5m and 0.84m telescopes at the Observatorio Astronomico Nacional at San Pedro Martir. Results. From time-resolved photometry of the system obtained during six nights (covering more than twenty primary eclipse cycles in more than three years), we show that this binary presents a strong primary and a weak secondary modulation. Our light curve analysis shows that only two fundamental frequencies are present, corresponding to the orbital period and a modulation with twice this frequency. We determine the accurate ephemeris of the system to be HJD(eclipse)= 2454967.6750(1) + 0.06531866661(1) E. A double-hump orbital period modulation, a standing feature in several bounce-back systems at quiescence, is present at several epochs. However, we found no other evidence to support the hypothesis that this system belongs to the post-minimum orbital-period systems.
• ### Multisite spectroscopic seismic study of the beta Cep star V2052 Oph: inhibition of mixing by its magnetic field(1208.4250)

Aug. 21, 2012 astro-ph.SR
We used extensive ground-based multisite and archival spectroscopy to derive observational constraints for a seismic modelling of the magnetic beta Cep star V2052 Ophiuchi. The line-profile variability is dominated by a radial mode (f_1=7.14846 d^{-1}) and by rotational modulation (P_rot=3.638833 d). Two non-radial low-amplitude modes (f_2=7.75603 d^{-1} and f_3=6.82308 d^{-1}) are also detected. The four periodicities that we found are the same as the ones discovered from a companion multisite photometric campaign (Handler et al. 2012) and known in the literature. Using the photometric constraints on the degrees l of the pulsation modes, we show that both f_2 and f_3 are prograde modes with (l,m)=(4,2) or (4,3). These results allowed us to deduce ranges for the mass (M \in [8.2,9.6] M_o) and central hydrogen abundance (X_c \in [0.25,0.32]) of V2052 Oph, to identify the radial orders n_1=1, n_2=-3 and n_3=-2, and to derive an equatorial rotation velocity v_eq \in [71,75] km s^{-1}. The model parameters are in full agreement with the effective temperature and surface gravity deduced from spectroscopy. Only models with no or mild core overshooting (alpha_ov \in [0,0.15] local pressure scale heights) can account for the observed properties. Such a low overshooting is opposite to our previous modelling results for the non-magnetic beta Cep star theta Oph having very similar parameters, except for a slower surface rotation rate. We discuss whether this result can be explained by the presence of a magnetic field in V2052 Oph that inhibits mixing in its interior.
• ### Astroclimate at San Pedro M\'artir I: 2004-2008 Seeing Statistics from the TMT Site Testing Data(1206.3319)

Aug. 21, 2012 astro-ph.IM
We present comprehensive seeing statistics for the San Pedro M\'artir site derived from the Thirty Meter Telescope site selection data. The observations were obtained between 2004 and 2008 with a Differential Image Motion Monitor (DIMM) and a Multi Aperture Scintillation Sensor (MASS) combined instrument (MASS--DIMM). The parameters that are statistically analised here are: whole atmosphere seeing -measured by the DIMM-; free atmosphere seeing --measured by the MASS--; and ground-layer seeing (GL) --difference between the total and free-atmosphere seeing--. We made a careful data coverage study along with statistical distributions of simultaneous MASS--DIMM seeing measurements, in order to investigate the nightly, monthly, seasonal, annual and global behaviour, as well as possible hourly seeing trends. Although this campaign covers five years, the sampling is uneven, being 2006 and 2007 the best sampled years in terms of seasonal coverage. The overall results yield a median seeing of 0.78 (DIMM), 0.37 (MASS) and 0.59 arcsec (GL). The strongest contribution to the whole atmosphere seeing comes, therefore, from a strong ground layer. We find that the best season is summer, while the worst one is winter, in accordance with previous studies. It is worth noting that the best yearly results are correlated with the best sampled years. The hourly analysis shows that there is no statistically significant tendency of seeing degradation towards dawn. The seeing values are slightly larger than those reported before. This may be caused by climate changes.
• ### Doppler Tomography in Cataclysmic Variables: an historical perspective(1201.3075)

Jan. 15, 2012 astro-ph.SR
To mark the half-century anniversary of this newly-born field of Cataclysmic Variables, a special emphasis is made in this review, on the Doppler Effect as a tool in astrophysics. The Doppler Effect was in fact, discovered almost 170 years ago, and has been since, one of the most important tools which helped to develop modern astrophysics. We describe and discuss here, its use in Cataclysmic Variables which, combined with another important tool, the tomography, first devised for medical purposes 70 years ago, helped to devise the astronomical Doppler Tomography, developed only two decades ago. A discussion is made since the first trailed spectra provided a one dimensional analysis of these binaries; on the establishment of a 2D velocity profiling of the accretion discs; and unto modern techniques, which include Roche Tomography, time modulation and 3D imaging.
• ### SDSS J0018+3454: An Extremely Long Orbital Period CV(1009.5803)

Sept. 29, 2010 astro-ph.SR
We present here results of an optical spectroscopic study of a new Cataclysmic Variable SDSS J001856.93+345444.3. We demonstrate that the most probable value of the orbital period of the system is Porb = 0.6051 \pm 0.022 days (=14.5226 hours), based on the measurements of radial velocity of a complex of absorption features emanating from the K2-K4V type secondary component. However, the radial velocity measurements from the emission lines are best folded with the period Pem = 0.5743day (=13.78 hours). The gamma-velocity of the emission lines varies significantly from epoch to epoch. There is an underlying broader and weaker component to the emission lines, which we could not resolve. Based on the appearance of the emission lines, the presence of very strong He II lines and the moderate polarization detected by Dillon et al. (2008), we conclude that SDSS J0018+3454 is an asynchronous magnetic CV (Polar).
• ### G 112-29 (=NLTT 18149), a Very Wide Companion to GJ 282 AB with a Common Proper Motion, Common Parallax, Common Radial Velocity and Common Age(0910.2956)

Oct. 15, 2009 astro-ph.GA
We have made a search for common proper motion (CPM) companions to the wide binaries in the solar vicinity. We found that the binary GJ 282AB has a very distant CPM companion (NLTT 18149) at a separation $s=1.09 \arcdeg$. Improved spectral types and radial velocities are obtained, and ages determined for the three components. The Hipparcos trigonometric parallaxes and the new radial velocities and ages turn out to be very similar for the three stars, and provide strong evidence that they form a physical system. At a projected separation of 55733AU from GJ 282AB, NLTT 18149 ranks among the widest physical companions known.
• ### High Dispersion Absorption-line Spectroscopy of AE Aqr(0804.0291)

April 2, 2008 astro-ph
High-dispersion time-resolved spectroscopy of the unique magnetic cataclysmic variable AE Aqr is presented. A radial velocity analysis of the absorption lines yields K_2 = 168.7+/- 1 km/s. Substantial deviations of the radial velocity curve from a sinusoid are interpreted in terms of intensity variations over the secondary star's surface. A complex rotational velocity curve as a function of orbital phase is detected which has a modulation frequency of twice the orbital frequency, leading to an estimate of the binary inclination angle that is close to 70^o. The minimum and maximum rotational velocities are used to indirectly derive a mass ratio of q= 0.6 and a radial velocity semi-amplitude of the white dwarf of K_1 = 101+/-3 km/s. We present an atmospheric temperature indicator, based on the absorption line ratio of Fe I and Cr I lines, whose variation indicates that the secondary star varies from K0 to K4 as a function of orbital phase. The ephemeris of the system has been revised, using more than one thousand radial velocity measurements, published over nearly five decades. From the derived radial velocity semi-amplitudes and the estimated inclination angle, we calculate that the masses of the stars are M_1 = 0.63+/-0.05M_sun; M_2 = 0.37+/-0.04 M_sun, and their separation is a = 2.33+/-0.02R_sun. Our analysis indicates the presence of a late-type star whose radius is larger, by a factor of nearly two, than the radius of a normal main sequence star of its mass. Finally we discuss the possibility that the measured variations in the rotational velocity, temperature, and spectral type of the secondary star as functions of orbital phase may, like the radial velocity variations, be attributable to regions of enhanced absorption on the star's surface.
• ### Determination of the basic parameters of the dwarf nova EY Cygni(astro-ph/0611267)

Nov. 8, 2006 astro-ph
High-dispersion spectroscopy of EY Cyg obtained from data spanning twelve years show, for the first time, the radial velocity curves from both emission and absorption line systems, yielding semi-amplitudes K_{em}=24+/- 4 km s^-1 and K_{abs}=54+/- 2 km s^-1. The orbital period of this system is found to be 0.4593249(1)d. The masses of the stars, their mass ratio and their separation are found to be M_1 sin^3 i = 0.015+/-0.002 M_sun, M_2 sin^3 i = 0.007+/-0.002 M_sun, q = K_1/K_2 = M_2/M_1 = 0.44+/-0.02 and a sin i = 0.71+/-0.04 R_sun. We also found that the spectral type of the secondary star is around K0,consistent with an early determination by Kraft(1962). From the spectral type of the secondary star and simple comparisons with single main sequence stars, we conclude that the radius of the secondary star is about 30 per cent larger than a main sequence star of the same mass. We also present VRI CCD photometric observations, some of them simultaneous with the spectroscopic runs. The photometric data shows several light modulations, including a sinusoidal behaviour with twice the frequency of the orbital period, characteristic of the modulation coming from an elongated, irradiated secondary star. Low and high states during quiescence are also detected and discussed. From several constrains, we obtain tight limits for the inclination angle of the binary system between 13 and 15 degrees, with a best value of 14 degrees obtained from the sinusoidal light curve analysis. From the above results we derive masses M_1 = 1.10+/-0.09 M_sun, M_2 = 0.49+/-0.09 M_sun, and a binary separation a = 2.9+/- 0.1 R_sun.
• ### The kinematics of the most oxygen-poor planetary nebula PN G135.9+55.9(astro-ph/0308225)

Aug. 13, 2003 astro-ph
PN G135.9+55.9 is a compact, high excitation nebula that has been identified recently as the most oxygen-poor halo planetary nebula. Given its very peculiar characteristics and potential implications in the realms of stellar and Galactic evolution, additional data are needed to firmly establish its true nature and evolutionary history. Here we present the first long-slit, high spectral resolution observations of this object in the lines of H$\alpha$ and He II 4686. The position-velocity data are shown to be compatible with the interpretation of PN G135.9+55.9 being a halo planetary nebula. In both emission lines, we find the same two velocity components that characterize the kinematics as that of an expanding elliptical envelope. The kinematics is consistent with a prolate ellipsoidal model with axis ratio about 2:1, a radially decreasing emissivity distribution, a velocity distribution that is radial, and an expansion velocity of 30 km/s for the bulk of the material. To fit the observed line profiles, this model requires an asymmetric matter distribution, with the blue-shifted emission considerably stronger than the red-shifted emission. We find that the widths of the two velocity components are substantially wider than those expected due to thermal motions, but kinematic structure in the projected area covered by the slit appears to be sufficient to explain the line widths. The present data also rule out the possible presence of an accretion disk in the system that could have been responsible for a fraction of the H$\alpha$ flux, further supporting the planetary nebula nature of PN G135.9+55.9.


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The Seifert-van Kampen Theorem

# The Seifert-van Kampen Theorem

We are now ready to state a very important theorem for determining the fundamental groups of certain spaces. This theorem is called the Seifert-van Kampen theorem. We first state the simplest version of the theorem and then state a more general version.

 Theorem 1 (The Seifert-van Kampen Theorem): Let $X$ be a topological space and let $\{ U_1, U_2 \}$ be an open cover of $X$. Let $\pi_1(U_1, x) = \langle A_1: \mathcal R_1 \rangle$ and $\pi_1(U_2, x) = \langle A_2 : \mathcal R_2 \rangle$. If: 1) $U_1$, $U_2$, and $U_1 \cap U_2$ are path connected. 2) $x \in U_1 \cap U_2$. Then $\pi_1(X, x)$ is the free product of $\pi_1(U_1, x)$ and $\pi_1(U_2, x)$ with amalgamation by $\pi_1(U_1 \cap U_2, x)$ by $i_{1*} : \pi_1(U_1 \cap U_2, x) \to \pi_1(U_1, x)$ and $i_{2*} : \pi_1(U_1 \cap U_2, x) \to \pi_1(U_2, x)$, that is, $\pi_1(X, x) = \langle A_1 \cup A_2 : \mathcal R_1 \cup \mathcal R_2 \cup \{ i_{1*}(h) = i_{2^*}(h) : h \in \pi_1(U_1 \cap U_2, x) \} \rangle$.

We denote the inclusion map of $U_1 \cap U_2$ in $U_1$ by $i_1 : U_1 \cap U_2 \to U_1$, and we denote the inclusion map of $U_1 \cap U_2$ in $U_2$ by $i_2 : U_1 \cap U_2 \to U_2$. Hence, $i_{1*} : \pi_1(U_1 \cap U_2, x) \to \pi_1(U_1, x)$ and $i_{2*} : \pi_1(U_1 \cap U_2, x) \to \pi_1(U_2, x)$ are the induced mapping as seen on The Induced Mapping from the Fundamental Groups of Two Topological Spaces page, and they are also group homomorphisms.

 Theorem 2 (The General Seifert-van Kampen Theorem): Let $X$ be a topological space and let $\{ U_j : j \in J \}$ be an open cover of $X$. Let $\pi_1(U_j, x) = \langle A_j, \mathcal R_j \rangle$ for each $j \in J$. If: 1) Each intersection of four or less $U_j$s is path connected. 2) $\displaystyle{x \in \bigcap_{j \in J} U_j}$. Then $\pi_1(X, x)$ is the free product of the $\pi_1(U_j, x)$s with amalgamation by the maps $\phi_{ij*}^{i} : \pi_1(U_i \cap U_j, x) \to \pi_1(U_i, x)$ and $\phi_{ij*}^{j} : \pi_1(U_i \cap U_j, x) \to \pi_1(U_j, x)$, that is, $\pi_1(X, x) = \biggr \langle \bigcup_{j \in J} A_j : \bigcup_{j \in J} \mathcal R_j \cup \bigcup_{i, j \in J} \{ \phi_{ij*}^{i}(h) = \phi_{ij*}^{j}(h) : h \in \pi_1(U_i \cap U_j, x) \} \biggr \rangle$.

Here, $\phi_{ij}^{i} : U_i \cap U_j \to U_i$ and $\phi_{ij}^{j} : U_i \cap U_j \to U_j$ are the usual inclusion maps.

We look at some examples of applying the Seifert-van Kampen theorem on the following pages:

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# Proof of quasi-concavity by induction or direct proof

I’m having a hard time trying to prove that a univariate function is (strictly) quasi-concave (sorry for the long post).

I think the main reason for my struggle is:

(1) that the function is quite complex (not complicated but very large) and

(2) That I can’t find a good example where someone is actually going through a proof of quasi-concavity (except for something like $f(x,y)=x^2+y^2$).

Therefore even an actual example on how to prove quasi-concavity would be quite useful for me.

In the next few lines I will present the function and its properties.

I have a function $f(x)$ that consists of three linear functions $g(x), h(x), k(x)$ and that looks like this: $$f(x)= \sum_{i=1}^n g(x)(1-h(x))^i \left ( {(1+k(x)^n -(1+k(x))^{i-1} \over (1+k(x))^n-1} k(x)\right ) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -m\sum_{i=1}^ng(x)(1-h(x))^{i-1} \left ( {(1+k(x)^n \ -(1+k(x))^{i-1} \over (1+k(x))^n-1}\right )~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-\sum_{i=1}^ng(x)h(x)(1-h(x))^{i-1} \left ( {(1+k(x)^n -(1+k(x))^{i-1} \over (1+k(x))^n-1} \right )~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$ with $n \ge1, n \in \mathbb{N}$ and $x \in \mathbb{R}^{+}$, as well as $f(x)\ge 0$.

All the sub functions of $f(x)$ are linear functions with the following properties (one can also assume that they are continuous and differentiable): $$g(x)>0; ~ g'(x)>0; g''(x)=0$$ $$0<h(x)<1; ~h('x)>0; h''(x)=0$$ $$0<k(x)<1; ~k'(x)<0; k''(x)=0$$

Using some properties of the sub functions one can see that:

$0<(1-h(x))^i<1$ and because of $0<h(x)<1$ the term $0<(1-h(x))^i<1$ is always decreasing.

It is also clear that $( {(1+k(x)^n -(1+k(x))^{i-1} \over (1+k(x))^n-1} )$ is decreasing with an increase in $x$ which is also true for $( {(1+k(x)^n -(1+k(x))^{i-1} \over (1+k(x))^n-1} )k(x)$ using the product rule.

Apart from that all of the three expressions after the sigma signs are positive (with this I mean that I'm only interested in positive values). Using this I want to show that $f(x)$ is either decreasing over $x$ or is inverted U-shaped and therefore is strictly quasi concave.

In the next part I will give my impression on $f(x)$ with two ideas on how to proof the statement.

Looking at strictly quasi concavity there are two statements that I think could be useful in proving that $f(x)$ actually is strictly quasi concave:

(1) $f(\lambda x_1+(1-\lambda)x_2) > min {f(x_1);f(x_2)}$ with $0<\lambda<1$

or

(2) $f'(x_2)(x_1-x_2)>0$ for $f(x_1)>f(x_2)$

One idea I had is to proof the statement by induction:

1. Use $n=1$ and show that in this case the function is quasi-concave
2. Assume that for n=l the function is quasi-concave
3. Proof that the function is quasi-concave for l+1 using the hypothesis from 2.

Step 1: For n=1 $f(x)$ is simply:

$$f(x)=g(x)((1-h(x))k(x)-mg(x)-g(x)h(x)$$

Using the statement (1) for strict quasi-concavity one can see that this is true because of the linearity of $g(x)$, $h(x)$, and $k(x)$ (But this is again a point where an example of someone actually doing something similarly would be nice).

While step 2 is just $f(x)$ with $l$ I have some trouble going through step 3.

Another solution might be to use the linear properties of $g(x), h(x)$ and $k(x)$. The proof goes something like this: Because $g(x)$ is linear $g(\lambda x_1 + (1-\lambda) x_2) > min (g(x_1); g(x_2))$ is always true. Apart from that it is clear that because of $(1-h(x))$ $f(x)$ can not be increasing all the time is has to decrease at a certain point. The same statement can be made about $h(x)$ as well as $k(x)$.

Therefore the first expression $$\sum_{i=1}^n g(x)(1-h(x))^i \left ( {(1+k(x)^n -(1+k(x))^{i-1} \over (1+k(x))^n-1} k(x)\right )$$ has the same property of being bigger than the $min f(x_1), f(x_2)$ and is therefore strictly quasi concave.
But I don't think this is a definite proof. Apart from that I have some trouble trying to wrap my head about the connection between the three expressions (what is if they are all three strictly quasi concave? Does this also mean the the sum of them is quasi concave?)

Sorry for the long and messy post. Any help is appreciated.


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# KINDERGARTEN-SAMPEL PAPER FMO 2021

Berikut ini adalah soal beserta kunci jawaban Fermat Mathematic Olympiad (FMO) 2021  (Sc: Edukultur Indonesia)

A. Warm-up (4 points per question / No points deducted for wrong answers)

1. One car has 4 wheels. How many wheels do two cars have in total?
A) 2
B) 8
C) 4
D) 6
E) 12

2. Find the missing piece.

A)

B)

C)

D)

E)

3. Ken connects each animal with its shadow as in the figure below. How many
of them are WRONG?
A) 1
B) 3
C) 5
D) 2
E) 4

4. There is a queue of 20 people waiting to get in the restaurant. There are 10
people behind Anna. How many people does Anna have to wait until her
turn?
A) 11
B) 8
C) 4
D) 7
E) 3

5. The numbers of stuffed animals of five children are recorded in the following
table. How many stuff animals does Paul have more than Sara?
A) 5
B) 11
C) 4
D) 7
E) 3

B. Speed-up (6 points per question / No points deducted for wrong answers)

6. How many circles are there in the figure below?
A) 7
B) 6
C) 9
D) 8
E) 10

7. Peter draws a sequence with pattern as follow. How many are there in
the first 11 figures counting from the left?
A) 11
B) 6
C) 8
D) 7
E) 9

8. A pencil costs 2 dollars. If you buy 4 pencils, you get 1 eraser for free. Lucy
got 2 erasers for free, at least how much did she pay for the pencils?
A) $12 B)$8
C) $10 D)$16
E) \$18

9. Lucy has a sticker. She can cut her sticker into two identical pieces below.
Which answer CANNOT be the shape of Lucy’s sticker?
A)

B)

C)

D)

E)

10. There are 12 books on 3 shelves as below. At least how many books have to
be moved so that the number of books on every shelf is the same?
A) 5
B) 1
C) 2
D) 4
E) 3

C. Challenge (8 points per question / No points deducted for wrong answers)

11. It is Wednesday on 15th December 2021. Which day of the week does the
Christmas this year fall on?
A) Friday
B) Thursday
C) Saturday
D) Wednesday
E) Sunday

12. When Fred was born, his mother was 30 years old. His sister was born 4
years after Fred was born. His father is 5 years older than Fred’s mother.
When Fred’s father is 65 years old, how old is Fred’s sister?
A) 34
B) 20
C) 25
D) 26
E) 35

13. Jennie stacks bricks to get a tower. If she makes a 4-level tower, she will
need 10 bricks. If she makes a 5-level tower, she will need 15 bricks. How
many bricks does she need if she makes a 6-level tower?
A) 19
B) 21
C) 20
D) 22
E) 18

14. Given that each figure appears once in every row or every column. Choose a
figure to fill in the cell with question mark.
A)

B)

C)

D)

E) No possible answer

15. By putting two coins each time in the arcade machine, Lucy can play in 12
turns. She wants to play 50 turns. At least how many coins should she buy?

A) 6
B) 38
C) 62
D) 4
E) 5


text:
Neutral to charged kaon yield fluctuations in Pb$-$Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76TeV

The collaboration
Phys.Lett.B 832 (2022) 137242, 2022.

Abstract
We present the first measurement of event-by-event fluctuations in the kaon sector in Pb-Pb collisions at $\sqrt {s_{\rm NN}}=$ 2.76 TeV with the ALICE detector at the LHC. The robust fluctuation correlator $\nu_{\rm dyn}$ is used to evaluate the magnitude of fluctuations of the relative yields of neutral and charged kaons, as well as the relative yields of charged kaons, as a function of collision centrality and selected kinematic ranges. While the correlator $\nu_{\rm dyn}[\rm K^+,\rm K^-]$ exhibits a scaling approximately in inverse proportion of the charged particle multiplicity, $\nu_{\rm dyn}[\rm K_S^0,\rm K^{\pm}]$ features a significant deviation from such scaling. Within uncertainties, the value of $\nu_{\rm dyn}[\rm K_S^0,\rm K^{\pm}]$ is independent of the selected transverse momentum interval, while it exhibits a pseudorapidity dependence. The results are compared with HIJING, AMPT and EPOS-LHC predictions, and are further discussed in the context of the possible production of disoriented chiral condensates in central Pb-Pb collisions.

• #### Table 1

Data from Figure 2a top

10.17182/hepdata.131260.v1/t1

• #### Table 2

Data from Figure 2a bottom

10.17182/hepdata.131260.v1/t2

• #### Table 3

Data from Figure 2b top

10.17182/hepdata.131260.v1/t3

• #### Table 4

Data from Figure 2b bottom

10.17182/hepdata.131260.v1/t4

• #### Table 5

Data from Figure 3

10.17182/hepdata.131260.v1/t5

• #### Table 6

Data from Figure 3

10.17182/hepdata.131260.v1/t6

• #### Table 7

Data from Figure 3

10.17182/hepdata.131260.v1/t7

• #### Table 8

Data from Figure 4

10.17182/hepdata.131260.v1/t8

• #### Table 9

Data from Figure 4

10.17182/hepdata.131260.v1/t9

• #### Table 10

Data from Figure 4

10.17182/hepdata.131260.v1/t10

• #### Table 11

Data from Figure 5a

10.17182/hepdata.131260.v1/t11

• #### Table 12

Data from Figure 5a

10.17182/hepdata.131260.v1/t12

• #### Table 13

Data from Figure 5b

10.17182/hepdata.131260.v1/t13

• #### Table 14

Data from Figure 5b

10.17182/hepdata.131260.v1/t14


text:
Soon, we will launch a new and updated customer portal, which is an important step toward providing our customers with one place to learn, interact, and get help.

# How to Draw Specific Rays in Zemax Layouts

• How to draw desired rays in the Zemax OpticStudio Layouts via the RAYLIST
• What the RAYLIST is
• The difference between IMPLICIT and EXPLICIT methods for defining rays in the RAYLIST
• How to use each format with visual aids
Dan Hill
10/14/2015
Analysis Features
OpticStudio

# What is the RAYLIST?

The normalized field and pupil coordinates (Hx, Hy, Px and Py respectively) of a ray are often used to indicate to Zemax OpticStudio exactly which ray is desired for analysis. For example, the Single Ray Trace calculation (via The Analyze Tab > Image Quality Group > Rays and Spots > Single Ray Trace) requires the specification of Hx, Hy, Px, and Py, as do many of the Merit Function operands.

Although the single ray trace calculation and merit function operands can return very useful information about a given ray, it is also helpful to have a qualitative feel for how a specific ray traverses through your optical system. This very concept may be applied to the various layout plots in Zemax OpticStudio via the RAYLIST.

The layouts in Zemax OpticStudio (excluding the 2D Layout) support a number of different Ray Patterns, including XY Fan, X Fan, Y Fan, Ring, Random, Grid, and List. By choosing List as the “Ray Pattern”, the rays within the layout are traced according to a simple text file.

This text file is entitled, “RAYLIST.TXT,” and is located in the <Documents>\Zemax\Miscellaneous directory. Note that the filename and directory MUST remain the same. Any changes made to the filename or the directory in which it is saved will not be recognized by Zemax OpticStudio.

The RAYLIST.TXT file is simply of ASCII format, and supports two distinct methods for defining rays, summarized below and discussed in more detail in the next chapter.

 FORMAT SUMMARY IMPLICIT Rays are defined by the normalized pupil coordinates EXPLICIT Rays are defined by their starting coordinates, direction cosines, and wavelength

# The IMPLICIT Format

IMPLICIT:  The implicit format for the RAYLIST.TXT file consists of two numbers on each line: one for the Px and one for the Py normalized pupil coordinates. The syntax is as follows:

Px Py

For example, in a rotationally symmetric system, the two meridional marginal rays are defined in the RAYLIST.TXT file by:

These specified rays are traced at each defined field and wavelength in the settings of the respective layout plot.

Take the “Double Gauss 28 degree field.ZMX” sample file (Zemax\Samples\Sequential\Objectives directory) as an example. The Double Gauss design includes three fields (0, 10, and 14 degrees) and three wavelengths (F, d, and C wavelengths representing the visible spectrum).

If we choose to display the two defined meridional rays at the primary wavelength (wavelength 2) and from All fields, a total of 6 rays will be drawn.  That is, 2 marginal rays * 1 wavelength * 3 fields. Remember to select List as the “Ray Pattern”!

# The EXPLICIT Format

EXPLICIT:  The explicit format consists of the word EXPLICIT on the very first line of the RAYLIST.TXT file, followed by the values x, y, z, l, m, n, and wavenumber; where x, y, and z are the ray starting coordinates, l, m, n are the direction cosines of the ray, and wavenumber is the integer indicating the wavelength to use.

All coordinates are in object space. If the system is at infinite conjugates, then the spatial coordinates are relative to Surface 1. If the object in not at infinity, then the coordinates are with respect to Surface 0 (the OBJ surface).

The syntax of the explicit format is as follows:

EXPLICIT
x y z l m n wavenumber

Using the “Double Gauss 28 degree field.ZMX” once again as an example, edit the RAYLIST.TXT file to trace the rays defined by the following starting coordinates, direction cosines, and wavelength:

When an EXPLICIT trace is defined, the field and wavelength settings are ignored, and only those rays listed in the file are traced. In the current example, the three above listed rays are traced as follows:

# Summary

The RAYLIST is a feature within Zemax OpticStudio which provides and easy way to visualize the propagation of any number of rays via the layout plots in Zemax OpticStudio (excluding the 2D Layout). Rays may be defined one of two ways in the RAYLIST.TXT file:

• IMPLICITLY:  rays are defined by their normalized pupil coordinates, Px and Py
• EXPLICITLY:  rays are defined by their starting coordinates, direction cosines, a wavenumber

# References

Zemax OpticStudio Help files, Zemax LLC


text:
Markets can be volatile, ripping higher over a few weeks before accumulating a series of down days. Timing these momentum trades is never going to be perfect, but indicators like the Parabolic Stop-and-Reverse (Parabolic SAR or PSAR) are designed to help you find the trend and ride it to the top (or bottom) and get out with a tidy profit.

The PSAR is predicated off of the sequence of highs and lows that a stock makes. If the highs keep getting higher, it's accelerating and the indicator will be flashing BUY! It has a short lookback period, so it will signal an exit when the trend stops, and give you a short signal when it reverses (the Stop-and-Reverse in the name makes a bit more sense now, doesn't it?).

In a previous post, we walked through a detailed explanation of how the PSAR is calculated with plenty of example code and diagrams. If you missed that, go check it out, otherwise we're going to press on and provide three example backtests that you can trade.

## TL;DR

We provide backtests and code for three distinct trading strategies using the PSAR alone as well as with other indicators. These tests are designed to give you a feel for the indicator and running backtests. If you want something faster and more complete, check out our free, no-code platform here to get started today!

## Trend Following with the PSAR

The PSAR is a trend following indicator - it's designed to identify periods of up trends, down trends, and when they stop. The most straightforward way to apply this is simply to buy when the PSAR indicates a stock is going up and short it when it says the trend is going down.

Here's the code for the strategy:

def PSARTrendStrategy(data: pd.DataFrame, init_af: float=0.02,
max_af: float=0.2, af_step: float=0.02,
shorts: bool=True):
indic = PSAR(init_af, max_af, af_step)
data['PSAR'] = data.apply(
lambda x: indic.calcPSAR(x['High'], x['Low']), axis=1)
data['Trend'] = indic.trend_list

# Long when trend is up, short when trend is down
data['position'] = 0
data['position'] = np.where(data['Trend']==1, 1,
data['position'])
if shorts:
data['position'] = np.where(data['Trend']==0, -1,
data['position'])
return calcReturns(data)

We initialize our PSAR class with an initial acceleration factor and set the associated parameters, then apply that to our data to calculate the PSAR. The thing we're going to be looking at is the Trend value for making our decisions. We use this to determine our position (1 = long, 0 = neutral, -1 = short) and calculate our returns using the helper functions here.

Now, we just need some data to test our simple strategy!

We'll sample from the S&P 500 and get our data using the yfinance package.

import pandas as pd
import yfinance as yf
import numpy as np
import matplotlib.pyplot as plt

'https://en.wikipedia.org/wiki/List_of_S%26P_500_companies')
df = table[0]
syms = df['Symbol']

# Sample symbols
ticker = np.random.choice(syms.values)
# ticker = 'ABMD' # Uncomment this line to re-create my example
print(f"Ticker Symbol: {ticker}")
start = '2000-01-01'
end = '2021-12-31'

# PSAR Trend Strategy
yfObj = yf.Ticker(ticker)
data = yfObj.history(start=start, end=end)
data.drop(['Open', 'Volume', 'Dividends', 'Stock Splits'],
axis=1, inplace=True)
Ticker Symbol: ABMD

My sample chose ABMD, some biomedical stock that appears to have taken off in recent years. Go ahead and pass this data to our strategy and plot it to see what happens.

df_psar = PSARTrendStrategy(data.copy())

colors = plt.rcParams['axes.prop_cycle'].by_key()['color']

fig, ax = plt.subplots(2, figsize=(12, 8), sharex=True)

ax[0].plot(df_psar['strat_cum_returns']*100, label='PSAR Strategy')
ax[0].legend()
ax[0].set_ylabel('Returns (%)')
ax[0].set_title('PSAR Strategy Cumulative Returns')

ax[1].plot(df_psar['Close'], label='Close')
ax[1].set_xlabel('Date')
ax[1].set_ylabel('Price ($)') ax[1].set_title(f'{ticker} Price') plt.tight_layout() plt.show() Our strategy had some major volatility! But, it outperformed the buy and hold strategy by a bit. We can use our getStratStats helper function to look at some of the key performance metrics. bh_stats = pd.DataFrame(getStratStats(df_psar['log_returns']), index=['Buy and Hold']) psar_stats = pd.DataFrame(getStratStats(df_psar['strat_log_returns']), index=['PSAR']) stats = pd.concat([bh_stats, psar_stats]) stats So this strategy did outperform our baseline with some healthy, annual returns, but the equity curve does have some odd lumps in it. For example, wtf happened in 2012? It was doing great - then dropped off a cliff! At this time scale, we can't really see the moves of the underlying very well, so let's zoom in on the year. # What happened in 2012? df_psar['year'] = df_psar.index.map(lambda x: x.year) df_sub = df_psar.loc[df_psar['year']==2012] # Reset cumulative returns df_sub['strat_cum_returns'] = np.exp(df_sub['strat_log_returns'].cumsum()) - 1 df_sub['strat_peak'] = df_sub['strat_cum_returns'].cummax() psar_bull = df_sub.loc[df_sub['Trend']==1]['PSAR'] psar_bear = df_sub.loc[df_sub['Trend']==0]['PSAR'] fig, ax = plt.subplots(2, figsize=(12, 8), sharex=True) ax[0].plot(df_sub['Close'], label='Close') ax[0].scatter(psar_bull.index, psar_bull, color=colors[1], label='PSAR Up') ax[0].scatter(psar_bear.index, psar_bear, color=colors[3], label='PSAR Down') ax[0].set_ylabel('Price ($)')
ax[0].set_title(f'Price and PSAR for {ticker}')

ax[1].plot(df_sub['strat_cum_returns'] * 100)
ax[1].set_xlabel('Date')
ax[1].set_ylabel('Returns (%)')
ax[1].set_title('2012 Cumulative Returns')
plt.tight_layout()
plt.show()

The strategy peaked in 2012 on June 1 at 69% YTD - then it was all downhill from there (which, I suppose is the definition of a peak). The strategy was caught short with a big move up, reversing the trend - but it was too late and it wound up closing out the month of June "only" up 32% YTD. If that was it, nobody would complain, but it got worse from there.

You can see in the chart above that the strategy was consistently a day or two behind with big, choppy moves pushing it to the wrong side as it slowly gave up equity. The worst of it came on November 1.

ABMD closed out October about 20% off of its highs for the year after a 4-month down trend. The price finished up on October 31st, closing nearly 5% up vs the day before. This caused the PSAR to flip from short to long - the lone blue dot from early October to late November in the chart above - expecting a trend reversal. It couldn't have been more wrong. November 1 closed with ABMD losing 37% in a single day causing the strategy to go a YTD winner to down 15% so far in 2012.

It reversed to go short again a day too late and got whipped around for the last two months of the year to finish down 41% for all of 2012.

The same story played out in 2015 when the underlying hit a 33% single day loss on October 29, 2015, reversing a brief uptrend during a longer, down-trend.

These equity curves look so choppy because there is no risk management at play in these simplified strategies. The algo puts 100% of its capital at risk on each and every position, so a 30% move in the ABMD is going to cause the cumulative returns to drop 30% as well. So if the strategy had racked up a whopping 7,000% cumulative return over 15 years, that one bad move would see that track record eviscerated by 30% as well.

Most traders combine the PSAR with another indicator to look for confirmation to avoid being whipped in and out of positions. We'll look at some of these strategies next.

## Trading the PSAR with the RSI

A classic combination is to use the PSAR in conjunction with the Relative Strength Index (RSI).

Most frequently, the RSI is used in mean reversion strategies to identify overbought or oversold positions in preparation for the price to move the other direction. It can also be used as a momentum or trend following indicator, which is how we'll employ it here.

The RSI measures the strength of a price move on a scale of 0-100. If it is over 50, it is interpreted as being in a rising trend, under 50 a downward or falling trend. The more extreme it gets, the stronger the trend. An easy momentum strategy with the RSI is to go long if it moves above 50 and short if it's under 50. In this case, we're going to combine it with the PSAR, so we'll go long/short only if the PSAR and RSI are in agreement, otherwise, we'll be neutral.

This is going to cut down on the number of trades and may lead to longer times out of the market, but the hope is that we'll have a more reliable indicator to work with.

Let's get to the code!

First, here's the function to calculate RSI - which is explained in detail here.

def calcRSI(data, P=7):
data['diff_close'] = data['Close'] - data['Close'].shift(1)
data['gain'] = np.where(data['diff_close']>0,
data['diff_close'], 0)
data['loss'] = np.where(data['diff_close']<0,
np.abs(data['diff_close']), 0)
data[['init_avg_gain', 'init_avg_loss']] = data[
['gain', 'loss']].rolling(P).mean()
avg_gain = np.zeros(len(data))
avg_loss = np.zeros(len(data))

for i, _row in enumerate(data.iterrows()):
row = _row[1]
if i < P - 1:
last_row = row.copy()
continue
elif i == P-1:
avg_gain[i] += row['init_avg_gain']
avg_loss[i] += row['init_avg_loss']
else:
avg_gain[i] += ((P - 1) * avg_gain[i-1] + row['gain']) / P
avg_loss[i] += ((P - 1) * avg_loss[i-1] + row['loss']) / P
last_row = row.copy()
data['avg_gain'] = avg_gain
data['avg_loss'] = avg_loss
data['RS'] = data['avg_gain'] / data['avg_loss']
data['RSI'] = 100 - 100 / (1 + data['RS'])
return data

Now we will write a function that will take our data and parameters, calculate our indicators, then determine the position our algorithm will take based on the rules we outlined above.

def RSITrendStrategy(data: pd.DataFrame, init_af: float=0.02, max_af: float=0.2,
af_step: float=0.02, rsi_p: int=7, centerline: float=50,
shorts: bool=True):
'''
Enter when RSI moves above/below centerline and PSAR is in agreement with the
move. Exit when either reverses.
'''
# Calculate indicators
data = calcRSI(data, rsi_p)
indic = PSAR(init_af, max_af, af_step)
data['PSAR'] = data.apply(
lambda x: indic.calcPSAR(x['High'], x['Low']), axis=1)
data['Trend'] = indic.trend_list

data['position'] = np.nan
data['position'] = np.where(
(data['RSI']>centerline) & (data['Trend']==1), 1, data['position'])
if shorts:
data['position'] = np.where(
(data['RSI']<centerline) & (data['Trend']==0), -1, data['position'])

data['position'] = data['position'].fillna(0)
return calcReturns(data)

Finally, let's test our strategy and plot the results.

df_rsi = RSITrendStrategy(data.copy()) #, rsi_p=20)

fig, ax = plt.subplots(2, figsize=(12, 8), sharex=True)

ax[0].plot(df_rsi['strat_cum_returns']*100, label='RSI Trend Strategy')
ax[0].legend()
ax[0].set_ylabel('Returns (%)')
ax[0].set_title('Cumulative Returns')

ax[1].plot(df_rsi['Close'], label='Close')
ax[1].set_ylabel('Price ($)') ax[1].set_title(f'{ticker} Price') ax[1].set_xlabel('Date') plt.tight_layout() plt.show() rsi_stats = pd.DataFrame(getStratStats(df_rsi['strat_log_returns']), index=['PSAR and RSI']) stats = pd.concat([bh_stats, psar_stats, rsi_stats]) stats Adding the RSI does increase our time out of market from 2 days (for initialization) to 1,123 days. Unfortunately, adding the RSI doesn't prevent us from getting caught off guard on those key days that hurt our original PSAR strategy. Let's zoom in on that gnarly 2015 again, this time we'll overlay the PSAR indicator with the RSI and mark trade entry/exit points on the price to get a better feel for what's going on. You can see that again, the combined indicators went long (just barely) at exactly the wrong time in October. Even though we have two indicators, this new system still was caught on the wrong side of the major moves. This could possibly be ameliorated by adjusting the RSI lookback period or changing some other parameters such as the long/short thresholds to be stronger before reversing. Of course, always be careful that you don't run too many backtests and overfit your data (I shudder to think how many traders are doing this). Let's turn to one more combo where we use the PSAR with the Trend Intensity Indicator. ## Trading the PSAR and Trend Intensity Indicator The Trend Intensity Indicator (TII or sometimes called the Trend Intensity Index) is another measure of the strength of a given trend. Like the RSI, this indicator is scaled from 0-100 with 50 being a neutral point, values greater than 50 indicating up-trends, and values below 50 indicating down trends. The farther you get away from 50, the stronger the trend. If you read the previous section, this should all be familiar. Here's the code to calculate it, and, of course, all of the details for the TII calculation are given here. def calcTrendIntensityIndex(data, P): data[f'SMA_{P}'] = data['Close'].rolling(P).mean() diff = data['Close'] - data[f'SMA_{P}'] pos_count = diff.map( lambda x: 1 if x > 0 else 0).rolling(int(P/2)).sum() data['TII'] = 200 * pos_count / P return data There are multiple ways to trade the Trend Intensity Indicator, either as a trend following/momentum indicator or in a mean reversion strategy. We're going to stick to our momentum theme here by combining the PSAR with the TII to see if we can get some additional alpha out of it. The TII is commonly a slower indicator than something like the RSI, so the hope is that we can combine this with the PSAR in order to avoid those few catastrophic days that demolished our returns before. We'll set this up with the same basic rules as the RSI model, buy if PSAR is moving up and the TII is above 50, short if the values are flipped. Here's the code for the TII strategy: def TrendIntensityPSARStrategy(data: pd.DataFrame, init_af: float=0.02, max_af: float=0.2, af_step: float=0.02, tii_p: int=30, centerline: float=50, shorts: bool=True): ''' Long if PSAR is in up trend and TII > centerline. Short/sells if PSAR is in down trend and TII < centerline. ''' # Calculate indicators data = calcTrendIntensityIndex(data, tii_p) indic = PSAR(init_af, max_af, af_step) data['PSAR'] = data.apply( lambda x: indic.calcPSAR(x['High'], x['Low']), axis=1) data['Trend'] = indic.trend_list data['position'] = np.nan data['position'] = np.where( (data['TII']>centerline) & (data['Trend']==1), 1, data['position']) if shorts: data['position'] = np.where( (data['TII']<centerline) & (data['Trend']==0), -1, data['position']) data['position'] = data['position'].fillna(0) return calcReturns(data) And to test it... df_tii = TrendIntensityPSARStrategy(data.copy()) fig, ax = plt.subplots(2, figsize=(12, 8), sharex=True) ax[0].plot(df_tii['strat_cum_returns']*100, label='TII Strategy') ax[0].plot(df_tii['cum_returns']*100, label='Buy and Hold') ax[0].legend() ax[0].set_ylabel('Returns (%)') ax[0].set_title('Cumulative Returns') ax[1].plot(df_tii['Close'], label='Close') ax[1].set_ylabel('Price ($)')
ax[1].set_title(f'{ticker} Price')
ax[1].set_xlabel('Date')

plt.tight_layout()
plt.show()
tii_stats = pd.DataFrame(getStratStats(df_tii['strat_log_returns']),
index=['PSAR and TII'])
stats = pd.concat([bh_stats, psar_stats,
rsi_stats, tii_stats])
stats

This model did seem to cut down on some of the volatility significantly and kept us from getting wrecked by some of the sudden downward moves, yielding the lowest volatility and max drawdown of all the models. That difficult 2015 was even tamed with a 90% return for the year!

Maybe 90% isn't that impressive in a year where the underlying returned over 240%...but still, this was a highly volatile year and was very difficult for the other models.

Unfortunately, this model was more conservative than all the others yielding the lowest total and risk adjusted returns.

## How to Trade with the PSAR

We looked at a few ways to work with the PSAR, but we never looked at changing its parameters. Slowing it down a bit may help it stay in some of those trends a bit longer without getting knocked out on the reversal, which could be key into juicing some alpha out of your strategy. There are a million ways to combine this with other indicators and rules - we didn't even try to work in a mean reversion model - and its up to you to figure out an approach you're comfortable with.

Of course, one of the biggest limitations of any of these simplified tests is the lack of any risk management techniques: diversification, position sizing, stops, or position management. Perhaps the raw PSAR would do great on its own if only it was diversified across a variety of instruments, or positions were sized according to some optimality condition, or we adjusted our position size as a function of volatility.

The trouble is, a lot of these things are difficult and time consuming to test. At Raposa, we want to handle that for you so you can spend more time designing strategies and no time coding them. With just a few clicks, you can get a custom trading bot built and deploy it to work in the markets for you.