text:
[OS X TeX] toolbar

Robert Bruner robert.bruner at wayne.edu
Sun Jun 21 13:45:28 EDT 2020

\begin{rant}

This is an interesting philosophical issue on which I am completely at odds with the software culture.    The version of TeXShop that I have does everything I need.    I am not interested in spending time reconfiguring things and learning new ways to do the same things I already know how to do.    I would rather do mathematics than fine tune my computer's software.

But, I have no choice:  if I don't constantly update things I get hit by issues like this.

In mathematics, once something is proved, it is proved and we can move on to new issues.    Of course, we return to issues and refine our understanding, clean up proofs, find new and better axioms, etc., but the pace is slower, and the old proofs are still valid if they ever were.

Would it be possible to design software for similar backward compatibility.    Experience suggests not, but 'provably correct' software might enable this.

\end{rant}

\begin{counter-rant}

TeXShop is great and I am earnestly grateful for the effort that has gone into making it such an excellent tool.

\end{counter-rant}

Regards,
Bob Bruner

PS:   2016 was only 4 years ago.

________________________________________
From: MacOSX-TeX <macosx-tex-bounces at email.esm.psu.edu> on behalf of Herbert Schulz <herbs at wideopenwest.com>
Sent: Sunday, June 21, 2020 10:55 AM
To: List TeX on Mac OS X Mailing
Subject: Re: [OS X TeX] toolbar

> On Jun 21, 2020, at 8:31 AM, Robert Bruner <robert.bruner at wayne.edu> wrote:
>
> Mac OS 10.15.4
> TeXShop 3.6.2
> Yes, I copied my files from my existing, 2013 vintage, MacBookPro.
> Not sure about the single/multiple window mode:   I have one
> window with the .tex file and a separate preview window for the pdf.
> The .tex window is the one which keeps coming up w/o the typeset button. If this can't be fixed, I will probably just convert to always using Cmd-T to typeset, which works fine.
>
> I was vague in my first post because i didn't want to admit how old my TeXShop was :-(.
>
> Bob
>
>
> ________________________________________
> From: MacOSX-TeX <macosx-tex-bounces at email.esm.psu.edu> on behalf of Herbert Schulz <herbs at wideopenwest.com>
> Sent: Sunday, June 21, 2020 9:20 AM
> To: List TeX on Mac OS X Mailing
> Subject: Re: [OS X TeX] toolbar
>
>> On Jun 21, 2020, at 7:34 AM, Robert Bruner <robert.bruner at wayne.edu> wrote:
>>
>> I just moved to a new MacBook Pro.   Now, when TeXShop starts,
>> the 'Typseset' button is missing.    I can go to Window > Customize Toolbar, to add 'Typeset', but next time I start TeXShop, the 'Typeset'
>> button is again missing.
>>
>> What might I be doing wrong?
>>
>> Bob Bruner
>> Math
>> Wayne State
>
> Howdy,
>
> I think it would be helpful knowing the following information:
>
> what version of TeXShop you are using;
> what macOS version you are using (can I assume Catalina?);
> are you using Single or Multiple Window Mode; and,
> did you carry the preferences over from another computer.
>
> Good Luck,
>
> Herb Schulz
> (herbs at wideopenwest dot com)
>

Howdy,

Really, TeXShop 3.6.2? TeXShop is presently at version 4.44! Please go to <https://pages.uoregon.edu/koch/texshop/texshop.html> and download and install the latest version. There are probably a bunch of things that have changed so you may have to do some re-configuring.

Good Luck,

Herb Schulz
(herbs at wideopenwest dot com)

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text:
## anonymous one year ago Use congruence theory to prove that $\frac{1}{5}n^5+\frac{1}{3}n^3+\frac{7}{15}n$ is an integer for every integer n.

1. ganeshie8

familiar with fermat's little thm ?

2. anonymous

@ganeshie8 I have read about it but I am not familiar with it. Any hints would be helpful

3. ganeshie8

For all integers $$a$$ and primes $$p$$, we have $a^p\equiv a\pmod{p}$

4. ganeshie8

$n^5\equiv n\pmod{5} \implies n^5 = n+5k$ $n^3\equiv n\pmod{3} \implies n^3 = n+3l$ plug them int he given expression and simplify

5. anonymous

$\frac{1}{5}n^5+\frac{1}{3}n^3+\frac{7}{15}n=\frac{ 1 }{ 15 } (3n^5+5n^3+7n)$ Plugging into the equation to get $n^5=3(n+5k)+7n=10n+15k$ $n^3=5(n+3l)+7n=12n+15l$ @ganeshie8 Am I doing this correct?

6. ganeshie8

simply replace n^5 by n+5k and n^3 by n+3l

7. ganeshie8

$\frac{1}{5}n^5+\frac{1}{3}n^3+\frac{7}{15}n = \frac{1}{5}(n+5k)+\frac{1}{3}(n+3l)+\frac{7}{15}n \\~\\=k+l+\frac{n}{5}+\frac{n}{3}+\frac{7n}{15}$ simplify

8. anonymous

@ganeshie8 Thanks a bunch

9. ganeshie8

yw

format: plain


text:
Hardcover | $85.00 X | £70.95 | 672 pp. | 8 x 9 in | August 2005 | ISBN: 9780262201629 eBook |$85.00 X | August 2005 | ISBN: 9780262332750
Mouseover for Online Attention Data

## Overview

Probabilistic robotics is a new and growing area in robotics, concerned with perception and control in the face of uncertainty. Building on the field of mathematical statistics, probabilistic robotics endows robots with a new level of robustness in real-world situations. This book introduces the reader to a wealth of techniques and algorithms in the field. All algorithms are based on a single overarching mathematical foundation. Each chapter provides example implementations in pseudo code, detailed mathematical derivations, discussions from a practitioner's perspective, and extensive lists of exercises and class projects. The book's Web site, www.probabilistic-robotics.org, has additional material. The book is relevant for anyone involved in robotic software development and scientific research. It will also be of interest to applied statisticians and engineers dealing with real-world sensor data.


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# The necessity to choose causes the effects of reward on saccade preparation

## Abstract

When humans have to choose between different options, they can maximize their payoff by choosing the option that yields the highest reward. Information about reward is not only used to optimize decisions but also for movement preparation to minimize reaction times to rewarded targets. Here, we show that this is especially true in contexts in which participants additionally have to choose between different options. We probed eye movement preparation by measuring saccade latencies to differently rewarded single targets (single-trial) appearing left or right from fixation. In choice-trials, both targets were displayed and participants were free to decide for one target to receive the corresponding reward. In blocks without choice-trials, single-trial latencies were not or only weakly affected by reward. With choice-trials present, the influence of reward increased with the proportion and difficulty of choices and decreased when a cue indicated that no choice will be necessary. Choices caused a delay in subsequent single-trial responses to the non-chosen option. Taken together, our results suggest that reward affects saccade preparation mainly when the outcome is uncertain and depends on the participants’ behavior, for instance when they have to choose between targets differing in reward.

## Introduction

Humans frequently decide where to look next. We shift our gaze 2–3 times a second by saccadic eye movements, each time choosing a different region or object of the visual scene for high acuity processing. This qualifies the oculomotor system as a suitable candidate to study decision-making in humans and other primates1,2. The selection of a particular target over others as well as the time required to initiate an eye movement (latency) are both informative about the underlying decision process.

Saccade latencies are not only influenced by low-level stimulus features3,4, but also by motivational factors like reward: Monkeys initiate saccades earlier and with higher peak-velocities when they expect a reward compared to non-rewarded saccades and reduced latencies are preceded by a modulated discharge rate of neurons in several brain areas5,6,7,8,9. To maximize outcome in decision-making, an option’s expected value (EV), the combination of reward magnitude and probability, has to be considered. Indeed, neural activity in the lateral intraparietal area (LIP) covaries with both reward magnitude and probability10. In humans, EV scales with activity of frontal areas11,12,13.

Despite this clear neurophysiological evidence that brain activity scales with reward, there are contradictory findings about its influence on eye movement preparation. Some studies did not find effects on saccade latencies in monkeys10,14 or reported changes in peak-velocity rather than latency when investing the effect of reward in humans15,16. However, there is also evidence favoring a modulation by reward. Two studies17,18 investigated whether saccades are influenced by a target’s EV. They showed that when two targets were presented (two-target trial), humans17 and monkeys18 more frequently chose the highly rewarded target. When one target was presented (single-target trial), latencies were affected by reward magnitude, but showed a stronger linear relationship with EV. This led to the conclusion that a representation of EV is incorporated in saccade preparation.

Where might these contradictory findings come from? A specific feature of the studies reporting an influence of EV on saccade preparation17,18 was the combined recording of several different trial types in the same experiment. Whereas latency analyses were based on responses to single-targets, additional trials were recorded in which participants had to choose among two targets (two-target trials) or trials with a distractor flashed before onset of the saccade target (distractor trials). These different trial types might have interacted: there is ample evidence that inter-trial priming can affect saccade metrics, especially when a competition between several targets is involved19,20,21.

Here, we investigated the hypothesis that effects of reward on saccade preparation are modulated or caused by interleaved choices between multiple targets. We measured saccade preparation by means of saccade latencies to single-targets (single-trials) and varied the proportion of interleaved choices (choice-trials) in a block. It is important to note that choice-trials were only included as independent variable: All results are based on latencies in single-trials. Differences in latencies to less and highly rewarded targets were present in blocks with interleaved choices – and mostly absent in blocks where participants never made a choice. The magnitude of this effect increased with increasing proportion and difficulty of choices. Choices caused a delay in subsequent saccade responses to the non-chosen target. Modelling latency distributions suggested that this delay was due to a reduced baseline level in the response signal.

## Results

### Latency differences between less and highly rewarded targets

In Experiment 1, we tested the hypothesis that saccade preparation in response to rewarded single-targets is modulated by the presence of choices which participants have to make in a block. To this end, we measured single-trial saccade latencies (Fig. 1a) in blocks without choice-trials (0%) or in blocks with different proportions of choice-trials randomly interleaved (25%, 75%). In single-trials, one target appeared at 15° eccentricity either left or right from fixation. Participants had to saccade to the target within 500 ms to receive the reward. In choice-trials, both targets were displayed and participants were free to decide for one target to obtain the corresponding reward. In every block, each hemifield was assigned either a highly or a less rewarded target. Across blocks, the difference in reward magnitude between the opposite hemifields could be either small (4 vs 6) or large (1 vs 9).

Saccade latencies from single-trials are shown in Fig. 1b. With an increasing proportion of choice-trials (0%, 25%, 75%) latency differences between less and highly rewarded targets increased for the large (2, 29 and 52 ms) and small reward difference (8, 22 and 45 ms), F(2,48) = 49, p < 0.001 (interaction proportion choice-trials × reward magnitude). This was mainly because latencies to less rewarded targets increased linearly with an increasing proportion of choice-trials, F(1,24) = 83.86, p < 0.001. Without choice-trials, latencies between less and highly rewarded targets did not differ significantly for the large reward difference: t(24) = 0.43, p = 0.671, but they did for the small if not Bonferroni-corrected: t(24) = 2.12, p = 0.045 (α’ = 0.05/6 [3 proportion choice-trials × 2 reward differences] = 0.0083). The corresponding Bayes factor (BF) favored the null hypothesis (i.e. reward does not influence latencies) for the large, BF = 0.23, but there was no conclusive evidence for the small reward difference, BF = 1.38. With 25% choice-trials, however, latency differences were significantly larger than without choice-trials, large: t(24) = 6.76, p < 0.001, small: t(24) = 4.02, p < 0.001. Compared to 25%, latency differences were even more pronounced with 75% choice-trials for the small reward difference, t(24) = 3.29, p = 0.003, but failed to reach significance for the large reward difference, t(24) = 2.29, p = 0.031, BF = 1.88. We found no evidence for an effect of reward difference (all ps > 0.4).

In Experiment 1, we mainly found differences in saccade latencies between less and highly rewarded targets when choices were interleaved. Because participants consistently chose highly rewarded targets, this observation could arise due to the choices themselves or because higher choice-trial proportions also implied lower saccade frequencies to the less rewarded (i.e. non-chosen) target. In Experiment 2, we eliminated this imbalance in saccade frequency by altering the frequency of single-trials to each target so that participants moved equally often to both targets if they always chose the highly rewarded target in choice-trials. Even with equated saccade frequency, participants still showed longer latencies to less rewarded single-targets, that is, non-chosen targets, F(1,7) = 123.97, p < 0.001 (Fig. 2a; main effect reward magnitude). Latency differences were 29 ms for the large, t(7) = 6.30, p < 0.001, and 17 ms for the small reward difference, t(7) = 4.27, p = 0.004, and thus similar to Experiment 1. Like in Experiment 1, we did not find evidence that reward differences affected latencies (all ps > 0.1).

We compared latency differences from Experiment 2 and the 25% choice-trial condition in Experiment 1. Experiments are identical with regard to choice-trial probability, but differ in saccade frequency. A 2 × 2 ANOVA with the factors reward difference (within) and experiment (between) revealed no significant main effect of experiment, F(1,31) = 0.39, p = 0.565, BF = 0.31. In a similar ANOVA, we compared latency differences from Experiment 2 with the 0% choice-trial condition in Experiment 1. Here, conditions from both experiments include the same saccade frequency, but differ with respect to choice-trial probability. Latency differences were larger in Experiment 2, F(1,31) = 24.61, p < 0.001, BF = 29.72. This suggests that latency differences between less and highly rewarded single-targets in blocks with interleaved choices occur even when overall saccade frequencies are matched for both targets.

To examine whether choices modulated the reward effects on saccade preparation or whether they caused them, we performed Experiment 3 where choice- and single-trial rewards were either incongruent or congruent, or where choice-trials were absent. In the congruent condition, highly rewarded targets for single- and choice-trials were presented in the same hemifield (equivalent to Experiment 1), whereas in the incongruent condition highly rewarded single- and choice-trials targets were presented in opposite hemifields. If the presence of choice-trials caused latency differences in single-trials, then single-trial latencies should only depend on which target is preferred in choice-trials and should be independent of the actual single-trial reward. Figure 2b shows mean and individual latencies for the different congruency conditions. Without choice-trials, latencies in both reward conditions perfectly coincided (196 ms) and did not differ significantly, t(7) = 0.06, p = 0.956, but the corresponding BF did not provide conclusive evidence, BF = 0.37. Instead, the effect of reward depended on the level of congruency, F(2,14) = 21.54, p < 0.001 (interaction reward magnitude × congruency). With congruent choice-trials present, latencies to less rewarded single-targets were increased by 29 ms (SD = 13 ms), t(7) = 6.2, p < 0.001. This pattern was reversed with incongruent choice-trials (M = −57 ms, SD = 39 ms), t(7) = 4.11, p = 0.005. Increased latencies in single-trials thus did not depend on single-trial reward itself, but on reward in choice-trials and therefore on which target was chosen. It thus seems that choices caused rather than modulated, the observed reward effects in single-trials.

### The non-chosen target is inhibited in the subsequent single-trial

Confronted with a choice, one could either increase saccade preparation towards the highly rewarded and thus chosen target or one could inhibit the less rewarded and thus non-chosen target (or any combination of more mechanisms, see discussion). The former case predicts lower latencies when highly rewarded single-trials follow a choice-trial, whereas the latter case predicts increased latencies when less rewarded single-trials follow a choice-trial. The latter case appears more likely, given that we mainly found increased latencies for the non-chosen target. To test whether inter-trial effects contributed to our results, we reanalyzed the 25% choice-trial condition of Experiment 1 with regard to previous trial effects. We compared single-trials following a choice-trial with single-trials following a single-trial (Fig. 3). After choice-trials, saccades were initiated later when the upcoming trial was a single-trial to the non-chosen target, F(1,24) = 11.82, p = 0.002 (main effect reward magnitude). This cannot be attributed to a change in saccade direction, because there is no such difference when a previous single-trial was directed in the other or in the same direction, F(1,24) < 0.01, p = 0.954. An ANOVA which included both trial sequences, revealed an interaction of trial sequence with reward magnitude, F(1,24) = 7.13, p = 0.013 (i.e., an interaction of the black and red line in Fig. 3). Thus, in single-trials with less rewarded targets of either 1, t(24) = 4.77, p < 0.001, or 4 points, t(24) = 3.98, p < 0.001, saccades were significantly slower after a choice-trial. This suggests that the non-chosen target is inhibited in choice-trials, affecting the subsequent single-trial. We found no evidence that this effect increased with reward difference, F(1,24) = 4.01, p = 0.057.

### Adaptive inhibition of the non-chosen target

Is the delay of saccades to the non-chosen target an adaptive behavior? If yes, then it should scale with the necessity to inhibit the less rewarded target in choice-trials. One possibility to manipulate the necessity for inhibition would be to change the relative salience of both choice targets. When the less rewarded target has a higher contrast than the highly rewarded one, stronger inhibition is required to make an optimal saccade and thus obtain the high reward. Any location-based inhibition should then propagate to single-trials and lead to larger latency differences. The opposite pattern should be observed when the highly rewarded target is more salient. A second possibility to manipulate the required inhibition would be cueing the upcoming trial type. If participants know that the next trial will be a single-trial, they can refrain from maintaining inhibition and rely on a purely visually evoked saccade instead. This, however, would require that the inhibition could be modulated by top-down control. We tested these two possibilities in Experiment 4 and 5.

In Experiment 4, we aimed to assess whether single-trial latency differences increase with the difficulty to saccade to highly rewarded targets in choice-trials. To this end, we changed the contrast of both choice-trial targets so that the contrast of the highly rewarded target was lower (difficult condition), higher (easy condition) or identical (medium condition). Beforehand, we measured two control conditions as a manipulation check (Fig. 4a). First, in the choice control task, participants had to choose one out of two targets which were either identical or different in contrast without receiving a reward. The probability of choosing targets on the right was lowest, when left targets had higher contrasts (M = 0.21, SD = 0.17), it was around chance when both contrasts were identical (M = 0.47, SD = 0.09) and highest when right targets had higher contrasts (M = 0.74, SD = 0.23), χ²(2) = 12.17, p = 0.002. Second, in the latency control task, we measured latencies to single-targets of different contrasts. Latencies decreased from 229 ms (low contrast) over 200 ms (medium contrast) to 196 ms (high contrast), F(2,22) = 15.91, p < 0.001. Compared to medium contrasts, latencies were increased for lower contrasts, t(11) = 4.7, p = 0.001, but they were not significantly decreased for high contrasts, t(11) = 0.81, p = 0.433.

Figure 4b shows individual and mean latencies for less and highly rewarded targets and for the three difficulty levels. Again, we found higher latencies to less rewarded targets, F(1,11) = 23.22, p = 0.001. Latency differences between less and highly rewarded targets were modulated by difficulty, F(2,22) = 7.24, p = 0.011. Compared to medium difficulty, latency differences were increased for the difficult condition, t(11) = 2.23, p = 0.047, and decreased for the easy condition, t(11) = 2.35, p = 0.038. Two separate ANOVAs suggested that difficulty affected latencies to less rewarded targets, F(2,22) = 8.39, p = 0.002, but not to highly rewarded targets, F(2,22) = 0.29, p = 0.751, BF = 0.2. Moreover, the probability to miss less rewarded single-trials increased with choice-trial difficulty, from 5% (easy), over 8.6% (medium) to 23.3% (difficult), χ²(2) = 15.2, p = 0.001. Misses were either due to too late (42.2%), too early (22.6%) or wrong saccades (35.2%). There was only one missed trial (<0.1%) in highly rewarded single-trials.

To test whether this behavior is not only adaptive with regard to low-level stimulus features, but also with regard to top-down processes, we cued half of the single-trials in Experiment 5. If there is a contribution from a top-down component, for example a preparation for an upcoming choice-trial, then differences in single-trial latencies between the chosen and non-chosen targets should be reduced by cueing. Figure 4c shows differences in saccade latencies, for cued compared to uncued single-trials. Latency differences were 37 ms without and 27 ms with cue. Wilcoxon signed-rank tests revealed that the latency difference was above 0 in both conditions, Z = −2.52, p = 0.012, but reduced by the presence of a cue, Z = −2.1, p = 0.036. This indicates that there is a voluntary component contributing to the observation of delayed saccades, yet it cannot fully account for it. In sum, the delay reduction by cueing (Experiment 5) and the delay increase with increasing difficulty (Experiment 4) point out that the inhibition of the non-chosen target is an adaptive behavior influenced by top-down and bottom-up factors.

### Decreased baseline level for the non-chosen target in single-trials

In order to identify likely neural mechanisms which can explain the delay of saccades to single-targets due to interleaved choices, we recorded the whole latency distribution for two participants (Experiment 6) and fitted the LATER model22,23 to the single-trial data. The LATER model is helpful in pointing out potential neural mechanisms of motor responses and decision making, on the basis of reaction time distributions. It assumes that for every response (here single-trial saccade) at stimulus onset, evidence is accumulated starting from a baseline level $${\theta }_{0}$$ with an average rate of rise µ until a response threshold $${\theta }_{T}$$ is reached. Within one trial the accumulation rate rises constantly but varies across trials with a Gaussian standard deviation σ. Several studies identified such evidence accumulation in the primate brain24,25 that can account for saccade latency distributions. With behavioral data however, it is only possible to obtain information about the threshold height, that is, the difference between baseline level and response threshold, θ = θ T  − θ 0 . Since there is physiological evidence that the baseline firing rate in saccade related areas represents economic decision variables as reward and target probability10,25 and saccades are initiated once the neural activity reaches a constant threshold24, we fixed the response threshold to an arbitrary value. The three remaining parameters are the baseline level, θ 0 , the accumulation rate, µ, and its variability, σ.

To find out which of these three parameters can most likely explain the latency differences between conditions, we abided by the following procedure: For every individual, we applied a bootstrap procedure with 100 iterations. For every iteration, we fitted three versions of the model in which we allowed one of the parameters to vary across conditions while the remaining two were kept identical across conditions. We then used information weights26, derived from the Bayesian information criterion (BIC) to compare the three model versions and thus to identify which parameter is best in explaining the latency differences across conditions. Information weights can range from 0 to 1 and higher values speak in favor of a particular model.

With regard to average latencies, we replicated our main findings also with the more extensive measurements with these two participants: Without interleaved choices, average single-trial latencies were M = 192 ms for the less and M = 187 ms for the highly rewarded target. With choice-trials present, latencies were M = 192 ms for the highly and M = 228 ms for the less rewarded target. For both participants, information weights (Fig. 5a) were highest for the $${\theta }_{o}$$ parameter (baseline level). Thus, changes in $${\theta }_{o}$$ were best in explaining differences in latency distributions between conditions. Cumulative probability plots of latency distributions together with model fits are plotted in Fig. 5b. Without choice-trials, baseline levels for less rewarded single-trials were reduced by 12 and 17% relative to baseline levels for highly-rewarded targets. With choice-trials present, baseline levels were reduced by 82% (Fig. 5c) for both participants. Technically, this suggests that either a lower baseline level, an increased response threshold or both are most likely to explain delayed saccades to non-chosen targets.

## Discussion

In this study we investigated whether saccade preparation to single-targets is influenced by interleaved choices among two targets differing in reward and if this is able to account for differential previous results on the modulation of saccade latencies by reward. In blocks without choices (Experiment 1 and 3), we only found a comparatively small effect of reward on saccade latencies that was only significant in one (only without correction for multiple testing) out of three cases. When choices were present, reactions to less rewarded single-targets were delayed and the magnitude of this delay increased significantly with the proportion of choice-trials, both for saccades (Fig. 1b) and button presses27. When changing the reward congruency between choice- and single-trials, latency differences in single-trials depended on the reward assignments in choice- rather than in single-trials (Fig. 2b). Moreover, latency differences were adaptive because they scaled with the necessity to inhibit saccades which do not maximize reward during choices (Fig. 4b) and decreased when upcoming single-trials were cued in advance (Fig. 4c), suggesting the contribution of both, bottom-up and top-down factors. Increased latencies to less rewarded single-targets can be explained in terms of a reduced baseline level (Fig. 5). Although a difference in response threshold could technically also account for the observed latency difference, this is unlikely given that saccades are executed at a constant threshold24.

Taken together, our results suggest that information about reward might not always be incorporated for the preparedness of motor responses like saccadic eye movements. This does neither suggest that it is not represented in the brain, nor that it does not affect behavior. Rather, it suggests that reward affects preparation of saccades mostly when it is behaviorally relevant as in choice-trials and less so when it is behaviorally irrelevant as in single-trials. When responding to single-targets without strong temporal urgency, there is no necessity to optimize behavior, for instance, by preferring one target location over the other. Thus, the modulation of latencies in single-trials appears to be a direct effect of target selection and mostly no (or only an indirect) effect of reward per se.

Many studies have shown that reward influences oculomotor behavior. Monetary and non-monetary reward alters eye movement behavior, by changing saccade latencies5,28,29,30, kinematics15,31,32 and target selection33,34,35. Most of these studies however have compared rewarded to unrewarded behavior and did not include different levels of reward. When rewards of different magnitudes can be obtained, saccade endpoints are closer to high than to low reward targets36 and maximize gain33, the microsaccade rate scales with value37 and saccade vigor decreases with advanced discounting of rewards16,38. Here, looking at saccade latencies without interleaved choices (Experiment 1 & 3), we found no significant evidence for a direct influence of value in two out of three conditions. Bayesian analyses provided evidence for the notion that reward magnitude does not affect latencies in one out of three cases and inconclusive evidence in the remaining two. In Experiment 6, reward influenced baseline levels even without choice-trials. This might point out that latency distributions are more sensitive to reward than average latencies. However, congruent with the average latency differences in the other experiments, modulations of baseline levels were much larger with choice-trials. Thus, in total this suggests that reward alone influences latencies only weakly or not at all. Moreover, the magnitude of reward differences did not modulate latencies (Experiment 1 & 2). This, together with the observation that response delays to less rewarded single-targets can be varied by the amount of inhibition required to perform a reward-maximizing choice (Experiment 4), suggests that participants tried to make an optimal choice, no matter how big the gain or loss.

A previous study17 reported a linear relation of saccade latency and EV. However, because choice- and single-trials were mixed in this study, it is unclear whether this link would persist in the absence of choices. The here reported inter-trial dependency might also have affected oculomotor and neural findings in monkeys18,39. A recent study tested whether microsaccade behavior also varies as a function of EV37. The authors reanalyzed their previously collected monkey data mixing choice and single responses18 and recorded new human data for single-trials only. Both, humans’ and monkeys’ microsaccades were biased by the subjective target value. This points out that microsaccades seem to represent value irrespective of whether choices are interleaved or not. Unfortunately, the authors did not report saccade latencies for the human data. This could have been an indication whether or not EV can affect saccade preparation in the absence of interleaved choices.

Our results suggest that target selection modifies subsequent saccade preparation. There is converging evidence that attentional control is not only influenced by stimulus properties (bottom-up) or current goals (top-down), but also by a bias to attend previously selected items40,41. For example, inter-trial priming effects seem to require attentional selection42 and can either be facilitating or inhibitory. Facilitating effects can be observed in visual search when the search target, distractors or their particular features are identical to the preceding trial, leading to shorter reaction times43,44,45. Inhibitory effects occur in conjunction with distractors, for example, saccades curve away from previous distractor locations19 or in the negative priming paradigm, when the identities of target and distractor are exchanged between trials46. The present study extends findings on inter-trial priming by showing that a selection between two differentially rewarded targets does not facilitate a subsequent response to the chosen one but inhibits a response to the non-chosen one.

We interpret our results in terms of an inhibition of the less rewarded target. Theoretically, the fact that latencies to less rewarded targets increased with an increasing proportion of choices in Experiment 1 does not necessarily imply that these targets are selectively inhibited. Other combinations of several inhibitory and facilitating mechanisms could also explain this pattern: the presence of choices might generally slow down latencies and, simultaneously, selectively speed up responses to highly rewarded targets. In this case, these two mechanisms might cancel each other out for highly rewarded targets, whereas delays towards the less rewarded one would become observable. However, this alternative interpretation seems unlikely because of two other findings: first, the analysis of inter-trial effects showed that responses to less rewarded targets were slowed down after choice-trials, but responses to highly rewarded targets remained unaffected. However, a potential facilitation for highly rewarded targets should have been observable here. Second, the same argument is true for the findings of Experiment 4, where choice-trial difficulty selectively modulated latencies to less but not to highly rewarded targets (Fig. 4b). These two findings favor a selective inhibition of less rewarded targets, but we cannot rule out that other mechanisms are also involved.

Differences between conditions with different proportions of choice-trials (Experiment 1) could theoretically be explained by changes in saccade frequency. Although we cannot dismiss this interpretation, we consider it unlikely, given that latency delays with 25% choice-trials had the same magnitude when we equated saccade frequencies in both directions (Experiment 2). Moreover, studies showing influences of probability on saccade latencies23,47 employ hundreds of trials for every probability condition and dismiss the first 100 trials or more, whereas our blocks in Experiment 1 consisted of only 80 trials.

In Experiment 5 we tested whether our results are influenced by the expectation of an upcoming choice-trial. We cued single-trials to eliminate the expectation of an upcoming choice-trial. If expectation could fully explain our data, latency imbalances should have completely disappeared when upcoming single-trials were cued. However, latency differences were only reduced but not eliminated, suggesting that expectation can only partially explain our findings. Because we only manipulated expectation on a short-term timescale (trial-wise), we cannot exclude the possibility that expectation operating on longer timescales (block-wise) influenced our data but was unaffected by cueing. Nonetheless, our findings cannot be explained by (long-term) expectation alone, given that we found strong inter-trial effects within the same block. However, expectation (short-term and long-term) will have likely added up with inter-trial effects and resulted in delayed saccades to the less rewarded target.

In conclusion, our findings suggest that there is no or only a weak direct connection between reward and saccade preparation to single-targets. A decision between two reward-associated targets leads to a subsequent delay in responses to the non-chosen option. The amount of delay depends on the difficulty to make an optimal, that is, reward-maximizing decision in choice-trials. We propose that these changes in saccade preparation occur due to the subsequent inhibition of the non-chosen target and the expectation of an upcoming choice-trial. This is reflected by a reduced baseline level in the response signal. These results suggest that reward affects saccade preparation particularly if it is behaviorally relevant, for instance if a choice has to be made.

## Methods

### Participants and apparatus

In total, 47 students from Marburg University aged 19–29 years (M = 23 years) participated in this study (30 female, 17 male). All of them had normal or corrected to normal vision and gave prior informed consent. Participants were paid for participation (8€/h) and received additional reward based on their performance. All experiments were conducted in accordance with the ethical standards laid down in the 1964 Declaration of Helsinki and were approved by the local ethics committee LEK FB06 at Giessen university (proposal number 2013–0020). We recorded 25 participants for Experiment 1, 8 participants for Experiment 2, 3 and 5, 12 participants for Experiment 4 and 2 for Experiment 6. Experiments were conducted using the Psychtoolbox48 in MATLAB (The Mathworks, Natick, MA, USA) and presented on a VIEWPixx monitor (VPixx Technologies Inc., Saint-Bruno, Quebec, Canada) at a viewing distance of 60 cm. The monitor had a spatial resolution of 1920 × 1080 pixel and a size of 51.5 × 29 cm. We recorded eye movements of the right eye using a desktop mounted EyeLink 1000 (SR Research Ltd., Ontario, Canada) with a sampling rate of 1000 Hz and the Eyelink Toolbox49.

### General methods

At the beginning of each trial, a black fixation cross with a diameter of 0.5° appeared at screen center on a gray background (Fig. 1). Participants could start trials by pushing the space bar on a keyboard while maintaining fixation. Two crosses (placeholders) with a diameter of 0.25° appeared both left and right from fixation at an eccentricity of 15°. After a random interval (500–1000 ms), the central fixation cross changed its size to 0.25° indicating the onset of the target after additional 600 ms. Targets were dots with a radius of 0.25° and were presented for 500 ms. In single-trials, one dot replaced one of the placeholders, whereas in choice-trials both placeholders were replaced by dots. Participants were instructed to maintain fixation until target appearance and then saccade to a target while it was presented. If they succeeded, their reward for that trial was shown at the target location after target offset. If participants did not make saccades or made saccades to placeholders, they received no reward. Rewards were score points (1, 4, 6 or 9) which were converted into monetary reward at the end of the experiment (1€ for 500 points). At the beginning of each block, participants were informed about the distribution of reward to each hemifield and the relative probability of choice and single-trials. For every experiment, the order of blocks was balanced across participants.

### Experiment 1

Experiment 1 tested the hypothesis that the effect of reward on saccade latencies in single-trials is modulated by the presence of interleaved choice-trials. We varied the proportion of choice-trials within one block (0%, 25%, 75%). In every block, a fixed reward was assigned to each target/hemifield and rewards were identical for choice- and single-trials. Rewards summed up to 10 score points with one target receiving a higher reward (6 or 9, ‘highly rewarded target’) than the other (4 or 1, ‘less rewarded target’). The reward difference between the two hemifields could be large (1 vs 9) or small (4 vs 6). The experiment thus comprised the three factors (i) choice-trial probability (0%, 25%, 75%), (ii) reward magnitude (highly or less rewarded) and (iii) reward difference (large or small). The trial order was randomized and single-trials to both hemifields appeared equally often. Every combination of choice-trial probability and reward difference was recorded in a block of 80 trials. In total, every participant completed 480 trials and could receive up to 5.60€ reward. The experiment lasted 60–90 minutes.

### Experiment 2

To show that latency differences caused by choice-trials cannot be explained by a higher saccade probability to highly rewarded targets, we increased the single-trial probability to the less rewarded side. Every participant completed two blocks, one for a small (4 vs 6) and one for a large (1 vs 9) reward difference. Blocks consisted of 120 trials and contained 30 choice-trials (25%). The remaining 90 trials were single-trials, 30 to the highly rewarded and 60 to the less rewarded side. Consequently, participants would saccade equally often to both hemifields if they always chose the highly rewarded target in choice-trials. If they did not, the saccade probability to the less rewarded target would be even higher than 50%.

### Experiment 3

To test whether the presence of choice-trials modulates or causes the effects of reward on saccade preparation, we changed the reward correspondence between choice- and single-trials. Every participant completed three blocks of 120 trials, all with a high reward difference (1 vs 9). In one block, the highly rewarded side for choice- and single-trials was identical (congruent condition), like in Experiment 1. In another block, the highly rewarded side for choice-trials was the less rewarded side in single-trials (incongruent condition). Both, the congruent and incongruent condition contained 75% of choice-trials. In a third block there were only single-trials.

### Experiment 4

In order to assess whether the latency modulation due to choices is adaptive, we varied the choice difficulty by changing the contrast of both targets. All targets were darker than the background and Michelson contrasts were 0.5 (black), 0.2 and 0.08. In the difficult condition, the contrast of the highly rewarded target was 0.08 while the other had a contrast of 0.5. It was the other way round for the easy condition. In the medium condition, both targets had identical contrasts (0.2). The same contrast of 0.2 applied to all fixation crosses, placeholders and targets in single-trials. To make the transition from placeholder to target less salient for the low contrast condition, placeholders remained visible on top of the target during the whole trial for all conditions. Every condition comprised 120 trials. As a manipulation check, we additionally recorded a choice control task and a latency control task. The choice control task consisted of 60 choice-trials without reward but with either the same (0.2) or a different contrast (0.08 vs 0.5). The latency control task consisted of 120 unrewarded single-trials of the three different contrast levels.

### Experiment 5

To determine whether the effects observed in the previous experiments are caused by the expectation of an upcoming choice-trial, we cued half of the single-trials. The cue was a “1” displayed 1.3° above the central fixation cross. It appeared together with the peripheral placeholders and vanished after 200 ms. The whole experiment consisted of 280 trials, with 50% choice-trials and 25% of cued and uncued single-trials each.

### Experiment 6

To determine likely neural mechanisms for the interaction of choice- and single-trials, we measured latency distributions to single-trials with (50%) and without choice-trials interleaved and fitted the LATER model22,23 to the data. Blocks consisted of 100 trials and participants completed 10 blocks without and 20 blocks with choice-trials (3000 trials in total).

### Data and statistical analysis

We used the EyeLink 1000 algorithm to determine saccade onsets. Latencies were defined as the first saccadic sample with respect to target onset and successful target choice was defined as the first sample where the gaze was within a square region of 2° around the target. Trials with saccades initiated earlier than 100 ms or later than 450 ms after target onset were not considered for the final analysis of latencies. Across all experiments (apart from Experiment 4 where missing the target was a dependent variable), this happened in 1.59% of trials. Due to technical issues, some eye movement traces could not be saved in 2.99% of trials. These recording errors were evenly distributed across all experiments and conditions.

Normality of the data was assessed by Kolmogorov-Smirnov tests and by visually inspecting Q-Q-plots. Statistical tests on saccade latencies in Experiment 1–4 were done using repeated-measures ANOVA and post-hoc t-tests with Bonferroni-corrected α level. If sphericity was violated, we report corrected p-values according to Greenhouse-Geisser. We supplemented our analyses with Bayes factors50 (BF) when non-significant results were crucially relevant for interpreting the data. BFs were computed in R (3.3.2; R Development Core Team, 2016) using the BayesFactor package with default priors. BFs smaller one favor the null hypothesis and values greater one favor the alternative hypothesis. Evidence is stronger, the further BFs deviate from 1, with values between 0.33 and 3 being considered inconclusive evidence51. In Experiment 5, we compared latency differences using Wilcoxon signed-rank tests, because the data were not normally distributed. Performance values in Experiment 4 were compared using the non-parametrical Friedman test. Analyses were carried out in MATLAB, R and SPSS (Version 22, IBM Corp., Armonk, NY).

### Choice-trial behavior

In all experiments, we varied the presence of choice-trials as independent variable without being interested in the participants’ behavior in these trials. In choice-trials, participants almost always chose the target with the higher reward (e.g. in Experiment 1: M = 95.3%, SD = 2.5%; Experiment 3: M = 95%, SD = 3.1%) with similar latencies as in single-trials without choice-trials (Experiment 1: M = 214 ms, SD = 22 ms) or slightly elevated (Experiment 3: M = 224, SD = 22 ms).

### Data Availability

Data are publicly available at the doi:10.5281/zenodo.343881.

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## Acknowledgements

This work was supported by DFG grant SFB/TRR 135. We thank Amanda Kelch, Felix Jung and the participants of the NP-3 seminar 2016 for help with data collection.

## Author information

Authors

### Contributions

C.W., A.H., A.S. and A.C.S. conceived and designed the research. C.W and A.C.S. analyzed the data. C.W. and A.C.S. wrote the first draft of the manuscript. All authors revised the manuscript.

### Corresponding author

Correspondence to Christian Wolf.

## Ethics declarations

### Competing Interests

The authors declare that they have no competing interests.

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## Rights and permissions

Reprints and Permissions

Wolf, C., Heuer, A., Schubö, A. et al. The necessity to choose causes the effects of reward on saccade preparation. Sci Rep 7, 16966 (2017). https://doi.org/10.1038/s41598-017-17164-w

• Accepted:

• Published:

• ### Vision as oculomotor reward: cognitive contributions to the dynamic control of saccadic eye movements

• Christian Wolf
•  & Markus Lappe

Cognitive Neurodynamics (2021)

• ### Reward expectation facilitates context learning and attentional guidance in visual search

• Nils Bergmann
• , Dennis Koch
•  & Anna Schubö

Journal of Vision (2019)

• ### The possibility to make choices modulates feature-based effects of reward

• Anna Heuer
• , Christian Wolf
• , Alexander C. Schütz
•  & Anna Schubö

Scientific Reports (2019)

• ### The Limitations of Reward Effects on Saccade Latencies: An Exploration of Task-Specificity and Strength

• Stephen Dunne
• , Amanda Ellison
•  & Daniel T. Smith

Vision (2019)

• ### The relationship between reward and probability: Evidence that exploration may be intrinsically rewarding

• Rosie Clark
•  & Iain D. Gilchrist

Visual Cognition (2018)


text:
Outlook: PSC INSURANCE GROUP LIMITED is assigned short-term Ba1 & long-term Ba1 estimated rating.
Time series to forecast n: 31 Jan 2023 for (n+8 weeks)
Methodology : Deductive Inference (ML)

## Abstract

PSC INSURANCE GROUP LIMITED prediction model is evaluated with Deductive Inference (ML) and Wilcoxon Sign-Rank Test1,2,3,4 and it is concluded that the PSI stock is predictable in the short/long term. According to price forecasts for (n+8 weeks) period, the dominant strategy among neural network is: Buy

## Key Points

1. How can neural networks improve predictions?
2. Is Target price a good indicator?
3. What are the most successful trading algorithms?

## PSI Target Price Prediction Modeling Methodology

We consider PSC INSURANCE GROUP LIMITED Decision Process with Deductive Inference (ML) where A is the set of discrete actions of PSI stock holders, F is the set of discrete states, P : S × F × S → R is the transition probability distribution, R : S × F → R is the reaction function, and γ ∈ [0, 1] is a move factor for expectation.1,2,3,4

F(Wilcoxon Sign-Rank Test)5,6,7= $\begin{array}{cccc}{p}_{a1}& {p}_{a2}& \dots & {p}_{1n}\\ & ⋮\\ {p}_{j1}& {p}_{j2}& \dots & {p}_{jn}\\ & ⋮\\ {p}_{k1}& {p}_{k2}& \dots & {p}_{kn}\\ & ⋮\\ {p}_{n1}& {p}_{n2}& \dots & {p}_{nn}\end{array}$ X R(Deductive Inference (ML)) X S(n):→ (n+8 weeks) $\stackrel{\to }{S}=\left({s}_{1},{s}_{2},{s}_{3}\right)$

n:Time series to forecast

p:Price signals of PSI stock

j:Nash equilibria (Neural Network)

k:Dominated move

a:Best response for target price

For further technical information as per how our model work we invite you to visit the article below:

How do AC Investment Research machine learning (predictive) algorithms actually work?

## PSI Stock Forecast (Buy or Sell) for (n+8 weeks)

Sample Set: Neural Network
Stock/Index: PSI PSC INSURANCE GROUP LIMITED
Time series to forecast n: 31 Jan 2023 for (n+8 weeks)

According to price forecasts for (n+8 weeks) period, the dominant strategy among neural network is: Buy

X axis: *Likelihood% (The higher the percentage value, the more likely the event will occur.)

Y axis: *Potential Impact% (The higher the percentage value, the more likely the price will deviate.)

Z axis (Grey to Black): *Technical Analysis%

## IFRS Reconciliation Adjustments for PSC INSURANCE GROUP LIMITED

1. If, in applying paragraph 7.2.44, an entity reinstates a discontinued hedging relationship, the entity shall read references in paragraphs 6.9.11 and 6.9.12 to the date the alternative benchmark rate is designated as a noncontractually specified risk component for the first time as referring to the date of initial application of these amendments (ie the 24-month period for that alternative benchmark rate designated as a non-contractually specified risk component begins from the date of initial application of these amendments).
2. Paragraph 6.3.4 permits an entity to designate as hedged items aggregated exposures that are a combination of an exposure and a derivative. When designating such a hedged item, an entity assesses whether the aggregated exposure combines an exposure with a derivative so that it creates a different aggregated exposure that is managed as one exposure for a particular risk (or risks). In that case, the entity may designate the hedged item on the basis of the aggregated exposure
3. Credit risk analysis is a multifactor and holistic analysis; whether a specific factor is relevant, and its weight compared to other factors, will depend on the type of product, characteristics of the financial instruments and the borrower as well as the geographical region. An entity shall consider reasonable and supportable information that is available without undue cost or effort and that is relevant for the particular financial instrument being assessed. However, some factors or indicators may not be identifiable on an individual financial instrument level. In such a case, the factors or indicators should be assessed for appropriate portfolios, groups of portfolios or portions of a portfolio of financial instruments to determine whether the requirement in paragraph 5.5.3 for the recognition of lifetime expected credit losses has been met.
4. For the purpose of applying paragraph 6.5.11, at the point when an entity amends the description of a hedged item as required in paragraph 6.9.1(b), the amount accumulated in the cash flow hedge reserve shall be deemed to be based on the alternative benchmark rate on which the hedged future cash flows are determined.

*International Financial Reporting Standards (IFRS) adjustment process involves reviewing the company's financial statements and identifying any differences between the company's current accounting practices and the requirements of the IFRS. If there are any such differences, neural network makes adjustments to financial statements to bring them into compliance with the IFRS.

## Conclusions

PSC INSURANCE GROUP LIMITED is assigned short-term Ba1 & long-term Ba1 estimated rating. PSC INSURANCE GROUP LIMITED prediction model is evaluated with Deductive Inference (ML) and Wilcoxon Sign-Rank Test1,2,3,4 and it is concluded that the PSI stock is predictable in the short/long term. According to price forecasts for (n+8 weeks) period, the dominant strategy among neural network is: Buy

### PSI PSC INSURANCE GROUP LIMITED Financial Analysis*

Rating Short-Term Long-Term Senior
Outlook*Ba1Ba1
Income StatementCaa2B2
Balance SheetBaa2B3
Leverage RatiosB3Ba2
Cash FlowBa2Baa2
Rates of Return and ProfitabilityB3Baa2

*Financial analysis is the process of evaluating a company's financial performance and position by neural network. It involves reviewing the company's financial statements, including the balance sheet, income statement, and cash flow statement, as well as other financial reports and documents.
How does neural network examine financial reports and understand financial state of the company?

### Prediction Confidence Score

Trust metric by Neural Network: 89 out of 100 with 767 signals.

## References

1. Matzkin RL. 2007. Nonparametric identification. In Handbook of Econometrics, Vol. 6B, ed. J Heckman, E Learner, pp. 5307–68. Amsterdam: Elsevier
2. Athey S, Blei D, Donnelly R, Ruiz F. 2017b. Counterfactual inference for consumer choice across many prod- uct categories. AEA Pap. Proc. 108:64–67
3. V. Mnih, A. P. Badia, M. Mirza, A. Graves, T. P. Lillicrap, T. Harley, D. Silver, and K. Kavukcuoglu. Asynchronous methods for deep reinforcement learning. In Proceedings of the 33nd International Conference on Machine Learning, ICML 2016, New York City, NY, USA, June 19-24, 2016, pages 1928–1937, 2016
4. Arjovsky M, Bottou L. 2017. Towards principled methods for training generative adversarial networks. arXiv:1701.04862 [stat.ML]
5. N. B ̈auerle and J. Ott. Markov decision processes with average-value-at-risk criteria. Mathematical Methods of Operations Research, 74(3):361–379, 2011
6. Matzkin RL. 2007. Nonparametric identification. In Handbook of Econometrics, Vol. 6B, ed. J Heckman, E Learner, pp. 5307–68. Amsterdam: Elsevier
7. Y. Chow and M. Ghavamzadeh. Algorithms for CVaR optimization in MDPs. In Advances in Neural Infor- mation Processing Systems, pages 3509–3517, 2014.
Frequently Asked QuestionsQ: What is the prediction methodology for PSI stock?
A: PSI stock prediction methodology: We evaluate the prediction models Deductive Inference (ML) and Wilcoxon Sign-Rank Test
Q: Is PSI stock a buy or sell?
A: The dominant strategy among neural network is to Buy PSI Stock.
Q: Is PSC INSURANCE GROUP LIMITED stock a good investment?
A: The consensus rating for PSC INSURANCE GROUP LIMITED is Buy and is assigned short-term Ba1 & long-term Ba1 estimated rating.
Q: What is the consensus rating of PSI stock?
A: The consensus rating for PSI is Buy.
Q: What is the prediction period for PSI stock?
A: The prediction period for PSI is (n+8 weeks)


text:
[AAR] BLAPMerlinLowsecRoam - 2nd Jan 2019

Assistant Teaching Manager

[AAR] BLAPMerlinLowsecRoam - 2nd Jan 2019

Roam members (13)
SPOILER WARNING!
Alf Saissore - PunisherAres Dodekatheon - MerlinB'aldrick Aivoras - SlasherDrault Sarn - MerlinDurnik Risalo - BurstEsca Sinak - MerlinKirsi Isrik - MerlinKordun Naskingar - MerlinMax Tsero - Burst, MerlinMike Kingswell - Merlin, PunisherMopfog - MerlinShauny Tsero - Burst, MerlinTurlough Dominian - Merlin

This morning I woke up and felt the urge to roam. The call went out and 12 people answered! Shoutout to Esca for their first PVP fleet and not blowing up!

Kills and Losses

(19:26:30) Ouelletta
Heron +2.09m
We hadn't event started yet (were about to do our broadcast check) when Alf the Bloodthristy got his hands on a newbro Heron in Ouelletta. Killed it, SRP'ed his ship and send him the standard "nothing personal - learn about what you did wrong in Eve Universtiy!".

(19:45:46) Abune
Atron +4.22m
Scout Alf cought a Atron in Abune. Killed it, SRP'ed his ship and send him the standard "nothing personal - learn about what you did wrong in Eve Universtiy!".

(20:00:07) Tama
Caracal +27.87m
Capsule +6.71m
We had a Jaguar at sun in Tama. He jumped away and a Caracal jumped into our fleet, said "hi guys" and let us kill him without fighting back...Well I wont complain about a free kill.

(20:11:45) Hikkoken
Algos +4.64m
Vexor +23.07m
IMO best fight of the night. Alf reported 2 Algoses and a Vexor in a PLEX. Vanguard proceded to go in. 1 Algos managed to break away and get away while the other Algos and Vexor got killed. Logi managed to keep EVERYONE(including the scout!) from exploding! This was a classic "good fight" and great fun.

(20:28:35) Nennamaila
Burst -4.32m
Jaguar +58m
Damavik, Jaguar and FedNavy Comet kept playing range games with us. They got a Burst, we got a Jaguar. The rest of them took off.

(20:37:49) Nennamaila
Burst -2.89m
Burst -3.41m
Merlin -3.31m
Experimental fight#1 for tonight: Alf found a Armageddon anfter first hesitation I remembered the FCC rule:"If in doubt fight it". We lost 3 of us and then broke off since his reps could withstand our damage.

(20:58:39) Eha
Imperial Navy Slicer +20.11m
Thrasher +1.65m

(21:08:31) Alsavoinon
Merlin -5.71m
Merlin -9.74m
Experiment#2: Alf followed a Drake and baited him a bit. Was too much for us (mainly due to gateguns). Additionally FC got got by Gateguns - not Killreport for that though(didn't get one?).
Ares took the fleet out and on the way home.

(21:27:09) Renarelle
Punisher -7.83m
Capsule -0.01m
On the way to rejoin the fleet with a quick reship I got got again.

Stats
ISK Destroyed: 148,371,502.85
ISK Lost: 37,213,878.59
ISK Delta: 111,157,624.26
Efficiency: 79.948%

Overall evaluation
Good turnout on (yet another) impromtu fleet
good fights and fun was had
Logi did a great job <3
Alf did a great job <3
Even the expremental fights didn't welp us and we/I learned something from them. IMO well worth the risk
The 2. Algos got away since we didn't have anyone calling targets for scout+vanguard while FC was not on. Next time either have scout call targets until FC gets there or have clearer instructions by FC beforehand.

Shoutout to B'aldrick for having reships in Ichoria!
Last edited by Mike Kingswell on 2019.01.02 22:22, edited 4 times in total.
Assistant Teaching Manager

Retired Director

Re: [AAR] BLAPMerlinLowsecRoam - 2nd Jan 2019

ty for fleet o7 much appreciated

Retired Assistant Personnel Manager,Retired Director of special projects, Retired Assistant Guidance Manager, Retired Events Manager,Retired senior Personnel Officer Graduate,

Member

Re: [AAR] BLAPMerlinLowsecRoam - 2nd Jan 2019

Very fun fleet! Thanks!
[url=http://killfeed.eveuniversity.org?a=pilot_detail&plt_id=154107]
[/url]

Member

Re: [AAR] BLAPMerlinLowsecRoam - 2nd Jan 2019

Again, a massive thanks to the logi for this one. I was flying a really tanky (20+ehp) Punisher as a tackle, which allowed me to tank gate guns and do all sorts of silly stuff tackle usually avoids. But even with that I would have died like 10+ times without the logi. That just kept the content coming...

Member

Re: [AAR] BLAPMerlinLowsecRoam - 2nd Jan 2019

Great looking fleet Mike.
Well done.

Experimental fight#1 for tonight: Alf found a Armageddon anfter first hesitation I remembered the FCC rule:"If in doubt fight it". We lost 3 of us and then broke off since his reps could withstand our damage.

This was my favorite part of the AAR. Not only because you took the fight you didnt know if you could win, but that you figured out you couldnt win it with limited losses and bugged out. And a bit later tried another thing you weren't sure about. I find this to be the thing I struggle with most, engaging when I honestly have no idea.

Member

Re: [AAR] BLAPMerlinLowsecRoam - 2nd Jan 2019

Was a nice fleet - we got a few engagements and my Slasher survived - great fun flying with you as always Mike o7

FCC Manager
|
Graduate | Assistant Orientation Manager | Senior Personnel Ofiicer

Assistant Teaching Manager

Re: [AAR] BLAPMerlinLowsecRoam - 2nd Jan 2019

Glad you had fun! As always it is the Friend Ship that made this fleet good! Pure pleasure flying with!
Assistant Teaching Manager

Member

Re: [AAR] BLAPMerlinLowsecRoam - 2nd Jan 2019

https://zkillboard.com/kill/74367867/ So I looked up the armageddon you lost some stuff too, and I think you could have killed it eventually with the right tatics. What you could have done here is defang it, then the cruises are quite limited impact vs frigates, and it would have eventually run out of cap to run the repper as it is no where near cap stable. even if it just runs reactive and a single rep, it is only repping 514 hp/s before drugs/implants/drugs/heat, and with your 10 man gang with the 2-3 logi,you should be pumping out somewhere around 1k dps assuming each fit can do 150 dps, which is not unreasonable. Defang the drones, then the only dps is the cruise missiles which should be not super high, though the primary would do well to burn outside of scram/grappler range so they can bail if needed. Eventually he would run out of 3600s and die.

As far as the drake engagement goes, main mistake was engaging him on the gate, I looked him up, and holy tank batman: https://zkillboard.com/kill/74442199/ but he had no web, which would have given you guys a decent chance to kill him as he was firing hams vs frigates with no webs. Also you can shot at someone at a gate, bounce, then warp back in and the gate guns won't shoot you for whatever target (s) you have aggressed on in that period. Key for the primary would be to keep up speed as high as possible while being shot at.
Champion of Tash-Murkon

Assistant Teaching Manager

Re: [AAR] BLAPMerlinLowsecRoam - 2nd Jan 2019

Thanks for those insigts Kelon! Really appreciate it
Especially the Armagedon fight! I can definatly follow your reasoning about defanging and then witteling it down. Didn't even occure to me that slow and steady might win the race since he seemed to hold his tank quite securly and even baited us into staying by playing a bit with it.

This just shows that no matter how one sees a specific engagement happening there is always more happening behind the scenes that you have to consider and weigh up against what might also be happening!
Assistant Teaching Manager

Member

Re: [AAR] BLAPMerlinLowsecRoam - 2nd Jan 2019

Sounds like a great fleet.

I love that you tested out your fleet through some experimental fights, I think trying something out like that is the best way to learn and a good way to see what your fleet can do.

On the Drake fight, I’m usually hesitant when on a normal roam the FC wants to engage a non-flashy on gate. Chances are the majority of newbros and those that don’t roam often don’t completely understand the gate gun mechanics and how to bounce/rotate tackle till everyone has shed the gate gun aggro, it looks like this may have been the case here. Unfortunately a couple good gate gun hits can blap a frig pretty quick if they don’t know how to shed them.

I also really like the fact you did some recruitment as you went. That’s something I do on my solo stuff as well.you get a few interested people here and there plus some entertaining salt. I think sending these out to npc corp kills should be SOP for FCCs on fleets as it’s a fantastic way to leverage our interactions with other players to possibly recruit some.
Graduate xxxxxx Former FCC Staff xxxxxx Content Creator


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# tikz-qtree and tikz trees not playing nice with each other

I have the following problem: I have tikz with tikz-qtree installed. Hence, if I draw a tree with q-tree it works perfectly. However, I have to recycle one drawing done without q-tree, so that the code looks like

\documentclass{article}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}[every node/.style={circle,draw},level 1/.style={sibling distance=30mm},level 2/.style={sibling distance=10mm}]
\node {A}
child { node {B}
child { node {E} }
child { node {F} child { node {K} } child { node {L} } child { node {M}}}
child { node {G} }}
child { node {C}
child {node {H}}}
child { node {D} child {node {I}} child {node {J} child {node {N}} child  {node {O}}}}
;
\end{tikzpicture}
\end{document}

This usually works if you don't have qtree installed, but if you do, then the level distance is ignored and the third level nodes z1 and t1 will overlap with each other. Is there any quick solution for that?

TIA!

-
Welcome to TeX.sx! Please add a minimal working example (MWE) that illustrates your problem. I had to make several changes to your code to get it to compile and when I did then I found no difference in placement of the nodes. It's more than likely that the changes I made were significant so a compilable example would really help here. –  Andrew Stacey Jun 11 '12 at 9:04
First you need to replace \node by node –  Alain Matthes Jun 11 '12 at 9:32
I'm without a compiler at the moment, but I just copypasted a standard example here from SE, since I tried, and it doesn't work if you have \usepackage{tikz-qtree}. Then you remove it, and everything magically works. Sorry for this. –  Alessio Jun 11 '12 at 9:49
Not a problem! It's just so that folk can best help. Which version of tikz-qtree are you using? Also, now that you have 11 reputation here you can upload images - that would be extremely useful as I'm still getting no difference in the positioning. –  Andrew Stacey Jun 11 '12 at 9:56


text:
# Math Help - Intro to Analysis

1. ## Intro to Analysis

I just started this class and for some reason the material isn't clicking at all. I was wondering if anyone could give me some sort of hint on how to approach this problem. I would GREATLY appreciate it.

2. Originally Posted by haikuambulance
I just started this class and for some reason the material isn't clicking at all. I was wondering if anyone could give me some sort of hint on how to approach this problem. I would GREATLY appreciate it.

I will do the first bit:

$x = qx + (1 - q) x < qx + (1 - q)y$ for $x < y$.

$y = qy + (1 - q) y > qx + (1 - q)y$ for $x < y$.

Therefore $x < qx + (1 - q)y < y$ for $x < y$.

3. Here's how I would do it. Since x< y, y- x> 0. Since q< 1, subtracting q from each part, 0< 1-q. Since 0< q, -q< 0 and 1- q< 1. Setting p= 1- q, we also have 0< p< 1. multiplying each part by the positive number y- x, 0(y-x)< p(y-x)< 1(y-x) or 0< py- px< y- x. Adding x to each part, x< py- px+ x< y. -px+ x= (1- p)x= qx and py= (1-q)y so that is x< (1- q)y+ qx< y.

(I first did that using "q" rather than "p" and arrived at x< qy+ (1-q)x< y which is true but not what is wanted. That when I switched to p= 1- q and did the same proof.)


text:
SMS scnews item created by Munir Hiabu at Tue 4 Jun 2019 1519
Type: Seminar
Distribution: World
Expiry: 21 Jun 2019
Calendar1: 21 Jun 2019 1400-1500
CalLoc1: Carslaw 373
CalTitle1: Low-rank tensor decompositions for sampling of high-dimensional probability distributions
Auth: munir@pmunir.pc (assumed)

# Low-rank tensor decompositions for sampling of high-dimensional probability distributions

### Friday June 21, 2pm, Carslaw 373

Sergey Dolgov ( University of Bath, Department of Mathematical Sciences)

Title: Low-rank tensor decompositions for sampling of high-dimensional probability distributions

Uncertainty quantification and inverse problems in many variables are pressingly needed tasks, yet high-dimensional functions are notoriously difficult to integrate in order to compute desired quantities of interest. Functional approximations, in particular the low-rank separation of variables into tensor product decompositions, have become popular for reducing the computational cost of high-dimensional integration down to linear scaling in the number of variables. However, tensor approximations may be inefficient for non-smooth functions. Sampling based Monte Carlo methods are more general, but they may exhibit a very slow convergence, overlooking a hidden structure of the function. In this talk we review tensor product approximations for the problem of uncertainty quantification and Bayesian inference. This allows efficient integration of smooth PDE solutions, posterior density functions and quantities of interest. Moreover, we can use the low-rank approximation of the density function to construct efficient proposals in the MCMC algorithm, as well as in the importance weighting. This combined tensor approximation - MCMC method is more accurate also if the quantity of interest is not smooth, such as the indicator function of an event.

Actions:


text:
# The history of Proper Forcing

What were the initial motivations of the use of the proper forcing.?

-
Harrington once told me that Shelah called it proper forcing cause he thought that was the proper way to do forcing. –  Amit Kumar Gupta Nov 26 '10 at 0:32
@Amit: Haha. That sure sounds like Shelah. –  Andres Caicedo Nov 26 '10 at 1:31
Gapo: I think your question, as stated, is a bit too ambitious for this site. You are asking us for something that could easily end up being a short paper if answered properly. Would you mind saying something about what you are looking for, where in the literature you have looked, that sort of thing? It may help us. –  Andres Caicedo Nov 26 '10 at 1:33
I believe that the initial motivations for proper forcing are well explained throughout Shelah's book. If you want a brief look at the evolution of the subject, a good place to start reading at is the "Proper forcing" chapter in the handbook of set theory (written by Uri Abraham). Regarding the open problems, you can check this paper: shelah.logic.at/files/666.pdf –  Haim Nov 26 '10 at 1:59
@Andreas: I did not expect to have one good answer to my question but a sort of collective answer where every one could bring one original information as indeed Amit Haim and Stefan did. It was my first question I'm sorry if is not in the spirit of mathoverflow next time I will ask thing more precisely. If I should be more precise now I would say that initial motivation was the most important part of the question for me so that Stefan answer me. –  gapo Nov 26 '10 at 11:08

I agree with Andres that this is a very ambitious question. Let me throw in a tiny little bit of information:

As far as I know, Jensen's construction of a model of CH without Souslin trees was one of the first uses of both countable support iteration and master conditions (conditions generic over a countable elementary submodel of a sufficiently large initial part of the universe).

One of the first publications using proper forcing seems to be Shelah 100, with the hilarious title "Independence results" (JSL 45 (1980), 563-573).
This paper not only introduces proper forcing (without proofs of the the iteration theorem and so on), but also oracle-ccc forcing and oracle-proper forcing.

-

What I remember is that first Laver solved the Borel conjecture by an countable support iteration which added a real at every stage (1976). That such an iteration can be of any use, was very surprising at the time. Then Baumgartner introduced the very general notion of property A, which included most (all?) standard forcings known then which added reals and showed that countable support iteration of them behaves nicely (1978). Then came Shelah, who gave the proper definition (1980). This was again a surprising thing, as Baumgartner's definition was combinatorial (containing combinatorial properties of the poset, eh, almost) while Shelah simply required that P should preserve all stationary subsets of all sets of the form $[A]^{\aleph_0}$, that is, a semantic definition. Notice that Shelah's new theory gave new and elegant proofs to old theorems, as Baumgartner's consistency of that any two $\aleph_1$-dense sets of reals are isomorphic or Mitchell's consistency of the tree property of $\aleph_2$.

-

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This would have been my answer too, Assaf! –  Todd Eisworth Mar 19 '12 at 17:15
What a fun article! Thanks. –  Andres Caicedo Mar 19 '12 at 17:19
Thanks to Assaf for pointing out Andrzej's paper, and to Andrés for linking this on twitter! –  Andrés Villaveces Mar 19 '12 at 17:48

My understanding is that part of the early motivation was the observation of the attractive features of two main classes of forcing:

• ccc forcing. Preserves all cardinals. Preserves stationary subsets of $\omega_1$. Closed under finite support iterations.
• countably closed forcing. Preserves $\omega_1$. Preserves stationary subsets of $\omega_1$. Closed under countable support iterations.

And what was wanted was a class of forcing notions in which these two kinds of forcing could be mixed together in iterations, while still preserving $\omega_1$. That is, what is wanted is a class of forcing that contains all ccc forcing, all countably closed forcing, which all preserve $\omega_1$ and which support some kind of iteration theorem. The full class of all $\omega_1$-preserving forcing does not fit the bill, since it is not closed under iterations. But meanwhile, the class of proper forcing does have the desired features...

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I think you have a typo in the last paragraph. $\omega-1$ instead $\omega_1$. –  Asaf Karagila Mar 18 '12 at 16:48
Thanks, Asaf, it's fixed now. –  Joel David Hamkins Mar 18 '12 at 18:54


text:
How can I find the simplest curve that gives me a scroll like this?

This is where I am at the moment and doing it by hand is becoming more and more cumbersome.

First, take a drawing of the scroll since the outline is already processed:

Then, define a spiral in a parametric plot with several parameters that move things around:

Manipulate[
Show[ParametricPlot[
1/d (1 + a Sin[ϕ/2 + f]/4 + a Sin[ϕ/4 + f]/4) Norm[
2 + ϕ^1.1] {Sin[-ϕ], 0.075 + Cos[ϕ]}, {ϕ, 0,
5.25 π}, PlotRange -> {-1, 1}],
Epilog -> Disk[{0, 0}, .075],
Prolog -> {Texture[img],
Polygon[{Scaled[{0, 0}], Scaled[{1, 0}], Scaled[{1, 1}],
Scaled[{0, 1}]},
VertexTextureCoordinates -> {{0 + b, c}, {1 + b, c}, {1 + b,
1 + c}, {0 + b, 1 + c}}]}], {a, .5,
2}, {b, .091}, {c, .148}, {d, 30, 40}, {f, 0, π}]

(b and c are there to center the image only, could not find a nicer way)

What I tried to do was to increase the radius of the spiral as a function of $\phi$ but I have the impression a computer would be smarter than me in finding this. This is the best I can do atm.

I know there is a new FindFormula but cannot figure out how to make it work for parametric functions.

I would very much like something that does not involve BSplineCurve. A piecewise function on $\phi$ is welcome. Violin shapes were made with compass and rulers so I imagine that there must be a simple function for the curvature and how it changes as the scroll turns. With simple function I mean something like this:

In this case the radius of each segment on the scroll is given by the projection from the lower triangle. This can be taken to the continuous case as a simple function.

Using the method provided by nikie I can obtain different scrolls varying the functions I use for the fit. This is the one I most like fitFn = a + b theta + c theta^2 + d /theta + e/theta^2 + h Sin[ f (-theta)^.5 + g];

Maybe you can find out to which one it corresponds

• This may be helpful. – xslittlegrass Apr 22 '16 at 1:29
• Okay I'm confused: splines are still piecewise functions after all, so you seem to have conflicting desires. Also, "with compass and rulers" can be taken to mean that the arcs could very well be made up of circle arcs and line segments suitably pasted together. – J. M. will be back soon Apr 22 '16 at 1:46
• Maybe I should rephrase it as "don't use BSplineCurve" which would be easy enough to fit with enough points. Let me try to edit the question – tsuresuregusa Apr 22 '16 at 1:50
• With respect to "simple": the lituus seems to be a good starting point as any for a volute. – J. M. will be back soon Apr 22 '16 at 2:08
• @J.M. I tried a few Archimedes' spirals but it doesn't really work, you need to add some kind of "asymmetry" (for lack of a better word), that's what my 1 + a Sin[\[Phi]/2 + f]/4 + a Sin[\[Phi]/4 + f]/4 is doing, modulating the radius of the spiral to try to fit it the real scroll. I thought about subtracting a simple spiral to the edge and fitting the difference to a function with FindFormula but don't really know yet how to yet. – tsuresuregusa Apr 22 '16 at 2:38

Step 1: find the curve in the black and white image. For this, I will use a shortest path search, and to make sure it finds the path I'm looking for (instead of one of the short cuts through the labels), I will give it a few "path markers" along the way that the path should visit in order:

First, load the image, get the black pixels:

img = Import["http://i.stack.imgur.com/XBEkk.png"];
blackPts = PixelValuePositions[Binarize[img], 0];
nearestFn = Nearest[blackPts -> Range[Length[blackPts]]];

Next, enter rough path markers using a LocatorPane

pathMarkers = {{373, 5}, {290, 595}, {475, 588}, {495, 371}, {382, 476}};
LocatorPane[Dynamic[pathMarkers],
Dynamic@Show[img,
Graphics[{Red,
Line[blackPts[[First /@ nearestFn /@ pathMarkers]]]}]]]

Then, turn the black pixels into a graph, with point distances as edge weight:

vertices = Range[Length[blackPts]];
edges = Select[
#[[1]] < #[[2]] &];
edgeWeights =
edges /. {i_ <-> j_ :> Norm[N@blackPts[[i]] - blackPts[[j]]] - .1};
g = Graph[vertices, edges, EdgeWeight -> edgeWeights];

(Don't try to evaluate g in a notebook. It's a very large graph, displaying it crashed the kernel on my machine. But we don't have to display it, we just want to find paths in it, which is very fast.)

Then find the shortest path that touches out path markers:

pathPoints =
Flatten[FindShortestPath[g, ##] & @@@
Partition[First /@ nearestFn /@ pathMarkers, 2, 1]];
path = blackPts[[pathPoints]];
HighlightImage[img, {Thick, Line[path]}]

That looks promising.

Step 2: transform to polar coordinates.

I find handling polar coordinates easier working in the complex plane:

center = {402, 465};
pathComplex = N[(# - center & /@ path).{1, I}];

radii = Abs[pathComplex];

The branch cut takes a bit more effort:

first, multiply each point with the conjugate of the next point:

pointPairs =
pathComplex[[2 ;;]]}];

Now, the phase of pointPairs is the angle between each point pairs. And since the points are close to each other, this will not contain any discontinuities due to the branch cut:

anglesBetweenPointPairs = Im@Log[pointPairs];

I just have to sum these angles up to get the original phase again:

angles = Prepend[Accumulate[anglesBetweenPointPairs], 0] +
Im[Log[pathComplex[[1]]]];

We can check that it's the same path:

ListPolarPlot[{angles, radii}\[Transpose], Joined -> True]

And that there are no path cuts:

ListLinePlot[{angles, radii}\[Transpose]]

You can even move the center around and watch as the polar curve changes dynamically to see where the best center point might be:

center = {402, 465};
Row[{
LocatorPane[Dynamic[center], Image[img, ImageSize -> All]],
Dynamic@
angles},
pathComplex = N[(# - center & /@ path).{1, I}];
pointPairs =
pathComplex[[2 ;;]]}];
anglesBetweenPointPairs = Im@Log[pointPairs];
angles =
Prepend[Accumulate[anglesBetweenPointPairs], 0] +
Im[Log[pathComplex[[1]]]];
]}]

(I have no idea what kind of function might have been used for the radius/angle relationship, so I'm not going to attempt the actual curve fitting. IMHO, if you don't know anything about a functional relationship, a spline usually is the best guess.)

FWIW, I get a somewhat decent approximation using these terms:

Clear[g, eq]
fitFn = a + b theta + c theta^2 + d/theta + e/theta^2;
eq[theta_] =
fitFn, {a, b, c, d, e, f, g, h}, theta];

Show[
Plot[eq[theta], {theta, Min[angles], Max[angles]},
PlotStyle -> Directive[Dashed, Red], PlotRange -> All]]

HighlightImage[img,
Line[Array[center + ReIm[Exp[#*I + Log[eq[#]]]] &, 500,
MinMax[angles]]]]

• Beautiful. At the moment I;m using this to fit: a + b theta + c Cos[theta + e] + f theta^2 + g Sqrt[Abs[theta]]  I don't know either what function was, but I guess it should be easy to obtain with compass and ruler via different projections. Now, which kind of projections are those... that's another question. Thanks a lot! – tsuresuregusa Apr 22 '16 at 19:58
• I've added a linear fit. No idea if this can be constructed using ruler and compass. – Niki Estner Apr 22 '16 at 20:17
• This might be useful for taking care of the branch cuts in the angles. – J. M. will be back soon Apr 22 '16 at 22:30
• One of the most beautiful things of the scroll is that has a certain "movement". If you plot the difference between your last fit and the curve you obtain a very marked oscillation with a defined frequency during at least 2 cycles. I think that's what make the scrolls alive somehow. I added some figures obtained with your code. Thanks again. – tsuresuregusa Apr 22 '16 at 23:37
• I thought the "oscillation" was because I missed the actual center point... – Niki Estner Apr 23 '16 at 7:06

I didn't get a good answer, but this is how I would do it and probably how I would evaluate the quality of an answer. Much of what I do below involves manual work, but it could be automated:

First, we need some points along the curve. I selected them manually

If I hadn't manually selected them, I would have used some image processing including a thinning transform and then I would have used FindCurvePath.

We also want to select a point that seems like the "center" of the spiral.

center = {{165.92578125, 147.453125}};
pts = {{183.65234375, 292.625}, {201.4765625,
287.3671875}, {215.21484375, 282.234375}, {226.21484375,
277.7578125},.....};

And we want to normalize the pts so the "center" is the origin:

normalizedPts = (# - First@center) & /@ pts;

We want to work with this in polar coordinates:

ToPolarCoordinates@normalizedPts

But there's a problem. There's a branch cut. I've manually edited the results so that there isn't a branch cut. This involves subtracting or adding 2Pi to the results so that they form a continuous curve:

arc1 = {{146.25014293696273,
1.4492904217728535}, .... {82.16481360563434, -3.113682087252578}}

arc2 = {{79.36095240482918, 2.9819303345670916},... {39.305506825650895, -2.883525354070561}}

arc3 = {{36.32153376025393, 3.0092472204025533}, {19.075687437056864, -2.8414075099042826}}

branchCutsRemoved = Reverse/@
Join[arc1, (# - 2 Pi) & /@ arc2, (# - 4 Pi) & /@ arc3];

I reversed all the points so that they're in {arg, abs} form. This means we are looking for the absolute value of the point as a function of arg. I use theta for arg or the angle.

We now have a fairly reasonable function of theta to try and approximate:

ListPlot[branchCutsRemoved, PlotRange -> All]

You can try to fit to this:

eq = Normal@
NonlinearModelFit[branchCutsRemoved,
a + b theta + c  Cos[d theta + e] + f  Cos[d theta/2 + e], {a, b,
c, d, e, f, g, h}, theta]

The fit seems... okayish:

Show[ListPlot[sorted, PlotStyle -> Red], Plot[eq, {theta, -16, 5}]]

• thanks a lot, I didn't know about the coordinate tool, quite useful. Is there a way to automatically fix the branches? Will leave the question open for a bit to see if another answer appears but this is kinda exactly what I was looking for, now I will check the coefficients for different scrolls and see how the different "schools" behave. – tsuresuregusa Apr 22 '16 at 18:56
• Yes. You could write a function that goes through a list and when it sees a value drop by 2Pi adds 2Pi to all subsequent values. – Searke Apr 22 '16 at 19:17


text:
# Spring constants

Nanabit
A block of mass 12.0 kg slides from rest down a frictionless 37.0 degree incline and is stopped by a strong spring with k=3.50e4 N/m. The block slides 3.00 m from the point of release to the point where is comes to a rest against the spring. When the block comes to rest, how far has the spring been compressed?

I really wanted to use 1/2kx^2 = W to get this, but I know I didn't have all of the values. I thought of using KE = 1/2 mv^2, but I don't know how to get velocity without a time value. So I ended up using an equation for the incline, g sin theta = acceleration, got 5.90 m/s^2 for that, plugged mass and acceleration into f=ma to get a force of 70.8 N, plugged values into W=F S Cos theta to get work as 170 J, and finally plugged values into 1/2kx^2 = W to get x as .0984, but this is wrong. For every other equation I try to use, I'm missing 2 values and don't know how to get either of them. Where did I go wrong?

arcnets
You're overcomplicating.
Use conservation of energy.
When an object of mass m goes down a vertical distance (height) h, then it gains the energy E = mgh (g is gravitational acceleration).
This energy goes into the spring, which will thus compress by a length x, where E = kx. OK?

Nanabit
But it's not exactly a vertical distance since it's on an incline, is it?

arcnets
Come on. You know the distance, and you know the angle...

Nanabit
So I took the sin of 37 times 3 to get the height. I got 1.805. Then I multiplied that times 12.0 and 9.8 to get E. I got 212.3. Then I divided that by 3.50e4. Got .00607. And it isn't right. I've been doing physics since 8 AM so I'm really tired, sorry. What am I missing?

Homework Helper
So I took the sin of 37 times 3 to get the height. I got 1.805.
Yes, that is the correct height (in meters).

Then I multiplied that times 12.0 and 9.8 to get E. I got 212.3.
Yes, that is the change in potential energy (in Joules).

Then I divided that by 3.50e4. Got .00607.
Why? Did you forget the formula 1/2kx^2 = W that you gave before?

x2= 2W/k or, using the figures you got
x2= 2(212.3)/(35000). Solve for x.

Nanabit
thank you!


text:
# The Japanese Theorem for Nonconvex Polygons - Carnot's Theorem

Author(s):
David Richeson

### Carnot's Theorem

For proofs of the quadrilateral case of the Japanese theorem or the polygonal case with all diagonals eminating from a single vertex see [AUM1, AUM2, FP, FR, Gr, Hay, Jo1, Mi, Yo]. We will not prove those separately, but instead prove the more general Japanese theorem for cyclic polygons. Our proof, which is in essence Honsberger's proof ([Ho]), relies on a classical theorem discovered by Lazare Nicolas Marguérite Carnot (1753–1823) that exhibits a relationship between a triangle, its inradius, and its circumradius.

The signed distance of a side of a convex cyclic polygon to the circumcenter is the distance, unless the segment joining the cirucumcenter to the side lies completely outside the polygon, in which case it is the negative of the distance. For example, for the triangles in Figures 4(a) and 4(b), all the signed distances $a, b,$ and $c$ are positive except $b$ in Figure 4(b). The sign contributions of the cyclic hexagon in Figure 4(c) are indicated.

Figure 4

Carnot's theorem. If $a$, $b$, and $c$ are the signed distances from the circumcenter of a triangle to the sides of the triangle, $R$ is the circumradius, and $r$ is the inradius, then $R + r = a + b + c$.

Figure 5

See [AN] for a short proof of Carnot's theorem for acute triangles (in this case $a$, $b$, and $c$ are all positive); the given argument can be modified to cover the non-acute case.

To see the theorem in action, move the vertices of the triangle below and see the equality of the two sums.

David Richeson, "The Japanese Theorem for Nonconvex Polygons - Carnot's Theorem," Convergence (December 2013)