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# Ecosystem-based fisheries management forestalls climate-driven collapse

Sep 11, 2020

### Regionally-downscaled climate change projections

We used a management strategy evaluation (MSE) applied to ensemble projections of a climate-enhanced multispecies stock assessment within the integrated modeling framework of the Alaska Climate Change Integrated Modeling project (ACLIM)19. For this, six high resolution downscaled projections of oceanographic and lower trophic level conditions in the Bering Sea (using the Regional Ocean Modeling System49,50) were coupled to the BESTNPZ nutrient-phytoplankton-zooplankton model51; we refer to this model complex throughout this paper as the Bering10K ROMSNPZ, or just ROMSNPZ, model. Boundary conditions were driven by three global general circulation models (GFDL-ESM2M52, CESM153, and MIROC-ESM54) projected (2006–2099) under the high-baseline emission scenario Representative Concentration Pathway 8.5 (RCP 8.5) and midrange global carbon mitigation (RCP 4.5; note, that for CESM under RCP 4.5, projections from 2080–2100 were unavailable so conditions from 2080–2099 were held constant at 2080 conditions for that scenario only) future scenarios from the Coupled Model Intercomparison Project phase 5 (CMIP5)29,55. Hermann et al.30,56 also report on downscaled hindcasts of oceanographic and lower trophic conditions in the EBS from 1970–2012 (see refs. 30,56,57 for detailed descriptions of model evaluation and performance). For each downscaled model simulation, we replicated the National Marine Fisheries Service Alaska Fisheries Science Center annual summer bottom-trawl survey in time and space in the ROMSNPZ model (using historical mean survey date at each latitude and longitude of each gridded survey station) to derive estimates of sea surface and bottom temperatures (Fig. 1). We additionally used a polygon mask of the survey area to estimate the average zooplankton abundance in the system during spring, summer, winter, and fall months. These indices were derived for each climate projection scenario, as well as a persistence scenario where conditions were held constant at the average of those for 2006–2017 from a hindcast simulation. All index projections were bias corrected to the 2006–2017 hindcast period using the delta method assuming unequal variance in the GCM projections and hindcast58 such that:

$$T_{mathop {{{mathrm{fut}}}}limits^prime ,y} = bar T_{{mathrm{hind}},overrightarrow {scriptstyle{mathrm{ref}}} } + frac{{sigma _{{mathrm{hind}},overrightarrow {scriptstyle{mathrm{ref}}} }}}{{sigma _{{mathrm{fut}}, overrightarrow{scriptstyle{mathrm{ ref}}} }}}left( {T_{{mathrm{fut}},y} – bar T_{{mathrm{fut}},overrightarrow {scriptstyle{mathrm{ref}}} }} right)$$

(1)

where (T_{mathop {{{mathrm{fut}}}}limits^prime ,y}) is the bias-corrected projected timeseries, (T_{{mathrm{fut}},y}) is the raw projected timeseries, (bar T_{{mathrm{hind}},overrightarrow {scriptstyle{mathrm{ref}}} }) is the mean of the hindcast during the reference years (overrightarrow {{mathrm{ref}}}) (2006–2017), (bar T_{{mathrm{fut}},overrightarrow {scriptstyle{mathrm{ref}}} }) is the mean of the raw projected timeseries during the reference years (overrightarrow {{mathrm{ref}}}), (sigma _{{mathrm{hind}},overrightarrow {scriptstyle{mathrm{ref}}} }) is the standard deviation of the hindcast during the reference years (overrightarrow {{mathrm{ref}}}), (sigma _{{mathrm{fut}},overrightarrow {scriptstyle{mathrm{ref}}} }) is the standard deviation of the raw projection timeseries during the reference years (overrightarrow {{mathrm{ref}}}).

### Climate-enhanced multispecies stock assessment model

Bias-corrected indices were then used as covariates in the climate-enhanced multispecies stock assessment model for the Bering Sea (hereafter CEATTLE)59 to evaluate the performance of alternative management approaches on future fish biomass and catch. CEATTLE is a climate-enhanced multispecies statistical age-structured assessment model with parameters for growth that are functions of temperature (i.e., temperature-specific average weight-at-age) and predation that are functions of temperature (via a bioenergetics-based predation sub-model)59,60,61. Since 2016, the model has been used operationally in the Bering sea as a supplement to the annual BSAI pollock stock assessment61. Various configurations of CEATTLE are possible; for this study we chose one where temperature-specific predator and prey interactions influenced natural mortality, temperature influenced weight-at-age, and the spawner-recruit relationship was a function of physical and biological future conditions as well as random variability (i.e., a climate-informed multispecies model). We fit the model using penalized maximum likelihood to survey biomass, diet, and fishery harvest data for three groundfish species pollock, Pacific cod, and arrowtooth flounder from the EBS in the EBS over the period 1979–2017. We also used the Bering10K ROMSNPZ model to produce detailed hindcasts of temperature for the period 1970–2017. We used hindcast-extracted timeseries from the ROMSNPZ model and CEATTLE model estimates of recruitment ((R_{i,y,l})) and spawning biomass ((B_{i,y – 1})) in hindcast year y for each species i to fit a climate-enhanced logistic recruitment per spawner model36, such that:

$$ln left( {hat R_{i,y}} right) = alpha _i – beta _{0,i}B_{i,y – 1} + ln left( {B_{i,y – 1}} right) + {{{bf{B}}}}_i{{{bf{X}}}} + varepsilon _{i,y}$$

(2)

where ({{{bf{B}}}}_i{{{bf{X}}_l}}) is the summed product of each covariate parameter (beta _{ij}) and the corresponding environmental covariate (X_{j,y}) for each bias-corrected environmental index (j = ( {1,2…n_j} )). We selected indices representative of ecological conditions important for groundfish recruitment in the Bering sea39; spring and fall large zooplankton abundances, survey replicated bottom temperature, and extent of the residual cold pool of extremely dense and cold sea water that persists across the EBS shelf following spring sea ice retreat. We assumed normally distributed (in log space) residual errors for each species ((varepsilon _{i,y} sim Nleft( {0,sigma _i^2} right))). The CEATTLE model was then projected forward where ROMSNPZ indices from individual projections drove growth, predation, and recruitment in each future simulation year36,62.

### Evaluation of harvest management approaches

Previous authors have defined EM (i.e., the incorporation of ecosystem information into marine resource management) as a continuum between two paradigms of management and focus18. On one end is within-sector single-species management that considers ecosystem information (EAFM) and on the other is cross-sectoral whole of ecosystem management (i.e., EBM). EBFM is intermediate between these and is defined by quantitative incorporation of ecosystem interactions into assessment models and target setting (EBFM). Most fisheries management in the Bering Sea can be characterized as EBFM or EAFM, with increasing trends toward cross-sectoral coordination at the scale of EBM. Here we focus on one aspect on this scale of potential management options, operational EBFM and EAFM as captured through the CEATTLE multispecies stock assessment model and harvest policies decisions made annually under the constraint of the 2 MT cap (modeled via the ATTACH model).

MSE is a process of “assessing the consequences of a range of management strategies or options and presenting the results in a way which lays bare the tradeoffs in performance across a range of management objectives”63. MSE has been frequently used to evaluate alternative management strategies based on single-species estimation methods64. It is increasingly used to evaluate ecosystem management performance, although these evaluations are far less commonplace due to the complexity of modeling and assessing the performance of ecosystem level metrics64. Importantly, MSE “does not seek to proscribe an optimal strategy or decision”63, rather it aims to describe the uncertainty and tradeoffs inherent in alternative strategies and scenarios. In this case, through a series of workshops, we worked with managers and stakeholders to identify priority scenarios and outputs19. From this, risk, sensitivity, and uncertainty under contrasting climate scenarios were requested outputs of the analysis, as was the performance of current climate-naive EBFM policies.

A key component of MSE is identifying and quantifying uncertainty (i.e., process, observation, estimation, model, and implementation error) and representing it using an operating model. In the case of this MSE, the focus was on process error uncertainty due to variation in recruitment about the fitted stock-recruitment relationship, one major source of model error in the form of alternative climate scenarios, and implementation error. The MSE does not account for estimation error (uncertainty in the parameters of the operating model) nor observation error. This is because the estimates of recruitment and spawning biomass from CEATTLE for the BSAI are very precise (see Fig. 10 in ref. 60) and the estimation and operating models are therefore very similar. Thus, CEATTLE is the operating model for this MSE and implicitly the estimation method. In this approach we assume that while allowing for observation error would have increased overall error, the effect would have been minor compared to the investigated uncertainties. Future analyses using a full MSE (i.e., separate operating and estimation models) could evaluate the effect of observation error, but perhaps more importantly, the potential for model error, whereby the population dynamics model (on which the estimation method is based) differs from that of the operating model such that the estimates on which management decisions are made are biased relative to the true values in the operating model.

Given this we summarized the relative change in catch and biomass for the three species in the model under the following fishing scenarios (Fig. 1): (a) projections without harvest ((F_{i,y} = 0)) in each year y of scenario l for each species i, (b) projections under target harvest rate (Supplementary Fig. 7 left) and with a sloping harvest control rule (HCR) (Supplementary Fig. 7 right), (c) as in 2 but with the constraint of a 2 MT cap applied dynamically to the three focal species only.

Under the North Pacific Fishery Management Council (NPFMC) constraint of the 2 MT cap on cumulative total annual catch, realized harvest (i.e., catch) and specification of individual species harvest limits known as Total Allowable Catch (TAC; metric tons) are a function of the acceptable biological catch (ABC) for the given species, as well as ABC of other valuable species in the aggregate complex19,34 (https://github.com/amandafaig/catchfunction). TAC must be set at or below ABC for each species, therefore TAC of individual species are traded-off with one another to avoid exceeding the 2 MT cap. From 1981 to 1983, the TAC of pollock was reduced significantly below the ABC and in 1984 the 2 MT cap became part of the BSAI fishery management plan21,34,65. Pacific cod regulations have changed markedly over recent decades and it was only in the 1990s that in many years the catch and TAC approached its ABC. Thus, we used the socioeconomic ATTACH model (the R package ATTACHv1.6.0 is available with permission at https://github.com/amandafaig/catchfunction34) to model realized catch in each simulation year as a function of CEATTLE assessment estimates of ABC (tons) for pollock, Pacific cod, and arrowtooth flounder under future projections (2018–2100). This entailed three steps for each future simulation year (y):

1. 1.

project the population forward from (y – 1) to (y) using estimated parameters from the multispecies mode of the CEATTLE model fit to data from 1979 to 2017 and recruitment based on biomass in simulation year (y) and future environmental covariates from the ROMSNPZ model downscaled projections (see “Methods” above) to determine ({mathrm{ABC}}_{i,y,l}) for each species (i) under each scenario (l) given the sloping harvest control rule for pollock, Pacific cod, and arrowtooth flounder in each simulation year (y);

2. 2.

use ({mathrm{ABC}}_{i,y,l}) of each species from step 1 as inputs to the ATTACH model in order to determine the North Pacific Marine Fishery Council Total Allowable Catch (TACi,y,l) for the given simulation l year y;

3. 3.

use TACi,y,l from step 2 to estimate catch (tons) in the simulation year (Fig. 1); remove catch from the population and advance the simulation forward 1 year.

### Determine the annual ABC

We used end-of-century projections (2095–2099) to derive a maximum sustainable yield (MSY) proxy for future harvest recommendations (ABCi,y,l) for each scenario l. To replicate current management, we used a climate-specific harvest control rule that uses climate-naive unfished and target spawning biomass reference points ((B_{0,i}) and (B_{{mathrm{target},i}}), respectively) and corresponding harvest rates ((F_{i,y} = 0) and (F_{i,y} = F_{{mathrm{target}}}) and (B_{i,y,l}) in each simulation l year (y) for each species (i)

$${mathrm{ABC}}_{i,y,l} = mathop {sum }limits_a^{A_i} left( {frac{{S_{i,a}F_{{mathrm{ABC}},i,y,l}}}{{Z_{i,a,y,l}}}left( {1 – e^{ – Z_{i,a,y,l}}} right)N_{i,a,y,l}W_{i,a,y,l}} right)$$

(3)

where (W_{i,a,y,l}), (N_{i,a,y}), and (Z_{i,a,y,l}) is the climate-simulation specific annual weight, number, and mortality (i.e., influenced through temperature effects on recruitment, predation, and growth) at age (a) for (A_i) ages in the model, and (S_{i,a}) is the average fishery age selectivity from the estimation period 1979–201759,60. (F_{{mathrm{ABC}},i,y,l}) and was determined in each simulation timestep using an iterative approach66 whereby we: (i) first determined average (B_{0,i}) values in years 2095–2099 by projecting the model forward without harvest (i.e., (F_{i,y} = 0)) for each species under the persistence scenario. We then (ii) iteratively solved for the harvest rate that results in an average spawning biomass ((B_{i,y})) during 2095–2099, that is, 40% of (B_{0,i}) (i.e., (F_{{mathrm{target}},i})) for pollock and Pacific cod simultaneously, with arrowtooth flounder (F_{i,y}) set to the historical average (as historical F for arrowtooth flounder (F_{40% })); once (F_{{mathrm{target}},i}) for pollock and Pacific cod were found, we then iteratively solved for (F_{{mathrm{target}},i}) for arrowtooth flounder (Supplementary Fig. 7 left panel)59,60. Last, (iii) to derive a climate-informed ({mathrm{ABC}}_{i,y,l}) in each simulation year, the North Pacific Marine Fisheries Council (hereafter, “Council”) Tier 3 sloping harvest control rule with an ecosystem cutoff at (B_{20% }) was applied to adjust (F_{{mathrm{ABC}},i,y,l}) lower than (F_{{mathrm{target}},i}) if the simulation specific (climate-informed) (B_{i,y,l}) was lower than 40% of the climate-naive (B_{0,i}) at the start of a given year or set to 0 if (B_{i,y,l} ,<, 20% B_{0,i}); (F_{{mathrm{ABC}},i,y} = F_{{mathrm{target}},i,y}) for the remainder of the simulations where (B_{i,y,l} ge 40% B_{0,i}))21.

This approach follows the status quo Council reviewed multispecies assessment methodology60 and represents a precautionary approach that minimizes inflation of ABC due to predator release59 and also minimizes potential non-intuitive compound effects of climate change and fishing under declining conditions (i.e., whereby a climate-informed (B_0) declines with climate change and produces a lower target ((B_{{mathrm{40% }}})) biomass). While beyond the scope of this study, future simulations might further explore the performance of alternative (B_0) approaches (e.g., periodically updated climate-informed (B_0), annually varying (B_0) with climate-penalized (B_{{mathrm{40% }}})).

### Simulate TAC

ATTACH estimates the TAC that would be set by the North Pacific Fishery Management Council, and the subsequent catch that would be harvested by the commercial fishery, based on historical data and assuming current policies and priorities remain relatively unchanged in projections. The impacts of existing policies such as Amendment 80 (i.e., A80, which created multispecies cooperatives), the American Fisheries Act (i.e., AFA, which created pollock cooperatives), and large spatial closures are included and evaluated in the retrospective analysis. Future scenarios will explore relaxation or alteration of these underlying assumptions19. ATTACH first estimates the TAC from the specified ABC through an ensemble of three log-linear regressions and seemingly unrelated regressions (SUR) where normally distributed error terms (varepsilon _{i,y}^{{mathrm{TAC}}}) are independent across time, but may have cross-equation contemporaneous correlations67. Specifically, the models are statistically fit to historical ABC and TAC data from 1992 to 2017 such that:

$$ln left( {{mathrm{TAC}}_{i,y}} right) = , alpha _i + beta _i {mathrm{ln}}left( {{mathrm{ABC}}_{i,y}} right) + mathop {sum }limits_{j = 1}^{N_j} gamma _{ij}{mathrm{ABC}}_{j,y} + mathop {sum }limits_{k = 1}^{N_k} theta _{ik}I_k + varepsilon _{i,y}^{{mathrm{TAC}}}, \ qquad{mathrm{where}},varepsilon _{i,y}^{{mathrm{TAC}}}sim Nleft( {0,sigma _i^{{mathrm{TAC}}}} right)$$

(4)

where the harvest limit for species (i) in a given historical year (y) (({mathrm{TAC}}_{i,y})) is a function of the assessment model-based ABC (({mathrm{ABC}}_{i,y}), in metric tons) for the species (i), (alpha _i) is the log-linear intercept and (beta _i), (gamma _{ij}), and (theta _{ik}) are coefficients for ABC and policy covariates (I_k) (e.g., closures, A80, AFA). (varepsilon _{i,y}^{{mathrm{TAC}}}) is the residual error and is log-normally distributed. The residuals of equations estimated as a SUR system are assumed to be correlated, and this is used to more efficiently estimate the regression coefficients. This parameterization assumes exogenous shocks affect all included species to varying degrees. The mean relationship between TAC and ABC (Eq. 4) was used to simulate TACi,y,l in each projection year using inputs of ABCi,y,l from the CEATTLE model (Eq. 3). Historically, in all but two years the sum of TAC across species has equaled 2 MT exactly, thus we imposed an addition constraint; if the cumulative predicted TACs from the ensemble exceeded 2 MT, the ({mathrm{TAC}}_{i,y,l}) of each species was proportionally reduced to satisfy the constraint of a 2 MT limit on the sum of all ({mathrm{TAC}}_{i,y,l}). Of note, this step simulates the current management regime, and is not an optimization.

### Simulate annual harvest (catch)

Similarly, ATTACH uses an ensemble of three models to estimate catch biomass ((C_{i,y}); tons) for a given target species (i) as a function of TACi,y and sometimes the TACj,y of 1 to 2 additional species j, as well as relevant policy/events in a given year:

$$ln left( {C_{i,y}} right) = , alpha _i + beta _iln left( {{mathrm{TAC}}_{i,y}} right) + mathop {sum }limits_{j = 1}^{N_j} gamma _{ij}{mathrm{TAC}}_{j,y} + mathop {sum }limits_{k = 1}^{N_k} theta _{ik}I_k + varepsilon _{i,y}^C, \ qquad {mathrm{where}},varepsilon _{i,y}^Csim Nleft( {0,sigma _i^C} right)$$

(5)

The three models in the ensemble differ in their error structure for catch: model 1 assumes each of the log-linear equations are independent, model 2 has two groups of linked SUR (representing species that are caught concurrently), and model 3 has three groups of linked SUR (representing a third group, on top of the two of model 2, of species whose catches are linked). As in (Eq. 4), (alpha _i) is the log-linear intercept and (beta _i), (gamma _{ij}), and (theta _{ik}) are coefficients for TAC and policy covariates (I_k) and (varepsilon _{i,y}^C) is the residual error and is log-normally distributed and assumed to be correlated across linked SUR.

The authors of the ATTACH model evaluated the performance of the TAC and catch prediction models using leave-one-out cross-validation (LOO-CV)34. For each year, they estimated the coefficients of each model using data from all but that year, and then used those estimators to predict the TAC or catch of the omitted year. They then calculated the difference between the predicted and actual catch for each year (1992–2017) to evaluate the models using a variety of metrics: simple sum of differences, sum of percent differences, sum of squared differences (weighted by value, ABC, TAC, and catch, respectively), the sum of squared percent differences. They also evaluated those metrics for the final models (trained on the entire data set). They found that the ATTACH model performed best for species targeted by directed fisheries, as is the case for the species in this study, while predictive skill of the model was weaker for less economically valuable species. In all species, however, the ATTACH model performed better than assuming catch is equal to ABC (Supplementary Fig. 8).

For the MSE application in this study ({mathrm{ABC}}_{i,y,l}) and ({mathrm{ABC}}_{j,y,l}) of pollock, Pacific cod, and arrowtooth flounder output from the CEATTLE model were used as inputs into the ATTACH function for each simulation year, while the ({mathrm{ABC}}_{j,y}) of the remaining non-CEATTLE species were set to their respective historical averages (1992–2017). Pollock, Pacific cod, and arrowtooth flounder ABCs were used to predict ({mathrm{TAC}}_{i,y,l}) for each species, which in turn was used to predict catch (C_{i,y,l}) using estimated coefficients from the regressions described above (Eq. 45). The (C_{i,y,l}) of pollock, Pacific cod, and arrowtooth were then removed from the population of each species in the simulation year (y) (by calculating the effective harvest rate (F_{i,y,l}) given the input catch (C_{i,y,l})) and the simulation was rolled forward 1 timestep.

Under this approach, TAC and catch estimates from ATTACH change solely in response to changes in pollock, Pacific cod, and arrowtooth flounder input ABCs. This assumption is significant but difficult to assess without considering alternative models that resolve the population dynamics of the remaining species managed under the 2 MT cap along with their biological interactions and potential future changes in relative harvest values. That said, the three species in CEATTLE are known to strongly interact, account for a large proportion of total EBS fish biomass, and strongly drive TAC allocation. Future sensitivity analyses, possibly using more speciose food web models32,68 would be useful to quantify this sensitivity.

### Evaluating management performance and risk

We evaluated the performance of these various management strategies under moderate and high-baseline climate futures. We explored temporal patterns in risk defined as the probability of a 10% decline in biomass or catch relative to the persistence baseline scenario for each harvest scenario. We also evaluated the probability of severe decline and collapse, defined as a greater than 50 and 80% decline (respectively) in biomass or catch in climate change scenarios (relative to the persistence scenario).

### Threshold and tipping point analysis

We use ecosystem threshold analyses69,70 to identify thresholds and tipping points for each species in the non-linear relationship between catch (response) and temperature (pressure). Using the multispecies model estimates of recruitment for each species ((i)) for each future year (y) of the simulation and each future scenario ((l)) we drew random samples from the log-normally distributed parameter estimate for climate-enhanced recruitment relationships and projected the model forward under the 2 MT cap management scenario. This formed 100 replicates of each future scenario for each species under no fishing and no cap simulations, and 30 replicates under the 2 MT cap. We calculated the change in catch (({Delta}C_{i,y,l})) relative to the persistence scenario ((l = 1)) as:

$${Delta}C_{i,y,l} = left( {C_{i,y,l} – C_{i,y,l = 1}} right)/C_{i,y,l = 1}$$

(6)

We then pooled the full set of ({Delta}C_{i,y,l}) and corresponding bottom temperatures (T) to estimate the threshold for the ({Delta}C_{i,y,l}sim fleft( T right)) relationship. Following Samhouri et al.69, we fit general additive models using thin plate regression spline smoothing terms, using the mgcv71 package in R, with the basis dimension set to 4 to avoid overfitting. We then calculated the first and second derivative of the smoothing function s(x), as well as the 95% CI (as the 2.75 and 0.975 quantiles) of the smoothing and its second derivative by bootstrapping (n = 1000) the residuals. Last, the inflection or tipping point was defined as the temperature whereby the second derivative (s(x)”) changed sign and the 95% CI of the second derivative (smoothed with a local polynomial regression smoother using the loess() function in R with the span set to 10%) was most different from zero69.

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.


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1

### AIPMT 2001

The beans are cooked earlier in pressure cooker because
A
boiling point increases with increasing pressure
B
boiling point decreases with increasing pressure
C
extra pressure of pressure cooker softens the beans
D
internal energy is not lost while coocking in pressure cooker.

## Explanation

When water pressure increases in the pressure cooker, water boils at lower temperature and the beans in pressure cooker are cooked earlier.
2

### AIPMT 2000

Which of the following expressions correctly represents the relationship between the average molar kinetic energy, KE, of CO and N2 molecules at the same temperature ?
A
KECO = KEN2
B
KECO > KEN2
C
KECO < KEN2
D
cannot be predicted unless volumes of the gases are given

## Explanation

Average molar kinetic energy = ${3 \over 2}nRT$

As temperature is same hence average kinetic energy of CO and N2 will be same.

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# Why do we get only one frequency as output in oscillators?

I am just into oscillators where I learned $$\AB=1\$$ for sustaining oscillations in positive feedback. Since $$\A\$$ and $$\B\$$ are both frequency-dependent, $$\AB=1\$$ is true only for a particular frequency.

1. What happens to those frequencies for which $$\AB>1\$$ holds??

2. Will these frequencies keep on getting amplified until the limiter circuit limits them?

3. Then why don't we get those frequencies in our output??

• I don't think "AB" is standard terminology. I'm guessing it means loop gain? – Hearth Dec 1 '18 at 13:46
• Yes ! sorry about that. – Souhardya Mondal Dec 1 '18 at 13:54
• @Hearth A is gain and B is the feedback fraction. When their product is one the denominator of the transfer function is zero. – user110971 Dec 1 '18 at 13:55
• I think, we should be more specific: When the loop gain approaches unity (real, with zero phase shift) the CLOSED-LOOP function has a denominator that approaches zero. – LvW Dec 2 '18 at 10:13

Why do we get only one frequency as output in oscillators?

Oscillators work at one frequency by ensuring two things: -

• The signal fed back to sustain oscillations is exactly in phase with the signal it is trying to sustain. Think about lightly tapping a swinging pendulum at exactly the right place and, in the right direction.
• The loop-gain is slightly more than unity. This ensures that a sinewave is produced without too much distortion and it is "sustained". If the loop-gain were less than 1, then it can't "sustain" an oscillation.

So, if we design a phase-shifting network that has a unique phase-shift for each frequency it handles, we will get an oscillator but, only if the signal fed back is sufficient in amplitude to sustain oscillation.

However, some phase shifting networks can produce a phase shift that is a multiple of the basic oscillation frequency. In other words if 1 MHz produces a phase shift of 360 degrees, maybe some higher-frequency might produce 720 degrees (2 x 360). This could potentially give rise to a sustained oscillation at two frequencies (usually deemed undesireable).

So, we design the phase-shifting network to ensure that the higher-frequency "in-phase" candidate is much lower in amplitude than the "basic" candidate and, given that we only allow the gain to be unity or slightly higher (to accommodate losses in the phase shift network) for the frequency we want, the higher-frequency candidate will not cause oscillation.

The above is also referred to as the Barkhausen criteria.

• So what happens to those frequencies which have AB>1?? – Souhardya Mondal Dec 1 '18 at 16:33
• They won't sustain oscillation because they do not feedback a signal that is precisely in phase. Think of a pendulum; if you hit it (slightly) at exactly the point where it is starting to swing back you would not effect it's oscillating frequency AND you would sustain oscillations. – Andy aka Dec 1 '18 at 16:36
• If the gain is greater than just more than unity at the oscillation frequency then the oscillator’s amplitude rises until it can’t rise any more because of supply voltage limitations or slew rate limitations. In other words the amplitude clips. – Andy aka Dec 1 '18 at 18:29

So what happens to those frequencies which have AB>1??

Saturation.

Let's say there are several frequencies with loop gain $$\AB \ge 1\$$ and $$\n2\pi\$$ phase shift, but let's call the one with the highest loop gain $$\f_x\$$. For $$\f_x\$$, $$\AB > 1\$$ and you might expect it to produce an oscillation with amplitude increasing in time. But no real circuit can have its output increase in amplitude indefinitely. There is usually some saturation behavior that limits the output amplitude.

And when this happens, it tends to reduce the gain for all frequencies, not just the one that had the super-unity loop gain. So accounting for saturation, this frequency $$\f_x\$$ will end up with $$\AB=1\$$ and all the other frequencies that linear analysis told you had $$\AB \ge 1\$$ but less than at $$\f_x\$$, now have $$\AB < 1\$$, so they no longer oscillate indefinitely.

• The Photon...may I ask you: Have you ever seen a circuitry with a real loop gain >1 (zero phase !) at "several frequencies"? More than that, I have some severe doubts if the 1st sentence of the second paragraph in your answer is correct. The loop gain is frequency-dependent - and when it is reduced (due to some non-linearity) at one frequency, it will not automatically reduced for other frequencies with other amplitudes (because the non-linearity is by its nature amplitude-dependent). – LvW Dec 1 '18 at 21:25
• I suppose you are referring to integrator-based oscillators, correct? But the magnitude condition is fulfilled at one single frequency only!! – LvW Dec 1 '18 at 21:33
• I wonder if it’s viable to design an oscillator that naturally oscillates at two different sinewave frequencies? Reading your answer it shouldn’t be possible by mistakingly trying to design a regular oscillator but it has got me thinking. I think I remember messing around with a transmission line oscillator that produced two disparate sinewaves but I didn’t explore things. – Andy aka Dec 2 '18 at 12:36

A short answer from my side:

You must not think in magnitude terms only. Don`t forget the phase. The product AB must be a REAL one . A frequency-selective circuitry has a magnitude as well as a phase which is a function of frequency. And - for a correct design - there will be only one single frequency which can fulfill both conditions at the same time (Barkhausens oscillation criterion with loop gain AB=1):

• |A*B|=1 (for practical reasons somewhat larger than "1", for example "1.2") and

• phaseshift exp(j*phi)=1 (phi=0).

For this purpose, most known oscillators use lowpass, highpass or bandpass filters as feedback elements. But there are also other (more advanced) topologies.

• @ LvW Can you provide one or two examples (links) to more advanced topologies? please. – analogsystemsrf Dec 2 '18 at 2:47
• Examples: (a) Two integrators in series (inv./non-inv), (b) notch-filter, (c) double-T-topology, (d) Allpass, (e) Active negative-resistance structure, (f) GIC resonator (FDNR resonance). – LvW Dec 2 '18 at 8:51
• Assuming you mean a classic crystal oscillators (XOs) with a square wave output (either series or parallel mode).

When saturation occurs, the loop gain (GH or AB) drops to zero, except during the linear transition of the output. The crystal acts as a bandpass filter to produce a sine wave at the input which may also contain harmonics, but the slew rate of the square wave output is generally much faster than the sine wave input, so the harmonic energy has insufficient outline linear time to amplify when it is not saturated and the gain is zero, thus suppressed.

• However in linear oscillators the harmonic content may contribute to phase noise, so those with the lowest phase noise have the highest Q at the fundamental, such as SC-cut crystals e.g. 10 MHz oven-controlled crystal oscillators (OCXOs) vs. standard AT cuts commonly used everywhere. That's all I'll say about this for now.

However, for smaller crystal structures >= 33 MHz resonance the gain of the harmonics tends to be higher than the fundamental. Thus you will find these classified as "overtone crystals".

For CMOS feedback oscillators, often a series R (3 kΩ ~ 10 kΩ) from the output is used to limited uW power dissipation in microslice crystals AND in high frequency >> 10 MHz also create additional attenuation of harmonics from the RC effects with the first load capacitor. The most common is third harmonic or "overtone", but higher overtones are used >> 150 MHz.

But when selective harmonics are desired for oscillation (3, 5, 7, etc.) then either how the crystal is processed or additional passive LC tuning helps to boost the harmonic of choice.

The most common warning for XO designs "Never use a buffered inverter" (three linear gain stages vs. one) to avoid amplification of spurious harmonics. When they saturate the inverter and the gain drops to zero, they suppress the fundamental frequency except for a short transition interval. They can behave like an injection locked loop (ILL) where it may randomly oscillate at the fundamental or harmonic depending relative gains and startup conditions. But with a buffered inverter there is more chance during the output transition time to cause spurious harmonic glitches on the transitions and lock onto harmonics.

However, those who successfully used a buffered inverter (myself included) for an XO can now understand that the type of crystal and relative lower gain of the harmonic protected the XO from locking onto the desired fundamental frequency. In some cases, this can be an advantage, but that's a different question.

Although all answers are correct, I believe these are all missing the spirit of your question.

The term "oscillator" generally applies to a circuit specifically designed to produce an AC waveform at a specific frequency. This entails some design choices intended to minimize unwanted effects. This is particularly true for linear oscillators (which is the loop-gain case stated in your question).

You specifically design the gain to be slightly larger than 1 at a specific frequency and you design/rely on non-linearities in the system to keep the oscillation stable. If you allow the gain to be much larger than 1 then you stop having a linear oscillator.

However, this useful engineering simplification comes from having the loop gain only be slightly larger than one which allows you to treat it as a linear oscillator, when in reality it is not. What you actually have is the simplified border case of a non-linear dynamical system with a stable periodic orbit that approaches a sinusoid.

If you further develop that dynamical system (for example by making AB>>1) you can reach another extreme, a very non-linear but stable relaxation oscillator or in intermediate cases you will find a period doubling sequence that creates a chaotic oscillator such as Chua's circuit or a Van Der Pol oscillator.

This image is from an implementation of Chua's circuit you can see that it behaves somewhat as a combination relaxation oscillator/linear oscillator. But the "relaxation component" is non-periodic and long-term unpredictable.

There are uses for all of those alternatives, but linear oscillator theory specifically stays away from those conditions.

• Relaxation Effects are due to nonlinear negative resistance also such as tiny hysteresis which is positive AC feedback together with DC negative feedback. This effect is common in cascaded Buck PWM then Boost-PFM converter control system noise, an example causing chaos theory noise. – Tony Stewart Sunnyskyguy EE75 Dec 1 '18 at 18:50
• @TonyEErocketscientist those are all "conceptual buckets" that we use to make the effects easier to understand, analyze, and design around. But in reality these are all particular cases of more generalized non-linear dynamical systems. Note that you can tune Chua's circuit to present all of those behaviors just by tweaking the non-linear element characteristics. – Edgar Brown Dec 1 '18 at 18:55
• Sorry, which examples are "those". Not familiar with Chua's publications except for the name, as I discovered how to make stable-linear low THD sine oscillators with nonlinear ccts long before Chua in the 70's. My 90's cascade Buck>Boost inductor sounded like bubbling water in the lab from the piezo acoustics, until I fixed it. – Tony Stewart Sunnyskyguy EE75 Dec 1 '18 at 19:11
• @TonyEErocketscientist all behaviors from linear oscillator, through chaotic oscillator, to relaxation oscillator. Although Chua's circuit is the simplest possible physical dynamical system to produce chaos, in essence it is nothing more than a third-order transfer function attached to a non-linear negative resistor. – Edgar Brown Dec 1 '18 at 19:17
• Yes of course. I see. Because all behaviours with harmonics are due to nonlinearities, even the structural properties of Xtals, buildings. So the "conceptual buckets" refers to specific linear approximations. I have learnt how to use linear piecewise approximations for non-linear theory for good uses like when the bulk Rs ( or ESR as I call it) exceeds the nonlinear incremental resistance with rising current in LED's or the soft limiting of sine wave Osc. to attenuate harmonics and raise Q as gain converges on unity. – Tony Stewart Sunnyskyguy EE75 Dec 1 '18 at 19:41

The Barkhausen stability criterion says: $$\|A\ \beta| = 1\$$ and $$\\angle A\ \beta = 0\$$.

Where $$\A\$$ = amplifier gain,

and $$\\beta\$$ = feedback attenuation.

So if $$\|A\ \beta| = 1\$$, then the oscillator will be stable. The feedback loop feeds a portion of the output $$\v_o\$$ back to the input, $$\v_f\$$. The amplifier amplifies the input $$\v_f\$$ to make a larger output, $$\v_o\$$.

If $$\|A\ \beta| > 1\$$, then the oscillator will drive itself into saturation and clip the output waveform. The amplifier is an operational amplifier with $$\\pm\$$ power rails. The amplifier cannot drive the output past the power rails.

The gain and attenuation are not stable and the amplifier output increases to the power rails of the amplifier. If it is a sine wave oscillator, the output increases until the amplifier saturates, and it is no longer a sine wave. Tops get clipped.

If $$\|A\ \beta| < 1\$$ the oscillations will fade out. This is called damping.

Given that: an oscillator circuit is designed to oscillate at a fixed frequency if $$\|A\ \beta| = 1\$$ and $$\\angle A\ \beta = 0\$$ (phase angle is 0$$\^\circ\$$).

So the crux of your question is: Why don't oscillators oscillate at other frequencies? This is governed by the components used (resistors, capacitors, inductors, and amplifiers).

• Yes - I agree to the (short) explanation. Just one further comment: Because it is impossible to achieve per design a loop gain that is exactly "one", we always realize a loop gain somewhat larger than "1" (at one single frequency) and make use of an automatic amplitude regulation mechanism (non-linearity, diodes , NTC, FET as resistor,...), which can bring the loop gain back to "1" before clipping occurs. – LvW Dec 2 '18 at 10:18
• @LvW I'd go for simple. When I answer a question, I look at 4 criteria. Can I answer the question? What is the level of OP? Do answers meet that level? What misconceptions must be corrected? We have an OP who doesn’t know his Greek $\beta$ from B. So I create a simple answer with graphics to not overload the cognitive load of the OP. Then I compete with better quality answers to see if I can connect with the OP. I'm happy if the OP understands $\beta$. – StainlessSteelRat Dec 2 '18 at 22:49
• Again - I agree to your approach for answering questions like this. May I add another comment (correction)? The criterion named after Heinrich Barkhausen is not a "stability criterion" (such a criterion was formulated by Strecker and in parallel by Nyquist). Barkhausens criterion is a so called "oscillation condition" - to be more specific: It is just a "necessary" condition for a circuit to oscillate - not a sufficient one (Wikipedia is not always correct). – LvW Dec 3 '18 at 8:07
• @LvW Thanks. That was an edit by another. I'll tweak it. I'm not a fan of wiki links. – StainlessSteelRat Dec 3 '18 at 8:14


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Home
MAY 2021

# AI-Assisted Tic-Tac-Toe

The interactive app lets you play the game with AI assistance, illustrating which moves can guarantee a win: Red fields allow $X$ to win, blue fields allow $O$ to win. Note that the player who is given the theoretical guarantee to win still has to realize this guarantee by following one of the sequences of best moves, i.e. by adhering to the respective color. Then, you may notice, the other player will not have any other choice (all possible moves will be marked in the opponent's color).

The game tic-tac-toe is simple, but it is also sufficiently complex to illustrate a few core concepts of artificial intelligence. In tic-tac-toe, two players, $X$ and $O$, mark the board with their respective symbol in turns. The first player to form a row, column, or diagonal of three identical symbols wins. Three-in-a-row board games can be traced back thousands of years.

The game is simple, but its game tree, i.e. the set of possible paths, is sufficiently complex: There are $9!$ possible sequences of moves. A play is one specific sequence of moves. However, not all sequences are valid: Once the game has ended, no more moves can be made. This reduces the number of possible plays to 255,168. The number of plays for different scenarios is discussed here.

Therefore, the game tree is too complex to be read manually. In practice, though, due to symmetry of moves, the number of distinct strategies is much lower. The minimax algorithm offers an approach to analyze the game tree recursively in order to identify the outcome of a specific move, given a certain board state and given the assumption that the other player plays perfectly. Minimax assigns each leaf of the game tree a value, e.g. $+1$ when player $X$ wins, $-1$ when $O$ wins, and $0$ when the game is a draw.

In order to determine the best possible move for a player at a certain state of the game, results are recursively propagated upwards, given the assumption that both players will always choose one of their best moves. The following minimax implementation is part of the script that computes the complete game tree with all sequences of plays, including the expected outcomes for each board state. Here, bx and bo are sets of numbers from zero to eight, denoting the positions of symbols on the board for both players $X$ and $O$.

def minimax(bx, bo):
if is_over(bx, bo):
return board_value(bx, bo)
if turn(bx, bo) == "X":
v = -math.inf
for cx, co in next_boards(bx, bo):
v = max(v, minimax(cx, co))
return v
if turn(bx, bo) == "O":
v = math.inf
for cx, co in next_boards(bx, bo):
v = min(v, minimax(cx, co))
return v

The minimax algorithm is an example of how a simple idea can solve a comparatively complex problem through recursion. However, minimax is not always feasible. The number of plays in a chess game is so large that traversing the tree is practically impossible. The optimization approaches that exist, such as alpha-beta pruning, reduce the number of nodes in the game tree that have to be evaluated, but the game tree still has to be traversed at least partially. Play the game here.


text:
# Solution 2 Computational Intelligence Lab 2011

## Problem 1

1. 6 x 6, correlation between movies
2. 5 x 5, correlation between users
3. U = 6 x 6, D = 6 x 5, V = 5 x 5
4. Identify Concepts and find hidden structures. Due to the insertion of the average rating we introduce some noise. After the decomposition we hope to find the "true" values by hoping that the data is explained by less dimmensions (${\displaystyle k)
5. The 3 or 4 highest ones
6. Affinity of the movies to the concepts
7. Affinity of the users to the concepts
8. Plot. Interpretation: "Natural" clustering around the mean of the values of the first principal component (${\displaystyle {\bar {u_{1}}}=-0.4}$)
9. See above.
10. Strengths/expressiveness of each concept
11. ${\displaystyle A_{3}=U_{3}*D_{3}*V_{3}^{T}=\sum _{i=1}^{3}d_{i}*{\vec {u}}_{i}*{\vec {v}}_{i}^{T}}$
12. ${\displaystyle \|A-A_{3}\|_{2}=\sigma _{4}=2.75}$, ${\displaystyle \|A-A_{3}\|_{F}={\sqrt {\sigma _{4}^{2}+\sigma _{5}^{2}}}={\sqrt {(2.75)^{2}+(0.67)^{2}}}}$
13. ${\displaystyle Bob=[1,*,*,6,*,10]^{T}}$
1. (a) Transform into 2 dimensional representation without doing an SVD again: ${\displaystyle A_{2}=U_{N,3}*D_{2,2}*V_{2,N}^{T}\leftrightarrow V_{2,N}^{T}=D_{2,2}^{-1}*U_{N,3}^{T}*A_{2}}$
2. (a) Fill in the missing values with the average: ${\displaystyle Bob=[1,5.5,5.5,6,5.5,10]^{T}}$
3. (a) Representation in the "new coordinates": ${\displaystyle Bob_{2,1}^{T}=D_{2,2}^{-1}*U_{N,3}^{T}*Bob_{2,1}=D_{2,2}^{-1}*U_{N,3}^{T}*[1,5.5,5.5,6,5.5,10]^{T}=[-0.46,0.36]^{T}}$
4. (b) Find the nearesst neighbour (scaled with ${\displaystyle D}$)
14. As long we do not take the new user-data into account, it would not affect our prediction system. We just use our existing decomposition.


text:
BatchToSpace

Versioned name: BatchToSpace-2

Category: Data movement

Short description: The BatchToSpace operation reshapes the "batch" dimension 0 into N - 1 dimensions of shape block_shape + [batch] and interleaves these blocks back into the grid defined by the spatial dimensions [1, ..., N - 1] to obtain a result with the same rank as data input. The spatial dimensions of this intermediate result are then optionally cropped according to crops_begin and crops_end to produce the output. This is the reverse of the SpaceToBatch operation.

Detailed description:

The BatchToSpace operation is similar to the TensorFlow* operation BatchToSpaceND

The operation is equivalent to the following transformation of the input tensors data with shape [batch, D_1, D_2 ... D_{N-1}] and block_shape, crops_begin, crops_end of shape [N] to Y output tensor.

note: B_0 is expected to be 1.
x' = reshape(data, [B_1, ..., B_{N - 1}, batch / (B_1 * ... B_{N - 1}), D_1, D_2, ..., D_{N - 1}]), where B_i = block_shape[i]

x'' = transpose(x', [N, N + 1, 0, N + 2, 1, ..., N + N - 1, N - 1])

x''' = reshape(x'', [batch / (B_1 * ... * B_{N - 1}), D_1 * B_1, D_2 * B_2, ... , D_{N - 1} * B_{N - 1}])

Crop the start and end of dimensions according to crops_begin, crops_end to produce the output of shape: note: crops_begin[0], crops_end[0] are expected to be 0. y = [batch / (B_1 * ... * B_{N - 1}), crop(D_1 * B_1, crops_begin[1], crops_end[1]), crop(D_2 * B_2, crops_begin[2], crops_end[2]), ... , crop(D_{N - 1} * B_{N - 1}, crops_begin[N - 1], crops_end[N - 1])]

Attributes

No attributes available.

Inputs

• 1: data - input N-D tensor [batch, D_1, D_2 ... D_{N-1}] of T1 type with rank >= 2. Required.
• 2: block_shape - input 1-D tensor of T2 type with shape [N] that is equal to the size of data input shape. All values must be >= 1.block_shape[0] is expected to be 1. Required.
• 3: crops_begin - input 1-D tensor of T2 type with shape [N] that is equal to the size of data input shape. All values must be non-negative. crops_begin specifies the amount to crop from the beginning along each axis of data input . It is required that crop_start[i] + crop_end[i] <= block_shape[i] * input_shape[i]. crops_begin[0] is expected to be 0. Required.
• 4: crops_end - input 1-D tensor of T2 type with shape [N] that is equal to the size of data input shape. All values must be non-negative. crops_end specifies the amount to crop from the ending along each axis of data input. It is required that crop_start[i] + crop_end[i] <= block_shape[i] * input_shape[i]. crops_end[0] is expected to be 0. Required.

Outputs

• 1: N-D tensor with shape [batch / (block_shape[0] * block_shape[1] * ... * block_shape[N - 1]), D_1 * block_shape[1] - crops_begin[1] - crops_end[1], D_2 * block_shape[2] - crops_begin[2] - crops_end[2], ..., D_{N - 1} * block_shape[N - 1] - crops_begin[N - 1] - crops_end[N - 1] of the same type as data input.

Types

• T1: any supported type.
• T2: any supported integer type.

Example

<layer type="BatchToSpace" ...>
<input>
<port id="0"> <!-- data -->
<dim>48</dim> <!-- batch -->
<dim>3</dim> <!-- spatial dimension 1 -->
<dim>3</dim> <!-- spatial dimension 2 -->
<dim>1</dim> <!-- spatial dimension 3 -->
<dim>3</dim> <!-- spatial dimension 4 -->
</port>
<port id="1"> <!-- block_shape value: [1, 2, 4, 3, 1] -->
<dim>5</dim>
</port>
<port id="2"> <!-- crops_begin value: [0, 0, 1, 0, 0] -->
<dim>5</dim>
</port>
<port id="3"> <!-- crops_end value: [0, 0, 1, 0, 0] -->
<dim>5</dim>
</port>
</input>
<output>
<port id="3">
<dim>2</dim> <!-- data.shape[0] / (block_shape.shape[0] * block_shape.shape[1] * ... * block_shape.shape[4]) -->
<dim>6</dim> <!-- data.shape[1] * block_shape.shape[1] - crops_begin[1] - crops_end[1]-->
<dim>10</dim> <!-- data.shape[2] * block_shape.shape[2] - crops_begin[2] - crops_end[2] -->
<dim>3</dim> <!-- data.shape[3] * block_shape.shape[3] - crops_begin[3] - crops_end[3] -->
<dim>3</dim> <!-- data.shape[4] * block_shape.shape[4] - crops_begin[4] - crops_end[4] -->
</port>
</output>
</layer>


text:
# The Partial Elekes-Sharir Framework

A while ago, I had a post about lower bounding the minimum number of distinct distances between points on two lines in ${\mathbb R}^2$. That post showed how such an upper bound can be obtained by applying the Elekes-Sharir framework (a pdf file of my guide to the Elekes-Sharir framework can be found here). Since then, we noticed that a much simpler technique can be used to obtain the same bound. I like to call this simpler technique the partial Elekes-Sharir framework since the analysis starts in the same way, but the more technical parts of the original framework are replaced with a simpler analysis. More specifically, the partial framework leads to a planar incidence problem, instead of to one in ${\mathbb R}^3$. Since Micha Sharir, József Solymosi, and myself introduced this idea in the spring of 2013 (later on, we discovered that a similar idea was also used by Elekes and Szabó), it was applied and extended in a few additional papers (see here, here, and here). Moreover, I know of at least a couple of additional manuscripts that apply the same ideas for problems not directly involving distinct distances.

The purpose of this post is to present the partial Elekes-Sharir framework, by applying it to obtain a bound for the problem concerning distinct distances on two lines. First, we recall the problem.

Consider a set ${\cal P}_1$ of $n$ points on a line $\ell_1$, and a set ${\cal P}_2$ of $n$ points on a line $\ell_2$ (such that $\ell_1$ and $\ell_2$ are distinct) and let $D({\cal P}_1,{\cal P}_2)$ denote the number of distinct distances between pairs of points from ${\cal P}_1\times{\cal P}_2$. Consider first the “balanced” case, where $|{\cal P}_1|=|{\cal P}_2|=n$. When the two lines are parallel or orthogonal, the points can be arranged so that $D({\cal P}_1,{\cal P}_2) = \Theta(n)$. For example, see the following figure.

Purdy conjectured that if the lines are neither parallel nor orthogonal then $D({\cal P}_1,{\cal P}_2) = \omega(n)$. Elekes and Rónyai proved that the number of distinct distances in such a scenario is indeed superlinear. They did not give an explicit bound, but a brief analysis of their proof shows that $D({\cal P}_1,{\cal P}_2) = \Omega(n^{1+\delta})$ for some $\delta>0$. Elekes derived the improved and concrete bound $D({\cal P}_1,{\cal P}_2) = \Omega(n^{5/4})$ (when the lines are neither parallel nor orthogonal) and gave a construction, reminiscent of the one by Erdős, with $D({\cal P}_1,{\cal P}_2) = O\left(n^2/\sqrt{\log n}\right)$, in which the angle between the two lines is $\pi/3$. The unbalanced case, where $|{\cal P}_1|\ne |{\cal P}_2|$, has recently been studied by Schwartz, Solymosi, and de Zeeuw, who have shown, among several other related results, that the number of distinct distances remains superlinear when $|{\cal P}_1| = n^{1/2 + \varepsilon}$ and $|{\cal P}_2| = n$, for any $\varepsilon>0$.

Our result, taken from this paper is:

Theorem 1. Let ${\cal P}_1$ and ${\cal P}_2$ be two sets of points in ${\mathbb R}^2$ of cardinalities $n$ and $m$, respectively, such that the points of ${\cal P}_1$ all lie on a line $\ell_1$, the points of ${\cal P}_2$ all lie on a line $\ell_2,$ and the two lines are neither parallel nor orthogonal. Then

$\displaystyle D({\cal P}_1,{\cal P}_2)=\Omega\left( \min\left\{n^{2/3}m^{2/3},n^2,m^2 \right\}\right)$.

Notice that Theorem 1 immediately improves Elekes’s lower bound from $\Omega(n^{5/4})$ to $\Omega(n^{4/3})$. Moreover, the theorem is a generalization of the aformentioned bound by Schwartz, Solymosi, and de Zeeuw.

Consider a $\sqrt{n}\times\sqrt{n}$ integer grid ${\cal G}$ in ${\mathbb R}^2$. It is easy to prove that any constant-degree algebraic curve passes through $O(\sqrt{n})$ points of ${\cal G}$. This bound is tight, since a line can pass through $\Theta(\sqrt{n})$ points of ${\cal G}$. Somewhat surprisingly, every other constant-degree algebraic curve passes through an asymptotically smaller number of grid points.
Lemma 1. Let ${\cal G}$ be a $\sqrt{n}\times\sqrt{n}$ integer grid in ${\mathbb R}^2$.
(a) Let $C$ be a constant-degree algebraic curve which is not a line. Then $C$ contains $o(\sqrt{n})$ points of ${\cal G}$ (recall that little-$o()$ notation implies asymptotically smaller).
(b) Let $C$ be a strictly convex curve (not necessarily algebraic). Then $C$ contains $O(n^{1/3})$ points of ${\cal G}$.


text:
# What are these new LibraryLink functions and datatypes for?

Looking at the LibraryLink examples, there are several new functions and objects, as well as example programs, that appeared in version 9. Are these documented? If yes, where?

If would be nice to have a summary of the new features available, and a summary of how and when to use them, so people can more easily decide whether there is something that's useful for their project.

• MInputStream and MOutputStream types, WolframStreamLibrary.h header
• DataStore type
• WolframIOLibrary_Functions (comparable to WolframLibraryData, contains function pointers)
• all the async-*.c examples which use these features

It should be possible to figure much of this out using the (uncommented) example programs, but I'm hoping that someone has already done this and can contribute and answer.

I believe that some of these (DataStore and the async-*.c examples) are related to the Asynchronous Tasks functionality, such as URLFetchAsynchronous.

-
I think these items are largely to do with the "data" (Hadoop, etc.) functionality being written by Taliesin Beynon and presumably to appear in version 10. (Other traces of this project are also visible already in version 9.) It wouldn't surprise me if they weren't documented until then, and as such, it may be that nobody can answer the question due to NDA. OTOH, maybe Joel Klein can comment at least on the stream functionality. –  Oleksandr R. Sep 4 '13 at 9:33
@OleksandrR. I was thinking they might also be related to the AsynchronousTasks functionality of v9, and the DataStore might provide a way to temporarily store the results before they're ready to be returned. –  Szabolcs Sep 4 '13 at 14:41


text:
My Math Forum What is the minimum number of days until the expected value hits $1,000,000?  User Name Remember Me? Password  Economics Economics Forum - Financial Mathematics, Econometrics, Operations Research, Mathematical Finance, Computational Finance February 8th, 2018, 02:32 PM #1 Newbie Joined: Aug 2013 Posts: 18 Thanks: 0 What is the minimum number of days until the expected value hits$1,000,000?

Given the left page scenario, what is the minimum number of days until the expected value rebalances to $1,000,000? The diagram on the right demonstrates pictorially how expected values are calculated per day iteration Attached Images  Harvesting Volatility.jpg (94.9 KB, 9 views)  February 10th, 2018, 08:39 PM #2 Newbie Joined: Jan 2018 From: somewhere Posts: 14 Thanks: 2 Math Focus: Algebraic Number Theory / Differential Fork Theory By expected value are you referring to the probabilitic meaning or do you mean the point at which you are guaranteed to make$1,000,000? I ask because the latter is possible to never come.
 February 12th, 2018, 06:24 PM #3 Newbie   Joined: Aug 2013 Posts: 18 Thanks: 0 Hi, I solved it. Thanks!
 February 12th, 2018, 06:55 PM #4 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,788 Thanks: 1037 Math Focus: Elementary mathematics and beyond Would you mind posting your solution? Thanks from Joppy
 February 12th, 2018, 08:00 PM #5 Senior Member   Joined: Aug 2012 Posts: 1,854 Thanks: 513 Forget it. It's too late to become a bitcoin billionaire.

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# boolean algebra calculator symbolab

Expressions inside brackets () are always evaluated … Message received. Algebra טרום אלגברה סדר פעולות חשבון גורמים משותפים וראשוניים שברים חיבור, חיסור, כפל, חילוק ארוך מספרים עשרוניים חזקות ושורשים מודולו Over the next few weeks, we'll be showing how Symbolab... Middle School Math Solutions – Inequalities Calculator. Algebra Calculator - get free step-by-step solutions for your algebra math problems ... Boolean Algebra; ... Over the next few weeks, we'll be showing how Symbolab... Read More. Middle School Math Solutions – Inequalities Calculator. Over the next few weeks, we'll be showing how Symbolab... Middle School Math Solutions – Inequalities Calculator. Free Boolean Algebra calculator - calculate boolean logical expressions step-by-step. (~A*B)+(B*~C)+(A*B) is OK 2. By using this website, you agree to our Cookie Policy. To create your new password, just click the link in the email we sent you. Operations and constants are case-insensitive. Operations and constants are case-insensitive. See {{ ext_info ? Operations with the same precedence are executed from left to right. Welcome to our new "Getting Started" math solutions series. Variables are case sensitive, can be longer than a single character, can only contain alphanumeric Use ~ * + to represent NOT AND OR respectively. By using this website, you agree to our Cookie Policy. Do not omit the * operator for an AND operation. Boolean Algebra Calculator is a free online tool that displays the truth table, logic circuit and Venn diagram for the given input. Next up in our Getting Started maths solutions series is help with another middle school algebra topic - solving... simplify\:\frac{2}{3}-\frac{3}{2}+\frac{1}{4}. This website uses cookies to ensure you get the best experience. Variables are case sensitive, can be longer than a single character, can only contain alphanumeric characters, digits and the underscore character, and cannot begin with a digit. Middle School Math Solutions – Equation Calculator. 1. IMPLIES, PROVIDED and EQUIV. This website uses cookies to ensure you get the best experience. BYJU’S online boolean algebra calculator tool makes the conversion faster and it displays the output in a fraction of seconds. characters, digits and the underscore character, and cannot begin with a digit. 'less' : 'more' }} information. To create your new password, just click the link in the email we sent you. Thanks for the feedback. This website uses cookies to ensure you get the best experience. Supported operations are AND, OR, NOT, XOR, (~AB)+(B~C)+(AB) will return an error 2. Detailed steps, K-Map, Truth table, & Quizes Next up in our Getting Started maths solutions series is help with another middle school algebra topic - solving... simplify\:\frac{2}{3}-\frac{3}{2}+\frac{1}{4}. Boolean algebra has many properties (boolen laws):. An error occurred when parsing the expression: Enter a boolean expression such as A ^ (B v C) in the box and click Parse. Middle School Math Solutions – Equation Calculator. To execute a particular operation first, surround it with parenthesis ( ). Please try again using a different payment method. Welcome to our new "Getting Started" math solutions series. Boolean operations follows a precedence order of NOT AND OR. Learn more Accept. Find more Computational Sciences widgets in Wolfram|Alpha. Operations are executed in order of precedence, from higher to lower. Free Boolean Algebra calculator - calculate boolean logical expressions step-by-step This website uses cookies to ensure you get the best experience. Boolean Algebra simplifier & solver. Use the calculator to find the reduced boolean expression or to check your own answers.Please check your entries Your answer Notes: 1. 1 - Identity element : $0$ is neutral for logical OR while $1$ is neutral for logical AND $$a + 0 = a \\ a.1 = a$$ 2 - Absorption : $1$ is absorbing for logical OR while $0$ is absorbing for logical AND Get the free "Boolean Algebra Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle.


text:
# Select the number from among the given options that can replace the question mark (?) in the following series. 1, 9, 25, 49, 81, ?

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closed

Select the number from among the given options that can replace the question mark (?) in the following series.

1, 9, 25, 49, 81, ?

1. 91
2. 111
3. 121
4. 94

by (111k points)
selected

Correct Answer - Option 3 : 121

The logic is:

12 = 1

32 = 9

52 = 25

72 = 49

92 = 81

112 = 121

Hence, ‘121’ is the correct answer.


text:
# How have I to understand in what those quantum numbers refer for ortho and para water?

I need to make an English presentation at my School (Graduate school) about something people may don't know. I chose to speak about the separation of ortho and para water. I first chat here with @Orthocresol about questions I have but this is really tricky and I'm not understanding that so much. So I chose to ask the question here, it might be helpful for other people and I hope at least for me.

At first my question is to understand what the notations $\lvert0_{00}0 \rangle$ and $\lvert1_{01}M \rangle$ in the article mention before about different states of water refers too?

What are those four numbers if the ket notation, from where they come from and what does they mean? I know it makes a lot of questions but maybe are they linked?

I read this article on wikipedia and this one too, and I also found that but I don't understood a lot of things. Maybe that $J$ is a quantum number which quantify the rotation energy of a molecule. But I have a lot of troubles with the projection and I don't know what $M$ is.

I would be glad if the answer was in a part like popularisation because I guess I'll have some troubles to understand it. But I have good skills in algebra, I had a twenty hours introduction in quantum chemistry, the hardest thing I did was a presentation of the spin-orbital coupling if it can help you to jugde my level on the subject. Thank you in advance for your answer.

## 1 Answer

It's explained in the first link you gave in the previous paragraph:

"The rotational quantum states of the asymmetric rotor, i. e., water, can be classified by $J_{K_{a}K_{c}}M$, with the total angular momentum quantum number $J$, the projection labels $K_{a}$ and $K_{c}$ onto the molecule-fixed a and c axes as defined in Figure 1 respectively, and the projection quantum number $M$ onto the space-fixed $Z$ axis."

Do you know how angular momentum in quantum mechanics works? You can look at the projection of the angular momentum onto a specific axis and the total angular momentum. Turns out the Heisenberg Uncertainty Principle prevents you from knowing the angular momentum projections along more than one axis at once...

• Why need I these representation of the states? Why only J is not enough, why to add its projections? And what M is .. ? Are we speaking about only nuclear spin or might it be also electron spin? – Hexacoordinate-C Nov 25 '16 at 2:12
• I think it's the molecular rotation. Otherwise, I don't think they would refer to it as an asymmetric rotor. Rotation about the a and c axes are such that the molecule is not symmetric about the axis. It is about the b axis though. Honestly, this is beyond the QM I know, and I would be hard pressed to explain it in clearer detail. – Zhe Nov 25 '16 at 2:59


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# Find the Mean of the Following Data X 19 21 23 25 27 29 31 F 13 15 16 18 16 15 13 - CBSE Class 10 - Mathematics

#### Question

Find the mean of the following data:-

 x 19 21 23 25 27 29 31 f 13 15 16 18 16 15 13

#### Solution

 x f fx 19 13 247 21 15 315 23 16 368 25 18 450 27 16 432 29 15 435 31 13 403 N = 106 sumfx = 2650

Mean (barx)=(sumfx)/N=2650/106=25

Is there an error in this question or solution?

#### Video TutorialsVIEW ALL [6]

Solution Find the Mean of the Following Data X 19 21 23 25 27 29 31 F 13 15 16 18 16 15 13 Concept: Mean of Grouped Data.
S


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# 5 - (4 - (3 - (2 - 1))) =?

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5 - (4 - (3 - (2 - 1))) =? [#permalink]  19 May 2018, 14:11
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Question Stats:

90% (00:23) correct 10% (00:11) wrong based on 30 sessions
What is the value $$5 - (4 - (3 - (2 - 1)))$$ ?

[Reveal] Spoiler: OA
3

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Last edited by Carcass on 27 May 2018, 08:53, edited 2 times in total.
Edited by Carcass
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Re: 5 - (4 - (3 - (2 - 1))) =? [#permalink]  28 May 2018, 13:04
Expert's post
Explanation

Make sure to begin with the innermost parentheses:

$$5 - (4 - (3 - (2 - 1))) =$$
$$5 - (4 - (3 - 1)) =$$
$$5 - (4 - 2) =$$
$$5 - (2) =$$
$$3$$

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Re: 5 - (4 - (3 - (2 - 1))) =? [#permalink]  09 Jun 2018, 00:08
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Re: 5 - (4 - (3 - (2 - 1))) =?   [#permalink] 09 Jun 2018, 00:08
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# [Tex/LaTex] Changing the font in the ToC

I am using the package memoir and I redefined the fonts related to the chapters headings:

\renewcommand\chapnamefont{\normalfont\Large\sffamily\scshape\raggedleft\so}
\renewcommand\chaptitlefont{\normalfont\huge\sffamily\bfseries\scshape\color{darkBlue}}

But altough I am using a sans-serif font for the chapters, the ToC use a serif version of LM. That's not a major problem except that I have this warning:

LaTeX Font Warning: Font shape T1/lmss/bx/sc' undefined

So Latex is complaining that it can't find a small-caps and bold version of LM sans-serif.

And the table of content:

So I am a bit confused:

• My chapters appear as I want (sans serif font)
• The ToC looks like it is using serifed fonts
• Yet it is complaining not to find a SS font
• I have this strange problem with viii
• LM SS seems to have a SC bold version: Latex font catalogue

It seems that I don't know enough about Latex and fonts …

LaTeX tells you that it is find a font. It also tells you which font it uses instead:

 LaTeX Font Warning: Font shape T1/lmss/bx/sc' undefined
(Font)              using T1/lmss/bx/n' instead on input line 8.
`

Substituations are normal. As you can see here a sans serif font is used as substitute.

\chapnamefont defines the font of the vertical chapter beside the large "2" not the font in the toc.

For the problem with "viii": http://www.tex.ac.uk/cgi-bin/texfaq2html?label=tocloft


text:
# Arm Orientation Estimation Method with Multiple Devices for NUI/NUX

Yunsick Sung* , Ryong Choi* and Young-Sik Jeong*

## Abstract

Abstract: Motion estimation is a key Natural User Interface/Natural User Experience (NUI/NUX) technology to utilize motions as commands. HTC VIVE is an excellent device for estimating motions but only considers the positions of hands, not the orientations of arms. Even if the positions of the hands are the same, the meaning of motions can differ according to the orientations of the arms. Therefore, when the positions of arms are measured and utilized, their orientations should be estimated as well. This paper proposes a method for estimating the arm orientations based on the Bayesian probability of the hand positions measured in advance. In experiments, the proposed method was used to measure the hand positions with HTC VIVE. The results showed that the proposed method estimated orientations with an error rate of about 19%, but the possibility of estimating the orientation of any body part without additional devices was demonstrated.

Keywords: Bayesian Probability , HTC VIVE , Motion Estimation , NUI/NUX , Thalmic Myo

## 1. Introduction

Given the recent various applications of virtual reality and mixed reality [ 1 ], the importance of the Natural User interface/Natural User Experience (NUI/NUX) has increased for improved realism of games and control accuracy. Motions in games can be utilized as commands. Therefore, motion estimation techniques are a key NUI/NUX technology.

Traditional motion estimation approaches utilize Kinect data [ 2 ], anatomic information [ 3 ], and vision-based calibration [ 4 , 5 ] to support and improve the accuracy of signals for motion estimation. For example, the arm movements of users can be estimated by using optical markers such as the Perception Neuron and Xsens MVN or several tangible devices such as the HTC VIVE and Thalmic Lab Myo. However, such devices are inconvenient because users have to wear one or more devices. To solve this problem, the Bayesian probability is utilized to estimate unmeasured motions based on data from a Myo, which can reduce the number of tangible devices that should be attached to users for accurate motion measurement. However, motion estimation approaches based on heterogeneous devices are required to increase the motion estimation accuracy.

This paper proposes an orientation estimation method based on the positions of hands and orientations of arms by using multiple devices and the Bayesian probability. Estimating the arm orientations should make it possible to realize more realistic virtual characters. This paper is constructed as follows. Section 2 describes traditional motion estimation approaches, and Section 3 describes the proposed method. Section 4 describes the experimental results, and Section 5 concludes the paper.

## 2. Traditional Motion Estimation Approaches

Wang et al. [ 4 ] developed a computer vision-based marker-less motion capture method that uses two optical cameras. The joint kinematics data of a person estimated by this method are almost like that estimated by a marker-based motion capture method. Mehrizi et al. [ 5 ] developed a vision-based calibration method for dual remote center-of-motion (RCM)-based robot arms in a human–robot collaborative minimally invasive surgery (MIS) scenario.

Kim et al. [ 6 ] introduced a method for estimating upper arm motions by forearm motions. This allow the number of Myos attached to a user to be reduced. Given that motions are expressed by orientations, they utilized two Myos to collect the orientations of a forearm and upper arm. They calculated the Bayesian probability based on the two kinds of measured orientations. The motions corresponding to the orientations with the maximum probabilities were selected as the estimated motions. Thus, only one Myo is required to estimate the forearm and upper arm based on the calculated Bayesian probability. However, they classified orientations with a fixed interval considering the possible ranges of orientations and then calculated the Bayesian probability with the classified orientations. Therefore, some Bayesian probabilities could not be calculated because the associated orientations were not measured. Lee et al. [ 7 ] introduced an approach to optimize the intervals of the motion orientations. Instead of considering the possible range of orientations, they considered the range between the minimum and maximum orientations among the measured orientations. By reducing the size of the considered range, they increased the accuracy of the Bayesian probability. Chae et al. [ 8 ] improved the accuracy of the estimated motions by utilizing a genetic algorithm to calculate the weights for each type of orientation. They utilized the products of the estimated orientations and calculated weights as the final estimated orientations. By considering the importance of each type of orientations, the motion estimation accuracy was increased. In another study, Chae et al. [ 9 ] used not only orientations but also electromyography (EMG) to improve the motion estimation accuracy. They calculated the Bayesian probability for each motion by utilizing both signals. The motions with a higher probability according to the estimated final motions of the two signals were utilized. By referencing motions with a higher probability, the motion estimation accuracy was increased.

Kim et al. [ 10 ] estimated the motions of forearms based on EMG of the biceps and triceps and inertial measurement unit (IMU) values of the upper arm. Especially, they focused on the mean, sum, and velocity of sequences based on the measured signals of both types.

## 3. Orientation Estimation Based on Positions

The proposed method estimates the orientations of an arm based on the positions of a hand using Bayesian probability. Dependent orientations are estimated by independent positions. Set the independent positions [TeX:] $$p _ { t } ^ { I } = \left[ x _ { t } ^ { I } , y _ { t } ^ { I } , z _ { t } ^ { I } \right]$$ and the dependent orientation [TeX:] $$o _ { t } ^ { D } = \left[ x ^ { D } _ { t } , y ^ { D } _ { t } , z _ { t } ^ { D } \right]$$ at time t .

Set [TeX:] $$\delta ^ { x , \zeta } = \frac { \left( \operatorname { Max } \left( x ^ { I } _ { t } \right) - \operatorname { Min } \left( x ^ { I } _ { t } \right) \right) } { \zeta },$$ [TeX:] $$\delta ^ { y , \zeta } = \frac { \left( \operatorname { Max } \left( y ^ { I } _ { t } \right) - \operatorname { Min } \left( y ^ { I } _ { t } \right) \right) } { \zeta },$$ [TeX:] $$\delta ^ { z , \zeta } = \frac { \left( \operatorname { Max } \left( z _ { t } ^ { I } \right) - \operatorname { Min } \left( z ^ { I } _ { t } \right) \right) } { \zeta }$$

where ζ is the number of sections.

Set [TeX:] $$\theta ^ { x , \zeta } = \frac { \left( \operatorname { Max } \left( x ^ { D } _ { t } \right) - \operatorname { Min } \left( x ^ { D } _ { t } \right) \right) } { \zeta },$$ [TeX:] $$\theta ^ { y , \zeta } = \frac { \left( \operatorname { Max } \left( y ^ { D } _ { t } \right) - \operatorname { Min } \left( y ^ { D } _ { t } \right) \right) } { \zeta },$$ [TeX:] $$\theta ^ { z , \zeta } = \frac { \left( \operatorname { Max } \left( z ^ { D } _ { t } \right) - \operatorname { Min } \left( z ^ { D } _ { t } \right) \right) } { \zeta }.$$

Set [TeX:] $$x ^ { D } _ { t , \zeta } = \underset { \arg \max }{ \left| \frac { x ^ { D } } { \theta ^ { x , \zeta } } \right| } p \left( \left[ \frac { x ^ { D } _ { i } } { \theta ^ { x , \zeta } } \right] | \left[ \frac { x ^ { I } _ { t } } { \delta ^ { x , \zeta } } \right] , \left[ \frac { y ^ { I } _ { t } } { \delta ^ { y , \zeta } } \right] , \left[ \frac { z ^ { I } _ { t } } { \delta ^ { z , \zeta } } \right] \right),$$ [TeX:] $$y ^ { D } _ { t , \zeta } = \underset { \arg \max }{ \left| \frac { x ^ { D } } { \theta ^ { x , \zeta } } \right| } p \left( \left[ \frac { y ^ { D } _ { i } } { \theta ^ { y , \zeta } } \right] | \left[ \frac { x ^ { I } _ { t } } { \delta ^ { x , \zeta } } \right] , \left[ \frac { y ^ { I } _ { t } } { \delta ^ { y , \zeta } } \right] , \left[ \frac { z ^ { I } _ { t } } { \delta ^ { z , \zeta } } \right] \right),$$ and [TeX:] $$z ^ { D } _ { t , \zeta } = \underset { \arg \max }{ \left| \frac { x ^ { D } } { \theta ^ { x , \zeta } } \right| } p \left( \left[ \frac { z ^ { D } _ { i } } { \theta ^ { z , \zeta } } \right] | \left[ \frac { x ^ { I } _ { t } } { \delta ^ { x , \zeta } } \right] , \left[ \frac { y ^ { I } _ { t } } { \delta ^ { y , \zeta } } \right] , \left[ \frac { z ^ { I } _ { t } } { \delta ^ { z , \zeta } } \right] \right)$$

where i is the index of dependent orientations, probability [TeX:] $$\mathrm { p } \left( x ^ { D } t , \zeta \right) = \max p \left( \left[ \frac { x ^ { D } _ { i } } { \theta ^ { x , \zeta } } \right] | \left[ \frac { x ^ { I } _ { t } } { \delta ^ { x , \zeta } } \right] , \left[ \frac { y ^ { I } _ { t } } { \delta ^ { y , \zeta } } \right] , \left[ \frac { z ^ { I } _ { t } } { \delta ^ { z , \zeta } } \right] \right),$$ probability [TeX:] $$p \left( y ^ { D } _ { t , \zeta } \right) = \max p \left( \left[ \frac { y ^ { D } _ { i } } { \theta ^ { y , \zeta } } \right] | \left[ \frac { x ^ { I } _ { t } } { \delta ^ { x , \zeta } } \right] , \left[ \frac { y ^ { I } _ { t } } { \delta ^ { y , \zeta } } \right] , \left[ \frac { z ^ { I } _ { t } } { \delta ^ { z , \zeta } } \right] \right),$$ and probability [TeX:] $$\mathrm { p } \left( z ^ { D } _ { t , \zeta } \right) = \max p \left( \left[ \frac { z ^ { D } _ { i } } { \theta ^ { z , \zeta } } \right] | \left[ \frac { x ^ { I } _ { t } } { \delta ^ { x , \zeta } } \right] , \left[ \frac { y ^ { I } _ { t } } { \delta ^ { y , \zeta } } \right] , \left[ \frac { z ^ { I } _ { t } } { \delta ^ { z , \zeta } } \right] \right).$$

The error rate is effected by the section ζ. Therefore, dependent orientations are determined considering multiple orientations calculated by multiple section ζs. Set [TeX:] $$x ^ { D } _ { t } = \underset { \arg \max } {x ^ { D }_{t,\zeta} } p \left( x^{D} _ { t, \zeta }\right)$$ , [TeX:] $$y ^ { D } _ { t } = \underset { \arg \max }{ y ^ { D }_{t,\zeta} } p \left( y^ { D }_{ t , \zeta } \right)$$ , [TeX:] $$z ^ { D } _ { t } = \underset { \arg \max }{ z ^ { D }_{t,\zeta} } p \left( z^ { D }_{ t , \zeta } \right)$$

## 4. Experiments

The proposed method was validated on a computer equipped with Windows 10 Pro, Intel i7-7700 (3.6 GHz), NVIDIA GeForce GTX 1050 2G, and RAM 16G (DDR4). One VIVE and one Myo were utilized for motion learning and motion estimation, as shown in Fig. 1. In the experiment, the VIVE headset was only used to view the boxing contents, not to measure the location of the headset. First, seven boxing motions (jab, straight, hook, uppercut, left weaving, right weaving, and guard) were defined, as shown in Fig. 1.

Seven boxing motions (jab, straight, hook, uppercut, left weaving, right weaving, and guard).

Each motion was measured three times at 30 fps for motion learning. The motions (jab, straight, hook, uppercut, left weaving, right weaving, and guard) were measured one by one. To train the orientations with the Bayesian probability, the minimum and maximum of the x and y positions for the hands and x, y, and z orientations for the arms were obtained. The differences were divided into five increments for orientation and position quantization. Fig. 2 shows the obtained positions and orientations of each motion, and Fig. 3 shows the quantized positions and orientations of each motion.

The quantized positions and orientations of each motion were utilized to calculate the Bayesian probability. To verify the proposed method, each boxing motion were measured twice more and then quantized. The measured and quantized positions of the hands are shown in Figs. 4 and 5, respectively. The quantized orientations of the arms were estimated according to the Bayesian probability, as shown in Fig. 6.

Measured positions and orientations.
Quantized positions and orientations.
Measured positions for validation.
Quantized positions for validation.
Quantized and estimated orientations for verification.
Measured orientation with the Myo.
Quantized orientation with the Myo.
Sums of the measured orientations.
Cumulative error rates.

To verify the estimated orientations, the estimated orientations were compared with the actual orientations of the arms as measured by a Myo, as shown in Figs. 7 and 8.

Fig. 9 shows the accumulated sum of the measured orientations. Fig. 10 shows the cumulative error rate, which was calculated by the rate of the difference between the estimated and quantized orientations and the measured and quantized orientations.

Error rate of the orientation with 5 steps.
Error rate of the orientation with 10 steps.

Figs. 11 and 12 show the error rate for each type of orientation. The average error rate was about 26% with the section ζ by 10. The average error rate was effected by the section ζ. Therefore, the section ζ was set by 5. In the results, the average error rate with the section ζ by 5was about 20%. When both sections were considered to validate the proposed method, the average error rate was about 19%.

## 5. Conclusion

This paper proposes a method to estimate the arm orientations based on the hand positions by using the Bayesian probability. One VIVE and one Myo were utilized to collect the hand positions and arm orientations. The Bayesian probability was then calculated based on the collected data during orientation learning. The unmeasured arm orientations were estimated using the positions measured with VIVE based on the calculated Bayesian probability during the orientation estimation.

In the experiments, seven boxing motions were defined to validate the proposed method. Each boxing motion was performed three times for orientation learning and two times for orientation estimation. The estimated boxing orientations were compared with the orientations measured by a Myo and quantized. The average error rate was 19%, but the possibility of estimating the orientation of any body part without additional devices was demonstrated.

## Acknowledgement

This research was supported by the Ministry of Science, ICT and Future Planning (MSIP), Korea, under the Information Technology Research Center support program (No. IITP-2017-2013-0-00684) supervised by the Institute for Information & Communications Technology Promotion (IITP).

## Biography

##### Yunsick Sung
https://orcid.org/0000-0003-3732-5346

He is currently an Assistant Professor in the department of Multimedia Engineering at Dongguk University, Seoul, Korea. He received the B.S. degree in Division of Electrical and Computer Engineering from Pusan National University, Busan, Korea, in 2004, the M.S. degree in Computer Engineering from Dongguk University, Seoul, Korea, in 2006, and the Ph.D. degree in Game Engineering from Dongguk University, Seoul, Korea, in 2012. He was employed as a Member of the Researcher at Samsung Electronics in Korea, between 2006 and 2009.

## Biography

##### Ryong Choi
https://orcid.org/0000-0003-3044-7877

He received B.S. degrees in Department of Computer Engineering from Soongsil University in 2017. Since August 2017, he is with the Department of Multimedia Engineering from Dongguk University, Seoul as a MS candidate. His current research interests include Artificial intelligence and Immersive SW.

## Biography

##### Young-Sik Jeong
https://orcid.org/0000-0002-7421-1105

He is a professor in the Department of Multimedia Engineering at Dongguk University in Korea. His research interests include multimedia cloud computing, information security of cloud computing, mobile computing, IoT (Internet of Things), and wireless sensor network applications. He received his B.S. degree in Mathematics and his M.S. and Ph.D. degrees in Computer Science and Engineering from Korea University in Seoul, Korea in 1987, 1989, and 1993, respectively. He was a professor in the Department of Computer Engineering at Wonkwang University in Korea from 1993 to 2012. He worked and researched to Michigan State University and Wayne State University as visiting professor in 1997 and 2004, respectively. He is also an Executive Editor of Journal of Information Processing Systems, an Associate Editor of Journal of Supercomputing (JoS) and International Journal of Communication Systems (IJCS), and an Editor of Journal of Internet Technology (JIT), finally an Associate Editor of Journal of Human-centric Computing (HCIS). In addition, he has been serving as a Guest Editor for international journals by some publishers: Springer, Elsevier, John Wiley, Oxford University Press, Hindawi, Emerald, Inderscience, and so on. He is also a member of the IEEE.

## References

• 1 G. Bang, J. Yang, K. Oh, I. Ko, "Interactive experience room using infrared sensors and user’s poses," Journal of Information Processing Systems, 2017, vol. 13, no. 4, pp. 876-892. doi:[[[10.3745/jips.02.0068]]]
• 2 W . Song, L. Liu, Y . Tian, G. Sun, S. Fong, K. Cho, "A 3D localisation method in indoor environments for virtual reality applications," Human-centric Computing and Information Sciencesarticle no. 39, 2017, vol. 7, no. article 39. doi:[[[10.1186/s13673-017-0120-7]]]
• 3 J. Kim, D. Chung, I. Ko, "A climbing motion recognition method using anatomical information for screen climbing games," Human-centric Computing and Information Sciencesarticle no. 25, 2017, vol. 7, no. article 25. doi:[[[10.1186/s13673-017-0106-5]]]
• 4 Z. Wang, Z. Liu, Q. Ma, A. Cheng, Y. H. Liu, S. Kim, et al., "Vision-based calibration of dual RCM-based robot arms in human-robot collaborative minimally invasive surgery," IEEE Robotics and Automation Letters, 2017, vol. 3, no. 3, pp. 672-679. doi:[[[10.1109/LRA.2017.2737485]]]
• 5 R. Mehrizi, X. Peng, X. Xu, S. Zhang, D. Metaxas, K. Li, "A computer vision based method for 3D posture estimation of symmetrical lifting," Journal of Biomechanics, 2018, vol. 69, pp. 40-46. doi:[[[10.1016/j.jbiomech.2018.01.012]]]
• 6 P. Y. Kim, Y. Sung, J. H. Park, "Bayesian probability-based motion estimation method in ubiquitous computing environments," in Advances in Computer Science and Ubiquitous Computing. Singapore: Springer2015,, pp. 593-598. doi:[[[10.1007/978-981-10-0281-6_84]]]
• 7 S. G. Lee, Y. Sung, and J. H. Park, Multimedia Tools and Applications, 2016., https://doi.org/10.1007/s11042-016-3843-y
• 8 J. Chae, Y . Jin, Y . Sung, K. Cho, "Genetic algorithm-based motion estimation method using orientations and EMGs for robot control," Sensorsarticle no. 183, 2018, vol. 18, no. article 183. doi:[[[10.3390/s18010183]]]
• 9 J. Chae, Y. Jin, M. Wen, W . Zhang, Y. Sung, K. Cho, "Genetic algorithm-based adaptive weight decision method for motion estimation framework," The Journal of Supercomputing2018. https://doi.org/10.1007/ s11227-018-2247-3. doi:[[[10.1007/s11227-018-2247-3]]]
• 10 H. J. Kim, Y . S. Lee, D. Kim, "Arm motion estimation algorithm using MYO armband," in Proceedings of IEEE International Conference on Robotic Computing, Taichung, Taiwan, 2017;pp. 376-381. doi:[[[10.1109/IRC.2017.32]]]
Seven boxing motions (jab, straight, hook, uppercut, left weaving, right weaving, and guard).
Measured positions and orientations.
Quantized positions and orientations.
Measured positions for validation.
Quantized positions for validation.
Quantized and estimated orientations for verification.
Measured orientation with the Myo.
Quantized orientation with the Myo.
Sums of the measured orientations.
Cumulative error rates.
Error rate of the orientation with 5 steps.
Error rate of the orientation with 10 steps.


text:
Free Version
Easy

# Set Theory: Union of Sets, Definition and Examples

ABSALG-UJJA0U

The union of two sets $A$ and $B$ is the set of elements that are either in $A$ or in $B$ or in both $A$ and $B$.

In other words, the unions of two sets $A$ and $B$ is the set of elements that are in at least one of $A$ and $B$.

The notation for the union of two sets $A$ and $B$ is $A\cup B$ or $A\bigcup B$ (just a bigger $\cup$ symbol). In set notation, we have:

$$A\cup B=\{x\mid x\in A \;{\rm or}\;x\in B\}$$

Which of the following is FALSE?

A

The set of all odd integers equals the union

$\{x\in{\mathbb Z}\mid\,x=4k+1, \;{\rm for\;some\;integer}\;k\}\cup\{x\in{\mathbb Z}\mid\,x=4k+3, \;{\rm for\;some\;integer}\;k\}$

B

The set of all odd integers is a subset of the union

$\{x\in{\mathbb Z}\mid\,x=2k+1, \;{\rm for\;some\;integer}\;k\}\cup\{x\in{\mathbb Z}\mid\,x=4k+2, \;{\rm for\;some\;integer}\;k\}$

C

The set of all even integers equals the union

$\{x\in{\mathbb Z}\mid\,x=4k+3=4\ell+1, \;{\rm for\;some\;integers}\;k,\ell\}\cup\{x\in{\mathbb Z}\mid\,x=2k, \;{\rm for\;some\;integer}\;k\}$

D

The set $S$ of titles of the albums by Sugababes. The union of the set $S_1$ of all movies starring Morgan Freeman and the set $S_2$ of all movies starring both Morgan Freeman and Brad Pitt equals the set $S_2=S_1\cup S_2$ of all movies starring both Morgan Freeman and Brad Pitt.

E

The union of the set $S_1$ of all movies starring Morgan Freeman and the set $S_2$ of all movies starring Brad Pitt equals the set $S=S_1\cup S_2$ of all movies starring either Morgan Freeman or Brad Pitt or both Morgan Freeman and Brad Pitt.


text:
## The LSND and MiniBooNE Oscillation Searches at High $Δm^2$    [PDF]

Janet M. Conrad, William C. Louis, Michael H. Shaevitz
This paper reviews the results of the LSND and MiniBooNE experiments. The primary goal of each experiment was to effect sensitive searches for neutrino oscillations in the mass region with $\Delta m^2 \sim 1$ eV$^2$. The two experiments are complementary, and so the comparison of results can bring additional information with respect to models with sterile neutrinos. Both experiments obtained evidence for $\bar \nu_\mu \rightarrow \bar \nu_e$ oscillations, and MiniBooNE also observed a $\nu_\mu \rightarrow \nu_e$ excess. In this paper, we review the design, analysis, and results from these experiments. We then consider the results within the global context of sterile neutrino oscillation models. The final data sets require a more extended model than the simple single sterile neutrino model imagined at the time that LSND drew to a close and MiniBooNE began. We show that there are apparent incompatibilities between data sets in models with two sterile neutrinos. However, these incompatibilities may be explained with variations within the systematic error. Overall, models with two (or three) sterile neutrinos seem to succeed in fitting the global data, and they make interesting predictions for future experiments.
View original: http://arxiv.org/abs/1306.6494


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# Worksheet on Conversion of Fractions

In worksheet on conversion of fractions covers questions on conversion of fractions into fractions having same denominator and also on conversion of fractions into its smallest and simplest form. This exercise sheet on conversion of fractions can be practiced by the students to get more ideas on fractions.

1. (i) Change the fractions 2/5, 7/10 and 11/15 into fractions having 30 as denominator.

(ii) Change the fractions 2/3, 5/6, 7/9 and 9/24 into fractions having 72 as denominator.

2. Change the given fractions into fractions of the same denominator.

(i) 1/4 and 5/12

(ii) 4/5 and 7/15

(iii) 3/4 and 11/18

(iv) 2/8, 7/12 and 13/24

(v) 2/5, 3/7 and 27/35

(vi) 5/8, 3/4 and 11/12

(vii) 3/4 7/12 and 17/24

(viii) 5/8, 7/16 and 9/16

(ix) 4/7, 5/28 and 9/28

3. Which is the greater fraction?

(i) 2/9 or 5/12 ⇒ (8/36 or 15/36) so 15/16 > 8/36

(ii) 3/4 or 11/18

(iii) 11/16 or 13/24

(iv) 5/2 or 11/3

(v) 1 11/12 or 1 11/15

4. Which is the smaller fraction?

(i) 4/7 or 2/3 ⇒ (12/21 or 14/21) so, 12/21 < 14/21

(ii) 3/7 or 5/9

(iii) 11/16 or 13/20

(iv) 5/14 or 10/21

(v) 7/16 or 19/24

5. Fill in the blanks with the symbol > or <:

(i) 1/3 ………. 1/5

(ii) 5/12 ………. 7/8

(iii) 11/12 ………. 5/6

(iv) 2/3 ………. 14/15

(v) 11/12 ………. 15/16

(vi) 5/12 ………. 9/16

6. Arrange the given fractions in ascending order by changing them into fractions of the same denominator:

(i) 2/3, 3/4, 5/6

(ii) 5/9, 4/5, 11/15

(iii) 5/8, 3/4, 11/12

(iv) 3/2, 5/4, 2/3

(v) 1/3, 3/4, 2/5

(vi) 2/5, 3/7, 27/35

7. Arrange the given fractions in descending order by changing them into fractions of the same denominator:

(i) 5/11, 9/22, 8/33

(ii) 5/7, 2/3, 9/14

(iii) 13/15, 13/30, 14/45

(iv) 1/4, 3/8, 3/16

(v) 3/4, 3/8, 7/16

(vi) 8/15, 9/20, 7/12

8. Change the fractions into their simplest form:

(i) 30/40

(ii) 8/24

(iii) 45/60

(iv) 36/180

(v) 75/80

(vi) 48/72

(vii) 72/180

(viii) 144/192

9. Which is greater?

(i) 1 3/5 kg of apple or 1 2/3 kg of chikoo?

(ii) 3/5 part of an iron rod or 3/4 part of the same iron rod?

10. Find the reciprocal of the following by interchanging the numerators and denominators:

(i) 3/5

(ii) 17/18

(iii) 6/17

(iv) 17/5

(v) 25/30

(vi) 36/118

(vii) 115/18

(viii) 3/26

If students have any queries regarding the Worksheet on expanded form of a number questions please fill-up the below comment box so that we can help you.

However, suggestions for further improvement, from all quarters would be greatly appreciated.

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