Complex (Cognitive) Systems
work in progress...
This marvelous course by Prof. Markus Müller revealed like a hidden treasure in the most unexpected of places: a social-sciences and humanities-oriented postgrad program, in an unimportant university of a less-than-blooming country (the cognitive science masters degree from the Autonomous State University of Morelos, Mexico). The aim of self-publishing notes and assignments, and assembling them into a coherent whole, goes beyond giving everyone brave enough1 a shot to enjoy it. It also sits as a personal reminder of my interest at the intersection of mathematical modeling, biophysics and the mind.
the brain meets the operational definition of what a complex2 system is:
- hierarchical organisation, scale-free structure, many-to-many relations among components, interconnectedness (recurrence)
- non-linear behaviour, emergent properties (hard to determine from the observation of components alone, which number in the millions)
- chaotic to initial conditions (slight differences lead to widely different states)
- non-stationary measurements (dynamical system whose model constants/parameters actually vary; system doesn't stay in an attractor)
- phase transitions, equilibrium depends on critical states
- noisy measurements, plus some rather stochastic processes (spiking), whose measurable fluctuations may be difficult to distinguish from chaotic determinism
• dynamical systems basics
we'll distinguish between two kinds of systems:
- dynamical (deterministic). e.g. the harmonic oscillator:
• harmonic oscillator
consider the mechanics of a mass attached to a spring (no other forces involved, e.g. friction, gravity). let's say this is a very simple brain model (or of one of its components thereof) whose activity behaves like an oscillating variable:
from Newton's second law and Hooke's law:
in order to solve the second-order linear ordinary homogeneous differential equation, we partition it into a system of two first-order equations. let , therefore:
notice that with the last substitutions both equations are the same, except in terms of different variables. both sin(ω0t) and cos(ω0t) satisfy it, because:
more generally, solutions will conform to the linear superposition principle:
any possible function of position on time will look sinusoidal, of varying frequency (depending on ), amplitude and phase (a and b together). we rewrite it as:
this has the property that , and .
let be the initial phase. then
• initial conditions
finally, let's say the spring is totally relaxed at t=0, so . this implies that :
• complex polar notation
prove that is the unit circle.
solutions of the form include the aforementioned trigonometric solutions when λ, A ∈ ℂ. so the #original differential equation can be rewritten as:
the general solution becomes:
if we restrict x(t) to the type ℝ→ℝ (position is real), it follows that x(t) equals its complex conjugate.
moreover, . this implies that and . so:
compare this expression against the one using trigonometric functions.
• energy loss
(also see #dissipative systems and attractors)
now we will damp the oscillator by adding a term for friction, whose force is proportional to velocity (but in opposite direction):
assume a family of solutions of the form . therefore:
plug that back into the general solution:
asymptotically, the system will inevitably evolve to the most entropic dynamic state. let's show this on a case-by-case basis, taking note on the different (non)monotonic dynamics:
• underdamped: ω0 > β
by restricting x(t) to ℝ→ℝ, and from our previous knowledge of the simple harmonic oscillator solution:
• ω0 = β
we leverage the fact that amplitude isn't constant anymore:
and the damped harmonic oscillator equation becomes:
i.e., amplitude is a linear function. so the solution must be of the form:
graphically, the spring oscillates only once:
• overdamped: ω0 < β
because , is overshadowed by . the plot never gets to complete an oscillation:
• energy input
the damped model will get augmented with the influence of external stimuli, driving our cognitive-unit model to activity. this is known as a driven harmonic oscillator. let's go with an engine rotating at frequency and amplitude F, which in turn pushes and pulls the rest of the system:
the equation isn't homogeneous anymore. the solution is the sum of the general and particular solutions: the homogeneous one is of the type and eventually dissipates. the particular one oscillates forever:
starting from an exploration of the steady state ("final conditions"), we will show that amplitude is a function of the engine's frequency, and that there are privileged frequencies for which steady-state amplitude forms maxima or singularities: resonance frequencies.
so we rewrite the full differential equation:
(...magic happens here...)
• phase space
phase space contains all information of a dynamical system, with a different axis for each variable of interest. each point in it represents a state of the whole system, and a trajectory in this virtual space corresponds to the evolution of the system.
signal: disregarding noise, taking a series of measurements is equivalent to observing variation in a single dimension of phase space. graphically, this is a projection of the system upon some vector in phase space.
• Fourier analysis
• linear independence of Fourier terms
let with be functions on the interval . prove that , given the proper definition of the dot product. is Dirac's delta distribution.
let be the scalar product for functions of the vector basis defined by .
proof by modus ponens follows:
- restatement of conditional premise
given the above definitions; if , then is an orthogonal basis. that is:
- proof of the antecedent premise
from Euler's formula:
from the integration of the odd function over a zero-centered interval, :
let ; then .
note that .
is an orthogonal basis.
• Fourier transform
the Fourier series represents a periodic function as a discrete linear combination of sines and cosines . on the other hand, the Fourier transform is capable of representing a more general set of functions, not restricted to being periodic. how is that achieved?
(for convenience, we will start from the series in complex exponential notation)
if weren't periodic, the series would be representing it only at . nonetheless, we can make the interval arbitrarily large:
let , then . substituting in the last expression yields:
which is a Riemman sum for variable :
which symmetry properties does H(f) has if h(t) is:
- odd real
- odd imaginary
- odd complex
and which symmetry properties does h(t) has if H(f ) is:
- odd real
- even complex
- even imaginary
- odd real h(t)
we get rid of the imaginary part of H(f):
from integration of an odd function (r(t)cos(2πft)) at a symmetric interval:
is odd. is odd and imaginary.
- odd imaginary h(t)
is odd. is odd and real.
- odd h(t)
is odd. is odd and complex.
- odd real H(f)
is odd. is imaginary and odd.
- complex even H(f)
is even. is even and complex.
- imaginary even H(f)
is even. is even and imaginary.
• uncertainty principle
the more short-lived the wave in the time domain, the more widespread its power spectrum (a.k.a. frequency spectrum). vice versa.
• convolution theorem
is the convolution of functions x(t) y h(t). F[y(t)] denotes its Fourier transform:
time-domain convolution is equivalent to frequency-domain product!
• cross-correlation theorem
is the correlation (covariance, actually) of functions x(t) y h(t) as a function of a time lag between them. F[z(t)] denotes its Fourier transform:
cross-correlation is equivalent to frequency domain product (with conjugate X(f)).
• discrete Fourier transform
the computation of a DFT is done segment by segment. the signal is multiplied by some "window" function (e.g. a square pulse) which is zero outside the window. this is known as windowing.
the effect of using a finite window is that unexisting frequencies appear at the power spectrum (around the original frequencies). this phenomenon is known as leakage. the explanation is evident from the #convolution theorem: one can use similar reasoning to show that . therefore, multiplication by a window will induce a convolution in the frequency domain.
low frequencies (and therefore #non-stationary systems) are the most problematic to compute a DFT for, since the window may not be long enough to capture them. there's no single #test for stationarity. it can be tested for nonetheless, measuring the effect of varying window sizes on different parameters (centrality and spread statistics, Lyapunov exponent, etc.).
• Whittaker–Shannon (sinc) interpolation
what function should we convolve the signal's Fourier transform with in order to obtain the same result as windowing it with a rectangular function? we will analyze the closely-related case of a square pulse window:
calculate the first three terms of the Fourier series for function , with
in terms of the Fourier series:
from the integration of odd functions ( and ) at symmetrical interval ( ):
we will consider the interval , which includes a full period of for any n. therefore:
f(x) could be just a discrete sample of a continuous signal, but because convolution with sinc(x) produces a continuous function (a sum of sine waves), the operation will fill in all the missing time points.
• Nyquist-Shannon sampling theorem
continuous functions of bounded frequency bandwidth ("band-limited") contain a bounded amount of information. they can be perfectly represented using a function of countable (i.e. discrete) sets and an interpolation operation.
the Nyquist frequency is the maximum sampling frequency at which the digitization of a continuous signal still losses fidelity. that is, aliasing occurs:
sampling frequency should be greater than twice the maximum frequency in the power spectrum of the original signal.
undersampling results in the unconsidered high frequencies aliasing (adding up) over at the frequency domain. oversampling is harmless, beyond being a waste of disk space.
• testing for undersampling
what if the maximum frequency isn't known in advance? it is possible to test that the maximum frequency has been considered, because aliasing will make two power spectra of the same signal look dissimilar under two different sampling frequencies:
- register using two sampling frequencies, one greater than the one to test for the Nyquist criterion.
- compute both power spectra ( ).
- if they are simply scaled versions of one another, aliasing didn't occur. i.e. the ratio between two features (e.g. maxima) should stay constant.
• other time-series topics
• stochastic correlations
suppose the signal is generated by an stochastic process. if low frequencies dominate the power spectrum, then the process is non-stationary (time-series mean value will be trending over long periods of time) and autocorrelation will be large.
corollary: even two independent sources of noise can display high statistical dependence.
averaging the time series would be no good. prediction should take autocovariance times into account, detrending methods are also available.
strict/particular stationarity: system's parameters are truly constant. too hard a requirement, since we must know the system's differential equations in advance.
weak/general stationarity: empirical transition probabilities in the F map stay constant. affected by low-band frequencies. alternatively, 1st moment and autocovariance stay constant and finite. this requires greater data availability, because short acquisition times may not be enough to establish statistical relevance.
• linear prediction
suppose there's a time series whose value we want to predict. in a linear prediction algorithm we define as a linear combination of the last m points:
it's clear that there are as many linear combinations of the last m points as there are permutations (with substitution) of coefficients . in order to find the optimal coefficients we try to minimize the mean square error predicting the previous m - 1 values:
- in-sample accuracy: estimated from predictions made for the same data segment used during training/optimisation phase. it doesn't reflect the algorithm's true predictive power (overestimate), since tests aren't predictions but "post-dictions".
- out-sample accuracy: error estimated from true predictions, using the values of data never "seen" before by our predictive model.
• phase-space methods
• dissipative systems and attractors
dissipative system: on average, the volume of the manifold describing the state trajectory contracts under dynamics, after certain initial conditions. alternatively, it can be said that #energy loss creates an attractor for that system, reducing the set of possible states that it can take with time. nonetheless, the system still operates far from thermodynamic equilibrium.
in order to verify the presence of an attractor, the L1 metric of the determinant of the system's Jacobian (i.e., of the state transition function's (F) partial derivatives) must be smaller than 1, on average:
this means that the transformation's scaling factor produced by change on the different directions is a contracting one. visually, if phase space contains an attractor then the system is dissipative.
• visual inspection
- Poincaré section: allows to tell whether the trajectory behavior is classically deterministic, chaotic deterministic or random. it amounts to finding the intersection between trajectories and a secant plane.
- stroboscopic map: graphical representation of Poincaré's section technique for attractors. section is adjusted until the scatter plot is maximally decorrelated. also used for qualitative detection of strange/fractal attractors.
- "phase portrait": also known as delay representation. it is a bidimensional representation of any m-dimensional phase space. each point is given by a pair of subsequent states in phase space (see also #Takens-Whitney embedding theorem). it eases the visualisation of multidimensional systems and helps identify determinism, periodicity, linearity and chaos.
• Takens-Whitney embedding theorem
it is possible to reconstruct a phase space manifold that is isomorphic to the original one, even when all information we have is a time series, through the method known as delay reconstruction.
- stationary data
- low-dimensional system
- negligible noise
the basic idea is that for each subset of m equidistant points in the signal, there's an associated unique point in m-dimensional phase space. each value in this subset of the time series becomes a component of the state vector:
note that the closer and are, the more correlated the reconstructed attractor; rendering it less useful:
the spacing parameter among points in the time series is called the time lag τ. to set an adequate value for τ, we want τ > L; with L being the autocovariance length. that is, we want the time between two measurements to be long enough so that they aren't correlated anymore.
finding the right number m of dimensions involves increasing the number of embedding components per point, until the manifold can't unfold anymore. we say there are false neighbors (and therefore that the embedding isn't high-dimensional enough) whenever an increase in m separates seemingly close points considerably in reconstructed phase space.
• non-linear prediction
unlike the #linear prediction algorithm for time series; a non-linear algorithm doesn't use the last m points combined. rather, it builds upon the intuition that the most relevant states for making a prediction some time Δn after the present are the ones that most resembled the current state. it then averages those similar states after Δn:
where denotes the cardinality of the current state vector's neighborhood (in phase space).
• as stationarity test
divide the series in intervals so short so as to have lineal segments.
predict and cross-validate using all possible segment pairs.
if some training segments display unusually different performance, then they don't generalise to the whole series. therefore we can conclude that the attractor changed as we swapped training segments; and the time series isn't stationary.
• as a noise smoother
suppose the measured time series is the sum of its true value and noise:
where is the random component. also suppose and are independent, and that temporal correlation is short.
we want to substitute with . starting from the #non-linear prediction algorithm:
in phase space, use trajectories which are close to the current one (both in the past and future portions), since the points inside that neighborhood belong to an n-dimensional sphere with center at the vector state under consideration.
apply the nonlinear prediction algorithm afterwards: average the vectors in the neighborhood. thus the stochastic component will be largely cancelled out.
• chaos theory
• Lyapunov coefficient
quantifies chaos levels in a dynamical system. it is the exponent in the exponential divergence measurement of phase space. the system has as many Lyapunov exponents as dimensions, positive ones are chaotic dimensions:
|Lyapunov coefficient||kind of attractor|
|λ < 0||stable fixed point|
|λ = 0||stable limit cycle|
|λ > 0||chaos|
|λ → ∞||noise|
• estimation from time series
it is advisable to perform a #phase portrait beforehand, so as to rule out that the system is nondeterministic.
- embedding: if the #embedding dimension is too low divergence will be underestimated because of false neighbors.
- noise level: overestimated because of noise
- #stochastic correlations: if autocorrelation is big, the attractor will look compressed along a diagonal, underestimating true divergence
• fractal dimension
strange attractors are the flagship of chaos: trajectories are still deterministic (no crossroads) yet non-cyclic (exact knowledge is required to tell accurately which trajectory the system is in). since some chaotic systems are also #dissipative, all of this implies that a subset of infinitely-dense phase space is being relentlessly filled by a line of infinite detail.
• box counting
box counting is one of many methods to assess scaling property of fractals.
let ε be the side of each box (n-dimensional squares), which are in turn used to segment the figure. then let N(ε) be the number of boxes. note that:
D being the box-counting fractal dimension.
• correlation dimension
the correlation dimension can be used as an estimator of an attractor's dimension.(under certain circumstances), including fractal dimensions.
the idea now isn't to take the ratio of the number of boxes and their length ε in the limit to infinity. instead, we take the ratio of points (among all possible valid pairs) which fall within the neighborhood (defined by a radius of length ε) of some other point:
where C(N, ϵ) is the correlation sum:
• estimation from time series
given the lack of a phase space, it is necessary to #reconstruct it first. the right choice of delay τ during reconstruction of embedded vectors provides a suitable delay to estimate correlation dimension.
the aforementioned estimator of C(m, ϵ) is biased to underestimate the dimension, because data that are close in a trajectory of phase space tend to retain some temporal correlation.
the corrected estimator of C(m, ϵ) excludes sufficiently-close vector pairs (in phase space); which usually correspond to subsequent vectors in the time series:
with . doesn't necessarily have to be the average correlation time.
the following plots from Kantz & Schreiber's book5 come from a map-like system. for a only 2.5% of all pairs is lost in the corrected correlation dimension estimator. this is statistically negligible.
possible error sources:
- the correlation is a well-defined construction (for finite data sets), whereas the correlation dimension is just an extrapolation that must be handled with care.
- it isn't enough to observe fractal scaling for a single choice of m. saturation should also occur for many values.
- if measurements are too noisy, a scaling regime may never be observed. the dimension estimate can be as big as m whenever ϵ is small enough, so that only noise is captured by it.
- noise affects scales 3 orders of magnitude as big as its own level. in order to observe a scaling region, noise should be at most 1/8 of the whole attractor.
- it is therefore advisable to use a #noise reduction algorithm in advance.
- certain stochastic systems may pass for low-dimension deterministic systems.
- low-resolution digitization will make the estimate look smaller
- self-similarity may never be observed, even for good-quality deterministic data, depending on the manifold's large-scale structure.
• minimum delay
a space-time separation plot shows the proportion of points inside the neighborhood as a function of Δt and ϵ. we hope to find some value of Δt after which increments in ϵ don't add extra points. such delay marks the ideal choice of Δt when computing the correlation sum.
this is more rigorous a technique than using the autocorrelation distance.
• as a noise meter
figures 19 and 20 introduced the value ε that marks the limit between noise and scaling regimes. we found a resolution limit in the data, when neighborhoods were so tiny that they only capture randomly distributed points in all directions.
in a nutshell, given that ε is a distance in phase space units, we can define the noise level as the ratio between ε and the attractor's "size".
• neuroscience applications
- how not to use it:
Lehnertz and Elger (1998)6 analysed localized intracranial EEG data from 16 epilepsy patients. they claimed that a loss of complexity in brain activity should result from the synchronization of pathological cells before and during seizures.
linear techniques can only predict seizures a few seconds in advance. here the intention is to use the correlation dimension estimate as a complexity detector, even though it cannot actually estimate the fractal dimension for these data.
a state called state 2 coincides with pre-ictal synchronization some 10 to 30 minutes in advance to the seizure. nothing is mentioned with regard to confounding sources of complexity reduction, though.
quasi-scaling regions were observed for state 2 (D2eff = 3.7) and the seizure state (D2eff = 5.3), but not the control state. in other words, complexity apparently increased, counter to theory, and there's no base level to compare it with.
plots don't really show a clear power law either that would justify the reliability of their estimates. nor was any statistical hypothesis test performed using surrogate data.
- a more credible one?
Martinerie et al (1998)7 attempted to predict seizures by looking at phase transitions between two subsections in reconstructed phase space, also built from intracranial EEG series. it is claimed that for most cases (17 out of 19) it is possible to foretell seizure onset 2 to 6 seconds in advance, and that all of them follow a similar trajectory in phase space.
the alternative hypothesis is actually explored with surrogate data, and authors made sure to smooth noise out previous to the embedding. the number of dimensions (16) may be too unwieldy, though.
• hypothesis testing
the maximal Lyapunov coefficient and the correlation dimension come handy in detecting non-linear dynamics, however they cannot establish such a fact by themselves. it is necessary to observe a clear scaling region.
a stricter analysis would test the hypothesis that observed data are due to a linear process. this implies that certain parameter's conditional distribution (given H0) has to be empirically estimated, so that it can be contrasted with the real data.
• surrogate data
surrogate data is a akin to resampling methods (permutation, Monte Carlo, bootstrapping) for time series. surrogate data share many similarities with true data for some respects, while consistent with H0 for others. when testing for non-linearity, Fourier phases are generated randomly while the power spectrum is retained, since the imaginary part is where non-linearity could emerge from.
the general procedure for generating surrogate data is as follows:
decompose the signal in amplitudes and phases using the (fast, discrete) Fourier transform:
substitute the alleged non-linear properties with random numbers drawn from the uniform distribution ranging from 0 to 2π radians (trying to make , so that the imaginary part being odd will result in a substitute series belonging to ℝ. see the #Fourier transform properties section):
wrap the surrogate series back with the inverse Fourier transform:
test the series out (dimension estimation, Lyapunov coefficient, autocorrelation, non-linear prediction error, etc.).
rinse and repeat till the parameter's distribution given H0 is dense enough to yield good type-I error estimates.
• statistical significance
as in standard hypothesis testing, for a significance level of α (type I error: probability that H0 is erroneously rejected) we would like at most α% of the area under the surrogate data curve to overlap the original measurements, or be equal or more extreme than our point measurement. such a rank-based test is a non-parametric way of deriving a p-value.
for instance, in order to reject H0 with 95% of success, it would be enough to perform tests (for a one-side test. for the two-tail case).
• as a determinism detector
does passing the test also mean that the system is deterministic (since we tested for non-linear determinism)?
partially yes, with some probability. recall that all statistical inference is based on limited evidence and inductive reasoning:
|accept H0||reject H0|
|H0 is true||1-α (confidence)||α (type I error)|
|H0 is false||β (type II error)||1-β (power)|
- we have assumed that the available power spectrum captures at least one whole period in the time domain (i.e. no systemic sampling bias due to long-term process). otherwise, random events in a non-stationary process could be misinterpreted as deviations from H0.
- the process may not follow the conjectured probability distribution (this problem is mitigated by non-parametric tests).
- multiple comparisons problem: performing many independent tests overblows type I error. for those cases, significance levels should be made more stringent according to some correction method (Bonferroni e.g.)
• as a phase transition detector
the same non-linearity tests have been used to distinguish states of a system.
compute the same statistic for different time segments and compare with one another. the reasoning follows what we did in the #stationarity test.
complexity doesn't imply advanced math, for it can emerge from very simple descriptions (see rule 110 for instance). that said, this post relies on calculus (including some differential equations), inferential statistics (also the fundamentals of information theory), bits of graph theory and linear algebra; most (or all) of which should be familiar to STEM undergraduates. topics like Fourier analysis and chaos theory are introduced from there. ↩
not to be confused with "complicated" or "difficult". ↩
, because ↩
Holger Kantz, Thomas Schreiber (2003). Nonlinear Time Series Analysis. Cambridge University Press. ↩
Klaus Lehnertz, Christian E. Elger. Can Epileptic Seizures be Predicted? Evidence from Nonlinear Time Series Analysis of Brain Electrical Activity (1998). Phys. Rev. Lett. 80, 5019. https://doi.org/10.1103/PhysRevLett.80.5019 ↩
Martinerie, J., Adam, C., Le Van Quyen, M., Baulac, M., Clemenceau, S., Renault, B., & Varela, F. J. (1998). Epileptic seizures can be anticipated by non-linear analysis. Nature medicine, 4(10), 1173. https://doi.org/10.1038/2667 ↩


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After the course, attendees will be able to apply various statistical methods for both continuous and discrete data, and be able to interpret the statistical output generated in R.
The course will cover both descriptive and graphical data exploration techniques, hypothesis testing, sampling methods and the comparison of two groups of data. It will cover:
This course is for statisticians, data analysts and scientists wishing to explore and analyse data using R. The course is appropriate for those making use of statistics in all industries.
Basic knowledge of the R language
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Inverse Toeplitz Preconditioners for Ill-Posed Problems
M. Hanke and J. Nagy
It has been shown
recently that iterative regularization using
conjugate gradient type methods for image restoration problems
can be effectively preconditioned with circulant approximations.
Here it is shown that the theoretical properties of this
approach are not restricted to circulant matrices.
a Toeplitz approximate inverse preconditioning scheme for
discrete ill-posed problems is considered.
It is proved that the preconditioned system approximates the
prolate matrix, and that this property implies
that fast convergence of conjugate gradient type methods
can be expected. In addition, it is shown that
these results can be generalized to two-dimensional problems.
An image restoration application is used to demonstrate the
properties of the preconditioner.


URI online judge 1002 solution: Area of a Circle
URI 1002 Problem: Program to Print Area of a Circle
URI 1002 Problem solution: In this problem, you have to write a program that takes the radius as input and print the Area of a Circle as output.
According to URI Online judge, problem 1002 is given below:
The formula to calculate the area of a circumference is defined as A = π . R2. Considering to this problem that π = 3.14159:
Calculate the area using the formula given in the problem description.
The input contains a value of floating-point (double precision), that is the variable R.
Present the message "A=" followed by the value of the variable, as in the example below, with four places after the decimal point. Use all double precision variables. Like all the problems, don't forget to print the end of the line after the result, otherwise, you will receive a "Presentation Error".
|Input Sample||Output Sample|
URI 1002 Solution in C: Area of a Circle
This is a test input in C: 2.00
This is the output of test input: A=12.5664
To run this code (URI 1002 Solution/Answer) on your Laptop or PC, download Code::blocks, and to run in your android mobile download CppDroid - C/C++ IDE from play store or you can simply search on google Online C Compiler or Online C++ Compiler, you'll get plenty of online compilers! Try them!


Steam Boiler Capacity Calculation--ZBG
28/02/2019 · Formula for calculating fuel: Fuel Consumption = steam quantity* latent heat / Efficiency * Calorific Value Boiler Calculations KTH Another tool used in boiler calculation is the pressure-enthalpy diagram for steam/water (figure 5).
02/11/2017 · W = boiler capacity (Btu/h) hg = enthalpy steam (Btu/lb) hf = enthalpy condensate (Btu/lb) m = steam evaporated (lb/h) Calculate the boiler capacity for example: Boiler capacity = Burner capacity * efficiency. Example: efficiency = 95% & Burner capacity =11,110 kW. boiler capacity =11,110 kW
26/04/2021 · Boiler sizes, or outputs, are measured in Kilowatts, or kW for ease.. For an interactive way of calculating the size of boiler that your home requires, use our 60 second online quoting tool.You'll see fixed prices on boilers which are the perfect power
How Much Gas Does a Boiler Use Per Hour? | Viessmann
How to calculate the power of a gas boiler for a private house, a calculator, recommendations and formulas. Competent choice of the boiler will allow you to maintain a comfortable air temperature in the room during the winter season. A large
Gas rate = 3600/122 x 0.088 x 10.76 = 27.94kW gross, or 27.94/1.11 = 25.17kW net. The manual gives 24.49kW net maximum pretty close, but is the boiler slightly over-gassed (under is better than over)? If we stop at stage 2 above, we simply leave out the 10.76 figure: Gas flow rate = 3600/122 x 0.088 = 2.597m 3
calculating fuel consumptionn on boiler Industrial Boiler Supplier


How To Convert Meters To Feet: A Detailed Step-by-Step Guide
The metric system is a system of measurements used by scientists and other professionals throughout the world. It is the most widely used system of measurements. In this system, one unit of length is the meter. However, many people use the imperial system of measurements, which includes the foot. Converting meters to feet is a common task that can be done in a variety of ways. This guide will explain how to convert meters to feet in a step-by-step fashion.
Step-By-Step Guide To Converting Meters To Feet
- Determine the number of meters. The first step in converting meters to feet is to determine the number of meters that need to be converted. This can be done by measuring the object or distance in meters using a ruler or other measuring tool. Another way to determine the number of meters is to use a map or other visual representation of distances to calculate the number of meters between two points.
- Understand the metric system. Before attempting to convert meters to feet, it is important to understand the basics of the metric system. In this system, one meter is equal to one thousandth of a kilometer. One foot is equal to 0.3048 meters. Knowing these measurements can come in handy when converting meters to feet.
- Multiply the meters by 3.28084. To convert meters to feet, the number of meters must be multiplied by 3.28084. This will give the number of feet. For example, if the distance is 10 meters, then 10 multiplied by 3.28084 equals 32.8084 feet.
- Round the number of feet. After multiplying the meters by 3.28084, the answer should be rounded to the nearest whole number. If the number after the decimal point is between 0.5 and 1, then it should be rounded up to the nearest whole number. If the number after the decimal point is between 0 and 0.4, then it should be rounded down to the nearest whole number.
- Check the results. After rounding the number of feet, the results should be checked to ensure they are accurate. If they are not accurate, then the steps should be repeated until the desired results are achieved.
Converting meters to feet is a relatively straightforward task that can be done in a few simple steps. First, the number of meters should be determined. Then, the metric system should be understood. After that, the meters should be multiplied by 3.28084. Finally, the number of feet should be rounded to the nearest whole number and the results checked. With these steps, anyone can easily convert meters to feet.


Originally from England, Clare has over 10 years’ experience working in education as a teacher, tutor, university lecturer and researcher in various STEM subjects including math, physics, robotics, environmental science, citizenship, engineering, urban environments, and renewable energy. Clare's academic research has included work on learning processes and motivations as well as common barriers and challenges for learners. Through her research into how children learn, Clare found herself drawn to alternative models of education that offer more research-backed approaches, such as Acton Academy. She has a Masters degree in Engineering from Cambridge University, a PhD in Renewable Energy (Social Science / Engineering combined) and a UK teaching qualification, specializing in math. Outside of academics, Clare has brought her passion for teaching to other pursuits including teaching improvised comedy, music and coaching women's soccer.
Clare believes that math should be relevant, hands-on and fun, focusing on problem-solving and collaboration. At AASV, Clare facilitates immersive math workshops that give learners the opportunity to translate story-based, real-world challenges into Mathematical problems.


LITTLE ROCK, Ark — A lot of us are spending more time than ever at home. So why not use that time to invest in your biggest investment?
Last week, we worked on updating old counter tops and cabinets. This week, we are adding the wow factor to a room for less than $50!
“My tools are already out, right?” Rob Evans said.
So why not prove we’re Pinterest pros? Make a statement wall.
With seven 1x3’s and one 1x4. First, mark your studs.
Then outline where your boards will go.
“For all of you doing math homework with the kids? You’ll be doing that same math if you build this wall.”
Rob didn’t pay attention in math class apparently, “I messed up,” he said, trying to figure out why the dimensions and his math didn’t add up. After he figured it out, he picked out the boards.
“This is MDF board. It’s very soft.”
Using a 15-gauge nail gun and adhesive, he attached the boards to the wall. Added wood putty to cover the holes, caulk to merge the gaps, and sanded down the wood putty.
Then he repaired damaged wood trim from a dog attack with wood putty, and decided to repaint all the crown molding and baseboards a different shade of white (Natural Choice).
Then on went the new color (Venus Teal) on the walls, and for $35 worth of boards, a gallon of paint, about a day’s worth of labor, you’ve got your wow factor.
And it’s on to the next room.


If you realize that you are struggling with a issue, try a number of these tips. In the majority of activities, a incorrect answer is accepted in just the exact same way as a perfect one. If you don’t wish to go all of the way, you may still partially derive me.
The standards are made to make sure that Indiana students are ready to enter and successfully complete postsecondary education, and that they’re ready for long-term, economically-viable career opportunities. General Awareness To ace this section you have to read extensively and take note of what is happening within this time. NCERT Solutions class 11 maths provided here have been collected by our subject experts who have plenty of experience in the area.
If you’re in the math education world I wonder in the event you too have sensed a developing angst concerning the content of what constitutes school mathematics. Ensuring your children succeed in school is crucially important. It’s highly important that students consult with their advisor to make sure that they are following the correct mathematics path required for their chosen degree.
In reality, forget you even understand what a number is. To make certain it works, we will need to model how diseases spread. REM sleep, specifically, is important to your capacity to form connections between unrelated ideas.
Elements of Mathematics and Elements of Mathematics – The Perfect Combination
Check out whether you’re able to turn the page with some arrow keys or click a specific part of the screen, aside from using the mouse to deal with everything. Many logic or puzzle games require that you work a problem from several sides and have numerous methods of locating the solution. Let’s try yet another example to be sure we understand the way to use the area formula for parallelograms.
How to Choose Elements of Mathematics
A mathematical model supplies an answer. NCERT solutions supply you with a opportunity to check your expertise and improve in weak places. Throughout your studies, you’re going to be immersed in a diverse number of quantitative topics.
Last, technology is driving a shift in that which we can, and so ought to do in the mathematical classroom, NOW! Concentrate on the facets of life that you may control instead of focusing on what you are unable to control. Thus the solutions offer you comprehensive learning and also helps students develop much better math skills together with logical and reasoning abilities.
The Bad Side of Elements of Mathematics
One particular special sort of polygons is referred to as a parallelogram. Russell has an issue with superimposing both triangles. The transpose of a vector is, thus, a matrix with just one row.
Many tools are also studied to compute the several elements. In the procedure for rehashing familiar concepts, you might trick your brain into forming the comprehensive solution. It is not hard to think about the earth as being solid dirt through and through, but in fact it’s composed of several layers.
It’s experienced in the social context in addition to in academia. Put simply, innovation demands teamwork. Because of the simulative character of remediation, each illustration of remediation includes a perspective on narrative.
This program is beautifully structured to secure you to comprehend what’s really going on with matrices and makes it possible to become very acquainted with the terminology for linear algebra. German mathematician G. Cantor introduced the idea of sets. This concept of proof still dominates pure mathematics in today’s world.
What You Don’t Know About Elements of Mathematics
So let’s connect the 3 elements. It really just is based on the set! Given an unsorted variety of positive integers.
Facts, Fiction and Elements of Mathematics
A fantastic benefit is that you can finish this section in no moment. To understand that the important point is true requires a small argument. Calculating the area, nevertheless, is somewhat tougher.
How to Get Started with Elements of Mathematics?
Some lessons offer detailed instructions or recommendations but it’s important to say that the lessons aren’t scripts but rather they ought to be viewed as guides so the reader can imagine how classroom instruction could look. Regrettably, it’s quite possible that the price of publishing Byrne’s book proved to be a important reason for the firm’s bankruptcy in 1853. Many studies have proven that the action of writing kinds of cement not merely new ideas into your brain, but in addition generates new connections and creative ideas.
There is a particular quality to every number. The quantity of choice increases each year and following Year 1 you can pick the range of choices you study. Modern-day experiments with computer music are just the latest example.
In the event the side of the parallelogram include not just the segment, but a prolongation, that part that’s developed on the extension is known as the excess. The base is just the period of the side on the ground,” or at the base of the figure. Therefore, the diameter of a circle is twice provided that the radius.
Dymocks Online will do their very best to guarantee the information you’ve got input is accurate. You should find out how to fix complicated problems without following a particular set of actions you’ve memorized. Let’s take this up a level.


SAT scores are weird. You get a number, ending in a zero, on a scale of 200 to 800, twice: one for reading & writing, one for math. You get a total score between 400 and 1600...except, of course, for those years when the writing was separate and you got somewhere between 600 and 2400. You're allowed to take the test multiple times and combine your highest reading & writing score with your highest math score, giving you a "superscore" that's higher than the total scores you got any of the individual times you took the test.
And then what? What does that number even mean?


f(n+2) = f(n+1) + f(n)
where n begins with 1, f(1) = 1, and f(2) =1
The series is named for Leonardo Fibonacci, an Italian mathematician who first posed a mathematical problem (based on reproduction patterns in rabbits) the answer to which is the sequence that would later bear his name. (See, generally, wikipedia.org/wiki/Fibonacci.) Fibonacci lived from around 1170 to 1250, and he is widely considered to be the greatest western mathematician of the Middle Ages. He haled from Pisa -- hence his nickname: Leonardo of Pisa.
Well, the Fibonacci Sequence is absolutely everywhere. It proves, definitively, that if God is not a lawyer, s/he is a mathematician. I need one quick detour to back up that conclusion. It concerns the "Golden Ratio."
If you take two consecutive Fibonacci numbers and divide the later one by the earlier one, a pattern quickly emerges:
1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.666666....
8/5 = 1.6
13/8 = 1.625
21/13 = 1.615384...
What you see is that the ratios approach a special number called the Golden Ratio, which is approximately 1.618. (The actual number is 1 plus the square root of 5 where the entire sum is divided by 2.) A rectangle where the ratio of length to width is 1.618 is considered to be most pleasing to the eye and is called a "golden rectangle." The Greeks knew this, so the Parthenon is a "golden" rectangle. Da Vinci knew this, so he used golden rectangles everywhere including in the Mona Lisa. The same goes for other famous works of art that are too numerous to mention here.
We find the golden ratio and Fibonacci numbers in music too, most famously in the work of Bela Bartok.
And nature. The spiral of the chambered nautilus spins out in a path that tracks the golden ratio and the Fibonacci Sequence. (See, e.g., 2muchfun.info/nautilusshell.html.) Likewise for the spirals on pine cones and on the faces of sunflowers. (See, e.g., io9.com/5985588/15-uncanny-examples-of-the-golden-ratio-in-nature.) Reproductive patterns in some species -- like bees -- follow the Fibonacci Sequence. (Id.) Even the patterns of spots on Dalmatians are organized according to Fibonacci numbers. (OK, I totally made that last one up; just wanted to see if anyone was still reading.)
But, seriously, other examples from the arts, nature, aesthetics, sex, computer science, physics, and other branches of mathematics abound. I could go on and on. And I did -- this was a project for three straight years in high school, and even in one class during college.
So, the point here is that Pisa remains proud of its greatest mathematical son. In the Camposanto, which is one of the 4 buildings that constitute the Campo dei Miracoli, stands a statue of Fibonacci. Seeing it live was a thrill for me and I'm sure it would be the same for you. The crowd around the statue of Fib was not quite as huge as the line for the Leaning Tower. But it was close.
There is actually more. According to a website I read on the train ride up to Pisa, as well as the map on my iPhone, there is a street in Pisa named after Fibonacci. It's a little bit out of the way and completely on the other side of town from the Leaning Tower. But we trekked over there. I mean, how many times will you get to see the intersection of 2 streets where one is named after Galileo and one is named after Fibonacci?!?!? The only problem is that the street sign for Rue de Fibonacci was missing. It was cute to see Galileo Avenue though.
|This is where the street sign for Leonardo Fibonacci is supposed to go.|


How Do Casinos EARN MONEY?
Many people that play casino games have no idea just how much there’s to learn. In fact, they may not even understand how a simple strategy for a certain kind of casino game works. Learning several basic strategies can make all the difference in whether you earn at gambling.
For instance, in roulette there are the quantity of winning cards, the minimum and maximum bet, and the payout odds. Casinos use all of these factors to look for the odds that a person has of winning when they place their bets on a specific casino game. The payout chances determine whether or not you will walk away successful or if you will lose out. Knowing precisely how these odds are figured out can help you decide what odds to bet on when you place bets on casino game titles.
One of the most important considerations for gamblers may be the standard deviation. A standard deviation is an estimate of the value of the random variable, in this 마카오 갤럭시 카지노 슬롯 the casino video game table. The standard deviation is simply the difference between your expected and your actual outcome. When discussing standard deviation, you should talk regarding tails and heads. A standard binomial distribution uses a binomial distribution with mean and normal deviation values, which makes things a little easier to recognize.
Standard deviation is not the only thing that helps gamblers choose what chances to bet on casino games. Another consideration may be the simple game concept, which states that casino games are based on random prospect and that any player’s likelihood of winning are exactly the same irrespective of where they sit at the table. House advantage refers to the difference between the probability of a person player at any given desk against the probability of all players at that desk, and this can cause gamblers to vary their bets depending on which factors give them an edge.
Residence advantage could make the difference between a win and a loss. Because of this the same amount of people at a table could have the same odds of hitting on a jackpot but someone will always win more than they lose. One example of using random number games to greatly help gamblers decide what chances to bet on casino games would be the number of people at a casino who has ever won or lost, a single card game. These people can then be used to greatly help gamblers figure out if casino games have a larger house advantage than other games. For instance, someone at a progressive slot machine game may have the chances of hitting a crimson jackpot one in three million, but this person has never won a single red or black jackpot in their entire lifetime. Using these folks to help with the mathematics can considerably improve gamblers chances of striking the jackpot.
The ultimate area of the casino game may be the wheel. The wheel takes the winning amounts from the basic roulette system and then places them in specific jobs on the wheel. These opportunities will always change because the odds of all the different combinations that come off the wheel are never exactly the same. If a casino were to offer the customer the choice of having the steering wheel reset to any placement they desired them to place a bet on, it might very well end up changing the odds dramatically. This is due to the casino can change the odds depending on how much cash is in the pot, how many players are left in the overall game and any other number of aspects.
Roulette, slots and poker game titles all have random elements that make them unique. Due to this they are able to cope with roulette, slots and poker by the use of a wheel. In order to know how gambling machines and roulette do the job, you must know how the odds of every combination is figured out. To carry out that, you should look at a few of the math engaged.
Just about all casinos have a standard deviation that is necessary when they calculate the odds of the various outcomes. Basically, it is the standard deviation that is the deviation of the average outcome that occurs throughout a random walk from one value to another. When a casino uses normal deviation in calculating their odds, they are basically determining what is normal behavior. Standard deviation is used in many various kinds of gambling games including slot machines, roulette and blackjack among numerous others. It is the one factor that makes all these games what they are and just why they continue being popular among people even with all these a long time.


Hello parents and students,
Today we started math 🙂 This year we will be using the Jump Math book(s) for most of our math program. Since the books are not yet at school, I gave the students a photocopy for today. Please take the opportunity to help/review with your child the concepts discussed/learned in class. Also, it is always a good habit to check over math homework at home (even if competed in class) just to make sure things are going well. Thanks!
Keep all of the forms coming to me… I am happy with the progress so far… way to go parents!!! The students loved to hear me say that you have homework too 🙂
Today I sent home September scholastic book orders. This is an optional book ordering program but from the sales we get coupons to use for the class. September Scholastic orders will be due no later than Tuesday, September 19.
Have a wonderful evening!
Math – Jump Math Package #1 p. 1-4 – Thursday
Study for Spelling Test #1 – Friday
School/Jiu Jitsu/Agenda Forms and # – ASAP


MATH 155 – Introduction to Statistical Analysis
An introductory statistics course with an emphasis on multivariate modeling. Topics include descriptive statistics, experiment and study design, probability, hypothesis testing, multivariate regression, single and multi-way analysis of variance, logistic regression.
MATH 135 – Applied Multivariable Calculus I
This course focuses on calculus useful for applied work in the natural and social sciences. There is a strong emphasis on developing scientific computing and mathematical modeling skills. The topics include functions as models of data, differential calculus of functions of one and several variables, integration, differential equations, and estimation techniques. Case studies are drawn from varied areas, including biology, chemistry, economics, and physics.
MATH 136 – Discrete Mathematics
Discrete mathematics studies collections of distinct, separate objects and is complementary to calculus (which studies continuous phenomena). This course introduces techniques for analyzing arrangements of objects and the relationships between them. The material emphasizes problem solving and logical argumentation. Topics include basic counting principles, induction, logic, recurrence relations, number theory, and graph theory.
MATH 236 – Linear Algebra
This course covers a central point of contact between mathematics and computer science. Many of the computational techniques important in science, commerce, and statistics are based on concepts from linear algebra: subspaces, projection, matrix decompositions, etc. The course reviews these concepts, adopts them to large scales, and applies them in the core techniques of scientific computing; solving systems of linear and nonlinear equations, approximation and statistical function estimation, optimization, interpolation, Monte Carlo techniques. Applications throughout the sciences and statistics.
MATH 237 – Multivariable Calculus
This course focuses on calculus useful for the mathematical and physical sciences. Topics include: scalar and vector-valued functions and derivatives; parameterization and integration over regions, curves, and surfaces; the divergence theorem; and Taylor series. Attention is given to both symbolic and numerical computing. Applications drawn from the natural sciences, probability, and other areas of mathematics.
PHYS 194 – Nanotechnology
Nanoscience is the emerging field of science concerned with the control of matter on the atomic and molecular scale. This interdisciplinary field sits at the convergence of Physics, Chemistry, Biology, Materials Science and Electrical Engineering. Our course will introduce science at the nanometer length scale, the fabrication of nano-scale systems and some of their technological applications.
COMP 124 – Object-Oriented Programming and Data Structures
This course introduces the principles of software design and development using the object-oriented paradigm (OOP) and the Java programming language. Students will learn to use data structures such as lists, trees and hash tables and they will compare the efficiency of these data structures for a particular application. Students will learn to decompose a project using OOP principles. They will work with integrated development environments (IDEs) and version control systems. Students will practice their skills by creating applications in areas such as graphics, games, simulations, and natural language processing.
COMP 365 – Computational Linear Algebra
A mix of applied linear algebra and numerical analysis, this course covers a central point of contact between mathematics and computer science. Many of the computational techniques important in science, commerce, and statistics are based on concepts from linear algebra, such as subspaces, projections, and matrix decompositions. The course reviews these concepts, adopts them to large scales, and applies them in the core techniques of scientific computing. These include solving systems of linear and nonlinear equations, approximation and statistical function estimation, optimization, interpolation, eigenvalue and singular value decompositions, and compression. Applications throughout the natural sciences, social sciences, statistics, and computer science
POLI 120 – International Politics
This course has three broad goals. The first is to develop the foundational knowledge and conceptual literacy necessary to engage with International Relations’ multidimensional concerns. These include issues such as world order, power, hierarchy, political violence, international law, development, religion, human rights, gender, humanitarianism and international organizations (such as the United Nations). The second is to introduce students to the different perspectives or intellectual frameworks for making sense of international relations (also known as global or world politics), including realist, liberal, constructivist, historical materialist, postcolonial and feminist approaches. The third is to encourage students to reflect on some of the ethical issues inherent in both the study and practice of international politics. Emphasis will also be placed on developing a range of critical, analytical, research and writing skills required for the further study of international politics. The course is thus intended to prepare students for advanced work in the field, although it is also appropriate for those merely seeking to satisfy an interest in the study of global politics.
POLI 222 – Regional Conflict and Security in the Middle East
This course is intended to introduce students to the military, political, economic, cultural and/or diplomatic dimensions of various regional conflicts or “security complexes.” The specific region to be covered is the Middle East.
POLI 202 – Political Participation: Politics and Mathematics of Elections
This course focuses on the various ways in which mathematics and political science interact. Topics covered will include the role of elections and representative government in the United States, comparison of electoral systems used around the world, the apportionment problem, redistricting and gerrymandering, weighted voting systems and voting power, the costs and benefits associated with political participation, and predicting electoral outcomes.
POLI 216 – Legislative Politics
This course explores legislative politics through a combination of academic theory and focused field experiences. Each student must simultaneously enroll in a credit-bearing internship at the Minnesota State Legislature. The class examines the basic structures, players and forces that shape legislative decision-making, the motivation or individual legislators, and their interactions with other political actors and institutions.
POLI 394 – Policymaking in the 4th Branch
This course examines the role that members of the federal bureaucracy, like Secretary Clinton, play in setting policy agendas, writing rules and regulations that have the force of law, implementing laws on the books, enforcing compliance with existing laws, and expanding opportunities for Americans to participate in governance.
ART 1401 – Drawing I
This course introduces basic drawing concepts such as line, value, gesture, proportion, composition, and space; and techniques using traditional and contemporary drawing media. A variety of subjects from still life, architectural forms, nature and the human figure are used as inspiration for the student’s drawings. Students will also be introduced to the art of important artists who have used drawing successfully in their work.
HISP 111 – Accelerated Elementary Portuguese
Intensive instruction in speaking, understanding, reading and writing Portuguese. Brazilian usage emphasized.
Wayzata High School
AP Computer Science
An introductory Java course in computer science which required me to: design and implement solutions to problems by writing, running, and debugging computer programs; develop and select appropriate algorithms and data structures to solve problems.
Introductory course to C++ which included creating variables, methods, and classes, and using Eclipse and Visual Studio. · Software Engineering Course topics included creating apps for mobile devices using Android Studio, automating tasks in a variety of languages – including Python (using Canopy), finding patterns in data, interpreting simulations, and weighing the ethical and societal issues of technology and data storage.


Meet the Math Department
Mr. Gregery Holland, 7th Grade Math Teacher; Math Department Chair
Mr. Holland has taught for 17 years at the elementary and middle school level. He teaches Math Transitions, is the Assistant Golf Coach, and the current Math Department Chair. Mr. Holland received his B. S. and M. A. in elementary education from the University of New Mexico.
Ms. Audra Garcia, 6th Grade Math Teacher
Mrs. Garcia absolutely loves being a Coyote! Her favorite thing about her job is using her passion for math to inspire the next generation. She teaches the 6th grade Math Foundations class. When she’s not busy teaching her favorite subject, she enjoys playing board games, spending time with her friends and family, and hanging out with her two corgis, Whiskey and Suki.
Mr. David Meidinger, Algebra 1 and Algebra 2 Teacher
David taught all levels of math in Arizona for 27 years prior to moving to Albuquerque. He moved here a year ago with his wife, Laura, of (now) 32 years to pastor a small Christian congregation. Now in his second year at Cottonwood, David teaches Algebra I and Advanced Algebra.
Mrs. Ashley Weaver, Algebra 1 Teacher and Communications Officer
Mr. Jordan VanLief, Geometry Teacher
Jordan received his Bachelor of Science in Physics from the University of New Mexico, and his Alternative Teacher Licensure Certification through Central New Mexico Community College. Formerly, he worked in software engineering and non-destructive testing, but ultimately decided that his career interests were in education. Currently, he teaches all 9th grade geometry classes at CCPS. In his free time he enjoys spending time with his wife and family, playing competitive video games, playing guitar, exercising, and reading about math and philosophy.
Ms. Patricia Spitz, Algebra 2 Teacher and Testing Coordinator
Mrs. Spitz has been a mathematics teacher at Cottonwood since 2011. She teaches Advanced Algebra 2. Mrs. Spitz has a Master’s Degree in Educational Administration. She is certified in Secondary Mathematics, Language Arts, Social Studies and Health. She also holds a Level 3B Administrative License. She has also trained in current IB Math and was an IB Math instructor for several years. She currently serves as the Testing Coordinator and is in charge of all testing. Her favorite quote from the Dalai Lama is “Be kind whenever possible. It is always possible.”
Ms. Quilimaca Padilla, IB Math Teacher and BYU Credit Recovery Facilitator
Quilimaca has been teaching mathematics since 2005. She has her Level III teaching license and endorsed in Secondary Mathematics. Also, endorsed in Modern and Classical Languages primarily Spanish. She enjoys spending time with her family. She likes to run on her spare time and play sand volleyball once a week with her teammates. Her current goal right now is to be a National Board Certified teacher in Mathematics within the next 5 years. She really enjoys teaching IB Math: Applications and Interpretation Year 1 and Year 2 courses which is 11th and 12 grade level. Her advisory class is Year 1 and they are beginning on working on the CAS experiences along with their interviews. In her teaching she uses a variety of teaching strategies, for example, calculator, collaboration with their peers, challenging tasks, real life application problems, small groups, seminars, guided notes, and practice in and out of class. In Year 1, the students work on Topic 1 through Topic 4 IB content with practicing their math internal assessments throughout the year. In Year 2, students finish up Topic 5 content and work on writing their math internal assessment. Ms. Padilla is also a BYU credit recovery facilitator to the school. She assists students in enrolling them to the course they need to recover credits for class(es).
Mr. John Tennison, IB Math Teacher
John is in his 5th year teaching at CCPS and he loves math! He’s all about puzzles, games, and solving problems. This year he is teaching Analysis and Approaches years 1 and 2 along with Advanced Topics. If you have a math question, he is your guy. In his free time he likes running and watching movies. Feel free to come nerd out with him anytime!


Always happy to help, the White Rose Maths Team has prepared a series of five maths lessons for each year group from Year 1-6. We will be adding five more each week for the next few weeks. Every lesson comes with a short video showing you clearly and simply how to help your child to complete the activity successfully. Our pupils follow the White Rose Planning when they are in school, so please continue to use the resources when learning from home.
You can either print the sheets out, or get the children to follow the lessons on screen and write their answers in the books sent home as part of their home working pack.
Year 1 Week 1: https://whiterosemaths.com/homelearning/year-1/


Rate Group Test Worksheet Report
Normal and MVAL EBARS
The normal accrual rate for an employee for the plan year is the increase in the employee’s accrued benefit (within the meaning of section 411(a)(7)(A)(i) during the measurement period.
The most valuable optional form of benefit used to determine the most valuable accrual rate reflects the value of all Optional Forms of Benefits accrued or treated as accrued that are payable in any form and at any time under the plan.
What is a Rate Group?
A Rate Group consists of one HCE and all other employees whose Normal and MVAL EBARS are equal to or greater than the EBARS of the HCE.
In order to determine whether a plan satisfies the general test, the plan is broken down into rate groups, or “mini plans”. Each HCE who receives an allocation or an accrual rate forms a rate group. Every other participant who has an equal or greater allocation rate or accrual rate than the HCE is a member of that rate group.
In our example, the first Rate group consists of Brian HCE3 only because there are no other employees with a Normal and MVAL EBARS greater than Brian.
Rate Group 2 consists of Kyle HCE2 and Brian HCE3. That’s because Brian’s Normal and MVAL EBARS are greater than Kyle.
Rate Group 3 consists of Chad HCE1 and Jamie, Kyle and Brian.
To simplify things, this report, is sorted by both Normal and MVAL EBARS in descending order.
Now, it should be easy to see that Rate Group 4 consists of Rick HCE4 and all the employees above Rate Group 4.
Finally, Rate Group 5 consists of Frank HCE5 and all the other employees.
What is a Ratio Percentage?
A Ratio Percentage must be calculated for each Rate Group. This calculation is done in three steps:
Step 1: Count the number of NHCE's in each Rate Group and divide this by the total NHCE's in the Plan.
Step 2: Count the number of HCE's in the Rate Group and divide this by the total HCE's in the Plan.
Step 3: The Ratio Percentage for a Rate Group is Step 1 divided by Step 2.
Look at Rate Group 3 in the example below:
The Ratio Percentage of Rate Group 3 is 1/3 (one out of 3 NHCE's) divided by 3/5 (3 HCE's out of 5 HCE's) equals 55.56%.
If the Ratio Percentage for all Rate Groups is 70% or more, the plan automatically passes the Rate Group test.
In this example, our Ratio Percentages are all lower than 70%.
However, if the Plan passes the Average Benefit Percentage Test, the Ratio Percentages must only be greater than the Mid-Point Percentage of 45% to pass.
But since Rate Group 1 and Rate Group 2 are both zero, we fail.
The worksheet below, shows that we only need 1 NHCE to have Normal & MVAL EBAR greater than Brian & Kyle’s EBARS to pass. You should then go to the Classes and increase the Profit sharing for the NHCE's until it passes the Rate Group Test.


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How do you calculate a quadratic equation.
- To solve a quadratic equation, use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
What is the quadratic formula?
- The quadratic formula gives solutions to the quadratic equation ax^2+bx+c=0 and is written in the form of x = (-b ± √(b^2 - 4ac)) / (2a)
Does any quadratic equation have two solutions?
- There can be 0, 1 or 2 solutions to a quadratic equation. If the discriminant is positive there are two solutions, if negative there is no solution, if equlas 0 there is 1 solution.
What is quadratic equation in math?
- In math, a quadratic equation is a second-order polynomial equation in a single variable. It is written in the form: ax^2 + bx + c = 0 where x is the variable, and a, b, and c are constants, a ≠ 0.
How do you know if a quadratic equation has two solutions?
- A quadratic equation has two solutions if the discriminant b^2 - 4ac is positive.
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This online calculator is a quadratic equation solver that will solve a second-order polynomial equation such as ax 2 + bx + c = 0 for x, where a ≠ 0, using the quadratic formula .
The calculator solution will show work using the quadratic formula to solve the entered equation for real and complex roots. Calculator determines whether the discriminant \( (b^2 - 4ac) \) is less than, greater than or equal to 0.
When \( b^2 - 4ac = 0 \) there is one real root.
When \( b^2 - 4ac > 0 \) there are two real roots.
When \( b^2 - 4ac < 0 \) there are two complex roots.
The quadratic formula
is used to solve quadratic equations where a ≠ 0 (polynomials with an order of 2)
Examples using the quadratic formula
Example 1: Find the Solution for \( x^2 + -8x + 5 = 0 \), where a = 1, b = -8 and c = 5, using the Quadratic Formula.
The discriminant \( b^2 - 4ac > 0 \) so, there are two real roots.
Simplify the Radical:
Simplify fractions and/or signs:
Example 2: Find the Solution for \( 5x^2 + 20x + 32 = 0 \), where a = 5, b = 20 and c = 32, using the Quadratic Formula.
The discriminant \( b^2 - 4ac < 0 \) so, there are two complex roots.
calculator updated to include full solution for real and complex roots
Cite this content, page or calculator as:
Furey, Edward " Quadratic Formula Calculator " at https://www.calculatorsoup.com/calculators/algebra/quadratic-formula-calculator.php from CalculatorSoup, https://www.calculatorsoup.com - Online Calculators
Quadratic Equation Solver
Is it quadratic.
Only if it can be put in the form ax 2 + bx + c = 0 , and a is not zero .
The name comes from "quad" meaning square, So, the biggest clue is that highest power must be a square (in other words x 2 ).
These are all quadratic equations in disguise:
How does this work?
The solution(s) to a quadratic equation can be calculated using the Quadratic Formula :
The "±" means you need to do a plus AND a minus, so there are normally TWO solutions !
The blue part ( b 2 - 4ac ) is called the "discriminant", because it can "discriminate" between the possible types of answer. If it is positive, you will get two normal solutions, if it is zero you get just ONE solution, and if it is negative you get imaginary solutions.
Quadratic Formula Calculator
Enter the equation you want to solve using the quadratic formula.
The Quadratic Formula Calculator finds solutions to quadratic equations with real coefficients. For equations with real solutions, you can use the graphing tool to visualize the solutions.
Quadratic Formula : x = − b ± b 2 − 4 a c 2 a
Click the blue arrow to submit. Choose "Solve Using the Quadratic Formula" from the topic selector and click to see the result in our Algebra Calculator !
Solve Using the Quadratic Formula Apply the Quadratic Formula
Solve Using the Quadratic Formula x 2 + 5 x + 6 = 0 Solve Using the Quadratic Formula x 2 - 9 = 0 Solve Using the Quadratic Formula 5 x 2 - 7 x - 3 = 0 Apply the Quadratic Formula x 2 - 14 x + 49 Apply the Quadratic Formula x 2 - 18 x - 4
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Quadratic Formula Calculator
The calculator below solves the quadratic equation of ax 2 + bx + c = 0 .
In algebra, a quadratic equation is any polynomial equation of the second degree with the following form:
ax 2 + bx + c = 0
where x is an unknown, a is referred to as the quadratic coefficient, b the linear coefficient, and c the constant. The numerals a , b , and c are coefficients of the equation, and they represent known numbers. For example, a cannot be 0, or the equation would be linear rather than quadratic. A quadratic equation can be solved in multiple ways, including factoring, using the quadratic formula, completing the square, or graphing. Only the use of the quadratic formula, as well as the basics of completing the square, will be discussed here (since the derivation of the formula involves completing the square). Below is the quadratic formula, as well as its derivation.
Derivation of the Quadratic Formula
From this point, it is possible to complete the square using the relationship that:
x 2 + bx + c = (x - h) 2 + k
Continuing the derivation using this relationship:
Recall that the ± exists as a function of computing a square root, making both positive and negative roots solutions of the quadratic equation. The x values found through the quadratic formula are roots of the quadratic equation that represent the x values where any parabola crosses the x-axis. Furthermore, the quadratic formula also provides the axis of symmetry of the parabola. This is demonstrated by the graph provided below. Note that the quadratic formula actually has many real-world applications, such as calculating areas, projectile trajectories, and speed, among others.
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Quadratic Formula Calculator & Solver
Calculator for solutions to any quadratic equation..
The calculator uses the quadratic formula to find solutions to any quadratic equation .
The quadratic formula calculator below will solve any quadratic equation that you type in. Simply type in a number for 'a', 'b' and 'c' then hit the 'solve' button.
Quadratic Formula Calculator and Solver
The Actual Solutions
Parabola Animated Gifs
More Quadratic Gifs
The calculator on this page shows how the quadratic formula operates, but if you have access to a graphing calculator you should be able to solve quadratic equations, even ones with imaginary solutions.
- Step 1) Most graphing calculators like the TI- 83 and others allow you to set the "Mode" to "a + bi" (Just click on 'mode' and select 'a+bi').
- If you can set your calculator's mode to a + bi you should be able to even calculate imaginary solutions.
- The rest of the steps just involve typing in a,b and c. Make sure that you divide the entire numerator by 2a, just use parentheses.
- Quadratic formula worksheets (several free printable pdfs with answer keys on the quadratic formula)
Create the graph of any Parabola
Save Graph as Image to your desktop
- Quadratic Equations
- Other Free Math Calculators
- All roots calculated
Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there!
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Shows you the step-by-step solutions using the quadratic formula! ... Solve an equation of the form a x 2 + b x + c = 0 by using the quadratic formula:.
Quadratic Equation Solver · Step-By-Step Example. Learn step-by-step how to solve quadratic equations! · Example (Click to try) · Choose Your Method. There are
To solve a quadratic equation, use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a). What is the quadratic formula? The quadratic formula gives solutions
This online calculator is a quadratic equation solver that will solve a second-order polynomial equation such as ax2 + bx + c = 0 for x, where a ≠ 0, using the
Quadratic Equation Solver. We can help you solve an equation of the form "ax2 + bx + c = 0" Just enter the values of a, b and c below:.
Quadratic Equation Solver. If you have an equation of the form "ax2 + bx + c = 0", we can solve
Enter the equation you want to solve using the quadratic formula. The Quadratic Formula Calculator finds solutions to quadratic equations with real coefficients
A quadratic equation can be solved in multiple ways, including factoring, using the quadratic formula, completing the square, or graphing. Only the use of the
The calculator uses the quadratic formula to find solutions to any quadratic equation. ... The quadratic formula calculator below will solve any quadratic
Inuyasha The Final Act (Subbed) · ALEKS - Drawing the Haworth projection of a ketose from its Fisher projection · Solving Linear Equations Using


In Wednesdayâ€™s class we began by pairing up and experimenting with long ropes in order to visualize the vibration of a string. One person stood holding their end in front of them while the opposite person swung the rope at different speeds. We attempted to create, at first, one broad swing of the rope (like you might see in a game of jump-rope). Then we doubled the speed so that the rope was divided into two equal parts, each rotating conversely (while one side swung upwards, the other rotated downwards). This increase in speed was continued until it wasn’t possible to divide the rope into any smaller sections (usually occuring around five divisions of the rope). Each dividing point between rotating sections is considered a ‘node’, or a place where the vibration is zero.
We then gathered into a circle in the classroom and used a monochord (an instrument consisting of a single string) to discover the specific ratios that create each interval above the tonic pitch. We began by splitting the chord in half (done by lightly touching in the center of the vibrating string) so that each section of the string was vibrating at twice its original speed. This is the same as what we had just experienced with the rope when we doubled our initial speed in order to create two vibrating sections. This time with the string of the monochord, an octave occured above the original pitch (shown by the ratio 2:1, where the higher pitch is vibrating two times for each one vibration in the lower note). We continued to use this same method to achieve the 5th (ratio of 3:2), the 4th (4:3), and so on through each of the twelve intervals. We discussed that frequency ratios always come in pairs that add up to an octave. For instance, the ratio 3:2 will be paired with the ratio 4:3 (a 5th plus a 4th equaling an octave).
The class reminded me of Stuart Isacoffâ€™s book â€œTemperamentâ€ which addresses the history, problems, and evolution of tempering the Western scale. After the class, I went back and read the section concerning Pythagoras and his original discovery of the geometry of music. Pythagoras, who invented the monochord, stated that â€œmusicâ€™s rules are simply the geometry governing things in motion: not only vibrating strings but also celestial bodies and the human soul.â€ Pythagoras believed that the most pleasing of harmonies arose from the simplest of proportions and that complexity would insight chaos. What is fascinating about this is that behind his discoveries of pure musical geometry there lies a forbidden and volatile darkness. He found that pure octaves and fifths, according to his ratios, are incommensurate (also referred to in Greek as â€˜alogonâ€™ meaning â€˜the unutterableâ€™). Fifths will never complete a perfect circle (as suggested by the widely accepted circle-of-fifths), but will reach toward infinity in an unending spiral. This essentially boils down to the fact that octaves are based upon multiples of 2 (2:1) while fifths are based upon multiples of 3 (3:2). In this case, no multiple of 2 will ever meet a multiple of 3. If one were to compare the pitch achieved by an octave and that achieved from the completion of a circle of fifths, they would be very similar yet â€œout of tuneâ€. This spiraling phenomenon hints at a more complex mathematic sequence, that of the golden ratio. Even so, these simple ratios were believed to be an expression of the divine. It is easy to find similar ratios present within nature. Saint Augustine, in fact, believed that churches and cathedrals were to be more than just shrines, and instructed that proper proportions were to be used in their construction. Thus the heights, lengths, and depths of the structures formed the proportions of Pythagorasâ€™s â€œcelestial harmoniesâ€ (1:1, 2:1, 2:3, and 3:4).
So what difference does this make to us, as musicians and as people? What effect does this really have on our performance? I think it is crucial to understand the fundamentals of the creation of sound, of pitch, especially when such things are taken for granted everyday. I remember the feeling I had when I first discovered the ratios involved in music. Once I got past the initial migraine acquired from my first lecture on equal temperament, I began to look a bit into proportions. It made perfect sense (and also supplied an interesting and practical perspective to my high school math classes). This is the real foundation of what I do every day, of each note I play. It is a fundamental that comes before technique, before fingerings and musicality. In a sense it is the DNA of music (more specifically of pitch). Yet as crucial as these fundamentals are, an understanding of them is not essential for the enjoyment of music. Recently, Warren mentioned a workshop that he was conducting years ago. During the course of the class, he plucked two notes on a string, the second a fifth higher than the first. Soon after, a young boy came running into the room exclaiming “What was that beautiful music?!”. Like the young boy, a single, simple fifth can produce a level of joy bordering on ecstasy. Warren also noted that infants are particularly drawn to simple intervals. This has been quite a meal for my thoughts (even just thinking back to our class sends my head spinning!). Every time I try to find a solution to these musical systems I find that I develop more and more questions. It is truely amazing how much chaos lies within order!


estimating where the where the movement will eventually reach. We could do this with other numbers in the Fibonacci sequence as well. Important note: in this Fibonacci tutorial.618 extension level (as well.000,.618 levels) are calculated in relation to the point B, which means that B point represents a 0 extension. The most common.618 extension level, but when the price shows good potential to reach next.000 or even.618 level, you can leave your trade to get that target too. Fibonacci Extension Levels are:.618,.000,.618 three the most important levels. Because of all the people who use the Fibonacci tool, those levels become self-fulfilling support and resistance levels. For example, 34 divided by 55 equals.618. Divide by 2nd following Divide by 2nd preceding Divide by 3rd following Divide by 3rd preceding.4 5.5 2.236 As you can see, we could get many different numbers by just taking numbers within the Fibonacci sequence and developing a divisory pattern. Highest and lowest swings in the chart formation (A, B points). Dont worry, well explain retracements, extensions, and most importantly, how to grab some pips using the Fibonacci tool in the following lessons.
By the way, all Forex brokers, the list of which can be found here: m/ will always have Fibonacci tool available within bitcoin news rss their trading platforms. Thus moving further from B would be 1xx. Downtrend, now, lets see how we would use the. However, there are two things that jump to the forefront of our discussion in todays world. The retracement levels are.3933 (23.6.3983 (38.2.4023 (50.0.4064 (61.8) and.4114 (76.4). As you can see, we found our Swing High.4195 on January 25 and our Swing Low.3854 a few days later on February.


Virat Kohli Reviews on getting 3/100 in mathematics in school days
The Indian skipper Virat Kohli recently revealed that he was never very studious or “smarter” in his school days, but he always managed to understand things quickly. However, one topic that always kept Kohli alert was mathematics. By his admission, Kohli has never worked hard even for cricket, as he did to pass math on his class 10 exams.
Kohli about Mathematics in School
“So in math, we used to have exams and the maximum grades we could get at 100, right. I used to get 3, and it was so good. I didn’t understand why anyone would want to learn math,” Virat Kohli said on an Internet sports program, “in-depth with Graham Bensinger.” “I couldn’t understand the complications.
“I couldn’t understand the formulas and believe me; I’ve never used those formulas in my life.” I just wanted to pass my tenth-grade exam, because they used to happen at the state level and after that, you could choose whether you wanted to continue with the math or not. I’m telling you, I’ve never worked as hard in cricket as I did to pass that exam so that I can avoid the subject forever,” Kohli added.
The Skipper of the Indian cricket team Virat Kohli can have all the answers to what the bowlers throw at him, whether rhythm or turn. Virat Kohli works tirelessly on networks for hours so it can be useful for India when it matters most. From a young Delhi cricketer during his U-19 days, Virat Kohli has now become one of the best Batmen in the world. He is also one of India’s fittest athletes.
Today, however, statisticians around the world have to return to numbers at the time Kohli takes the fold, as there will always be the ability to create records. So Kohli, who hated numbers and formulas in mathematics, is seen today in amazement after the number of runs he has scored.
Virat Kohli’s next assignment will be the local series against South Africa from 15 September. India recently began its Test Championships campaign with a 2-0 series victory over the West Indies. He would certainly like the team to keep the same winning streak.
Read More: Shadab Khan Performance in Recent CPL Match


One of these ground breaking features is cas functionality. The test center won’t loan you one, and students aren’t allowed to share.
They can edit each step and calculate them again.
Best scientific calculator for sat. This model is the latest and most advanced scientific calculator manufactured by the company until the date and can make all science or maths operations easy. You can solve all the problems on the sat math with calculator section with a pretty simple calculator, so don’t stress about purchasing the “correct” calculator. Trusted and widely used by sat students, it has proven its worth for a long time.
You definitely need to bring your own calculator. If you’re planning on taking the sat this year, you’re probably wondering what the best calculator for the sat math section is. It’s a great option for engineers, surveyors, scientists, medical professionals, and college students.
Our team unanimously chooses the casio fx115esplus as the #1 scientific calculator for a chemistry subject. You can only use your calculator on the “math test—calculator” section of the sat. The quick functions of your calculator will help you be more efficient in the exam, so you can focus on answering the questions better.
It also comes in three colors: This guide will show you which sat calculators are the best to choose from. When you get to the math sections of the sat exam, you are allowed to use a calculator to make things easier for you.
Can i use a scientific calculator on the sat? Because of the way that this calculator can be used for basic math, algebra, science, trigonometry, and geometry, this is one of the best calculators for engineering students. However, if you’re more comfortable with scientific devices, the first two products on the following list fit the bill.
When it comes to shopping for the best sat calculators, there are two brands to look out for. ( check today’s price on amazon) 6.5/10. If you have the budget for it, and the need for a scientific calculator far more powerful than any of the others we’ve seen, the hp 35s may be the one for you.
Without a doubt, texas instruments and casio make the best devices on the market. Although a scientific calculator can also do the math needed for trigonometry, calculus, physics, and engineering, if you need to see a visual representation of an answer in the form of a graph, you will need to use a graphing calculator instead. The best calculator to use on the sat test is the one you’re familiar with and is allowed during the test.
This device is one of the older releases on our list, but when it was released it had many ground breaking features. Whether you are looking for a scientific calculator or graphing calculator, we’ve got the best calculator for the sat to fit your needs. Top 5 best sat approved calculators.
As mentioned, its best upside is that it is affordable. The 5 best calculators allowed for sat. It may not be as advanced as other sat test approved calculators, but it will be useful for you on the sat.
Glancing takes less than a second, and so many times i caught myself typing (425+25) instead of (424+25) or reversing a. The best calculator for the sat won’t help you if you’re struggling to use it. Kkmoon scientific calculator counter 401 functions matrix dot vector equation calculate solar and battery dual powered 2 line display business office middle high school student sat/ap test calculate.
This is the scientific calculator you’ll need if you’re doing the ged exams, but not the best for college. The online visualization service of this device allows you to immediately see a graphical representation of your data by generating a qr code and scanning it. You can also use this calculator on the sat, psat/nmsqt, and act college entrance exams.


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The BYU Salt Lake Center offers online tutoring through BYU Independent Study to help students in the following subjects: Accounting, Biology, Business, Chemistry, Economics, English, Finance, History, Math, Management Communication, Physics, Statistics, World Languages, and Writing. To set an appointment, click on the following link: https://is.byu.edu/support/students/tutoring. All tutoring must be arranged 24 hours in advance. Please let the tutor know that you are taking your class at the BYU Salt Lake Center. If you have any problems setting up a tutoring time, contact Customer Support. They are open Monday - Friday between 8 am and 5 pm. The local phone number is 1-801-422-2868 and the toll free phone number is 1-800-914-8931.
Information on the most heavily used tutoring services can be found below. Information on all other subjects can be found at https://is.byu.edu/support/students/tutoring.
Serivces are on a first-come, first-serve basis for no more than 30 minutes at a time, up to ONE hour per week.
Students may meet with a tutor up to ONE hour per week. Tutoring in math is by appointment only.
Students may meet with a tutor one-on-one up to ONE hour per week.
Students may meet with a tutor up to ONE hour per week. Tutoring in economics is by appointment only.