The fractured rock mass is complex in structure and variable in morphological parameters. Due to lack of effective techniques for direct observation of internal water flow in fractures, accurate calculation of three-dimensional permeability parameters of rock mass still remains a scientific problem. The single fracture seepage is related to the crevice fluctuation and the opening degree. The seepage in the spatial fracture network is related to the fracture communication. Based on the actual measurement of fracture morphology, this study proposes a multi-fractal model for analyzing seepage parameters of fractured rock mass. Firstly, the single-fracture wall is scanned by a three-dimensional laser to obtain the fractal dimension of the closed or open fracture surface morphology. Combined with the cubic law and Darcy's law, the single-fracture water flow is analyzed, the fractal dimension of the water flow path is calculated, and the fractal relationship between the single fracture morphology and the permeability coefficient is obtained. Then the three-dimensional fracture communication network is established by Monte Carlo method, and the two-dimensional fracture communication network in different directions is obtained through orthogonal slice. The fracture communication is described by the fractal dimension of the fracture density, and the fractal dimension of the two-dimensional and three-dimensional fracture network communication rate is established. Finally, the two-dimensional fracture seepage path and permeability coefficient are analyzed according to the undirected graph adjacency matrix and continuity equation of seepage, and the three-dimensional seepage flow coefficient of the fractured rock mass is solved by the fractal dimension of the two-dimensional and three-dimensional connectivity. The model realizes the analysis of the permeability parameters of the fractured rock mass by the fractal dimension of the fracture and the seepage analysis of complex fracture networks is conducted by step-by-step dimensionality reduction from three dimension to one dimension, which provides a new idea for the study of fractured rock mass hydraulics.


It aims to optimize the total resistance and the pitch amplitude of the KCS sailing in waves by adding two kinds of appendage, a pair of anti-pitch-fins at the bow and a wave suppressor at the stern, to the KCS and changing their size.Firstly, numerical calculations are performed on KCS with just a pair of anti-pitch-fins, just a wave suppressor, and both kinds of the appendage separately to study the effect on the hull of the appendages. Secondly, 30 KCS ships with deformed appendages are obtained by changing the width of the anti-pitch-fins and the length of the wave suppressor. The calculation uses the in-house viscous flow solver naoe-FOAM-SJTU. Thirdly, a Kriging surrogate model is established and applied to a multi-objective genetic algorithm to obtain the Pareto front of the hull with optimal appendages. Three KCS ships with optimal appendages are selected from the Pareto front to do further analysis.In the process of the pitch, appendages will generate resorting moment to lessen the pitch amplitude of the hull, the draft of the bow and stern. So the pressure of the hull will decrease, and reduce the pressure resistance and finally reduce the total resistance.


Adaptemy is an on-line assessment and learning platform, covering all topics on the both the Junior and Leaving cert. Maths syllabus. Rice College are one of the participating pilot schools using the programme. Ms. Grimes’s 2nd yr. Maths class have been using the programme for one class period per week since September.
It is working extremely well as a revision tool. It assesses each individual and continuously adapts the level of questioning to the individual. It ensures the student is challenged at all times.
Reports can be compiled showing the progress of the students on each topic.
The following students were the most productive workers and achieved the highest results using the programme over the past number of weeks.
Well done boys – keep up the good work.


Greetings friends! This is what I did this week.
- No school Monday.
- No school Tuesday.
- On Wednesday I did change-up meeting played some Clash Royale with friends, did some IXL, played Bedwars and caught up on my geometry work.
- On Thursday I did geometry, and in geometry we learned a useful formula to measure area and perimeter. I also played Totally accurate battle simulator, played more Bedwars and went outside and played basketball
- On Friday I played MORE Bedwars, I also drew a little bit and ate lunch.
Yeahhh thats kinda it.


EducationMaster degree computer science university of Queensland 2007.
HobbiesWatching soccer and reading.
10th Grade math, Developmental Algebra, Differential Equations, Electrical and Computer Engineering, Graphic Design, HTML, IB Information Technology in a Global Society HL, IB Information Technology in a Global Society SL, Java, Math, Operating Systems, PHP, Probability, Programming, Programming VB.NET, Python, Software Engineering, Web Design, Data Structures, Computer Skills, Computer Science, 1st Grade math, 3rd Grade math, 4th Grade math, 5th Grade math, 6th Grade math, 7th Grade math, 8th Grade math, 9th Grade math, Algebra 1, Algebra 2, Algebra 3/4, Algorithms, AP Computer Science, AP Statistics, Arabic, C++, Computer Programming, Web Development
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Applied Mathematics Seminar 15: A Partially Functional Linear Modeling Framework for Integrating Genetic, Imaging, and Clinical Data
Speaker: Ting Li, Shanghai University of Finance and Economics
Time：Friday, December 9th, 2022, 16:15-17:15
Location：RS408; Online, Tencent Meeting
Abstract: This paper is motivated by the joint analysis of genetic, imaging, and clinical (GIC) data collected in many large-scale biomedical studies, such as the UK Biobank study and the Alzheimer's Disease Neuroimaging Initiative (ADNI) study. We propose a regression framework based on parti...


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Continuing from the point at which we stopped in the first part of this series, we will now model a digital PID with a slightly different approach than the one used to design the compensator described in the previous article, the idea here is to provide a PID compensator that can be implementable in a way simple in a general purpose processor (as discussed in part I of this series, general purpose processor is meant any microcontroller that has at least the hardware multiplication instruction).
Shall we recap?
In part I of this series, it was explained what is a closed-loop control system, the reason of its use, as well as the PID (Proportional – Integral – Derivative) type compensator as a versatile type of controller applied in large control systems of the most varied types.
We still remember that its expression takes the form of an equation, as the one exemplified below:
- Kp: Proportional action coefficient;
- Ki: Coefficient of integral action;
- Kd: Derivative action coefficient
- t: State of the state to be processed;
- u (t): System output signal at time t;
- and (t): Error signal at controller input at time t.
From this equation followed by the use of the well – known Laplace transform, a much simpler equation was obtained than the one exemplified above, so that an electronic circuit could be designed to reproduce the behavior of a PID compensator.
In contrast, in order to implement a component of these software paths (in a microcontroller), both one equation and the other, are very difficult to implement, so, using an approach based on numerical methods, we were able to design an equation easily programmable on a processor:
Where, the parameters (constants) can be calculated as follows:
In addition we have:
ti = integration time
td = Time of differentiation.
Also adopted was a name called the rectangular approximation of the integral term, so it could translate this simple equation into a program that can be embedded in the processor to be used.
Problems with the rectangular approximation
Although the rectangular approximation algorithm of the integral term is versatile and serves a good range of applications, some problems begin to emerge as the complexity of the system being controlled increases.
The first and great problem is the convergence of the integral of the error function to the correct value when the previous value of this error function is equal to zero, or when the difference between the current and the previous error becomes very large, this problem could be corrected by increasing the number of iterations for calculating the integral term (and not based on two points), but at the expense of execution efficiency.
Another solution would be to approximate the integral only by accumulating its value, each time the PID is computed, this maintains the efficiency of the code execution, and causes the same effect of the increase of iterations, however the accumulation would occur only once in each value of the computed PID, returning the problem of slow convergence in the results of the integral in the first values calculated by the PID, as the process advances this effect is being minimized.
The other big problem of this algorithm can be elucidated with a practical example, considering a system where the proportional gain is something high, and the accumulated error begins to grow or to decrease for long periods of time, this can fatally lead the system in control to have overshoot with high values, and in more severe cases, lead the system to a permanent oscillation. This phenomenon is known by the name of wind-up, and can be minimized using low values for the integral gain, added to techniques of saturation of the integrator (accumulator in the case of integral approximation), but thus, we return to the problem of delay of convergence, since the saturation can be understood as zeroing the integrator (or accumulator).
We then have some drawbacks with this algorithm, so what can be done to minimize the effects of convergence and wind-up? There are several ways to model algorithms of digital compensators from
Digital PID Compensator 2 poles / 2 zeros
Knowing the problems with the first algorithm we can proceed with the modeling of a more precise digital PID controller. For this we will need a mathematical tool well known to those who have studied digital control (or digital signal processing), the Z transform.
This tool allows us to translate equations of difference (in the domain of time), ie those in which two events are separated by an interval whose value is finite and numerically representable, and to obtain an expression that takes a form very similar to equations obtained by the Laplace transform for continuous time systems.
Its definition is exemplified in the equation below:
Equation 1.2 can intimidate at first, but observing with attention we can note that the Z transform by definition does nothing more than take a sequence of pulses whose capture instants are spaced by a time interval, the independent variable n denotes the instant of sampling, so that a discrete time signal can be described as the sequence below:
Thus, once passed to the domain Z each instant of the signal is accompanied by the complex operator (“z”), which denotes the displacement of that signal in time, simplistically the representation of the operator z serves to say if the sample is a value current, a future value (predictors and estimators) or an earlier value, the latter of our interest, let us see:
X (z) z-n: Sample delayed in time;
X (z) z0. ‘. X (z): Current sample;
X (z) zn: Future sample
Although the theory behind the Z transform involves much more concepts, if we look at it in a simplistic way, we can use this simple property (referred to as time shift) to realize a new digital PID controller with better performance than the previous article.
Now that we already know what z operators mean, let’s change a little subject and return to our problem, a way to better approximate the integral term. In addition to the accumulation, there are several numerical methods that allow us to solve integrals with considerable precision and much smaller convergence intervals. For this purpose, let’s select the rule of the trapezoids, with which we can solve an integral according to the following equation:
A much simpler expression to solve, in addition to the rule of trapezoids by approaching the difference between two values of the function f (t) by a trapezoidal aspect, if the sampling rate of the system is well chosen, the convergence for the desired result is much faster and does not suffer from the problem presenting in the first algorithm developed.
We already have a new solution for our integral term, since the derivative does not need modifications, why use the Z-transform, why not go straight to the accounts? With the use of Z transform, we can further reduce the final equation, pre – compute its coefficients, and we can also use its Z – transfer function for analysis in a simulator such as MATLAB.
We know that numerical methods by nature work in a domain of discrete intervals, which is perfect for our application, means that we can take the Z-transform of equation 1.3? Yes we can, but first let’s change the notations to become more familiar:
The above equation is in a much more convenient notation for discrete time system. But now
Taking the Laplace transform and then the Z transform on both sides of the equation we have (Note that we are computing two derivatives in the domain of continuous time):
Thus we can relate the complex variable s to the discrete time variable z. But before applying this to our PID control, let’s see how to do for the term derivative. Following the line of numerical reasoning to compute a derivative, we have:
As we did for the integral, let’s change the notation for the discrete time domain:
To facilitate computation we will delay both samples in one unit and representing x [n], thus we obtain:
Let us now take the Laplace’s transform then the Z transform in the same way as we did for the integral:
Algebra, algebra, algebra, eis that leads to:
Here we now have an approximation for the integral term, and another for the derivative term. But note that they are in the frequency domain, and our PID is in the time domain, we cover equation 1:
This is not a big problem, first, let’s modify the equation above to the form of a transfer function, ie the quotient of the output by the input of the controller, hence this leads to:
And finally we take the Laplace’s transform, from the above equation:
Our PID has a much friendlier equation but it still finds itself in the continuous time domain, we now need to translate this equation into the discrete time domain, but as was done in 1.5 and 1.6 we now have terms in Z that represent S , so we can take the Z-transform of equation 1.5 substituting the integral and derivative terms:
And it is done, a digital PID compessor, but to the reader I say, do not be scared, although the equation does not seem to make much sense, even more if it is the implementation in a processor, it will begin to make sense as we better order the terms, before anything, we will disappear with the two denominators that does not contain any z, this will help facilitate the accounts. I will add two operators, h1 and h2 where:
h1 = Δt / 2;
h2 = 1 / Δt.
Here the new transfer function becomes:
From this point I will make the necessary mathematical passages until we obtain the final transfer function of our compensator, I ask the reader to follow the passages calmly:
Here we have our transfer function, it is much more friendly, and allows to work on it to form an equation of differences. But before proceeding, let’s make it even more intuitive for the reader, let’s note:
We will replace these coefficients in the transfer function, and we will note that our digital PID is now nothing more than a special case of IIR filter, having two poles and two zeros, from which comes the origin of the algorithm name:
Look at how interesting math is, we start with a differential equation, we put a trapezoidal integral, a numerical derivative, we use two transforms, to find that our best approximated PID is a digital filter of infinite impulsive response. In other words, there are many ways to implement this in a microcontroller, this same equation may already be put into a simulator like MATLAB to evaluate the response of the system to be controlled.
Placing inside a microcontroller
We have the perfectly usable Z-transfer function to simulate its behavior in a MATLAB or any other system simulator, but this function of the form that it is is not very useful, to implement in any processor whatsoever, we must first compute a difference equation from the inverse transform Z, for this we will return to equation 1.7:
We have the input and output of the system, now let’s output according to the input:
It has improved, right? Let’s move a little bit by factoring U (z) and E (z) to “inside” the polynomials we will get:
But: a2 = 0, this simplifies the expression in:
In function of U (z) we obtain:
Interestingly, at this point we can take the inverse transform Z, and surprise the equation of differences becomes this simple expression:
As? Yes my friend, all that hassle of transforming here, transforming there, and using numeric method, will generate this trivial expression, proof that to get things done simple is necessary to think. Equation 1.8 is perfectly implementable, the coefficients are easily calculable, so we can translate this equation as, to compute a new compensator output, we need the previous output sample, and the last three calculated error values.
For the closer reader, this one must have realized that the full term windup problem does not even exist, one less worry, just choose whether to work with fixed or floating point and put it inside your processor.
Let’s test our example, we will ship a small firmware to control the brightness of an RGB LED by closed loop.
Example in test
And finally we come to the most fun session of the article, the practical implementation, we will ship our test firmware in the following development kit:
Again we will not go into details about the Kit or microcontroller, more information can be found on the Texas website, the link is available at the end of the article.
We will use our digital PID to control a string of LEDs, through the operation of the microcontroller in a DC-DC converter, in fact one of the most interesting parts of this kit in particular. As a simple example, I opted for the use of only the control of the chain of red LEDs, using the control only by the output voltage of the DC-DC converter (yes it is a boost), consulting the datasheet of these leds, arrived at the value for obtaining a current of 25 [mA], by means of experimental adjustment, the following coefficients were reached: Kp = 50, Ki = 15, Kd = 100, the sampling rate was placed at 58,000 samples per second and a carrier frequency of 150 [KHz] allows the PWM module of the microcontroller to operate at 9-bit resolution.
Let’s see the image below, we have the bus voltage perfectly controlled by the microcontroller, let’s see her stable behavior:
Below we have a photo of the bench, where we have the LEDs being controlled by the microcontroller, one of the things that you write is more interesting is the possibility of the microcontroller itself become a digital control unit, flexible according to the application.
In spite of the simplicity of the example, the entire project is in a repository in GitHub, to you, reader, I leave the freedom to use in the C2000 itself or carry the PID.c and PID.c files for the microcontroller of your interest, I suggest only the alteration of the types according to the microcontroller used, in case of use of more compact platforms such as AVR, and microcontrollers based on the Cortex-M0 core. And for you most curious reader, why not explore the equation 1.7 and implement in your processor on your own? Writing code in assembly can also be an extremely pleasurable task.
So we come to the end of this small series of articles and hopefully have demystified some concepts on the implementation of digital compensators in microcontrollers of almost any nature. We also hope that the examples provided will be useful to those who wish to venture into the world of digital control. We have also been able to model a simple PID compensator using the Z transform in a simple way without worrying too much about the mathematics involved.
Example Project: https://github.com/uLipe/PID_Embarcados_C2000
Microcontrollers C2000 : http://www.ti.com/product/tms320f28027
Led Booster Pack, Kit used in the example: http://www.ti.com/tool/boostxl-c2kled
CRENSHAW, Jack W. : Math toolkit for real time programming, 2001
MEYRATH, Todd : Multipurpose analog PID controller, 2005
POLEY, Richard : Introduction to digital control using digital signal processors, 2003
STARR, Gregory P. : Introduction to applied digital control , 2006


My master thesis (Mar 1995)
There are several definitions for randomness of infinite binary sequences.
One definition(Chaitin randomness) uses Kolmogorov complexity,
and another(Martin-L\"of randomness) uses tests.
It is a well known result that
Chaitin randomness and Martin-L\"of randomness are equivalent.
So, infinite binary sequences having high Kolmogorov complexity
are characterized by tests.
In this paper, we show one characterization of
infinite binary sequences having partial randomness
that is similar to Martin-L\"of randomness.
We study the relationship between this characterization
and the characterization that uses
Kolmogorov complexity of infinite binary sequences.
My master thesis (Dvi file, in Japanese)
- Kolmogorov complexity
- Martin-L\"of randomness
- Recursion theory
is home page of Kobayasi labo (in Japanese).
is home page of Kobayasi labo (in English).
Home page of Dept. of Information Science (in English).


Co-authored with Katharina Boudgoust, Adeline Roux-Langlois, and Weiqiang Wen. This paper introduces new results and generalizes the ones published at Asiacrypt 2020  and CT-RSA 2021 .
Published in IACR Journal of Cryptology.
Get paper on IACR ePrint or here.
The Module Learning With Errors problem (M-LWE) is a core computational assumption of lattice-based cryptography which offers an interesting trade-off between guaranteed security and concrete efficiency. The problem is parameterized by a secret distribution as well as an error distribution. There is a gap between the choices of those distributions for theoretical hardness results (standard formulation of M-LWE, i.e., uniform secret modulo \(q\) and Gaussian error) and practical schemes (small bounded secret and error). In this work, we make progress towards narrowing this gap. More precisely, we prove that M-LWE with \(\eta\)-bounded secret for any \(1 \leq \eta \ll q\) and Gaussian error, in both its search and decision variants, is at least as hard as the standard formulation of M-LWE, provided that the module rank \(d\) is at least logarithmic in the ring degree \(n\). We also prove that the search version of M-LWE with large uniform secret and uniform \(\eta\)-bounded error is at least as hard as the standard M-LWE problem, if the number of samples \(m\) is close to the module rank \(d\) and with further restrictions on \(\eta\). The latter result can be extended to provide the hardness of M-LWE with uniform \(\eta\)-bounded secret and error under specific parameter conditions. Overall, the results apply to all cyclotomic fields, but most of the intermediate results are proven in more general number fields.


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Marked June 7th
Linear Relations page by: Jas,
What is Linear Relations?
Linear Relations is the relationship between two variables that appears as a straight line when graphed. Linear relations is broken down into linear and non-linear, and direct and partial variation. These can be identified using
, graphing, and first differences.
Definitions and Equations
The steepness determined by vertical change and horizontal change
where the line crosses the y axis
where the line crosses the x axis Linear Relation
the study of lines
a straight line
not a straight line
- An e
quation written in Ax+By+C=0 format
When using standard form there are three rules that must be followed at all times.
All terms must be on the left side
The first term must be positive
There can't be any fractions or decimals in the formula
An example of a proper equation that has been converted into the standard form would be:
From y=mx+b form to Standard Form:
y =3x+12 ...........ymx+b form to
Linear or Non-Linear
Examine the equation, if there are any exponents 2 and over it is
You can check by looking at the graph, if the line is straight then it is linear, if it is not straight then it is not linear
You can also check by looking at the first differences. If the First differences are the same or follow a pattern then it is linear, if the first differences are all different or follow no pattern then it is non - linear.
Using First Differences:
You must first make sure that your "x" column in your table of values increases by the same amount each time. If your table does not do that then you have made a mistake
To find first differences you must subtract the "x" column from the "y" column. This will give you your difference between the two columns.
If the first differences are all the same or have the same pattern then you know that your line is linear. If your first differences are different or have an irregular pattern then you know that the line is non-linear.
3 – 1 =
5 – 2 =
7 – 3 =
9 – 4 =
11 – 5 =
This is an example of a linear line. We can tell that this is a linear line because the first differences in the chart are in a constant pattern of increasing by one.
Height of Building (m)
Number of Stairs
1000 – 800 =
700 – 600 =
400 – 250 =
100 – 90 =
60 – 30 =
Formula's And When To Use Them
There are three major formula's that we will be focusing on in grade 9. The first one is Y=MX+B. This one is the most commonly used out of the three and is very useful when graphing. In this formula, M repersent the slope, and B repersents the y-intercept. When asked in a question to find the equation of a line, the first thing you would write down is
The next formula is the standard form formula,Ax+By+C=0, which isn't used much in grade 9 math but more in grade 10. The only thing we would use it for in grade 9 is if you where asked to convert a Y=MX+B formula into standard form or if you wanted to find the Y and X intercepts of a line.
The last formula is Y2-Y1/X2-X1 which would be use if you were given 2 coordinates and asked to find the slope. If you are given 2 coordinates and asked to find the equation of a line you first write y=m
x+b. Then you need to find M and as you already know, M is the slope. To find M when given 2 coordinates use the formula M=Y2-Y1/X2-X1. Once you have solved that you would want to find b. to find b use the formula y=mx+b and use one of the coordinates. Sub in the y coordinate to y and the x coordinate to x, than solve you should get b. Then sub M and B into y=mx+b and that is how to find the equation of line.
The last formula is RISE/RUN. This formula is used when you are given a line on a graph and you are trying to find the slope. Simply find the measurments it takes to get from on point to the other, going across and up/down. For example,
We will be doing some examples for each formula starting with M=Y2-Y1/X2-X1.
Example 1. Find the slope of a line that passes through points (5,3) and (4,9).
So the first thing you would write down is the formula. M=Y2-Y1/X2-X1.
Then you would label one coordinate A and the other B. A(5,3) and B(4,9)
After that get the Y coordinate from coordinates B, sub it into Y2. Get the Y coordinate from coordinates A, sub it into -Y1. M=4-5/X2-X1
Now do the same thing with the two X coordinates. M=4-5/9-3
Last solve. M=-1/6 is the slope of the line
In this example we will be using the formula Y=MX+B.
Example 2. Write the equation of a line that goes through the point (0,-3) and has a slope of 5.
For this all you need to do is fill in M and B, M is the slope so M is 5. Y=5X+B
Then fill in B which is the y-intercept (the y coordinate). Y=5X-3 is the equation of this line.
For this example there isn't much to show but we will be using the formula Ax+By+C=0.
Example 3. Convert this into standard form Y=7X-9.
First you need to move every thing to the left, and make sure you change the sign when you move it. -7X+Y+9=0
Then if the first term on the left is negative you need to multiply the whole equation by negative one. -1(-7X+Y+9)=-1(0)
Once you have multiplied the formula by negative 1 the final answer will look like 7X-Y-9=0
For this example we will be using RISE/RUN.
Example 4. Find the slope of the black line.
So after you have got your line, you will find the 2 points, put one at the bottom of your line and one at the top.Count the number of points it takes to get from the bottom point to the top point going upwards therefore this is the
Once you have found your rise you must find your run, to do this you will have to start at the bottom points and count how many points it takes to get to your other line going across therefore this is called
Put them in your formula and do the math, lets say that your rise was 6 and your run was 2 using the formula
your answer would be 3.
There is one more example for the formula Ay+By+C=0 and this is how to find the Y and X intercepts of a line.
In Real Life
Unlike some other units in math, Linear relations is actually very common in everday life. Infact, most people use it without even knowing. In application to the real world, linear relations usually reffers to the relation between time and something. That somthing could be distance,earnings,pionts totaled, etc. It basically shows that progress of a variable over a period of time. For example:
Kevin Garnett earned 20 million dollars over one season in the NBA. He lost 5 million dollars to the NBA after he was fined for misconduct to a fan. He earns a bonus 15 million in endorsments over his whole careers regardless. The Boston Celtics promise him 20 million dollar each year he decideds to play for them. How much will Garnett earn in his remaining years before he is retired?
This is a perfect example of a linear relation, so lets put it into Y=MX+B format. Wheather Garnett decides to play or not, he will earn atleast 15 million dollars more in endorsments buy the end of his career, but because of the fine he losses 5 million, so he is garunteed 10 million(10,000,000). This is were Garnetts salary will start at no matter what, and therefore it is the Y intercept, and our B (in Y=MX+B). From that point on, Garnetts salary can increase on a 20 million dollar for each year he plays, so the slope is 20 million (20,000,000). X repersents how many more years he plays. When the numbers are plugged in you get Y=20,000,000X+10,000,000. If Garnett played 4 more years, this is how you would find his earnings:
Therefore Garnett would have earned $90,000,000 more dollars in 4 years before he retired.
Linear relation graphed.
FOR FINDING THE Y-INTERCEPT
Finding the y-intercept in a Y=MX+B formula is simple, as the B is the y-intercept (remebering that if there is no B then the y-intercept is therefore 0).
Although finding the y-intercept when the Linear eqaution is written in standard form is not as easy. Lets use the following equation to help explain the procedure,
. To find the y-intercept, you have to solve the equation for y (meaning you must isolate y and find out what it equals). The first thing you do to find y is make X=0. X must always equal zero when trying to find y, because if you solve for one variable while there is still another variable in the equation you will get a number attached to the other unsolved variable, which dosen't give you an exact answer. Therefore X=0. Lets see how this is done on our equation
.<--- notice how the X has been repalced with a 0.
.<---after multilpying 4 and 0 you get 0, which dosen't need to be written.
The next step is to isolate y. To isolate y we must first get rid of the C (16).
.<---to get rid of the 16, we subtract 16 ffrom that side and the other side, because whatever you do to one side of a equation you must do to the other.
.<---after subtacting 16 from both sides we are left with this equation.
The last step is to split the 8 and the y. To do this we have to subtract both sides by 8.
.<---dividing both sides by 8 to isolate 8.
.<---after divding bith sides by by 8 we are left with this.
Now that y is completly isolated, and solved, you have the y-intercept, which equals -2.
FOR FINDING THE X-INTERCEPT
Follow the same steps for finding the Y-intercept, except now Y=0 and you isolate for X.
The last aspect to Linear relations is
. With graphs, it goes either one of two ways; your either making the graph, or you use the graph to solve questions regerding it. The reason the graph is important is because it helps to visually see the linear relations, and provides other options, such as a line of best fit, which help you further understand the linear relation.
Graphs help us do so many things in various ways, but we're just going to focus on how to graph linear realtions, and understand them.
When it comes to graphing Linear relations,what you are going to graph is the corridinites or the LOBF (
it). Graphing this line is as simple as just connecting the dots in a stright line or makeing a line that is as close as possible to all sets of data. Although, the tough part is to find the co-ordinates and get them graphed. To find the coordinates you have to fill in the blanks to the Y=MX+B formula, giving you the slope and y-intercept. Once you have that you can graph the y-intercept and then using the slope graph the co-ordinate giving you the line on the graph.
The difficult parts of graphing are using equations, formula's, and finding the linear relation. The rest of graphing is the basics things to remeber when making the graph. They are:
Rember your titles. One title for the graph, and one for the x intercpt and y intercept if required. Also label the axis (x and y).
Label the lines on the graph with there equations (ex.Y=5x+3)
If there is a graph were you have to label the axis, always remeber that time is on the x-axis.
Make sure all lines are straight (if not they wouldn't be linear therefore there not linear relations).
Draw arrows at the end of lines to indicate that the line keeps going.
Here are you examples of linear relations being graphed. The first is on a cartasian plain, the second is showing the relation ship between time and distance on a standard graph.
Equation for first Linear relation graph-Y=2X+4.
This equation and graph is showing the Linear relation between time and distancewith the equation Y=10X
The slope of a horizontal line is zero
The slope of a verticle line is undefined
If a line is steeper it will have a larger slope
Hendricks, J.D. (2008) MPM 1D1 - Course Notes
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Most of us have heard of the famous Normal Distribution (a.k.a Gaussian Distribution, a.k.a “Bell’s Curve). What some of us may have forgotten, is that Normal Distribution’s popularity is powered by the Central Limit Theorem, definition below:
“Central Limit Theorem states that the sampling distribution of the sample means approaches a Gaussian Normal distribution as the sample size gets larger — no matter what the shape of the population distribution.”
In plain English, CLT implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of non-normal distributions.
The only caveat(s)? Your sample data must meet all 3 conditions below:
Randomized Sampling: The data must be sampled randomly. Is one of the good sampling methodologies discussed in the chapter “Sampling and Data” being used?
Independence: The sample values must be independent of each other. This means that the occurrence of one event has no influence on the next event.
Sample Size: The sample size must be sufficiently large.
#2 can indeed be tested through various statistical techniques, one of which being ANOVA (Analysis of Variance).
#3 is where it gets really vague. What defines “sufficiently large”? From prevailing literature, it appears a “sufficiently large” range can stretch from 30 to Infinity.
Yet, despite meeting Conditions 1 and 2, and Condition 3 (with very large sample sizes, certainly not infinity!) non-normal behaviour is exhibited through a wide range of physical and social phenomena. Some examples are listed below:
Geology: Concentration of Elements in Earth’s core
Medicine: Survivor times after cancer diagnosis
Ecology: Inheritance of fruit and petal sizes
Environment: Amount of Rainfall
Linguistics: Length of spoken words in phone conversations
Business: Call Centre Service Times
Incidentally, because Lognormal distribution describes all the above phenomena very well, let’s use this as an illustrative comparison to Normal Distribution.
Below is a graphical comparison of a Normal Distribution (in yellow square), compared with Lognormal Distributions:
The Normal Distribution has a mean of 100 and standard deviation of 20
The other curves, which are lognormal, share a mean of 100. However, their standard deviations are varied
From the chart, we can make a few observations:
The more Coefficient-of-Variation (COV) approaches 0, the more a Lognormal Distribution resembles a Normal Distribution.
A Lognormal Distribution with a COV of just 0.2 already differs significantly from a visual standpoint
Holding only the mean constant and changing only standard deviations, a Normal Distribution overestimates values that are most likely the occur (as measured by the peak points on the curves – which have the highest likelihood of occurrence)
So what does all of these have to do with Supply Chain Inventory Planning?
Here is the inventory planning (Re-Order Point) equation still largely in use by most organisations today:
We can clearly see that Normal Distribution assumptions are embedded in the common Inventory Planning equations, because of:
Zα is meant for standard deviations Eg. Zα = 1.645 when cumulative probability of a normal distribution is 95%
BUT what if Demand does NOT follow a Normal Distribution, but Lognormal (Like many things in nature)? What will it mean for Inventory Planning?
Firstly, there is a basic reason why most SKUs probably do not follow a Normal Distribution: Going by the Pareto Rule, around only around 20% of SKUs would generate 80% of demand (Low Mix/High Volume) and are more likely to demonstrate stable demand Hence, remaining 80% of SKUs would have significantly less demand (High Mix/Low Volume) and are more likely to demonstrate volatile demand
Secondly, estimated demand can vary significantly between the 2 distributions (with the same Mean and Standard Deviation)
The below table shows projected stocks required to cover a client’s peak demand, at a desired confidence level %. Both distributions assume a mean of 100, and standard deviation of 250 (volatile demand)
As one can see, Normal Distribution tends to over-estimate the required stocks until around 98% confidence level %. For context, most service level agreements stipulate a minimum service level of 95%.
Then, why was there such a reliance on Normal Distribution?
Mainly due to 3 reasons:
Normal distribution is considerably easier to compute. In a bygone era of low processing power, usage of normal distribution could quickly reap significant benefits delivered by statistical techniques
Normal distributions are more “intuitive” to understand due to inherent symmetry
History of distributions: Normal distributions have been applied twice as long as it’s log-normal sister distribution (and even longer relative to other distributions like Weibull)
So how can Supply Chain Organisations choose the appropriate Distribution for their supply chain demand - not only for today, but ongoingly?
In addition to lognormal distribution, consider also other statistical distributions such as Rayleigh, Weibull, Gamma, etc. Remember that Lognormal is just used as an example alternative to Normal distribution in this article
Consider Multi-Echelon Inventory Optimization (MEIO), and segmentize SKUs according to different criteria such as demand volatility, volume, profit margin, et cetera. To each SKU segmentation, deploy a different forecasting and inventory policy.
Given big leaps in computing power, consider leveraging software and smart analytics tools that will dynamically advise on historical trends and supplement MEIO (it could recommend an SKU to be moved to another segment, with an ongoing change in forecasting and inventory policy) Gartner’s 2018 Quadrant for Supply Chain Planning software (which includes inventory planning) is as below. Highlighted in red is the Leaders’ Quadrant indicating High execution ability and High completeness of vision.
On top of analytical tools and SKU segmentation, set up a rigorous periodic process to review exceptional SKUs within SKU segments (which again, can be supplemented by software and tools)
Johnson Chua - Forizons Hong Kong Director


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The payout ratio of a Real Estate Investment Trust (REIT) is a financial metric that measures the percentage of the REIT’s earnings that are distributed to shareholders as dividends.
In general, REITs are required by law to distribute at least 90% of their taxable income to shareholders in the form of dividends, which means that their payout ratio is usually quite high. However, the actual payout ratio can vary depending on the specific REIT and its financial performance.
To calculate the payout ratio of a REIT, you would divide the total dividends paid by the REIT over a given period (usually a year) by its net income over the same period. The resulting percentage is the payout ratio. For example, if a REIT paid out $2 per share in dividends over the course of a year and had a net income of $2.50 per share during the same period, its payout ratio would be 80% ($2 divided by $2.50).


Symmetric-key cryptosystems order levitra generic use the same key
for encryption and decryption of a message, though a message or group of messages may have a different key than others. A significant disadvantage of symmetric ciphers is the key management necessary to use them securely. Each distinct pair of communicating parties must, ideally, share a different key, and perhaps viagra naturel maca each ciphertext exchanged as well. The number of keys required increases as the square of the number of network members, which very quickly requires complex key management schemes to keep them all consistent and secret. The difficulty of securely establishing a secret key between two communicating parties, when a secure channel does not already exist between them, also presents a chicken-and-egg problem which is a considerable practical obstacle for cryptography users in the real world.
In a groundbreaking 1976 paper, Whitfield Diffie and Martin Hellman proposed the notion of public-key (also, more generally, called asymmetric key) cryptography in which two different
but mathematically related keys are used—a public key and a private key. A public key system is so constructed that calculation of one key (the ‘private key’) is
computationally infeasible from the other (the ‘public key’), tadalafil vasodilator even though they are necessarily related. Instead, both keys are generated secretly, as an interrelated pair. The historian David Kahn described public-key cryptography as “the most revolutionary new concept in the
field since polyalphabetic substitution emerged
in the Renaissance”.
At CRG, interesting areas of this field including signcryption, identity based encryption, digital signatures and elliptic curve cryptography are currently being researched by the group members.
A cryptographic hash function is a hash function which is considered practically impossible to invert, that is, to recreate the input data from its hash value alone. The input data is often called https://www.acheterviagrafr24.com/viagra-100mg/ the message, and the hash value is often called the message digest or simply the digest.
The ideal cryptographic hash function has four main properties:
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as checksums to detect accidental data corruption. Indeed, in information security contexts, cryptographic
hash values are sometimes called (digital) fingerprints, checksums, or just hash values.
At CRG, research on time complexity of hash functions and use of hash functions for compression is being https://www.acheterviagrafr24.com/viagra-en-france/ conducted by the group members.
A protocol between two parties Alice and Bob is zero-knowledge (from Alice’s point of view), if it does not leak any information to Bob. Zero-knowledge is a viagra naturel maca fundamental https://www.acheterviagrafr24.com/viagra-prix-en-pharmacie/ notion in cryptography and has important applications. For example, Alice can prove to Bob that she knows a secret key corresponding to a given public
https://www.acheterviagrafr24.com/achat-viagra-cialis-levitra/ key (e.g., for identifying herself to Bob) without leaking any information whatsoever about the secret key.
A zero-knowledge proof must satisfy three properties:
- Completeness: if the statement is true, the honest verifier (that is, one following the protocol properly) will be convinced of this fact by an honest prover.
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The first two of these are properties of more general interactive proof systems. The third is what makes the proof zero-knowledge.
At CRG, researchers are currently working on finding feasible solutions for using zero knowledge proofs
for authentication and access control.


Dr. Tim Can Read Your Mind!
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|Learning Strategies - Mathematics
To support skills
to execute problem solving, children need to have basic mathematic
fact and computation knowledge. Besides the below suggestions,
check our Games and Activities under Educational Resources
for additional ideas.
Activities to support fact knowledge:
- Play games to support computation
fluency. Card games such as War, Addition
War, and 21 provide practice with these facts.
The book Card
Games for Kids is a valuable
resource for games of math skills.
- Domino games can be used
to practice addition and subtraction as well
as used for sorting, forming number sets, and
- Other games to support basic mathematic
Quizmo The Quizmo game series has versions to practice
math skills in areas such as number recognition,
money, telling time, and fractions, as well
as basic addition, subtraction, multiplication,
Trilemma and Equate provide practice with math computation and forming
- Use computer games to practice math
facts and skills. Suggestions for computer
games that focus on math include Mia’s
Math Adventure, Learn
to Play Chess with Fritz and Chester, Genius, Crazy
Machines and Wacky Contraptions, and Math
- Teach older children strategies to
assist with fact recall. For
example, children can remember easily a number
added to 10. Children can add to ten,
but remember to +1 to find the sum of a number
added to 11 or -1 to remember the sum of a number
added to 9. Once children learn to use this
strategy, they can apply it to other sums.
children sort coins according to their amounts.
This will help them recognize coins and provide
exposure to their value.
- Play money exchange games to help students
work with place value, equivalent amounts, and
carrying and borrowing.
- To practice with number recognition and basic counting skills for
young children, play games like Chutes
and Ladders, Hi
Ho Cherry-O, Memory, Go
- Drilling with flash cards can help with math fact knowledge. To
support children’s generation of the equation,
present a card with a signal digit written on
it such as 9. Then ask the child to compute
a problem, such as "multiply by 3".
Have the child state the answer. This
will help children generate the equation they
- Have children complete origami, make paper airplanes, and participate in paper folding activities, like wrapping presents, to recognize common fractions terms, size estimation, and spatial skills. Have children participate in tasks that use math, such as cooking or woodworking.
- Play checkers or chess with children to explore patterns and spatial skills.
The games Master
Mind and Battleship also provide practice with patterning, logic,
and spatial skills.
- Encourage children to build with a variety
of materials such as legos, or build
domino rallies to practice with spatial relationships.
- Have children explore tessellations
or complete tangram or pentomino puzzles to explore visual patterns. Have children make
tessellation stencils to replicate the patterns
or color the patterns in different ways. One
source of materials is www.deltaeducation.com
- Teach children finger-string games such as Cat’s Cradle. This helps
with spatial skills, following directions, and
sequencing the steps to complete the form. The
Cradle: A Book of String Figures is a valuable resource for teaching children
- To minimize visual distractions and confusion, limit the number of problems presented on a page or worksheet. Try to keep 4-5 problems on a worksheet.
Keep worksheets free of unnecessary decorations or information. This will help to maintain focus to the direction of the task.
Division Cue Card
- Teach students to make cue cards to help remember the steps to complete computation.
- To support fact knowledge in the classroom, provide access to number charts, multiplication grids, and calculators, such as the See N’Solve Calculator. This calculator shows the problem as the child enters the information. It is available at www.deltaeducation.com.
- Use grid paper to help children line up problems and limit visual confusion.


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CPM, or College Preparatory Arithmetic, is an instructional program designed to instruct learners not only how to resolve mathematical problems but also how to consider critically and work collaboratively. CPM math courses are extensively utilized in faculties throughout the United States, and the curriculum can be demanding. This is the place CPM Math Research Assist comes in helpful.
One particular of the primary benefits of CPM Math Research Support is that it gives learners with extra help outside the house of the classroom. Whether you are battling to comprehend a certain principle, require assist with a demanding problem, or basically want to reinforce your understanding of the content, CPM Math Homework Assist delivers the guidance and sources you require. With access to seasoned tutors and a prosperity of apply issues and explanations, you can tackle your research assignments with confidence.
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In the book Innumeracy, John Allen Paulos describes our problems in dealing with big numbers. I agree that we can, and should, learn to distinguish easily between a million, a billion, and a trillion. However, some numbers are just too large to grasp. We can’t really comprehend a number as large as a googol , or 10^100. And I don’t really know of any useful analogies that would allow me to compare the quantity represented by a googolplex, 10^(10^100), to that of a googol. These quantities, and the difference between them, are just so vast that we can’t really wrap our minds around them. There are only 10^78 or so atoms in the universe, and even that is a number which we cannot really fathom. But large numbers do pop up even when you ask relatively simple questions in mathematics.
For instance, in how many ways can you arrange a full deck of cards? You can easily show that the answer is 52 x 51 x … x 1, or 52! We can use Stirling’s approximation to estimate this number as sqrt(2*pi*52)(52/e)^52 which is approximately 8 x 10^67. This number is getting close to the number of atoms in the universe. Indeed, whenever you shuffle a deck of cards well, you can be nearly certain that nobody in the history of the world has played with the same arrangement before.
Large numbers appear frequently whenever we count the number of possible ways of arranging different objects. You have probably heard that if you put a number of monkeys in a room and let them pound away on typewriters, they will eventually produce all the works of Shakespeare. This may be true, but the amount of time it would take is not really comprehensible.
Let us look at a related question: In his story “The Library of Babel“, the Argentinian writer Luis Borges described a strange universe that consists of hexagonal cells lined by shelves full of books. Each book is unique and consists of different permutations of 25 letters and symbols. The narrator reveals that any possible arrangement of these 25 characters is contained somewhere in the library.
Borges writes that therefore:
Everything: the minutely detailed history of the future, the archangels’ autobiographies, the faithful catalogues of the Library, thousands and thousands of false catalogues, the demonstration of the fallacy of those catalogues, the demonstration of the fallacy of the true catalogue, the Gnostic gospel of Basilides, the commentary on that gospel, the commentary on the commentary on that gospel, the true story of your death, the translation of every book in all languages, the interpolations of every book in all books.
Taking into account Borges’ description of each book we can compute exactly how many of them are contained in the library: Each book has 410 pages, each page 40 lines, and line 80 characters. Therefore there are 410 x 40 x 80 = 1,312,000 characters in each book. The Library of Babel therefore consists of 25^(410x40x80) = 25^(1,312,000) books. (You can also check here for a more detailed explanation, or go here to download Borges’ number). I was trying to come up with a way to illustrate how large the Library or Babel. But I don’t think there is a useful analogy – the best we can do with numbers this big is to write them down. Perhaps my imagination is lacking, but I don’t see any way to illustrate their vastness other than essentially saying “This number is really, really big” in a different way.
This is pretty weird stuff. But these numbers are not really all that big. You can easily come up with much bigger numbers yourself. For example, you can order operations in an ascending order as follows: Repeated summation leads to multiplication, and repeated multiplication leads to exponentiation. Next repeated exponentiation give you towers of powers. You can keep on going, and soon you’ll reach operations that quickly produce immense integers (check out Ackerman’s notation to get a handle on how to write such numbers).
Still we have not come even close to infinity. Here is one way to see how far we are: We know that irrational numbers have a series of non-repeating digits. But there are different types of irrational numbers. For instance, you can just arrange 1s and 2s in a non-repeating pattern. Or you can take .1010010001000001000001… Since this expansion is not eventually periodic, this is an irrational number. However, the arrangement of digits in this number seems non-random. When we think of an arbitrary irrational numbers, we have a notion that the appearance of any digit, and group of digits, should be equally likely. Irrational numbers that have this property are called normal (I will actually consider numbers that are normal in any base. These are sometimes called absolutely normal numbers). More precisely, in the decimal expansion of a normal irrational number, each digit, each pair of digits, each triplet and so on appear with equal frequency. Interestingly, while we do know that almost all irrational numbers are normal, we don’t know whether specific irrational numbers are normal. Nobody knows whether pi or sqrt(2) are normal.
As a consequence, any and all sequences of digits appear in the expansions of normal numbers: Suppose that you have a normal number in base 25. We can map each of the numbers in this base to one of 25 characters that fill the books in the Library of Babel. That means that any normal number contains the text of all books in the Library of Babel. To see this, expand the number in base 25, convert the numbers to the corresponding characters, and break the expansion into blocks of 1,312,000 letters. Each such block corresponds to a book. Indeed, each book from the Library, and the entire Library itself appear in the expansion infinitely many times.
Let me finish with a last point that I owe to Lawrence Krauss. I recommend his book entitled A Universe from Nothing. Krauss talks a lot about quantum fluctuations which are present even in a vacuum. They allow particles to pop in and out existence all the time. Indeed, Krauss proposes that our entire universe may have simply popped into existence out of nothingness. Even more bizarrely, it is possible that any arrangements of particles may pop into existence out of nothingness, and disappear in the next instant. If you allow for infinitely many possibilities of this type, it means that we, along with all our memories, could be nothing more than a quantum fluctuation that exists only for a moment and disappears. Indeed, this may be a much more likely possibility than the alternative that we truly exist continuously in time.


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One of the most common questions when buying wall paper is to calculate the number of rolls required. In most cases, wall paper vendors seem to be doing complex calculations in calculating the amount of paper required, but you can easily count the number of rolls of wall paper you need by doing a few simple measurements and calculations.
۱ First, measure the height of the wall. (The height of each wall is typically 2.7 to 3 meters). Note that false ceilings and ceilings are not counted.
۲ Measure the widths of the walls you want and fit the sizes together. The walls in the window and the window, the size of the window and the window of the calculations is low.
۳ Now you have the length and height of a large rectangle that you want to cover with a wall paper.
۴ Each roll has a special width and length of the wallpaper, the size of each wall of the wallpaper will be examined according to the background characteristics of the paper.
The size of the usual rolls on the market:
Width 53 cm in 10 meters
Width of 70 cm along 8.5 meters
Width 90 cm 12.5 meters long
Width of 106 cm in 10 meters
Width of 64 cm in 10 meters
Width of 70 cm in 10 meters
Width 90 cm 12.5 meters long
Width: 106 centimeters across 15.5 meters
In the installation of wall paper, it is always tried to use a piece of paper that looks both prettier and less likely to be slipped off, from the top of the wall to the bottom, using a pad.
۵ Given the width of each roll, we divide the length of the rectangle into the roll width. By doing so, we’ll get the number of sip plates, that is, we’ll find that there are several lines of paper along the wall.
۶ Now we have to calculate how each roll fills a few tiles. We divide the length of the roll over the height of the room and round the resulting number downwards. The number of rolls will be obtained by dividing the total number of slabs by the number of rolls per roll. Round up the resulting number.
If your wall is 7.5 meters long, you will need 5 rolls of wallpaper up to 53 cm long, and for wall papers up to 70 cm, 4 rolls will be required, which will add 1 teaspoon.
Do not worry about excess rolls. In many cases, the amount of extra paper is used for some margins or the matching between the rolls of wallpaper. In any case, keep the amount of wallpaper to be used in the future in case of breakage or tear of paper.
In the figure below, the height of the room is 2.85 meters and the total length of the walls is 13.9 meters. You have selected a wall paper with width of 53 cm and a length of 10 meters. The number of slots you need is equal to 13.9 by 0.53, which is 26.3. We sum up this number up, equivalent to 27 beds for the whole room.
We calculate the number of swords in a roll, is equal to the result of the success of 10 to 2.85, which is 3.5. We sum up this number downwards, and weigh the equivalent of 3 troughs per roll. From the result of dividing the total number of slats (27 slabs) into a roll of a roll (3 slabs), the number of rolls is obtained, which is equal to 9 rolls.


Origami Assays Game
The following game uses a small design to illustrate how a pooled assay works. The game is played as follows:
Choose either 1 or 2 numbers between 1 and 13. Write them down.
Select which of the 10 wells contain your number.
The decoder guesses what numbers you have by highlighting them in green . Near matches will be highlighted in gold.
Why is this impressive?
While this is a small example, stop to consider what it is doing. You are allowed to pick 0, 1, or 2 positive samples and the design is able to decode these all with just 10 assays. If we count them there are:
0: 1 If you choose none, then this is 1 configuration.
1: 13 If you choose one, then there are 13 configurations.
2: 72 If you choose two, then there are 13*12/2=72 configurations.
This means that with just 10 assays we can uniquely identify any one of 86 possible configurations.
What about errors?
This small example also shows what happens if there is an assay failure. Try toggling one well to add in a false positive or false negative. You may find a few interesting outcomes:
No green entries, just gold:
This outcome would suggest an assay error because there are positive results but those results are not explained by one or more of the samples being positive.
Too many green entries:
If you choose 3 or 4 positive samples, the design becomes overloaded and starts to call many more samples as positive. These extra positive results are false positives.
Positive assays not associated with green entries:
There are cases where we have green and gold entries, but some of the gold entries are not "explained away" by the other assay results. This state also indicates an assay error.


Published in Cancer Weekly, January 10th, 2006
"We present a mathematical formulation to evaluate the effects of gestational mutations on cancer risk. The hazard or incidence function of cancer is expressed in terms of the Probability Generating Function (PGF) of the number of normal and mutated cells at birth," scientists in the United States reported.
"Using Filtered Poisson Process Theory, we obtain the PGF for several models for the accumulation of gestational mutations. In particular, we develop expressions for the hazard function when one or two successive mutations could...
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Learn more about a six-week, no-risk free trial of Cancer Weekly
NewsRx also is available at LexisNexis, Gale, ProQuest, Factiva, Dialog, Thomson Reuters, NewsEdge, and Dow Jones.


- Enter an amount of money (with no $ or ,) in the field below.
- Select one of the options below to let the calculator know how often one would receive this amount.
- Click Calculate.
I first created this wage calculator when I was job searching and wanted to know how much money I would earn per hour, week, month, and year, but didn't want to calculate them over and over. This calculator can help you determine your salary, budget, and plan for the calendar year. Please see the instructions below to continue.


Herman K. Fulmer was a professor of mathematics at Georgia Tech and acting head of the mathematics department in 1952. Born in Como, Mississippi, he received his B. S. degree from the University of Mississippi in 1921 and his Masters degree from Columbia University in mathematics in 1922. Subsequently, he joined the Georgia Tech faculty as an instructor in mathematics. Prof. Fulmer is a veteran of World War I and a member of the American Mathematical Society, Mathematical Association of America, Phi Kappa Phi, Phi Eta Sigma, Omicron Delta Kappa, ANAK, and the Georgia Education Association. He also was a member of the attendance standing committee of faculty with Skiles and Field. He taught college algebra and analytical geometry and published a book in college algebra in 1951 along with Walter Reynolds, another professor.
- Georgia Education Association


The Modern Slot Machine
Most modern slot machines are designed to look and feel like the old mechanical
models, but they work on a complete different principle. The outcome of each
pull is actually controlled by a central computer inside the machine, not by
the motion of the reels.
The computer uses step motors to turn each reel and stop it at the predetermined
point. Step motors are driven by short digital pulses of electricity controlled
by the computer, rather than the fluctuating electrical current that drives
an ordinary electric motor. These pulses move the motor a set increment, or
step, with great precision (see this page to find out more about step motors).
But even though the computer tells the reels where to stop, the games are not
pre-programmed to pay out at a certain time. A random number generator at the
heart of the computer ensures that each pull has an equal shot at hitting the
Whenever the slot machine is turned on, the random number generator is spitting
out whole numbers (typically between 1 and several billion) hundreds of times
a second. The instant you pull the arm back (or press the button), the computer
records the next few numbers from the random number generator. Then it feeds
these numbers through a simple program to determine where the reels should stop.
Here's how the complete process plays out in a typical three-reel machine.
You pull the handle, and the computer records the next three numbers from the
random number generator. The first number is used to determine the position
of the first reel, the second number is used for the second reel and the third
number is used for the third reel. For this example, let's say the first number
To determine the position of the first reel, the computer divides the first
random number by a set value. Typically, slot machines divide by 32, 64,128,
256 or 512. In this example, we'll say the computer divides by 64.
When the computer divides the random number by the set value, it records the
remainder of the quotient. In our example, it finds that 64 goes into 123,456,789
a total of 1,929,012 times with a remainder of 21.
Obviously, the remainder can't be more than 64 or less than 0, so there are
only 64 possible end results of this calculation. The 64 possible values act
as stops on a large virtual reel.
Each of the 64 stops on the virtual reel corresponds to one of the 22 stops
on the actual reel. The computer consults a table that tells it how far to move
the actual reel for a particular value on the virtual reel. Since there are
far more virtual stops than actual stops, some of the actual stops will be linked
to more than one virtual stop.
Computer systems have made slot machines a lot more adaptable. For example,
players can now bet money straight from a credit account, rather than dropping
coins in for every pull. Players can also keep track of their wins and losses
more easily, as can the casinos. The operation is also simpler in modern machines
-- if they want to, players can simply press a button to play a game, rather
than pull the handle.
For the manufacturers and slot proprietors, one of the main advantages of the
computer system is that they can easily configure how often the machine pays
out (how loose or tight it is).
So that leads us to the next section, that of .


Hey. Just learned about a cool data structure for computational geometry called, KD-Trees. Yes, it's sort of embarrassing that I've just now learned about them, but what can I say? I still think they are cool. Anyway, KD-Trees are like binary trees but for points in space. They make it really easy (well, fast) to do things like search for the nearest point in a set to some given point.
SO, I quickly hacked up a nearest neighbor application. It generates a bunch of random points, and then when you click highlights the nearest point in red. Yay.
Click here to download the JAR if you want to run it. It's on of those runnable Jars... remember those? Ah, Java... I tried to make it an Applet, but that process was even more ridiculous.


Spruce Mountain Middle School
After School Activities
Once again, ALL Spruce Mountain Middle School students have access to an online Math program called ALEKS (www.aleks.com). All students have login names and passwords. The program is web based, which means students can access it from any computer connected to the internet. Students will be using the ALEKS program in their math class and it creates an individualized math program for each student. After taking an assessment, ALEKS provides lessons on concepts that the student is ready to learn next. The program is a great way for students to work on math that is tailored to their individual needs. Many students use ALEKS at home and make great strides in their understanding of math. Students that spend 3-5 hours per week on ALEKS usually see improvement in their math performance. To maximize the effectiveness of ALEKS, using it at home at least an hour a week is recommended. The more students use ALEKS, the more growth they will see.
There is an after school ALEKS Math Club that meets on Monday and Study Buddies which meets Monday through Thursday available for students to work on ALEKS as well. Both programs are part of the Spruce Mountain Middle School After School Program. Math teachers, college students from UMF, and peer tutors will be available to work with students during these times. Students must sign up for the late bus before the close of school if they wish to stay after for either program.
New Dates To Be Set Soon
(FUN AND GAMES AFTER SCHOOL) WILL BE OFFERED TO ALL ELIGIBLE STUDENTS/ALL GRADES (No field hockey, soccer and cross country) on the following dates.
Activties will be offered from 2:10-3:10 and students will need to either ride the 3:10 late bus or have a ride home. They may also choose to stay and watch any home games that follow on that date. Students that need academic help should stay for Study Buddies before attending intramurals. Students that have ANY form of detention or other disciplinary consequences on the following dates are NOT allowed to attend.


Engineering Math - Matrix
Hermitian Matrix is a special type of matrix, which is same as its conjugate transpose as expressed below.
In orther words, a Hermitian Matrix has following properties
One example of Hermintian Matrix is as follows. In this example, if you conjugate the matrix A and then transpose it, the result is same as the original matrix (A). So you can say the matrix 'A' is a Hermitian matrix.
One of the important characteristics of Hermitian Matrix is that Eigenvalues of all Hermitian Matrix are all real as shown in the following example.
Related Reading : Hermitian Conjugate, Conjugate Transpose


The Marine Sciences and Applied Mathematics (MARMAT) double major program gives you the opportunity to earn two undergraduate degrees in four years: a Bachelor of Science degree in Marine Sciences and a Bachelor of Science degree in Applied Mathematics. The field emphasizes computer-based mathematical modeling and statistical analysis to address such problems as managing fish populations and ensuring a sustainable future for these species.
In this unique double major program, you'll spend the first two years completing fundamental mathematics and marine science coursework, leading to a culminating applied student research project in your senior year. UNE offers a large university marine science program at a student-centered university right on the coast. You'll also benefit from the Department of Mathematical Sciences’ integrated curricular approach that combines the latest technology with theoretical and applied mathematics to teach you the real-world applications of mathematics.


For additional digital leasing and purchase options contact a media consultant at 800-257-5126 (press option 3) or firstname.lastname@example.org.
The Standard Deviants travel from top to bottom and from high to low, all in search of the math concepts you need to know: tree diagrams and square roots, greatest common factors and least common multiples. Plus, get ready for some prime-time prime factorization.
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Infinity is a minimalistic looking calculator with an opinion. While providing all the basic functionality, this calculator tries to be helpful at the same time. Type in 3.14, and Infinity will fill in the rest of the decimals. Start with the Euler number, and the same thing happens. Try to divide by zero, and… oh no, actually, don’t try that!Details
Some of the projects that I've been working on lately - unfortunately I cannot share yet. What you see here are the ones that I've started myself, or I am free to publish.


Inspiring National Statistics Day Quotes to celebrate the day. National Statistics Day is celebrated on the 29th of June every year in India. The day commemorates the contributions of Prof. Prasanta Chandra Mahalanobis in the field of statistics. The objective of the day is to create awareness about the importance of statistics in our everyday lives. This day is all about celebrating the joys of stats and data. In honor of this special day, here are some of our favorite National Statistics Day Quotes.
These National Statistics Day Quotes have been said about statistics and its importance in our world. The day is celebrated to spread awareness about the importance of statistics in our everyday lives. The theme for this year’s National Statistics Day is not announced yet. In this blog post, we’ll be exploring some of the best National Statistics Day Quotes and their importance.
- Related: Statistics Albert Einstein Quotes
National Statistics Day Quotes
National Statistics Day Quotes that will make you think. National Statistics Day was created to acknowledge the important role that statistics play in our society. On this day, we reflect on the progress that has been made in the field of statistics and celebrate the contributions of those who have made statistical advancements possible. Here we gather the best National Statistics Day Quotes to increase your knowledge about statistics.
“Facts are stubborn things, but statistics are pliable.” ― Mark Twain
“Statistically speaking, there is a 65 percent chance that the love of your life is having an affair. Be very suspicious.”
“All the statistics in the world can’t measure the warmth of a smile.”
“A single death is a tragedy, a million deaths is a statistic.”
“If your experiment needs a statistician, you need a better experiment.”
“There are three types of lies, damn lies, and statistics.”
“Facts are stubborn things, but statistics are pliable.”
“Raise awareness about the critical role of high-quality official statistical information in our economy and celebrate this day.”
“If your experiment needs a statistician, you need a better experiment.” ― Ernest Rutherford
“99 Percent of all statistics only tell 49 percent of the story.”
National Statistics Day Message
National Statistics Day Messages you need to see for this modern life. From providing a snapshot of the country’s social and economic health to helping assess progress towards the Sustainable Development Goals, official statistics play a vital role in decision-making, policy-making, and planning. Statistical data plays an important role in our daily lives. It helps us to understand the world around us and make informed decisions. To mark this occasion, we’ve put together the most interesting Quotes about statistics.
“The greatest value of a picture is when it forces us to notice what we never expected to see.”
“Find the truth of your life. Logically calculate and analyze to understand and get a clearer picture of your life.”
“Statisticians like artists have the bad habit of falling in love with their models.” Happy National Statistics Day.
“Statistics is a catalyst for the modern life and the basis of government and community planning. Maintain it.”
“Bring everyone together, raise awareness about the importance of statistics and create a happier and prosperous future with better data.”
“For an ability to read write and properly understand, statistical knowledge is important. So, educate yourself and celebrate.”
“Let the logistics and number take control of your life, educate yourself about the statistics and make your life easier and happier.”
“Remember the importance of statistical information for analysis and informed policy decision-making in support of sustainable development and root for it.”
National Statistics Day Wishes
National Statistics Day Wishes to wish your friends who are related to this field. National Statistic Day is celebrated every year. From the mathematical to the personal, statistics play a role in our understanding of the world and the universe. Let’s take a moment to celebrate some of the most interesting statistics out there. Statistics play a vital role in They help us to make sense of the patterns and relationships in data. Wishing the men and women who have dedicated their lives to this field of study. Here are some National Statistic Day Wishes that can help you to wish others on this day.
“Experience your life with a series of calculations and a total of statistics and celebrate this occasion of World Statistics Day.”
“For a better life for the people, get reliable statistics. Educate yourself and raise awareness all across the globe.”
“Prediction about everything is difficult, especially about the future. So, make life easier by being aware and logical.”
“Statisticians like artists have the bad habit of falling in love with their models.”
“Better data leads to better lives for everyone. Best wishes to you on this occasion of World Statistics Day.”
“Facts are stubborn things, but statics are pliable.”
“Statistics are the triumph of the quantitative method, and the quantitative method is the victory of sterility and death.” Happy National Statistics Day.
History of National Statistic Day of India?
National Statistics Day is celebrated in India on the 29th of June. The day commemorates the birth anniversary of the late Prof. Prasanta Chandra Mahalanobis, fondly remembered as the father of Indian statistics. The Department of Statistics, Ministry of Statistics and Programme Implementation, Government of India, has been celebrating this day since the year 2009. The purpose of this day is to create awareness about the importance of statistics in our daily lives.
Importance of National Statistic Day of India?
The day provides an opportunity to create public awareness about the importance of statistics in our daily lives and its role in the development of the country. The purpose of this day is to encourage better understanding and use of statistics in all walks of life. On this day, we honor the contributions of statisticians to the field of science and society as a whole.
This blog post is a collection of National Statistics Day Quotes. National Statistics Day is an important day to recognize the significance of statistics in our lives. The quotes provided in this article show the value that statistics have in helping us understand the world around us. Like, comment, and share this article to spread awareness of the importance of statistics. Follow us for more content like this!


I am a data scientist working in finance on the development of state-of-the-art portfolio construction algorithms and trading signal generation. I hold a PhD in Applied Mathematics from Western University where I developed algorithms for Bohemian matrices.
I completed my PhD under the supervision of Prof. Robert M. Corless and Prof. Marc Moreno Maza. My thesis developed new algorithms for computations on matrices with parameters. These algorithms were implemented in the Maple computer algebra system and are available in the ParametricMatrixTools package. My thesis also laid ground work for research on Bohemian matrices. Further information on Bohemian matrices can be found at BohemianMatrices.com. My thesis can be found here.
When I'm not learning new things I enjoy long-distance running, camping, woodworking, rock climbing and cooking.


Learn how to draw a hexagon in two ways. Construct a precise hexagon using a compass or learn how to draw hexagon from six triangles by hand.
The first way how to draw a hexagon is to construct it precisely using a compass and ruler. That way we get a geometrically perfect hexagon.
Our second hexagon is drawn by hand and built up from six triangles of the exact same shape. This is an easy technique how to draw a hexagon with a reasonably regular shape by hand.
These are the favorites we use for drawings on Let's Draw That that we love and recommend:
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Love drawing? Never miss a tutorial!
How to draw a hexagon with a compass
Our first method of how to draw a hexagon uses a compass and a ruler for precise construction – we get a geometrically perfect shape. All sides of the hexagon are the same length – exactly half of the diameter of the circle it is inscribed in. That makes it easy to split the circle regularly and mark the required points on the circle with a compass. Let’s get into it.
Hexagon drawing with a compass
Step 1: Draw a circle
First draw a cross – two guidelines of the same length, crossing at a right angle (90 degrees). These will form the base of our construction.
Then take your compass and draw a circle from the centre where the two lines cross.
If you are aiming to construct a hexagon with a given side length, the radius of the circle, i.e. the opening of your compass, is the given length of the side of a completed hexagon.
Step 2: Mark the top two corners
Next, we will mark the top two corners. Leave your compass set to the same radius, and draw a semicircle from the top intersection of our circle and the vertical cross line.
If you are aiming to erase all these construction lines later, just mark the spots where the semicircle intersects with the circle.
We have drawn the full half-circle here just to show the fact that we still draw with the same radius, and to make the explanation a bit more obvious.
Step 3: Repeat and mark the bottom two corners
We repeat and draw the half-circle with exactly the same radius from the bottom intersection of our circle and the vertical cross line. That marks the remaining two points of our hexagon at the bottom half of the circle.
Again, feel free to just mark the spots, rather than drawing the full semicircle.
Step 4: Connect the marked points to draw a hexagon
On this picture we have highlighted the six intersecting points that will make the corners of our hexagon.
All that is left is to connect them all with straight lines and we have constructed a perfect regular hexagon.
Enjoying the geometry? Here is another technical drawing for you:
How to draw a hexagon with a compass: finished drawing
How to draw a hexagon with a compass: completed perfect hexagon
Here is the complete geometrically perfect hexagon.
If you would want the hexagon to sit on the flat side, rather than stand on its corner, you can use the same technique. Just draw the half circles from the left and right, rather than the top and bottom intersection of the cross and circle.
And finally, if you do not have a compass on hand: You can get pretty close using anything round, such as the top of a glass, to draw your circles. Just try to make the top and bottom semicircle meet as closely as possible to the centre of the full circle.
Now you have learned an easy method of how to draw a hexagon using the compass to construct a perfectly regular hexagon.
How to draw a hexagon by hand (without a compass)
In a regular hexagon, a length from the centre to any corner of the hexagon is the same length as any side. Because of this, a regular hexagon can be thought of as being made of six equilateral triangles – triangles with all three sides of the same length. Let’s use that knowledge to learn how to draw a hexagon by hand.
Drawing hexagon by hand
Step 1: Draw a triangle
So let’s draw the first triangle. As mentioned, all three sides are the same length. The internal angle of a equilateral triangle is 60 degrees. That makes our triangle a nice and symmetrical pyramid shape.
Step 2: Add two more upside down triangles
Next we add two more triangles of the same size and shape . These will be drawn upside down, one on each side of our starting triangle.
Probably the easiest way is to draw a horizontal line through the top of the starting triangle. Then extend the sides, until they meet with the horizontal line.
Try to mirror the triangles as close as possible to the original – size and all angles.
Step 3: And two more triangles on top
Let’s add two more triangles – by now you are getting professional at drawing equilateral triangles.
These two stand back on the flat side, with their points up again, same as our first triangle.
The way to draw these is to extend the side lines of our first triangle past the centre of the hexagon. Then draw the outer sides. Again, try to mirror the bottom triangles as closely as possible.
As you can see, after I drew the outer sides, I had to adjust the inner ones a bit. Now the top points of my triangles line up nicely with the mirror triangles at the bottom half of the hexagon.
Step 4: Complete the hexagon
Well, this is the easy bit. To close the last triangle, we just add a simple line at the top. That completes the hexagon drawing.
Here is another shape to construct next:
How to draw a hexagon by hand: finished drawing
Completed hand-drawn hexagon
Finally, let’s outline our hexagon. You can erase the triangle guidelines if you plan to use your hexagon in some further drawing.
As you can see, while the shape is not as perfect as the one constructed with the compass, we have still drawn a pretty regular hexagonal shape just by hand.
And if you feel like practising your hexagons on some real drawing, how about drawing a turtle? The entire turtle shell is made of hexagons!


Maths Curriculum Statement
The 2014 National Curriculum for Maths aims to ensure that all children:
- Become fluent in the fundamentals of Mathematics
- Are able to reason mathematically
- Can solve problems by applying their Mathematics
At St Andrew’s Infant School these skills are embedded within Maths lessons and developed consistently over time. We aim for each child to be confident in the yearly objectives and develop their ability to use this knowledge to solve varied fluency problems as well as problem solving and reasoning questions, equipping them with the skills they will need in life beyond school.
We want all children to enjoy Mathematics and to experience success in the subject.
Our implementation is developed through secure understanding of the curriculum and subject area.
- Long term: National Curriculum.
- Medium term: Lancashire Planning Disc provides a Yearly overview.
- Short term: - Short term planning is supported by the use of the Lancashire Planning Disk materials, our school calculation policy and Year 1 are currently trialing Maths Mastery 1 materials from the Lancashire Maths team. Online resources are also used to ensure a curriculum that is specific to each child’s needs.
Daily lessons are taught across EYFS and KS1.
Planning using the Lancashire Maths disc and Maths Mastery materials ensures that time is built in for regular revisiting of learning throughout the year and there is a good balance of content between e.g. number, geometry, statistics and measurement.
Children explore maths using concrete materials. Pupils are encouraged to physically represent mathematical concepts. Objects and pictures are used to demonstrate and visualise abstract ideas, alongside numbers and symbols.
Concrete – children have the opportunity to use concrete objects and manipulatives to help them understand and explain what they are doing.
Pictorial – children then build on this concrete approach by using pictorial representations, which can then be used to reason and solve problems.
Abstract – With the foundations firmly laid, children can move to an abstract approach using numbers and key concepts with confidence.
Through our teaching we continuously monitor pupils’ progress against expected attainment for their age, making formative assessment notes where appropriate and using these to inform our discussions in termly Pupil Progress Meetings and update our summative school tracker. The Lancashire tests are used in Key Stage 1 at the end of each term to support teacher judgements. The main purpose of all assessment is to always ensure that we are providing excellent provision for every child.
Children in Years 1 and 2 complete their homework activities using the online homework resource SUMDOG which aims to build pupil engagement and consolidate maths knowledge. Children also use the SUMDOG online resource to play games and enter competitions both at home and within school to develop a love of maths.