Yozshuzil Cofactors and Minors 1. Ivan Cherevko rated it really liked it Aug 04, The author begins with elementary material and goes easily into the advanced areas, covering all the standard topics of an advanced undergraduate or beginning graduate course. Ordinary Differential Equations M. No trivia or quizzes yet.
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INTERMEDIATE ALGEBRA 8TH EDITION AUFMANN AND LOCKWOOD PDF
Donald Steiny rated it really liked it Feb 08, This was a solid text for an introduction to linear algebra. Preview — Linear Algebra by Georgi E. Practical Evaluation of Determinants 1. Refresh and try again.
LINEAR ALGEBRA GEORGIE.SHILOH PDF
Tygojinn The Algebra of Polynomials 6. Other books in this series. Transformation of the Components of a Vector 5. There are no discussion topics on this book yet. The Case of Complete Algebras A. Canonical Form of the Matrix of an Arbitrary Operator 6. Dustin Burt rated it really liked it Jun 21, Shilov, Professor of Mathematics at the Moscow State University, covers determinants, linear spaces, systems of linear equations, linear functions of a vector argument, coordinate transformations, the canonical form of the matrix of a linear operator, bilinear and quadratic forms, Euclidean spaces, unitary spaces, quadratic forms in Euclidean and unitary spaces, finite-dimensional algebras and their representations, with an appendix on categories of finite-dimensional spaces.


Playing online slot machines is now increasingly popular, as online casinos have become in popularity. This growth in online gaming has seen a rise in the amount of players looking for a simple way to hit the million jackpots and become among the few high rollers who flourish in online slots. The majority are tempted to get an on the web slot system which claims to be able to make the purchaser regular huge profits. The fact of online slot machine systems however, is that the claims don’t match the hype. Slot machines remain games of chance, and the same as roulette and craps, there is no system that may guarantee you regular jackpots. Don’t buy an on the web slot machine system. Continue reading and find out why!
Fact: You Cannot Apply a System to Online Slots to Make Regular Income
There’s no way to create guaranteed profits from mathematically detrimental games, and online slot machines are such games. In mathematics, you know joker slot exactly what’ll happen. Games of chance are the actual opposite. You never know what’ll happen next. In the event that you did, then obviously, it would not be considered a game of chance. Online slots really are a game of chance, so mathematical systems can’t be applied. Period.
Online Slots Do Work To A Mathematical Formula!
The winning combinations produced by online slot machines are generated by a Random Number Generator (RNG). In online slot machines, RNG’s aren’t truly random, since they are the result of a mathematical process. In the event that you knew the formula utilized in any online casino slot machine and the value of the last random number generated, you would be able to calculate the next random number that could be generated, but obviously, you can’t. Why? This is because the speed at that the RNG calculates winning combinations. The RNG is truly a number of codes written into the program of the game chip. It generates numbers and it will it very quickly. Actually, at least 100 numbers every second may be generated. In an on the web casino slot machine, each one of those numbers corresponds to a result on the reels. The aftereffect of this for the gamer is just a random choice from a subject of numbers that may determine the results of the play.
Why You Can’t BEAT Online Casino Slot Machines
Online slot machines RNG’s create a random generation of lots from the field of numbers in this system, at least every one-hundredth of a second. The RNG is definitely generating numbers even though it’s idle. Even if the programmer of the internet slot machine knew the sequence in that the numbers are now being generated, by the time he calculates what the next number is the device can have managed to move on, as we all know all computers can crunch numbers quicker than any person. Although it isn’t totally random by the type of its programming, a programmer even though he knew the sequence would not find a way keep up with the device, just what exactly chance would a new player have?


Support vector machines present a unique combination of several key concepts :
- Maximum margin hyperplane to find a linear classifier through optimization
- Kernel trick to expand up from linear classifier to a non-linear one
- Soft-margin to cope with the noise in the data.
This chapter describes the derivation of the support vector machines as the maximum margin hyperplane using the Lagrangian function to find a global extreme of the problem. First, hard-margin SVM applicable only to linearly separable data is derived. Primal and dual form of the Lagrangian optimization problem is formulated. The reason, why the dual representation poses significant advantage for further generalization is explained. Further, a non-linear generalization using kernel functions is described and the “soft-margin” variation of algorithm, allowing for noise, errors and misclassification, is finally derived.
Maximum margin hyperplane
Let us start with the assumption, that the training data set is linearly separable. We are looking for the hyperplane parameters , so that the distance between the hyperplane and the observations is maximized. We shall call the Euclidean distance between the point and the hyperplane as geometric margin.
Note, that the equation (1) really does correspond to the perpendicular distance of the point,8 only the multiplier seems to be superfluous. But recall that assigns either +1 or -1. Its only purpose in the equation is to assure the distance to be a non-negative number.
Definition 2.2. Given the training set D, the geometric margin of a hyperplane with respect to D is the smallest of the geometric margins on the individual training observations:
If we are looking for the optimal separating hyperplane, our goal should be to maximize the geometric margin of the training – i.e. to place the hyperplane such that its distance from the nearest points will be maximized. However, this is hard to do directly as such an optimization problem is a non-convex one.  Therefore, the problems needs to be reformulated before we can move forward.
Definition 2.3. The functional margin of an individual observation is given by
Definition 2.4. Given a training set D, the functional margin of with respect to D is the smallest of the functional margins on the individual training observations.
Also note, that the geometric margin is scaling invariant – replacing the parameters with , for instance, will not change the results. The opposite is true for the functional margin. Therefore we may impose an arbitrary scaling constraint on conveniently to suit our needs. We will set the scaling so that the functional margin is 1 and maximize the geometric margin under this constraint.
When we want to maximize the geometric margin, from (5) it follows that we need to maximize the term . Such a task is equal to the minimization of in the denominator. Yet this task is still quite a difficult one to solve, as by (1.1) it would involve the terms in a square root.
Nevertheless, it is possible to replace this term by without changing the optimal solution.9 The multiplier is added for the mathematical convenience in later steps and again, does not affect the position of the extreme.
Finally, applying (1.3), the objective function to be minimized is
It is necessary to ensure that the equation (4) holds when the objective function is minimized. The functional margin of each individual observation must be equal to or larger than the functional margin of the whole dataset.
By substituting (6), we have
The solution of this problem will be a linear decision surface with the maximum possible margin (given the training dataset). The objective function is clearly a convex function which implies that we are able to find its global maximum. The entire optimization problem consists of a quadratic objective function (7) and linear constraints (9) and is therefore solvable directly by the means of the quadratic programming. [28, p. 82]
Lagrangian methods for optimization
Since the means of the quadratic programming are usually computationally inefficient when dealing with large data sets10, we will continue solving this optimization problem using the Lagrangian methods.11
Let us start more generally and then apply the mathematical theory to the problem of finding the maximum-margin hyperplane.
Consider the optimization problem
The minimization is constrained by an arbitrary number of equality and inequality constraints.12 One method to solve such a problem consists in formulating a derivative problem, that has the same solution as the original one. Define the generalized Lagrangian function of (10) as
This allows as to formulate the so called primal optimization problem, denoted by subscript on that account:
Note that the function value assumes plus infinity, in case any constraint from (10) does not hold. If for any , (12) can be maximized by simply setting . If on the other hand , (12) is maximized by setting to plus or minus infinity (depending on the sign of the function ).
If all the constraints hold, takes the same value as the objective function in the original problem, . Provided both constraints hold, the third equation term in (11) is always zero and the second term is also maximized when it is set to zero by appropriate setting of weights . Thus we have
We could solve the primal problem, but when applied to our task, the properties of the algorithm would not bring much benefits. Instead, for the sake of the forthcoming solutions, let us define a different optimization problem by switching the order of minimization and maximization.
The subscript stands for dual and it will be be used to formulate the so called dual optimization problem:
The value of the primal problem and the dual problem is not necessarily equal. There is a generally valid relationship between maximum of minimization and minimum of maximization
The difference between primal and dual solution, is called the duality gap. As follows from (17), the duality gap is always greater than or equal to zero.
Certain conditions have been formulated that (if met) guarantee the duality gap to be zero, i.e. under such conditions, the solution of the dual problem equals to the solution of the primal one.
Definition 2.5. Suppose and all functions are convex and all are affine. Suppose that the constraints are strictly feasible (i.e. there exists some so that for all ). Then there exist , , so that is the solution to the primal problem, , is the solution to the dual problem, and satisfy following Karush-Kuhn-Tucker (KKT) conditions
Remark: For its specific importance, KKT condition (20) is often referred as complementary condition.
Application to maximum margin hyperplane
Substituting these equations into (24) and simplifying the result
This equation was obtained by minimizing with respect to and .  According to (16), this expression needs to be maximized now. Putting it together with (22) and (26), the dual optimization problem is
One interesting property should be pointed out at the moment. The constraint (9) actually enforces the fact that the functional margin of each observation should be equal to or greater than the functional margin of the dataset (which we set by (6) to one).
It will be exactly one only for the points closest to the separating hyperplane. All other points will lie further and therefore their functional margin will be higher than one. By the complementary condition (20) the corresponding Lagrangian multipliers for such points must be equal to zero. [14, p. 97]
Therefore the solution of the problem, the weight vector in (25), depends only on a few points with non-zero that lie closest to the separating hyperplane. These points are called support vectors. [14, p. 97]
The solution of the dual problem (28) leads to an optimal vector , which can be immediately used to calculate the weight vector of the separating hyperplane using (25). Since most of the coefficients will be zero, the weight vector will be a linear combination of just a few support vectors.
Having calculated the optimal , the bias must be found making use of the primal constraints, because the value does not appear in the dual problem: [14, p. 96]
From (1.9) this leads to a decision function that can be used to classify new out-of-sample observation :
As stated above, this classifier is determined completely only by the support vectors. Because of this property, it is called a support vector machine. To be more precise, a linear support vector machine, since it is based on a linear decision surface. [28, p. 102] It is able to classify linearly separable data only. On the other hand, unlike some other classifiers, it is guaranteed to converge and find the hyperplane with the maximum margin. Since the optimization problem is a convex and KKT conditions are met, there is always a unique solution and it is a global extreme.14
There is one remarkable property in the decision function (30) that is worthy of our special attention. The observations only appear there as dot products between the input vectors. This opens a door to the use of a special technique that researcher usually refer as “kernel trick”. Owing to it, the linear support vector machine derived in this section can be extended and generalized to fit almost any non-linear decision surface.
Kernels and kernel trick
The problem with the real life data sets is that the relationship among the input variables are rarely simple and linear. In most cases some non-linear decision surface would be necessary to correctly separate the observations into two groups. This necessity, on the other hand, noticeably increases the complexity and computational difficulty of such a task.
Instead, the support vector machines implement another conceptual idea: the input vectors are mapped from an original input space into a high-dimensional feature space through some non-linear mapping function chosen a priori. The linear decision surface is then constructed in this feature space.
The underlying justification can be found in Cover’s theorem , which may be qualitatively expressed as:
A complex pattern-classification problem, cast in a high-dimensional space nonlinearly, is more likely to be linearly separable than in a low-dimensional space, provided that the space is not densely populated. [34, p. 231]
Because now we want to operate in the feature space rather than in the original input space, we can slightly modify the original equation (28) and replace the plain dot products with . By we denote some appropriately chosen mapping function.
Consider that the transformation may be very complex, complicated. The results could belong to high dimensional, in some cases even infinitely dimensional spaces. Such transformations would be computationally very expensive or even impossible to do.
The key idea of this concept is contained in the fact, that to find the maximum margin hyperplane in the feature space, one actually need not to calculate the transformed vectors explicitly, since it is present only as a dot product in the training algorithm.
It would be very wasteful to spend a precious computational time to perform these complex transformations only to aggregate them immediately in a dot product, one single number.
It appears that for our needs it is sufficient to use a so called kernel function which is defined as a dot product of the two transformed vectors and which may be calculated very inexpensively.
Definition 2.6. Given an appropriate mapping with , the functions of the form
where are called kernel functions or just kernels. [28, p. 107]
The process of mapping the data from the original input space to the feature space is called kernel trick. [28, p. 103] The kernel trick is rather universal approach, its application is not solely limited to the support vector machines. In general, any classifier that depends merely on the dot products of the input data may capitalize on this technique.
Let us rephrase the support vector machines using the kernel functions. The training algorithm 28 can be now expressed as
while the decision surface (30) in terms of kernel function is defined as
There are several well known kernel functions that are commonly used in connection with support vector machines.
The first, rather trivial, example is a linear kernel. It is defined simply as
In this case, the feature space and the input space are same and we get back to the solution derived in the previous Section, where we discussed linear support vector machines.
I mention the linear kernel here just to emphasize the fact, that the substitution with the kernel function does not change the original algorithm entirely. It merely generalizes it.
The polynomial kernels are another example of frequently used kernel functions. When Vapnik and Cortes in their essential paper  achieved a great shift in the optical character recognition of handwritten numbers, it was the polynomial kernel that showed its strength for this particular task.
The homogeneous polynomial kernel of degree () is defined as
The more general non-homogeneous polynomial kernel () is
Note, that the latter maps the original input space to the feature space of the dimensionality , where is the degree of the kernel and is the dimension of the original input space. However, despite working in this -dimensional space, computation of consumes only time.
Gaussian kernel (Radial basis function)
The gaussian kernel is probably the mostly used kernel function and is defined as
It maps the input space to the infinitely dimensional feature space. Thanks to this property, it is very flexible and is able to fit a tremendous variety of shapes of the decision border. The gaussian kernel is also often referred as radial basis function (RBF). In some instances is parametrized in a slightly different manner:
There are many other well researched kernel functions. An extensive, although probably not exhaustive, list of other commonly used kernel functions is available in . Most of them are, however, used only for research purposes or under the special circumstances and have no use in the credit scoring models.
For any function K to be a valid kernel, a necessary and sufficient condition has been derived. This condition is usually referred as Mercer’s theorem, which is formulated in a slightly simplified form below .
Definition 2.7. Let a finite set of points be given and let a square -by- matrix be defined so that entry is given by . Then such a matrix is called Kernel matrix.
Theorem 2.1. (Mercer’s theorem)
Let be given. Then for to be a valid kernel, it is necessary and sufficient that for any , the corresponding kernel matrix is symmetric positive semi-definite.
There are no general rules concerning the selection of a suitable kernel. It highly depends on the character of the task being solved, the data set and the relationships among the variables play a major role, too. In case of simple jobs with plain linear relationships, the linear kernel may be sufficient and other kernels will not bring any improvement.
In other applications, where the data are complex and interdependent, more advanced kernels may bring a major leap in the classification performance. This may be the case in the aforementioned optical character recognition (OCR), face recognition, natural language processing or genetics.
The credit scoring lies kind of in the middle of these extremes. Some papers suggest, that the consumer credit scoring data tend to be linear or just mildly non-linear. Therefore, the linear support vector machines predominate in the literature, often accompanied with the Gaussian kernel to seek for the non-linearities in the data (see  for comprehensive list of papers). The polynomial kernels are rarely represented in the credit scoring literature (see ,  for examples) as they seem to give the mixed results.
Soft margin classifiers
Until now we have assumed that our training data are linearly separable – either directly in the input space or in the feature space. In other words we have supposed only perfect data sets. In reality the data sets are often far from perfect. The real data are noisy, contain misclassified observations, random errors etc.
We need to generalize the algorithm to be able to deal with the non-separable data. Surprisingly enough, the basic idea of the generalization is very simple. We just need to allow some observations to end up on the “wrong” side of the separating hyperplane.
The required property is accomplished by introducing a slack variable and rephrasing the original constraint (9)
where . This way we allow the functional margin of some observations to be less than 1.
The slack variable works like a corrective equation term here. For the observations located on the correct side of the hyperplane, the respective and the (39) is then same as (9). For noisy or misclassified observations appearing on the wrong side of the hyperplane, the respective , as shown in Figure 5.
It is necessary that with each misclassification we pay some price, since in general we want to maximize the margin while simultaneously minimize the adverse effect of the wrong observations. Otherwise the trivial solutions where all data are marked as wrong would always prevail. This is why we have to expand the original objective function by the penalty term
where is the cost parameter, which determines the importance of wrong observations. Small values of C lead to more benevolent solutions while the larger values of C would lead to solutions similar to the hard margin classifier from the previous sections.
This time due to the limited space I will not go again through the whole process of deriving the dual of this problem (for the step by step solution see [28, p. 118-121]. It turns out, that the dual of the soft margin classifier simplifies to a very convenient form:
Note that the newly inserted cost term does not appear in the objective function and poses only a restriction for the values of the individual coefficients.
Thanks to this property and combined with the kernel trick, the soft margin support vector machines are very easily implemented and solved efficiently on the computers without requiring a tremendous computational power.
The results, again, will be the decision function (30). This function produces results on the simple yes/no scale. In credit scoring we often need more than that: we need to quantify the quality of the debtor as a real-valued number which is then called score in this context. This may come handy in tasks like calculating the area under curve or finding the optimal cut-off value, which both will be discussed later in the text.
For this purpose, the functional margin of the observation can be used. Thus, the score of the observation will be simply defined as
where , are the parameters of the optimal hyperplane calculated by the support vector machines, , are the individual observations from the training set and is the number of observations in the training data set.
The simplest approach to solve a convex optimization problem numerically on computers is the gradient ascent, sometimes known as the steepest ascent algorithm. The method starts at some arbitrary initial vector (e.g. rough estimate of the possible solution) and proceeds iteratively, updating the vector in each step in the direction of the gradient of the objective function at the point of the current solution estimate. [14, p. 129]
But using the gradient methods to solve (41) brings problems which are hard to be circumvented. The main trouble is caused by the fact, that the constraint must hold on every iteration. From this constraint it follows, that the elements of vector of solutions, are interconnected and cannot be updated independently. It also implies, that the smallest number of multipliers that can be optimised at each step is 2. [14, p. 137]
This key fact is used by the algorithm Sequential Minimal Optimisation (SMO) which, in the course of time, was established as the optimal way to solve the problem (41). At each iteration, the SMO algorithm chooses exactly 2 elements of the optimised vector, and , finds the optimal values for those elements given that all others are fixed and updates the vector accordingly. [14, p. 137]
In each step, the vector and are chosen by the heuristics described in  by Platt, who first introduced the SMO algorithm; the optimisation itself is however solved analytically in every iteration. The introduction of SMO removed the main barrier for support vector machines to be widely adopted by the researchers, as it sped up the training process at least 30 times against the methods used before.
Since then, the SMO efficiency was further improved and it quickly became a standard for the support vector machines training.
Because of the fact, that the actual process of implementation of the training algorithm is rather tedious, prone to errors and represents more just a technicality rather than actual research value, the standardized packages and libraries to support the SVM training and evaluation were created.
Among them, LIBSVM  is the most prominent one, with the longest history, continuous updates, associated research paper and data sets and widest support for different development environments (IDEs). Other libraries, like SVMlight15, mySVM16 or the specialised toolbox for R17, Torch18 or Weka19 do exist.
Matlab, which was used for the practical part of my thesis, has its own support for SVM. However, its performance and the possibilities it offers are not astonishing. For that reason, I decided to use the LIBSVM library as an extension for the standard Matlab functions.
8 See [48, p. 161] for details and proof.
9 Optimization over the positive values is invariant under the transformation with the square function. In general, is a monotonic function for . Therefore optimizing over is same as optimizing over . [28, p. 81]
10 thousands or tens of thousands of observations
11 This whole section of my work is based mostly on the online videolecture of professor Andrew Ng from the Stanford University  who (along with many other lecturers from the most prominent universities) made his knowledge freely available under the Creative Commons license to people from all over the world. For that I express him (them) my deepest gratitude.
12 Note, that (10) embodies a very broad class of problems. For example, any maximization problem can be transformed to the minimization, since or , if defined. Also any constrained in the form can be converted to the form of (10) multiplying by (-1) and every constraint in the form can be transformed by subtracting to the desirable form.
13 Note, that there are no equality constraints in this task, so there will be no terms.
14 Even some advanced classification algorithms, like neural networks, must face the problem that they may get “stuck” in the local extreme and find only a suboptimal solution.
17 SVM in R, http://cran.r-project.org/src/contrib/Descriptions/e1071.html
18 SVMTorch, http://www.idiap.ch/learning/SVMTorch.html


Fast Fourier Transform (FFT) in IDL
When processing a time series of data, one of the main analysis consists of
finding whether the series presents some periodic behaviour. One way to analyze
the periodicity is to do a spectral analysis. In this work, I try to understand
some of the features presented by the IDL FFT routine and give some tentative
notes on its use.
IDL provides a build-in function called FFT to be used as
result = FFT(series)
A time series result of a sum of various sinusoidal functions of different
frequencies is generated.
"generate_sin_data.pro" is a simple IDL routine which generates a long time series
with sampling every half an hour (30 min). The series is made of 4 sinusoidal
frequencies at 0.5, 1, 10, and 36 day period (2, 1, 0.1, and 0.0278 cycle per day).
Series of 120 days and of 360 days are studied in this work.
The 120-day series as well as its corresponding power spectrum are illustrated in
Figure 1. In this case, both spectrum with and without Hanning windowing are shown.
Clear narrow peaks corresponding to 2 frequencies (2 and 1 cycles per day) are observed.
Another frequency 0.1 cycle per day is also present but with a certain degree of
spread. Those three frequencies correspond to 12, 24, and 240 hour (0.5, 1, and 10 day)
period. Finally, there is the very spread peak at 0.025 frequency (or 40 day period).
As we can see, the first 2 frequencies are well defined and set to the right
original frequencies, the third is also defined at the right original frequency;
however, it starts showing some leakage phenomenon. The last one is defined with
more leakage than the previous spike, its period is actually shifted by 4 days.
From this data, we can see that a series of 120 days allows to clearly extract
periods of less than few days. At 10 day period, the spectrum starts to show
leakage. The period which corresponds to 1/3rd of the length of the series are
there but not well defined anymore.
When processing a series of 360 days of data, the corresponding power spectrum
graph is given in Figure 2. One can see that some noises show up at high frequencies.
In addition, the low frequency peak now shows up at 0.0278 cycle per day (~ 35.98
days closer to the real frequency). The way this frequency is spread is the same
as in Fig. 1 for frequency at 10 days.
The following conclusion is drawn by looking at the result. A furher and more
in-depth investigation might be necessary to confirm it.
We are tempted to conclude that for a series of data, periods above 1/12 of
the length of the data will start to show up with some sort of leakage
(illustrated as spreading in the power spectrum graph).
The noises at high frequencies can be an artifact of the IDL FFT routine. Even by
low-pass filtering the data, we still have almost the same power spectrum graph.
The following IDL routines were written and used to carry out the above study:


Converting between different units of measurement can be a tricky task, but fortunately, it’s not as complicated as it may seem at first glance. If you’ve ever wondered how many ounces there are in .6 pounds or need to make this conversion for a recipe or other purpose, you’re in the right place.
First off, let’s clarify what these units represent. “Lb” is an abbreviation for “pounds,” a unit of weight commonly used in the United States and some other countries. “Oz” stands for “ounces,” another unit of weight that is frequently used for smaller items like spices, grains, and liquids.
So how many ounces are in .6 pounds? The answer is 9.6 ounces. This conversion is based on the fact that there are 16 ounces in one pound, which means that .6 pounds is equal to .6 multiplied by 16 (or 9.6) ounces.
Check out our next post for more interesting content!
.6 lb to oz
When it comes to converting .6 lb to oz, the process is quite simple. A pound is a unit of weight in the imperial and US customary systems of measurement, while an ounce is a smaller unit of weight commonly used for measuring smaller commodities such as spices or precious metals.
To convert .6 lb to oz, all you need to do is multiply the value in pounds by 16. With .6 representing the number of pounds, we can easily multiply it by 16 to get the number of ounces. The calculation looks like this:
Converting .6 lb to oz: A Quick Guide
So, .6 lb is equivalent to 9.6 oz. That’s all there is to it. The process involves a simple multiplication operation that can be done quickly and easily using a calculator or even manually.
It’s important to note that in some cases, you might need to round up or down the number of ounces depending on the context in which the measurement is being used. In certain industries, for example, precise measurements are very important, and rounding off may not be advisable. However, for general purposes, rounding off is acceptable.
It’s also worth noting that the conversion factor of 16 is based on the fact that there are 16 ounces in a pound. This conversion factor is used to convert larger units of weight, such as pounds, to smaller units, like ounces.
In conclusion, converting .6 lb to oz is a basic arithmetic operation that involves multiplying the number of pounds by 16. The result is the equivalent weight in ounces. This operation is especially important for people who need to make quick and accurate measurements of weight in both pounds and ounces.
Converting .6 lb to oz may seem like a daunting task, but it’s actually quite simple. All you need is the right formula and a calculator, and you’ll have your answer in no time.
The formula for converting pounds to ounces is straightforward: 1 lb equals 16 oz. So, to find out how many ounces are in .6 lb, all you need to do is multiply .6 by 16.
Here’s how the math looks:
.6 lb * 16 oz/lb = 9.6 oz
That’s it – .6 lb is equivalent to 9.6 oz. It’s a good idea to check your work, though, just to make sure you didn’t make any mistakes along the way.
Converting between units of measurement can be tricky, so it’s important to remember the formula and take your time with the calculations. But don’t worry – with a little practice, and you’ll be a pro at converting .6 lb to oz (and other units of measurement) in no time!


Computer scientific disciplines and encoding is a discipline of study with many distinct applications. The field has become incredible since the initial stored-program computer systems were made in the 1950s. Since then, computer structure seems to have advanced noticeably. Today’s computers can handle many different types of tasks and tend to be capable more tips here of time-sharing and data transfer.
A key theory in laptop science and programming is certainly computational intricacy. The complexness of algorithms depends on how much computer power they need. Some algorithms require simply linear calculation time, whilst others require quadratic or exponential computation. Computer scientists study the appropriate info structures as well as the computational restrictions of algorithms to create one of the most efficient devices. In addition for their research in the area of computational difficulty, computer scientists also review the use of info structures in computer software.
Study regarding mathematical logic is another essential tool in computer scientific disciplines. This branch of mathematics is used to model scientific data using algorithmic methods. Computational reasoning has applications in databases, automatic theorem-proving, and distributed devices. In addition , algorithmic theory of point lattices is used in cryptography. It really is used in cryptanalysis and broadcast-RSA.
Computer science continues to be a very math-intensive field. For that reason, students who have choose to analysis computer science at a university will have to complete relevant mathematics and engineering classes in order to enter into the field. Some of the most popular pc science and information technology level programs might also require learners to comprehensive an internship or a job.


Pocklington Students into National Finals of Maths Competition
10:53am 29th April 2013
On Tuesday 16 April, four of keen mathematicians from Pocklington School took part in the UKMT Team Maths Challenge in Hull. They were competing against 12 other teams and emerged the regional winners. The team will now go through to the national final at the Royal Horticultural Halls, London, on Monday 17 June.
The pupils who made up the team were James Laudage aged 14, Tom Baarda aged 14, Beth Noble aged 13 and Louisa Cullen aged 12. They were prepared for the competition by Mrs Vicky Bell.
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An Invitation to Biomathematics
By Raina Robeva, James Kirkwood, Robin Davies, Leon Farhy, Boris Kovatchev, Martin Straume and Michael Johnson
An Invitation to Biomathematics PDF. Essential for all biology and biomathematics courses, this textbook provides students with a fresh perspective of quantitative techniques in biology in a field where virtually any advance in the life sciences requires a sophisticated mathematical approach. An Invitation to Biomathematics, expertly written by a team of experienced educators, offers students a solid understanding of solving biological problems with mathematical applications. This text succeeds in enabling students to truly experience advancements made in biology through mathematical models by containing computer-based hands-on laboratory projects with emphasis on model development, model validation, and model refinement.
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Regulatory Document Type
Use of Formulas: Annual Interest Securities
Use of formulas: annual interest securities. I am writing in response to your letter of June 1, 1983 regarding the appropriate method of calculating yield and dollar price on periodic-interest municipal securities which pay interest on an annual, rather than the more customary semi-annual, basis. You note in your letter that Board rule G-33 requires the use for purposes of computations of yield and dollar price on such securities of a formula which presumes semi-annual payment of interest (i.e., that formula set forth in subparagraph (b)(i)(B)(2) of the rule). You suggest that the rule should be amended to require the use of a formula that recognizes the annual interest payment cycle on the securities.
As I indicated to you in our previous telephone conversation on this subject, the industry has traditionally disregarded the unusual nature of the interest payment cycle on these securities when computing yields and dollar prices on them, and has followed the practice of using the standard formula for computing yield and dollar price on a security paying interest on a semi-annual basis for these purposes. As a result of this traditional practice, all of the calculators presently available for use by industry members when computing yields and dollar prices have been designed in accordance with the assumption that all periodic-interest municipal securities pay interest on a semi-annual basis; these calculator models cannot be used to compute yields and dollar prices on such securities on any other basis. Therefore, the adoption of a requirement that yields and dollar prices on securities which pay interest on an annual basis be computed by means of a formula which recognizes the annual nature of the interest payment cycle, such as you suggest, would render all of the existing calculator models obsolete, and require that all industry members incur the cost of purchasing new calculator equipment capable of performing such computations (equipment which does not, to my knowledge, exist as of yet).
It is because of the substantial compliance expense that would have been imposed on the industry that the Board declined to adopt a requirement such as you suggest at the time rule G-33 was promulgated, even though it recognized that the requirement that was adopted mandated the use of a formula that would produce slightly less accurate results. MSRB interpretation of June 6, 1983.


A Few Papers by Arthur Jaffe in PDF file format.
You can get Adobe Reader here:
Replica Condensation and Tree Decay (with David Moser)
Reflection Positivity and Monotonicity (with Gordon Ritter), submitted for publication.
Quantum Field Theory on Curved Backgrounds. II. Spacetime Symmetries (with Gordon Ritter), to appear in Comm. Math. Phys.
Quantum Field Theory on Curved Backgrounds. I. The Euclidean Functional Integral (with Gordon Ritter), published in Comm. Math. Phys. 270 (2007), 545—572.
An Exchange Identity for Non-Linear Fields (with Christian Jäkel), published in Comm. Math. Phys. 264 (2006), 283—289.
Quantum Yang-Mills Theory (with Edward Witten), published in The Millennium Prize Problems, American Mathematical Society 2006.
Derivatives with Twists, published in Reviews in Mathematical Physics, 14 (2002), 887—895.
The Elliptic Genus and Hidden Symmetry
Twist Fields, the Elliptic Genus, and Hidden Symmetry
Twist Fields and Broken Super-symmetry (with Olivier Grandjean), published in Journal of Mathematical Physics 41 (2000) 3698-3763.
Twist Positivity for Lagrangian Symmetries (with Olivier Grandjean and Jon Tyson)
The Holonomy Expansion, Index Theory, and Approximate Super-symmetry
Quantum Harmonic Analysis and Geometric Invariants, published in Advances in Mathematics 143 (1999) 1-110.
Scalar Bosons. Course Notes for Quantum Field Theory 253a, Fall 2007 (work in progress).
Quantum Theory and Relativity, Dedicated to George Mackey, to appear. This review focuses on the questions: Does physics suggest a way to combine quantum theory, relativity, and interaction? To what extent can one formulate such quantum physics as mathematics? What is the present status of quantum field theory? What new mathematics emerges? What new insight does one apparently require to make further progress?
Constructive Jürg: A Personal Overview of Constructive Quantum Field Theory This is a talk given 3 July 2007 at the E.T.H. Zurich duirng the meeting entitled "The Inexhaustible Electron."
Lunch with George published in Notices of the American Mathematical Society, August 2007, Volume 54, pages 833-836.
BBC Interveiw on the Solution of the Poincaré Conjecture "The World Today," 16 August 2006 .
The Millennium Grand Challenge in Mathematics, published in Notices of the American Mathematical Society, June-July 2006, Volume 53, pages 652-660.
Witten Citation IAMP Meeting Rio, 9 August 2006.
Introduction to Quantum Field Theory, Spring 2005 Lecture Notes (work in progress).
Constructive Quantum Field Theory, published in Mathematical Physics 2000, pages 111-127, edited by A. Fokas, A. Grigoryan, T. Kibble, and B. Zegarlinski, Imperial College Press, London 2000
Reflections and Twists, written for the celebration of the 40th annivarsary of the IHES.
Roland L. Dobrushin (with J. Lebowitz and Ya. G. Sinai)
Theoretical Mathematics (with Frank Quinn)


Applied mathematics involves using mathematics to solve real-world problems, in particular, challenges faced in scientific research and in the development of new technology.
Learn how to use mathematics to grow your problem-solving skills and expand your knowledge in the STEM field.
This bachelor’s program in applied mathematics provides students with a well-rounded education. With a heavy emphasis on the STEM education and a liberal arts background of Spring Hill College, students will learn to become well-rounded professionals. Students will learn not only mathematical principles and sequences, but also how to utilize these in the workplace.
Graduates of this program will be able to work in several different STEM professions including technology development and research. The problem-solving skills developed in this program mixed with the liberal arts background of Spring Hill College will lead to a well-rounded graduate with highly-demanded skills and experiences.
“I have always loved math and declared my major day one of freshman year. The SHC professors really want to see us succeed and do well. Our small class sizes have enabled us to develop a relationship with our instructors as well as our classmates.” — Mary Catherine McKinley
Visiting Assistant Professor
Assistant Professor; Department Chair, Mathematics and Computer Science
Learn more about our admissions process and policies, including our test-optional policy, at www.shc.edu/admissions
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Apply for FREE to one of the most affordable and accessible private colleges in the country!
Learn more about our beautiful campus, involved student clubs and organizations, and all you need to know about life on The Hill!


View and download inferential statistics essays examples also discover topics, titles, outlines, thesis statements, and conclusions for your inferential statistics essay. descriptive statistics essay descriptive statistics is the analysis of data that summarize data in a way such that, meaningful patterns emerge from the data inferential statistics basic statistics for business and economics. Descriptive and inferential statistics psy 315 descriptive and inferential statistics statistics play a vital role in research performed today with the. This guide explains the properties and differences between descriptive and inferential statistics.
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Boundary value problems for a class of elliptic operator pencils
R. Denk, R. Mennicken, and L. Volevich
Abstract. In this paper operator pencils depending polynomially on the spectral parameter are studied which act on a manifold with boundary and satisfy the condition of N -ellipticity with parameter, a generalization of the notion of ellipticity with parameter as introduced by Agmon and Agranovich-Vishik. Sobolev spaces corresponding to a Newton polygon are defined and investigated; in particular it is possible to describe their trace spaces. With respect to these spaces, an a priori estimate holds for the Dirichlet boundary value problem connected with an N-elliptic pencil, and a right parametrix is constructed.
Integral Equations Operator Theory 38 (2000), 410-436.
The paper is available here: pdf


- Session type
- Technical level
To know how scientific & mathematical principles can be applied for creating/analyzing lyrics. To see demo of a few soft tools developed for engineering lyrics. To explore the research topics in the area of lyric engineering.
I see a lyric as a building, a bridge or a software, in short, a process.
I envision a process applying scientific and mathematical principles to develop effective lyrics.
Lyric Engineering is my pet research topic.
Lyric Mining, Lyricist Modeling, Rhyme Analysis, Pleasantness, Freshness, Semantic Analysis, Lyric Visualisation are a few interesting topics in the field of Lyric Engineering.


Today we will be focusing on the Weird Mystical Numbers you get from this workout…
This Pick 3 lottery system takes a little bit to get whats going on so pay close attention here.
Here is the Chart we will be working with.
Take the last draw and separate the digits. From our sample history Jan 17th, 487 hit. Assign each of the digits to the following letters.
4 = A
8 = B
7 = C
Next take the chart and add these in
Now above the A-1 box (4-1) you would put 3
In the B-1 box (8-1) you would put 7
And in the C+1 box (7+1) you would put 8
Now take all of the bottom row digits and add them together to get the D1,D2.
3+4+7+7+8 = 29
So D1 would be 2 (The first digit) and D2 would be 9 (The second digit)
Ok, from here you simply wheel the highlighted numbers from the chart.
That would be the 3, 8, 8, 2 and 9
388, 382, 389
Eliminate the duplicates and you have
388, 382, 389, 329, 882, 889, 829
Filter by removing doubles and that leaves
382, 389, 329, 829
The following Draw on Jan 18th was 923 Weird Mystical? Maybe.


Celestial Navigation for Cuttermen Video Series
Episode 1 video under construction.
Episode 2 video under construction.
The following videos are under construction.
Episode 1 (Tools of the Trade - Basic) introduces the series, describes the USCG Navigation Standards and the basic tools required for solving problems.
Episode 2 (Tools of the Trade - Intermediate) discusses the marine sextant and its required corrections.
Episode 3 (Tools of the Trade - Advanced) discusses the software STELLA, the System for Estimating Longitude and Latitude Astronomically.
Episode 4 (Time of Phenomenon - Basic) introduces the nautical almanac and the principles behind sunrise and sunset calculations.
Episode 5 (Time of Phenomenon - Intermediate) discusses more difficult sunrise and sunset problems, as well as moonrise and moonset problems.
Episode 6 (Time of Phenomenon - Advanced) discusses more complicated moonrise/moonset problems, calculations for moving ships, and time of local apparent noon.
Episode 7 (Amplitude - Basic) introduces the concepts of declination, amplitudes, and the process of how to calculate gyro compass error at sunrise or sunset using an amplitude of the Sun.
Episode 8 (Amplitude - Advanced) discusses more complicated Amplitude problems using the Sun.
Episode 9 (LAN - Sun Meridian Passage) discusses the basics of how to calculate latitude using the sun at meridian passage (Local Apparent Noon).
Episode 10 (Polaris Latitude) introduces the concepts of GHA, LHA, and how to use the altitude of Polaris to calculate Latitude at night.
Episode 11 will discuss basic azimuth problems.
Episode 12 will discuss advanced azimuth problems and how to calculate azimuth directly using trigonometry.
Episode 13 will show how to calculate an LOP of the Sun.
Episode 14 will show how to calculate advanced LOPs from planets and stars.
Episode 15 will show how to calculate LOPs from the Moon.
Episode 16 will show an example of a Sun/Moon/Planet/Star sight to reduce a 4-body fix.


Title: Adjacency Labelling of Planar Graphs
Adjacency labelling schemes, which have been studied since the 1980s, ask for short labels for n-vertex graphs G such that the labels of two vertices u and v are sufficient to determine (quickly) if uv is an edge of G. One of the long-standing problems in the area was the optimal length of labels for planar graphs. The problem is closely related to the size of the smallest universal graph for all n-vertex planar graphs. In this talk I will show how we resolved this problem (up to lower order terms) with the help of a new graph theoretic tool: a product-structure theorem for planar graphs. This new tool and our result are applicable not only to planar graphs but also to bounded genus graphs, apex-minor-free graphs, bounded-degree graphs from minor closed families, and k-planar graphs.
Title: Algorithms for Explainable Clustering
An important topic in current machine learning research is to explain and/or interpret how models actually make their decisions. Motivated by this, Moshkovitz, Dasgupta, Rashtchian, and Frost recently formalized the problem of explainable clustering. A k-clustering is said to be explainable if it is given by a decision tree where each internal node splits data points with a threshold cut in a single dimension (feature), and each of the k leaves corresponds to a cluster.
In this talk, we see an algorithm that outputs an explainable clustering that loses at most a factor of O(log2 k) compared to an optimal (not necessarily explainable) clustering for the k-medians objective, and a factor of O(k log2 k) for the k-means objective. This improves over the previous best upper bounds of O(k) and O(k2), respectively, and nearly matches the previous Ω(log k) lower bound for k-medians and our new Ω(k) lower bound for k-means. Moreover, the algorithm is remarkably simple and, given an initial not necessarily explainable clustering, it is oblivious to the data points and runs in time O(dk log2 k), independent of the number of data points n.
This is joint work with Buddhima Gamlath, Xinrui Jia, and Adam Polak.


Blaise Pascal was a French mathematician, scientist, and philosopher. Although he is best known to online roulette players as the inventor of the roulette wheel, that was actually only one of his many brilliant accomplishments.
Pascal was born in 1623 in Clermont, France. He was the third of four children, and the only boy in the family. His mother died when Pascal was only three, and his father, Etienne, decided to home-school his son. At first, Etienne forbade young Blaise to study mathematics until he reached age 15, preferring that he concentrate on studying Greek and Latin. Nevertheless, Blaise learned geometry clandestinely until his father, impressed by the child's brilliance, agreed to allow him to continue his mathematical studies. When he was still only 16, Blaise presented a paper with a number of projective geometry theorems that impressed the leading mathematicians of Paris.
Pascal's contributions to mathematics continued into his adulthood, and they are still taught in maths classes to this day. He is most famous today for Pascal's Triangle, which illustrates the expansion of binomial coefficients. (By the way, you don't have to understand binomial coefficients to play online roulette. It is just interesting to find out about the many ways in which the genius of this great mind expressed itself.)
Pascal also distinguished himself in mathematics by inventing the theory of probabilities. This came about through a problem posed by Pascal's friend, the Chevalier de Mere, an inveterate gambler. The problem: two players must abort their game before it is finished, with one player ahead at the time. How, then, to divide the stakes? On the one hand, it would be unfair to divide them equally, because that does not give sufficient credit to the player who is ahead. On the other hand, it would also be unfair to give the entire pot to the player who is ahead at the time, because his ultimate victory is not certain.
Pascal therefore devised a mathematical formula for determining the probability that each player would have won if the game had been played to its conclusion. This is the essence of probability theory: establishing the numerical odds of a future event occurring or not occurring with mathematical precision. Pascal's theory of probability is used today in virtually every area of modern life.
In the physical sciences, Pascal developed Pascal's Law, which states that pressure applied to a confined liquid at any point is transmitted undiminished throughout the fluid in all directions equally. Practical applications of Pascal's law are still seen today in hydraulic machines and in the barometer.
Pascal got involved in a heated dispute about the empty space above the column of mercury in a barometer. Pascal said it was a vacuum. Descartes, the leading intellectual of the day, did not believe a vacuum could exist, and said that Pascal must have a vacuum in his head. Today, of course, we know that Pascal was correct.
More than any other figure in roulette history, it is Blaise Pascal who made the game of roulette what it is today. It was in the course of trying to develop a perpetual motion machine that Pascal invented the roulette wheel. In a sense, we should be grateful that his experiments failed. If perpetual motion had succeeded, the wheel would never stop spinning and there would never be a winner in roulette. Sometimes it can truly be said that there's no success like failure.
Pascal died in 1662 at the young age of 39. More than 350 years later, we would be wise to pay homage to this great thinker every time we play a game of roulette online. If there had been no Pascal, there would be no roulette, and we would be living in a sadder, duller, less enjoyable world. So let us all praise Blaise Pascal, scientific genius and inventor of the roulette wheel.


An actuarial science profession deals with the studying of mathematical and statistical methods to the financial assessments in the fields of insurance, retirement and other investment plans. Where there are risk strategies in the terms of finance, then this profession comes into the consideration. A bachelor’s degree in the same profession will also be very helpful. This profession requires good analytical skills and good conceptual skills. The pay scale varies with the variation in the industry standards. Certified courses and institutes promote this course as a course of interest.
You can find information about degree, diploma and certificate courses for Actuarial Science. You can also fine colleges and universities offering these courses.
Degree Course in Actuarial Science
Diploma Course in Actuarial Science
Certificate Course in Actuarial Science


Visual Identity Development + Graphic Design + Creative Management
The STEAM exhibition explored the intersections of science, technology, engineering, math and creativity. Inspired by the STEM to STEAM dialogue in education, this exhibition presented artists using the interdisciplinary concepts of STEM to explore how science, technology and art overlap, interact, and innovate.
My role was to create a visual brand and implement across different media/mediums for the new exhibition. It was important to reflect all themes of STEAM and be accessible to all ages.


Sun is a very interesting interlocking puzzle designed by Jos Bergmans. Two separate pieces which form almost closed loops require to be connected in such a way that the two semi-circles end up fitting together to form a completed circle, the sun.
The copy I have is made by Eric fuller and was offered in his most recent set of puzzles at Cubic Dissection. Made from Sapele with a Maple veneer for the sun the puzzle measures 2.6″ x 3″ x 3″ so it’s a good size and the construction is superb. Eric hand glued each joint, including a couple of triple mitre joints using a granite plate and a machinists square to ensure that every joint was at exactly at right angles, and it really shows in the finished puzzle. Attention to detail is superb. Three different wood combinations were available when Eric produced the Sun, including the inverse of my copy, Maple with a Sapele sun, and Walnut with a Maple sun. As always Eric has signed and dated the puzzle. 36 copies were made available.
The goal of the puzzle is to interleave the two pieces of the puzzle, to create one solid object where the two halves of the sun line up. Looking at the pieces, they initially seem like closed loops so joining them together at all seems impossible. With a little study though, there is one place where the two pieces will allow for an initial move to link them together, and that’s where the fun starts.
Normally with this sort of rectilinear puzzle, some of the pieces have rounded edges, or parts of the edges have been taken away to allow the pieces to move past each other, such as the Cast Coil from Hanayama. This puzzle however has no corners or edges rounded, and all the moves, slides and rotations happen because there’s just enough room for the pieces to move past each other, making this as close to a perfect example of this style of puzzle as is possible.
Once you have the two pieces intertwined, there’s a couple of easy rotations, then you find yourself at a dead end. (At least I seem to every time I solve this puzzle!) I’ve solved it a good few times now, and each time I spend several minutes trying to figure out where to go next. There are a good few dead ends in the puzzle, so the challenge is fairly good. I seem to go though the same set of wrong moves at the start each time before finding that magic move and from there the solution seems to just flow, and I have it solved very quickly thereafter. I took around 20 minutes to solve this for the first time, but each time after that it still takes me 10 minutes, so I’d say is has some good replay value.
Taking the two pieces apart seems much easier than putting them together, so I’m glad that Eric shipped the puzzle in its unsolved state. It’s a really fun puzzle, and very well made. I’m happy to have been able to get a copy of this for my collection.
As I mentioned earlier, the detail and quality of the work in this copy of the puzzle is superb. Just look at the triple mitre joint above, and you’ll see what I’m talking about. Despite Eric commenting that the construction was painstaking, I think he’s right to be very pleased with how these turned out!


Right so I always thought Sims 4's days were too fast and unimpactful. The consensus for a long time has been 1 Sim minute = 1 real second.
That's not actually true. In the Sims 4 (and you can time it yourselves). 1 Sim minute = 1.5 seconds. Meaning days are 36 Minutes long.
Personally I like GTA V's time scale of 1 day = 48 minutes. So here is how to change that.
With MCCC click on a computer and open the MC command center panel.
- Go to MCCC settings
- Click "Gameplay Settings"
- Click "Game Time Speed"
Right so the way this works is the value is = to the amount of milliseconds, per Sim second that passes.
Eg default is 25 milliseconds per Sim second. Long story short there are 60 Sim seconds in a Sim minute (just like in real life).
So to save you the math headache here is how you calculate that.1st we need to convert milliseconds to seconds.
Eg 25 (default value). 1 second = 1000 milliseconds so 25/1000 = 0.025.Now that we have that we can calculate how many real life seconds pass for every Sim minute.
Take your value, in this case 0.025 and x it by 60 (Minutes). This = 1.5. Meaning for each Sim Minute that passes, 1.5 real Seconds pass.
Now x that by 24 (hours) and you get 36 Minutes for each Sim day.
To get GTA V's time or how ever long you
want a day to last we can work just backwards to make it easy.
Take the amount of time in minutes you want a day to last, in this case I want a day to be 48 minutes long.48
/24(hours) = 2 (Minutes) for every Sim hour that passes.
2/60(Minutes) = 0.0333333
0.033333 x 1000 (milliseconds) = 33.333Unfortunately
we can't add decimals so lets round it down to 33
. This is the value you put in
A value of 33 means that for every Sim Minute, 1.98 seconds pass (which is close enough to 2).
A value of 33 means that 47 Minutes and 31 seconds will pass for each Sim day, which is what I want.


Properties of Multiplication of Integers Wehave, (—48) ÷ 8= -(48 ÷ 8) = -6 4. Integers questions for class 7 – Question 1-5 . It can be observed from the following table: (ii) Commutativity: We know that division is not commutative for whole numbers. Students are advised to solve the Integers Multiple Choice Questions of Class 7 Maths to know different concepts. In general, for any integer a, a + 0 = a = 0 + a. We know that the addition of two whole numbers is again a whole number. Answer the following: 1. In general, for any two integers a and b, a + b is an integer. We hope the given CBSE Class 7 Maths Notes Chapter 1 Integers Pdf free download will help you. (i) Radhika scored 20 marks. We cannot find a pair of integers whose addition is not an integer. Yes. In general, for any integer a, we have, In general, for any two integers a and b, a × b = b × a, The product of a negative integer and zero is zero a × 0 = 0 × a=0, 1 is the multiplicative identity for integers. Associative property of Addition of Integers. 1. (-45) ÷ (-9) = 5 = 45 ÷ 9 (i) Mohan gets four correct and six incorrect answers. What is the vertical distance between them? For this, he jumps 4 steps up and then jumps back 2 steps down in every move. 6. 2. Because these questions are important from an exam perspective. The given equation is in the form of Distributive law of Multiplication over Subtraction. Since subtraction of integers gives integers, we can say integers are closed under subtraction. a) 10 b) 12 c) 9. Closure Under Multiplication NCERT Solutions for Class 7 Maths Chapter 1 Integers: Summary. (-32) ÷ (-8) = 4 = 32 ÷ 8 Overview. How many questions has she attempted incorrectly? We cannot find any pair of integers for which the sum is different when the order is changed. Division of a negative integer by a negative integer So, at midnight temperature will be 14oC below 0. 6. Representation of integers on the number line. ∵Any integer is multiplied with zero and the answer is zero itself. (c) Write a negative integer and a positive integer whose difference is – 3. CBSE Class 7 Maths Notes Chapter 1 Integers Pdf free download is part of Class 7 Maths Notes for Quick Revision. 3. Multiplication and Division of two Integers. 4. (-11) ÷ 1 = -11 These examples show that zero is an additive identity for negative integers also. 2. Integers are closed under subtraction. CBSE CLASS 7 MATHEMATICS INTEGERS – Chapter 1 / MCQ Multiple Choice Questions. Integers Class 7 MCQs Questions with Answers. In a class test containing 10 questions, 3 marks are awarded for every correct answer and \[(-1)\] mark is awarded for every incorrect answer and 0 for the questions not attempted. One such pair is (6, –2) because 6 ÷ (–2) = (–3). Let the number of white cement bags be x. Integers worksheet for class 7 in PDF for free download. For example: 16 × (10 + 2) = (16 × 10) + (16 × 2). NCERT Exemplar Problems for Class 7 Integers for all topics, Download Exemplar Solutions for Class 7 Integers and download in pdf free. These free printable Integers practice sheets are prepared by subject experts. Free PDF Download of CBSE Maths Multiple Choice Questions for Class 7 with Answers Chapter 1 Integers. Find the product, using suitable properties: The given equation is in the form of Distributive law of Multiplication over Addition. a × 1 = 1 × a = a, Multiplication with (-1): Let us observe the following table showing the product of an integer and (-1). We can say that if dividend and divisor are of same signs, then the quotient is a positive integer. Temperature at the beginning i.e., at 12 noon = 10oC, Rate of change of temperature = – 2oC per hour, Temperature at 1 PM = 10 + (-2) = 10 – 2 = 8oC, Temperature at 2 PM = 8 + (-2) = 8 – 2 = 6oC, Temperature at 3 PM = 6 + (-2) = 6 – 2 = 4oC, Temperature at 4 PM = 4 + (-2) = 4 – 2 = 2oC, Temperature at 5 PM = 2 + (-2) = 2 – 2 = 0oC, Temperature at 6 PM = 0 + (-2) = 0 – 2 = -2oC, Temperature at 7 PM = -2 + (-2) = -2 -2 = -4oC, Temperature at 8 PM = -4 + (-2) = -4 – 2 = -6oC, Temperature at 9 PM = -6 + (-2) = -6 – 2 = -8oC, ∴At 9 PM the temperature will be 8oC below zero, The temperature at mid-night i.e., at 12 AM, Change in temperature in 12 hours = -2oC × 12 = – 24oC, So, at midnight temperature will be = 10 + (-24). (-8) ÷ 1 = -8 What was the temperature of Srinagar on Tuesday? Hence, the temperature difference between the hottest and the coldest place is 30oC. Commutative 3. Then, multiply the product 6 with 5 to get 30, i.e., [(-3) × (-2)] × 5 = 6 × 5 = 30. So, the temperature of Srinagar and Shimla taken together is less than the temperature at Shimla. For example: integers). (-32) ÷ 4 = -8 = -(32 ÷ 4) 4. 1. You should get good marks in Class 7 examinations as it will always help you to get good rank in school. By observing the number line, we can find the temperature of the cities as follows. In general, for any two integers a and b, we can say that a + b = b + a. Commutativity of Subtraction: We know that the subtraction is not commutative for whole numbers. Let us observe the following table showing the product of a negative integer and 1. (ii) After drinking water, he wants to go back. Company must sell to have neither profit nor loss. So, zero is an additive identity for whole numbers. and, -9 × 10 – (-9) × (-3) = -90 – 27 = -117 This page consists of CBSE Class 7 Maths Worksheet for chapter-Integers prepared by Entrancei. The NCERT Textbook Solutions for the chapter have been designed accurately by mathematics experts at BYJU’S. 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Serge A.Yagunov -- Mathematics page
Serge A.Yagunov --- Mathematics Page
• Motivic Steenrod operations and Characteristic classes
• Poincare Duality for Algebraic Varieties
• Cohomology of Linear Groups
• Bloch groups
• Friedlander-Milnor conjecture
• Suslin's bi-Grassmannian conjecture.
• Some notes of recently given courses • Video records
• Some conference information • Conference webpages • Pictures.
Math references and webpages
• Useful Math related web-sites.
Math misc (page is in Russian)
• " Mathematician's miscellany " (c) Littlewood.
• Texts and problems related mostly to the recreational Math
• Some high-school level stuff.


Do you ever need to measure out a specific amount of liquid, but don’t have a measuring cup handy? If so, you can use a standard drinking glass to measure the fluid volume. In this blog post, we’ll show you how to convert ounces to cups so that you can easily measure out your desired amount of liquid.
Cups and Ounces Units of Measurement
Cups and Ounces Units of Measurement
Before we can answer how many cups is 32 oz, we need to understand the equivalence of cups and ounces. There are 16 tablespoons in a cup and 8 liquid ounces in a liquid tablespoon, so if you have 32oz you can multiply it by either one or both of these values. If we say that 1 cup is equal to 8 liquid ounces, then we can also say that 2 cups are equal to 1 pint and 4 cups are equal to 1 quart.
A standard cup, which can be either made of metal or plastic and holds 240 milliliters (ml), weighs around 250 grams (g). So if one cup is equal to about 1/2 pound in weight, then 32oz equals 16 cups.
If we say that 1 cup is equal to 16 tablespoons, then we can also say that 2 cups are equal to 1/2 pound and 1/4 cup is equal to 2 ounces. Using the same logic as above, if you multiply 32oz by 16, you get 512 fluid ounces. If you multiply it by 8, you get 256 fluid ounces. And of course, if you multiply it by 4 or 2, you will get the same results.
For following recipes, it’s always best to weigh the ingredients if you can or to put them in cups. One cup is equal to 1/4 pound so if your recipe calls for 3 cups of ingredients it’s equivalent to 2/3 lb (650g) and so on. Using this logic, if your recipe calls for one cup of an ingredient, then just weigh out 85g.
If you’re measuring flour by volume, this is usually sifted before you measure it. If your recipe calls for 4 cups of flour and you’re using a cup measurer (or even worse, a spoons measure), then after sifting the flour you will be left with much less than 4 cups. So if we say that 1 cup is equal to 85g, then after sifting your flour it will be equal to around 50g.
Other common fluid volume measurements are listed in the table below:
Gallons | Quarts | Pints | Cups | Fluid Ounces | Tablespoons
1 US gallon = 4 quarts = 8 pints = 32 cups = 128 fluid ounces = 3.75 liters
1 US gallon = 4 quarts = 8 pints = 32 fluid ounces = 128 fluid drams
1 quart = 2 pints = 16 cups = 64 fluid ounces 1 quart = 2 pints = 32 fluid ounces
1 cup = 8 fluid ounces
8 tablespoons (Tbsp) or 16 teaspoons (tsp) 1 US cup = 8 US fluid ounces
1 US tablespoon = 3 teaspoons
3 teaspoons or 1/2 fluid ounce or 15 milliliters
The differences among US customary cup, US legal cup, metric cup, Canadian cup, and Imperial cup
Measuring cup in baking
1 US customary cup = 8 US fluid ounces (8 fl oz)
1 US legal cup = 16 US fluid ounces (16 fl oz)
1 metric cup = 250 mL (millilitres) or 8.45 U.S. fluid ounces (8.45 fl oz)
1 Canadian cup = 10 imperial fluid ounces (10 fl oz or 284.5 mL). It is often rounded to 11 imperial fluid ounces (11 fl oz or 326 mL).
1 Imperial cup = 8 imperial fluid ounces (8 fl oz)
A US customary cup weighs approximately 8.3 ounces.
A US customary cup of water weighs approximately 8.345 ounces at 62 °F (17 °C). An imperial cup of water weighs 8 ounces. A US legal cup of water weighs 7.99788 Oz at 62° F (US Customary) or 8 Oz at 59° F (US Legal). A metric cup of water weighs approximately 236.588 grams at 62 °F (17 °C).
So How Many Cups Is 32 oz?
One way to decide is to pour one cup of water from a container with a known measurement on it, such as the measuring cup, into the same type of container with markings on it. Then you would have two containers, each measured at 8 ounces. This leaves us with 24 ounces of water to pour out. We can do this because there are 16 ounces in a cup and we have two cups of water, so, twice times 16 is 32.
The second way would be to use the measuring device that measures cups as your measuring tool, then you would pour out 32 ounces of water into the measuring device with the markings on it. Then you would have to do some math to find out how many cups are in 32 ounces.
Quick Trick for Quick Measurement
A Cup for 32 oz
While most conversions are simple, there is a way to remember this rule.
To calculate ounces, divide the number of cups by 8, and then subtract that from 32. This will be the number of fluid ounces.
32 oz divided by 8 is 4 cups
4 cups minus 32 is 28 fluid ounces
Here’s another example:
36 oz divided by 8 is 4 and ½ cups. 4 and ½ cups minus 32 equals 22 and ½ fluid ounces
This trick works because:
1 cup = 8 ounces (fluid) OR 2/3 cup = 6 ounces (dry)
2/3 cup + 1/3 cup + 2/3 cup = 1 1/3 cups
1 cup = 8 ounces (fluid) OR 1/2 cup + 1/3 cup + 1/3 cup = 2/3 of a cup
By dividing 32 by 3, you get approximately 11. So there are about 11 cups in 32 oz.
32 divided by 3 is 10 about 10 and 2/3 cups
10 and 2/3 cups – 32 oz = 22 and 1/3 fluid ounces
There are approximately 5 cups in a quart, so there would be about 22 and 1/3 ounces in a cup.
Although this trick works most of the time, it doesn’t take into consideration that different types of liquids have different weights. Milk, for example, weighs approximately 8.8 pounds per gallon. You can see that to convert from cups to gallons, a different method should be used.
How to measure without any measuring device?
No one likes to drink water from a shallow pie pan, but pie pans do come in handy for occasional baking. If you need to fill a pie pan with liquid without any measuring device on hand, you can use the following simple formula:
There are 8 cups in 1 gallon, so 32 oz is equal to 2 gallons. Dividing by 8 gets you 2 cups.
In other words, 32 oz of water is equal to 2 US pints or 1/2 a US gallon. This should be enough for a standard 9×13″ cake pan filled to the top with batter. If you also need a crust, increase the recipe by half again and suit yourself as to whether you need a second pie pan.
If you have a little more space to work with, 32 oz of water is equal to 1 US quart, which is the same as 2 pints. This will fill a 9×13″ cake pan about half full and leave enough room for the crust on top. If you need even more room, divide 64 oz by 8 to get a whopping 8 cups or 1 US gallon. In this case, you may need to make a second recipe of cake batter because the amount of water is equal to half a standard 9×13″ cake pan.
Remember that if you choose to measure liquids by weight instead of volume, 1 cup weighs about 8 oz. The total weight of a full cake pan filled with batter will be about 2 lbs 8 oz.
Water is not the only liquid you might need to measure by the pint or quart without any measuring tools available. Milk, cream, and buttermilk are also sold in pints and quarts for very reasonable prices at most grocery stores. If you know how much a pint or quart weighs, you can use the same simple formula above to measure out any amount of liquid.
In these cases, it’s not about exact measurements—it’s about getting enough liquid in your recipe without following measuring spoons and cups. If you do have measuring tools available, please look up accurate weight or volume conversions for liquids so you can measure appropriately.
Measurement conversion can be difficult, but with practice, it becomes easier. In this blog post, we provide the answer to “How Many Cups Is 32 oz?” and a few tips to help you quickly convert measurements so that you can focus on the recipe at hand. By following these tips, you will be able to breeze through your next baking project!


xitorch: differentiable scientific computing library¶
xitorch (pronounced “sigh-torch”) is a library based on PyTorch that provides differentiable operations and functionals for scientific computing and deep learning. xitorch provides analytic first and higher order derivatives automatically using PyTorch’s autograd engine. It is inspired by SciPy, a popular Python library for scientific computing.
Example operations available in xitorch:
xitorch.linalg.symeig: symetric eigendecomposition for large sparse matrix or implicit linear operator,
xitorch.optimize.rootfinder: multivariate root finder, and
xitorch.integrate.solve_ivp: initial value problem solver or commonly known as ordinary differential equations (ODE) solver.
Why use xitorch:
contains differentiable functionals;
provides 1st, 2nd, and higher order gradients of functionals;
enables the use of functionals in the object-oriented way.
Source code: https://github.com/xitorch/xitorch/


MATH 8 (CORE)
This preparatory course is designed for those students who have displayed weaknesses in Math. The course covers Math fundamentals. Upon completion of this course, students may enroll in regular Math 8, or consider Math 9 (Core.)
The new Math 8 course consists of five major strands: number operations, probability and data analysis, algebra, geometry and measurement. Students are expected to use a calculator to assist them with their calculations for most topics in this course.
Textbook: Mathpower Eight, McGraw-Hill Ryerson
MATHEMATICS 8A (Accelerated)
Students in MA 8A will briefly cover most MA 8 topics and then extend them into MA 9. All MA 9 topics will be covered thoroughly. Students are expected to participate in various math contests at the appropriate level. On completion of MA 8A, students will be placed in MA l0 or MA l0E.
MATHEMATICS 9/9E (Enriched)
This course reviews arithmetic skills, provides for additional understanding of real numbers and the associated algebra, graph-ing in two dimensions, solving equations and formulas, extends geometric concepts, examines applications of numbers, algebra, and geometry, and has emphasis on problem-solving. Students in the Enriched level are expected to complete enrichment work and to participate in various Math contests. Students who achieve A or B in Math 8 may elect to take .Math 9E.
Prerequisite: MA 8
MATHEMATICS 9 (CORE)
The course will cover many of the same topics as Mathematics 9 however at a lower level of understanding. All basic number operations will be reviewed and consolidated. A student who successfully completes Math 9 (Core) may enroll in Essentials of Math 10
PRINCIPLES OF MATHEMATICS 10/10E (Enriched)
Topics covered include number sequences, real numbers, exponents and radicals, coordinate geometry, functions, polynomials, rational expressions, deductive reasoning, trigonometry, statistics and probability. Students in the enriched program are expected to complete enrichment work and to participate in various math contests.
ESSENTIALS OF MATH 10
Topics include problem-solving, personal banking, wages, salaries and expense, spreadsheets, rate, ratio and proportion, trigonometry, measurement, geometry project, probability and sampling. Students successfully completing Essentials of Math 10 will continue with Essentials of Math 11.
PRINCIPLES OF MATHEMATICS 11/11E (Enriched)
Topics covered include mathematics of finance, quadratic functions, polynomial and rational functions, analysis of equations and inequalities, systems of equations and inequations, reasoning and geometry, properties of angles, chords and circles and coordinate geometry and trigonometry. Students in the enriched program are expected to complete enrichment work and to participate in various math contests.
ESSENTIALS OF MATHEMATICS 11
Topics include problem solving, relations and formulas, income and debt, data analysis, measurement technology, owning and operating a vehicle, personal income tax, applications of probability and preparing a business plan. The course is consumer-oriented and is designed to assist students in day-to-day mathematical tasks.
PRINCIPLES OF MATHEMATICS 12/12E (Enriched)
Topics covered include transformations of functions, exponential and logarithmic functions, trigonometric functions, trigonometric equations and identities, permutations and combinations, probability, statistics and conic sections.
Students in the enriched program are expected to complete enrichment work and to participate in various math contests. Principles of Math 12 is a provincially examinable course.
Topics covered include problem-solving, overview and history of Calculus, functions, graphs and limits, derivatives, antiderivatives and applications of derivatives and antiderivatives. Students who plan on taking Calculus 12 (which is only offered 2nd semester) must have completed Principles of Math 12 in the 1st semester.


Calculating your Screen Dimensions (Part 1)
By Dave Rodgers (Marketing Manager of Elite Screens & EPV® Screens)
Garden Grove, CA – October 1, 2015 – There are many online links that can help you do this but if you pride yourself on being a screen-guy/gal, it’s a good idea to know how to do the calculations yourself. These easy calculations will enable you to calculate the height, width and the diagonal measurements.
Look at the image above. Identify the sides to be measured. Notice that it forms a triangle.
- X= width
- Y= height
- C= diagonal measurement
An aspect ratio is always laid out as (x:y)
Ex. A 4:3 (NTSC) aspect ratio indicates that there are 4 measured (x) units of width for every 3 measured (y) units of height.
- The height of the 4:3 screen is ¾ or (.75) of the width.
- The width of the 4:3 screen is 4/3 or (1.3333) of the height.
Calculate more Aspect Ratios:
This means there will be 16 units of horizontal (x) measurements to every 9 units of vertical (y) measurement.
The Width(x) = 16/9 or 1.78 times the height
Ergo, 1.78 * (height) = (width)
Example. A 100” high screen will have a width of 178”
The height (y) = 9/16 or .5625 times the width
Ergo, .5625 *(width) = (height)
Example. A 100” wide screen will have a height of 56.25” (or) ≈56”
The same rules apply with other aspect ratios.
- Width = 2.35 * height
- Height = .426 * width
- Width = 1.6 * height
- Height = .625 * width
- Calculate The Diagonal
Think way back to the high school math days. Do you remember something called the Pythagorean Theorem? Even if you don’t, the formula is mercifully easier than the name is. All you need is the width and height of your projection screen and this calculation does the rest.
Here are some points to remember:
- X = the screen width
- Y = the screen height
- C = the screen diagonal
- X² = a number squared (a number times itself). Example: 4² is 4*4 = 16
- √ = a square root ( the opposite of a number squared). Example: √16 = 4 (Because 4*4 = 16)
To calculate, you need to get the square of both the width and height; add them together; and then (using a calculator is fastest) get the square root.
Pythagorean Theorem (determine diagonal screen size)
- x² + y² =c²
Example: Find the diagonal of a screen that is 87” wide and 49” high
Step 1. Replace “x” with the width measurement Step 2. Replace “y” with the height measurement
- c² = (87² + 49² ) = (7569 + 2401 ) = 9970
*Eliminate (x²) with a square root (√).
- c = √ 9970≈ 100
Answer is 100” diagonal.
Try this on any other measurements. All that is needed is either the length or width and the aspect ratio.


School Coin Stamps
Perfect for teaching children how to add, subtract and count numbers, you can purchase these coin stamps individually or as a full set of 8. This set of 8 UK coin stamps save you almost 50% on the individual prices. As the stamps are self-inking they’ll produce thousands of impressions, making them ideal for schools. These stamps are available in black ink, and if you place your order before 12 noon we’ll dispatch the same working day.
Full Set of 8 Coin Stamps