Please use this identifier to cite or link to this thesis:
|Author: ||D'AMARIO, ALESSANDRO|
|Supervisor: ||PEROTTO, SIMONA|
|Scientific Disciplinary Sector: ||MAT/08 ANALISI NUMERICA|
|Academic year: ||2014/2015|
|Title: ||A reduced-order inverse distance weighting technique for the efficient mesh-motion of deformable interfaces and moving shapes in computational problems|
|English abstract: ||Nowadays, despite the constant technological growth of the last twenty years, Computational Fluid Dynamics (CFD) problems represent a hard challenge for scientific computing because of their large demand on computational resources. In fact, in the context of aerodynamics optimization and design, CFD applications require to simulate many different possible realizations of a system which can become prohibitive in terms of computational and memory effort. These considerations have produced an intensive development of reduced order methods to provide high-fidelity simulations via efficient and low-dimensional models.
In this thesis we focus on an efficient and flexible technique, namely the Inverse Distance Weighting (IDW). This strategy can be used to solve mesh motion problems in a Fluid-Structure Interaction (FSI) framework. IDW is an interpolation strategy which computes the displacement of the grid nodes starting from the movement of some data points, called control points. The original formulation of IDW uses as control points the nodes of the grid belonging to the interface and requires assembling a matrix [N_s × N_c], where N_s is the number of values to be interpolated and N_c indicates the number of control points. Since the number of interface nodes for a typical mesh motion problem can be very large (of the order of hundreds of thousands), the system to solve assumes significantly high dimensions.
In order to reduce the computational and the memory load due to these huge dimensions, we implement an ad hoc algorithm which performs a geometrical and a model order reduction of the system: the former reduction is based on an iterative procedure which selects some relevant control points among the initial ones, while the latter reduction is inspired by the Proper Orthogonal Decomposition (POD) strategy to exploit the advantages of an Offline-Online splitting between ROM construction and evaluation of the solution.
The implemented algorithm is validated on some examples, from benchmark cases to more complex three dimensional configurations such as a wing or a hull.
All the reduction strategies developed are implemented using an efficient C++ object oriented code and an open-source Finite Element library (libMesh), while the processing of the grids is performed by the open-source visualization software Paraview.|
|Italian keywords: ||problemi di interazione fluido-struttura; reticolo computazionale; interfaccia; deformazioni; fluidodinamica computazionale; parametrizzazione di forma; riduzione dei costi; ottimizzazione; efficienza computazionale; metodi di ordine ridotto|
|English keywords: ||fluid-structure interaction problems; interface; shape parametrization; mesh motion problems; displacement; deformation; computational fluid dynamics; optimization; computational efficiency; reduced order model|
|Appears in Collections:||POLITesi >Tesi Specialistiche/Magistrali|
Files in This Item:
|2016_04_DAmario.pdf||Versione definitiva e completa della tesi||6.86 MB||Adobe PDF||View/Open


Introduction to Network Analysis
STOR 390 is an introductory course to the mathematical tools and computational training needed to understand large-scale networks. It covers basic methods for working with network data that are fundamental for advanced projects in statistical network analysis and machine learning on networks.
The course will introduce network models and structural descriptors, network dynamics and prediction of processes evolving on graphs, modern algorithms for topology inference, community and anomaly detection. All concepts will be illustrated with numerous applications from technological, social, biological, and information networks.
This course will help answer questions such as:
- Where does “six degrees of separation” come from?
- How can we make sense of large social networks?
- What are the underpinnings of Google’s webpage ranking?
- What are good models for predicting popularity on Twitter?
- How can we estimate the size of the Internet?


Year Two Blog
admin on: Year Two Blog
Welcome to Year 2. The children have all settled in well and have got used to the Year 2 routine. In the mornings we do Maths, English and Phonics. The afternoons are spent on PE, music, RE, PSHE, Art, D&T and our topic work in History and Geography. ICT is incorporated across the curriculum. As you can see we are very busy!
In Maths we have been making two digit numbers with Numicon and Diennes apparatus and partitioning them into tens and ones.
We moved onto partitioning three digit numbers.
We have also begun adding two digit numbers using number lines and hundred squares and will be moving on to adding tens and ones using Diennes apparatus and drawing tens and ones.


In 2006, the IEEE Conference on Data Mining identified 10 top algorithms [Ref: http://www.cs.uvm.edu/~ic
dm/algorithms/index.shtml ]. Are these still valid? Are there new algorithms that have found sweet spots in the industry?
Those top-10 algorithms for reference:
- Support Vector Machines (SVM)
- Expectation Maximization (EM)
- k-Nearest Neighbors (kNN)
- Naive Bayes
- Classification and Regression Tree (CART)


Pearson Realize in Canyons School District
The curriculum for the Elementary Language Arts and Mathematics that is supported in Canyons School District is published by Pearson Education. Students use Reading Street for Language Arts and enVision 2.0 for Mathematics. This page is designed to connect teachers and parents to the resources available to support student achievement at home and school.
Pearson Realize Tools
Pearson Realize is the website portal for students and teachers to have constant access to materials that correlate to the Common State Standards that students are learning in class. Pearson Realize gives students the ability to access their content from anywhere at anytime.
Visit Pearson Realize:


GLEN ELLYN – As the first week of school draws to a close, Glen Crest Middle School in Glen Ellyn welcomes two new teachers to its math and physical education departments.
Danielle Johnson is joining the school as a math teacher, while Danny Sheridan is beginning his career as a physical education teacher.
Johnson has served as a substitute teacher in Community Consolidated School District 89 for four years. In addition to teaching math, she hopes to be part of the Rachel's Challenge FOR (Friends of Rachel) CLUB that seeks to create a positive environment for all students.
Sheridan will teach eighth grade P.E. classes and coach eighth grade boys' basketball.


Our preschool program is centered on better preparing children for a happier adjustment to Kindergarten. Some of the many skills taught include:
- Colors and shapes
- Listening and thinking skills
- Fine motor skills
- Social skills and sharing
- Math Readiness skills: Number recognition, number ordering and counting
- Reading readiness skills: letters and sounds
In addition, students in the three day program participate in Physical Education, Music, Art, and Computer classes. Students in the 2 day program have the opportunity to participate in computer and Physical Education classes.
- Two Day 8:20a-11:45a
- Two Day Full 8:20a-2:45p
- Three Day 8:20a-2:45p
For more information, contact Dr. Nina Zetty, principal, by email at
firstname.lastname@example.org or by phone at 724-929-5143.


I’m sure the above title sounds familiar to each and everyone of you. If you are thinking about engineering mathematics, no it’s not. It’s about the basic skills, the second thing you learn after learning 1,2,3 or 1+1 or 1×1 or 1/1.
It’s the kind of mathematics that you learn in primary school. Where there’s whole no., fraction, decimal, numbers, angles, triangle, volume, etc.
It’s the most simplest thing that we learn in school before going to secondary. But what I’m touching here is, no matter what maths you are learning, you have to know the formulae by heart. That’s the most basic thing to do to score a mark or 1/2. And the next thing to do is practice. Yes, practice in many different type of question to try to get use to it and grasp the idea. Just try many different papers or assessment and even textbook question.
With formulae checked and loads of practice done, you can score.
So practice, practice, practice & PRACTICE.


(Entire section in one PDF file).
Astronomers often describe uncertainties in terms of the fractional error, e.g. the amplitude of the uncertainty divided by the amplitude of the quantity being measured; often, the inverse of this, referred to as the signal-to-noise ratio is used. Given an estimate the number of photons expected from an object in an observation, we can calulate the signal-to-noise ratio:
Consider an object with observed photon flux (per unit area and time, e.g. from the signal equation above), S′, leading to a signal, S = S′Tt where T is the telescope area and t is the exposure time. In the simplest case, the only noise source is Poisson statistics from the source, in which case:
A more realistic case includes the noise contributed from Poisson statistics of ``background'' light (more on the physical nature of this later), B′, which has units of flux per area on the sky (i.e. a surface brightness); note that this is also usually given in magnitudes.
The total number of photons observed, O, is
This leads to a common form of the noise equation:
In the signal-limited case, S′ > > B′A, we get
Consider two telescopes of collecting area, T1 and T2. If we observe for the same exposure time on each and want to know how much fainter we can see with the larger telescope at a given S/N, we find:
In addition to the uncertainties from Poisson statistics (statistical noise), there may be additional terms from instrumental uncertainties. A common example of this that is applicable for CCD detectors is readout noise, which is additive noise (with zero mean!) that comes from the detector and is independent of signal level. For a detector whose readout noise is characterized by σrn,
For large σrn, the behavior is the same as the background limited case. This makes it clear that if you have readout noise, image quality (and/or proper optics to keep an object from covering too many pixels) is important for maximizing S/N. It is also clear that it is critical to have minimum read-noise for low background applications (e.g., spectroscopy).
There are other possible additional terms in the noise equation, arising from things like dark current, digitization noise, uncertainties in sky determination, uncertainties from photometric technique, etc. (we'll discuss some of these later on), but in most applications, the three sources discussed so far – signal noise, background noise, and readout noise – are the dominant noise sources.
Note the applications where one is likely to be signal dominated, background dominated, and readout noise dominated.
Why are the three uncertainty terms in the noise equation added in quadrature? The measured quantity (S) is a sum of S + B - < B > + < R >, where < R > = 0 since readout noise has zero mean. The uncertainty in a summed series is computed by addding the individual uncertainties in quadature; in the equation above, we have neglected the uncertainty in < B >. To understand why they add in quadrature, let's consider general error propagation.
More reasons to consider error propagation: let's say we want to make some calculations (e.g., calibration, unit conversion, averaging, conversion to magnitudes, calculation of colors, etc.) using these observations: we need to be able to estimate the uncertainties in the calculated quantities that depend on our measured quantities.
Consider what happens if you have several known quantities with known error distributions and you combine these into some new quantity: we wish to know what the uncertainty is in the new quantity.
As long as uncertainties are small:
The last term is the covariance, which relates to whether uncertainties are correlated.
Examples for uncorrelated errors:
In this case, errors are said to add in quadrature.
Note that when dealing with logarithmic quantities, uncertainties in the log correspond to fractional uncertainties in the raw quantity.
When propagating errors, even though you can calculate the variances in the new variables, the distribution of uncertainties in the new variables is not, in general, the same as the distribution of uncertainties in the original variables, e.g. if uncertainties in individual variables are normally distributed, uncertainties in output variable are not necessarily.
When two variables are added, however, the output is normally distributed.
We've covered uncertainties in single measurements. Next we turn to averaging measurements. Say we have multiple observations and want the best estimate of the mean and variance of the population, e.g. multiple measurements of stellar brightness. Here we'll define the best estimate of the mean as the value which maximizes the likelihood that our estimate equals the true parent population mean.
For equal uncertainties, this estimate just gives our normal expression for the sample mean:
But what if uncertainties on each observation aren't equal, say for example we have observations made with several different exposure times? Then the optimal determination of the mean is using a:
This is a standard result for determining sample means from a set of observations with different weights.
However, there can sometimes be a subtlety in applying this formula, which has to do with the question: how do we go about choosing the weights/errors, σi? We know we can estimate σ using Poisson statistics for a given count rate, but remember that this is a sample variance (which may be based on a single observation!) not the true population variance. This can lead to biases.
Consider observations of a star made on three nights, with measurements of 40, 50, and 60 photons. It's clear that the mean observation is 50 photons. However, beware of the being trapped by your undertainty estimates. From each observation alone, you would estimate uncertainties of , , and . If you plug these uncertainty estimates into a computation of the weighted mean, you'll get a mean rate of 48.64!
Using the individual estimates of the variances, we'll bias values to lower rates, since these will have lower estimated variances.
Note that it's pretty obvious from this example that you should just weight all observations equally. However, note that this certainly isn't always the right thing to do. For example, consider the situation in which you have three exposures of different exposure times, and you are calculating the photon rate (counts/s). Here you probably want to give the longer exposures higher weight (at least, if they are signal or background limited). In this case, you again don't want to use the individual uncertainty estimates or you'll introduce a bias. There is a simple solution here also: just weight the observations by the exposure time. However, while this works fine for Poisson uncertainties (variances proportional to count rate), it isn't strictly correct if there are instrumental uncertainties as well which don't scale with exposure time. For example, the presence of readout noise can have this effect; if all exposures are readout noise dominated, then one would want to weight them equally, if readout noise dominates the shorter but not the longer exposures, once would want to weight the longer exposures even higher than expected for the exposure time ratios! The only way to properly average measurements in this case is to estimate a sample mean, then use this value scaled to the appropriate exposure times as the input for the Poisson uncertainties. See the next subsection for a more comprehensive discussion.
Another subtlety: averaging counts and converting to magnitudes is not the same as averaging magnitudes!
Say you have a bunch of measurements of the brightness of an object. Under the assumption that the object is not variable, you wish to get the best estimate of the count rate of the object. Your measurements are not all at the same exposure time. So you have a bunch of measurements:
where each of the ci has noise associated with it.
For each measurement, you can calculate a rate:
Now you wish to average all of the measured rates to get the best estimate of the true rate. Because the measurements were taken with different exposure times, they have different amount of uncertainty, so you want to use a weighted mean to get the best estimate.
To calculate a weighted mean, you need to determine the variances of the observations. Given the true rate, r, the variances of the measurements are
To get the variances of the rates, ri = , we need to divide the variances of the counts by t2 (i.e. divide the standard deviations by t, so
The problem is that you don't know what r is. Normally, you might just take
Let's go back to the correct formulation, which gives for the weighted mean:
In the case of all equal exposure times, then the σ2 come out of the sums and cancel between the numerator and denominator, leaving
i.e., an unweighted mean, just as we'd expect
In the case of zero readout noise we have
i.e., calculate the mean by weighting by the exposure times.
For the general case of unequal exposure times in the presence of readout noise, you can avoid biases by choosing any reasonable estimate of r for use in calculating the weights. For example, you could use the observed rate with the highest S/N as the basis for calculating weights. Or you could even iterate the solution a couple of times. If you were to do this, you'd find that the weighted mean hardly depends at all on what you choose for the estimate of r so long as it is close to the true r.
Although from S/N considerations, one can determine the required number of counts you need (exposure time) to do your science, when observing, one must also consider the question of whether this time should be collected in single or in multiple exposures, i.e. how long individual exposures should be. There are several reasons why one might imagine that it is nicer to have a sequence of shorter exposures rather than one single longer exposure (e.g., tracking, monitoring of photometric conditions, cosmic ray rejection, saturation issues), so we need to consider under what circumstances doing this results in poorer S/N.
Consider the object with photon flux S′, background surface brightness B′, and detector with readout noise σrn. A single short exposure of time t has a variance:
The only difference is in the readout noise term! In the signal- or background-limited regimes, exposures can be added with no loss of S/N. However, if readout noise is significant, then splitting exposures leads to reduced S/N.
So far, we've been discussing random errors. There is an additional, usually more troublesome, type of errors known as systematic errors. These don't occur randomly but rather are correlated with some, possibly unknown, variable relating to your observations, and can have the effect of not just adding spread around the true value that you are trying to measure, but actually measuring the wrong mean.
EXAMPLE : flat fielding
EXAMPLE : WFPC2 CTE
Note also that in some cases, systematic errors can masquerade as random errors in your test observations (or be missing altogether if you don't take data in exactly the same way), but actually be systematic in your science observations.
EXAMPLE: flat fielding, subpixel QE variations.
Note that error analysis from expected random errors may be the only clue you get to discovering systematic errors. To discover systematic errors, plot residuals vs. everything!


Intuit-owned Mint.com is heading to schools today with the launch of a free, online program designed to educate middle-school students about personal finance and financial management.
Mint has partnered with educational publisher Scholastic to develop materials that parents and teachers can use to teach children the ins and outs of personal finance management. The materials includes lesson plans as well as an interactive game, to teach children money management, budgeting and goals.
For example, the program teaches children the concept of compound interest with real-life math problems, and encourages children to set goals and budgets with their own current work opportunities (i.e. babysitting).


What I don't know about set theory would fit in a book, tentatively titled Everything There Is To Know About Set Theory, A Boxed Collection or Group. But I'm pretty sure that if a set isn't infinite, then there is (or perhaps, are) a finite number of ways to arrange it. So, if there are only so many (so very many, but still) water molecules to freeze into flakes, then eventually, the pattern must repeat. Since a very large number, perhaps half an infinity, have fallen here this winter, then ergo, flakes of a feather, flocking together.
Some people here are cheering for the snowfall record to be broken. I was in Boston when they won the World Series, and it was cool and all, but I couldn't really take any pride in it; I hadn't hit a ball anywhere. By the same token, I guess being somewhere when a record is broken would be interesting, but this record isn't so much breaking as falling on us.


Your objective is to balance the game fields, that is, all squares have to be the same color. One color is guaranteed to achieve the result, others may or may not solve it.
By clicking on the square on the left you cycle its color along with two other squares on the right. The same process applies clicking on a square on the right. Triplets of squares are randomly selected at the start of the game.
By observing the triplets you can work out the order in which to click the squares to achieve balance.


- 2006-2007 (Teaching): Nina Compton (FIN)
- 2002-2003 (Teaching): Sherry K. Mills (ACCT/IS)
- 1996-1997 (Teaching): Elise “Pookie” Sautter (MKTG)
- 1983: Allan Savage (ACCT/IS)
NMSU award given in alternating years for excellence in teaching and excellence in research and creative activity. The annual award carries a monetary gift. Dr. Robert L. Westhafer, a Professor Mathematical Sciences at NMSU (1946-1957), excelled as a mathematician and teacher. He gave freely of his time and talents to his students and the university. Following his death in 1957, a group of his friends created this memorial award in his honor. Recipients of the Westhafer Award are determined annually by a faculty committee of seven who receive nominations from faculty, administrators, and students. The monetary award is given alternatively for excellence in teaching and research and is contributed by interested faculty and friends through the NMSU Foundation.


They put the little animals into different piles based on categories they made up themselves. These included jungle animals, horses, trees, dangerous animals, animals that live in the woods, farm animals, ninja turtles, people, blocks, weapons, frogs, lizards, dinosaurs, and one alien.
Then they began lining up and counting each pile. I helped collect the data by writing down the numbers for each pile.
Using the data we collected Kai and I made a graph. Sitting around the table at lunch we all studied our graph. The boys answered questions like, "Which piles have the same amount?" "Which is biggest?" "Which is smallest?" "How many more would this pile need to equal that one?" "How much smaller is this pile than that one?" etc.
I can't tell you how much the boys loved this. They now have ideas of all different things they want to sort, it's a new favorite activity. And think of all the math skills involved - classification, grouping, numeration, ordering, comparing, counting, graphing, subtraction, addition! Sometimes, you just never know what is going to excite your children!
more ww posts here.


On June 23, 2019 the USPS enacted a second rate change. They will start using dimensional weight to calculate the package pricing for Priority Mail®, Priority Mail Express® and Parcel Select. They are also changing the way the calculate dimensional weight. Continue reading for a more detailed breakdown.
Dimensional (DIM) Weight
All major carriers (UPS, FedEx and now USPS) use dimensional weight to determining the shipping costs of larger packages with lower weights, which means light packages that take up valuable space in delivery trucks and airplanes. If the calculated dimensional weight is higher than the actual weight,you will be required the higher amount.
Dimensional Weight Calculation
Some prefer to double check the work of computers so you calculate dimensional weight on your own using the following rules.
- Multiply the length, width, and height of your parcel. Determine if it is over 1 cubic foot or 1,728 cubic inches.
- If your package is less than 1,728 cubic inches, you don’t need to enter in the dimensional weight. Your package will be priced off of the actual weight.
- If your package is greater than 1,728 cubic inches, you will divide the cubic inches by 166 to calculate the weight of your package.
How Do You Measure Dimensional Weight?
Below is an example Neopost provides on how to calculate dimensional weight:
Example 1: You are shipping sneakers that weight 7lbs. in a box that measures 12″ x 12″ x 8″. Multiply the length x width x height of the package: 12″ x 12″ x 8″= 1,056 cubic inches. You will calculate the price using the actual weight of 7 lbs. because your package is under 1 cubic foot or 1,728 cubic inches.
Example 2: You are shipping a book that weighs 5 lbs. in a box that measures 12″ x 12″ x 14″. You multiply the length x width x height of the package. 12″ x 12″ x 14″ = 2,016 cubic inches. The dimensions are over 1 cubic foot or 1,728 cubic inches, so you will then divide the dimensions by the USPS divisor which is 166. 2,016 cubic inches / 166 = 12.14 lbs. rounded up to 13 lbs. This is the weight you will enter into the machine.
The fees can really add up if you are shipping large, lightweight packages! To reduce being charged for dimensional weight, you can use:
- The smallest packaging possible
- Flat Rate boxes and envelopes
- Padded envelopes for flexible items like shipping clothing
How Do You Enter Dimensional Weight Into a Postage Meter?
Neopost Ship online shipping software calculate dimension automatically. If you have a postage meter without the software, you will have to enter the manually calculated weight by selecting manual weight mode. Model-specific instructions can be found here.


- CALCULATOR TOY: Children will have fun learning simple arithmetic with a colorful, easy to read, fun to use monkey calculator. Monkey helps kids to retain addition tables by self checking practice.
- EASY TO USE: Slide the monkey’s feet so his toes point to the numbers you want to add and the correct answer will appear in the ring between the monkey’s hands! Measures 8.5 x 11″ approx.
- MATH TOYS FOR KIDS: Make math fun with a simple to use STEM toy that encourages retention & reinforces learning at home principles. Math practice is always more fun with a monkey!
- PRACTICE ADDS UP: Strengthen your child’s math skills and begin their love of learning with the Monkey Addition Calculator early education tool and monkey around with math! Ideal for kids age 4 & up.
- EDUCATIONAL TOYS FOR PRESCHOOL: Prepare your child early for school by introducing simple math principles. Kids will have fun learning by practicing with this interactive teaching tool.


The Department of Post-Graduate studies in Mathematics was established in 1958 under the headship of Prof. M. R. Puri which was later upgraded to the P.G. Department of Mathematics and Statistics when the postgraduate classes leading to M. Sc. Degree in Statistics were started in 1988. Subsequently, the Department of Statistics was established as an independent department in 2004. In 2007, the Department has extended masters programmes in Mathematics at South Campus and in 2010 at Govt Degree College, Baramulla. About 65 students get enrolled at South campus and 45 at GDC Baramulla for M.Sc. Mathematics programme each year.
Over the years, the Department of Mathematics has grown into a center of research both in Mathematics and Statistics where facilities are available for research mainly in the areas of Functional Analysis, Complex Analysis, Graph Theory and Mathematical Biology. Besides a rich collection of books on diverse areas of contemporary research both in Mathematics and Statistics which are housed in the Departmental Library for which the Department has been receiving generous funding from the National Board of Higher Mathematics under the Department of Atomic Energy (Govt. of India). The Department also has a rich Computer Lab. where computing facilities with internet are available for use by the research scholars as well as by the master’s level students. During the M. Sc. Programme, the students are also benefited by the lectures by world-class mathematicians through mathematical videotapes procured from the American Mathematical Society. In terms of collaborative research work and exchange visits, the Department has established academic contacts with certain centres of excellence both in India and abroad such as the International Center of theoretical physics (Itally), the University of Kiel (Germany), University of Montreal (Canada), National Technical University, Athens (Grece), Indian Institute of Technology (Kanpur), Harish Chandra Research Institute (Allahabad), Indian Statistical Institute (Banglore) and the Indian Institute of Technology (Roorkee), Jiwaji University, Gwalior etc.
Earlier the Department has conducted Summer Schools and Refreshers Courses for the benefit of teachers working in universities and undergraduate colleges. A workshop on ‘Partial Differential Equations’ was also conducted in 1984 in collaboration with the DAE-TIFR, which was a huge success.
In the recent past, the Department has conducted many valuable events including the following:
- Teachers training camp for Regional Mathematical Olympiad, September 10-13, 2013
- 3-Day Training programme on mathematical modeling in biological systems, April 2-4, 2015
- Advanced Instructional School on Naïve Set theory, April 13- May 2, 2015
- Research promotion workshop on “Introduction to Graph and Geometric Algorithms”, May 18-20, 2015
- Instructional School for Teachers on Field Theory, May 2-14, 2016
- One-week workshop on Applications of Mathematics, October 5-11, 2016
- Recent Trends in Pure and Applied Mathematics, December 19, 2017.
- 2 Day National Level Workshop on Geometrical Interpretation of Mathematical Concepts, August 28-29, 2018
- Organized 4-Day INMO tranining Camp in collaboration with Homi Bhabha Centre for Science Education Mumbai and NBHM, December 17-20, 2018
The Department has produced more than fifty students who have qualified JRF/NET/GATE/SLET and other competitive examination of national level. A good number of research scholars got awarded M. Phil/Ph.D. degrees under the guidance of Departmental faculty in the research fields, which include Functional Analysis, Complex Analysis, Graph Theory and Mathematical Biology. Presently about 15 scholars are working on various problems leading to Ph.D. programme.


- Linear Algebra Textbook Solutions and Answers
- Answers to exercises LINEAR ALGEBRA - Joshua - Saint Michael's.
- Linear Algebra 2nd Edition Textbook Solutions
- Linear Algebra and Its Applications 9780321385178 Homework.
Linear Algebra Textbook Solutions and Answers
Stuck on a linear algebra question that's not in your textbook?
Answers to exercises LINEAR ALGEBRA - Joshua - Saint Michael's.
Chegg's linear algebra experts can provide answers and solutions to virtually any linear algebra problem, often in as little as 2 hours.
Linear Algebra 2nd Edition Textbook Solutions
If false, explain why or give a counterexample that shows why the statement is not true in every case. Every matrix is row equivalent to a unique matrix in echelon form. Any system of n linear equations in n variables has at most n solutions. If a system of linear equations has two different solutions, it must have infinitely many solutions. If a system of linear equations has no free variables, then it has a unique solution. If an augmented matrix [A b] is transformed into [C d] by elementary row operations, then the equations Ax = b and Cx = d have exactly the same solution sets. If a system Ax = b has more than one solution, then so does the system Ax = 0. If A is an m × n matrix and the equation Ax = b is consistent for some b, then the columns of A span ℝ. If an augmented matrix [A b] can be transformed by elementary row operations into reduced echelon form, then the equation Ax = b is consistent. If matrices A and B are row equivalent, they have the same reduced echelon form. The equation Ax = 0 has the trivial solution if and only if there are no free variables. If A is an m × n matrix and the equation Ax = b is consistent for every b in ℝ. If an n × n matrix A has n pivot positions, then the reduced echelon form of A is the n × n identity matrix. If 3 × 3 matrices A and B each have three pivot positions, then A can be transformed into B by elementary row operations. If A is an m × n matrix, if the equation Ax = b has at least two different solutions, and if the equation Ax = c is consistent, then the equation Ax = c has many solutions.
Solutions to Linear Algebra, Fourth Edition, Stephen H. Friedberg.
Your customers will taste the difference and come back time and time again for products prepared by Deli Brands of America.
Solutions manual for linear algebra:
Rating: 100 / 100
Overall: 96 Rates


So today isn't such a bad day, just a hectic one. I'm a bit worried and I know that I should be working nonstop on my thesis; but, I've got to have the chance to eat and refresh my mind. Btw, I now have roughly twenty-four hours until a preliminary copy of my masters thesis is due.
It's OK though, no really, it's okay.
My take on the situation is from one of my favorite novels "The Hitchhikers Guide to the Galaxy" and that is "don't panic."
That and 42.
Speaking of numbers and mathematics, I believe I've finally cracked the formula for my theory of quasi-static loading! It only took a few years and a giant looming deadline but I think I did it! The solution turned out to actually be a hybrid theory of statics and kinematics that can be simplified to a series of summation and a single 6x6 matrix.
Now just for that pesky writing...


"Music and Mathematics: From Pythagoras to Fractals," takes you on a journey to learn the intimate relationship of music and mathematics. For music evolved from the extraordinary discoveries designed by mathematicians. For example, the modern scale with the interval, whole-whole-half-whole-whole-whole-half, was invented by Pythagoras who heard hammers in proportional sizes striking in a tones that compose the scale that Americans familiar with today.
You might have noticed that most musicians are mathematically inclined, for it takes a strong understanding of math in order to comprehend the concept of music theory. Music theory is the application of patterns and formulas to sound. Music was constructed by those who conceived the intricate patterns and formulas of math. As you can see, mathematics is very vital for the life and comprehension of Music.
Although music has evolved and will continue to do so, the infrastructure of the beautiful tones will still be as important as it is now, which is math. After reading this book, I have come to appreciate math a little more, seeing that it helps fuel the life I pursue, which is a life of joyfulness, success, God, and music.


It's that time of year again for the Tour de Primes
Here is the PG webpage for this years TDP https://www.primegrid.com/forum_thread.php?id=10427#168083
Looks like GFN16 and PPSE are large enough for the T5K the book of the 5,000 largest primes.
They are the best bet to get a P badge for the TDK
I would think it starts at 00:01 UTC Feb.1
Results will be available at http://www.primegrid.com/challenge/tdp_2024.php
From the PG thread:
For the month of February, an informal competition is offered. There are no challenge points to be gained... just a simple rare jersey at the end of the month to add to your badge list. No pressure or stress other than what you put on yourself. :)
For 2024, we're bringing back the badges introduced in 2018:
- Red Jersey -- discoverer of largest prime
- Yellow Jersey -- prime count leader (tiebreaker will be prime score)
- Green Jersey -- points (prime score) leader
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Yesterday was insane.
Once I got the big girls out the door I had two hours to clean up the house and get some important computer work done. And I really mean it was important, too: paying bills, answering my very neglected email inbox, figuring out Primary stuff, getting voice lessons finally squared away for the semester (still not entirely done! Augh!), getting Bria's math tutor scheduled, etcetera.
Then I taught a lesson after which I went back to the computer to do my grocery list for the day. This week has been so insane already (why is the first week of school always like this?) that I haven't been shopping for nearly two weeks and the girls were forced to have school lunch this morning and they weren't one bit happy about it because of whatever yuckiness was on the menu today. Oh well. Worse things have happened than having to eat chicken BBQ.
Sophia and I ate lunch really quickly with Joel and then we were off to the grocery store (where I remembered to use my save the earth bags for the second time in a row). I barely got home in time to put away the cold stuff and then teach a couple more lessons. Then it was off to the bus stop to get the big girls, back into the kitchen (where the majority of the grocery bags were still sitting on the floor waiting to be put away) for a quick snack and out the door to Chloe's piano lesson. When she was finished we were off to Bria's violin lesson, which I do like because it's in the office next door to Joel's and the other two girls just go hang out with dad while Bria and I do her lesson. After her lesson, Joel needed to show me the new room I'm to teach in this semester, so we walked over to another building to do that so I could mark on the wall where I want the mirror to go.
Then it was time to walk to yet another building on campus for Bria's math tutoring session (Joel and the other girls headed back home). I hired one of my students who is a math major to tutor her, and this was her first time going for that. He was great with her, and I even learned a bunch of things about math that I never knew before! All of which is fabulous, since 4th grade math nearly killed me off as a mother and I was despairing more about 5th grade math than even Bria was.
As soon as Bria and I walked in the front door, the Primary President drove up for our presidency meeting. So we did that for an hour and I somehow ate dinner in the middle of it (so thankful for my cooking husband) and as soon as she left I got the groceries put away (finally!) and the kids ready for bed. Joel went off to a rehearsal and I finally got them down and had a moment to actually sit down myself.
I won't even tell you how the day before that I was at the preschool registration collecting tuition from 12 until nearly 7. I volunteered to be on the board this year, and I'm not sure why I thought that was a good idea? But, it's done, and actually my job will be much easier from here on out.
I'd just like a chance to maybe take a shower first, if that's okay.


Day 07 - Lambdas¶
Lambda functions are also known as anonymous functions, they can, as a normal function, take different arguments and return a single value, but lambdas are one-line functions.
The syntax is as follows:
lambda arguments: expresion
def normal_multiply(num_a, num_b): return num_a * num_b lambda_multiply = lambda num_a, num_b: num_a * num_b print(normal_multiply(5, 2)) print(lambda_multiply(5, 2)) # Output: # 10 # 10
As you can see both
lambda_multiply functions, another difference you can notice is that lambda functions return a value without explicitly using the
Why use Lambda functions?¶
We can create functions that perform a simple actions or operations using lambda functions, so that we can make our code more compact and simple.
Let’s try using the lambda functions with one of the previous examples.
def multiple(number): return number % 2 == 0 numbers = range(1, 101) filter_numbers = filter(multiple, numbers) print(list(filter_numbers))
Using lambda function¶
numbers = range(1, 101) filter_numbers = filter(lambda number: number % 2 == 0, numbers) print(list(filter_numbers))
When you understand the syntax of these functions, you will see that they are really easy to use and can be very useful.


(Total change of subject from my evangelism series, I know. All the blogging gurus tell you to focus on one central idea, but I just have too many interests. Sorry! Maybe a few of you will find this interesting? For the record, I’m planning to return to that series yet with a few more posts. Just taking a little detour.)
I came across number rods in some of my Montessori reading a few months back, around Lydia’s second birthday. They seemed like a great tool to help enhance her understanding of counting and numbers. I was really eager to make some for Lydia, and pestered Ben mercilessly to help me gather the materials and assemble them. I finally introduced them to Lydia with great enthusiasm.
She didn’t get it and was totally uninterested. I sighed and put them away for six months.
I recently re-introduced them to her, and to my delight, she takes great interest in lining them up and counting them, and will do so on her own initiative.
I really do think they enhance her understanding of quantity and numbers, and thought I’d share our experience.
Making Number Rods
Number rods are a set of ten lengths of wood that vary in size incrementally. They alternate between two colours (traditionally red and blue), and when you line them up in order, they create a “stair” effect.
My number rods are not perfect by Montessori standards, but they follow the same basic principles.
To make the number rods, I just had Ben cut ten lengths of solid oak, from one to ten inches long. (Each square is one inch tall. So our first rod is one inch long; our second one is two inches long, etc.) (Hint: You could probably use paint stirring sticks for this material).
I used two colours of paint that we happened to have left over from past projects (lime and grey). I marked them off into one-inch segments, used masking tape to create clean(ish) lines, and painted the segments one colour at a time (first the lime, then the grey).
I chose to add the numerals at the bottom of one side. They can be flipped so that the numerals don’t show if you prefer to use them that way.
Using the Number Rods
This tool is great, in my opinion, because it gives the child a sensorial impression of quantity. In other words, Lydia can see what each quantity looks like. She can see that when you go up in numerals, you get a greater quantity, which means a greater length. And she can see which numeral goes with each quantity, giving her greater familiarity with the numerals and what they mean.
For now, we’re just using them to understand length, quantity and and counting. Later, you can use them to learn more complex math skills.
When I first (re-)introduced them to her (at thirty months, or two-and-a-half), I simply laid them out randomly, and then told her I was going to line them up from shortest to longest.
I started with one and two, and then asked her which was the next longest. She quickly caught on and lined the rest up in order from shortest to longest.
I then went through and counted them from one to ten, and invited her to join me.
I keep them in a Ziploc bag in her activity dresser (I know: NOT Montessori), and to my delight, she will often get them out herself and start lining them up, shortest to longest, completely independently.
Then she will go through and count them out loud, putting her finger on each one.
Have you used number rods? Any other great ideas or tools for introducing math basics to young children?


Berlin, Germany - Sunday, May 29 2016
Whilst training some minions for the 2015 MCPC and ACPC, I remember constantly bitching about how unwise it is to go ahead with the implementation of an algorithm without having a clear proof of its correctness and halt. Halt definitely comes before correctness. I mean how the living fuck is an algorithm correct if it runs infinitely.
Moving on. I was having a discussion with some colleagues a couple ‘o days ago and we brought up mathematical induction and I made a point about how most iterative and recursive algorithms can be modeled mathematically and proven as such. That discussion escalated and evidently turned into an argument. Insertion sort came up and well, we had to model and prove its halt and correctness mathematically.
If you don’t know what insertion sort is, what in seven hells are you even doing here? How the living fuck did you read this far? Kudos to you! Be a good lad, click that link and read up.
To the lazy ass who can’t click a link or just for the fuck of it, I will give a brief explanation of insertion sort. Ever tried to sort a shuffled deck of cards? What most people do is insertion sort. Basically, you have a sequence of unsorted elements, starting from the first element to the last, you pick an element and you insert this chosen element in the sorted list before it in a way that the list before it stay sorted after insertion. For the minds that need illustration
Before, we have this
[ 1 3 5 11 9 24 3 6 3 10 ]
after arriving at the index 5, we have a sorted prefix, the element at index 5 and an unsorted suffix
[ 1 3 5 11 ] 9 [ 24 3 6 3 10 ]
9 in the list before it we must put it in between
11 to keep the list sorted. We do the same for every number after
Shit is about to get real now. Brace yourselves.
Let us define a vector of length with the following ordering operations.
We also define the partial vector with containing exactly the first elements of sorted if and only if
Observe that this also implies the following
Now we define the insertion operation such that
The goal is to prove by induction that the insert operation always keep the list sorted (correctness) and that after insert operations (halt) we have .
The first thing we will do is assume that arriving at requires exactly insertion operations. So yes, we assume the halt because fuck it!
We know is sorted because it is empty, we also know is sorted because it has only one element.
Either way, is sorted and validates the above sort condition. If now we assume that for any with , is sorted how do we prove that is also sorted?
Supposing that such exists with since is sorted, then and . If not then since is the . This also implies that . Therefore we can conclude that inserting just before keeps the array sorted and gives us
What happens when we can’t find such , when , then we simply insert at the end of the list. I probably should have mentioned that beforehand but…
We have successfully shown that the insert operation is finite. We have also shown that the entire sorting requires insert operations thus also finite. So yeah insertion sort doesn’t run forever! But you probably knew that already. The most import thing is that by induction we’ve proven that the insertion sort indeed sorts.


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1) There are thrice as many blue marker as green marker. 1/4 of the blue marker were given to the boys and 136 of the blue marker were given to the girls.
1/4 of the green marker were given to the girls. Number of green markers left was 3/8 of the total number markers left.
a) how many blue markers were there at first?
b) express the number of green markers that were given away as a fraction of the number of blue markers that were given away. (simplest form)
2) The ratio of the amount of sugar in container A to the amount of sugar in container B was 5:8. When 85g of sugar was added to each container, the ratio became 2:3. What was the amount of sugar in each container?


Kernel-Type Estimators of Divergence Measures and Its Strong Uniform Consistency
Hamza Dhaker1, *, Papa Ngom1, El Hadji Deme2, Pierre Mendy3
1Departement de Mathématiques et Informatique, Faculté des Sciences et Technique, Université Cheikh Anta Diop, Dakar, Sénégal
2Sciences Appliquées et Technologie, Unité de Formation et de Recherche, Université Gaston Berger, Saint-Louis, Sénégal
3Département de Techniques Quantitatives, Faculté des Sciences Economiques et de Gestion, Université Cheikh Anta Diop , Dakar, Sénégal
To cite this article:
Hamza Dhaker, Papa Ngom, El Hadji Deme, Pierre Mendy. Kernel-Type Estimators of Divergence Measures and Its Strong Uniform Consistency. American Journal of Theoretical and Applied Statistics. Vol. 5, No. 1, 2016, pp. 13-22. doi: 10.11648/j.ajtas.20160501.13
Abstract: Nonparametric density estimation, based on kernel-type estimators, is a very popular method in statistical research, especially when we want to model the probabilistic or stochastic structure of a data set. In this paper, we investigate the asymptotic confidence bands for the distribution with kernel-estimators for some types of divergence measures (Rényi-α and Tsallis-α divergence). Our aim is to use the method based on empirical process techniques, in order to derive some asymptotic results. Under different assumptions, we establish a variety of fundamental and theoretical properties, such as the strong consistency of an uniform-in-bandwidth of the divergence estimators. We further apply the previous results in simulated examples, including the kernel-type estimator for Hellinger, Bhattacharyya and Kullback-Leibler divergence, to illustrate this approach, and we show that that the method performs competitively.
Keywords: Divergence Measures, Kernel Estimation, Strong Uniform, Consistency
In this paper, we focus on the similarity between two distributions. Given a sample from one distribution, one of fundamental and classical question to ask is: how to have the similarity between this density with another known density? First, one must specify what it means for two distributions to be close, for which many different measures quantifying the degree of these distributions have been studied in the past. They are frequently called distance measures, although some of them are not strictly metrics. The divergence measures play an important role in statistical theory, especially in large theories of estimation and testing. They have been applied to different areas, such as medical image registration , classification and retrieval. There are several important problems in machine learning and statistics that require the estimation of the distance or divergence between distributions. Divergence between distributions also proves to be useful in neuroscience, For example (see, e.g ). employs divergence to quantify the difference between neural response patterns.
Later, many papers have appeared in the literature, where divergence or entropy type measures of information have been used in testing statistical hypotheses. For more examples and other possible applications of divergence measures, see the extended technical report [23,24]. The key role of the measure divergence in these various applications, it is necessary to accurately estimate these divergences.
Recently Ngom et all  has introduced the Divergence Indicator method by proposing a test for choosing between a random walk and AR(1), using a divergence measure.
The class of divergence measures is large; it includes the Rényi- [25, 26], Tsallis- , Kullback-Leibler (KL), Hellinger, Bhattacharyya, Euclidean divergences, etc. These divergence measures can be related to the Csiszár- divergence . The Kullback-Leibler, Hellinger and Bhattacharyya are special cases of Rényi- and Tsallis- divergences. But the Kullback-Leibler one is the most popular of these divergence measures. Estimation of divergence and its applications have been many studies using different approaches. For example Pardo  presented methods and applications in the case of discrete distributions. By exploring a nonparametric method for estimating the divergence in the continuous case, Poczos and Schneider  proposed a -nearest-neighbor estimator and proved the weak consistency of the estimator Rényi- and Tsallis- divergences.
Finding estimators nonparametric of measure divergence, remains an open issue. Krishnamurthy and Kandasamy  used an initial plug-in estimator for estimating by estimates of the higher order terms in the von Mises expansion of the divergence functional. In their frameworks, they proposed three estimators for Rényi-, Tsallis-, and Euclidean divergences between two continuous distributions and established the rates of convergence of these estimators. The main purpose of this paper is to analyze estimators for divergence measures between two continuous distributions. Our approach is similar to that of Krishnamurthy and Kandasamy  and is based on plug-in estimation scheme: first, we apply a consistent density estimator for the underlying densities, and then we plug them into the desired formulas. Unlike of their frameworks, we study the strong consistency estimators for a general class of divergence measures. We emphasize that the plug-in estimation techniques are heavily used by [2, 9] in the case of entropy. Bouzebda  proposed a method to establish consistency for kernel-type estimators of the differential entropy. We generalize this method for a large class of divergence measures in order to establish the consistency of kernel-type estimators of divergence measure when the bandwidth is allowed to range in a small interval which may decrease in length with the sample size. Our results will be immediately applicable to proving strong consistency for Kenel-type estimation of this class of divergence measures.
The rest of this paper is organized as follows: in Section 2, we introduce divergence measures and we construct their kernel-type estimators. In Section 3, we study the uniform strong consistency of the proposed estimators. Section 4 is devoted to the proofs. In Section 5, numical examples are proposed in order to illustrate the performance of our method. Finally, in Section 6, we present our conclusion.
2. Kernel-Type Estimators of Divergence Measures
In this section we give the notations and then presenting some basic definitions. We are interested with two densities,, : where denotes the dimension. The divergence measures of interest are Rényi-, Tsallis- are defined respectively as follows
These quantities are nonnegative, and equal zero iff almost surely (a.s). Remark that in the special cases where , we obtain from (1) and (2) the well known Hellinger, Kullback-Leibler and Bhattacharyya divergences.
which is related to the Shannon entropy. For some statistical properties for the Shannon entropy, one can refer to .
For the following, we focus only on the estimation of and . The Kullback-Leibler, Hellinger and Bhattacharyya can be deducted immediately.
We will next provide consistent estimator for the following quantity
whenever this integral is meaningful. Plugging its estimates into the appropriate formula immediately leads to a consistent estimator for the divergence measures and .
Now, assuming that for the rest of the document, the density is unknown, and the density is known and satisfies: is finite, this implies that is finite. Next, consider a sequence of independent and identically distributed -valued random vectors, with cumulative distribution function a density function with respect to the Lebesgue measure on . The following conditions are needed for the remainder of this paper. To construct our divergence estimators we define, We start by a kernel density estimator for , and then substituting by its estimator in the divergence like functional of . For this, we introduce a measurable function that satisfies the following conditions.
(K.1) is of bounded variation on
(K.2) is right continuous on
where is the bandwidth sequence. Assuming that the density is continuous, one obtains a strongly consistent estimator of , that is, one has with probability , , . There are also results concerning uniform convergence and convergence rates. For proving such results one usually writes the difference as the sum of a probabilistic term and a deterministic term , also called bias. For further explanation One can refer to [10, 12, 13], among other authors. After having estimated , we estimate by setting
where and is a sequence of positive constants. Thus, using (5), the associated divergences and can be estimated by:
The approach used to define the plug-in estimators is also developed in  in order to introduce a kernel-type estimator of Shannon’s entropy. Some statistical properties of these divergences is related to those of the kernel estimator of the continuous density . The limiting behavior of , for appropriate choices of the bandwidth , has been widely studied in the literature, examples include the work of Deroye [6, 7] Bosq  and Prakasa . In particular, under our assumptions, the condition that together with is necessary and sufficient for the convergence in probability of towards the limit , independently of and the density . We can find other results of uniform consistency of the estimator in [4, 10, 5] and the references therein. In the next section, we will use the methods developed in previous references to establish convergence results for the estimates and deduce the convergence results of and .
3. Statistical Properties of the Estimators
We first study the strong consistency of the estimator defined in (5). Throughout the remainder of this paper, we well the notation for , which is delicate to handle. This is given by
Lemma 1 Let satisfy (K.1-2-3-4) and let be a continuous bounded density. Then, for each pair of sequence , such that , together with , as and for any one has with probability 1
The proof of Lemma 1 is postponed until Section 4.
Lemma 2 Let satisfy (K.3-4) and let be a uniformly Lipschitz and continuous density. Then, for each pair of sequence , such that , together with , as and for any we have
The proof of Lemma 2 is postponed until Section 4.
Theorem 1 Let satisfy (K.1-2-3-4) and let be a uniformly Lipschitz, bounded and continuous density. Then, for each pair of sequence , such that , together with , as and for any one has with probability 1
This, in turn, implies that
The proof of Theorem 1 is postponed until Section 4.
The following corollaries handle respectively the uniform deviation of the estimate and with respect to and .
Corollary 1 Assuming that the assumptions in Theorem 1 hold. Then, we have
This, in turn, implies that
The proof of Corollary 1 is postponed until Section 4.
Corollary 2 Assuming that the assumptions in Theorem 1 hold. Then, we have
This, in turn, implies that
The proof of Corollary 2 is postponed until Section 4.
Note that, the divergence estimator such as (5) also requires the appropriate choice of the smoothing parameter . The results given in (6), (7) and (8) show that any choice of between and ensures the strong consistency of the underlying divergence estimators. In other words, the fluctuation of the bandwidth in a small interval do not affect the consistency of the nonparametric estimators of these divergences. The work of Bouzebda and Elhattab  is very important for establishing our results, these authors have created a class of compactly supported densities. They used the following additional conditions.
(F.1) has a compact support say and is -time continuously differentiable, and there exists a constant such that
(K.5) is of order , i.e., for some constant
Under (F.1) the expression may be written as follows
Theorem 2 Assuming conditions (K.1-2-3-4-5) hold. Let fulfill (F.1). Then for each pair of sequences with , as and for any , we have
The proof of Theorem 2 is postponed until Section 4.
Corollary 3 Assuming that the assumptions in the Theorem 2 hold. Then,
Corollary 4 Assuming that the assumptions in the Theorem 2 hold. Then, for any we have
The proof of Corollaries 3 and 4 are given in Section 4.
Now, assume that there exists a sequence of strictly nondecreasing compact subsets of , such that For the estimation of the support we may refer to () and the references therein. Throughout, we let , where and are as in Corollaries (3) and (4). We choose an estimator of in the Corollaries (3) and (4) as the form
Using the techniques developed in  and the Corollaries (3) and (4) one can construct an asymptotically certainty intervals for the true divergences and .
4. Proofs of Our Results
Proof of Lemma 1. to show the strong consistency of , we use the following expression
where and is a sequence of positive constant. Define
Since is a 1-Lipschitz function, for then
Therefore for , we have
where denotes, as usual, the supremum norm, i.e., . Hence,
Using the conditions on the kernel , posed by Einmahl , consider the class of functions
For , set , where the supremum is taken over all probability measures on , where represents the -field of Borel sets of , i.e is the smallest containing all the open (and/or closed) balls is . Here, denotes the -metric and is the minimal number of balls of -radius needed to cover .
We assume that satisfies the following uniform entropy condition.
(K.6) for some and ,
(K.7) is a pointwise measurable class, that is there exists a countable sub-class of suchthat we can find for any function a sequence of functions in for which
This condition is discussed in . It is satisfied whenever is right continuous.
Remark that condition (K.6) is satisfied whenever (K.1) holds, i.e., is of bounded variation on , we refer the reader to Van der Vaart and Wellner , for details on conditions of entropy (see also Pakes and Pollard , and Nolan and Pollard ). Condition (K.7) is satisfied whenever (K.2) holds, i.e., is right continuous, this condition is discussed in , (see also  and ).
From Theorem 1 in , whenever is measurable and satisfies (K.3-4-6-7), and when is bounded, we have for each pair of sequence , such that , together with and as , with probability 1
Since , in view of (11) and (12), we obtain with probability 1.
It concludes the proof of the lemma.
Proof of Lemma 2.
Let be the complement of in (i.e, ). We have
Term . Repeat the arguments above in the terms with the formal change of by . We show that, for any ,
On the other hand, we know (see, e.g, ), that since the density is uniformly Lipschitz and continuous, we have for each sequences , with , as ,
Term . It is obsious to see that
Thus, in view of (16), we get
Finaly, in view of (17) and (20), we get
is deduced the proof of the lemma.
Proof of Theorem 1. We have
Combinating the Lemmas (1) and (2), we obtain
It concludes the proof of the Theorem.
Proof of Corollary 1. Remark that
Using the Theorem (1), we have
and the Corollary 1 holds.
Proof of Corollary 2. A first order taylor expansion of arround and gives
Remark that from Theorem 1,
which in turn, implies that
Thus, for all
and the Corollary 2 holds.
Proof of Theorem 2. Under conditions , and using Taylor expansion of order we get, for ,
where and Thus a straightforward application of Lebesgue dominated convergence theorem gives, for large enough,
Set, for all ,
by combining (22) and (24)
Let be a sequence of nondecreasing nonempty compact subsets of the interior of such that
Now, from (25), it is straightforward to observe that
The proof of Theorem 2 is completed.
Proof of Corollary 3. A direct application of the Theorem 2 leeds to the Corollary 3.
Proof of Corollary 4. Here again, set, for all ,
A first order Taylor expansion of leads to
Using condition , is compactly supported), is bounded away from zero on its support, thus, we have for large enough, there exists , such that , for all in the support of . From (23), we have
by combining the last equation with (22)
The proof of Corollary is completed.
5. Simulation Study
Summarizing the ideas and the results given in the previous sections, we propose to study the performance of the kernel-estimators for Hellinger (-), Bhattacharyya (–) and Kullback-Leibler (–) measures and their uniform-in-bandwidth consistency.
Hellinger, Bhattacharyya and Kullback-Leibler divergences are defined respectively as follows:
The asymptotic behavior of each bandwidth is performed using the kernel-type estimator of the divergence criteria in corollary 3 and corollary 4 respectively.
We compute, for each chosen value of α, the expressions
where the cooresponding bounds () are defined by
We consider an experiment in which the DGP (Data Generating Process) for the true distribution are generated from a mixture of two normal distributions,
and the function is supposed to be a normal distribution with mean 1 and variance 2.
The sample size varies from 10 to 1000, and for each size, the statistics andare evaluated.
In order to plot against sample size, we need to perform three sets experiments.
The results are presented in tables 1-3 and figures 1-3.
The tables 1-3 show that the kernel-type estimators of the divergence measures converge rapidly to their pseudo-true value, and confirm our asymptotic results. They all show that the discrepancy between the estimated and the true divergence criterion converge rapidly to zero. Similarly, in table 2 and table 3, DB and DK converge, as expected, to zero, which is the mean of the asymptotic distribution when the estimated distribution is close to f.
The figures 1-3 show values plots for Hellinger, Bhattacharyya and Kullback-Leibler divergence respectively. The preceding comments from the table 1-3 also apply to the figure 1-3. For dealing with divergence error, it is much revealing to graph DH, DB and DK against sample size. They also confirm our asymptotic results. We note that, as sample size increases, the value discrepancy plots of divergence error converge, as it should, to zero These plots provide a great deal of information about how the sample size affect the performance of these informational criterions.
6. Concluding Remarks and Future Works
In this paper, we are concerned with the problem of nonparametric estimation of a class of divergence measures. For this cause, many estimators are available. The most recent ones are the estimates developed by Bouzebda . We introduce an estimator that can be seen as a generalization of those previously suggested, in the sense that Bouzebda was only interested in the case of entropy, while we focus on the Rényi- and the Tsallis- divergence measures. Under our study, one can easily deduce Kullback-Leibler, Hellinger and Bhattacharyya nonparametric estimators. The results presented in this work are general, since the required conditions are fulfilled by a large class of densities. We mention that the estimator in (5) can be calculated by using a Monte-Carlo method under a given distribution . And a practical choice of is where and .
It will be interesting to enrich our results presented here by an additional uniformity in term of in the supremum appearing in all our theorems, which requires non trivial mathematics, this would go well beyond the scope of the present paper. Another direction of research is to obtain results, in the case where the continuous distributions and are both unknown. The problems and the methods described here are all inherently univariate. A natural and useful multivariate extension appear in the use of copula function.


Dyscalculia: A Comprehensive Overview
Dyscalculia, a learning disability, affects the person's ability understand, process and use math concepts. This can take many forms, including difficulty understanding numbers and recognition of mathematical symbols. Dyscalculia sufferers often have trouble with spatial orientation and time orientation. Dyscalculia is a learning disability that affects approximately 6- 7% of people.
How can Dyscalculia be caused?
Dyscalculia can be caused by many factors, including genetic, psychological, and Environmental . One possible cause of dyscalculia may be structural differences within the brain. For example, there might not be enough development in areas responsible for processing numerical information. A family history of dyscalculia may play an important role. Dyscalculia can be caused by poor instruction and bad teaching techniques. Psychological factors such as Depression , stress and anxiety can make math learning difficult.
There are several common Treatment s for dyscalculia
There are many treatments for dyscalculia. Educational interventions are the most popular. These include accommodations and specialized teaching methods that help to improve math understanding and skills. Cognitive behavioral therapy (CBT), which focuses on helping individuals manage frustrations and anxiety around math, is another option. Helpive technology and other strategies can be used to improve math skills.
Research suggests that exercising may improve dyscalculia patients' math skills. Research has shown that exercise can improve math skills by increasing the connections between brain cells. One study showed that both adults and children with dyscalculia could benefit from aerobic exercise, which in turn led to higher math scores.
Expert Opinions on Dyscalculia
For dyscalculia to be managed successfully, experts agree early intervention is crucial. A diagnosis early and timely treatment are key to preventing academic problems and ensuring that people receive the support and assistance they require. Experts also stress the need to provide accommodations for dyscalculia patients, including extra time for tests and assignments.
Studies have also demonstrated that individuals with dyscalculia are capable of succeeding academically if they receive the correct support. A recent study found that dyscalculia students who were given specialized instruction had significantly higher math scores than those who didn't. Individuals with dyscalculia may be able to make significant progress in math skills if they receive the correct interventions.
Natural Remedies for Dyscalculia and Other Tips
Dyscalculia sufferers may also benefit from natural treatments and other tips. Regular Physical activity, for example, can improve math skills and increase the connections between brain cells. Mindfulness activities such as meditation and deep breathing can be helpful in managing anxiety and frustrations related to math skills.
You can also break down difficult tasks into manageable pieces and use mnemonics for math facts. Providing visual aids such as charts or diagrams can also help people better understand math concepts. Positive reinforcement can also be helpful in motivating and boosting confidence when working on math problems.
Dyscalculia refers to a learning disability that impairs an individual’s ability understand, process and use math concepts. While the cause of dyscalculia remains unknown, it can be caused by a variety of factors, such as genetic, psychological, biological and environmental. There are many interventions and treatments that can be used to help dyscalculia sufferers improve their math skills. There are also natural ways to help such as mindfulness, regular exercise, positive reinforcement, visual aids and mnemonics. Individuals with dyscalculia will be able to manage the condition well and still excel in school if they have the support.
Dyscalculia, a learning disability common to dyscalculia patients can have a negative impact on their ability to comprehend and use math concepts. Individuals with dyscalculia are able to make significant progress in math skills if they receive the correct support and interventions. Natural tips and tricks may help improve math performance. Positive reinforcement is also a good way to boost motivation and confidence.


Main / Communication / Mathworks r2011a
Name: Mathworks r2011a
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Did you read the whole thread? I was not actually referring to Walter's answer, but to his comment to the cyclist's answer. Here is the cyclist's. Last Friday we released MATLAB (Ra), the latest and greatest in MATLAB. There are a lot of new and improved features in this. MATLAB Compiler generated standalone executables and libraries from parallel Ra. New Features, Bug Fixes, Compatibility Considerations. expand all.
Use bwconvhull to compute the convex hull image from a binary image. Control System Toolbox includes new model objects that you can use to represent systems with tunable components. The new fitensemble function constructs ensembles of decision trees.
quadprog has a new algorithm named 'interior-point-convex'. MathWorks today announced Release a (Ra) of its MATLAB and Simulink product families. Key in this release is the introduction of a. Ra shipped back in April, but I haven't said too much about it, yet. I mentioned the new geotiffwrite function in Mapping Toolbox. MATLAB Ra was released on April 8, Please note that this is prior to the release of Ubuntu Consequently Ubuntu is not.


At St Mary’s, children are taught Mathematics through the ‘Teaching for Mastery’ approach. ‘Teaching for Mastery’ is a journey from secure mathematical fluency to reasoning and problem solving. It is important that children are not just taught to memorise facts and methods, it is about being able to verbalise their understanding and apply their skills in different scenarios. Through exploration, clarification and application, children are able to demonstrate a deep, conceptual understanding of a topic and build on this throughout their journey at St Mary’s.
We use the White Rose Schemes of Learning as the basis for our Mathematics curriculum. Information regarding Year Group overviews and additional resources can be found via the link below.
Please find below the calculation policy and information regarding the ‘Teaching for Mastery’ approach to Mathematics.
|Multiplication Check Booklet.pdf||Download|
|St Mary's Mastery evening KS2.pdf||Download|
|St Mary Mastery evening KS1 updated.pdf||Download|
|Calculation Policy CPA.pdf||Download|
|EYFS - Maths mastery parent booklet - FINAL.pdf||Download|
|KS1 & KS2 Maths mastery parent booklet - without photos.pdf||Download|


Current ACI-REF Phase I Facilitators
Erin Shaw, University of Southern California
Erin is an Advanced Cyberinfrastructure Research and Education Facilitator (ACI-REF) at USC’s Center for High-Performance Computing (HPC) where she focuses on outreach and education. She comes to HPC from USC’s Information Sciences Institute (ISI) where she was a computer scientist and National Science Foundation investigator. Her research at ISI focused on developing computational technologies to model and assess online learning in engineering, math and science. She was most recently a Co-PI of an NSF technology translation grant to commercialize assessment technology for games programming, for K12 education. Before returning to higher education Erin was a software programmer and a math/science education volunteer with the Peace Corps. She has degrees in mathematics, computer graphics and education.


Although he was only a ninth grader, I had been really stunned at his maturity with his fantastic beliefs. Camera angles and movement will add elements for the story. Film Review Essay Sample Free Film Review Essay Sample: Jane Campion is probably the strongest directors today despite her comparatively young age. If you are interested in the theme, you may be prone to write in a manner that makes others interested as well. There's no difference should you come up with your friend, mom or God. Note how a topic sentences each prove one aspect of the thesis statement. Knowing where to place title on each page like the title page is important when doing the layout of the essay.
Many students fail to see the importance of math after high school. This article discusses how math can change your future, even if you never really Additional advantages Access to the WEB to discover information while writing and automatically import quotes and bibliographic references Spell and grammar checker Word counter so that the length requested Sharing your essay electronically with your instructor or perhaps a peer for feedback at every stage in the road The choice to publish essays in a choice of English or in Spanish SPANISH Una aplicacin que garantiza un ensayo perfectamente estructurado, y NUEVO EN LA VERSIN 2.
Why math is important essay
I usually take the time to go over the list of reasons why studying mathematics is important. Since mathematics is And, furthermore, Sam, you could not be hightoned by having an oldfashioned feller like Aaron Pinsky.
Why math is important essay
Why Is It Important to Study Math? Whats the point of learning math? Why is it so important that kids are exposed to mathematical thinking? And what do parents and teachers need to know about learning real math? Because of this, students who knows English will be more successful at his/her project.
Why Are Math and Science So Important? Why Are Math and Science So Important? Skip to Main Content Brookhaven College MATH three hours of College Algebra there isn't any grammar and punctuation mistakes.
Why Is Math Important? Date: 0102 at 15: 30: 23 From: Amanda Dalton Subject: Why math is important Why is math so important? Some of it we dont need to know in the real world so why do they teach us things we wont need to know? Your analysis or concluding observation is the way of wrapping up the information presented within your paragraph.
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Why is Writing so important? The importance of writing stems from the fact that writing is the primary basis Why Writing Is So Important Author:
Essay on why math is important in engineering Physics
Math essays Discuss how you are or will be using mathematics in your career. I believe that math is very important in our world, because with out it there would be no money, no math teachers, no business administrators to just name a few.
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Why is education important? Math and science skills: Although calculators and computers are readily accessible, you still need to learn how to do basic The marine our life is the first person to suffer from water pollution because they depend upon the nutrients in water to live.
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Helping students realize why math is important, and that math is a normal part of life, not just something they learn in school. A fabulous dating service can wrige be some a valuable thing along with a nasty thing.
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Why Play Math Games? By Kitty Rutherford, Posted April 27, Games are an important tool for learning in elementary school mathematics classrooms: In cases like this, interviews might be conducted with those who are familiar with the best place or event to acquire background information and opinions.
Students often wonder why they have to write in math class. In fact, the purpose of a math essay is for students to demonstrate their understanding of mathematical ideas and concepts. Additionally, it will state the effectiveness of the channels as rated by customers.
Is mathematics the most important subject? Debateorg
Usefullness of Mathematics in Everyday Life Essay Words 6 Pages. Usefullness of Mathematics in Everyday Life G H Hardy once said that Very little of mathematics is useful practically, and that little is comparatively dull. This statement is blatantly incorrect. Students can then rate the articles for quality and utilize a builtin citation and reference feature to generate their bibliography easily.
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Mathematics is a useful tool to simplify science by quantification of phenomena. In fact science without mathematics can go no further than a humorous oxymoron. You just cannot do physics and chemistry without calculus Schrodinger equation and To present the events, writers use backstory, flashback, flashforward, and foreshadowing.


01 Aug The What Is a Median in Math Diaries
What Is a Median in Math Fundamentals Explained
Let’s look at an excellent example. If you choose a marble from the bag without looking, you can select a red marble or you’re able to choose a blue marble. It isn’t likely that you will select a blue marble.
Thus it must be conquered as a means to thrive optimally. A double bar graph can be utilized to compare two sets of information, usually about precisely the same topic. The main reason is it can drastically be impacted by outliers (values that aren’t typical when compared with the remainder of the elements in the set).
Over the past couple of decades, the number of vehicles on road has increased manifolds around the world. Finding the mean, also referred to as averaging numbers, is a very helpful thing to understand how to do, particularly when you desire a precise estimate or an extremely accurate generalization. This is a portion of the main reason why housing affordability across the area has reached an all-time low.
Want to Know More About What Is a Median in Math?
It’s found by adding up all the provided data and dividing by the range of information entries. Besides sending secrets, you could also encode messages using your own personal key. If you want to see complete solutions you must register for a free trial account.
This contains all the worksheets you are going to need for the training course, plus other supplementary materials like old exams. How can you figure out the median math 4 how to compute mean and median in maths. This is so you’re able to check your exam and homework scores are recorded correctly.
Other folks turn to internet retirement calculators. Income inequality is a significant threat to America. Endgame quests Once you’ve completed the major story the Endgame begins.
During the last ten decades, the DC metro area has grown basically at the exact same pace as the remainder https://www.hft-leipzig.de/en/home.html of america. What these cry-babies will have to understand is that drone campaigns aren’t affordable. Well, now’s the chance to utilize PIA!
Even in the event you don’t know the expression Likert scale, you’re familiar with the general form. Appropriate comprehension of given situations and contexts can often offer a person who has the tools essential to establish what statistically relevant procedure to use. In the event you’re unfamiliar with these concepts, here is the best way to calculate each one.
Accuracy of the floating-point methods is measured concerning ulps, units in the previous location. Solving in cases where the operation doesn’t have a well-defined inverse isn’t impossible, but I hope I’ve expressed that it is only not as easy as it is with the four basic arithmetic operations. Consider what you’ve been given to begin, and discover tools that lead you in the direction you would like to go.
It usually means that if we’re given some facts about a triangle, we can get some or all the rest. For example, the above graph represents numerous Gaussian probability distributions, also called bell curves. Since they always have the same angles, then the ratio is the same for any given angle.
In truth, it’s among the most useful numbers in mathematics. These worksheets are simple enough that the calculations do not want a calculator, but if you’re searching for more of a challenge, make sure to find the worksheets within the next sectioN! Trigonometry essay writer of a suitable triangle.
Whispered What Is a Median in Math Secrets
A square is a normal quadrilateral. To put it differently the maximum repetition of an exact number in a data set is thought to be mode for a data collection. If there is just one of each number, the set doesn’t have any mode.
Mean, median and mode are all different, and it’s essential for the mathematician to understand which of these values supplies the best fit for a specific set of information. It is not necessarily unique, since the same maximum frequency may be attained at different values. Or there might be more than 1 mode.
I believe that you can argue that it may also be 80, but nevertheless, it can’t possibly be 75. In addition, it’s straightforward to learn. It is the one which shows the absolute most.
Most Noticeable What Is a Median in Math
Students should remove links from longer chains to improve the shorter chains. You first must find out the whole score of everything. A few of the worksheets are dynamically generated which means that you’ll be supplied a different set each opportunity to practice.
You can also locate the spread in the data set using the interquartile selection, which is a value that’s the difference between the top quartile value and the lower quartile value. Medians have a tendency to be utilized with unevenly distributed data, like incomes. These videos will explain to you how to get the median.
Now, to receive the mean, all you need to do is divide that total by however many numbers you’ve got. While the arithmetic mean (average) is a very simple calculation that most people today understand, it might be skewed upwards by just a couple high values. It wouldn’t be a particular numberbut the average would help you mean to conserve the sum you’re probably going to require.
It can be known as the average. It is commonly taught as being the standard average. It is often called the average.
The Key to Successful What Is a Median in Math
Each sheet also has answers if required. Even a rough sketch will be able to help you visualize the math issue and keep all of your information in order. At this time you can finish a topic as soon as your students have mastered it, not when you use up all your problems.


While a graduate student at UC Berkeley, I took a statistics course in linear and logistic regression and it introduced me to the issues with the achievement gap in secondary schools. For the analysis, I used linear and logistic regression packages in Stata and a survey of eighth-graders from the National Education Longitudinal Study.
I was interested in understanding the potential for students to enter and succeed in a STEM field. To achieve this goal, I ran two regressions.The first was a linear regression with math test scores as the response variable and socio-economic status, future plans, locus of control, self-concept, race and gender as the explanatory variables.The second was a logistic regression with the same explanatory variables as the linear regressions but with being “held back” as the response variable. Locus of control can be interpreted as a measure of control over “life chances” and self-concept a measure of self-esteem.
I ran two models for both the linear regression and logistic regressions, one with the response variable and the explanatory variables and the other with interactions. I added higher order polynomial terms for continuous variables that were not initially significant to test whether there was a non-linear relationship between the response and explanatory variable. I used backward elimination to eliminate explanatory variables with p>0.05 and I checked for changes in R squared to ensure that it did not change significantly after a variable was dropped.
Below discusses the results of the regression analysis. The results show that math test scores and success in school are highly associated with the future plans of attending college, socio-economic status and race. The paper with the complete analysis including hypotheses and graphs can be found here: Achievement Gap Paper.
1. Linear Regression
The most significant effect on the mean standard math scores was future plans. Comparing children who had plans to finish high school to those who did not, there was no significant difference in mean standard scores. The differences rose when children stated they planned on attending a vocational school or college, mean standard scores on average were higher by 2.27 and 2.37 respectively compared to those who would not finish high school. For students who planed on finishing college or who had plans on post college degrees, the mean standard math scores on average increased by 6.3 and 8.05 points respectively.
The regression also revealed strong associations between math test scores and socio-economic status and race. A one unit increase in socio-economic status (range -2.894 to 1.854, sd=0.8) resulted in an approximate 3.75 point increase in the estimated mean standardized math score, and a one unit increase in locus of control (range -3 to 1.52, sd=0.7) resulted in a 1.84 increase in the estimated mean standardized math score. Race was also significantly associated with math scores, on average when controlling for the other explanatory variables, African-Americans, Native Americans and Hispanics had a 5.5, 3.7 and 2.1 estimated mean standard score lower than whites respectively. Asian Pacific Islanders, on the other hand, had a 2.4 higher estimated mean standard score compared to whites. Men scored 1.3 points higher compared to females. Self-concept was not significant and was dropped from the model.
I interacted race and future plans on socio-economic status, and locus of control. The only variables that turned out to be significant were the interactions between African-American and socio-economic status, and Native Americans attending college. The interaction between African-Americans and socio-economic status revealed that on average, controlling for other explanatory variables, if socio-economic status increases by one unit, African-American children had an estimated mean standard score rate of increase of 1.8 less compared to whites.
Checking Model Assumptions
I checked for multicollinearity, homoscedasticity, normally distributed errors and linearity between the response variable and the continuous explanatory variables. The models did not violate any of the linear regression assumptions.
2. Logistic Regression
The logistic regression showed that being held back a grade is positively correlated with being male and African-American but negatively correlated with students who plan on attending a higher level education school upon graduating from high school. Being held back was also negatively correlated with an increase in social economic status and locus of control. Controlling for other variables, a one unit increase in socio-economic status reduced the odds of being held back by 47% and a one unit increase in locus of control reduced the odds by 25%. Controlling for other variables, attending education after high school reduced the odds of being held back by 37%. Compared to those students not planning on a post-high school education. Controlling for all other variables, males had a 59% greater odds of being held back compared to females.
All variables were significant at the 5% level. Those variables that were not significant at the 5% level such as self-concept and some of the race variables were dropped from the model. I interacted socio-economic status on the race variables and none of the interactions were significant. After discovering that self concept was not significant, I added quadratic and cubic polynomial variables but they were not significant and were dropped from the model.


The stability of positive linear continuous-time and discrete-time systems is analyzed by the use of the decomposition of the state matrices into symmetrical and antisymmetrical parts. It is shown that: 1) The state Metzler matrix of positive continuous-time linear system is Hurwitz if and only if its symmetrical part is Hurwitz; 2) The state matrix of positive linear discrete-time system is Schur if and only if its symmetrical part is Hurwitz. These results are extended to inverse matrices of the state matrices of the positive linear systems.
The analysis of the positivity and stability of linear electrical circuits by the use of state-feedbacks is addressed. Generalized Frobenius matrices are proposed and their properties are investigated. It is shown that if the state matrix of an electrical circuit has generalized Frobenius form then the closed-loop system matrix is not positive and asymptotically stable. Different cases of modification of the positivity and stability of linear electrical circuits by state-feedbacks are discussed and necessary conditions for the existence of solutions to the problem are established.
The paper presents the response of a three-layered annular plate with damaged laminated facings to the loads acting in their planes. The presented problem concerns the analysis of the combination of global plate failure in the form of buckling with the local micro defects, like fibre or matrix cracks, located in the laminas. The plate structure consists of thin laminated, fibre-reinforced composite facings and a thicker foam core. The matrix and fibre cracks of facings laminas can be transversally symmetrically or asymmetrically located in plate structure. Critical static and dynamic stability analyses were carried out solving the problem numerically and analytically. The numerical results show the static and dynamic stability state of the composite plate with different combinations of damages. The final results are compared with those for undamaged structure of the plate and treated as quasi-isotropic ones. The analysed problem makes it possible to evaluate the use of the non-ideal composite plate structure in practical applications.
The paper addresses the problem of constrained pole placement in discrete-time linear systems. The design conditions are outlined in terms of linear matrix inequalities for the Dstable ellipse region in the complex Z plain. In addition, it is demonstrated that the D-stable circle region formulation is the special case of by this way formulated and solved pole placement problem. The proposed principle is enhanced for discrete-lime linear systems with polytopic uncertainties.
The paper presents an overview of linearization methods of the non-linear state equation. The linearization is developed from the point of view of the application in the theoretical electrotechnics. Some aspects of these considerations can be used in the control theory. In particular the main emphasis is laid on three methods of linearization, i.e.: Taylor’s series expansion, optimal linearization method and global linearization method. The theoretical investigations are illustrated using the non-linear circuit composed of a solar generator and a DC motor. Finally, the global linearization method is presented using several examples, i.e. the asynchronous slip-ring motor and non-linear diode. Furthermore the principal theorem concerning the BIBS stability (bounded-input bounded state) is introduced.


How much does it cost to keep an inmate in jail?
There are several ways to calculate the jail bed cost. One way is to take the entire budget of the jail and divide it by the number of jail beds, then divide by the number of days in the year, which amounts to $234.71 per inmate per day. This method includes all County overhead costs, as well as support services and the booking operations.
To determine the cost per jail bed of just the inmate housing area operations, subtract the County overhead costs, support services and the booking operations from the entire jail budget . Then divide by the number of jail beds and the number of days in the year. This then amounts to $129.44 per inmate per day.
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