The recent death of Apple founder Steve Jobs from pancreatic cancer at the young age of fifty-six highlights the dismal progress in the War on Cancer, despite over $200 billion, over one million published research papers, and the efforts of hundreds of thousands of highly qualified, hard working, committed researchers since 1971. Steve Jobs inspiring commencement address to Stanford University is also a poignant reminder of the ephemeral nature of words like “cured” and “curable” in cancer research and treatment. Steve Jobs may well have believed his rare form of pancreatic cancer was “cured” or “curable” as he claimed his doctors told him. Steve Jobs death also highlights the limited benefits of today’s extremely powerful computers and electronics in fields outside of computers and electronics. Despite the frequently hyped promise of multi-Gigahertz and multi-core CPUs, these impressive chips have rarely translated into substantial progress in medicine, power, propulsion, and other essential areas. One need only consider the many tragic deaths from cancer, the current rising energy prices, and the seeming wars over dwindling supplies of inexpensive oil and natural gas that plague the world today. Steve Jobs and his team at Apple have created many impressive gadgets such as the iPhone and iPad, but they were unable to exploit their computer expertise to defeat cancer. Is there a better way? Can we harness the unused power of today’s computers to solve these pressing problems? The enormous power of today’s computers is useless without concepts, mathematics, and algorithms that use this power to solve real problems. There has been impressive progress in some areas including video compression such as the H.264 and related standards used by YouTube, Skype and many other tools, audio compression such as MP3, image compression such as the widely used JPEG standard, computer generated images for movies, television, and computer games, the Global Positioning System or GPS that tells people where they are, and even speech recognition which is slowly finding some practical use despite many difficulties. There is currently a fad to develop and implement recommendation engines such as Netflix’s Cinematch system to recommend purchases to customers using advanced statistical methods. At best recommendation engines can increase sales by only a tiny amount, a few percent, and can never solve critical, trillion dollar market size, problems such as cancer, the diseases of old age, and energy shortages. Improved video compression in the form of video conferencing tools such as Skype may well help solve the current energy crisis. Video conferencing, however, cannot substitute for most energy needs. Other advances are needed. As Steve Jobs death shows, many major problems have not been solved at all. This article discusses some ways that math and computers might be used to develop a cure for cancer. It is a follow-on article to the previous article Can Mathematics Cure Cancer? This article discusses ways that mathematics might be used to identify and selectively destroy cancer cells. It discusses a specific approach and algorithm, “The Bathtub Mechanism,” that may be able to selectively kill cells with an abnormal number of chromosomes, a common feature of many cancers, and presents a sketch of some ways this algorithm might be implemented using cellular and molecular building blocks that may be known to present day biology, avoiding the need to construct nanorobots, something still far in the future.

All About Cancer

The current prevailing theory of cancer is the oncogene or “cancer gene” theory. This is viewed as a proven fact by many molecular biologists. Cancer is now said to be hundreds, even thousands of different diseases. While a medical doctor or pathologist may identify something as “breast cancer” or “skin cancer” or a similar general category, at a molecular and genetic level, “breast cancer” is actually many different diseases. It is thought that cancer is caused by the accumulation of many mutations of many different oncogenes and tumor suppressor genes that control complex networks of proteins that direct the growth, functioning, and differentiation of cells. In biology, differentiation refers to the process by which cells “differentiate” during growth into various specialized types of cells such as neurons in the brain, blood cells, and skin cells with different specific properties and functions. One type of breast cancer may have genes A,B,C, and D mutated while another has genes W, X, Y, and Z mutated. Not only this, but the cancers are thought to be continually mutating and evolving in the body, developing immunity to chemotherapy drugs for example. Thus, there does not seem to be a common molecular target that an anti-cancer drug can target in the way that penicillin or other antibiotics can kill a wide range of different bacteria, for example. There may be system level features of cancer cells that identify them. Traditional chemotherapy drugs were designed to kill dividing cells on the theory that cancer cells divide rapidly. However, healthy cells divide also and traditional chemotherapy has very limited benefits for most cancers. Only surgical removal of a tumor before it spreads — becomes metastatic in cancer jargon — appears to be able to cure cancer using the common sense definition of “cure”. While targeting cell division largely does not work, targeting other system level characteristics of cancer may work. It may be possible, with great difficulty, to produce a small system of interacting drugs that perform a mathematical or logical calculation in the cell and selectively kill cancer cells or probable cancer cells while sparing normal cells. It is here that mathematics may be of use. To achieve success in the near future, the simpler the mathematics the better. Even engineering a single molecule such as genetically engineered insulin for diabetics is a daunting task at present. So a system of even a few molecules would be a substantial and difficult undertaking.

The Selective Destruction of Cells with Abnormal Numbers of Chromosomes

(NOTE: This section largely repeats the section with the same title in the previous article Can Mathematics Cure Cancer? If you are familiar with the concept, you may skip this section and jump to the following section which discusses how to implement the bathtub mechanism.) One common characteristic of many cancers is an abnormal number of chromosomes, known as aneuploidy. This is often an excess number of chromosomes. A normal healthy human cell has forty-six (46) chromosomes. Cancer cells often have more than forty-six chromosomes. This was discovered long before the modern genetic era. One historical theory, now out of favor, is that the abnormal number of chromosomes causes cancer. This theory is usually credited to the German biologist Theodor Boveri. The most prominent modern advocate of the role of aneuploidy and chromosomes in cancer is the extremely controversial researcher Peter Duesberg who has published some articles on his theories in cancer research journals and a popular article in Scientific American in 2007 (“Chromosomal Chaos and Cancer”, Scientific American, May, 2007). A number of other researchers such as Angelika Amon at MIT have been investigating the role of chromosomes and aneuploidy in cancer in recent years; references are given in the previous article Can Mathematics Cure Cancer?. The abnormal number of chromosomes or the other chromosomal anomalies often seen in a wide range of cancers may be a system-level characteristic of cancer that could be targeted despite the extreme variation in gene-level mutations (part-level characteristics of cancer). Even though there are over one-million research papers on cancer, it is difficult to get a clear picture of the role of aneuploidy in cancer. Most modern cancer research is conducted within the framework of the oncogene theory and an implicit assumption that the way to cure or treat cancer is to target either a protein generated by a cancer gene or the gene directly. Chromosomal anomalies, both abnormal numbers of chromosomes and the rearrangements of chromosomes that are common in many cancers, are usually discussed as an aside to the putative cancer genes. This translocation of chromosome X mutated the key cancer gene ABC, or the duplication of chromosome X resulted in two copies of the key cancer gene ABC. It could be that killing cancer cells with the wrong number of chromosomes would have no effect on the disease. It would simply result in a cancer with the correct number of chromosomes in the surviving cancer cells. It could slow the disease if the abnormal number of chromosomes is related to the malignancy of the cancer cells. In the best case, it might cure the disease, if the abnormal number of chromosomes is either the cause of cancer, essential to the malignant nature of the cancer cells, or simply always associated with malginancy for some other reason. It may be possible to kill cells with an abnormal number of chromosomes using a system of five molecules: a harmless precursor A, a source catalyst S, a cell killer B, a drain catalyst D, and a neutralized cell killer C that the cell can safely digest or excrete. The source catalyst S is inactive until it bonds to a numerical or quantitative feature on the chromosomes such as the telomeres at the ends of the chromosomes or the centromeres at the center. It becomes an active catalyst S* when it bonds to the chromosomes. Then the activated catalyst S* catalyzes the conversion of a harmless precursor A into a cell killer B. The activated catalyst S* has a maximum throughput. If the concentration of the precusor A is high enough in the cells, the catalyst S* will add the cell killer to the cell at a rate proportional to the number of chromosomes in the cell. The cell killer is relatively harmless in low concentrations. It needs to build up to a high level to kill the cell. So far, this will happen in all cells. However, if there is a drain catalyst D that bonds to a numerical feature in the cell that is the same in both normal cells and abnormal cells (cancer cells) and becomes an active drain catalyst D* that removes the cell killer B by converting it to the neutralized cell killer C, then the concentration of B can be engineered to rise to lethal levels only in cells with too many chromosomes. This system of drugs is like a bathtub with several running faucets, one for each chromosome, and a single drain. If there are too many faucets, chromosomes, the water level, the concentration of the cell killer B, will rise and overflow the bathtub. If there are the right number, forty-six, or too few, less than forty-six, faucets, the drain can remove the water being added and the water level never rises. The water level remains almost zero; the concentration of the cell killer B is way too low to harm the cell. One can kill cells with too few chromosomes (less than forty-six) by swapping the roles of the drain and the source. The drain catalyst bonds to the chromosomes. The source catalyst bonds to the constant numerical feature of the cells. Thus, if there are too few chromosomes, there are not enough activated drains to remove the cell killer B produced by the source catalyst. The bathtub has one big faucet and many small drains, one for each chromosome. In principle, one could eliminate all cells with either too many or too few chromosomes by first treating the patient with a system of drugs that kills cells with too many chromosomes and then a system of drugs that kills cells with too few chromosomes. Cancer cells are frequently reported to have too many chromosomes, but sometimes too few is also reported. A computational system of this type would now (2011) be easy to implement using mechanical components like the gears and springs used in traditional mechanical clocks, vacuum tubes and other traditional analog electronics components, or an integrated circuit. The problem is that as simple as such a computational system is, it is extremely challenging to implement using our current ability to engineer proteins and molecular biological systems in the cell.

How to Implement the Bathtub Mechanism

The bathtub mechanism requires two features in the cell: a numerical or quantitative feature that is proportional to the number of chromosomes and a feature that is constant in all cells, both normal and cancerous. It is sometimes reported that cancer cells have abnormal numbers of antigens on the membranes of the cells. Hence, the bathtub mechanism may not require a feature that varies with the number of chromosomes, but this article is about targeting abnormal numbers of chromosomes rather than antigens. Some obvious features that probably vary with the number of chromosomes are the telomeres at the end of the chromosomes and the centromeres at the center of the chromosomes. These are both involved in cell division. There should be concern that the source or drain catalyst binding to the telomere or centromere may interfere with cell division. The bathtub mechanism must kill all the cancer cells and spare most or all of the healthy cells. It may be possible to use the telomeres or centromeres, but it could be impossible. A more promising feature may be some of the non-coding sequences in the chromosome DNA, the so-called “junk DNA.” It is currently thought that the vast majority of DNA in the chromosome has no function. On theoretical grounds, the author finds this implausible as do many. However, the genes that appear to code for the proteins in the body seem to comprise only a few percent of the DNA in the chromosomes. The rest seems to do nothing. Sequences of non-coding DNA are used in DNA profiling, for example. Depending on the actual function of the junk DNA, if any, it may be possible to safely bind a source or drain catalyst to non-coding sequences that vary in quantity with the number of chromosomes. There are many molecular structures in the chromosomes and associated with the chromosomes. It seems probable, although not certain, that one can find a numerical or quantitative feature that varies with the number of chromosomes that could be used. A more serious problem with the bathtub mechanism is the constant feature that is the same in both healthy cells and cancer cells, especially since cancer cells are thought to be constantly mutating and changing. This may be a show-stopper. Since the cancer cells may be mutating, it may be impossible to find a constant feature in the cancer cells. The feature could disappear entirely or change in size or number. There is at least one possible way to add such a feature artificially to the cells, both healthy and malignant.

 A bacteriophage is a kind of virus that attaches to the exterior membrane of a cell and injects its genetic material into the cell. The bacteriophage’s genetic material then takes over the machinery of the cell and directs it to make more bacteriophages. The bacteriophage consists of a protein sheath that looks something like a science fiction bug (see pictures) with several arms that grab the surface of the cell and a polygonal chamber that carries the genetic material.

In principle, one could modify the genetic material of the bacteriophage to create cells (the commonly used E. Coli bacteria, for example) that make not the virus, but the protein sheath with a payload of other proteins. These pseudo-bacteriophages would inject their protein payloads into cells instead of the genetic material of the naturally occurring bacteriophage. They would not be infectious like a normal virus. If, and this is a big if, one could modify the protein sheath so it would only inject the protein payload into a cell without an inhibitor protein I that is part of the payload, one could inject a payload that contained an artificial constant feature F and the inhibitors I into the cell. Once the new feature that the drain or source catalysts would bind to was added to the cell, the pseudo-bacteriophages would stop injecting payloads into the cell because it now also contained the inhibitors. Thus, a constant number of features could be added to each cell, both healthy and cancerous.

Math and Computers

This is a simplified sketch of the bathtub mechanism, a basic concept. Many technical details and difficulties have been omitted to present the idea. While it might be possible to research and develop the bathtub mechanism entirely empirically at a laboratory bench through massive trial and error, it should be possible to substantially accelerate the development process by simulating the molecular mechanisms using today’s powerful computers. In practice, it would probably require careful tuning of the chemical reaction rates in the cell to produce the desired selective destruction of cells with abnormal numbers of chromosomes or other features associated with cancer. One should not expect the computer simulations to be perfect. They would probably be far from perfect at first. Rather, the use of mathematical models and computers should be part of an iterative process in which the models and simulations are continuously compared to laboratory bench experiments and improved. The basic concept may also need to be modified iteratively as new data is collected. This has been the usual process in most genuine breakthroughs.

Conclusion

It may be possible to cure or effectively treat cancer with a system of smart drugs that perform a simple mathematical or logical calculation to selectively destroy cancer cells or probable cancer cells while sparing normal healthy cells. These systems of smart drugs may be able to identify system level features of cancer cells independent of the confusing plethora of cancer genes and tumor suppressor genes. The bathtub mechanism discussed in this article is one possible example of such a system of smart drugs. Mathematics and computers can enable or greatly accelerate the development of such systems of smart drugs. The author suggests that cancer researchers, business leaders, and policy makers should direct a significant amount of time and resources to the investigation of such systems of smart drugs. This should be a diversified effort not focusing on any one particular approach such as the bathtub mechanism. While there should be some redundancy, there is probably no point in having dozens of competing research groups all trying the same basic approach as seems to be the case with the current attempts to apply differential equations to modeling the growth and spread of cancer, the major current example of applying mathematics to cancer research and treatment. A more diverse effort that is willing and able to question more assumptions is more likely to succeed based on the history of scientific research and technological development. The successful application of mathematics and computers to cancer and biology requires a professional working relationship based on mutual respect between experts in several fields: computers, mathematics, physics, and traditional biology. The recent appearance of extremely powerful computers presages a sea change in biology and many other fields where computers and mathematics play a much more important role than in the past. Computer experts, mathematicians, and physicists need to respect the hard earned experience of traditional biologists. There is no way the bathtub mechanism could be implemented successfully, if possible, without the expertise of molecular biologists, cell biologists, organic chemists, and others familiar with the detailed structure and function of the chromosomes and cells in the human body. The same can be said of other possible systems of smart drugs and algorithms that may be able to selectively kill cancer cells. So too, biologists need to respect the expertise of computer experts, mathematicians, and physicists. Successful mathematical modeling is usually a tedious, time consuming process taking months or years, typically longer than many quick few week biology experiments. Even a Nobel Prize in molecular biology or other impressive credentials does not make one an expert in mathematical modeling or other techniques that will be needed to apply mathematics and computers successfully to cancer and other problems. Management level issues such as technical feasibility, scope, difficulty, and complex technical issues will arise in a collaboration between biologists and mathematicians. These will need to be discussed freely in an adult manner to succeed. There are many pressing problems in the world today like cancer. As current headlines attest, we are doing a poor job solving many of these problems. For the most part, the enormous power of today’s computers has not been applied successfully to these problems. In some cases, there has been no attempt. In other cases, the favored approaches have failed despite decades of effort and genuinely new or simply unpopular ideas should be tried. The War on Cancer is probably an example of the latter case. Steve Jobs will be remembered for entertaining gadgets like the iPad, the iPhone, and the Macintosh. What an accomplishment it would be if these gadgets went on to successfully solve major problems like the cancer that felled their creator.

© 2011 John F. McGowan About the Author John F. McGowan, Ph.D. solves problems by developing complex algorithms that embody advanced mathematical and logical concepts, including video compression and speech recognition technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech

There are a number of meanings for the word “arc” in mathematics. In general, an arc is any smooth curve joining two points. The length of an arc is known as its arc length.

In a graph, a graph arc is an ordered pair of adjacent vertices. In particular, an arc is any portion (other than the entire curve) of the circumference of a circle. An arc corresponding to the central angle L AOC is denoted arc AC . Similarly, the size of the central angle subtended by this arc (i.e., the measure of the arc) is sometimes (e.g., Rhoad et al. 1984, p. 421) but not always (e.g., Jurgensen 1963) denoted m arc AC .

The center of an arc is the center of the circle of which the arc is a part.

An arc whose endpoints lie on a diameter of a circle is called a semicircle.

For a circle of radius , the arc length subtended by a central angle is proportional to , and if is measured in radians, then the constant of proportionality is 1, i.e.,

As Archimedes proved, for chords and which are perpendicular to each other,

(Wells 1991).

An arc of a topological space is a homeomorphism , where is a subspace of . Every arc is a path, but not conversely. Very often, the name arc is given to the image of .

The prefix “arc” is also used to denote the inverse functions of trigonometric functions and hyperbolic functions. Finally, any path through a graph which passes through no vertex twice is called an arc (Gardner 1984, p. 96).

A Woo circle is an Archimedean circle with center on the Schoch line and tangent to certain other circles. An applet for investigating Woo circles and Schoch lines has been prepared by Schoch.

How to construct a Woo circle?

Very easy! Chose any nonnegative real number m. Draw two circles (magenta) centered on the arbelos ground line (blue) tangent to each other in point C (blue), and with radius m times the radius of the corresponding small arbelos arc.  The circle (red) centered on the Schoch line and externally tangent to these circles is a Woo circle.

How to use the SketchPad diagram?

The applet above has two draggable points. The blue one is point C and can be moved along the arbelos ground line. The red one is used to adjust the radius multiplier m, the value of which is displayed bottom left. (The scale may help at this.) Even though the value of m may be zero (the Woo circle then is circle W11), its upward movability is limited to prevent the draggable points from overlapping.  The applet starts with the radius multiplier set to 2, and the Woo circle is Schoch circle W15. Setting it to 1 the Woo circle will be Schoch circle W16.

References:

Okumura, H. and Watanabe, M. “The Archimedean Circles of Schoch and Woo.” Forum Geom.

In the arbelos especially in 5th grade math , consider the semicircles K1 and K2 with centers A and C passing through B. The Apollonius circle K3 of K1,  K2 , and the large semicircle of the arbelos is an Archimedean circle A1 . This circle has radius

p=½r(1-r) 

(as it must), and center

A1=(r(1-r)√((1+r)(2-r)), ½r(1+3r-2r²)) 

The line perpendicular to AB  and passing through the center of A1 is called the Schoch line.

Now let Ka and Kc be two semicircles through C with radii proportional to AC and BC  respectively. The circle tangent to Ka and Kc with its center on the Schoch line is an Archimedean circle. These circles are called Woo circles.

Let be the radical axis of the great semicircle of the arbelos and K1 . From a point on consider the tangents to the circle on diameter BC . The circle with center on the Schoch line and tangent to these tangents is a Woo circle (Okumura and Watanabe 2004).

An applet for investigating Woo circles and Schoch lines has been prepared by Schoch.

References:

Dodge, C. W.; Schoch, T.; Woo, P. Y.; and Yiu, P. “Those Ubiquitous Archimedean Circles.”

Okumura, H. and Watanabe, M. “The Archimedean Circles of Schoch and Woo.”

Hello Students!

Today we’ll be speak about bankoff circle. Only here you can get help from free online math tutor Jimmy.

The circle through the cusp of the arbelos and the tangent points of the first Pappus circle, which is congruent to the two Archimedes’ circles. If  AB = r and AC = 1 , then the radius of the Bankoff circle is

R=½r(1-r),

REFERENCES:

• Bankoff, L. “Are the Twin Circles of Archimedes Really Twins?” Math. Mag. 47, 214-218, 1974.
• Gardner, M. “Mathematical Games: The Diverse Pleasures of Circles that Are Tangent to One Another.” Sci. Amer. 240, 18-28, Jan. 1979

Draw the perpendicular line from the intersection of the two small semicircles in the arbelos. The two circles and tangent to this line, the large semicircle, and each of the two semicircles are then congruent and known as Archimedes’ circles.

For an arbelos with outer semicircle of unit radius and parameter , Archimedes’ circles have radii

p=½r (1-r)

and centers :

C1=(½r(1+r),r√r-1)

C2= ( ½r(3-r), (1-r)√r)

Here you can find free math home work and free online math tutors for students. If you want to get news about new free math courses and lessons please subscribe on it!

An Archimedean circle is a circle defined in the arbelos in a natural way and congruent to Archimedes’ circles, i.e., having radius

                                             p= ½*r*(1-r)

for an arbelos with outer semicircle of unit radius and parameter .

See also: Arbelos, Archimedes’ Circles, Bankoff Circle, Schoch Line

REFERENCES:

• Bankoff, L. “Are the Twin Circles of Archimedes Really Twins?” Math. Mag. 47, 214-218, 1974.
• Dodge, C. W.; Schoch, T.; Woo, P. Y.; and Yiu, P. “Those Ubiquitous Archimedean Circles.” Math. Mag. 72, 202-213, 1999.
• Okumura, H. and Watanabe, M. “The Archimedean Circles of Schoch and Woo.” Forum Geom. 4, 27-34, 2004
• Okumura, H. and Watanabe, M. “A Generalization of Power’s Archimedean Circles.” Forum Geom. 6, 103-105, 2006.
• Power, F. “Some More Archimedean Circles in the Arbelos.” Forum Geom. 5, 133-134, 2005.
• Schoch, T. “A Dozen More Arbelos Twins.”
• Van Lamoen, F. “Archimedean Adventures.” Forum Geom. 6, 77-96, 2006.

CapJaxMathFax

Our online tutors provides 5th Grade Homework help for students. Homework is done by tutors who are subject experts and well experienced in teaching. The students can get all help regarding their topic queries and also obtain help with solving their homework problems. All important topics under the 5th grade math help are provided with explanations step by step, solved examples and practice problems.
Get 5th Grade Math Homework Help Online

Students can also get 5th grade math homework help online from Tutornext tutorial. The tutors provide the solutions and the methodology to solve the problems online. To get your homework done, you can upload your assignments to our website so that our tutors will work on it and provide you a step by step process of solving problems with clear explanations. You can avail homework help any time as per your requirement. Our homework help is available for 24 hours.

Online Tutoring on 5th Grade Homework

Get 5th grade homework help online to solve your math assignments online. You can make the learning process easier with the help of the expert tutors available online for your assistance. You can learn about the steps involved and the concepts in depth to improve your grades in the examinations too. Our tutors are trained in the syllabus of all states’ syllabus. The basic concepts are taught in a simple way with clear examples in an easier way to understand for 5th grade students.

5th Grade Homework Help for Free

Students can avail Free 5th grade math homework help. They can get help for working with numbers, understand measurements and solving probability problems, rational and irrational numbers, constructions in geometry, patterns and word problems in algebra and word problems solving. The tutors will provide all the required help to students to solve and learn problem solving in mathematics. Upload your assignments any time, our tutors will get you back with solved answer with explanations.

In fact, in anything. Free online math tutors – it is the same teacher who helps the student in the shortest time to learn a math or algebra and geometry on an individually chosen for its program, but learning takes place through the Internet by Skype (Skype). That is, the student will not need to walk or ride, no need to learn to program a template and adapt to the opportunities of other students. You yourself now can choose when, where and how much time to spend on learning a mathematics, and most importantly what to study. More about free online math tutors learning via the Internet

On-line coach – is a certified instructor with extensive experience in teaching, which deals with you personally. He chooses for you the most optimal program, adjusting the material is under your desires and capabilities. Distance learning classes are similar to traditional math courses, with the only difference being that they are individually and in real time on the Internet. If, for example, do you think that some possess math, algebra and geometry, but you have a little difficulty with solving tasks, correct use of tenses, the teacher in training will focus specifically on the grammatical aspect of language and help you in the shortest possible time to master all the subtleties and difficulties in the target math grammar.

Benefits of free online math tutors!

• Time savings (no need to waste time on crossings)
• Saving money (remote to learn a language is cheaper than in a language course in groups)
• Classes are held face-to-face with a personal tutor
• Free schedule of classes (You choose when and how to do)
• Individual program of study (teacher develops lessons and activities based on your wishes)
  as a result of learning a foreign language is much faster and more efficiently

Free online math tutoring for You:

• if your working day is planed by the minute
• if you have a “floating” schedule
• if you have only a short time for training
• if you can do only at certain times
• if you are in constant traveling, but you definitely need to maintain the level of language proficiency
• If you do not like to study in a group and depend on the most
• If you are not satisfied with the pace set by learning in groups, classes, courses
• If you’re a young mother and now a lot of time to stay at home, why not take your education?
• if you can not be away from home and go to class
• if you go on vacation and need to quickly learn the basics of the math

And all this is much cheaper, faster and more efficiently than any math tutors course in every group!