shah_nosrat

Reducing this leads to:

 

[latex]2xy^2 - y^2 + y = 2x^2 - x^2 + x[/latex]

 

 

Of course, I need to show that [latex]x = y[/latex] but I'm not sure how to reduce this equality any further. Any ideas?

 

Thanks

There's a mistake in your equality; it's supposed to be [latex]2xy^2 - y^2 + y = 2yx^2 - x^2 + x [/latex], factoring both terms on each side of the equality we have: [latex] y(2xy -y + 1) = x(2xy -x + 1) [/latex], which we can argue for x to equal to y we should have [latex]2xy -y + 1 = 2xy - x + 1[/latex], which reduces to [latex]x=y[/latex] as required, hence, your function is injective.

From what I studied in discrete mathematics, relations have matrix representations, and besides the operations we are used in linear algebra, there are also boolean operations that can be performed on matrices.

We had our first test and I'm trying to understand what it is I'm missing. I list each of the questions below, followed by what I answered.

 

1. Let v1=[ 1 ] and v2 = [ 1 ] --- Find a nonzero vector w that exists in R^3 such that {v1, v2, w} is linearly independent.

[ 1 ] [ 2 ]

[ 1 ] [ 3 ]

 

ans: w = [ 1 ] this was assuming that as long as the vector was a multiple of the of the vectors then the set would be linearly independent.

[ 4 ]

[ 6 ]

I don't understand your vector representations. But from the definition of linear independence, it says the following, that the linear combination of the vectors; k_{1}v_{1} + k_{2}v_{2} + k_{3}w = 0, if k_{1}=k_{2}=k_{3}=0 solving this equation will give your solution.

So why call it linear algebra? It is supposed to be in contrast to "abstract algebra" ?

Just to answer your last statement. Abstract Algebra deals with the study of mathematical structures called groups. To give an example between LA (Linear Algebra) and AL (Abstract Algebra) on their similarity (not in property, but on the approach, since it's algebra).

In LA we have basis sets that spans a particular vector space, and how an entire vector space can be constructed by the basis set. Similarly in AL we have cyclic sets that generates an entire group! So their approach is similar, but as John puts it:

Linear algebra develops from techniques used to solve systems of linear equations. Building on these methods, linear algebra gets into the study of vector spaces, which are sets of vectors combined with two operations such that certain requirements (which we call axioms) are met.

that's why is called Linear Algebra.

Besides, LA has immense applications; handwriting analysis, solving linear first-order differential equations, search engines use the mathematics from LA, Differential Geometry: representation of the coefficients of the first-fundamental form is in matrix form.

Hope this clarifies things.

I'm reading Comprehensive Mathematics for Computer Scientists 1, has anyone read it?

I haven't read the above book. But I do know that any computer scientist needs to have knowledge of Discrete Mathematics: As this will teach you naive set theory, logic, counting principles, Relations, Digraphs, Graph theory, Languages and finite - state machines and much more. Then you could complement it with the above mentioned book.

I have a doubt on how it's reading should proceed. There are lots of axioms, remarks, proofs, sorites, etc.

 

 

Should i memorize each one of them? What would be a effective way of reading it?

 

Thanks in advance.

When dealing with Definitions, axioms, theorems, and proofs. It's always a good idea to understand what a particular definition, axiom or theorem is saying and then going on to reading the associated proof to get a complete picture of whats going on.

Memorizing is never a good idea.

In your opinion, who are some of the top 5 mathematicians, currently

1. Edward Witten
2. Michio Kaku
3. Andrew Wiles
4. Stephen Hawking
5. Roger Penrose

Yes some of the above are Theoretical Physicists, but as ajb put it, they did spur modern mathematics.

I was thinking that what we consider to be alive has to have intellectual movements, and that we humans are at the top of this pyramid, but if by this definition of being alive wouldn't an intellectual robot be consider alive as well and if we are able to make a robot smart enough to replicate then wouldn't they be considered a species? if this is true then wouldn't sending robots out into space to repopulate another world be considered as the only way for intellectual species of this world to survive after the sun destroys everything?

I don't know if considering robots as its own species would be a good idea, not to mention it being an intellectual being/entity. We would then have to consider their robotic rights as well, and would give rise and debate to ethical considerations of how to deal with these new robotic beings or their species as a whole.

But I do know that Japanese scientists creating realistic humanoid robots to assist us in our daily chores or life for that matter, they consider it as being the next evolution of the so-called "Personal Computer".

Now to consider something as alive, they would have to satisfy certain prerequisites (Which I'm not really sure of) but I'm sure it exists.

I was wondering if any of you guys knew of any books that focus on the best ways to teach children about critical thinking, the scientific method, etc. Not books that focus on teaching specific things, or forcing your child to learn or memorize, but just a general kind for the best approaches to basically equipping a young child (4-5 years old) with the kind of mindset that will allow him to think, develop ideas and hypotheses, investigate things, etc.

 

I already do things like this with my nephew, so he can have a good foundation in critical thinking, but I didn't know if maybe there were other ways I can try, or games or experiments I could do to show him these things in action.

 

Thanks.

Hi, try some books by Stephen Hawking: George's Secret Key to The Universe, and other science books for Kids. I don't know if they improve critical thinking, but it sure will encourage imagination and creative thinking in kids, which is also important in any field of science.

Hi, I came across this theorem and decided to prove it, as follows:

Theorem: A set [math]A \subseteq R[/math] is bounded if and only if it is bounded from above and below.

I would like the prove the converse of the above statement; If a set [math]A \subseteq R[/math]is bounded from above and below, then it is bounded.

Let [math]M = |M_{1}| + |M_{2}|[/math]and using this preliminary result I proved earlier [math]-|a| \leq a \leq |a|[/math].

Now, [math]\forall a \in A[/math] we have [math] a \leq M_{1}[/math] ---> definition of bounded from above.

and [math] M_{2} \leq a[/math] ---> definition of bounded from below.

Using the result: [math]-|a| \leq a \leq |a|[/math]. Since [math]M = |M_{1}| + |M_{2}|[/math] is the sum of absolute value of [math]|M_{1}|[/math] and [math]|M_{2}|[/math], it is a big number. (trying to convince myself).

Also, [math]-M = -(|M_{1}| + |M_{2}|)[/math], This is on the opposite of the spectrum.

Now, [math]-M \leq -|M_{2}| \leq M_{2} \leq -|a| \leq a \leq |a| \leq M_{1} \leq |M_{1}| \leq M [/math].

[math]M = |M_{1}| + |M_{2}| \geq |M_{1} + M_{2}| \geq 0 [/math] ---> Which shows that M is positive by using the triangle inequality.

Hence, [math]-M \leq a \leq M[/math].

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

I'm excited for this proof, hopefully it's correct.

Your help is once again appreciated.

Yes, it really is just this simple. If as least element exist it is unique simply because any two least elements are equal.

Thank you Dr.Rocket. You are giving me confidence in art of proofing.

Yes, it really is just this simple. If as least element exist it is unique simply because any two least elements are equal.

Thank you Dr.Rocket. You are giving me confidence in art of proofing.

This theorem has nothing to do with well-ordering. It only requires linear ordering.

 

The point is that no set can have two or more "least elements".

 

You only need well-ordering in order to conclude that a non-empty set has at least one least element, but existence of a least element is not required for the theorem as stated.

Okay, taking your advice. Forget about my previous attempt at the proof. Using Linear (total) ordering.

If we take [math]\mathbf{U} \subseteq W[/math] ---> I need to invoke existence for it to make sense (To me anyway)

suppose [math]a, b \in \mathbf{U}[/math] with the property of being the least elements for all elements in [math]\mathbf{U}[/math].

Now, because of the antisymmetry property of the linear ordering then, as follows:

if aRb and bRa then a = b ---> Does this conclude the least element in the subset if it exist is unique?

Your help is once again appreciated

Seattle's Calling - by Burn The Charts

Hi,

This is the question that needs a proofs, as follows: Show that the smallest element of a nonempty subset of [math]\mathbf{W}[/math] is unique.

My attempt at the proof, as follows:

Let [math]\mathbf{U} \subseteq \mathbf{W}[/math], by the well ordering principle (WOP) we have that [math]a \in \mathbf{U}[/math] such that [math]a \leq x [/math] [math]\forall x \in \mathbf{U}[/math].

Now suppose [math]b \in \mathbf{U}[/math] such that [math]b \leq x [/math] [math]\forall x \in \mathbf{U}[/math].

Since [math]0 \leq x - a[/math] and [math]0 \leq x - b[/math] by definition.

Now, [math]0 \leq x + x - (a + b)[/math]

[math]a+ b \leq x + x = 1\cdot x + 1\cdot x = (1+ 1)\cdot x = 2x[/math]

[math]a+ b \leq 2x[/math]

[math]\frac{a + b}{2} \leq x[/math] . The only way that this inequality will hold [math] \forall x \in \mathbf{U}[/math] is when [math]a = b[/math]

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Is the above reasoning and proof correct?

Is there any point in even attempting to grasp higher mathematics in my late 20s if I do have a below average IQ (since I don't think there's any evidence that IQ can be significantly altered past a certain age or if one doesn't have a genetic disposition toward higher intelligence)?

 

Thanks for the help

First of all keep the doubts and negative thinking aside. I believe that anyone is capable of great things as long as they're interested. If you are interested in Higher Mathematics then that's the first step to learning Higher Mathematics.

Let me tell you this, that Higher Mathematics trains you to think analytically and critically about any problem presented to you.

And remember to always have fun

Of the few books I keep around are some chemistry books and some math books (algebra/stats), because I might need a refresher on those topics in order to solve some problems, and I find the Internet to be inadequate.

 

But as of late, I've considered just tossing all of my books to save space around my place. Anyway, I know where to retrieve similar books if ever necessary.

 

What do some of you do with your college books after passing a class?

Sell? Toss? Keep? Reasons?

I keep my books, because I can always use them for reference later on.

I totally agree that much of the packaging these days seems excessive. Right now the manufacturers are more concerned with the things I've mentioned rather than how much is going into the landfills. If we want to change all that, I think the best way to accomplish the goal is the same way recycling got a foothold. If consumers start purchasing more products that offer biodegradable or less bulky packaging, the markets will start leaning that way. Then we have to make sure the corporate lobbyists don't arrange to have the legal definition of "biodegradable" changed to the point where it's meaningless.

I appreciate your feedback, and I agree it starts with the consumers. I think the best way to do this is by proper education, and make the general public aware of the damage waste is doing to our planet. We really need to be aware of this fact, because it is our environment that supports us, not the other way around.

Hello, my teacher gave us an assignment to ask someone with a science degree questions to put in a portfolio project that we are doing. I would greatly appreciate it if one..or a few...of you could answer the questions below. thanks in advance.

 

• 1. What motivated you to pursue a career in thefield of science?
• 2. What advice do you have for people planning topursue a career in the field of science?
• 3. Your favorite science related experience?
• 4. Your favorite element?
• 5. What do you think will be the next greatadvancement in science?
• 6. Anything else you would like to add?

Hi, I am currently completing my Undergraduate degree in BSc(Mathematics), but I'm glad to help.

1. My motivation was simple, I always was intrigued by the Mathematics.
2. To pursue what they are most interested in the field of science, and to always keep an open mind.
3. Using the techniques and methods of differential equations to develop a mathematical model, and it's qualitative analysis, for example Lokta-Volterra model.
4. Hmmmm, don't have one. But if I had to choose it would K (potassium).
5. I follow achievements in Mathematics, and thus, The Mathematician Andrew Wiles proved Fermat's Last Theorem; which was first conjectured by Pierre de Fermat in 1637.
6. To always believe in your capabilities, and never take no for an answer.

Hope this helps

Hey guys, Im lost a bit:D I wana write a mathematical model for line balancing... i need help, do you know where should i ask this question?!

You can ask the question under the Mathematics section; I presume your question deals with a problem in Applied Mathematics, post your question under that heading in the Mathematics section.

Do this by starting a topic!

I have been trying to find a good course for me to take at university for a few months and earlier i stumbled upon a 'physics and philosophy' course at oxford. Whilst i appreciate the course requires three A's at a2 and is likely to be very costly i was wondering if, discounting those factors, the course would be a good way to spend my time at uni. Physics and Philosophy are studied in parallel during the first three years. The physics corresponds to the more theoretical side of the standard three-year Oxford physics course while the philosophy focuses on modern philosophy and particularly on metaphysics and the theory of knowledge. Students who complete the first three years can if they wish leave with a BA degree. Students going on to the MPhysPhil in the fourth year may specialise in either physics or philosophy, or continue in their study of both disciplines and their interrelations.The bridging subject, philosophy of physics, is studied in each of the first three years, and is an option in the fourth year. Specialist lectures are given in this subject together with tutorials and classes. Other final year options include a physics project or philosophy thesis. Thanks, Drew.

I wouldn't know about the courses at Oxford University, but I do know that Physics and Philosophy are a good combination for a degree. Physics courses are pretty standard around the world with Universities offering the following at 3rd year:

1. Quantum Mechanics
2. Statistical and Thermal Physics
3. Nuclear and Atomic Physics
4. Solid State Physics
5. Computational Physics
6. and the required laboratory sessions

As to regards to Philosophy you will be introduced to subjects such as:

1. Critical Reasoning and Argumentation
2. The Philosophy of Science

....etc. The above mentioned courses in Philosophy is important for the following reasons, respectively:

To acquire critical thinking, problem-solving methods and skills in argumentation by identifying fallacies and obstacles to reasoning, by constructing, analysing and critically evaluating arguments.

To acquaint students with the nature of scientific reasoning, the status of scientific in terms of their relations(s) to reality, and connections between the theories and practice of science.

Also courses in Theoretical and Applied Ethics is useful.

The above quotes are course descriptions offered by the University of South Africa.

Regards.

I am 15 year old student and I am planning to take Biochemistry as a proffesion in future. My reason to make this topic is that I need advice from you what to take for GCSE, on which subjects should I concentrate more and what countries are the best for biochemistry. Since this educational year I am getting ready for GCSEs it is good idea to start thinking about it now. I really appreciate your help.

The following are a required, as follows:

1. Biology (Extended)
2. Chemistry (Extended)
3. Physics (Extended)
4. Mathematics (Extended)
5. Additional Mathematics

Also try completing A-Levels in the following subject, as follows:

1. Mathematics
2. Biology
3. Chemistry

A-Levels gives you strong grounding and makes the transition to first year University smooth.

Hope this helps and Best of Luck in you endeavors

I am not sure what books would be worth my time in these topics and what books would be a waste of money could you please help me?

A good book would be Fundamentals of Physics by Jearl Walker, and will accompany you up to first year Physics at University. The book comes in a bundle which covers: Mechanics, Electromagnetism, Heat and Modern Physics.

The book requires working knowledge of Differential and Integral Calculus, which can be acquired using the book Calculus Concepts and Contexts by James Stewart, which will also accompany up to first year University.

I would also like to learn about aerodynamics and meteorology, and would like to hear about good books on those topics

I wouldn't know much about aerodynamics books, but if you intend to learn the more advanced aerodynamics theory, you would need a strong background in Mathematics in topics such as; Multivariate Calculus, Linear Algebra, Differential Equations.

My suggestion would be to first learn the qualitative knowledge of aerodynamics, more like what pilots are introduced to while training for their PPL (Private Pilots License), for example what an airfoil is, its camber, leading and trailing edge, flaps spoilers. Also how the airfoil produces lift relative to moving air, the Venturi effect etc. It will benefit you as it will give a strong conceptual understanding of aerodynamics.

Hope this helps and have fun

Hallo my friends from Crete,

 

I want to continue my studies with a master and Ph.D in Mathematical Biology or Computational/ Mathematical Neurosciences.

I want to tell me, if you know, about the graduate programs in this area of Mathematics.

I know about the Honors/MSc and PhD Programs at the University of Cape Town, South Africa. There is a group within the Department of Mathematics and Applied Mathematics called MARAM (Marine Resource Assessment and Management Group)

Research focus

The focus of the Group's work is the assessment and management of renewable marine resources. 'Assessment' relates to the evaluation of the present size (in particular in relation to pre-exploitation levels) and the productivity of a resource, while 'management' pertains to the translation of this information into scientific recommendations on appropriate limitations for harvest levels. Most of the methods used at present to lead to such recommendations for South Africa's major commercial fisheries have been developed by the Group. In particular, the adoption of automated feedback-control management procedures has been successfully promoted for some of these fisheries (this is an area where South Africa is regarded as a world leader).

 

They also offer courses in Biological, Ecology and Environmental Modeling.

http://www.mth.uct.ac.za/maram/

What are the requirements of the entrance in these programs (GRE test, IELTS test ext.).

IELTS is usually an overall score of 7.0

it's just that I enjoy it SO LITTLE that it makes it hard to concentrate and reason out the logic behind the code. Is this supposed to be so incredibly boring/challenging for me?

Let me be frank, I'm taking a course in Programming this semester, and let me tell you it is boring! As you mentioned, that getting around solving problems isn't the issue, it's just that it's boring.

Or is computer science a long-term pay off where I have to eat a lot of dirt before I actually enjoy it? Thanks

Also if you feel as though you're not enjoying it, switch majors and do something you love, and that's what you'll excel at. If you like Mathematics, then pursue it.

Mathematics is my major and I'm loving every minute of it .

Hope this helps, and all the best.

Regards.

You can, of course define e in any fashion that results in the number that is universally recognized as e.

 

However the usual definition comes in terms of the exponential function which is defined by the power series

 

[math] exp(x) = \sum_{n=0}^\infty \frac {x^n}{n!}[/math] and [math] e = exp(1)[/math]

Thanks!

Tons of ways, but it won't work. Plain packaging or no packaging will sell x amount of product. Branded packaging (even when factoring in the extra costs) sells 4x amount of product (some a little lower, some much higher). Packaging makes lots more money than it costs.

I'm not discouraging branding your products.

Some packaging is necessary just to protect the product. Who would want their shoes shipped from wherever if they were loose and scuffing up against all the other shoes? Boxes help prevent that, and make it easier to stack and ship.

Of course, no one would want their new shoes or any product they buy to be destroyed by shipping, I was suggesting of finding new ways to package the products we buy by making the packaging more compact and less excessive.

I think recycling is the best solution. Make sure the stuff you buy is in recyclable packaging.

Yes, I agree recycling is one of the solutions (I was reading up on Japan's recycling policies, and I think it should be a model to adopt by the rest of the world), but there are also biodegradable packaging solutions as well.

I appreciate your feedback and opinion

There is some movement in this direction in the UK.

 

An example that jumps to mind is instant coffee...

 

Interesting video, nice to see some initiative being taken.

Thank you for your feedback!