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Jun
16
comment Linear algebra: Proving uniqueness and existence of a function
I do not have the time right now to write an answer. Look at the Gramm-Schmidt process. In that process the initial operations are only operations of type 2) , so there is no change in the value of the function. Changes come when you normalize, whcih is an operation of type 3). If you write things down and use 4, you get an explicit formula, which gives unicity.
Jun
16
comment Linear algebra: Proving uniqueness and existence of a function
You're probably on the right track with that function. The hints are valid though. You can prove existence and uniqueness by using the properties given without using the determinant. In fact, this is now one can prove that the function determinant exists without really writing the formula.
Jun
5
comment Proof of $\sum_{x = 1}^\infty \frac{1}{x}$'s divergence by absurdity?
Yes, the proof is valid (as far as I can tell). The only troubling part is the rearrangement of the terms of a series, grouping and splitting, which cannot be done in general. But assuming convergence, you have, in fact, absolute convergence, and the operations you make are valid.
Jun
1
comment How many 3 digit even numbers can be formed by using digits 1,2,3,4,5,6,7 ,if no digits are repeated
That depends on how you look at things. :)
May
17
comment Is Lipschitz's condition necessary for existence of unique solution of an I.V.P.?
Continuity alone is not sufficient. Continuity plus two different solutions give an infinity of solutions. Think that $f$ Lipschitz (which is continuous) implies unique solution.
May
17
comment Show that $ \int_{-\infty}^{\infty} \frac{x^3}{(x^2+4)(x^2+1)}\, dx$ does not converge
If $a=b$ you integrate an odd function on $[-a,a]$ so you get $0$.
May
16
comment Is Lipschitz's condition necessary for existence of unique solution of an I.V.P.?
I do not get the question regarding the sufficiency. Can you please detail what you mean?
May
16
comment Is Lipschitz's condition necessary for existence of unique solution of an I.V.P.?
@S.Panja-1729: Wikipedia is only as right as the people contributing to it. So the question "is wikipedia wrong?" can well be answered "Yes" in plenty of situations. You can see in the comments to your questions that the equation you mention has infinitely many solutions.
May
16
comment Is Lipschitz's condition necessary for existence of unique solution of an I.V.P.?
Two solutions means exactly that. One may be trivial, there's no problem.
May
15
comment Accelerating approximations for arccos
Could you develop what you mean by the fact that you accelerated drastically the accuracy of $\arccos$? What was the rate of convergence before? What is your rate of convergence?
May
15
comment Are my results realistic or is there an error somewhere?
You know, you could include all the details of your problem in your question. If someone is willing to help, making him/her go through another question (without details), which links to a book, is not really appealing. Make your question self-contained, so that someone looking at it can understand what you ask without going through other sites/books.
May
14
comment Linear independence of $\sin(x)$, $\sin(2x)$, $\sin(3x)$ in Map($\mathbb{R},\mathbb{R}$)
Yes, but in your case, the family is not independent. Assume that there are $a,b,c$, such that the relation you wrote is true for all $x \in \Bbb{R}$. Now you choose three values for $x$, in order to find a system of equations satisfied by $a,b,c$. You can choose whatever values you like, since the relation is valid for all $x \in \Bbb{R}$. If the family is not dependent, you'll find, after a few tries, that $a=b=c=0$, and this means that you have independence.
May
11
comment Limit of sequence - hard one
I have put a factor $1/n$ before the sum.
Apr
27
comment Cavalieri’s Principle for calculating volume.
Yes, I did replace $z$ with height, since they mean the same thing (if you orient your $z$ axis up)
Apr
27
comment Does a bounded countably infinite union of sets with volume have volume?
What do you mean by "$A$ has volume"? Do you mean it is measurable? If so, then countable union of measurable sets is measurable.
Apr
19
comment inverse of sum of diagonal matrix and eigendecomposition
I don't think that you can compute the inverse of a sum, in terms of the inverses of the two matrices.
Apr
12
comment Set of points at which a function coincides with its convexification is compact?
Yes, I guess lower semicontinuous could work.
Apr
12
comment $x\over(1-x)$ $y\over(1-y)$ $z\over(1-z)$ >= 8 when $x ,y ,z $ are positive proper fractions and $x+y+z = 2$
What does proper fractions mean? Are they rational numbers?
Apr
11
comment Proving properties about matrix $A$ s.t. $A^2 = -I$
The equation you get for $\lambda$ is $\lambda^2 = -1$. This has roots $\pm i$. The sum of the eigenvalues is equal to the trace (the sum of the elements on the diagonal), and since $A$ has real entries, the trace is real.
Apr
11
comment For $n \ge 3$, every subgroup of $A_n$ with index $n$, is isomorphic to $A_{n-1}$
I guess you could use the fact that for $n \geq 5$, normal subgroups of $S_n$ are $\{e\},A_n,S_n$. Prove that a subgroup of $A_n$ with index $n$ is a normal subgroup of a group isomorphic to $S_{n-1}$. Then its cardinality implies that it can only be $A_{n-1}$.