George Stoica

11 Questions 140 Answers 0 Followers

Questions related from George Stoica

Any G_delta set (on the real axis, say) contains a closed set?

Maybe in general (on metric spaces, etc).

29 October 2023 6,512 10 View

For any given function f : [a, b] → R, is there a sequence of polynomial functions converging to f at each point where f is continuous?

Note. The sequence of polynomials should be the same for all the continuity points; yet the convergence does not have to be uniform of the continuity set. Comment. Looks and sounds like "déjà...

10 October 2023 5,308 11 View

Is there such a function? Is it unique?

Let a ∈ ℝ and b > 0 be fixed. Find all functions f : [0, ∞) → ℝ satisfying the differential equation f'(x) = -a + b/f(x) for x > 0 and f(0) = 0.

22 January 2021 9,225 13 View

Is this true for all values c > 0?

Let 0 < xn ↗ ∞ such that xn+1 - xn → 0 as n → ∞ . Then, for every 0 < c < 1, there exists a subsequence k(n) such that xk(n) - xn → c as n → ∞ . Is the problem true if c ≥ 1?

26 December 2020 3,903 45 View

Are those two limits complicated to prove? Are their results correct?

I have some complicated hints and clues, but I think their solutions should be much simpler.

13 September 2020 6,780 4 View

Are the power functions uniquely determined in the following context?

Let T denote the circle group, that is, the multiplicative group of all complex numbers with absolute value 1. Let f : T → T be a (sequentially) continuous map, and such that f(z2 ) = f2 (z) for...

25 July 2020 2,200 3 View

Are there Lebesgue measurable functions f : [0,∞) → R such that, for any a > 0, the function x → f(x) · f(a − x) is constant on the interval [0, a] ?

If it makes easier, assume that f is continuous on [0,∞).

25 July 2020 2,984 3 View

Is the following version of the Weierstrass approximation theorem true?

For any given function f : [a, b] → R, there exists a sequence of polynomial functions converging to f at each point where f is continuous. (Note that we did not ask the convergence to be uniform).

25 July 2020 9,109 30 View

Let x(n) be a non-decreasing sequence of real numbers, uniformly distributed modulo 1. Is it true that the sequence x(n)/log n is unbounded?

If the answer is yes, can one replace log n by another sequence that approaches ∞ faster than log n ?

22 July 2020 7,680 4 View

Can we find all continuous and periodic real-valued functions f satisfying the equation f(x) + af(x + b)=0 for x∈ℝ?

To avoid trivial solutions, assume that a and b are non-zero real numbers. My feeling is that, for certain values of a and b, there are no such functions.

14 July 2020 4,880 24 View

A colleague asked me what can be said about the convergence/divergence of the following sequences: n·|sin(n)|; n·|cos(n)|; n·sin(n); n·cos(n) ?

I was able to prove (only) that neither sequence has a finite limit; probably anyone saw that coming, but it does not answer the question in full. Am I missing something, or is this hard to get?...

04 July 2020 3,889 99 View