Convergence of the bisection, the Secant method and Newton in the absence of root

If there is no root in a function, the method of bisection, secant or Newton converges, p. Ex. F (x) = 1 / x in the interval (-1,1). How did you show why converge or do not converge? If it converges, what to do?

langlands conjectures – Defining the shape of the cusp points in $ L ^ 2 $ and convergence on $ N _ { mathbb Q} backslash N _ { mathbb A} $

Let $ G $ to be a semi-simple group on $ mathbb Q $ with parabolic subgroup $ P = $ MN in good position compared to a compact subgroup $ U = prod limits_v K_v $ of $ G ( mathbb A) $. Let $ L $ to be the space of the square functions integrable on $ G ( mathbb Q) backslash G ( mathbb A) $ which are invariant right by $ U $. In Euler products, Langlands defines a form of cusp in $ L $ to be an element $ phi $ satisfactory

$$ int limits_ {N ( mathbb Q) backslash N ( mathbb A)} phi (ng) dn = 0 tag {1} $$

for almost everyone $ g in G ( mathbb A) $. However, I do not understand why the left converges at all. All we know is that

$$ int limits_ {G ( mathbb Q) backslash G ( mathbb A)} | phi (g) | ^ 2 dg < infty $$
Using Iwasawa's decomposition, we can write $ G ( mathbb A) = N ( mathbb A) M ( mathbb A) K $, so that, at least formally,

$$ int limits_ {G ( mathbb A)} phi (g) dg = int limits_ {M ( mathbb A)} int limits_ {N ( mathbb A)} int limits_K phi (nmk) delta_P (m) dk dm dn = operatorname {vol} (K) int limits_ {M ( mathbb A)} int limits_ {N ( mathbb A)} phi (nm) delta_P (m) dndm $$

We can probably find something here like:

$$ int limits_ {G ( mathbb Q) backslash G ( mathbb A)} | phi (g) | ^ 2 dg = operatorname {vol} (K) int limits_ {M ( mathbb Q) backslash M ( mathbb A)} int limits_ {N ( mathbb Q) backslash N ( mathbb A)} | phi (nm) | ^ 2 delta_P (m) dn dm $$

from which we should have

$$ int limits_ {N ( mathbb Q) backslash N ( mathbb A)} | phi (ng) | ^ 2 dn < infty $$
for almost everyone $ g in $ G. However, this does not say anything about the convergence of $ n mapsto phi (ng) $, only $ n mapsto | phi (ng) | ^ 2 $.

low convergence in $ L ^ 2 $ and integral convergence involving test functions

Let $ Omega $ to be a bounded set of $ mathbb {R} ^ n $ and $ (f_n) _n subset L ^ 2 ( Omega) $ such as $ f_n to f in L ^ 2 ( Omega) $ weakly $ L ^ 2 ( Omega) $. Then for a given test function $ phi in C ^ infty_c ( Omega) $, do we have the following convergent property:
$$
int_ Omega | u_n | phi , dx to int_ Omega | u | phi , dx, quad textrm {as $ n to infty $.}
$$

pr.probability – Convergence of GC Developments for Random Variables in the Total Variation Distance

Suppose that a random variable $ Y $ can be written as $ Y = g (Z) $, or $ g $ is a function and $ Z $ is a random variable. When $ Z $ is a continuous random variable with finite absolute moments, we consider a sequence of orthogonal polynomials with respect to the density function $ f_Z $, $ { phi_m (Z) } _ {m = 0} ^ infty $which is called the base of generalized polynomial chaos (CPG). then $ Y $ to the following extension:
Y = sum_ {m = 0} ^ inftyle y_m phi_m (Z), quadri y_m = frac {E (f (Z) phi_m (Z))} {E ( phi_m (Z) ^ 2)}. $$
These extensions can be generalized to random vectors $ Z $ and have many applications for solving stochastic systems. An introduction is presented in the book Numerical methods for stochastic calculations, a spectral method approach, by Dongbin Xiu (2010).

As stated in the book, the convergence of CPG extensions applies in the square sense when the support of $ Z $ is delimited. Moreover, this convergence is spectral (the most smooth $ g $ that is, convergence is faster; if $ g $ is analytic, the exponential convergence is valid). In the newspaper On the convergence of developments of generalized polynomial chaos, ESAIM: M2AN 46 (2012) 317-339, it is proved that the mean quadratic convergence holds when the problem of the moment $ Z $ is only solvable.

My question is to know if there are theoretical results in the literature that guarantee the convergence of GC developments in the total variation distance.

Two definitions of convergence – MathOverflow

I'm trying to show that the following two definitions of convergence are equivalent:

1] $ forall epsilon> 0 exists N in mathbb {N} forall n geq N: mid s_ {n} -L mid < epsilon $

2] $ forall m in mathbb {Z} _ {+} exists N in mathbb {R} forall n geq N: mid s_ {n} -L mid < frac {1} {m } $

Could someone provide assistance in the right direction? Thank you.

Digital Integration – How to solve the problems of slow convergence and highly oscillatory integrand?

I'm trying to numerically solve an integral in a specific region, and then visualize it as follows.

RegionPlot3D[
 NIntegrate[1/Sqrt[r] - 1/Sqrt[l + r Sin
  ImplicitRegion[r + l Sin
  0, 5}, {l, 0, 10}, {t, 0, pi/4}]

However, Mathematica complains that

NIntegrate::slwcon: Digital integration converging too slowly; suspect a
of the following: singularity, the value of integration is 0, highly
oscillatory integrand, or WorkingPrecision too small.

I've basically tried to remove the potential singularities by taking into account this specific integration region. Yet I have no idea of ​​the slow convergence of highly oscillatory integrands.

How can I fix this error?

Build series of i.i.d. random variables all having a different distribution with a convergence towards the standard normal.

I am asked to build a series of i.i.d. Random variables $ X_k, k in mathbb {N} $ all with different distributions, such as

$$ N ^ {- frac {1} {2}} sum_ {k = 1} ^ NX_k rightarrow_d X sim mathcal {N} (0,1) $$

as $ N rightarrow infty $. I thought I'd have them all-normally distributed with 0 waiting and different variance. That the sum of each $ 2 ^ k, k in mathbb {N} $ random variables are once again standard distributed standard, but I'm not sure how to choose the variance. Since they all the distributions must be different.

real analysis – Uniform convergence of a family of functions continuously parameterized

Let $ (X, d) $ to be a metric space, and let $ F: X times (0, B) to mathbb {R} $ to be a function. My question is the following:

What does it mean for $ F ( cdot, t) $ converge uniformly $ t to B $?

Equally, we can think of $ F $ as a family of functions $ f_t: X to mathbb {R} $ Defined by
begin {align}
f_t (x): = F (x, t), qquad forall x in X
end {align}

continuously set by $ t in (0, B) $. If I am correct, then the question is equivalent to asking about

What does it mean for the family $ {f_t } _ {t in (0, B)} $ converge uniformly $ t to B $?

I ask this question because uniform convergence is usually defined for a sequence of functions $ {f_n } _ {n in mathbb {N}} $and, as far as I could find, there is no reference that has an explicit answer to my questions above. I wonder if this is because such a generalization (from a sequence of functions to a continuous family of functions) is actually easy and left to readers.

Keeping this in mind, I therefore make the following assumption: Choose a sequence $ {t_n } _ {n in mathbb {N}} $ in $ (0, B) $ such as $ t_n to B $ as $ n to infty $. So let's consider the sequence $ f_n: = f_ {t_n} $. We say that $ {f_t } _ {t in (0, B)} $ converges uniformly $ t to B $ if $ {f_n } _ {n in mathbb {N}} $ converges uniformly $ n to mathbb {B} $.

I am not convinced of that, however. A major problem to be solved is to show that the definition is independent of the choice of the sequence. $ {t_n } $. However, even if it is not a big problem, I feel that something is missing.

All comments, suggestions and answers are welcome and are greatly appreciated. In particular, if there is a reference that explicitly gives the definition, please also let me know.

homological algebra – conditional convergence spectral sequences with outgoing and incoming differentials

I have to deal with unbounded filtrations and I want to use the conditional convergence of the spectral sequences and the results of

(1): J. Michael Boardman, Conditionally Convergent Spectral Sequences, March 1999 (http://hopf.math.purdue.edu/Boardman/ccspseq.pdf)

The article uses cohomological spectral sequences derived from the exact pair resulting from a cochain complex $ C $ and decreasing filtration $ F $ of $ C $. The system of inclusions is $$ A ^ s: = H (F_s C) leftarrow A ^ {s + 1} $$ and the pages are noted by $ E ^ s_r $ for $ s in mathbb {Z} $ and $ r in mathbb {N} $ ($ r $ is the page number and $ s $ the "degree of filtration"). The symbol $ A ^ infty $ denotes the limit and the symbol $ A ^ {- infty} $ the colimit. The symbol $ RA ^ infty $ denotes the derived module right of the limit. I work mainly on $ mathbb {R} $.

Here are the two theorems (or their parts) of (1) that interest me:

Theorem 6.1 (p.19): Let $ C $ to be a filtered cochaine complex. Assume that begin {equation} label {Eq: Exit} tag {C1} E ^ s = 0 quad text {for all}
s> 0. end {equation}
Yes $ A ^ infty = 0 $, then the spectral sequence
converges strongly towards $ A ^ {- infty} $.

Theorem 7.2 (p.21): Let $ f: C rightarrow bar {C} $ to be a morphism of filtered cochain complexes and suppose that $ E ^ s $, resp. $ bar {E} ^ s $
conditionally converge to $ A ^ {- infty} $, resp. $ bar {A} ^ {- infty} $.
Suppose further that begin {equation} tag {C2} E ^ s = bar {E} ^ s =
0 quad text {for all} s <0. end {equation}
Yes $ f $ induces the
isomorphisms $ E ^ infty simeq bar {E} ^ infty $ and $ RE ^ infty simeq
R bar {E} ^ infty $
, then he induces isomorphism $ H (C) simeq H ( bar {C}) $.

Let me introduce the standard bigrading (staggered degree) on $ E_r $ and visualize $ E_r ^ {s, d} $ as sitting at the coordinate $ (s, d) $ by plane. The differentials are then
$$ d_r: E_r ^ {s, d} rightarrow E_r ^ {s + r, d-r + 1}. $$
My questions are:

  1. How does Theorem 6.1 generalize if (C1) is replaced by the following status of existing differentials?
    $$ E_r text {sit down in a half-plane and correct any coordinates} (s, d), text {then all but a lot}} d_r text {from} (s, d) text {leave the half plane} $$

  2. How does Theorem 7.2 generalize if (C2) is replaced by the following differential input condition?
    $$ E_r text {sit down in a half-plane and correct any coordinates} (s, d), text {then all but a finite number} d_r text {ending in} (s, d) text {start outside the half-plane.} $$

The author of (1) answers the following questions:

  1. On page 19, chapter 6 in parentheses just before Theorem 6.1:

    …The
    the results are generalized appropriately because all the arguments can be treated in degrees; the
    The main difficulty is to find a notation that would help rather than hinder the exposure

  2. At p.20, chapter 7 brackets a few paragraphs before Theorem 7.2:

    … The results remain valid when they are modified appropriately, like all arguments
    can be done in degrees; the difficulty is finding the notation
    help rather than embarrassment.

How do these theorems become generalized precisely? Has it been done anywhere? Thank you!

P.S. I come from differential geometry and I do not know the proof methods for the spectral sequences at all. I just use it as a black box.

weak convergence – Proof: the weakly convergent sequence is bounded

In class, we had to prove:

Let $ X $ and $ Y $ to be normed vector spaces and $ T: X → Y $ an operator.
Yes $ (x_n) _n $ is a weakly convergent sequence $ X $ it follows that $ (x_n) _n $ is delimited.

Although I misunderstand the evidence we have made in class (use of isometric canonical incorporation and Hahn-Banach), I also wish to test another approach with a separation corollary of Hahn-Banach:

Let $ X $ to be a normed non-trivial vector space.
Then for everything $ x ∈ X $ there is a functional $ x ^ * ∈ X ^ $ with $ || x ^ * || = $ 1 and
with $ x ^ * (x) = || x || $.

Proof for which I need validation:

Yes $ (x_n) _n $ converges weakly towards $ x $ in $ X $so for everything $ n $ we have that $ x_n – x $ is also in $ X $.
Therefore, by the corollary of separation, there is a functional function. $ x ^ * $ in $ X ^ * $ such as $ x ^ * (x_n-x) = || x_n-x || $.

If we take the standard from both sides we get

$ || x_n-x || = || x ^ * (x_n-x) || $

And using the linearity of $ x ^ * $:

$ || x_n-x || = || x ^ * (x_n-x) || = || x ^ * (x_n) -x ^ * (x) || $

Since $ (x_n) _n $ converges weakly towards $ x $ we have this for everything $ epsilon> $ 0 there is a $ N $ as for all $ N leq n $ we have

$ || x ^ * (x_n) -x ^ * (x) || < epsilon $

and so too

$ || x_n-x || = || x ^ * (x_n) -x ^ * (x) || < epsilon $

So, $ x_n $ converges to $ x $ and so $ x_n $ must be delimited.