## 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.