complex analysis – A sequence $a_1=f'(0),a_2=f”(0),…$

I am working on this problem from my past Qual

“Give a sequence s.t. there is no analytic function $f:Dto mathbb{C}$ s.t. $a_1=f'(0),a_2=f”(0),…$” where $D$ is the unit disk.”

The only thing I can think of the Cauchy’s integral formula$$f^{(n)}(0)=frac{n!}{2pi i} int frac{f(w)}{w^n}dw$$
But that’s it. I don’t see a relation between these to construct a counterexample. How do I proceed?

pr.probability – Show that : “${g_m}$ is uniformly integrable sequence.”

Let $(E,mathcal{A},mu)$ be a finite measure space and ${f_n}subset L^1$, such that:
$$
sup_{n}{int_{E}{|f_n(t)|dmu(t)}}<infty
$$

Such that, there exists a subsequence ${g_{m}}$ of ${f_n}$, such that for all subsequence ${g_{m_i}}$ of ${g_m}$
$$
{g_{m_i}1_{|g_{m_i}|leq i})}text{ is uniformly integrable,} qquad (1)
$$

$$
sum_{igeq 1}{mubig({tin E~:~|g_{m_i}(t)|>i }big)}<+infty,qquad (2)
$$

Show that : “${g_m}$ is uniformly integrable sequence.”

My effort:

Let $epsilon>0$ and $Ain cal{A}$. Put
$$A_i = {x in E: |g_{i}(x)|> i}.$$

From (2), $mu(A_i) to 0$ as $i to infty$. Choose $N in mathbb{N}$ such that $i geq N$ implies
$$mu(A_i) < frac{epsilon}{2sup_n int_E |f_n| , d mu}.$$

From (1), choose $delta >0$ such that $i in mathbb{N}$ and $mu(A) < delta$ implies
$$
int_A |g_i| cdot 1_{Esetminus A_i}, dmu < frac{epsilon}{2}.
$$

Then $i geq N$ and $mu(A)<delta$ implies

begin{align*}
int_A |g_i| , dmu & = int_{Acap A_i} |g_i| , dmu + int_{Acap (Esetminus A_i)} |g_i| , dmu\
& = int_{E} |g_i|cdot 1_{Acap A_i} , dmu + int_{A} |g_i| cdot 1_{ Esetminus A_i}, dmu\
& leq mu(A_i) cdot sup_n int_E|f_n| , dmu +int_A|g_i| cdot 1_{Esetminus A_i}, dmu\
& < frac{epsilon}{2} + frac{epsilon}{2}= epsilon.
end{align*}

This proves that the subsequence ${g_m}_{m=N}^infty$ of ${f_n}$
is uniformly integrable.

Can I write $int_{E} |g_i|cdot 1_{Acap A_i} , dmu leq mu(A_i) cdot sup_n int_E|f_n| , dmu $?

Efficient way to import a sequence of data labeled with a parameter value

Suppose we have a number of data in .m format, which are labeled with the value of a parameter $c$, for example, c=0data.m, c=0.2data.m, c=0.4data.m, c=0.6data.m, c=0.8data.m, c=1data.m, …

Now, we need to import these data and assign them to solc0, solc2, solc4,…, solc10 for batch processing. I use the following Do loop with ToString:

Do[solc<>ToString[i] = << "C:\Users\c="<>ToString[i/10]<>data.m", {i, 0, 10, 2}]

which generates lots of errors. Can anyone point out what I overlooked? Thank you.

transactions – Sequence valid before time

I’m studying sequence, and I set a transaction valid after 512 seconds.
First of all I use regtest and I start from clean blockchain, after that I mine 114 blocks.
At this point miner creates a transaction and tries to send it

My decode transaction

{
  "txid": "59ff4adafb47a5b22c6434af38f6e138c9008356778ed8b308c48029d7d4032f",
  "hash": "1bef48f96d18f1c78021b7e4b0a7d5285f6f7d2cc8c7adc31671820e367c3d70",
  "version": 2,
  "size": 191,
  "vsize": 110,
  "weight": 437,
  "locktime": 0,
  "vin": (
    {
      "txid": "88fb1408675c774c36692f170c6122af49cab7a5bab336272f0e4d8c0ef8c89a",
      "vout": 0,
      "scriptSig": {
        "asm": "",
        "hex": ""
      },
      "txinwitness": (
        "3044022070b753c99e2b6d241fd8a4ebe26f53644ef3eea58fff743e51f7a69e1ee7a99602204cbe9a60fc25be24baa3e4613f27a3e51df4a285174e001b62de18e1da2e42fd01",
        "03494041191fd2b02579fd49877755e55d9d451a37c6d0bbe04aab8ef507a78b19"
      ),
      "sequence": 4194305
    }
  ),
  "vout": (
    {
      "value": 49.99100000,
      "n": 0,
      "scriptPubKey": {
        "asm": "0 18363770025baac1ebdb99a948eab2776d9568ae",
        "hex": "001418363770025baac1ebdb99a948eab2776d9568ae",
        "reqSigs": 1,
        "type": "witness_v0_keyhash",
        "addresses": (
          "bcrt1qrqmrwuqztw4vr67mnx55364jwake269w8hl5fe"
        )
      }
    }
  )
}

I Give an error when I try to use sendrawtransaction

error code: -26
error message:
non-BIP68-final (code 64)

And it’s correct because my UTXO (88fb1408675c774c36692f170c6122af49cab7a5bab336272f0e4d8c0ef8c89a) come from the block with height 2 and 113 confirmations, and its median time is 1590276467 (2020-05-24 01:27:47 CET/CEST) and My transaction is valid after 512 seconds (Date:2020-05-24 01:41:23 CET/CEST) and the best block has that value 2020-05-24 01:28:06 CET/CEST (it’s not useful)

Now, If I create another transaction with the TXID comes from block with height 1 and 114 confirmations it works, I can send it.
Below my transaction and details.

{
  "txid": "f98d3ef70ca2c9d797bf7fff2e96e07f5ba10a280186a2132c6a902eebcab31e",
  "hash": "5776ad750fe759411a1ed5aa10e759b33e52b2935555d90b997fa788c9e25405",
  "version": 2,
  "size": 191,
  "vsize": 110,
  "weight": 437,
  "locktime": 0,
  "vin": (
    {
      "txid": "e1ee4602a78ab4f5f58705a75405d2f223307989950e1cbe03a4518cf23b7914",
      "vout": 0,
      "scriptSig": {
        "asm": "",
        "hex": ""
      },
      "txinwitness": (
        "304402200641ef29e3f0c8ef0c55dbb4a86a752e3655dd0c928c4e2e6659d3beeaf3f3870220026821b9ba043894cb20c7d7a378eaae76f56400755bf412f8ab1524a9a238b301",
        "03494041191fd2b02579fd49877755e55d9d451a37c6d0bbe04aab8ef507a78b19"
      ),
      "sequence": 4194305
    }
  ),
  "vout": (
    {
      "value": 49.99100000,
      "n": 0,
      "scriptPubKey": {
        "asm": "0 18363770025baac1ebdb99a948eab2776d9568ae",
        "hex": "001418363770025baac1ebdb99a948eab2776d9568ae",
        "reqSigs": 1,
        "type": "witness_v0_keyhash",
        "addresses": (
          "bcrt1qrqmrwuqztw4vr67mnx55364jwake269w8hl5fe"
        )
      }
    }
  )
}

The median time of block 1 and 114 confirmations is 1590276486 (2020-05-24 01:28:06 CET/CEST) and my transaction should be valid after 512 seconds (2020-05-24 01:45:05 CET/CEST), and my best block median time is 2020-05-24 01:28:06 CET/CEST

Below, the details of block 2 with 114 confirmations

{
  "hash": "5f769f610f29057577611868a660b353bea06e51d94905f3dbf7fb93e60a3d30",
  "confirmations": 1,
  "strippedsize": 214,
  "size": 250,
  "weight": 892,
  "height": 114,
  "version": 536870912,
  "versionHex": "20000000",
  "merkleroot": "1699f1cebdde9f4da1211f948f2ec194f7820fcfbab80c93bb5ea486e6837be7",
  "tx": (
    "1699f1cebdde9f4da1211f948f2ec194f7820fcfbab80c93bb5ea486e6837be7"
  ),
  "time": 1590276487,
  "mediantime": 1590276486,
  "nonce": 0,
  "bits": "207fffff",
  "difficulty": 4.656542373906925e-10,
  "chainwork": "00000000000000000000000000000000000000000000000000000000000000e6",
  "nTx": 1,
  "previousblockhash": "36f373a49886a31eff05ff8de39722f589cff103bed47715dd697d14350539ce"
}

Now, I know the sequence (00000000010000000000000000000001) is checked on median time of UTXO’s block, But block 2 (with 114 confirmations) and block 3 (with 113 confirmations) are very similar and very close, and I don’t understand why with block height 2 I’m able to send the transaction.

replace by fee – Can inputs to a transaction have different values for sequence?

I noticed that the sequence field is used to signal opt-in RBF. I would think that signaling RBF applies to the full transaction rather than particular inputs.

If my transaction has two inputs, and I signal opt-in RBF on one of these inputs, am I still signaling that my transaction might be replaced? Or do I need to signal it for both inputs? Is it valid to signal RBF on one input but not another? Does the sequence field need to match for all inputs in the transaction?

How can I find the explicit formula and 8th term of 2,1/2,1/8,1/32 ? and how can I tell if it’s a geometric sequence?

how can I find the explicit formula and 8th term of 2,1/2,1/8,1/32 ?

functional analysis – Show that a sequence is eventually null

Let H be a separable Hilbert space and ${e_n}$ and ${f_n}$ be orthonormal basis in H. Define the linear operator:

begin{equation*}
Sx = sum_{n=1}^{infty} mu_n (x,e_n)f_n.
end{equation*}

I have to prove that if the dimension of Im S is finite, then ${mu_n}$ is eventually null.

This is my attempt.

Let’s suppose that $rank; T=k$.

$$Sx= sum_{n=1}^infty alpha_n(x|e_n)f_n= sum_{n=1}^k alpha_n(x|e_n)f_n+ sum_{n=k+1}^infty alpha_n(x|e_n)f_n= sum_{n=1}^k alpha_n(x|e_n)f_n$$

So we have:

begin{equation} label{0}
sum_{n=k+1}^infty alpha_n(x|e_n)f_n=0
end{equation}

As ${f_n}$ is a basis:
$$ alpha_n(x|e_n) =0 quad forall ngeq k+1 $$

I don’t know how to conclude rigorously that $alpha_n=0$.

I’d appreciate if someone could revise this.

Thank you.

Nesting a sequence of functions

Suppose I have two functions, f and g. I would like to arbitrarily produce a new function like for example f[g[g[f[g[]]]]]. If it was just a function, I could use Nest. Any way to do this procedure correctly?

design: CRUD use case and relative sequence diagram

I am building my project with Visual Paradigm and I have some use cases that implement the CRUD pattern.

As specified in the book Use cases: patterns and plans, Övergaard and Palmkvist suggest implementing a single use case as one of the best ways to handle this type of use case. There is a different flow for each action: one operation is considered as the main flow, the others as extended flows.

The question is: considering that I am using Visual Paradigm, what is the best way to write the relative sequence diagram of a similar use case?

Functional analysis: how to find a sequence that converges with the fixed random variable

Suppose $ mathscr {A} $ is a convex set $ L ^ infty (P) $, $ mathscr {B} $ is the closing of $ mathscr {A} $ under the topology $ σ (L ^ infty (P), L ^ 1 (P)) $. For each $ X in mathscr {B} $, we can find a sequence $ {X_n } _ {n geq 1} subseteq mathscr {A} $ S t. $ X_n xrightarrow {L_1} X $?

I thought about the problem a lot but still had no answer, so I ask for help here. Can you give me proof of existence or a counterexample? Thank you.