## Explicit expression of sequence and rationality

We have : $$phi_ {1,1} = 1$$, and $$phi_ {a, b} = frac { phi_ {a-1, b-1}} {b (b-1)}$$ with $$a in ( mspace {-2 mu} (2, infty ( mspace {-2 mu} ($$ and $$b in ( mspace {-2 mu} (2, a) mspace {-2 mu})$$ and we have $$phi_ {n, 1} = sum_ {k = 2} ^ n (-1) ^ k phi_ {n, k}$$ with $$n in ( mspace {-2 mu} (2, infty ( mspace {-2 mu} ($$.

The question is: find the explicit expression of $$( phi_ {a, b})$$ and prove that all the elements of $$( phi_ {a, b})$$ is a rational number.

## functional analysis – Sequence of operator matrices in Hilbert Schmidt's space.

Consider the Banach space $$mathcal K = S_2 (H)$$ operators of Hilbert Schmidt on a Hilbert space $$H$$. I'm looking for an example of two pairs of sequences $${T ^ {(i)} }, { tilde T ^ {(j)} }$$ and $${S ^ {(i)} }, { tilde S ^ {(j)} }$$ in the unit ball $$mathcal K$$ and an anti-linear operator $$phi: mathcal K to mathcal K$$ so that the two iterated boundaries exist but $$lim_i lim_j sum_ {r, s} T ^ {(i)} _ {rs} tilde T ^ {(j)} _ {rs} phi (S ^ {(i)} star S ^ {(j)}) _ {rs} neq lim_j lim_i sum_ {r, s} T ^ {(i)} _ {rs} tilde T ^ {(j)} _ {rs} phi (S ^ {(i)} star tilde S ^ {(j)}) _ {rs}$$

Or $$T_ {rs}$$ denote the $$r times s$$ entry in the matrix of $$T$$ and "$$star$$"denotes the Schur product of operators (input product of matrices).

(Or else, prove that these limits are always equal regardless of the choice of sequences in unit of ball and $$phi$$).

## functions – Sequence of repeating numbers

I want to generate the sequence with the following model.

1 2 3 1 1 2 2 3 3 1 1 1 1 2 2 2 2 3 3 3 3 ….

Where the lower number is 1, the upper number is 3. Each time the numbers start at 1 and each number is repeated `2 ^ n` times, with n starting with 0.

## How to find the missing sequence number in Oracle (the sequence number can be recycled in the same or the following date)

i face the same problem already archieved one.

Can you please provide the solution for the problem below?

I have below the table group master

groupid, from_no, to_no
1,00,99
2000999
3,0000,9999

history_table groupid, date_timestamp (date || hh24: mi: ss), sequenceno
1.20200220 20: 10: 01.00
1.20200220 20: 10: 02.01
1.20200220 20: 10: 03.02
1.20200220 20: 10: 05.04
.
.
1.20200220 20: 10: 05.99
(03) missing sequence number
1.20200220 20: 11: 02.01
1.20200220 20: 11: 05.04
. .
1.20200220 20: 11: 05.99

(00.02.02) missing sequence number

2.20200220 20: 10: 02.001
2.20200220 20: 10: 03.002
2.20200220 20: 10: 05.004
. .
2.20200220 20: 10: 05.099

(000.003) missing sequence number

2.20200220 20: 11: 02.001
2.20200220 20: 11: 05.004
. .
2.20200220 20: 11: 05.099
(000.02.02) missing sequence number

another important example
3.20200220 23: 59: 57,0001
3.20200220 23: 59: 58,0002
3.20200220 23: 59: 59,0004
3.20200221 00: 59: 59,0008 (the next day)
3.20200221 01: 59: 59.0010 3,
20200221 02: 59: 59.0011
. .
3.20200221 04: 10: 05.0099
(0000,0003,0005,0006,0007) missing sequence number
3.20200221 05: 11: 02.0001
3.20200221 05: 11: 06,0004
. .
3.20200221 14: 11: 05.0099

(0000,0002,0003) missing sequence number

The real problem is that I should identify the missing sequence number. the sequence group can repeat itself on the same day or fall on another day.

Oracle 10G

## real analysis – Given the sequence, how can I show that this series diverges?

I currently have some problems with this math exercise. I have to show that the series $$sum_ {n = 1} ^ infty b_n$$ with $$b_n = sum_ {k = n + 1} ^ {2n} frac {1} {k ^ 2}$$ diverges.

I think I have problems with this problem because I don't really know how to apply the quotient rule or the comparison test to the given sequence.

Could someone help me a little?

Thank you!

## magento2 – Magento Framework Config Dom ValidationException: element & # 39; argument & # 39: duplicate key sequence

I have an error when developer mode is active.
The exception is thrown when entering a product detail page.
I use magento 2.3.4

``````1 exception(s):
``````

Exception # 0 (Magento Framework Config Dom ValidationException): Element & # 39; argument & # 39: duplicate key sequence (& # 39; title & # 39;) in key identity constraint & # 39; & # 39 ;. blockArgumentName
Line: 1388

Exception # 0 (Magento Framework Config Dom ValidationException): Element & # 39; argument & # 39: duplicate key sequence (& # 39; title & # 39;) in key identity constraint & # 39; & # 39 ;. blockArgumentName
Line: 1388

```#1 MagentoFrameworkConfigDom->__construct() called at (vendor/magento/framework/ObjectManager/Factory/AbstractFactory.php:121)
#2 MagentoFrameworkObjectManagerFactoryAbstractFactory->createObject() called at (vendor/magento/framework/ObjectManager/Factory/Compiled.php:108)
#3 MagentoFrameworkObjectManagerFactoryCompiled->create() called at (vendor/magento/framework/ObjectManager/ObjectManager.php:56)
#4 MagentoFrameworkObjectManagerObjectManager->create() called at (vendor/magento/framework/Config/DomFactory.php:43)
#5 MagentoFrameworkConfigDomFactory->createDom() called at (vendor/magento/framework/View/Model/Layout/Update/Validator.php:141)
#6 MagentoFrameworkViewModelLayoutUpdateValidator->isValid() called at (vendor/magento/framework/View/Model/Layout/Merge.php:512)
#7 MagentoFrameworkViewModelLayoutMerge->_validateMergedLayout() called at (vendor/magento/framework/View/Model/Layout/Merge.php:488)
#11 MagentoFrameworkViewLayoutBuilder->build() called at (vendor/magento/framework/View/Layout.php:257)
#12 MagentoFrameworkViewLayout->build() called at (vendor/magento/framework/View/Layout.php:882)
#13 MagentoFrameworkViewLayout->getBlock() called at (generated/code/Magento/Framework/View/Layout/Interceptor.php:414)
#14 MagentoFrameworkViewLayoutInterceptor->getBlock() called at (vendor/magento/module-cms/Helper/Page.php:215)
#15 MagentoCmsHelperPage->prepareResultPage() called at (vendor/magento/module-cms/Controller/Noroute/Index.php:47)
#16 MagentoCmsControllerNorouteIndex->execute() called at (generated/code/Magento/Cms/Controller/Noroute/Index/Interceptor.php:24)
#17 MagentoCmsControllerNorouteIndexInterceptor->execute() called at (vendor/magento/framework/App/Action/Action.php:108)
#18 MagentoFrameworkAppActionAction->dispatch() called at (vendor/magento/framework/Interception/Interceptor.php:58)
#19 MagentoCmsControllerNorouteIndexInterceptor->___callParent() called at (vendor/magento/framework/Interception/Interceptor.php:138)
#20 MagentoCmsControllerNorouteIndexInterceptor->MagentoFrameworkInterception{closure}() called at (vendor/magento/framework/Interception/Interceptor.php:153)
#21 MagentoCmsControllerNorouteIndexInterceptor->___callPlugins() called at (generated/code/Magento/Cms/Controller/Noroute/Index/Interceptor.php:39)
#22 MagentoCmsControllerNorouteIndexInterceptor->dispatch() called at (vendor/magento/framework/App/FrontController.php:159)
#23 MagentoFrameworkAppFrontController->processRequest() called at (vendor/magento/framework/App/FrontController.php:98)
#24 MagentoFrameworkAppFrontController->dispatch() called at (vendor/magento/framework/Interception/Interceptor.php:58)
#25 MagentoFrameworkAppFrontControllerInterceptor->___callParent() called at (vendor/magento/framework/Interception/Interceptor.php:138)
#26 MagentoFrameworkAppFrontControllerInterceptor->MagentoFrameworkInterception{closure}() called at (vendor/magento/module-store/App/FrontController/Plugin/RequestPreprocessor.php:99)
#27 MagentoStoreAppFrontControllerPluginRequestPreprocessor->aroundDispatch() called at (vendor/magento/framework/Interception/Interceptor.php:135)
#28 MagentoFrameworkAppFrontControllerInterceptor->MagentoFrameworkInterception{closure}() called at (vendor/magento/module-page-cache/Model/App/FrontController/BuiltinPlugin.php:73)
#29 MagentoPageCacheModelAppFrontControllerBuiltinPlugin->aroundDispatch() called at (vendor/magento/framework/Interception/Interceptor.php:135)
#30 MagentoFrameworkAppFrontControllerInterceptor->MagentoFrameworkInterception{closure}() called at (vendor/magento/framework/Interception/Interceptor.php:153)
#31 MagentoFrameworkAppFrontControllerInterceptor->___callPlugins() called at (generated/code/Magento/Framework/App/FrontController/Interceptor.php:26)
#32 MagentoFrameworkAppFrontControllerInterceptor->dispatch() called at (vendor/magento/framework/App/Http.php:116)
#33 MagentoFrameworkAppHttp->launch() called at (vendor/magento/framework/App/Bootstrap.php:261)
#34 MagentoFrameworkAppBootstrap->run() called at (pub/index.php:40)
```

## real analysis – Show that a subsequence is a subset of the original sequence

My main question is the 4th point, but I hope you can clarify some things for me along the way.

The definition of a sequence says that a function $$a: mathbb {N} to S$$ is a sequence on a set $$S$$, denoted $$(a_n)$$.

1. Can i freely restrict the area of ​​function $$a$$ and still call it a sequence? In particular, a) is it valid to define $$a_n$$ on a finite subset of $$mathbb {N}$$ b) on an infinite subset of $$mathbb {N}$$?

Let's say that at each term in the sequence $$(a_n) _ {n in mathbb {N}}$$, I have to define a new sequence from there. I first define a sequence $$(m_k)$$ who maps $${k in mathbb {N}: k geq n } to mathbb {N}, forall n in mathbb {N}$$ with $$m_k (assuming affirmative on question 1b because the domain is an infinite subset of $$mathbb {N}$$). Then, like $$(a_ {m_k})$$ is the composition of the sequence $$(a_n)$$ and the increasing sequence $$(m_k)$$, by definition $$(a_ {m_k})$$ is a subsequence of $$(a_n)$$.

1. Is it a good way to show that these new sequences $$(a_ {m_k})$$ are subsequences?

A set of points defined for the sequence $$(a_n)$$ East $$left {a_n: n in mathbb {N} right }$$.

1. How do you define a set of points for a subsequence $$(a_ {m_k})$$? East $$forall n in mathbb {N}, left {a_ {m_k}: k geq n right }$$ well?

Assuming yes on question 3, the main question is:

1. How to prove this $$forall n in mathbb {N}, left {a_ {m_k}: k geq n right } subseteq left {a_n: n in mathbb {N} right }$$? In other words, that the set of points defined for a subsequence is a subset of the set of points in the original sequence. I think i should take a term off $$left {a_ {m_k}: k geq n right }$$ and deduce that its also in the whole $$left {a_n: n in mathbb {N} right }$$, but I don't know how to do it rigorously.

To give you context on my questions, I want to show that $$sup { left {a_n: n in mathbb {N} right }} geq sup { left {a_ {k}: k geq n right }}$$. Given that $$(a_n)$$ is bounded, defines $$left {a_n: n in mathbb {N} right }$$ and $$left {a_ {k}: k geq n right }$$ are delimited. Having $$left {a_ {k}: k geq n right } subseteq left {a_n: n in mathbb {N} right }$$ I would prove the supremums as in the question Prove the supremum of a subset is smaller than the supremum of the whole.

## Determine the convergence of the sequence

I have to prove that the next recursively defined sequence is convergent and find its limit.

$$a_1 = frac14$$

$$a_ {n + 1} = frac {{a_ {n}} ^ 2} 6 (n + 5) int_0 ^ { frac 3n} e ^ {- 2 {x ^ 2}} dx$$

Assuming that it is convergent, I have proved that the limit is $$0$$ or $$6$$. I would like to determine that it is $$0$$ showing that the sequence is decreasing but I couldn't do that. Help? Thank you

## functional analysis – Double stereotype of sequence spaces \$ ell_p \$

The stereotyped topology on the dual of a topological vector space is defined as the topology of uniform convergence on totally bounded sets. To help understand this concept, it would be helpful to have some simple examples.

Is there a simple characterization of the double stereotype of $$ell_p$$ sequence spaces? Or other simple examples?

## similar seed words minor sequence difference

I created a wallet a few weeks ago and I apparently restored the seed to my 2 other devices. Recently, I noticed that I had mistakenly logged into another wallet with identical starting words but a different word sequence. Unfortunately, when I noticed the difference, I couldn't say which seed I generated and which I came across. From a security perspective, should I abandon the two wallets? or is there a way to secure both. Note I have funded each other is empty.