## Subset of the matrix S4 in R

I have a matrix (33694 x 7047) that I named expr_matrix:

``````matrix.path <- paste0(matrix_dir, "matrix.mtx")
matrix.path

#> class (expr_matrix)
(1) "dgTMatrix"
attr(,"package")
(1) "Matrix"
``````

But when I try to define it:

``````expr_matrix.s1 <- subset.matrix(expr_matrix, colnames(expr_matrix) %in% barcode_file.s1\$barcode)
``````

I get a matrix (0x7047) instead of one (33694 x 1101) despite the presence of 1101 values ​​in barcode_file.s1 \$ barcode. Does anyone know why?

Thank you so much,

Abigail

## \$ gamma \$ is an invertible matrix such as \$ a = b gamma \$, so why do we have \$ oplus_ {i = 1} ^ n Ra_i subset oplus_ {i = 1} ^ n Rb_i \$

Let $$a = (a_1, a_2, … a_n)$$ and $$b = (b_1, b_2, … b_n)$$ be line vectors with entries in a ring $$R$$. Yes $$gamma$$ is an invertible matrix such that $$a = b gamma$$, so why do we have $$oplus_ {i = 1} ^ n Ra_i subset oplus_ {i = 1} ^ n Rb_i$$

## 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.

## javascript – Read a large JSON file and return a subset for the map

Fortran and Python are already great tools for calculating numbers, and, if you know you don't want to display all the markers at once, you will have "filtering" rules based on your entries, right? They probably have streams or buffers that can allow you to read a file by not loading it at the same time.

How about doing raw preprocessing in a way that allows you to partition your JSON tag file in bulk based on entries or "ranges of entries"?

I'm talking about doing this "offline" processing, so let's say you know roughly a range of "zoom" values ​​and which points to include in each:

You can create a map or an associative list like:

• Zoom level between 1 and 5: JSON ((list of corresponding points for this range))
• Between 6 and 8: JSON ((filtered list))
• Greater than 8: JSON ((remaining list))

So if you have zoom 3, you fall on the first category, zoom 8 on the second etc.

The list search for the initial compartment into which to fall is easy to do and for cases where these lists become large, if they have a beginning and an end, the search takes only O (1), then it is is to work with a pre-treated list.

This may perhaps give you an idea of ​​the procedure to follow.

## T or F: the subset of a regular set on {a, b, c} consisting only of strings that do not use the symbol c is regular.

i think this is true. A subset of any regular set must always be recognized by certain DFAs, and is it therefore regular? Is it correct? I don't know if I understand the question. Thank you for any clarification.

## general topology – The existence of a countable dense subset of a non-empty compact network of \$ mathbb {R} ^ {n} \$

In the proof of Lemma 2.13, the author said that

Let $$D = left {d_ {1}, ldots, d_ {n}, ldots right }$$ be a dense (finite or) countable subset of $$A$$.

Could you explain the existence of $$D$$? Thank you so much!

## Real analytical function: the zero of the gradient is a subset of the zero of the function

I got this question when I read the famous article by Bierstone and Milman "Semianalytic and subanalytic sets". In their proof of the inequality of the Łojasiewicz gradient (proposition 6.8 of the article), they used the following: for any function $$g$$ real analytics in a neighborhood of origin such as $$g ( mathbf {0}) = 0$$, there is a small ball $$K$$ centered at $$mathbf {0}$$ in which the zero set of the gradient $$nabla g$$ is contained in the zero set of $$g$$. Does anyone know how to prove this statement or can provide a reference? Thank you!

## combinatorial optimization – Finding an efficient way to maximize "pair scores" for a subset of 30 selected from 50 to 10,000 objects

Context: I have a mosaic program which uses a first directed width search algorithm. It is "led" by what I call "pairwise scoring". There are N polyforms (pieces) used in tiling. I have an NxN table with a score at each location. While tiling, I try to maximize the sums of pairs of unused pieces. At the start, I add up all the scores, find the average score, and subtract the average of all the scores so that they are zero. As I tile, I can find the new total just by adjusting the current total for the node by deleting the scores belonging to the last piece placed.

I generate the table NxN & # 39; even score & # 39; in mosaic of a small shape with about 6 to 10 pieces, for each mosaic found, I look at the border cells of each room & # 39; a & # 39; in the mosaic, and wherever it touches the part & # 39; b & # 39;, I increment (a, b) and (b, a). And wherever it touches the edge of the shape, I increase (a, a).

It works fairly well, but I suspect I could further optimize the “ final game '' & # 39; by encouraging a tendency to end with one of the “ local maxima '', being the group (s) of coins that “ mark the best as '' together & # 39; & # 39; .

In order to know if my algorithm finds such maxima, I must first find them myself so that I know when the node scores miss these maxima.

So my question is, how can I efficiently find the highest score sets from 1 to M on N pieces, where M goes up to around 30 and N up to 1000 or more?

There is some redundancy available in the table, I am currently repeating (a, b) in (b, a). I could use (a, b) for the score of the pair when a <b and (b, a) for something else. I already use (a, a) to keep track of the number of times a piece hits an edge square in a tessellation of smaller shape. I can adjust the importance of this edge score using a simple "edge score factor".

It would be nice to have an algorithm fast enough to track these local maxima as they change due to the parts used, without affecting the tiling speed too much. Then I would know fairly quickly that I had "gone wrong", that is to say that even if I had a good high knot score, I had used a set of pieces that would reduce my best possible end game, and I could use it to direct the BFS in addition to the single knot score.

Or is it the case that the node score already gives me that?

## real analysis – The restriction of the non-metric topology to the dense subset is not metriable

Let $$(X, tau)$$ be a non-metriable topological space that is not accountable first and leave $$emptyset neq Y subset X$$ be a suitable dense subset. Is it possible for $$(Y, tau_Y)$$ (or $$tau_Y$$ is the relativization of $$tau$$ at $$Y$$) also non-metric?

## Discrete Mathematics – Is the sum of the members of a subset of \$ mathbb {Z} / p mathbb {Z} \$ known? \$ c \$ in: \$ sum_ {n = 1} ^ { text {first} n text {with} g ^ n = 1} (g ^ n bmod P) = c cdot P \$

Let $$P$$ to be a prime ($$> 2$$) and $$g$$ a value between $$2$$ and $$P$$.
Let $$M$$ be the set of numbers that can be generated with $$g$$:
$$M = {g ^ n bmod P, text {with} 0
Yes $$g$$ is a main root of $$P$$ all values $$1$$ at $$P-1$$ can be generated.
The sum of these would be:
$$S = sum M = sum_ {n = 1} ^ {P-1} (g ^ n bmod P) = sum_ {n = 1} ^ {P-1} n = (P / 2) cdot (P + 1)$$

Is there also a formula for values $$g$$ not a main root of $$P$$?

L & # 39; together $$M$$ can also be written:
$$M = {g ^ n bmod P, text {with} 0
$$1 equiv g ^ k bmod P$$
For all $$g$$ there is an exhibitor $$k$$ which is limited to the factorization factors of $$P-1$$ or products outside of these raw values, so that the order of $$g$$ East $$k$$.

Main question: What is the exact sum of such a subset with $$k$$ a prime number greater than $$2$$ (and given appropriate $$g$$)?

Partial solution:
During the tests, I noticed a factor of this sum $$S$$ seems to be $$P$$.
Therefore
$$S = sum M = c cdot P$$

Any way to calculate this factor $$c$$?

I did a related question in the cryptography stack. There only a partial answer was given for even $$k$$ Which one is $$c = k / 2$$ and only an approximation for odd $$k$$ Which one is $$c about k / 2$$. But I'm interested in the main values ​​of $$k$$ and an exact solution. I made a new post here because I think the math expertise is higher here and an answer to that might be known.

Example: $$g = 13, P = 23$$
With $$g = 13$$ only half the numbers on $$mathbb {Z} / 23 mathbb {Z}$$ can be generated:
$$M = {13,8,12,18,4,6,9,2,3,16,1 }$$
sum $$S = sum M = 92$$, Which one is $$4 cdot P$$
Why $$4$$ time? Any way to calculate this factor?