## terminology – Name for specific cycles in graphs

Is there an established name for cycles? $$C subseteq G (V, E)$$ with the property that
$$lbrace u, v rbrace subseteq C cap V implies mathrm {dist} _ {| C} (u, v) le mathrm {dist} _ {| G} (u, v) PS$$

I would be tempted to call them facets because the vertices and edges that constitute the boundary of a facet of a polyhedron are prototypical examples of such cycles.

## graphics: find the longest route by number of edges, excluding cycles

I need to analyze a directed graph, but I don't know the name of the algorithm I would need to use. The graph has many cycles.

My desired behavior is: given a graphical source and a graphical sink, find the longest path by number of edges, excluding cycles.

By graphic font, I mean a vertex with one or more edges to other vertices and no incoming edges, and the opposite to sink. If there is better terminology let me know about it.

By excluding cycles, this could involve not traversing an edge of the previously traversed process.

Do you recognize this algorithm and could you tell me the name please?

## design: I often create class cycles using the Observer Pattern. How I can avoid this?

I often find myself creating class cycles using the observer pattern. Consider the following scenario:

• I have a centrally accessible global data source (Subject)
• The data source is reflected in many GUI components (watchers) in various ways (for example, different states for buttons, size indicators, numbers, etc.)
• Some of those GUI components (like buttons) may update the global data source (for example, a button may remove something from the data source or be shown disabled if the data source is empty)

So now I have a loop between some GUI components and the data source.

Abusing the pattern of observation here for something it's not intended for? Is there another way to solve this?

## cpu – how many clock cycles

It actually differs from one CPU to another, but it is possible to choose a conventional CPU technology used in moderate servers or in home computers.

How many clock cycles does it take to read a 50 KB file from the Ethernet socket and then forward this file to another IP address without saving the file to disk? I suppose this operation can be done in RAM. If not, how many clock cycles are required to store a copy of this 50 Kb file on disk (7200 rpm)?

Thank you.

## Can we construct infinite finite symmetric graphs with at least one simple cycle of length \$ k \$ but without cycles of lesser length?

For any $$k geq 3$$, Is it possible to construct infinite finite symmetric graphs with at least one simple cycle in length $$k$$ but there are no cycles of length smaller than $$k$$ ?

To clarify,

Yes $$k = 3$$, any full graph with at least $$3$$ vertices satisfies this definition

Yes $$k = 4$$, the cube works

Yes $$k = 5$$, dodecahedron and Petersen graph work

Can we build for each $$k geq 3$$, Infinites of these examples (that is to say, where the number of vertices tends to infinity)?

Is there a name for such graphics?

## Without mirror: Do Olympus OM-D cameras have two exposure shutter cycles in one shot?

Inspired by "Does live viewing increase the number of shutter activations?"

This question should apply to most cameras in the mirrorless system, but I would like to see an answer that applies to the Olympus OM-D series, as I am preparing to buy an OM-D E-M1 Mark III.

As I understand it, the mechanical shutter of a typical mirrorless camera requires two activations for each exposure in single shot mode: the shutter closes to end the live view, then opens and closes for the exposure, then opens again to resume live viewing. (This may vary depending on the shooting mode; for example, the burst shot at the highest speed allowed by the mechanical shutter only requires one cycle for each shot after the first one, and the electronic front curtain mode shooting requires only one exposure cycle).

Do Olympus OM-D cameras correctly count this as two shutter drives in the internal shutter count? This is important because the shutter count would otherwise be inaccurate. (The shutter on the E-M1 Mark III is rated for 400k drives).

## sql server: delete similar cycles in the directed chart

I have the following data:

Table:

``````CREATE TABLE tblLoop
(
person1 varchar(20),
person2 varchar(20),
ColDate date,
);

INSERT INTO tblLoop VALUES('A','B','2020-01-01'),('A','C','2020-01-01'),('A','D','2020-01-01'),
('B','E','2020-01-02'),('B','F','2020-01-02'),
('D','G','2020-01-03'),('D','H','2020-01-03'),
('F','i','2020-01-04'),
('G','J','2020-01-05'),
('i','A','2020-01-06'),
('J','D','2020-01-07'),
('X','Y','2020-01-08'),('X','Z','2020-01-08'),
('Z','X','2020-01-09'),
('Y','W','2020-01-09');
``````

The records look like this:

Requirement: I need to find the people who form a cycle. For an example in the given data, we found 3 cycles:

Cycle 1: `A` connected with `B` connected with `F` connected with `i` connected with `A`.

Cycle 2: `A` connected with `D` connected with `G` connected with `J` connected with `D`.

Cycle 3: `X` connected with `Z` connected with `X`.

Expected result:

``````LoopFound
--------------------
A->B->F->i->A
A->D->G->J->D
X->Z->X
``````

My attempt:

``````;WITH CTE AS
(
SELECT Person1, Person2,
CONVERT(VARCHAR(MAX), (','+ Person1+ ','+ Person2+ ',')) AS nodes, 1 AS lev,
(CASE WHEN Person1 = Person2 THEN 1 ELSE 0 END) AS has_cycle
FROM tblLoop e
UNION ALL
SELECT cte.Person1, e.Person2,
CONVERT(VARCHAR(MAX), (cte.nodes+ e.Person2+ ',')), lev + 1,
(CASE WHEN cte.nodes LIKE ('%,'+ e.Person2+ ',%') THEN 1 ELSE 0 END) AS has_cycle
FROM CTE
JOIN tblLoop e ON e.Person1 = cte.Person2
WHERE cte.has_cycle = 0
)
SELECT *
FROM CTE
WHERE has_cycle = 1;
``````

NOTE: Obtaining multiple combinations of cycles from the previous query.

## co.combinatorics – Limit number of cycles k in a graph

Fix any $$k geq 3$$, and suppose I have a simple non-directed graph $$G = (V, E)$$. I want a limit on the number of $$k$$ cycles in $$G$$ as a function of $$| E |$$. In particular, I would like to try the following "conjectured" statement:

W.T.S: Given any $$G = (V, E)$$, The number of $$k$$-cycles in $$G$$ it is at most $$O (| E | k / 2)$$.

Clearly $$binom {| V |} {k}$$ is adjusted if we only want a limit in terms of $$| V |$$, which comes from the full graph in $$| V |$$ vertices However, it is not so clear how to demonstrate a tight upper limit in terms of $$| E |$$. The intuition of the conjecture, of course, is that, up to a constant, it is optimal to organize the edges in a complete graph (if $$k$$ it is even, you earn a constant by taking a complete bipartite graph instead of a complete graph).

There seems to be some evidence of this fact for triangles (k = 3), for example https://math.stackexchange.com/questions/823481/number-of-triangles-in-a-graph-based-on- number of edges However, they are not clearly generalized to large $$k$$. In addition, I don't know any cycle enumeration algorithm k for k> 3 that is parameterized by $$| E |$$ (whose runtime would provide an upper limit).

For my application, I really just need limits for odd $$k$$, but it seems that any such test should be generalized to any $$k$$. Also, to be clear, I'm only interested in asymmetric limits here in terms of $$| E |$$ (getting tight constants seems much harder). Any suggestions or references that I missed would be greatly appreciated.

## optimization: cover of vertex of minimum weight in \$ G \$ connected with cycles of maximum length \$ 3 \$

Leave $$G = (V, E)$$ be a non-directed weight chart $$w: V rightarrow (0, infty)$$. We want to find an algorithm that finds a vertex cover (that is, a set of vertices so that each edge contains an element of that set)
$$U$$ from $$G$$ minimizing the amount $$w (U) = sum_ {u in U} w (u)$$ Dice
every single cycle in $$G$$ It has a length of at most $$3$$.

We are supposed to use the following fact: given an expansion tree $$T$$ from $$G$$, for all $$uv in E$$, the length of the road in $$T$$ since $$u$$ to $$v$$ it is $$1$$ or $$2$$.

I thought about using a more sophisticated version of the well-known greedy algorithm that finds a vertex cover for a tree without weights (in each iteration, find a leaf and eliminate its father, including all his children, marking the father). However, I could not generalize the underlying principles.

## logic – validity of the incompleteness theorem in statements without referential cycles

Popularized exposures of Gödel's incompleteness theorem generally show that the theorem is based on a self-referential statement that cannot be proven from existing axioms.

It is known that self-referential statements are problematic since they can lead to verbal paradoxes like Russell's paradox

Therefore, this leads me to the question: consider the subset of mathematical assertions that have referential cycles (that is, their assertions have references that can lead to the original assertion), I am not sure if this subset has a usual name in the field, but let's call it a set of autocyclical statements $$bf {SC}$$. Now consider the complement of this set ($$bf {SC ^ complement}$$), which may include assertions that have references that can be assigned to any directed acyclic graph.

Questions in my mind:

1 is $$bf {SC ^ complement}$$ a well defined set? one in which one or more assertions may or may not belong in a verifiable manner

2) Does the incompleteness theorem still apply to assertions in $$bf {SC ^ complement}$$?