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Agile Algebraic Geometry – Some questions about the algebraic cycles and the conjectures of Beilinson and Tate

I have three slightly related questions about algebraic cycles that I will just list. I would appreciate to get your answers:

1) Is there an example of a smooth projective diversity? $ X $ over a finite field that is of dimension $ n> 1 $ and at least for some $ i> 1 $. $ CH ^ i (X) $ the chow group of the co-dimension $ i $ Cycles is of rank greater than 1?

2) Leave $ X $ be like above. It is known if it is possible to couple the intersection $ CH ^ i (X) otimes mathbb {Q} $ and $ CH ^ {n-i} (X) otimes mathbb {Q} $ can be completely zero? The conjecture of Beilinson on the coincidence of numerical and rational equivalences in this situation predicts that this can not happen. I'm just wondering if you can conclude without assuming this assumption. (For example, if there is a line bundle whose top self-intersection point is not zero, this can not happen.)

3) The Beilinson Tate conjecture predicts the same thing $ X $. $ c ^ i: CH ^ i (X) otimes mathbb {Q} rightarrow H ^ {2i} (X, mathbb {Q} _ { ell} (i)) $ is injective. Is it known if this can not be completely zero?

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finite groups – expander graphs with many 4 cycles

The question is not well defined. But it works like this:

Could you find an infinite family of diagrams? $ G_i $these are all $ epsilon $ Expander, but have many 4-cycles?

$ epsilon $ should be as large as possible as well as the number of 4 cycles.
The chart family should be as wide as possible.


That is the basic question. A trivial good example would be a Cartesian chart product $ G times H $,

If we think in Cayley diagrams:

Suppose our artwork is $ Cay (G, S) $ and S is symmetrical.
A 4-stroke results from 4 generators a, b, c, d s.t. ab = cd.
It is trivial whether either (c = b and d = a) or (c = a and d = b).

I think many trivial cycles would hurt expansion, as Abelian groups are poor expander according to Alon-Roichman's theorem.

I think the goal should be to find groups and generator sets that have many non-trivial 4-cycles.

Many Thanks.

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Agile Algebraic Geometry – Are Chern Classes of Holomorphic Vector Bundles Enough to Create Hodge Cycles?

To let $ X $ ba a smooth projective variety of dimensions $ n $, Hodge conjecture states that every hodge cyclists in $ Hdg ^ k (X, mathbb {Q}) $ comes from a Chern class of the codimension $ k $ in the $ CH ^ k (X, mathbb {Q}) $, Now the $ k $-th Chern class of holomorphic vector bundles generates a subgroup $ CH ^ k_ {vec} (X, mathbb {Q}) $, Is it possible that
every Hodge goes with it $ Hdg ^ k (X, mathbb {Q}) $ comes from $ CH ^ k_ {vec} (X, mathbb {Q}) $? Are there counterexamples or results?

Networking – How Do I Support Multiple Platform Patch Cycles in a Client / Server Multiplayer Game?

I'm doing a network multiplayer game that can be played with Crossplay from multiple platforms. The game is written in c ++ and uses normal TCP sockets for network communication. Currently there is the game under Windows, but ideally under ios / android / linux / mac. The server is a single process written to run on Ubuntu Linux. Currently, clients find the server through a hard-coded URL that resolves to the server IP and a hard-coded port.

This is fine as long as the clients / servers are developed / delivered in perfect lockstep. If I change the client or network code in any way, I can only release the client update and run the new server version at the same time.

The problem is that once I've published on multiple platforms, I'll never be able to send an update to all clients at once. In addition, old versions of the client will exist forever in the wild.

I can add a version handshake at the beginning of every connection to the server. A client would know if it is outdated and prompt a user to update (and disable all multiplayer features until then).

I could run multiple servers concurrently and hardcode only one new port if updates lead to important changes.

What are the typical solutions to this problem? I know that games like Hearthstone accomplish this, but I assume they have a massive server infrastructure / operations team to help with that.

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