... which numbers? For example, the tree-height function h(age) = age×20 makes no sense for an age less than zero. ... it could also be letters ("A"→"B"), or ID codes ("A6309"→"Pass") or stranger things.So we
Chemically plausible RNs allow neither a perpetuum mobile, i.e., a “futile cycle” of reactions with non-vanishing energy production, nor the creation or annihilation of mass. Such RNs are said to be thermodynamically sound and conservative. For finite RNs, both conditions can be expressed equi...
Journal of Cheminformatics (2022) 14:63 https://doi.org/10.1186/s13321-022-00621-8 Journal of Cheminformatics RESEARCH ARTICLE Open Access What makes a reaction network "chemical"? Stefan Müller1 , Christoph Flamm2 and Peter F. Stadler2,3,4,5,6,7* ...
We start with a seemingly trivial shift in perspective. Instead of asking whether we can find four points forming a rectangle, we ask whether it’s possible to find two distinct pairs of points which form a rectangle. Considering the points as diagonal pairs makes the problem easier. We’ll...
This makes the subject of analysis considerably more “algebraic” in nature, as the “epsilon management” that is so prevalent in orthodox analysis is now performed much more invisibly. For instance, as we shall see, in the nonstandard framework, orders of infinity acquire the algebraic ...
Let c1 be the distributed certificate that makes A accept (G,x) at all nodes, for all certificates c2. Let c1′ be the distributed certificate for (G′,x′) defined by c1′(u)=c1(ϕ(u)). Since, with certificate c1, A accepts at all nodes of G, for every certificate c2, and...
A basic tool for this purpose is the invariance of domain theorem: Theorem 3 (Brouwer invariance of domain theorem) Let be an open subset of , and let be a continuous injective map. Then is also open. We prove this theorem below the fold. It has an important corollary: Corollary 4 (...
The assumption that the support of ν is symmetric around 0 is not essential; however, the assumption that the support is bounded is important as it makes it possible for players to distinguish between zero and non-zero types.15 Given the signals, si and sj, the players update their ...
Once this is done, one invokes a change-of-rings theorem that makes calculating the -page of (4.1) much easier. For several great examples of this technique, see [Cam17, BC18]. Unfortunately, this strategy is not available for spin- bordism. The reason is that , thought of as a map...