89 research outputs found

### Effect algebras with the maximality property

The maximality property was introduced in [9] in orthomodular posets as
a common generalization of orthomodular lattices and orthocomplete orthomodular
posets. We show that various conditions used in the theory of e ect
algebras are stronger than the maximality property, clear up the connections
between them and show some consequences of these conditions. In particular,
we prove that a Jauch{Piron e ect algebra with a countable unital set of states
is an orthomodular lattice and that a unital set of Jauch{Piron states on an
e ect algebra with the maximality property is strongly order determining

### IST Austria Thesis

In this thesis we study certain mathematical aspects of evolution. The two primary forces that drive an evolutionary process are mutation and selection. Mutation generates new variants in a population. Selection chooses among the variants depending on the reproductive rates of individuals. Evolutionary processes are intrinsically random – a new mutation that is initially present in the population at low frequency can go extinct, even if it confers a reproductive advantage. The overall rate of evolution is largely determined by two quantities: the probability that an invading advantageous mutation spreads through the population (called fixation probability) and the time until it does so (called fixation time). Both those quantities crucially depend not only on the strength of the invading mutation but also on the population structure. In this thesis, we aim to understand how the underlying population structure affects the overall rate of evolution. Specifically, we study population structures that increase the fixation probability of advantageous mutants (called amplifiers of selection). Broadly speaking, our results are of three different types: We present various strong amplifiers, we identify regimes under which only limited amplification is feasible, and we propose population structures that provide different tradeoffs between high fixation probability and short fixation time

### Concrete quantum logics with generalised compatibility

summary:We present three results stating when a concrete (=set-representable) quantum logic with covering properties (generalization of compatibility) has to be a Boolean algebra. These results complete and generalize some previous results [3, 5] and answer partiallz a question posed in [2]

### Robust Draws in Balanced Knockout Tournaments

Balanced knockout tournaments are ubiquitous in sports competitions and are
also used in decision-making and elections. The traditional computational
question, that asks to compute a draw (optimal draw) that maximizes the winning
probability for a distinguished player, has received a lot of attention.
Previous works consider the problem where the pairwise winning probabilities
are known precisely, while we study how robust is the winning probability with
respect to small errors in the pairwise winning probabilities. First, we
present several illuminating examples to establish: (a)~there exist
deterministic tournaments (where the pairwise winning probabilities are~0 or~1)
where one optimal draw is much more robust than the other; and (b)~in general,
there exist tournaments with slightly suboptimal draws that are more robust
than all the optimal draws. The above examples motivate the study of the
computational problem of robust draws that guarantee a specified winning
probability. Second, we present a polynomial-time algorithm for approximating
the robustness of a draw for sufficiently small errors in pairwise winning
probabilities, and obtain that the stated computational problem is NP-complete.
We also show that two natural cases of deterministic tournaments where the
optimal draw could be computed in polynomial time also admit polynomial-time
algorithms to compute robust optimal draws

### Atomistic and orthoatomistic effect algebras

We characterize atomistic effect algebras, prove that a weakly orthocomplete
Archimedean atomic effect algebra is orthoatomistic and present an example of
an orthoatomistic orthomodular poset that is not weakly orthocomplete.Comment: 6 page

### Strong Amplifiers of Natural Selection: Proofs

We consider the modified Moran process on graphs to study the spread of
genetic and cultural mutations on structured populations. An initial mutant
arises either spontaneously (aka \emph{uniform initialization}), or during
reproduction (aka \emph{temperature initialization}) in a population of $n$
individuals, and has a fixed fitness advantage $r>1$ over the residents of the
population. The fixation probability is the probability that the mutant takes
over the entire population. Graphs that ensure fixation probability of~1 in the
limit of infinite populations are called \emph{strong amplifiers}. Previously,
only a few examples of strong amplifiers were known for uniform initialization,
whereas no strong amplifiers were known for temperature initialization.
In this work, we study necessary and sufficient conditions for strong
amplification, and prove negative and positive results. We show that for
temperature initialization, graphs that are unweighted and/or self-loop-free
have fixation probability upper-bounded by $1-1/f(r)$, where $f(r)$ is a
function linear in $r$. Similarly, we show that for uniform initialization,
bounded-degree graphs that are unweighted and/or self-loop-free have fixation
probability upper-bounded by $1-1/g(r,c)$, where $c$ is the degree bound and
$g(r,c)$ a function linear in $r$. Our main positive result complements these
negative results, and is as follows: every family of undirected graphs with
(i)~self loops and (ii)~diameter bounded by $n^{1-\epsilon}$, for some fixed
$\epsilon>0$, can be assigned weights that makes it a strong amplifier, both
for uniform and temperature initialization

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