3,487,183 research outputs found
Understanding Fixed Point Theorems
Fixed point theorems are the standard tool used to prove
the existence of equilibria in mathematical economics. This paper
shows how to prove a slight generalization of Brouwer's and
Kakutani's fixed point theorems using the familiar
techniques of drawing and shifting curves in the plane and is,
therefore, intelligible without advanced knowledge of topology.
This makes proofs of fixed point theorems accessible to a broader
audience.Kurvenverschieben in der Ebene' beweisen
lassen.
Das wichtigste Instrument für Existenzsätze in der mathematischen
Volkswirtschaftslehre sind Fixpunkttheoreme. Dieses Papier zeigt,
wie sich Brouwers Fixpunktsatz und einige geringfĂĽgige
Verallgemeinerungen durch 'Kurvenverschieben in der Ebene' beweisen
lassen. Weil keine tiefen topologischen Vorkenntnisse notwendig sind,
ist der grafische Beweis fĂĽr einen breiten Adressatenkreis
zugänglich
The Mermin fixed point
The most efficient known method for solving certain computational problems is
to construct an iterated map whose fixed points are by design the problem's
solution. Although the origins of this idea go back at least to Newton, the
clearest expression of its logical basis is an example due to Mermin. A
contemporary application in image recovery demonstrates the power of the
method.Comment: Contribution to Mermin Festschrift; 8 pages, 5 figure
Fixed-point Factorized Networks
In recent years, Deep Neural Networks (DNN) based methods have achieved
remarkable performance in a wide range of tasks and have been among the most
powerful and widely used techniques in computer vision. However, DNN-based
methods are both computational-intensive and resource-consuming, which hinders
the application of these methods on embedded systems like smart phones. To
alleviate this problem, we introduce a novel Fixed-point Factorized Networks
(FFN) for pretrained models to reduce the computational complexity as well as
the storage requirement of networks. The resulting networks have only weights
of -1, 0 and 1, which significantly eliminates the most resource-consuming
multiply-accumulate operations (MACs). Extensive experiments on large-scale
ImageNet classification task show the proposed FFN only requires one-thousandth
of multiply operations with comparable accuracy
Layered Fixed Point Logic
We present a logic for the specification of static analysis problems that
goes beyond the logics traditionally used. Its most prominent feature is the
direct support for both inductive computations of behaviors as well as
co-inductive specifications of properties. Two main theoretical contributions
are a Moore Family result and a parametrized worst case time complexity result.
We show that the logic and the associated solver can be used for rapid
prototyping and illustrate a wide variety of applications within Static
Analysis, Constraint Satisfaction Problems and Model Checking. In all cases the
complexity result specializes to the worst case time complexity of the
classical methods
Fixed Point and Aperiodic Tilings
An aperiodic tile set was first constructed by R.Berger while proving the
undecidability of the domino problem. It turned out that aperiodic tile sets
appear in many topics ranging from logic (the Entscheidungsproblem) to physics
(quasicrystals) We present a new construction of an aperiodic tile set that is
based on Kleene's fixed-point construction instead of geometric arguments. This
construction is similar to J. von Neumann self-reproducing automata; similar
ideas were also used by P. Gacs in the context of error-correcting
computations. The flexibility of this construction allows us to construct a
"robust" aperiodic tile set that does not have periodic (or close to periodic)
tilings even if we allow some (sparse enough) tiling errors. This property was
not known for any of the existing aperiodic tile sets.Comment: v5: technical revision (positions of figures are shifted
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