67 research outputs found
Nested recursions with ceiling function solutions
Consider a nested, non-homogeneous recursion R(n) defined by R(n) =
\sum_{i=1}^k R(n-s_i-\sum_{j=1}^{p_i} R(n-a_ij)) + nu, with c initial
conditions R(1) = xi_1 > 0,R(2)=xi_2 > 0, ..., R(c)=xi_c > 0, where the
parameters are integers satisfying k > 0, p_i > 0 and a_ij > 0. We develop an
algorithm to answer the following question: for an arbitrary rational number
r/q, is there any set of values for k, p_i, s_i, a_ij and nu such that the
ceiling function ceiling{rn/q} is the unique solution generated by R(n) with
appropriate initial conditions? We apply this algorithm to explore those
ceiling functions that appear as solutions to R(n). The pattern that emerges
from this empirical investigation leads us to the following general result:
every ceiling function of the form ceiling{n/q}$ is the solution of infinitely
many such recursions. Further, the empirical evidence suggests that the
converse conjecture is true: if ceiling{rn/q} is the solution generated by any
recursion R(n) of the form above, then r=1. We also use our ceiling function
methodology to derive the first known connection between the recursion R(n) and
a natural generalization of Conway's recursion.Comment: Published in Journal of Difference Equations and Applications, 2010.
11 pages, 1 tabl
Solving Non-homogeneous Nested Recursions Using Trees
The solutions to certain nested recursions, such as Conolly's C(n) =
C(n-C(n-1))+C(n-1-C(n-2)), with initial conditions C(1)=1, C(2)=2, have a
well-established combinatorial interpretation in terms of counting leaves in an
infinite binary tree. This tree-based interpretation, which has a natural
generalization to a k-term nested recursion of this type, only applies to
homogeneous recursions, and only solves each recursion for one set of initial
conditions determined by the tree. In this paper, we extend the tree-based
interpretation to solve a non-homogeneous version of the k-term recursion that
includes a constant term. To do so we introduce a tree-grafting methodology
that inserts copies of a finite tree into the infinite k-ary tree associated
with the solution of the corresponding homogeneous k-term recursion. This
technique can also be used to solve the given non-homogeneous recursion with
various sets of initial conditions.Comment: 14 page
Pacifying the Fermi-liquid: battling the devious fermion signs
The fermion sign problem is studied in the path integral formalism. The
standard picture of Fermi liquids is first critically analyzed, pointing out
some of its rather peculiar properties. The insightful work of Ceperley in
constructing fermionic path integrals in terms of constrained world-lines is
then reviewed. In this representation, the minus signs associated with
Fermi-Dirac statistics are self consistently translated into a geometrical
constraint structure (the {\em nodal hypersurface}) acting on an effective
bosonic dynamics. As an illustrative example we use this formalism to study
1+1-dimensional systems, where statistics are irrelevant, and hence the sign
problem can be circumvented. In this low-dimensional example, the structure of
the nodal constraints leads to a lucid picture of the entropic interaction
essential to one-dimensional physics. Working with the path integral in
momentum space, we then show that the Fermi gas can be understood by analogy to
a Mott insulator in a harmonic trap. Going back to real space, we discuss the
topological properties of the nodal cells, and suggest a new holographic
conjecture relating Fermi liquids in higher dimensions to soft-core bosons in
one dimension. We also discuss some possible connections between mixed
Bose/Fermi systems and supersymmetry.Comment: 28 pages, 5 figure
On Dynamics of Integrate-and-Fire Neural Networks with Conductance Based Synapses
We present a mathematical analysis of a networks with Integrate-and-Fire
neurons and adaptive conductances. Taking into account the realistic fact that
the spike time is only known within some \textit{finite} precision, we propose
a model where spikes are effective at times multiple of a characteristic time
scale , where can be \textit{arbitrary} small (in particular,
well beyond the numerical precision). We make a complete mathematical
characterization of the model-dynamics and obtain the following results. The
asymptotic dynamics is composed by finitely many stable periodic orbits, whose
number and period can be arbitrary large and can diverge in a region of the
synaptic weights space, traditionally called the "edge of chaos", a notion
mathematically well defined in the present paper. Furthermore, except at the
edge of chaos, there is a one-to-one correspondence between the membrane
potential trajectories and the raster plot. This shows that the neural code is
entirely "in the spikes" in this case. As a key tool, we introduce an order
parameter, easy to compute numerically, and closely related to a natural notion
of entropy, providing a relevant characterization of the computational
capabilities of the network. This allows us to compare the computational
capabilities of leaky and Integrate-and-Fire models and conductance based
models. The present study considers networks with constant input, and without
time-dependent plasticity, but the framework has been designed for both
extensions.Comment: 36 pages, 9 figure
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