Randomly Charged Polymers, Random Walks, and Their Extremal Properties

Motivated by an investigation of ground state properties of randomly charged polymers, we discuss the size distribution of the largest Q-segments (segments with total charge Q) in such N-mers. Upon mapping the charge sequence to one--dimensional random walks (RWs), this corresponds to finding the probability for the largest segment with total displacement Q in an N-step RW to have length L. Using analytical, exact enumeration, and Monte Carlo methods, we reveal the complex structure of the probability distribution in the large N limit. In particular, the size of the longest neutral segment has a distribution with a square-root singularity at l=L/N=1, an essential singularity at l=0, and a discontinuous derivative at l=1/2. The behavior near l=1 is related to a another interesting RW problem which we call the "staircase problem". We also discuss the generalized problem for d-dimensional RWs.Comment: 33 pages, 19 Postscript figures, RevTe

Berezinians, Exterior Powers and Recurrent Sequences

We study power expansions of the characteristic function of a linear operator $A$ in a $p|q$-dimensional superspace $V$. We show that traces of exterior powers of $A$ satisfy universal recurrence relations of period $q$. `Underlying' recurrence relations hold in the Grothendieck ring of representations of \GL(V). They are expressed by vanishing of certain Hankel determinants of order $q+1$ in this ring, which generalizes the vanishing of sufficiently high exterior powers of an ordinary vector space. In particular, this allows to explicitly express the Berezinian of an operator as a rational function of traces. We analyze the Cayley--Hamilton identity in a superspace. Using the geometric meaning of the Berezinian we also give a simple formulation of the analog of Cramer's rule.Comment: 35 pages. LaTeX 2e. New version: paper substantially reworked and expanded, new results include

Hydrogeological Investigations in the Pampa of Argentina

The author has identified the following significant results. Satellite imagery in combination with ground investigations allows the identification and delineation of the near surface ground water (depth to ground water, salinity). The degree of precision achieved is greater than that obtainable by conventional ground survey methods alone

Collapse of Randomly Self-Interacting Polymers

We use complete enumeration and Monte Carlo techniques to study self--avoiding walks with random nearest--neighbor interactions described by $v_0q_iq_j$, where $q_i=\pm1$ is a quenched sequence of ``charges'' on the chain. For equal numbers of positive and negative charges ($N_+=N_-$), the polymer with $v_0>0$ undergoes a transition from self--avoiding behavior to a compact state at a temperature $\theta\approx1.2v_0$. The collapse temperature $\theta(x)$ decreases with the asymmetry $x=|N_+-N_-|/(N_++N_-)$Comment: 8 pages, TeX, 4 uuencoded postscript figures, MIT-CMT-

A coding problem for pairs of subsets

Let $X$ be an $n$--element finite set, $0<k\leq n/2$ an integer. Suppose that $\{A_1,A_2\}$ and $\{B_1,B_2\}$ are pairs of disjoint $k$-element subsets of $X$ (that is, $|A_1|=|A_2|=|B_1|=|B_2|=k$, $A_1\cap A_2=\emptyset$, $B_1\cap B_2=\emptyset$). Define the distance of these pairs by $d(\{A_1,A_2\} ,\{B_1,B_2\})=\min \{|A_1-B_1|+|A_2-B_2|, |A_1-B_2|+|A_2-B_1|\}$. This is the minimum number of elements of $A_1\cup A_2$ one has to move to obtain the other pair $\{B_1,B_2\}$. Let $C(n,k,d)$ be the maximum size of a family of pairs of disjoint subsets, such that the distance of any two pairs is at least $d$. Here we establish a conjecture of Brightwell and Katona concerning an asymptotic formula for $C(n,k,d)$ for $k,d$ are fixed and $n\to \infty$. Also, we find the exact value of $C(n,k,d)$ in an infinite number of cases, by using special difference sets of integers. Finally, the questions discussed above are put into a more general context and a number of coding theory type problems are proposed.Comment: 11 pages (minor changes, and new citations added

Collineation group as a subgroup of the symmetric group

Let $\Psi$ be the projectivization (i.e., the set of one-dimensional vector subspaces) of a vector space of dimension $\ge 3$ over a field. Let $H$ be a closed (in the pointwise convergence topology) subgroup of the permutation group $\mathfrak{S}_{\Psi}$ of the set $\Psi$. Suppose that $H$ contains the projective group and an arbitrary self-bijection of $\Psi$ transforming a triple of collinear points to a non-collinear triple. It is well-known from \cite{KantorMcDonough} that if $\Psi$ is finite then $H$ contains the alternating subgroup $\mathfrak{A}_{\Psi}$ of $\mathfrak{S}_{\Psi}$. We show in Theorem \ref{density} below that $H=\mathfrak{S}_{\Psi}$, if $\Psi$ is infinite.Comment: 9 page

New control strategies for neuroprosthetic systems

The availability of techniques to artificially excite paralyzed muscles opens enormous potential for restoring both upper and lower extremity movements with\ud neuroprostheses. Neuroprostheses must stimulate muscle, and control and regulate the artificial movements produced. Control methods to accomplish these tasks include feedforward (open-loop), feedback, and adaptive control. Feedforward control requires a great deal of information about the biomechanical behavior of the limb. For the upper extremity, an artificial motor program was developed to provide such movement program input to a neuroprosthesis. In lower extremity control, one group achieved their best results by attempting to meet naturally perceived gait objectives rather than to follow an exact joint angle trajectory. Adaptive feedforward control, as implemented in the cycleto-cycle controller, gave good compensation for the gradual decrease in performance observed with open-loop control. A neural network controller was able to control its system to customize stimulation parameters in order to generate a desired output trajectory in a given individual and to maintain tracking performance in the presence of muscle fatigue. The authors believe that practical FNS control systems must\ud exhibit many of these features of neurophysiological systems