arXiv subjects
Publications and source records attributed to Carsten Peterson.
We prove that for almost all symmetric spaces $X$ and for any sequence of compact locally symmetric spaces $Y_n$ which is uniformly discrete, has a uniform spectral gap, and converges in the sense of Benjamini--Schramm to $X$, the joint eigenfunctions of all invariant differential operators on $Y_n$ delocalize on average when their spectral parameters are taken to lie in a fixed spectral window.
In analogy with the spectral theory of geometrically finite hyperbolic manifolds, we initiate the study of resonances on geometrically finite (q+1)-regular graphs of groups. We prove the meromorphic continuation of the resolvent of the adjacency operator on such spaces and give a geometric characterization of the resonant states. In contrast to the hyperbolic surfaces setting, geometrically finite graphs have only finitely many resonances, and these resonances may be computed explicitly, yet they exhibit many of the same qualitative phenomena as in the hyperbolic manifolds setting. Particularly interesting examples arise from algebraic curves over finite fields.
Let $\mathfrak{p} \subset V$ be a polytope and $ξ\in V_{\mathbb{C}}^*$. We obtain an expression for $I(\mathfrak{p}; α) := \int_{\mathfrak{p}} e^{\langle α, x \rangle} dx$ as a sum of meromorphic functions in $α\in V^*_{\mathbb{C}}$ parametrized by the faces $\mathfrak{f}$ of $\mathfrak{p}$ on which $\langle ξ, x \rangle$ is constant. Each term only depends on the local geometry of $\mathfrak{p}$ near $\mathfrak{f}$ (and on $ξ$) and is holomorphic at $α= ξ$. When $\langle ξ, \cdot \rangle$ is only constant on the vertices of $\mathfrak{p}$ our formula reduces to Brion's formula. Suppose $\mathfrak{p}$ is a rational polytope with respect to a lattice $Λ$. We obtain an expression for $S(\mathfrak{p}; α) := \sum_{λ\in \mathfrak{p} \cap Λ} e^{\langle α, λ\rangle}$ as a sum of meromorphic functions parametrized by the faces $\mathfrak{f}$ on which $e^{\langle ξ, x \rangle} = 1$ on a finite index sublattice of $\text{lin}(\mathfrak{f}) \cap Λ$. Each term only depends on the local geometry of $\mathfrak{p}$ near $\mathfrak{f}$ (and on $ξ$ and $Λ$) and is holomorphic at $α= ξ$. When $e^{\langle ξ, \cdot \rangle} \neq 1$ at any non-zero lattice point on a line through the origin parallel to an edge of $\mathfrak{p}$, our formula reduces to Brion's formula, and when $ξ= 0$, it reduces to the Ehrhart quasi-polynomial. Our formulas are particularly useful for understanding how $I(\mathfrak{p}(h); ξ)$ and $S(\mathfrak{p}(h); ξ)$ vary in a family of polytopes $\mathfrak{p}(h)$ with the same normal fan. When considering dilates of a fixed polytope, our formulas may be viewed as polytopal analogues of Laplace's method and the method of stationary phase. Such expressions naturally show up in analysis on symmetric spaces and affine buildings.
With a view towards studying the multitemporal wave equation on affine buildings recently introduced by Anker-Rémy-Trojan [arXiv:2312.06860], we systematically develop the basic properties of the discrete wave equation on $\mathbb{Z}$ and use this to explain existing results about the wave equation on regular graphs. Furthermore, we explicitly compute the incoming and outgoing translation representations and the scattering operator, in the sense of Lax-Phillips, for regular and biregular trees. Finally, we use the wave equation on biregular graphs to extend a result of Brooks-Lindenstrauss about delocalization of eigenfunctions on regular graphs to the setting of biregular graphs.
Quantum annealing has shown promise for finding solutions to difficult optimization problems, including protein folding. Recently, we used the D-Wave Advantage quantum annealer to explore the folding problem in a coarse-grained lattice model, the HP model, in which amino acids are classified into two broad groups: hydrophobic (H) and polar (P). Using a set of 22 HP sequences with up to 64 amino acids, we demonstrated the fast and consistent identification of the correct HP model ground states using the D-Wave hybrid quantum-classical solver. An equally relevant biophysical challenge, called the protein design problem, is the inverse of the above, where the task is to predict protein sequences that fold to a given structure. Here, we approach the design problem by a two-step procedure, implemented and executed on a D-Wave machine. In the first step, we perform a pure sequence-space search by varying the type of amino acid at each sequence position, and seek sequences which minimize the HP-model energy of the target structure. After mapping this task onto an Ising spin glass representation, we employ a hybrid quantum-classical solver to deliver energy-optimal sequences for structures with 30-64 amino acids, with a 100% success rate. In the second step, we filter the optimized sequences from the first step according to their ability to fold to the intended structure. In addition, we try solving the sequence optimization problem using only the QPU, which confines us to sizes $\le$20, due to exponentially decreasing success rates. To shed light on the pure QPU results, we investigate the effects of control errors caused by an imperfect implementation of the intended Hamiltonian on the QPU, by numerically analyzing the Schrödinger equation. We find that the simulated success rates in the presence of control noise semi-quantitatively reproduce the modest pure QPU results for larger chains.
We study joint eigenfunctions of the spherical Hecke algebra acting on $L^2(Γ_n \backslash G / K)$ where $G = \text{PGL}(3, F)$ with $F$ a non-archimedean local field of arbitrary characteristic, $K = \text{PGL}(3, O)$ with $O$ the ring of integers of $F$, and $(Γ_n)$ is a sequence of torsion-free lattices. We prove a form of equidistribution on average for eigenfunctions whose spectral parameters lie in the tempered spectrum when the associated sequence of quotients of the Bruhat-Tits building Benjamini-Schramm converges to the building itself. This result is a higher rank non-archimedean analogue of existing results for graphs and locally symmetric spaces. A recurring theme in the proof is the reduction of many computations to computing the sum of an exponential function over lattice points in a polytope; such expressions can subsequently be simplified using Brion's formula. Along the way of proving our main result we prove several other results which may be of independent interest including a "degenerate" version of Brion's formula which "interpolates" between the usual Brion's formula and the Ehrhart polynomial, an effective rate of convergence for the distribution of spectral parameters to the Plancherel measure under Benjamini-Schramm convergence, and a classification of relative positions of triples of points in buildings of type $\tilde{A}_2$.
Quantum annealing is a promising approach for obtaining good approximate solutions to difficult optimization problems. Folding a protein sequence into its minimum-energy structure represents such a problem. For testing new algorithms and technologies for this task, the minimal lattice-based HP model is well suited, as it represents a considerable challenge despite its simplicity. The HP model has favorable interactions between adjacent, not directly bound hydrophobic residues. Here, we develop a novel spin representation for lattice protein folding tailored for quantum annealing. With a distributed encoding onto the lattice, it differs from earlier attempts to fold lattice proteins on quantum annealers, which were based upon chain growth techniques. With our encoding, the Hamiltonian by design has the quadratic structure required for calculations on an Ising-type annealer, without having to introduce any auxiliary spin variables. This property greatly facilitates the study of long chains. The approach is robust to changes in the parameters required to constrain the spin system to chain-like configurations, and performs very well in terms of solution quality. The results are evaluated against existing exact results for HP chains with up to $N=30$ beads with 100% hit rate, thereby also outperforming classical simulated annealing. In addition, the method allows us to recover the lowest known energies for $N=48$ and $N=64$ HP chains, with similar hit rates. These results are obtained by the commonly used hybrid quantum-classical approach. For pure quantum annealing, our method successfully folds an $N=14$ HP chain. The calculations were performed on a D-Wave Advantage quantum annealer.
Given a recurrence sequence $H$, with $H_n = c_1 H_{n-1} + \dots + c_t H_{n-t}$ where $c_i \in \mathbb{N}_0$ for all $i$ and $c_1, c_t \geq 1$, the generalized Zeckendorf decomposition (gzd) of $m \in \mathbb{N}_0$ is the unique representation of $m$ using $H$ composed of blocks lexicographically less than $σ= (c_1, \dots, c_t)$. We prove that the gzd of $m$ uses the fewest number of summands among all representations of $m$ using $H$, for all $m$, if and only if $σ$ is weakly decreasing. We develop an algorithm for moving from any representation of $m$ to the gzd, the analysis of which proves that $σ$ weakly decreasing implies summand minimality. We prove that the gzds of numbers of the form $v_0 H_n + \dots + v_\ell H_{n-\ell}$ converge in a suitable sense as $n \to \infty$, furthermore we classify three distinct behaviors for this convergence. We use this result, together with the irreducibility of certain families of polynomials, to exhibit a representation with fewer summands than the gzd if $σ$ is not weakly decreasing.
A bidirectional ballot sequence (BBS) is a finite binary sequence with the property that every prefix and suffix contains strictly more ones than zeros. BBS's were introduced by Zhao, and independently by Bosquet-M{é}lou and Ponty as $(1,1)$-culminating paths. Both sets of authors noted the difficulty in counting these objects, and to date research on bidirectional ballot sequences has been concerned with asymptotics. We introduce a continuous analogue of bidirectional ballot sequences which we call bidirectional gerrymanders, and show that the set of bidirectional gerrymanders form a convex polytope sitting inside the unit cube, which we refer to as the bidirectional ballot polytope. We prove that every $(2n-1)$-dimensional unit cube can be partitioned into $2n-1$ isometric copies of the $(2n-1)$-dimensional bidirectional ballot polytope. Furthermore, we show that the vertices of this polytope are all also vertices of the cube, and that the vertices are in bijection with BBS's. An immediate corollary is a geometric explanation of the result of Zhao and of Bosquet-M{é}lou and Ponty that the number of BBS's of length $n$ is $Θ(2^n/n)$.
A more sums than differences (MSTD) set $A$ is a subset of $\mathbb{Z}$ for which $|A+A| > |A-A|$. Martin and O'Bryant used probabilistic techniques to prove that a non-vanishing proportion of subsets of $\{1, \dots, n\}$ are MSTD as $n \to \infty$. However, to date only a handful of explicit constructions of MSTD sets are known. We study finite collections of disjoint intervals on the real line, $\mathbb{I}$, and explore the MSTD question for such sets, as well as the relation between such sets and MSTD subsets of $\mathbb{Z}$. In particular we show that every finite subset of $\mathbb{Z}$ can be transformed into an element of $\mathbb{I}$ with the same additive behavior. Using tools from discrete geometry, we show that there are no MSTD sets in $\mathbb{I}$ consisting of three or fewer intervals, but there are MSTD sets for four or more intervals. Furthermore, we show how to obtain an infinite parametrized family of MSTD subsets of $\mathbb{Z}$ from a single such set $A$; these sets are parametrized by lattice points satisfying simple congruence relations contained in a polyhedral cone associated to $A$.
A method for estimating nonlinear regression errors and their distributions without performing regression is presented. Assuming continuity of the modeling function the variance is given in terms of conditional probabilities extracted from the data. For N data points the computational demand is N2. Comparing the predicted residual errors with those derived from a linear model assumption provides a signal for nonlinearity. The method is successfully illustrated with data generated by the Ikeda and Lorenz maps augmented with noise. As a by-product the embedding dimensions of these maps are also extracted.
We determine stability and attractor properties of random Boolean genetic network models with canalyzing rules for a variety of architectures. For all power law, exponential, and flat in-degree distributions, we find that the networks are dynamically stable. Furthermore, for architectures with few inputs per node, the dynamics of the networks is close to critical. In addition, the fraction of genes that are active decreases with the number of inputs per node. These results are based upon investigating ensembles of networks using analytical methods. Also, for different in-degree distributions, the numbers of fixed points and cycles are calculated, with results intuitively consistent with stability analysis; fewer inputs per node implies more cycles, and vice versa. There are hints that genetic networks acquire broader degree distributions with evolution, and hence our results indicate that for single cells, the dynamics should become more stable with evolution. However, such an effect is very likely compensated for by multicellular dynamics, because one expects less stability when interactions among cells are included. We verify this by simulations of a simple model for interactions among cells.
The recently measured yeast transcriptional network is analyzed in terms of simplified Boolean network models, with the aim of determining feasible rule structures, given the requirement of stable solutions of the generated Boolean networks. We find that for ensembles of generated models, those with canalyzing Boolean rules are remarkably stable, whereas those with random Boolean rules are only marginally stable. Furthermore, substantial parts of the generated networks are frozen, in the sense that they reach the same state regardless of initial state. Thus, our ensemble approach suggests that the yeast network shows highly ordered dynamics.
A mean field feedback artificial neural network algorithm is developed and explored for the set covering problem. A convenient encoding of the inequality constraints is achieved by means of a multilinear penalty function. An approximate energy minimum is obtained by iterating a set of mean field equations, in combination with annealing. The approach is numerically tested against a set of publicly available test problems with sizes ranging up to 5x10^3 rows and 10^6 columns. When comparing the performance with exact results for sizes where these are available, the approach yields results within a few percent from the optimal solutions. Comparisons with other approximate methods also come out well, in particular given the very low CPU consumption required -- typically a few seconds. Arbitrary problems can be processed using the algorithm via a public domain server.
Background: Designing amino acid sequences that are stable in a given target structure amounts to maximizing a conditional probability. A straightforward approach to accomplish this is a nested Monte Carlo where the conformation space is explored over and over again for different fixed sequences, which requires excessive computational demand. Several approximate attempts to remedy this situation, based on energy minimization for fixed structure or high-$T$ expansions, have been proposed. These methods are fast but often not accurate since folding occurs at low $T$. Results: We develop a multisequence Monte Carlo procedure, where both sequence and conformation space are simultaneously probed with efficient prescriptions for pruning sequence space. The method is explored on hydrophobic/polar models. We first discuss short lattice chains, in order to compare with exact data and with other methods. The method is then successfully applied to lattice chains with up to 50 monomers, and to off-lattice 20-mers. Conclusions: The multisequence Monte Carlo method offers a new approach to sequence design in coarse-grained models. It is much more efficient than previous Monte Carlo methods, and is, as it stands, applicable to a fairly wide range of two-letter models.
A new method for sequence optimization in protein models is presented. The approach, which has inherited its basic philosophy from recent work by Deutsch and Kurosky [Phys. Rev. Lett. 76, 323 (1996)] by maximizing conditional probabilities rather than minimizing energy functions, is based upon a novel and very efficient multisequence Monte Carlo scheme. By construction, the method ensures that the designed sequences represent good folders thermodynamically. A bootstrap procedure for the sequence space search is devised making very large chains feasible. The algorithm is successfully explored on the two-dimensional HP model with chain lengths N=16, 18 and 32.
Folding properties of a two-dimensional toy protein model containing only two amino-acid types, hydrophobic and hydrophilic, respectively, are analyzed. An efficient Monte Carlo procedure is employed to ensure that the ground states are found. The thermodynamic properties are found to be strongly sequence dependent in contrast to the kinetic ones. Hence, criteria for good folders are defined entirely in terms of thermodynamic fluctuations. With these criteria sequence patterns that fold well are isolated. For 300 chains with 20 randomly chosen binary residues approximately 10% meet these criteria. Also, an analysis is performed by means of statistical and artificial neural network methods from which it is concluded that the folding properties can be predicted to a certain degree given the binary numbers characterizing the sequences.
The question of whether proteins originate from random sequences of amino acids is addressed. A statistical analysis is performed in terms of blocked and random walk values formed by binary hydrophobic assignments of the amino acids along the protein chains. Theoretical expectations of these variables from random distributions of hydrophobicities are compared with those obtained from functional proteins. The results, which are based upon proteins in the SWISS-PROT data base, convincingly show that the amino acid sequences in proteins differ from what is expected from random sequences in a statistical significant way. By performing Fourier transforms on the random walks one obtains additional evidence for non-randomness of the distributions. We have also analyzed results from a synthetic model containing only two amino-acid types, hydrophobic and hydrophilic. With reasonable criteria on good folding properties in terms of thermodynamical and kinetic behavior, sequences that fold well are isolated. Performing the same statistical analysis on the sequences that fold well indicates similar deviations from randomness as for the functional proteins. The deviations from randomness can be interpreted as originating from anticorrelations in terms of an Ising spin model for the hydrophobicities. Our results, which differ from previous investigations using other methods, might have impact on how permissive with respect to sequence specificity the protein folding process is -- only sequences with non-random hydrophobicity distributions fold well. Other distributions give rise to energy landscapes with poor folding properties and hence did not survive the evolution.