This work investigates the emergence of oscillations in one of the simplest cellular signaling networks exhibiting oscillations, namely, the dual-site phosphorylation and dephosphorylation network (futile cycle), in which the mechanism for phosphorylation is processive while the one for dephosphorylation is distributive (or vice-versa). The fact that this network yields oscillations was shown recently by Suwanmajo and Krishnan. Our results, which significantly extend their analyses, are as follows. First, in the three-dimensional space of total amounts, the border between systems with a stable versus unstable steady state is a surface defined by the vanishing of a single Hurwitz determinant. Second, this surface consists generically of simple Hopf bifurcations. Next, simulations suggest that when the steady state is unstable, oscillations are the norm. Finally, the emergence of oscillations via a Hopf bifurcation is enabled by the catalytic and association constants of the distributive part of the mechanism: if these rate constants satisfy two inequalities, then the system generically admits a Hopf bifurcation. Our proofs are enabled by the Routh-Hurwitz criterion, a Hopf-bifurcation criterion due to Yang, and a monomial parametrization of steady states.
Our work joins a growing number of works that harness steady-state parametrizations. Such results include criteria for when such parametrizations exist @cite_19 @cite_37 and methods for using them to determine whether a network is multistationary @cite_20 @cite_23 @cite_27 @cite_2 . Going further, steady-state parametrizations can also be used to find a witness to multistationarity or even the precise parameter regions that yield multistationarity @cite_13 @cite_39 . In this work, we use a steady-state parametrization in a novel way: to study oscillations via Hopf bifurcations. (Our approach is similar in spirit to using Clarke's convex parameters together with a Hopf-bifurcation criterion @cite_31 @cite_15 @cite_22 @cite_10 ).
@cite_37 @cite_22 @cite_15 @cite_10 @cite_39 @cite_19 @cite_27 @cite_23 @cite_2 @cite_31 @cite_13 @cite_20
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We study the approximability of multiway partitioning problems, examples of which include Multiway Cut, Node-weighted Multiway Cut, and Hypergraph Multiway Cut. We investigate these problems from the point of view of two possible generalizations: as Min-CSPs, and as Submodular Multiway Partition problems. These two generalizations lead to two natural relaxations that we call respectively the Local Distribution LP, and the Lovasz relaxation. The Local Distribution LP is generally stronger than the Lovasz relaxation, but applicable only to Min-CSP with predicates of constant size. The relaxations coincide in some cases such as Multiway Cut where they are both equivalent to the CKR relaxation. We show that the Lovasz relaxation gives a (2--2 k)-approximation for Submodular Multiway Partition with k terminals, improving a recent 2-approximation [2]. We prove that this factor is optimal in two senses: (1) A (2--2 k -- e)-approximation for Submodular Multiway Partition with k terminals would require exponentially many value queries (in the oracle model), or imply NP = RP (for certain explicit submodular functions). (2) For Hypergraph Multiway Cut and Node-weighted Multiway Cut with k terminals, both special cases of Submodular Multiway Partition, we prove that a (2--2 k -- e)-approximation is NP-hard, assuming the Unique Games Conjecture. Both our hardness results are more general: (1) We show that the notion of symmetry gap, previously used for submodular maximization problems [19, 6], also implies hardness results for submodular minimization problems. (2) Assuming the Unique Games Conjecture, we show that the Local Distribution LP gives an optimal approximation for every Min-CSP that includes the Not-Equal predicate. Finally, we connect the two hardness techniques by proving that the integrality gap of the Local Distribution LP coincides with the symmetry gap of the multilinear relaxation (for a related instance). This shows that the appearance of the same hardness threshold for a Min-CSP and the related submodular minimization problem is not a coincidence. Bifurcation theory is one of the most widely used approaches for analysis of dynamical behaviour of chemical and biochemical reaction networks. Some of the interesting qualitative behaviour that are analyzed are oscillations and bistability (a situation where a system has at least two coexisting stable equilibria). Both phenomena have been identified as central features of many biological and biochemical systems. This paper, using the theory of stoichiometric network analysis (SNA) and notions from algebraic geometry, presents sufficient conditions for a reaction network to display bifurcations associated with these phenomena. The advantage of these conditions is that they impose fewer algebraic conditions on model parameters than conditions associated with standard bifurcation theorems. To derive the new conditions, a coordinate transformation will be made that will guarantee the existence of branches of positive equilibria in the system. This is particularly useful in mathematical biology, where only positive variable values are considered to be meaningful. The first part of the paper will be an extended introduction to SNA and algebraic geometry-related methods which are used in the coordinate transformation and set up of the theorems. In the second part of the paper we will focus on the derivation of bifurcation conditions using SNA and algebraic geometry. Conditions will be derived for three bifurcations: the saddle-node bifurcation, a simple branching point, both linked to bistability, and a simple Hopf bifurcation. The latter is linked to oscillatory behaviour. The conditions derived are sufficient and they extend earlier results from stoichiometric network analysis as can be found in (Aguda and Clarke in J Chem Phys 87:3461-3470, 1987; Clarke and Jiang in J Chem Phys 99:4464-4476, 1993; in J Symb Comput 40:1361-1382, 2005). In these papers some necessary conditions for two of these bifurcations were given. A set of examples will illustrate that algebraic conditions arising from given sufficient bifurcation conditions are not more difficult to interpret nor harder to calculate than those arising from necessary bifurcation conditions. Hence an increasing amount of information is gained at no extra computational cost. The theory can also be used in a second step for a systematic bifurcation analysis of larger reaction networks. Abstract We consider boundary value problems for semilinear hyperbolic systems of the type ∂ t u j + a j ( x , λ ) ∂ x u j + b j ( x , λ , u ) = 0 , x ∈ ( 0 , 1 ) , j = 1 , … , n , with smooth coefficient functions a j and b j such that b j ( x , λ , 0 ) = 0 for all x ∈ [ 0 , 1 ] , λ ∈ R , and j = 1 , … , n . We state conditions for Hopf bifurcation, i.e., for existence, local uniqueness (up to phase shifts), smoothness and smooth dependence on λ of time-periodic solutions bifurcating from the zero stationary solution. Furthermore, we derive a formula which determines the bifurcation direction. The proof is done by means of a Liapunov–Schmidt reduction procedure. For this purpose, Fredholm properties of the linearized system and implicit function theorem techniques are used. Mulmuley [Mul12a] recently gave an explicit version of Noether’s Normalization lemma for ring of invariants of matrices under simultaneous conjugation, under the conjecture that there are deterministic black-box algorithms for polynomial identity testing (PIT). He argued that this gives evidence that constructing such algorithms for PIT is beyond current techniques. In this work, we show this is not the case. That is, we improve Mulmuley’s reduction and correspondingly weaken the conjecture regarding PIT needed to give explicit Noether Normalization. We then observe that the weaker conjecture has recently been nearly settled by the authors ([FS12]), who gave quasipolynomial size hitting sets for the class of read-once oblivious algebraic branching programs (ROABPs). This gives the desired explicit Noether Normalization unconditionally, up to quasipolynomial factors. As a consequence of our proof we give a deterministic parallel polynomial-time algorithm for deciding if two matrix tuples have intersecting orbit closures, under simultaneous conjugation. We also study the strength of conjectures that Mulmuley requires to obtain similar results as ours. We prove that his conjectures are stronger, in the sense that the computational model he needs PIT algorithms for is equivalent to the well-known algebraic branching program (ABP) model, which is provably stronger than the ROABP model. Finally, we consider the depth-3 diagonal circuit model as defined by Saxena [Sax08], as PIT algorithms for this model also have implications in Mulmuley’s work. Previous work (such as [ASS12] and [FS12]) have given quasipolynomial size hitting sets for this model. In this work, we give a much simpler construction of such hitting sets, using techniques of Shpilka and Volkovich [SV09]. We introduce a novel embedding method for knowledge base completion task. Our approach advances state-of-the-art (SOTA) by employing a convolutional neural network (CNN) for the task which can capture global relationships and transitional characteristics. We represent each triple (head entity, relation, tail entity) as a 3-column matrix which is the input for the convolution layer. Different filters having a same shape of 1x3 are operated over the input matrix to produce different feature maps which are then concatenated into a single feature vector. This vector is used to return a score for the triple via a dot product. The returned score is used to predict whether the triple is valid or not. Experiments show that ConvKB achieves better link prediction results than previous SOTA models on two current benchmark datasets WN18RR and FB15k-237. We show an exponential separation between two well-studied models of algebraic computation, namely read-once oblivious algebraic branching programs (ROABPs) and multilinear depth three circuits. In particular we show the following: 1. There exists an explicit n-variate polynomial computable by linear sized multilinear depth three circuits (with only two product gates) such that every ROABP computing it requires 2^ Omega(n) size. 2. Any multilinear depth three circuit computing IMM_ n,d (the iterated matrix multiplication polynomial formed by multiplying d, n * n symbolic matrices) has n^ Omega(d) size. IMM_ n,d can be easily computed by a poly(n,d) sized ROABP. 3. Further, the proof of 2 yields an exponential separation between multilinear depth four and multilinear depth three circuits: There is an explicit n-variate, degree d polynomial computable by a poly(n,d) sized multilinear depth four circuit such that any multilinear depth three circuit computing it has size n^ Omega(d) . This improves upon the quasi-polynomial separation result by Raz and Yehudayoff [2009] between these two models. The hard polynomial in 1 is constructed using a novel application of expander graphs in conjunction with the evaluation dimension measure used previously in Nisan [1991], Raz [2006,2009], Raz and Yehudayoff [2009], and Forbes and Shpilka [2013], while 2 is proved via a new adaptation of the dimension of the partial derivatives measure used by Nisan and Wigderson [1997]. Our lower bounds hold over any field. We consider homogeneous factor models on uniformly sparse graph sequences converging locally to a (unimodular) random tree T, and study the existence of the free energy density ϕ, the limit of the log-partition function divided by the number of vertices n as n tends to infinity. We provide a new interpolation scheme and use it to prove existence of, and to explicitly compute, the quantity ϕ subject to uniqueness of a relevant Gibbs measure for the factor model on T. By way of example we compute ϕ for the independent set (or hard-core) model at low fugacity, for the ferromagnetic Ising model at all parameter values, and for the ferromagnetic Potts model with both weak enough and strong enough interactions. Even beyond uniqueness regimes our interpolation provides useful explicit bounds on ϕ. In the regimes in which we establish existence of the limit, we show that it coincides with the Bethe free energy functional evaluated at a suitable fixed point of the belief propagation (Bethe) recursions on T. In the special case that T has a Galton–Watson law, this formula coincides with the nonrigorous “Bethe prediction” obtained by statistical physicists using the “replica” or “cavity” methods. Thus our work is a rigorous generalization of these heuristic calculations to the broader class of sparse graph sequences converging locally to trees. We also provide a variational characterization for the Bethe prediction in this general setting, which is of independent interest. The objective of this paper is to study the performance of a multicell vector perturbation (MVP) precoding technique under practical situations in a network multiple-input multiple-output (MIMO) scheme employing joint transmission. The conventional perturbation strategy that minimizes the total power is considered, and the power at each base station (BS) is properly scaled to enforce per-BS power constraints. In our scenario, we consider multiple-antenna users and use block diagonalization (BD) as the linear front-end precoder for VP. The sum rate for the MVP in the case of uniformly distributed input and an asymptotic upper bound on the sum rate at high signal-to-noise ratios (SNRs) are derived. In addition, using the asymptotic upper bound on the individual user rates, we propose a proportionally fair (PF) user scheduling algorithm of lower complexity and better performance compared with the benchmark fair semiorthogonal user selection (SUS) algorithm. As opposed to the PF-SUS, the proposed PF scheduling algorithm requires no predefined correlation threshold. Furthermore, we study the impact of backhaul delay on the performance of both VP and BD by deriving bounds on the sum rate. The numerical results show that MVP in the case of perfect channel state information (CSI) outperforms multicell BD. In the presence of a backhaul delay, the performance of MVP significantly degrades, but the upper bound on the sum rate for MVP is still higher than for multicell BD. This paper studies the mutual information improvement attained by rotated multidimensional (multi-D) constellations via a unitary precoder G in Rayleigh fading. At first, based on the symmetric cut-off rate of the N-D signal space, we develop a design criterion with regard to the precoder G. It is then demonstrated that the use of rotated constellations in only a reasonably low dimensional signal space can significantly increase the mutual information in high-rate regimes. Based on parameterizations of unitary matrices, we then construct good unitary precoder G in 4-D signal space using a simple optimization problem, which involves only four real variables and it is applicable to any modulation scheme. To further illustrate the potential of multi-D constellation and to show the practical use of mutual information improvement, we propose a simple yet powerful bit-interleaved coded modulation (BICM) scheme in which a (multi-D) mapping technique employed in a multi-D rotated constellation is concatenated with a short-memory high-rate convolutional code. By using extrinsic information transfer (EXIT) charts, it is shown that the proposed technique provides an exceptionally good error performance. In particular, both EXIT chart analysis and simulation results indicate that a turbo pinch-off and a bit error rate around 10-6 happen at a signal-to-noise ratio that is well below the coded modulation and BICM capacities using traditional signal sets. For example, with code rates ranging from 2 3 to 7 8, the proposed system can operate 0.82 dB-2.93 dB lower than the BICM capacity with QPSK and Gray labeling. The mutual information gain offered by rotated constellations can be therefore utilized to design simple yet near Shannon limit systems in the high-rate regions. This paper proposes a simple method for estimating dense and accurate optical flow field. It revitalizes an early idea of piecewise parametric flow model. A key innovation is that, we fit a flow field piecewise to a variety of parametric models, where the domain of each piece (i.e., each piece's shape, position and size) is determined adaptively, while at the same time maintaining a global inter-piece flow continuity constraint. We achieve this by a multi-model fitting scheme via energy minimization. Our energy takes into account both the piecewise constant model assumption and the flow field continuity constraint, enabling the proposed method to effectively handle both homogeneous motions and complex motions. The experiments on three public optical flow benchmarks (KITTI, MPI Sintel, and Middlebury) show the superiority of our method compared with the state of the art: it achieves top-tier performances on all the three benchmarks. We investigate random interlacements on ℤd, d ≥ 3. This model, recently introduced in [8], corresponds to a Poisson cloud on the space of doubly infinite trajectories modulo time shift tending to infinity at positive and negative infinite times. A nonnegative parameter u measures how many trajectories enter the picture. Our main interest lies in the percolative properties of the vacant set left by random interlacements at level u. We show that for all d ≥ 3 the vacant set at level u percolates when u is small. This solves an open problem of [8], where this fact has only been established when d ≥ 7. It also completes the proof of the nondegeneracy in all dimensions d ≥ 3 of the critical parameter u* of [8]. © 2008 Wiley Periodicals, Inc. Most methods proposed to uncover communities in complex networks rely on their structural properties. Here we introduce the stability of a network partition, a measure of its quality defined in terms of the statistical properties of a dynamical process taking place on the graph. The time-scale of the process acts as an intrinsic parameter that uncovers community structures at different resolutions. The stability extends and unifies standard notions for community detection: modularity and spectral partitioning can be seen as limiting cases of our dynamic measure. Similarly, recently proposed multi-resolution methods correspond to linearisations of the stability at short times. The connection between community detection and Laplacian dynamics enables us to establish dynamically motivated stability measures linked to distinct null models. We apply our method to find multi-scale partitions for different networks and show that the stability can be computed efficiently for large networks with extended versions of current algorithms. While sensor networks are going to be deployed in diverse application specific contexts, one unifying view is to treat them essentially as distributed databases. The simplest mechanism to obtain information from this kind of a database is to flood queries for named data within the network and obtain the relevant responses from sources. However, if the queries are (a) complex, (b) one-shot, and (c) for replicated data, this simple approach can be highly inefficient. In the context of energy-starved sensor networks, alternative strategies need to be examined for such queries. We propose a novel and efficient mechanism for obtaining information in sensor networks which we refer to as ACtive QUery forwarding In sensoR nEtworks (ACQUIRE). The basic principle behind ACQUIRE is to consider the query as an active entity that is forwarded through the network (either randomly or in some directed manner) in search of the solution. ACQUIRE also incorporates a look-ahead parameter d in the following manner: intermediate nodes that handle the active query use information from all nodes within d hops in order to partially resolve the query. When the active query is fully resolved, a completed response is sent directly back to the querying node. We take a mathematical modelling approach in this paper to calculate the energy costs associated with ACQUIRE. The models permit us to characterize analytically the impact of critical parameters, and compare the performance of ACQUIRE with respect to other schemes such as flooding-based querying (FBQ) and expanding ring search (ERS), in terms of energy usage, response latency and storage requirements. We show that with optimal parameter settings, depending on the update frequency, ACQUIRE obtains order of magnitude reduction over FBQ and potentially over 60–75 reduction over ERS (in highly dynamic environments and high query rates) in consumed energy. We show that these energy savings are provided in trade for increased response latency. The mathematical analysis is validated through extensive simulations. � 2003 Elsevier B.V. All rights reserved. This paper describes a probabilistic multiple-hypothesis framework for tracking highly articulated objects. In this framework, the probability density of the tracker state is represented as a set of modes with piecewise Gaussians characterizing the neighborhood around these modes. The temporal evolution of the probability density is achieved through sampling from the prior distribution, followed by local optimization of the sample positions to obtain updated modes. This method of generating hypotheses from state-space search does not require the use of discrete features unlike classical multiple-hypothesis tracking. The parametric form of the model is suited for high dimensional state-spaces which cannot be efficiently modeled using non-parametric approaches. Results are shown for tracking Fred Astaire in a movie dance sequence. A modification of the classical technique of Prony for fitting sums of exponential functions to data is considered. The method maximizes the likelihood for the problem (unlike the usual implementation of Prony’s method, which is not even consistent for transient signals), proves to be remarkably effective in practice, and is supported by an asymptotic stability result. Novel features include a discussion of the problem parametrization and its implications for consistency. The asymptotic convergence proofs are made possible by an expression for the algorithm in terms of circulant divided difference operators. Institute for Theoretical Physics, University of Bremen, Otto-Hahn-Allee, D-28359 Bremen, Germany(Dated: February 3, 2008)Starting from a general ansatz, we show how community detection can be interpreted as ﬁnding theground state of an inﬁnite range spin glass. Our approach applies to weighted and directed networksalike. It contains the at hoc introduced quality function from [1] and the modularity Q as deﬁnedby Newman and Girvan [2] as special cases. The community structure of the network is interpretedas the spin conﬁguration that minimizes the energy of the spin glass with the spin states being thecommunity indices. We elucidate the properties of the ground state conﬁguration to give a concisedeﬁnition of communities as cohesive subgroups in networks that is adaptive to the speciﬁc class ofnetwork under study. Further we show, how hierarchies and overlap in the community structure canbe detected. Computationally eﬀective local update rules for optimization procedures to ﬁnd theground state are given. We show how the ansatz may be used to discover the community around agiven node without detecting all communities in the full network and we give benchmarks for theperformance of this extension. Finally, we give expectation values for the modularity of randomgraphs, which can be used in the assessment of statistical signiﬁcance of community structure. Abstract We present a novel convolutional neural network (CNN) based pipeline which can effectively fuse multi-level information extracted from different intermediate layers generating hybrid convolutional features (HCF) for edge detection. Different from previous methods, the proposed method fuses multi-level information in a feature-map based manner. The produced hybrid convolutional features can be used to perform high-quality edge detection. The edge detector is also computationally efficient, because it detects edges in an image-to-image way without any post-processing. We evaluate the proposed method on three widely used datasets for edge detection including BSDS500, NYUD and Multicue, and also test the method on Pascal VOC’12 dataset for object contour detection. The results show that HCF achieves an improvement in performance over the state-of-the-art methods on all four datasets. On BSDS500 dataset, the efficient version of the proposed approach achieves ODS F-score of 0.804 with a speed of 22 fps and the high-accuracy version achieves ODS F-score of 0.814 with 11 fps . In this paper, we present a new approach to automatically verify multi-threaded programs which are executed by an unbounded number of threads running in parallel. The starting point for our work is the problem of how we can leverage existing automated verification technology for sequential programs (abstract interpretation, Craig interpolation, constraint solving, etc.) for multi-threaded programs. Suppose that we are given a correctness proof for a trace of a program (or for some other program fragment). We observe that the proof can always be decomposed into a finite set of Hoare triples, and we ask what can be proved from the finite set of Hoare triples using only simple combinatorial inference rules (without access to a theorem prover and without the possibility to infer genuinely new Hoare triples)? We introduce a proof system where one proves the correctness of a multi-threaded program by showing that for each trace of the program, there exists a correctness proof in the space of proofs that are derivable from a finite set of axioms using simple combinatorial inference rules. This proof system is complete with respect to the classical proof method of establishing an inductive invariant (which uses thread quantification and control predicates). Moreover, it is possible to algorithmically check whether a given set of axioms is sufficient to prove the correctness of a multi-threaded program, using ideas from well-structured transition systems. This paper examines techniques for finding falsifying trajectories of hybrid systems using an approach that we call trajectory splicing. Many formal verification techniques for hybrid systems, including flowpipe construction, can identify plausible abstract counterexamples for property violations. However, there is often a gap between the reported abstract counterexamples and the concrete system trajectories. Our approach starts with a candidate sequence of disconnected trajectory segments, each segment lying inside a discrete mode. However, such disconnected segments do not form concrete violations due to the gaps that exist between the ending state of one segment and the starting state of the subsequent segment. Therefore, trajectory splicing uses local optimization to minimize the gap between these segments, effectively splicing them together to form a concrete trajectory. We demonstrate the use of our approach for falsifying safety properties of hybrid systems using standard optimization techniques. As such, our approach is not restricted to linear systems. We compare our approach with other falsification approaches including uniform random sampling and a robustness guided falsification approach used in the tool S-Taliro. Our preliminary evaluation clearly shows the potential of our approach to search for candidate trajectory segments and use them to find concrete property violations. We review the method of Parallel Factor Analysis, which simultaneously fits multiple two-way arrays or ‘slices’ of a three-way array in terms of a common set of factors with differing relative weights in each ‘slice’. Mathematically, it is a straightforward generalization of the bilinear model of factor (or component) analysis (xij = ΣRr = 1airbjr) to a trilinear model (xijk = ΣRr = 1airbjrckr). Despite this simplicity, it has an important property not possessed by the two-way model: if the latent factors show adequately distinct patterns of three-way variation, the model is fully identified; the orientation of factors is uniquely determined by minimizing residual error, eliminating the need for a separate ‘rotation’ phase of analysis. The model can be used several ways. It can be directly fit to a three-way array of observations with (possibly incomplete) factorial structure, or it can be indirectly fit to the original observations by fitting a set of covariance matrices computed from the observations, with each matrix corresponding to a two-way subset of the data. Even more generally, one can simultaneously analyze covariance matrices computed from different samples, perhaps corresponding to different treatment groups, different kinds of cases, data from different studies, etc. To demonstrate the method we analyze data from an experiment on right vs. left cerebral hemispheric control of the hands during various tasks. The factors found appear to correspond to the causal influences manipulated in the experiment, revealing their patterns of influence in all three ways of the data. Several generalizations of the parallel factor analysis model are currently under development, including ones that combine parallel factors with Tucker-like factor ‘interactions’. Of key importance is the need to increase the method's robustness against nonstationary factor structures and qualitative (nonproportional) factor change.