Monomial Testing and Applications

In this paper, we devise two algorithms for the problem of testing $q$-monomials of degree $k$ in any multivariate polynomial represented by a circuit, regardless of the primality of $q$. One is an $O^*(2^k)$ time randomized algorithm. The other is an $O^*(12.8^k)$ time deterministic algorithm for the same $q$-monomial testing problem but requiring the polynomials to be represented by tree-like circuits. Several applications of $q$-monomial testing are also given, including a deterministic $O^*(12.8^{mk})$ upper bound for the $m$-set $k$-packing problem.Comment: 17 pages, 4 figures, submitted FAW-AAIM 2013. arXiv admin note: substantial text overlap with arXiv:1302.5898; and text overlap with arXiv:1007.2675, arXiv:1007.2678, arXiv:1007.2673 by other author

Distributing the burdens of climate change

Global climate change raises many questions for environmental political theorists. This article focuses on the question of identifying the agents that should bear the financial burden of preventing dangerous climate change. Identifying in a fair way the agents that should take the lead in climate mitigation and adaptation, as well as the precise burdens that these parties must bear, will be a key aspect of the next generation of global climate policies. After a critical review of a number of rival approaches to burden sharing, the paper argues that only a principled and philosophically robust reconciliation of three approaches to burden sharing (‘contribution to problem’, ‘ability to pay’ and ‘beneficiary pays’) can generate a satisfactory mix of theoretical coherence and practical application

Evaluation of the Multiplane Method for Efficient Simulations of Reaction Networks

Reaction networks in the bulk and on surfaces are widespread in physical, chemical and biological systems. In macroscopic systems, which include large populations of reactive species, stochastic fluctuations are negligible and the reaction rates can be evaluated using rate equations. However, many physical systems are partitioned into microscopic domains, where the number of molecules in each domain is small and fluctuations are strong. Under these conditions, the simulation of reaction networks requires stochastic methods such as direct integration of the master equation. However, direct integration of the master equation is infeasible for complex networks, because the number of equations proliferates as the number of reactive species increases. Recently, the multiplane method, which provides a dramatic reduction in the number of equations, was introduced [A. Lipshtat and O. Biham, Phys. Rev. Lett. 93, 170601 (2004)]. The reduction is achieved by breaking the network into a set of maximal fully connected sub-networks (maximal cliques). Lower-dimensional master equations are constructed for the marginal probability distributions associated with the cliques, with suitable couplings between them. In this paper we test the multiplane method and examine its applicability. We show that the method is accurate in the limit of small domains, where fluctuations are strong. It thus provides an efficient framework for the stochastic simulation of complex reaction networks with strong fluctuations, for which rate equations fail and direct integration of the master equation is infeasible. The method also applies in the case of large domains, where it converges to the rate equation results

Approximating Multilinear Monomial Coefficients and Maximum Multilinear Monomials in Multivariate Polynomials

This paper is our third step towards developing a theory of testing monomials in multivariate polynomials and concentrates on two problems: (1) How to compute the coefficients of multilinear monomials; and (2) how to find a maximum multilinear monomial when the input is a $\Pi\Sigma\Pi$ polynomial. We first prove that the first problem is \#P-hard and then devise a $O^*(3^ns(n))$ upper bound for this problem for any polynomial represented by an arithmetic circuit of size $s(n)$. Later, this upper bound is improved to $O^*(2^n)$ for $\Pi\Sigma\Pi$ polynomials. We then design fully polynomial-time randomized approximation schemes for this problem for $\Pi\Sigma$ polynomials. On the negative side, we prove that, even for $\Pi\Sigma\Pi$ polynomials with terms of degree $\le 2$, the first problem cannot be approximated at all for any approximation factor $\ge 1$, nor {\em "weakly approximated"} in a much relaxed setting, unless P=NP. For the second problem, we first give a polynomial time $\lambda$-approximation algorithm for $\Pi\Sigma\Pi$ polynomials with terms of degrees no more a constant $\lambda \ge 2$. On the inapproximability side, we give a $n^{(1-\epsilon)/2}$ lower bound, for any $\epsilon >0,$ on the approximation factor for $\Pi\Sigma\Pi$ polynomials. When terms in these polynomials are constrained to degrees $\le 2$, we prove a $1.0476$ lower bound, assuming $P\not=NP$; and a higher $1.0604$ lower bound, assuming the Unique Games Conjecture

Preferential effect of isoflurane on top-down vs. bottom-up pathways in sensory cortex

The mechanism of loss of consciousness (LOC) under anesthesia is unknown. Because consciousness depends on activity in the cortico-thalamic network, anesthetic actions on this network are likely critical for LOC. Competing theories stress the importance of anesthetic actions on bottom-up “core” thalamo-cortical (TC) vs. top-down cortico-cortical (CC) and matrix TC connections. We tested these models using laminar recordings in rat auditory cortex in vivo and murine brain slices. We selectively activated bottom-up vs. top-down afferent pathways using sensory stimuli in vivo and electrical stimulation in brain slices, and compared effects of isoflurane on responses evoked via the two pathways. Auditory stimuli in vivo and core TC afferent stimulation in brain slices evoked short latency current sinks in middle layers, consistent with activation of core TC afferents. By contrast, visual stimuli in vivo and stimulation of CC and matrix TC afferents in brain slices evoked responses mainly in superficial and deep layers, consistent with projection patterns of top-down afferents that carry visual information to auditory cortex. Responses to auditory stimuli in vivo and core TC afferents in brain slices were significantly less affected by isoflurane compared to responses triggered by visual stimuli in vivo and CC/matrix TC afferents in slices. At a just-hypnotic dose in vivo, auditory responses were enhanced by isoflurane, whereas visual responses were dramatically reduced. At a comparable concentration in slices, isoflurane suppressed both core TC and CC/matrix TC responses, but the effect on the latter responses was far greater than on core TC responses, indicating that at least part of the differential effects observed in vivo were due to local actions of isoflurane in auditory cortex. These data support a model in which disruption of top-down connectivity contributes to anesthesia-induced LOC, and have implications for understanding the neural basis of consciousness

Effects of liraglutide versus placebo on cardiovascular events in patients with type 2 diabetes mellitus and chronic kidney disease

BACKGROUND: LEADER trial (Liraglutide Effect and Action in Diabetes: Evaluation of CV Outcome Results) results demonstrated cardiovascular benefits for patients with type 2 diabetes mellitus at high cardiovascular risk on standard of care randomized to liraglutide versus placebo. The effect of glucagon-like peptide-1 receptor agonist liraglutide on cardiovascular events and all-cause mortality in patients with type 2 diabetes mellitus and chronic kidney disease is unknown. Liraglutide's treatment effects in patients with and without kidney disease were analyzed post hoc. METHODS: Patients were randomized (1:1) to liraglutide or placebo, both in addition to standard of care. These analyses assessed outcomes stratified by baseline estimated glomerular filtration rate (eGFR; <60 versus ≥60 mL/min/1.73 m2) and baseline albuminuria. The primary outcome (composite of cardiovascular death, nonfatal myocardial infarction, or nonfatal stroke) and secondary outcomes, including all-cause mortality and individual components of the primary composite outcome, were analyzed using Cox regression. RESULTS: Overall, 2158 and 7182 patients had baseline eGFR <60 or ≥60 mL/min/1.73 m2, respectively. In patients with eGFR <60 mL/min/1.73 m2, risk reduction for the primary composite cardiovascular outcome with liraglutide was greater (hazard ratio [HR], 0.69; 95% CI, 0.57-0.85) versus those with eGFR ≥60 mL/min/1.73 m2 (HR, 0.94; 95% CI, 0.83-1.07; interaction P=0.01). There was no consistent effect modification with liraglutide across finer eGFR subgroups (interaction P=0.13) and when analyzing eGFR as a continuous variable (interaction P=0.61). Risk reductions in those with eGFR <60 versus ≥60 mL/min/1.73 m2 were as follows: for nonfatal myocardial infarction, HR, 0.74; 95% CI, 0.55-0.99 versus HR, 0.93; 95% CI, 0.77-1.13; for nonfatal stroke, HR, 0.51; 95% CI, 0.33-0.80 versus HR, 1.07; 95% CI, 0.84-1.37; for cardiovascular death, HR, 0.67; 95% CI, 0.50-0.90 versus HR, 0.84; 95% CI, 0.67-1.05; for all-cause mortality, HR, 0.74; 95% CI, 0.60-0.92 versus HR, 0.90; 95% CI, 0.75-1.07. Risk reduction for the primary composite cardiovascular outcome was not different for those with versus without baseline albuminuria (HR, 0.83; 95% CI, 0.71-0.97; and HR, 0.92; 95% CI, 0.79-1.07, respectively; interaction P=0.36). CONCLUSIONS: Liraglutide added to standard of care reduced the risk for major cardiovascular events and all-cause mortality in patients with type 2 diabetes mellitus and chronic kidney disease. These results appear to apply across the chronic kidney disease spectrum enrolled. CLINICAL TRIAL REGISTRATION: URL: https://www.clinicaltrials.gov/ . Unique identifier: NCT01179048

Life at high Deborah number

In many biological systems, microorganisms swim through complex polymeric fluids, and usually deform the medium at a rate faster than the inverse fluid relaxation time. We address the basic properties of such life at high Deborah number analytically by considering the small-amplitude swimming of a body in an arbitrary complex fluid. Using asymptotic analysis and differential geometry, we show that for a given swimming gait, the time-averaged leading-order swimming kinematics of the body can be expressed as an integral equation on the solution to a series of simpler Newtonian problems. We then use our results to demonstrate that Purcell's scallop theorem, which states that time-reversible body motion cannot be used for locomotion in a Newtonian fluid, breaks down in polymeric fluid environments