Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length

We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class $\operatorname{PTIME}$ of languages computable in polynomial time in terms of differential equations with polynomial right-hand side. This result gives a purely continuous (time and space) elegant and simple characterization of $\operatorname{PTIME}$. This is the first time such classes are characterized using only ordinary differential equations. Our characterization extends to functions computable in polynomial time over the reals in the sense of computable analysis. This extends to deterministic complexity classes above polynomial time. This may provide a new perspective on classical complexity, by giving a way to define complexity classes, like $\operatorname{PTIME}$, in a very simple way, without any reference to a notion of (discrete) machine. This may also provide ways to state classical questions about computational complexity via ordinary differential equations, i.e.~by using the framework of analysis

On the complexity of solving polynomial initial value problems

International audienceIn this paper we prove that computing the solution of an initial-value problem $\dot{y}=p(y)$ with initial condition $y(t_0)=y_0\in\R^d$ at time $t_0+T$ with precision $e^{-\mu}$ where $p$ is a vector of polynomials can be done in time polynomial in the value of $T$, $\mu$ and Y=\sup_{t_0\leqslant u\leqslant T}\infnorm{y(u)}. Contrary to existing results, our algorithm works for any vector of polynomials $p$ over any bounded or unbounded domain and has a guaranteed complexity and precision. In particular we do not assume $p$ to be fixed, nor the solution to lie in a compact domain, nor we assume that $p$ has a Lipschitz constant

On the functions generated by the general purpose analog computer

PreprintWe consider the General Purpose Analog Computer (GPAC), introduced by Claude Shannon in 1941 as a mathematical model of Differential Analysers, that is to say as a model of continuous-time analog (mechanical, and later one electronic) machines of that time. The GPAC generates as output univariate functions (i.e. functions f:R→R). In this paper we extend this model by: (i) allowing multivariate functions (i.e. functions f:Rn→Rm); (ii) introducing a notion of amount of resources (space) needed to generate a function, which allows the stratification of GPAC generable functions into proper subclasses. We also prove that a wide class of (continuous and discontinuous) functions can be uniformly approximated over their full domain. We prove a few stability properties of this model, mostly stability by arithmetic operations, composition and ODE solving, taking into account the amount of resources needed to perform each operation. We establish that generable functions are always analytic but that they can nonetheless (uniformly) approximate a wide range of nonanalytic functions. Our model and results extend some of the results from [19] to the multidimensional case, allow one to define classes of functions generated by GPACs which take into account bounded resources, and also strengthen the approximation result from [19] over a compact domain to a uniform approximation result over unbounded domains.info:eu-repo/semantics/acceptedVersio

Polynomial time corresponds to solutions of polynomial ordinary differential equations of polynomial length

The outcomes of this article are twofold. Implicit complexity. We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class P of languages computable in polynomial time in terms of differential equations with polynomial right-hand side. This result gives a purely continuous elegant and simple characterization of P. We believe it is the first time complexity classes are characterized using only ordinary differential equations. Our characterization extends to functions computable in polynomial time over the reals in the sense of Computable Analysis. Our results may provide a new perspective on classical complexity, by giving a way to define complexity classes, like P, in a very simple way, without any reference to a notion of (discrete) machine. This may also provide ways to state classical questions about computational complexity via ordinary differential equations. Continuous-Time Models of Computation. Our results can also be interpreted in terms of analog computers or analog models of computation: As a side effect, we get that the 1941 General Purpose Analog Computer (GPAC) of Claude Shannon is provably equivalent to Turing machines both in terms of computability and complexity, a fact that has never been established before. This result provides arguments in favour of a generalised form of the Church-Turing Hypothesis, which states that any physically realistic (macroscopic) computer is equivalent to Turing machines both in terms of computability and complexity

Interaction between galectin-3 and cystinosin uncovers a pathogenic role of inflammation in kidney involvement of cystinosis.

Inflammation is involved in the pathogenesis of many disorders. However, the underlying mechanisms are often unknown. Here, we test whether cystinosin, the protein involved in cystinosis, is a critical regulator of galectin-3, a member of the β-galactosidase binding protein family, during inflammation. Cystinosis is a lysosomal storage disorder and, despite ubiquitous expression of cystinosin, the kidney is the primary organ impacted by the disease. Cystinosin was found to enhance lysosomal localization and degradation of galectin-3. In Ctns-/- mice, a mouse model of cystinosis, galectin-3 is overexpressed in the kidney. The absence of galectin-3 in cystinotic mice ameliorates pathologic renal function and structure and decreases macrophage/monocyte infiltration in the kidney of the Ctns-/-Gal3-/- mice compared to Ctns-/- mice. These data strongly suggest that galectin-3 mediates inflammation involved in kidney disease progression in cystinosis. Furthermore, galectin-3 was found to interact with the pro-inflammatory cytokine Monocyte Chemoattractant Protein-1, which stimulates the recruitment of monocytes/macrophages, and proved to be significantly increased in the serum of Ctns-/- mice and also patients with cystinosis. Thus, our findings highlight a new role for cystinosin and galectin-3 interaction in inflammation and provide an additional mechanistic explanation for the kidney disease of cystinosis. This may lead to the identification of new drug targets to delay cystinosis progression

Computability of ordinary differential equations

In this paper we provide a brief review of several results about the computability of initial-value problems (IVPs) defined with ordinary differential equations (ODEs). We will consider a variety of settings and analyze how the computability of the IVP will be affected. Computational complexity results will also be presented, as well as computable versions of some classical theorems about the asymptotic behavior of ODEs.info:eu-repo/semantics/publishedVersio

In Situ Dividing and Phagocytosing Retinal Microglia Express Nestin, Vimentin, and NG2 In Vivo

BACKGROUND: Following injury, microglia become activated with subsets expressing nestin as well as other neural markers. Moreover, cerebral microglia can give rise to neurons in vitro. In a previous study, we analysed the proliferation potential and nestin re-expression of retinal macroglial cells such as astrocytes and Müller cells after optic nerve (ON) lesion. However, we were unable to identify the majority of proliferative nestin(+) cells. Thus, the present study evaluates expression of nestin and other neural markers in quiescent and proliferating microglia in naïve retina and following ON transection in adult rats in vivo. METHODOLOGY/PRINCIPAL FINDINGS: For analysis of cell proliferation and cells fates, rats received BrdU injections. Microglia in retinal sections or isolated cells were characterized using immunofluorescence labeling with markers for microglia (e.g., Iba1, CD11b), cell proliferation, and neural cells (e.g., nestin, vimentin, NG2, GFAP, Doublecortin etc.). Cellular analyses were performed using confocal laser scanning microscopy. In the naïve adult rat retina, about 60% of resting ramified microglia expressed nestin. After ON transection, numbers of nestin(+) microglia peaked to a maximum at 7 days, primarily due to in situ cell proliferation of exclusively nestin(+) microglia. After 8 weeks, microglia numbers re-attained control levels, but 20% were still BrdU(+) and nestin(+), although no further local cell proliferation occurred. In addition, nestin(+) microglia co-expressed vimentin and NG2, but not GFAP or neuronal markers. Fourteen days after injury and following retrograde labeling of retinal ganglion cells (RGCs) with Fluorogold (FG), nestin(+)NG2(+) microglia were positive for the dye indicating an active involvement of a proliferating cell population in phagocytosing apoptotic retinal neurons. CONCLUSIONS/SIGNIFICANCE: The current study provides evidence that in adult rat retina, a specific resident population of microglia expresses proteins of immature neural cells that are involved in injury-induced cell proliferation and phagocytosis while transdifferentiation was not observed