Weak Kernels Research Articles

Abelian envelopes of exact categories and highest weight categories

We define admissible and weakly admissible subcategories in exact categories and prove that the former induce semi-orthogonal decompositions on the derived categories. We develop the theory of thin exact categories, an exact-category analogue of triangulated categories generated by exceptional sequences. The right and left abelian envelopes of exact categories are introduced, an example being the category of coherent sheaves on a scheme as the right envelope of the category of vector bundles. The existence of right (left) abelian envelopes is proven for exact categories with projectively (injectively) generating subcategories with weak (co)kernels. We show that highest weight categories are precisely the right/left envelopes of thin categories. Ringel duality on highest weight categories is interpreted as a duality between the right and left abelian envelopes of a thin exact category. A duality for thin exact categories compatible with Ringel duality is introduced by means of derived categories and Serre functor on them.

Hopf bifurcation mechanism analysis in an improved cortex-basal ganglia network with distributed delays: An application to Parkinson’s disease

In this paper, we introduce general distributed delays (in pathways between different neurons) and self-feedback distributed delays (in self-feedback pathways) into an improved cortex-basal ganglia (BG) network to study possible Hopf bifurcation mechanisms for Parkinson’s oscillations. Hopf bifurcation critical conditions for discrete delays are obtained, which compare well to numerical simulations. General distributed delays inhibit oscillations in BG, but self-feedback distributed delays promote oscillations. The different effects of strong and weak kernels on Hopf bifurcations mainly depend on self-feedback distributed delays, strong kernels are more conducive to the generation of oscillations when self-feedback distributed delays exceed a certain value. The excitatory feedback and inhibitory feedback from cortex to BG have different effects on oscillation. We define four different states (absolutely stable, conditional stable, conditional oscillation and absolutely oscillation) in this model, which can explain different mechanisms of oscillation origin. An increase in distributed delays can induce the transition between supercritical (SPH) and subcritical (SBH) Hopf bifurcations, which in turn may cause the transition of oscillations in different frequency bands, such as beta oscillations and alpha oscillations. Near the SBH, frequencies increase with an increase in discrete delays; and the trend is opposite at the SPH. In general, the amplitude of oscillation increases with the increase of firing activation level. Discrete delays can improve the firing activation level of the BG and suppress oscillations in its adjacent circuits. With the increase of distributed delays, oscillatory regions evolve regularly and the roles of general distributed delays and self-feedback distributed delays are contrary. Different types of delay have different effects on oscillation frequency. Discrete delay and self-feedback distributed delay have similar effects on oscillation frequency, but are different from general distributed delay. Whether the frequency will change with an increase in delay depends on coupling weights.

Stationary distribution, extinction and density function for a stochastic HIV model with a Hill-type infection rate and distributed delay

<abstract><p>In this article, we investigate the dynamics of a stochastic HIV model with a Hill-type infection rate and distributed delay, which are better choices for mass action laws. First, we transform a stochastic system with weak kernels into a degenerate high-dimensional system. Then the existence of a stationary distribution is obtained by constructing a suitable Lyapunov function, which determines a sharp critical value $ R_0^s $ corresponding to the basic reproduction number for the determined system. Moreover, the sufficient condition for the extinction of diseases is derived. More importantly, the exact expression of the probability density function near the quasi-equilibrium is obtained by solving the Fokker-Planck equation. Finally, numerical simulations are illustrated to verify the theoretical results.</p></abstract>

Tensor products of finitely presented functors

We provide explicit constructions for various ingredients of right exact monoidal structures on the category of finitely presented functors. As our main tool, we prove a multilinear version of the universal property of so-called Freyd categories, which in turn is used in the proof of correctness of our constructions. Furthermore, we compare our construction with the Day convolution of arbitrary additive functors. Day convolution always yields a closed monoidal structure on the category of all additive functors. In contrast, right exact monoidal structures for finitely presented functor categories are not necessarily closed. We provide a necessary criterion for being closed that relies on the underlying category having weak kernels and a so-called finitely presented prointernal hom structure. Our results are stated in a constructive way and thus serve as a unified approach for the implementation of tensor products in various contexts.

BIFURCATIONS AND AMPLITUDE DEATH FROM DISTRIBUTED DELAYS IN COUPLED LANDAU STUART OSCILLATORS AND A CHAOTIC PARAMETRICALLY FORCED VAN DER POL-RAYLEIGH SYSTEM

Distributed delays modeled by 'weak generic kernels' are introduced in the well-known coupled Landau-Stuart system, as well as a chaotic van der Pol-Rayleigh system with parametric forcing. The systems are close via the 'linear chain trick'. Linear stability analysis of the systems and conditions for Hopf bifurcation which initiates oscillations are investigated, including deriving the normal form at bifurcation, and deducing the stability of the resulting limit cycle attractor. The value of the delay parameter $a=a_{Hopf}$ at Hopf bifurcation picks out the onset of Amplitude Death(AD) in all three systems, with oscillations at larger values (corresponding to weaker delay). In the Landau-Stuart system, the Hopf-generated limit cycles for $a>a_{Hopf}$ turn out to be remarkably stable under very large variations of all other system parameters beyond the Hopf bifurcation point, and do not undergo further symmetry breaking, cyclic-fold, flip, transcritical or Neimark-Sacker bifurcations. This is to be expected as the corresponding undelayed systems are robust oscillators over wide ranges of their respective parameters. Numerical simulations reveal strong distortion and rotation of the limit cycles in phase space as the parameters are pushed far into the post-Hopf regime, and reveal other features, such as how the oscillation amplitudes and time periods of the physical variables on the limit cycle attractor change as the delay and other parameters are varied. For the chaotic system, very strong delays may still lead to the cessation of oscillations and the onset of AD (even for relatively large values of the system forcing which tends to oppose this phenomenon). Varying of the other important system parameter, the parametric excitation, leads to a rich sequence of dynamical behaviors, with the bifurcations leading from one regime (or type of attractor) into the next being carefully tracked.

A Constructive Approach to Freyd Categories

We discuss Peter Freyd’s universal way of equipping an additive category mathbf {P} with cokernels from a constructive point of view. The so-called Freyd category mathcal {A}(mathbf {P}) is abelian if and only if mathbf {P} has weak kernels. Moreover, mathcal {A}(mathbf {P}) has decidable equality for morphisms if and only if we have an algorithm for solving linear systems X cdot alpha = beta for morphisms alpha and beta in mathbf {P}. We give an example of an additive category with weak kernels and decidable equality for morphisms in which the question whether such a linear system admits a solution is computationally undecidable. Furthermore, we discuss an additional computational structure for mathbf {P} that helps solving linear systems in mathbf {P} and even in the iterated Freyd category construction mathcal {A}( mathcal {A}(mathbf {P})^{mathrm {op}} ), which can be identified with the category of finitely presented covariant functors on mathcal {A}(mathbf {P}). The upshot of this paper is a constructive approach to finitely presented functors that subsumes and enhances the standard approach to finitely presented modules in computer algebra.

Bifurcations in an economic growth model with a distributed time delay transformed to ODE

In this paper, we consider a model of economic growth with a distributed time-delay investment function, where the time-delay parameter is a mean time delay of the gamma distribution. Using the linear chain trick technique, we transform the delay differential equation system into an equivalent one of ordinary differential equations (ODEs). Since we are dealing with weak and strong kernels, our system will be reduced to a three- and four-dimensional ODE system, respectively. The occurrence of Hopf bifurcation is investigated with respect to the following two parameters: time-delay parameter and rate of growth parameter. Sufficient criteria on the existence and stability of a limit cycle solution through the Hopf bifurcation are presented in case of time-delay parameter. Numerical studies with the Dana and Malgrange investment function show the emergence of two Hopf bifurcations with respect to the rate growth parameter. In this case, we have been able to detect the existence of stable long-period cycles in the economy. According to the time-delay and adjustment speed parameters, the range of admissible values of the rate of growth parameter breaks down into three intervals. First, we have stable focus, then the limit cycle and finally again the stable solution with two Hopf bifurcations. Such behavior appears for some middle interval of the admissible range of values of the rate of growth parameter.

A Dynamic Model of Unemployment with Migration and Delayed Policy Intervention

The purpose of this paper is to build and analyse a model of unemployment, where jobs search is open to both natives and migrant workers. Markets and government intervention respond jointly to unemployment when creating new jobs. Full employment of resources is the focal point of policy action, stimulating vacancy creation. We acknowledge that policy is implemented with delays, and capture labour market outcomes by building a non-linear dynamic system. We observe jobs separation and matching, and extend our model to an open economy with migration and delayed policy intervention meant to reduce unemployment. We analyse the stability behaviour of the resulting equilibrium for our dynamic system, including models with Dirac and weak kernels. We simulate our model with alternative scenarios, where policy action towards jobs creation considers both migration and unemployment, or just unemployment.

Radiation hybrid map construction problem parameterized

In this paper, we study the Radiation hybrid map construction ( $$\mathsf{{RHMC} }$$ ) problem which is about reconstructing a genome from a set of gene clusters. The problem is known to be $$\mathsf{{NP} }$$ -complete even when all gene clusters are of size two and the corresponding problem ( $$\mathsf{{RHMC}_2 }$$ ) admits efficient constant-factor approximation algorithms. In this paper, for the first time, we consider the more general case when the gene clusters can have size either two or three ( $$\mathsf{{RHMC}_3 }$$ ). Let $${p\text{- }\mathsf {RHMC} }$$ be a parameterized version of $$\mathsf{{RHMC} }$$ where the parameter is the size of solution. We present a linear kernel for $${p\text{- }\mathsf {RHMC}_3 }$$ of size $$22k$$ that when combined with a bounded search-tree algorithm, gives an FPT algorithm running in $$O(6^kk+n)$$ time. For $${p\text{- }\mathsf {RHMC}_3 }$$ we present a bounded search tree algorithm which runs in $$O^*(2.45^k)$$ time, greatly improving the previous bound using weak kernels.

Boosted kernel for image categorization

Recent machine learning techniques have demonstrated their capability for identifying image categories using image features. Among these techniques, Support Vector Machines (SVM) present good results for example in Pascal Voc challenge 2011 [8], particularly when they are associated with a kernel function [28, 35]. However, nowadays image categorization task is very challenging owing to the sizes of benchmark datasets and the number of categories to be classified. In such a context, lot of effort has to be put in the design of the kernel functions and underlying semantic features. In the following of the paper we call semantic features the features describing the (semantic) content of an image. In this paper, we propose a framework to learn an effective kernel function using the Boosting paradigm to linearly combine weak kernels. We then use a SVM with this kernel to categorize image databases. More specifically, this method create embedding functions to map images in a Hilbert space where they are better classified. Furthermore, our algorithm benefits from boosting process to learn this kernel with a complexity linear with the size of the training set. Experiments are carried out on popular benchmarks and databases to show the properties and behavior of the proposed method. On the PASCAL VOC2006 database, we compare our method to simple early fusion, and on the Oxford Flowers databases we show that our method outperforms the best Multiple Kernel Learning (MKL) techniques of the literature.

Direction and stability of bifurcating periodic solutions of a chemostat model with two distributed delays

We have considered a chemostat model with two distributed delays in a recent paper [Chaos, Solitons & Fractals 2004;20:995–1004], where, using the average time delay corresponding to the growth response as a bifurcation parameter, it is proven that the model undergoes Hopf bifurcations for two weak kernels. This article is a sequel to the previous work. The direction and stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. The results are consistent with the numerical results in [Chaos, Solitons & Fractals 2004;20:995–1004].

Weak subobjects and the epi-monic completion of a category

The notion of weak subobject, or variation, was introduced by Grandis (Cahiers Topologie Géom. Différentielle Catégoriques 38 (1997) 301–326) as an extension of the notion of subobject, adapted to homotopy categories or triangulated categories, and well linked with their weak limits. We study here some formal properties of this notion. Variations in the category X can be identified with (distinguished) subobjects in the Freyd completion Fr X , the free category with epi-monic factorisation system over X , which extends the Freyd embedding of the stable homotopy category of spaces in an abelian category (Freyd, in: Proceedings of Conference on Categ. Algebra, La Jolla, 1965, Springer, Berlin, 1966, pp. 121–176). If X has products and weak equalisers, as Ho Top and various other homotopy categories, Fr X is complete; similarly, if X has zero-object, weak kernels and weak cokernels, as the homotopy category of pointed spaces, then Fr X is a homological category (Grandis, Cahiers Topologie Géom. Différentielle Catégoriques 33 (1992) 135–175); finally, if X is triangulated, Fr X is abelian and the embedding X → Fr X is the universal homological functor on X , as in Freyd's original case. These facts have consequences on the ordered sets of variations.

A numerical method for solving the Fredholm integral equation of the second kind

The numerical method is used to solve the Fredholm integral equation of the second kind with weak singular kernels using the Toeplitz matrices. The solution has a computing time requirement ofO(N2), where 2N + 1 is the number of discretization points used. Also, the error estimate is computed. Some numerical examples are computed using the MathCad package.

Periodic orbits arising from Hopf bifurcations in a Volterra prey-predator model

In this paper we derive a formula which enables the stability of periodic solutions to a Volterra integro-differential system to be determined. This system which has been studied by Cushing [1], models a predator-prey interaction with distributed delays. The results are obtained by using the algorithm developed by Kazarinoff, Wan and van den Driessche [2] based on the centre manifold formulas of Hassard and Wan [3]. We discuss an example of the formula for the case of weak kernels and show that under certain conditions stable periodic solutions arising from Hopf bifurcations at different critical values of the parameters can exist together.

Some remarks on weak sufficiency

Reduction by sufficiency of statistical procedures such as testing and estimation has proved to be of significant importance in the applications. To replace this strong concept by weaker ones has been suggested my several authors. Here the interrelationship between the various sufficiencies is studied by a systematic use of weak kernels.

Self-sharpening distributions revisited—polydispersity in growth by monomer addition

Using the population balance method we investigated the dependence of polydispersity on the growth law that governs the size increase of particles. Using a power-law kernel K i ∼ i α (where i is the mass of the particle) explicit results are derived for two limiting cases of constant kernel α = 0.0 and proportional kernel α = 1.0. For the constant kernel the normalized variance decreases in time. For the proportional kernel the variance increases during growth. In the two transition regions of weak kernels α < 0.5 and strong kernels α > 0.5, the polydispersity approaches the asymptotic values of the constant and proportional kernels. In the transition regions the growth law cannot be deduced from the polydispersity.

On the Quality of Rice Kernel in Early Season Cultivation

The quality of rice kernel in early season cultivation is studied in comparison with that in usual or late season cultivation. In early season cultivation, rice kernels mature in haste and the accumulation of storage matter into the inner parts of kernel is abundant at early stage of maturation, therefore the arrangement of starch cells seems to be somewhat radial on the cross section of kernel. On the other hand, the weak kernels on panicle become frequently into milk-white kernels and chalky kernels on account of an unbalance between demand and supply of storage matter about 10 days after flowering, and some of the other kernels become into partially-chalky kernels and white-belly kernels owing to a tendency. to premature. On the kernels in early season cultivation number of aleurone cells and thickness of bran layer increase and embryos fall off more easily from kernels in process of cleaning, accordingly the cleaning-loss is relatively high as compared with that of the kernels in usual or late season cultivation.

Effects of Temperature in the Ripening Periods upon the Development and Qualities of Lowland Rice Kernels

Lowland rice plants (Norin-17) of early and sermi-late cultivations, and the ears of the same variety of late cultivation were treated during night time with high (30°C) and low (20°C) temperature (15°C -30°C for the ear treatment) for certain periods in various stages of kernel development. Investigations on maturation, fruiting and specific gravities of the seeds, and some microscopic observations on bran layer and starchy tissue of the kernels were conducted. The results are summmarized as follows: 1) High night temp. accelerated the kernel development and maturation. 2) Chalky kernels and partially-chalky kernels markedly occured in the plots of high night temp. around 1O days after flowering. That was presumably due to an unbalance in supply and demand of nutrients, and to increased consumption of storage foods in the weak kernels, which were caused by high temp. 3) Both high and low night temp. 5 to 15 days after flowering caused milky-white kernels which considered to be due to the temporary hindrance of ripening. Many of milky-white kernels due to hindrance in early ripening stages had an opaque part (white-core-like but different from that of white-core kernel in shape and location) in their tissues. 4) Measurements on specific gravities of paddy, brown rice and parts of kernel revealed insufficiency of storage matter accumulation in the outer parts of kernel under high night temp. conditions. But as for the inner parts, they had larger s.g. values than those from low temp. plots, indicating good accumulation in these parts in earlier stages of seed development. 5) High night temp. in early stages of seed growth increased aleurone cells and thickness of bran layer on back side of kernel. Low temp. decreased these measurements. 6) On the cross sections of kernels from high night temp. plots a well-ar-ranged radial starch tissue (characteristic of Koshitsu-Mai) was observed. 0n the other hand, remarkable development of flatish cells along the dorso-ventral diameter of kernel (typical for Nanshitsu-Mai) was noticed in the kernels from low night temp. plots