Search

We establish the Trudinger-Moser inequality on weighted Sobolev spaces in the whole space, and for a class of quasilinear elliptic operators in radial form of the type L u : = − r − θ ( r α | u ′ ( r ) | β u ′ ( r ) ) ′ , where θ , β ≥ 0 and α > 0 , are constants satisfying some existence conditions. It is worth emphasizing that these operators generalize the p-Laplacian and k-Hessian operators in the radial case. Our results involve fractional dimensions, a new weighted Polya-Szego principle, and a boundness value for the optimal constant in a Gagliardo-Nirenberg type inequality.

In 1975, Lieb and Thirring derived a semiclassical lower bound on the kinetic energy for fermions, which agrees with the Thomas-Fermi approximation up to a constant factor. Whenever the optimal constant in their bound coincides with the semiclassical one is a long-standing open question. We prove an improved bound with the semiclassical constant and a gradient error term which is of lower order., 6 pages, comments and references added

We perform a detailed analysis of the solvability of linear strain equations on several non-characteristic regions to obtain L 2 regularity solutions. As an application, the rigidity results on the strain tensor of the middle surface are implied by the L 2 regularity for non-characteristic regions. Finally, we obtain the optimal constant in the first Korn inequality scales like h 4 / 3 for hyperbolic shells, generalizing the assumption that the middle surface of the shell is given by a single principal system in the literature.

Let 0 1p>1 be such that ps0S≡S(N,p,s,β)>0. (3) If β≡N−ps2, as a consequence of the improved Hardy inequality, we obtain that for all q

In this paper, we investigate the complete weight enumerators of two classes of irreducible cyclic codes. We present the explicit complete weight enumerator of the irreducible cyclic codes. Furthermore, we obtain a class of optimal constant composition codes from irreducible cyclic codes.

We focus on the study of ground-states for the system of M coupled semilinear Schrodinger equations with power-type nonlinearities and couplings. General results regarding existence and characterization are derived using a variational approach. We show the usefulness of such a characterization in several particular cases, including those for which uniqueness of ground-states is already known. Finally, we apply the results to find the optimal constant for the vector-valued Gagliardo–Nirenberg inequality and we study global existence, L 2 -concentration phenomena and blowup profile for the evolution system in the L 2 -critical power case.

This paper is devoted to rigidity results for some elliptic PDEs and to optimal constants in related interpolation inequalities of Sobolev type on smooth compact connected Riemannian manifolds without boundaries. Rigidity means that the PDE has no other solution than the constant one at least when a parameter is in a certain range. The largest value of this parameter provides an estimate for the optimal constant in the corresponding interpolation inequality. Our approach relies on a nonlinear flow of porous medium / fast diffusion type which gives a clear-cut interpretation of technical choices of exponents done in earlier works on rigidity. We also establish two integral criteria for rigidity that improve upon known, pointwise conditions, and hold for general manifolds without positivity conditions on the curvature. Using the flow, we are also able to discuss the optimality of the corresponding constants in the interpolation inequalities.

This paper analyzes optimal contracting when an agent has private information before contracting and exerts hidden effort that stochastically affects the output. Additionally, the contract is constrained to satisfy the agentʼs ex post participation. We highlight three features of this model. First, the agent faces countervailing incentives. Second, the separation of types is never optimal. Third, the optimal constant bonus rewarding success is distorted downward below its efficient level.

In this paper, we determine the optimal constant in the estimate of Ohsawaʼs generalized L 2 extension theorem. The result holds for holomorphic vector bundles on a class of complex manifolds including both Stein manifolds and complex projective algebraic manifolds. As an application, we obtain a solution to a related conjecture of Ohsawa.

We deal with domains with infinite inner radius. More precisely, we introduce a new geometric assumption on an exterior domain Ω⊂Rn; n⩾3 (i.e. complement of smooth compact domain not containing the origin). Under this assumption, we prove the Hardy inequality with optimal constant involving the distance to the boundary. In addition, in the case n⩾4, we improve this inequality by adding a critical Sobolev norm. Furthermore, we investigate the singular case n=3 and we show that, under some additional geometric assumption on Ω, the Hardy inequality can be improved by adding a Sobolev type term with critical exponent. Also, we prove some Hardy–Sobolev type inequalities without any geometric assumptions on Ω, which are of independent interest. Finally, we prove Harnack inequality up to the boundary for the positive solutions of the problem ut=Δu+14udist2(x,∂Ω) and we prove heat kernel estimates for small times.

We examine the optimal customization of a financial derivative in the presence of a background risk. This problem includes the model of finding the optimal constant amount of a given pecuniary risk as a degenerated case. We show the importance of this perspective with a preference-free solution for the pecuniary background risk case and a general solution for any given utility function. The latter solution allows us to measure the incompleteness premium demanded by risk-averse agent for a market where only constant amount of risk is allowed in comparison with a complete and arbitrage-free market where the customization is unrestricted.

In this Note, we solve the optimal constant problem in the L 2 -extension theorem with negligible weight on Stein manifolds. As an application, we prove the Suita conjecture on arbitrary open Riemann surfaces.

Let d be a given positive integer and let { R j } j = 1 d denote the collection of Riesz transforms on R d . For any K > 2 / π we determine the optimal constant L such that the following holds. For any locally integrable Borel function f on R d , any Borel subset A of R d and any j = 1 , 2 , … , d we have ∫ A | R j f ( x ) | d x ⩽ K ∫ R d Ψ ( | f ( x ) | ) d x + | A | ⋅ L . Here Ψ ( t ) = ( t + 1 ) log ( t + 1 ) − t for t ⩾ 0 . The proof is based on probabilistic techniques and the existence of certain special harmonic functions. As a by-product, we obtain related sharp estimates for the so-called re-expansion operator, an important object in some problems of mathematical physics.

In this paper, we consider a modified two-component Camassa–Holm (MCH2) system which arises in shallow water theory. We analyze the wave breaking mechanism by establishing some new blow-up criteria for this system formulated either on the line or with space-periodic initial condition.

We compute the optimal constant for some weighted Poincare inequalities obtained by Fausto Ferrari and Enrico Valdinoci in [F. Ferrari, E. Valdinoci, Some weighted Poincare inequalities, Indiana Univ. Math. J. 58 (4) (2009) 1619–1637].

Cultivation of microalgae Chlorella sp. was performed in draft-tube photobioreactors. Effect of light intensity on the microalgae growth performance was conducted under a light intensity range of 82–590 μmol/m2 s. A lumostatic strategy was proposed based on the light distribution profiles obtained by image analysis and specific chlorophyll a content. The proposed lumostatic strategy allowed a maximum biomass dry weight of 5.78 g/L and a productivity of 1.29 g/L d, which were 25.7% and 74.3% higher than that achieved by the optimal constant light intensity, respectively. A comparison with other lumostatic strategies reported in the literature indicated that the proposed lumostatic strategy in the current study can be a promising approach in improving the growth of microalgae.

In this paper, we consider the periodic weakly dissipative Dullin–Gottwald–Holm equation. The present work is mainly concerned with blow-up phenomena for the Cauchy problem for this new kind of equation. We apply the optimal constant to give sufficient conditions via an appropriate integral form of the initial data, which guarantee the finite-time singularity formation for the corresponding solution.

We address the question of Lipschitz stability results in inverse source problems for the heat equation perturbed by a singular inverse-square potential − μ / | x | 2 when μ ⩽ μ ⋆ ( N ) : = ( N − 2 ) 2 / 4 where μ ⋆ ( N ) is the optimal constant in the so-called Hardy inequality. Following Immanuvilov and Yamamoto (1998) [9] , our proof is based on Carleman inequalities like those developed by Fursikov and Immanuvilov (1996) [8] for the classical heat equation. However, we need here to take into account the singularity. Therefore, the first step of the proof consists in some improvements of the Carleman inequalities specifically developed for equations with inverse-square potentials by Vancostenoble and Zuazua (2008) [15] and next Ervedoza (2008) [7] . Major steps rely on various improved forms of the Hardy inequality.

In this article, we consider an optimal proportional reinsurance with constant dividend barrier. First, we derive the Hamilton-Jacobi-Bellman equation satisfied by the expected discounted dividend payment, and then get the optimal stochastic control and the optimal constant barrier. Secondly, under the optimal constant dividend barrier strategy, we consider the moments of the discounted dividend payment and their explicit expressions are given. Finally, we discuss the Laplace transform of the time of ruin and its explicit expression is also given.

Fish stocking is an important element of aquaculture management. A constant stocking rate is a convenient strategy in intensive aquaculture, being also the best under steady physical and economic conditions. If, however, these conditions vary with time, the best stocking rate may also vary. This problem has been addressed in a previous paper, where water temperature, fish price and/or market demand were allowed to vary sinusoidally over the annual cycle, while a constant upper limit on the water-treatment capacity constrained the process. Considering only situations where feeding is always to satiation, sinusoidal stocking rates were shown to perform often significantly better than the best constant stocking-rate strategy. In this paper a further step is taken in that over-stocking (more than hitherto) for certain periods of time is allowed. This leads to restricted feeding at a later time (to avoid overload of the water-treatment system) and to some loss of feeding efficiency. Often, however, the loss in feeding efficiency is more than compensated for by better utilization of the rearing space. Optimal constant and sinusoidal over-stocking strategies were applied to eight sample combinations of sinusoidal temperature and market conditions. Restricted over-stocking often resulted in a considerable improvement of economic performance over the previous results. The potential for increase of yield and profit by over-stocking depends on the amount of under-utilization of the rearing space by the previous solution, namely when only feeding to satiation is allowed. On average (for the system under consideration) out of 10% of under-utilized space when feeding to satiation, 4.4% (almost half) can be recovered by properly applied over-stocking. Expressed in terms of feed processing capability, this 4.4% recovery is equivalent to about 16 kg[feed]/(m 3 [tank]y), and to an additional production of about 6.6 kg[fish]/(m 3 [tank]y). The gain in profit is of the order of 25 $/(m 3 [tank]y) per 10% of previously unutilized capacity.

The complex pattern of mines made by leafminers is postulated to hinder the host-searching behaviour of parasitoids, and if this is true, parasitoids should have evolved strategies of searching mines in a way that will achieve the highest efficiency. We investigated the possible deterrent effect of mine pattern complexity on a parasitoid searching for a host within a patch, and the searching behaviour used by the parasitoid to shorten search time. For this, we used the leafminer Liriomyza trifolii (Diptera: Agromyzidae) and its generalist parasitoid Hemiptarsenus varicornis (Hymenoptera: Eulophidae). Observations of the parasitoids showed that searching time increased on mines with complex patterns and crosses. We also found that the parasitoids used a specific searching strategy: they made multiple trackings to find the host and each tracking was terminated with a constant mine-leaving rate per unit time irrespective of mine complexity. With an individual-based model, we also found that while search time to host encounter increased with mine complexity, the optimal mine-leaving rate that attained the most efficient search did not vary with mine complexity, indicating that making multiple trackings with an optimal constant rate was advantageous in terms of search time for any search on a mine over a range of levels of complexity. Furthermore, the mine-leaving rate estimated from the behavioural observations was consistent with the optimal rate obtained in the simulations. Multiple-tracking behaviour by H. varicornis is the best way for foraging for hosts whose feeding pattern varies from simple to more tortuous.

We prove optimal constant over root $n$ upper bounds for the maximal probabilities of $n$th convolution powers of discrete uniform distributions., Comment: 6 pages

We give improved results on the existence of positive solutions for a nonlinear differential equation with nonlocal boundary conditions, that arise as a model of a thermostat. We obtain some optimal criteria for the existence of one positive solution which involve the principal eigenvalue of a related linear operator. We also determine optimal values of some other constants that are useful in obtaining existence of multiple positive solutions.

Cross-correlated relaxation rates Gamma are commonly obtained from constant time experiments by measuring the effect of the desired cross-correlated relaxation on an appropriate coherence during the constant time T. These measurements are affected by systematic errors, which derive from undesired cross-correlated relaxation effects taking place before and after the constant time period T. In this paper we discuss the sources and the size of these errors in an example of two pulse sequences. Higher accuracy of the measured data can be obtained by recording a set of experiments with different T values. Cross-correlated relaxation rates are measured in constant time experiments either from the differential relaxation of multiple components (J-resolved Gamma experiments) or from the efficiency of magnetization transfer between two coherences (quantitative Gamma experiments). In this paper we calculate analytically the statistical errors in both J-resolved and quantitative Gamma experiments. These formulae provide the basis for the choice of the most efficient experimental approach and parameters for a given measurement time and size of the rate. The optimal constant time T for each method can be calculated and depends on the relaxation properties of the molecule under investigation. Moreover, we will show how to optimize the relative duration of cross and reference experiments in a quantitative Gamma approach.

Abstract. By combining elementary orientation mechanisms with a path-integration process, two spatial memory-based searching mechanisms were modelled. These mechanisms make it possible to account for the behaviour of many animals which locally increase their search effort in the vicinity of a given memorized location to discover with maximal efficiency a small target which is likely to be close to this location. Common examples are provided by desert arthropods searching for their burrow entrance or predators searching for clustered prey items. The spatial distributions of the search effort generated by the two spatial memory-based searching mechanisms were derived using computer simulations. The efficiency of a predator searching for clustered prey items with a mechanism of this type was computed in four habitat types with differing grains and/or heterogeneity levels. A predator exhibiting optimal spatial memory-based area-concentrated searching behaviour was able to harvest about 1·6 times more prey items (whatever the habitat type) than if it exhibited optimal area-concentrated searching behaviour without referring to a spatial memory, and almost three to five (depending on the habitat type) times more than if it did not exhibit any area-concentrated searching behaviour but moved in a straight line with an optimal constant speed.

we examine harvest strategies for two models of an exploited ecological system. For the model with constant growth and death rates, an optimal constant harvest strategy is obtained. For the model with uncertainty occurring in the system parameters, a harvest strategy is proposed which stabilizes the biomass of the ecological system about a desired level, namely, the level generated by the optimal constant harvest strategy in the absence of uncertainty. Keywords-Ecological modelling, Optimal harvest strategies. 1. INTRODUCTION In [l], Lee and Ang investigated a logistic type seaweed growth model in which the growth and death rates of seaweed are known periodic functions of time, and they deduced an optimal periodic harvesting strategy which maximizes the average accumulated yield of an unspecified but periodic seaweed biomass having the same period as that of the growth and death rates. They noted, however, that in practice, a constant harvest rate is more practical, and hence, preferable to a time-varying one. In this note we consider first an ecological system with constant growth and death rates, and obtain an optimal constant harvest rate that maximizes the average accumulated yield of the biomass of the ecological system. Then, uncertainty is introduced into the model via the growth and death rates of the ecological system, and we address the following question. Suppose that we do not know the instantaneous growth and death rates of the ecological system except for the bounds of their possible variations. What should our harvest strategy be if we desire to maintain the biomass of the ecological system at or near the level that corresponds to maximum average accumulated yield in the absence of uncertainty? Clearly, when uncertain disturbances occur in both the growth and death rates of the ecological system, utilization of a constant harvest strategy can no longer assure a desired constant biomass of the ecological system.

Results of fundamental research demonstrate that the dynamics of longterm membrane fouling is affected by the applied settings of operation variables. As a consequence, the operation efficiency will be improved by the application of optimal trajectories of the operation variables. For both a precipitation model and a gel layer model specific trajectories are calculated by dynamic optimization and utilization of an objective function based on the cash flow of the operation. The relevance of dynamically optimal operations is demonstrated by a comparison with statically optimal operations where the optimal constant values of operation variables are applied. For a case study on reverse osmosis of cheese whey (precipitation model), the cash flow rises by up to 25% as a result of improved membrane performance (10%) and reduced energy costs (26%), while for gel layer fouling significant energy savings can be realized (35–55%).

This paper considers due-date determination and sequencing of n jobs on a single-machine where each job is given a constant flow allowance. The objective is to determine the optimal value of the flow allowance and the optimal job sequence to minimize a cost function based on the flow allowance and the job earliness and tardiness values. We first propose a linear programming (LP) formulation of the problem and then derive the optimal constant flow allowance via considering the LP dual problem. We show that the optimal constant due-date value is independent of job sequence. After the theoretical treatment, a numerical example is presented for discussion.

The problem considered here is an infinite stage inventory decision process in which the penalty cost for failing to have in stock sufficient items to satisfy demand at the next stage includes a fixed administrative cost (red tape) and a cost proportional to the deficiency in stock. It is shown here that when the demand follows a Weibull distribution, optimal constant stock level solution exists. A simple computer program is given to calculate this optimal solution.

This paper considers the problem of optimal constant due-date assignment and sequencing of jobs in a single-machine shop. We formulate the problem as a general constrained optimization problem and apply the Kuhn-Tucker conditions to find the optimal solution which is shown to be independent of the job sequence.

This paper develops general overtaking techniques for studying the asymptotic properties of portfolio policies optimal with respect to a terminal utility valuation. For a restricted class of utility functions the sequence of optimal constant (non-revised) portfolio policies formed as the horizon recedes into the future is shown to converge. Furthermore, for utility functions unbounded above and below, this turnpike policy need not be the policy associated with the minimal constant relative risk aversion function that bounds the valuation function from above. Finally, an analogy between the portfolio turnpike problem and the turnpike problem of growth theory is studied.

A model for cyclical pressure freeze drying has been solved numerically and the results used to evaluate the effectiveness of the technique when compar However, all the cyclical policies tried were inferior (although only slightly in some cases) to a near optimal constant pressure policy developed prev

The optimal constant gain output feedback control of linear time-invariant discrete systems with respect to an averaged performance criterion is discussed. The algebraic conditions necessary for minimizing the averaged performance cost are derived and a method for computing the constant optimal feedback gain is presented.

One of the basic problems in Periodic Optimization is to ascertain by simple calculations whether the optimal constant control of a given plant can be improved by cycling or not. In this direction, one of the most powerful results is a frequency domain condition usually referred to as Π-criterion. The main limitation to the use of such a criterion is the assumption that the optimal constant control be an interior point of the admissible control set. In this paper, an extension of the Π-criterion to the case of boundary optimal constant control is presented and a particular case discussed in which the condition above can be given the useful form of a circle criterion.

Suppose prices p are adjusted in accordance with excess demand z by Δp = k 1 z . We determine the optimal constant k 1 given that demand is the sum of a linear function of price and a Markov disturbance. Supply is fixed.