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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.

Let \(\Delta _0\) be the Laplace–Beltrami operator on the unit sphere \(\mathbb {S}^{d-1}\) of \({\mathbb R}^d\). We show that the Hardy–Rellich inequality of the form $$\begin{aligned} \mathop \int \limits _{\mathbb {S}^{d-1}} \left| f (x)\right| ^2 \mathrm{d}{\sigma }(x) \le c_d \min _{e\in \mathbb {S}^{d-1}} \mathop \int \limits _{\mathbb {S}^{d-1}} (1- {\langle }x, e {\rangle }) \left| (-\Delta _0)^{\frac{1}{2}}f(x) \right| ^2 \mathrm{d}{\sigma }(x) \end{aligned}$$ holds for \(d =2\) and \(d \ge 4\) but does not hold for \(d=3\) with any finite constant, and the optimal constant for the inequality is \(c_d = 8/(d-3)^2\) for \(d =2, 4, 5,\) and, under additional restrictions on the function space, for \(d\ge 6\). This inequality yields an uncertainty principle of the form $$\begin{aligned} \min _{e\in \mathbb {S}^{d-1}} \mathop \int \limits _{\mathbb {S}^{d-1}} (1- {\langle }x, e {\rangle }) |f(x)|^2 \mathrm{d}{\sigma }(x) \mathop \int \limits _{\mathbb {S}^{d-1}}\left| \nabla _0 f(x)\right| ^2 \mathrm{d}{\sigma }(x) \ge c'_d \end{aligned}$$ on the sphere for functions with zero mean and unit norm, which can be used to establish another uncertainty principle without zero mean assumption, both of which appear to be new.

In this article we consider the following family of nonlinear elliptic problems, &nbsp &nbsp &nbsp &nbsp $-\Delta (u^m) - \lambda \frac{u^m}{|x|^2} = |Du|^q + c f(x). $ We will analyze the interaction between the Hardy-Leray potential and the gradient term getting existence and nonexistence results in bounded domains $\Omega\subset\mathbb{R}^N$, $N\ge 3$, containing the pole of the potential. Recall that $Λ_N = (\frac{N-2}{2})^2$ is the optimal constant in the Hardy-Leray inequality. &nbsp 1.For $0 q_+$, the above equation has no positive distributional solution. If $q < q_+$ we find solutions by using different alternative arguments. Moreover if $q = q_+ > 1$ we get the following alternative results. (a) If $m < 2$ and $q=q_+$ there is no solution. &nbsp (b) If $m = 2$, then $q_+=2$ for all $\lambda$. We prove that there exists solution if and only if $2\lambda\leq\Lambda_N$ and, moreover, we find infinitely many positive solutions. 2. If $m > 2$ we obtain some partial results on existence and nonexistence. We emphasize that if $q(\frac{1}{m}-1)

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, a new construction of zero-difference balanced functions defined on is given, where is an odd positive integer. Based on the generic constructions proposed by Ding, optimal constant composition codes and perfect difference systems of sets with new parameters can be generated from the zero-difference balanced functions constructed in this paper.

This article proposes a simple and reliable damping strategy for wave powerfarm operation of small-scale point-absorber converters. The strategy is based on passiverectification onto a constant DC-link, making it very suitable for grid integration of the farm.A complete model of the system has been developed in Matlab Simulink, and uses real sitedata as input. The optimal constant DC-voltage is evaluated as a function of the significantwave height and energy period of the waves. The total energy output of the WEC is derivedfor one year of experimental site data. The energy output is compared for two cases, onewhere the optimal DC-voltage is determined and held constant at half-hour basis throughoutthe year, and one where a selected value of the DC-voltage is kept constant throughout theyear regardless of sea state.

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 consider the weighted parabolic problem of the type $$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} u_t-\mathrm{div}(\omega _2(x)|\nabla u|^{p-2} \nabla u )= \lambda \omega _1(x) |u|^{p-2}u,&{} x\in \Omega ,\\ u(x,0)=f(x),&{} x\in \Omega ,\\ u(x,t)=0,&{} x\in \partial \Omega ,\ t>0,\\ \end{array}\right. \end{aligned} \end{aligned}$$ for quite a general class of possibly unbounded weights $$ \omega _1,\omega _2$$ satisfying the Hardy-type inequality. We prove existence of a global weak solution in the weighted Sobolev spaces provided that $$\lambda >0$$ is smaller than the optimal constant in the inequality. The domain is assumed to be bounded or quasibounded. The obtained solution is proven to belong to $$\begin{aligned} L^p({{\mathbb {R}}}_+; W_{(\omega _1,\omega _2),0}^{1,p}(\Omega ))\cap L^\infty ({{\mathbb {R}}}_+; L^2(\Omega )). \end{aligned}$$

We give a necessary and sufficient condition for the precompactness of all optimizing sequences for the Stein–Tomas inequality. In particular, if a well-known conjecture about the optimal constant in the Strichartz inequality is true, we obtain the existence of an optimizer in the Stein–Tomas inequality. Our result is valid in any dimension.

Constant composition codes (CCCs) are a special class of constant-weight codes. They include permutation codes as a subclass. The construction of CCCs with parameters achieving certain bounds has been an interesting research topic in coding theory. Recently, Ding established a bridge from zero difference balanced (ZDB) functions to CCCs with parameters meeting the Luo-Fu-Vinck-Chen bound. This provides a new approach for obtaining optimal CCCs. One objective of this paper is to present two new classes of ZDB functions whose parameters have been not covered in the literature. Another objective of this paper is to introduce two classes of CCCs meeting the Luo-Fu-Vinck-Chen bound from these new ZDB functions.

A new and simple proof of the embedding of the Hardy--Hilbert space of Dirichlet series into a conformally invariant Hardy space of the half-plane is presented, and the optimal constant of the embedding is computed., This note has been accepted for publication in Proceedings of the AMS

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.

We discuss the existence of breathers and lower bounds on their power, in nonlinear Schr\"odinger lattices with nonlinear hopping. Our methods extend from a simple variational approach to fixed point arguments, deriving lower bounds for the power which can serve as a threshold for the existence of breather solutions. Qualitatively, the theoretical results justify non-existence of breathers below the prescribed lower bounds of the power which depend on the dimension, the parameters of the lattice as well as of the frequency of breathers. In the case of supercritical power nonlinearities we investigate the interplay of these estimates with the optimal constant of the discrete interpolation inequality. Improvements of the general estimates, taking into account the localization of the true breather solutions are derived. Numerical studies in the one dimensional lattice corroborate the theoretical bounds and illustrate that in certain parameter regimes of physical significance, the estimates can serve as accurate predictors of the breather power and its dependence on the various system parameters., Comment: To be published in Journal of Nonlinear Science

We study the following coupled Schrodinger system which has appeared as several models from mathematical physics: $$\left\{\begin{array}{ll}-\Delta u + \lambda_1 u = \mu_1 u^3 + \beta uv^2, \quad x \in \mathbb{R}^N,\\-\Delta v + \lambda_2 v = \mu_2 v^3 + \beta vu^2, \quad x \in \mathbb{R}^N,\\u \geq 0, v \geq 0 \,\,{\rm in}\mathbb{R}^N, \quad u, v \in H^1(\mathbb{R}^N).\end{array}\right.$$ Here, N = 2, 3, and λ1, λ2, μ1, μ2 are all positive constants. In [Ambrosetti and Colorado in C R Acad Sci Paris Ser I 342:453–438, 2006], Ambrosetti and Colorado showed that, there exists β0 > 0 such that this system has a nontrivial positive radially symmetric solution for any \({\beta \in (0, \beta_0)}\). Later in [Ikoma and Tanaka in Calc Var 40:449–480, 2011], Ikoma and Tanaka showed that solutions obtained by Ambrosetti and Colorado are indeed least energy solutions for any \({\beta \in (0, {\rm min}\{\beta_0, \sqrt{\mu_1\mu_2}\})}\) . Here, in case λ1 = λ2 and μ1 ≠ μ2, we prove the uniqueness of the positive solutions for min{μ1, μ2} − β > 0 sufficiently small. In case λ1 ≠ λ2 and (λ2 − λ1)(μ2 − μ1) ≤ 0, we prove that \({\beta_0 0 such that this system has no nontrivial nonnegative solutions for any \({\beta \in ({\rm min}\{\mu_1, \mu_2\} - \delta,\, \max\{\mu_1, \mu_2\} + \delta)}\) . This answers an open question of [Sirakov in Commun Math Phys 271:199–221, 2007] partially, and improves a result of [Sirakov in Commun Math Phys 271:199–221, 2007]. The asymptotic behavior of the least energy solutions is also studied as \({\beta \nearrow \beta_0}\) .

We determine the best (optimal) constant in the $L^2$ Folland-Stein inequality on the quaternionic Heisenberg group and the non-negative functions for which equality holds., Corrected typos and improved the presentation

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.

In this article the authors study analytically the impact of the optimal constant feedback gain of the steady-state linear-quadratic optimal control problem to the corresponding finite-time linear-quadratic optimal control problem. Contributions of the study are 3-fold: (i) conditions for applicability of this suboptimal procedure are presented; (ii) a formula for the performance loss because of this practical implementation (use of the constant instead of the time-varying gain) is derived; and (iii) a technique based on the prescribed degree of stability that can be successfully used in practice is discussed. Examples are included to demonstrate the performance loss owing to this simplified approximate (suboptimal) linear-quadratic finite-horizon optimal control problem implementation.

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.

For any > we determine the optimal constant for which the following holds. If , are conjugate harmonic functions on the unit disc with , then We also establish a related estimate for orthogonal harmonic functions given on Euclidean domains as well as an extension concerning orthogonal martingales under differential subordination.

In this note we obtain the best constant for the Bergman projection from L 1 onto the Bloch space. We also consider the question for the projection from C(D) onto the little Bloch space.

For any , we determine the optimal constant such that the following holds. If is the Haar system on [0,1], then for any vectors ak from a separable Hilbert space and , we have This is generalized to the sharp weak-type inequality where X, Y stand for -valued martingales such that Y is differentially subordinate to X.

We study a multilinear analogue of the Hilbert transform. As can be expected, the finiteness of the form depends on cancellation properties in the kernel and care must be taken in the definition of the form. We show how to define the form in terms of distributions and prove L bounds for that form. In the second part, we study an analogous form on the level of fractional integration. This has been studied in one form by Drury. We note the L bounds for it and find the optimal constant for this bound in the case with the most symmetries. We also determine all functions which are optimisers for this inequality. Finally, we consider analogues of the fractional integration form in directions similar to those of Beckner’s approach for multilinear multilinear Hardy–Littlewood–Sobolev inequalities.

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.

For any 1 ≤ p > ∞ 1\leq p>\infty , we determine the optimal constant C p C_p such that the following holds. If ( h k ) k ≥ 0 (h_k)_{k\geq 0} is the Haar system, then for any vectors a k a_k from a separable Hilbert space H \mathcal {H} and θ k ∈ { 0 , 1 } \theta _k\in \{0,1\} , k = 0 , 1 , 2 , … k=0,\,1,\,2,\,\ldots , we have \[ | | ∑ k = 0 n θ k a k h k | | p , ∞ ≤ C p | | ∑ k = 0 n a k h k | | p . \left |\left |\sum _{k=0}^n \theta _ka_kh_k\right |\right |_{p,\infty }\leq C_p\left |\left |\sum _{k=0}^n a_kh_k\right |\right |_p. \] This is generalized to the weak type inequality \[ | | g | | p , ∞ ≤ C p | | f | | p , ||g||_{p,\infty }\leq C_p||f||_p, \] where f f is an H \mathcal {H} -valued martingale and g g is its transform by a predictable sequence taking values in [ 0 , 1 ] [0,1] . We extend this further to the estimate \[ | | Y | | p , ∞ ≤ C p | | X | | p , ||Y||_{p,\infty }\leq C_p||X||_p, \] valid for any two H \mathcal {H} -valued continuous-time martingales X X , Y Y , such that ( [ Y , X − Y ] t ) ([Y,X-Y]_t) is nondecreasing and nonnegative as a function of t t .

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.

Let K q (n, w, t, d) be the minimum size of a code over Z q of length n, constant weight w, such that every word with weight t is within Hamming distance d of at least one codeword. In this article, we determine K q (n, 4, 3, 1) for all n ? 4, q = 3, 4 or q = 2 m + 1 with m ? 2, leaving the only case (q, n) = (3, 5) in doubt. Our construction method is mainly based on the auxiliary designs, H-frames, which play a crucial role in the recursive constructions of group divisible 3-designs similar to that of candelabra systems in the constructions of 3-wise balanced designs. As an application of this approach, several new infinite classes of nonuniform group divisible 3-designs with block size four are also constructed.

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.

The Specht ratio S(h) is the optimal constant in the reverse of the arithmetic-geometric mean inequality, i.e., if 0 0, μ ∈ (0,1) ,w hereh = b and r = min{μ,1 − μ}. In this paper, we improve it by virtue of the Kantorovich constant, utilizing the refined scalar Young inequality we establish a weighted arithmetic-geometric-harmonic mean inequality for two positive operators. In the remainder of this work we focus on extending the refined weighted arithmetic-harmonic mean inequality to an operator version for another type of improvement.

McGilliard, C. R., Hilborn, R. MacCall, A., Punt, A. E., and Field, J. C. 2011. Can information from marine protected areas be used to inform control-rule-based management of small-scale, data-poor stocks? – ICES Journal of Marine Science, 68: 201–211. Many small-scale, nearshore fisheries lack the historical catch and survey information needed for conventional stock-assessment-based management. The potential use of the ratio of the density of fish outside a marine protected area to that inside it each year (the density ratio, DR) in a control rule is evaluated to determine the direction and magnitude of change in fishing effort in the next year. Management strategy evaluation was used to evaluate the performance of this DR control rule (DRCR) for a range of movement rates of larvae and adults and other biological scenarios, and the parameters of the control rule that maximized cumulative catch (over 95 years) for each scenario were found. The cumulative catch under the optimal DRCR was 90% of the cumulative catch from an optimal constant effort rule (CER). A small range of parameter values for the DRCR produced 75% or more of the cumulative catch produced from optimal CERs for a variety of assumptions about biology and initial stock status. The optimal DRCR was most sensitive to the movement patterns of larvae and adults and survey variability.

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.

We prove the neo-classical inequality with the optimal constant, which was conjectured by T. J. Lyons [Rev. Mat. Iberoamericana 14 (1998) 215-310]. For the proof, we introduce the fractional order Taylor's series with residual terms. Their application to a particular function provides an identity that deduces the optimal neo-classical inequality., 11 pages, 1 figure

We introduce some sequences of binary constant weight codes which are constructed from the affine constructions of balanced incomplete block designs. They give the values for the function [Formula: see text] of constant weight codes, and they are optimal since those values reach the Johnson Bound. We focus on the values obtained by this method.

In the paper we determine, for any K>0 and α∈[0,1], the optimal constant L(K,α)∈(0,∞] for which the following holds: If X is a nonnegative submartingale and Y is α-strongly differentially subordinate to X, then $$\sup_t\mathbb{E}|Y_t|\leq K\sup_t\mathbb{E}X_t\log^+X_t+L(K,\alpha).$$ Related sharp inequalities for martingales are also established. As an application, we obtain logarithmic estimates for smooth functions on Euclidean domains.

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.

We analyze a periodic review inventory system where inventory of a single item is used to serve high and low priority customers. High priority customers demand a higher service level than low priority customers and both types of customers can have quite general discrete demand distributions. The inventory is controlled by a (S, CL)-policy, where S denotes the base stock level and CL specifies the critical level, i.e., the amount of inventory reserved for high priority demand. We consider a policy, where the critical level is constant over time, which is not optimal, but makes the model analytically tractable. Unlike previous research, we clear backorders optimally and find the optimal solution. We also derive structural results.

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

In the cone over a cubic three-torus T3, balls about the vertex are isoperimetric if the volume of T3 is less than π/16 times the volume of the unit three-sphere. The conjectured optimal constant is 1.

Generalized Steiner systems GS (3, 4, v, 2) were first discussed by Etzion and used to construct optimal constant weight codes over an alphabet of size three with minimum Hamming distance three, in which each codeword has length v and weight four. Not much is known for GS (3, 4, v, 2)s except for a recursive construction and two small designs for v = 8,10 given by Etzion. In this paper, more small designs are found by computer search and also given are direct constructions based on finite fields and rotational Steiner quadruple systems and recursive constructions using three-wise balanced designs. Some infinite families are also obtained.

We consider the rate of piecewise constant approximation to a locally stationary process \(X(t),t\in [0,1]\), having a variable smoothness index \(\alpha (t)\). Assuming that \(\alpha (\cdot )\) attains its unique minimum at zero and satisfies $$ \alpha (t)=\alpha _0+b t^\gamma + \mathrm{o}(t^\gamma ) \quad \text{ as } t \rightarrow 0, $$ we propose a method for construction of observation points (composite dilated design) such that the integrated mean square error $$ \int _0^1 {\mathbb E}\{(X(t)-X_n(t))^2\} dt \sim \frac{K}{n^{\alpha _0}(\log n)^{(\alpha _0+1)/\gamma }} \qquad \text{ as } n\rightarrow \infty , $$ where a piecewise constant approximation \(X_n\) is based on \(N(n)\sim n\) observations of \(X\). Further, we prove that the suggested approximation rate is optimal, and then show how to find an optimal constant \(K\).

We study concentration inequalities for Lipschitz functions on graphs by estimating the optimal constant in exponential moments of subgaussian type. This is illustrated on various graphs and related to various graph constants. We also settle, in the affirmative, a question of Talagrand on a deviation inequality for the discrete cube.

In this paper, a construction of optimal constant composition codes is developed, and used to derive some series of new optimal constant composition codes meeting the upper bound given by [13].

We study analytically and numerically a model describing front propagation of a KPP reaction in a fluid flow. The model consists of coupled one-dimensional reaction-diffusion equations with different drift coefficients. The main rigorous results give lower bounds for the speed of propagation that are linear in the drift coefficient, which agrees very well with the numerical observations. In addition, we find the optimal constant in a functional inequality of independent interest used in the proof., Comment: 13 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.