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For any positive integer m > 2 and an odd prime p, let $\mathbb {F}_{p^{m}}$ be the finite field with pm elements and let $ \text {Tr}^{m}_{e}$ be the trace function from $\mathbb {F}_{p^{m}}$ onto $\mathbb {F}_{p^{e}}$ for a divisor e of m. In this paper, for the defining set $D=\{x\in \mathbb {F}_{p^{m}}:\text {Tr}^{m}_{e}(x)=1\text { and } \text {Tr}^{m}_{e}(x^{2})=0\}=\{d_{1}, d_{2}, \ldots , d_{n}\}$ (say), we define a pe-ary linear code $\mathcal {C}_{D}$ by $$ \mathcal{C}_{D}=\{\textbf{c}_{a} =\left( \text{Tr}^{m}_{e}(ad_{1}), \text{Tr}^{m}_{e}(ad_{2}),\ldots,\text{Tr}^{m}_{e}(ad_{n})\right) : a\in \mathbb{F}_{p^{m}}\}. $$ Then we determine the complete weight enumerator and weight distribution of the linear code $\mathcal {C}_{D}$ . The presented code is optimal with respect to the Griesmer bound provided that $\frac {m}{e}=3$ . In fact, it is MDS when $\frac {m}{e}=3$ . This paper gives the results of S. Yang, X. Kong and C. Tang (Finite Fields Appl. 48 (2017)) if we take e = 1. In addition to the generalization of the results of Yang et al., we study the dual code $\mathcal {C}_{D}^{\perp }$ of the code $\mathcal {C}_{D}$ as well as find some optimal constant composition codes.

We consider the scaling of the optimal constant in Korn’s first inequality for elliptic and parabolic shells which was first given by Grabovsky and Harutyunyan with hints coming from the test functions constructed by Tovstik and Smirnov on the level of formal asymptotic expansions. Here, we employ the Bochner technique in Remannian geometry to remove the assumption that the middle surface of the shell is given by one single principal coordinate, in particularly, including closed elliptic shells.

Let m, p, q ∈ (0, ∞) and let u, v, w be nonnegative weights. We characterize validity of the inequality $$\left(\int_{0}^{\infty} w(t)\left(f^{*}(t)\right)^{q} \mathrm{d} t\right)^{\frac{1}{q}} \leq C\left(\int_{0}^{\infty} v(t)\left(\int_{t}^{\infty} u(s)\left(f^{*}(s)\right)^{m} \mathrm{d} s\right)^{\frac{p}{m}} \mathrm{d} t\right)^{\frac{1}{p}}$$ for all measurable functions f defined on ℝn and provide equivalent estimates of the optimal constant C > 0 in terms of the weights and exponents. The obtained conditions characterize the embedding of the Copson—Lorentz space CLm,p(u, v), generated by the functional $$\|f\|_{C L^{m, p}(u, v)}:=\left(\int_{0}^{\infty} v(t)\left(\int_{t}^{\infty} u(s)\left(f^{*}(s)\right)^{m} \ \mathrm{d} s\right)^{\frac{p}{m}} \mathrm{d} t\right)^{\frac{1}{p}},$$ into the Lorentz space Λq(w). Moreover, the results are applied to describe the associated space of the Copson—Lorentz space CLm,p(u, v) for the full range of exponents m, p ∈ (0, ∞).

The purpose of this note is to expose a short proof of Hardy’s inequality in the sequence case. The proof is straightforward and provides the optimal constant $$p'$$.

We obtain sharp estimate on p-spectral gaps, or equivalently optimal constant in p-Poincare inequalities, for metric measure spaces satisfying measure contraction property. We also prove the rigidity for the sharp p-spectral gap.

Recently, linear codes over finite fields with a few weights have been extensively studied due to their applications in secret sharing schemes, authentication codes, constant composition codes. In this paper, for an odd prime p, the complete weight enumerator of a class of p-ary linear codes based on defining sets are determined. Furthermore, from the explicit complete weight enumerator of linear codes, a new class of optimal constant composition codes and several classes of asymptotically optimal systematic authentication codes are obtained.

Let $$k\ge 2, n\ge 1$$ be integers. Let $$f: {\mathbb {R}}^{n} \rightarrow {\mathbb {C}}$$. The kth Gowers–Host–Kra norm of f is defined recursively by $$\begin{aligned} \Vert f\Vert _{U^{k}}^{2^{k}} =\int _{{\mathbb {R}}^{n}} \Vert T^{h}f \cdot {\bar{f}} \Vert _{U^{k-1}}^{2^{k-1}} \, \text {d}h \end{aligned}$$with $$T^{h}f(x) = f(x+h)$$ and $$\Vert f\Vert _{U^1} = | \int _{{\mathbb {R}}^{n}} f(x)\, \text {d}x |$$. These norms were introduced by Gowers (Geom Funct Anal 11:465–588, 2001) in his work on Szemeredi’s theorem, and by Host and Kra (in Ann Math 161:398–488, 2005) in ergodic setting. These norms are also discussed extensively in Tao and Vu (in Additive combinatorics, Cambridge University Press, 2016). It is shown by Eisner and Tao (in J Anal Math 117:133–186, 2012) that for every $$k\ge 2$$ there exist $$A(k,n)< \infty $$ and $$p_{k} = 2^{k}/(k+1)$$ such that $$\Vert f\Vert _{U^{k}} \le A(k,n)\Vert f\Vert _{p_{k}}$$, for all $$f \in L^{p_{k}}({\mathbb {R}}^{n})$$. The optimal constant A(k, n) and the extremizers for this inequality are known [9]. In this dissertation, it is shown that if the ratio $$\Vert f \Vert _{U^{k}}/\Vert f\Vert _{p_{k}}$$ is nearly maximal, then f is close in $$L^{p_{k}}$$ norm to an extremizer.

Zero-difference balanced (ZDB) function plays an important role in communication field. In this paper, we propose a generic method to construct ZDB functions on generic algebraic rings. Using this method, we construct many new ZDB functions and retrieve some existing ZDB functions in a much simpler way. Moreover, new applications of the constructed ZDB functions, such as constructing optimal constant weight codes and optimal frequency-hopping sequences, are presented.

Carrying on the discussion initiated in (Dovetta-Tentarelli'18), we investigate the existence of ground states of prescribed mass for the $L^2$-critical NonLinear Schr\"odinger Equation (NLSE) on noncompact metric graphs with localized nonlinearity. Precisely, we show that the existence (or nonexistence) of ground states mainly depends on a parameter called reduced critical mass, and then we discuss how the topological and metric features of the graphs affect such a parameter, establishing some relevant differences with respect to the case of the extended nonlinearity studied by (Adami-Serra-Tilli'17). Our results rely on a thorough analysis of the optimal constant of a suitable variant of the $L^2$-critical Gagliardo-Nirenberg inequality., Comment: 22 pages, 7 figures. Keywords: metric graphs, NLS, ground states, localized nonlinearity, $L^2$-critical case. Some minor revisions have been made with respect to the previous version. Accepted for publication by Calc. Var. Partial Differential Equations

Partitioned difference families are an interesting class of discrete structures which can be used to derive optimal constant composition codes. There have been intensive researches on the construction of partitioned difference families. In this paper, we consider the combinatorial approach. We introduce a new combinatorial configuration named partitioned relative difference family, which proves to be very powerful in the construction of partitioned difference families. In particular, we present two general recursive constructions, which not only include some existing constructions as special cases, but also generate many new series of partitioned difference families. As an application, we use these partitioned difference families to construct several new classes of optimal constant composition codes.

A new family of effectively nine stages, ninth-order hybrid explicit Numerov-type methods is presented for the solution of some special second order Initial Value Problem. After dealing with a reduced set of order conditions, we derive an optimal constant coefficients method along with a similar kind of method with reduced phase errors. We proceed with numerical tests using quadruple precision arithmetic on some well-known problems from the relevant literature. Finally, in the appendices, we list Mathematica packages implementing the corresponding algorithms.

We consider the minimizers for the biharmonic nonlinear Schrodinger functional $$\begin{aligned} \mathcal {E}_a(u)=\int \limits _{\mathbb {R}^d} |\Delta u(x)|^2 \mathrm{d}x + \int \limits _{\mathbb {R}^d} V(x) |u(x)|^2 \mathrm{d}x - a \int \limits _{\mathbb {R}^d} |u(x)|^{q} \mathrm{d}x \end{aligned}$$ with the mass constraint $$\int |u|^2=1$$ . We focus on the special power $$q=2(1+4/d)$$ , which makes the nonlinear term $$\int |u|^q$$ scales similarly to the biharmonic term $$\int |\Delta u|^2$$ . Our main results are the existence and blowup behavior of the minimizers when a tends to a critical value $$a^*$$ , which is the optimal constant in a Gagliardo–Nirenberg interpolation inequality.

By exploring some geometry of Segre varieties and Veronese varieties, new families of non linear maximum rank distance codes and optimal constant rank codes are provided.

To estimate the optimal constant in Hardy-type inequalities, some variational formulas and approximating procedures are introduced. The known basic estimates are improved considerably. The results are illustrated by typical examples. It is shown that the sharp factor is meaningful for each finite interval and a classical sharp model is re-examined., Comment: 29 pages, 7 figures in Acta Mathematica Sinica, English Series 2015

The mixed principal eigenvalue of p -Laplacian (equivalently, the optimal constant of weighted Hardy inequality in Lp space) is studied in this paper. Several variational formulas for the eigenvalue are presented. As applications of the formulas, a criterion for the positivity of the eigenvalue is obtained. Furthermore, an approximating procedure and some explicit estimates are presented case by case. An example is included to illustrate the power of the results of the paper.

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.

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 \({\beta \nearrow \beta_0}\) .

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 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.

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.

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 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 obtained that for any n ∈ N, C = 1 is the smallest constant for which the inequality ||B n (f) - f|| ≤ C ⋅ ω2(f, 1/√n) holds on the class of continuous functions f, as well as on the class of bounded functions f, where B n is the Bernstein operators of degree n, ω2 is the second order modulus and || ⋅ || is the sup-norm.

In this paper we study optimal harvesting of the Norwegian spring-spawning herring stock. The biological model is described by a discrete time age-structured model. The optimal harvesting patterns are studied numerically and the results show that when using a linear cost function and constant price in the optimisation model, the optimal harvesting pattern is pulse fishing. However, optimal constant fishing effort gives only slightly lower profit. Moreover, when price is made responsive to harvest the optimal harvesting strategy is substantially smoothed.

The study of a class of optimal constant weight codes over arbitrary alphabets was initiated by Etzion, who showed that such codes are equivalent to special GDDs known as generalized Steiner systems GS(t,k,n,g) Etzion. This paper presents new constructions for these systems in the case t=2, k=3. In particular, these constructions imply that the obvious necessary conditions on the length n of the code for the existence of an optimal weight 3, distance 3 code over an alphabet of arbitrary size are asymptotically sufficient.

In this article we prove the equivalence between the strong isoperimetric inequality #A ≤ α#∂A, for any subset A of countable graph cG, and the inequality $$||f||_1 $$ ≤ $$ alpha |f|_{ operatorname{var} } $$ for any function with finite variation on cG and null at infinity, with optimal constant. More generally, we prove the equivalence between the isoperimetric inequality # A cP -1(1/# A)≤ α # A and the inequality || $$f $$ ||cM ≤ α | $$f $$ |var, where cM is a Young function and cP its conjugate, and we also obtain an isoperimetric inequality in $$ mathbb{Z}^n $$ as an application.

We consider the problem of optimal assignment of NOP due-dates ton jobs and sequencing them on a single machine to minimize a penalty function depending on the values of assigned constant waiting allowance and maximum job tardiness. It is shown that the earliest due date (EDD) order is an optimal sequence. For finding optimal constant waiting allowance, we reduce the problem to a multiple objective piecewise linear programming with single variable. An efficient algorithm for optimal solution of the problem is given.