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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.

The problem of equivalent parameters and the best constant factor for the existence of quasi-homogeneous half-discrete Hilbert type inequality $ \int_{R_ + ^m} {\sum\limits_{n = 1}^\infty G } \left( {{n^{{\lambda _1}}}/\left\| x \right\|_{m, \rho }^{{\lambda _2}}} \right){a_n}f(x){\rm{d}}x \le M{\left\| {\tilde a} \right\|_{p, \alpha }}{\left\| f \right\|_{q, \beta }} $ is discussed, and their applications in the study of operator boundedness and norm are also considered.

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.

The existence of an extremal in an exponential Sobolev type inequality, with optimal constant, in Gauss space is established. A key step in the proof is an augmented version of the relevant inequality, which, by contrast, fails for a parallel classical inequality by Moser in the Euclidean space., Comment: 28 pages

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.

In the article, we consider the multilinear fraction maximal operators and the multilinear fraction integrals with rough kernels, we introduce a new class for (m+1)‐weights tuple (u,ω), and prove the optimal constant estimates of the sharp multi‐weighted bounds for both operators. We also show the sharp multi‐weighted bounds for the Cohen–Gosselin type multi‐commutator of the both operators.

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, ∞).

Sepsis is one of the biggest risks to patient safety, with a natural mortality rate between 25% and 50%. It is difficult to diagnose, and no validated standard for diagnosis currently exists. A commonly used scoring criteria is the quick sequential organ failure assessment (qSOFA). It demonstrates very low specificity in ICU populations, however. We develop a method to personalize thresholds in qSOFA that incorporates easily to measure patient baseline characteristics. We compare the personalized threshold method to qSOFA, five previously published methods that obtain an optimal constant threshold for a single biomarker, and to the machine learning algorithms based on logistic regression and AdaBoosting using patient data in the MIMIC-III database. The personalized threshold method achieves higher accuracy than qSOFA and the five published methods and has comparable performance to machine learning methods. Personalized thresholds, however, are much easier to adopt in real-life monitoring than machine learning methods as they are computed once for a patient and used in the same way as qSOFA, whereas the machine learning methods are hard to implement and interpret.

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.

The $L^2 \to L^p$ adjoint Fourier restriction inequality on the $d$-dimensional hyperboloid $\mathbb{H}^d \subset \mathbb{R}^{d+1}$ holds provided $6 \leq p < \infty$, if $d=1$, and $2(d+2)/d \leq p\leq 2(d+1)/(d-1)$, if $d\geq2$. Quilodr\'{a}n recently found the values of the optimal constants in the endpoint cases $(d,p)\in\{(2,4),(2,6),(3,4)\}$ and showed that the inequality does not have extremizers in these cases. In this paper we answer two questions posed by Quilodr\'{a}n, namely: (i) we find the explicit value of the optimal constant in the endpoint case $(d,p) = (1,6)$ (the remaining endpoint for which $p$ is an even integer) and show that there are no extremizers in this case; and (ii) we establish the existence of extremizers in all non-endpoint cases in dimensions $d \in \{1,2\}$. This completes the qualitative description of this problem in low dimensions., Comment: 32 pages, 7 figures

Let $\Omega$ be a domain in $R^d$ and $d_\Gamma$ the Euclidean distance to the boundary $\Gamma$. We investigate whether the weighted Hardy inequality \[ \|d_\Gamma^{\delta/2-1}\varphi\|_2\leq a_\delta\,\|d_\Gamma^{\delta/2}\,(\nabla\varphi)\|_2 \] is valid, with $\delta\geq 0$ and $a_\delta>0$, for all $\varphi\in C_c^1(\Gamma_r)$ and all small $r>0$ where $\Gamma_r=\{x\in\Omega: d_\Gamma(x)1$ but if $\delta\in[0,1\rangle$ then $a_\delta(\Gamma)$ can be strictly larger than $2/|\delta-1|$. Finally we use these results to establish self-adjointness criteria for degenerate elliptic diffusion operators., Comment: This version differs from the earlier one by the correction of various typos, an extended Section 7 and an additional reference

We improve the discretization technique for weighted Lorentz norms by eliminating all "non-degeneracy" restrictions on the involved weights. We use the new method to provide equivalent estimates on the optimal constant $C$ such that the inequality $$\left( \int_0^L (f^*(t))^{p_2} w(t)\,\mathrm{d}t \right)^\frac 1{p_2} \le C \left( \int_0^L \left( \int_0^t u(s)\,\mathrm{d}s \right)^{-\frac {p_1}\alpha} \left( \int_0^t (f^*(s))^\alpha u(s) \,\mathrm{d}s \right)^\frac {p_1}\alpha v(t) \,\mathrm{d}t \right)^\frac 1{p_1}$$ holds for all relevant measurable functions, where $L\in(0,\infty]$, $\alpha, p_1, p_2 \in (0,\infty)$ and $u$, $v$, $w$ are locally integrable weights, $u$ being strictly positive. It the case of weights that would be otherwise excluded by the restrictions, it is shown that additional limit terms naturally appear in the characterizations of the optimal $C$. A weak analogue for $p_1=\infty$ is also presented., Comment: 23 pages

We state the following weighted Hardy inequality: $$\begin{aligned} c_{o, \mu }\int _{{\mathbb {R}}^N}\frac{\varphi ^2 }{|x|^2}\, \mathrm{d}\mu \le \int _{{\mathbb {R}}^N} |\nabla \varphi |^2 \, \mathrm{d}\mu + K \int _{\mathbb {R}^N}\varphi ^2 \, \mathrm{d}\mu \quad \forall \, \varphi \in H_\mu ^1, \end{aligned}$$in the context of the study of the Kolmogorov operators: $$\begin{aligned} Lu=\Delta u+\frac{\nabla \mu }{\mu }\cdot \nabla u, \end{aligned}$$perturbed by inverse square potentials and of the related evolution problems. The function $$\mu $$ in the drift term is a probability density on $$\mathbb {R}^N$$. We prove the optimality of the constant $$c_{o, \mu }$$ and state existence and nonexistence results following the Cabre–Martel’s approach (Cabre and Martel in C R Acad Sci Paris 329 (11): 973–978, 1999) extended to Kolmogorov operators.

In this work, we obtain an existence of nontrivial solutions to a minimization problem involving a fractional Hardy-Sobolev type inequality in the case of inner singularity. Precisely, for $\lambda>0$ we analyze the attainability of the optimal constant $$ \mu_{\alpha, \lambda}(\Omega):=\inf\left\{ [u]^2_{s,\Omega}+\lambda\int_{\Omega}|u|^2 \, dx \colon u\in H^s(\Omega), \, \int_{\Omega} \frac{|u(x)|^{2_{s,\alpha}}}{|x|^{\alpha}} \, dx=1 \right\}, $$ where $04s, 0, Comment: 11 pages

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.

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.

Gogatishvili, Amiran/0000-0003-3459-0355; Yildiz, Tugce Unver/0000-0003-0414-8400; Gogatishvili, Amiran/0000-0003-3459-0355; Mustafayev, Rza/0000-0002-2806-9646 WOS: 000416445500016 In this paper, characterizations of the embeddings between weighted Copson function spaces Cop(p1,q1)(u(1),v(1)) and weighted Cesaro function spaces Ces(p2,q2) (u(2) , v(2)) are given. In particular, two-sided estimates of the optimal constant c in the inequality (integral(infinity)(0) (integral(t)(0) f(tau)(p2)v2(tau)d tau)(q2/p2) u2(t)dt)(1/q2)& para;& para

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.

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.

Let $D \subset \mathbb{C}$ be a domain with $0 \in D$. For $R>0$, let ${{\hat \omega }_D}\left( {R} \right)$ denote the harmonic measure of $ D \cap \left\{ {\left| z \right| = R} \right\}$ at $0$ with respect to the domain $ D \cap \left\{ {\left| z \right| < R} \right\} $ and ${\omega_D}\left( {R} \right)$ denote the harmonic measure of $\partial D \cap \left\{ {\left| z \right| \ge R} \right\}$ at $0$ with respect to $D$. The behavior of the functions ${\omega_D}$ and ${{\hat \omega }_D}$ near $\infty$ determines (in some sense) how large $D$ is. However, it is not known whether the functions ${\omega_D}$ and ${{\hat \omega }_D}$ always have the same behavior when $R$ tends to $\infty$. Obviously, ${\omega_D}\left( {R} \right) \le {{\hat \omega }_D}\left( {R} \right)$ for every $R>0$. Thus, the arising question, first posed by Betsakos, is the following: Does there exist a positive constant $C$ such that for all simply connected domains $D$ with $0 \in D$ and all $R>0$, \[{\omega_D}\left( {R} \right) \ge C{{\hat \omega }_D}\left( {R} \right)? \] In general, we prove that the answer is negative by means of two different counter-examples. However, under additional assumptions involving the geometry of $D$, we prove that the answer is positive. We also find the value of the optimal constant for starlike domains., Comment: 22 pages, 13 figures

The main results in the paper are the weighted multipolar Hardy inequalities \begin{document}$ \begin{equation*} c\int_{\mathbb{R}^N}\sum\limits_{i = 1}^n\frac{\varphi^2}{|x-a_i|^2}\,\mu(x)dx \leq\int_{\mathbb{R}^N}|\nabla \varphi |^2\mu(x)dx+ K\int_{\mathbb{R}^N} \varphi^2\mu(x)dx, \end{equation*} $\end{document} in \begin{document}$ \mathbb{R}^N $\end{document} for any \begin{document}$ \varphi $\end{document} in a suitable weighted Sobolev space, with \begin{document}$ 0 , \begin{document}$ a_1,\dots,a_n\in \mathbb{R}^N $\end{document} , \begin{document}$ K $\end{document} constant. The weight functions \begin{document}$ \mu $\end{document} are of a quite general type. The paper fits in the framework of Kolmogorov operators defined on smooth functions \begin{document}$ \begin{equation*} Lu = \Delta u+\frac{\nabla \mu}{\mu}\cdot\nabla u, \end{equation*} $\end{document} perturbed by multipolar inverse square potentials, and related evolution problems. Necessary and sufficient conditions for the existence of exponentially bounded in time positive solutions to the associated initial value problem are based on weighted Hardy inequalities. For constants \begin{document}$ c $\end{document} beyond the optimal Hardy constant \begin{document}$ c_{o,\mu} $\end{document} we are able to show nonexistence of positive solutions.

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

In this paper we provide an explicit formula for the optimal lower bound of Donaldson's J-functional, in the sense of finding explicitly the optimal constant in the definition of coercivity, which always exists and takes negative values in general. This constant is positive precisely if the J-equation admits a solution, and the explicit formula has a number of applications. First, this leads to new existence criteria for constant scalar curvature K\"ahler (cscK) metrics in terms of Tian's alpha invariant. Moreover, we use the above formula to discuss Calabi dream manifolds and an analogous notion for the J-equation, and show that for surfaces the optimal bound is an explicitly computable rational function which typically tends to minus infinity as the underlying class approaches the boundary of the K\"ahler cone, even when the underlying K\"ahler classes admit cscK metrics. As a final application we show that if the Lejmi-Sz\'ekelyhidi conjecture holds, then the optimal bound coincides with its algebraic counterpart, the set of J-semistable classes equals the closure of the set of uniformly J-stable classes in the K\"ahler cone, and there exists an optimal degeneration for uniform J-stability., Comment: Final version, to appear in Adv. Math

In 1989 Almgren and Lieb proved a rearrangement inequality for the Sobolev spaces of fractional order $W^{s,p}$. The case $p = 2$ of their result implies the nonlocal isoperimetric inequality \[ \frac{P_s(E)}{|E|^{\frac{N-2s}N}} \ge \frac{P_s(B_1)}{|B_1|^{\frac{N-2s}N}},\ \ \ \ \ \ \ 0, Comment: This submission represents an entirely revised version, with different title and abstract, of a previous submission, entitled:"Observations on a theorem of Almgren and Lieb"

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.

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.

We revisit the celebrated family of BDG-inequalities introduced by Burkholder, Gundy (Acta Math. 124 (1970) 249–304) and Davis (Israel J. Math. 8 (1970) 187–190) for continuous martingales. For the inequalities $\mathbb{E}[\tau^{\frac{p}{2}}]\leq C_{p}\mathbb{E}[(B^{*}(\tau))^{p}]$ with $0

We obtain the sharp factor of the two-sides estimates of the optimal constant in generalized Hardy's inequality with two general Borel measures on $\mathbb{R}$, which generalizes and unifies the known continuous and discrete cases.

Let P : C n → C be an m-homogeneous polynomial given by P ( x ) = ∑ 1 ≤ j 1 ≤ … ≤ j m ≤ n c j 1 … j m x j 1 … x j m . Defant and Schluters defined a non-symmetric associated m-form L P : ( C n ) m → C by L P ( x ( 1 ) , … , x ( m ) ) = ∑ 1 ≤ j 1 ≤ … ≤ j m ≤ n c j 1 … j m x j 1 ( 1 ) … x j m ( m ) . They estimated the norm of L P on ( C n , ‖ ⋅ ‖ ) m by the norm of P on ( C n , ‖ ⋅ ‖ ) times a ( c log ⁡ n ) m 2 factor for every 1-unconditional norm ‖ ⋅ ‖ on C n . A symmetrization procedure based on a card-shuffling algorithm which (together with Defant and Schluters' argument) brings the constant term down to ( c m log ⁡ n ) m − 1 is provided. Regarding the lower bound, it is shown that the optimal constant is bigger than ( c log ⁡ n ) m / 2 when n ≫ m . Finally, the case of l p -norms ‖ ⋅ ‖ p with 1 ≤ p 2 is addressed.

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.

Inspired by Clarkson's inequalities for $L^p$ and continuing work from \cite{CR}, this paper computes the optimal constant $C$ in the weak parallelogram laws $$ \|f + g \|^r + C\|f - g\|^r \leq 2^{r-1}\big( \|f\|^r + \|g\|^r \big), $$ $$ \|f + g \|^r + C\|f- g \|^r \geq 2^{r-1}\big( \|f\|^r + \|g \|^r \big)$$ for the $L^p$ spaces, $1 < p < \infty$., 10 pages

In this paper we obtain the best constants in some higher order Sobolev inequalities in the critical exponent. These inequalities can be separated into two types: those that embed into L ∞ ( R N ) and those that embed into slightly larger target spaces. Concerning the former, we show that for k ∈ { 1 , … , N − 1 } , N − k even, one has an optimal constant c k > 0 such that ‖ u ‖ L ∞ ≤ c k ∫ | ∇ k ( − Δ ) ( N − k ) ∕ 2 u | for all u ∈ C c ∞ ( R N ) (the case k = N was handled in Shafrir, 2018). Meanwhile the most significant of the latter is a variation of D. Adams’ higher order inequality of J. Moser: For Ω ⊂ R N , m ∈ N and p = N m , there exists A > 0 and optimal constant β 0 > 0 such that ∫ Ω exp ( β 0 | u | p ′ ) ≤ A | Ω | for all u such that ‖ ∇ m u ‖ L p ( Ω ) ≤ 1 , where ‖ ∇ m u ‖ L p ( Ω ) is the traditional semi-norm on the space W m , p ( Ω ) .

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.

Frame difference families, which can be obtained via a careful use of cyclotomic conditions attached to strong difference families, play an important role in direct constructions for resolvable balanced incomplete block designs. We establish asymptotic existences for several classes of frame difference families. As corollaries new infinite families of 1-rotational $(pq+1,p+1,1)$-RBIBDs over $\mathbb{F}_{p}^+ \times \mathbb{F}_{q}^+$ are derived, and the existence of $(125q+1,6,1)$-RBIBDs is discussed. We construct $(v,8,1)$-RBIBDs for $v\in\{624,1576,2976,5720,5776,10200,14176,24480\}$, whose existence were previously in doubt. As applications, we establish asymptotic existences for an infinite family of optimal constant composition codes and an infinite family of strictly optimal frequency hopping sequences., arXiv admin note: text overlap with arXiv:1702.07500

Th e eff ects of alternating and constant temperatures and light regimes, osmotic and salt stress and pH were tested on seed germination in four populations of Eryngium caeruleum . Laboratory experiments revealed that the four populations exhibited diff erent responses to alternating temperature and light conditions. Alternating temperature and photoperiod had a greater positive eff ect on seed germination compared to complete darkness. Th e optimal constant temperature within 10 oC to 15 oC for seed germination of each population was determined in a light/dark photoperiod. Seed germination severely decreased under water stress and was completely inhibited at -0.8 MPa osmotic potential. Saline stress sharply decreased germination percentage. Germination was restricted by pH lower and higher than 5 and 8, respectively. Th e information obtained from this study helps to fi ll the gap of knowledge about seed germination requirements of E. caeruleum and enhance our understanding of this species distribution and its potential to develop in stressful and/or new habitats.

An inequality of Hardy and Littlewood for m-homogeneous polynomials on ℓp spaces is valid for p > m: In this note, among other results, we present an optimal version of this inequality for the case p = m and obtain the optimal constant, when restricted to the case of 2-homogeneous polynomials on ℓ2(R2). In an Appendix we justify why, curiously, the optimal exponents of some Hardy-Littlewood type inequalities do not behave smoothly.Mathematics Subject Classication (2010): 47B10, 26D15, 46B25.Key words: Hardy and Littlewood inequalities, multiple summing operators.

We study the large time behavior of solutions v : Ω × ( 0 , ∞ ) → R of the PDE ∂ t ( | v | p − 2 v ) = Δ p v . We show that e ( λ p / ( p − 1 ) ) t v ( x , t ) converges to an extremal of a Poincare inequality on Ω with optimal constant λ p , as t → ∞ . We also prove that the large time values of solutions approximate the extremals of a corresponding “dual” Poincare inequality on Ω. Moreover, our theory allows us to deduce the large time asymptotics of related doubly nonlinear flows involving various boundary conditions and nonlocal operators.

International audience; This paper is devoted to the Moser-Trudinger-Onofri inequality on smooth compact connected Riemannian manifolds. We establish a rigidity result for the Euler-Lagrange equation and deduce an estimate of the optimal constant in the inequality on two-dimensional closed Riemannian manifolds. Compared to existing results, we provide a non-local criterion which is well adapted to variational methods, introduce a nonlinear flow along which the evolution of a functional related with the inequality is monotone and get an integral remainder term which allows us to discuss optimality issues. As an important application of our method, we also consider the non-compact case of the Moser-Trudinger-Onofri inequality on the two-dimensional Euclidean space, with weights. The standard weight is the one that is computed when projecting the two-dimensional sphere using the stereographic projection, but we also give more general results which are of interest, for instance, for the Keller-Segel model in chemotaxis.

The search of the optimal constant for a generalized Wirtinger inequality in an interval consists in minimizing the $p$-norm of the derivative among all functions whose $q$-norm is equal to~1 and whose $(r-1)$-power has zero average. Symmetry properties of minimizers have attracted great attention in mathematical literature in the last decades, leading to a precise characterization of symmetry and asymmetry regions. In this paper we provide a proof of the symmetry result without computer assisted steps, and a proof of the asymmetry result which works as well for local minimizers. As a consequence, we have now a full elementary description of symmetry and asymmetry cases, both for global and for local minima. Proofs rely on appropriate nonlinear variable changes., Comment: 17 pages

In this paper our main results are the multipolar weighted Hardy inequality c ∑ i = 1 n ∫ R N φ 2 | x − a i | 2 d μ ≤ ∫ R N | ∇ φ | 2 d μ + K ∫ R N φ 2 d μ , c ≤ c o , where the functions φ belong to a weighted Sobolev space H μ 1 , and the proof of the optimality of the constant c o = c o ( N ) : = ( N − 2 2 ) 2 . The Gaussian probability measure dμ is the unique invariant measure for Ornstein–Uhlenbeck type operators. This estimate allows us to get necessary and sufficient conditions for the existence of positive solutions to a parabolic problem corresponding to the Kolmogorov operators defined on smooth functions and perturbed by a multipolar inverse square potential L u + V u = ( Δ u + ∇ μ μ ⋅ ∇ u ) + ∑ i = 1 n c | x − a i | 2 u , x ∈ R N , c > 0 , a 1 , … , a n ∈ R N .

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.

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.