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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.

U ovom diplomskom radu smo izveli Hilbertovu nejednakost te pokazali neke njene primjene. U prvome dijelu rada smo izveli Hilbertovu nejednakost u diskretnom i integralnom obliku te odredili da je optimalna konstanta u oba slučaja jednaka π. Za nejednakost u diskretnom obliku smo ponudili i alternativni dokaz pomoću geometrijske interpretacije. Također, izveli smo poopćenu Hilbertovu nejednakost koja vrijedi za konjugirane potencije p, q > 1 te pokazali da, uz različitu optimalnu konstantu, mogu vrijediti i slične nejednakosti, npr. varijanta s maksimumom koja nastaje zamjenom izraza m + n u nazivniku s max (m, n). Na kraju smo pokazali da je lijeva strana Hilbertove nejednakosti nenegativna. U drugome dijelu smo se bavili primjenama Hilbertove nejednakosti. Za primjene su nam najbitnije Hilbertove matrice pa smo im, nakon što smo ih definirali, iskazali i neka od njihovih važnijih svojstava, poput (anti)simetričnosti i regularnosti. Zatim smo uveli trigonometrijske polinome pomoću kojih smo dokazali još jedan oblik Hilbertove nejednakosti te odredili normu Hilbertovih matrica. Na kraju smo Hilbertovu nejednakost primijenili na integrale polinoma. In this thesis, we derive Hilbert’s inequality and present some of its applications. In the first chapter, we derive Hilbert’s inequality in its discrete and integral forms and determine that in both cases, π is the optimal constant. Additionally, we offer an alternative proof of discrete Hilbert’s inequality using a geometric interpretation. We also derive a generalization of the discrete Hilbert’s inequality for conjugate exponents p, q > 1 and show that similar inequalities, such as the maximum variant where the expression m + n in the denominator on the left side is replaced with max (m, n), also hold. In the end, we prove that the left-hand side of the discrete Hilbert’s inequality is non-negative. In the second chapter, after once again reviewing the key notions, we define Hilbert matrices and state some of their properties, such as (skew)symmetry and regularity. Next, we introduce trigonometric polynomials and use them to prove another form of Hilbert’s inequality, as well as determine the norm of Hilbert matrices. In the end, we apply Hilbert’s inequality to integrals of polynomials.

Numerous empirical studies have witnessed a plastic increase in meiotic recombination rate in organisms experiencing physiological stress due to unfavourable environmental conditions. Yet, it is not clear enough which characteristics of an ecological factor (intensity, duration, variability, etc.) make it stressogenic and therefore recombinogenic for an organism. Several previous theoretical models proceeded from the assumption that organisms increase their recombination rate when the environment becomes more severe, and demonstrated the evolutionary advantage of such recombination strategy. Here we explore another stress-associated recombination strategy, implying a reversible increase in recombination rate each time when the environment alternates. We allow such plastic changes in the organisms, grown in an environment different from that of their parents, and, optionally, also in their offspring. We show that such shift-inducible recombination is always favoured over intermediate constant optimal recombination. Besides, it sometimes outcompetes also zero and free optimal constant recombination, therefore making selection on recombination less polarized. Shift-inducible strategies with a longer, transgenerational plastic effect, are favoured under slightly stronger selection and longer period. These results hold for both panmixia and partial selfing, although selfing makes the dynamics of recombination modifier alleles faster. Our results suggest that epigenetic factors, presumably underlying the environmental plasticity of recombination, may play an important evolutionary role.

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 $$p'$$.

The problem of minimum-time, constant-thrust orbital transfer between coplanar circular orbits is revisited using a relative motion approach in curvilinear coordinates for the dynamics and an indir...

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

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.

We survey some classical norm inequalities of Hardy, Kallman, Kato, Kolmogorov, Landau, Littlewood, and Rota of the type \[ \|A f\|_{\mathcal{X}}^2 \leq C \|f\|_{\mathcal{X}} \big\|A^2 f\big\|_{\mathcal{X}}, \quad f \in dom\big(A^2\big), \] and recall that under exceedingly stronger hypotheses on the operator $A$ and/or the Banach space $\mathcal{X}$, the optimal constant $C$ in these inequalities diminishes from $4$ (e.g., when $A$ is the generator of a $C_0$ contraction semigroup on a Banach space $\mathcal{X}$) all the way down to $1$ (e.g., when $A$ is a symmetric operator on a Hilbert space $\mathcal{H}$). We also survey some results in connection with an extension of the Hardy-Littlewood inequality involving quadratic forms as initiated by Everitt., Comment: 28 pages, some updates added

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

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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

Recently, we have developed optimal harvesting policies based on profit optimization in random varying environments. Namely, we have considered a logistic stochastic differential equation growth model, with the purpose of discussing the use of variable versus constant effort harvesting policies in terms of the expected accumulated discounted profit during a finite time interval. Using realistic parameters, we have concluded that there is only a slight reduction in profit when choosing the applicable constant effort policy instead of the variable effort policy, which presents strong disadvantages. Here, we apply a logistic growth model and a more general profit structure to present alternative policies based on variable effort, named stepwise policies, where the harvesting effort is determined, under the optimal variable effort policy, at the beginning of each year (or of each biennium) but is kept constant during that year (biennium). Replacing the optimal variable effort policy by these stepwise non-optimal policies has the advantage of applicability but, at best, considerably reduces the already small profit advantage the optimal variable effort policy has over the optimal constant effort sustainable policy.

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.