Universal extrapolation spaces for \hbox {C}_{0}-semigroups

The classical theory of Sobolev towers allows for the construction of an infinite ascending chain of extrapolation spaces and an infinite descending chain of interpolation spaces associated with a given \(C_0\) -semigroup on a Banach space. In this note we first generalize the latter to the case of a strongly continuous and exponentially equicontinuous semigroup on a complete locally convex space. As a new concept—even for \(C_0\) -semigroups on Banach spaces—we then define a universal extrapolation space as the completion of the inductive limit of the ascending chain. Under mild assumptions we show that the semigroup extends to this space and that it is generated by an automorphism of the latter. Dually, we define a universal interpolation space as the projective limit of the descending chain. We show that the restriction of the initial semigroup to this space is again a semigroup and always has an automorphism as generator.     

Measures and submeasures on $$\hbox {d}_{0}$$-algebras

The structure of \(\hbox {d}_{0}\)-algebra is a generalization of a D-lattice. We extend to this structure the definitions of measure and submeasure, and investigate their properties. We also investigate the relationships between uniformities and measures (or submeasures).     

On the Diophantine equation $$X^{2N}+2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5$$

We prove that for each prime p, positive integer \(\alpha \), and non-negative integers \(\beta \) and \(\gamma \), the Diophantine equation \(X^{2N} + 2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5\) has no solution with N, X, \(Z\in \mathbb {Z}^+\), \(N > 1\), and \(\gcd (X,Z) = 1\).     

On Projective Modules in Category \mathcal{O}_{int} of Quantum \mathfrak{sl}_{2}

The restriction of a Verma module of ${\bf U}(\mathfrak{sl}_3)$ to ${\bf U}(\mathfrak{sl}_2)$ is isomorphic to a Verma module tensoring with all the finite dimensional simple modules of ${\bf U}(\mathfrak{sl}_2)$ . The canonical basis of the Verma module is compatible with such a decomposition. An explicit decomposition of the tensor product of the Verma module of highest weight 0 with a finite dimensional simple module into indecomposable projective modules in the category $\mathcal O_{\rm{int}}$ of quantum $\mathfrak{sl}_2$ is given.     

Unified optimization of \hbox {H}_{\infty } index and upper stability bound for singularly perturbed systems

In this paper, unified optimization problem for the upper stability bound \(\varepsilon ^{*}\) and the \(\hbox {H}_{\infty }\) performance index \(\gamma \) based on state feedback is considered for singularly perturbed systems. First, a sufficient condition for the existence of state feedback controller is presented in terms of linear matrix inequalities such that the resulting closed-loop system is asymptotically stable if \(0<\varepsilon <\varepsilon ^{*}\) and also guarantees \(\hbox {H}_{\infty }\) performance index. Furthermore, a new algorithm to optimize these two indices simultaneously is proposed based on Nash game theory which transfers multi-objective problem into a single objective problem as well we determines the objective weights. Then an optimal state feedback controller can be derived. Finally, some numerical examples are provided to demonstrate the effectiveness and correctness of the proposed results.     

On integral graphs which belong to the class\overline {\alpha K_{a,a} \cup \beta {\rm K}_{b,b} }

LetG be a simple graph and letG denote its complement. We say thatG is integral if its spectrum consists of integral values. In this work we establish a characterization of integral graphs which belong to the class $\overline {\alpha K_{a,a} \cup \beta {\rm K}_{b,b} } $ wheremG denotes them-fold union of the graphG.     

On the Convergence of Products γ^-sh1hn in the Adams Spectral Sequence

Abstract Let A be the mod p Steenrod algebra and S the sphere spectrum localized at p, where p is an odd prime. In 2001 Lin detected a new family in the stable homotopy of spheres which is represented by （b0hn-h1bn-1）∈ ExtA^3,（p^n＋p）q（Zp,Zp） in the Adams spectral sequence. At the same time, he proved that i.（hlhn） ∈ExtA^2,（p^n＋P）q（H^＊M, Zp） is a permanent cycle in the Adams spectral sequence and converges to a nontrivial element ξn∈π（p^n＋p）q-2M. In this paper, with Lin＇s results, we make use of the Adams spectral sequence and the May spectral sequence to detect a new nontrivial family of homotopy elements jj′j^-γsi^-i′ξn in the stable homotopy groups of spheres. The new one is of degree p^nq ＋ sp^2q ＋ spq ＋ （s - 2）q ＋ s - 6 and is represented up to a nonzero scalar by hlhnγ-s in the E2^s＋2,＊-term of the Adams spectral sequence, where p ≥ 7, q = 2（p - 1）, n ≥ 4 and 3 ≤ s 〈 p.     

Isometric embeddings of \mathbb{Z}_{p^k} in the Hamming space \mathbb{F}_{p}^{N} and \mathbb{Z}_{p^k}-linear codes

Isometric embeddings of $\mathbb{Z}_{p^n+1}$ into the Hamming space ( $\mathbb{F}_{p}^{p^n},w$ ) have played a fundamental role in recent constructions of non-linear codes. The codes thus obtained are very good codes, but their rate is limited by the rate of the first-order generalized Reed–Muller code—hence, when n is not very small, these embeddings lead to the construction of low-rate codes. A natural question is whether there are embeddings with higher rates than the known ones. In this paper, we provide a partial answer to this question by establishing a lower bound on the order of a symmetry of ( $\mathbb{F}_{p}^{N},w$ ).     

Analysis of $$\hbox {M}^{\mathrm {x}}/\hbox {G}/1$$ queues with impatient customers

This paper considers a batch arrival \(\hbox {M}^{\mathrm {x}}/\hbox {G}/1\) queue with impatient customers. We consider two different model variants. In the first variant, customers in the same batch are assumed to have the same patience time, and patience times associated with batches are i.i.d. according to a general distribution. In the second variant, patience times of customers in the same batch are independent, and they follow a general distribution. Both variants are related to an M/G/1 queue in which the service time of a customer depends on its waiting time. Our main focus is on the virtual and actual waiting times, and on the loss probability of customers.     

Higher-order $$(\Phi , \rho )\hbox {-}V$$-invexity and duality for vector optimization problems

In this paper, we introduce the concept of higher-order \((\Phi , \rho )\hbox {-}V\)-invex function and prove duality theorems for higher-order Wolfe and Mond–Weir type duals of the vector optimization problem using aforesaid assumptions.     

On the difference equationx_{n + 1} = \alpha + \frac{{x_n^p }}{{x_{n - 1}^p }}

We study, firstly, the dynamics of the difference equation $x_{n + 1} = \alpha + \frac{{x_n^p }}{{x_{n - 1}^p }}$ , withp ∈ (0,1) and α∈ [0, ∞). Then, we generalize our results to the (k + 1)th order difference equation $x_{n + 1} = \alpha + \frac{{x_n^p }}{{x_{n - k}^p }}$ ,k = 2, 3,... with positive initial conditions.     

On the recursive sequencex_{n + 1} = \alpha + \frac{{x_{n - 1}^p }}{{x_n^p }}

The boundedness, global attractivity, oscillatory and asymptotic periodicity of the positive solutions of the difference equation of the form $$x_{n + 1} = \alpha + \frac{{x_{n - 1}^p }}{{x_n^p }}, n = 0,1,...$$ is investigated, where all the coefficients are nonnegative real numbers.     

On the Dynamics of x_{\bf{n} + \bf{1}} = {{ \mgreek{a} + \mgreek{g} x_{n-1} + \mgreek{d} x_{n-2}} \over {A + x_{n-2}}}

We investigate the global stability, the periodic character and the boundedness nature of solutions of the equation in the title for all admissible nonnegative values of the parameters and the initial conditions. We show that the solutions exhibit a trichotomy character depending on how the parameter γ compares to the sum of the parameters δ and A.     

On the continuity of \text{ SLE }_{\kappa } in \kappa

We prove that for almost every Brownian motion sample, the corresponding SLE \(_\kappa \) curves parameterized by capacity exist and change continuously in the supremum norm when \(\kappa \) varies in the interval \([0,\kappa _0)\) , where \(\kappa _0=8(2-\sqrt{3})\approx 2.1\) . We estimate the \(\kappa \) -dependent modulus of continuity of the curves and also give an estimate on the modulus of continuity for the supremum norm change with \(\kappa \) .     

On Quadratic Differentials and Twisted Normal Maps of Surfaces in {\mathbb{S}^{2} \times\mathbb{R}} and {\mathbb{H}^{2} \times\mathbb{R}}

Given a Lie group G with a bi-invariant metric and a compact Lie subgroup K, Bittencourt and Ripoll used the homogeneous structure of quotient spaces to define a Gauss map ${\mathcal{N}:M^{n}\rightarrow{\mathbb{S}}}$ on any hypersupersurface ${M^{n}\looparrowright G/K}$ , where ${{\mathbb{S}}}$ is the unit sphere of the Lie algebra of G. It is proved in Bittencourt and Ripoll (Pacific J Math 224:45–64, 2006) that M ⁿ having constant mean curvature (CMC) is equivalent to ${\mathcal{N}}$ being harmonic, a generalization of a Ruh–Vilms theorem for submanifolds in the Euclidean space. In particular, when n = 2, the induced quadratic differential ${\mathcal{Q}_{\mathcal{N}}:=(\mathcal{N}^{\ast}g)^{2,0}}$ is holomorphic on CMC surfaces of G/K. In this paper, we take ${G/K={\mathbb{S}}^{2}\times{\mathbb{R}}}$ and compare ${\mathcal{Q}_{\mathcal{N}}}$ with the Abresch–Rosenberg differential ${\mathcal{Q}}$ , also holomorphic for CMC surfaces. It is proved that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ , after showing that ${\mathcal{N}}$ is the twisted normal given by (1.5) herein. Then we define the twisted normal for surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ and prove that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ as well. Within the unified model for the two product spaces, we compute the tension field of ${\mathcal{N}}$ and extend to surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ the equivalence between the CMC property and the harmonicity of ${\mathcal{N}.}$