Boundedness and exponential stability for periodic time dependent systems

The time dependent $2$-periodic system \begin{equation*} \dot x{(t)} = A(t)x{(t)} , \ t\in \mathbb{R}, \ \ x(t) \in\mathbb{C}^{n}\tag{A(t)} \end{equation*} is uniformly exponentially stable if and only if for each real number $\mu$ and each $2$-periodic, $\mathbb{C}^{n}$-valued function $f,$ the solution of the Cauchy Problem \begin{equation*} \left\{\begin{split} \dot y{(t)} &= A(t) y{(t)} + e^{i \mu t}f(t),\ \ t\in \mathbb{R}_+, \ y(t) \in \mathbb{C}^{n} \\ y(0) &= 0 \end{split}\right. \end{equation*} is bounded. In this note we prove a result that has the above result as an immediate corollary. Some new characterizations for uniform exponential stability of $(A(t))$ in terms of the Datko type theorems are also obtained as corollaries

Spectral characterizations for Hyers-Ulam stability

First we prove that an $n\times n$ complex linear system is Hyers-Ulam stable if and only if it is dichotomic (i.e. its associated matrix has no eigenvalues on the imaginary axis $i\mathbb{R}$). Also we show that the scalar differential equation of order $n,$ $$\begin{split} x^{(n)}(t)=a_1x^{(n-1)}(t)+\ldots+a_{n-1}{x}'(t)+a_nx(t),\quad t\in\mathbb{R}_+:=[0, \infty), \end{split}$$ is Hyers-Ulam stable if and only if the algebraic equation $$\begin{split} z^n=a_1z^{n-1}+\cdots +a_{n-1}z+a_n, \end{split}$$ has no roots on the imaginary axis

Connections between exponential stability and boundedness of solutions of a couple of differential time depending and periodic systems

Among others, we prove that the vectorial time dependent $q$-periodic differential system $$\dot x(t)=A(t)x(t),\quad t\in\mathbb{R}, \quad x(t)\in\mathbb{C}^n\tag{A(t)}$$ is uniformly exponentially stable (i.e. all its solutions decay exponentially at infinity) if and only if for each vector $b\in \mathbb{C}^n$, the solution of the Cauchy Problem $$\dot{y}(t)=A(t)y(t)+e^{i\mu t}b,\quad t\ge 0,\quad b\in\mathbb{C}^n,\quad y(0)=0$$ is bounded on $\mathbb{R}_+,$ uniformly in respect with the parameter $\mu$ on the entire real axis. As a consequence, we get that the system $(A(t))$ is uniformly exponentially stable if and only if for each vector $x\in \mathbb{C}^n,$ the map $$t\mapsto\int\limits_0^t||ds$$ is bounded on $\mathbb{R}_+.$ This latter result is a weak version of the Barbashin theorem which seems to be new. Here $\Phi(t)$ is the fundamental matrix associated to the system $(A(t)).

Connections between the stability of a Poincare map and boundedness of certain associate sequences

Let $m\ge 1$ and $N\ge 2$ be two natural numbers and let ${\mathcal{U}}=\{U(p, q)\}_{p\ge q\ge 0}$ be the $N$-periodic discrete evolution family of $m\times m$ matrices, having complex scalars as entries, generated by ${\mathcal{L}}(\mathbb{C}^m)$-valued, $N$-periodic sequence of $m\times m$ matrices $(A_n).$ We prove that the solution of the following discrete problem $$y_{n+1}=A_ny_n+e^{i\mu n}b,\quad n\in\mathbb{Z}_+,\quad y_0=0$$ is bounded for each $\mu\in\mathbb{R}$ and each $m$-vector $b$ if the Poincare map $U(N, 0)$ is stable. The converse statement is also true if we add a new assumption to the boundedness condition. This new assumption refers to the invertibility for each $\mu\in\mathbb{R}$ of the matrix $V_{\mu}:=\sum\nolimits_{\nu=1}^NU(N, \nu)e^{i\mu \nu}.$ By an example it is shown that the assumption on invertibility cannot be removed. Finally, a strong variant of Barbashin's type theorem is proved

Influence of sulphide Cu (I) promoting additives concentration on acid and catalytic properties of high-silica zeolites in straight-run gasoline conversion

In present article the influence of Cu[2]S promoting additives concentration on acid and catalytic properties of high silica MFI-type zeolites is investigated in the process of conversion of straight-run gasoline fractions of gas condensate into high octane components of motor fuels. It was shown that zeolite modified with 1% of Cu[2]S nanoscaled powder possesses the highest acid centers concentration and highest catalytic activity

Switching to Once-Daily Liraglutide From Twice-Daily Exenatide Further Improves Glycemic Control in Patients With Type 2 Diabetes Using Oral Agents

OBJECTIVETo evaluate efficacy and safety of switching from twice-daily exenatide to once-daily liraglutide or of 40 weeks of continuous liraglutide therapy.RESEARCH DESIGN AND METHODSWhen added to oral antidiabetes drugs in a 26-week randomized trial (Liraglutide Effect and Action in Diabetes [LEAD]-6), liraglutide more effectively improved A1C, fasting plasma glucose, and the homeostasis model of β-cell function (HOMA-B) than exenatide, with less persistent nausea and hypoglycemia. In this 14-week extension of LEAD-6, patients switched from 10 μg twice-daily exenatide to 1.8 mg once-daily liraglutide or continued liraglutide.RESULTSSwitching from exenatide to liraglutide further and significantly reduced A1C (0.32%), fasting plasma glucose (0.9 mmol/l), body weight (0.9 kg), and systolic blood pressure (3.8 mmHg) with minimal minor hypoglycemia (1.30 episodes/patient-year) or nausea (3.2%). Among patients continuing liraglutide, further significant decreases in body weight (0.4 kg) and systolic blood pressure (2.2 mmHg) occurred with 0.74 episodes/patient-year of minor hypoglycemia and 1.5% experiencing nausea.CONCLUSIONSConversion from exenatide to liraglutide is well tolerated and provides additional glycemic control and cardiometabolic benefits

Poling effect on distribution of quenched random fields in a uniaxial relaxor ferroelectric

The frequency dependence of the dielectric permitivity's maximum has been studied for poled and unpoled doped relaxor strontium barium niobate $Sr_{0.61}Ba_{0.39}Nb_{2}O_{6}:Cr^{3+}$ (SBN-61:Cr). In both cases the maximum found is broad and the frequency dispersion is strong. The present view of random fields compensation in the unpoled sample is not suitable for explaining this experimental result. We propose a new mechanism where the dispersion of quenched random electric fields, affecting the nanodomains, is minimized after poling. We test our proposal by numerical simulations on a random field Ising model. Results obtained are in agreement with the polarization's measurements presented by Granzow et al. [Phys. Rev. Lett {\bf 92}, 065701 (2004)].Comment: 7 pages, 4 figure

Administration of Endotoxin, Tumor Necrosis Factor, or Interleukin 1 to Rats Activates Skeletal Muscle Branched-Chain α-Keto Acid Dehydrogenase

Protein catabolic states (i.e., sepsis and trauma) are thought to be associated with accelerated oxidation of branched-chain amino acids (BCAA). Branched-chain α-keto acid dehydrogenase (BCKAD), the rate-limiting enzyme for BCAA oxidation by muscle, is regulated by phosphorylation/dephosphorylation. Skeletal muscle BCKAD was only 2-4% active in control rats. Intravenous injection of Salmonella enteritidis endotoxin (0.25-10 mg/kg) did not change total BCKAI) activity, but increased the percent active enzyme in muscle three- to four-fold in 4-6 h. Identical results were observed in adrenalectomized rats pretreated with one dose of α-methylprednisolone (2.5 mg/kg i.p.) 30-60 min before saline or endotoxin injection, indicating that endotoxin\u27s effect was not mediated by hypersecretion of adrenal hormones. Cortisone pretreatment of normal rats (100 mg/kg per d) for 2 d prevented endotoxin-induced activation of muscle BCKAD, suggesting that endogenous secretion products mediated BCKAD activation by endotoxin. Human recombinant tumor necrosis factor-α and/or IL-1β or α (50 μg/kg) increased muscle BCKAD activation two- to fourfold in normal rats 4-6 h after intravenous injection. We conclude that cytokine-mediated activation of muscle BCKAD may contribute to accelerated BCAA oxidation in septicemia

The polaroid image as photo-object

This article is part of a larger project on the cultural history of Polaroid photography and draws on research done at the Polaroid Corporate archive at Harvard and at the Polaroid company itself. It identifies two cultural practices engendered by Polaroid photography, which, at the point of its extinction, has briefly flared into visibility again. It argues that these practices are mistaken as novel but are in fact rediscoveries of practices that stretch back as many as five decades. The first section identifies Polaroid image-making as a photographic equivalent of what Tom Gunning calls the ‘cinema of attractions’. That is, the emphasis in its use is on the display of photographic technologies rather than the resultant image. Equally, the common practice, in both fine art and vernacular circles, of making composite pictures with Polaroid prints, draws attention from image content and redirects it to the photo as object