The existence of asymptotically stable solutions for a nonlinear functional integral equation with values in a general Banach space

Motivated by the recent known results about the solvability and existence of asymptotically stable solutions for nonlinear functional integral equations in spaces of functions defined on unbounded intervals with values in the n-dimensional real space, we establish asymptotically stable solutions for a nonlinear functional integral equation in the space of all continuous functions on R₊ with values in a general Banach space, via a fixed point theorem of Krasnosel’skii type. In order to illustrate the result obtained here, an example is given.     

Random walks with different directions

As an extension of Polya’s classical result on random walks on the square grids (\({\mathbf {Z}}^d\)), we consider a random walk where the steps, while still have unit length, point to different directions. We show that in dimensions at least 4, the returning probability after n steps is at most \(n^{-d/2 - d/(d-2) +o(1) }\), which is sharp. The real surprise is in dimensions 2 and 3. In dimension 2, where the traditional grid walk is recurrent, our upper bound is \(n^{-\omega (1) }\), which is much worse than in higher dimensions. In dimension 3, we prove an upper bound of order \(n^{-4 +o(1) }\). We find a new conjecture concerning incidences between spheres and points in \({\mathbf {R}}^3\), which, if holds, would improve the bound to \(n^{-9/2 +o(1) }\), which is consistent to the \(d \ge 4\) case. This conjecture resembles Szemerédi-Trotter type results and is of independent interest.     

A strongly regularN-full graph of small order

For every positive integern we show the construction of a strongly regular graph of order at most 2 ⁿ⁺² which contains every graph of ordern as a subgraph. The estimation concerning the construction is best possible.     

Molality as a unit of measure for expressing 1H MRS brain metabolite concentrations in vivo

Absolute concentrations of cerebral metabolite in in vivo 1H magnetic resonance spectroscopy studies (1H-MRS) are widely reported in molar units as moles per liter of tissue, or in molal units as moles per kilogram of tissue. Such measurements require external referencing or assumptions as to local water content. To reduce the scan time, avoid assumptions that may be invalid under specific pathologies, and provide a universally accessible referencing procedure, we suggest that metabolite concentrations from 1H-MRS measurements in vivo be reported in molal units as moles per kilogram of tissue water. Using internal water referencing, a two-compartment water model, a simulated brain spectrum for peak identification, and a spectroscopic bi-exponential spin-spin relaxation segmentation technique, we measured the absolute concentrations for the four common 1H brain metabolites: choline (Cho), myo-inositol (mIno), phosphocreatine + creatine (Cr), and N-acetyl-aspartate (NAA), in the hippocampal region (n = 26) and along the Sylvian fissure (n = 61) of 35 healthy adults. A stimulated echo localization method (20 ms echo time, 10 ms mixing time, 4 s repetition time) yielded metabolite concentrations, uncorrected for metabolite relaxation or contributions from macromolecule resonances, that were expectantly higher than with molar literature values. Along the Sylvian fissure the average concentrations (coefficient of variation (CV)) in mmoles/kg of tissue water were 17.6 (12%) for NAA, 14.2 (9%) for Cr, 3.6 (13%) for Cho, and 13.2 (15%) for mIno. Respective values for the hippocampal region were 15.7 (20%), 14.7 (16%), 4.6 (19%), and 17.7 (26%). The concentrations of the two regions were significantly different (p 相似文献   

Frequency modules and nonexistence of quasi-periodic solutions of nonlinear evolution equations

This paper gives lower estimates for the frequency modules of almost periodic solutions to equations of the form , where A generates a strongly continuous semigroup in a Banach space , F(t,x) is 2π-periodic in t and continuous in (t,x), and f is almost periodic. We show that the frequency module ℳ(u) of any almost periodic mild solution u of (*) and the frequency module ℳ(f) of f satisfy the estimate e ^{2π iℳ(f)}⊂e ^{2π iℳ(u)}. If F is independent of t, then the estimate can be improved: ℳ(f)⊂ℳ(u). Applications to the nonexistence of quasi-periodic solutions are also given.     

Existence and Exponential Decay of Solutions for a Wave Equation with Integral Nonlocal Boundary Conditions of Memory Type

In this paper, we consider a wave equation with integral nonlocal boundary conditions of memory type. First, we establish two local existence theorems by using Faedo–Galerkin method and standard arguments of density. Next, we give a su?cient condition to guarantee the global existence and exponential decay of weak solutions. Finally, we present numerical results.     

Simultaneous Determination of Paracetamol,Ibuprofen, and Caffeine in Tablets by Molecular Absorption Spectroscopy Combined with Classical Least Square Method

In this paper, the classical least-squares (CLS) method with molecular absorption spectrophotometric measurement was used to determine simultaneously paracetamol (PAR), ibuprofen (IBU), and caffeine (CAF) in tablets. The absorbance spectra of the standard solutions and samples were measured over a wavelength from 220 to 300 nm with a 0.5 nm step. The concentration of PAR, IBU, and CAF in the sample solutions was calculated by using Visual Basic for Applications (VBA) and a program called CLS-Excel written in Microsoft Excel 2016. The method and the CLS-Excel program were tested on mixed standard laboratory samples with different PAR, IBU, and CAF concentration ratios, and they showed only small errors and a satisfying repeatability. An analytical procedure for tablets containing PAR, IBU, and CAF was developed. The reliability of the procedure was proved via the recovery and repeatability of the analysis results with an actual tablet sample and by comparing the mean contents of active substances in the tablets obtained from the analytical procedure with the HPLC method. The procedure is simple with a reduced cost compared with the HPLC standard method.     

Decarboxylative cross-couplings of 2-aminopyrimidine-5-carboxylic acids

Decarboxylative CC cross-couplings of 2-aminopyrimidine-5-carboxylic acids under a Pd/Ag-based catalytic system opens a new platform for the introduction of diverse C5 substituents. The reaction methods proceeded efficiently with a wide range of the acids and the coupling partners of aryl iodides, alkenes, bromoalkynes, and azoles. Considering ready availability of 2-aminopyrimidine-5-carboxylic acid from the oxidative dehydrosulfurative CN cross-coupling of the 3,4-dihydropyrimidin-1H-2-thiones, this reaction method unambiguously pave a shortcut to densely substituted 2-aminopyrimidine derivatives with unprecedented diversity.     

Effect of potassium addition on bimetallic PtSn/θ-Al<Subscript>2</Subscript>O<Subscript>3</Subscript> catalyst for dehydrogenation of propane to propylene

PtSn/θ-Al₂O₃ catalysts with different amounts of K (0.14, 0.22, 0.49, 0.72, and 0.96 wt%) are prepared to investigate the K effects on the PtSn catalyst in propane dehydrogenation (PDH). KPtSn catalyst with 0.xx wt% K, 0.5 wt% Pt and 0.75 wt% Sn is designated as xx-KPtSn. PDH was performed at 873 K and a gas hourly space velocity (GHSV) of 53,000 mL/g_cat h. The temperature-programmed desorption (NH₃-TPD), temperature-programmed reduction (TPR) and CO chemisorption of the KPtSn catalysts with K added revealed the potassium addition blocked the acid sites, promoted the reduction of Sn oxide and decreased the Pt dispersion. The formations of cracking products and higher hydrocarbons on acid sites were suppressed by the K effect of blocking the acid sites. In contrast, K addition at more than 0.72 wt% rather increased cracking products and the amount of coke, resulting in the severe deactivation of catalysts. The high cracking products on the KPtSn catalysts with the high amount of K should not be related to the acid sites, because the acid sites were monotonously decreased with an increase in the amount of K. Instead, the potassium affected the characteristics of PtSn. The interaction between Pt and Sn could be weakened by enriching the reduced Sn, because the K component promoted the reduction of Sn oxide in the TPR experiments. Therefore, the 14-KPtSn catalyst with the low amount of K exhibits the highest stability and selectivity among the prepared KPtSn catalysts due to the compromise of the advantageous (blocking the acid sites) and bad (weakening the interaction between Pt and Sn) effects of the K addition in PDH.