The sensory acceptance scores for FFWMP and SMP remained stable throughout storage in all conditions, despite the increased perception of some LO products. The IMF sample was perceived negatively in each storage condition and at each time point. Overall increases in hexanal, heptanal, and pentanal correlated with “painty”, “oxidised”, “cooked”, and “caramelised” attributes in all samples. The concentration of some LO volatiles in the IMF was far in excess of those in FFWMP and SMP. High levels of LO volatiles in IMF were presumably due to the addition of polyunsaturated fatty acids (PUFA) in the formulation.Lipid oxidation (LO) is a recognised problem in dairy powders due to the formation of volatile odour compounds that can negatively impact sensory perception. Three commercial dairy powders, fat-filled whole milk powder (FFWMP), skim milk powder (SMP), and infant milk formula (IMF), stored under different conditions , were evaluated by consumer acceptance studies, ranked descriptive sensory analysis, and LO volatile profiling using headspace solid phase microextraction gas chromatography mass spectrometry (HS-SPME GCMS) over 16 weeks. Significant ( Lipid oxidation (LO) is a well-documented cause of quality deterioration in dairy powders . OxidatiThree batches of FFWMP and SMP were obtained from local suppliers as commercial products, while three batches of IMF were purchased from local retailers. FFWMP, SMP, and IMF samples were manufactured on the following dates: December 2017, May 2018, and October 2017, respectively, each with a 24 month shelf-life. The FFWMP, SMP, and IMF were 11, 3, and 10 months into their shelf-life, respectively, at the beginning of the study. For each sample type , the three batches were mixed together thoroughly to remove the batch effect and to create a bulk sample of each. The bulk sample was subsequently separated into four 1.5 kg lots and placed in light-omitting sealed bags at the beginning of the study (T0). One bag of each sample was stored in one of the four storage treatments; –18 °C (control), 21 °C (ambient), 37 °C (accelerated), and 25 °C with humidity controlled at 50% (humidity) resulting in 12 samples (3 × 4) in total. Samples were stored at –18 °C (freezer) and 21 °C and 37 °C (incubation rooms) at the Teagasc Food Research Centre , and samples kept at 25 °C with humidity controlled at 50% were stored in a Binder KBF P Series Humidity Test Chamber  located at University College Cork . Ambient is denoted as AM, control as CON, accelerated as ACC, and humidity as HUM throughout the study, unless otherwise stated. Volatile data were undertaken at T0, T2, T4, T6, T8, T10, T12, T14, and T16, which represented the number of weeks of storage. Sensory analysis was carried out at T4, T8, T12, and T16 for practical purposes, and sensory and volatile data were correlated at T4, T8, T12, and T16. An additional six IMF samples were purchased from local retail units and immediately analysed for LO volatiles (in triplicate) for comparative purposes, as it was necessary to get a better understanding of the potential LO range in these products as little or no data were available.Thirteen volatile aromatic compounds including seven aldehydes, hexanal, pentanal, heptanal, octanal, (E)-2-nonenal, 2,4-decadienal, and undecanal, four ketones, 2-heptanone, 2-nonanone, 2-pentanone, and 3-octen-2-one, and two alcohols, 1-heptanol and 1-pentanol, known to be important for the sensory perception of dairy products, were selected for quantification based on current literature ,18,21. A2O) 24 h prior to analysis. Samples were heated to ~40 °C immediately before sampling. Results were expressed as the averages of 2 replicates.Each sample was analysed for fat, protein, lactose, true protein, and casein content using a Bentley DairySpec FT . Samples were reconstituted to 10% total solids for SMP and 13% for FFWMP and IMF, as per the manufacturer’s instructions using distilled water . One millilitre of each dilution was pipetted onto sterile petri dishes and covered with warm (45 ± 2 °C), sterile milk plate count agar (15 mL) . The mixture was allowed to cool and solidify, and plates were incubated for 72 h at 30 °C. Analysis was performed in duplicate.The pour plate method was used to estimate the total bacterial count of the reconstituted milk powder samples prior to each sensory evaluation session. Serial dilutions from 102O 24 h prior to analysis and chilled at 4 °C. Approximately 2 mL of sample were placed in a spectrophotometric cuvette 1 h prior to analysis. Results were expressed as the average of triplicate measurements of each liquid sample [Colour measurements were performed on each of the 12 milk powder samples according to the CIE Lab system , using a Minolta colorimeter . Samples were reconstituted in dHd sample .Lipid extraction and methyl ester derivatisation of triglycerides were carried out as per De Jong and Badings  and O‘CaA 37 component fatty acid methyl ester (FAME) reference mix containing C4:0 to C24:0 (Part Number 35077)  was analysed as an in-run quality control sample, with the FAME present at a 60–180 ppm concentration. This was used to ensure accurate quantitation was achieved throughout sample analysis. The FAME mix was analysed once every five samples in the sequence. Accuracy was monitored by comparing the measured concentration of the FAME mix against its true concentration.w/v) in dH2O were added along with 3.5 mL dH2O. A calibration curve was prepared by spiking a set of the hydrated FFWMP samples with varying levels of the external standard mixture (13 compounds of interest prepared at 0.004% (w/v) in dH2O). Matrix (control) samples (FFWMP sample + dH2O only) were also included in each run. Samples were extracted for 45 min at a temperature of 43 °C. Each sample was subjected to a 10-min pre-extraction incubation time at 43 °C with pulsed agitation of 5 s at 500 rpm, automated by a Bruker CombiPal autosampler . Each sample was analysed in triplicate every two weeks during the 16 week storage period.Headspace solid-phase microextraction (HS-SPME) analysis was carried out using an HS-SPME method optimised for the detection and quantification of LO compounds in WMP as per Clarke et al. . BrieflyIncubation, extraction, and injection processes were implemented using a Bruker CombiPal autosampler . A mid-polar DB 624 UI column (60 m × 0.32 mm × 1.80 μm)  and a 2 cm, 50/30 μm, Divinylbenzene/Carboxen/Polydimethylsiloxane (DVB/CAR/PDMS) Stableflex SPME fibre  were used for the duration of the study. Following extraction, the SPME fibre was retracted and injected into the split/splitless 1177 GC inlet for 5 min at 250 °C in split mode at a ratio of 10:1 followed by 2 min at 270 °C in a bake-out station to minimise carry-over of compounds. The column oven was held at 65 °C for 10 min, then increased to 240 °C at a rate of 10 °C/min, and held for 4 min. Helium was used as the carrier gas with a constant flow rate of 1.0 mL/min.2O based on the fat and protein content as per the equation outlined by the International Dairy Federation (IDF) [Powder samples were reconstituted in dHon (IDF) , to a toStatistical analysis for data relating to colour, composition, and microbial analysis was carried out using one-way ANOVA with post-hoc Tukey tests using the Statistical Package for the Social Sciences (SPSS) software, Version 24 . Pearson correlation analysis was carried out on the sensory and volatile data using SPSS. Principal component analysis (PCA) biplots of the volatile vs. sensory data were used to demonstrate correlations between the volatile compounds and the sensory attributes. These were constructed using the “factoextra” and “FactoMinoR” packages in R (v 3.4.1) .p ≤ 0.05) between the powders. As expected, the FFWMP contained the highest fat content (3.77%) and the SMP the lowest (0.02%). The IMF contained the highest level of lactose across all the storage conditions (7.53–7.89%). True protein and casein values were higher in SMP  than FFWMP and IMF.Compositional results for the 12 reconstituted milk powder samples  are outlined in No significant differences were observed between the total bacterial counts for the 12 reconstituted milk powder samples  over the 16 week storage period (data not shown). Total bacteria counts were below the limit for powdered milk and milk based products intended for human consumption within Europe at all-time points .p ≤ 0.001) between the samples (p ≤ 0.001) higher in FFWMP samples than in SMP and IMF samples regardless of the storage treatments. The a* values were negative across all samples with significantly (p ≤ 0.001) higher values observed for SMP than FFWMP and IMF samples across all treatments with the highest a* value observed in the SMP stored at 37 °C (–5.83). As expected, significantly (p ≤ 0.001) higher b* (yellowness) values were observed in FFWMP samples, due to the increased fat content [p ≤ 0.001) higher a* value (green-to-red colour) observed in SMP samples analysed in this study could be due to the Tyndall effect; the scattering of light as it passes through a colloid. The higher casein content in SMP  scattered more blue light than red. Furthermore, β-carotene is lost when milk is skimmed, removing the source of the yellow colour observed in FFWMP and IMF samples [p ≤ 0.001) different between the samples. Several other factors such as the modification of particle size, Maillard reactions forming brownish pigments during heat treatment [The L* (lightness), a* (green-to-red colour), and b* (blue-to-yellow colour) values were statistically different , as shown in via uptake of existing FA from the diet (and rumen bio-hydrogenation) or through de novo synthesis [p ≤ 0.05) higher proportions of palmitic acid (C16:0) than the SMP and IMF samples. The IMF sample contained higher proportions of linoleic acid (C18:2 n6c) and α-linolenic acid (C18:3 n3) than the FFWMP, likely due to fortification. The FA C20:2, C20:3 n6, C24:1 n9, and C20:5 were identified only in the IMF sample. Some of these FA are found in fish oils [The FA composition of milk is generally derived from two primary sources, ynthesis . C4:0-C1ynthesis . The remynthesis . Bovine ynthesis ,34, and ynthesis . The FA ish oils  and vegeish oils ,38, and Volatile analysis was carried out on each of the milk powder samples by HS-SPME GCMS every two weeks  over the 16 week storage period. The 13 selected volatile aromatic compounds were quantified at each time point . As expeMuch less variability in the levels of aldehyde and ketone compounds was observed in the SMP samples . HoweverWhen compared to FFWMP and SMP, increased levels of some oxidation products were observed in IMF samples at T0 and throughout the study at all-time points and in all storage conditions  . HexanalThe concentrations of the aldehydes hexanal, pentanal, and heptanal increased throughout storage in the both the FFWMP and IMF samples stored in ACC . The sensory attributes associated with these compounds have been described as “painty”, “cardboard-like”, and “grassy” ,18. HexaNo significant differences were observed between the FFWMP or SMP samples for hedonic assessment over the storage period. Significant differences were found between “liking of flavour” and “overall acceptability” for the IMF samples. The IMF samples scored lowest for “overall acceptability” across all time points and storage treatments. At T4, significant differences were observed between the FFWMP, SMP, and IMF samples for “liking of appearance”, “liking of aroma”, “liking of flavour”, “overall acceptability”, “colour”, “creamy aroma”, “oxidised aroma”, “painty aroma”, “powdery texture”, “oxidised flavour”, and “off-flavour”. Again, at T8, differences between the samples were observed for “liking of appearance”, “liking of aroma”, “liking of flavour”, and “overall acceptability”, with IMF samples scoring the lowest for “liking of aroma”, “liking of flavour”, and “overall acceptability” and highest for “liking of appearance”. SMP samples scored highest for “liking of aroma”, “liking of flavour”, and “overall acceptability”. In the RDA, differences were again observed for “oxidised aroma”, “painty aroma”, “powdery texture”, and “off-flavour” in addition to “rancid butter flavour” and “painty flavour”, all of which were more correlated with IMF samples except for “powdery texture”, which was more correlated with FFWMP samples. Finally, at T16, significant differences were observed for “creamy aroma”, “oxidised aroma”, “painty aroma”, “painty flavour”, and “off-flavour” . This stThe odour threshold of many volatile compounds is greater in oil and fat matrices when compared with water and air, generally due to the complexity of the matrix and possible matrix binding . In the Many significant differences were observed for the hedonic and RDA scores, particularly between SMP and IMF samples, but also within SMP and IMF samples stored in different conditions at different time points. FFWMP remained stable and acceptable throughout the 16 week storage period despite increases in LO volatile compounds. This suggested that although these compounds were present at levels above their odour thresholds, and thus potentially perceivable, they did not adversely impact sensory perception, presumably because they were not concentrated enough in the FFWMP matrix. SMP samples remained acceptable throughout storage, also scoring highest for “overall acceptability” across the storage treatments. The IMF samples were found to be unacceptable at each time point. Levels of the oxidation products hexanal, pentanal, and heptanal were present at 5986, 1209, and 861 ppm, respectively in IMF samples at T0 with significant increases over storage. As previously mentioned, the level of sensory acceptability for IMF samples remained the same from T4 to T16, which suggested that once LO products reached certain levels above their odour thresholds, panellists deemed the product as “unsatisfactory”. Many of the descriptors commonly used to describe off-flavours associated with LO in dairy products were most correlated with IMF samples . CorrelaThe ability of the sensory panel to perceive differences in many sensory attributes in SMP was unexpected as it is typically less susceptible to LO due to its low-fat content. However, the lipid phase of WMP and other high-fat dairy powders could act as a solvent for LO compounds ; thus, tr = 0.74) and 1-pentanol (r = 0.74), and “off-flavours” were positively correlated with (E)-2-nonenal. “Overall acceptance” was negatively correlated with “oxidised aroma” (–0.53) in IMF samples, and there was no correlation between “overall acceptability” and sensory attributes in the FFWMP and SMP samples. Lloyd et al. [Correlation relationships were observed between volatile compounds and sensory perception and between the concentrations of individual LO volatile compounds. The main correlation relationships observed in FFWMP samples are outlined in d et al.  reportedd et al.  concludeIn the present study, freshly opened WMP and SMP remained acceptable to the sensory panel for the whole 16 weeks of storage despite levels of primary oxidation products being present above their odour thresholds. Storing samples in ACC versus AM accelerated the formation of LO compounds, which contradicted the results of Cesa et al. , which sThe amount of total fat as outlined by the manufacturers did not vary significantly between the IMF retail brands ; howeverp ≤ 0.001) higher in Brand 1 compared to the other four brands of IMF powdered samples (Brand 2–5); the same was observed for the alcohol compounds 1-heptanol and 1-pentanol. Trans-2-nonenal and 2,4–decadienal were significantly higher in Brands 3, 4, and 5 when compared to Brands 1 and 2. 2-Heptanone was significantly higher in Brand 2 compared with Brands 1, 3, 4, and 5. Levels of undecanal, 2-nonanone, 2-pentanone, and 3-octen-2-one did not vary significantly between the samples  (24.36 ± 1.72) and oleic acid (C18:1) n9c (30.36 ± 1.52). Brands 2, 3, and 4 contained significantly higher proportions of lauric acid (C12:0) compared to Brands 1 and 5. The total FA contents of IMF Brands 1–5 are shown in The increased concentrations of certain volatile compounds observed in the six retail IMF samples likely related to the significant differences in FA profile. The FA potentially originating from the addition of fish and vegetable oils  were likely some of the main contributors to the observed oxidative state of the IMF powders. The significantly higher levels of primary aldehyde and secondary alcohol compounds in Brand 1 compared to the other brands indicated issues in relation to the fat component of this product, which resulted in more LO products present immediately after manufacture. The level of long-chain PUFA was slightly higher in Brand 1, however unlikely to be responsible for the significantly higher levels of short- and medium-chain aldehydes. The compounds –2,4-heptadienal, –2,4-nonadienal (derived from linolenic and linoleic acid degradation), and 2-pentylfuran (derived from linoleic acid degradation) were only identified in Brand 1. These compounds have been shown to be high-impact flavour compounds in edible oils with relative odour activity values ≥ 1 . CompounThe antioxidant capacity of dairy powders is mainly determined by the levels of sulphur containing amino acids such as cysteine, but also levels of phosphate, carotenoids, zinc, selenium, and vitamins A and E . DependiThe FA profile varied significantly between the samples and some long-chain FA were identified only in IMF samples . The likely source for these FA in the IMF was fortification with vegetable and fish oils during manufacture followed by thermal processing, which could initiate the LO process, resulting in increased susceptibility to LO. The volatile profile of the FFWMP, SMP, and IMP also varied significantly, which was related to differences in FA profile, and possibly due to the origin of the milk, i.e., bovine diet and presence or absence of pro- and anti-oxidants. Overall, increases in the concentrations of hexanal, heptanal, and pentanal were good indicators of LO occurring in FFWMP. Fewer significant increases in LO compounds were evident in SMP; however, changes in the concentrations of hexanal, 1-heptanol, and 1-pentanol were evident. Regarding IMF, significant increases in specific aldehydes hexanal, pentanal, heptanal, and octanal were good indicators of LO. The concentrations of (E)-2-nonenal and 2,4-decadienal were also higher in IMF samples compared to FFWMP and SMP. The sensory acceptance scores for FFWMP and SMP remained stable throughout storage, despite some LO compounds being perceived by the panellists. The IMF sample was perceived negatively from the start of storage due to high levels of numerous LO compounds present. “Oxidised” and “painty” attributes were correlated with increased concentrations of hexanal and heptanal and were particularly evident in IMF samples. In addition to the main experiment, the FA content and volatile profile of six widely available IMF brands (five powdered and one ready-to-use UHT product) were evaluated. The proportions of 20 of the 25 FA identified varied significantly between the powdered IMF brands, while the concentrations of nine of the 13 volatile compounds also varied significantly. The concentrations of volatile compounds, in particular aldehydes, were also very high in the six IMF brands in comparison to the FFWMP and SMP samples.


Reactive oxygen species (ROS) are continuously generated as a by-product of normal aerobic metabolism. Elevated ROS formation leads to potential damage of biological structures and is implicated in various diseases. Astaxanthin, a xanthophyll carotenoid, is a secondary metabolite responsible for the red-orange color of a number of marine animals and microorganisms. There is mounting evidence that astaxanthin has powerful antioxidant, anti-inflammatory, and antiapoptotic activities. Hence, its consumption can result in various health benefits, with potential for therapeutic application. Astaxanthin contains both a hydroxyl and a keto group, and this unique structure plays important roles in neutralizing ROS. The molecule quenches harmful singlet oxygen, scavenges peroxyl and hydroxyl radicals and converts them into more stable compounds, prevents the formation of free radicals, and inhibits the autoxidation chain reaction. It also acts as a metal chelator and converts metal prooxidants into harmless molecules. However, like many other carotenoids, astaxanthin is affected by the environmental conditions, e.g., pH, heat, or exposure to light. It is hence susceptible to structural modification, i.e., via isomerization, aggregation, or esterification, which alters its physiochemical properties. Here, we provide a concise overview of the distribution of astaxanthin in tissues, and astaxanthin structures, and their role in tackling singlet oxygen and free radicals. We highlight the effect of structural modification of astaxanthin molecules on the bioavailability and biological activity. These studies suggested that astaxanthin would be a promising dietary supplement for health applications. Carotenoids are a class of bioactive natural products synthesized by plants and certain photosynthetic microorganisms. Many carotenoids are directly involved in the photosynthesis process, while others protect the host from photooxidation and related damage . Astaxanin vitro and in vivo studies of the biological role of astaxanthin are presented in Astaxanthin supplementation has been investigated in a broad range of clinical applications in humans and was shown to exert multiple pharmacological effects , 6. The Reactive oxygen species (ROS) have attracted attention as novel signal mediators involved in the modulation of cell survival, cell death, differentiation, cell signaling, and inflammation-related factor production . Large qβ-carotene, and 100-fold higher than that of α-tocopherol . Concerning reactivity, HO• is the most reactive and dangerous species among ROS. ROO• is significantly less reactive than HO•, which allows them to diffuse to remote cellular locations    has a higher oxidation potential than other carotenoids, such as β-carotene [E1°(Car/Car•+) = 0.634 V], zeaxanthin [E1°(Car/Car•+) = 0.63 V], and lycopene [E1°(Car/Car•+) = 0.60 V]  . Further and HO• , 140. Th and HO• . After a and HO• .in vitro studies, astaxanthin esters act as powerful quenchers of singlet oxygen under both hydrophilic and hydrophobic conditions [1O2) by carotenoids and food extracts in solution revealed that carotenoids with 11 carbon atoms involved in the π-conjugation (n), such as astaxanthin, quench 1O2 most effectively [3Car)  are higher than those in micelle (71.1 × 108 M–1 s–1) and liposome (0.19–5.9 × 108 M–1 s–1). The rate constants in heterogeneous environment are lower than those in organic solvents, which can be explained by a slower diffusion of singlet oxygen from the aqueous phase to the micelle or lipid membrane in these environments [α-tocopherol [The antioxidant capacity of astaxanthin can be determined from the quenching rate constant of singlet oxygen. A fast quenching rate constant indicates efficient singlet oxygen quenching. The quenching rate constants of singlet oxygen for astaxanthin determined by various methods and in various environments are shown in ronments . In addicopherol .α-hydroxyketones and hydroxyquinones [•) via Fenton and photo-Fenton reactions in vitro can be completely inhibited by the chelation of metal ions Fe2+ or Cu2+ [2+, Cu2+, Pb2+, Zn2+, Cd2+, and Hg2+ [2+, Zn2+, and Fe2+, was evaluated by mass spectrometry and nuclear magnetic resonance. The analysis revealed that the two oxygen atoms at the terminal cyclohexene rings chelate the metal to form 1 : 1 complexes at low Ca(ClO4)2, Zn(ClO4)2, and Fe(ClO4)2 salt concentrations; at high salt concentrations (>0.2 mM), a 2 : 1 salt to astaxanthin complex is formed [4)2 was calculated to be K1 = 3000 M–1, whereas for 1 : 1 complex with Ca(ClO4)2, the stability constants are K1 = 23,000 M–1 and K2 = 5000 M–1. The ability to form chelate complexes with metals could be related to the efficiency with which astaxanthin acts as a protective agent, particularly in inhibiting hydroxyl radicals [The molecular structure of astaxanthin endows it with unique properties. Hydroxyl and keto groups are present at positions C3 (C3′) and C4 (C4′), respectively, on each ionone ring. The two adjacent oxygen atoms on the cyclohexene ring enable the formation of stable complexes with metal ions , as is aquinones , 158. Th or Cu2+ . Accordiand Hg2+ . Indeed,s formed . Furtherradicals .• scavenging capacity were then calculated using α-tocopherol as a reference compound. Among the carotenoids studied, all-trans astaxanthin was identified as a highly efficient ROO• scavenger  [cis isomers of astaxanthin, especially the 9-cis isomer, showed a higher antioxidant potential in vitro than all-trans astaxanthin [cis isomer also most effectively inhibits the generation of ROS induced by 6-hydroxydopamine in human neuroblastoma SH-SY5Y cells, as well as the degradation of collagen type II induced by docosahexaenoic acid (DHA) and linoleic acid hydroperoxides [cis astaxanthin had higher antioxidant activity than all-trans and 9-cis in oxygen radical absorbing capacity assay for lipophilic compounds, photochemiluminescence, and cellular antioxidant activity assay [cis and 13-cis astaxanthin at pH 2. Different methods of antioxidant evaluation and antioxidant activity of astaxanthin and cis-trans isomers are summarized in The geometrical isomers of astaxanthin play an important role in its antioxidant activity. A microassay evaluating the peroxyl radical scavenging capacity of 15 carotenoid standards has been developed and validated, specifically to study the structure–activity relationship . The valelative) . As deteaxanthin . Furthereroxides . The 13-ty assay . It was in vitro and in vivo. However, there is a lack of research on astaxanthin and its metabolism in biological systems. Future research should focus on the physicochemical properties of different astaxanthin structures, their uptake mechanisms, and the facility of incorporation into metabolic pathways. Molecular studies involving in vitro and in vivo models should also be performed to investigate the nutraceutical and pharmaceutical applications of various astaxanthin isomers.In the last decade, studies on astaxanthin as a potent therapeutic agent have shown encouraging results. Astaxanthin can potentially be used to address various human health issues, including cancer, cardiovascular and neurodegenerative diseases, and aging. These diseases are mostly associated with inflammation caused by an interaction of nucleic acids and proteins with harmful reactive species. These effects are related to the unique properties of astaxanthin molecular structure that allow it to scavenge reactive species and quench singlet oxygen. Recent experiments have uncovered the contribution of different astaxanthin isomers to antioxidant activities, both


This paper studies uniqueness of solutions for a nonlinear Hadamard-type integro-differential equation in the Banach space of absolutely continuous functions based on Babenko’s approach and Banach’s contraction principle. We also include two illustrative examples to demonstrate the use of main theorems. Butzer amard in –16 studiamard in , 18. In amard in  studied 8, Matar  applied 8, Matar  introduc8, Matar  defined 8, Matar  further 8, Matar  studied α and type β given by There are new studies on fixed point theorems for different operators on metric spaces –28, as wc2≠0. Li  obtainedAssume thatandforandare arbitrary complex numbers, andIn addition, we letandwhereThen equationhas a unique solutionin the spacet and belongs to Applying the operator entclass1pt{minimamentclasspt{minimaentclass1pt{minimaIndeed, Clearly, the uniqueness follows immediately from the fact that the integro-differential equation (i)It follows from Theorem 5.3 in  that for(ii)u is stable.A solution of equation  in the s(iii)The multivariate Mittag-Leffler function was initially introduced by Hadid and Luchko , who useLet relation  \documenLet entclass1pt{minimaBy Theorem equation , \documeThe following theorem shows the uniqueness of equation .Assume thatis a continuous function, and there exists a constantCsuch thatfor allandFurthermore, Then equation (has a unique solution in the spacefor everymentclass2pt{minimT on T is a mapping from T is contractive. Indeed, T is contractive. This completes the proof of Theorem Let equality , we claiC is to be determined.Let R and satisfies C such that Clearly, Using Babenko’s approach and Banach’s contraction principle, we have derived the uniqueness of solutions for the new nonlinear Hadamard-type integro-differential equation for all


This paper provides readers with three partial results that are mutually connected. Firstly, the gallery of the so-called constant phase elements (CPE) dedicated for the wideband applications is presented. CPEs are calculated for 9°  and 10° phase steps including ¼, ½, and ¾ orders, which are the most used mathematical orders between zero and one in practice. For each phase shift, all necessary numerical values to design fully passive RC ladder two-terminal circuits are provided. Individual CPEs are easily distinguishable because of a very high accuracy; maximal phase error is less than 1.5° in wide frequency range beginning with 3 Hz and ending with 1 MHz. Secondly, dynamics of ternary memory composed by a series connection of two resonant tunneling diodes is investigated and, consequently, a robust chaotic behavior is discovered and reported. Finally, CPEs are directly used for realization of fractional-order (FO) ternary memory as lumped chaotic oscillator. Existence of structurally stable strange attractors for different orders is proved, both by numerical analyzed and experimental measurement. If speaking in terms of module frequency response, admittance linearly increases , namely with slope 20α dB per frequency decade. Besides fractal capacitor, we can find FO integrator, i.e., two-port where voltage transfer function has FO character. Higher non-integer orders can be implemented easily by a cascading two or more FO integrators. It is much more transparent than utilization of the general immittance converters to create FO immittance with an order higher than one. Some interesting structures of immittance converters capable to create arbitrary FO can be found in papers T, xe2 = T, xe3 = T, xe4 = T, and xe5 = T. In each segment of vector field, local behavior is uniquely determined by eigenvalues, i.e., roots of characteristic polynomialng is slope of n-th PWL function in the investigated segment of vector field.Using these auxiliary numbers, positions of the equilibrium points (if exist) are c1 = 10 F, c2 = 6 F, l = 100 mH, g1inner = −20 S, g1outer = 8 S, β1 = 200 mV, g2inner = −15 S, g2outer = 18 S, and β2 = 400 mV. All state trajectories plotted in this section were numerically integrated using Mathcad 15 and build-in fourth order Runge-Kutta method having fixed step size. The type of the dynamical behavior of the memory strongly depends on the shapes of both PWL functions. For example, numerically observed attractors for different slope of outer segments associated with second RTD are demonstrated in g2outer = 18 S. Here, final time for numerical integration was set to 200 and time step 0.01. Further experimentations reveal that funnel and double-scroll chaotic attractor can be also robust solution of analyzed set of differential equations, namely for normalized values c2 = 4.5 F, l = 150 mH and c2 = 6 F, l = 170 mH, g2outer = 20 S, respectively. Remaining internal parameters of memory system are unchanged. Position of these attractors within state space is visualized by means of 4, time step 0.1 and initial conditions were x0 = T for single-spirals and x0 = T for funnels. Due to vector field symmetry, two lateral strange attractors can merge, forming large attracting set that enters all state space segments.Several methods of how to distinguish between regular and irregular behavior of arbitrary order mathematical model have been developed and published. Some of them are based on calculation of flow quantifier such as the largest Lyapunov exponent (LLE), metric dimensions, or by using return maps. Interesting reading about this topic is provided in paper  and refe4 initial conditions were integrated with a final time of 100 and time step of 0.1 (ending state is plotted). Each group is generated in the close neighborhood of some fixed point (black dots) distinguished by colors  using normal distribution with mean deviation 10−3. Note that self-excitation process of the limit cycle and both mirrored single-spiral attractors is verified.A key feature of chaos is the extreme sensitivity of the system behavior to the tiny changes of initial conditions. This unique property is proved in Let’s see what kind of vector field geometry forms double-scroll attractor newly presented in this paper. This attractor occupies all affine segments of the state space, i.e., dynamics of memory is uniquely determined by eigenvalues and eigenspaces associated with all fixed points. For numerical set of parameters given above, formula (11) returns the following results: saddle-focus with unstable eigenplane in blue segments in the sense of g2outer, namely 18 S and 20 S, graphical visualization of BA is provided by means of x = 0, y = 0, and z = 0 axis. Note that, in the case of g2outer = 18 S, geometrical structures of individual BA seem to be quite simple. On the other hand, transconductance slope equal to g2outer = 20 S leads to a much more complicated snake-like regions ending into periodic solution. It should be noted that the chaotic attractors discovered in this paper are to self-excited. However, the existence of the hidden chaotic attractors is not excluded since, in the sense of initial conditions, investigated space is too small and grid large. Remember that, even in the case of the “old” and well-known Chua´s oscillator, which was analyzed more than three decades, hidden strange attractors were discovered quite recently [As nicely demonstrated by the chaotic Chua´s oscillator  or memorrecently .Y1,2 is the pseudo-capacitance of first and second FO capacitor, respectively. If the capacitor is replaced by the approximation circuit depicted in v and i is external voltage and current across CPE and kv are the internal nodes of CPE practically invisible to the rest of circuit. Note that state vector associated with memory changes from basic set x = T into more complex form x = T. It means that each FO capacitor increases in order of final mathematical model by number equivalent to order of CPE approximation. For definition of individual state variables and complete schematic of FO memory, see dx/dt = A·x + f(x), where entries of state matrix An+32, n = 7 is order of CPE approximation, aref are f1(v1) and f2(v2) given by PWL function (9). Numerical values of components paR, paC, pbR, pbC, akR, akC, bkR, bkC for k = 1, 2, …, 7 can be adopted directly from By introducing FO derivatives to differential equations that describe voltage vs. current flowing through capacitors, we getr is the main parameter of the numerical algorithm, which measures and quantify similarity patterns in the data sequence of the increasing length (up to the self-comparison)—see tutorial paper [Both al paper  for a beg1inner∈, g1outer∈, g2inner∈, g2outer∈ and resolution 201 × 201 × 201 × 201 points. For this calculation, the fourth-order Runge-Kutta method in Matlab. Final time for integration was set to 1000 with transient behavior omitted.5 and 104, respectively. Thus, real-valued integer-order capacitors are C1 = 10 nF, C2 = 6 nF and inductor is L = 10 mH. Remaining circuit components of this IO memory are nR1 = 3 k∧, nR2 = 1 k∧, nR3 = 180 ∧ and nR4 = 1.5 k∧. Computer-aided analysis of this dynamical system in the time domain is given in α = β = 9/10, see α = β = 4/5. In this case, impedance norms were chosen to be equal to 15 and 55. Additionally, the famous double-scroll strange attractor can be generated by FO active memory. Corresponding proof can be found inside For practical experiments, fundamental frequency and impedance norm was chosen to be 10From the perspective of the reader, the orientation of this manuscript is towards three problems. Firstly, it brings thorough investigation of research and review papers focused on applications of FO circuits in analog design engineering . Considerable attention is paid on the various implementations of CPE. This section can help curious reader to find specific topic for their own research, to develop new application with promising properties, or to fix engineering problem unsolvable with IO circuits.A wide spectrum of potential applications with wideband CPEs are proposed in Each CPE provided in This paper brings a rich gallery of high-precision CPEs dedicated for wideband signal processing. Readers can pick and use proposed CPEs directly, without the need of additional calculations. Individual designed CPEs have reasonable values of circuit components that can be found commonly in stocks of markets. Individual outputs resulting from this paper attract a wide spectrum of enthusiasts, electronic engineers, and design specialist to construct linear and nonlinear systems described by FO dynamics. Moreover, using designed CPEs, existing structures of FO filters, harmonic oscillators and arbitrary waveform generators , phase correctors, PID controllers, regulators, models of dynamical systems, etc. can be simulated again, and associated results can be polished.From a nonlinear dynamics point of view, this paper demonstrates that FO analog memory can be chaotic, even if real CPEs are included both into mathematical model and real fabricated circuit. This is a new and so far unpublished reality, proved by means of numerical calculations, computer-aided analysis of memory circuit, as well as experimental outputs.


Purpose: Optical coherence elastography (OCE) is a promising technique for high-resolution strain imaging in ocular tissues. A major strain-inducing factor in the eye is intraocular pressure (IOP), with diurnal physiological fluctuations reaching up to 5 mmHg. We study herein low-amplitude IOP modulation to assess local corneal strain patterns.Methods:Ex vivo porcine eye globes were adjusted to an initial IOP of 15 mmHg and subsequently 25 mmHg. Corneal strain was induced by two subsequent pressure cycles, in which IOP was first increased and then decreased, each by a total of 5 mmHg. Two-dimensional optical coherence tomography (2D-OCT) B-scans were recorded after each loading step. Axial strain maps were obtained from magnitude and phase changes and supra-pixel displacements from cross-correlation. The strain detection sensitivity was evaluated in an isotropic material.Results: Deformations arising from a single 1-mmHg step could be resolved. The largest strain amplitudes (5.11·10−3) were observed in the posterior stroma at a low initial IOP. Strain amplitude was 1.34 times higher at 15 mmHg than at 25 mmHg (p = 0.003). Upon IOP increase, the anterior cornea was compressed, whereas the posterior cornea showed axial expansion. Both morphological images and strain maps were sensitive to postmortem time. Strains that are larger than 2.44·10−5 could be reliably measured.Conclusions: Low-amplitude IOP modulation, similar to diurnal physiological changes, induced measurable deformations in corneal tissue. Axial strain maps permit a localized comparison of the corneal biomechanical response. Small-strain OCE can likely be extended to other domains. Intraocular pressure (IOP) is the principal source of mechanical stress in ocular tissues. IOP undergoes diurnal physiologic fluctuations of ~5 mmHg in healthy individuals, with highest values in the first half of the day /666 μm (in vivo), a mean keratometry value of 40.0 diopters, and a diameter of 23.9 mm  spectral domain optical coherence tomography (SD-OCT) system with a central wavelength of 878 nm and a bandwidth of 62.5 nm was used. Images were recorded with an A-line rate of 10 kHz, an integration time of 90 μs, and an intensity of 1.62 mW at the sample surface. The OCT system had an axial sampling precision of 4.48 μm, corresponding to its axial resolution in air, and a spot size of 12.5 μm, corresponding to its lateral resolution. All samples were imaged over a lateral range of 14 mm with 1,000 A-lines/B-scan. A single OCT measurement set consisted of six B-scans taken in series each with a size of 2,048 axial × 1,000 lateral pixels, whereas the length of the reference arm was modulated  with a pThe raw SD-OCT signal was subjected to basic signal processing, including background removal, remapping, and dispersion compensation using procedures described earlier :z ∈ ℕm≤ × x ∈ ℕn≤ → ℂ, with m = 2,048 and n = 1,000 representing the complex OCT signal in spatial domain, z and x indicating the axial and lateral positions, and * indicating the complex conjugate. j ∈ ℕjmax≤ indicates the number of steps for reference arm modulation with jmax = 6. Next, to compute the deformation induced by IOP modulation, the spatially resolved phase difference map ΔΦiop between the images of two subsequent pressure steps was determined using an approach similarly to that described above, with the difference that the summation was performed over a window of size 7 × 7 pixels with the purpose of noise removal and missing data fill-in. ΔΦiop corresponds to the angle of Ciop in the following equation:Following the complex-valued vector summation approach described by Zaitsev et al.  ΔΦstep rwz = 3 and wx = 3 pixels. To quantify supra-pixel displacements (bulk motion), an axial grid search was performed over p and the magnitude of Ciop was used—similarly to standard cross-correlation speckle tracking—to extract macroscopic absolute axial supra-pixel displacements zsp for aligning two subsequent B-scans during phase difference computation:where Bulk motion compensation during processing is particularly necessitated for measurements with a low initial IOP. Note that bulk motion does not compromise strain analysis as long as displacement between two adjacent axial pixels is lower than λ/4, which corresponds to a strain of:mean is the central wavelength of the OCT (878 nm), ncornea is the refractive index of corneal tissue (1.375), and asu is the axial sampling unit in air (4.48 μm). This strain threshold is more than 10 times larger than our experimentally measured strains. For enhanced interpretation of axial displacement, a parabola was fitted to the critical point Δzsp and its two neighbors (p = Δzsp ± 1) to determine subpixel shifts in addition to bulk motion. After correcting for bulk motion , the axial strain εz was determined from the axial gradient of ∠Ciop by summing the complex vector differences between the original image and a copy of it shifted by 1-pixel in z-direction, within a window of size 17 × 3 pixels, that is:where λz′ = 8 and wx′ = 1 pixels andwith wvia Equation 3a were directly used for strain interpretation without any further post-processing. It should be noted that a phase unwrapping algorithm or a least-squares estimator is not required with this vector-based phase gradient approach . The same approach was applied to retrieve axial strain. In consequence, strain maps computed E · ε, where σ represents the stress tensor, ε the strain tensor, and E the stiffness tensor (or elastic modulus). Assuming plane stress is applied in the x-direction, the stress tensor can be reduced to:For a ground-truth controlled reference measurement of the error inherent to strain measurements in the current study, a homogeneous isotropic material [polydimethylsiloxan (PDMS)] with an elastic modulus of 200 kPa was stepwise laterally compressed while axial strain was measured by the OCT. Using the Cartesian coordinate system, Hook's law for an isotropic material Bayoumi,  can be eThe strain tensor then results in:andPoisson's ratio is defined as the ratio between the axial strain and lateral strain:PDMS = 0.5 refers to the Poisson's ratio of PDMS. In combination with a macroscopic estimate of the lateral strain according to:where νzz,GT relates as follows to the lateral displacement x:the ground-truth axial strain εx0 refers to the initial lateral length of the sample, which was ~5 mm. The entire sample had dimensions of ~5 × 5 × 5 mm. Lateral displacement was incremented nearly logarithmically from 3.85 to 38.5 μm using a discrete piezoelectric stack actuator . This displacement range corresponds to an axial strain range of 0.385–3.85‰, which matches well with the observed strain amplitude observed in corneal tissue (0.00–5.11‰). For statistical analysis, six arbitrary locations within the B-scan of a given lateral deformation were selected, which were spaced by more than the accumulated processing windows applied for phase and strain computations. The same relative locations were then tracked across B-scans taken at different lateral deformations. Mean values and standard deviation were calculated to determine strain measurement precision at a single pixel, which in turn was used for comparison with the experimentally determined interspecimen standard deviation in porcine corneas. These parameters were independent of measurement location.where r and circumferential σθ = σφ stress components  are given by the Lame's equation:The stress distribution imposed by the IOP does not have a homogeneous distribution. Assuming that the eye is a thick-walled spherical pressure vessel, the radial σrpost represents the posterior radius of the curvature, rant is the anterior radius of the curvature, Piop is the IOP, Patm is the atmospheric pressure, and r is the radial distance from the origin. Patm was set to zero, as IOP is defined as the pressure difference between IOP and extraocular pressure.where In corneal tissue, ~250 collagen lamellae Meek,  are stacθ = εϕϕ = εθθ and εr = εrr, Hook's law for a transversely isotropic material  and radially , respectively; νθ and νrθ represent Poisson's ratio, respectively, circumferentially and radially. Applied to the current experimental setting, the following equation is relevant:where Erpost≈ 6.4 mm, rant≈ 7.2 mm), σθ dominates over σr. However, the circumferential E-modulus (Eθ) in the direction of the collagen lamellae is considerably higher than the radial E-modulus (Er) along corneal thickness. A recent study  and across five subsequent pressure steps for each individual sample. To assess axial apex displacement, vertical displacements in the central region were averaged across the entire corneal depth for each sample. We also computed the maximal strain amplitude and hysteresis for each sample. A paired two-tailed Student's GTzz, − εzz,exp|/εGTzz,, where εzz,exp represents the experimentally determined axial strain value for different levels of lateral deformation of the PDMS sample. Relative deviation from ground truth was lower for larger deformations. For strains larger than ~2.44·10−5, the OCT-derived strain did match acceptably well with the ground-truth strain  and 7 pixels (~98.0 μm), respectively. h strain  imposed p < 0.001) and central (p = 0.016) corneas did in response to IOP modulation and presented a residual displacement of the opposite sign than the anterior (p = 0.004) and central (p = 0.015) cornea. p = 0.003) larger (3.80 ± 0.98‰) compared to a higher initial IOP (2.48 ± 0.75‰) (± 0.75‰) .p = 0.054)  . Table 1pearson = 0.917, p = 0.010) but not at 25 mmHg . Anterior strain also correlated significantly with postmortem time at 25 mmHg  but not at 15 mmHg . A significant correlation of corneal strain between 15 and 25 mmHg was observed, both in the anterior  and posterior  tissues. In contrast, posterior strain was not significantly affected by postmortem time or corneal thickness.We observed a substantial change of axial strain amplitude and distribution with increasing experimental time and hence with longer postmortem time. 4 times lower stresses than those described before in extensometry  observed in corneal tissue and by factor 2 less than the interspecimen standard deviation (0.70‰) encountered. Spatial resolution of strain maps is currently limited to 71.7 μm axially and 98.0 μm laterally. Further refinement of processing windows, for example, by using an adaptive windowing approach, could locally increase strain resolution and improve epithelial strain retrieval. Without this tool, interpretation of structures of dimensions exceeding 20 μm axially and laterally can be performed on the phase map, which however is less intuitive. Nonetheless, a distinct sign of strain of epithelium and anterior stroma can be recognized by visual inspection of the phase map, that is, by taking the vertical gradient, in A significantly larger strain amplitude and larger hysteresis were observed at the lower compared to the higher initial IOP. The stress–strain curve  did showin situ is a strongly hydrated tissue and has a swelling pressure  being observed in the posterior stroma and lower strains toward the anterior stroma . Such a The observation that corneal strain, both in the anterior and posterior corneas, continued to increase even when the applied stress (IOP) was released corroborates the above hypothesis. Although the vertical apex displacement recovered after reducing the IOP, the osmotic swelling could have further progressed and potentially caused the anterior compression. To estimate the extent of osmotic swelling during the measurement, consider the following: When storing enucleated porcine eyes for 24 h (without IOP control), they typically swell by ~300 μm, corresponding to a rate of 3.5 nm/s, assuming swelling at a constant speed. A similar speed (3.6 nm/s) is obtained when computing the slope of corneal thickness across time in the current experiment. With a temporal separation between subsequent measurement steps of 14 s, an axial strain contribution of 49 nm/800 μm = 0.06‰ can be expected. This strain originating from osmotic swelling is substantially lower than posterior strain values (~1‰). An alternative interpretation to explain that strain continued to increase during pressure decrease considers two additional factors: (1) corneal tissue has viscoelastic properties and (2) enucleated eyes lose IOP with time. Potentially, adjusting the initial IOP to 15 or 25 mmHg could have induced a creep that masked the subsequent pressure modulation. A limitation of the current study arising from these hypotheses is that neither osmotic swelling nor creep could be separated from IOP-induced deformations.ex vivo study, at the same time, the strong dependency, particularly of the sign of the induced axial strain, on hydration could be an interesting feature for the diagnosis of Fuchs dystrophy, in which endothelial pump function is impaired  conditions resulting from the IOP are considered in finite element simulations (Elsheikh et al., In the current study, small strains were induced by means of minimal IOP changes. However, this does not restrict OCT-based small strain imaging to be transferred to other areas. In the domain of mechanotransduction, potentially compatible mechano-coupling techniques for subjecting cells, or tissue samples, to microscale mechanical strains have already been developed (Quinn et al., The raw/processed data required to reproduce these findings are available from the corresponding author on request.SK perceived and designed the study, performed the experiments, processed, analyzed and interpreted the data, drafted the manuscript, and obtained funding. HK adapted the experimental setup, performed the experiments, and revised the manuscript. OG contributed to image processing algorithms, biomechanical interpretation, and revised the manuscript.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Chinese-speaking parents are believed to use less cognitive mental-state-talk than their English-speaking counterparts on account of their cultural goals in socializing their children to follow an interdependence script. Here, we investigated bilingual English–Mandarin Singaporean mothers who associate different functions for each language as prescribed by their government: English for school and Mandarin for in-group contexts. English and Mandarin maternal mental-state-talk from bilingual English–Mandarin mothers with their toddlers was examined. Mothers produced more ‘’cognitive’’ terms in English than in Mandarin and more ‘’desire’’ terms in Mandarin than in English. We show that mental-state-talk differs between bilingual parents’ languages, suggesting that mothers adjust their mental-state-talk to reflect the functions of each language. In toddlerhood, children’s primary socialization models are their parents . ThroughParental mental-state-talk affects several aspects of a child’s social development, including but not limited to sociocultural values and norms, socioemotional competence, and theory of mind reasoning, even after controlling for age and language ,7. To daIn daily interactions, parents often highlight the thoughts  and wishes  of others through their mental-state-talk. Mothers frequently talk about both desires and cognitions in child-directed speech ,15,16,17Across cultures, one prominent difference in parental mental-state-talk is the frequency of cognitive mental-state-talk noted between English-speaking and Chinese-speaking parents ,22,23,24Some studies have examined differences in Chinese-speaking and English-speaking parents’ mental-state-talk ,10, but Singapore is a diverse society that prides itself in striking a balance between adopting modern Western values while retaining traditional Eastern values ,33,34. AFor the present study, we explored how bilingual mental-state-talk may differ between two languages. To elicit maternal mental-state-talk in bilinguals, mothers were asked to engage in two free-play sessions, where they were instructed to exclusively use English for one of the sessions and Mandarin for the other session. Free-play sessions were used over story-telling sessions for three reasons. First, meta-analysis indicates that there are no quantitative differences in maternal mental state references between story-telling and naturalistic interactions . Second,We had two predictions. First, we expected to replicate previous findings showing that mothers frequently use desire mental-state-talk when engaging with their 18-month-old toddlers. Thus, regardless of language, mothers will produce more expressions of desire than expressions of cognition. Second, we expected the frequency of mental-state-talk to differ depending on the language mothers spoke. Specifically, because English is promoted as the lingua franca for business and education, and mother tongues are used as the language for home and in-group interactions in Singapore, mothers should produce more cognitive expressions in English than in Mandarin and more desire expressions in Mandarin than in English, reflecting each language’s functions and roles in Singaporean society.M = 33 years, SD = 3 years, range = 29 years to 40 years) and their toddlers  participated. Mothers were recruited online through Facebook pages and were asked several screening questions, including whether the mother would be able to spend 30 min alone with her toddler and whether the mother would be comfortable with speaking exclusively English and speaking exclusively Mandarin for 15 min each. Mothers had either a university degree (70%), a post-graduate degree (10%), a junior college/polytechnic diploma (13%), or completed secondary school (7%). Most mothers reported that they were currently employed (77%). The remaining mothers reported that they were on leave from work (3%), currently unemployed (7%), or a homemaker (13%). Most mothers reported that their child’s preferred language was English (73%), while the rest either reported that their child’s preferred language was Mandarin (20%) or had no preference (7%). Toddlers’ average daily percentages for hearing each language were also collected. Two mothers reported percentages that summed to greater than 100%. Excluding these mothers, mothers reported that their toddlers heard more English on average  than Mandarin . Including the two mothers whose summed percentages exceeded 100% produced a similar result . Mothers were compensated with a small monetary amount, and toddlers received a certificate for their participation. The study was approved by the Institutional Review Board of Nanyang Technological University (NTU-IRB-2014-11-010). All data have been made available at the following URL: https://doi.org/10.21979/N9/1KTUHC.Thirty Singaporean-Chinese English and Mandarin bilingual mothers , a toy cellphone, a train, a foam ball, five nesting barrels, and two storybooks, ‘’Guess How Much I Love You’’ and ‘’The Very Hungry Caterpillar’’. The language of the storybooks matched the language of the free-play session . Mothers first completed a demographic questionnaire and two vocabulary checklists, one in English and one in Mandarin. Specifically, the short-form versions of the MacArthur–Bates Communicative Development Inventory  and the They were then asked to engage their toddler in two 10 min free-play sessions, where they were asked to speak exclusively in English for one session and exclusively in Mandarin for the other session. Language order for the two free-play sessions was counterbalanced across participants. Free-play sessions were video-recorded. Speech from the two free-play sessions was transcribed by one of four English–Mandarin bilingual research assistants using the Computerized Language Analysis (CLAN) program . Each trMental state words were taken from existing studies on maternal mental-state-talk ,32 and wMothers infrequently code-switched between languages . Including English types and Mandarin types outside of their designated language session, mothers produced less than 5% of English types in the Mandarin-designated session and less than 1% of Mandarin types in the English-designated session. In total, there were eight occurrences of mental-state-talk outside of their designated language session. There were three occurrences of “要” (“want”) across English sessions and five occurrences of “want” across Mandarin sessions. To capture maternal mental-state-talk, researchers have calculated either a sum to measure absolute frequency of mental-state-talk  or a proEnglish and Mandarin mental state words were first coded by two independent coders. A third coder coded both sets of transcripts. All coders reviewed all extracted words with the transcript and the accompanying video to examine how each targeted word was used. Cohen’s kappas, 0.87 for English and 0.73 for Mandarin, indicated high interrater reliability for both English (92% agreement) and Mandarin (84% agreement) mental state word coding. The following analyses used the first coders’ mental state frequency counts. Mental state type was calculated as frequency for each mental state word category.n = 5868 English utterances, n = 5857 Mandarin utterances), and their toddlers produced 2206 utterances . There was no difference between the numbers of utterances produced in the English sessions and the Mandarin sessions for both mothers, t(29) = 0.04, p = 0.97, d < 0.01, or toddlers, t(29) = −0.19, p = 0.85, d = 0.04. Speech richness between the two languages was also examined by conducting t-tests on type-token ratios. Type-token ratio was calculated by dividing the number of types produced in one language by the number of tokens produced in the same language. Higher type-token ratios indicate more complex speech. Type-token ratios did not differ between languages for either mothers, t(29) = −0.22, p = 0.83, d = 0.04, or toddlers, t(29) = 1.78, p = 0.08, d = 0.324.Overall, mothers produced a total of 11,725 utterances  was larger than their reported Mandarin vocabulary , t(29) = 4.42, p < 0.01, d = 0.81. There were no significant correlations between toddlers’ age, maternal educational level, and the frequency of cognitive and desire maternal mental-state-talk. There were significant positive relationships between toddlers’ age and their reported English vocabulary, r(28) = 0.75, p < 0.01, as well as their reported Mandarin vocabulary, r(28) = 0.41, p < 0.05.Toddlers’ reported English vocabulary (t(29) = −0.48, p = 0.63, d = 0.15. A 2 (Language) × 2  repeated-measures ANOVA revealed a main effect of Mental State Type, F = 58.52, p < 0.01, ηges2 = 0.42. Overall, mothers used more desire  than cognitive  words. There was also an interaction between language and mental state type, F = 8.54, p < 0.01, ηges2 = 0.03. Follow-up pairwise t-test comparisons revealed that mothers used more cognitive words in English  than in Mandarin , t(29) = 2.98, p < 0.01, d = 0.54. There was also a marginal difference in the frequency of maternal desire mental-state-talk, t(29) = −2.01, p = 0.054, d = 0.37; mothers produced more desire words in Mandarin  than in English . See Total maternal mental-state-talk as the sum of cognitive and desire mental-state-talk for each language was calculated. Total maternal mental-state-talk did not differ between languages, n = 25 total occurrences), mothers produced the cognitive words, “forget” (n = 1 occurrence), “guess” (n = 1 occurrence), “pretend” , “remember” (n = 11 occurrences), “think” (n = 8 occurrences), and “understand” (n = 1 occurrence). The average frequency of each mental state produced by mothers is illustrated in Upon closer examination of the frequency of mental-state words produced, we found that mothers’ desire production was dominated by the production of “want.” In fact, the only other desire word produced, “care”, was produced only once by a mother in the English session; no other desire words were produced in the Mandarin session. Overall, the cognitive mental state word, “know”  was produced most frequently in each language. For Mandarin, only one other cognitive mental state, “pretend,” was produced and only once across sessions. There was more variation in English cognitive mental state production. Aside from “know” (n = 1) and a third person’s knowledge state (n = 2). In most instances of “remember,” mothers were asking their toddler if the toddler could recall a previous event except for two instances where mothers used the plural first person, “we.” Similarly, “forget,” “guess,” “understand,” and “pretend” were used to refer to the child’s cognitive mental state. However, unlike other cognitive mental states, “think” was used primarily to refer to the mother’s own cognitive state (7 out of 8 occurrences).Most cognitive expressions were used to refer to the child’s cognitive state rather than the mother’s or a third person’s cognitive state. Specifically, all occurrences of “know” in Mandarin were mothers asking their toddler about their (the toddler’s) knowledge capacity. For example, one mother asked her toddler, “懂 什么 是 青色 吗” (“Know what is green?”). Similarly, in English, mothers primarily (apart from three instances of “know”) asked their toddlers about their knowledge capacity. For example, one mother asked, “does xxx know how to jump?” where “xxx” is the toddler’s name. For the other instances of “know” mothers referred to their own knowledge state  as the language for cultural and heritage-related communications, highlighting interdependence ,44, it iExtensive research has examined the significance of maternal mental-state-talk in children’s development, but with bilingualism becoming increasingly common, it is also critical to examine the characteristics of bilingual mothers’ mental-state-talk. In Singapore, bilingualism is encouraged, but each language’s functions differ. English is deemed as a common language that unites Singaporeans, and Singaporeans to Westerners, in public business and educational spaces. Singaporean mother tongues, such as Mandarin, are languages to be used at home or with an individual’s ethnic group. In the present study, bilingual English–Mandarin Singaporean mothers produced mostly words referring to desires. There were also cross-linguistic differences in mental-state-talk. Mothers used more cognitive mental state words in English than in Mandarin, and more desire mental state words in Mandarin than in English. This cross-linguistic difference cannot be explained by the mother’s perception of their child’s language preference or dominance. The government’s demarcation of social values and norms for each language may have contributed to variations in how the Singaporean mothers use mental state words in each language, and thereby, socialize children to understand the different roles that each language plays in their society.


EPSPS. In horseweed (Conyza canadensis), this p106 mutation has only been reported in Canada. Here, we sampled seeds from one plant (= biotype) at 24 sites in Ohio and 20 in Iowa, screened these biotypes for levels of resistance, and sequenced their DNA to detect the p106 mutation. Resistance categories were based on 80% survival at five glyphosate doses: S (0×), R1 (1×), R2 (8×), R3 (20×), or R4 (40×). The p106 mutation was not found in the19 biotypes scored as S, R1, or R2, while all 25 biotypes scored as R3 or R4 had the same proline-to-serine substitution at p106. These findings represent the first documented case of target-site mediated glyphosate resistance in horseweed in the United States, and the first to show that this mutation was associated with very strong resistance. We hypothesize that the p106 mutation has occurred multiple times in horseweed and may be spreading rapidly, further complicating weed management efforts.Documenting the diversity of mechanisms for herbicide resistance in agricultural weeds is helpful for understanding evolutionary processes that contribute to weed management problems. More than 40 species have evolved resistance to glyphosate, and at least 13 species have a target-site mutation at position 106 of  Arguably, no herbicide has had a larger impact on agricultural weed management practices in the last few decades or been more widely applied than glyphosate, the active ingredient in RoundUp3. As a non-selective, post-emergent herbicide, glyphosate inhibits 5-enolpyruvylshikimate-3-phosphate synthase (EPSPS), a key enzyme in the shikimate acid pathway required for aromatic amino acid synthesis2. The majority of all maize, soybean, cotton, sugar beet, and canola planted in the United States is glyphosate resistant3. The total amount of glyphosate applied in the United States has risen drastically over the past two decades, and the amount of glyphosate applied per hectare in soybean fields more than doubled from 1995 to 20153.Widespread application of herbicides has spurred the rapid and repeated evolution of herbicide resistance in agricultural weeds, and at least 43 weed species have evolved glyphosate resistance5. Non-target site mechanisms involve metabolism, altered translocation, and vacuolar sequestration, while target-site mechanisms generally include amplification of EPSPS, overexpression of EPSPS, and amino acid substitutions in EPSPS5. Point mutations conferring glyphosate resistance have been well-documented in several weed species; the most common substitution is a change of a conserved proline at position 106 (p106) of EPSPS to one of four other amino acids  by Baerson et al.6, who noted that it lies within a highly conserved region of amino acids found within all plants’ and most bacterial EPSPS genes. For E. indica, Baerson et al.6 aligned the sequence of amino acids in this region with the numbering system for Escherichia coli and other microbes, rather than the true sequence for E. indica  was found as the sole glyphosate resistance mechanism within individuals of Amaranthus hybridus L. from Argentina12.In addition to glyphosate resistance mutations at p106, there are reports of additional point mutations occurring at p102 and p103. For example, a double point mutation (threonine to isoleucine at p102 and proline to serine at p106) was found in individuals of Conyza canadensis, known as horseweed or marestail, was the first broadleaf weed to evolve glyphosate resistance, first observed in 2000 in no-tillage soybean fields in Delaware, USA13. Since then, glyphosate resistance in horseweed has been documented in 25 states within the USA and 12 countries worldwide1. As one of the first glyphosate-resistant weeds identified, C. canadensis has been the focus of many studies on the specific mechanisms of glyphosate resistance. Early reports indicated that glyphosate resistance was controlled by an incompletely dominant, single-locus nuclear allele15. Initial reports, which were later confirmed by other studies, determined that horseweed was able to absorb glyphosate and rapidly sequester it within the vacuoles of its leaves, and this was determined to be “the horseweed resistance mechanism”17. Indeed, non-target site resistance mechanisms  have been found in horseweed samples from Arkansas, California, Delaware, Iowa, Mississippi, Ohio, Virginia, and Washington in the USA, as well as in Canada, Spain, and Greece27. Several studies have linked vacuolar sequestration to an active process involving transporter genes that are induced by glyphosate treatment28.C. canadensis. A study by Ge et al.29 reported an unknown resistance mechanism in a biotype from Delaware conferring low-levels of resistance. González-Torralva et al.22 documented horseweed biotypes in Spain that metabolized glyphosate. In addition to having reduced translocation, horseweed biotypes from Ohio, Delaware, Arkansas, and Virginia were found to have 2–3-fold greater levels of basal EPSPS mRNA in glyphosate resistant biotypes19. Similarly, Tani et al.28 reported 2–3-fold greater EPSPS expression in a resistant biotype from Greece. However, another study from Greece30 did not find increased EPSPS transcript levels in two resistant biotypes, and Peng et al.25 did not detect multiple copies of EPSPS in a resistant biotype from Tennessee, USA.However, there also have been scattered reports of other non-target and target-site mechanisms in EPSPS genes , and several studies have attempted to identify a point mutation conferring glyphosate resistance in horseweed. The first target-site mediated glyphosate resistance in horseweed was reported in Canada in 2018 by Page et al.23. Page and colleagues identified the proline to serine mutation at p106 of EPSPS2 in 21 glyphosate resistant populations sequenced at EPSPS2. However, other researchers did not detect this point mutation in horseweed populations sampled from Crete and Lakonia in Greece, from Delaware in the United States, or from the Beijing and Shandong provinces in China31.Horseweed has three Conyza canadensis across northcentral Ohio and southern Iowa. In these screenings, we found that glyphosate resistance to up to 40x the manufacturer’s recommended application rate (1x= 840 g ae ha−1) was quite common in both states32. These very high levels of glyphosate resistance, along with the recent finding of a point mutation in Canada, prompted us to sequence our horseweed biotypes from Ohio and Iowa to search for a similar point mutation. Unlike previous studies investigating point mutations in horseweed, including Page et al.23, we characterized different levels of glyphosate resistance based on 80% survival at various dosages: 0x(S = Susceptible), 1x(R1), 8x(R2), 20x(R3), and 40x(R4), and we sequenced a large number of biotypes.Many of the studies cited above examined relatively few biotypes and did not account for the broad range of glyphosate resistance levels found within horseweed. We had previously screened single-plant seed accessions (=biotypes) of Does a point mutation occur in horseweed biotypes originating from northcentral Ohio and southern Iowa?If so, is the same point mutation (Pro-106-Ser) found in biotypes from Ohio, Iowa, and Canada?Is the point mutation found in biotypes with all levels of glyphosate resistance (R1–R4)?The main questions addressed by the study were:EPSPS2 gene, while all biotypes from Ohio and Iowa that were scored as either R3 or R4 did have the proline to serine substitution at p106 , R1, or R2 had a point mutation at p106 of the 26.While we did not test the Ohio and Iowa biotypes for altered translocation and vacuolar sequestration, we presume that biotypes at all levels of resistance (R1–R4) would most likely possess one or more non-target site mechanisms of resistance. We base this assertion on the fact that every previous study which has looked for altered translocation and vacuolar sequestration within glyphosate resistant horseweed populations has found ite.g.4, although there have been reports of a p106 point mutation working synergistically with non-target or other unknown mechanisms to provide increased levels of glyphosate resistance35. In the present study, the Pro-106-Ser mutation was associated with extreme levels of glyphosate resistance  is noteworthy. Individually, target-site mechanisms are typically considered to confer low levels of glyphosate resistance in other specieset al.37 estimated that heavily infested fields may potentially disperse horseweed seeds ~1–5 km per year. For non-target site glyphosate resistance, Okada et al.38 detected multiple origins in horseweed populations in California, where resistance was first reported in 20061. For reference, glyphosate resistance was first reported in Ohio, Iowa, and Canada in 2002, 2011, and 2010, respectively1. Seed collections for the current study were carried out in 2015  and 2011 and 2012 (Canada).The broad geographic distribution of the point mutation across northcentral Ohio, southern Ontario, and southern Iowa Fig. , suggestet al.23 did not characterize the levels of glyphosate resistance in their study, they did sequence a sufficiently large sample (N = 28 glyphosate-resistant populations), which may have improved the likelihood of detecting the point mutation. Studies that use small sample size comparisons and only low-level resistant (R1 or R2) biotypes might not be expected to detect a point mutation, given that the point mutation was associated with the extremely resistant (R3 and R4) biotypes in the present study  and used extremely small sample sizes for comparisons (1–2 susceptible populations and 1–2 resistant populations). While Page EPSPS at p106, p102, and, in one case p103. Further research and literature reviews on target-site mediated glyphosate resistance would benefit from clear and consistent numbering of amino acid positions in EPSPS. In general, references to the p106, p102, and p103 mutations have become standard practice in research on herbicide resistance mechanisms, but this is not always the case. For example, some authors have referred to the p106 mutation as p182 based on alignment with Arabidopsis thaliana, GenBank:CAA29828.139. Figure Eleusine indica.As shown in Table et al.6, which aligns the conserved region with microbial sequences, while also acknowledging the true position of a given substitution based upon species-specific sequences . Soybean-producing counties were selected for sampling in northcentral Ohio and southern Iowa, with a minimum of 1.6 km between collection sites in both states. Briefly, we collected seeds from one mature plant in each population during fall 2015 . We refer to these samples as accessions, rather than biotypes, because they represent bulked samples from populations that may or may not have a mixture of susceptible and resistant plants.This study used a subset of samples that were characterized for glyphosate resistance in two previous studies  tray, which had 2 plants per accession. As a method of quality control and reproducibility, we included three previously screened biotypes from Ohio while screening the Canadian accessions.Methods for glyphosate resistance screening for biotypes originating from Ohio and Iowa are described in Beres www.fafard.com) mixed with a slow-release fertilizer  to minimize nutrient deficiencies, based on manufacturer’s recommendations for annuals (~360 grams/2.8 ft3). Tray positions were randomized weekly to minimize environmental variation in the greenhouse. The greenhouse was maintained at 25/18 °C day/night with a 14-hour photoperiod, and plants were watered as needed. Seedlings were thinned 7 days after planting to leave 4 (2 for control) uniformly-sized and spaced plants within each tray.Seeds from each biotype or accession were germinated in trays (12 cm ×24 cm) with potting soil  and non-ionic surfactant  at 5% and 0.5% (v/v), respectively. When rosettes were 4–6 cm in diameter, the plants were treated with one of five glyphosate doses: 0x(ammonium sulfate solution and surfactant only), 1x, 8x, 20x, or 40x. This rosette size range at application was similar to previous studies of GR horseweed45. Treatments were applied using a pneumatic track sprayer equipped with an even, flat-spray tip  calibrated to apply 140 L ha−1 of spray solution at 3.5 km hr−1. Plants were returned to the greenhouse and maintained for 3 weeks as previously described.All spray treatments included ammonium sulfate solution  to 10 (dead), with 9 being ~90% dead tissue. Proportion survival was calculated for each biotype/accession at each dosage. The proportion surviving at each dosage was used to characterize each biotype or accession into resistance categories. Biotypes with less than 80% survival at 1x were classified as “Susceptible.” Resistant biotypes with 80% survival at 1x but less than 80% survival at 8x were classified as “R1.” Likewise, biotypes with >80% survival at 8x were classified as “R2,”>80% survival at 20x as “R3,” and >80% survival at 40x as “R4.”Horseweed seeds from the same biotypes or accessions described above were germinated during spring 2018 in 3” round pots filled with moistened Fafard no. 2 soil. Plants were thinned one week after planting to leave one seedling per pot . The greenhouse was maintained at 18–21/23–26 °C (night/day), and supplemental lights  were used for 14-hour days.After ~3 weeks in the greenhouse, 2 leaves from each plant were harvested (~200 μg of leaf tissue), snap frozen in 2-mL tubes in liquid nitrogen, and stored at −80 °C. One leaf was used to collect DNA while the second leaf remained in storage as a back-up sample. Tissue samples were ground in 2-mL tubes by shaking with two 5-mm glass beads at 30-Hz for 1 minute with a Qiagen Tissue Lyser in Tissue Lyser plates that had been pre-chilled to −80 °C. Genomic DNA was then extracted using a DNeasy Plant Mini Kit from Qiagen (#69104), quantified with a NanoDrop™ spectrophotometer, and stored at −20 °C until use.et al.23, we focused on sequencing DNA from exon 2 in EPSPS2. PCR was used to amplify a 986-bp region (GenBank Accession: AY545667.1) around and including the expected single nucleotide polymorphism (SNP) at p106  of the EPSPS2 gene. The designed primers ensured coverage of the p102 and p106 sites because these are the most common positions for a point mutation conferring glyphosate resistance. The forward primer had a sequence of ggactactgttgtagacaacttg, and the reverse primer was gtgggcagtttgtaccgaga. The PCR reactions were run under standard conditions with Taq from New England Biolabs (#M0267L). The thermal cycler program started with an initial denaturation at 95 °C for 2 min, followed by 35 cycles of 95 °C for 20 sec, 58 °C for 30 sec, and 68 °C for 1 min. After cycling was complete a final extension at 72 °C for 5 min was performed to end the PCR reaction. PCR products were held at −20 °C until they were sequenced.Based on findings of Page Conyza canadensis EPSPS2 cDNA was scored as either “C” encoding proline or “T” encoding serine at amino acid position 106  to remove excess reagents and primers. The OSU Genomics Shared Resource at The Ohio State University performed sequencing of the clean PCR products. The sequencing primer had a sequence of gaatcctcctactcatataattgtg. Sequence data was analyzed using SnapGene viewer (version 3.0.1) to identify any point mutations, especially a proline to serine substitution. A single nucleotide polymorphism (SNP) at position 106 of 04 to remSupplementary Information.


Correction to: Alz Res Therapy 12, 5 (2020)https://doi.org/10.1186/s13195-019-0571-3Following publication of the original article , the aut1. The sentence “TST was lower in CN, SCI, and MCI compared to mAD patients who in turn showed a lower TST than msAD patients.” under subsection “PSG data ”should be replaced with:TST was HIGHER in CN, SCI, and MCI compared to mAD patients who in turn showed a HIGHER TST than msAD patients.2. The sentence “Was excluded as it presented split loadings on all factors, and SE and TIB were excluded as they were computed from other polysomnographic parameters.” under subsection “Principal component analysis” should be removed.


Sleep difficulties are common among older adults and are associated with cognitive decline. We used data from a large, nationally representative longitudinal survey of adults aged older than 50 in the United States to examine the relationship between specific sleep difficulties and cognitive function over time.n = 16 201). Sleep difficulty measures included difficulty initiating sleep, nocturnal awakenings, early morning awakenings, and waking up feeling rested from rarely/never (1) to most nights (3). The modified Telephone Interview for Cognitive Status was used to measure cognitive function. Generalized linear mixed models were used with time-varying covariates to examine the relationship between sleep difficulties and cognitive function over time.Longitudinal data from the 2004–2014 waves of the Health and Retirement Study were used in the current study. We examined sleep difficulties and cognitive function within participants and across time  as was difficulty waking up too early “most nights” . In covariate-adjusted analyses, compared to “never” reporting waking up feeling rested, cognitive function was higher among those who reported waking up feeling rested “some nights” .In covariate-adjusted models, compared to “never” reporting sleep difficulty, difficulty initiating sleep “most nights” was associated with worse cognitive function over time (Year 2014: Our findings highlight an association between early morning awakenings and worse cognitive function, but also an association between waking up feeling rested and better cognitive function over time. Translational Significance: Sleep difficulties are common among older adults yet reduce the quality of life and also contribute to the development of and potentially accelerate cognitive decline. This study examines specific sleep difficulties  and their unique relationship to cognition over time among older adults in the United States. The primary aim of this work is to illuminate the specific sleep difficulties that are most concerning from the standpoint of cognitive impairment so as to inform the design of future tailored sleep improvement programs for older adults.Sleep difficulties  are more common among older adults (65 years or older) than any other age group in the United States . ParticuResearch has examined sleep and cognition among older adults and employed a range of sleep parameters in doing so. In research conducted among a nationally representative sample of men using both objective and self-reported measures, researchers found a proxy for sleep difficulty  and objectively recorded nighttime awakenings were associated with cognitive decline in the follow-up period, which was between 3 and 5 years . AnotherThe literature to date is limited in several ways. One clear critique of the literature is that the conceptualization of sleep difficulty has ranged widely from one study to another. Indeed, as noted above, research has employed a range of sleep difficulty measures, ranging from sleep apnea to daytime sleepiness. Furthermore, existing literature often develops summary scores for sleep difficulty as opposed to measures of specific sleep difficulties  ,9. Also,The present study addressed these gaps in the literature by examining specific symptoms of sleep difficulty across time and their association to cognitive function using 10 years of longitudinal, national data from the Health and Retirement Study (HRS). We hypothesized that each sleep difficulty  will be associated with worse cognitive function over time in covariate-adjusted analyses.We analyzed data from the HRS, an ongoing nationally representative longitudinal survey of adults aged older than 50 in the United States. The study began in 1992, with a core interview administered every 2 years. In 1998, HRS instituted a steady-state design, enrolling a new birth cohort every 6 years. The Health and Retirement Study uses a multistage area probability sampling design that includes geographic stratification, clustering, and oversampling of African American, Hispanic, and minority households . During n = 293). The final analytic sample was 16,201 respondents (88.4% of the population). The Health and Retirement Study collected information on sleep difficulties biennially until 2006, then in alternate waves thereafter. A new birth cohort was enrolled in 2010, but we excluded these new participants from our sample to maintain a consistent follow-up interval for all respondents. Thus, our analytic sample included all respondents surveyed in 2004 and at follow-up .We began with 18,327 respondents who participated in the 2004 wave of the HRS. While the sleep difficulty variables were available beginning in the prior wave in 2002, we chose 2004 as our baseline in order to maximize our sample for longitudinal analysis because 2004 was the enrollment year for the Early Baby Boomer cohort . We selected 17 592 respondents who demonstrated no significant signs of cognitive impairment at baseline, as evidenced by a score of 7 or higher on the validated Telephone Interview for Cognitive Status (TICS-m) during the baseline year for this analysis (2004) representing 96.0% of respondents. We then selected 16,494 respondents (90.0% of the population) who were aged older than 50 years. We excluded those respondents who were represented by a proxy (Sleep difficulty was measured with four items. Participants were asked to report the frequency of the following: (i) difficulty initiating sleep, (ii) difficulty maintaining sleep , (iii) early morning awakenings, and (iv) waking up feeling rested. Response options included 1 (most nights), 2 (sometimes), and 3 (rarely/never). We reverse-scored the items so that higher scores indicated more sleep difficulty  with the exception of feeling rested, for which higher values indicated more feelings of being rested. Finally, we created a new variable to indicate those who experienced concurrent sleep difficulties. Specifically, we identified those individuals who responded either “most nights” to two or more difficulties in a single wave  or “rarely/never” to the question regarding waking up and feeling refreshed.Cognitive function was measured using the TICS-m, which included immediate and delayed recall items, serial 7s subtraction, and counting backward. Immediate list recall included presenting 10 English nouns audibly to participants at a 2 seconds/word rate . ParticiWe controlled for a number of covariates that are known to be associated with cognitive function  and sleep value less than .05, and different models were compared based on Akaike’s information criterion. Model fit was assessed using the Wald statistic to test the null hypothesis, which was that the coefficients are simultaneously equal to zero. All analyses were performed using Stata software, version 16 .Baseline sample characteristics are reported in SD = 10.8 years). Participants reported a mean total asset value of $409,730 . Among participants, self-reported health was rated “excellent” by 7.0%, “very good” by 28.3%, “good” by 34.1%, “fair” by 22.2%, and “poor” by 8.3%. Among participants, 80.4% were married, 80.8% were white/Caucasian, and 9.3% were Hispanic/Latino. The average BMI of the participants was 27.4. Approximately one quarter (26.1%) reported one chronic medical condition and another quarter (25.5%) reported two chronic conditions.Summary statistics for all demographic and sleep difficulty variables at baseline are given in SD = 3.9).Regarding sleep difficulty, the majority of individuals reported “never” having difficulty initiating sleep (51.4%), followed by those reporting experiencing difficulty “some nights” (34.3%) and “most nights” (14.3%). Regarding difficulty with nocturnal awakenings, the majority responded “some nights” (38.3%), followed by “never” (34.9%) and “most nights” (26.8%). Regarding difficulty waking too early and not being able to return to bed, the majority of participants reported “never” experiencing this difficulty (52.5%), followed by “some nights” (34.8%) and “most nights” (12.7%). Regarding waking and feeling rested, the majority of the participants reported “never” waking and feeling rested (57.6%), followed by “some nights” (29.7%) and “most nights” (12.7%). Overall, the average cognitive function score was 15.6  = 1.53, p < .001; nocturnal awakenings: F = 4.19, p = .015; waking too early: F = 229.02, p < .001; feeling rested: F = 49.94, p < .001). Post-hoc comparisons examining pairwise relationships using the Tukey’s honestly significant difference revealed that cognitive function was significantly different between those who “never” experienced difficulty initiating sleep and those who reported “sometimes” experiencing difficulties initiating sleep (p < .001) and “most nights” experiencing difficulty initiating sleep (p < .001). Additional post-hoc pairwise comparisons revealed that cognitive function was significantly different between those who “never” experienced nocturnal awakenings and those who experienced difficulty with nocturnal awakenings “sometimes” (p < .05), but not those who experienced nocturnal awakenings “most nights” (p = .933). Pairwise post-hoc comparisons revealed that cognitive function was significantly different between those who “never” experienced waking too early and those who experienced difficulty waking too early “sometimes” (p < .001) and “most nights” (p < .001). Post-hoc comparisons also revealed that cognitive function was significantly different between those who reported “never” waking and feeling rested and those who reported “sometimes” waking and feeling rested (p < .001) and those who reported “most nights” waking and feeling rested (p < .001).Using analysis of variance, we found significant associations between each sleep difficulty and cognitive function  and in 2014 . In Model 2, partially adjusted analyses, compared to “never” experiencing difficulty initiating sleep, difficulty waking up too early “most nights” was associated with worse cognitive function in 2006  and in 2014 . In Model 3, fully adjusted analyses, compared to “never” experiencing difficulty initiating sleep, difficulty “most nights” was associated with worse cognitive function in 2014 . The overall test of significance suggested that the coefficients were not simultaneously equal to zero .b = −0.38, 95% CI: −0.62 to −0.13, p < .01). In Model 2, partially adjusted analyses, compared to “never” waking up too early and not being able to fall back asleep, difficulty “most nights” was associated with worse cognitive function in 2014 . In Model 3, fully adjusted analyses, compared to “never” waking up too early and not being able to fall back asleep, difficulty “most nights” was associated with worse cognitive function in 2014 . The overall test of significance suggested that the coefficients were not simultaneously equal to zero .b = 0.20, 95% CI: 0.00 to 0.39, p < .05). In Model 3, fully adjusted analyses, compared to those who reported “never” waking up feeling rested, those who reported “sometimes” waking up feeling rested were more likely to have a higher cognitive function in 2010 . The overall test of significance suggested that the coefficients were not simultaneously equal to zero .b = −0.71, 95% CI: −1.32 to −0.12, p < .05). In Model 2, partially adjusted analyses, compared to those not reporting concurrent difficulties, those who reported these difficulties demonstrated lower cognitive function in 2010 . The overall test of significance suggested that the coefficients were not simultaneously equal to zero .We examined specific sleep difficulties and their relationship to cognitive function during a 10-year period using nationally representative, longitudinal data collected among U.S. adults aged 51 or older between 2004 and 2014. Findings from our study demonstrate that, after adjusting for demographic and health condition covariates, two of the four sleep difficulties measured in this study were associated with cognitive function over time. First, early morning awakenings were associated with worse cognitive function over time. Specifically, for those individuals who reported experiencing early morning awakenings “most nights,” compared to never reporting such difficulties, there was a significant decline in scores on the validated instruments assessing cognitive function. Second, in adjusted analyses, those who reported feeling rested “some nights” had better cognitive function over time. With an aging population in the United States, understanding the contributing factors to cognitive impairment and the initiation and worsening of AD is imperative , followed by approximately one third who reported “sometimes” (29.7%) and a significantly smaller proportion reporting waking and feeling rested “most nights” (12.7%). In either case, our results showed that, even in fully adjusted models, this effect persists.Although we included a broad array of sleep difficulties in our analysis , only early morning awakenings and waking up feeling rested were significantly associated with cognitive function over time. The lack of significant findings for the other sleep difficulty items may be understood through the lens of previous literature. First, extensive literature documents that sleep is fundamentally more challenging among older adults compared to their younger counterparts . Thus, lSleep difficulty is more prevalent among older adults than younger adults  and in pThe longitudinal design of the HRS data is a significant strength of this study. They enabled our analysis of cognitive function and sleep difficulty over time. Cross-sectional studies, by comparison, are unable to address the core question of this study that examined the variability in cognitive function due to specific sleep difficulties over time. Unfortunately, while the HRS has been collecting data for nearly one decade prior to the waves selected for this study, questions on sleep were not included until 2004. Furthermore, beginning in the 2006 wave of data collection, the sleep measures were offered only in alternate waves, making for uneven follow-up intervals.The sleep difficulty measures employed in this study included simplistic measures of frequency . Also, previous research that has examined self-reported sleep difficulty and objectively recorded sleep (i.e., wrist-worn actigraphy  and polyAn additional limitation is that our regression models used time as a categorical predictor rather than a continuous one. Based on the trends of cognition over time shown in , we did Several consensus statements call for increased attention to nature and methods for attenuating sleep difficulties among older adults, including a National Institutes of Health consensus statement, which recently addressed the diagnosis, risks, consequences, and treatment of chronic insomnia among adults . The AmeOur results also identify a number of opportunities for future research. For instance, research to better identify the causal pathway between sleep difficulty and the onset and development of AD and related dementia is one opportunity for future research. Future research may also consider how to design programs to attenuate sleep difficulty among older adults. For instance, evidence-based programs such as Cognitive Behavioral Therapy for Insomnia  may be tSleep disturbance is more prevalent among older adults relative to their younger counterparts and is associated with cognitive decline and risk for AD. We used a large, national data set to examine sleep difficulties and cognitive function across a 10-year period among older adults in the United States. Our results show that early morning awakenings were associated with worse cognitive function, whereas reports of waking up feeling rested were associated with better cognitive function, over time. Future research may examine methods for attenuating sleep difficulties among older adults.