The metallography of aluminum     

Robert J. Anderson 《Journal of The Franklin Institute》1919,187(1):1-47

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Development of vertical SU-8 microneedles for transdermal drug delivery by double drawing lithography technology     

Zhuolin Xiang  Hao Wang  Aakanksha Pant  Giorgia Pastorin  Chengkuo Lee 《Biomicrofluidics》2013,7(6)

Polymer-based microneedles have drawn much attention in transdermal drug delivery resulting from their flexibility and biocompatibility. Traditional fabrication approaches are usually time-consuming and expensive. In this study, we developed a new double drawing lithography technology to make biocompatible SU-8 microneedles for transdermal drug delivery applications. These microneedles are strong enough to stand force from both vertical direction and planar direction during penetration. They can be used to penetrate into the skin easily and deliver drugs to the tissues under it. By controlling the delivery speed lower than 2 μl/min per single microneedle, the delivery rate can be as high as 71%.Microelectromechanical systems (MEMS) technology has enabled wide range of biomedical devices applications, such as micropatterning of substrates and cells,¹ microfluidics,² molecular biology on chips,³ cells on chips,⁴ tissue microengineering,⁵ and implantable microdevices.⁶ Transdermal drug delivery using MEMS based devices can delivery insoluble, unstable, or unavailable therapeutic compounds to reduce the amount of those compounds used and to localize the delivery of potent compounds.⁷ Microneedles for transdermal drug delivery are increasingly becoming popular due to their minimally invasive procedure,⁸ promising chance for self-administration,⁹ and low injury risks.¹⁰ Moreover, since pharmaceutical and therapeutic agents can be easily transported into the body through the skin by microneedles,^{11, 12} the microneedles are promising to replace traditional hypodermic needles in the future. Previously, various microneedles devices for transdermal drug delivery applications have been reported. They have been successfully fabricated by different materials, including silicon,¹³ stainless steel,¹⁴ titanium,¹⁵ tantalum,¹⁶ and nickel.¹⁷ Although microneedles with these kinds of materials can be easily fabricated into sharp shape and offer the required mechanical strength for penetration purpose, such microneedles are prone to be damaged¹⁸ and may not be biocompatible.¹⁹ As a result, polymer based microneedles, such as SU-8,^{20, 21} polymethyl meth-acrylate (PMMA),^{22, 23} polycarbonates (PCs),^{24, 25} maltose,^{26, 27} and polylactic acid (PLA),^{28, 29} have caught more and more attentions in the past few years. However, in order to obtain ultra-sharp tips for penetrating the barrier layer of stratum corneum,³⁰ conventional fabrication technologies, for instances, PDMS (Polydimethylsiloxane) molding technology,^{31, 32} stainless steel molding technology,³³ reactive ion etching technology,³⁴ inclined UV (Ultraviolet) exposure technology,³⁵ and backside exposure with integrated lens technology³⁶ are time-consuming and expensive. In this paper, we report an innovative double drawing lithography technology for scalable, reproducible, and inexpensive microneedle devices. Drawing lithography technology³⁷ was first developed by Lee et al. They leveraged the polymers'' different viscosities under different temperatures to pattern 3D structures. However, it required that the drawing frames need to be regular cylinders, which is not proper for our devices. To solve the problem, the new double drawing lithography is developed to create sharp SU-8 tips on the top of four SU-8 pillars for penetration purpose. Drugs can flow through the sidewall gaps between the pillars and enter into the tissues under the skin surface. The experiment results indicate that the new device can have larger than 1N planar buckling force and be easily penetrated into skin for drugs delivery purpose. By delivering glucose solution inside the hydrogel, the delivering rate of the microneedles can be as high as 71% when the single microneedle delivery speed is lower than 2 μl/min.An array of 3 × 3 SU-8 supporting structures was patterned on a 140 μm thick, 6 mm × 6 mm SU-8 membrane (Fig. ​(Fig.1a).1a). Each SU-8 supporting structure included four SU-8 pillars and was 350 μm high. The four pillars were patterned into a tubelike shape on the membrane (Fig. ​(Fig.1b).1b). The inner diameter of the tube was 150 μm, while the outer diameter was 300 μm. SU-8 needles of 700 μm height were created on the top of SU-8 supporting structures to ensure the ability of transdermal penetration. Two PDMS layers were bonded with SU-8 membrane to form a sealed chamber for storing drugs from the connection tube. Once the microneedles entered into the tissue, drugs could be delivered into the body through the sidewall gaps between the pillars (Fig. ​(Fig.1c1c).Open in a separate windowFigure 1Schematic illustration of the SU-8 microneedles. (a) Overview of the whole device; (b) SU-8 supporting structures made of 4 SU-8 pillars; and (c) enlarged view of a single SU-8 microneedle.The fabrication process of SU-8 microneedles is shown in Fig. ​Fig.2.2. SU-8 microneedles fabrication started from a layer of Polyethylene Terephthalate (PET, 3M, USA) film pasted on the Si substrate by sticking the edge area with kapton tape (Fig. ​(Fig.2a).2a). The PET film, a kind of transparent film with poor adhesion to SU-8, was used as a sacrificial layer to dry release the final device from Si substrate. A 140 μm thick SU-8 layer was deposited on the top of this PET film. To ensure a uniform surface of this thick SU-8 layer, the SU-8 deposition was conducted in two steps coating. After exposed under 450 mJ/cm² UV, the membrane pattern could be defined (Fig. ​(Fig.2b).2b). In order to ensure an even surface for following spinning process, another 350 μm SU-8 layer was directly deposited on this layer in two steps without development. With careful alignment, an exposure of 650 mJ/cm² UV energy was performed on this 350 μm SU-8 layer to define the SU-8 supporting structures (Fig. ​(Fig.2c).2c). The SU-8 structure could be easily released from the PET substrate by removing the kapton tape and slightly bending the PET film. Two PDMS layers were bonded with this SU-8 structure by a method reported by Zhang et al.³⁸ (Fig. ​(Fig.2d2d).Open in a separate windowFigure 2Fabrication process for SU-8 microtubes. (a) Attaching a PET film on the Si substrate; (b) exposing the first layer of SU-8 membrane without development; (c) depositing and patterning two continuous SU-8 layers as sidewall pillars; (d) releasing the SU-8 structure from the substrate and bonding it with PDMS; (e) drawing hollowed microneedles on the top of supporting structures; (f) baking and melting the hollowed microneedles to allow the SU-8 flow in the gaps between pillars; and (g) drawing second time on the top of the melted SU-8 flat surface to get microneedles.In our previous work,³⁹ we used one time stepwise controlled drawing lithography technology for the sharp tips integration. However, since the frame used to conduct drawing process in present study is a four-pillars structure rather than a microtube, the conventional drawing process can only make a hollowed tip but not a solid tip structure (Fig. ​(Fig.3).3). This kind of tip was fragile and could not penetrate skin in the practical testing process. To solve the problem, we developed an innovative double drawing lithography process. After bonding released SU-8 structure with PDMS layers (Fig. ​(Fig.2d),2d), we used it to conduct first time stepwise controlled drawing lithography³⁷ and got hollowed tips (Fig. ​(Fig.2e).2e). Briefly, the SU-8 was spun on the Si substrate and kept at 95 °C until the water inside completely vaporized. Device of SU-8 supporting structures was fixed on a precision stage. Then, the SU-8 supporting structures were immersed into the SU-8 by adjusting the precision state. The SU-8 were coated on the pillars'' surface. Then, the SU-8 supporting structures were drawn away from the interface of the liquid maltose and air. After that, the temperature and drawing speed were increased. Since the SU-8 was less viscous at higher temperature, the connection between the SU-8 supporting structures and surface of the liquid SU-8 became individual SU-8 bridge, shrank, and then broke. The end of the shrunk SU-8 bridge forms a sharp tip on the top of each SU-8 supporting structure when the connection was separated. After the hollowed tips were formed in the first step drawing process, the whole device was baked on the hotplate to melt the hollowed SU-8 tips. Melted SU-8 reflowed into the gaps between four pillars and the tips became domes (Fig. ​(Fig.2f).2f). Then, a second drawing process was conducted on the top of melted SU-8 to form sharp and solid tips (Fig. ​(Fig.2g).2g). The final fabricated device is shown in Fig. ​Fig.44.Open in a separate windowFigure 3A hollowed SU-8 microneedle fabricated by single drawing lithography technology (scale bar is 100 μm).Open in a separate windowFigure 4Optical images for the finished SU-8 microneedles.During the double drawing process, as long as the heated time and temperature were controlled, the SU-8 flow-in speed of SU-8 inside the gaps could be precisely determined. The relationship between baking temperature and flow-in speed was studied. As shown in Fig. ​Fig.5,5, the flow-in speed is positive related to the baking temperature. The explanation for this phenomena is that the SU-8''s viscosity is different under different baking temperatures.⁴⁰ Generally, baked SU-8 has 3 status when temperature increases, solid, glass, and liquid. The corresponding viscosity will decrease and the SU-8 can also have higher fluidity. When the baking temperature is larger than 120 °C, the flow-in speed will increase sharply. But, if the baking temperature is higher, the SU-8 will reflow in the gaps too fast, which makes the flow-in depth hard to be controlled. There is a high chance that the whole gaps will be blocked, and no drugs can flow through these gaps any more. Considering that the total SU-8 supporting structure is only 350 μm high, we choose 125 °C as baking temperature for proper SU-8 flow-in speed and easier SU-8 flow-in depth control.Open in a separate windowFigure 5The relationship between flow-in speed and baking temperature.To ensure the adequate stiffness of the SU-8 microneedles in vertical direction, Instron Microtester 5848 (Instron, USA) was deployed to press the microneedles with the similar method reported by Khoo et al.⁴¹ As shown in Fig. ​Fig.6a,6a, the vertical buckling force was as much as 8.1N, which was much larger than the reported minimal required penetration force.⁴² However, in the previous practical testing experiments, even though the microneedles were strong enough in vertical direction, the planar shear force induced by skin deformation might also break the interface between SU-8 pillars and top tips. In our new device with four pillars supporting structure, the SU-8 could flow inside the sidewall gaps between the pillars to form anchors. These anchors could enhance microneedles'' mechanical strength and overcome the planar shear force problems. Moreover, the anchors strength could be improved by controlling the SU-8 flow-in depth. Fig. ​Fig.77 shows that the flow-in depth increases when the baking time increases as the baking time increases at 125 °C. Fig. ​Fig.6b6b shows that the corresponding planar buckling force can be improved to be larger than 1 N by increasing flow-in depth. Some sidewall gaps at bottom are kept on purpose for drugs delivery; hence, the flow-in depth is chosen as 200 μm.Open in a separate windowFigure 6(a) Measurement of the vertical buckling force. (b) The planar buckling force varies under different flow-in depth (I, II, III, and IV corresponding to the certain images in Fig. ​Fig.77).Open in a separate windowFigure 7Different flow-in depth inside the gaps between SU-8 pillars. (a) 0 μm; (b) 100 μm; (c) 200 μm; and (d) 350 μm (scale bar is 100 μm).The penetration capability of the 3 × 3 SU-8 microneedles array is characterized by conducting the insertion experiment on the porcine cadaver skin. 10 microneedles devices were tested and all of them were strong enough to be inserted into the tissue without any breakage. Histology images of the skin at the site of one microneedle penetration were derived to prove that the sharp conical tip was not broken during the insertion process (Fig. ​(Fig.8).8). It also shows penetrated evidence because the hole shape is the same as the sharp conical tip.Open in a separate windowFigure 8Histology image of individual microneedle penetration (scale bar is 100 μm).In order to verify that the drug solution can be delivered into tissue from the sidewall gaps of the microneedles, FITC (Fluorescein isothiocyanate) (Sigma Aldrich, Singapore) solution was delivered through the SU-8 microneedles after they were penetrated into the mouse cadaver skin. The representative results were then investigated via a confocal microscope (Fig. ​(Fig.9).9). The permeation pattern of the solution along the microchannel created by microneedles confirmed the solution delivery results. The black area was a control area without any diffused florescent solution. In contrast, the illuminated area in Fig. ​Fig.99 indicates the area where the solution has diffused to it. These images were taken consecutively from the skin surface down to 180 μm with 30 μm intervals. The diffusion area had a similar dimension with the inserted microneedles. It has proved that the device can be used to deliver drugs into the body.Open in a separate windowFigure 9Images of confocal microscopy to show the florescent solution is successfully delivered into the tissue underneath the skin surface. (a) 30 μm; (b) 60 μm; (c) 90 μm; (d) 120 μm; (e) 150 μm; and (f) 180 μm (scale bar is 100 μm).Due to the uneven surface of deformed skin, there is always tiny gap happened between tips of some microneedles and local surface skin. The microneedles could not be entirely inserted into the tissue. Drugs might leak to the skin surface through the sidewall gaps under certain driven pressure. Hydrogel absorption experiment was conducted to quantify the delivery rate (i.e., the ratio of solution delivered into tissues in the total delivered volume) and to optimize the delivery speed. Using hydrogel as the tissue model for quantitative analysis of microneedle releasing process was reported by Tsioris et al.⁴³ The details are shown here. Gelatin hydrogel was prepared by boiling 70 ml DI (Deionized) water and mixing it with 7 g of Knox^TM original unflavored gelatin powder. The solution was poured into petri dish to 1 cm high. Then, the petri dish was put into a fridge for half an hour. Gelatin solution became collagen slabs. The collagen slabs were cut into 6 mm × 6 mm sections. A piece of fully stretched parafilm (Parafilm M, USA) was tightly mounted on the surface of the collagen slabs. This parafilm was used here to block the leaked solution further diffusing into the collagen slab in the delivery process. Then, the microneedles penetrated the parafilm and went into the collagen slab. Controlled by a syringe pump, 0.1 ml–0.5 mg/ml glucose solution was delivered into the collagen slab under different speeds. Methylene Blue (Sigma Aldrich, Singapore) was mixed into the solution for better inspection purpose (Fig. 10a). Then, the collagen slabs was digested in 1 mg/ml collagenase (Sigma Aldrich, Singapore) at room temperature (Fig. 10b). It took around 1 h that all the collagen slabs could be fully digested (Fig. 10d). The solution was collected to measure the glucose concentration with glucose detection kit (Abcam, Singapore). Briefly, both diluted glucose standard solution and the collected glucose solution were added into a series of wells in a well plate. Glucose assay buffer, glucose enzyme, and glucose substrate were mixed with these samples in the wells. After incubation for 30 min, their absorbance were examined by using a microplate reader at a wavelength of 450 nm. By comparing the readings with the measured concentration standard curve (Fig. 11a), the glucose concentration in the hydrogel, the glucose absorption rate in the hydrogel, and the solution delivery rate by microneedles could be measured and calculated. As shown in Fig. 11b, when the delivering speed of a single microneedle increased from 0.1 μl/min to 2 μl/min, the glucose absorption rate also increased. Most of the glucose solution from microneedles could go into the hydrogel. The delivered rate could be as high as 71%. The rest solution leaked from sidewall gaps and blocked by parafilm. However, when the delivered speed for a single microneedle was larger than 2 μl/min, the hydrogel absorption rate was saturated. More and more solution could not go into the hydrogel but leak from the sidewall gaps. Then, the delivered rate decreased. Therefore, 2 μl/min was chosen as the optimized delivery speed for the microneedle.Open in a separate windowFigure 10Glucose solution could be delivered into the hydrogel, and the collagen stabs were dissolved by collagenase.Open in a separate windowFigure 11(a) Standard curve for glucose detection; (b) glucose absorption rate and solution delivery rate in a single needle corresponding to different delivery speed.In conclusion, a drug delivery device of integrated vertical SU-8 microneedles array is fabricated based on a new double drawing lithography technology in this study. Compared with the previous biocompatible polymer-based microneedles fabrication technology, the proposed fabrication process is scalable, reproducible, and inexpensive. The fabricated microneedles are rather strong along both vertical and planar directions. It is proved that the microneedles were penetrated into the pig skin easily. The feasibility of drug delivery using SU-8 microneedles is confirmed by FITC fluorescent delivery experiment. In the hydrogel absorption experiment, by controlling the delivery speed under 2 μl/min per microneedle, the delivery rate provided the microneedle is as high as 71%. In the next step, the microneedles will be further integrated with microfluidics on a flexible substrate, forming a skin-patch like drug delivery device, which may potentially demonstrate a self-administration function. When patients need an injection treatment at home, they can easily use such a device just like using an adhesive bandage strip.  相似文献   

Some transmission line transients. Graphical evaluation by Fourier's series     

J.A. Strelzoff 《Journal of The Franklin Institute》1934,217(3):353-372

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Croatian Society of Medical Biochemistry and Laboratory Medicine     

《Biochemia medica : ?asopis Hrvatskoga dru?tva medicinskih biokemi?ara / HDMB》2013,23(2):A56-A61

Introduction:

Intensive exercising may significantly damage muscles which is reflected in pain, fatigue and the increase of muscle proteins concentrations in blood such are creatinin kinase (CK), lactic dehydrogenase (LD), myoglobin (MB) and other biochemical parameters including urea serum concentration (SU). Biochemical markers vary with age, sex, race, muscle mass, physical activity and climate conditions. They also assist us in determining the limit between the capacity for adaptation to given training process which results in supercomepensation and in condition of overtraining (OT), in the case of load that exceeds the physiologic potential of regeneration. Concerning the problem of diagnosis and explanation of the symptoms of overtraining, markers that can apply reliably and with sufficient sensitivity and simplicity of interpretation in the praxis are sought. It is critical to take into account difference among individuals and groups that could hamper the interpretation.

The most frequently used markers:

The most frequently used biomarkers that provide us with the information on physical activity and on the amount of load through exercise are CK, SU and LD. Level of serum retaining kinas has been measured and interpreted for years as part of different scientific and professional investigations and presents one of basic parameters for determining the level of muscle damage. It reaches maximal concentration of the fourth day of exercising which depends on the type of exercise and the nature of stress triggered by exercise but also on individual characteristics.The level of serum urea presents marker of nitric compounds metabolism and is the principle chemical substance in the urine of mammals. It is thus possible to draw a parallel between the increases of serum urea concentration on increased degradations of proteins. Significant fall of serum amino acid levels occurs after sixty to seventy minutes of exercising with the increase of urea and free tyrosine and these changes have high correlation with the duration and intensity of.LD changes are important index of well-trained sportsmen and their capability to withstand the pace and force during strain in the training process. The level of LD is a good index of exercise intensity and marker of metabolic exchange in tissues whose concentration in serum is dependent of cell damage.

Conclusion:

There is not a single, unique parameter that would provide enough valuable information for the estimation of the quality of exercising, amount of load and identification of overtraining. Delayed measurement of biomarkers is far from ideal, but it is obvious that the amount of stress/ load in training is the most important factor for the development of state of overtraining. Daily body weight control, diet, biochemical indices values and the input of water should be known and standardized before measurements. For the most of parameters determination of basal levels are needed in specific populations for more accurate interpretation and evaluation of results. The sampling process itself should be under the most strict conditions of standardization by repeating measurement at least every third day. Dependence of mentioned parameters (SU, CK, LD) on exercise intensity varies among individuals and without these additional measurements and subpopulation evaluations it is difficult to come to conclusions with certainty as well as to come to conclusions on causative correlations of training load and dynamic in biochemical parameters.

• Biochem Med (Zagreb). 2013 Jun; 23(2): A56–A61.
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• Common sports injuries

Biochem Med (Zagreb) 2013 Jun; 23(2): A57–A58. Published online 2013 Jun 15. doi: 10.11613/BM.2013.027

Common sports injuries

Miljenko FranićAuthor information Copyright and License information DisclaimerDubrava University Hospital, ZagrebCorresponding author: rh.dbk@cinarfm©Copyright by Croatian Society of Medical Biochemistry and Laboratory MedicineThis is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.Sports injuries are injuries that occur in athletic activities and can be broadly classified as either traumatic or overuse injuries. Traumatic injuries because of the dynamic and high collision are nature of some sports. Overuse injuries cause wear and tear on the body, particularly on joints subjected to repeated activity.At every age, competitive and recreational athletes sustain a wide variety of soft tissue, bone, ligament, tendon and nerve injuries, caused by direct trauma or repetitive stress. Different sports are associated with different patterns and types of injuries, whereas age, gender and type of activity influence the prevalence of injuries. Sports trauma commonly affects joints of the extremities or the spine.The hip, knee and ankle are at risk of developing osteoarthritis (OA) after injury or in the presence of malalignment, especially in association with high impact sport. Spine pathologies are associated more commonly with certain sports. Upper extremity syndromes caused by a single stress or by repetitive micro-trauma occur in a variety of sports.Random control trials expose some subjects, but not others, to an intervention. This is more clinical in nature and not typically appropriate for the study of injury patterns. Cohort studies monitor both injured and non-injured athletes, thereby providing results on the effects of participation. Case-control studies monitor only those athletes who suffered an injury. The Ideal study would be Cohort design conducted over several teams, with longitudinal prospective data collection and one recorder where possible, as well as uniformity of injury definition across sports so comparisons between studies can be made accurately.Physical injury is an inherent risk in sports participation and, to a certain extent, must be considered an inevitable cost of athletic training and competition. Injury may lead to incomplete recovery and residual symptoms, drop out from sports, and can cause joint degeneration in the long term.Advances in arthroscopic techniques allow operative management of most intraarticular post-traumatic pathologies in the lower and upper limb joints, but long-term outcomes are not available yet. It is important to balance the negative effects of sports injuries with the many benefits that a serious commitment to sport brings.

• Biochem Med (Zagreb). 2013 Jun; 23(2): A56–A61.
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• Determination of sample size and number of study groups in sport studies

Biochem Med (Zagreb) 2013 Jun; 23(2): A58–A59. Published online 2013 Jun 15. doi: 10.11613/BM.2013.027

Determination of sample size and number of study groups in sport studies

Mladen PetrovečkiAuthor information Copyright and License information DisclaimerDepartment of Laboratory Diagnosis, Dubrava University Hospital, Zagreb, Croatia, and Department of Medical Informatics, Rijeka University School of Medicine, RijekaCorresponding author: rh.irdem@pnedalm;©Copyright by Croatian Society of Medical Biochemistry and Laboratory MedicineThis is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.  相似文献   

You cannot tell a book by looking at the cover: Cryptic complexity in bacterial evolution     

Qiucen Zhang  Julia Bos  Grigory Tarnopolskiy  James C. Sturm  Hyunsung Kim  Nader Pourmand  Robert H. Austin 《Biomicrofluidics》2014,8(5)

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Hierarchically porous materials and green chemistry—an interview with Ming-Yuan He     

Li-Hua Chen 《国家科学评论(英文版)》2020,7(11):1759

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Effect of Copper on <Emphasis Type="SmallCaps">l</Emphasis>-Cysteine/<Emphasis Type="SmallCaps">l</Emphasis>-Cystine Influx in Normal Human Erythrocytes and Erythrocytes of Wilson’s Disease     

Nabarun Mandal  Debojyoti Bhattacharjee  Jayanta Kumar Rout  Anindya Dasgupta  Gorachand Bhattacharya  Chandan Sarkar  Prasanta Kumar Gangopadhyaya 《Indian journal of clinical biochemistry : IJCB》2016,31(4):468-472

Wilson’s disease is a disease of abnormal copper metabolism in which free serum copper level is raised. The objective of the study was to determine, whether in Wilson disease, l-cysteine/l-cystine influx into RBC was decreased or not and the specific amino acid transporter affected by copper in normal human RBC. For l-cysteine/l-cystine influx, ten untreated cases, ten treated cases and ten age and sex matched healthy controls were recruited. To study the effect of copper on l-cysteine/l-cystine influx in RBC, 15 healthy subjects were selected. RBC GSH and l-cysteine/l-cystine influx were estimated by Beautler’s and Yildiz’s method respectively. In untreated cases, l-cysteine/l-cystine influx and erythrocyte GSH level were decreased showing that elevated level of free copper in serum or media decreased l-cysteine/l-cystine influx in human RBC. Copper treatment inhibited L amino acid transporter in normal RBC specifically.  相似文献   

A graphical method for the construction of electrostatic fields     

Sylvan J. Crooker 《Journal of The Franklin Institute》1918,185(1):131-135

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An adaptive online learning algorithm for distributed convex optimization with coupled constraints over unbalanced directed graphs     

Chuanye Gu  Jueyou Li  Zhiyou Wu 《Journal of The Franklin Institute》2019,356(13):7548-7570

This paper investigates a distributed optimization problem over multi-agent networks subject to both local and coupled constraints in a non-stationary environment, where a set of agents aim to cooperatively minimize the sum of locally time-varying cost functions when the communication graphs are time-changing connected and unbalanced. Based on dual decomposition, we propose a distributed online dual push-sum learning algorithm by incorporating the push-sum protocol into dual gradient method. We then show that the regret bound has a sublinear growth of O(T^p) and the constraint violation is also sublinear with order of $O\left(T^{1?p/2}\right),$ where T is the time horizon and 0 < p ≤ 1/2. Finally, simulation experiments on a plug-in electric vehicle charging problem are utilized to verify the performance of the proposed algorithm. The proposed algorithm is adaptive without knowing the total number of iterations T in advance. The convergence results are established on more general unbalanced graphs without the boundedness assumption on dual variables. In addition, more privacy concerns are guaranteed since only dual variables related with coupled constraints are exchanged among agents.  相似文献   

The vibration spectrum of the ammonia molecule     

Joseph W. Ellis 《Journal of The Franklin Institute》1929,208(4):507-519

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