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Hoarding disorder is characterized by difficulty parting with possessions because of strong urges to save the items. Difficulty discarding often includes items others consider to be of little value and results in accumulation of a large number of possessions that clutter the home. Cognitive-behavioral therapy (CBT) with exposure and response prevention and selective serotonin reuptake inhibitor medications traditionally used to treat obsessive-compulsive disorder are generally not efficacious for people with hoarding problems. A specialized CBT approach for hoarding has shown progress in reaching treatment goals and has been modified to be delivered in group, peer-facilitated, and virtual models. Research on hoarding remains in the early phases of development. Animal, attachment, and genetic models are expanding. Special populations, such as children, older adults, and people who do not voluntarily seek treatment need special consideration for intervention. Community-based efforts aimed at reducing public health and safety consequences of severe hoarding are needed.

In this report, 20MnCr5 gear is characterized and analyzed under two carburizing heat treatment processes by optical microscope (OM) analysis, scanning electron microscope (SEM) analysis, X-ray diffraction (XRD) and Vickers hardness test. Results show that spherical and angular carbides can be obtained by a high concentration carburizing process for 20MnCr5 gear. Reheating and subsequent low-temperature tempering help refine the grains of the infiltration layer, reduce the residual austenite content in the permeable layer, and increase the thickness of the effective hardening layer.

Compared to aluminum alloy wheels, steel wheels have been widely used in the automobile industry due to the low cost and high plasticity and toughness. In this report, wind-cooling quenching of the wheels made of 20Mn2CrSi steel is studied. The CCT curves of 20Mn2CrSi are firstly drawn based on the thermal expansion experiment. Then the samples with different thicknesses (4mm, 6mm, 11mm) are tested under air cooling and wind cooling conditions to find out their optimum transfer time of quenching, and the cooling rates of the surfaces and internal of the sample under wind-cooling condition. The Vickers hardness of the samples after wind-cooling quenching are also measured by a Vickers hardness meter. The results show that the cooling rate is slow when the air volume during cooling is 900m3/h, making the hardness of the samples fail to meet the use standard. Therefore, in order to improve the cooling rate and reduce the thermal stress and structural stress during quenching, water atomization quenching method is proposed, which can provide a certain reference value for the quenching process of 20Mn2CrSi steel wheels.

Mutations in the CYP1B1 gene were detected in primary open-angle glaucoma (POAG) patients. However, the association between these mutations and the incidence of POAG remains to be elucidated. Here, we have conducted a meta-analysis to analyze this correlation, using relevant studies obtained from an extensive search of various electronic databases, including EMBase, Web of Science, and PubMed. The extracted studies were selected for the meta-analysis based on the inclusion and exclusion criteria. The quality of each included study was assessed by the Newcastle-Ottawa scale (NOS), and the I2 value was calculated to evaluate the heterogeneity between studies. The combined effect size was presented as the odds ratio (OR), and confidence intervals (CI) were used to assess the association between POAG and CYP1B1 mutations. Eight studies, each with a high NOS score, were included in the analysis. Compared to the mutant allele, the wild-type allele of the rs180040 polymorphism in POAG patients showed a 12% decrease in OR (OR = 0.88, 95%CI = 0.76- 1.00); also, the wild-type allele of rs1056827 showed a 23% decrease in OR of the incidence of POAG (OR = 0.77, 95%CI = 0.60-0.99). However, the latter result was controversial. Polymorphisms at rs1056836, rs10012, and rs1056837 were not correlated with the incidence of POAG (using any evaluation model). In conclusion, three of the tested SNPs in the CYP1B1 gene were correlated with POAG; however, the SNPs rs180040 and rs1056827 showed an association with risk of POAG. These results must be further validated with larger-scale evaluations.

Liver cancers are characterized by high morbidity and mortality owing to few effective drugs for its treatment. Wilfortrine has several pharmacological effects, including an inhibitory effect on liver cancer cell proliferation. However, whether wilfortrine can induce liver cancer cell apoptosis has not been elucidated. We investigated the role of wilfortrine on liver cancer cell HepG2 apoptosis and analyzed its possible mechanisms to provide a theoretical basis for clinical analysis of liver cancer pathogenesis. The liver cancer cell line HepG2 was treated with 40 mM wilfortrine for 48 h. Flow cytometry was applied to detect HepG2 cell apoptosis and cell cycle changes. Western blot was used to analyze Bcl-2 and Bax expression. The HepG2 cell apoptosis rate increased significantly after treatment with wilfortrine, especially the early apoptosis rate (P < 0.05). However, wilfortrine did not change the cell cycle of HepG2 cells. After wilfortrine treatment, Bcl- 2 expression decreased significantly (P < 0.05); on the contrary, Bax expression increased noticeably compared with the control group (P < 0.05). Wilfortrine can induce liver cancer cell HepG2 apoptosis, but with no effect on the cell cycle, mainly by promoting Bax expression and inhibiting anti-apoptotic protein Bcl-2 expression.

Specific AA affect rates of milk protein synthesis in the mammary glands of lactating cows. The objective of this study was to quantify the rate of αS1-casein synthesis in response to Ile, Leu, Met, and Thr supplementation, and to test the single-limiting AA theory for milk protein synthesis by exploring interactions among these AA. Effects of Ile, Leu, Met, and Thr were studied in vitro with a composite design containing a central point repeated 4 times, with 2 axial points per AA and a complete 2(4) factorial. Other AA were at the concentration in Dulbecco's modified Eagle medium/F12 medium (DMEM). The experiment was replicated with mammary tissue from 5 lactating cows. Mammary tissue slices (0.12 ± 0.02 g) were incubated for 4h at 37°C in 5 mL of treatment medium containing (2)H5-Phe. Caseins were precipitated from cell homogenate supernatants. Enrichment with (2)H5-Phe of the N[34]LLRFFVAPFPE αS1 peptide was determined by matrix-assisted laser desorption/ionization-tandem time-of-flight (MALDI-TOF-TOF), which was used to determine enrichment of Phe in the transfer (t)RNA pool and αS1-casein fractional synthesis rates (CFSR). Data were analyzed with a polynomial mixed model containing linear, quadratic, and 2-factor interactions for Ile, Leu, Met, and Thr, and cow and residual as random factors. Interactions were not significant at P

Improved representation of postabsorptive N metabolism in lactating dairy cows requires a better understanding of protein synthesis regulation in the mammary glands. This study aimed to determine the quantitative effects of Ile, Leu, Met, and Thr on the phosphorylation state of signaling proteins that regulate protein synthesis. The experiment used a composite design with a central point, 2 axial points per AA, and a complete 2(4) factorial. All of the other AA were provided at the concentrations in Dulbecco's modified Eagle's medium. The experiment was replicated with tissues from 5 lactating cows. Mammary tissue slices (0.12 ± 0.02 g) were incubated for 4h. Total and site-specific phosphorylated mammalian target of rapamycin (mTOR; Ser2448), eukaryotic elongation factor (eEF) 2 (Thr56), ribosomal protein S6 (Ser235/236), and eukaryotic initiation factor 2α (Ser51) were determined by western immunoblotting. Tissue concentrations of the 4 AA studied responded linearly to media supply. Addition of Ile, Leu, Met, or Thr had no effect on eukaryotic initiation factor 2α phosphorylation. Isoleucine and Thr positively affected mTOR phosphorylation. However, the 2 AA had an antagonistic relationship. Similarly, Ile linearly increased ribosomal protein S6 phosphorylation, and Thr inhibited the Ile effect. In addition, eEF2 phosphorylation was linearly decreased by Ile and Leu. Threonine curvilinearly decreased eEF2 phosphorylation, Ile and Leu negatively interacted on eEF2, and Thr tended to inhibit Leu effects on eEF2. This work demonstrated saturable responses and interactions between AA on activation of the mTOR pathway. Incorporation of these concepts into milk protein response models will help to improve milk and milk protein yield predictions and increase postabsorptive N efficiency and reduce N excretion by dairy cows.

To investigate responses of milk protein synthesis and mammary AA metabolism to a graded decrease of postruminal Lys supply, 4 lactating goats fitted with jugular vein, mammary vein, and carotid artery catheters and transonic blood flow detectors on the external pudic artery were used in a 4 × 4 Latin square experiment. Goats were fasted for 24 h and then received a 9-h intravenous infusion of an AA mixture plus glucose. Milk yield was recorded and samples were taken in h 2 to 8 of the infusion period; a mammary biopsy was performed in the last hour. Treatments were graded decrease of lysine content in the infusate to 100 (complete), 60, 30, or 0% as in casein. Lysine-removed infusions linearly decreased milk yield, tended to decrease lactose yield, and tended to increase milk fat to protein ratio. Milk protein content and yield were linearly decreased by graded Lys deficiency. Mammary Lys uptake was concomitantly decreased, but linear regression analysis found no significant relationship between mammary Lys uptake and milk protein yield. Treatments had no effects on phosphorylation levels of the downstream proteins measured in the mammalian target or rapamycin pathway except for a tended quadratic effect on that of eukaryotic initiation factor 2, which was increased and then decreased by graded Lys deficiency. Removal of Lys from the infusate linearly increased circulating glucagon and glucose. Removal of Lys from the infusate linearly decreased arterial and venous concentrations of Lys. Treatments also had a significant quadratic effect on venous Lys, suggesting mechanisms to stabilize circulating Lys at a certain range. The 2 infusions partially removing Lys resulted in a similar 20% decrease, whereas the 0% Lys infusion resulted in an abrupt 70% decrease in mammary Lys uptake compared with that of the full-AA mixture infusion. Consistent with the abrupt decrease, mammary Lys uptake-to-output ratio decreased from 2.2 to 0.92, suggesting catabolism of Lys in the mammary gland could be completely prevented when the animal faced severe Lys deficiency. Mammary blood flow was linearly increased, consistent with the linearly increased circulating nitric oxide by graded Lys deficiency, indicating mechanisms to ensure the priority of the mammary gland in acquiring AA for milk protein synthesis. Infusions with Lys removed increased mammary clearance rate of Lys numerically by 2 to 3 fold. In conclusion, the decreased milk protein yield by graded Lys deficiency was mainly a result of the varied physiological status, as indicated by the elevated circulating glucagon and glucose, rather than a result of the decreased mammary Lys uptake or depressed signals in the mTOR pathway. Mechanisms of Lys deficiency to promote glucagon secretion and mammary blood flow and glucagon to depress milk protein synthesis need to be clarified by future studies.

In this paper, numerical solution of the singular integral equation for the multiple curved branch-cracks is investigated. If some quadrature rule is used, one difficult point in the problem is to balance the number of unknowns and equations in the solution. This difficult point was overcome by taking the following steps: (a) to place a point dislocation at the intersecting point of branches, (b) to use the curve length method to covert the integral on the curve to an integral on the real axis, (c) to use the semi-open quadrature rule in the integration. After taking these steps, the number of the unknowns is equal to the number of the resulting algebraic equations. This is a particular advantage of the suggested method. In addition, accurate results for the stress intensity factors (SIFs) at crack tips have been found in a numerical example. Finally, several numerical examples are given to illustrate the efficiency of the method presented.

A hypersingular integral equation for the curved crack problems of an elastic half-plane is introduced. Formulation of the equation is based on the usage of a modified complex potential. The potential is generally expressed in the form of a Cauchy-type integral. The modified complex potential is composed of the principal part and the complementary part. The principal part of the complex potential is actually equivalent to the original complex potential for the curved crack in an infinite plate. The role of the complementary part is to eliminate the boundary traction along the boundary of the half-plane caused by the principal part. From the assumed boundary traction condition, a hypersingular integral equation is obtained for the curved crack problems of an elastic half-plane. The curve length coordinate method is used to obtain a final solution. Several numerical examples are presented that prove the efficiency of the suggested method.

In this paper, a successive integration technique is suggested for solving the Duffing oscillator equation with damping and excitation. In the technique, one performs integrations from some initial values over a wider range. Some of them may even be divergent; however, some of them will give a convergent result such that the periodic condition of motion is satisfied. In fact, the convergent result represents a stable periodic solution for the motion. If a convergent result or a stable periodic solution is obtained, the stability test is passed. A harmonic balance method in conjunction with the successive integration technique (abbreviated as HBMSIT) is also suggested. In the method, the initial values are obtained from the harmonic balance method. Therefore, the HBMSIT belongs to the successive integration techniques. Many examples with calculated results are presented.

In this paper, elastic analysis for a thick cylinder made of functionally graded materials (FGMs) is carried out. The property of FGMs is assumed to be exponential function form. The problem is reduced to solve an ordinary differential equation numerically. Stress distributions along the radial direction are studied. The obtained result shows that the property of FGMs has a significant influence to the stress distribution along the radial direction. Numerical results are given which are useful for engineer to design a cylinder made of FGMs. Similar analysis for spherical pressure vessel made of FGMs is presented.

T-stress dependence on the loading along the crack face is an important topic in fracture analysis. It is found that a statement for T-stress evaluation in Engng Fract Mech 69, 959–981 (2002) is incorrect. A rigorous derivation for T-stress for a semi-infinite crack is presented.

This paper investigates the dependence of the stress intensity factor (abbreviated as SIF) and T-stress from the imposed boundary conditions. A rectangular cracked plate is subject to a tension in the vertical direction. In the first boundary condition, the tension loading is applied at the tops of plate. In the second boundary condition, the relevant assumed displacement is applied on the edges of plate. In the study, the complex potentials are assumed in the series form with undetermined coefficients, and those coefficients can be evaluated by using the variational principle. The aim of study is to obtain the SIF and T-stress at the crack tips for two different boundary conditions. Finally, numerical results are presented. The obtained results may have their application in the fatigue test of a cracked component.

In this paper, a convenient technique for evaluating angular frequency in some nonlinear oscillations is proposed. It is well known that once the restoring force function is given beforehand, the period of motion can be determined by an integral. The angular frequency has a relation with the period of motion as well as the integral. One makes a little modification for the integrand in the integral and a change of variable. It is found that if the three divisions are chosen on the integration interval and the trapezoid quadrature rule is used, a higher accurate result for the angular frequency can be achieved. For the restoring force being an odd function, three numerical examples are presented. The eardrum-type oscillation is studied as well. Higher accurate results for the angular frequency are obtained in all those examples.

In this paper, elastic analysis for an edge‐cracked plate of functionally graded materials (FGMs) is carried out. The cracked plate is subject to a longitudinal tension. The property of FGMs is assumed to be exponential function form in both x‐ and y‐directions. The finite element method is suggested to solve the boundary value problem. An indirect method, the energy release method, is developed to evaluate the stress intensity factors (SIFs) at the crack tip. Under the applied loading, the amount of the release energy from the crack length “a” to “ a + Δa ” can be evaluated from relevant displacements at the top face of plate. The SIFs can be obtained from a relation between the energy release rate and SIF. The obtained result shows that the property of FGMs has a significant influence to the value of SIF at crack tip. Numerical results are given which are useful for engineer to assess the safety of the cracked plate of FGMs.

Three numerical solution techniques for nonlinear eardrum-type oscillations with an even function in the restoring force are studied. The first suggested technique is named the target function technique. In the technique, the second zeros of the target function is the circular frequency of motion. The second suggested technique is named the multiple-parameter technique, and the involved parameters are evaluated from the governing equation, the initial conditions, and properties of motion. The third suggested technique is named the direct integration technique. All techniques depend on the computer computation significantly and provide very accurate numerical results. Finally, numerical examples are given.

In plane elasticity, a general expression for a mutual work difference integral (MWDI) derived from two stress fields is introduced. Once two physical stress fields are known beforehand, the relevant MWDI can be evaluated exactly from the coefficients in the complex potentials. A biaxial tension model for evaluating defect energy is introduced. A particular MWDI from two fields, one is for the damaged medium under remote biaxial tension and other is for an infinite perfect plate under the same remote biaxial tension, can be defined as a suitable measure of stiffness for the damaged medium, which is called the defect energy ( E (a) ). The suggested model can deal with the cracks, holes, and elastic inclusions in a unique way. The model can also evaluate the defect energies for different damages exactly without dependence on the orientation of damages. Physically, the higher is the defect energy achieved, the more are the involved damages in the medium. The defect energy may be negative, which means a more ...

This paper investigates periodic group crack problems in an infinite plate. The periodic group crack is composed of infinite groups with numbering from j = −∞, …, −2, −1, 0, 1, 2, …, to j = ∞, and the groups are placed periodically. The same loading condition and the same geometry are assumed for cracks in all groups. A singular integral equation is used to solve the problems. The singular integral equation is formulated on cracks of the 0th group (or the central group) with the collection of influences from the infinite groups. The influences of many neighboring groups to the central group are evaluated exactly. Meantime, the influences of many remote groups to the central group can be summed up into one term approximately. The stress intensity factors at crack tips can be evaluated from the solution of the singular integral equation. It is found from some sample problems that the obtained results are very accurate. Finally, several numerical examples are presented and interaction among the group cracks is addressed.

This paper describes a natural-gradient field tracer test to characterise solute-transport properties in a sand and gravel aquifer in the Hebei Province, northern China. Some laboratory-scale column tests on aquifer material and a local-scale field borehole-dilution test have been conducted previously, but the field test reported herein represents the only large-scale tracer test in the aquifer, which is the sole water supply to the city of Shi Jiazhuang and which is threatened by urban pollution. The aim of the study was to quantify the transport behaviour of nonreactive pollutants in this aquifer. Little quantitative data are available concerning its solute-transport properties; thus, the results of the tracer test are significant and critical for understanding pollutant transport and fate. The in-situ tracer test was carried out in the aquifer using a slug injection of the geochemically conservative, radioactive iodine tracer I-131. Th, longitudinal (alpha (L)) and transverse (alpha (T)) hydrodynamic dispersivities for solute transport in the field are 1.72 and 0.0013 m, respectively. The ratio of longitudinal dispersivity alpha (L) and the flow length at the field scale is 1:10. The ratio between alpha (L) and alpha (T) from the in-situ test (similar to1,300:1) demonstrates a dominant longitudinal dispersion in this fluvial sand and gravel aquifer. The tracer test further indicates a relatively short transit time for the aquifer (linear velocities similar to 13 m/d) under natural-gradient conditions.

Groundwater is an important water resource and its management is vital for integrated water resources development in semiarid catchments. The River Shiyang catchment in the semiarid area of northwestern China was studied to determine a sustainable multi-objective management plan of water resources. A multi-objective optimization model was developed which incorporated water supplies, groundwater quality, ecology, environment and economics on spatial and temporal scales under various detailed constraints. A calibrated groundwater flow model was supplemented by grey simulation of groundwater quality, thus providing two lines of evidence to use in the multi-objective water management. The response matrix method was used to link the groundwater simulation models and the optimization model. Multi-phase linear programming was used to minimize and compromise the objectives for the multi-period, conjunctive water use optimization model. Based on current water demands, this water use optimization managemen...

We employ transient photocurrent decay measurements to determine the distribution of conduction band tail states in hydrogenated amorphous silicon. It is found that these experimental results suggest the possibility of deviations from a single-exponential functional form. This is consistent with other more recent experimental determinations of the distribution of conduction band tail states in hydrogenated amorphous silicon.

Four types (DD1, DD2, FD1 and FD2) of complex potential are introduced in this paper. All types are obtained by distributing the dislocation doublet or force doublet along a plane curve. If the axis of dislocation doublet is placed in the tangential (normal) direction of curve, the DD1 (DD2) type of complex potential is obtainable. Similarly, if the axis of force doublet is placed in the tangential (normal) direction of curve, the FD1 (FD2) type of complex potential is obtainable. The continuous and discontinuous behaviors of the displacement and the resultant force function for four types of complex potential are discussed. It is found that: (a) the complex potential of the type DD1 can model the curve crack problem, (b) the complex potential of the type FD1 can model the curve rigid line problem, (c) the complex potential of the type DD1 can be obtained by a superposition of the types FD1 and FD2, (d) the complex potential of the type FD1 can be obtained by a superposition of the types DD1 and DD2. In this paper, the dislocation doublet (the force doublet) is abbreviated as DD (FD), respectively.

Dairy cattle only capture about 25% of dietary N in milk protein, with the remainder being excreted. By coordinating AA supply with demand, mammary cells reduce recycling of AA to the splanchnic tissues, thereby increasing efficiency. Several essential AA regulate the rate of synthesis of casein by altering the phosphorylation state of intracellular signaling proteins that affect the activity of initiation (eIF) and elongation (eEF) factors (Appuhamy et al., 2012). Therefore, supplementing the specific AA that exert the most influence on pathways controlling eIF and eEF should allow a reduction in CP supply to the animal, which will increase post-absorptive efficiency and reduce N excretion. The objective of this study was to determine the quantitative responses in milk protein synthesis to individual essential AA in mammary tissue.

Research studies have uncovered the unfair WLAN bandwidth distribution between uplink and downlink TCP flows in the infrastructure WLAN using the distributed coordination function mode. Different mechanisms at or above the MAC layer have been proposed for handling this problem. This paper presents a simple but effective approach in order to achieve the fair distribution by using an adaptive queue management mechanism to manage the queue between the logic link and MAC layers at the access point. Extensive simulation results show that this approach outperforms other mechanisms discussed in the literature over a wide range of network scenarios in terms of fair distribution, small delay, and high WLAN bandwidth utilization

This paper presents a novel shape based genetic algorithm template matching (GATM) method for the automated detection of lung nodules. The GA process is employed as an optimisation method to effectively search for the location of nodule candidates within the lung area. To define the fitness function for GATM, 3D geometric shape feature is calculated at each voxel and then combined into global nodule intensity distribution. Lung nodule phantom images are used as reference images for template matching. The proposed method has been validated on 70 clinical thoracic CT scans that contain 178 nodules as a gold standard. 151 nodules were detected by the proposed method, a detection rate of 85%, with the number of false positives (FP) at approximately 14.0/scan. This high detection performance provides a good basis for a computer-aided detection (CAD) system for lung nodules.

Previous experimental results showed that seriously contaminated films were found on silver contact surfaces which caused electric contact failure. To investigate the tarnished film formed by natural air exposure, silver plated coupons were exposed indoors for 18 months. Tarnished ring patterns with various sizes were observed. The thickness of the film on ring area was as high as 1500 A and a polarity effect was also found. The composition mainly seems to be copper sulfide. The contact resistance on outside ring areas appeared lower than that on the ring areas but they were still beyond the normal value. The inside ring areas looked to be least corroded. Sliding contact behavior showed high resistance and only part of the film was peeled away. The existence of copper on the surface is believed to result from the corroded copper products which have crept through the tiny holes and micro-cracks in the silver plating and nickel underplates. A circular ring stain pattern was formed probably due to water droplets. The tarnished film was mainly composed of oxides, sulfides, and chlorides of silver, copper and nickel. Even it was formed at moderate indoor conditions, experimental results showed that serious failure can occur in static contacts, sliding and fretting contacts as well.

This paper presents a hybrid scheme for lossless compression of the X-ray non-destructive testing (NDT) images of aircraft components. The method combines predictive coding and integer wavelet transform (IWT). Furthermore, with the aid of component CAD models to divide the X-ray images of aircraft components into different regions based on the material structures, the design of the predictors and the choice of the IWT are optimised according to the specific image features contained in each region having the same material structure. Using a real X-ray image of a practical aircraft component, the proposed hybrid scheme is presented and shown to offer a significantly higher compression ratio than other lossless compression schemes.

Citrus is a major fruit crop in the acid red soils (Ultisol) of hilly areas in Southeast China. These soils are normally deficient in P, K, and other elements. Integrated nutrient management is important for sustainable production of citrus in these areas. In this study, a systematic approach was used to identify the limiting factors for plant growth, using sorghum as a test species. Long-term field experiments were conducted with seven different P and K supply levels to determine optimal application rates for citrus (cv. Ponkan), following alleviation of other limiting factors. The primary nutritional limitations to plant growth in red acid soils included: severe deficiencies in N, P, and K, and moderate deficiencies in Ca, Zn, and B. With increasing application of P and K to field soil, N concentrations in citrus leaves decreased up to 60% due to dilution from increased growth, whereas P and K concentrations increased 2-3 fold. After 2 years of fertilizer application, the N: P: K ratio in leaves reached 1:0.5:1 for the optimal P and K treatment. The available P and K in the soils, measured after harvest each year, increased with increasing P and K application rates. However, within each treatment, increase in P and K with additional years of fertilization was modest. Citrus fruit yields generally increased with increasing P and K and reached a maximum at P and K rates of 125 kg P2O5/ha and 500 kg K2O/ha. In 3 years of successive field experiments, the highest net income was obtained by a balanced NPK fertilization practice using N: P2O5: K2O input of 450: 125: 500 kg/ha per year.

This paper deals with three boundary value problems of the Laplace equation: 1. (1) potential distribution of a doubly-connected region with inner circular or elliptic boundary—a problem in electrostatics. 2. (2) Saint-Venant torsion of a bar with same region—a problem in elasticity. 3. (3) conformal mapping of the same region into a ring region—a problem in complex variable function. It is proved here that the first is a Dirichlet problem while the second and third are modified Dirichlet problems of Laplace's equation, and a modified Dirichlet problem can be easily reduced to two Dirichlet problems. In order to solve the Dirichlet problems, an eigenexpansion form is proposed. The eigenexpansion form always satisfies the Laplace equation and a particular condition along the inner boundary. The undetermined coefficients in the eigenexpansion form are determined by the use of the variational principle. Several numerical examples and calculated results are given.