Search

Lions (1959, Bull. Soc. Math. France, \textbf{87}, 245--273) introduced the Navier-Stokes equations with a viscous diffusion in the form of a fractional Laplacian; subsequently, he (1969, Dunod, Gauthiers-Villars, Paris) claimed the uniqueness of its solution when its exponent is not less than five quarters in case the spatial dimension is three. Following the work of Hofmanov$\acute{\mathrm{a}}$, Zhu and Zhu (2019, arXiv:1912.11841 [math.PR]), we prove the non-uniqueness in law for the three-dimensional stochastic Navier-Stokes equations with the viscous diffusion in the form of a fractional Laplacian with its exponent less than five quarters., This is the version accepted by Stochastic Processes and their Applications

Femtosecond laser processing is a non-thermal process with less heat effect on a target material and is applied to dielectric and difficult-to-machine materials, such as titanium alloy and diamond. Femtosecond laser processing has been used for 3-dimensional shape generation in recent years, and increasing accuracy and productivity are required for such generation by femtosecond lasers. However, a 2.5-dimensional computer-aided manufacturing (CAM) system, which is mainly used in femtosecond laser processing, does not satisfy the requirement for complicated 3-dimensional shape generation. Therefore, there is a need for the development of a 3-dimensional CAM system for high-precision and productive 3-dimensional processing. Furthermore, for the 3-dimensional CAM system, quantitative identification of a 3-dimensional removal shape through ablation is necessary. This study proposes the development of a simulation method to accurately predict the 3-dimensional removal shape through femtosecond laser processing, including the piecewise approximation of the ablation rate and absorptance correction. At the same time, a similar 3-dimensional shape as that found through the experimental results was obtained during the simulation by considering the mechanical errors of the laser processing machine in the simulation model. Furthermore, an on-machine measurement system was developed to efficiently identify the ablation parameters.

Many ladybird species are known to have an elytral colour polymorphism, which indicates geographical variation. The ladybird beetle Cheilomenes sexmaculata (Fabricius) exhibits elytral colour polymorphism and has expanded its distribution from 33°N to 36°N in Japan over 100 years since 1900. The mitochondrial COI gene haplotypes were integrated into two haplotype groups, with one group existing at higher frequencies in lower latitudes, the other group appearing at higher frequencies in higher latitudes. In addition, the dark morph types of this species increase with latitude, whereas the light types appear at higher relative frequencies in lower latitudes. In the present study, we first determined the morph types of individuals and examined the mitochondrial DNA COI gene. Second, we investigated the relationship between the genetic population structure based on the mitochondrial DNA COI gene and the morph types’ geographical variation. Results indicated that the mtCOI genetic structure was associated with the morph types by latitude; specifically, the haplotype group existing at higher frequencies in lower latitudes tended to be light morph types. In contrast, the haplotype dominant in higher latitudes more frequently exhibited dark morph types, indicating that dark morph types in the higher latitude genetic group may have led the distributional expansion toward higher latitudes since 1900 rather than the lower latitude light morph types.

Directed Energy Deposition (DED) is a metal additive manufacturing process where feedstock (metal powder in this study) is delivered to a meltpool on the surface of a metallic part simultaneously with an energy source (a laser in this study) that causes the local melting. DED process productivity is measured using the Material Addition Rate (MAR) which is a measure of the amount of material that is added to the part per interval of time, while process efficiency is measured with the powder catchment efficiency which is the ratio of powder deposited into the part and powder fed through the nozzle during the deposition cycle. If clad geometry can be maintained while deposition head feed rate is increased, then MAR is increased. As DED functionality has been integrated into industrial CNC machining centers published deposition head feed rates in literature have not kept up with machine feed rate capabilities. This research demonstrates a method of using energy and powder fluence into the meltpool to scale up deposition head feed rate in a DED equipped CNC machining center to improve MAR while maintaining part quality and geometric accuracy.

We propose and prove several regularity criteria for the 2D and 3D Kuramoto–Sivashinsky equation, in both its scalar and vector forms. In particular, we examine integrability criteria for the regularity of solutions in terms of the scalar solution $$\phi $$ ϕ , the vector solution $$u\triangleq \nabla \phi $$ u ≜ ∇ ϕ , as well as the divergence $$\text {div}(u)=\Delta \phi $$ div ( u ) = Δ ϕ , and each component of u and $$\nabla u$$ ∇ u . We also investigate these criteria computationally in the 2D case, and we include snapshots of solutions for several quantities of interest that arise in energy estimates.

The efficiency of powder-based directed energy deposition (DED) nozzle is dependent on its ability to direct the pneumatically conveyed powder into the meltpool. Computational fluid dynamics (CFD) with discrete phase modeling (DPM) has been used to investigate the optimization of DED nozzle geometry and DED parameter selection, however, the effect of material choice for nozzle fabrication has not been fully investigated. To explore the effects of the nozzle material on powder efficiency a CFD DPM model was created and analyzed in ansys fluent. Various nozzle materials were simulated using statistical models for the coefficient of restitution (COR) between the powder and nozzle wall from the literature. The results of the CFD DPM model aligned well with experimental data for a 316L stainless steel prototype nozzle. CFD DPM results indicated that using a material with a lower mean COR value improved the powder efficiency of the nozzle. Powder efficiency improved because the component of powder velocity normal to the direction of gas flow was reduced in nozzles made from materials with lower COR values, which in turn led to fewer impacts between the particles and the nozzle walls.

The magnetohydrodynamics system forced by space–time white noise has been proposed and investigated by physicists and engineers, although without rigorous mathematical proofs of its solution’s fundamental properties. Applying the theory of regularity structures, we prove its well-posedness and strong Feller property. The proof requires a careful treatment of nonlinear terms which are sensitive to specific components of the solution that is a six-dimensional vector field, of which the first three and the last three are velocity and magnetic fields, respectively.

Successful and efficient hybrid manufacturing consisting of Directed Energy Deposition (DED) followed by milling as a postprocess requires that material deposited during the DED step be geometrically consistent and slightly larger than the desired final, machined dimensions. DED clad width is directly correlated to meltpool energy, while powder fluence directly correlated to DED clad cross-sectional area. Due to heat accumulation into the DED part during deposition, meltpool energy does not remain constant without modification of DED process parameters over the deposition cycle. Meltpool energy is typically managed by decreasing the laser power to reduce the energy fluence into the meltpool – though this technique limits DED productivity. If the working distance and laser optics within the deposition head are static, energy fluence reduction can also be achieved by increasing deposition head feedrate. Modifying powder mass flowrate through the deposition nozzle to maintain powder fluence into the meltpool allows for both the clad geometry to be maintained while also increasing the productivity of the DED process. Numerical simulation of the heat flow in a DED thin wall part was used to derive laser energy fluence values to maintain a constant meltpool energy. Derived values were tested using both the laser power and feedrate to moderate energy fluence into the meltpool. Results are compared to a thin wall deposited without meltpool energy control. Clad geometry variation and cycle time are both shown to reduce when using a feedrate-based meltpool energy control method as compared to laser power-based meltpool energy control or no meltpool energy control, increasing the geometric accuracy for postprocess machining and productivity.

We study the Boussinesq system in a two-dimensional domain. In case only the velocity equation is forced by a white-in-time multiplicative noise, we prove the global existence of a martingale solution in the following two cases: positive horizontal viscous diffusion but zero vertical viscous diffusion and zero thermal diffusion; positive horizontal thermal diffusion but zero viscous diffusion and zero vertical thermal diffusion. Finally, in case both velocity and temperature equations are forced by a white-in-time multiplicative noise, and the velocity equation has zero viscous diffusion while the temperature equation has full thermal diffusion, the global existence of a martingale solution was shown in Yamazaki (Stoch Anal Appl, 34:404–426, 2016); in this manuscript we additionally prove significantly higher regularity.

One advantage of a femtosecond laser is that it enables the processing of materials that are otherwise difficult to process via conventional methods such as cutting, grinding, electric discharge machining, etc. However, it is difficult to predict a three dimensional shape processed by the laser. To achieve required three-dimensional shapes using femtosecond lasers with high accuracy and efficiency, it is essential to develop a computer-aided manufacturing (CAM) system that generates processing data (NC data). In this paper, a simulation method is proposed for femtosecond laser processing, which is the basis for the development of a CAM system. A binderless nano-polycrystalline diamond is used as the workpiece material to test the proposed method. Based on a comparison with the experimental results, it is confirmed that the processing shape could be simulated at a high accuracy for fixed-point irradiation as well as scanning irradiation.

In order to achieve high accuracy of computer numerical controlled (CNC) machine tools, various error compensation functions are implemented in the CNC control system. However, these error compensations are not designed to track the dynamic change in accuracies, such as thermal distortion or aging deterioration, during a machining process. To compensate for the dynamic change in accuracy, a built-in automatic measurement and compensation system based on dual linear scales was proposed, and its performance requirement was analyzed. Based on the analysis, a physical prototype was developed, and its feasibility was investigated by a series of experiments. The results showed that the measurement with the linear scale system agreed with a conventional measurement method within the uncertainty estimated at the design stage. In a subsequent experiment, the accuracy compensation was performed based on the measurement value. With compensation, the maximum accuracy deterioration was successfully suppressed from 22 to 3 μm.

Directed energy deposition (DED) is an additive manufacturing (AM) process where a laser is used to fuse a metal powder onto a substrate. The powder is deposited in a layer-by-layer fashion, and each deposited track is referred to as the clad. Large thermal gradients that occur during DED can cause changes in the clad geometry as well as deformation of the substrate and final part. Thermal behavior of DED manufactured parts and mitigation of thermal distortion has been addressed in the literature, but these studies mainly focused on material properties while neglecting the effect of thermal distortion on part geometry. One of the mitigation strategies that has been widely tested and accepted is substrate preheating. This study uses analytical and experimental models to observe changes in clad geometry when energy input to the part is varied by adjusting process parameters and preheating the substrate. A method to quantify substrate energy, or the amount of heat energy the substrate holds, is found as well as a relationship between the substrate energy and clad geometry. The geometry of the clad can be controlled by controlling the total energy added to the clad.

The directed energy deposition (DED) process enables the production of near net shape parts in the starting form of either powder or wire melted by a focused energy source, such as laser. The target geometry is then built layer by layer. In theory, engineering components with complex shapes can be fabricated with a single setup using a hybrid CNC machine tool, which has capabilities of both DED and traditional subtractive machining processes. However, the issue of inaccurate DED deposited geometry and insufficient material deposition results in the repetition of the hybrid process to achieve the goal geometric dimension. This paper presents a generalized strategy for different materials that plan DED toolpath for the planar geometric feature to achieve its required final geometric dimension by using only a single hybrid manufacturing cycle based on simulated results and validate the strategy experimentally by using a hybrid CNC machine.

Calibration of linear encoders has conventionally been performed using an optical interferometer for reference. Calibration using the optical interferometer, however, cannot fulfill the accuracy and stability requirements when the linear encoders exhibit sub-nanometer resolution. This study describes a new calibration method developed using the picometer resolution laser holographic scale as a reference, employing a range-dependent and minimum error-variance algorithm. The method has been evaluated via comparison tests with conventional optical-interferometer-based methods. Moreover, it has also been verified to achieve higher resolution, accuracy, and higher stability during the measurements, even in the atmospheric environment.

Ablation processing using a femtosecond laser has the advantage of processing materials that are impossible to machine by conventional methods. However, it is difficult to predict the three-dimensional removal shape owing to a femtosecond laser irradiation and the current processing method, repeating through trial and error, is inefficient and expensive. For this reason, for processing materials to desired shapes using a femtosecond laser, development of a simulation system and a CAM (Computer-Aided Manufacturing) system are indispensable. In this paper, a simulation method for processing materials using a femtosecond laser is proposed and conducted to nano-polycrystalline diamond (NPD). It is confirmed that the proposed method can precisely simulate the femtosecond laser processing for three-dimensional shape generation. Therefore, the proposed method enables to predict the removal shape identified as the virtual tool shape for calculating a tool path on a CAM system. In addition, an automation process method including an on-machine real-time measurement system of the proposed method is described.

Sapphire capillaries manufactured with high efficiency and precision are needed for use in laser–plasma accelerators. A hybrid manufacturing process combining femtosecond laser machining and diamond tool micromilling was applied. By using a femtosecond laser mounted on a 5-axis machine, a capillary with a groove width of 630 µm and length of 90 mm was fabricated on a sapphire plate within 5 h. The surface roughness of the bottom of the cylinder groove was finished to 7.1 nm in Sa by milling with a polycrystalline diamond end mill.

Inspired by the system of ordinary differential equations in Agusto et al. (2017) that models Zika virus dynamics by taking into account of both sexual and vector-borne transmissions, we furthermore add diffusive terms in order to capture the movement of human hosts and mosquitoes, considering the unique threat of the sexual transmission route of Zika virus. We conduct complete theoretical analysis. In particular, we show that every initial data that is continuous and non-negative admits a unique continuous and non-negative solution for all positive times. Moreover, we derive the basic reproduction number and when it is beneath one, we prove that the disease-free-equilibrium is globally attractive. Finally, when the basic reproduction number is above one, and any of the exposed males or the exposed females or the exposed mosquitoes is not identically zero, we prove the existence of a positive asymptotic lower bound for every component of the solution which in particular implies the uniform persistence of the disease, as well as the existence of at least one positive steady state.

In this manuscript, we discuss two examples of the results on the systems of partial differential equations in fluid mechanics obtained via stochastic analysis, for which deterministic case has no counterpart. Firstly, it is well-known that for a deterministic system of equations in fluid mechanics, a lack of diffusion creates a significant obstacle in proving its global well-posedness for a sufficiently small initial data. Nevertheless, it may be proven that for the Hall-magnetohydrodynamics system forced by L e vy noise, such a result may be obtained even with zero viscous diffusion. Secondly, we may extend the classical Kelvin’s conservation of circulation flow, as well as the Cauchy’s vorticity transport formula, on the Euler equations to show that the solution to the damped Euler equations enjoys the exact same identities except with an exponential decay in time. Such identities are known to completely fail once we add diffusion and consider the damped Navier–Stokes equations. Nevertheless, discovering a stochastic Lagrangian formulation, we are able to show that such identities indeed do continue to hold informally on average, i.e. via a mathematical expectation, over the random ensemble at earlier time. The stochastic Lagrangian formulation and these identities have also been obtained for the solution to the Boussinesq system. These results are rigorously proven in Mohan and Yamazaki (0000) and Yamazaki (0000). In this manuscript we will not pursue complete proofs, which were in particular mostly inspired by Kim (2011) and Iyer (2006), but rather elaborate on motivations, difficulty and remaining open problems for future works.

Phenotypic polymorphisms are found in a wide array of taxa, and unravelling the mechanisms that maintain them is of great interest to evolutionary and ecological biologists. Temporal environmental heterogeneity may play a role in the maintenance of polymorphisms but is poorly understood. In the present study, we analysed trends in intergenerational elytral colour morph frequencies in relation to changes in fitness and life history traits (i.e. body size, mortality, fecundity, hatching rate and mate preference) in the ladybird beetle Cheilomenes sexmaculata (Coleoptera: Coccinellidae). A long-term field survey spanning nine years showed that the frequency of dark morphs increases over winter and then decreases in spring. Dark morphs may have an advantage in winter due to their higher tolerance of low temperatures compared with light morphs. Light-morph females were heavier in winter than dark-morph females. They also mated more frequently and had higher hatching rates, potentially causing an increase in light morphs in spring. These results suggest that fluctuations in morph frequencies resulting from the conflicting directions of selection pressures between overwintering and spring generations may help to maintain genetic polymorphism.

The magnetohydrodynamics system consists of the Navier-Stokes equations from fluid mechanics, coupled with the Maxwell’s equations from electromagnetism through multiples of non-linear terms involving derivatives. Following the approach of [1], we prove the existence of a unique invariant measure in case the forcing terms consist of the cylindrical Wiener processes with only low modes. Its proof requires taking advantage of the structure of the non-linear terms carefully and is extended to various other related models such as the magnetohydrodynamics-Boussinesq system from fluid mechanics in atmosphere and oceans, as well as the magneto-micropolar fluid system from the theory of microfluids.

The magnetohydrodynamics system has found rich applications in applied sciences and has history of intense investigation by researchers including mathematicians. On the other hand, the Hall-magnetohydrodynamics system consists of the former system added by a Hall term which is very singular, and the mathematical development on this system has seen its significant progress only in the last several years. In short, the magnetohydrodynamics system, similarly to the Navier–Stokes equations, is semilinear, while the Hall-magnetohydrodynamics system is quasilinear. In this manuscript, we consider the three-dimensional magnetohydrodynamics, as well as the Hall-magnetohydrodynamics systems, and prove that they have Markov selections. In the case of the magnetohydrodynamics system, we prove furthermore a weak–strong uniqueness result and that any Markov solution has the strong Feller property for regular initial conditions. Consequently, it is deduced that if the magnetohydrodynamics system is well posed starting from one initial data, then it is well posed starting from any initial data, verifying a sharp contrast to the deterministic case in which the well posedness for all time with small initial data is well known. In the case of the Hall-magnetohydrodynamics system, we are also able to prove the weak–strong uniqueness result; however, the proof of strong Feller property seems to break down due to the singularity of the Hall term, about which we elaborate in detail.

Pineapple lilies Eucomis autumnalis (Mill.) Chitt. and Eucomis comosa Houtt. ex. Wehrh. are endemic herbaceous plants in southern Africa but are planted as ornamental bulbous plants globally. The flowers are known to be primarily pollinated by pompilid wasps in their native range. In a Japanese botanical garden, Blattella nipponica Asahina, 1963 cockroaches visited the inflorescences of the two Eucomis species, fed on the floral nectar, and used the comae (i.e., leaf-like tufts on the top of inflorescences) as physical shelters and transported the pollen, thereby resulting in a fruit set. Ants, flies, and beetles were also observed to visit the flowers but did not touch the anthers or stigmas. These observations suggest that ground-dwelling cockroaches can act as pollinators of Eucomis flowers in an ex-situ setting.

Non-uniqueness of three-dimensional Euler equations and Navier-Stokes equations forced by random noise, path-wise and more recently even in law, have been proven by various authors. We prove non-uniqueness in law of the three-dimensional Navier-Stokes equations forced by random noise and diffused via a fractional Laplacian that has power between zero and one half. The solution we construct has H$\ddot{\mathrm{o}}$lder regularity with a small exponent rather than Sobolev regularity with a small exponent. For the power sufficiently small, the non-uniqueness in law holds at the level of Leray-Hopf regularity. In particular, in order to handle transport error, we consider phase functions convected by not only a mollified velocity field but a sum of that with a mollified Ornstein-Uhlenbeck process if noise is additive and a product of that with a mollified exponential Brownian motion if noise is multiplicative.

A beam delivery system (BDS) fibered with hollow core photonic crystal fibre (HC-PCF) has been identified as very promising for ultra-short pulse (USP) laser micromachining for it allows flexible, secured and robust laser beam delivery to the work piece. These features are of paramount importance for vertical markets such as automotive or consumer electronics. We report on a system that integrates a BDS, a laser beam pointing stabilization module and a 50 W power, 250 fs pulsewidth Yb-based USP laser. The BDS comprises a low loss, low dispersion and high damage threshold Inhibited-Coupling (IC) HC-PCF. The latter is ruggedized with a semi-rigid industrial cable with ends attached respectively to an injection head for ease of laser beam coupling, and to an output connector for beam delivery to the work piece. The injection head is aligned with a special module for laser beam jitter stabilization and beam-shape control and monitoring. Also, it exhibits the necessary gas and thermal handling to minimize parasitic optical nonlinear effect or photoionization and power induced heating. To test the endurance of this fibered-USP laser system, it was continuously run over 17 hours. The results show exceptional integrity in the power transmission (86 ±1%), spectral and temporal structure and beam quality (M2~1.1). The BDS output beam shows a pointing stability of only 0.7±0.1 μrad. These results represent an important milestone towards the industrialization of fiber delivered USP laser-based machines.

Non-uniqueness in law for three-dimensional Navier-Stokes equations forced by random noise was established recently in Hofmanov$\acute{\mathrm{a}}$ et al. (2019, arXiv:1912.11841 [math.PR]). The purpose of this work is to prove non-uniqueness in law for the Boussinesq system forced by random noise. Diffusion within the equation of its temperature scalar field has a full Laplacian and the temperature scalar field can be initially smooth., The range of the exponent of a fractional Laplacian in the 3D case has been expanded. Some other structural changes have been made. This preprint has not undergone peer review or any post-submission improvements or corrections

Laser metal deposition (LMD) is an additive manufacturing technique, whose performance can be influenced by a considerable number of factors and parameters. Typically, a powder is carried by an inert gas and sprayed by a nozzle, with a coaxial laser beam passing through the nozzle and overlapping the powder flow, thereby generating a molten material pool on a substrate. Monitoring the evolution of this process allows for a better comprehension and control of the process, thereby enhancing the deposition quality. As the metal additive manufacturing mechanism has not yet been elucidated, it is not clear how process parameters affect material properties, molding accuracy, and molding efficiency. When cladding is performed under uncertain conditions, a molded part with poor material properties and dimensional accuracy is created. In this paper, we propose a method for high efficiency molding by controlling the distance between the head nozzle and the molten pool in real time. The distance is identified by an originally developed sensor based on a triangulation method. According to the distance, the head nozzle is automatically controlled into the optimum position. As a result, an ideal molding process can be generated, so that high efficiency molding and high-quality material properties can be obtained. Experimental results show that continuing deposition at the optimum distance assists in achieving deposition efficiency and dimensional accuracy. According to the specific experimental results of this method, the modeling efficiency was increased by 27% compared to the method without correction, and the modeling was successful with an error within 1 mm.

We study the two-dimensional Navier-Stokes equations forced by random noise with a diffusive term generalized via a fractional Laplacian that has a positive exponent strictly less than one. Because intermittent jets are inherently three-dimensional, we instead adapt the theory of intermittent form of the two-dimensional stationary flows to the stochastic approach presented by Hofmanov$\acute{\mathrm{a}}$, Zhu $\&$ Zhu (2019, arXiv:1912.11841 [math.PR]) and prove its non-uniqueness in law., arXiv admin note: text overlap with arXiv:2006.11861. This is the version that was accepted by SIAM Journal on Mathematical Analysis (SIMA), before the final edit by the journal

We study the Navier–Stokes equations with a dissipative term that is generalized through a fractional Laplacian in any dimension higher than two. We extend the horizontal Biot–Savart law beyond dimension three. Using the anisotropic Littlewood–Paley theory with which we distinguish the first two directions from the rest, we obtain a blow-up criteria for its solution in norms which are invariant under the rescaling of these equations. The proof goes through for the classical Navier–Stokes equations if dimension is three, four or five. We also give heuristics and partial results toward further improvement

We establish the existence and uniqueness of a local smooth solution to the Cauchy problem for the Hall-magnetohydrodynamics system that is inviscid, resistive, and forced by multiplicative L $$\acute{\mathrm {e}}$$ vy noise in the three dimensional space. Moreover, when the initial data is sufficiently small, we prove that the solution exists globally in time in probability; that is, the probability of the global existence of a unique smooth solution may be arbitrarily close to one given the initial data of which its expectation in a certain Sobolev norm is sufficiently small. The proofs go through for the two and a half dimensional case as well. To the best of the authors’ knowledge, an analogous result is absent in the deterministic case due to the lack of viscous diffusion, exhibiting the regularizing property of the noise. Our result may also be considered as a physically meaningful special case of the extension of work of Kim (J Differ Equ 250:1650–1684, 2011) and Mohan and Sritharan (Pure Appl Funct Anal 3:137–178, 2018) from the first-order quasilinear to the second-order quasilinear system because the Hall term elevates the Hall-magnetohydrodynamics system to the quasilinear class, in contrast to the Naiver–Stokes equations that has most often been studied and is semilinear.

Yamazaki, Kazuo (2018): The Picture of Dorian Gray: shell corrosion allows freshwater and brackish-water gastropods to masquerade as empty shells. Journal of Natural History 52 (35-36): 2331-2338, DOI: 10.1080/00222933.2018.1537408, URL: http://dx.doi.org/10.1080/00222933.2018.1537408

Magnetohydrodynamics system consists of a coupling of the Navier-Stokes and Maxwell's equations and is most useful in studying the motion of electrically conducting fluids. We prove the existence of a unique invariant, and consequently ergodic, measure for the Galerkin approximation system of the three-dimensional magnetohydrodynamics system. The proof is inspired by those of \cite{EM01, R04} on the Navier-Stokes equations; however, computations involve significantly more complications due to the coupling of the velocity field equations with those of magnetic field that consists of four non-linear terms., Comment: This is a pre-print version, and the revised version was accepted by Stochastics: An International Journal of Probability and Stochastic Processes. Its journal reference to be updated later

The ladybird beetle, Menochilus sexmaculatus (Fabricius), has a remarkable elytral colour polymorphism composed of black and red. In the present study, we investigated the effect of temperature on growth from the first instar larva to the pupal stage, as well as maternal morph types on the phenotypic expression of the elytral colour morph in a polymorphic population from Osaka, Japan. Female individuals of three different elytral colour morphs were collected from a wild population, and hatchlings from each female were divided into three groups, which were reared at three constant temperatures: 20 °C, 25 °C, and 30 °C. The phenotypic frequency of F 1 adults indicated that the elytral morph type was determined by genetic factors, but not by growth temperatures. Namely, type A (almost black morph) was the most abundant in F 1 from type A mothers (Male: 52.6%; Female: 32.3%); and types B (four small-dotted morph) and F (four medium-dotted morph) were the most abundant from type B (Male: 56.7%; Female: 53.3%) and type G (four larger-dotted morph) mothers (Male: 33.3%; Female: 31.3%), respectively. Therefore, the expression of elytral colour polymorphism in the Osaka, Japan population is likely to have a genetic basis contingent on parental morphs, rather than a phenotypic plasticity associated with growth temperatures.