Dirichlet series of Rankin-Cohen Brackets

Given modular forms $f$ and $g$ of weights $k$ and $\ell$, respectively, their Rankin-Cohen bracket $[f,g]^{(k, \ell)}_n$ corresponding to a nonnegative integer $n$ is a modular form of weight $k +\ell +2n$, and it is given as a linear combination of the products of the form $f^{(r)} g^{(n-r)}$ for $0 \leq r \leq n$. We use a correspondence between quasimodular forms and sequences of modular forms to express the Dirichlet series of a product of derivatives of modular forms as a linear combination of the Dirichlet series of Rankin-Cohen brackets

PROJECTIVE STRUCTURES AND AUTOMORPHIC PSEUDODIFFERENTIAL OPERATORS

Automorphic pseudodifferential operators are pseudodifferential operators that are invariant under an action of a discrete subgroup Γ of SL(2,ℝ), and they are closely linked to modular forms. In particular, there is a lifting map from modular forms to automorphic pseudodifferential operators, which can be interpreted as a lifting morphism of sheaves over the Riemann surface X associated to the given discrete subgroup Γ. One of the questions raised in a paper by Cohen, Manin, and Zagier is whether the difference in the images of a local section of a sheaf under such lifting morphisms corresponding to two projective structures on X can be expressed in terms of certain Schwarzian derivatives. The purpose of this paper is to provide a positive answer to this question for some special cases

The reduction of solutions of some integrable partial differential equations

AbstractWe reduce the finite-zone solutions of the Korteweg-de Vries equation, sine-Gordon equation and nonlinear Schrödinger equation associated to a hyperelliptic curve of genus four

A Study on Adhesive Strength of Co-Cured CFRP-Metal Multi-Material Joints and Joint Failure Detection Using Electrical Resistance Measurement

Department of Mechanical EngineeringCarbon-Fiber-Reinforced Plastics (CFRPs) are composite materials, consisting of carbon fibers and polymeric matrices. Depending on the types of carbon fiber and polymer used, CFRP can have a variety of properties. Generally, CFRP show high specific strength and stiffness, so it is regarded as a substitute material for existing structural materials, such as metals. In addition, CFRP can have high temperature or corrosion resistance based on the type of matrix used. For this reason, despite the high price of carbon fiber, it is widely applied to the aerospace industry and has gradually expanded into the automotive industry in recent years. Despite their advantages in terms of weigh-saving, it is not possible to replace all the metal parts, especially ultra-high-strength steels, etc., with CFRPs due to their limitations in intrinsic properties. This has led to the ???multi-material design??? concept, in which hetero-junctions between composites and metals have become an important issue. Typical methods for multi-material joining include mechanical joining and adhesive bonding. Mechanical joining, e.g., riveting, mechanical fastening, etc., leads to high stress concentration due to the pre-drilled holes, and it has to bear additional weight of inserts, such as bolts and rivets. Adhesive bonding, on the other hand, requires time of adhesive curing in addition to matrix curing, which has detrimental effects on manufacturing time and costs. To overcome these drawbacks, the co-curing method, in which the infused resin serves as the adhesive and therefore, the additional adhesive curing time can be omitted, has been considered as an alternative cost-effective adhesive joining method. Although multi-material joining using the co-curing method results in a lower adhesive strength than adhesive-bonded joints, this method can reduce the curing time since adhesive and CFRP curing proceed simultaneously and makes possible real-time health monitoring of the joints using electrical resistance measurement because carbon fiber directly contacts the metal surface, both of which are electrically conductive. In this study, we showed that structural health monitoring using electrical resistance measurement at the junction between metals and CFRPs joined by co-curing is feasible, and its effectiveness was studied as compared to the case where a conductive epoxy was used as the adhesive. Also, we measured the adhesive strength and determined the possibility of failure detection when a steel bushing, which is one of metal inserts, was joined by co-curing with CFRP. In addition, the interfacial strength between metal and polymer resin was enhanced by atmospheric plasma surface treatment since aluminum-CFRP co-cured joints initially had poor lap shear strength. CFRP was fabricated by plain-woven carbon fibers and unsaturated polyester resin, and stainless steel and aluminum sheets were used as the metals for multi-material joining. To detect the failure at the junction between CFRP and metal, co-curing was adopted rather than an epoxy adhesive containing dispersed carbon nanotubes (CNTs). In the co-curing process, conductive carbon fiber and metals directly contacted each other, so electric current can flow through both materials. As the initial load increased, the resistance gradually decreased, and then increased drastically due to de-bonding at the co-cured joints. Electrical resistance was increased when the contact area between carbon fiber and metal surface were decreased, so it can monitor the failure detection at the multi-material joints. Single-lap shear test was performed for each joint, and four-wire Kelvin resistance measurement was adopted to measure the change in resistance during the test. To apply this research, we manufactured steel bushing-inserted CFRPs joined by co-curing method. Push-out tests were performed to measure the adhesive strength between the inserts and CFRPs. Next, we demonstrated the proof-of-concept of health monitoring at the co-cured joints between steel bushings and CFRPs using electrical resistance measurements. In the case of aluminum-CFRP co-cured joints, the adhesive strength was about 30% compared to the other joints, so we applied atmospheric plasma to the metal surfaces such as steel, aluminum and steel bushings. Upon plasma treatment, the adhesive strength of aluminum-CFRPs co-cured joint was increased by 300%. After plasma treatment, the number of hydrogen bonds increased between the unsaturated polyester and the metal surfaces as the metal surfaces were getting more hydrophilic. Wettability was increased due to the increase of -OH functional groups on the metal surfaces, which led to the enhancement of the interfacial adhesive strength between polyester and metal surfaces bonded through the co-curing process. As the adhesive strength increased with plasma treatment, it was shown that the gradient of the resistance rate decreases prior to the complete destruction at the joints. For this reason, it is important to identify the optimized point to secure failure strength and predict joint failure. Based on the experimental results, it is feasible to monitor failures in multi-material joints between CFRPs and conductive metals real-time by measuring the change in resistance. This ensures the safety of various CFRP-metal multi-material structures, including aircraft, automotive parts, civil structures, sporting goods, electronic modules, and biomedical devices.ope

Differential operators on modular forms associated to Jacobi forms

Given Jacobi forms, we determine associated Jacobi-like forms, whose coefficients are quasimodular forms. We then use these quasimodular forms to construct differential operators on modular forms, which are expressed in terms of the Fourier coefficients of the given Jacobi forms

Mixed Jacobi-like forms of several variables

We study mixed Jacobi-like forms of several variables associated to equivariant maps of the Poincaré upper half-plane in connection with usual Jacobi-like forms, Hilbert modular forms, and mixed automorphic forms. We also construct a lifting of a mixed automorphic form to such a mixed Jacobi-like form