A proof of the four-colour theorem

The four-colour problem remained unsolved for more than a hundred years has played a role of the utmost importance in the development of graph theory. The four-colour theorem was confirmed in 1976, which is not completely satisfied due to: i) part of the proof using computers cannot be verified by hand; ii) even the part, supposedly hand-checkable, is extraordinarily complicated and tedious, and as far as we know, no one has entirely verified it. Seeking a hand-checkable proof of the four-colour theorem is one of world-interested problems, which is addressed in this paper. A necessary and sufficient condition for n-colour theorem in a space is: there exists a largest n-complete graph base in the same space. Examples are given to illustrate applications

An investigation into natural vibrations of fluid-structure interaction systems subject to Sommerfeld’s radiation condition

A fluid-structure interaction system subject to Sommerfeld’s condition is defined as a Sommerfeld system which is divided into three categories: Fluid Sommerfeld (FS) System, Solid Sommerfeld (SS) System and Fluid Solid Sommerfeld (FSS) System of which Sommerfeld conditions are imposed on a fluid boundary only, a solid boundary only and both fluid and solid boundaries, respectively. This paper follows the previous initial results claimed by simple examples to further mathematically investigate the natural vibrations of generalized Sommerfeld systems. A new parameter representing the speed of radiation wave for generalized 3-D problems with more complicated boundary conditions is introduced into the Sommerfeld condition which allows investigation of the natural vibrations of a Sommerfeld system involving both free surface and compressible waves. The mathematical demonstrations and selected examples confirm and reveal the natural behaviour of generalized Sommerfeld systems defined above. These generalized conclusions can be used in theoretical or engineering analysis of the vibrations of various Sommerfeld systems in engineering

Polarization-Independent Metasurface Lens Based on Binary Phase Fresnel Zone Plate

Based on the binary phase Fresnel zone plate (FZP), a polarization-independent metasurface lens that is able to focus incident light with any polarization state, including circular, linear, and elliptical polarizations, has been proposed and investigated. We demonstrate that the metasurface lens consisting of metal subwavelength slits can operate in a wide bandwidth in the visible range, and has a higher focusing efficiency than that of an amplitude FZP lens without phase modulation. A multi-focus FZP metasurface lens has also been designed and investigated. The proposed lens can provide potential applications in integrated nanophotonic devices without polarization limitations

A DNA Vaccine Encoding the VAA Gene of Vibrio anguillarum Induces a Protective Immune Response in Flounder

Vibrio anguillarum is a pathogenic bacterium that infects flounder resulting in significant losses in the aquaculture industry. The VAA protein previously identified in flounder is associated with a role in immune protection within these fish. In the present study, a recombinant DNA plasmid encoding the VAA gene of V. anguillarum was constructed and its potential as a DNA vaccine, to prevent the infection of V. anguillarum in flounder fish, investigated. We verified the expression of the VAA protein both in vitro in cell lines and in vivo in flounder fish. The protective effects of pcDNA3.1-VAA (pVAA) were analyzed by determination of the percentage of sIgM+, CD4-1+, CD4-2+, CD8β+ lymphocytes, and the production of VAA-specific antibodies in flounder following their immunization with the DNA vaccine. Histopathological changes in immune related tissues, bacterial load, and relative percentage survival rates of flounder post-challenge with V. anguillarum, were all investigated to assess the efficacy of the pVAA DNA vaccine candidate. Fish intramuscularly immunized with pVAA showed a significant increase in CD4-1+, CD4-2+, and CD8β+ T lymphocytes at days 9, 11, and 14 post-vaccination, reaching peak T-cell levels at days 11 or 14 post-immunization. The percentage of sIgM+ lymphocytes reached peak levels at weeks 4–5 post-immunization. Specific anti-V. anguillarum or anti-rVAA antibodies were induced in inoculated fish at days 28–35 post-immunization. The liver of vaccinated flounder exhibited only slight histopathological changes compared with a significant pathology observed in control immunized fish. Additionally, a lower bacterial burden in the liver, spleen, and kidney were observed in pVAA protected fish in response to bacterial challenge, compared with pcDNA3.1 vector control injected fish. Moreover, the pVAA vaccine confers a relative percentage survival of 50.00% following V. anguillarum infection. In summary, this is the first study indicating an initial induction of the T lymphocyte response, followed by B lymphocyte induction of specific antibodies as a result of DNA immunization of flounder. This signifies the important potential of pVAA as a DNA vaccine candidate for the control of V. anguillarum infection

Crystal Structure Manipulation of the Exchange Bias in an Antiferromagnetic Film

Exchange bias is one of the most extensively studied phenomena in magnetism, since it exerts a unidirectional anisotropy to a ferromagnet (FM) when coupled to an antiferromagnet (AFM) and the control of the exchange bias is therefore very important for technological applications, such as magnetic random access memory and giant magnetoresistance sensors. In this letter, we report the crystal structure manipulation of the exchange bias in epitaxial hcp Cr2O3 films. By epitaxially growing twined (10-10) oriented Cr2O3 thin films, of which the c axis and spins of the Cr atoms lie in the film plane, we demonstrate that the exchange bias between Cr2O3 and an adjacent permalloy layer is tuned to in-plane from out-of-plane that has been observed in (0001) oriented Cr2O3 films. This is owing to the collinear exchange coupling between the spins of the Cr atoms and the adjacent FM layer. Such a highly anisotropic exchange bias phenomenon is not possible in polycrystalline films.Comment: To be published in Scientific Reports, 12 pages, 6 figure

Pressure-induced spin reorientation transition in layered ferromagnetic insulator Cr2Ge2Te6

Anisotropic magnetoresistance (AMR) of Cr2Ge2Te6 (CGT), a layered ferromagnetic insulator, is investigated under an applied hydrostatic pressure up to 2 GPa. The easy axis direction of the magnetization is inferred from the AMR saturation feature in the presence and absence of the applied pressure. At zero applied pressure, the easy axis is along the c-direction or perpendicular to the layer. Upon application of a hydrostatic pressure>1 GPa, the uniaxial anisotropy switches to easy-plane anisotropy which drives the equilibrium magnetization from the c-axis to the ab-plane at zero magnetic field, which amounts to a giant magnetic anisotropy energy change (>100%). As the temperature is increased across the Curie temperature, the characteristic AMR effect gradually decreases and disappears. Our first-principles calculations confirm the giant magnetic anisotropy energy change with moderate pressure and assign its origin to the increased off-site spin-orbit interaction of Te atoms due to a shorter Cr-Te distance. Such a pressure-induced spin reorientation transition is very rare in three-dimensional ferromagnets, but it may be common to other layered ferromagnets with similar crystal structures to CGT, and therefore offers a unique way to control magnetic anisotropy

Transcriptome analysis reveals molecular mechanisms of lymphocystis formation caused by lymphocystis disease virus infection in flounder (Paralichthys olivaceus)

Lymphocystis disease is frequently prevalent and transmissible in various teleost species worldwide due to lymphocystis disease virus (LCDV) infection, causing unsightly growths of benign lymphocystis nodules in fish and resulting in huge economic losses to aquaculture industry. However, the molecular mechanism of lymphocystis formation is unclear. In this study, LCDV was firstly detected in naturally infected flounder (Paralichthys olivaceus) by PCR, histopathological, and immunological techniques. To further understand lymphocystis formation, transcriptome sequencing of skin nodule tissue was performed by using healthy flounder skin as a control. In total, RNA-seq produced 99.36%-99.71% clean reads of raw reads, of which 91.11%-92.89% reads were successfully matched to the flounder genome. The transcriptome data showed good reproducibility between samples, with 3781 up-regulated and 2280 down-regulated differentially expressed genes. GSEA analysis revealed activation of Wnt signaling pathway, Hedgehog signaling pathway, Cell cycle, and Basal cell carcinoma associated with nodule formation. These pathways were analyzed to interact with multiple viral infection and tumor formation pathways. Heat map and protein interaction analysis revealed that these pathways regulated the expression of cell cycle-related genes such as ccnd1 and ccnd2 through key genes including ctnnb1, lef1, tcf3, gli2, and gli3 to promote cell proliferation. Additionally, cGMP-PKG signaling pathway, Calcium signaling pathway, ECM-receptor interaction, and Cytokine-cytokine receptor interaction associated with nodule formation were significantly down-regulated. Among these pathways, tnfsf12, tnfrsf1a, and tnfrsf19, associated with pro-apoptosis, and vdac2, which promotes viral replication by inhibiting apoptosis, were significantly up-regulated. Visual analysis revealed significant down-regulation of cytc, which expresses the pro-apoptotic protein cytochrome C, as well as phb and phb2, which have anti-tumor activity, however, casp3 was significantly up-regulated. Moreover, bcl9, bcl11a, and bcl-xl, which promote cell proliferation and inhibit apoptosis, were significantly upregulated, as were fgfr1, fgfr2, and fgfr3, which are related to tumor formation. Furthermore, RNA-seq data were validated by qRT-PCR, and LCDV copy numbers and expression patterns of focused genes in various tissues were also investigated. These results clarified the pathways and differentially expressed genes associated with lymphocystis nodule development caused by LCDV infection in flounder for the first time, providing a new breakthrough in molecular mechanisms of lymphocystis formation in fish