From rr-Spin Intersection Numbers to Hodge Integrals

Generalized Kontsevich Matrix Model (GKMM) with a certain given potential is the partition function of $r$-spin intersection numbers. We represent this GKMM in terms of fermions and expand it in terms of the Schur polynomials by boson-fermion correspondence, and link it with a Hurwitz partition function and a Hodge partition by operators in a $\widehat{GL}(\infty)$ group. Then, from a $W_{1+\infty}$ constraint of the partition function of $r$-spin intersection numbers, we get a $W_{1+\infty}$ constraint for the Hodge partition function. The $W_{1+\infty}$ constraint completely determines the Schur polynomials expansion of the Hodge partition function.Comment: 51 pages, 1 figur

Evolution of host support for two ancient bacterial symbionts with differentially degraded genomes in a leafhopper host.

Plant sap-feeding insects (Hemiptera) rely on bacterial symbionts for nutrition absent in their diets. These bacteria experience extreme genome reduction and require genetic resources from their hosts, particularly for basic cellular processes other than nutrition synthesis. The host-derived mechanisms that complete these processes have remained poorly understood. It is also unclear how hosts meet the distinct needs of multiple bacterial partners with differentially degraded genomes. To address these questions, we investigated the cell-specific gene-expression patterns in the symbiotic organs of the aster leafhopper (ALF), Macrosteles quadrilineatus (Cicadellidae). ALF harbors two intracellular symbionts that have two of the smallest known bacterial genomes: Nasuia (112 kb) and Sulcia (190 kb). Symbionts are segregated into distinct host cell types (bacteriocytes) and vary widely in their basic cellular capabilities. ALF differentially expresses thousands of genes between the bacteriocyte types to meet the functional needs of each symbiont, including the provisioning of metabolites and support of cellular processes. For example, the host highly expresses genes in the bacteriocytes that likely complement gene losses in nucleic acid synthesis, DNA repair mechanisms, transcription, and translation. Such genes are required to function in the bacterial cytosol. Many host genes comprising these support mechanisms are derived from the evolution of novel functional traits via horizontally transferred genes, reassigned mitochondrial support genes, and gene duplications with bacteriocyte-specific expression. Comparison across other hemipteran lineages reveals that hosts generally support the incomplete symbiont cellular processes, but the origins of these support mechanisms are generally specific to the host-symbiont system

Product Construction of Affine Codes

Binary matrix codes with restricted row and column weights are a desirable method of coded modulation for power line communication. In this work, we construct such matrix codes that are obtained as products of affine codes - cosets of binary linear codes. Additionally, the constructions have the property that they are systematic. Subsequently, we generalize our construction to irregular product of affine codes, where the component codes are affine codes of different rates.Comment: 13 pages, to appear in SIAM Journal on Discrete Mathematic

Reconstruction of Cosmological Initial Density Field with Observations from the Epoch of Reionization

Initial density distribution provides a basis for understanding the complete evolution of cosmological density fluctuations. While reconstruction in our local Universe exploits the observations of galaxy surveys with large volumes, observations of high-redshift galaxies are performed with a small field of view and therefore can hardly be used for reconstruction. Here we propose to reconstruct the initial density field using the H I 21 cm and CO line intensity maps from the epoch of reionization. Observations of these two intensity maps provide complementary information of the density field -- the H I 21 cm field is a proxy of matter distributions in the neutral regions, while the CO line intensity maps are sensitive to the high-density, star-forming regions that host the sources for reionization. Technically, we employ the conjugate gradient method and develop the machinery for minimizing the cost function for the intensity mapping observations. Analytical expressions for the gradient of cost function are derived explicitly. We show that the resimulated intensity maps match the input maps of mock observations using semi-numerical simulations of reionization with an rms error $\lesssim 7\%$ at all stages of reionization. This reconstruction is also robust at the same level of accuracy when a noise at the level of $\lesssim 1\%$ of the standard deviation is applied to each map. Our proof-of-concept work demonstrates the robustness of the reconstruction method, thereby providing an effective technique for reconstructing the cosmological initial density distribution from high-redshift observations.Comment: 12 pages, 8 figures, 2 table

Physics-informed neural networks with residual/gradient-based adaptive sampling methods for solving PDEs with sharp solutions

We consider solving the forward and inverse PDEs which have sharp solutions using physics-informed neural networks (PINNs) in this work. In particular, to better capture the sharpness of the solution, we propose adaptive sampling methods (ASMs) based on the residual and the gradient of the solution. We first present a residual only based ASM algorithm denoted by ASM I. In this approach, we first train the neural network by using a small number of residual points and divide the computational domain into a certain number of sub-domains, we then add new residual points in the sub-domain which has the largest mean absolute value of the residual, and those points which have largest absolute values of the residual in this sub-domain will be added as new residual points. We further develop a second type of ASM algorithm (denoted by ASM II) based on both the residual and the gradient of the solution due to the fact that only the residual may be not able to efficiently capture the sharpness of the solution. The procedure of ASM II is almost the same as that of ASM I except that in ASM II, we add new residual points which not only have large residual but also large gradient. To demonstrate the effectiveness of the present methods, we employ both ASM I and ASM II to solve a number of PDEs, including Burger equation, compressible Euler equation, Poisson equation over an L-shape domain as well as high-dimensional Poisson equation. It has been shown from the numerical results that the sharp solutions can be well approximated by using either ASM I or ASM II algorithm, and both methods deliver much more accurate solution than original PINNs with the same number of residual points. Moreover, the ASM II algorithm has better performance in terms of accuracy, efficiency and stability compared with the ASM I algorithm.Comment: 22 pages, 9 figure

Importance of Symbol Equity in Coded Modulation for Power Line Communications

The use of multiple frequency shift keying modulation with permutation codes addresses the problem of permanent narrowband noise disturbance in a power line communications system. In this paper, we extend this coded modulation scheme based on permutation codes to general codes and introduce an additional new parameter that more precisely captures a code's performance against permanent narrowband noise. As a result, we define a new class of codes, namely, equitable symbol weight codes, which are optimal with respect to this measure