Normalized Wolfe-Powell-type local minimax method for finding multiple unstable solutions of nonlinear elliptic PDEs

The local minimax method (LMM) proposed in [Y. Li and J. Zhou, SIAM J. Sci. Comput., 23(3), 840--865 (2001)] and [Y. Li and J. Zhou, SIAM J. Sci. Comput., 24(3), 865--885 (2002)] is an efficient method to solve nonlinear elliptic partial differential equations (PDEs) with certain variational structures for multiple solutions. The steepest descent direction and the Armijo-type step-size search rules are adopted in [Y. Li and J. Zhou, SIAM J. Sci. Comput., 24(3), 865--885 (2002)] and play a significant role in the performance and convergence analysis of traditional LMMs. In this paper, a new algorithm framework of the LMMs is established based on general descent directions and two normalized (strong) Wolfe-Powell-type step-size search rules. The corresponding algorithm framework named as the normalized Wolfe-Powell-type LMM (NWP-LMM) is introduced with its feasibility and global convergence rigorously justified for general descent directions. As a special case, the global convergence of the NWP-LMM algorithm combined with the preconditioned steepest descent (PSD) directions is also verified. Consequently, it extends the framework of traditional LMMs. In addition, conjugate gradient-type (CG-type) descent directions are utilized to speed up the NWP-LMM algorithm. Finally, extensive numerical results for several semilinear elliptic PDEs are reported to profile their multiple unstable solutions and compared for different algorithms in the LMM's family to indicate the effectiveness and robustness of our algorithms. In practice, the NWP-LMM combined with the CG-type direction indeed performs much better than its known LMM companions.Comment: 27 pages, 9 figures; Accepted by SCIENCE CHINA Mathematics on January 17, 202

Nonmonotone local minimax methods for finding multiple saddle points

In this paper, by designing a normalized nonmonotone search strategy with the Barzilai--Borwein-type step-size, a novel local minimax method (LMM), which is a globally convergent iterative method, is proposed and analyzed to find multiple (unstable) saddle points of nonconvex functionals in Hilbert spaces. Compared to traditional LMMs with monotone search strategies, this approach, which does not require strict decrease of the objective functional value at each iterative step, is observed to converge faster with less computations. Firstly, based on a normalized iterative scheme coupled with a local peak selection that pulls the iterative point back onto the solution submanifold, by generalizing the Zhang--Hager (ZH) search strategy in the optimization theory to the LMM framework, a kind of normalized ZH-type nonmonotone step-size search strategy is introduced, and then a novel nonmonotone LMM is constructed. Its feasibility and global convergence results are rigorously carried out under the relaxation of the monotonicity for the functional at the iterative sequences. Secondly, in order to speed up the convergence of the nonmonotone LMM, a globally convergent Barzilai--Borwein-type LMM (GBBLMM) is presented by explicitly constructing the Barzilai--Borwein-type step-size as a trial step-size of the normalized ZH-type nonmonotone step-size search strategy in each iteration. Finally, the GBBLMM algorithm is implemented to find multiple unstable solutions of two classes of semilinear elliptic boundary value problems with variational structures: one is the semilinear elliptic equations with the homogeneous Dirichlet boundary condition and another is the linear elliptic equations with semilinear Neumann boundary conditions. Extensive numerical results indicate that our approach is very effective and speeds up the LMMs significantly.Comment: 32 pages, 7 figures; Accepted by Journal of Computational Mathematics on January 3, 202

Convergence analysis of a spectral-Galerkin-type search extension method for finding multiple solutions to semilinear problems

In this paper, we develop an efficient spectral-Galerkin-type search extension method (SGSEM) for finding multiple solutions to semilinear elliptic boundary value problems. This method constructs effective initial data for multiple solutions based on the linear combinations of some eigenfunctions of the corresponding linear eigenvalue problem, and thus takes full advantage of the traditional search extension method in constructing initials for multiple solutions. Meanwhile, it possesses a low computational cost and high accuracy due to the employment of an interpolated coefficient Legendre-Galerkin spectral discretization. By applying the Schauder's fixed point theorem and other technical strategies, the existence and spectral convergence of the numerical solution corresponding to a specified true solution are rigorously proved. In addition, the uniqueness of the numerical solution in a sufficiently small neighborhood of each specified true solution is strictly verified. Numerical results demonstrate the feasibility and efficiency of our algorithm and present different types of multiple solutions.Comment: 23 pages, 7 figures; Chinese version of this paper is published in SCIENTIA SINICA Mathematica, Vol. 51 (2021), pp. 1407-143

On the two-distance embedding in real Euclidean space of coherent configuration of type (2,2;3)

Finding the maximum cardinality of a $2$-distance set in Euclidean space is a classical problem in geometry. Lison\v{e}k in 1997 constructed a maximum $2$-distance set in $\mathbb R^8$ with $45$ points. That $2$-distance set constructed by Lison\v{e}k has a distinguished structure of a coherent configuration of type $(2,2;3)$ and is embedded in two concentric spheres in $\mathbb R^8$. In this paper we study whether there exists any other similar embedding of a coherent configuration of type $(2,2;3)$ as a $2$-distance set in $\mathbb R^n$, without assuming any restriction on the size of the set. We prove that there exists no such example other than that of Lison\v{e}k. The key ideas of our proof are as follows: (i) study the geometry of the embedding of the coherent configuration in Euclidean spaces and to drive diophantine equations coming from this embedding. (ii) solve diophantine equations with certain additional conditions of integrality of some parameters of the combinatorial structure by using the method of auxiliary equations

Development and validation of prognostic index based on purine metabolism genes in patients with bladder cancer

BackgroundBladder cancer (BLCA) is a prevalent malignancy affecting the urinary system and is associated with significant morbidity and mortality worldwide. Dysregulation of tumor metabolic pathways is closely linked to the initiation and proliferation of BLCA. Tumor cells exhibit distinct metabolic activities compared to normal cells, and the purine metabolism pathway, responsible for providing essential components for DNA and RNA synthesis, is believed to play a crucial role. However, the precise involvement of Purine Metabolism Genes (PMGs) in the defense mechanism against BLCA remains elusive.MethodsThe integration of BLCA samples from the TCGA and GEO datasets facilitated the quantitative evaluation of PMGs, offering potential insights into their predictive capabilities. Leveraging the wealth of information encompassing mRNAsi, gene mutations, CNV, TMB, and clinical features within these datasets further enriched the analysis, augmenting its robustness and reliability. Through the utilization of Lasso regression, a prediction model was developed, enabling accurate prognostic assessments within the context of BLCA. Additionally, co-expression analysis shed light on the complex relationship between gene expression patterns and PMGs, unraveling their functional relevance and potential implications in BLCA.ResultsPMGs exhibited increased expression levels in the high-risk cohort of BLCA patients, even in the absence of other clinical indicators, suggesting their potential as prognostic markers. GSEA revealed enrichment of immunological and tumor-related pathways specifically in the high-risk group. Furthermore, notable differences were observed in immune function and m6a gene expression between the low- and high-risk groups. Several genes, including CLDN6, CES1, SOST, SPRR2A, MYBPH, CGB5, and KRT1, were found to potentially participate in the oncogenic processes underlying BLCA. Additionally, CRTAC1 was identified as potential tumor suppressor genes. Significant discrepancies in immunological function and m6a gene expression were observed between the two risk groups, further highlighting the distinct molecular characteristics associated with different prognostic outcomes. Notably, strong correlations were observed among the prognostic model, CNVs, SNPs, and drug sensitivity profiles.ConclusionPMGs have been implicated in the etiology and progression of bladder cancer (BLCA). Prognostic models corresponding to this malignancy aid in the accurate prediction of patient outcomes. Notably, exploring the potential therapeutic targets within the tumor microenvironment (TME) such as PMGs and immune cell infiltration holds promise for effective BLCA management, albeit necessitating further research. Moreover, the identification of a gene signature associated with purine Metabolism presents a credible and alternative approach for predicting BLCA, signifying a burgeoning avenue for targeted therapeutic investigations in the field of BLCA

PD-1/PD-L1 Axis, Rather Than High-Mobility Group Alarmins or CD8+ Tumor-Infiltrating Lymphocytes, Is Associated With Survival in Head and Neck Squamous Cell Carcinoma Patients Who Received Surgical Resection

In current studies, the influence of tumor immune microenvironment on tumorigenesis and tumor progression has been widely explored. In the present study, we investigated the expression and significance of high mobility group box 1 (HMGB1), HMG nucleosome-binding protein 1 (HMGN1), the receptor programmed cell death 1 (PD-1) and its ligand programmed cell death ligand 1 (PD-L1) in head and neck squamous cell carcinoma (HNSCC). We explored whether HMGB1 and HMGN1 take part in recruiting T cells to HNSCC microenvironment. Furthermore, we assessed the prognostic value of HMG proteins, TILs, and PD-1/PD-L1 in postoperative patients. Tumor tissue sections were collected from 81 cases of patients with resectable HNSCC. All patients' information was integrated with clinical and pathological records, as well as follow-up data. We used immunohistochemistry to examine the subcellular localization and expression levels of HMGB1 and HMGN1, as well as tumor CD3+, CD8+, FOXP3+ lymphocyte infiltration, and the expression of immune inhibiting molecules PD-1/PD-L1. Results showed that there was no significant difference in the number of CD8+ and FOXP3+ T cells between the two groups with or without HMGB1 cytoplasmic expression in tumor tissues. The number of CD3+ T cells in HMGB1 cytoplasmic expression group (339.39 ± 230.76) was more than that in group without HMGB1 cytoplasmic expression (233.30 ± 230.91, P < 0.05). The number of CD3+, CD8+, and FOXP3+ T cells in HMGN1 cytoplasmic expression group [400.74 ± 224.04, 158.10 ± 112.10, 36.00(15.00, 69.00)] was more than that in group without HMGN1-cytoplasmic expression [222.84 ± 217.78, P < 0.01; 105.10 ± 108.25, P < 0.05; 13.00(6.75, 32.25), P < 0.01]. The positive rates of PD-1 and PD-L1 in tumor tissues were 29.6 and 67.9%, respectively. Multivariate analysis suggested that tumor expression of PD-L1 was an independent prognostic factor and PD-L1 overexpression indicated a poor overall survival (OS) and disease-free survival (DFS). Taken together, we concluded that HMGB1 and HMGN1 secreted by cancer cells may relate to recruitment of tumor infiltrating lymphocytes (TILs) in HNSCC. PD-1/PD-L1 axis, rather than HMG proteins or CD8+ tumor-infiltrating lymphocytes, has a critical role in tumor immune microenvironment and could predict the outcome of HNSCC patients who received surgical resection

Multi-strategy engineering greatly enhances provitamin A carotenoid accumulation and stability in Arabidopsis seeds

Staple grains with low levels of provitamin A carotenoids contribute to the global prevalence of vitamin A deficiency and therefore are the main targets for provitamin A biofortification. However, carotenoid stability during both seed maturation and postharvest storage is a serious concern for the full benefits of carotenoid biofortified grains. In this study, we utilized Arabidopsis as a model to establish carotenoid biofortification strategies in seeds. We discovered that manipulation of carotenoid biosynthetic activity by seed-specific expression of Phytoene synthase (PSY) increases both provitamin A and total carotenoid levels but the increased carotenoids are prone to degradation during seed maturation and storage, consistent with previous studies of provitamin A biofortified grains. In contrast, stacking with Orange (ORHis), a gene that initiates chromoplast biogenesis, dramatically enhances provitamin A and total carotenoid content and stability. Up to 65- and 10-fold increases of β-carotene and total carotenoids, respectively, with provitamin A carotenoids composing over 63% were observed in the seeds containing ORHis and PSY. Co-expression of Homogentisate geranylgeranyl transferase (HGGT) with ORHis and PSY further increases carotenoid accumulation and stability during seed maturation and storage. Moreover, knocking-out of β-carotene hydroxylase 2 (BCH2) by CRISPR/Cas9 not only potentially facilitates β-carotene accumulation but also minimizes the negative effect of carotenoid over production on seed germination. Our findings provide new insights into various processes on carotenoid accumulation and stability in seeds and establish a multiplexed strategy to simultaneously target carotenoid biosynthesis, turnover, and stable storage for carotenoid biofortification in crop seeds

Improving Generalization in Visual Reinforcement Learning via Conflict-aware Gradient Agreement Augmentation

Learning a policy with great generalization to unseen environments remains challenging but critical in visual reinforcement learning. Despite the success of augmentation combination in the supervised learning generalization, naively applying it to visual RL algorithms may damage the training efficiency, suffering from serve performance degradation. In this paper, we first conduct qualitative analysis and illuminate the main causes: (i) high-variance gradient magnitudes and (ii) gradient conflicts existed in various augmentation methods. To alleviate these issues, we propose a general policy gradient optimization framework, named Conflict-aware Gradient Agreement Augmentation (CG2A), and better integrate augmentation combination into visual RL algorithms to address the generalization bias. In particular, CG2A develops a Gradient Agreement Solver to adaptively balance the varying gradient magnitudes, and introduces a Soft Gradient Surgery strategy to alleviate the gradient conflicts. Extensive experiments demonstrate that CG2A significantly improves the generalization performance and sample efficiency of visual RL algorithms.Comment: accepted by iccv202

A Stress Induced Source of Phonon Bursts and Quasiparticle Poisoning

The performance of superconducting qubits is degraded by a poorly characterized set of energy sources breaking the Cooper pairs responsible for superconductivity, creating a condition often called "quasiparticle poisoning." Recently, a superconductor with one of the lowest average quasiparticle densities ever measured exhibited quasiparticles primarily produced in bursts which decreased in rate with time after cooldown. Similarly, several cryogenic calorimeters used to search for dark matter have also observed an unknown source of low-energy phonon bursts that decrease in rate with time after cooldown. Here, we show that a silicon crystal glued to its holder exhibits a rate of low-energy phonon events that is more than two orders of magnitude larger than in a functionally identical crystal suspended from its holder in a low-stress state. The excess phonon event rate in the glued crystal decreases with time since cooldown, consistent with a source of phonon bursts which contributes to quasiparticle poisoning in quantum circuits and the low-energy events observed in cryogenic calorimeters. We argue that relaxation of thermally induced stress between the glue and crystal is the source of these events, and conclude that stress relaxation contributes to quasiparticle poisoning in superconducting qubits and the athermal phonon background in a broad class of rare-event searches.Comment: 13 pages, 6 figures. W. A. Page and R. K. Romani contributed equally to this work. Correspondence should be addressed to R. K. Roman

Flutter kick technique and its effect on overall front crawl velocity

Purpose: To investigate the changes in flutter kicking frequency, knee angle and ankle plantarflexion angle at different free front crawl swimming velocities. Methods: Nine participants (six males and three females) swam 25 m of front crawl at three different velocities – 50%, 70% and 90% of their maximum velocity. Underwater video recordings of participants were analysed and recorded via Kinovea, and data collected were analysed using JASP Version 0.14. Results: A negative correlation was observed between kicking frequency and velocity during trials of V50 (p .05) and V90 (p > .05) respectively. Knee angles positively correlated with velocity (p > .05), while a large disparity in correlation was observed between velocity and ankle plantarflexion (p > .05). A significant difference in kicking frequencies and knee angles between different velocities was observed (p .05). Conclusions: Flutter kick frequency increased significantly as task demand increases. A higher swimming velocity may result in smaller degrees of knee flexion, but no significant differences were observed in this study. No clear relationship between velocity and ankle plantarflexion was observed possibly due to experimental errors.Bachelor of Science (Sport Science and Management