To Prove Four Color Theorem

In this paper, we give a proof for four color theorem(four color conjecture). Our proof does not involve computer assistance and the most important is that it can be generalized to prove Hadwiger Conjecture. Moreover, we give algorithms to color and test planarity of planar graphs, which can be generalized to graphs containing $K_x(x>5)$ minor. There are four parts of this paper: Part-1: To Prove Four Color Theorem Part-2: An Equivalent Statement of Hadwiger Conjecture when $k=5$ Part-3: A New Proof of Wagner's Equivalence Theorem Part-4: A Geometric View of Outerplanar GraphComment: The paper is further reduced, and each part is more self-contained, is the fina

Brain perfusion patterns are altered in chronic knee pain:a spatial covariance analysis of arterial spin labelling MRI

Chronic musculoskeletal pain is a common problem globally. Current evidence suggests that maladapted central pain pathways are associated with pain chronicity, for example, in postoperative pain after knee replacement. Other factors such as low mood, anxiety, and tendency to catastrophize are also important contributors. We aimed to investigate brain imaging features that underpin pain chronicity based on multivariate pattern analysis of cerebral blood flow (CBF), as a marker of maladaptive brain changes. This was achieved by identifying CBF patterns that discriminate chronic pain from pain-free conditions and by exploring their explanatory power for factors thought to drive pain chronification. In 44 chronic knee pain and 29 pain-free participants, we acquired both CBF and T1-weighted data. Participants completed questionnaires related to affective processes and pressure and cuff algometry to assess pain sensitization. Two factor scores were extracted from these scores representing negative affect and pain sensitization. A spatial covariance principal component analysis of CBF identified 5 components that significantly discriminated chronic pain participants from controls, with the unified network achieving 0.83 discriminatory accuracy (area under the curve). In chronic knee pain, significant patterns of relative hypoperfusion were evident in anterior default-mode and salience network hubs, while hyperperfusion was seen in posterior default mode, thalamus, and sensory regions. One component correlated positively with the pain sensitization score (r = 0.43, P = 0.006), suggesting that this CBF pattern reflects neural activity changes encoding pain sensitization. Here, we report a distinct chronic knee pain-related representation of CBF, pointing toward a brain signature underpinning central aspects of pain sensitization

a generalized framework for bdd-based replanning a* search

IEEE Comp Soc, ACISRecently, it has been suggested that BDD-based Re-Planning A* (BDDRPA*), a BDD-based incremental version of A*, might be an efficient search method for solving path-planning problems in artificial intelligence. BDDRPA* combines ideas of BDD-based search and incremental search to repeatedly find shortest paths from a start vertex to a goal vertex while the topology of the graph changes. However BDDRPA* only works well when vertices art, added or deleted but doest consider the weighted edges. When the edge costs are changed, it doesnt work, and moreover, in BDDRPA*, the heuristic function h is set to 0, so BDDRPA* is degenerated to BDD-based incremental breadth-first search. In this article, we consider BDD-based weighted and heuristic search methods and generalize BDDRPA* to be a real BDD-based incremental heuristic search algorithm (GBDDRPA*). We then show experimentally that GBDDRPA* indeed speeds BDDRPA* up on gridworlds and thus promises to provide a good foundation for building incremental heuristic BDD-search-based replanners

the bdd-based dynamic a* algorithm for real-time replanning

Finding optimal path through a graph efficiently is central to many problems; including route planning for a mobile robot. BDD-based incremental heuristic search method uses heuristics to focus their search and reuses BDD-based information from previous searches to find solutions to series of similar search problems much faster than solving each search problem from scratch. In this paper, we apply BDD-based incremental heuristic search to robot navigation in unknown terrain, including goal-directed navigation in unknown terrain and mapping of unknown terrain. The resulting BDD-based dynamic A* (BDDD*) algorithm is capable of planning paths in unknown, partially known and changing environments in an efficient, optimal, and complete manner. We present properties about BDDD* and demonstrate experimentally the advantages of combining BDD-based incremental and heuristic search for the applications studied. We believe that our experimental results will make BDD-based D* like replanning algorithms more popular and enable robotics researchers to adapt them to additional applications

Genetic Basis of Drought Resistance at Reproductive Stage in Rice: Separation of Drought Tolerance From Drought Avoidance

Drought tolerance (DT) and drought avoidance (DA) are two major mechanisms in drought resistance of higher plants. In this study, the genetic bases of DT and DA at reproductive stage in rice were analyzed using a recombinant inbred line population from a cross between an indica lowland and a tropical japonica upland cultivar. The plants were grown individually in PVC pipes and two cycles of drought stress were applied to individual plants with unstressed plants as the control. A total of 21 traits measuring fitness, yield, and the root system were investigated. Little correlation of relative yield traits with potential yield, plant size, and root traits was detected, suggesting that DT and DA were well separated in the experiment. A genetic linkage map consisting of 245 SSR markers was constructed for mapping QTL for these traits. A total of 27 QTL were resolved for 7 traits of relative performance of fitness and yield, 36 QTL for 5 root traits under control, and 38 for 7 root traits under drought stress conditions, suggesting the complexity of the genetic bases of both DT and DA. Only a small portion of QTL for fitness- and yield-related traits overlapped with QTL for root traits, indicating that DT and DA had distinct genetic bases

Associations of Muscle Strength with Central Aspects of Pain: Data from the Knee Pain and Related Health in the Community (KPIC) Cohort

Knee pain is associated with lower muscle strength, and both contribute to disability. Peripheral and central neurological mechanisms contribute to OA pain. Understanding the relative contributions of pain mechanisms to muscle strength might help future treatments. The Knee Pain and related health In the Community (KPIC) cohort provided baseline and year 1 data from people with early knee pain (n = 219) for longitudinal analyses. A cross-sectional analysis was performed with baseline data from people with established knee pain (n = 103) and comparative data from people without knee pain (n = 98). Quadriceps and handgrip strength indicated local and general muscle weakness, respectively. The indices of peripheral nociceptive drive were knee radiographic and ultrasound scores. The indices associated with central pain mechanisms were Pressure Pain detection Threshold (PPT) distal to the knee, and a validated self-report Central Aspects of Pain Factor (CAPF). The associations were explored using correlation and multivariable regression. Weaker quadriceps strength was associated with both high CAPF and low PPT at baseline. Year 1 quadriceps weakness was predicted by higher baseline CAPF (β = −0.28 (95% CI: −0.55, −0.01), p = 0.040). Weaker baseline and year 1 handgrip strength was also associated with higher baseline CAPF. Weaker baseline quadriceps strength was associated with radiographic scores in bivariate but not adjusted analyses. Quadriceps strength was not significantly associated with total ultrasound scores. Central pain mechanisms might contribute to muscle weakness, both locally and remote from the knee