Approximate Solutions of Polynomial Equations

AbstractIn this paper, we introduce “approximate solutions" to solve the following problem: given a polynomial F(x, y) over Q, where x represents an n -tuple of variables, can we find all the polynomials G(x) such that F(x, G(x)) is identically equal to a constant c in Q ? We have the following: let F(x, y) be a polynomial over Q and the degree of y in F(x, y) be n. Either there is a unique polynomial g(x) ∈Q [ x ], with its constant term equal to 0, such that F(x, y) =∑j=0ncj(y−g(x))jfor some rational numbers cj, hence, F(x, g(x) +a) ∈Q for all a∈Q, or there are at most t distinct polynomials g1(x),⋯ , gt(x), t≤n, such that F(x, gi(x)) ∈Q for 1 ≤i≤t. Suppose that F(x, y) is a polynomial of two variables. The polynomial g(x) for the first case, or g1(x),⋯ , gt(x) for the second case, are approximate solutions of F(x, y), respectively. There is also a polynomial time algorithm to find all of these approximate solutions. We then use Kronecker’s substitution to solve the case of F(x, y)

Promoter methylation of the hMLH1 gene and protein expression of human mutL homolog 1 and human mutS homolog 2 in resected esophageal squamous cell carcinoma

ObjectiveAberrant expression of mismatch repair genes, such as human mutL homolog 1 (hMLH1) and human mutS homolog 2 (hMSH2), are common in some human cancers, and promoter methylation is believed to inactivate expression of hMLH1. We investigated whether promoter methylation is involved in loss of hMLH1 protein and whether aberrant expression of hMLH1 and hMSH2 protein is related to prognosis after resection for esophageal squamous cell cancer.MethodsWe analyzed promoter methylation of hMLH1 using methylation-specific polymerase chain reaction and hMLH1 and hMSH2 protein by using immunohistochemistry in 60 resected tumor specimens. The Pearson χ2 test was used to compare expression of hMLH1 and hMSH2 protein among patients with different clinicopathologic parameters. Concordance analysis was performed between hMLH1 methylation and its protein expression.ResultsLoss of hMLH1 and hMSH2 protein was found in 43 (72%) and 39 (65%, P = .06) of 60 resected specimens, respectively. hMLH1 protein correlated well with tumor staging (P < .0001), depth of tumor invasion (P = .008), and nodal involvement (P < .0001) but not with distant metastasis, whereas hMSH2 did not show correlation with any of these parameters. A concordance rate of 83.3% was present between expression of hMLH1 protein and its promoter methylation (P < .001).ConclusionsAberrant expression of hMLH1 and hMSH2 protein is frequently associated with the presence of esophageal squamous cell carcinoma, and expression of hMLH1 protein is a better prognostic predictor than is expression of hMSH2 protein. Promoter methylation is one of the mechanisms responsible for loss of hMLH1 protein in esophageal squamous cell cancer

Granulocyte Colony-Stimulating Factor Activating HIF-1α Acts Synergistically with Erythropoietin to Promote Tissue Plasticity

Stroke and peripheral limb ischemia are serious clinical problems with poor prognosis and limited treatment. The cytokines erythropoietin (EPO) and granulocyte-colony stimulating factor (G-CSF) have been used to induce endogenous cell repair and angiogenesis. Here, we demonstrated that the combination therapy of EPO and G-CSF exerted synergistic effects on cell survival and functional recovery from cerebral and peripheral limbs ischemia. We observed that even under normoxic conditions, G-CSF activates hypoxia-inducible factor-1α (HIF-1α), which then binds to the EPO promoter and enhances EPO expression. Serum EPO level was significantly increased by G-CSF injection, with the exception of Tg-HIF-1α+f/+f mice. The neuroplastic mechanisms exerted by EPO combined with G-CSF included enhanced expression of the antiapoptotic protein of Bcl-2, augmented neurotrophic factors synthesis, and promoted neovascularization. Further, the combination therapy significantly increased homing and differentiation of bone marrow stem cells (BMSCs) and intrinsic neural progenitor cells (INPCs) into the ischemic area. In summary, EPO in combination with G-CSF synergistically enhanced angiogenesis and tissue plasticity in ischemic animal models, leading to greater functional recovery than either agent alone

Honokiol Eliminates Human Oral Cancer Stem-Like Cells Accompanied with Suppression of Wnt/ β

Honokiol, an active compound of Magnolia officinalis, exerted many anticancer effects on various types of cancer cells. We explored its effects on the elimination of cancer stem-like side population (SP) cells in human oral squamous cell carcinoma SAS cells. The sorted SP cells possessed much higher expression of stemness genes, such as ABCG2, ABCC5, EpCAM, OCT-4, CD133, CD44, and β-catenin, and more clonogenicity as compared with the Non-SP cells. After 48 h of treatment, honokiol dose dependently reduced the proportion of SP from 2.53% to 0.09%. Apoptosis of honokiol-treated SP cells was evidenced by increased annexin V staining and cleaved caspase-3 as well as decreased Survivin and Bcl-2. Mechanistically, honokiol inhibited the CD44 and Wnt/β-catenin signaling of SP cells. The Wnt signaling transducers such as β-catenin and TCF-4 were decreased in honokiol-treated SP cells, while the β-catenin degradation promoting kinase GSK-3α/β was increased. Consistently, the protein levels of β-catenin downstream targets such as c-Myc and Cyclin D1 were also downregulated. Furthermore, the β-catenin-related EMT markers such as Slug and Snail were markedly suppressed by honokiol. Our findings indicate honokiol may be able to eliminate oral cancer stem cells through apoptosis induction, suppression of Wnt/β-catenin signaling, and inhibition of EMT

Polynomial time algorithms for sentences over number fields

AbstractWe call ϕ a ∀∃ sentence if and only if ρ is logically equivalent to a sentence of the form ∀x∃yψ(x, y), where ψ(x, y) is a quantifier free formula constructed with logical and arithmetical symbols. Now let ϕ be a ∀∃ sentence in conjunctive or disjunctive normal form. We show that given an arbitrary algebraic number field K there is a polynomial time algorithm to decide whether ϕ is true in K or not. We also show that ther are polynomial time algorithms to decide whether or not ϕ is true in every algebraic number field or every radical extension field of Q