Spectrum of Linear Difference Operators and the Solvability of Nonlinear Discrete Problems

Let T∈ℕ with T>5. Let 𝕋:={1,…,T}. We study the Fučik spectrum Σ of the discrete problem Δ2u(t-1)+λu+(t)-μu-(t)=0, t∈𝕋, u(0)=u(T+1)=0, where u+(t)=max⁡{u(t),0}, u-(t)=max⁡{-u(t),0}. We give an expression of Σ via the matching-extension method. We also use such discrete spectrum theory to study nonlinear boundary value problems of difference equations at resonance

Dynamic joint sentiment-topic model

Social media data are produced continuously by a large and uncontrolled number of users. The dynamic nature of such data requires the sentiment and topic analysis model to be also dynamically updated, capturing the most recent language use of sentiments and topics in text. We propose a dynamic joint sentiment-topic model (dJST) which allows the detection and tracking of views of current and recurrent interests and shifts in topic and sentiment. Both topic and sentiment dynamics are captured by assuming that the current sentiment-topic specific word distributions are generated according to the word distributions at previous epochs. We study three different ways of accounting for such dependency information, (1) Sliding window where the current sentiment-topic-word distributions are dependent on the previous sentiment-topic specific word distributions in the last S epochs; (2) Skip model where history sentiment-topic-word distributions are considered by skipping some epochs in between; and (3) Multiscale model where previous long- and shorttimescale distributions are taken into consideration. We derive efficient online inference procedures to sequentially update the model with newly arrived data and show the effectiveness of our proposed model on the Mozilla add-on reviews crawled between 2007 and 2011

Spectrum of Discrete Second-Order Difference Operator with Sign-Changing Weight and Its Applications

Let > 1 be an integer, and letT = {1, 2, . . . , }. We discuss the spectrum of discrete linear second-order eigenvalue problems 0) = ( + 1) = 0, where ̸ = 0 is a parameter, : T → R changes sign and ( ) ̸ = 0 on T. At last, as an application of this spectrum result, we show the existence of sign-changing solutions of discrete nonlinear second-order problems by using bifurcate technique

Spectrum of Discrete Second-Order Neumann Boundary Value Problems with Sign-Changing Weight

We study the spectrum structure of discrete second-order Neumann boundary value problems (NBVPs) with sign-changing weight. We apply the properties of characteristic determinant of the NBVPs to show that the spectrum consists of real and simple eigenvalues; the number of positive eigenvalues is equal to the number of positive elements in the weight function, and the number of negative eigenvalues is equal to the number of negative elements in the weight function. We also show that the eigenfunction corresponding to the th positive/negative eigenvalue changes its sign exactly times

Existence of Positive Solutions of a Discrete Elastic Beam Equation

Let T be an integer with T≥5 and let T2={2,3,…,T}. We consider the existence of positive solutions of the nonlinear boundary value problems of fourth-order difference equations Δ4u(t−2)−ra(t)f(u(t))=0, t∈T2, u(1)=u(T+1)=Δ2u(0)=Δ2u(T)=0, where r is a constant, a:T2→(0,∞),  and  f:[0,∞)→[0,∞) is continuous. Our approaches are based on the Krein-Rutman theorem and the global bifurcation theorem

Existence of Solutions of a Discrete Fourth-Order Boundary Value Problem

Let a,b be two integers with b-a≥5 and let 𝕋2={a+2,a+3,…,b-2}. We show the existence of solutions for nonlinear fourth-order discrete boundary value problem Δ4u(t-2)=f(t,u(t), Δ2u(t-1)), t∈𝕋2, u(a+1)=u(b-1)=Δ2u(a)=Δ2u(b-2)=0 under a nonresonance condition involving two-parameter linear eigenvalue problem. We also study the existence and multiplicity of solutions of nonlinear perturbation of a resonant linear problem

Serum peptidome profiling for the diagnosis of colorectal cancer: Discovery and validation in two independent cohorts

Colorectal cancer (CRC) is one of the most common malignant neoplasms worldwide. Except for the existing fecal occult blood test, colonoscopy and sigmoidoscopy, no widely accepted in vitro diagnostic methods have been available. To identify potential peptide biomarkers for CRC, serum samples from a discovery cohort (100 CRC patients and 100 healthy controls) and an independent validation cohort (91 CRC patients and 91 healthy controls) were collected. Peptides were fractionated by weak cation exchange magnetic beads (MB-WCX) and analysed by matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDITOF MS). Five peptides (peaks at m/z 1895.3, 2020.9, 2080.7, 2656.8 and 3238.5) were identified as candidate biomarkers for CRC. A diagnostic panel based on the five peptides can discriminate CRC patients from healthy controls, with an accuracy of 91.8%, sensitivity of 95.6%, and specificity of 87.9% in the validation cohort. Peptide peaks at m/z 1895.3, 2020.9 and 3238.5 were identified as the partial sequences of complement component 4 (C4), complement component 3 (C3) and fibrinogen a chain (FGA), respectively. This study potentiated peptidomic analysis as a promising in vitro diagnostic tool for diagnosis of CRC. The identified peptides suggest the involvement of the C3, C4 and FGA in CRC pathogenesis