Eigenvalue inequalities for positive block matrices with the inradius of the numerical range

We prove the operator norm inequality, for a positive matrix partitioned into four blocks in [Formula: see text], [Formula: see text] where [Formula: see text] is the diameter of the largest possible disc in the numerical range of [Formula: see text]. This shows that the inradius [Formula: see text] satisfies [Formula: see text] Several eigenvalue inequalities are derived. In particular, if [Formula: see text] is a normal matrix whose spectrum lies in a disc of radius [Formula: see text], the third eigenvalue of the full matrix is bounded by the second eigenvalue of the sum of the diagonal block, [Formula: see text] We think that [Formula: see text] is optimal and we propose a conjecture related to a norm inequality of Hayashi.

On the multiplicity of the second eigenvalue of the Laplacian in non simply connected domains – with some numerics –

We revisit an interesting example proposed by Maria Hoffmann-Ostenhof, the second author and Nikolai Nadirashvili of a bounded domain in R 2 for which the second eigenvalue of the Dirichlet Laplacian has multiplicity 3. We also analyze carefully the first eigenvalues of the Laplacian in the case of the disk with two symmetric cracks placed on a smaller concentric disk in function of their size.

Second eigenvalue of the CR Yamabe operator

Let [Formula: see text] be a compact, connected, strictly pseudo-convex CR manifold. In this paper, we give some properties of the CR Yamabe Operator [Formula: see text]. We present an upper bound for the Second CR Yamabe Invariant, when the First CR Yamabe Invariant is negative, and the existence of a minimizer for the Second CR Yamabe Invariant, under some conditions.

Generalized Group–Subgroup Pair Graphs

A regular finite graph is called a Ramanujan graph if its zeta function satisfies an analog of the Riemann Hypothesis. Such a graph has a small second eigenvalue so that it is used to construct cryptographic hash functions. Typically, explicit family of Ramanujan graphs are constructed by using Cayley graphs. In the paper, we introduce a generalization of Cayley graphs called generalized group–subgroup pair graphs, which are a generalization of group–subgroup pair graphs defined by Reyes-Bustos. We study basic properties, especially spectra of them.

Dimensionality assessment of binary response test items: A non-parametric approach of bayesian item response theory measurement

This study investigated dimensionality of Binary Response Items through a non-parametric technique of Item Response Theory measurement framework. The study used causal comparative research type of non-experimental design. The sample consisted of 5,076 public senior secondary school examinees (SSS3) between the age of 14-16 years from 45 schools, which were drawn randomly from three senatorial districts of Osun State, Nigeria. Instrument used for this study was 2018 Osun State unified multiple-choice mathematics achievement test items with empirical reliability coefficient of 0.82. Data obtained were analysed using Non-linear factor analysis, Stout’s Test of Essential Unidimensionality (STEU), Factor Analysis (FA), Full Information Factor Analysis (FIFA) and Bootstrap Modified Parallel Analysis Test (BMPAT). Results showed that both the BMPAT and STEU ascertained violation of unidimensionality assumption of the test items (the observed difference in the second eigenvalue of the observed data and that of second eigenvalue of the simulated data was statistically significant, p = 0.0099; Stout’s test rejected the assumption of essential unidimensionality, T = 10.6260, p<0.05). Non-linear factor analysis and full information factor analysis revealed that four dimensions embedded in the test items and loadings of the items showed within-item multidimensionality respectively. The authors’ concluded that modeling examinees’ performance with unidimensional model when it was actually multidimensional in nature would affect performance of examinees adversely and could lead to blur judgment. Consequently, it is recommended that unidimensional scoring method of Osun State unified mathematics achievement test implicit in Classical Test Theory should be jettisoned and an appropriate scoring model (multidimensional) should be embraced.

Summarizing Complex Graphical Models of Multiple Chronic Conditions Using the Second Eigenvalue of Graph Laplacian: Algorithm Development and Validation (Preprint)

BACKGROUND It is important but challenging to understand the interactions of multiple chronic conditions (MCC) and how they develop over time in patients and populations. Clinical data on MCC can now be represented using graphical models to study their interaction and identify the path toward the development of MCC. However, the current graphical models representing MCC are often complex and difficult to analyze. Therefore, it is necessary to develop improved methods for generating these models. OBJECTIVE This study aimed to summarize the complex graphical models of MCC interactions to improve comprehension and aid analysis. METHODS We examined the emergence of 5 chronic medical conditions (ie, traumatic brain injury [TBI], posttraumatic stress disorder [PTSD], depression [Depr], substance abuse [SuAb], and back pain [BaPa]) over 5 years among 257,633 veteran patients. We developed 3 algorithms that utilize the second eigenvalue of the graph Laplacian to summarize the complex graphical models of MCC by removing less significant edges. The first algorithm learns a sparse probabilistic graphical model of MCC interactions directly from the data. The second algorithm summarizes an existing probabilistic graphical model of MCC interactions when a supporting data set is available. The third algorithm, which is a variation of the second algorithm, summarizes the existing graphical model of MCC interactions with no supporting data. Finally, we examined the coappearance of the 100 most common terms in the literature of MCC to validate the performance of the proposed model. RESULTS The proposed summarization algorithms demonstrate considerable performance in extracting major connections among MCC without reducing the predictive accuracy of the resulting graphical models. For the model learned directly from the data, the area under the curve (AUC) performance for predicting TBI, PTSD, BaPa, SuAb, and Depr, respectively, during the next 4 years is as follows—year 2: 79.91%, 84.04%, 78.83%, 82.50%, and 81.47%; year 3: 76.23%, 80.61%, 73.51%, 79.84%, and 77.13%; year 4: 72.38%, 78.22%, 72.96%, 77.92%, and 72.65%; and year 5: 69.51%, 76.15%, 73.04%, 76.72%, and 69.99%, respectively. This demonstrates an overall 12.07% increase in the cumulative sum of AUC in comparison with the classic multilevel temporal Bayesian network. CONCLUSIONS Using graph summarization can improve the interpretability and the predictive power of the complex graphical models of MCC.