a. N1 = {(i1, i2, . . . , in) : ik ∈ Ik for all k ∈ {1, 2, . . . , n}} and b. N2 = {(x1, x2, . . . , xn) : ∑n k=1 xk = 0}. Proof. To prove (a), it suffices to show, by Proposition 1, that N1 is nonempty and x + ry ∈ N1 for all r ∈ R and all x, y ∈ N1. For the first condition, (0, 0, . . . , 0) ∈ N1 since Ik is a subgroup of R containing the additive identity 0 for all k ∈ {1, 2, . . . , n}. Tha...

The location and motion of sounds in space are important cues for encoding the auditory world. Spatial processing is a core component of auditory scene analysis, a cognitively demanding function that is vulnerable in Alzheimer’s disease. Here we designed a novel neuropsychological battery based on a virtual space paradigm to assess auditory spatial processing in patient cohorts with clinically ...

A 30-year-old male patient, RVD-reactive on treatment, presented with a fast-growing, painful swelling involving the mandible of unknown duration. panoramic radiograph (PR) and cone-beam computed tomography (CBCT) imaging were performed. What are pertinent radiological features your diagnostic hypothesis?

We previously reported that v-fos transfer to a src-transformed rat 3Y1 cell line enhanced lung metastasis. To clarify the mechanism of this enhancement, we compared various biological factors related to metastatic potential between a fos-transferred highly metastatic cell line (fos-SR-3Y1-202) and the control cell line transferred with genetic marker (pSV2-neo) plus pBR322, neo-SR-3Y1-200. Lun...

Insects found associated with corpse can be used as one of the indicators in estimating postmortem interval (PMI). The objective of this study was to compare the stages of decomposition and faunal succession between a partially burnt pig (Sus scrofa Linnaeus) and natural pig (as control). The burning simulated a real crime whereby the victim was burnt by murderer. Two young pigs weighed approxi...

Exercise 10.3.2. Let R be a commutative ring with identity. For all positive integers n and m, R ∼= R if and only if n = m. Proof. Let φ : R → R be an isomorphism of R-modules and let I E R be a maximal ideal. Then the map φ̄ : R → R/IR given by φ̄(α) = φ(α) is a morphism of R-modules. Moreover ker φ̄ = {α ∈ R | φ̄(α) = 0} = {α ∈ R | φ(α) ∈ IR} = φ−1(IRm) = IR. Therefore by the first isomorphism th...