Localization in dimension theory

AbstractSullivan (1970, 1974) pointed out the availability and applicability of localization methods in homotopy theory. We shall apply the method to dimension theory and analyze covering dimension and cohomological dimension from the viewpoint. The notion of localized dimension with respect to prime numbers shall be introduced as follows: the P-localized dimension of a space X is at most n (denoted by dimp X ⩽ n) provided that every map f : A → Snp of a closed subset A of X into a P-localized n-dimensional sphere Snp admits a continuous extension over X.The main results are: 1.(1) Let P1 ⊆ P2 ⊆ P. Then dimp1, X ⩽ dimp2 X (Theorem 1.1).2.(2) Let X be a compactum. Then the following conditions are equivalent: 2.1.(a) dimX < ∞;2.2.(b) for some partition P1,…, Ps of P, max{dimpi X: i = 1,…, s} < ∞;2.3.(c) for any partition P1,…, Ps of P, max{dimpi; X: i = 1,…, s} < ∞ (Theorem 1.2).3.(3) Let X be a compactum, G an Abelian group. We have that sup{c-dimGp X: p ϵ P} = c-dimGX (Theorem 1.4)

The construction of P-expansive maps of regular continua: A geometric approach

AbstractIn this paper, we prove the following: Let G be a graph, f:G→G a continuous map and P a finite subset of G such that f(P)⊂P. Then there exist a regular continuum Z, a continuous map g:Z→Z and a semi-conjugacy π:G→Z such that(1) g is π(P)-expansive, and(2) if p,q∈P and Q is a subset of P with A∩Q≠∅ for any arc A in G between p and q, then A′∩π(Q)≠∅ for any arc A′ in Z between π(p) and π(q).In addition, f is point-wise P-expansive if and only if π|P is one-to-one.In this paper we are especially interested in the geometrical structure of Z. Actually we can see the complicated construction of Z

Monoclonal gammopathy of renal significance (MGRS)-related AL amyloidosis complicated by amyloid myopathy: a case report

BACKGROUND: Lately, monoclonal gammopathy of renal significance (MGRS) has been defined as a group of renal disorders that are strongly associated with monoclonal protein, including amyloid immunoglobulin light chain (AL) amyloidosis. Amyloid myopathy is rare (1.5% of all patients with amyloidosis) and the prognosis is poor. Furthermore, only approximately 20% of patients with amyloid myopathy are reported to have renal involvement, indicating a lack of data in the literature. CASE PRESENTATION: Here, we report a rare case of MGRS-related AL amyloidosis complicated by amyloid myopathy that presented with muscle weakness in the upper and lower limbs, neck and fingers, and nephrotic syndrome. Blood, urine, and bone marrow examination revealed monoclonal gammopathy of undetermined significance (MGUS) (Bence Jones protein-lambda). Muscle biopsy of the vastus lateralis muscle demonstrated amyloid proteins in the sarcolemma and in the blood vessel walls on Congo red staining, suggesting amyloid myopathy, and tiny inclusions in fibers on modified Gomori trichrome stain. Although we thought they were reminiscent of nemaline bodies, we could not confirm the nature of this structure. Renal biopsy demonstrated amyloid proteins in the mesangial region, part of the capillary walls, and the blood vessel walls on direct fast scarlet staining. As these amyloid proteins were positive for p-component staining and negative for amyloid A staining, β2-microglobulin, and pre-albumin, and as lambda light chains were positive in the mesangial region, we diagnosed the patient with MGRS-related AL amyloidosis. Although he was treated with melphalan and dexamethasone, his symptoms did not improve. CONCLUSIONS: AL amyloidosis involving the kidneys and muscles has a poor prognosis, and a delayed diagnosis of amyloid myopathy is common because of its rarity and frequent misdiagnosis, which increases organ function deterioration. Therefore, early detection, therapeutic intervention, and careful follow-up are crucial