Retraction map categories and their applications to the construction of lambda calculus models

This paper deals with categorical models of the λ-calculus. We generalize the inverse limit method Scott used for his construction of D∞, and introduce order-enriched ccc's, retraction map categories and ɛ-categories. An order-enriched ccc is a cartesian closed category C equipped with a partial order relation ⩽ on the set of the arrows. A retraction map category of C is R=(R, ⩽, i, j), where ⩽ is a partial order relation on the set |C| of all the objects of C, R is the category of the poset (|C|, ⩽), and i and j are functors from R to C and from Rop to C that satisfy the conditions: (1) j a, b ∘ i a, b ⩾ ida and (2) i a, b ∘ j a, b ⩽ idb for every arrow a, b: a → b in R (i.e., a⩽b). The ɛ-category E=E(C, R) of C w.r.t. R is the category whose objects are ideals of (|C|, ⩽) and whose arrows are ideals of (C, ⊑), where ⩽ is the partial order relation in R and ⊑ is the partial order relation defined by f ⊑ g iff dom(f)⩽dom(g), cod(f)⩽cod(g) in R and f⩽j a, b ∘ g ∘ i(a, b in C. We show that every ɛ-category E=E(C, R) is also an order-enriched ccc. Moreover when E and R satisfy a particular condition, E(C, R) has a reflexive object. For example, if there is an ideal U of (|C|, ⩽) satisfying the following conditions, then U is isomorphic to UU in E and a λ-algebra is constructed from E and U: (1) for every pair of a, b ∈ U, U contains ba, and (2) for every c ∈ U, there are a, b ∈ U such that c ∈ ba. We reconstruct Pω and D∞ using ɛ-categories

Methyl halides in surface seawater and marine boundary layer of the northwest Pacific

The partial pressures of methyl halides (CH3X; X = Cl, Br, or I) and of CHClF2 (HCFC‐22), which are all volatile organic compounds (VOCs), were measured in the air of the marine boundary layer (pVOCair) and in surface seawater (pVOCwater) during a cruise from the subarctic to subtropical regions of the northwest Pacific in summer of 2008. In the northern transition water (TWN) with high biological activity, high levels of the three CH3Xs in surface seawater were frequently observed, probably owing to their enhanced production by phytoplankton. Supersaturation of CH3Br was only present in TWN water, with a saturation anomaly (SCH3Br) of 0.95 [SCH3X = (pCH3Xwater − pCH3Xair)/pCH3Xair]. The highest saturation anomalies for CH3Cl (SCH3Cl = 1.6) and CH3I (SCH3I = 91) were found in the southern subtropical water (STS) with low biological production south of the subtropical front. We found that the molar concentrations of CH3Cl (CCH3Cl) and CH3I (CCH3I) sharply increased with increasing sea surface temperature (SST) in the subtropical waters. The maximum CCH3Cl (144 pmol l−1) was present in STS water at SST = 30°C and is 1.5 times the value extrapolated from the previously reported relationship between CCH3Cl and SST. Photochemical production might have contributed to the production of CH3Cl and CH3I in STS water

Early complete remission of osteoid osteoma with conservative medical management

Osteoid osteoma is a benign bone tumor and causes persistent pain that is usually treated by surgery or ablation therapy. Conservative management with non-steroidal anti-inflammatory drugs (NSAIDs) is also used to avoid the morbidity associated with surgery or ablation therapy; however, it usually takes several years for the condition to resolve using conservative treatment. Our patient, a 10-year-old boy, presented with a 3-month history of a painful lesion in his leg. Plain radiography, bone scanning, computed tomography and magnetic resonance images showed the presence of a lesion with radiological features consistent with an osteoid osteoma of the cortex in the tibial diaphysis. The patient was treated with a usual dose of ibuprofen for 3 weeks. Within 3 weeks, his symptoms were almost completely resolved; he no longer needed NSAIDs and returned to normal life. Repeat imaging studies showed complete disappearance of the nidus within 2.5 years after the resolution of symptoms

Doppler ultrasound for diagnosis of soft tissue sarcoma: efficacy of ultrasound-based screening score

Background. The utility of ultrasound imaging in the screening of soft-part tumours (SPTs) has been reported. We classified SPTs according to their blood flow pattern on Doppler ultrasound and re-evaluated the efficacy of this imaging modality as a screening method. Additionally, we combined Doppler ultrasound with several values to improve the diagnostic efficacy and to establish a new diagnostic tool