1.1.8. In any vector space V , show that (a+ b)(x+y) = ax+ay+ bx+ by for any x, y ∈ V and any a, b ∈ F . Proof. Let x, y ∈ V and a, b ∈ F . Note that (a + b) ∈ F (it is a scalar). By (VS7 ), we have that (a + b)(x + y) = (a + b)x + (a + b)y. By (VS8 ), (a + b)x = ax + bx. Likewise, (a + b)y = ay + by. Thus, (a+ b)x+ (a+ b)y = ax+ bx+ ay + by. Finally, by (VS1 ), we have (a+ b)(x+ y) = ax+ ay + ...

The CADE ATP System Computer (CASC) evaluates the performance of sound, fully automatic, classical logic, ATP systems. The evaluation is in terms of the number of problems solved, the number of acceptable proofs and models produced, and the average runtime for problems solved, in the context of a bounded number of eligible problems chosen from the TPTP problem library, and a specified time limi...

Interleukin-22 (IL-22) is a pluripotent T cell-derived cytokine which is a member of IL-10 cytokine family. It is the only interleukin produced by immune cells but does not target immune system components. IL-22 is mainly produced by dendritic cells (DCs) and TH17, TH22, NK, and NKT cells and targets a number of body tissues including liver, pancreas, and other epithelial tissues. It provokes a...

Iron is a trace element important for the proper folding and function of various proteins. Physiological regulation of iron stores is of critical importance for RBC production and antimicrobial defense. Hepcidin is a key regulator of iron levels within the body. Under conditions of iron deficiency, hepcidin expression is reduced to promote increased iron uptake from the diet and release from ce...

Here, we report on a rare case of a giant invasive sacral schwannoma. The patient was a 58-year-old woman who had a 6-year history of non-specific buttock pain. Histological investigation confirmed the diagnosis of cellular schwannoma. The following numerical aberration was detected using the GTG-banding method for karyotypes: 47,XX,-14,+18,+22. Cytogenetic studies of schwannomas have indicated...

IL-22 is a cytokine recently discovered to be produced by TH17 cells. Its receptor is expressed in the skin, liver, the digestive and the respiratory tract where IL-22 conveys signals of the immune system directly to the tissue. IL-22 is involved in the host defense against extracellular pathogens as well as in autoimmune inflammatory diseases and exerts, depending on the tissue and inflammator...