Computing simplicial representatives of homotopy group elements

A central problem of algebraic topology is to understand the homotopy groups $\pi_d(X)$ of a topological space $X$. For the computational version of the problem, it is well known that there is no algorithm to decide whether the fundamental group $\pi_1(X)$ of a given finite simplicial complex $X$ is trivial. On the other hand, there are several algorithms that, given a finite simplicial complex $X$ that is simply connected (i.e., with $\pi_1(X)$ trivial), compute the higher homotopy group $\pi_d(X)$ for any given $d\geq 2$. %The first such algorithm was given by Brown, and more recently, \v{C}adek et al. However, these algorithms come with a caveat: They compute the isomorphism type of $\pi_d(X)$, $d\geq 2$ as an \emph{abstract} finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of $\pi_d(X)$. Converting elements of this abstract group into explicit geometric maps from the $d$-dimensional sphere $S^d$ to $X$ has been one of the main unsolved problems in the emerging field of computational homotopy theory. Here we present an algorithm that, given a~simply connected space $X$, computes $\pi_d(X)$ and represents its elements as simplicial maps from a suitable triangulation of the $d$-sphere $S^d$ to $X$. For fixed $d$, the algorithm runs in time exponential in $size(X)$, the number of simplices of $X$. Moreover, we prove that this is optimal: For every fixed $d\geq 2$, we construct a family of simply connected spaces $X$ such that for any simplicial map representing a generator of $\pi_d(X)$, the size of the triangulation of $S^d$ on which the map is defined, is exponential in $size(X)$

Innate Lymphoid Cells in Skin Homeostasis and Malignancy

Innate lymphoid cells (ILCs) are mostly tissue resident lymphocytes that are preferentially enriched in barrier tissues such as the skin. Although they lack the expression of somatically rearranged antigen receptors present on T and B cells, ILCs partake in multiple immune pathways by regulating tissue inflammation and potentiating adaptive immunity. Emerging evidence indicates that ILCs play a critical role in the control of melanoma, a type of skin malignancy thought to trigger immunity mediated mainly by adaptive immune responses. Here, we compile our current understanding of ILCs with regard to their role as the first line of defence against melanoma development and progression. We also discuss areas that merit further investigation. We envisage that the possibility to harness therapeutic potential of ILCs might benefit patients suffering from skin malignancies such as melanoma.publishedVersio

Innate lymphoid cells and tumor-derived lactic acid: novel contenders in an enduring game

Aerobic glycolysis, also known as the Warburg effect, has for a prolonged period of time been perceived as a defining feature of tumor metabolism. The redirection of glucose utilization towards increased production of lactate by cancer cells enables their rapid proliferation, unceasing growth, and longevity. At the same time, it serves as a significant contributor to acidification of the tumor microenvironment, which, in turn, imposes substantial constraints on infiltrating immune cells. Here, we delve into the influence of tumor-derived lactic acid on innate lymphoid cells (ILCs) and discuss potential therapeutic approaches. Given the abundance of ILCs in barrier tissues such as the skin, we provide insights aimed at translating this knowledge into therapies that may specifically target skin cancer

ILC2s and Adipose Tissue Homeostasis: Progress to Date and the Road Ahead

Group 2 innate lymphoid cells (ILC2s) were initially identified as a new type of lymphocytes that produce vigorous amounts of type 2 cytokines in adipose tissue. Subsequent studies revealed that ILC2s are present not only in adipose tissue but also in various other tissues such as lung and skin. ILC2s are generally recognized as tissue-resident immune cells that regulate tissue homeostasis. ILC2s express receptors for various humoral factors and thus can change their functions or distribution depending on the environment and circumstances. In this review, we will outline our recent understanding of ILC2 biology and discuss future directions for ILC2 research, particularly in adipose tissue and metabolic homeostasis.publishedVersio

Electroreduction of Benzalacetone in N,N -dimethylformamide

The electrochemical reduction of benzalacetone (BA) has been studied at mercury cathodes in N,N-dimethylformamide solutions by the techniques of cyclic voltammetry and controlled potential coulometry. The radical anion (BA\u27-) forms ion pairs with tetraethylammonium cations and then disappears in oligomerization processes. Atentative structureof the dimer BA- BA"-... (C2H5)4N+is suggested on the basis of quantum mechanical callations

Udział wojska litewskiego w wojnie polsko-tureckiej w latach 1684-1699

The article constitutes an attempt to describe the participation of Lithuanian forces in the warfare with Turks and Tatars from 1684 to 1699, i.e. from the formation of The Holy League until the Karłowice Treaty. Author describes and analyzes In particular, the author describes the campaigns in which Lithuanian forces took part in Podole, Ukraine, and Northern Moldavia and then analyzes them in a broader political and social context of Polish - Lithuanian State at the end of 17th century

Brothers Berens. A contribution to the history of Polish Lutheranism in the 17th century

The Lutheran family of Berens (Behrens) came from Denmark, probably had peasant or bourgeois roots, while in the 16th century its members lived in the area of Ducal Prussia. In the mid-17th century, four brothers were identified – Jan, Jakub, Jerzy, also Paweł, who served in the army of the Polish-Lithuanian Commonwealth and who, thanks to their own abilities and royal protection, reached high military levels