This paper presents an empirical verification of the effectiveness and usefulness of investment diversification using the main stock exchange indices and Bitcoin. The objective is to determine the effects applying the Markowitz portfolio optimization theory, i.e., the advantages of applying the modern portfolio theory for institutional investors. The research offers an answer to the following question: what are the advantages and disadvantages of using Bitcoin in portfolio optimization? The paper contributes to the representation of the reach and limitations of the modern portfolio theory for institutional investors. The conclusion is that rational behaviour of institutional investors requires consideration of portfolio optimization using the Markowitz model, because it is possible to create portfolios which, on the basis of historical returns, provide desired returns alongside certain risks. The methodology includes the analysis of high frequency data, i.e., daily trading data were used. The results indicate that the use of the Markowitz portfolio selection method, with all its limitations, is desirable, possible and applicable, but that it entails serious flaws in the sense of neglecting transaction costs, foreign exchange differences and the real value in the stock market. The results of the research show that Bitcoin is a good source of diversification in a portfolio that contains traditional financial instruments both for the risk-averse investor as well as for those investors who have a greater appetite for risk. The conclusion is that rational behavior of institutional investors requires consideration of investing in Bitcoin using the Markowitz model. However, given the high degree of volatility, investors should be very careful when making decisions about including Bitcoin in the portfolio.

Traditional drug discovery strategies are usually focused on occupancy of binding sites that directly affect functions of proteins. Hence, proteins that lack such binding sites are generally considered pharmacologically intractable. Modulators of protein activity, especially inhibitors, must be applied in appropriate dosage regimens that often lead to high systemic drug exposures in order to maintain sufficient protein inhibition in vivo. Consequently, there is a risk of undesirable off-target drug binding and side effects. Recently, PROteolysis TArgeting Chimera (PROTAC) technology has emerged as a new pharmacological modality that exploits PROTAC molecules for induced protein degradation. PROTAC molecule is a heterobifunctional structure consisting of a ligand that binds a protein of interest (POI), a ligand for recruiting an E3 ubiquitin ligase (an enzyme involved in the POI ubiquitination) and a linker that connects these two. After POI-PROTAC-E3 ubiquitin ligase ternary complex formation, the POI undergoes ubiquitination (an enzymatic post-translational modification in which ubiquitin is attached to the POI) and degradation. By merging the principles of photopharmacology and PROTAC technology, photocontrollable PROTACs for spatiotemporal control of induced protein degradation have recently emerged. The main advantage of photocontrollable over conventional PROTACs is the possible prevention of off-target toxicity thanks to local photoactivation.

Abstract This paper is concerned with a recently introduced graph invariant, namely the Sombor index. Some bounds on the Sombor index are derived, and then utilized to establish additional bounds by making use of the existing results. One of the direct consequences of one of the obtained bounds is that the cycle graph Cn attains the minimum Sombor index among all connected unicyclic graphs of a fixed order n ≥ 4. Graphs having the maximum Sombor index are also characterized from the classes of all connected unicyclic, bicyclic, tricyclic, tetracyclic, and pentacyclic graphs of a fixed order, and a conjecture concerning the maximum Sombor index of graphs of higher cyclicity is stated. A structural result is derived for graphs with integer values of Sombor index. Several possible directions for future work are also indicated.

It is well known that the Golden Section plays an important role in the geometry of several polygons and polyhedra; the best known example is the length of a diagonal in the regular pentagon with unit side. In this contribution we show how the Golden Section appears as the solution of an enumerative problem connected with heptagons, more precisely, with heptagonal tilings of the hyperbolic plane. The results are then generalized by investigating whether it also appears in other types of hyperbolic tilings.

The Sombor index is a recently introduced graph-theoretical invariant of the bond-additive type. It is known that it takes integer values for bipartite semi-regular graphs whose degrees appear as two smaller elements in a Pythagorean triple. In this note, we show that it can have integer values also for graphs with more complicated structure and construct infinite families of graphs with integer Sombor indices.

Monitoring programs should generate high-quality data on the concentrations of substances and other pollutants in the aquatic environment to enable reliable risk assessment. Furthermore, the need for comparability over space and time is critical for analysis of trends and evaluation of restoration of natural environment