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The Tree Packing Conjecture (TPC) by Gyárfás states that any set of trees \(T_2,\dots,T_{n-1}, T_n\) such that \(T_i\) has \(i\) vertices pack into \(K_n\). The conjecture is true for bounded degree trees, but in general, it is widely open. Bollobás proposed a weakening of TPC which states that \(k\) largest trees pack. The latter is true if none tree is a star, but in general, it is known only for \(k=5\). In this paper we prove, among other results, that seven largest trees pack.

Objectives: The purpose of the research was to perform a stakeholder analysis for the Research and Development project of Mediprintic sp. z o.o. in order to confirm correctly identified parties for cooperation within the 1.1.2 PARP Start-up Platforms project. Methods: For the assessment of the selected stakeholders, the following tools were used: a tool to identify stakeholders and their interests, a tool to identify the importance and impact of stakeholders, a risk analysis for the undertaken activities of the project implementer with stakeholders together with the establishment of the necessary indicators to assess the achievement of the milestone. Results: From the research carried out, 14 key stakeholders for the success of the project were selected which are important for Mediprintic sp. z o.o. to be able to launch its product on the medical market. Conclussions: Completing a stakeholder analysis facilitates the decision-making process for management in order to prioritise the actions to be taken, taking into account the risks of cooperation with individual business partners.

Versatile devices with tunable capabilities for controlling lasing wavelength and intensity are in high demand. Liquid crystals (LCs) exhibit immense potential for such applications, offering fine-tuning possibilities through external factors. On the other hand, laser technology is currently a research hotspot in optoelectronics, and also in practical applications. The recent market introduction of laser television marks a significant stride toward making such advanced technology accessible in every household. In synergy with laser pumping, the LCs unquestionably can be rediscovered. This study presents a comprehensive investigation into the crystallization phenomenon within a host-guest device, compact in size, free from moving parts, and integrating the liquid crystalline (LC) matrix doped with 3,4,9,10-tetra-(3-alcoxy-carbonyl)-perylene (THCP) dye. The focus lies in examining the influence of varying dye concentrations on multicolor fluorescence, lasing behavior, and device morphology. The systematic analysis of Random Lasing (RL) energy thresholds and the impact of DC voltage on light intensity modulation is demonstrated. Morphological changes were monitored in real-time using optical microscopy techniques, including crossed polarizer, and fluorescence imaging under 450 nm excitation. Utilizing advanced Transmission Electron Microscopy (TEM) techniques, we explore exceptional insights into our set of devices, providing novel information about the THCP crystallization process for the first time in the literature. To gain a comprehensive understanding of the crystal forming and molecular geometry we examined additionally the THCP dye, using X-ray diffraction and Raman spectroscopy. Furthermore, we showcase that varying the pumping energy enables multicolor tuning in the fabricated systems, presenting an attractive feature in the context of laser display technologies.

Underwater wireless communication remains a challenging topic, particularly for applications such as wreck penetration where multipath and Doppler effects are very intense. These effects are becoming even more difficult to mitigate for fast data transmission systems that utilize wideband signals. Due to the low propagation speed of acoustic wave in the water, there is a significant difference between the Doppler shift for lower and upper frequencies of the utilized spectrum. To address these challenges, this paper describes various methods for determining the Doppler frequency shift for MFSK signals, including cross-correlation, double FFT, pilots, and additional Up-Down chirp signals. The reception quality of the transmitted data in a real environment was used as an evaluation criterion for each method. The tests were carried out in motion within the towing tank for different movement speeds of the transmitter relative to the receiver. The tank’s limited dimensions created conditions for multipath signal propagation. Under very difficult multipath signal propagation conditions, the pilots method was found to be the most effective. It gave over two times lower BER than the well-known Up-Down chirp method.

This research paper aims to explore the mechanical characteristics of polyamide PA12 (PA12) as a 3D material printed utilizing Selective Laser Sintering (SLS) and HP MultiJet Fusion (HP MJF) technologies in order to design and manufacture forearm orthoses. The study assessed the flowability of the materials used and compared the mechanical performance of PA12 with each other using tensile, flexure, and impact tests in five different fabrication orientations: X, Y, Z, tilted 45° XZ, and tilted 45° YZ. The results of the study provide, firstly—the data for testing the quality of the applied polyamide powder blend and, secondly—the data for the design of the orthosis geometry from the aspect of its strength parameters and the safety of construction. The mechanical parameters of SLS specimens had less variation than MJF specimens in a given orientation. The difference in tensile strength between the 3D printing technologies tested was 1.8%, and flexural strength was 4.7%. A process analysis of the forearm orthoses revealed that the HP MJF 5200 system had a higher weekly production capacity than the EOS P396 in a production variance based on obtaining maximum strength parameters and a variance based on maximizing economic efficiency. The results suggest that medical device manufacturers can use additive manufacturing technologies to produce prototypes and small-batch parts for medical applications. This paper pioneers using 3D printing technology with Powder Bed Fusion (PBF) methods in designing and manufacturing forearm orthoses as a low- to medium-volume product. The applied solution addresses the problem of medical device manufacturers with regard to the analysis of production costs and mechanical properties when using 3D printing for certified medical devices.