###### The sparing number of certain graph powers

## Abstract

Let ℕ_{0} be the set of all non-negative integers and 𝒫(ℕ_{0}) be its power set. Then, an integer additive set-indexer (IASI) of a given graph G is an injective function f : V(G) → P(ℕ_{0}) such that the induced function f^{+} : E(G) → 𝒫(ℕ_{0}) defined by f^{+}(uv) = f(u) + f(v) is also injective. An IASI f is said to be a weak IASI if |f^{+}(uv)| = max(|f(u)|, |f(v)|) for all u, v ∈ V(G). A graph which admits a weak IASI may be called a weak IASI graph. The set-indexing number of an element of a graph G, a vertex or an edge, is the cardinality of its set-labels. The sparing number of a graph G is the minimum number of edges with singleton set-labels, required for a graph G to admit a weak IASI. In this paper, we study the admissibility of weak IASI by certain graph powers and their corresponding sparing numbers.

###### Subgroups of 3-Factor Direct Products

## Abstract

Extending Goursat’s Lemma we investigate the structure of subdirect products of 3-factor direct products. We construct several examples and then provide a structure theorem showing that every such group is essentially obtained by a combination of the examples. The central observation in this structure theorem is that the dependencies among the group elements in the subdirect product that involve all three factors are of Abelian nature. In the spirit of Goursat’s Lemma, for two special cases, we derive correspondence theorems between data obtained from the subgroup lattices of the three factors (as well as isomorphisms between arising factor groups) and the subdirect products. Using our results we derive an explicit formula to count the number of subdirect products of the direct product of three symmetric groups.

###### Synovitis in Osteoarthritic Patients: Morphological and Virological Evidence of its Contribution to Development of the Disease

## Abstract

The role of inflammation in the development of osteoarthritic joint degeneration is not completely understood. Recent data suggest that processes that cause and orchestrate inflamed synovial lesions may be implicated in the development of the disease. The morphological changes of the synovium in patients with osteoarthritis (OA), as well as the level of synovial inflammation cautiously graded, in association to the presence of human parvovirus B19 (B19V) infection markers, were evaluated. Qualitative and quantitative detection of B19V genomic sequence was performed in OA and rheumatoid arthritis (RA) groups. The expression of CD68, S100 (Ca^{2+} binding proteins soluble in 100% ammonium sulfate) and B19 VP1/VP2 capsid proteins found in the synovium were investigated by single and double immunolabeling, whereas fine features of synoviocytes — by electron microscopy. One-third of OA and RA patients demonstrated synovial expression of B19V antigen, which was confirmed in both types of synoviocytes. The overall expression of B19V in OA patients was weaker than that found in RA subjects. Positive correlation between B19V-positive vascular endothelial cells, sublining infiltrating lymphocytes, macrophages, and B19V-positive synoviocytes was established. No correlation between synovitis score indices as well as the expression of S100 and expression of B19V was found. The results suggest that the synovial membrane maintains local joint homeostasis, and that virus mediated synovitis is implicated in the development of OA.

###### Tabu Cryptanalysis of VMPC Stream Cipher

## Abstract

In the era of global informatization, transmitting and storing information in digital form it is very important to ensure an adequate level of security of ciphers used. Cryptanalysis deals with studying the level of security, thus exposing the weakness of theoretical and implemented cryptographic solutions. In this paper cryptanalysis of stream cipher VMPC using Tabu Search is shown. From estimates made on a full version of VMPC cipher we concluded that about 2^{157} possibilities needs to be checked in order to find the proper one, which would be the best attack known so far.

###### Vaccination Against Influenza: Attitude of Health Care Personnel in a Multidisciplinary Hospital in Latvia

## Abstract

Health care personnel (HCP) are at high risk of acquiring influenza due to exposure to patients. However, vaccination in HCP is lower than 40% for most European countries. The aim of this study was to determine the attitude towards influenza vaccination and possible reasons for this attitude in HCP. A cross-sectional study was performed in a multidisciplinary hospital of Latvia. The sample (n = 1099) included doctors (239), nurses (418), care services (236), administrative staff (108), and technical support staff (98). Five questions addressed vaccination of planned patients and HCP, knowledge of etiological anti-influenza drugs, and their storage at the hospital for immediate use. The results revealed that the level of regular vaccination against influenza in HCP was relatively low (14%). This contrasted with a more positive attitude towards vaccination of patients (53%) and personnel (60%). This contrast provided evidence for a low level of proactive action. High expectations regarding medications covered by the hospital (82%) indicated transferring of part of personal responsibility to the organisation. Doctors demonstrated a better understanding of the problem and a higher level of vaccination. However, some of doctors’ attitudes showed underestimation of influenza-associated risk.

###### Zero forcing number of degree splitting graphs and complete degree splitting graphs

## Abstract

A subset ℤ ⊆ V(G) of initially colored black vertices of a graph G is known as a zero forcing set if we can alter the color of all vertices in G as black by iteratively applying the subsequent color change condition. At each step, any black colored vertex has exactly one white neighbor, then change the color of this white vertex as black. The zero forcing number ℤ (G), is the minimum number of vertices in a zero forcing set ℤ of G (see [11]). In this paper, we compute the zero forcing number of the degree splitting graph (𝒟𝒮-Graph) and the complete degree splitting graph (𝒞𝒟𝒮-Graph) of a graph. We prove that for any simple graph, ℤ [𝒟𝒮(G)] k + t, where ℤ (G) = k and t is the number of newly introduced vertices in 𝒟𝒮(G) to construct it.

###### Acta Marisiensis. Seria Technologica

###### Influence of Different Training Systems on the Beginning of Domestic Plum Production and Yield

## Abstract

All growers prefer to obtain a first harvest as soon as possible. The aim of this study was to identify an effective tree training system for new Latvian plum cultivars, which provides the highest yield and fruit quality. The trial was established in 2012 at the Institute of Horticulture (formerly Latvia State Institute of Fruit-Growing) and included cultivars ‘Ance’, ‘Adelyn’, and ‘Sonora’; and the controls ‘Victoria’ and ‘Jubileum’. Planting distances were 4 × 2.5 m, and the rootstock was *Prunus cerasifera*. Tree training was done using four systems: two systems with branch bending (Heka espalier and spindle); and two systems without bending (standard (round) and flat crown). Evaluation of average fruit weight and yield per trunk cross section area (TCSA) was done from 2015 to 2017. The Heka espalier and spindle systems had first yield in 2015, but systems without bending had first yield only in 2016. Average yield per TCSA was significantly higher for the Heka espalier system (0.49 kg·cm^{−2}) than for the spindle system (0.24 kg·cm^{−2}), standard crown system (0.17 kg·cm^{−2}), and flat crown system (0.30 kg·cm^{−2}). Cultivar ‘Victoria’ had the highest yield among all systems. Average fruit size showed the same results: the biggest fruits were obtained with the Heka espalier system, and the smallest for systems without bending in the standard crown.