<p> This thesis deals with combinatorial properties of hyperplane arrangements. In particular, we address a conjecture of Deza, Terlaky and Zinchenko stating that the largest possible average diameter of a bounded cell of a simple hyperplane arrangement is not greater than the dimension. We prove that this conjecture is asymptotically tight in fixed dimension by constructing a family of hyperplane arrangements containing mostly cubical cells. The relationship with a result of Dedieu, Malajovich and Shub, the conjecture of Hirsch, and a result of Haimovich are presented.</p> <p> We give the exact value of the largest possible average diameter for all simple arrangements in dimension two, for arrangements having at most the dimension plus two hyperplanes, and for arrangements having six hyperplanes in dimension three. In dimension three, we strengthen the lower and upper bounds for the largest possible average diameter of a bounded cell of a simple hyperplane arrangements.</p> <p> Namely, let ΔA(n, d) denote the largest possible average diameter of a bounded cell of a simple arrangement defined by n hyperplanes in dimension d. We show that • ΔA(n, 2) = 2[n/2] / (n-1)(n-2) for n ≥ 3, • ΔA(d + 2, d) = 2d/d+1, • ΔA(6, 3) = 2, • 3 - 6/n-1 + 6([n/2]-2) / (n-1)(n-2)(n-3) ≤ ΔA(n, 3) ≤ 3 + 4(2n^2-16n+21) / 3(n-1)(n-2)(n-3) • ΔA (n, d) ≥ 1 + (d-1)(n-d d)+(n-d)(n-d-1) for n ≥ 2d. We also address another conjecture of Deza, Terlaky and Zinchenko stating that the minimum number Φ0A~(n, d) of facets belonging to exactly one bounded cell of a simple arrangement defined by n hyperplanes in dimension d is at least d (n-2 d-1). We show that • Φ0A(n, 2) = 2(n - 1) for n ≥ 4, • Φ0A~(n, 3) ≥ n(n-2)/3 +2 for n ≥ 5. We present theoretical frameworks, including oriented matroids, and computational tools to check by complete enumeration the open conjectures for small instances. Preliminary computational results are given.</p> / Thesis / Master of Science (MSc)

Baigiamajame darbe tiriama grynų ir mišrių maumedynų našumas ir paplitimas Lietuvoje (pagal Lietuvos miškų kadastro elektroninę duomenų bazę, 2010). Darbo objektas – Lietuvos miškuose augantys maumedžiai bei gryni ir mišrūs maumedynai (pagal Lietuvos miškų kadastro elektroninę duomenų bazę, 2010). Darbo tikslas – ištirti svetimkraštės medžių rūšies maumedžio bendrijų našumą, paplitimą, taksacinius ypatumus ir juos palyginti su kitų Lietuvos spygliuočių medžių rūšių (paprastosios pušies ir paprastosios eglės) medynų parametrais. Darbo metodai – monografinis, dokumentų ir taksacinių normatyvų analizės bei taikomosios statistikos metodai. Darbo rezultatai. Atlikus maumedynų tyrimus nustatyta, kad gryni ir mišrūs maumedynai Lietuvoje užima 748,8 ha plotą. Turime 727 miško sklypų su grynais ir mišriais maumedynais. Bendras miškų, kuriuose auga maumedžiai, plotas lygus 2337,7 ha. Miškininkai neįvardina maumedynų rūšies tikslumu (kaip ir Lietuvos beržynų, kuriuos sudaryti gali karpotojo beržo arba plaukuotojo beržo medžiai). Lietuvoje dominuoja Europinio maumedžio medynai. Visi Lietuvos maumedynai kultūrinės kilmės. Jau 50 metų amžiuje (Ld augavietėje) maumedynai pasiekia 380 m3/ha, o brandžiuose ir perbrendusiuose maumedynuose tūris siekia net 800 – 1300 m3/ha. Maumedžiai Lietuvos miškuose pasiekia rekordinį 40 – 45 m aukštį, o savo gimtajame areale - net 55 m aukštį. Lietuvos teritorijoje dominuoja jauni maumedžio miškai (Lietuvos maumedynų vidutinis amžius 36 metai). Bendras... [toliau žr. visą tekstą] / In master's thesis were investigated Lithuanian Larch stands and their productivity. Object of the research work – Larch forests in Lithuania. The aim of the work – is to determine Larch prevalence in Lithuanian state and private forests. Analyze the estimation of forest parameters and compare with local coniferous tree species (Scots Pine and Norway Spruce) stands that grow in similar conditions. Methods of the research work - documents analysis, electronics database analysis The results of the work. Analysis showed that in Lithuania Larch stands occupies 748,8 ha. Best performance was in Ld forest site at age of 50 Larch stand reaches average 380 m3/ha stand volume. Old Larch Stands reach 1300 m3/ha.

<p>This thesis deals with combinatorial and geometric aspects of linear optimization, and consists of two parts.</p> <p>In the first part, we address a conjecture formulated in 2008 and stating that the largest possible average diameter of a bounded cell of a simple hyperplane arrangement of n hyperplanes in dimension d is not greater than the dimension d. The average diameter is the sum of the diameters of each bounded cell divided by the total number of bounded cells, and then we consider the largest possible average diameter over all simple hyperplane arrangements. This quantity can be considered as an indication of the average complexity of simplex methods for linear optimization. Previous results in dimensions 2 and 3 suggested that a specific type of extensions, namely the covering extensions, of the cyclic arrangement might achieve the largest average diameter. We introduce a method for enumerating the covering extensions of an arrangement, and show that covering extensions of the cyclic arrangement are not always among the ones achieving the largest diameter.</p> <p>The software tool we have developed for oriented matroids computation is used to exhibit a counterexample to the hypothesized minimum number of external facets of a simple arrangement of n hyperplanes in dimension d; i.e. facets belonging to exactly one bounded cell of a simple arrangement. We determine the largest possible average diameter, and verify the conjectured upper bound, in dimensions 3 and 4 for arrangements defined by no more than 8 hyperplanes via the associated uniform oriented matroids formulation. In addition, these new results substantiate the hypothesis that the largest average diameter is achieved by an arrangement minimizing the number of external facets.</p> <p>The second part focuses on the colourful simplicial depth, i.e. the number of colourful simplices in a colourful point configuration. This question is closely related to the colourful linear programming problem. We show that any point in the convex hull of each of (d+1) sets of (d+1) points in general position in R^d is contained in at least (d+1)²/2 simplices with one vertex from each set. This improves the previously established lower bounds for d>=4 due to Barany in 1982, Deza et al in 2006, Barany and Matousek in 2007, and Stephen and Thomas in 2008.</p> <p>We also introduce the notion of octahedral system as a combinatorial generalization of the set of colourful simplices. Configurations of low colourful simplicial depth correspond to systems with small cardinalities. This construction is used to find lower bounds computationally for the minimum colourful simplicial depth of a configuration, and, for a relaxed version of the colourful depth, to provide a simple proof of minimality.</p> / Doctor of Philosophy (PhD)