On matroids that are transversal and cotransversal

<p>There are distinct differences between classes of matroids that are closed under principal extensions and those that are not Finite-field-representable matroids are not closed under principal extensions and they exhibit attractive properties like well-quasi-ordering and decidable theories (at least for subclasses with bounded branch-width). Infinite-field-representable matroids, on the other hand, are closed under principal extensions and exhibit none of these behaviours. For example, the class of rank-3 real representable matroids is not well-quasi-ordered and has an undecidable theory. The class of matroids that are transversal and cotransversal is not closed under principal extensions or coprincipal coextentions, so we expect it to behave more like the class of finite-field-representable matroids. This thesis is invested in exploring properties in the aforementioned class. A major idea that has inspired the thesis is the investigation of well-quasi-ordered classes in the world of matroids that are transversal and cotransversal. We conjecture that any minor-closed class with bounded branch-width containing matroids that are transversal and cotransversal is well-quasi-ordered. In Chapter 8 of the thesis, we prove this is true for lattice-path matroids, a well-behaved class that falls in this intersection. The general class of lattice-path matroids is not well-quasi-ordered as it contains an infinite antichain of so-called ‘notch matroids’. The interesting phenomenon that we observe is that this is essentially the only antichain in this class, that is, any minor-closed family of lattice-path matroids that contains only finitely many notch matroids is well-quasi-ordered. This answers a question posed by Jim Geelen. Another question that drove the research was recognising fundamental transversal matroids, since these matroids are also cotransversal. We prove that this problem in general is in NP and conjecture that it is NP-complete. We later explore this question for the classes of lattice-path and bicircular matroids. We are successful in finding polynomial-time algorithms in both classes that identify fundamental transversal matroids. We end this part by investigating the intersection of bicircular and cobicircular matroids. We define a specific class - whirly-swirls - and conjecture that eventually any matroid in the above mentioned intersection belongs to this class.</p>

On matroids that are transversal and cotransversal

<p>There are distinct differences between classes of matroids that are closed under principal extensions and those that are not Finite-field-representable matroids are not closed under principal extensions and they exhibit attractive properties like well-quasi-ordering and decidable theories (at least for subclasses with bounded branch-width). Infinite-field-representable matroids, on the other hand, are closed under principal extensions and exhibit none of these behaviours. For example, the class of rank-3 real representable matroids is not well-quasi-ordered and has an undecidable theory. The class of matroids that are transversal and cotransversal is not closed under principal extensions or coprincipal coextentions, so we expect it to behave more like the class of finite-field-representable matroids. This thesis is invested in exploring properties in the aforementioned class. A major idea that has inspired the thesis is the investigation of well-quasi-ordered classes in the world of matroids that are transversal and cotransversal. We conjecture that any minor-closed class with bounded branch-width containing matroids that are transversal and cotransversal is well-quasi-ordered. In Chapter 8 of the thesis, we prove this is true for lattice-path matroids, a well-behaved class that falls in this intersection. The general class of lattice-path matroids is not well-quasi-ordered as it contains an infinite antichain of so-called ‘notch matroids’. The interesting phenomenon that we observe is that this is essentially the only antichain in this class, that is, any minor-closed family of lattice-path matroids that contains only finitely many notch matroids is well-quasi-ordered. This answers a question posed by Jim Geelen. Another question that drove the research was recognising fundamental transversal matroids, since these matroids are also cotransversal. We prove that this problem in general is in NP and conjecture that it is NP-complete. We later explore this question for the classes of lattice-path and bicircular matroids. We are successful in finding polynomial-time algorithms in both classes that identify fundamental transversal matroids. We end this part by investigating the intersection of bicircular and cobicircular matroids. We define a specific class - whirly-swirls - and conjecture that eventually any matroid in the above mentioned intersection belongs to this class.</p>

Obstructions for Bounded Branch-depth in Matroids

DeVos, Kwon, and Oum introduced the concept of branch-depth of matroids as a natural analogue of tree-depth of graphs. They conjectured that a matroid of sufficiently large branch-depth contains the uniform matroid Un;2n or the cycle matroid of a large fan graph as a minor. We prove that matroids with sufficiently large branch-depth either contain the cycle matroid of a large fan graph as a minor or have large branch-width. As a corollary, we prove their conjecture for matroids representable over a fixed finite field and quasi-graphic matroids, where the uniform matroid is not an option.

Typical features of morphometric parameters of the mandible in adults

Objective: to develop a classification of mandibular forms and to study typical features of the morphometric characteristics of this bone in adults. Materials and methods. The study was conducted on 150 lower jaws of adults. To determine the shape of the lower jaw, four morphometric parameters were measured: angular width, projection length from the corners, branch height, smallest branch width, and three morphometric indexes were introduced: 1 - the long-length longitudinal index of the lower jaw; 2 - longitude latitudinal index of the body of the lower jaw; 3 - latitudinal-altitude index of the branches of the lower jaw. According to these indices, 9 groups of jaws with different shapes were identified. In these groups, the values of 35 morphometric parameters of the body and branches of the lower jaw were studied. Results. It was found that statistically significant differences (p <0.05) between the groups of jaws, determined by the value of the altitude-longitude index of the lower jaw and the longitude-latitude index of the body of the lower jaw, exist between the same morphometric parameters: angular width, projection length from the corners and chin angle , and most of the morphometric parameters of the body and branches of the lower jaw do not statistically significantly differ between the extreme forms (dolicho- and brachi, lepto- and eurimandibular). There are statistically significant differences between the jaw groups, systematized by the latitude-altitude index of the branch of the lower jaw (p <0.05) for most of the studied indicators of the branch of the lower jaw: branch height, smallest branch width, notch width, notch angle, base of the coronoid process , the base of the condylar process, the distance from the front edge of the lower jaw branch to the opening of the lower jaw, the distance from the notch of the lower jaw to the opening of the lower jaw, the distance from the angle of the lower jaw to the opening of the lower jaw. It has been proved that in the lower jaws with a hypsiramimandibular form, the values of the smallest branch width, the base length of the coronoid and condylar processes, as well as the distance from the front edge of the branch to the opening of the lower jaw are significantly smaller, however, the values of the branch height, notch angle, notch width, notch distance the angle of the lower jaw to its opening is larger compared to the platyramimandibular form (p˂0.05). Conclusion. The greatest number of differences in the value of morphometric parameters is observed during the systematization of the lower jaw according to the shape of its branch. This can be explained by the fact that it is under the direct influence of the masticatory muscles, performing not only supporting, protective, but also motor function.

T- and Y-Branched Three-Terminal Junction Graphene Devices

Heteroepitaxial graphene on semiinsulating silicon carbide was used to fabricate nanoelectronic devices. T- and Y-branched graphene three-terminal junction devices were realized. Room temperature electrical measurements demonstrate pronounced nonlinear electrical properties of the devices. Voltage rectification at room temperature was observed. Increasing branch width reduces the curvature of the voltage rectification response curve of the three-terminal junc¬tions.