A Linear Program is a problem where the goal is to maximize a linear function subject to a set of linear inequalities. Geometrically, this can be rephrased as finding the highest point on a polyhedron. The Simplex method is a commonly used algorithm to solve Linear Programs. It traverses the vertices of the polyhedron, and in each step, it selects one adjacent better vertex and moves there. There can be multiple vertices to choose from, and therefore the Simplex method has different variants deciding how the next vertex is selected. One of the most natural variants is Random Edge, which in each step of the Simplex method uniformly at random selects one of the better adjacent vertices. It is interesting and non-trivial to study the complexity of variants of the Simplex method in the number of variables, d, and inequalities, N. In 2011, Friedmann, Hansen, and Zwick found a class of Linear Programs for which the Random Edge algorithm is subexponential with complexity 2^Ω(N^(1/4)), where d=Θ(N). Previously all known lower bounds were polynomial. We give an improved lower bound of 2^Ω(N^(1/2)), for Random Edge on Linear Programs where d=Θ(N). Another well studied variant of the Simplex method is Random Facet. It is upper bounded by 2^O(N^(1/2)) when d=Θ(N). Thus we prove that Random Edge is not faster than Random Facet on Linear Programs where d=Θ(N). Our construction is very similar to the previous construction of Friedmann, Hansen and Zwick. We construct a Markov Decision Process which behaves like a binary counter with linearly many levels and linearly many nodes on each level. The new idea is a new type of delay gadget which can switch quickly from 0 to 1 in some circumstances, leading to fewer nodes needed on each level of the construction. The key idea is that it is worth taking a large risk of getting a small negative reward if the potential positive reward is large enough in comparison. / Ett linjärt program är ett problem där målet är att maximiera en linjär funktion givet en mängd linjära olikheter. Geometriskt kan detta omformuleras som att hitta den högsta punkten på en polyeder. Simplexmetoden är en algoritm som ofta används för att lösa linjära program. Den besöker hörnen i polyedern, och i varje steg väljer den ett närliggande bättre hörn och flyttar dit. Det kan finnas flera hörn att välja mellan, och därför finns det olika varianter av simplexmetoden som bestämmer hur nästa hörn ska väljas. En av de mest naturliga varianterna är Random Edge, som i varje steg av simplexmetoden, uniformt slumpmässigt väljer ett av de närliggande bättre hörnen. Det är intressant och icke-trivialt att studera komplexiteten av olika varianter av simplexmetoden i antalet variabler, d, och olikheter N. År 2011 hittade Friedmann, Hansen och Zwick en familj av linjära program där Random Edge är subexponentiell med komplexitet 2^Ω(N^(1/4)), där d=Θ(N). Innan dess var alla kända undre gränser polynomiska. Vi ger en förbättrad undre gräns på 2^Ω(N^(1/2)), för Random Edge på linjära program där d=Θ(N). En annan välstuderad variant av simplexmetoden är Random Facet. Dess komplexitet har en övre gräns på 2^O(N^(1/2)) när d=Θ(N). Alltså bevisar vi att Random Edge inte är snabbare än Random Facet på linjära program där d=Θ(N). Vår konstruktion är väldigt lik den tidigare konstruktionen av Friedmann, Hansen och Zwick. Vi konstruerar en Markov-beslutsprocess som beter sig som en binär räknare med linjärt många nivåer och linjärt många noder på varje nivå. Den nya idén är en ny typ av försenings-multinod som kan byta snabbt från 0 till 1 i vissa fall, vilket leder till att det behövs färre noder på varje nivå av konstruktionen. Nyckelidén är att det är värt att ta en stor risk att få en liten negativ poäng om den potentiella positiva poängen är stor nog i jämförelse.

In this dissertation, we exhibit two instances of polyhedra in combinatorial convex geometry. The first instance arises in the context of Ehrhart theory, and the polyhedra are the central objects of study. The second instance arises in algebraic statistics, and the polyhedra act as a conduit through which we study a nonpolyhedral problem. In the first case, we examine combinatorial and algebraic properties of the Ehrhart h*-polynomial of the r-stable (n,k)-hypersimplices. These are a family of polytopes which form a nested chain of subpolytopes within the (n,k)-hypersimplex. We show that a well-studied unimodular triangulation of the (n,k)-hypersimplex restricts to a triangulation of each r-stable (n,k)-hypersimplex within. We then use this triangulation to compute the facet-defining inequalities of these polytopes. In the k=2 case, we use shelling techniques to devise a combinatorial interpretation of the coefficients of the h*-polynomials in terms of independent sets of certain graphs. From this, we then extract some results on unimodality. We also characterize the Gorenstein r-stable (n,k)-hypersimplices, and we conclude that these also have unimodal h*-polynomials. In the second case, for a graph G on p vertices we consider the closure of the cone of concentration matrices of G. The extreme rays of this cone, and their associated ranks, have applications in maximum likelihood estimation for the undirected Gaussian graphical model associated to G. Consequently, the extreme ranks of this cone have been well-studied. Yet, there are few graph classes for which all the possible extreme ranks are known. We show that the facet-normals of the cut polytope of G can serve to identify extreme rays of this nonpolyhedral cone. We see that for graphs without K5 minors each facet-normal of the cut polytope identifies an extreme ray in the cone, and we determine the rank of this extreme ray. When the graph is also series-parallel, we find that all possible extreme ranks arise in this fashion, thereby extending the collection of graph classes for which all the possible extreme ranks are known.

Master of Science / Department of Industrial and Manufacturing Systems Engineering / Todd Easton / Graph theory is a widely researched topic. A graph contains a set of nodes and a set of edges. The nodes often represent resources such as machines, employees, or plant locations. Each edge represents the relationship between a pair of nodes such as time, distance, or cost. Integer programs are frequently used to solve graphical problems. Unfortunately, IPs are NP-hard unless P = NP, which implies that it requires exponential effort to solve them. Much research has been focused on reducing the amount of time required to solve IPs through the use of valid inequalities or cutting planes. The theoretically strongest cutting planes are facet defining cutting planes. This research focuses on the node packing problem or independent set problem, which is a combinatorial optimization problem. The node packing problem involves coloring the maximum number of nodes such that no two nodes are adjacent. Node packings have been applied to airline traffic and radio frequencies. This thesis introduces a new class of graphical structures called suns. Suns produce previously undiscovered valid inequalities for the node packing polyhedron. Conditions are provided for when these valid inequalities are proven to be facet defining. Sun valid inequalities have the potential to more quickly solve node packing problems and could even be extended to general integer programs through conflict graphs.

M.Tech. / Purpose: To determine what effects flexion distraction will have on males with chronic lumbar facet syndrome, in reference to intersegmental range of motion and pain. Method: Thirty male participants with chronic lumbar facet syndrome received flexion distraction technique. The trail consisted of seven visits over two to three weeks. Treatment was performed on the first to sixth visit, readings and questionnaires were completed on the first, fourth and seventh visits. Data collection was concluded before treatment on the mentioned visits. Subjective data consisted of the Oswestry Disability Index (ODI) and the Short-Form McGill Pain Questionnaire-2 (SF-MPQ-2). Objective data included global lumbar range of motion measurements using the Baseline Digital Inclinometer, and segmental lumbar range of motion by means of the Radiographical Midplane Angle method. Results: Objectively, clinical significance of the intersegmental lumbar range of motion increased from the L1/L2 to the L4/L5 level (2.2% - 17.7%), but showed no statistical significance. In relation to the maximal motion possible at these levels, the increase in motion escalated from 0.8% at the L1/L2 segment to 9.2% at the L4/L5 segment. The global lumbar range of motion showed clinical significant increase in all of the ranges except for extension, however statistical significance was found in right lateral flexion only (p = 0.045). The greatest increase in motion was observed in left lateral flexion (15.72%) and a decrease in extension (3.72%) was illustrated. Subjectively, the group showed statistical significant improvement in both the SF-MPQ-2 (p = 0.000) and ODI (p = 0.000). Clinical meaningful change was noted throughout the subjective data, which resultantly showed a 66% and a 67% change in pain respectively. Conclusion: Flexion distraction has shown to have clinical significant effects on segmental and thus global range of motion, and great clinical meaningful change in pain levels and pain perception. Thus specific segmental mobilisation does affect the segmental motion.

M.Tech. / Purpose: The cervical facet joints have attracted relatively little attention as possible sources of neck pain and referred pain. Multiple authors have described the management of cervical facet joint pain but not the cause (Manchikanti et al., 2002). Method: This study consisted of one group of 100 participants. The participants were between the ages of 18 and 40 years. Potential participants were examined and accepted based on the inclusion and exclusion criteria. All the participants received a cervical spine adjustment. Objective and subjective readings were taken. Procedure: The participants were seen only once. The Visual Analogue Scale was completed by each participant before treatment. Algometer readings were taken over the most restricted cervical spine facet joint and cervical spine range of motion (CROM) machine readings were taken measuring the ranges of motion of the cervical spine. The participants received an adjustment to the most restricted cervical spine facet joint as determined by motion palpation. The CROM machine and algometer readings were taken again immediately after the adjustment and the algometer readings were taken again 10 minutes later. Results: In terms of subjective measurements based on the Visual Analogue Scale, all participants experienced clinically significant pain before starting the trial.In terms of objective measurements based on algometer readings, a clinically significant difference was found as the pressure pain threshold increases over a period of time. In terms of the CROM machine readings there was a clinical improvement from the pretreatment ranges of motion to the post-treatment ranges of motion. The algometer and CROM readings were statistically incomparable to begin with. Conclusion: The results proved that there was a statistical significant noted immediately after the adjustment and 10 minutes later, however, this does not mean much as thegroups were not comparable to begin with. A statistically significant difference was noted for all ranges of cervical spine motion (flexion, extension, right and left lateral flexion and rotation), thus showing that the cervical spine adjustment was successfully delivered to the restricted segments.

A lombalgia e a cervicalgia possui grande incidência em toda a população mundial. Segundo a Organização Mundial de Saúde (OMS), em média, cerca de 80% da população adulta apresenta, em algum momento da vida, pelo menos uma queixa de dor na coluna, onde a degeneração da articulação do processo articular é frequentemente apontada como a etiologia. Diversos estudos vêm sendo realizados, na tentativa de se compreender como ocorre o processo degenerativo patológico dessas articulações, onde, frequentemente são usados indivíduos sintomáticos. Deste modo, avaliou-se, com o uso de técnicas macroscópicas, de microscopia de luz e de microscopia eletrônica, a organização estrutural da cartilagem dos processos articulares (CPas) das vértebras cervicais (C) e lombares (L) de indivíduos jovens (GJ) e idosos (GI), presumivelmente assintomáticos. Foram utilizados blocos vertebrais obtidos de cadáveres necropsiados no Serviço de Verificação de Óbitos da Capital do Estado de São Paulo, onde familiares de todos os indivíduos forneceram informações que permitiram incluir ou excluí-los da pesquisa. Os resultados mostram que ocorrem alterações na superfície da CPa do segmento L no GJ. A degeneração da CPa ocorre de forma heterogênea entre os indivíduos do GI e as característica do grau 2 podem ser admitidas como decorrentes do processo de envelhecimento normal, não havendo diferenças entre os segmentos C e L. / Back pain and neck pain have a high occurrence in populations worldwide. According to the World Health Organization (WHO), approximately 80% of adults have at some instance in life pain in the spine with an etiology frequently indicated to be the degeneration of the articular process. Several studies have been undertaken to understand how the pathological degenerative process occurs and symptomatic subjects are frequently used for this end. Macroscopic and light and electron microscopy techniques have been employed to assess the structural organization of the cartilage of the articulation processes (CAP) of the cervical (C) and lumbar (L) vertebrae of presumably asymptomatic young (Y) and elderly (E) people. Samples, retrieved from routinely necropsied corpses by the Death Verification Service of the Capital City of the State of São Paulo (SVOC/SP) and the family of all individuals provided information that enabled include or exclude them from search. Results show that changes in the CAP surface of segment L of Y occur. CAP degradation occurs heterogeneously among elderly people, whereas second degree characteristics are caused by normal aging without any difference between the C and L segments.

The study of how personality traits relate to psychopathology has flourished in the past three decades, with strong evidence for systematic links between personality and psychological disorders. Great progress has been made in our understanding of the associations between broad traits and the mood and anxiety disorders (or internalizing disorders). In particular, it is clear that the broad trait neuroticism/negative emotionality (N/NE; stress reactivity and a tendency to experience negative emotions) is moderately to strongly associated with all of the internalizing disorders, both concurrently and longitudinally. However, researchers have noted the relative dearth of studies that examine associations with more narrow facet-level traits. The current study examined the relations of N/NE facets with six of the internalizing disorders (i.e., depression, GAD, PTSD, social anxiety, panic, and OCD). The above symptoms were expected to load on to two higher order factors (fear and distress). Based on pilot analyses, a five-factor model for N/NE was hypothesized, consisting of sadness, anxiety, angry hostility, mistrust, and dependency. In addition, stress vulnerability marked the shared variance among these facets. I also examined associations between the disorders and four clinical traits (i.e., anxiety sensitivity, experiential avoidance, perfectionism, and intolerance of uncertainty) that are related to N/NE. Finally, I analyzed the associations of the N/NE facets and clinical traits with heterogeneous symptom dimensions within PTSD and OCD. Self-report and clinical interview data were collected from a college student sample (N = 373) and a psychiatric outpatient sample (N = 252; an additional 44 patients completed self-report measures only), with multiple measures of each internalizing disorder and personality trait described above. Structural equation modeling was used to remove shared variance among the six disorders and among the traits, allowing for the examination of relations across the unique variances of each construct. The hypothesized N/NE model provided a good fit to the data in both samples, as did the hypothesized psychopathology structure in the patient sample. However, markers of depression, panic, PTSD, and GAD were indistinguishable in the student sample and were therefore collapsed into a single factor. The results of the current study delineated unique patterns of association for each of the internalizing symptoms (as well as symptom dimensions within OCD and PTSD) in reference to the N/NE facets and clinical traits, highlighting shared and specific trait contributors. There was also evidence that all four clinical traits (as well as their subscales) are not redundant with N/NE and are differentially associated with the internalizing psychopathology examined here. The results of the current study helped clarify personality-psychopathology relations within a large network of traits and symptoms, while also controlling for the extensive overlap among these constructs. As such, implications for taxonomy, differential assessment, and structural models in these domains are discussed. Future research should focus on expanding this model to other traits and disorders, utilizing other methods of assessment such as informant data, and striving to delineate underlying mediating factors that may account for the pattern of associations found between traits and symptoms in the current study.

Radiofrequency neurotomy is a pain intervention procedure designed to coagulate nerves that innervate a specific area of spinal vertebrae known as the facet joint. Despite moderate to strong research support for the efficacy of radiofrequency neurotomy to improve short-term subjective pain levels, much of the literature to date has used strict selection criteria and has not focused on functional and quality of life outcomes. Moreover, few studies have examined outcomes in worker's compensation patients or considered biopsychosocial predictive variables for the procedure. The current study aimed to characterize injured workers who have undergone radiofrequency neurotomy across a number of pre and post-procedural variables, evaluate multidimensional functional and quality of life outcomes, and examine biopsychosocial variables predictive of success and failure in this sample. The current study comprised 101 injured workers who had undergone at least one radiofrequency neurotomy of the spine (cervical, thoracic, or lumbar) in the past 11 years. Participants were solicited through the Worker's Compensation Fund of Utah computerized database. Employing a retrospective cohort design, patients' medical charts were reviewed and various preprocedural variables were coded for analysis including age at the time of the first neurotomy, history of depression, lawyer involvement in the claim, prior back and neck surgical history, and quantity of other compensation claims. Of the total sample, 56 patients (55.4%) were contacted and completed outcome surveys that assessed patient satisfaction, functional impairment, disability status, pain catastrophization, and general physical and mental health functioning. Findings revealed a moderate proportion of patients with total disability (40%), poor back/neck specific functioning (63%), and dissatisfaction with their current back/neck condition (75%). A multivariate regression model was consistently predictive of patient outcomes. Specifically, litigation status was a robust predictor of multidimensional outcomes, while depression and age retained slightly less predictive power. Results of descriptive, correlational, and regression analyses are compared to existing data for radiofrequency neurotomy and other spine procedures with similar populations. Limitations of the study are discussed, such as the retrospective design, lack of matched controls, and small sample size.

1. INTRODUCTION Injuries of the lower cervical spine can be among the most devastating injuries of the musculoskeletal system because of the increased risk of the injury to the spinal cord, and also because they so often occur to the younger members of the population (Jones A.A.M. et al., 2003; Sekhon H.S.L. et al., 2001; Ball P.A., 2001). The cervical spine is the most vulnerable spinal segment (Sekhon H.S.L. et al., 2001). The mechanism of cervical spine trauma is defined by the direction and magnitude of the forces that have been applied externally to the head and neck complex resulting in injury (Allen B.L.Jr., 1982). Common injury vectors include flexion, compression, rotation and extension (Allen B.L.Jr., 1982). The pattern of injury is related not only to the external applied force, but also to the initial position or posture of the head and neck at the time of injury (Allen B.L.Jr., 1982). One pattern of these injuries of the lower cervical spine is a facet dislocations (Allen B.L.Jr., 1982). The facet dislocation of the cervical spine result from a hyperflexion injury of the neck (Allen B.L.Jr., 1982). These injuries are characterized radiographically by anterolisthesis of one cervical vertebrae over the other and include the slide anteriorly of the inferior facet of the upper dislocated vertebra over the superior facet of the vertebra below (Allen B.L.Jr., 1982; Razack N. et al., 2000). The facet dislocations of the lower cervical spine represent from 4% to 50% of... [to full text]

Semiconductive metal oxides are of great importance in environmental remediation and electronics because of their ability to generate charge carriers when excited with appropriate energy. The electronic structure, light absorption and charge transport properties have made the transition metal oxides an attractive material as photocatalyst. Recently, facet-engineering by morphology control has been intensively studied as an efficient approach to further enhance their photocatalytic performance. However, various processing steps and post-treatments used in the preparation of facet-engineered particles may generate different surface active sites which may affect their photocatalysis. Moreover, many traditional techniques (PL, EPR and XPS) used for materials characterization (oxygen vacancy, hydroxyl group, cation, etc.) are not truly surface specific but analyzing a range from surface few layers to bulk. Accordingly, they can only provide very limited information on chemical states of the surface active features and their distribution among facets, causing difficulties to unambiguously correlate facet-dependent results with activity. As a result, this often leads to different interpretations amongst researchers during the past decades. As the publications of titanium and zinc ranked top two among studies of first row of transition oxides in the past decades, this thesis will firstly review on the disagreements generated among researchers when they correlated the performance of ZnO and TiO₂ with their facet activities based on traditional techniques. As there are shortcomings of these techniques in producing truly facet-dependent features, some results can be misleading and with no cross-literature comparison. To address these issues, we have developed a new technique "probe-molecule-assisted NMR" which allows a genuine differentiation of surface active sites from various facets. This surface-fingerprint technique has been demonstrated to provide both qualitative (chemical shift) and quantitative (peak intensity) information on the concentration and distribution of truly surface features among facets. In light of the new technique, this thesis will revisit the facet-dependent photocatalytic properties and shed light on these issues.