The Philosophical magazine : or Annals of chemistry, mathematics, astronomy, natural history and general science.

2nd ser. : v.3 (1828)

The Philosophical magazine : or Annals of chemistry, mathematics, astronomy, natural history and general science.

2nd ser. : v.4 (1828)

influence of different forms of outcome information on the neural substrates of the acquisition and representation of categories

Magdeburg, Univ., Fak. für Naturwiss., Diss., 2012

A curious example of two model categories and some associated differential graded algebras

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Homotopy limits for 2-categories

We study homotopy limits for 2-categories using the theory of Quillen model categories. In order to do so, we establish the existence of projective and injective model structures on diagram 2-categories. Using these results, we describe the homotopical behaviour not only of conical limits but also of weighted limits. Finally, pseudo-limits are related to homotopy limits.

Model structures on the category of small double categories

In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorification-nerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2-monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofi brant objects. As part of this program we give explicit descriptions and discuss properties of free double categories, quotient double categories, colimits of double categories, and several nerves and categorifications.

Derived categories and Grothendieck duality

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On the applicability of mathematics : philosophical and historical perspectives

The present thesis is a contribution to the debate on the applicability of mathematics; it examines the interplay between mathematics and the world, using historical case studies. The first part of the thesis consists of four small case studies. In chapter 1, I criticize "ante rem structuralism", proposed by Stewart Shapiro, by showing that his so-called "finite cardinal structures" are in conflict with mathematical practice. In chapter 2, I discuss Leonhard Euler's solution to the Königsberg bridges problem. I propose interpreting Euler's solution both as an explanation within mathematics and as a scientific explanation. I put the insights from the historical case to work against recent philosophical accounts of the Königsberg case. In chapter 3, I analyze the predator-prey model, proposed by Lotka and Volterra. I extract some interesting philosophical lessons from Volterra's original account of the model, such as: Volterra's remarks on mathematical methodology; the relation between mathematics and idealization in the construction of the model; some relevant details in the derivation of the Third Law, and; notions of intervention that are motivated by one of Volterra's main mathematical tools, phase spaces. In chapter 4, I discuss scientific and mathematical attempts to explain the structure of the bee's honeycomb. In the first part, I discuss a candidate explanation, based on the mathematical Honeycomb Conjecture, presented in Lyon and Colyvan (2008). I argue that this explanation is not scientifically adequate. In the second part, I discuss other mathematical, physical and biological studies that could contribute to an explanation of the bee's honeycomb. The upshot is that most of the relevant mathematics is not yet sufficiently understood, and there is also an ongoing debate as to the biological details of the construction of the bee's honeycomb. The second part of the thesis is a bigger case study from physics: the genesis of GR. Chapter 5 is a short introduction to the history, physics and mathematics that is relevant to the genesis of general relativity (GR). Chapter 6 discusses the historical question as to what Marcel Grossmann contributed to the genesis of GR. I will examine the so-called "Entwurf" paper, an important joint publication by Einstein and Grossmann, containing the first tensorial formulation of GR. By comparing Grossmann's part with the mathematical theories he used, we can gain a better understanding of what is involved in the first steps of assimilating a mathematical theory to a physical question. In chapter 7, I introduce, and discuss, a recent account of the applicability of mathematics to the world, the Inferential Conception (IC), proposed by Bueno and Colyvan (2011). I give a short exposition of the IC, offer some critical remarks on the account, discuss potential philosophical objections, and I propose some extensions of the IC. In chapter 8, I put the Inferential Conception (IC) to work in the historical case study: the genesis of GR. I analyze three historical episodes, using the conceptual apparatus provided by the IC. In episode one, I investigate how the starting point of the application process, the "assumed structure", is chosen. Then I analyze two small application cycles that led to revisions of the initial assumed structure. In episode two, I examine how the application of "new" mathematics - the application of the Absolute Differential Calculus (ADC) to gravitational theory - meshes with the IC. In episode three, I take a closer look at two of Einstein's failed attempts to find a suitable differential operator for the field equations, and apply the conceptual tools provided by the IC so as to better understand why he erroneously rejected both the Ricci tensor and the November tensor in the Zurich Notebook.

Cluster algebras, quiver representations and triangulated categories

This is an introduction to some aspects of Fomin-Zelevinsky’s cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria)and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences.

On the homotopy theory of spectral categories

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A Thomason model structure on the category of small n-fold categories

We construct a cofibrantly generated Thomason model structure on the category of small n-fold categories and prove that it is Quillen equivalent to the standard model structure on the category of simplicial sets. An n-fold functor is a weak equivalence if and only if the diagonal of its n-fold nerve is a weak equivalence of simplicial sets. We introduce an n-fold Grothendieck construction for multisimplicial sets, and prove that it is a homotopy inverse to the n-fold nerve. As a consequence, the unit and counit of the adjunction between simplicial sets and n-fold categories are natural weak equivalences.

Finiteness obstructions and Euler characteristics of categories

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Robust discrimination between EEG responses to categories of environmental sounds in early coma.

Humans can recognize categories of environmental sounds, including vocalizations produced by humans and animals and the sounds of man-made objects. Most neuroimaging investigations of environmental sound discrimination have studied subjects while consciously perceiving and often explicitly recognizing the stimuli. Consequently, it remains unclear to what extent auditory object processing occurs independently of task demands and consciousness. Studies in animal models have shown that environmental sound discrimination at a neural level persists even in anesthetized preparations, whereas data from anesthetized humans has thus far provided null results. Here, we studied comatose patients as a model of environmental sound discrimination capacities during unconsciousness. We included 19 comatose patients treated with therapeutic hypothermia (TH) during the first 2 days of coma, while recording nineteen-channel electroencephalography (EEG). At the level of each individual patient, we applied a decoding algorithm to quantify the differential EEG responses to human vs. animal vocalizations as well as to sounds of living vocalizations vs. man-made objects. Discrimination between vocalization types was accurate in 11 patients and discrimination between sounds from living and man-made sources in 10 patients. At the group level, the results were significant only for the comparison between vocalization types. These results lay the groundwork for disentangling truly preferential activations in response to auditory categories, and the contribution of awareness to auditory category discrimination.

Dukes B colorectal cancer: distinct genetic categories and clinical outcome based on proximal or distal tumor location.

PURPOSE: The aim of this study was to determine whether tumor location proximal or distal to the splenic flexure is associated with distinct molecular patterns and can predict clinical outcome in a homogeneous group of patients with Dukes B (T3-T4, N0, M0) colorectal cancer. It has been hypothesized that proximal and distal colorectal cancer may arise through different pathogenetic mechanisms. Although p53 and Ki-ras gene mutations occur frequently in distal tumors, another form of genomic instability associated with defective DNA mismatch repair has been predominantly identified in the proximal colon. To date, however, the clinical usefulness of these molecular characteristics remains unproven. METHODS: A total of 126 patients with a lymph node-negative sporadic colon or rectum adenocarcinoma were prospectively assessed with the endpoint of death by cancer. No patient received either radiotherapy or chemotherapy. p53 protein was studied by immunohistochemistry using DO-7 monoclonal antibody, and p53 and Ki-ras gene mutations were detected by single strand conformation polymorphism assay. RESULTS: During a mean follow-up of 67 months, the overall five-year survival was 70 percent. Nuclear p53 staining was found in 57 tumors (47 percent), and was more frequent in distal than in proximal tumors (55 vs. 21 percent; chi-squared test, P < 0.001). For the whole group, p53 protein expression correlated with poor survival in univariate and multivariate analysis (log-rank test, P = 0.01; hazard ratio = 2.16; 95 percent confidence interval = 1.12-4.11, P = 0.02). Distal colon tumors and rectal tumors exhibited similar molecular patterns and showed no difference in clinical outcome. In comparison with distal colorectal cancer, proximal tumors were found to be statistically significantly different on the following factors: mucinous content (P = 0.008), degree of histologic differentiation (P = 0.012), p53 protein expression, and gene mutation (P = 0.001 and 0.01 respectively). Finally, patients with proximal tumors had a marginally better survival than those with distal colon or rectal cancers (log-rank test, P = 0.045). CONCLUSION: In this series of Dukes B colorectal cancers, p53 protein expression was an independent factor for survival, which also correlated with tumor location. Eighty-six percent of p53-positive tumors were located in the distal colon and rectum. Distal colon and rectum tumors had similar molecular and clinical characteristics. In contrast, proximal neoplasms seem to represent a distinct entity, with specific histopathologic characteristics, molecular patterns, and clinical outcome. Location of the neoplasm in reference to the splenic flexure should be considered before group stratification in future trials of adjuvant chemotherapy in patients with Dukes B tumors.