MAT 116 Introduction to Calculus - Course Material

Updated addendum to MAT 116 (Introduction to Calculus) syllabus, updated course timeline (both before and after the storms since many students lost power) and a contingency plan for their course team

Sperner\u27s Lemma, The Brouwer Fixed Point Theorem, the Kakutani Fixed Point Theorem, and Their Applications in Social Sciences

Can a cake be divided amongst people in such a manner that each individual is content with their share? In a game, is there a combination of strategies where no player is motivated to change their approach? Is there a price where the demand for goods is entirely met by the supply in the economy and there is no tendency for anything to change? In this paper, we will prove the existence of envy-free cake divisions, equilibrium game strategies and equilibrium prices in the economy, as well as discuss what brings them together under one heading. This paper examines three important results in mathematics: Sperner’s lemma, the Brouwer fixed point theorem and the Kakutani fixed point theorem, as well as the interconnection between these theorems. Fixed point theorems are central results of topology that discuss existence of points in the domain of a continuous function that are mapped under the function to itself or to a set containing the point. The Kakutani fixed point theorem can be thought of as a generalization of the Brouwer fixed point theorem. Sperner’s lemma, on the other hand, is often described as a combinatorial analog of the Brouwer fixed point theorem, if the assumptions of the lemma are developed as a function. In this thesis, we first introduce Sperner’s lemma and it serves as a building block for the proof of the fixed point theorem which in turn is used to prove the Kakutani fixed point theorem that is at the top of the pyramid. This paper highlights the interdependence of the results and how they all are applicable to prove the existence of equilibria in fair division problems, game theory and exchange economies. Equilibrium means a state of rest, a point where opposing forces balance. Sperner’s lemma is applied to the cake cutting dilemma to find a division where no individual vies for another person’s share, the Brouwer fixed point theorem is used to prove the existence of an equilibrium game strategy where no player is motivated to change their approach, and the Kakutani fixed point theorem proves that there exists a price where the demand for goods is completely met by the supply and there is no tendency for prices to change within the market

Maximization of seed yield in transplanted fennel (Foeniculum vulgare Mill.)

A field experiment was conducted at Udaipur (Rajasthan) to study the effect of three plantdensities, four fertilizer treatments and two growth regulators on yield maximization oftransplanted fennel (Foeniculum vulgare).  Planting of fennel at a density of 27,770 plants ha-1recorded significantly higher yield attributes (umbels plant-1, seed yield plant-1 and test weight),while seed yield ha-1 was higher with a density of 55,550 plants ha-1 (20.89 q ha-1).  Applicationof 90 kg N + 40 kg P2O5 + 20 kg K2O + 20 kg S + 5 kg Zn ha-1 recorded significantly higher yieldattributes and yield of fennel during both the years.  Application of naphthalene acetic acid100 ppm recorded significantly higher yield attributes and seed yield. &nbsp

The role of high density lipoproteins in the biodistribution of two radioiodinated probes in the rat

Two radioiodinated probes, 125I-cholesteryl oleate (125I-CO), a derivative of a natural constituent of lipoproteins, and 1-(2-chlorophenyl)-1-(4[125I]iodophenyl)-2,2-dichloroethane (125I-DDD), an analog of the adrenolytic drug o,p'-DDD (mitotane), were selected to study the role of lipoproteins in drug disposition and to examine the ability of these vehicles to direct foreign molecules to specific tissues. In vivo and in vitro techniques were utilized to associate these probes with rat high density lipoproteins (HDL). Tissue distribution studies indicated that prior incorporation of 125I-CO into rat HDL increased the uptake of 125I-CO by rat adrenal, which was dramatically enhanced when this preparation was administered to animals made hypolipidemic with 4-aminopyrazolo-(3,4-d)-pyrimidine (4-APP). Acetylation of HDL labeled with 125I-CO provided evidence that the observed uptake into the adrenal was via a receptor-mediated process. In contrast with these results, prior association of 125I-DDD with rat HDL failed to alter the ability of this compound to accumulate in adrenal tissue of normal or hypolipidemic animals. Polyacrylamide gel electrophoresis (PAGE) was utilized to examine the stability of the association of 125I-CO and 125I-DDD with rat HDL. These results suggested that 125I-CO was associated with the lipophilic core of HDL, whereas 125I-DDD appeared to be partially associated with the suface components of HDL. Saturation of surface components with stable o,p'-DDD offered data to suggest that this binding to apoproteins may disrupt the normal receptor-mediated uptake process. These studies indicate that lipoproteins may effect the distribution and tissue uptake of lipophilic compounds and, conversely, lipophilic molecules can effect the metabolic fate of lipoproteins. The overall result is dependent upon the nature of the association of these lipophilic compounds with lipoproteins which is difficult to predict on the basis of molecular structure alone.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/25808/1/0000371.pd

Annealed Silver-Island Films for Applications in Metal-Enhanced Fluorescence: Interpretation in Terms of Radiating Plasmons

The effects of thermally annealed silver island films have been studied with regard to their potential applicability in applications of metal-enhanced fluorescence, an emerging tool in nano-biotechnology. Silver island films were thermally annealed between 75 and 250°C for several hours. As a function of both time and annealing temperature, the surface plasmon band at ≈420 nm both diminished and was blue shifted. These changes in plasmon resonance have been characterized using both absorption measurements, as well as topographically using Atomic Force Microscopy. Subsequently, the net changes in plasmon absorption are interpreted as the silver island films becoming spherical and growing in height, as well as an increased spacing between the particles. Interestingly, when the annealed surfaces are coated with a fluorescein-labeled protein, significant enhancements in fluorescence are osbserved, scaling with annealing temperature and time. These observations strongly support our recent hypothesis that the extent of metal-enhanced fluorescence is due to the ability of surface plasmons to radiate coupled fluorophore fluorescence. Given that the extinction spectrum of the silvered films is comprised of both an absorption and scattering component, and that these components are proportional to the diameter cubed and to the sixth power, respectively, then larger structures are expected to have a greater scattering contribution to their extinction spectrum and, therefore, more efficiently radiate coupled fluorophore emission. Subsequently, we have been able to correlate our increases in fluorescence emission with an increased particle size, providing strong experiment evidence for our recently reported metal-enhanced fluorescence, facilitated by radiating plasmons hypothesis