The most important prognostic factor in breast cancer is the axillary lymph node status. The sentinel node biopsy (SNB) is reported to stage the axilla with an accuracy > 95 % in early breast cancer. Tumour-related perturbation of T-cell function has been observed in patients with malignancies, including breast cancer. The down-regulation of the important T-cell activation molecules CD3-ζ and CD28 may cause T-cell dysfunction, anergy, tolerance and deletion. The expression of CD3-ζ and CD28 was evaluated in 25 sentinel node biopsies. The most pronounced down-regulation was seen in the paracortical area, where the best agreement between both parameters was observed. CD28 expression was significantly more suppressed in CD4+ than in CD8+ T-cells. From the Swedish sentinel node database, 109 patients with breast cancer > 3 cm planned for both SNB and a subsequent axillary dissection were identified. The false negative rate (FNR) was 12.5%. Thirteen cases of tumour multifocality were detected on postoperative pathology. The FNR in this subgroup was higher (30.8%) than in patients with unifocal disease (7.8%; P = 0.012). From the Swedish SNB multicentre cohort trial, 2246 sentinel node-negative patients who had not undergone further axillary surgery were selected for analysis. After a median follow-up time of 37 months (range 0-75), 13 isolated axillary recurrences (13/2246; 0.6%) were found. In another 14 cases, local or distant failure preceded or coincided with axillary relapse (27/2246; 1.2%). In conclusion, the immunological analysis of the sentinel node might provide valuable prognostic information and aid selection of patients for immunotherapy. SNB is encouraged in breast cancer larger than 3 cm, if no multifocal growth pattern is present. The axillary recurrence rate after a negative SNB in Sweden is in accordance with international figures. However, a longer follow-up is mandatory before the true failure rate of the SNB can be determined.

This thesis consists of an introduction and four papers. All four papers are devoted to problems in Number Theory. In Paper I, a special class of local ζ-functions is studied. The main theorem states that the functions have all zeros on the line Re(s)=1/2.This is a natural generalization of the result of Bump and Ng stating that the zeros of the Mellin transform of Hermite functions have Re(s)=1/2.In Paper II and Paper III we study eigenfunctions of desymmetrized quantized cat maps.If N denotes the inverse of Planck's constant, we show that the behavior of the eigenfunctions is very dependent on the arithmetic properties of N. If N is a square, then there are normalized eigenfunctions with supremum norm equal to <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?N%5E%7B1/4%7D" />, but if N is a prime, the supremum norm of all eigenfunctions is uniformly bounded. We prove the sharp estimate <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5C%7C%5Cpsi%5C%7C_%5Cinfty=O(N%5E%7B1/4%7D)" /> for all normalized eigenfunctions and all $N$ outside of a small exceptional set. For normalized eigenfunctions of the cat map (not necessarily desymmetrized), we also prove an entropy estimate and show that our functions satisfy equality in this estimate.We call a special class of eigenfunctions newforms and for most of these we are able to calculate their supremum norm explicitly.For a given <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?N=p%5Ek" />, with k&gt;1, the newforms can be divided in two parts (leaving out a small number of them in some cases), the first half all have supremum norm about <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?2/%5Csqrt%7B1%5Cpm%201/p%7D" /> and the supremum norm of the newforms in the second half have at most three different values, all of the order <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?N%5E%7B1/6%7D" />. The only dependence of A is that the normalization factor is different if A has eigenvectors modulo p or not. We also calculate the joint value distribution of the absolute value of n different newforms.In Paper IV we prove a generalization of Mertens' theorem to Beurling primes, namely that \lim_{n \to \infty}\frac{1}{\ln n}\prod_{p \leq n} \left(1-p^{-1}\right)^{-1}=Ae^{\gamma}<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%5Cfrac%7B1%7D%7B%5Cln%20n%7D%5Cprod_%7Bp%20%5Cleq%20n%7D%0A%5Cleft(1-p%5E%7B-1%7D%5Cright)%5E%7B-1%7D=Ae%5E%7B%5Cgamma%7D," />where γ is Euler's constant and Ax is the asymptotic number of generalized integers less than x. Thus the limit <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?M=%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cleft(%5Csum_%7Bp%5Cle%20n%7Dp%5E%7B-1%7D-%5Cln(%5Cln%20n)%5Cright)" />exists. We also show that this limit coincides with <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Clim_%7B%5Calpha%5Cto%200%5E+%7D%0A%5Cleft(%5Csum_p%20p%5E%7B-1%7D(%5Cln%20p)%5E%7B-%5Calpha%7D-1/%5Calpha%5Cright)" /> ; for ordinary primes this claim is called Meissel's theorem. Finally we will discuss a problem posed by Beurling, namely how small |N(x)-[x] | can be made for a Beurling prime number system Q≠P, where P is the rational primes. We prove that for each c&gt;0 there exists a Q such that |N(x)-[x] | / QC 20100902

Probiotics and prebiotic are becoming increasingly important to consumers to alleviate issues surrounding gut health, despite the lack of definitive efficacy studies to support health claims. The addition of both probiotics and prebiotics to foods is challenging due to the harsh environmental conditions within the food itself and during transit through the gastrointestinal (GI) tract. To circumvent these challenges encapsulation technology is being explored to protect sensitive ingredients and to control their release within the lower intestines thereby maximizing the health benefiting effects. The overall goal of this research was to design a protein delivery capsule using phase separated pea protein isolate (PPI)-alginate (AL) mixtures for the entrapment of the synbiot which includes the probiotics, Bifidobacterium adolescentis, and the prebiotic, fructooligosaccharides (FOS), such that the capsule design provides highly effective protection and release within the GI tract. Research was carried out in three studies.<p> In study 1, PPIn (native isolate) and AL interactions were studied in dilute aqueous solutions as a function of pH and biopolymer mixing ratio. Turbidimetric analysis and electrophoretic mobility during an acid titration was used to determine conditions where phase separation occurred. Critical structure forming events associated with the formation of soluble and insoluble complexes in a 1:1 PPIn-AL mixture were found to occur at pH 5.00 and 2.98, respectively, with optimal interactions occurring at pH 2.10. As the PPIn-AL ratio increased, critical pH values shifted towards higher pH until a mixing ratio between 4:1 and 8:1was reached, above which structure formation became independent of the ratios through to ratios of 20:1. Electrophoretic mobility measurements showed a similar trend, where the isoelectric point (pI) shifted from pH 4.00 (homogeneous PPIn) to pH 1.55 (1:1 PPIn-AL). As the ratio increased towards 8:1 PPIn-AL, net neutrality values shifted to higher pHs (~3.80) before becoming constant at higher ratios. Maximum coacervate formation occurred at a mixing ratio of 4:1. Based on these findings, capsule design by segregative phase separation was only used in future studies, due to the acidic nature associated with associative phase separation.<p> In study 2, capsule formation using a native and commercial PPI was studied, and showed no difference between the two formulations during challenge experiments in simulated gastric juice (SGJ). As a result study 3 focused on optimization and characterization of capsules prepared using the commercial PPI. Capsule designs were investigated as a function of protein concentration, prebiotic level, and extrusion conditions (20 vs. 27 G needle) in order to determine protective ability for B. adolescentis within SGJ. Capsule designs were also measured in terms of protein and prebiotic retention during the encapsulation process, geometric mean diameter and size distribution, swelling behaviour and release characteristics within simulated intestinal fluids (SIF). All capsules provided adequate protection over the 2 h duration within SGJ. Capsule breakdown and release was similar for all designs within SIF, with a release mechanism believed to be tied to enzymatic degradation of the PPI material within the wall matrix and/or the amount of excessive Na+ present in the SIF. Capsule size was found to be dependent only on the needle gauge used in the extrusion process. Swelling behaviour of the capsules with SGJ was also found to be dependent only on the protein concentration, where capsules shrank once immersed in SGJ.<p> A 2.0% PPI-0.5% AL capsule without FOS and extruded through a 20 G needle represents the best and most cost effective design for entrapping, protecting and delivering probiotic bacteria. Future work to establish the role FOS could play post-release as the entrapping probiotics colonize the GI tract, and the protective effect of the capsules wall on FOS structure during transit is recommended.

Electroosmotic flow is widely used as a solution pumping method in numerous microfluidic applications. This type of flow has several advantages over other pumping techniques, such as the fast response time, the ease of control and integration in different microchannel designs. The flow utilizes the scaling of channel dimensions, which enhances the effects of the electrostatic forces to create flow in microchannels under an electrical body force. However, the electrostatic properties of the solution/wall material pairings are unique and must be experimentally measured. As a consequence, accurate knowledge about the electrostatic properties of the solution and wall material pairings is important for the optimal design of microfluidic devices using electroosmotic flow. Moreover, the introduction of new solutions and new channel materials for different applications is common in the microfluidics area. Therefore, any improvement on the experimental techniques used to examine the electrostatic properties of microchannels is beneficial to the research community. In this work, an improvement to the current-monitoring technique for studying the electrokinetic properties of microchannels is achieved by replacing the conventional straight channel design with a new Y-channel design. The errors from both the undesired pressure driven flow and solution electrolysis were addressed and significantly reduced. The new design offers high accuracy in finding the electrokinetic properties of microchannels. The experimental outcome from the new channel design is better compared to the outcomes of the straight channel, which helps in distinguishing the important electroosmotic pumping regions from the current-time plot. Moreover the time effectiveness in performing the experiments with the new channel design is better compared to that for the straight channel design. A modified analysis approach is also presented and validated for finding the electrokinetic properties from the outcomes of the current-monitoring technique, which is called the current-slope method. This approach is validated by comparing its findings with the results of the conventional length method. It was found for most situations that the discrepancy between the two methods, the current-slope and total length method, are within the uncertainty of the experimental measurements, thus validating the new analysis approach. In situations where it is hard to distinguish the start and end of solution replacement from the current-time plot of the current-monitoring technique, the current-slope method is advised. With the new design, different parametric studies of electroosmotic flow in PDMS based microchannels are estimated. At first the zeta potential of biological buffers are studied. Moreover the effect of continuous electroosmotic pumping, the chip substrate structure, and temperature on the average zeta potential of microchannels are examined. It was found that for air plasma treated PDMS microchannels the chip substrate material does not have an effect on the average zeta potential of the microchannels. The following chemical treatments are attempted with the aim of improving the surface and electrostatic properties of PDMS based microchannels: prepolymer additive with acrylic acid, extraction of PDMS, and both heat and plasma induced HEMA (Hydroxyethyl methacrylate) grafting on the surface of PDMS. Extensive characterization is performed with different experimental methods. The stability of the artificial hydrophilic properties of the PDMS microchannels with time was improved with both the extraction and HEMA grafting techniques. On the other hand, there was no evidence of any improvement in the zeta potential of microchannels with the surface treatments.

In recent years, the moments of L-functions has been a topic of growing interest in the field of analytic number theory. New techniques, including applications of Random Matrix Theory and multiple Dirichlet series, have lead to many well-posed theorems and conjectures for the moments of various L-functions. In this thesis, we theoretically and numerically examine the integral moments of quadratic Dirichlet $L$-functions. In particular, we exhibit and discuss the conjectures for the moments which result from the applications of Random Matrix Theory, number theoretic heuristics, and the theory of multiple Dirichlet series. In the case of the cubic moment, we further numerically investigate the possible existence of additional lower order main terms.

Electroosmotic flow is widely used as a solution pumping method in numerous microfluidic applications. This type of flow has several advantages over other pumping techniques, such as the fast response time, the ease of control and integration in different microchannel designs. The flow utilizes the scaling of channel dimensions, which enhances the effects of the electrostatic forces to create flow in microchannels under an electrical body force. However, the electrostatic properties of the solution/wall material pairings are unique and must be experimentally measured. As a consequence, accurate knowledge about the electrostatic properties of the solution and wall material pairings is important for the optimal design of microfluidic devices using electroosmotic flow. Moreover, the introduction of new solutions and new channel materials for different applications is common in the microfluidics area. Therefore, any improvement on the experimental techniques used to examine the electrostatic properties of microchannels is beneficial to the research community. In this work, an improvement to the current-monitoring technique for studying the electrokinetic properties of microchannels is achieved by replacing the conventional straight channel design with a new Y-channel design. The errors from both the undesired pressure driven flow and solution electrolysis were addressed and significantly reduced. The new design offers high accuracy in finding the electrokinetic properties of microchannels. The experimental outcome from the new channel design is better compared to the outcomes of the straight channel, which helps in distinguishing the important electroosmotic pumping regions from the current-time plot. Moreover the time effectiveness in performing the experiments with the new channel design is better compared to that for the straight channel design. A modified analysis approach is also presented and validated for finding the electrokinetic properties from the outcomes of the current-monitoring technique, which is called the current-slope method. This approach is validated by comparing its findings with the results of the conventional length method. It was found for most situations that the discrepancy between the two methods, the current-slope and total length method, are within the uncertainty of the experimental measurements, thus validating the new analysis approach. In situations where it is hard to distinguish the start and end of solution replacement from the current-time plot of the current-monitoring technique, the current-slope method is advised. With the new design, different parametric studies of electroosmotic flow in PDMS based microchannels are estimated. At first the zeta potential of biological buffers are studied. Moreover the effect of continuous electroosmotic pumping, the chip substrate structure, and temperature on the average zeta potential of microchannels are examined. It was found that for air plasma treated PDMS microchannels the chip substrate material does not have an effect on the average zeta potential of the microchannels. The following chemical treatments are attempted with the aim of improving the surface and electrostatic properties of PDMS based microchannels: prepolymer additive with acrylic acid, extraction of PDMS, and both heat and plasma induced HEMA (Hydroxyethyl methacrylate) grafting on the surface of PDMS. Extensive characterization is performed with different experimental methods. The stability of the artificial hydrophilic properties of the PDMS microchannels with time was improved with both the extraction and HEMA grafting techniques. On the other hand, there was no evidence of any improvement in the zeta potential of microchannels with the surface treatments.

In recent years, the moments of L-functions has been a topic of growing interest in the field of analytic number theory. New techniques, including applications of Random Matrix Theory and multiple Dirichlet series, have lead to many well-posed theorems and conjectures for the moments of various L-functions. In this thesis, we theoretically and numerically examine the integral moments of quadratic Dirichlet $L$-functions. In particular, we exhibit and discuss the conjectures for the moments which result from the applications of Random Matrix Theory, number theoretic heuristics, and the theory of multiple Dirichlet series. In the case of the cubic moment, we further numerically investigate the possible existence of additional lower order main terms.

Probiotics and prebiotic are becoming increasingly important to consumers to alleviate issues surrounding gut health, despite the lack of definitive efficacy studies to support health claims. The addition of both probiotics and prebiotics to foods is challenging due to the harsh environmental conditions within the food itself and during transit through the gastrointestinal (GI) tract. To circumvent these challenges encapsulation technology is being explored to protect sensitive ingredients and to control their release within the lower intestines thereby maximizing the health benefiting effects. The overall goal of this research was to design a protein delivery capsule using phase separated pea protein isolate (PPI)-alginate (AL) mixtures for the entrapment of the synbiot which includes the probiotics, Bifidobacterium adolescentis, and the prebiotic, fructooligosaccharides (FOS), such that the capsule design provides highly effective protection and release within the GI tract. Research was carried out in three studies.<p> In study 1, PPIn (native isolate) and AL interactions were studied in dilute aqueous solutions as a function of pH and biopolymer mixing ratio. Turbidimetric analysis and electrophoretic mobility during an acid titration was used to determine conditions where phase separation occurred. Critical structure forming events associated with the formation of soluble and insoluble complexes in a 1:1 PPIn-AL mixture were found to occur at pH 5.00 and 2.98, respectively, with optimal interactions occurring at pH 2.10. As the PPIn-AL ratio increased, critical pH values shifted towards higher pH until a mixing ratio between 4:1 and 8:1was reached, above which structure formation became independent of the ratios through to ratios of 20:1. Electrophoretic mobility measurements showed a similar trend, where the isoelectric point (pI) shifted from pH 4.00 (homogeneous PPIn) to pH 1.55 (1:1 PPIn-AL). As the ratio increased towards 8:1 PPIn-AL, net neutrality values shifted to higher pHs (~3.80) before becoming constant at higher ratios. Maximum coacervate formation occurred at a mixing ratio of 4:1. Based on these findings, capsule design by segregative phase separation was only used in future studies, due to the acidic nature associated with associative phase separation.<p> In study 2, capsule formation using a native and commercial PPI was studied, and showed no difference between the two formulations during challenge experiments in simulated gastric juice (SGJ). As a result study 3 focused on optimization and characterization of capsules prepared using the commercial PPI. Capsule designs were investigated as a function of protein concentration, prebiotic level, and extrusion conditions (20 vs. 27 G needle) in order to determine protective ability for B. adolescentis within SGJ. Capsule designs were also measured in terms of protein and prebiotic retention during the encapsulation process, geometric mean diameter and size distribution, swelling behaviour and release characteristics within simulated intestinal fluids (SIF). All capsules provided adequate protection over the 2 h duration within SGJ. Capsule breakdown and release was similar for all designs within SIF, with a release mechanism believed to be tied to enzymatic degradation of the PPI material within the wall matrix and/or the amount of excessive Na+ present in the SIF. Capsule size was found to be dependent only on the needle gauge used in the extrusion process. Swelling behaviour of the capsules with SGJ was also found to be dependent only on the protein concentration, where capsules shrank once immersed in SGJ.<p> A 2.0% PPI-0.5% AL capsule without FOS and extruded through a 20 G needle represents the best and most cost effective design for entrapping, protecting and delivering probiotic bacteria. Future work to establish the role FOS could play post-release as the entrapping probiotics colonize the GI tract, and the protective effect of the capsules wall on FOS structure during transit is recommended.

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