On integer partitions corresponding to numerical semigroups

Numerical semigroups are cofinite additive submonoids of the natural numbers. In 2011, Keith and Nath illustrated an injection from numerical semigroups to integer partitions. We explore this connection between partitions and numerical semigroups with a focus on classifying the partitions that appear in the image of the injection from numerical semigroups. In particular, we count the number of partitions that correspond to numerical semigroups in terms of genus, Frobenius number, and multiplicity, with some restrictions

Hook length biases and general linear partition inequalities

Motivated in part by hook-content formulas for certain restricted partitions in representation theory, we consider the total number of hooks of fixed length in odd versus distinct partitions. We show that there are more hooks of length $2$, respectively $3$, in all odd partitions of $n$ than in all distinct partitions of $n$, and make the analogous conjecture for arbitrary hook length $t \geq 2$. We also establish additional bias results on the number of gaps of size $1,$ respectively $2$, in all odd versus distinct partitions of $n$. We conjecture similar biases and asymptotics, as well as congruences for the number of hooks of fixed length in odd distinct partitions versus self-conjugate partitions. An integral component of the proof of our bias result for hooks of length $3$ is a linear inequality involving $q(n)$, the number of distinct partitions of $n$. In this article we also establish effective linear inequalities for $q(n)$ in great generality, a result which is of independent interest. Our methods are both analytic and combinatorial, and our results and conjectures intersect the areas of representation theory, analytic number theory, partition theory, and $q$-series. In particular, we use a Rademacher-type exact formula for $q(n),$ Wright's circle method, modularity, $q$-series transformations, asymptotic methods, and combinatorial arguments.Comment: 36 page

Aggressive mammary carcinoma progression in Nrf2 knockout mice treated with 7,12-dimethylbenz[a]anthracene

<p>Abstract</p> <p>Background</p> <p>Activation of nuclear factor erythroid 2-related factor (Nrf2), which belongs to the basic leucine zipper transcription factor family, is a strategy for cancer chemopreventive phytochemicals. It is an important regulator of genes induced by oxidative stress, such as glutathione S-transferases, heme oxygenase-1 and peroxiredoxin 1, by activating the antioxidant response element (ARE). We <it>hypothesized </it>that (1) the citrus coumarin auraptene may suppress premalignant mammary lesions via activation of Nrf2/ARE, and (2) that Nrf2 knockout (KO) mice would be more susceptible to mammary carcinogenesis.</p> <p>Methods</p> <p>Premalignant lesions and mammary carcinomas were induced by medroxyprogesterone acetate and 7,12-dimethylbenz[a]anthracene treatment. The 10-week pre-malignant study was performed in which 8 groups of 10 each female wild-type (WT) and KO mice were fed either control diet or diets containing auraptene (500 ppm). A carcinogenesis study was also conducted in KO vs. WT mice (n = 30-34). Comparisons between groups were evaluated using ANOVA and Kaplan-Meier Survival statistics, and the Mann-Whitney U-test.</p> <p>Results</p> <p>All mice treated with carcinogen exhibited premalignant lesions but there were no differences by genotype or diet. In the KO mice, there was a dramatic increase in mammary carcinoma growth rate, size, and weight. Although there was no difference in overall survival, the KO mice had significantly lower mammary tumor-free survival. Also, in the KO mammary carcinomas, the active forms of NF-κB and β-catenin were increased ~2-fold whereas no differences in oxidized proteins were observed. Many other tumors were observed, including lymphomas. Interestingly, the incidences of lung adenomas in the KO mice were significantly higher than in the WT mice.</p> <p>Conclusions</p> <p>We report, for the first time, that there was no apparent difference in the formation of premalignant lesions, but rather, the KO mice exhibited rapid, aggressive mammary carcinoma progression.</p

Conditioned Place Preference as a Means to Measure Analgesic Drug Effects on Cancer Pain in Murine Models

Pain associated with cancer is unpredictable and may reflect an inability to quantify it in animals. We hypothesize that Conditioned Place Preference (CPP) testing is sensitive to pain relieving drugs in animal models of cancer pain and that CPP scores will reflect varying degrees of pain relief To test this, mice were assigned to one of four groups: tumor or non-tumor groups conditioned to saline or morphine during preference testing. After baseline preference testing in the CPP apparatus, animals received either saline or morphine in the S+ chamber (initially non-preferred chamber). All animals received saline in the S- chamber (initially preferred chamber). Preference scores for non-tumor groups showed that mice receiving saline in both S+ and S- chambers show no particular compartment preference, indicating that this apparatus and test procedure does not show any compartment bias. For non-tumor animals administered morphine, the preference score was higher than their saline control groups. We interpret this increase in preference scores to be associated with morphine’s pleasurable side effects, which known to underlie its abuse. Tumor saline mice also showed no particular compartment preference. Mice with tumors administered morphine showed an increase in preference scores over the last six trials prior to euthanasia. We interpret this increase in preference scores to be due to the increasing amounts of pain associated with tumor growth. We interpret that an increase in S+ compartment preference in tumor morphine animals reflects drug-seeking behavior that was previously associated with pain relief in this chamber. These findings show that CPP is capable of quantifying drug-seeking behavior in tumor models

Combinatorial approaches to problems about integer partitions and q-series

In this thesis, we study three types of partition-theoretic objects. The first objects studied are odd Ferrers diagrams, which we use in new combinatorial proofs of two general q-series identities containing false theta function and mock theta function identities as special cases. Additionally, we introduce a Gaussian coefficient analogue for restricted odd Ferrers diagrams. The second set of objects we study is a class of core partitions. We prove a recursive formula for the generating polynomial for the number of parts of s-core partitions into d-distinct parts with a given maximum hook length. Applications of the main result of this section include formulas for the maximum number of parts of simultaneous s- and t- core partitions for certain values of t. In the final section, we introduce a new class of objects called copartitions, which generalize a special class of partitions where all even parts are smaller than all odd parts. A simple graphical representation immediately follows from our definition. We prove an in finite product generating function for cp(n), the number of copartitions of size n. Special cases of this function include sums of the ordinary partition function p(n), a new combinatorial interpretation of the Rogers-Ramanujan functions, and quotients involving theta functions and eta functions. Furthermore, we explore the parity of cp(n) and consider a weighted count of copartitions.U of I OnlyAuthor requested U of Illinois access only (OA after 2yrs) in Vireo ETD syste

Welfare Assessment following Heterotopic or Orthotopic Inoculation of Bladder Cancer in C57BL/6 Mice

<div><p>Few studies have assessed whether mice used as cancer models experience pain. Despite this possibility, the usual practice is to withhold analgesics as these are generally viewed as confounding. However, pain also alters cancer progression, so preventing it might not only be beneficial to welfare but also to study validity. Establishing the extent to which different cancer models result in pain is an important first step towards their refinement. We used conditioned place preference (CPP) testing and body-weight and behaviour analyses to evaluate the assumption that heterotopically implanted tumours result in less pain and fewer welfare concerns than those implanted orthotopically. C57Bl/6 mice received MB49^Luc luciferase expressing bladder cancer cells or saline implanted subcutaneously or into the bladder. These tumour-bearing or control groups underwent 2 daily 45 minute conditioning trials to saline or morphine (2mg/kg) and then a 15 minute drug-free preference test on day 3 of a 3 day cycle, continuing until the study ended. Tumours were imaged and behaviour data obtained following preference tests. Development of preference for the morphine-paired chamber (morphine-seeking) was determined over time. Heterotopic tumour development had no effect on morphine-seeking, and although the restraint used for heterotopic inoculation caused greater initial weight losses than anaesthesia, these mice steadily gained weight and behaved comparatively normally throughout the study. Orthotopic tumour inoculation caused no initial weight losses, but over the final 7 days these mice became less active and lost more body weight than cancer-free controls. This indicated orthotopic implantation probably caused a more negative impact on welfare or conceivably pain; but only according to the current test methods. Pain could not be confirmed because morphine-seeking in the tumour-bearing groups was similar to that seen in controls. Imaging was not found to be an effective method of monitoring tumour development surpassing manual tumour inspection.</p></div

Behaviour results.

<p>The mean frequency of rearing (±SEM) in the orthotopic (left axis; circles) and heterotopic study (right axis; triangles) in tumour bearing and control groups (filled versus open symbols); showing the significantly greater decline in activity in the tumour-bearing mice in the orthotopic study from CPP test 3 (P_3) to just prior to euthanasia (P_1).</p

Imaging results.

<p>(a) the mean total flux (TF ±SEM) of bioluminescent signals emanating from tumours implanted orthotopically (Orth; left axis, circles) or heterotopically (Het; right axis, triangles) in mice conditioned to morphine (Mor; closed symbols) or saline (Sal; open symbols) and imaged every 3 days (beginning on day 3) for 18 days (Day_CPP Test); (b) Mean TF (±SEM) during the final 6 CPP tests (P_6 to P_1; 18 days) before euthanasia; (c) Caliper measurements showing the mean tumour surface area (mm² ±SEM) of heterotopically implanted tumours over the first 18 post-inoculation days (6 CPP tests) and (d) last 16 days (6 CPP tests) in mice conditioned to morphine or saline.</p