Abstract:
We analyze the large $N$ spectrum of chiral primary operators of three dimensional fixed points of the renormalization group. Using the space-time picture of the fixed points and the correspondence between anti-de Sitter compactifications and conformal field theories we are able to extract the dimensions of operators in short superconformal multiplets. We write down some of these operators in terms of short distance theories flowing to these non-trivial fixed points in the infrared.

Abstract:
Five dimensional field theories with exceptional gauge groups are engineered from degenerations of Calabi-Yau threefolds. The structure of the Coulomb branch is analyzed in terms of relative K\"ahler cones. For low number of flavors, the geometric construction leads to new five dimensional fixed points.

Abstract:
Sigma models describing low energy effective actions on D0-brane probes with N=8 supercharges are studied in detail using a manifestly d=1, N=4 super-space formalism. Two 0+1 dimensional N=4 multiplets together with their general actions are constructed. We derive the condition for these actions to be N=8 supersymmetric and apply these techniques to various D-brane configurations. We find that if in addition to N=8 supersymmetry the action must also have Spin(5) invariance, the form of the sigma model metric is uniquely determined by the one-loop result and is not renormalized perturbatively or non-perturbatively.

Abstract:
In this note we discuss D-branes on T^4/Z_2 using the boundary states formalism. Explicit formulas for the untwisted boundary states inherited from the underlying T^4 and twisted states corresponding to branes wrapping collapsed 2-cycles at the orbifold singularities are given. The exact CFT description of the orbifold makes it possible to study how the boundary states transform under R_i -> 1/R_i transformation on all directions of the underlying T^4. We compare their transformation law with results obtained from world volume considerations.

Abstract:
One of the key activities of any client is contractor selection. Without a suitable and precise method for selecting the best contractor, the completion of a project will likely be affected. In this study, we examine the use of the analytical hierarchy process (AHP) as a decision-support model for contractor selection. This model can assist project management teams in identifying contractors who are most likely to deliver satisfactory outcomes in a selection process that is not based simply on the lowest bid. In this study, an AHP-based model is tested using a hypothetical scenario in which candidate contractors are evaluated. Six criteria for the primary objective are evaluated. The criteria used for contractor selection in the model are identified, and the significance of each criterion is determined using a questionnaire. Comparisons are made by ranking the aggregate score of each candidate based on each criterion, and the candidate with the highest score is deemed the best. This study contributes to the construction sector in two ways: first, it extends the understanding of selection criteria to include degrees of importance, and second, it implements a multi-criteria AHP approach, which is a new method for analyzing and selecting the best contractor.

Abstract:
We prove an analogue of the classical Bateman-Horn conjecture on prime values of polynomials for the ring of polynomials over a large finite field. Namely, given non-associate, irreducible, separable and monic (in the variable $x$) polynomials $F_1,\ldots,F_m\in\mathbf{F}_q[t][x]$, with $q$ odd, we show that the number of $f\in\mathbf{F}_q[t]$ of degree $n\ge\max(3,\mathrm{deg}_t F_1,\ldots,\mathrm{deg}_t F_m)$ such that all $F_i(t,f)\in\mathbf{F}_q[t],1\le i\le m$ are irreducible is $$\left(\prod_{i=1}^m\frac{\mu_i}{N_i}\right) q^{n+1}\left(1+O_{m,\,\max\mathrm{deg} F_i,\,n}\left(q^{-1/2}\right)\right),$$ where $N_i=n\mathrm{deg}_xF_i$ is the generic degree of $F_i(t,f)$ for $\mathrm{deg} f=n$ and $\mu_i$ is the number of factors into which $F_i$ splits over $\bar{\mathbf{F}_q}$. Our proof relies on the classification of finite simple groups. We will also prove the same result for non-associate, irreducible and separable (over $\mathbf{F}_q(t)$) polynomials $F_1,\ldots,F_m$ not necessarily monic in $x$ under the assumptions that $n$ is greater than the number of geometric points of multiplicity greater than two on the (possibly reducible) affine plane curve $C$ defined by the equation $$\prod_{i=1}^mF_i(t,x)=0$$ (this number is always bounded above by $\left(\textstyle\sum_{i=1}^m\mathrm{deg} F_i\right)^2/2$, where $\mathrm{deg}$ denotes the total degree in $t,x$) and $$p=\mathrm{char}\,\mathbf{F}_q>\max_{1\le i\le m} N_i,$$ where $N_i$ is the generic degree of $F_i(t,f)$ for $\mathrm{deg} f=n$.

Abstract:
For a function field $K$ and fixed polynomial $F\in K[x]$ and varying $f\in F$ (under certain restrictions) we give a lower bound for the degree of the greatest prime divisor of $F(f)$ in terms of the height of $f$, establishing a strong result for the function field analogue of a classical problem in number theory.

Abstract:
We give a new derivation of an identity due to Z. Rudnick and P. Sarnak about the $n$-level correlations of eigenvalues of random unitary matrices as well as a new proof of a formula due to M. Diaconis and P. Shahshahani expressing averages of trace products over the unitary matrix ensemble. Our method uses the zero statistics of Artin-Schreier L-functions and a deep equidistribution result due to N. Katz and P. Sarnak.

Abstract:
We study the distribution of the zeroes of the L-functions of curves in the Artin-Schreier family. We consider the number of zeroes in short intervals and obtain partial results which agree with a random unitary matrix model.