Abstract:
Fix a Galois extension E/F of totally real number fields such that the Galois group G has exponent 2. Let S be a finite set of primes of F containing the infinite primes and all those which ramify in E, let S_E denote the primes of E lying above those in S, and let O_E^S denote the ring of S_E-integers of E. We then compare the Fitting ideal of K_2(O_E^S) as a Z[G]-module with a higher Stickelberger ideal. The two extend to the same ideal in the maximal order of Q[G], and hence in Z[1/2][G]. Results in Z[G] are obtained under the assumption of the Birch-Tate conjecture, especially for biquadratic extensions, where we compute the index of the higher Stickelberger ideal. We find a sufficient condition for the Fitting ideal to contain the higher Stickelberger ideal in the case where E is a biquadratic extension of F containing the first layer of the cyclotomic Z_2-extension of F, and describe a class of biquadratic extensions of F=Q that satisfy this condition.

Abstract:
Fix a relative quadratic extension E/F of totally real number rields and let G denote the Galois group of order 2. Let S be a finite set of primes of F containing the infinite primes and all those which ramify in E, let S_E denote the primes of E and let O_E^S denote the ring of S_E-integers of E. Assume the truth of the 2-part of the Birch-Tate conjecture relating the order of the tame kernel K_2(O_E^S) to the value of the Dedekind zeta function of E at s=-1, and assume the same for F as well. We then prove that the Fitting ideal of K_2(O_E^S) as a Z[G]-module is equal to a generalized Stickelberger ideal. Equality after tensoring with Z[1/2][G] holds unconditionally.

Abstract:
Suppose that $EE$ is a totally real number field which is the composite of all of its subfields $E$ that are relative quadratic extensions of a base field $F$. For each such $E$ with ring of integers $\O_E$, assume the truth of the Birch-Tate conjecture (which is almost fully established) relating the order of the tame kernel $K_2(\O_E)$ to the value of the Dedekind zeta function of $E$ at $s=-1$, and assume the same for $F$ as well. Excluding a certain rare situation, we prove the annihilation of $K_2(\Oc_EE)$ by a generalized Stickelberger element in the group ring of the Galois group of $EE/F$. Annihilation of the odd part of this group is proved unconditionally.

Abstract:
New Urban developments offer a physical form that differs considerably from the dominant pattern of suburbanism in North America. While theorists argue that New Urbanist principles must be adopted in their entirety, property developers often find that compromises must be made to obtain necessary government approvals. This results in “hybrid” developments that lack all of the features of true New Urbanism. Based on surveys of residents of two Canadian communities, it would appear that some of the touchstones of New Urbanism are not actually essential and that there are few significant differences in in resident satisfaction levels between residents of different types of New Urban communities.

Abstract:
We are given four cards, each containing four nonnegative real numbers, written one below the other, so that the sum of the numbers on each card is 1. We are allowed to put the cards in any order we like, then we write down the first number from the first card, the second number from the second card, the third number from the third card, and the fourth number from the fourth card, and we add these four numbers together. We wish to find real intervals $[a,b]$ with the following property: there is always an ordering of the four cards so that the above sum lies in $[a,b]$. We prove that the intervals $[1,2]$ and $[2/3,5/3]$ are solutions to this problem. It follows that $$[0,1], [1/3,4/3], [2/3,5/3], [1,2]$$ are the only minimal intervals which are solutions. We also discuss a generalization to $n$ cards, and give an equivalent formulation of our results in matrix terms.

Abstract:
Comparison of the thermodynamic entropy with Boltzmann's principle shows that under conditions of constant volume the total number of arrangements in simple thermodynamic systems with temperature-independent heat capacities is TC/k. A physical interpretation of this function is given for three such systems; an ideal monatomic gas, an ideal gas of diatomic molecules with rotational motion, and a solid in the Dulong-Petit limit of high temperature. T1/2 emerges as a natural measure of the number of accessible states for a single particle in one dimension. Extension to N particles in three dimensions leads to TC/k as the total number of possible arrangements or microstates. The different microstates of the system are thus shown a posteriori to be equally probable, with probability T-C/k, which implies that for the purposes of counting states the particles of the gas are distinguishable. The most probable energy state of the system is determined by the degeneracy of the microstates.

Abstract:
We introduce an infinite sequence of higher order Schwarzian derivatives closely related to the theory of monotone matrix functions. We generalize the classical Koebe lemma to maps with positive Schwarzian derivatives up to some order, obtaining control over derivatives of high order. For a large class of multimodal interval maps we show that all inverse branches of first return maps to sufficiently small neighbourhoods of critical values have their higher order Schwarzian derivatives positive up to any given order.

Abstract:
Side channel attacks have emerged as a serious threat to the security of both networked and embedded systems -- in particular through the implementations of cryptographic operations. Side channels can be difficult to model formally, but with careful coding and program transformation techniques it may be possible to verify security in the presence of specific side-channel attacks. But what if a program intentionally makes a tradeoff between security and efficiency and leaks some information through a side channel? In this paper we study such tradeoffs using ideas from recent research on declassification. We present a semantic model of security for programs which allow for declassification through side channels, and show how side-channel declassification can be verified using off-the-shelf software model checking tools. Finally, to make it simpler for verifiers to check that a program conforms to a particular side-channel declassification policy we introduce a further tradeoff between efficiency and verifiability: by writing programs in a particular "manifest form" security becomes considerably easier to verify.

Abstract:
Differentially private mechanisms enjoy a variety of composition properties. Leveraging these, McSherry introduced PINQ (SIGMOD 2009), a system empowering non-experts to construct new differentially private analyses. PINQ is an LINQ-like API which provides automatic privacy guarantees for all programs which use it to mediate sensitive data manipulation. In this work we introduce featherweight PINQ, a formal model capturing the essence of PINQ. We prove that any program interacting with featherweight PINQ's API is differentially private.