The significant increases in absorption, temperature elevation, and cell death with inclusion of SWNHs in laser therapy demonstrate the potential of their use as agents for enhancing photothermal tumor destruction.

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We derive a general upper bound for the number of incidences with k-dimensional varieties in R d . The leading term of this new bound generalizes previous bounds for the special cases of k = 1, k = d − 1, and k = d/2, to every 1 ≤ k < d. We derive lower bounds showing that this leading term is tight in various cases. We derive a bound for incidences with transverse varieties, generalizing a result of Solymosi and Tao. Finally, we derive a bound for incidences with hyperplanes in C d , which is also tight in some cases. (In both R d and C d , the bounds are tight up to sub-polynomial factors.)To prove our incidence bounds, we define the dimension ratio of an incidence problem. This ratio provides an intuitive approach for deriving incidence bounds and isolating the main difficulties in each proof. We rely on the dimension ratio both in R d and in C d , and also in some of our lower bounds.

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ABSTRACT. Many fundamental questions in additive number theory (such as Goldbach's conjecture, Fermat's last theorem, and the Twin Primes conjecture) can be expressed in the language of sum and difference sets. As a typical pair of elements contributes one sum and two differences, we expect that |A − A| > |A + A| for a finite set A. However, in 2006 Martin and O'Bryant showed that a positive proportion of subsets of {0, . . . , n} are sum-dominant, and Zhao later showed that this proportion converges to a positive limit as n → ∞. Related problems, such as constructing explicit families of sum-dominant sets, computing the value of the limiting proportion, and investigating the behavior as the probability of including a given element in A to go to zero, have been analyzed extensively.We consider many of these problems in a more general setting. Instead of just one set A, we study sums and differences of pairs of correlated sets (A, B). Specifically, we place each element a ∈ {0, . . . , n} in A with probability p, while a goes in B with probability ρ 1 if a ∈ A and probability ρ 2 if a ∈ A. If |A + B| > |(A − B) ∪ (B − A)|, we call the pair (A, B) a sum-dominant (p, ρ 1 , ρ 2 )-pair. We prove that for any fixed ρ = (p, ρ 1 , ρ 2 ) in (0, 1) 3 , (A, B) is a sum-dominant (p, ρ 1 , ρ 2 )-pair with positive probability, and show that this probability approaches a limit P ( ρ). Furthermore, we show that the limit function P ( ρ) is continuous. We also investigate what happens as p decays with n, generalizing results of Hegarty-Miller on phase transitions. Finally, we find the smallest sizes of MSTD pairs.