Equilibrium with strategy-dependent trembles

Trembling-hand arguments, introduced by Selten, where players make errors in implementing a strategy, are a successful and intuitively plausible approach to eliminating some equilibria. Typically, these errors are independent of the strategy the player aims to implement. In many contexts, however, when a player aims at one strategy, some errors are more likely than if a different strategy is the target. We introduce the notion of strategy-dependent trembles where each strategy intended by a player implies a different probability vector of errors over other strategies. This seemingly minor but plausible change makes a major difference. The focus becomes a player's own errors rather than the errors of others. A weakly dominated strategy is not necessarily eliminated if less costly errors arise when aiming at it than at the dominating strategy. Almost every Nash equilibrium in pure strategies is a strategy-dependent trembling-hand perfect equilibrium. We thank the associate editor, two referees, James Friedman, and audiences at the Public Choice Society Meetings and Southeastern Economic Theory Meetings for comments and the Public Utility Research Center and the College of Business Administration at the University of Florida for financial support.     

The impact of bubbles on measurement of drug release from echogenic liposomes

Echogenic liposomes (ELIP) encapsulate gas bubbles and drugs within lipid vesicles, but the mechanisms of ultrasound-mediated drug release from ELIP are not well understood. The effect of cavitation activity on drug release from ELIP was investigated in flowing solutions using two fluorescent molecules: a lipophilic drug (rosiglitazone) and a hydrophilic drug substitute (calcein). ELIP samples were exposed to pulsed Doppler ultrasound from a clinical diagnostic ultrasound scanner at pressures above and below the inertial and stable cavitation thresholds. Control samples were exposed to a surfactant, Triton X-100 (positive control), or to flow alone (negative control). Fluorescence techniques were used to detect release. Encapsulated microbubbles reduced the measured fluorescence intensity and this effect should be considered when assessing drug release from ELIP. The origin of this effect is not specific to ELIP. Release of rosiglitazone or calcein compared to the negative control was only observed with detergent treatment, but not with ultrasound exposure, despite the presence of stable and inertial cavitation activity. Release of rosiglitazone or calcein from ELIP exposed to diagnostic ultrasound was not observed, even in the presence of cavitation activity. Ultrasound-mediated drug delivery strategies with ELIP will thus rely on passage of the drug-loaded liposomes to target tissues.     

Surjectivity for Hamiltonian loop group spaces

Let G be a compact Lie group, and let LG denote the corresponding loop group. Let (X,) be a weakly symplectic Banach manifold. Consider a Hamiltonian action of LG on (X,), and assume that the moment map :XL ^* is proper. We consider the function ||²:X, and use a version of Morse theory to show that the inclusion map j:^-1(0)X induces a surjection j ^*:H _G ^*(X)H _G ^*(^-1(0)), in analogy with Kirwans surjectivity theorem in the finite-dimensional case. We also prove a version of this surjectivity theorem for quasi-Hamiltonian G-spaces.     

A three-dimensional steady-state tumor system

The growth of tumors can be modeled as a free boundary problem involving partial differential equations. We consider one such model and compute steady-state solutions for this model. These solutions include radially symmetric solutions where the free boundary is a sphere and nonradially symmetric solutions. Linear and nonlinear stability for these solutions are determined numerically.     

Normal surfaces as combinatorial slicings

We investigate slicings of combinatorial manifolds as properly embedded co-dimension 1 submanifolds. Focus is given to the case of dimension 3, where slicings are (discrete) normal surfaces. For the cases of 2-neighborly 3-manifolds as well as quadrangulated slicings, lower bounds on the number of quadrilaterals of slicings depending on its genus g are presented. These are shown to be sharp for infinitely many values of g. Furthermore, we classify slicings of combinatorial 3-manifolds which are weakly neighborly polyhedral maps.     

Rebuttal of Kowalenko’s Paper As Concerns the Irrationality of Euler’s Constant γ

We rebut Kowalenko??s claims in this journal that he proved the irrationality of Euler??s constant ??, and that his rational series for ?? is new.     

Variants of Dynamic Mode Decomposition: Boundary Condition, Koopman, and Fourier Analyses

Dynamic mode decomposition (DMD) is an Arnoldi-like method based on the Koopman operator. It analyzes empirical data, typically generated by nonlinear dynamics, and computes eigenvalues and eigenmodes of an approximate linear model. Without explicit knowledge of the dynamical operator, it extracts frequencies, growth rates, and spatial structures for each mode. We show that expansion in DMD modes is unique under certain conditions. When constructing mode-based reduced-order models of partial differential equations, subtracting a mean from the data set is typically necessary to satisfy boundary conditions. Subtracting the mean of the data exactly reduces DMD to the temporal discrete Fourier transform (DFT); this is restrictive and generally undesirable. On the other hand, subtracting an equilibrium point generally preserves the DMD spectrum and modes. Next, we introduce an ??optimized?? DMD that computes an arbitrary number of dynamical modes from a data set. Compared to DMD, optimized DMD is superior at calculating physically relevant frequencies, and is less numerically sensitive. We test these decomposition methods on data from a two-dimensional cylinder fluid flow at a Reynolds number of?60. Time-varying modes computed from the DMD variants yield low projection errors.     

T1Effects in Sequential Dynamic Susceptibility Contrast Experiments

Residual effects of an initial bolus of gadolinium contrast agent have been previously demonstrated in sequential dynamic susceptibility contrast MR experiments. While these residual effects quickly reach a saturation steady state, their etiology is uncertain, and they can lead to spurious estimates of hemodynamic parameters in activation experiments. The possible influence ofT₁effects is now investigated with experiments in whichT₁weighting is varied as well as with serial regionalT₁measurements. Little evidence for significant residualT₁effects is found, suggesting instead that susceptibility effects underlie these observations. An initial saturation dose of contrast agent minimizes this effect.     

On SA, CA, and GA numbers

Gronwall’s function G is defined for n>1 by $G(n)=\frac{\sigma(n)}{n \log\log n}$ where σ(n) is the sum of the divisors of n. We call an integer N>1 a GA1 number if N is composite and G(N)≥G(N/p) for all prime factors p of N. We say that N is a GA2 number if G(N)≥G(aN) for all multiples aN of N. In (Caveney et al. Integers 11:A33, 2011), we used Robin’s and Gronwall’s theorems on G to prove that the Riemann Hypothesis (RH) is true if and only if 4 is the only number that is both GA1 and GA2. In the present paper, we study GA1 numbers and GA2 numbers separately. We compare them with superabundant (SA) and colossally abundant (CA) numbers (first studied by Ramanujan). We give algorithms for computing GA1 numbers; the smallest one with more than two prime factors is 183783600, while the smallest odd one is 1058462574572984015114271643676625. We find nineteen GA2 numbers ≤5040, and prove that a GA2 number N>5040 exists if and only if RH is false, in which case N is even and >10⁸⁵⁷⁶.     

The -homology class of the Euler characteristic operator is trivial

On any manifold , the de Rham operator (with respect to a complete Riemannian metric), with the grading of forms by parity of degree, gives rise by Kasparov theory to a class , which when is closed maps to the Euler characteristic in . The purpose of this note is to give a quick proof of the (perhaps unfortunate) fact that is as trivial as it could be subject to this constraint. More precisely, if is connected, lies in the image of (induced by the inclusion of a basepoint into ).