More Like this

1. Sealing from iterability

   Sargsyan, Grigor; Trang, Nam (January 2021, Transactions of the American Mathematical Society)

   null (Ed.)

   We show that if V has a proper class ofWoodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy (as defined in the paper) then Sealing holds after collapsing the successor of the least strong cardinal to be countable. This result is complementary to other work by the authors where it is shown that Sealing holds in a generic extension of a certain minimal universe. The current theorem is more general in that no minimality assumption is needed. A corollary of the main theorem is that Sealing is consistent relative to the existence of a Woodin cardinal which is a limit of Woodin cardinals. This improves significantly on the first consistency of Sealing obtained by W.H. Woodin. The Largest Suslin Axiom (LSA) is a determinacy axiom isolated byWoodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let LSA-over-uB be the statement that in all (set) generic extensions there is a model of LSA whose Suslin, co-Suslin sets are the universally Baire sets. The other main result of the paper shows that assuming V has a proper class of inaccessible cardinals which are limit of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy, in the universe V [g], where g is V -generic for the collapse of the successor of the least strong cardinal to be countable, the theory LSA-over-UB fails; this implies that LSA-over-UB is not equivalent to Sealing (over the base theory of V [g]). This is interesting and somewhat unexpected, in light of other work by the authors. Compare this result with Steel’s well-known theorem that “AD in L(R) holds in all generic extensions” is equivalent to “the theory of L(R) is sealed” in the presence of a proper class of measurable cardinals. 

   more » « less
2. SEALING OF THE UNIVERSALLY BAIRE SETS

   https://doi.org/10.1017/bsl.2021.29  

   Sargsyan, Grigor; Trang, Nam (October 2021, The bulletin of symbolic logic)

   A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by set forcings. The 𝖫𝖺𝗋𝗀𝖾𝗌𝗍 𝖲𝗎𝗌𝗅𝗂𝗇 𝖠𝗑𝗂𝗈𝗆 ( 𝖫𝖲𝖠 ) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable surjections. Let 𝖫𝖲𝖠 - 𝗈𝗏𝖾𝗋 - 𝗎𝖡 be the statement that in all (set) generic extensions there is a model of 𝖫𝖲𝖠 whose Suslin, co-Suslin sets are the universally Baire sets. We outline the proof that over some mild large cardinal theory, 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 is equiconsistent with 𝖫𝖲𝖠 - 𝗈𝗏𝖾𝗋 - 𝗎𝖡 . In fact, we isolate an exact theory (in the hierarchy of strategy mice) that is equiconsistent with both (see Definition 3.1). As a consequence, we obtain that 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 is weaker than the theory “ 𝖹𝖥𝖢 + there is a Woodin cardinal which is a limit of Woodin cardinals.” This significantly improves upon the earlier consistency proof of 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 by Woodin. A variation of 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 , called 𝖳𝗈𝗐𝖾𝗋 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 , is also shown to be equiconsistent with 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 over the same large cardinal theory. We also outline the proof that if V has a proper class of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy, then 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 holds after collapsing the successor of the least strong cardinal to be countable. This result is complementary to the aforementioned equiconsistency result, where it is shown that 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 holds in a generic extension of a certain minimal universe. This theorem is more general in that no minimal assumption is needed. A corollary of this is that 𝖫𝖲𝖠 - 𝗈𝗏𝖾𝗋 - 𝗎𝖡 is not equivalent to 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 . 

   more » « less
3. On the Computational Complexity of Non-Dictatorial Aggregation

   https://doi.org/10.1613/jair.1.12476  

   Livieratos, John; Kolaitis, Phokion G.; Kirousis, Lefteris (September 2021, Journal of Artificial Intelligence Research)

   We investigate when non-dictatorial aggregation is possible from an algorithmic perspective, where non-dictatorial aggregation means that the votes cast by the members of a society can be aggregated in such a way that there is no single member of the society that always dictates the collective outcome. We consider the setting in which the members of a society take a position on a fixed collection of issues, where for each issue several different alternatives are possible, but the combination of choices must belong to a given set X of allowable voting patterns. Such a set X is called a possibility domain if there is an aggregator that is non-dictatorial, operates separately on each issue, and returns values among those cast by the society on each issue. We design a polynomial-time algorithm that decides, given a set X of voting patterns, whether or not X is a possibility domain. Furthermore, if X is a possibility domain, then the algorithm constructs in polynomial time a non-dictatorial aggregator for X. Furthermore, we show that the question of whether a Boolean domain X is a possibility domain is in NLOGSPACE. We also design a polynomial-time algorithm that decides whether X is a uniform possibility domain, that is, whether X admits an aggregator that is non-dictatorial even when restricted to any two positions for each issue. As in the case of possibility domains, the algorithm also constructs in polynomial time a uniform non-dictatorial aggregator, if one exists. Then, we turn our attention to the case where X is given implicitly, either as the set of assignments satisfying a propositional formula, or as a set of consistent evaluations of a sequence of propositional formulas. In both cases, we provide bounds to the complexity of deciding if X is a (uniform) possibility domain. Finally, we extend our results to four types of aggregators that have appeared in the literature: generalized dictatorships, whose outcome is always an element of their input, anonymous aggregators, whose outcome is not affected by permutations of their input, monotone, whose outcome does not change if more individuals agree with it and systematic, which aggregate every issue in the same way. 

   more » « less
4. Effects of swimming environment on bacterial motility

   https://doi.org/10.1063/5.0082768  

   Kim, Dokyum; Kim, Yongsam; Lim, Sookkyung (March 2022, Physics of Fluids)

   Swimming trajectories of bacteria can be altered by environmental conditions, such as background flow and physical barriers, that limit the free swimming of bacteria. We present a comprehensive model of a bacterium that consists of a rod-shaped cell body and a flagellum which is composed of a motor, a hook, and a filament. The elastic flagellum is modeled based on the Kirchhoff rod theory, the cell body is considered to be a rigid body, and the hydrodynamic interaction of a bacterium near a wall is described by regularized Stokeslet formulation combined with the image system. We consider three environmental conditions: (1) a rigid surface is placed horizontally and there is no shear flow, (2) a shear fluid flow is present and the bacterium is near the rigid surface, and (3) while the bacterium is near the rigid surface and is under shear flow, an additional sidewall which is perpendicular to the rigid surface is placed. Each environmental state modifies the swimming behavior. For the first condition, there are two modes of motility, trap and escape, whether the bacterium stays near the surface or moves away from the surface as we vary the physical and geometrical properties of the model bacterium. For the second condition, there exists a threshold of shear rate that classifies the motion into two types of paths in which the bacterium takes either a periodic coil trajectory or a linear trajectory. For the last condition, the bacterium takes upstream motility along the sidewall for lower shear rates and downstream motility for larger shear flow rates. 

   more » « less
5. Dynamics of a system of two coupled third-order MEMS oscillators

   Rand, R; Shayak, B; Bhaskar, A; Zehnder, A. (November 2020, International journal of engineering research and applications)

   In this work we present a systematic review of novel and interesting behaviour we have observed in a simplified model of a MEMS oscillator. The model is third order and nonlinear, and we expressit as a single ODE for a displacement variable. We find that a single oscillator exhibits limitcycles whose amplitude is well approximated by perturbation methods. Two coupled identicaloscillators have in-phase and out-of-phase modes as well as more complicated motions.Bothof the simple modes are stable in some regions of the parameter space while the bifurcationstructure is quite complex in other regions. This structure is symmetric; the symmetry is brokenby the introduction of detuning between the two oscillators. Numerical integration of the fullsystem is used to check all bifurcation computations. Each individual oscillator is based on a MEMS structure which moves within a laser-driven interference pattern. As the structure vibrates, it changes the interference gap, causing the quantity of absorbed light to change, producing a feedback loop between the motion and the absorbed light and resulting in a limit cycle oscillation. A simplified model of this MEMS oscillator, omitting parametric feedback and structural damping, is investigated using Lindstedt's perturbation method. Conditions are derived on the parameters of the model for a limit cycle to exist. The original model of the MEMS oscillator consists of two equations: a second order ODE which describes the physical motion of a microbeam, and a first order ODE which describes the heat conduction due to the laser. Starting with these equations, we derive a single governing ODE which is of third order and which leads to the definition of a linear operator called the MEMS operator. The addition of nonlinear terms in the model is shown to produce limit cycle behavior. The differential equations of motion of the system of two coupled oscillators are numerically integrated for varying values of the coupling parameter. It is shown that the in-phase mode loses stability as the coupling parameter is reduced below a certain value, and is replaced by two new periodic motions which are born in a pitchfork bifurcation. Then as this parameter is further reduced, the form of the bifurcating periodic motions grows more complex, with yet additional bifurcations occurring. This sequence of bifurcations leads to a situation in which the only periodic motion is a stable out-of-phase mode. The complexity of the resulting sequence of bifurcations is illustrated through a series of diagrams based on numerical integration. 

   more » « less