Martin’s maximum revisited

We present several results relating the general theory of the stationary tower forcing developed by Woodin with forcing axioms. In particular we show that, in combination with class many Woodin cardinals, the forcing axiom MM⁺⁺ makes the \({\Pi_2}\)-fragment of the theory of \({H_{\aleph_2}}\) invariant with respect to stationary set preserving forcings that preserve BMM. We argue that this is a promising generalization to \({H_{\aleph_2}}\) of Woodin’s absoluteness results for \({L(\mathbb{R})}\). In due course of proving this, we shall give a new proof of some of these results of Woodin. Finally we relate our generic absoluteness results with the resurrection axioms introduced by Hamkins and Johnstone and with their unbounded versions introduced by Tsaprounis.     

Martin’s axiom and the transitivity ofP c-points

It is shown to be consistent with Martin’s Axiom and >ℵ₁ that every twoP _c-points inβℕ\ℕ have the same topological type. This research was partially supported by NSERC. The author would also like to thank Saharon Shelah for making some enlightening remarks, on an earlier version of this paper, which resulted in Definition 2.3.     

Baumgartnerʼs conjecture and bounded forcing axioms

We study the spectrum of forcing notions between the iterations of σ-closed followed by ccc forcings and the proper forcings. This includes the hierarchy of α-proper forcings for indecomposable countable ordinals α, the Axiom A forcings and forcings completely embeddable into an iteration of a σ-closed followed by a ccc forcing. For the latter class, we present an equivalent characterization in terms of Baumgartner?s Axiom A. This resolves a conjecture of Baumgartner from the 1980s.     

Extension of Talenti’s Inequality and Maximum Values Relative to Rearrangement Classes

The article starts by revisiting and extending the Talenti’s inequality where the sharpness of the extended inequality is also addressed. The process leading to the extension comprises two steps. First, an observation that the Talenti’s inequality indeed can be formulated in terms of a rearrangement class. Second, proving that the inequality holds even when the rearrangement class is replaced by a much bigger (modulo trivial cases) set namely an appropriate closure of the class. The article then continues to introduce and explore a related maximization problem, associated to the classical Poisson equation, where the admissible set is the class of rearrangements of a given function. The article briefly explains the physical interest in this optimization problem. The existence of optimal solutions is proved and the optimality conditions they satisfy are explicitly derived. The particular case where the rearrangement class is built out of a characteristic function is also discussed.     

Distributive proper forcing axiom and a left-right dichotomy of Cichoń’s diagram

In this paper, we study distributive proper forcing axiom (DPFA) and prove its consistency with a dichotomy of the Cichoń’s diagram, relative to certain large cardinal assumption. Namely, we evaluate the cardinal invariants in Cichoń’s diagram with the first two uncountable cardinals in the way that the left-hand side has the least possible cardinality while the right-hand side has the largest possible value, and preserve the evaluation along the way of forcing DPFA.     

A Solution to Fuller’s Problem Using Constructions of Pontryagin’s Maximum Principle

The classical two-dimensional Fuller problem is considered. The boundary value problem of Pontryagin’s maximum principle is considered. Based on the central symmetry of solutions to the boundary value problem, the Pontryagin maximum principle as a necessary condition of optimality, and the hypothesis of the form of the switching line, a solution to the boundary value problem is constructed and its optimality is substantiated. Invariant group analysis is in this case not used. The results are of considerable methodological interest.     

A New Discrete Analogue of Pontryagin’s Maximum Principle

By introducing the concept of a γ-convex set, a new discrete analogue of Pontryagin’s maximum principle is obtained. By generalizing the concept of the relative interior of a set, an equality-type optimality condition is proved, which is called by the authors the Pontryagin equation.     

On Calabi’s Strong Maximum Principle via Local Semi-Dirichlet Forms

We give a stochastic proof of an extension of E. Calabi??s strong maximum principle under some geometric conditions in the framework of strong Feller diffusion processes associated to local regular semi-Dirichlet forms with lower bounds. As a corollary, our notion of subharmonicity implies a notion of viscosity subsolution in a stochastic sense. We can apply our result to singular geometric object like Alexandrov space, limit space under spectral distance of Riemannian manifolds with uniform lower Ricci curvature bound and so on.     

Bergmann’s Dilemma and Internalism’s Escape

Michael Bergmann has argued that internalist accounts of justification face an insoluble dilemma. This paper begins with an explanation of Bergmann??s dilemma. Next, I review some recent attempts to answer the dilemma, which I argue are insufficient to overcome it. The solution I propose presents an internalist account of justification through direct acquaintance. My thesis is that direct acquaintance can provide subjective epistemic assurance without falling prey to the quagmire of difficulties that Bergmann alleges all internalist accounts of justification cannot surmount.     

Warfield’s Duality and Malcev’s Matrix

In this work, we investigate relations between Malcev’s matrices of a torsion-free group G of finite rank and Malcev’s matrices of groups Hom(R,G) and Hom(G,R), where G is a locally free group and R is a torsion-free group of rank 1.     

Pontryagin’s Maximum Principle for Multidimensional Control Problems in Image Processing

In the present paper, we prove a substantially improved version of the Pontryagin maximum principle for convex multidimensional control problems of Dieudonné-Rashevsky type. Although the range of the operator describing the first-order PDE system involved in this problem has infinite codimension, we obtain first-order necessary conditions in a completely analogous form as in the one-dimensional case. Furthermore, the adjoint variables are subjected to a Weyl decomposition. We reformulate two basic problems of mathematical image processing (determination of optical flow and shape from shading problem) within the framework of optimal control, which gives the possibility to incorporate hard constraints in the problems. In the convex case, we state the necessary optimality conditions for these problems.     

Maximum modulus points of meromorphic functions and Valironʼs defect

The initial question of Paul Erdös concerned the existence of a non-polynomial entire function with the number of maximum modulus points tending to infinity. Later on the issue was considered also for meromorphic functions, leading to some interesting results on separated maximum modulus points and, in particular, their connection with Petrenkoʼs deviation and Valironʼs defect.