Let (M, g) be a compact Riemannian manifold without boundary, with dimM ≥ 3, and f : R→ R a continuous function which is sublinear at infinity. By various variational approaches, existence of multiple solutions of the eigenvalue problem −∆gω + α(σ)ω = K̃(λ, σ)f(ω), σ ∈M, ω ∈ H 1 (M), is established for certain eigenvalues λ > 0, depending on further properties of f and on explicit forms of the f...

This paper investigates some fundamental properties of 2-way alternating multi-counter automata (2amca’s) with only existential (universal) states which have sublinear space and 1 inkdot. It is shown that for any function s(n) ≥ log n such that log s(n) = o(log n), s(n) space-bounded 1-inkdot 2amca’s with only existential states are incomparable with the ones with only universal states, and the...

For sublinear-time algorithms, it is crucial how input is represented and accessed. Depending on input representation and queries allowed, we might or might not be able to solve a given problem in sublinear time. In addition, in the property testing model, the correctness guarantee of the algorithm depends on the distance between objects, which is typically normalized by the size of the input. ...

We study the existence of optimal actions in a zero-sum game infτ supP E P [Xτ ] between a stopper and a controller choosing a probability measure. This includes the optimal stopping problem infτ E(Xτ ) for a class of sublinear expectations E(·) such as the G-expectation. We show that the game has a value. Moreover, exploiting the theory of sublinear expectations, we define a nonlinear Snell en...

We consider dynamic sublinear expectations (i.e., time-consistent coherent risk measures) whose scenario sets consist of singular measures corresponding to a general form of volatility uncertainty. We derive a càdlàg nonlinear martingale which is also the value process of a superhedging problem. The superhedging strategy is obtained from a representation similar to the optional decomposition. F...

(f2) lim u→∞ f(u) u =f+ ¿ 0; lim u→−∞ f(u) u =f− ¿ 0: For the de5niteness, we assume f+ ¿f−, and when f+ =f−, we use f± to represent it. We will consider f being either superlinear or sublinear. f is said to be superlinear if f(u)=u is decreasing in (0;∞) and is increasing in (−∞; 0); and f is said to be sublinear if f(u)=u is increasing in (0;∞) and is decreasing in (−∞; 0): The semilinear equ...

We consider the domination number for on-line social networks, both in a stochastic network model, and for real-world, networked data. Asymptotic sublinear bounds are rigorously derived for the domination number of graphs generated by the memoryless geometric protean random graph model. We establish sublinear bounds for the domination number of graphs in the Facebook 100 data set, and these bou...