Proper Analytic Free Maps

This paper concerns analytic free maps. These maps are free analogs of classical analytic functions in several complex variables, and are defined in terms of non-commuting variables amongst which there are no relations - they are free variables. Analytic free maps include vector-valued polynomials in free (non-commuting) variables and form a canonical class of mappings from one non-commutative domain D in say g variables to another non-commutative domain D' in g' variables. As a natural extension of the usual notion, an analytic free map is proper if it maps the boundary of D into the boundary of D'. Assuming that both domains contain 0, we show that if f:D->D' is a proper analytic free map, and f(0)=0, then f is one-to-one. Moreover, if also g=g', then f is invertible and f^(-1) is also an analytic free map. These conclusions on the map f are the strongest possible without additional assumptions on the domains D and D'.Comment: 17 pages, final version. To appear in the Journal of Functional Analysi

Recent Issues in High-Level Perception

Recently, several theorists have proposed that we can perceive a range of high-level features, including natural kind features (e.g., being a lemur), artifactual features (e.g., being a mandolin), and the emotional features of others (e.g., being surprised). I clarify the claim that we perceive high-level features and suggest one overlooked reason this claim matters: it would dramatically expand the range of actions perception-based theories of action might explain. I then describe the influential phenomenal contrast method of arguing for high-level perception and discuss some of the objections that have been raised against this strategy. Finally, I describe two emerging defenses of high-level perception, one of which appeals to a certain class of perceptual deficits and one of which appeals to adaptation effects. I sketch a challenge for the latter approach

The convex Positivstellensatz in a free algebra

Given a monic linear pencil L in g variables let D_L be its positivity domain, i.e., the set of all g-tuples X of symmetric matrices of all sizes making L(X) positive semidefinite. Because L is a monic linear pencil, D_L is convex with interior, and conversely it is known that convex bounded noncommutative semialgebraic sets with interior are all of the form D_L. The main result of this paper establishes a perfect noncommutative Nichtnegativstellensatz on a convex semialgebraic set. Namely, a noncommutative polynomial p is positive semidefinite on D_L if and only if it has a weighted sum of squares representation with optimal degree bounds: p = s^* s + \sum_j f_j^* L f_j, where s, f_j are vectors of noncommutative polynomials of degree no greater than 1/2 deg(p). This noncommutative result contrasts sharply with the commutative setting, where there is no control on the degrees of s, f_j and assuming only p nonnegative, as opposed to p strictly positive, yields a clean Positivstellensatz so seldom that such cases are noteworthy.Comment: 22 page

Visually Perceiving the Intentions of Others

I argue that we sometimes visually perceive the intentions of others. Just as we can see something as blue or as moving to the left, so too can we see someone as intending to evade detection or as aiming to traverse a physical obstacle. I consider the typical subject presented with the Heider and Simmel movie, a widely studied ‘animacy’ stimulus, and I argue that this subject mentally attributes proximal intentions to some of the objects in the movie. I further argue that these attributions are unrevisable in a certain sense and that this result can be used to as part of an argument that these attributions are not post-perceptual thoughts. Finally, I suggest that if these attributions are visual experiences, and more particularly visual illusions, their unrevisability can be satisfyingly explained, by appealing to the mechanisms which underlie visual illusions more generally

If You Can't Change What You Believe, You Don't Believe It

I develop and defend the view that subjects are necessarily psychologically able to revise their beliefs in response to relevant counter-evidence. Specifically, subjects can revise their beliefs in response to relevant counter-evidence, given their current psychological mechanisms and skills. If a subject lacks this ability, then the mental state in question is not a belief, though it may be some other kind of cognitive attitude, such as a supposi-tion, an entertained thought, or a pretense. The result is a moderately revisionary view of belief: while most mental states we thought were beliefs are beliefs, some mental states which we thought were beliefs are not beliefs. The argument for this view draws on two key claims: First, subjects are rationally obligated to revise their beliefs in response to relevant counter-evidence. Second, if some subject is rationally obligated to revise one of her mental states, then that subject can revise that mental state, given her current psychological mechanisms and skills. Along the way to defending these claims, I argue that rational obligations can govern activities which reflect on one’s rational character, whether or not those activities are under one’s voluntary control. I also show how the relevant version of epistemic ‘ought’ implies ‘can’ survives an objection which plagues other variants of the principle

On real one-sided ideals in a free algebra

In classical and real algebraic geometry there are several notions of the radical of an ideal I. There is the vanishing radical defined as the set of all real polynomials vanishing on the real zero set of I, and the real radical defined as the smallest real ideal containing I. By the real Nullstellensatz they coincide. This paper focuses on extensions of these to the free algebra R of noncommutative real polynomials in x=(x_1,...,x_g) and x^*=(x_1^*,...,x_g^*). We work with a natural notion of the (noncommutative real) zero set V(I) of a left ideal I in the free algebra. The vanishing radical of I is the set of all noncommutative polynomials p which vanish on V(I). In this paper our quest is to find classes of left ideals I which coincide with their vanishing radical. We completely succeed for monomial ideals and homogeneous principal ideals. We also present the case of principal univariate ideals with a degree two generator and find that it is very messy. Also we give an algorithm (running under NCAlgebra) which checks if a left ideal is radical or is not, and illustrate how one uses our implementation of it.Comment: v1: 31 pages; v2: 32 page

Amodal completion and knowledge

Amodal completion is the representation of occluded parts of perceived objects. We argue for the following three claims: First, at least some amodal completion-involved experiences can ground knowledge about the occluded portions of perceived objects. Second, at least some instances of amodal completion-grounded knowledge are not sensitive, that is, it is not the case that in the nearest worlds in which the relevant claim is false, that claim is not believed true. Third, at least some instances of amodal completion-grounded knowledge are not safe, that is, it is not the case that in all or nearly all near worlds where the relevant claim is believed true, that claim is in fact true. Thus, certain instances of amodal completion-grounded knowledge refute both the view that knowledge is necessarily sensitive and the view that knowledge is necessarily safe

A Semidefinite Approach for Truncated K-Moment Problems

A truncated moment sequence (tms) of degree d is a vector indexed by monomials whose degree is at most d. Let K be a semialgebraic set.The truncated K-moment problem (TKMP) is: when does a tms y admit a positive Borel measure supported? This paper proposes a semidefinite programming (SDP) approach for solving TKMP. When K is compact, we get the following results: whether a tms y of degree d admits a K-measure or notcan be checked via solving a sequence of SDP problems; when y admits no K-measure, a certificate will be given; when y admits a K-measure, a representing measure for y would be obtained from solving the SDP under some necessary and some sufficient conditions. Moreover, we also propose a practical SDP method for finding flat extensions, which in our numerical experiments always finds a finitely atomic representing measure for a tms when it admits one