Second Order and Gaussian Fields

Author(s):  
Ulf Grenander ◽  
Michael I. Miller

This chapter studies second order and Gaussian fields on the background spaces which are the continuum limits of the finite graphs. For this random processes in Hilbert spaces are examined. Orthogonal expansions such as Karhunen–Loeve are examined, with spectral representations of the processes established. Gaussian processes induced by differential operators representing physical processes in the world are studied.

2019 ◽  
Vol 34 (6) ◽  
pp. 353-360
Author(s):  
Alisa M. Medvyatskaya ◽  
Vasily A. Ogorodnikov

Abstract We consider approaches to simulation of periodically correlated random processes based on the nonstandard spectral representation of the process with parameters periodically varying in time and also on spectral representations using the vector stationary Gaussian processes.


2017 ◽  
Vol 9 (3) ◽  
pp. 17-30
Author(s):  
Kelly James Clark

In Branden Thornhill-Miller and Peter Millican’s challenging and provocative essay, we hear a considerably longer, more scholarly and less melodic rendition of John Lennon’s catchy tune—without religion, or at least without first-order supernaturalisms (the kinds of religion we find in the world), there’d be significantly less intra-group violence. First-order supernaturalist beliefs, as defined by Thornhill-Miller and Peter Millican (hereafter M&M), are “beliefs that claim unique authority for some particular religious tradition in preference to all others” (3). According to M&M, first-order supernaturalist beliefs are exclusivist, dogmatic, empirically unsupported, and irrational. Moreover, again according to M&M, we have perfectly natural explanations of the causes that underlie such beliefs (they seem to conceive of such natural explanations as debunking explanations). They then make a case for second-order supernaturalism, “which maintains that the universe in general, and the religious sensitivities of humanity in particular, have been formed by supernatural powers working through natural processes” (3). Second-order supernaturalism is a kind of theism, more closely akin to deism than, say, Christianity or Buddhism. It is, as such, universal (according to contemporary psychology of religion), empirically supported (according to philosophy in the form of the Fine-Tuning Argument), and beneficial (and so justified pragmatically). With respect to its pragmatic value, second-order supernaturalism, according to M&M, gets the good(s) of religion (cooperation, trust, etc) without its bad(s) (conflict and violence). Second-order supernaturalism is thus rational (and possibly true) and inconducive to violence. In this paper, I will examine just one small but important part of M&M’s argument: the claim that (first-order) religion is a primary motivator of violence and that its elimination would eliminate or curtail a great deal of violence in the world. Imagine, they say, no religion, too.Janusz Salamon offers a friendly extension or clarification of M&M’s second-order theism, one that I think, with emendations, has promise. He argues that the core of first-order religions, the belief that Ultimate Reality is the Ultimate Good (agatheism), is rational (agreeing that their particular claims are not) and, if widely conceded and endorsed by adherents of first-order religions, would reduce conflict in the world.While I favor the virtue of intellectual humility endorsed in both papers, I will argue contra M&M that (a) belief in first-order religion is not a primary motivator of conflict and violence (and so eliminating first-order religion won’t reduce violence). Second, partly contra Salamon, who I think is half right (but not half wrong), I will argue that (b) the religious resources for compassion can and should come from within both the particular (often exclusivist) and the universal (agatheistic) aspects of religious beliefs. Finally, I will argue that (c) both are guilty, as I am, of the philosopher’s obsession with belief. 


Hypatia ◽  
2020 ◽  
pp. 1-21
Author(s):  
Paul-Mikhail Catapang Podosky

Abstract In what sense do people doubt their understanding of reality when subject to gaslighting? I suggest that an answer to this question depends on the linguistic order at which a gaslighting exchange takes place. This marks a distinction between first-order and second-order gaslighting. The former occurs when there is disagreement over whether a shared concept applies to some aspect of the world, and where the use of words by a speaker is apt to cause hearers to doubt their interpretive abilities without doubting the accuracy of their concepts. The latter occurs when there is disagreement over which concept should be used in a context, and where the use of words by a speaker is apt to cause hearers to doubt their interpretive abilities in virtue of doubting the accuracy of their concepts. Many cases of second-order gaslighting are unintentional: its occurrence often depends on contingent environmental facts. I end the article by focusing on the distinctive epistemic injustices of second-order gaslighting: (1) metalinguistic deprivation, (2) conceptual obscuration, and (3) perspectival subversion. I show how each reliably has sequelae in terms of psychological and practical control.


2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
E. I. Buchbinder ◽  
D. Hutchings ◽  
S. M. Kuzenko ◽  
M. Ponds

Abstract Within the framework of $$ \mathcal{N} $$ N = 1 anti-de Sitter (AdS) supersymmetry in four dimensions, we derive superspin projection operators (or superprojectors). For a tensor superfield $$ {\mathfrak{V}}_{\alpha (m)\overset{\cdot }{\alpha }(n)}:= {\mathfrak{V}}_{\left(\alpha 1\dots \alpha m\right)\left({\overset{\cdot }{\alpha}}_1\dots {\overset{\cdot }{\alpha}}_n\right)} $$ V α m α ⋅ n ≔ V α 1 … αm α ⋅ 1 … α ⋅ n on AdS superspace, with m and n non-negative integers, the corresponding superprojector turns $$ {\mathfrak{V}}_{\alpha (m)\overset{\cdot }{\alpha }(n)} $$ V α m α ⋅ n into a multiplet with the properties of a conserved conformal supercurrent. It is demonstrated that the poles of such superprojectors correspond to (partially) massless multiplets, and the associated gauge transformations are derived. We give a systematic discussion of how to realise the unitary and the partially massless representations of the $$ \mathcal{N} $$ N = 1 AdS4 superalgebra $$ \mathfrak{osp} $$ osp (1|4) in terms of on-shell superfields. As an example, we present an off-shell model for the massive gravitino multiplet in AdS4. We also prove that the gauge-invariant actions for superconformal higher-spin multiplets factorise into products of minimal second-order differential operators.


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