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Theory Reduction in Physics: A Model-Based, Dynamical Systems Approach
- Emery Hodges
- 2 years ago
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1 Theory Reduction in Physics: A Model-Based, Dynamical Systems Approach Joshua Rosaler Center for Philosophy of Science, University of Pittsburgh November 27, 2013 Abstract In 1973, Nickles identified two senses in which the term reduction is used to describe the relationship between physical theories: namely, the sense based on Nagel s seminal account of reduction in the sciences, and the sense that seeks to extract one physical theory as a mathematical limit of another. These two approaches have since been the focus of most literature on the subject, as evidenced by recent work of Batterman and Butterfield, among others. In this paper, I discuss a third sense in which one physical theory may be said to reduce to another. This approach, which I call dynamical systems (DS) reduction, concerns the reduction of individual models of physical theories rather than the wholesale reduction of entire theories, and specifically reduction between models that can be formulated as dynamical systems. DS reduction is based on the requirement that there exist a function from the state space of the low-level (more encompassing) model to that of the high-level (less encompassing) model that satisfies certain general constraints and thereby serves to identify quantities in the low-level model that mimic the behavior of those in the high-level model - but typically only when restricted to a certain domain of parameters and states within the low-level model. I discuss the relationship of this account of reduction to the Nagelian and limit-based accounts, arguing that it is distinct from both but exhibits strong parallels with a particular version of Nagelian reduction, and that the domain restrictions employed by the DS approach may, but need not, be specified in a manner characteristic of the limit-based approach. Finally, I consider some limitations of the account of reduction that I propose and suggest ways in which it might be generalised. I offer a simple, idealised example to illustrate application of this approach; a series of more realistic case studies of DS reduction is presented in another paper. 1
2 Contents 1 Introduction 3 2 Two Views of Reduction in Physics 4 3 Dynamical Systems in Physics 17 4 Dynamical Systems (DS) Reduction 22 5 Precursors to DS Reduction 37 6 DS Reduction and Nagelian Reduction 43 7 DS Reduction and Limits 47 8 Limitations of the DS Approach 48 9 Conclusion 49 2
3 1 Introduction Broadly speaking, reduction in physics can be understood as a relationship that obtains between two theories describing the same physical system or class of systems, such that the success of one theory in describing the system can be accounted for on the basis of the other, which is taken to be the more encompassing and more accurate of the two descriptions. Probing further, we can ask what it is specifically about the relationship between the two theories that allows this subsumption to occur. Over the past 40 years, the literature on reduction in physics has tended to revolve around two ways of addressing this question, which were first distinguished by Nickles in his widely cited 1973 paper, Two Concepts of Intertheoretic Reduction : first, the concept of reduction built around Nagel s seminal account first spelled out in Chapter 11 of The Structure of Science, and second, the concept that regards reduction as a matter of extracting one physical theory as a mathematical limit of another. In this paper, I elaborate a third sense in which one theory in physics may be said to reduce to another, one which I argue resolves some of the vagueness that afflicts both the Nagelian and limit-based accounts. It is important to note from the outset that this account concerns the reduction of individual models of physical theories, rather than the wholesale reduction of entire theories, and moreover, that it applies specifically to the reduction of models that can be formulated as dynamical systems - i.e., models that can be specified by some mathematical state space and some deterministic rule prescribing the time evolution of points in that space. Dynamical systems theory, I claim, allows the formulation of particularly natural and simple conditions for one dynamical systems model to reduce to another, and because many theories in physics permit formulation of their models in terms of dynamical systems, provides a general mathematical framework for describing a wide range of inter-theory relations in physics. For this reason, I call this approach the dynamical systems, or DS, approach to reduction. Insofar as one is inclined to speak of theories rather than models being reduced on this account, this occurs only piecemeal through the reduction of a theory s individual models. In section 2, I briefly review the Nagelian and limit-based approaches to reduction in physics and consider various precisifications of these approaches. In section 3, I introduce the concept of a dynamical systems model and discuss some of its applications in physics. In section 4, I give conditions for 3
4 the reduction of dynamical systems models in physics and provide a simple example to illustrate the application of these conditions. Section 5 reviews work by a number of authors that has served in one some way to anticipate or hint at the DS approach that I elaborate here. Section 6 provides a comparison between DS and Nagelian reduction, highlighting both parallels and differences. Section 7 comments on the role of limit-based results in DS reduction. Section 8 discusses some limitations of the DS view and ways in which it might be generalised. Section 9 is the conclusion. 2 Two Views of Reduction in Physics In [26], Nickles distinguishes two uses of the term reduction with regard to inter-theory relations in physics, one common to the philosophical literature and the other common to the physics literature on the subject. The philosopher s sense of reduction takes a high-level (i.e. less encompassing, less fundamental) theory T h to reduce to a low-level (i.e., more encompassing, more fundamental) theory T l, while the physicist s sense takes T l to reduce to T h. So on the philosopher s usage, Newtonian mechanics reduces to special relativity; on the physicist s usage, special relativity reduces to Newtonian mechanics; nevertheless, both usages presuppose that special relativity is the more accurate and more encompassing of the two theories, and that it is the success of Newtonian mechanics that must be accounted for on the basis of special relativity and not the other way around. Thus, the difference between the two usages is to some extent a matter of convention as to the direction in which reduction is taken to go. Yet the distinction between the two senses of reduction is not solely a matter of convention. Once the conventions are made to agree, there remains a substantive difference between the meaning of the term reduction as it is most commonly employed in the physics literature and its meaning as it is most commonly employed in the philosophy literature. The philosopher s notion is based on Nagel s account of reduction in the sciences while the physicist s notion views reduction in physics essentially as a matter of taking mathematical limits [25] 1. These two concepts of reduction, which 1 Of course, one may question whether it is entirely appropriate or fair to identify one sense of reduction as the physicist s and the other as the philosopher s. There are, after all, instances of physicists employing what is effectively reduction in the philosopher s sense (arguably, textbook proofs of the Ideal Gas Law on the basis of statistical mechanics are 4
5 Nickles identifies as reduction 1 and reduction 2, respectively, can be defined as follows: Nagelian Reduction: T h reduces 1 to T l iff the laws of T h can be derived from those of T l along with auxiliary assumptions (known bridge laws ) linking terms in T h foreign to T l with terms in T l. Limit-Based Reduction: T h reduces 2 to T l iff there exists some set of parameters {ɛ i } defined within T l such that lim {ɛi 0} T l = T h. [26],[3] 2 3 For reasons that I discuss shortly, both of these definitions are problematic as descriptions of actual inter-theory relations in physics and have been subject to various refinements as a result. Before moving on to discuss Nagelian and limit-based reduction in more detail, I should note that there also exist other influential philosophical accounts of reduction, such as Kim s functionalist model and Hooker s New Wave model [18], [17]. The relation of these accounts particularly to the Nagelian approach is a matter of some dispute; for example, Marras has argued that Kim s account is only superficially distinct from Nagel s, and Fazekas likewise has argued for a similar claim with regard to Hooker s New Wave model [23], [12]. Kim s and Hooker s accounts have been developed primarily within the context of discussions about reduction of the mental to the physical in philosophy of mind. Since it would take me too far afield to see how, if at all, they can be applied to reductions in physics, I will focus on the two approaches that Nickles discusses and that have been the main focus of the literature on theory reduction specifically in physics. 2.1 Nagelian Reduction According to the account of reduction that Nagel sets out in The Structure of Science, reductions can be broadly classified into two categories: homogeexamples of this [19]) and of philosophers employing reduction in the physicist s sense (see, for instance, [2]). Nevertheless, I will adhere to Nickles terminology. 2 Note that if one has lim {ɛi } T l = T h, or lim {ɛi a} T l = T h where 0 < a <, one can always redefine the parameters {ɛ i } so that each ɛ i approaches 0. 3 In Nickles original definition of reduction 2, the sense of reduction is the inverse of the one I give here, in that on Nickle s definition the superseding theory T 1 reduces 2 to the superseded theory T 2, rather than vice versa as in the definition that I provide. As discussed, this inversion merely reflects a difference of convention. 5
6 neous and inhomogeneous. In the former, the theory to be reduced contains no terms which are not contained in the reducing theory, while in the latter it does contain such terms. An example of a homogeneous reduction is the reduction of Kepler s theory of planetary motion to Newton s Theory of Gravitation, since Kepler s theory employs only concepts such as distance, time, and concepts derived from these, all of which are also contained in Newton s theory [9]. An example of a heterogeneous reduction is the reduction of thermodynamics, which employs the concept of temperature, to the Newtonian mechanics of microscopic particles, which contains no reference to temperature. Recognizing that no term can appear in the conclusion of a formal demonstration unless the term also appears in the premises, Nagel asserts that in the case of an inhomogeneous reduction, additional assumptions connecting terms in the high-level theory (which Nagel refers to as the secondary science ) with terms in the low-level theory (which Nagel calls the primary science ) must be introduced. Thus, he proposes two formal conditions - the condition of connectability and the condition of derivability - that must be satisfied in order to effect an inhomogeneous reduction of the higher- to the lower- level theory: (1) Assumptions of some kind must be introduced which postulate suitable relations between whatever is signified by A [a term in the secondary science] and traits represented by theoretical terms already present in the primary science. The nature of such assumptions remains to be examined; but without prejudging the outcome of further discussion, it will be convenient to refer to this condition as the condition of connectability. (2) With the help of these additional assumptions, all the laws of the secondary science, including those containing the term A, must be logically derivable from the theoretical premises and their associated coordinating definitions in the primary discipline. Let us call this the condition of derivability. ([25], Ch. 11) The additional premises furnished by the condition of connectability have come to be known as bridge laws, though occasionally are also referred to as bridge principles, bridge rules, coordinating definitions, and reduction functions. Their purpose is to link those terms in the high-level theory that do not occur in the low-level theory with terms in the low-level theory and thereby to facilitate derivation of the high-level theory s laws from the 6
7 low-level theory. The question as to the precise nature of these additional assumptions - whether they are in fact laws of nature in the same sense that, say, Newton s laws are, or whether they are mere coordinating definitions - remains a matter of controversy (see, for instance, [9] for further discussion of this point). I will abide by common usage here and refer to these additional assumptions as bridge laws, though the reader should not take this to entail any commitment to the view that they are laws of nature. The central example that Nagel employs to illustrate his account of reduction is the relation of the Ideal Gas Law (pv = nrt ), as understood in the context of classical thermodynamics, to the laws of Newtonian mechanics as applied to the microscopic constituents - hard sphere molecules - of an idealised gas. He notes that while the term temperature had an accepted meaning in the context of thermodynamics - given in terms of operational procedures employing thermometers and other devices - the term makes no appearance in the low-level theory, Newtonian mechanics. He then invokes a strategy employed by physicists in deriving the Ideal Gas Law from Newton s Law: namely, to associate the thermodynamical term temperature with a quantity, average molecular kinetic energy, that is defined within the framework Newtonian mechanics. More precisely, from Newton s laws and certain auxiliary assumptions concerning the position and velocity distributions of the molecules in the gas (such as uniformity of the distribution in position and isotropy in velocity), one can deduce that pv = 2 N K.E. (1) 3 where N is the number of molecules in the gas. Now the form of the Ideal Gas Law in thermodynamics is pv = nrt, (2) where n = N N A (with N A Avogadro s constant) is the number of moles in the gas and R = N A k B (with k B Boltzmann s constant) is the universal gas constant. If one makes the additional assumption, Bridge Law : T = 2 K.E. K.E. = 3 3 k B 2 k BT (3) one can deduce (2) from (1). Equation (3) has come to serve as the primary exemplar of a bridge law in Nagelian reduction. 7
8 2.1.1 Nagel s Account, Refined Schaffner, one of the first to expand on Nagel s account of reduction, observed that Nagel s account is, strictly speaking, too stringent since reductions in practice rarely if ever yield derivations of the higher level theory T h, but rather of some modified or corrected version T h of T h that employs the same vocabulary as T h. T h is sometimes referred to as the analogue theory of T h and is required to be strongly analogous to T h in the sense of approximating it closely. According to Schaffner, bridge laws can then be understood as enabling the derivation of T h - rather than T h - from T l [32], [33]. For example, a more precise treatment of the reduction just considered would take into account the fact that equation (1) holds only approximately (given the approximate nature of the assumptions concerning the distribution of molecules in position and velocity): Image Theory : pv 2 N K.E.. (4) 3 This relation, which is formulated in the language of the low-level theory, is an example of what is sometimes called an image theory, in the sense that it provides an image of laws in the high-level theory formulated in the language of the low-level theory (Hooker also employs the notion of an image theory in his New Wave account of reduction). Equations (4) and (3) jointly imply that Analogue Theory : pv nrt, (5) which is strongly analogous to the Ideal Gas Law. The refinement of Nagel s account that I consider here, dubbed the Generalized Nagel-Schaffner (GNS) model by Dizadji-Bahmani, Frigg and Hartmann in (see [9]), consolidates both Schaffner s and Nagel s insights. On this model, reduction can be understood as a three-step process, starting with the basic ingredients of a low-level theory T l, a high-level theory T h, and a set of bridge laws: 1. Derive the image theory Th for some restricted boundary or initial conditions within the low level theory T l, without employing bridge laws. 8
9 2. Use bridge laws to replace terms in Th, which belong to the vocabulary of the low level theory, with corresponding terms belonging to the high level theory. This yields the analogue theory T h. 3. If T h is strongly analogous to the high-level theory T h, the high-level theory has been reduced to T l. The strong analogy relation is sometimes also characterised as approximate equality, close agreement, or good approximation. Henceforth, when I speak of Nagelian reduction, I will construe it according to the GNS model, unless explicitly stated otherwise. Moreover, note that Nagel s connectability condition on this refinement consists of two connections : first, the bridge laws that link the image theory Th and the analogue theory T h, and second, the rather vaguely defined strong analogy relation that connects the analogue theory T h to the high-level theory T h. In what, precisely, does the approximate agreement typically used to characterise the strong analogy relation consist? Specifically which quantities between these two descriptions must approximately agree, and in what contexts? I argue below that the DS approach to reduction that I develop in this paper exhibits certain strong parallels with the GNS approach but also serves to resolve certain of its ambiguities, particularly relating to the notion of strong analogy and the nature of the linkages that must be established between the concepts of the reduced and reducing theories. 2.2 Limit-Based Reduction Unlike Nagelian reduction, the notion of a limit-based approach to reduction, as first explicitly identified by Nickles, seems to arise not from any clear-cut statement of the general conditions for this kind of reduction to take place, but rather from a plethora of highly suggestive mathematical results all of which involve or somehow imply the use of mathematical limits, and also from a certain manner of speaking that is often employed in talk about intertheory relations in physics, as exemplified by references to the nonrelativistic limit of special relativity, the classical limit of quantum mechanics, the thermodynamic limit of statistical mechanics, the geometric optics limit of wave optics, and so on. Batterman, Butterfield, Rohrlich, Berry, Ehlers, and Scheibe, among others, all have explored various facets of this approach to reduction from a general philosophical and methodological point of view as 9
10 well as in the context of particular case studies [7], [2], [30], [4], [11], [34], [35]. Talk of reduction in terms of limits also pervades discussion of inter-theory relations in many physics textbooks and journals. Despite the popularity of the limit-based approach to reduction, the vague and schematic relation lim {ɛi 0} T l = T h appears to be as close to a statement of general criteria for limit-based reduction as has been given in the literature. In this subsection, I will attempt to clarify the intended meaning of claims that one theory is a limit, or limiting case, of another by proposing a series of refinements to the definition given above that suggest general criteria for limit-based reduction. But first it will prove instructive to consider particular examples of the sort of result that motivates the idea of a limit-based approach to reduction in physics, of which there are many. Perhaps the simplest and most widely known example of such a result is given by the relativistic expressions for time dilation and length contraction between inertial reference frames: t 1 = t 1 v2 c 2 l = 1 v2 c l, 2 where t is the time between two events at the same location in some intertial lab frame and t the time between these same events as measured from an inertial frame moving with constant velocity v with respect to the lab frame, and likewise l is the length of an object at rest in the lab frame and l its length as measured from an inertial frame moving parallel to the line along which l is measured with constant velocity v relative to the lab frame. From these relations, it follows that (6) lim c t = t lim c l = l (7) so that in the limit c, time and length in the lab frame are equal, respectively, to time and length in the moving frame, as is the case in Newtonian mechanics. 10
11 As another example of limiting relations between theories in physics, this time concerning the quantum-classical correspondence, consider the quantum mechanical equation of motion for the Wigner function (a representation of the quantum state on phase space): ( ) 2i W (q, p, t) = t sin {, } (H, W ), (8) 2i where {H, W } is the classical Poisson bracket. From the expansion of the Moyal bracket, 2i sin ( {, }) (H, W ) = {H, W } V 3 W + O( 4 ) +..., 2i 24 q 3 p 3 it follows that ( ) 2i lim 0 sin {H, } W = {H, W }, (9) 2i so that in the limit 0, we retrieve the relation W (q, p, t) = {H, W }, (10) t the classical Liouville equation for the evolution of a probability distribution on phase space 4. There are many examples throughout physics similar to the two just described, where an equation of the form that occurs in one theory is retrieved from an equation of another theory by means of some limiting process. On the basis of results like these, classical mechanics is sometimes also characterised as the limit as N of quantum mechanics, where N is the energy quantum number, thermodynamics as the N of statistical mechanics, where N is now the number of degrees of freedom, and geometric optics as the λ 0 limit of wave optics, where λ is wavelength. The prevalence throughout physics of limiting relations like the ones just described initially seems to make a compelling case for the notion that reduction in physics is essentially a matter of taking mathematical limits. But if we are to countenance Nickles reduction 2 as furnishing a bona fide set of criteria for one physical theory to account for the success of another, we should be able to state precisely what these criteria are, rather than allowing them to remain implicit in the wide range of results that are taken to exemplify this kind of reduction. If the limit-based approach to reduction is to be more than a vague and merely suggestive manner of speaking about inter-theory 4 This example is drawn from Chapter 3 of [39]. 11
12 relations in physics, it is necessary first to clarify what is meant generally when one theory is characterised as a limit or limiting case of another, and then to explain why this relation enables the low-level theory to incorporate the successes of the high-level theory. What, then, is meant by the claim that one theory is a limit or limiting case of another? Let us start by considering an example before attempting to give a general answer: what, for instance, is meant by the claim that classical mechanics is the limit as 0 of quantum mechanics (which is taken to encompass more than simply the result quoted above)? On the most naive construal, one might interpret this as meaning that if one takes any quantity or relation in quantum mechanics and considers its 0 limit, one retrieves a corresponding quantity or relation in classical mechanics. Yet such a claim would be obviously false in this case (as well as in other purported cases of reduction 2 ), for if one considers the 0 limit of Schrodinger s equation, i 2 ψ(x, t) = t 2m 2 ψ(x, t) + V (x)ψ(x, t). (11) one obtains the non-sensical - and certainly not classical - result, 0 = V (x)ψ(x, t). (12) So we need to be more precise about what we mean by the 0 limit of quantum mechanics (where quantum mechanics is a theory which we might naturally regard as being embodied in some respects by the Schrodinger equation) and more generally what is meant by lim ɛ 0 T l, since taking the limit of just any part of the low-level theory is unlikely to yield an element of the high-level theory. Another potential pitfall in interpreting the relation lim {ɛi 0} T l = T h, already noted widely by a number of authors, occurs in cases (such as those discussed so far) where one of the parameters ɛ i is taken to be a constant of nature, as in the claims CM = lim 0 QM and NM = lim c SR (read CM: classical mechanics, QM: quantum mechanics, NM: Newtonian mechanics, SR: special relativity). Reduction, however precisely it is to be defined, requires that the domain of physical systems that the high-level theory describes well should be a subset of the domain of systems that the low-level theory describes well, and that in the domain of the high-level theory, the low-level theory provides a still more accurate approximation. So an account of the reduction of the high-level theory to the low-level theory should furnish a means of identifying that subset of theoretical descriptions within the 12
13 low-level theory that approximately agree with the high-level theory in cases where the high-level theory is approximately valid. Thus, it might seem natural at first to read the relation CM = lim 0 QM as implying that those quantum systems which exhibit approximately classical behavior are those for which = 0 or for which approaches zero, or that those relativistic systems which exhibit approximately Newtonian behavior are those for which c = or for which c approaches infiinity. But of course, all real physical systems possess the same fixed values for c and, so the physical significance of results that involve taking the limit 0 or c is unclear. Results that require us to posit values for and c other than those possessed by actual physical systems contribute an unhelpful layer of obscurity to the analysis of inter-theory relations in physics in that they fail to clearly identify the domain of real physical systems, as characterised by the low-level theory, for which the high-level theory provides an approximately accurate description. These considerations suggest the following revision to Nickles reduction 2 : Limit-Based Reduction (Second Pass): T h reduces 2 to T l iff there exists some set of parameters {ɛ i }, defined within T l, and which are not constants of nature, such that lim ɛi 0 T l = T h. Given this restriction, in the NM/SR case we may replace the limit c with the limit v 0, where v is some appropriate velocity characterising the system in question. With this revision, one retains the equality of times and of lengths in different reference frames in the relevant limit. Likewise, with regard to the CM/QM relation, a number of authors have tried to address this concern by claiming that strictly speaking the proper limit to take is not the 0 limit, but the limit where S cl is some measure of the typical classical action of the quantum system in question. With this revised understanding of these limits, we now have a hope of extracting some physical, rather than merely mathematical, significance for them since the relevant parameter now involves a quantity that varies from system to system rather than being the same for all systems. Yet once we have made made this last refinement to reduction 2 there remain still other potential pitfalls in the interpretation of the relation lim {ɛi 0} T l = T h. Consider the oft-cited result relating relativistic kinetic energy (γ 1)mc 2 to Newtonian kinetic energy 1 2 mv2 : S cl 0, or equivalently the limit S cl, 13
14 (γ 1)mc 2 = 1 2 mv mv (13) c2 While it is true that the Newtonian kinetic energy approximates the relativistic kinetic energy when v << c, the limit of this expression as the dimensionless parameter v 0 (or as v 0) is zero, not 1 c 2 mv2. More generally, lim v 0 SR (or lim v c 0 SR), if we are to understand the limit in terms of the mathematical notion of a limit, is a theory in which nothing moves, not Newtonian mechanics. Thus, when interpreting claims that one theory is a limit of another, in this and many other cases we must be careful not to understand limit literally to mean limit in the mathematical sense. There are also other important pitfalls that come with taking claims that one theory is a limit of another literally to mean that one theory is a mathematical limit of another. As Berry and Batterman have emphasised repeatedly, certain mathematical limits may fail to yield anything resembling the high-level theory because they are singular. However, they do not interpret this as a problem with the limit-based approach to reduction, but rather take such singularities to signify a failure of reducibility between the theories in question [2], [4]. As both Butterfield and Norton have observed, even in cases where the relevant limit is not singular, it still may yield unrealistic idealisations that do not correspond to any actual physical system, or may fail to yield any idealisation at all, as in certain classical- and statisticalmechanical examples where the number of degrees of freedom is taken to infinity. In cases where the limit leads to unrealistic idealisations, Butterfield suggests stopping before the limit in order to facilitate application of limit-based results to real physical systems. Norton is concerned to highlight the distinction between approximation and idealisation, and argues that in some cases where an infinite limit of well-behaved finite systems is taken, the limit system may fail to provide an idealisation of any real physical system [7], [28]. How are to we interpret the relation lim {ɛi 0} T l = T h, if not literally as a mathematical limit? Considering the example of Eq. (13), we might take the lim {ɛi 0} simply to signify the requirement that we restrict the values of ɛ i (in this case v) to be small. Such an interpretation is compatible with viewing T h as a first- or higher-order approximation rather than as the mathematical limit (which, in cases where the relevant function is continuous, is equal to a zeroth-order approximation). These considerations suggest the following further refinement of reduction 2 : 14
15 Limit-Based Reduction (Third Pass): T h reduces 2 to T l iff there exists some set of parameters {ɛ i } defined within T l which are not constants of nature, such that when {ɛ i } all are small, T h approximates T l. Once we adopt this refinement of limit-based reduction, it is important to keep in mind that reduction 2 is no longer strictly speaking about taking limits in the mathematical sense of the term, but about a certain way of specifying a particular domain within the low-level theory. Yet there still remain difficulties with this construal of reduction 2 in that if {ɛ i } are dimensionful quantities, the question of whether their numerical values are small will depend on an arbitrary choice of units. For this reason, when specifying that a dimensionful quantity is small, it is necessary to say small in comparison to what. For example, it is meaningless to characterise the velocity v as small unless one provides some other measure of velocity against which to compare it; for instance, in the NM/SR case, we may insist that v << c. Likewise in the CM/QM case we may insist that << S cl. Equivalently, we may recast these conditions by restricting the domain of an appropriate dimensionless parameter ɛ i so that ɛ i << 1; for instance, v << 1, and c S cl << 1. On this basis, we may propose one final revision to our definition of reduction 2 : Limit-Based Reduction (Fourth Pass): T h reduces 2 to T l iff there exists some set of dimensionless parameters {ɛ i } defined within T l, which are not constants of nature, such that when {ɛ i << 1} for all i, T h approximates T l. Again, it is important to emphasise that imposing the restriction ɛ i << 1 is not the same thing as taking the limit ɛ i 0. Henceforth, I shall construe limit-based reduction, or reduction 2, in the sense provided here. Note moreover, that this concept of limit-based reduction includes cases where the intended approximation is literally the mathematical limit or zerothorder approximation, though it is not restricted to such cases as a literal interpretation of the limit is. Yet it is also important to recognise the significant ambiguity that remains in the definition of limit-based reduction just provided. In particular, the meaning of the phrase T h approximates T l is still quite vague. How are we to assess in any given case whether T h approximates T l in the prescribed domain, given that T h and T l may be formulated in radically different conceptual 15
16 and mathematical frameworks? Precisely which quantities in T l and in T h ought we to be comparing in order to judge whether one theory approximates another? Simply restricting to the domain of T l in which ɛ i << 1 does not of itself return T h or an approximation to T h ; the resulting description is still formulated in the mathematical and conceptual framework of T l, not that of T h. For example, if we examine the quantum commutator in the domain of quantum theory where S cl << 1, the resulting quantity is an operator on Hilbert space, with no straightforward classical interpretation (it is not, for instance, approximately equal to the classical Poisson bracket, which is often cited as the classical counterpart to the commutator, since the Poisson bracket is a function on classical phase space, not an operator on any Hilbert space). Some means of translating between the framework of the high-level theory and that of the low-level theory - akin to Nagel s bridge laws - is needed before we can judge whether one theory approximates the other in the specified domain. Acknowledging this observation, I could continue in the attempt to resolve the various additional points of vagueness and ambiguity that afflict the limitbased approach to reduction. At this point, though, one can reasonably question whether the limit-based approach, in spite of its popularity, ever existed as anything more than a vague intuition (as oppposed a well-defined approach to reduction) given that we must do all the work of formulating reduction 2 ourselves in order to figure out precisely what is meant by it. Rather than continue in the attempt to give a precise meaning to Nickles reduction 2, I will simply state in the next two sections what I believe to be an appropriate set of general criteria for characterising the relationship between a wide range of physical theories, and specifically between models of these theories. In this subsection, I have highlighted a number of points of ambiguity in the limit-based approach to reduction and argued that in many purported instances of this type of reduction, a literal interpretation of the claim that one theory is a limit of another - i.e., in terms of mathematical limits - is simply not viable. As a consequence, the general appropriateness of approaching reduction in terms of limits, rather than in terms of domains - as one does for instance when one imposes the restriction ɛ i << 1 - should be called into question. If, on the other hand, we choose to associate limit-based reduction with conditions of the form ɛ i << 1 for some dimensionless parameters 16
17 ɛ i (though, again, the term limit in this case is arguably more misleading than descriptive), then this sort of of condition is naturally incorporated into various applications of the dynamical systems (DS) approach to reduction that I develop here, as I demonstrate in another paper where I consider numerous case studies of DS reduction. All of this is not to say that limits bear no relevance to reduction (for instance, they do play a useful role in the discussion of the thermodynamic limit), but only that the centrality of their role in reduction of physical theories has in places been both overstated and not made sufficiently precise. 3 Dynamical Systems in Physics Before laying out the conditions for dynamical systems reduction in the next section, I briefly introduce the concept of a dynamical system and discuss its relevance to physics, where it is has a very wide range of applicability. 3.1 Definition of a Dynamical System A dynamical systems model M consists of a state space S and a dynamical map D that prescribes the time evolution of states in S; for brevity, I will write M = (S, D). I assume here that S is endowed at least with the structure of a differentiable manifold and a norm, and that D is a differentiable function both of x, the state in S, and of the time t such that for every t, D specifies a one-to-one function from S onto itself, and such that D is the identity map on S when t = 0: D : R S S, (14) x(t) = D(x 0, t), (15) x 0 = D(x 0, 0). (16) The requirement that the dynamical map at fixed time be one-to-one ensures that the dynamics are deterministic. In most, and possibly all, cases of interest in physics, the dynamical map for the model is associated with the set of solutions to some set of first-order differential equations: 17
18 dx = f(x, t), (17) dt where f(x, t) is an arbitrary continuous function of the state x and of time t. In many cases, there is no explicit time dependence in f, so f(x, t) = f(x) (however, f may still depend on time indirectly through the time dependence of the state x). Since x(t) = D(x 0, t) is a solution to the above differential equation, we have the following relation between the functions f and D: f(x(t), t) = D(x t 0, t) Symmetries of Dynamical Systems The concept of a symmetry has numerous definitions depending on the particular context under consideration. Broadly speaking, it is a transformation on a system that leaves some relevant aspect of the system - which depends strongly on the particular context - unchanged (see, for instance, [37] for an extended philosophical discussion of symmetries and their relation to physical laws). In the context of a dynamical system, a symmetry may be understood as a one-to-one transformation of both the dependent and independent variables (i.e., x and t, respectively) in the equation of motion (17) that leaves the form of this equation invariant. The most general such transformation is one that transforms the variables (x, t) into some other variables (x, t ) - that is, that involves transformations of the time parameter as well as of the state x. A map x = s x (x, t), t = s t (x, t) is a symmetry of the model M with dynamical equation dx = f(x, t) if it is differentiable, one-to-one and dt satisfies the condition dx dt = f(x, t ), (18) 5 While conventional applications of dynamical systems models in physics involve equations whose solutions are deterministic, as a caveat it is important to note the work of Earman, Norton and others, which has served to underscore the fact that for certain forms of f, equations of this form do no always yield deterministic solutions, in the sense that they may admit solutions with the same initial condition that diverge in time, or solutions that differ at an earlier time but pass through the same point in state space at some later time [27], [10]. Whether this occurs will depends on the function f(x, t) obeying particular mathematical constraints that I do not discuss here. In the analysis that follows, I will assume that the function f obeys the constraints necessary for the dynamics to respect determinism; in particular, this will prove to be the case in the examples that I consider. 18
19 where f is the same function appearing in the untransformed equation (17). Henceforth, the reader should understand the term symmetry in this sense when I use it (see, for instance, [5] for further discussion of symmetries of dynamical systems in classical physics). It will prove worthwhile here to highlight certain subclasses of symmetries of dynamical systems. First, some symmetries, such as rotations, translations, and Galilean transformations, do not transform the time parameter, so that x = s x (x, t), t = t; let us designate such symmetries as invarianttime symmetries and all others as variant-time symmetries. A sub-class of invariant-time symmetries is the set of symmetries for which the state transformation does not depend on time, so that x = s x (x), t = t; this includes rotations and translations but not Galilean transformations; call these time-independent, invariant-time symmetries. Let us now examine two examples of dynamical systems in physics: first, a model of Hamiltonian classical mechanics, and second a model of nonrelativistic quantum mechanics in the Schrodinger picture. 3.3 Example 1: Hamiltonian Classical Mechanics A system of N particles in non-relativistic Hamiltonian classical mechanics can be modelled as a dynamical system whose state space is given by S = Γ N, (19) where Γ N is the phase space of N particles moving in 3-dimensional space, which is a 6N-dimensional symplectic manifold whose points (x, p) consist of the spatial positions x and canonically conjugate momenta p of all N particles (see, for instance, [16], or any other graduate-level text of classical mechanics, for detailed definition and discussion of phase space). The dynamical map D of this model furnishes solutions to the first-order equations, dx dt = H = {x, H} p dp dt = H = {p, H}, x (20) also known as Hamilton s equations, so that (x(t), p(t)) = D[(x 0, p 0 ), t], where (x 0, p 0 ) are the initial conditions (specified at t = 0), represents a 19
20 distinct solution for each (x 0, p 0 ). Such a solution is given formally by the expression D [ (x 0, p 0 ), t ] = (e {,H}t x x0,p0, e {,H}t p ) x0,p0, (21) where e {,H}t f(x, p) f(x, p)+{f(x, p), H}t+ 1 2! {{f(x, p), H}, H}t ! {{{f(x, p), H}, H}, H}t3 +..., and {, } denotes the Poisson bracket, defined by {f, g} x f p g x g p f, with f and g some arbitrary differentiable functions on phase space. Assume now that the Hamiltonian takes the form H = N i p 2 i 2m i i j V (x i x j ). Then it can be shown, for instance, that among other transformations, the transformation (x, p ) = s x ((x, p); t) = (x vt, p mv) t = s t ((x, p), t) = t, (22) for some constant velocity v, is a symmetry of the dynamics. 3.4 Example 2: Non-Relativistic Quantum Mechanics (NRQM) in the Schrodinger Picture A system of N spinless particles in non-relativistic quantum mechanics can be modelled as a dynamical system whose state space is given by S = H N, (23) the Hilbert space of N spinless particles moving in three dimensions. The dynamics of the system are furnished by the Schrodinger equation, i ψ = Ĥ ψ, (24) t where ψ H N. If Ĥ does not depend explicitly on time, as for example when it takes the common form Ĥ = ˆp2 +V (ˆx), the solution to Schrodinger s 2m equation can be written formally as D [ ψ 0, t ] = e iĥt ψ 0, (25) 20
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FLAHERTY, Justice.
This is an appeal from the order of the Commonwealth Court upholding a zoning ordinance of the Township of Hampton which prohibits a private enterprise from operating gas wells in residential districts but permits public operation of such wells, 158 Pa.Cmwlth. 681, 632 A.2d 1102. Appellant challenges the distinction between public and private operations, alleging that such a distinction is arbitrary and unreasonable and, hence, an invalid exercise of the township's police powers.
Appellant, Jack D. Mahony, owns property in a residential "A" district in Hampton Township. A gas well has existed on his property since 1931. The well had been plugged and abandoned in 1957, but continued to leak natural gas into the atmosphere. Because of the leak, appellant applied to the Department of Environmental Resources for a permit to refurbish the well, which was granted, whereupon appellant returned the well to production in October, 1986.
The township's zoning ordinance permits, in residential "A" districts, the use of municipal and public utility facilities to produce gas, but prohibits operation of gas wells by private entities. The zoning officer denied appellant's application for
The court of common pleas held, first, that the zoning ordinance's distinction between private operation and municipal or public utility operation of gas wells was a legitimate and reasonable distinction, and, second, that the prohibition against private gas production was not an unconstitutional taking of appellant's property without just compensation. The Commonwealth Court affirmed both holdings. We granted allocatur to consider the propriety of the Hampton Township zoning ordinance.
Appellant argues that the distinction in the zoning ordinance between public versus private gas production is arbitrary and unreasonable. His position is that no concerns related to the police power — protection of health, safety, morals, or general welfare — justify disparate treatment of public and private gas wells. The DER controls and regulates all gas wells identically regardless of ownership status, be it private, public utility, or municipal. Appellant argues further that the record establishes that there is no factual basis to distinguish between private and public operation of gas wells. Inasmuch as a restrictive zoning ordinance must be rationally related to the government's police power, appellant maintains that the arbitrary restriction in this case is invalid.
The township argues that "there are hazards associated with the operation of a gas well" introduced into evidence by appellant's own witness, so that appellant has failed to overcome the heavy presumption that a zoning ordinance is valid and constitutional.
In weighing these arguments, we begin with the following principles stressed by the parties. A restrictive zoning ordinance must bear a rational relationship to the health, safety, and general welfare of a community. Beaver Gasoline Co. v. Zoning Hearing Board, 445 Pa. 571, 285 A.2d 501 (1971). It is also true that a zoning ordinance is presumed
This court has never decided whether a municipality may validly enact a zoning ordinance creating a distinction based solely on public versus private ownership of an arguably noxious or hazardous use of land. Kavanagh v. London Grove Township, 486 Pa. 133, 404 A.2d 393 (1979), involved a zoning ordinance which permitted municipal operation of a sanitary landfill but prohibited private operation. The issue was not decided, as the Court was equally divided. In any event, Kavanagh would not necessarily control this case, for the Hampton Township ordinance, in addition to permitting municipal operation of a gas well, also permits some private operations, namely, operations by a public utility. A public utility, despite its name, is a private entity subject to extensive regulation; it is not a public entity. Thus, even if a public-private distinction were constitutionally permissible, the Hampton Township ordinance permits public operations and some private operations but bars other private operations.
This case lends itself to the more general test for the validity of a restrictive zoning ordinance expressed in Lutz v. Armour, 395 Pa. 576, 578-79, 151 A.2d 108, 110 (1959): "[A] law which purports to be an exercise of the police power must not be unreasonable, unduly oppressive or patently beyond the necessities of the case, and the means which it employs must have a real and substantial relation to the objects sought to be attained." (Emphasis in original.) We apply this test to the challenged ordinance which regulates on the basis of ownership, not on the basis of location or use. We are mindful of the common law maxim sic utere tuo ut alienum non lædas (one should use his own property in such a manner as not to injure that of another). This maxim encompasses two principles. First, one must use his own property in such a manner as not to injure that of another. Second, one may use his own property in any manner which does not injure that of another.
The justifications advanced by Hampton Township to support its permission of municipal or public utility operation and
Testimony was given in the trial court concerning various safety hazards and disturbances associated with a gas well, such as gas emerging from a hole, fire, explosion, and noise. These hazards and disturbances, by and large, are associated with drilling a well rather than its subsequent operation; gas production is not a noisy or unduly hazardous operation. Nonetheless, whatever its hazards, safety does not depend on ownership of the well. Safety hazards of gas wells might justify a zoning ordinance based on location or use, as by entirely prohibiting gas production in residential districts, but the Hampton Township ordinance does not do so. Instead, it permits gas wells in residential districts so long as they are operated by a municipality or public utility. There is not a scintilla of evidence, however, that a municipality or public utility can produce gas any more safely than a private operator can.
The township argued that public entities are more responsive and accountable to public pressure and concerns of the residents of the community and that public entities are more responsible fiscally than are private entities. However appealing these arguments might be, they are not supported by the record. The record discloses instead that all gas operations, municipal, public utility, and private, are subject to regulation by the DER under the Oil and Gas Act, 58 P.S. § 601.101 et seq., Coal and Gas Resource Coordination Act, 58 P.S. § 501 et seq., the Clean Streams Law, 35 P.S. § 691.1 et seq., and the Oil and Gas Conservation Law, 58 P.S. § 401 et seq. All gas wells, regardless of ownership, fall under the same environmental statutes and regulations, and DER monitors them uniformly without distinction as to status of owners or operators. If any regulations applicable only to public utilities add any safety controls to those enforced uniformly by the DER (though the township has failed to identify any such
It is clear that the Hampton Township zoning ordinance, which purports to be an exercise of the police power, is unreasonable, oppressive, beyond the necessities of the case, and its prohibition of private gas production bears no real and substantial relation to the health, safety, and welfare of the community. That being the case, it is an invalid exercise of the police power. The judgment of the Commonwealth Court must, therefore, be reversed.
Order of the Commonwealth Court is reversed and the case is remanded for entry of judgment in favor of appellant.
ZAPPALA, J., did not participate in the consideration or decision of this case.
CAPPY, J., files a dissenting opinion which is joined by NIX, C.J.
MONTEMURO, J., is sitting by designation.
CAPPY, Justice, dissenting.
I respectfully dissent and would adopt the rationale set forth in the opinion of the learned Commonwealth Court. In addition, I write separately to emphasize my concern that the majority posits its conclusion upon a factual basis that is dehors the record. The majority, in direct contradiction to
In addition, the uncontroverted testimony of record concerning various safety hazards and disturbances such as "gas emerging from a hole, fire, explosion and noises" (Maj. op. p. 196) cannot be dismissed summarily by an appellate court by means of a sweeping general statement, unsupported by the record, that "these hazards and disturbances, by and large, are associated with drilling a well rather than its subsequent operation; gas production is not a noisy or unduly hazardous operation." (Slip op. p. 5; emphasis mine.) In fact, on this record, the Commonwealth Court relied heavily upon the testimony of Appellant's own expert witness with regard to the issue of "hazards and disturbances." The expert, Mr. Weiss, testified that "[w]hen you are operating a well its (sic) in danger. Many of rigs has (sic) been burned up." Reproduced record at 18a (emphasis mine). Furthermore, Appellant's
Since I find more than adequate uncontradicted evidence of record to support the conclusions of both the trial court and the Commonwealth Court, I respectfully dissent.
NIX, C.J., joins this dissenting opinion.

subset: commoncrawl


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Our spiceworks help desk configuration has been working very well, but about a month ago, it stopped sending notifications to users. The admins get the notification emails about tickets being created and whether we reply by email or through the portal, it works properly on our side for notifications and logging, but the users do not get notified. We haven't changed anything that we can think of, it just stopped working. Obviously notifications as a whole aren't broken or we wouldn't be getting them, but the end users don't know when we update a ticket. Can anyone help? This is causing a serious issue with end users thinking we aren't responding to their tickets.
This topic was created during version 5.0.
The latest version is 7.5.00101.
14 Replies
Jan 10, 2011 at 10:31 UTC
I'm sure you have; however, have you checked your help desk settings and made sure that the check box for "comments made on ticket" for submitter and CC'd user are both checked?
I accidentally unchecked mine in the past and drove me crazy until I realized what I had done.
Jan 10, 2011 at 10:38 UTC
I wasn't going to rule it out so I just checked and they are checked properly.
Jan 10, 2011 at 10:47 UTC
Have you had the users check their junk email folder in Outlook to ensure the email isn't being seen as potential spam?
Also do you have any applications such as Symantec Brightmail that assists in filtering out junk mail?
Jan 10, 2011 at 10:56 UTC
Yeah, we are a Google Apps domain so I checked postini as well as her spam folder and could find no messages from spiceworks. And it's all web-based, so no Outlook allowed, which takes that out of the equation as well.
Jan 10, 2011 at 11:01 UTC
If you send a manual message from the address that you use to send notifications on does the recipient receive it? Might help to determine where the issue is originating from.
Jan 10, 2011 at 11:24 UTC
Just tested and it is working fine when sending from that address.
Jan 10, 2011 at 1:26 UTC
Have you changed the password on your google apps domain? I'm not exactly sure how google apps works with spiceworks; however, with an exchange server you have to provide the email address password for spiceworks to be able to access the email account.
Jan 10, 2011 at 1:33 UTC
Nope. It's all been the same. With the mail stuff, you just setup the auth, and it is sending notifications to admins properly, just not to end users. So the SMTP settings are set properly
Jan 10, 2011 at 1:40 UTC
Hmm. I am at a loss then. If it isn'tin your help desk notification settings and your passwords are correct on the email setup. I am not sure. I'm afraid you have me stumped. You did notice that the help desk settings section has two sections one for admin and one for users correct? That is the last thing I can think of to suggest atm.
Jan 10, 2011 at 1:41 UTC
Yep, looked through all of those and had no luck. Thanks for the suggestions though. Maybe someone else has had this issue before and can help out.
Feb 8, 2011 at 2:53 UTC
I'm sorry to hear you're having issues with the end user notifications. I'd like to pull this issue into the support queue. Please follow the below instructions to get me your log files so I can see what is going on:
1. Navigate to settings > Email Settings > Show Additional Settings > set Full Email Logging to On
2. Have a user create a ticket like they normally would
3. Comment or close the ticket
4. Verify no email is sent out.
5. email email@example.com, refer to this thread and we will send you a link to upload your logs to take a look. Thanks!
Feb 17, 2011 at 3:36 UTC
1st Post
We are also a Google Apps domain that uses postini (like the OP) and we have been experiencing the same issue as OP.
Has a solution to this problem been found yet?
Feb 17, 2011 at 3:51 UTC
Well, I think I have figured out the issue. As it turns out, the users who were having issues had reporting rights to spiceworks, and I had turned notifications off for the helpdesk because they were in the accounting department and they had no need to get ticket notifications.
So, i removed their accounts and they are now able to get the helpdesk notifications. It's still odd that they couldn't get notifications when they were the user, without getting admin notificiations.
But it looks like that's it.
Feb 17, 2011 at 4:49 UTC
Well, we fixed the problem. An email address needed certain privileges in postini.
This discussion has been inactive for over a year.
You may get a better answer to your question by starting a new discussion.


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Mon January 16, 2017 - 7:00 PM
Perth, Australia, 2017
Amplifier Bar
393 Murray St, Perth WA 6000
Perth, Australia
*EVERY TIME I DIE* and *LETLIVE.* are joining forces to play a string of national co-headline dates across Australia in January 2017 with very special guests, *COUNTERPARTS*. Buffalo, New York hardco...


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Black quilted satin shopper from Jack Gomme
Black Truffle is a great shop in Broadway Market, definitely a hot spot to get to know when you're in London. Great ranges, and if you need a gift for a vegan friend, then they have a small but beautiful range of vegan friendly bags. I like this one in particular from Jack Gomme. It's squash, squidgy, kind of cuddly, and with plenty of room for all the rubbish/essentials we girls carry around! And if you can't get to Broadway Market, then the Black Truffle website is pretty comprehensive.


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Fred Emney
The Italian Job
In director Peter Collinson's Golden Globe-nominated caper flick, enterprising swindler Charlie Croker (Michael Caine) hatches an ingenious plan to steal a huge cache of Chinese gold en route to Turin, Italy, as collateral for a new Fiat plant. The diversion for the heist: A gigantic traffic jam during an Italy-Great Britain soccer match. Incarcerated criminal genius Mr. Bridger (Noel Coward) helps Croker get financing -- from his jail cell.


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Smouha announce Momen Soliman departure
Smouha announced the departure of head coach Momen Soliman after his team...
Ahmed Hassan, the world's most capped international footballer, is set to be announced in Egypt's new management staff led by Shawky Gharib.
Amidst speculation of his retirement from football, 38-year-old Zamalek midfielder Ahmed Hassan, who signed a one-year contract extension with the Whites in the summer, is expected to be named Egypt's team director in the new management staff of recently-appointed coach Shawky Gharib.
Should he be confirmed as Egypt team director – the official announcement is set to be made on Thursday – the veteran midfielder would be expected to announce his retirement to a legendary 18-year career.
Egypt coach Gharib reportedly recommended to have his former Egypt team-mate Alaa Nabil as assistant manager, while including Osama Nabih (current Zamalek assistant manager) and ex-Benfica star Abdel Sattar Sabry for coaching roles, according to FilGoal.
The former Egypt captain eventually lost his spot with the national team under Bob Bradley; however, he remains the world's most capped player having amassed 184 games for the Pharaohs. Hassan has won four African Cup of Nations titles and was the tournament's best player in 2006 and 2010.
Ahmed Hassan spent 10 years playing in Europe where he flourished, winning the Belgian League with Anderlecht after which he returned to Ahly for a three-year spell. With Al Ahly, he won three Egyptian Premier League titles while helping the team lift the African Champions League trophy in 2008. He recently helped rivals Zamalek, whom he joined in 2011, win the Egypt Cup – their first trophy in five years.


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Travel Channel to Film in Santa Claus This Weekend
The Travel Channel will film in Santa Claus this weekend for a special featuring Christmas towns that will air this holiday season.
Locations that may be filmed include Santa Claus Museum & Village, Lake Rudolph Campground & RV Resort, Santa Claus Christmas Store, Holiday World & Splashin' Safari, and other Christmas-themed businesses.
Where to Go to Vote in Vanderburgh County
Today is primary election day. The voting centers allow you to cast your vote at whatever center is most convenient for you. Here's a list of the voting center locations:
Shrinersfest 2014 to Feature Planes Boats and Mike Harvey's Last Roadshow
The Shriners have announced plans for the 2014 Shrinersfest.
The big news is the addition of an exhibition of hydroplane boats. It won't exactly be 'Thunder on the Ohio' but it will be the first time in five years that we'll get to see the hydroplanes on the river...
CJ's Willow Bowling Center to Close
CJ's Willow Bowling Center is closing it's doors on May 18. The center has been run by the Rayburn family for 59 years.
The decision to close is because Mike Rayburn wishes to retire and his brother Patrick wants to spend more time focusing on his development business...
NBA Bans Clippers Owner Donald Sterling for Life
Donald Sterling is out as owner of the LA Clippers. The National Basketball Association announced today that Sterling is banned from basketball for life and fined him $2...
What You May Not Have Known About Easter
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More Fun at Walmart – Fight Over Entering Through the Exit
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Be On the Lookout for a Stolen Gazebo
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IU Decides to Locate School of Medicine Downtown
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Pat Shoulders made the motion to pass the proposal, calling the proposal "A win for Evansville...


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Ashley Johnston
April 30, 2012 at 12:52PM
Overton Park and Rust Hall will become a playground for creativity on Saturday, as the Memphis College of Art's Cinema 7 ensemble hosts an "art bike" parade and the (interactive) installation exhibit, ENTER=ACTIVE. Cinema 7 is a collective of MCA students comprised of an experimental video production class, taught by Associate Professor Jill Wissmiller.
Ashley Johnston
April 26, 2012 at 1:02PM
The Memphis Brooks Museum was recently awarded a $44,000 Art Works grant from the National Endowment for the Arts (NEA) to expand its unique Art Therapy Access program. Targeted to at-risk youth, adults, and families in partnership with local social service organizations, the year-long program combines the personal, yet connective outlet of creating art with the social aspects of a museum environment to facilitate interactive and ultimately therapeutic group experiences.
Ashley Johnston
April 19, 2012 at 6:23PM
Ashley Johnston
April 18, 2012 at 12:08PM
The Clough-Hanson Gallery at Rhodes hosted the college's Annual Juried Student Exhibition last Friday and Saturday, including a singularly entertaining performance piece from Johnathan Robert Payne that drew a small crowd of delighted students.
Ashley Johnston
April 12, 2012 at 2:44PM
While most of the selected filmmakers chose a more straightforward, traditional approach, with documentaries and narratives based inside the borders of a specific neighborhood, some branched out to the greater concept of the Memphis community as a whole. Out of eight hand-picked proposals, stretching across the city from Binghampton to Vance Avenue, the work of Nicki Newberger, Alan Spearman, Mark Adams, Morgan Jon Fox, and Sean Faust was chosen by the audience of more than 250 locals to receive a $5,000 grant - funded by ticket sales from the event as well as sponsor donations.
© 2017 Memphis Flyer


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Predictive Analytics – How to Conduct an Effective Safety Audit (South Bend, IN - 2016)
The Horton Group
Tuesday, October 11, 2016 from 8:30 AM to 11:30 AM (EDT)
Event Details
Registration is from 8:00 - 8:30 a.m. | Continental Breakfast for Attendees
Horton's predictive analytics platform compiles data from multiple sources to provide employers with actionable information to predict and prevent workplace injuries and improve safety outcomes.
These positive results have a ripple effect, reducing downtime and improving operational efficiencies. By integrating safety audits and reporting programs with a robust safety culture and analytical tools that are grounded in both leading and lagging indicators, you will be able to predict conditions where severe injuries are likely.
Gary Glader
President / Horton Safety Consultants / Bio
Gary Glader oversees a team of safety professionals dedicated to developing, implementing and managing effective safety and risk management programs. As a Certified Safety Professional, Gary consults with clients interested in reducing the frequency and severity of incidents and injuries in order to help drive the cost of insurance and risk down. Gary also provides clients with OSHA and other regulatory body compliance assistance.
Since 1980, Gary has built relationships with many clients, their supervisors, and employees. These relationships, coupled with Gary's extensive experience, offer strategic advantages to his clients in keeping employees safe, driving down costs, and ensuring compliance with regulatory requirements.
Return to website: http://www.thehortongroup.com/events/predictive-analytics-how-to-conduct-an-effective-safety-audit-south-bend-in
When & Where
Notre Dame / Morris Inn - Salon A & B
1399 N Notre Dame Ave
South Bend,
Tuesday, October 11, 2016 from 8:30 AM to 11:30 AM (EDT)
Add to my calendar
The Horton Group
The Horton Group is an insurance, employee benefits and risk advisory firm. We specialize in helping clients turn costs and risk into competitive advantage, and growing the business for tomorrow. Our business is about helping you achieve a higher level of performance. The Horton Group has the people, insight and innovation to get ahead of trends, and offer new ideas on critical challenges.
Check out our Upcoming Events, Industries, and Future Forum.


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Having trouble with Pivot Tables, this will make your day!. Can you do any better than this? … [Read more...]
Introduction To Calculations In Excel.
Performing calculations is one of the main uses of spreadsheets and Excel allows you to build quite complex formulae using arithmetic operators, but it also has a myriad of predefined functions for performing statistical analysis, conditional functions, scientific and financial calculations. A function is still a calculation, but it is simply a formula which is predefined in Excel. Formulae in Excel are made up of a number of different elements that when combined in a specific way ensure … [Read more...]