Product (mathematics)

About: Product (mathematics) is a research topic. Over the lifetime, 44382 publications have been published within this topic receiving 377809 citations.

Computing algebraic formulas using a constant number of registers

Abstract: It is shown that, over an arbitrary ring, the functions computed by polynomial-size algebraic formulas are also computed by polynomial-length algebraic straight-line programs that use only three registers. This was previously known for Boolean formulas [D. A. Barrington, J. Comput. System Sci., 38 (1989), pp. 150–164], which are equivalent to algebraic formulas over the ring $GF(2)$. For formulas over arbitrary rings, the result is an improvement over previous methods that require the number of registers to be logarithmic in the size of the formulas in order to obtain polynomial-length straight-line programs. Moreover, the straight-line programs that arise in these constructions have the property that they consist of statements whose actions on the registers are linear and bijective. A consequence of this is that the problem of determining the iterated product of $n3 \times 3$ matrices is complete (under P-projections) for algebraic $NC^1 $. Also, when the ring is $GF(2)$, the programs that arise in the c...

The basis of product emotions

Abstract: This paper introduces a conceptual model for the process underlying emotional responses that result from the perception of consumer products. The model distinguishes different kinds of emotions on the basis of eliciting conditions. It is based on the presumption that all emotional reactions result from an appraisal process in which the individual appraises the product as favouring or harming one or several of his concerns. In this process of appraisal, the personal concern gives the stimulus emotional relevance. The model describes the various ways in which products can act as emotional stimuli and the matching concerns that can either correspond or collide with these stimuli. Products can act as stimuli in three different ways: the product as such, the product (or designer) as an agent, and the products as a promise for future usage or ownership. The corresponding concerns that are addressed are respectively: attitudes, standards, and goals. By revealing the cognitive basis of product emotions, the model can be used to explain the nature and, often, mixed character of product emotions. The paper illustrates a possible application of the model in a tool for designers. INTRODUCTION After being neglected for many years, a sudden interest in product affect has emerged. The affective side of product experience has become a ‘hot topic’, which is probably best illustrated by this conference and similar events over the past few years (e.g., Overbeeke & Hekkert, 1999). A difficulty of affective concepts such as pleasure and emotion is that they are probably as intangible as they are appealing. Although some interesting and promising studies have been reported, the research field is still short of conceptual clarity and therefore lacks consensus on what the actual subject of study should be. In fact, both the concepts of pleasure and emotion are somewhat undifferentiated, they are used as collective nouns for all kinds of affective phenomena. Design literature tends to refer to these when studying anything that is so-called intangible, non-functional, non-rational, or, for that matter, non-cognitive. Some of the reported studies involve ‘experiential needs’ (Holbrook, 1982), ‘affective responses’ (Derbaix & Pham, 1991), ‘emotional benefits’ (Desmet, Tax & Overbeeke, 2000), ‘customer delight’ (Burns, Barrett & Evans, 2000) and ‘pleasure’ (Jordan & Servaes, 1995). Naturally, it is inherent to any newly emerging research field that the emulsion has not even started to crystallise. On the other hand, an adequate definition of the subject of study would probably facilitate fruitful discussions between researchers. In our view, a model of product emotions can help to get a grip on the concept of product pleasure and emotions. This paper introduces such a structured model that distinguishes different kind of product emotions on the basis of their eliciting conditions. The model adheres to the cognitive (functional) view on emotions and finds its roots in a structure developed by Ortony, Clore and Collins (1988). A first step in developing the model is to Desmet, P.M.A., & Hekkert, P. (2002). The basis of product emotions. In: W. Green and P. Jordan (Eds.), Pleasure with Products, beyond usability (60-68). London: Taylor & Francis. clarify the relationship between the concepts of pleasure and emotion. PLEASURE AND EMOTIONS The concept of affect refers to a large variety of psychological states such as emotions, feelings, moods, sentiments, and passions. Each of these affective states varies in duration, impact and eliciting conditions. Of these states, emotions are most relevant for product experience because only they imply a one-to-one relationship between the affective state and a particular object: one is afraid of something, angry at someone, happy about something, and so on (Frijda, 1986). The other affective states, such as feelings and moods, do not involve a specific object. For example, a moody person will find it difficult to pinpoint the exact cause of his mood. In the study of affective reactions to products, the object, i.e., the product, is the starting-point. Subsequently, the model of product affect presented in this paper focuses specifically on emotions. The place of pleasure in emotions is debatable. Both the propositions that pleasure is an emotion, and that it is not, are defensible. On the one hand, pleasure is an emotion if it is merely used as an equivalent of ‘fun’ or ‘enjoyment’. In this connotation pleasure is included in many of the taxonomies of emotions found in literature (e.g., Russell & Lanius, 1984). On the other hand, this view on pleasure seems to be rather narrow for the current application. Design research literature refers to pleasure as a product benefit that exceeds just proper functioning. In other words, pleasure is an emotional benefit that supplements product functionality. In this sense, pleasure covers all pleasant emotional reactions, of which the experience of fun is just one example. Valence (a bipolar ranging from pleasant to unpleasant) is a dimension frequently discovered in scaling procedures of emotion terms. If pleasure is regarded as a dimension of emotions, it can be used to describe emotions, but it is not an emotion as such. This notion befits everyday experience: one never feels pleasant as such. One feels happy, cheerful, surprised, inspired, etc. Although each of these emotions might be pleasant, that does not make pleasantness an emotion. Therefore, in the light of this paper, pleasure is defined as any pleasant emotional response elicited by product design. It might seem difficult, if not impossible, to find general relationships between product appearance and emotional responses because emotions are essentially personal. Nevertheless, although people differ in their emotional responses to products, general rules can be identified in the underlying process of emotion eliciting. A view that distinguishes such general rules is the cognitive view on emotions. THEORY OF EMOTION The cognitive, functionalist position on emotions posits that emotions serve an adaptive purpose. In this view, emotions are considered the mechanisms that signal when events are favourable or harmful to one’s concerns. This implies that in each emotion-eliciting stimulus some concern can be identified. These concerns are more or less stable preferences for certain states of the world; they are our personal motives in life (Frijda, 1986). Examples of human concerns are concerns for respect, safety, and self-esteem. For instance, we all have the concern of being treated with the respect we believe we deserve. When a person receives a degrading comment from a colleague, he will probably find this event conflicting with this concern for respect. Consequently, this person will experience a negative emotion such as shame or anger. The preceding example illustrates that the linking of the stimulus to the concern precedes the actual emotional response. This process of ‘signalling the emotional relevancy of an event’ is most commonly conceptualised as ‘a process of Desmet, P.M.A., & Hekkert, P. (2002). The basis of product emotions. In: W. Green and P. Jordan (Eds.), Pleasure with Products, beyond usability (60-68). London: Taylor & Francis. appraisal’ (e.g., Arnold, 1960; Frijda, 1986). Appraisal theories assert that it is not events per se that determine emotional responses, but evaluations and interpretations of events. Because emotions are intentional and essentially involve concerns, they seem to require an explanation that invokes these concerns. Moreover, as this paper focuses on emotions specifically elicited by products, the explanation should also include eliciting conditions. A good starting point for a model of product emotions is the model developed by Ortony et al. (1988) because it particularly focuses on this relationship between different types of concerns and the eliciting conditions. stimulus concern appraisal product as object attitude appealingness product as agent (action of) standard praiseworthyness product as event goal desirability Figure 1: Model of product emotions MODEL OF PRODUCT EMOTIONS According to the cognitive model of Ortony et al. (1988) there are three major aspects of the world we can focus on: events, agents, or objects. We focus on events (e.g., a football match) for their consequences (e.g., loss of your favourite team), we focus on agents (e.g., a dog) for their actions (e.g., barking at you), and we focus on objects (e.g., a painting) because we are interested in certain properties of them as such (e.g., its composition). Central to Ortony et al’s view is the position that emotions are valenced reactions to one of these perspectives on the world. Based on this division they developed a structure of emotion types that are logically related to one of these three aspects. In focusing on product emotions, at first side, it may seem tempting to restrict ourselves to the third class ‘products as objects’. Our major claim is however that all three perspectives are relevant when products are simply perceived with one of our senses, i.e., without requiring physical interaction with the product. What we will present here is an adjusted version of the original model in which those elements are adopted that cover emotions that may result from product perception. This adapted model is presented in Figure 1. Figures 2, 3, and 4 each show examples of one of the three classes of product emotions. Products as objects To start with the most obvious and simple branch of the structure in Figure 1, products are objects. Like all objects, products, or aspects of products, can be viewed as such in terms of their appealingness. Products are simply liked or disliked for their appearance, for the way they look. The emotional reactions are basically unstructured and comprise attraction emotions such as love, attracted-to, disgust, and boredom. attitude appealingness product as objec

Some remarks on commutators

Abstract: In a group the product of two commutators need not be a commutator, consequently the commutator group of a given group cannot be defined as the set of all commutators, but only as the group generated by these. There seems to exist very little in the way of criteria or investigations on the question when all elements of the commutator group are commutators. In the following it is shown that in the finite symmetric group 2n all elements of the alternating group are commutators; one can extend this and show that when n >5 all elements in the alternating group are commutators of elements in An. For the infinite symmetric group the situation is different since we obtain: Any one-to-one correspondence of an infinite set to itself is a commutator. 1. We shall begin by making a few general remarks which apply both to the finite and the infinite cases. Any one-to-one correspondence T of an arbitrary set S to itself can be written uniquely as a product of cycles which operate on disjoint sets of elements

Local description of quantum inseparability

Abstract: We show how to decompose any density matrix of the simplest binary composite systems, whether separable or not, in terms of only product vectors. We determine for all cases the minimal number of product vectors needed for such a decomposition. Separable states correspond to mixing from one to four pure product states. Inseparable states can be described as pseudomixtures of four or five pure product states, and can be made separable by mixing them with one or two pure product states.

Method and system for product configuration definition and tracking

Abstract: A method and system for the definition and tracking of multiple unique product configurations for configuration management in which customer contracted views of the product are defined by applying temporary changes to the base product configuration A product serial number effectivity is assigned to each unique customer specified product configuration and to the base product configuration in order to maintain multiple time-oriented versions of the product, thus supporting simultaneously both customer-specified and design-specified engineering changes to the base product A product configuration identifier provides non-redundant data storage for each unique product configuration and can refer to noncontiguous ranges of product serial numbers The configuration entry identifier in a product configuration table is used to peg detail component requirements to serially numbered products in order to correlate the product configurations as built with the product configuration as designed A component position identifier provides traceability of temporary changes made to the base product that result from customer contract specifications