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This chapter is about the prospect of proving that certain problems do not have polynomial-time algorithms (are not in P). Tractable problems (having low time complexity) are compared with seemingly intractable problems (having high time complexity). The class NP and NP-completeness are defined and elaborated. Several classic problems, including the seminal satisfiability problem, are proven NP-complete.
This chapter is about heuristics, which are algorithms that succeed on typical inputs (average-case) rather than on all conceivable inputs (worst-case). The chapter covers the average-case analogue of the P vs. NP question, worst-case to average-case reductions via error correcting codes, and hardness amplification.
This chapter proves one of the few known theorems toward separating P and NP: Satisfiability has no algorithm that runs in a certain modest polynomial amount of time and polylogarithmic amount of space. The proof serves as a unified excuse to present other important concepts, including padding, the time and space hierarchy theorems (showing that more time or space gives algorithms more power), and the polynomial hierarchy and its collapse property.
This chapter introduces the fundamental concepts of computational problems, algorithms, and time and space efficiency. These concepts are formally defined and illustrated with examples. The major time and space complexity classes P, EXP, L, and PSPACE are introduced. The chapter concludes by defining reductions, which are tools for relating complexities of different problems.
Medical waste management has become a critical sustainability challenge in developing countries due to increasing healthcare demand, inadequate waste treatment infrastructure, and weak regulatory implementation. Guided by circular economy theory, this study investigates the feasibility of adopting the 3R (Reduce, Reuse, Recycle) framework to improve medical waste management practices in Bangladesh. A qualitative research design was employed, involving in-depth interviews with 31 healthcare stakeholders, including cleaners, nurses, waste collectors, administrators, and waste management authorities. Data were analyzed using thematic analysis to explore existing waste management practices, implementation challenges, and opportunities for circular transformation. The outcomes reveal that healthcare facilities have partially adopted 3R practices through waste segregation, sterilization, resource recovery, wastewater treatment, and recycling initiatives. However, inadequate infrastructure, limited technical capacity, insufficient workforce training, and weak enforcement of existing regulations remain significant barriers to effective implementation. Despite these challenges, locally adapted circular practices demonstrate considerable potential to reduce environmental pollution, improve resource efficiency, and enhance healthcare sustainability. This study extends circular economy theory to healthcare waste management by introducing the concept of contextual circularity within resource-constrained settings. It offers practical recommendations for strengthening policy implementation, institutional capacity, and stakeholder collaboration while contributing to SDGs 3, 12, and 13. The quotes provide a scalable framework for advancing sustainable and climate-resilient healthcare waste management in developing economies.
This chapter examines Wilfrid Sellars’s idea of material inference – the view that there are valid inferences expressed in non-logical language that are irreducible to logical principles. Material inference can be characterised by three features: it is non-enthymemic, non-logical, and non-monotonic. However, the exact sense in which material inferences are valid remains unclear.After exploring various attempts to define the validity of material inference – such as through truth-preservation, analytic entailment, or reduction to general principles – I argue that material inference is not valid in the strict sense. Typically, validity refers to deductive validity, where the truth of the premises guarantees the truth of the conclusion.Nonetheless, pragmatists do not apply an external standard to our reasoning practices. Material inference can be understood not as valid, but as a legitimate and rationally authoritative form of reasoning. The key concept here is pragmatic propriety, which designates the social-normative entitlement to infer – rendering an inference either legitimate or illegitimate.
Psychiatry has a core goal of understanding the mechanisms that produce, underlie, and maintain psychiatric disorders. But what is a mechanism? And how should the answer to that question be justified? In this chapter Craver defends a practice-first direction of fit, which involves building a notion of “mechanism” for psychiatry by charitably reconstructing what the term appears to mean from the functions it serves for psychiatrists and researchers. Taking this approach, many philosophical objections to mechanism (i.e., that it is at odds with a process metaphysics and that it is committed to reductionist obsession with detail) turn out to be predicated on misunderstandings. Some directions for a future mechanistic philosophy of psychiatry are sketched out.
This paper describes a process of laryngeal reduction in San Martín Peras Mixtec (SMPM; ISO: jmx), an Otomanguean language spoken in Oaxaca and by diasporic communities throughout Mexico and the US. In this process, roots containing a laryngealized vowel often appear in a highly reduced form in fast speech. Laryngeal reduction is gradient, dependent on speech rate, and lacks a phonologically-defined conditioning environment, giving it the characteristics of a phonetic process. However, it is at least sometimes correlated with a phonological process of mora deletion, as evidenced by the fact that some highly reduced laryngealized roots—but no unreduced laryngealized roots—undergo a phonological tone sandhi alternation that applies only to mono-moraic rising tones. The phonological process of mora deletion is argued to be conditioned by the same phonetic factors that drive laryngeal reduction, constituting an instance of a phonological process triggered by purportedly phonetic factors.
This chapter focuses on how sounds can shift when they occur in particular environments. It introduces key concepts from the field of phonology, such as phonemes and allophones, and demonstrates how sounds commonly change during speech production. The major types of sound shifts discussed in this chapter include assimilation, deletion, insertion, and dissimilation. By the end of the chapter, you will be asked to apply phonological rules to a small data set and create a set of potential phonological shifts you can incorporate into your language.
We argue that logicism, the thesis that mathematics is reducible to logic and analytic truths, is true. We do so by (a) developing a formal framework with comprehension and abstraction principles, (b) giving reasons for thinking that this framework is part of logic, (c) showing how the denotations for predicates and individual terms of an arbitrary mathematical theory can be viewed as logical objects that exist in the framework, and (d) showing how each theorem of a mathematical theory can be given an analytically true reading in the logical framework.
Fulgurites are natural glasses that form when lightning strikes sand, soil, or rock and fuses the individual grains together to generate what is usually a tubular structure that follows the path of the strike. During this process, localised reducing conditions are conducive to forming rare minerals including iron silicides. This paper examines a fulgurite formed in Southwick, Massachusetts, USA, which displays an iron silicide that has a clearly defined reaction rim. The reaction rim demonstrates the production of a more silicon-rich rind consisting of Fe5Si3 on a core of Fe2Si, and the most likely route to forming this material is by reaction of silicon gas with Fe2Si at high temperature (>1000°C), with a reaction timescale of about one second. This reaction suggests the high temperature, reducing conditions of a lightning strike favour reactions of condensed matter (e.g. liquid or solid iron minerals) with gas that occurs rapidly during the lightning strike. The conditions necessary to form these minerals suggest that the fulgurite became more reducing over time, as more Si entered the solid phase, perhaps as oxygen left the system, either as CO2 or from the breakdown of SiO2 gas.
Structural Fe2+ in montmorillonite is readily oxidized by contact with water, salt solutions or on mild heating. This is shown clearly by the Mössbauer spectra and is associated with a sharpening of the infrared absorption near 880 cm−1. It was inferred that this band comprises the Fe2+—OH—Al and Fe3+—OH—Al deformations. The rate at which oxidation occurs depends on the exchangeable cations. High acidity of the interlayers is conducive to oxidation, as is contact with Cu2+-containing solutions or concentrated H2O2 solutions.
The results show clearly that any chemical treatment of montmorillonite causes changes in the oxidation state of structural iron.
Using Garfield, Washington, nontronite as the model mineral system, methods and apparatus were developed to prepare reduced suspensions in citrate-bicarbonate-dithionite (CBD) solution. These techniques were effective in removing excess, undesired solutes from reduced suspensions while maintaining a high Fe2+ content. They also enabled the preparation of dried, reduced films preferentially oriented with respect to the crystallographic c-axis. Supernatant solutions were collected and analyzed for Fe, Al, and Si, from which the extent of dissolution of the clay as a result of CBD treatment was assessed. Results indicated that very little Fe and Si were released to solution, but as much as about 8% of the total Al was solubilized. The highest levels of Al in solution were observed in CB treatments without dithionite.
The effects of reduction and reoxidation of octahedral Fe3+ on the exchange of structural hydrogen in nontronite were determined using tritium (3H) as a label element. The uptake of H from the surrounding solution of nontronite suspensions increased as the reduction of structural Fe3+ increased. Similarly, the loss of H from the structure increased as the reduction increased. The results are generally consistent with a reduction mechanism involving the loss of structural OH, leaving the affected Fe sites with less than six-fold coordination. The attenuation of increased negative charge on the clay layer, however, was less than predicted by such a mechanism.
During the reoxidation of reduced nontronite in suspension, about one-third of the H remaining as part of the structure following reduction was lost, whereas twice that amount of H was incorporated into the structure from the surrounding solution. A reoxidation mechanism is proposed whereby H2O from the surrounding solution is incorporated into the mineral structure followed by the elimination of a hydrogen ion, returning the Fe to six-fold coordination. This mechanism implies the reversibility of Fe reduction in nontronite.
A new type of environmental chamber for X-ray diffraction was designed that could sustain elevated, internal pressures of nitrogen or any other gas under oxygen-free conditions and that allowed the positions of the specimen and edge aperture to be adjusted by remote control. It was used to determine the values of the interlayer spacing, λ, of nontronite from Garfield, Washington, in different stages of reduction at different values of Π, the swelling pressure of the nontronite. At equilibrium, Π was equal to the pressure under which water was expressed from the clay. Both partially and fully expanded layers were found to exist in the reduced nontronite, the fraction of partially expanded layers increasing with increasing Π and Fe2+/Fe3+, the ratio of Fe2+ to Fe3+ in octahedral sites. Also, λ for the partially expanded layers was found to depend on Fe2+/Fe3+ but not on Π, and λ for the fully expanded layers was found to depend on Π but not on Fe2+/Fe3+. These findings were interpreted to mean that the reduction of Fe affected the short-range interlayer forces, but not the long-range ones.
An iron-rich chlorite, ripidolite, was oxidized by air-heating at 480°C, i.e., below the dehydroxylation temperature and subsequently reduced in hydrogen at the same temperature. On the basis of chemical, differential thermal, infrared, Mössbauer, and X-ray powder diffraction analyses, Fe(II) seems to be present only in the 2:1 layer of the original chlorite in a type of site similar to that of Fe(II) in biotite, with OH in cis-positions. These data also suggest that octahedral Al and Fe(III) are located in the hydroxide sheet of the original chlorite. The structural changes of the mineral due to the oxidation and the subsequent reduction appear limited to minor structural rearrangements and, perhaps, to the introduction of OH in both cis- and trans-positions. The results of the investigation are in agreement with a reaction of the form: [Fe(II)OH]+ ⇋ [Fe(III)O]+ + H(H+ + e−).
The reduction and reoxidation of three nontronite samples, GAN (API H-33a, Garfield, Washington), SWa-1 (ferruginous Washington smectite), and NG-1 (Hohen Hagen, Federal Republic of Germany) were studied with visible absorption and Mössbauer spectroscopy. The intensity of the intervalence electron transfer (IT) band at 730 nm in these nontronites was monitored during reduction and reoxidation at 277, 294, and 348 K. The results showed that the intensity of the band followed the number of Fe(II)-O-Fe(III) groups in the clay crystal, increasing to a maximum at about Fe(II): total Fe = 0.4; upon complete reduction, the band decreased to about the intensity of the unaltered, oxidized sample. With reoxidation of the sample with O2, the intensity of the band increased sharply, followed by a gradual decay back to the original, oxidized intensity. The ultimate level of Fe reduction achieved was at least 92%. Concomitantly, the color changed from yellow through green, blue-green, dark blue, light blue, and light gray as the Fe(II) content increased. The GAN nontronite was more difficult to reduce than the SWa-1 or NG-1 samples. The rate and level of reduction increased with the amount of reducing agent added.
The effects of chemical reduction of structural Fe3+ in nontronite SWa-1 (ferruginous smectite) on intervalence electron transfer (IT) and magnetic exchange were investigated. Visible absorption spectra in the region 800-400 nm of a chemical reduction series of the SWa-1 nontronite revealed an IT band near 730 nm (13,700 cm−1). Both the intensity and position of this band were affected by the extent of Fe reduction. The intensity increased until the Fe2+ content approached 40% of the total Fe, then decreased slightly with more Fe2+. The position of the band also shifted to lower energy as the extent of reduction increased.
Variable-temperature magnetic susceptibility measurements showed that the magnetic exchange in unaltered nontronite is frustrated antiferromagnetic, but ferromagnetic in reduced samples. Magnetic ordering temperatures are in the range 10–50 K, depending on the extent of reduction. The ferromagnetic component in the magnetization curve increased with increasing Fe2+ in the crystal structure. The positive paramagnetic interaction likely is due to electron charge transfer from Fe2+ to Fe3+ through such structural linkages as Fe2+-O-Fe3+ (perhaps following a double exchange mechanism), which is consistent with the visible absorption spectra.
A new iron oxide dissolution method designed to measure the abundance of “free” Fe oxide phases and associated elements in soils and sediments has been tested. The method employs a ternary complex of Ti(III), citrate, and ethylenediaminetetraacetate (EDTA) as a reductant and bicarbonate as a proton acceptor. The Ti(III)-citrate-EDTA-HCO3 method dissolved more synthetic amorphous ferric oxide and goethite, but less synthetic hematite, than the dithionite-citrate-HCO3 method of Mehra and Jackson. The production of acidity by the dissolution indicated that Ti(IV) is hydrolyzed to TiO2 during the extractions. The heated dithionite method dissolved 3–6 times more Al from kaolinite and nontronite standard clays than room temperature dithionite, and 4–6 times more Al than the Ti(III)-citrate-EDTA-HCO3 method. Furthermore, the release of Fe from the clay mineral samples consistently and rapidly reached a plateau during multiple extractions by the Ti(III)-citrate-EDTA-HCO3 method, indicating that a well-defined Fe oxide fraction was removed. Fe released by the dithionite method continued to increase with each extraction, suggesting that some release of structural Fe occurred. Tests on two natural sediments and one heavy mineral fraction from the Miocene Cohansey Sand in the New Jersey Coastal Plain suggested that the Ti(III)-citrate-EDTA-HCO3 method removed Fe oxides more effectively and more selectively than the dithionite method. The selectivity of the Ti(III)-citrate-EDTA-HCO3 method is enhanced by rapid extractions at room temperature and low free ligand concentrations.