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# The challenge of antimicrobial resistance: What economics can contribute

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Science  05 Apr 2019:
Vol. 364, Issue 6435, eaau4679
DOI: 10.1126/science.aau4679

## Incentivizing restraint in drug use

The accelerating tide of antimicrobial resistance (AMR) is a major worldwide policy concern. Like climate change, the incentives for individual decision-makers do not take into account the costs to society at large. AMR represents an impending “tragedy of the commons,” and there is an immediate need for collective action to prevent future harm. Roope et al. review the issues associated with AMR from an economics perspective and draw parallels with climate change. A major stumbling block for both challenges is to build consensus about the best way forward when faced with many uncertainties and inequities.

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## Structured Abstract

### BACKGROUND

Antimicrobial resistance (AMR) is increasing, driven by widespread antibiotic use. The wide availability of effective antibiotics is under threat, jeopardizing modern health care. Forecasts of the economic costs are similar to those of a 2°C rise in global average surface temperature, above preindustrial levels. AMR is becoming an urgent priority for policy-makers, and pressure is mounting to secure international commitments to tackle the problem.

Estimating the value of interventions to reduce antibiotic use requires predictions of future levels of antibiotic resistance. However, modeling the trajectory of antibiotic resistance, and how marginal changes in antibiotic consumption contribute to resistance, is complex. The challenge of estimating the resulting impact on health and the economy is similarly daunting. As with the cost of climate change, estimates of total AMR costs are fraught with uncertainty and may be far too low. Much of the uncertainty arises from the complexity of estimating the cost of changes in overall resistance levels. This cost depends on various factors: which drug and pathogen are involved, the mechanism of antibiotic resistance, the prevalence of that pathogen, the types of infections it causes and their level of transmissibility, the health burden of those infections, and whether alternative treatments are available.

Effective new antibiotics are urgently needed. However, without government intervention, R&D for antibiotics is rarely profitable, and most major pharmaceutical companies have left the field. New ways are needed to make antibiotic development profitable, decoupling profits from volumes sold.

### OUTLOOK

Analogies can be drawn between climate change and AMR, both of which have been described as a global “tragedy of the commons.” There is some consensus that we should treat carbon emissions reduction as an insurance policy against the possibility of a catastrophic climate outcome—and avoid waiting for a definitive optimum-abatement policy. A similar paradigm shift is needed to incentivize both the introduction and valuation of interventions to reduce antibiotic use and R&D of new antibiotics.

Rather than taxing the price and letting the market dictate the quantity of antibiotics supplied, an alternative may be to establish a regulatory body that issues prescribers tradable permits and to allow the market to determine the price. Such an approach could create a predictable revenue stream through more-foreseeable licensing fees for important antibiotics by decoupling the return on investment from the volume used. Approaches such as this could incentivize industry to develop new antibiotics for which there would otherwise be too small a market to provide a sufficient return on investment.

Reducing inappropriate antibiotic use while expanding essential access is a difficult challenge, especially in low- and middle-income countries. However, policy-makers and philanthropists are alert to the importance of AMR and increasingly are making substantial research funds available, with much of these funds devoted to the social sciences. We need economists, across many different fields, to engage with this pressing global problem.

## Abstract

As antibiotic consumption grows, bacteria are becoming increasingly resistant to treatment. Antibiotic resistance undermines much of modern health care, which relies on access to effective antibiotics to prevent and treat infections associated with routine medical procedures. The resulting challenges have much in common with those posed by climate change, which economists have responded to with research that has informed and shaped public policy. Drawing on economic concepts such as externalities and the principal–agent relationship, we suggest how economics can help to solve the challenges arising from increasing resistance to antibiotics. We discuss solutions to the key economic issues, from incentivizing the development of effective new antibiotics to improving antibiotic stewardship through financial mechanisms and regulation.

Before the discovery of penicillin by Alexander Fleming in 1928 and its subsequent development as a medicine by Howard Florey and Ernst Chain, death from infection could follow from something as minor as a scratch. Today, in high-income countries people take for granted the relative safety of procedures such as caesarean sections, joint replacements, and chemotherapy—all of which would be much more dangerous, perhaps prohibitively so, without effective antibiotics (13).

These gains in health care are now under threat from antimicrobial resistance (AMR). Here, we exclusively consider resistance against antibiotics used to treat bacterial infections. Resistance is largely driven by the ubiquitous use of antibiotics, often with little or no therapeutic benefit. For instance, antibiotics are commonly used for viral respiratory ailments (4) and are used nontherapeutically in agriculture to promote livestock growth (5). Additionally, widespread use of broad-spectrum antibiotics, which are effective against a variety of bacteria, has greater potential for selecting for extensive antibiotic resistance than drugs that target specific bacteria. However, it is important to acknowledge that even so-called “appropriate” use contributes to the development of AMR.

The potential costs to human life and to the economy are sobering. Recent U.K. government–commissioned reports (6, 7) estimate that if no action is taken, by 2050, AMR will cause up to 10 million annual deaths globally, reduce gross domestic product (GDP) by 2 to 3.5%, and cost US$100 trillion. The World Bank (8) has estimated that the impact on global GDP may be even greater, with an annual cost of up to US$3.4 trillion by 2030. Current estimates such as these are necessarily based on limited data and contestable assumptions, and some have questioned their utility (9). Notwithstanding their limitations, these projections exceed the current burden due to cancer, both in terms of mortality and cost (10). They also place the costs of AMR on par with the best current estimates of GDP loss associated with a 2°C rise above preindustrial levels in global average surface temperature (just under 3%) (11). In September 2016, AMR became only the fourth health issue—after HIV, noncommunicable diseases, and Ebola—to be discussed by the United Nations General Assembly. Pressure is mounting for an international commitment to tackle AMR (12), one that is analogous to the United Nations 2015 Paris Agreement to limit global temperature rise to 1.5°C.

Within economics, there has been some discussion of how AMR may be conceptualized and the difficulties of incorporating the large and uncertain future costs of AMR into assessments of health technologies (13, 14). Beyond this, however, discussion of AMR within economics has been limited. At the time of writing, only 55 of more than 1 million peer-reviewed articles in the EconLit database relate broadly to AMR. By contrast, 16,306 articles relate to climate change. We discuss what might be learned from this extensive economic literature on climate change to inform both current policy and future economic research on AMR. A key feature of these two global challenges is that antibiotic consumption and carbon use will bring about adverse future consequences, but there is substantial uncertainty over the timing and extent of these consequences. AMR also poses unique challenges that do not have a climate change equivalent. Here, we examine some of the concepts underlying these challenges and highlight prospects for future research on the economics of AMR.

## The scientific background

When Fleming, Chain, and Florey were awarded the Nobel Prize in 1945, they sounded an early warning that, owing to natural selection, bacteria could become resistant to antibiotics. They worried that subtherapeutic antibiotic doses would teach bacteria to resist antibiotics. Although this worry may still be relevant (15), most concerns currently center on common commensal bacteria being unnecessarily exposed to antibiotics in the treatment of bacterial (or even viral) infections. The more these commensal bacteria are exposed to antibiotics, the greater the risk of antibiotic resistance in subsequent infections. Nevertheless, the warning of future antibiotic ineffectiveness was prophetic and AMR is now associated with substantial morbidity and costs (16), which will continue to rise if resistance increases. Estimating the potential economically justified levels of investment in the development of new antibiotics and interventions that reduce unnecessary antibiotic use requires predictions of future levels of antibiotic resistance at the population level. However, modeling the trajectory of antibiotic resistance is complex (17).

### Uncertain future extent of AMR

The use of an antibiotic may not only select for resistance against itself but may also coselect for resistance against other antibiotics (18) or even result in susceptibility to other antibiotics (19). Thus, without detailed understanding of the underlying mechanisms and interactions, it is difficult to estimate the overall effect of changes in antibiotic consumption on resistance levels. Although the emergence of mutations conferring resistance to certain antibiotics is predictable, the spread of new resistance genes into mobile DNA or widespread bacterial strains is a so-called black swan event; such event will happen, but their timing, magnitude, and impact are difficult to predict (17, 20).

### Reversibility of AMR and renewability of antibiotics

Although increased selection pressure from rising antibiotic use makes greater resistance inevitable, it is not clear whether decreasing antibiotic use will necessarily be followed by a reduction in resistance (21). Most antibiotic resistance mechanisms are associated with a fitness cost, which reduces competitiveness in the absence of a drug and may allow susceptible bacteria to regrow. However, compensatory evolution, which reduces the fitness disadvantage of the resistance mechanism, is widespread among bacteria and limits the effectiveness of reducing antibiotic use (2123). Whether fitness costs or genetic compensation prevails in natural infections is poorly understood (24).

Another important parameter determining the potential impact of changing antibiotic prescribing is coselection of resistance to more than one antibiotic (21). Some empirical evidence suggests that the effect of reducing an antibiotic’s use depends on what, if any, antibiotic it is replaced by. In an effort to reduce resistance in pathogenic strains of Escherichia coli, use of trimethoprim and sulfonamide antibiotics was reduced in the United Kingdom and Sweden. Disappointingly, this particular policy was not followed by substantial reductions in antibiotic resistance (25, 26). However, the lack of effect was largely explained by the replacement of trimethoprim and sulfonamide with antibiotics that shared genetically linked resistance mechanisms and were therefore subject to coselection (18, 25, 26), resulting in AMR being maintained in the population.

This Review emphasizes the importance of understanding the precise mechanisms of resistance for each antibiotic before rolling out policies to reduce use of one in tandem with increased use of another (18, 27). It can be beneficial to have several types of antibiotics, which disable bacteria in different ways, available to treat a given infection. In this respect, antibiotics differ from most other drugs, for which there is generally little to no clinical benefit associated with producing a new drug that is only equally as effective as an existing one. A new antibiotic that matches the effectiveness of existing antibiotics but disables bacteria in a different way is valuable if it will remain effective in the future, once existing antibiotics become clinically useless. In this way, having access to a diverse set of antibiotics reduces the likelihood that infections will become untreatable as a result of universal resistance (17).

### Obstacles to the development of new antibiotics

As resistance to antibiotics accelerates, so too does the urgency of developing effective new antibiotics. Developing any new drug is expensive, risky, and highly likely to be unsuccessful. Unfortunately, the pace of development of antibiotics is slow, in part because of the challenging logistics and high costs of large clinical trials (28, 29). Even if successful, the need to conserve effective new antibiotics means that they will have restricted use, and their clinical utility will decline as resistance to them inevitably develops. All these factors will limit the volume of antibiotic sales and therefore the profits that pharmaceutical companies reap from their development. Without government intervention, the high cost of drug development and poor sales prospects mean that, on average, antibiotic R&D projects take a substantial loss, and few companies continue to pursue them (30). In July 2018, Novartis joined a growing list of major pharmaceutical companies that have abandoned antibiotic development owing to lack of financial incentives.

It is now widely recognized that better incentives are needed to encourage pharmaceutical companies to reengage with antibiotic development. Incentivization will require “push” incentives, such as research grants and tax credits, to bring down R&D costs. In addition, a variety of “pull” mechanisms aimed at providing sufficiently attractive returns on investment for developers have been proposed. A critical aspect is that the development of new antibiotics needs to be profitable regardless of prices and sales volume, as the existing model provides little incentive to produce a product that must be conserved (3134).

## Impact of AMR on health and the economy

Current worldwide deaths attributable to AMR, including antimalarial and antiviral resistance, have been estimated at ~700,000 per year, rising to 10 million per year by 2050 if present trends continue. Antibiotic resistance levels vary across the world (Fig. 1) (35). These estimates have many limitations that need to be addressed in future work to gain a more accurate picture of the extent of global AMR. For example, estimates are sensitive to frequency of blood culture sampling. In resource-poor settings, where cultures are often only obtained from patients who do not respond to empirical treatment, resistance estimates will be inflated (9).

### Direct and indirect consequences

AMR can result in treatment failures as well as normally uncomplicated infections becoming more complicated and severe (36). Increasing complication leads to increased risk of chronic conditions and death; longer hospital stays; even greater risks for patients with frequent infections, such as cystic fibrosis; and a need for more-toxic antibiotics if others fail owing to resistance. A review of papers that have tried to estimate the costs of AMR per patient episode (2) showed that additional costs varied from less than US$5 to more than US$55,000. Yet these papers estimated only the direct health care costs, such as additional hospitalizations.

In a world where antibiotic prophylaxis is no longer effective, there will be severe indirect consequences and costs, resulting from additional infections after invasive surgical procedures and immunosuppressive chemotherapy (3). It has recently been estimated that a 30% reduction in the efficacy of antibiotic prophylaxis for surgical procedures and chemotherapy would result in 120,000 additional infections and 6300 infection-related deaths per year in the United States (3).

### Wider economic impact of AMR beyond health care

In a recent review of estimates of the burden of AMR, only 11 of 214 studies considered the wider economic burden beyond the health care sector and only two did so at a global level (16). Measuring the economic impact of AMR solely with respect to health care misses broader social costs and benefits of interventions to stem AMR (37). The loss of effective antibiotics may have a substantial effect on the health and productivity of the workforce (3840). It has been estimated that caesarean sections, joint surgery, chemotherapy, and organ transplants contribute almost 4% to global GDP (6). This gives a rough indication of the scale of economic impact if AMR rendered these procedures so dangerous that they were abandoned.

Estimates of the total cost of AMR are fraught with uncertainty, and there is a risk of the cost being far higher than current best estimates. Much of the uncertainty arises from the complexity of estimating the cost of changes in overall resistance levels. For example, if there is a 1% increase in resistance to drug X, then it matters what pathogen is exhibiting that resistance, what the prevalence of that pathogen is, what type of infection the pathogen causes, the health burden of that infection, how transmissible the resistance gene and infection are (both between people and between other organisms), and whether there are alternative treatments available. Because the cost of increased resistance and the threat that it poses are uncertain, the investment that can be justified to control the threat and, ultimately, the value of antibiotics are also uncertain.

## Incentivizing socially desirable antibiotic consumption

To reduce antibiotic consumption to socially desirable levels, we must first consider the reasons why many people take antibiotics unnecessarily. One major factor is imperfect information, which might encourage use in cases where antibiotics are neither essential nor even beneficial. Second, how are people and organizations incentivized to prescribe, consume, and produce antibiotics? We must not lose sight of the fact that many people in low- and middle-income countries (LMICs) still struggle to access life-saving antibiotics. Economics has a number of concepts that can help us understand antibiotic consumption.

### The principal–agent problem, asymmetric information, and moral hazard

In health care, a doctor usually acts on behalf of the patient, who benefits from the doctor’s greater knowledge and skill. In economics, this scenario is known as the principal–agent relationship and applies to situations where a person or institution (the agent) makes decisions or takes action on behalf of (or that affect) another person or institution (the principal).

However, principal–agent relationships can give rise to a moral hazard if the agent’s own best interests do not align with, or if they conflict with, those of the principal. The principal–agent relationship between doctors and patients and, more broadly, between doctors and society can be problematic (41, 42). Patients are less likely than doctors to know if antibiotics are inappropriate, such as when a condition is caused by a virus rather than a bacterium or is a self-limiting illness. Patients are also less likely to realize that antibiotic use causes adverse consequences, for themselves and society, through increased resistance, i.e., a negative externality (43). When prescribing and dispensing are not separated, doctors may have financial incentives to prescribe. For example, in both China and Switzerland, countries with otherwise contrasting health care and social contexts, a lack of separation of prescribing and dispensing increases antibiotic prescribing (44, 45). Even without financial incentives, physicians are more likely to prescribe antibiotics if they think that their patients want them (46). If doctor and patient preferences for antibiotic use do align, they may not reflect the best interests of society more broadly (43).

The imbalance in the doctor–patient relationship could be relieved by altering incentive structures and through better education and public information campaigns for patients (47, 48). Alternatively, mirroring the system proposed for containing carbon emissions, a tradable permit system could be organized in which doctors or their institution are incentivized to reduce antibiotic prescriptions to a regulated optimum level (49).

Another major reason for unnecessary antibiotic use is the fact that physicians often work with imperfect information. Without reliable rapid diagnostic tests, physicians can sometimes only make educated guesses as to whether infections need antibiotics and, if so, whether they will be susceptible to first-line antibiotic treatment. As with antibiotic development, a combination of push and pull incentives could encourage the development of novel diagnostic tests (31). In addition to reducing use of antibiotics for viral or self-limiting bacterial infections, better diagnostic tests could improve the feasibility of R&D for innovative narrow-spectrum antibiotics. Empirical clinical use of such antibiotics might even be feasible for life-threatening infections if suitable rapid diagnostic tests become available (31, 50).

### Access versus excess

Brazil, Russia, India, China, and South Africa accounted for three-quarters of the increase in antibiotic consumption from 2000 to 2010 (51). Although there is substantial variation in consumption, even among countries in similar income groups, the overall consumption pattern has shifted toward broader-spectrum antibiotics (51, 52). Yet lack of, and delays in, access to any antibiotics still results in more deaths worldwide than does antibiotic resistance (53). More than 1 million children with untreated pneumonia and sepsis die annually (54). Across a sample of 101 countries (almost exclusively LMICs), it is estimated that universal provision of antibiotics could avert 75% of 590,000 annual deaths from pneumonia in children under the age of 5 (51). However, among neonates, increased resistance of bacterial pathogens to antibiotics jeopardizes improvements to child survival and causes an estimated 214,000 annual deaths globally (51).

Balancing the need to reduce overall antibiotic use with expanding essential access is challenging. Improving global access to antibiotics will increase the selective pressure for AMR. Conversely, reducing antibiotic use is necessary to stem resistance, but, perversely, this may reduce the incentives for developing new antibiotics, which then would impede access (55). This potential unintended consequence from antibiotic conservation emphasizes the importance of finding new ways for drug developers to profit from innovation, other than the current model based on drug prices and volumes.

## Learning from the economics of climate change

The scale and multifaceted nature of the challenges presented by AMR may seem overwhelming, yet AMR is not the first such challenge the international community has attempted to tackle. It is useful to reflect on what might be learned from other global challenges. There are some notable similarities between climate change and AMR (56, 57). Indeed, there may also be a direct link between the two phenomena; recent evidence shows that higher temperatures are associated with higher resistance levels in common pathogens (58). Both challenges have been described as a global “tragedy of the commons,” in which individuals, acting rationally and according to their self-interest, collectively damage public goods (5963). AMR and climate change are each driven by consumption of goods (carbon and antibiotics) that can provide people with valuable short-term benefits but that also impose long-term costs, such as existential threat from extreme weather or from life-threatening infections. Individuals may feel little incentive to change their course of action and forego short-term benefits because the costs are highly uncertain and harmful events may happen far in the future, which people typically discount (64). Moreover, the costs are unlikely to be avoided unless many other people also decide to reduce their carbon and antibiotic consumption.

The worst effects of both problems are likely to be distributed unequally across the world. In the most severe climate change scenarios, some of the small-island developing states, such as the Maldives and Tuvalu, could become uninhabitable (65). Likewise, increases in health care expenditures resulting from AMR are expected to be most severe in LMICs (66). Inadequate infection control systems within LMIC hospitals can lead to the spread of health care–associated infections and outbreaks caused by resistant pathogens, accompanied by marked increases in treatment costs, morbidity, and mortality (67). Within individual countries, the poor and vulnerable are also likely to be the most severely affected by climate shocks, for example, because of reliance on agricultural productivity for subsistence or employment (68). Similarly, because of its deleterious effect on labor and agricultural productivity, as well as on health care cost, AMR is likely to make it harder to reduce extreme poverty and may even increase the number of people living in extreme poverty (66).

Because of the time scale of cause and effect in climate change (69) and AMR (43), both raise questions of intergenerational equity, as future generations will not gain the benefits of carbon and antibiotic consumption but will face the brunt of the costs. Predicting the trajectory and extent of antibiotic resistance and temperature change is extremely difficult. On the one hand, increased temperatures resulting from carbon emissions are largely irreversible for 1000 years after emissions stop (70), but on the other hand, with the right approach, antibiotic resistance may be reversible within months to years. As LMIC economies grow, both carbon emissions and antibiotic consumption will rise (Fig. 2) (71), likely causing both temperatures and AMR to increase. The trajectory of both problems is strongly dependent on population levels and, in the case of AMR, population densities (72).

Despite similarities, AMR is arguably a more tractable problem than climate change. In many countries, doctors play a gatekeeping role with antibiotics, acting as agents for society. Though imperfect, this principal–agent relationship can be harnessed to steer antibiotic consumption toward optimum levels by providing doctors with appropriate incentives. Unfortunately, there is no comparable agent to act as a gatekeeper for carbon consumption. Reducing antibiotic use could potentially lower AMR in a much shorter time frame than the time frame over which reducing carbon consumption would lower temperatures. Because results from policy interventions may be faster for AMR than for climate change, it might be easier for policy-makers to agree to implement them. Successful cooperation over AMR could conceivably develop goodwill, or even infrastructures, for forging cooperation on climate policy.

Given the many parallels between AMR and climate change, it is instructive to consider the main economic approaches used to inform climate policy and the points of consensus and controversy regarding these approaches.

### Economic approaches to climate policy

Economists agree that the full cost to society of burning carbon is greater than its current market cost (typically measured in U.S. dollars per ton of CO2) (7375). The gap between the full cost and market cost is known as the social cost of carbon (SCC). In theory, if the market price could be increased by an amount equal to the SCC, the laws of supply and demand would result in the quantity of carbon burned falling to the socially optimum level, thus mitigating climate change. This forms the theoretical economic basis for introducing a carbon tax equal in size to the SCC. Economists broadly agree that global imposition of such a tax would be a sensible approach to tackle climate change. The leading alternative, of reducing emissions by issuing tradable emissions quotas, is similar in principle, even if mechanics differ. Instead of intervening in the market price of carbon to achieve the socially optimum quantity, “cap and trade” would instead enforce the socially optimum quantity directly. However, in theory, the price should then rise to the same level that a tax equal to the SCC would achieve. To date, such global approaches have eluded the international community for a variety of reasons, discussed below. More-local initiatives have shown mixed results: From 2003 to 2015, gasoline taxes rose in 83 countries, although they were reduced in another 46, while several other countries continued to subsidize gasoline prices (76).

There are major difficulties and controversies over the implementation of tax and quota schemes. In large part, the controversies arise over calculating the SCC. Estimates from leading experts differ by an order of magnitude (11), leading to profoundly different implications for public policy, ranging from quite modest (77) to a requirement for immediate and substantial cuts in emissions (69).

The standard approach to estimating the SCC has been to attempt a highly complex form of cost-benefit analysis using integrated assessment models (IAMs). An IAM attempts to monetize all the costs and benefits associated with carbon emissions to estimate the SCC. These costs and benefits are then discounted over some extended timeline to give the net present value (NPV) of burning a marginal ton of carbon. The marginal SCC is equal to this value minus its market price.

There are three major uncertainties in the inputs to an IAM. The first is climate sensitivity: the extent to which global temperatures would eventually rise if the atmospheric concentration of CO2 were to double. In AMR research, an equivalent uncertainty is the proportion of a bacterial population that will ultimately become resistant. The second uncertainty is the economic impact resulting from whatever climate change might occur, sometimes called the damage function. Both these sources of uncertainty have been described as not only unknown but unknowable (11, 78, 79). In AMR, the damage function reflects the health costs of future resistant infections, as well as the consequent economic costs. The third uncertainty is the choice of discount rate with which to perform NPV calculations, which requires a judgment of intergenerational equity. This applies to cost-benefit analyses of both climate change and AMR. It is important to remember that over long time frames, small differences in discount rates can make big differences to the NPV of distant outcomes. These uncertainties have led many economists, such as Stern (80), to caution against an overly narrow focus on cost-benefit approaches to climate change mitigation and to call for more drastic and imminent emission reductions as insurance against the non-negligible possibility of catastrophic outcomes (81, 82).

Theoretically, economists may largely agree that a global harmonized carbon tax, equal to the SCC, is an efficient response to climate change. However, lack of consensus regarding the size of the SCC is a major practical impediment. Instead, climate policy for reducing emissions has focused mainly on negotiating a limit for temperature increase on a country-by-country basis (11). The uncertainties are such that the choice of temperature target is highly contentious, on both environmental and economic grounds, but does offer a starting point for international negotiations on country-level emissions reductions.

### What can we learn from the economics of climate change?

Just as the full cost to society of burning a ton of carbon is much greater than its market cost, so too there is a social cost of antibiotics (SCA) that is not captured in market prices. The SCA could be estimated via detailed analyses akin to IAMs. The costs and benefits of antibiotic use can be monetized and discounted over time to obtain the NPV of consuming some unit of antibiotics. In essence, the many difficulties identified by Coast et al. (13, 14) result from a lack of knowledge of the SCA. Like the SCC, estimating the SCA presents analogous difficulties: How do increases or decreases in antibiotic use translate into changes in resistance levels? What is the impact of increased resistance levels on health and GDP (i.e., damage function)? How should the discount rate be chosen and what implications will this rate have for intergenerational equity? An important additional difficulty is that not all antibiotics, nor all resistances, are the same. Consequently, there is no fully satisfactory common unit of antibiotic quantity or, by extension, price in the way that there is for 1 ton of carbon (83). It may be more fruitful to model the costs and benefits, and resistance trajectories, associated with use of specific antibiotics for specific infections. Also, it might be valuable to gain a better understanding of the resistance trajectories of commensal bacteria, which are frequently exposed to antibiotics and may facilitate the spread of AMR (18).

Supposing that the social cost of using specific antibiotics for specific infections can be estimated, who carries the tax burden? If something analogous to the “polluter pays” principle is applied to antibiotic use, in the sense that those who cause externalities must pick up the bill (13, 84), is this the consumer, the prescriber, the national health care system, or the government? It has been argued that taxing patients for antibiotic consumption is unlikely to be effective, at least in high-income settings, due to lack of price sensitivity (7). However, antibiotics could also be taxed at a more aggregated level (e.g., general practice, local or national authority). A tax on each antibiotic prescribed might provide an effective incentive for reducing prescriptions, and the revenue raised could be invested in antibiotic development.

Rather than taxing the price and allowing the market to dictate the quantity of antibiotics, as we have discussed, an alternative may be to establish a regulatory body that gives prescribers permits (quotas) for prescribing, then lets the market determine the price (49). The quantity of permits available would be updated over time, according to resistance levels, and ideally the system would allow the permits to be traded (49). No such schemes have been implemented in practice, although since April 2015, the United Kingdom’s Quality Premium scheme has provided financial incentives to local Clinical Commissioning Groups (CCGs) (up to £140,000 for average-sized CCGs, which serve populations of around 280,000). Promisingly, this scheme has been credited with having reduced antibiotic prescribing in primary care by about 7% in 2015–2016 (85).

Similar approaches could be tailored to other health care systems. The U.S. Food and Drug Administration recently proposed a purchasable permit scheme for antimicrobials of last resort, where acute care institutions would pay a fixed licensing fee for the right to use a certain number of annual doses of a drug (86). The number of licensed doses could be tied to the number of beds or likelihood of encountering certain microbes. An attractive feature of this proposal is that it would create a predictable revenue stream, through more-foreseeable licensing fees, for important antibiotics, in a way that could potentially decouple the return on investment from the volume used. Assigning antibiotics an “option” rather than a “use” value could incentivize industry to develop important new antibiotics for which there would otherwise be too small a market to provide a sufficient return on investment. Another recent proposal for decoupling profits involves developers being paid an insurance premium to provide access to antibiotics, which could be renegotiated at regular intervals (32).

Taxes and quotas have also been considered as potential tools for discouraging unnecessary use of antibiotics in animals. Eighty percent of all antibiotics used in the United States (87) are employed in agriculture and aquaculture for routine nontherapeutic reasons, such as to promote growth, or as low-cost substitutes for hygiene measures to prevent infections (5). There is evidence that restricting antibiotic use in livestock is associated with a reduction in antibiotic-resistant bacteria in animals (88). The World Health Organization recently recommended a complete restriction on all antibiotic use for growth promotion and disease prevention in healthy livestock (89), and some European countries, including Denmark and Sweden, have imposed regulatory restrictions that have achieved substantial reductions in agricultural antibiotic use while maintaining productive livestock sectors (90).

The global average quantity of antibiotic administered per kilogram of animal (50 mg/kg) has been proposed as a possible regulatory target (7). This could reduce total consumption by 64% (5). Alternatively, a user fee of 50% of the current price on veterinary antimicrobials could reduce global consumption by 31% (5). User fees have several advantages (5, 87): They can be easily administered either at the manufacturing or importing stage, they would deter low-value antibiotic practices, and they would generate substantial annual revenues. User fees could be reinvested in R&D for new antibiotics or antibiotic stewardship programs and be used to compensate LMICs, which may be disproportionately affected by a user fee, by investing in strategies that reduce the spread of infection and improve veterinary services (5). In contrast, quantity restrictions would be difficult to enforce without adequate surveillance systems, which could be prohibitively expensive in LMICs (5). Whether through taxes or quotas, reducing predominantly nontherapeutic antibiotic use in agriculture is relatively uncomplicated compared with reducing human consumption.

Despite the shortcomings associated with the approach, combining biophysical and economic systems in climate IAMs is widely considered a worthy endeavor (80), even if the approach remains far from delivering convincing estimates of the SCC. Developing analogous models for antibiotics will be similarly valuable but will also have similar shortcomings. As for climate change, the magnitude of the uncertainties and potential risks has led to calls to try to reduce antibiotic use as a type of insurance policy (2). An analogy with fire prevention has recently been used to illustrate the insurance value of antibiotics (91, 92), in that the microbiology laboratory acts as the smoke detector and medical staff as the firefighters. Like climate change, rather than waiting for analyses of optimum policies to curb AMR, precaution necessitates setting ambitious but pragmatic targets for reducing antibiotic use on a country-by-country basis without delay.

## Outlook

Economics, as a discipline, has made valuable contributions to the debate on climate change mitigation. Economic analyses have successfully informed recommendations by the Intergovernmental Panel on Climate Change and decisions made by the international community, such as the Paris Agreement. We would like to see economists rise to the challenge of creating similar initiatives for AMR (table S1).

AMR presents particular challenges for LMICs, many of which already experience a high prevalence of AMR and its serious consequences. In LMICs, there is inadequate access to lifesaving antibiotics among the poor, coupled with often-substantial overconsumption of antibiotics among the middle classes. A lack of separation of prescribing and dispensing leads to supplier-induced demand, where patients consume more antibiotics than they would if they were better informed (32). The recent demonstration of substantial reductions in mortality from the mass administration of 6-monthly single doses of the antibiotic azithromycin in children (93) highlights this tension between access and excess. Thus, future research should consider mechanisms that reduce overall antibiotic consumption without restricting essential access (94). If the cost of antibiotics is to increase, via taxation or quotas, it will be vital to develop mechanisms that reduce the risk that they will only be taken by those who can afford them.

The challenge of dealing with AMR is sandwiched between two classic market failures. On the demand side is the tragedy of the commons, represented by the misuse of antibiotics as a public good, and on the supply side is the lack of incentive for firms to develop new antibiotics. A key challenge on the supply side is the need to hold back the distribution of new antibiotics until AMR renders existing ones ineffective. The question here is how to design incentives for production of a good that we must avoid using for as long as possible? It is now widely recognized that new product development requires the profits from innovation to be decoupled from prices and volumes (3234). One insight from game theory is that none of the players, neither individually nor collectively, should be able to gain by leaving a coalition. Developing an international agreement to limit antibiotic use is challenging because countries vary widely in their incomes, capacities, and objectives, and therefore in the costs they incur and benefits they derive from membership (95). Enshrined in the Paris Agreement is a principle of “common but differentiated responsibility and respective capabilities.” Similar consideration should be given to the different challenges AMR presents in different countries.

Like climate change, the potential costs of AMR are incredibly uncertain and potentially catastrophic. There is some consensus in the economics of climate change that we should treat the cutting of carbon emissions as an insurance policy. We suggest that a similar paradigm shift is needed for tackling AMR.

There is an urgent need to improve the supply of new antibiotics. At the 2019 World Economic Forum meeting in Davos, there were encouraging signs of recognition among policy-makers of the need to work with industry to change the way antibiotic innovation is rewarded (96). Policy-makers and philanthropists have recognized the importance of AMR and are increasingly making substantial research funds available, including to social scientists. A recent analysis of funding-organization databases, in 19 countries and at the EU level, found that from 2007 to 2013, a total of 1243 antibacterial resistance research projects were funded, at a total public investment of €1.3 billion (97). There is a great opportunity for economists, across many different fields, to engage with this pressing global problem.

## References and Notes

Acknowledgments: Funding: This research was funded by the National Institute for Health Research Health Protection Research Unit (NIHR HPRU) in Healthcare Associated Infections and Antimicrobial Resistance at the University of Oxford (grant HPRU-2012-10041) and in partnership with Public Health England (PHE). The support of the Economic and Social Research Council (UK) (grant ES/P008232/1) is acknowledged. L.S.J.R., L.A., A.S.W., and S.W. are supported by funding from the NIHR Oxford Biomedical Research Centre (BRC). L.A. and C.C.B. are supported by funding from the NIHR Community Healthcare MedTech and In-vitro Diagnostics Co-operative (MIC). The views expressed are those of the authors and not necessarily those of the NHS, the NIHR, or the Department of Health and Social Care. Competing interests: The authors declare no competing interests.
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## Linear Algebra and Its Applications, Exercise 2.2.7

Exercise 2.2.7. Consider the following system of linear equations:

$\begin{bmatrix} 1&0 \\ 0&1 \\ 2&3 \end{bmatrix} \begin{bmatrix} u \\ v \end{bmatrix} = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}$

Find the values of $b$ for which the system has a solution. What is the rank? How many basic variables and free variables are there?

Answer: We perform elimination by first subtracting 2 times the first row from the third row:

$\begin{bmatrix} 1&0 \\ 0&1 \\ 2&3 \end{bmatrix} \begin{bmatrix} u \\ v \end{bmatrix} = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} \Rightarrow \begin{bmatrix} 1&0 \\ 0&1 \\ 0&3 \end{bmatrix} \begin{bmatrix} u \\ v \end{bmatrix} = \begin{bmatrix} b_1 \\ b_2 \\ b_3 - 2b_1 \end{bmatrix}$

and then subtracting 3 times the second row from the third:

$\begin{bmatrix} 1&0 \\ 0&1 \\ 0&3 \end{bmatrix} \begin{bmatrix} u \\ v \end{bmatrix} = \begin{bmatrix} b_1 \\ b_2 \\ b_3 - 2b_1 \end{bmatrix} \Rightarrow \begin{bmatrix} 1&0 \\ 0&1 \\ 0&0 \end{bmatrix} \begin{bmatrix} u \\ v \end{bmatrix} = \begin{bmatrix} b_1 \\ b_2 \\ b_3 - 2b_1 -3b_2 \end{bmatrix}$

This completes elimination and produces the system $Ux = c$.

From the third equation we must have $b_3 -2b_1 -3b_2 = 0$ in order for the system to have a solution. The matrix $U$ has pivots in columns 1 and 2, so that the basic variables are $u$ and $v$; there are no free variables. The rank is 2 (the number of basic variables).

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

This entry was posted in linear algebra. Bookmark the permalink.

### 2 Responses to Linear Algebra and Its Applications, Exercise 2.2.7

1. mungaramanoj says:

could you please give how to get particular solutions

• hecker says:

The previous exercise, 2.2.6, shows an example of finding a particular solution to a system of linear equations. If you search for the word “particular” in the search box on this blog (in the upper right-hand corner) you may find more examples.

format: plain


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3 URL updated

Extremely long answer: The Clifford algebra and the Chevalley map - a computational approach Theorem 1 (or 2, or even 3). Actually the reason for writing this text was my disappointment with the wrong proof in Lawson-Michelson, and with the not sufficiently general proofs in the rest of literature. The heuristics for finding the proof are explained in §6.3 of my diploma thesis to be (unless I mess up the applied maths exam ;) ).

Yes, 90% of the proof are computations.

Now chances are high that you prefer a 1-page proof that works over $\mathbb R$ to a 40-pages one that works over any commutative ring, so you might be interested in the proof that uses orthogonal decomposition of the quadratic form (i. e., finding an orthogonal basis using Gram-Schmidt) and application of the fact that $\mathrm{Cl}\left(V\oplus W\right)\cong \left(\mathrm{Cl}V\right)\hat{\otimes}\left(\mathrm{Cl}W\right)$ (super-tensor product of superalgebras) for any two quadratic spaces $V$ and $W$. Such a proof can be found in Milne's ALA Theorem 18.18, and probably in many other places. As Guntram pointed out in the comment below, this proof only works for $V$ being finite-dimensional; however, the infinite-dimensional case follows from the fact that (unless I am mistaken) taking the Clifford algebra commutes with the direct limit. (Here we are using the fact that if $V$ and $W$ are two finite-dimensional quadratic spaces such that $V\subseteq W$, then the canonical map $\mathrm{Cl}V\to \mathrm{Cl}W$ is injective. This easily follows from the basis theorem for Clifford algebras of finite-dimensional quadratic spaces.)

2 fixed mistakes, thanks to Guntar

Extremely long answer: The Clifford algebra and the Chevalley map - a computational approach Theorem 1 (or 2, or even 3). Actually the reason for writing this text was my disappointment with the wrong proof in Lawson-Michelson, and with the not sufficiently general proofs in the rest of literature. The heuristics for finding the proof are explained in §6.3 of my diploma thesis to be (unless I mess up the applied maths exam ;) ).

Yes, 90% of the proof are computations.

Now chances are high that you prefer a 1-page proof that works over $\mathbb R$ to a 40-pages one that works over any commutative ring, so you might be interested in the proof that uses orthogonal decomposition of the quadratic form (i. e., finding an orthogonal basis using Gram-Schmidt) and application of the fact that $\mathrm{Cl}\left(V\oplus W\right)\cong \left(\mathrm{Cl}V\right)\otimes\left(\mathrm{Cl}W\right)$ left(\mathrm{Cl}V\right)\hat{\otimes}\left(\mathrm{Cl}W\right)$(super-tensor product of superalgebras) for any two quadratic spaces$V$and$W$. Such a proof can be found in Milne's ALA Theorem 18.18, and probably in many other places. As Guntram pointed out in the comment below, this proof only works for$V$being finite-dimensional; however, the infinite-dimensional case follows from the fact that (unless I am mistaken) taking the Clifford algebra commutes with the direct limit. (Here we are using the fact that if$V$and$W$are two finite-dimensional quadratic spaces such that$V\subseteq W$, then the canonical map$\mathrm{Cl}V\to \mathrm{Cl}W$is injective. This easily follows from the basis theorem for Clifford algebras of finite-dimensional quadratic spaces.) 1 Extremely long answer: The Clifford algebra and the Chevalley map - a computational approach Theorem 1 (or 2, or even 3). Actually the reason for writing this text was my disappointment with the wrong proof in Lawson-Michelson, and with the not sufficiently general proofs in the rest of literature. The heuristics for finding the proof are explained in §6.3 of my diploma thesis to be (unless I mess up the applied maths exam ;) ). Yes, 90% of the proof are computations. Now chances are high that you prefer a 1-page proof that works over$\mathbb R$to a 40-pages one that works over any commutative ring, so you might be interested in the proof that uses orthogonal decomposition of the quadratic form (i. e., finding an orthogonal basis using Gram-Schmidt) and application of the fact that$\mathrm{Cl}\left(V\oplus W\right)\cong \left(\mathrm{Cl}V\right)\otimes\left(\mathrm{Cl}W\right)$for any two quadratic spaces$V$and$W\$. Such a proof can be found in Milne's ALA Theorem 18.18, and probably in many other places.


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QUESTION

# In a right triangle $ABC$, the right angle at $C,M$ is the midpoint of hypotenuse $AB$. $C$ is joined to $M$ and produced to a point $D$ such that $DM = CM$. Point $D$ is joined to point $B$.A. $\Delta AMC \cong \Delta BMD$B. $\Delta DBC$ is a right angleC. $\Delta DBC \cong \Delta ACB$D. $CM = \dfrac{1}{2}AB$

Hint: In this question first of all draw the figure by using the given data which will give us a clear idea of what we have to find. By using SAS congruence rules, the triangles AMC and BMD are similar triangles. So, use this concept to reach the solution of the given problem.

Given $\Delta ABC$ is a right-angled triangle at $C$.
So, $\angle ACB = {90^\circ}$
$M$ is the midpoint of $AB$
So, $AM = BM...................................\left( 1 \right)$
$DM = CM..........................................\left( 2 \right)$
By using the given data, the diagram will be as shown in the figure:

Clearly, lines $CD \& AB$ intersect each other.
So, $\angle AMC = \angle BMD{\text{ }}\left( {{\text{Vertically opposite angles}}} \right){\text{ }}............\left( 3 \right)$
In $\Delta AMC$ and $\Delta BMD$
$\Rightarrow AM = BM{\text{ }}\left( {{\text{From }}\left( 1 \right)} \right) \\ \Rightarrow \angle AMC = \angle BMD{\text{ }}\left( {{\text{From }}\left( 3 \right)} \right) \\ \Rightarrow CM = DM{\text{ }}\left( {{\text{From }}\left( 2 \right)} \right) \\ \therefore \Delta AMC \cong \Delta BMD{\text{ }}\left( {{\text{SAS congruence rule}}} \right) \\$
Hence proved that $\Delta AMC \cong \Delta BMD$.
Thus, the correct option is A. $\Delta AMC \cong \Delta BMD$

Note: If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are said to be congruent by SAS similarity criterion. The vertically opposite angles are always equal.


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Today when I tried to connect to VcXsrv after running it with XLaunch it didn't work. I'd had it working for months and so was surprised it suddenly stopped working. The reason was simple; the IP subnet WSL2 had changed and so it was now being blocked by a firewall.

Annoyingly there is very little feedback as to why it can't connect to an XServer. I went back through my previous instructions of setting up an X server in WSL2, but noticed something. When I ran ip addr | grep eth0 it was in the 192.168.0.0/16 subnet. However the WSL2 X server firewall inbound rule I wrote whitelisted the 172.16.0.0/12 subnet. Apparently it can use either private subnet. I've now updated the instructions to include both. Because these are private subnets as long as you're only on networks you trust this should be fairly safe to do.

One upside is servers running newer builds of Windows (after 18945) then WSL2 servers can be accessed through localhost on Windows. This makes testing applications in a browser much easier.


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Browse Questions

# How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent.

$\begin{array}{1 1}(A)\;8.6C_4.7C_4\\(B)\;6.7.8C_4\\(C)\;6.8.7C_4\\(D)\;7.6C_4.8C_4\end{array}$

Toolbox:
• $C(n,r)=\large\frac{n!}{r!(n-r)!}$
• $n!=n(n-1)(n-2)(n-3)......(3)(2)(1)$
Given wird is MISSISSIPPI
Here I=4 times,S=4 times,P=2 times,M=1 time
$\therefore$ Required number of words =$8C_4\times \large\frac{7!}{4!2!}$
$\Rightarrow 8C_4\times \large\frac{7\times 6!}{4!2!}$
$\Rightarrow 7.8C_4.6C_4$
Hence (D) is the correct answer.

+1 vote


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# Finding some rational points on elliptic curves

If I am considering an elliptic curve, for example $$y^2=x^3-2$$ $$\text{Edit: and } y^2=x^3+2$$ over $\mathbb Q$, how to find rational points?

What possibilities do we have to calculate some of the rational points on it? Are there even possibilities for calculating integer points on the curve?

• Sorry, I wrote down another elliptic curve, then intendet. – Marc Jun 18 '15 at 14:47
• How about (3,5)? – Álvaro Lozano-Robledo Jun 18 '15 at 14:48
• I'm afraid not. Sometimes the "first" rational point on the curve (the smallest in absolute value) can be enormous. Obviously I answer in a general context. – Piquito Jun 18 '15 at 14:51
• I added another elliptic curve. Thats the curve I am dealing with (I wrote "-" instead of "+", I am sorry for that) – Marc Jun 18 '15 at 15:35
• Well... P=(-1,-1) works for the second one... That's a point of infinite order, and generates the group of rational points. For instance, 2P = (17/4, 71/8). – Álvaro Lozano-Robledo Jun 18 '15 at 15:47

The simplest way is to use existing methods in computer algebra systems, e.g. if you use the online Magma calculator here there are now awfully sophisticated algorithms there for this sort of thing. To learn more you could read the relevant section in the Magma handbook here

In the case of the first of your curves, if I put in the following

E:=EllipticCurve([0,0,0,0,-2]);

MordellWeilGroup(E);

RationalPoints(E : Bound:=1000);

then the output is

Abelian Group isomorphic to Z

Defined on 1 generator (free)

Mapping from: Abelian Group isomorphic to Z

Defined on 1 generator (free) to Set of points of E with coordinates in Rational Field given by a rule [no inverse]

true true

{@ (0 : 1 : 0), (3 : 5 : 1), (3 : -5 : 1), (129/100 : 383/1000 : 1), (129/100 : -383/1000 : 1) @}

Yes, there are methods to calculate points on elliptic curves. There are books dedicated to this topic... I'd recommend Silverman and Tate's "Rational Points on Elliptic Curves", for instance.

• I got this book. This was actually an exercise from this book. Could you tell me where the methods are explained? I cant find them.. – Marc Jun 18 '15 at 15:36
• I also added another elliptic curve above. In my opinion the new one doesnt have any integer points on it. I am also not able to find any rational points.. – Marc Jun 18 '15 at 15:37
• Chapter II for the points of finite order, and Chapter III for the points of infinite order (see Section III.6 in particular for examples). In general, finding all rational points is not a simple calculation. In many cases, it is beyond our computing capabilities... – Álvaro Lozano-Robledo Jun 18 '15 at 15:45


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# Tools of Algebra 2). Dedicated to Otto...

Algebra Level 5
1. $$\mathbb{K}$$ is a field, if and only if, the set of polynomials with coefficients in $$\mathbb{K}$$, ($$\mathbb{K}[x]$$), is an Euclidean Domain.
2. If $$\mathbb{K}$$ is a field, then $$\mathbb{K}[x]$$ is a field.
3. The polynomial $$2x + 2$$ is an irreducible polynomial over $$\mathbb{Q}[x]$$.
4. The polynomial $$2x + 2$$ is an irreducible polynomial over $$\mathbb{Z}[x]$$.

Which statement(s) above is/are true?

Clarification:

A polynomial with coefficients in an unique factorization domain $$\mathbb R$$, is said to be irreducible over $$\mathbb R$$ if it is an irreducible element of the polynomial ring, that is, it is not invertible, not zero, and cannot be factored into the product of two non-invertible polynomials with coefficients in $$\mathbb R$$.

Answer by entering the numbers of the statements you consider correct in increasing order. For example:

• If you only consider the statement 3 is correct, enter 3.
• If you only consider the statments 2 and 3 are correct, enter 23.
• If you only consider the statements 1, 2 and 3 are correct, type 123.
• If you consider all the statements are correct, type 1234.
• If you consider none of the statements is correct, enter 0.
×


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# Math Help - 4096 has 13 factors WHY

1. ## 4096 has 13 factors WHY

Why is 4096 the smallest number with 13 factors. What does 2^n have to do with it? I understand in order to get an odd number of factors you have to use a perfect square.

Thanks for anything you can add to the picture.

2. Originally Posted by reagan3nc
Why is 4096 the smallest number with 13 factors. What does 2^n have to do with it? I understand in order to get an odd number of factors you have to use a perfect square.

Thanks for anything you can add to the picture.

powers of 2 would yield the smallest number. since powers of 1 would only yield 1 and thus would not reach 4096, and powers of 3 (or more, or a product with numbers greater than 2) would be greater than corresponding powers of 2 and so would not be minimum

3. Hello, reagan3nc!

Why is 4096 the smallest number with 13 factors?
What does 2^n have to do with it?
I understand in order to get an odd number of factors you have to use a perfect square.
There is a theorem that might help explain it.

Given: . $N \:=\:p^a\cdot q^b\cdot r^c\,\cdots$ .
(prime factorization)

. . the number of factors of $N$ is: . $d(N) \:=\:(a+1)(b+1)(c+1)\cdots$

. . . .
Add one to each exponent and multiply.

Example: . $N \:=\:6125 \:=\:5^3\cdot7^2$

. . . $d(6125) \:=\:(3+1)(2+1) \:=\:12$ factors.

We want a number $K$ so that: . $d(K) \:=\: 13 \:=\:12 + 1$

Hence, $K$ has a prime factorization with an exponent of 12: . $K \:=\:p^{12}$

. . And the smallest $K$ occurs when $p = 2.$

Got it?

4. Thanks for all your help

5. Originally Posted by reagan3nc
Why is 4096 the smallest number with 13 factors. What does 2^n have to do with it? I understand in order to get an odd number of factors you have to use a perfect square.

Thanks for anything you can add to the picture.
Since any power of 1 will give only 1 as answer. After this number 2 comes.

$4096=2^{12}$

$4096=1.2.2.2.2.2.2.2.2.2.2.2.2$


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The fabric stiffness in terms of conventional flexural stiffness of the plain and coated fabrics with the lowest and highest amount of solids deposition (generated from a gap height of 5 and 200 ü1 1 1 Figure 6, showing average values in the warp and weft direction, as well as in the face up and down orientation. The first element of row one is occupied by the number 1 which belongs to row 1, column 1. - students will be able to characterize the types of materials for the matrix and the reinforcing phase, and methods for their preparation Skills outcomes: - students will be able to explain the interaction between matrices and reinforcement, and the effects on the properties of composite materials. The stiffness matrix is also symmetric with respect to the interchange of reduced indices I and J so that for the most general solid, 21 distinct entries (6 diagonal and 15 triangular entries) are required to completely characterize the solid. This corresponds to the fact that the element exhibits two nodal displacements (or degrees of freedom) and that the two displacements are not independent (that is, the body is continuous and elastic). Type 302 or 304 stainless steel is a good choice for Material 2. A square matrix is called diagonal if all its elements outside the main diagonal are equal to zero. CHAPTER 3 DIRECT STIFFNESS METHOD FOR TRUSSES: 3. matrix (and therefore no results): – νννν = 0. In this introduction, use simplifying assumptions to more easily convey the main ideas: at initial time the object is undeformed, and the material coordinates exactly match the space coordinates. void zeroMe Geometric Linear Transformation (2D) See also: Geometric Linear Transformation (3D). local or directional reinforcement of a component in order to improve properties like stiffness, strength, or toughness. Total number of DOFs of a member is the summation of DOFs of the two nodes. The resultant forces and moments are functions of the in-plane strains and curvatures (Berthelot, 1999). Example-SUDN)m 2 Gleb. For element 1, we divide the outside by 15 and multiply each element of the matrix by 15. Stiffness method of analysis of structure also called as displacement method. Modeling procedure. In a normal Beam, the bending stiffness is EI, where E is Young’s modulus, I is the “moment of inertia” of the beam cross-section. Very conveniently, the matrices themselves can be multiplied together to produce a third matrix (of constants) which performs the same transformation as the original 2 would perform in sequence. Direct Formulation Direct Stiffness matrix Global stiffness matrix Properties of Stiffness matrix. Human pericytes adopt myofibroblast properties in the microenvironment of the IPF lung Parid Sava, 1 Anand Ramanathan, 1 Amelia Dobronyi, 1 Xueyan Peng, 2 Huanxing Sun, 2 Adrian Ledesma-Mendoza, 1 Erica L. local or directional reinforcement of a component in order to improve properties like stiffness, strength, or toughness. By introducing small amount of inclusion phase to improved the bulk properties of the matrix. Pavan Naik often it's some material properties, perhaps you have some. The stiffness matrix is singular and is therefore non-invertible 2. If a structure is free to move. A widely used basic text by two recognized authorities. ppt - Free download as Powerpoint Presentation (. 0 Microsoft Equation Microsoft 方程式編輯器 3. 2 The Stiﬀness Matrix in the Literature. Classification of Engineering Materials. Answer to TABLE 1. Skip navigation Sign in. Carbon fiber is superior to E glass in fatigue and static strength as well as stiffness. The piezoelectric material properties are listed below. Both in-plane and out-of plane warping are permitted. stiffness matrix Sum of the internal forces should be equal to the external forces applied at each node k 1 u 1 –k 1 u 2 = F 1-k 1 u 1 + k 1 u 2 + k 2 u 2 –k 2 u 3 = F 2-k 2u 2 + k 2 u 3 = F 3 January 23, 2008 Rao, P. 2 The Stiﬀness Matrix in the Literature. Structural Dynamics Lecture 4 Outline of Lecture 4 Multi-Degree-of-Freedom Systems Formulation of Equations of Motions. Calculate reduced stiffness matrix Q ij. , square, triangular, hexagonal) and then adjusting its relative density by modifying the cell wall thickness. Lecture 2: The Direct Stiffness Method Throughout Chapters 2, 3, 4, and 5 we look at deriving the stiffness matrix for line elements. Uniaxial Bar. A Stiffness Matrix for the Analysis of. • Numerical methods are typically. edu , [email protected] 2) Damping: The natural viscous properties inherent to human muscles has been demonstrated [7] to be of great. About MYSTRAN. A FINITE-ELEMENT METHOD OF SOLUTION FOR STRUCTURAL FRAMES by Hudson Matlock Berry Ray Grubbs Research Report Number 56-3 Development of Methods for Computer Simulation of Beam-Columns and Grid-Beam and Slab Systems conducted for The Texas Highway Department in cooperation with the U. The geometric stiffness methods change the stiffness of the structure by altering the stiffness matrix to simulate the effects of the destabilizing gravity loads. The full stiffness matrix A is the sum of the element stiffness matrices. Method 2 The second method requires mass and stiffness matrices. 1 Formation and Structure The basic structure of plastics (or polymers) is given by macromolecule chains, formulated from monomer units by chemical reactions. Composites based on polymeric, metallic, and ceramic matrices are becoming increasingly important as aerospace and space materials. In this post, I would like to explain the step-by-step Figure 13: Formulation of stiffness matrix for element 3. Why the stiffness matrix method also called equilibrium method or. Answer to TABLE 1. elasticity = 30000 ksi. A diagonal matrix is called the identity matrix if the elements on its main diagonal are all equal to $$1. Matrix multiplication: if A is a matrix of size m n and B is a matrix of. Static parameter identificatio n metho d ca also be utilize for damage assessment of structures. The stiffness of unidirectional composites in the fibre direction is usually dominated by the fibre proper-ties while the strength in the transverse direction is domi-nated by the matrix properties. In short, a column of K matrix represent the nodal loads that needs to be applied to maintain a certain deformation - Let me explain it wit. 2) and ﬂexural stiff-ness (equation A. Such behavior exceeds classical bounds ( 14 ), in which composite properties cannot exceed the properties of the constituents. Consequently, some knowledge of its mechanical properties is of importance for an understanding of the mechanism and management of fractures, as well as the design of prosthetic or orthotic appliances and protective gear, e. The negative material properties lead to unusual physical phenomena. stiffness matrix [A] behaves like that of an isotropic material. Mostly the properties of interest in composites are the mechanical properties. ISO 10846-5, Acoustics and vibration - Laboratory measurement of vibro-acoustic transfer properties of resilient elements - Part 5: Driving point method for determination of the low-frequency transfer stiffness of resilient supports for translatory motion [5] R ubin, S. There is some suggestion that the stiffness of the Achilles tendon is not correlated to body weight. Improved stiffness, dimensional stability and reduced creep compared with 20% Talc filled PP. Stiffness Transfer Matrix Method (STMM) for Stable Dispersion Curves Solution in Anisotropic Composites Ayman Kamal, Victor Giurgiutiu [email protected] It is a specific case of the more general finite element method, and was in. Copper is chosen for Material 1 because of its excellent electrical conductivity. For this paper, the fibers are assumed to be arranged in a hexagonal arrangement in an epoxy matrix,. Derive the Element Stiffness Matrix and Equations-Define the stiffness matrix for an element and then consider the derivation of Then find the sum of matrix Aand matrix B. Answer to TABLE 1. The physical properties of tissue extracellular matrix (ECM), such as stiffness, regulate stem cell adhesion, proliferation, migration, differentiation, and fate [5,6,7]. These are the Direct Approach, which is the simplest method for solving discrete problems in 1 and 2 dimensions; the Weighted Residuals method which uses the governing differential equations directly (e. Static parameter identificatio n metho d ca also be utilize for damage assessment of structures. In short, a column of K matrix represent the nodal loads that needs to be applied to maintain a certain deformation - Let me explain it wit. Input consists of ply material properties, material strengths, ply fiber orientation and stacking sequence, mechanical loads and/or strains, and temperature and moisture loads. Formulation of the overall stiffness matrix After the element stiffness matrices in the global coordinates are formed, they are assembled to form the overall stiffness matrix. Remember that there was no stiffness associated with the local rotation degrees of freedom. 32 Furthermore, it is noted that the difference in the Achilles tendon mechanical properties between men and women seems to be correlated to the difference in muscle strength, rather than gender. 8 in4 29000 ksi. The mass properties can then be extracted using equations (1) and (5). • Numerical methods are typically. Young's modulus is a quantitative measure of stiffness of an elastic material. Fiberglass is lightweight, corrosion resistant, economical, easily processed, has good mechanical properties, and has over 50 years of history. the part is made of an elastomer (room temperature vulcanized). 5) denoted by the subscript m, gives rise to a composite with a stiffness E: The stiffness of such a composite is close to the Hashin. Effective Damping Value of Piezoelectric Transducer Determined by Experimental Techniques and Numerical Analysis Gilder Nader Department of Mechatronic and Mechanical Systems Engineering Escola Politécnica da Universidade de São Paulo Rua Prof. A derivative of spring force connecting two moving bodies is derived and it is applied to obtain the stiffness matrix of the mechanism. Pathological. Factors that contribute to arterial stiffness include inflammation, glycation, hypertension, and poor glucose control. Direct Formulation Direct Stiffness matrix Global stiffness matrix Properties of Stiffness matrix. the main advantages of composite ma-terials are their high strength and stiffness, com-. torsional stiffness, G J / L, where J is the polar moment of inertia of the cross‐sectional area (J = Iz = Ix + Iy). Fat Quarter Shop 296,376 views. A material's property (or material property) is an intensive property of some material, i. - Developed using assumptions on the process. – Often characterized by differential and/or integral equations. CIVL 7/8117 Chapter 3 - Truss Equations - Part 1 6/53. The stiffness at 45° to the fibres can be assumed to be just that of the matrix itself. We first calculate the stiffness properties of the shell and then apply them using the "SSPA" "SSPD" and "SSPE" commands, which is corresponding to command lines:. The 6x6 stiffness matrix pertaining to a rigid body mounted on a linearly elastic suspension is revisited here, with the aim of shedding light on its nature via its associated eigenvalue problem. Honeycomb is a very efficient structure for bending stiffness. The transfer function is defined over the complex Laplace plane, as a. The white dots are transversely-sectioned fibers, and the white highly-eccentric ellipses are longitudinally-sectioned fibers. The strength, stiffness, and density of the composite material is very dependent on the reinforcing material. Stiffness Matrix – Another Approach which gives or where element stiffness matrix Expression (**) is a general result which can be used for the construction of other types of elements. Given the matrix D we select any row or column. The stiffness matrix, Q, for plane stress is given by the matrix shown below, where is Poisson's ratio representing a strain in the "2" direction resulting from a stress applied in the "1" direction, i. ELASTIC FLEXURAL-TORSIONAL BUCKLING ANALYSIS USING FINITE ELEMENT METHOD AND OBJECT-ORIENTED TECHNOLOGY WITH C/C++ by Erin Renee Roberts B. Engineering Materials and Their Properties In this Chapter materials are classified and the most important properties of the engineering materials are listed with short explanations. The properties of both constant and configuration dependent stiffness matrices are presented with integration of work when manipulating along a closed path with no self-intersection. et al: rubber particles to improve the toughness of the polymer) 1. Effects of Matrix Stiffness on the Morphology, Adhesion, Proliferation and Osteogenic Differentiation of Mesenchymal Stem Cells. 1 Introduction 4. Chapter 14 Truss Analysis Using the Stiffness Method The space-truss analysis of electrical transmission towers can be. Further Restrictions on Material Properties The considerations above place restrictions on the properties of stable materials. Tissues stiffen during aging and during the pathological progression of cancer, fibrosis, and cardiovascular disease. Derive the Element Stiffness Matrix and Equations-Define the stiffness matrix for an element and then consider the derivation of the stiffness matrix for a linear-elastic spring element. As a result, the material becomes anisotropic—its properties differ between the direction it was rolled in and each of the two transverse directions. Masses, moments of inertia; Ixx, Iyy, Izz and. Each mode can be excited independently from the other modes. The method can be used for determination of displacements and internal forces due to external loads,. Now assemble the element matrices. Gonzalez 1. *This data applies to standard cantilever bearings loaded at the mid-point of the unsupported half, at 0 degrees rotation. Structural Dynamics Lecture 4 Outline of Lecture 4 Multi-Degree-of-Freedom Systems Formulation of Equations of Motions. 18Why is the stiffness matrix method also called equilibrium method or displacement method?. Stiffness Transfer Matrix Method (STMM) for Stable Dispersion Curves Solution in Anisotropic Composites Ayman Kamal, Victor Giurgiutiu [email protected] For stable structures, one of the important properties of flexibility and stiffness matrices is that the elements on the main diagonal (i) Of a stiffness matrix must be positive (ii) Of a stiffness matrix must be negative (iii) Of a flexibility matrix must be positive (iv) Of a flexibility matrix must be negative The correct answer is. 6 GPa, respectively (values given in the article "Influence of Material Properties Variability on Springback and Thinning in Sheet Stamping Processes: A Stochastic Analysis" (Intl. A diagonal matrix is called the identity matrix if the elements on its main diagonal are all equal to \(1. The axial stiffness of the member, S x. The full stiffness matrix A is the sum of the element stiffness matrices. The stiffness profiles of the pavement layers were back-calculated from the experimental dispersion curves using a forward-modelling approach based on a discrete stiffness matrix method. It is shown that the stiffness model cap-tures the intrinsic stiffness properties of the mechanism. increase in viscosity and stiffness, and to optimise the dispersion of TUBALL™ and the final properties. The stiﬀness matrix for this system is K + k −K −K K + k which is very close to K −K −K K. the main advantages of composite ma-terials are their high strength and stiffness, com-. stiffness matrix Sum of the internal forces should be equal to the external forces applied at each node k 1 u 1 –k 1 u 2 = F 1-k 1 u 1 + k 1 u 2 + k 2 u 2 –k 2 u 3 = F 2-k 2u 2 + k 2 u 3 = F 3 January 23, 2008 Rao, P. Toeplitz matrices emerge in many applications and the literature on them is immense. Articular cartilage functions: Increase load distribution area Allow movement while reducing friction and wear Articular Cartilage Composition Articular cartilage is multiphasic: Matrix of collagen and proteoglycan (25%) Free interstitial fluid (75%) Ion phase Collagen Gives a layered. Multiplying Matrices. The matrix forces load sharing among all the fibers, strengthening the material. 53 phr of oil (90%), then don't remove 1. 1 The Moment/Curvature Relation Just as we took the pure bending construction to be The bending stiffness is the resistance of a member against bending deformation. Flexural Modulus denotes the ability of a material to bend. Research on Vibration Behavior of the Plate-Like Joint Interfaces Based on Comprehensive Unit Stiffness Matrix Yichuan XIAO, Liangsheng WU, Jianfeng MA, Shuhui MA School of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Pingleyuan 100#, Chaoyang District, Beijing, 100124 , P. This flattens and stretches its grain structure. ppt from CE 351 at University of North Dakota. If a structure is stable (internally and externally), then its stiﬀness matrix is invertible. Depending on the element type, analysis type and loads, not all of the material properties may be required. The development of the finite element method was first introduced by Clough [2. A stiffness matrix that is not symmetric and cannot be made symmetric by multiplying one or more of its rows by constants indicates a system that is or can be dynamically unstable. Another example of an orthotropic material is sheet metal formed by squeezing thick sections of metal between heavy rollers. the coupling stiffness matrix [B] relates the resultant in‐plane forces to the curvatures and the resultant moments to the midplane strains. An orthotropic lamina is a sheet with unique and predictable properties and consists of an assemblage of fibers lying in the plane of the sheet and held in place by a matrix. From the minim-ization of potential energy, we get the formula: • As with the bar element, the strain energy of the element is given by. The properties of both constant and configuration dependent stiffness matrices are presented with integration of work when manipulating along a closed path with no self-intersection. the study shows a warning that the Assembled Stiffness Matrix has a negative diagona and later it fails showing the message in the picture i have attached. Derive the Element Stiffness Matrix and Equations-Define the stiffness matrix for an element and then consider the derivation of Then find the sum of matrix Aand matrix B. The global stiffness matrix is constructed by assembling individual element stiffness matrices. The The statement that the rank of product of two matrices cannot exceed the rank of either factor is a true statement. research, the effect of interphase relative stiffness to matrix (Ei/Em) is considered instead of interphase stiffness in order to understand interphase-matrix interaction more clearly. With this initial foundation, our main goal was to investigate the influence of effects likely present in 3D but generally not in 2D, such as steric (e. General Governing Equations of Inverse Method Stiffness matrix of a ply in on-axis coordinate is expressed 0as follow [5]: ÷ ÷ ÷ ø ö ç ç ç è æ. Properties of concrete and reinforcing steel, Philosophy, concept and. Finite elements analysis, please do Subscribe and share. Using the properties of this matrix, the number of operations needed for the matrix inversion is considerably reduced employing the decomposition of the block stiffness matrices. Transformation matrices satisfy properties analogous to those for rotation matrices. After this lecture, you'll be able to describe the use of uniaxial composites in wind turbine blades, calculate the stiffness and strength of a composite material, estimate the fatigue limit of a composite. The name in quality development sites & project marketing. • Numerical methods are typically. There will always be as many compatibility equations as redundants. This review summarizes recent data exploring the nature of matrix stiffness, mechanotransducers, and the many effects of changes in stiffness on cell function. In the process of programming this method, \(\mathbf{K}$$ is initialized as a zero matrix. Without sufficient vitamin K, matrix Gla-protein fails to become activated, resulting in calcium deposition and increased arterial stiffness. Elastic properties of fiber reinforced composite materials. , ignore axial member deformations. Density Density is fundamental physical property of oil. det I = 1 2. 9 1723 Aluminosilicate 12. Isoparametric derivation of bar element stiffness matrix: Matrix Where A = bar area and E = Modulus of elasticity are taken as constant; J = Jacobian Operator 3. The stiffness matrix is well known to give an ellipsoid shape distribution of the human Cartesian stiffness. Obtain K with volume integral (not necessary in this case, but for demo) 2. void zeroMe Geometric Linear Transformation (2D) See also: Geometric Linear Transformation (3D). Arterial stiffness is associated with increased risk of cardiovascular disorders, dementia, and death. An orthotropic lamina is a sheet with unique and predictable properties and consists of an assemblage of fibers lying in the plane of the sheet and held in place by a matrix. •The practical interest of VSCL is large, particularly in aeronautics, because it offers the possibility to reduce weight without compromising safety. 1Stiffness Matrix of a beam element The stiffness matrix of a beam element is formulated by assembling the matrix relation-ships for axial stiffness (equation A. Properties of the stiffness matrix Before evaluating the terms of the stiffness matrix, some observations can be made about its. For example, interactions between negative-stiffness inclusions and their positive-stiffness surrounding matrix may give rise to unbounded effective viscoelastic modulus and damping. Even if the spherical particles are perfectly rigid compared. Assuming that you have a background in FEM , I shall give you specific insights about the stiffness matrix. Stiffness is the rigidity of an object — the extent to which it resists deformation in response to an applied force. Undamped Eigenvibrations. Free Ansoff matrix for PowerPoint. • Computation of shape functions for 4-noded quad • Special case: rectangular element • Properties of shape functions • Computation of strain-displacement matrix • Example problem •Hint at how to generate shape functions of higher order (Lagrange) elements Finite element formulation for 2D:. fibres and matrix) and their corresponding stiffness. We observe the stiffness between inputs to regress to zero with the onset of overﬁtting, demonstrating the clear connection to generalization. Element stiffness is obtained with respect to its axes and then transformed this stiffness to structure axes. Flexural Modulus denotes the ability of a material to bend. keywords = "Cell adhesion, Chondrocyte, Extracellular matrix, Matrix rigidity, Mesenchymal stem cells, Smooth muscle cell" These results provide insights of how substrate stiffness differentially regulates stem cell differentiation, and have significant implications for the design of biomaterials with. The book may serve both as an introductory text and as a reference. Here, we show that ionizing radiation can reduce the stiffness of an ex vivo tumor and an in vitro collagen matrix. Learn about product property, Stiffness (Young's Modulus). The best stiffness for the phenotypic maintenance of chondrocytes is 5. Matrix of stiffness coefficients that corresponds to forces at specified degrees of freedom resulting from unit displacements at all the free degrees of freedom, while the specified displacements are held fixed at 0. Chapter 16 Composites With a knowledge of the various types of composites, as well as an understanding of the dependence of their behaviors on the characteristics, relative amounts, geometry/distribution, and properties of the con-stituent phases, it is possible to design materials with property combinations that are better than those. In contrast to metallic alloys, each material retains its separate chemical, physical, and mechanical properties. Matrix multiplication is really useful, since you can pack a lot of computation into just one matrix multiplication operation. This generalizes for higher-rank tensors such as elastic stiffness/compliance, or piezo-x effects, and all are discussed. dTkd 2 1 10. A new system identification (SI) scheme based on the finite element method (FEM) and the boundary element method (BEM) is proposed to determine geometric shapes and elastic material properties of inclusions in a 2-D finite body. These thermodynamic constraints are independent from, but combine with, any symmetry properties if the material in question.


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## anonymous one year ago Bayes Probability Question

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We are given $P(C|Q _{G}) = 0.72$ $P(C|Q _{OK}) = 0.40$ $P(C|Q _{B}) =0.00$ Using Bayes' Formula we know that $P(C|Q _{G}) = \frac{ P(C ∩ Q _{G}) }{ P(Q _{G}) }= 0.72$ $P(C|Q _{G}) = \frac{ P(C ∩ Q _{OK}) }{ P(Q _{OK}) }= 0.40$ $P(C|Q _{G}) = \frac{ P(C ∩ Q _{B}) }{ P(Q _{B}) }= 0.00$

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# Fast Multiplication of Large Integers: Implementation and Analysis of the DKSS Algorithm — Diploma Thesis

“Fast Multiplication of Large Integers: Implementation and Analysis of the DKSS Algorithm”, that is my diploma thesis. I just uploaded it to arXiv: http://arxiv.org/abs/1503.04955

Abstract: The Schönhage-Strassen algorithm (SSA) is the de-facto standard for multiplication of large integers. For N-bit numbers it has a time bound of $$O(N \cdot \log N \cdot \log \log N)$$. De, Kurur, Saha and Saptharishi (DKSS) presented an asymptotically faster algorithm with a better time bound of $$N \cdot \log N \cdot 2^{O(\log^∗ N)}$$. In this diploma thesis, results of an implementation of DKSS multiplication are presented: run-time is about 30 times larger than SSA, while memory requirements are about 3.75 times higher than SSA. A possible crossover point is estimated to be out of reach even if we utilized the whole universe for computer memory.

It contains not only what the title promises, but also a long presentation of my own endeavors regarding fast multiplication, from ordinary multiplication, Karatsuba, Toom-Cook 3-way and Schönhage-Strassen with theory and some code examples.

I’m happy for any commentary.


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## Algebra 1: Common Core (15th Edition)

In the equation $y=mx+c$, m is the slope. The slope is $-3$ in cases A and B. Substituting values, 2 for x and 5 for y, Case A: $5=-3(2)+17$ is not true, so (2,5) does not lie on that line. Case B: $5=-3(2)+11$ is true, so this line passes through (2,5). Answer: B