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# Whiting SJ & Gibson RS. Principles of Nutritional Assessment: Calcium

3rd Edition
March 2021

Abstract

Calcium is known for being a major constituent of bones and teeth. About 99% of the body's calcium is found in these hard tissues. Calcium is also an extracellular divalent cation in the blood, where it exerts functional roles such as cell signaling and membrane transport. Its concentration in blood is tightly regulated by many hormones including parathyroid hormone (PTH), calcitriol, and calcitonin. Blood levels of calcium are not useful in determining the nutritional status of calcium because of these hormonal influences. Dietary requirements for calcium are derived using calcium balance, which is negative when calcium intake is too low to maintain normal bone density and zero in adults when calcium intake is sufficient or positive during growth and pregnancy. Calcium balance can be estimated using dual-absorptiometry (DXA) or measuring urine and fecal losses. A calcium intake above requirement does not stimulate further increases in bone mass; instead excess intake can lead to adverse effects such as soft tissue calcification and kidney stone formation. When calcium intakes are low, and/or when vitamin D is inadequate to permit sufficient calcium absorption, bone health is impaired leading to rickets in children and osteomalacia or osteoporosis in adults. Other health benefits of calcium are under investigation.

CITE AS: Whiting, SJ & Gibson RS. Principles of Nutritional Assessment: Calcium https://nutritionalassessment.org/calcium/

## 23a.1 Calcium

Calcium is the most abundant mineral in the body, accounting for 10–20g per kg body weight in adults. Bone is the main reservoir for calcium. The remainder is located in body tissues (10g), blood, and the extracellular fluids (900mg). In contrast, a greater proportion of the distribution of phosphorus and magnesium, the other two bone minerals, is located in soft tissue (Table 23a.1).
Table 23a.1: Calcium, magnesium and phosphorus distribution in an adult weighing 70 kg
Element Mass (g) Percentage in
the skeleton
Percentage
in tissue
Calcium (Ca) 1200 99 1
Magnesium (Mg) 24 5540
Phosphorus (P) 9008012
All of these three minerals have two major roles —  as structural components in bone and regulatory agents in body fluids.

## 23a.2 Functions of calcium

Calcium has a structural role in bones and teeth, where 99% of the body's calcium exists primarily as calcium hydroxyapatite [Ca10(PO4)6(OH)2]. Bone is a dynamic tissue that undergoes remodeling throughout life except for the teeth. Bone formation is under the control of osteoblasts, the bone forming cells, whereas bone resorption is initiated by osteoclasts, the bone resorbing cells. During growth, bone formation exceeds resorption as bone modeling is predominant. In later life when only bone remodeling takes place, bone resorption begins to occur at a faster rate than bone formation and this results in age-related bone loss

The remaining 1% of total body calcium is in the soft tissues where it has a regulatory role in several metabolic processes, including enzyme activation, vascular contraction and vasodilation, muscle contractibility, nerve transmission, hormone function, and membrane transport. When the levels of absorbed calcium from the diet are insufficient to balance the obligatory fecal and urinary losses, calcium is drawn from the bone and blood calcium levels are maintained within narrow limits. This homeostatic regulation of the calcium level in the blood is achieved by the interaction of the parathyroid hormone (PTH), 1,25‑dihydroxyvitamin D (1,25(OH)2D), and calcitonin, the three major calcium-regulating hormones Phosphorus, protein, estrogen, and other hormones, including glucocorticoids, thyroid hormones, growth hormone, and insulin also have a role

## 23a.3 Absorption and metabolism of calcium

Calcium balance refers to the state of the calcium body stores (in bone) which are a function of dietary intake, intestinal absorption, renal excretion, and bone remodeling. A major control of calcium balance is the amount of calcium absorbed. Calcium is absorbed across the intestinal mucosa in two ways: active transport and passive para-cellular diffusion. Active absorption of calcium is influenced by calcium and vitamin D status of the individual, age, pregnancy, and lactation. Active absorption occurs when intakes of calcium are low relative to need, and is regulated primarily via 1,25(OH)2D and its intestinal receptors. Passive absorption is more important at high intakes of calcium

Efficiency of calcium absorption is highest when dietary calcium intakes are low or during periods of rapid growth in infancy, early childhood, and adolescence, and during pregnancy and lactation. Indeed, absorption is very high during infancy (about 60%) and about 34% during puberty stabilizing at about 25–30% in young adults. Subsequently, the intestinal absorption of calcium decreases progressively with age

Urinary calcium excretion is also regulated in response to need. Normally calcium re-absorption in the kidney nephrons is not 100%. When there is hypocalcemia to stimulate PTH release, one action of PTH is to increase efficiency of calcium re-absorption thus reducing calcium losses in urine Several dietary variables affect both calcium absorption and retention and, hence, calcium status. Diets high in protein increase the urinary excretion of calcium (hypercalciuria), which, however, is compensated by increased calcium absorption Dietary sodium also influences urinary calcium losses, with high sodium intakes increasing urinary calcium losses and lowering bone density in women Caffeine, like protein and sodium, also increases urinary calcium loss but has only a small negative effect on calcium retention, which can be offset by increasing dietary calcium intakes

High phosphorus intakes reduce urinary calcium losses (i.e., have a hypocalciuric effect), but because they increase losses of endogenous fecal calcium at the same time their net effect on calcium balance is still under debate. One might assume that there would be no effect except in conditions in which the coordination between calcium and phosphate homeostasis occurs through the actions of PTH, FGF‑23 and 1,25(OH)2D. The disruption of this coordination, however, can occur with disease states such as chronic kidney disease (CKD) However, increasing dietary phosphorus through inorganic phosphate additives may have detrimental effects on bone and mineral metabolism in humans and animals

Phytates and oxalates can inhibit the absorption of calcium by forming insoluble calcium complexes in the gastrointestinal tract Hence, they may have a negative effect on calcium status, although their overall effect may be small unless dietary calcium intakes are low. Some other dietary components, including lactose and inulin, enhance passive calcium absorption by increasing calcium solubility in the ileum

Several clinical conditions are associated with disturbances in calcium absorption. Calcium malabsorption is associated with gastrointestinal diseases such as Crohn's and celiac disease, and intestinal resection or bypass. Intestinal absorption of calcium is also impaired in patients with renal failure, caused by the reduced synthesis of 1,25‑dihydroxyvitamin D Excessive intestinal absorption of calcium and hypercalcemia occur in sarcoidosis, when excessive calcitriol appears in serum, and also in vitamin D intoxication.

## 23a.4 Dietary sources and intakes of calcium

Calcium is not widely distributed in all food groups, as shown in Table 23a.2.
Table 23a.2 Calcium bioavailability chart showing calcium content, estimated absorption, and net amount absorbed.
* Theoretical amount of calcium absorbed by an adult. Source: Weaver CM, Plawecki KL (1994). Dietary calcium: adequacy of a vegetarian diet. American Journal of Clinical Nutrition. 59: 1238S-1241S.
Food Serving
size
Calcium
Content
(mg)
Estimated
Absorp-
tion %
Net Ca*
Absor-
bed (mg)
Cow milk (fluid) 250mL 310 32 100
White beans (cooked) 125mL 85 22 18
Red beans (cooked) 125mL 26 24 6
Whole wheat bread 35g slice 26 82 21
Broccoli (cooked) 125mL 33 20 20
Spinach (cooked) 125mL 129 5 7
Almonds (raw, roasted) 60mL 97 21 21
Tofu (set with CaSO4) 85g 171 31 53
Soy “milk” (unfortified) 125mL 5 31 2
Milk and milk products (e.g., cheeses, yoghurt) are excellent sources of readily available (bioavailable) calcium. Plant-based beverages made from soy, almonds, rice and oats are available as substitutes for milk, but must be fortified. The availability of the calcium from these beverages is unknown, except for fortified soymilk for which the calcium availability is similar to cow milk, but only if calcium carbonate is used as a fortificant Unfortified plant-based beverages have little calcium. Canned fish with bones (e.g., salmon, sardines) are good sources of calcium. Leafy-green vegetables are also good sources, but absorption of calcium from some of these foods, especially spinach, may be low Ancient grains (e.g., teff, quinoa, amaranth) and nuts are good sources of calcium but highly refined or modern cereal grains such as wheat and maize are not.

Preparation of foods may also be a factor influencing their calcium content. For example, in Guatemala, when corn (maize) tortillas are prepared according to the Mayan tradition of cooking corn with limestone and leaving it to soak overnight in hot water, the calcium content rises from about 10mg to about 200mg of calcium per 100g In Sub-Saharan Africa, dietary calcium intake may be increased by using powdered eggshell added to foods

Foods fortified with calcium are available in many countries. In the European Union, the following forms of calcium are authorized for addition to foods and for use in food supplements: carbonate, chloride, citrate malate, gluconate, glycerophosphate, lactate, hydroxide, oxide, acetate, L‑ascorbate, bisglycinate, salts of citric acid, pyruvate, salts of orthophosphoric acid, succinate, L‑lysinate, malate, L‑pidolate, L‑threonate, sulphate Most of these forms are allowable in other countries. The citrate malate calcium salt is an allowable fortificant for orange juice in the United States The United Kingdom is the only country where calcium fortification is a mandatory addition to flour; 235–390mg of calcium carbonate are added per 100g of flour. This represents about 94–156mg of elemental calcium per 100g of flour Other countries permit calcium addition to certain foods, often proprietary packaged foods, a practice called discretionary fortification.

Supplemental sources of calcium are widely available in many dosages and in a variety of chemical forms in many countries. These pills, capsules and liquids are usually regulated by the government. For example, in the United States, the following forms of calcium are allowable as supplements: carbonate, phosphate, citrate, gluconate, and lactate Some supplements that are derived from oyster shell, algae, and in chewable drugs labeled as antacid such as TUMSTM, are mainly in the form of calcium carbonate, the form of calcium naturally occurring in rocks such as chalk and limestone. This form of calcium is generally the cheapest to buy and has the highest amount of calcium by weight (40%). However, calcium carbonate is poorly absorbed unless stomach acid is available for dissociation. Also, calcium carbonate is the form of calcium that causes the most side-effects such as bloating and constipation

Dietary calcium intakes vary across countries, ranging from 175–1233mg/d Countries in Asia have average dietary calcium intakes less than 500mg/d while in countries in Africa and South America, calcium intakes vary from 400–700mg/d. Only Northern European countries have a national calcium intake greater than 1000mg/d, reflecting a high dairy consumption. The average calcium intake is generally lower in women than men, although across the 74 countries included in an analysis by the International Osteoporosis Foundation no clear patterns regarding differences by age, sex, or socioeconomic status emerged.

## 23a.5 Nutrient reference values for calcium

New EAR, RDA and UL levels for calcium were published by the Institute of Medicine in 2011, as shown in Table 23a.3.
Table 23a.3 Estimated Average Requirement (EAR), Recommended Dietary Allowance (RDA, or Adequate Intake equivalent) and Upper level (UL) for calcium. * Applies to pregnant and lactating women. Source: Institute of Medicine (2011) and WHO (2018).
Dietary Reference Intakes Ca (mg/d) IOM  (2011)
Age/sex groups EAR RDA UL
0–6 mo 200 (AI) 1000
6–12 mo 260 (AI)1500
1–3y F & M5007002500
4–8y F & M800 10002500
9–18y F* & M110013003000
19–50y F* & M80010002500
51–70y F100012002000
51–70y M80010002000
71+y M & F100012002000
World Health Organization  (2018)
Pregnancy 1500
Calcium balance studies provided the evidence for values for ages 1–50y, whereas for those over 50y, values were set to prevent bone loss. For infants age 0–6mos, an AI is given based on the average intake of calcium from human milk, whereas at age 7–12mos, the AI assumes 120mg calcium from human milk plus 140mg calcium from complementary food. The values set for the AIs for infants provide enough absorbed calcium to meet the calcium requirements estimated by the factorial method. For children, estimates of desirable bone accrual were considered; EAR values, which start at 500mg for toddlers (age 1–3y), rise to 800mg for children, then peak at 1100mg for adolescents, were based on a new calculation of calcium retention data For adults, maintenance of bone through attaining neutral calcium balance is the goal. The EAR values are 800mg for men age 19–70y and for women 19–50y, and the RDA for these age groups is 1000mg. The difference in age cut-offs for men (i.e., 70y) and women (i.e., 50y) reflects data showing that women at 50y experience abrupt bone loss at menopause in contrast to men at this age where bone loss is slow. For women over 50y and men over 70y, EAR values are similar at 1000mg which translates into an RDA value of 1200mg. This higher level set for women over 50y and men over 70y was considered a justified public health measure in the face of supportive yet inconclusive data suggesting a higher intake of calcium would prevent bone loss and protect against fractures. Values during pregnancy and lactation are not different from other women in the same age group in view of the well documented ability of pregnant and lactating women to increase calcium absorption, and hence compensate for their increased needs

The IOM Upper Level (UL) for calcium for infants 0–1y was derived from feeding data that indicated no adverse effects on calcium excretion when calcium intakes as high as 1000–1500mg were provided. For all other age groups, the UL for calcium (Table 23a.3) was based on data related to the incidence of kidney stones, largely from the research conducted with post-menopausal women in trials where calcium supplements were provided. In contrast, the European Food Safety Authority retained their calcium UL of 2500mg for adults, including pregnant and lactating women, on the basis that the new data did not alter their previous adult UL estimates. Further, EFSA was not convinced there were sufficient data to set a UL for infants, children, or adolescents.

## 23a.6 Calcium deficiency

### 23a.6.1 Hormonal responses to calcium deficiency

Calcium homeostasis is under systemic (i.e., hormonal) control together with additional local regulating factors through intestinal absorption, influx and efflux from bone, and calcium excretion and re-absorption by the kidney. Serum calcium concentrations must be kept constant at approximately 2.5mmol/L, a concentration that is controlled by the interrelated action of three hormones: parathyroid hormone (PTH), 1,25(OH)2D, and calcitonin. The first two hormones determine how much calcium is retained in the body through control of absorption, bone turnover, and renal excretion. Calcitonin has a role in how calcium moves between the extracellular fluid and bone

Fasting, which represents acute calcium deficiency, causes a decrease in serum calcium concentrations, inducing the release of PTH via the calcium-sensing receptor (CaSR) located on the cell surface of the parathyroid glands In hypocalcemia, PTH has three actions which together bring serum calcium levels back to normal. The immediate response is bone resorption to release calcium into the blood. The second response is to increase renal re-absorption of calcium in kidney tubules in order to retain calcium. Finally, PTH stimulates 1,25(OH)2D synthesis in the kidney (Figure 23a.1) which serves to allow active absorption of calcium in enterocytes. Feedback control of this rise in serum calcium occurs as a normal serum concentration of calcium inhibits PTH secretion. In addition, higher than normal serum calcium stimulates calcitonin secretion by the para-follicular C cells of the thyroid gland, causing calcium to move into bone. Calcitonin's actual role in bone formation remains unclear, but it is a recognized treatment for hypercalcemia

### 23a.6.2 Bone health

Chronic calcium deficiency, arising from an habitually inadequate intake or poor absorption of calcium, is one of the many important factors associated with reduced bone mass. Bone mass in later life depends on the peak bone mass achieved during growth and the rate of the subsequent age-related bone loss. The age span over which peak bone mass is achieved is between the early teenage years and age 30, during which time 95% of bone mass is achieved. By 30y further gains are unlikely although exact timing varies with the skeletal site and sex. Optimization of bone mass is important because it reduces the risk of osteoporosis in later life.

Calcium intake and its association with bone health has been the subject of intense debate as epidemiological studies have not always found a consistent relationship between calcium intake and bone density. One reason is that calcium is a threshold nutrient so that when present in sufficient amounts, a higher calcium intake does not result in any further improvement in bone mineral accrual In most countries the Nutrient Reference Values (NRVs) have been set for maintenance of bone health. For example, in the United States and Canada, the Estimated Average Requirement (EAR) and Recommended Dietary Allowance (RDA) for calcium for all ages is based on maintaining a positive calcium balance in children and preventing a negative calcium balance in adults more details are described below.

Calcium-deficiency rickets and osteomalacia occur in many parts of the world. Case reports from the United States, South Africa, and Nigeria have described infants and children with radiological rickets, growth retardation, and biochemical signs of hyperparathyroidism but with normal vitamin D status. In all these cases of rickets, habitual calcium intakes were low but phosphorus intakes were adequate or high. Moreover, the children in South Africa responded to calcium-rich hospital diets whereas the Nigerian children responded more to calcium or a combination of calcium and vitamin D, and not to vitamin D alone Children with severe acute malnutrition experience hypocalcemia and many have accompanying rickets

Osteopenia and osteoporosis are conditions characterized by a reduced bone mass. In osteopenia, the T‑score is low, at −1. In osteoporosis, reduction in bone mass is greater, i.e., the T‑score is −2 and there is increased bone fragility and risk of fracture. The T‑score is a relative rating expressed in standard deviation units describing the bone mineral density (BMD) measured at the spine, hip, or forearm. A T‑score is similar to a Z‑score except it is not measured against a same age person but rather against a young adult with normal BMD. Measurement is done using a variety of methods described in Section 23a.9. Osteoporosis diagnosis, according to the International Osteoporosis Foundation, is done most often using BMD tests. For example, there is a Fracture Risk Assessment Tool for physicians to use with patients of both sexes called It is a web-based calculator to assess the ten-year risk of osteoporosis fracture based on specific individual risk factors, with or without BMD values There are separate country models for more than 60 individual countries, and it is available in more than 30 languages. The IOF supports the maintenance, development, and education about the use of FRAX® worldwide. The factors used in FRAX® to predict 10y fracture risk are listed in Table 23a.4.

Table 23.4: Risk factors for osteoporosis used in the FRAX® Fracture Risk Assessment Tool for physicians to assess 10-year risk of developing osteoporosis. Source: www.iofbonehealth.org
Risk factorsRisk factors used in FRAX®
to predict 10y fracture risk
AgeIncreasing risk 40–90y
SexFemale is risk factor
Weight and heightLow BMI is risk factor
Previous fracturePrevious fracture as adult is risk factor
Parent fractured hipIndicates genetic risk
SmokingCurrent use of tobacco is risk factor
Glucocorticoids ≥ 3mos prednisolone (≥ 5mg daily
or equivalent) is a risk factor
Rheumatoid
arthritis
Confirmed diagnosis of rheumatoid
arthritis is a risk factor
Secondary causes
of osteoporosis
imperfecta, untreated hyperthyroid-
menopause, chronic malnutrition,
malabsorption, chronic liver disease
Alcohol ≥ 3 or more units of alcohol daily is a
risk factor
Bone mineral
density (BMD)
Risk increases with decreased femoral
neck BMD (in g/cm2).

Dietary patterns and lifestyle factors such as sex, body composition, genetics, and physical activity contribute to bone health Dietary factors include high intakes of protein and phosphorus (both as positive and negative influences) and suboptimal intakes of vitamin D as well as other micronutrients including vitamin K, vitamin C, magnesium, and zinc. These micronutrients are related to calcium metabolism and/or bone or connective tissue metabolism. However, based on a recent systematic review, the Level of Evidence was designated as “Inadequate Evidence” of the benefit of micronutrients (with the exception of calcium and vitamin D) on bone during acquisition of peak bone mass In examining dietary patterns, Moderate Evidence was found to support a beneficial role for dairy products on bone, whereas for dietary fiber and fruit and vegetable intake the evidence on bone was deemed “Limited”. Likewise, the evidence for a detrimental effect of cola and caffeinated beverages on bone acquisition was also designated as “Limited”. For bone loss in older adults, a Canadian study concluded that the “Prudent” dietary pattern consisting of fruit, vegetables, whole grains, fish, and legumes reduced bone turnover compared to the Western diet of soft drinks, potato chips, french fries, meats, and desserts

### 23a.6.3 Non-bone effects of calcium deficiency

Calcium deficiency has been implicated in effects on health that are not bone-related. These include: reduction in hypertensive disorders of pregnancy, reduction in high blood pressure, reduction in risk of colorectal adenomas, and improvements in blood cholesterol in those at risk Further, calcium has been implicated in weight management The evidence for or against these effects is presented below. None of these non-bone effects were used to set Nutrient Reference Values (NRVs), yet each provides additional reasons for being concerned about calcium adequacy. Blood pressure and calcium intake are inversely related in studies in humans as well as animal models, suggesting a role for calcium in maintaining normal blood pressure Intracellular calcium regulates blood pressure in vascular smooth muscle cells, directly through vasoconstriction and indirectly through vascular volume control Hypertensive disorders of pregnancy, particularly eclampsia and pre-eclampsia, are a significant cause of severe morbidity, long-term disability and death among both mothers and their babies Using moderate-certainty evidence from intervention studies, WHO has made the following recommendation: In populations with low dietary calcium intake, daily calcium supplementation (1.5–2.0g oral elemental calcium) is recommended for pregnant women to reduce the risk of pre-eclampsia. Additional considerations included in this recommendation are that all of the calcium may be from foods, if possible; that the intake should be obtained in divided doses, not all at once; and that iron supplements should be taken at a different time from calcium supplements.

Reduction in risk of colorectal adenomas with dietary calcium was first postulated in 1984, whereby in the presence of calcium ions, the toxic effects of free fatty acids on colon epithelial cells could be reduced by conversion to insoluble calcium soaps Since that time, numerous studies have refined this hypothesis, and protective effects have been noted in animal studies. A systematic review and meta-analysis of randomized controlled trials regarding calcium supplementation for the prevention of colorectal adenomas found a modest protective effect of calcium (1200–2000mg doses of supplemental calcium) in prevention of adenomas (Relative Risk (RR) = 0.89, 95% CI: 0.82–0.96)

Blood cholesterol levels may be influenced by calcium intake. A systematic review reported that calcium supplementation reduced LDL cholesterol and increased HDL cholesterol through mechanisms involving suppression of calcium-regulating hormones (e.g., PTH) that subsequently reduced intracellular calcium in adipocytes, thus stimulating lipogenesis and lipid storage However, as described below in Section 23a.7, excess calcium intake may be implicated in promoting cardiovascular disease. Hence, to date there is no conclusion regarding the role of calcium intake in reducing heart disease risk through cholesterol metabolism. Body weight control by calcium has been debated for many decades. The initial suggestion arose from the finding that dairy intake facilitated weight loss better than low calorie diets without dairy, an effect that was attributed to calcium A recent trial compared the effect of supplements and dairy-containing foods and concluded that consuming 4 to 5 servings of low-fat dairy foods per day or taking calcium and vitamin D supplements in persons with borderline dietary intakes of these two nutrients is beneficial for weight loss and/or maintenance of weight loss in postmenopausal women

## 23a.7 Effects of high intakes of calcium

There are several health concerns regarding excess calcium intake over long periods of time. The risk of renal stones, risk of cardiovascular disease, and inhibition of iron absorption are current concerns Previously, recognition that high intakes of calcium caused calcification of soft tissue leading to “milk-alkali syndrome” was the rationale for setting the first Tolerable Upper Intake Level (UL) for calcium at 2500mg/d for all persons older than 1y, including pregnant and lactating women (IOM, 1997). More recently, the U.S. Food and Nutrition Board has set the Tolerable Upper Intake Level (UL) at 2000mg/d for adults over 50y due to reports of the increased incidence of kidney stones occurring in calcium supplementation trials A higher UL is set for other age groups in the United States and Canada as the risk of stone occurrence in younger ages is much lower. The European Food Safety Authority (EFSA) however, as noted earlier, ruled that there was not sufficient data to change from the previous UL of 2500mg derived as the amount showing no adverse effects with chronic intake

One concern that remains controversial is the effect of excess calcium on risk of cardiovascular disease (CVD). Many calcium trials were re‑analyzed for heart effects even though none were specifically designed to assess this effect. The most recent systematic review and meta-analysis of cohort studies and randomized controlled trials found that calcium intake from dietary sources did not increase the risk of CVD including coronary heart disease (CHD) and stroke. However, there was evidence that calcium supplements might raise CHD risk, especially myocardial infarction. It should be noted that many of these trials provided subjects (most often older women) with daily calcium supplements of 1000–1500mg without measuring dietary calcium intakes or accounting for the habitual calcium supplement use by subjects. Hence, calcium intakes may have exceeded 2000mg/d, the UL level set for adults over 50y by the Institute of Medicine

## 23a.8 Biochemical indices of calcium status

As calcium homeostasis is tightly regulated, there is no satisfactory single test to assess calcium status directly on a routine basis. Calcium is an important extracellular electrolyte (Ca++) and can be measured in serum or plasma. Normally values are reported as total calcium concentration, but ionized calcium can be an important measurement. Hypocalcemia and hypercalcemia (reduced and elevated serum calcium concentrations, respectively) are not likely to be caused by dietary intakes alone except on a transient basis. Urinary calcium excretion (calciuria) may give some clues as to calcium metabolism but as levels are equally likely to be from bone turnover or from dietary intakes, calciuria is difficult to interpret on its own. Instead, biochemical markers of bone formation and bone resorption are used as measures of calcium status as they reflect need for calcium (shown as excess bone resorption) and calcium adequacy (shown as appropriate bone formation)

### 23a.8.1 Serum calcium and its regulation

Serum calcium is not a test of nutritional status but rather reflects hormonal regulation of calcium, not calcium balance. Calcium is present in the blood in three different forms: as free Ca2+ ions, bound to protein (about 45%), and complexed to citrate, phosphate, sulfate and carbonate (about 10%). Calcium in the blood (and in extracellular fluid) is kept constant at 2.5mmol/L (range 2.25–2.6mmol/L), but it is ionized calcium (between 1.1–1.4mmol/L) that is the form that is actively transported across membranes Values decrease with age in men. Females usually have slightly lower concentrations (2.20–2.54mmol/L), associated with small differences in serum albumin content and the calcium-reducing effect of estrogens. Serum calcium decreases by 5–10% up to the end of the third trimester of pregnancy, after which concentrations rise. This decrease in total serum calcium may not be “true” hypocalcemia, as it is not due to a decrease in ionized calcium. However, changes in the serum chemistries and calcium-regulating hormones can be mistaken as a disorder of calcium

Hypocalcemia (low serum calcium) and hypercalcemia (elevated serum calcium) indicate serious disruption of calcium homeostasis but generally do not reflect calcium balance. In severe hypocalcemia, blood calcium is < 2.12mmol/L, and in severe hypercalcemia blood calcium is > 2.72mmol/L. Serum calcium levels must be corrected for the serum albumin concentration before confirming the diagnosis of hypercalcemia or hypocalcemia

Hypocalcemia is most commonly a consequence of vitamin D inadequacy or hypoparathyroidism, or a resistance to either of these hormones. Hypocalcemia has also been associated with the use of certain drugs including bisphosphonates, cisplatin, antiepileptics, diuretics, and proton pump inhibitors. Symptoms of hypocalcemia may include muscle spasms, cramps, tetany, numbness, and seizures, but also can be asymptomatic Hypocalcemia is common in the critically ill patient but the interpretation of the low serum calcium level is complicated by the co-existence of hypoalbuminemia and disorders of acid-base balance, thus in these patients hypocalcemia is defined in terms of ionized calcium concentration rather than total calcium concentration. Mechanisms causing hypocalcemia in critically ill patients include: hypoparathyroidism, severe vitamin D deficiency, hyperphosphatemia, infusion of chelating agents such as citrate, alkalosis caused by hyperventilation, and pancreatitis that causes a rise of free fatty acids (FFAs) that result in increased binding of calcium to albumin

Hypercalcemia occurs in association with hyperparathyroidism, hyperthyroidism, and sarcoidosis and when large parts of the body are immobilized (e.g., when a patient is hospitalized after spinal cord injury or major limb-bone fractures). In the latter cases, calcium from the rapidly atrophying bone is released into the circulating body fluids. In addition, hypercalcemia is one of the characteristic features of vitamin D intoxication. It arises from hyperabsorption of intestinal calcium and to a lesser degree from the release of calcium from bone Severe chronic hypercalcemia can result in nephrolithiasis (renal stones) and impairment of kidney function, as well as in calcification of soft tissues (e.g., nephrocalcinosis and vascular calcification), the latter when phosphorus concentrations in the blood are also high, as in renal insufficiency

Hypercalcemia may occur in advanced stages of the following cancers: multiple myeloma, breast cancer, parathyroid cancer, lung cancer, kidney cancer, lymphoma, leukemia, and bone metastases This hypercalcemia causes many unpleasant clinical symptoms such as vomiting, anorexia, constipation, frequent and increased urination, a patient may seek medical treatment at which time blood levels may be measured along with urinalysis and kidney function tests. Treatment would involve initially reducing hypercalcemia through fluid replacement followed by use of calcium-losing diuretics such as furosemide. As well, anti-resorptive medications such as bisphosphonates and denosumab, that reduce bone turnover, may be used to prevent calcium release from bone resorption.

### 23a.8.2 Measurement of serum calcium

Serum calcium levels have traditionally been determined by flame atomic absorption spectrophotometry (AAS) or by automated procedures that use the o‑cresolphthalein complexone method Earlier methods used flame photometry. For surveying the American population, the National Health and Nutrition Examination Survey (NHANES) uses a DxC800 system with ion-selective electrode methodology to measure calcium concentrations in serum, plasma, or urine. Only NHANES III provides ionized calcium values for the American population. Considerable differences in the quality of serum calcium measurements among laboratories may occur, emphasizing the necessity for rigorous quality-control practices. In general, the coefficient of variation for the AAS method is less than that for flame photometry Analytical variation for serum calcium via AAS ranges from 0.9%–3.04%

As such a large amount of calcium in serum is bound to albumin, it follows that concentrations can be influenced by changes in serum albumin levels. As a result, when serum albumin levels are outside the normal limits as may occur in liver disease or malnutrition, serum calcium values should be corrected, as noted earlier. The correction is approximate, and if there is uncertainty in the interpretation of the results, serum ionized calcium should be directly determined.

In many methods. serum rather than plasma should be used for calcium analysis as the former has no anticoagulant added. Most anticoagulants, with the notable exception of lithium heparin, interfere with the determination of calcium: they complex with or precipitate the calcium in the sample.

### 23a.8.3 Measurement of serum ionized calcium

Fasting blood samples are recommended for the measurement of ionized calcium in serum, to eliminate the effect of a recent meal. The samples should be stored anerobically at 0°C prior to the assay. Serum ionized calcium can be measured using commercially available ion-specific electrodes Care must be taken to ensure that the calcium standards used for the assay contain sodium and chloride at the same levels as in the test samples. Further details on preparing blood for ionized calcium are published Factors affecting serum ionized calcium levels include physical activity and circadian rhythm.

Many clinicians estimate ionized calcium using formulas that “adjust” total calcium with albumin and/or protein levels, and sometimes including other blood constituents such as phosphate There are over 30 published formulas, with the most widely used one being the Payne Formula: $\small \mbox{Adjusted calcium (mmol/L) = Total calcium (mmol/L) + 0.02 (40 − serum albumin (g/L))}$ However, caution is advised before applying this formula as it is not applicable in many situations.

The clinical utility of ionized calcium ranges from conditions with moderate evidence (drug-induced calcium disorders, cardiovascular outcomes), good evidence (post-thyroidectomy hypocalcemia), and very good to strong evidence (calcium disorders in critically ill patients, primary hyperparathyroidism, malignancy-related hypercalcemia). There is poor to fair evidence for use of ionized calcium in pre-eclampsia

#### Interpretive criteria

The concentration of serum ionized calcium is tightly maintained within a physiologic range of 4.4–5.4mg/dl (1.10–1.35mmol/L) Reported levels of serum ionized calcium in adults vary. In NHANES III, mean age-adjusted serum ionized calcium values were presented for male and female non-Hispanic Caucasians, African Americans, and Mexican Americans for two age groups: 25–59y and 60–89y; the reported mean age-adjusted value for serum ionized calcium was 1.237mmol/L for males and 1.232mmol/L for females Mean age-adjusted values for African Americans were slightly higher.

### 23a.8.4 Calcium balance

Calcium is lost daily through urinary and fecal excretion. In the gastrointestinal tract, calcium is lost through sloughing of cells as well as unabsorbed calcium in the feces. A small amount of calcium is lost in sweat. Replacement of these losses must occur through the diet A zero calcium balance means all losses have been replaced, and there is no net bone loss to provide serum calcium during periods of fasting. A positive calcium balance indicates that losses are less than intake, and implies calcium accretion into bones, an expected finding during growth and during pregnancy. A negative calcium balance means losses exceed intake, and the additional calcium lost is from bone. Over time a negative calcium balance indicates bone formation is less than bone resorption. Table 23a.5
Table 23a.5: Calcium balance, bone, and the life stages.
Ca Balance Bone Life Stage
Positive Accretion
Formation > Resorption
Childhood
Pregnancy
Neutral
(or zero)
Maintenance
Formation = Resorption
Negative Loss
Formation < Resorption
Post-menopausal
Women
provides a summary of these assumptions.

Measurement of calcium balance was a mainstay of nutrition research prior to the introduction of bone imaging equipment. These early experiments required facilities (metabolic units) where participants could receive all meals, sometimes for several months. A typical experimental protocol would adhere to the following steps:

1. participants ingest the experimental diet for at least one week adaptation;
2. in each 15d cycle participants would consume only those foods and beverages provided, and every day collect 24h urine samples;
3. additionally participants would collect all feces along with consumption of a fecal marker during these 15d;
4. study research personal prepare composites of food, urine, and feces representing each of three 5d cycles; and
5. calcium was measured in the food and fecal composites after dry ashing to remove the organic material, followed by quantitative dilution, while urine was acidified to prevent precipitation of calcium
Balance is calculated as Ca-intake minus Ca-losses. Unless participants resided in a metabolic unit, calcium losses through sweat were not taken into consideration. Overall, this method is expensive and requires exceptional participant cooperation.

### 23a.8.5 Urinary calcium as a biomarker

The amount of urinary calcium excreted is highly variable as it is dependent on the diet and the efficiency of intestinal absorption. Under conditions of bone accretion, one expects less calcium to be excreted as bones are forming, while during bone loss, more excretion might be observed. Measure of 24h urine calcium may provide some indication of calcium intake as extremely low excretion rates may indicate low calcium intake. However, extremely high excretion rates, (i.e., hypercalciuria), may be indicative of bone loss or clinical conditions unrelated to dietary intake.

For calcium balance, a 24h collection of urine is necessary to obtain the estimate of calcium excretion, yet this procedure requires enormous participant compliance. To gauge an indication of urinary calcium rate, some studies measure calcium in spot urine samples, usually after an overnight fast, when urinary calcium is expressed per mg or mmol of urinary creatinine. Using urinary creatinine in the denominator corrects for variability of urinary excretion as the amount of creatinine excreted in 24 hours is constant for individuals. Further, creatinine excretion can be used as an measure of muscle mass as it is made from creatine. Hence, when expressing urinary calcium excretion as a ratio of urinary creatinine excretion, one can equalize excretion between participants of dissimilar body mass. There is good correlation between excretion ratios of calcium expressed per mg or mmol of urinary creatinine in 24h urine collections and those based on spot urine collections

### 23a.8.6 Biomarkers of bone formation and resorption

Bone remodeling is an important process that continuously renews bone throughout life. It repairs micro-damage, maintains mineral homeostasis, and ensures mechanical proficiency by modifying the micro-architecture. Bone remodeling is regulated by a variety of systemic and local factors. Bone remodeling involves the following cycle of events: osteocyte signaling, recruitment of osteoclasts to begin resorption, degradation and removal of bone, reversal, formation of new bone by osteoblasts, and then a period of resting Each of the stages in bone remodeling can produce molecules, most of which are protein-derived, that enter the plasma. These biochemical markers of bone remodeling (see Table 23a.6)
Table 23.6 Proteins or protein fragments released from osteoclasts measure bone resorption while those from osteoblasts measure bone formation. Source: Bonjour et al., Nutrition Research Reviews, 2014, 27: 252–267.
Bone Resorption
(Osteoclast activity)
Osteoclast number
• Tartrate-resistant acid phosphatase (TRAP)
Bone matrix absorption
• Carboxy terminal telopeptide (CTX)
• terminal propeptide (PINP)
Bone Formation
(Osteoblast activity)
Osteoblast protein synthesis
• Procollagen type
Osteoblast activity
• Bone alkaline phosphatase (BALP)
• Osteocalcin (OC)
can be used to measure subtle changes in the rate of formation or degradation of the bone matrix brought about by dietary influences, such as availability of calcium, phosphate, and protein. When remodeling rates are changing, a combination of biomarkers, such as one formation marker and one resorption marker can provide more information than a single marker In addition, they can provide an earlier estimate on the effect of dietary interventions or drug treatments compared to measures of BMD. However, because of increases in bone turnover during the night, bone biomarker concentrations are affected by circadian rhythm. For all the bone biomarkers shown in Table 23a.6, their measurement is generally recommended using automated platforms which have improved technical performance

A description of the commonly used bone biomarkers shown in Table 23a.6 is outlined below. Of these, use of only two, namely P1NP (for formation) and CTX‑1 (for resorption) are recommended by the International Osteoporosis Foundation and the International Federation of Clinical Chemistry because they are well characterized for use in fracture risk prediction and for monitoring osteoporosis treatment

### 23a.8.6.1 Biomarkers of bone formation

Bone formation is determined by measuring proteins that increase in serum as a result of osteoblast activity during bone formation. These include bone alkaline phosphatase (BALP), which is measured as enzymatic activity, and the vitamin K‑dependent protein osteocalcin (OC). A third marker, procollagen type I N‑terminal propeptide (PINP) that reflects the protein synthesis capacity of the osteoblast can also be used. Early measurement of these three bone formation markers (OC, BALP and particularly PINP) has shown positive correlations with subsequent bone mineral density improvements in trials of teriparatide, a drug promoting bone formation Bone alkaline phosphatase (BALP), an isoenzyme also known as bone-specific alkaline phosphatase or skeletal alkaline phosphatase, is measured in serum, and is essential for normal bone mineralization. Several techniques have been developed to measure the activity of this enzyme. In adults approximately 50% of serum total alkaline phosphatase is present as bone alkaline phosphatase, whereas during childhood, the bone isoenzyme predominates. BALP may also be measured when assessing vitamin D deficiency (see chapter 18b.12.2) during which osteoblast activity is suppressed due to low calcium absorption. Thus, BALP is not specific to calcium deficiency and interpretation of its activity must be accompanied by other measurements.

Osteocalcin (OC), measured in serum, is also called a bone γ‑carboxyglutamic acid–containing protein because it contains up to three γ‑carboxyglutamic acid residues. This small non-collagenous protein is specific for bone tissue and dentin and comprises about 1–2% of total bone protein. During synthesis of osteocalcin, a small fraction is released directly into the blood. Osteocalcin is cleared by the kidneys so that serum levels depend on renal function. The half-life of serum osteocalcin is short (15–70min). Uncertainties persist as to the exact role of osteocalcin. Indeed, osteocalcin is secreted solely by osteoblasts, yet has only minor effects on bone mineralization. Instead, osteocalcin acts as a hormone-like factor in glucose homeostasis, brain development, cognition, and male fertility

Procollagen type I N‑terminal propeptide (PINP) measured in serum, is the preferred marker to measure bone formation Most of the organic matrix of bone is type I collagen associated with non-collagenous proteins, and Type I collagen is rich in the amino acid hydroxy-proline. During the conversion of procollagen to collagen, which occurs extracellularly, the procollagen propeptides, procollagen type I carboxy- terminal propeptide (PICP) and procollagen type I aminoterminal propeptide (PINP) are released into the circulation. Although both bone alkaline phosphatase and PINP are markers of bone formation, PINP is the reference biomarker for bone formation

### 23a.8.4.2 Biomarkers of bone resorption

Bone resorption markers include the pyridinium cross-links (PYR) subdivided into pyridinoline and deoxypyridinoline together with the associated telopeptides, i.e., carboxy terminal telopeptide (CTX) and N‑terminal telopeptide (NTX), all of which are released during collagen breakdown ( Table 23a.6). A second marker of bone resorption is tartrate-resistant acid phosphatase (TRAP), a measure of osteoclast number

Pyridinium cross-links (PYD) measured in urine were among the first bone biomarkers of resorption, but are rarely measured today. The crosslinks are hydroxylysylpyridinoline (known as pyridinoline, PYD) and lysylpyridinoline (known as deoxypyridinoline, DPD) and are important for the structural integrity of the collagen. Pyridinoline is widely distributed in both type I collagen of bone and type II collagen of cartilage, as well as in smaller amounts in the other connective tissues, and is thus considered non-specific. In contrast, deoxypyridinoline is found almost exclusively in type I collagen of bone Both the cross-links are released into the circulation after mature tissue collagen is degraded. They are not re-utilized or metabolized in the liver but, instead, are excreted in the urine unchanged, in peptide-bound and free forms. Neither PYD or DPD are commonly measured

Cross-linked N‑telopeptides (NTX) and C‑telopeptides (CTX) measured in serum or urine are now the most commonly measured bone biomarkers of resorption. C‑telopeptides (CTX ) are better characterized than NTX and are the preferred bone resorption marker, although both have similar properties and origins. CTX is the reference marker for bone resorption CTX and NTX are carboxy-terminal and amino-terminal fragments of collagen, respectively which are released into blood and subsequently excreted in the urine with cross-links attached However, serum CTX levels are decreased by food intake so blood withdrawal must take place in the fasting state Measurement of biomarkers can be conducted using more than one detection method for each biomarker. A complete list of all the biomarkers discussed above as well as others still under consideration is available combining the biomarkers to generate a biomarker risk score is under development to enhance the accuracy of predicting fractures

## 23a.9 Measurement of bone mineral content and bone mineral density

Noninvasive techniques can be used to provide an indirect assessment of calcium balance as well as calcium “stores” in bone by measuring the bone mineral content (BMC) or bone mineral density (BMD). These methods include dual X‑ray absorptiometry (DXA, also called DEXA) and quantitative computed tomography (QCT), both major diagnostic tools for osteoporosis. Ultrasound (US), although less accurate for medical diagnosis, can serve as a tool for screening and for field work. These technologies measure the mineral content of bone in the appendicular skeleton, the axial skeleton, or the total skeleton, measurements that relate directly to the calcium content of the skeleton. As almost all of the calcium in the body is stored in the skeleton, it follows that skeletal mineral content can be used as an indirect measure of body calcium stores. These same methods can also be used to monitor the response to changes in calcium intakes over relatively long time periods, i.e., months or even years.

Over the past 40 years technologies such as DXA and QCT have been developed and refined to measure calcium in hard tissues, and thus estimate of total body mineral content, which in turn can be used to calculate total body calcium. These technologies are also used to determine bone integrity for the diagnosis of osteoporosis and other bone diseases. They can also be used during treatment to evaluate efficacy as well as in research studies for the development of new drugs, exercise regimens, or nutrition interventions

The measurement of bone mineral using DXA is usually expressed as the bone mineral density, which corrects for projected bone area. However, the measurement is not a true density measurement, but instead, a measure of areal density (g/cm2) derived by dividing bone mineral content (g) by the scanned bone area. Bone mineral density is calculated in this way to reduce the biological variation observed in values for bone mineral content at all ages and thus to increase the statistical power of detecting abnormal values. Nevertheless, bone mineral density does not adequately correct for bone area and body size and thus must be interpreted carefully In contrast, QCT uses multiple slices to construct three-dimensional images, thus providing a true volumetric measure for BMD

Diagnosis for osteoporosis is based on the difference between the measured bone mineral density (BMD) of the participant and the mean BMD of healthy young adults, matched for sex and ethnicity. The difference is expressed relative to the young adult population standard deviation, and reported as a T‑score. has defined both osteoporosis and osteopenia based on T‑score values. An individual with a T‑score less than −2.5 at the spine, hip, or forearm is classified as having osteoporosis; a T‑score between −2.5 and −1 is classified as osteopenia; and a T‑score greater than −1 is healthy. These WHO definitions should not be applied to other bone mineral density measurement sites.

The International Society for Clinical Densitometry has established guidelines for bone density testing using DXA. In addition, they provide specific guidelines for diagnosing osteoporosis based on T‑score values for postmenopausal women, men (20y and older), pre-menopausal women (20y to menopause), and children (males or females < 20y). Persons using DXA technology should be certified, and facilities should be accredited; consult the ISCD for advice on certification and accreditation.

### 23a.9.1 Single-photon absorptiometry

Single-photon absorptiometry was the first practical noninvasive method that was developed to examine the peripheral skeleton although now it has mostly been replaced by DXA The original method was based on the assumption that the bone mineral content is inversely proportional to the amount of photon energy transmitted through the bone under study. A mono-energetic photon source is used, usually 125I or 241Am. The technique is fast, taking approximately 5min. With this method, bone mineral density is measured with a precision (%CV) of less than 2% and with an effective dose to the participant of only 0.1µSv. The site most frequently selected for measurements by single-photon absorptiometry is the lower radius, at approximately one-third of the distance from the styloid process to the olecranon. Single-photon absorptiometry is not suitable for measurements on the axial skeleton because the technique requires that a uniform thickness of soft tissue surrounds the bone Newer technologies have replaced SPA for human studies as SPA cannot measure those sites (hip or spine) needed for clinical diagnoses. In the past four decades SPA machines have been adapted and used for animal studies of bone health

### 23a.9.2 Dual X‑ray absorptiometry (DXA)

The first commercial DXA scanner was marketed in 1987. There are now several commercial versions, each using similar measurement procedures The original DXA method used an isotopic source, typically 153Gd , that emitted two low-energy gamma rays at 44Kev and 100KeV. The short half-life of the isotopic source 153Gd, however, limited the precision with which long-term changes in bone mineral content in the same participant could be made using this technique. This limitation led to the radioactive 153Gd source being replaced with an X‑ray tube behind a K‑edge filter. The filter converts the polychromatic X‑ray beam into one with two main energy peaks at 40Kev and 70KeV. These two congruent X‑ray beams are passed through the body, and the ratio of beam attenuation at the lower energy relative to that at the higher energy differentiates between bone mineral, bone-free mass, and the fat mass As a result, this modified instrument is termed a dual energy X‑ray absorptiometer, abbreviated as DXA (or DEXA).

The DXA instrument is used not only for bone mineral measures but also to assess body composition, as described in Chapter 14. Only the measurements of bone mineral mass (g), bone mineral content (g/cm), and “areal” bone mineral density (g/cm2) are discussed here. Using DXA, measurements of areal bone mass density (aBMD) can be made at the lumbar spine, femur, and forearm, as well as total body. Measurements of body mineral density at the lumbar vertebrae (L2 to L4) are often favored because they are sensitive to the changes associated with aging, disease, and therapy. Care must be taken when positioning the participant for the scan to ensure that precise and accurate data are obtained. Under some circumstances, it may be appropriate to make a whole-body scan. This is quick, taking from 3min to 35min, depending on the instrument used and the age of the participant, and requires very little cooperation from the participant. As a result, scans can be readily performed on young children, the elderly, and even persons who are sick, although pregnant women should be excluded

Dual-energy X‑ray absorptiometry is now the dominant method for measuring bone loss for clinical diagnosis of osteopenia and osteoporosis. In intervention studies care must be taken to ensure that both the length of the intervention and the size of the study population are sufficient to allow the detection of small changes in bone mineral density. A meta-analysis of trials reported that increasing calcium intake from dietary sources increased BMD by 0.6–1.0% at the total hip and total body at one year, and by 0.7–1.8% at these sites and the lumbar spine and femoral neck at two years Larger decreases in bone loss have been observed after treatment with estrogen, calcitonin, or biphosphates.

When comparing measurements using DXA based on different instruments and technologies, it is important to acknowledge the potential for systematic differences arising from methodological factors which could mask those associated with biological variation. Methodological factors may include differences in instrument calibration methods, whereas variations in body thickness, distribution of body fat, and the fat content of the bone marrow are major sources of biological variation. Of the latter, DXA is especially sensitive to discrepancies in body thickness, which can lead to systematic differences in the bone mineral estimates on thin and obese subjects. In a global survey of fracture liaison services, although access to DXA was reported to meet the needs only around 50% of institutions confirmed adherence to basic DXA quality and reporting procedures, and many required educating the operators/interpreters. Indeed, significant variability worldwide in both the quality and access to DXA services was reported in this survey.

Calcium balance using DXA may be determined by obtaining a total body (TB) DXA scan that provides bone mineral content (BMC). Scans of total body bone mineral content (TBBMC) can be used to calculate calcium requirements of specific life cycle groups, on the assumption that bone mineral is 32·2% calcium Use of this value is a better approximation of calcium in bone than simply dividing the atomic weight of calcium (40g/mol) by the molecular weight (i.e., 1005g/mol) of hydroxyapatite Ca10(PO4)6(OH)2 because the mineral in bone is hydrated so water molecules contribute to the weight of bone mineral content.

To derive the Estimated Average Requirement (EAR) for calcium for adolescents 9–18y, the U.S Food and Nutrition Board applied the factorial approach to estimate the physiological calcium requirements Estimates of calcium losses via urine, feces, and sweat were derived from metabolic balance studies to which was added the calcium accrued as bone mineral for growth (derived from TBBMC). The latter was measured in a cohort of Canadian males and females in two age groups, 9–13y and 14–19y as shown in Table 23a.7
Table 23.7. Determining calcium requirements during adolescence (age 9–18y) using total body bone mineral content (TBBMC). Source: Vatanparast et al., (2009). British Journal of Nutrition 103: 575–580.
Factorial Components Girls 9–18y
(mg/d)
Boys 9–18y
(mg/d)
Ca accretion from TBBMC 121 175
Urinary Ca Losses106127
Endogenous Fecal Ca112108
Sweat Ca Losses5555
Total Ca physiological need394465
Absorption (%)3838
Dietary Ca requirement (EAR)10371224
To translate the physiological requirements for absorbed calcium (from the factorial approach) into the EAR, it is necessary to take into account the proportion of calcium in the habitual diet that is absorbed by the intestine. Assuming an efficiency of absorption of 38% from an omnivorous diet, the equation used for the estimated Ca dietary requirement (i.e., the EAR) using the formula for adolescence which is a period of rapid growth:
EAR = Ca losses + (Ca needs for growth / 38%)

### 23a.9.3 Computerized tomography (CT) and peripheral quantitative CT (pQCT)

Computerized tomography (CT), often called quantitative CT (QCT) when referring to bone measurements, can be used to measure bone mass of both the appendicular and axial skeleton, although the method is not feasible for population studies. The advantage of QCT is that a true three-dimensional (i.e., volumetric) bone density can be obtained, so results can be expressed in g/cm3 rather than the two-dimensional areal density measured by DXA. Bone can also be identified as cortical or trabecular in nature on the basis of bone density measurements. This is an advantage because some forms of osteoporosis are predominantly trabecular in character. The equipment is not portable, and the radiation dose required per slice for imaging is relatively high (50–500µSv), although it can be reduced if scans are limited to regions of specific interest. The cost of the equipment and its maintenance is high.

The use of QCT technology applied to specific bone sites yields information about microarchitecture. Methods that can be used include micro‑CT and peripheral Quantitative CT (pQCT), with the latter having widespread use. The equipment for pQCT is small relative to DXA and CT, and provides imaging of both the arm at the distal radius and the lower leg (tibia) sites to assess bone microarchitecture. High-Resolution pQCT (HR‑pQCT) has been developed to improve the resolution problems of pQCT. Recently, second-generation HrpQCT has been introduced which differs from the first-generation HR‑pQCT in scan region, resolution, and morphological measurement techniques

### 23a.9.4 Quantitative ultrasound

Quantitative ultrasound (QUS) is used for measurements of the bone density of the peripheral skeleton such as the proximal arm (radius), lower leg (tibia) or heel. A wide variety of equipment is now available to measure the attenuation of a sonographic pulse as it passes through bone and is scattered and absorbed by the trabeculae. The heel is the most frequently used measurement site because it has a large volume of trabecular bone that is accessible for transmission measurements. However, there are ultrasound machines on the market that measure multiple skeletal sites. Quantitative Ultrasound has the advantage of being radiation-free and a portable system.

Quantitative ultrasound measures the speed of sound (SoS) in meters per second (m/s). In a study of different age groups using QUS, measures of SoS (m/s) varied according to life stages that reflected bone development For example, bone SoS accretion was found to begin 5y earlier in girls than boys (p < 0.05), a sex-related difference that mirrors results seen in DXA scans The maximal (peak bone) SoS was noted at 28y at the radius site and at 22y at the tibia site. Women who were postmenopausal (age 45–50y) showed a significant decrease in SoS measures at both these sites (p < 0.05) compared with men, as would be expected. Thus, while the precision of QUS is poor relative to DXA, the equipment may be suitable for characterizing bone in populations where DXA is not a feasible option. However, QUS has no clinical applications and is not used to define T‑scores for diagnosing osteoporosis or monitoring treatment. However, a systematic review has shown that quantitative ultrasound is an independent predictor of fracture for men and women when QUS values are low

## Acknowledgments

SJW would like to thank past and present Calcium collaborators. Both authors are grateful to Michael Jory for the HTML design and his tireless work in directing the translation to this HTML version.


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# Associated graded is integrally closed domain implies original ring was integrally closed

Let $$A$$ be a local ring (I am also happy to assume it is Noetherian) with maximal ideal $$\mathfrak{m}$$.

Claim: If the associated graded ring $$G_{\mathfrak{m}}(A)=\bigoplus_{n\geq 0} \mathfrak{m}^n/\mathfrak{m}^{n+1}$$ is an integrally closed domain, then $$A$$ is an integrally closed domain.

This claim appears without proof in Chapter 11 of Atiyah-MacDonald (page 123). I see why $$A$$ must be a domain, but have been unsuccessful in verifying the rest of the claim.

References for a proof or suggestions for how to proceed would be much appreciated.

• I am not sure the claim holds: take $A’=A\oplus Aq$ with $q^2=q$, $\mathfrak{m}’=\mathfrak{m}\oplus Aq$, then $G_{\mathfrak{m}}(A)=G_{\mathfrak{m}’}(A’)$, and $(A’,\mathfrak{m}’)$ is a local ring (I believe it is Noetherian) but not a domain. – Mindlack Jun 28 at 22:08
• @Mindlack Thanks - though I'm not sure I follow your comment. What exactly is $Aq$? If it is a nonzero ring (and $A$ is nonzero), then $A'$ is not local. In any event, it is true that if $G_\mathfrak{m}(A)$ is an integral domain, then $A$ is also an integral domain. This is Lemma 11.23 in Atiyah-MacDonald. I am interested in the corresponding claim for the property of being integrally closed. – 351910953 Jun 29 at 0:48
• See here. – user26857 Jun 29 at 21:19

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# Teaching and Learning Theorems

Theorems are carefully worded statements about mathematical facts that have been proved to be true. Important (and some not so important) ideas in calculus and all of mathematics are summarized as theorems. When you come across a theorem you need to understand it; the author of your textbook would not have included it and the AP Exams would not test it if it were not.  This post discusses some things about theorems in general. Students often do not realize these things; understanding them will help students understand a new theorem when it is presented to them.

Theorems have the form of IF one or more things are true (called the hypothesis), THEN some other thing is true (called the conclusion).  This is abbreviated $p\to q$ where p represents the hypothesis and q represents the conclusion. This is read as “if p, then q or “p implies q.” Theorems are also known as conditional statements.

In a certain instance, once you are sure all the conditions of the hypothesis are true, then you may be absolutely certain the conclusion is also true. When trying to determine if something is true about some function, check to see if the conditions of the hypothesis are all true.

People using a theorem need to know the hypotheses as well as the conclusion.

When teaching about a particular theorem, one thing that is often helpful to students is to “play” with the hypothesis and see how it affects the conclusion. For instance, if the hypothesis requires a function to be continuous, see what happens if the function is not continuous. If there are several parts to the hypothesis, see what happens if one or the other is changed. Hint: A change in part of the hypothesis, will make some difference – good theorems do not have extra, unneeded, or superfluous conditions.

To make them read better, some theorems are not stated in if …, then… form. If there is any confusion, restate the theorem in if…, then… form.

• The theorem often stated as, “Differentiability implies continuity,” really means: IF a function is differentiable at a point, THEN it is continuous at that point.
• The geometry theorem, “The diagonals of a rhombus are perpendicular,” really means: IF a quadrilateral is a rhombus, THEN its diagonals are perpendicular.

The Contrapositive

Since if p is true, q must be true, what happens if q is false? The answer is that p must also be false. This is a related conditional statement (theorem) called the contrapositive of the theorem. The contrapositive is abbreviated IF not q, THEN not p, or IF q is false, THEN p is false, or $\tilde{\ }q\to \tilde{\ }p$.  The contrapositive of a theorem is also true, always.

There are several theorems in the calculus where the contrapositive seems to be used more often than the theorem itself.

• Differentiability implies continuity. The contrapositive of this theorem is: IF a function is not continuous at a point, THEN it is not differentiable at that point. This is a quick way to tell if a function is not differentiable.
• IF an infinite series converges, THEN the limit as n goes to infinity of its nth term is zero. Here the contrapositive is IF the limit as n goes to infinity of its nth term of an infinite series is not zero, THEN the series does not converge. This is called the nth term test for divergence.
• IF the diagonals of a quadrilateral are not perpendicular, THEN the quadrilateral is not a rhombus.

The converse

The converse of a theorem is a statement formed by switching the hypothesis and conclusion of the theorem: IF q, THEN p (or $q\to p$). The converse is not necessarily true. It may be true, in which case it need to be proved as a separate theorem. Students (among others) often assume that the converse is true – this is called the fallacy of the converse.

• IF a function is continuous at a point, THEN it is differentiable there, is the converse of our previous example. It is false: a simple counterexample is f(x) = |x|. This function is continuous at the origin but not differentiable there.
• Another theorem states that IF the derivative of a function is positive on an interval, THEN the function is increasing on the interval. The converse is, IF a function is increasing on an interval, THEN its derivative is positive on the interval. The converse is false: for example, f(x) = x3 is increasing everywhere, yet its derivative at the origin is zero. (See Going up?)
• IF the diagonals of a quadrilateral are perpendicular, Then the quadrilateral is a rhombus, if false. The quadrilateral may have perpendicular diagonals, but unless they intersect at their midpoints the figure is not a rhombus (it is a kite shape).

The inverse

The inverse is the contrapositive of the converse or IF not p, THEN not q (or$\tilde{\ }p\to \tilde{\ }q$ ). The inverse will be true if the converse is true, and false if the converse is false.

• The inverse of our fist example is,IF a function is not differentiable on an interval, THEN it is not continuous there. This is false, since the function my fail to be differentiable even though it is continuous. An example is f(x) = |x|. again.
• IF a quadrilateral is not a rhombus, THEN he diagonals are not perpendicular (false – the kite again).

Biconditional statements

A biconditional theorem is a theorem whose converse is also true (and therefore its inverse and contrapositive are true). These are written in the form p if, and only if, q or $p\leftrightarrow q$ (or for that matter $q\leftrightarrow p$). It is equivalent to $p\to q\text{ and }q\to p$.

• An example from Geometry: IF two sides of a triangle are congruent, THEN the angles opposite them are congruent. The converse of this theorem is, if two angles of a triangle are congruent,THEN the sides opposite them are congruent. Since both the theorem and its converse (and its inverse, and its contrapositive) are true, you may write, “Two sides of a triangle are congruent if, and only if, the angles opposite them are congruent.”

Definitions are always biconditional statements. They are always true and do not need to be proved; in fact they cannot be proved.

• The definition of continuous at a point is, “A function is continuous at a point $\left( {a,f\left( a \right)} \right)$ if, and only if, $\underset{{x\to a}}{\mathop{{\lim }}}\,f\left( x \right)=f\left( a \right)$ and both the limit and value are finite.”
• A rectangle is defined as a quadrilateral with four right angles or A quadrilateral is a rectangle if, and only if, it has four right angles.

Which is which?

The theorem, its contrapositive, converse, and inverse are all theorems. Any of them could be taken as “the theorem” and the others would rearrange their names accordingly. (Which is good practice for your students.)

One other thing

What if the hypothesis of a theorem is false: can the conclusion still be true? The answer is, yes! The hypothesis of a theorem tells us that if true, the conclusion must be true. But the conclusion may be true anyway.

Consider the Mean Vale Theorem (MVT): IF a function, f, is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), THEN there exists at least one number c in the open interval (a, b) such that $\displaystyle {f}'\left( c \right)=\frac{{f\left( b \right)-f\left( a \right)}}{{b-a}}$.

• But consider the function $f\left( x \right)=\left\{ {\begin{array}{*{20}{c}} {{{x}^{2}}} & {-2\le x\le 1} \\ \text{4} & {1
• This function has a jump discontinuity at x = 1 and therefore is neither continuous nor differentiable on the interval [-2, 2]; the MVT does not apply. Yet $\displaystyle {f}'\left( 0 \right)=\frac{{f\left( 2 \right)-f\left( {-2} \right)}}{{2-\left( {-2} \right)}}=\frac{{4-4}}{4}=0$. (In fact, c could be 0 or any number between 1 and 2). In this example and not every example, the conclusion of the MVT is true even though the hypothesis is false.
• On the 2017 International exam AB 15 makes use of the idea that even though the theorem about limits that seems to apply doesn’t because the conditions are not met, but, nevertheless, the conclusion is true.

Proof

The proofs of all the important theorems are given in any good textbook. You study proofs for two reasons: (1) to see why a theorem is true, and (2) to learn how to write proofs of your own. If neither of these reasons concern you or your students (and they may not), then you still need to learn the hypothesis and conclusion, and how to apply the theorem. This cannot be avoided.

Some proofs are rather tricky. That is, the key step is not obvious. A beginning calculus student, should not expect to know how to prove most of the theorems; they should, however, be able to follow the proofs in the textbook. AP exam never ask for proof, per se, although they may ask you to justify a conclusion you make. The justification should show that the hypotheses are all true and state the name of the theorem that implies your conclusion.

I can recall only one time many years ago where students were asked to “prove” something on an AP Calculus Exam. The usual instruction is “Justify your answer” or “Explain your reasoning.” This means that students are supposed to cite the appropriate theorem and show that the hypotheses are met in the given situation. So, not quite ‘prove” but close. It’s not “prove” an original theorem, but rather determine which (unnamed) theorem applies (or does not apply) in a particular situation, and verify that the conditions are (or are not) met.

As always, look at a number of past exams and see just what is asked and how it is asked.

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text:
Browse Questions

# Show that the points A(1, – 2, – 8), B (5, 0, – 2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.

Toolbox:
• The position vector of a point which divides the line in the ratio $m:n$ internally is given by $\overrightarrow r=\large\frac{m\overrightarrow a+n\overrightarrow b}{m+n}$
Step 1:
Let the position vectors $\overrightarrow{OA}=\hat i-2\hat j-8\hat k$ and $\overrightarrow {OC}=11\hat i+3\hat j+7\hat k$ respectively.
Now let the point B divides the line AC in the ratio $\lambda:1$
Hence the position vector of $B$ is $\overrightarrow {OB}=\large\frac{\lambda(11\hat i+3\hat j+7\hat k)+1(\hat i-2\hat j-8\hat k)}{\lambda+1}$
Step 2:
But we know $\overrightarrow{OB}=5\hat i-2\hat k$
Therefore $5\hat i-2\hat k=\large\frac{\lambda(11\hat i+3\hat j+7\hat k)+1(\hat i-2\hat j-8\hat k)}{\lambda+1}$
$\Rightarrow 5\hat i-2\hat k=\hat i\large\frac{(11\lambda+1)}{\lambda+1}$$+\hat j\large\frac{(3\lambda-2)}{\lambda+1}$$+\large\frac{(7\lambda-8)}{\lambda+1}$
Step 3:
Now equating the coefficients of $\hat j$ on both sides we get
$0=\large\frac{3\lambda-2}{\lambda+1}$
$3\lambda=2$
Therefore $\lambda=\large\frac{2}{3}$
Step 4:
Now let us check if $A,B$ and $C$ are collinear.
$\large\frac{11\lambda+1}{\lambda+1}=\large\frac{11\big(\Large\frac{2}{3}\big)+1}{\Large\frac{2}{3}+1}$
$\qquad\quad=\large\frac{22+3}{2+3}=\frac{25}{5}$$=5$
Similarly $\large\frac{7\lambda-8}{\lambda+1}=\frac{7\times\Large\frac{2}{3}-8}{\Large\frac{2}{3}+1}$
$\qquad\qquad\qquad=\large\frac{14-24}{2+3}$
$\qquad\qquad\qquad=\large\frac{-10}{5}$
$\qquad\qquad\qquad=-2$
Step 5:
This proves $A,B$ and $C$ are collinear and B divides it in the ratio $2:3$


text:
Obtain Laplace transform given triangular signal.

Obtain the Laplace transform of the triangularsignal  by using differentiation theorem,the time-delay theorem, and expressing  in terms of unit steps.


text:
I need some good bargaining techniques

by flyingpig
Tags: bargaining, techniques
 P: 2,568 I recently bought an ODE text (Fundamentals of Diff Eqtns - Nagle 8ed) a few days ago. The book came and it was the wrong edition (6ed). I contacted the seller and now he is willing to return a portion of my money He mentioned at the end that "I think refund a portion of your money would be easiest. I don't need the textbook for anything so at least you can get some use out of it. The text cost me 18$to post. How much is it worth to you?" I paid$86 (including mailing fee) for the book, how much should I bargain to get the most out it for my book?
 P: 1,185 How did you buy it? A mention of "negative feedback" (or on the flip side, a promise of positive feedback if the situation is resolved satisfactorily) can work sometimes. Of course, this is assuming you bought it from eBay or Amazon or the like.
 P: 2,568 No I am willing to keep it, but I just want it cheaper. I kinda accidentally gave him a + feedback. Because I didn't even realize it was the 6ed (the cover looks EXACTLY the same)
PF Gold
P: 7,125

I need some good bargaining techniques

Did you buy it off ebay?
 P: 2,568 No I bought it from Amazon
P: 15,325
 Quote by flyingpig No I am willing to keep it, but I just want it cheaper. I kinda accidentally gave him a + feedback. Because I didn't even realize it was the 6ed (the cover looks EXACTLY the same)
I'm confused. You paid for it, but it's not what you wanted. But you want to keep it, you just want it cheaper. How is that the seller's problem?
 P: 1,035 Amazon has some sort of promisary (sp?) agreement with their sellers. You could get a full refund + shipping cost if you wanted.
 Sci Advisor PF Gold P: 2,793 If you don't want it, return it for a refund. If you do want it, keep it and stop trying it on.

 Related Discussions Calculus 0 Biology, Chemistry & Other Homework 4 Introductory Physics Homework 2 Computing & Technology 6


text:
# Probability of picking a uniquely colored ball

I came across an interesting problem on probability which is as follows:

Consider that there are two buckets A and B. A has N differently numbered balls {1,2,3,...,N} while B has a subset of balls that are already in A.

e.g. N = 5, A = {#1,#2,#3,#4,#5} and B = {#1,#4,#5}

What is the probability of picking d balls from bucket A such that exactly one ball is new to B

e.g. d = 2, Picked = {#2,#4} and #2 is unique

The answer can I could derive is $$\text{Prob} = \frac{\binom r{d-1}(N-r)}{\binom N{d}}$$

Now I wanted to generalize the above problem, but failed. The problem is as follows.

Consider that there are N different numbered balls that exist {1,2,...,N}.

Moreover, there are two buckets A having n and B having m uniquely numbered balls. A and B might have balls of the same color.

e.g. N = 10, A = {#1,#2,#3} and B = {#2,#3,#4}

What is the probability P that if we pick d balls from A, exactly 1 ball is absent in B.

e.g. d = 2. So we pick {#1,#2} and #1 is unique to bucket B.

The answer can I could derive is:

Given we know the number of balls common in A and B as k,

$$\text{Prob} = \frac{\binom k{d-1}(m-k)}{\binom n{d}}$$

Did I miss anything important?

• Where you have $\left(\frac r{d-1}\right)$ etc., it should say $\binom r{d-1}$ etc. You can get this either using \binom r{d-1} or using {r\choose d-1}. Also, if you have text like "probability" in an equation that shouldn't be italicised, you can use e.g. \text{probability}. – joriki Jul 4 '16 at 6:00
• Thanks @joriki for correcting my notations. Really appreciate the feedback :D – Mobashyr Mohammad Jul 5 '16 at 6:32

The problem isn't really any different than the one you already solved. Removing the balls in $B$ that aren't in $A$ and relabeling the balls transforms it into the first problem; the balls in $B$ that aren't in $A$ aren't playing any substantial role, and neither is the hypothetical limit $N$ on the numbers. So the result is
$$\frac{\binom k{d-1}\binom{n-k}1}{\binom nd}\;.$$
You can tell that your result can't be right e.g. by adding another ball in $B$ that wasn't in either bucket. Then your result involving $m$ would change, even though this clearly doesn't change the desired probability.


text:
## Functions Modeling Change: A Preparation for Calculus, 5th Edition

We first find an expression for the nth term: $$a_n=(2)^{n-1}$$ Thus, the 6th term is: $$(2)^{6-1} = 32$$


text:
# Correct model specification and pre-specification: Is this problem solved in Bayesian statistics?

In frequentist statistics, the validity of the inference depends on the assumption that the model is correctly specified as well as pre-specified. Violations of these assumptions (i.e. we only specify a linear relationship but the true relationship is non-linear or we perform model selection procedures) undermine the validity of inference. Are there similar assumptions in Bayesian statistics or how are they solved in Bayesian statistics? I assume the prior is our belief about the correct model and therefore there is no misspecification?

• I just found this comment from Andrew Gelman, which seems to discuss my problem but I still would appreciate some more formal explanations: andrewgelman.com/2013/03/14/… – Stats_Monkey Nov 9 '17 at 17:14

Model specification in the Bayesian paradigm is just as important and potentially just has harmful as in frequentist statistics. The main difference with Bayes is that you can incorporate uncertainties instead of picking a simple model and hoping you're right. Examples:

1. Instead of assuming normality, include a non-normality parameter in the model (indicating skewness, excess kurtosis, etc.)
2. Instead of choosing whether an interaction term is in or out of the model, put it "half in" by having a prior for the interaction that favors "no interaction" but allows larger samples to override this.
• Thank you for your helpful answer. I was asking this question because I was reading Richard Berk's comment about statistical inference with misspecified models. He explains that a parameter should be seen as the average parameter over realizations of the dataset and not as a fixed one as in frequentist statistics assumed. I tried to get my head around if this interpretation would be the same in a Bayesian approach but could not find anything about model misspecification. Reference: Berk et al 2017 Working with Misspecified Regression Models. DOI 10.1007/s10940-017-9348-7 – Stats_Monkey Nov 10 '17 at 15:01


text:
# Thread: Error-bound on Regula Falsi

1. ## Error-bound on Regula Falsi

Hi, in the lack of a Numerical Analysis sub forum, I will post this question here as it is quite "calculus-ish".

Does there exist a error-bound on Regula Falsi (Method of False Position) similar to the one that the Bisection method has? I would like to have an expression for the number of iterations necessary as a function of some prespecified error.

Thanks.

2. Originally Posted by Mollier
Hi, in the lack of a Numerical Analysis sub forum, I will post this question here as it is quite "calculus-ish".

Does there exist a error-bound on Regula Falsi (Method of False Position) similar to the one that the Bisection method has? I would like to have an expression for the number of iterations necessary as a function of some prespecified error.

Thanks.
Probably not, or if it does it is not very widely reported.

CB

3. Say that we have a function $f$ and we know that a root $r$ exists on $[a_1,b_1]$.
If we use the method of false position, our first approximation to this root will be,

$c_1 = b_1 - \frac{f(b_1)}{\frac{f(b_1)-f(a_1)}{b_1-a_1}}.$

The next one is,
$
c_2 = b_1 - \frac{f(b_1)}{\frac{f(b_1)-f(c_1)}{b_1-c_1}}$

and so on..

What if I use a similar bound to the one in the bisection method, and say that
$|r-c_n|\geq\frac{c_1}{2^n}$ such that

$n \geq log_2(\frac{c_1}{\epsilon_{step}})$

where $\epsilon_{step}=|r-c_n|$ ?

4. Originally Posted by Mollier
Say that we have a function $f$ and we know that a root $r$ exists on $[a_1,b_1]$.
If we use the method of false position, our first approximation to this root will be,

$c_1 = b_1 - \frac{f(b_1)}{\frac{f(b_1)-f(a_1)}{b_1-a_1}}.$

The next one is,
$
c_2 = b_1 - \frac{f(b_1)}{\frac{f(b_1)-f(c_1)}{b_1-c_1}}$

and so on..

What if I use a similar bound to the one in the bisection method, and say that
$|r-c_n|\geq\frac{c_1}{2^n}$ such that

$n \geq log_2(\frac{c_1}{\epsilon_{step}})$

where $\epsilon_{step}=|r-c_n|$ ?
Which depends of the function in question, unlike the bounds for bisection.

It would be interesting to see the error at the n-th step say in terms of bounds on the derivatives etc

(I don't think this impossible, just not worth the candle given false-positions other limitations and the nice properties of bisection - which does have its own problems when you have a double root)

CB

5. You mean we could perhaps bound the secant line from point $(a,f(a)) to (b,f(b))$?
I do not see what sort of bound we could use to do this though, other than

$tan^{-1}\left(\frac{f(b)-f(a)}{b-a}\right)\leq 90deg.$


text:
About the strength of representation-theoretic obstructions for orbit closure problems - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T09:40:43Z http://mathoverflow.net/feeds/question/106256 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106256/about-the-strength-of-representation-theoretic-obstructions-for-orbit-closure-pro About the strength of representation-theoretic obstructions for orbit closure problems Jesko Hüttenhain 2012-09-03T16:18:42Z 2012-09-05T09:49:47Z
Let $G$ be a reductive, affine, algebraic group over $\newcommand{\C}{\mathbb C}\C$. Let $X$ be a $G$-variety. For $x\in X$, we write $$G_x:=\{ g\in G\mid g.x=x\}$$ for its stabilizer and for any subgroup $H\subseteq G$, we write $$X^H:=\{x\in X\mid H.x=x\}$$ for the $H$-invariants of $X$. We say that $x\in X$ is characterized by its stabilizer if $X^{G_x}=\{x\}$. Let $\{V_\lambda\mid \lambda\in\Lambda\}$ be the irreducible $G$-modules.

Given two points $x,y\in X$, then $x\in\overline{G.y}$ implies $\overline{G.x}\subseteq\overline{G.y}$. Hence, $\C[\overline{G.y}]\twoheadrightarrow\C[\overline{G.x}]$ and thus,

$$\DeclareMathOperator{\mult}{mult}\forall \lambda\in\Lambda:\quad \mult\nolimits_\lambda(\C[\overline{G.x}])\le\mult\nolimits_\lambda(\C[\overline{G.y}])$$

Finding $\lambda\in\Lambda$ violating the above is therefore an "obstruction" for the inclusion of orbit closures.

My question now is the following: If $x$ and $y$ are characterized by their respective stabilizers, does the converse hold? I.e., does the above inequality imply that $x\in\overline{G.y}$? I have been trying to come up with a counterexample, but without success so far.

Intuition: If $G$ acts on a variety $Y$ and $y\in Y$ is characterized by its stabilizer, then you can very easily find counterexamples if you give up the condition that both points are characterized by their respective stabilizers: Consider $X:=Y\times\{z_1,z_2\}$ with $G$ acting trivially on $Z=\{z_1,z_2\}$. Now, the points $x_i:=(y,z_i)$ satisfy $x_1\notin\overline{G.x_2}$ and $\C[\overline{G.x_1}]\cong\C[\overline{G.x_2}]$. In the cases of interest to me, however, both points are characterized by their stabilizer and the question arises whether there are counterexamples under this additional condition.
 http://mathoverflow.net/questions/106256/about-the-strength-of-representation-theoretic-obstructions-for-orbit-closure-pro/106307#106307 Answer by Alexander Premet for About the strength of representation-theoretic obstructions for orbit closure problems Alexander Premet 2012-09-04T07:37:43Z 2012-09-05T09:32:07Z
Take the nilpotent cone $\mathcal N$ in $g={\rm Lie}(G)$ and the $G$-orbit of a regular semisimple element, $h$ say. The categorical quotient $g\rightarrow g//G\cong\mathbb{A}^l$, $l={\rm rk}(G),$ is equidimensional and each of its fibres is an trreducible complete intersection and contains a unique open $G$-orbit. Since the orbit $Gh$ is closed it coincides with one of the fibres and the algebra of regular functions $\mathbb{C}[Gh]$ is just a filtered deformation of the graded algebra $\mathbb{C}[\mathcal{N}]=\mathbb{C}[g]/(f_1,\ldots,f_l)$ whose defining ideal is generated by $f_1-\lambda_1,\ldots, f_l-\lambda_l$ for some $\lambda_i\in\mathbb{C}$ (here $f_1,\ldots, f_l$is a set of free homogeneous generators for $\mathbb{C}[g]^G$). Since $G$ is reductive, we are in characteristic $0$ and the action of $G$ on $\mathbb{C}[Gh]$ is rational, we have that $\mathbb{C}[\mathcal{N}]\cong \mathbb{C}[Gh]$ as $G$-modules. So all multiplicities will be the same in both cases. However, $Gh$ is not contained in $\mathcal N$ (and vice versa).

However this example does not answer the question as the stabilisers $G_x$ of regular elements $x\in g$ are not self-normalising (I have completely overlooked the extra condition on $x$ in the first reading, which implies that $N_G(G_x)=G_x$).


text:
Journal article

### Helicity of the W boson in lepton + jets tt̄ events

Abstract:

We examine properties of tt̄ candidate events in lepton + jets final states to establish the helicities of W bosons in t→W+b decays. Our analysis is based on a direct calculation of a probability density for each event to correspond to a tt̄ final state, as a function of the helicity of the W boson. Using the 125 events/pb of data collected by the DØ experiment at the Fermilab Tevatron pp̄ Collider at s=1.8 TeV, we obtain a longitudinal helicity fraction F0=0.56±0.31, consistent with the pred...

### Authors

Journal:
Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics
Volume:
617
Issue:
1-2
Pages:
1-10
Publication date:
2005-06-23
DOI:
ISSN:
0370-2693
Source identifiers:
178303
Language:
English
Pubs id:
pubs:178303
UUID:
uuid:8d272f42-b6be-4153-854c-bcabee04d93d
Local pid:
pubs:178303
Deposit date:
2012-12-19


text:
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

2020 has been a tough year, and I’ve been doing my best to keep busy (and distracted from all the insanity – both at the personal and worldwide levels). Earlier this year, I took a course in machine learning techniques and have been working on applying those techniques to work datasets, as well as fun sets through Kaggle.com.

Today, I thought I’d share another dataset I discovered through Kaggle: TV shows available on one or more streaming service (Netflix, Hulu, Prime, and Disney+). There are lots of fun things we could do with this dataset. Let’s start with some basic visualization and summarization.

setwd("~/Dropbox")

library(tidyverse)

## ── Attaching packages ────────────────────────────────────────────────────────── tidyverse 1.3.0 ──

## ✓ ggplot2 3.3.0     ✓ purrr   0.3.4
## ✓ tibble  3.0.0     ✓ dplyr   0.8.5
## ✓ tidyr   1.0.2     ✓ stringr 1.4.0
## ✓ readr   1.3.1     ✓ forcats 0.5.0

## ── Conflicts ───────────────────────────────────────────────────────────── tidyverse_conflicts() ──

## Warning: Missing column names filled in: 'X1' [1]

## Parsed with column specification:
## cols(
##   X1 = col_double(),
##   Title = col_character(),
##   Year = col_double(),
##   Age = col_character(),
##   IMDb = col_double(),
##   Rotten Tomatoes = col_character(),
##   Netflix = col_double(),
##   Hulu = col_double(),
##   Prime Video = col_double(),
##   Disney+ = col_double(),
##   type = col_double()
## )

First, we can do some basic summaries, such as how many shows in the dataset are on each of the streaming services.

Counts <- Shows %>%
summarise(Netflix = sum(Netflix),
Hulu = sum(Hulu),
Prime = sum(Prime Video),
Disney = sum(Disney+)) %>%
pivot_longer(cols = Netflix:Disney,
names_to = "Service",
values_to = "Count")

Counts %>%
ggplot(aes(Service,Count)) +
geom_col()

The biggest selling point of Disney+ is to watch their movies, though the few TV shows they offer can't really be viewed elsewhere (e.g., The Mandalorian). For the sake of simplicity, we'll drop Disney+, and focus on the big 3 services for TV shows.

The dataset also contains an indicator of recommended age, which we can plot.

Shows <- Shows %>%
mutate(Age = factor(Age,
labels = c("all",
"7+",
"13+",
"16+",
"18+"),
ordered = TRUE))

Shows %>%
ggplot(aes(Age)) +
geom_bar()

Many are 'NA' for age, though it isn't clear why. Are these older shows, added before these streaming services were required to add guidance on these issues? Is this issue seen more for a particular streaming site? Let's find out

Shows %>%
group_by(Age) %>%
summarise(Count = n(),
Year_min = min(Year),
Year_max = max(Year),
Prime = sum(Prime Video)/2144,
Netflix = sum(Netflix)/1931,
Hulu = sum(Hulu)/1754)

## Warning: Factor Age contains implicit NA, consider using
## forcats::fct_explicit_na

## # A tibble: 6 x 7
##   Age   Count Year_min Year_max    Prime Netflix   Hulu
##   <ord> <int>    <dbl>    <dbl>    <dbl>   <dbl>  <dbl>
## 1 all       4     1995     2003 0.000466 0.00155 0
## 2 7+     1018     1955     2020 0.0975   0.206   0.293
## 3 13+     750     1980     2020 0.0849   0.186   0.136
## 4 16+     848     1943     2020 0.104    0.155   0.208
## 5 18+     545     1932     2020 0.0896   0.0886  0.0906
## 6 <NA>   2446     1901     2020 0.623    0.363   0.272

It seems the biggest "offender" for missing age information is Prime - about 62% of the shows don't have an age indicator. More surprising, though, is the minimum year for some of these categories. I'm no expert in the history of TV, but I don't think any shows were being broadcast in 1901. What are these outliers?

YearOutliers <- Shows %>%
filter(Year < 1940)

list(YearOutliers\$Title)

## [[1]]
## [1] "Born To Explore"                    "The Three Stooges"
## [3] "The Little Rascals Classics"        "Space: The New Frontier"
## [5] "Gods & Monsters with Tony Robinson" "History of Westinghouse"
## [7] "Betty Boop"

Four of these entries are clearly in error - these are newer shows. This isn't important at the moment, but it's interesting nonetheless.

In terms of getting the most "bang for your buck," Amazon Prime has the most shows to offer (though if you're looking for data on recommended age for the TV show, Prime has the most missingness). But Hulu and Netflix, in terms of volume, are pretty comparable to Prime. What can be said about the quality of content on each of the 3?

The dataset offers some indicators of quality: IMDb rating and Rotten Tomatoes score. How do the 3 services measure up on these indicators?

Netflix <- Shows %>%
filter(Netflix == 1) %>%
select(IMDb, Rotten Tomatoes) %>%
mutate(Service = "Netflix")

Hulu <- Shows %>%
filter(Hulu == 1) %>%
select(IMDb, Rotten Tomatoes) %>%
mutate(Service = "Hulu")

Prime <- Shows %>%
filter(Prime Video == 1) %>%
select(IMDb, Rotten Tomatoes) %>%
mutate(Service = "Prime")

BigThree <- rbind(Netflix, Hulu, Prime)

BigThree <- BigThree %>%
mutate(RotTom = as.numeric(sub("%","",Rotten Tomatoes))/100)

BigThree %>%
ggplot(aes(Service, IMDb)) +
geom_boxplot()

## Warning: Removed 1194 rows containing non-finite values (stat_boxplot).

library(scales)

##
## Attaching package: 'scales'

## The following object is masked from 'package:purrr':
##

##
##     col_factor

BigThree %>%
ggplot(aes(Service, RotTom)) +
geom_boxplot() +
scale_y_continuous(labels = percent)

## Warning: Removed 4772 rows containing non-finite values (stat_boxplot).

It doesn't appear the 3 streaming services differ too much in terms of quality. But there's more analysis we can do of this dataset. More later.