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12900 | https://byjus.com/maths/trigonometric-functions-domain-range/ | All trigonometric functions are basically the trigonometric ratios of any given angle. For example if we take the functions, f(x)=sin x, f(z) = tan z, etc, we are considering these trigonometric ratios as functions. Since they are considered to be functions, they will have some domain and range. In the upcoming discussion, we shall figure out the domain and range of trigonometric functions.
Domain and Range of Trigonometric Functions (Sin, Cos, Tan)
To begin with, let us consider the simplest trigonometric identity:
sin2x + cos2 x = 1
From the given identity, the following things can be interpreted:
cos2x = 1- sin2 x
cos x = √(1- sin2x)
Now we know that cosine function is defined for real values therefore the value inside the root is always non-negative. Therefore,
1- sin2x ≥ 0
sin2x ≤ 1
sin x ∈ [-1, 1]
Hence, we got the range and domain for sine function.
Similarly, following the same methodology,
1- cos2x ≥ 0
cos2x ≤1
cos x ∈ [-1,1]
Hence, for the trigonometric functions f(x)= sin x and f(x)= cos x, the domain will consist of the entire set of real numbers, as they are defined for all the real numbers. The range of f(x) = sin x and f(x)= cos x will lie from -1 to 1, including both -1 and +1, i.e.
Now, let us discuss the function f(x)= tan x. We know, tan x = sin x / cos x. It means that tan x will be defined for all values except the values that will make cos x = 0, because a fraction with denominator 0 is not defined. Now, we know that cos x is zero for the angles π/2, 3 π/2, 5 π/2 etc. therefore,
Hence, for these values, tan x is not defined.
Domain and Range for Sec, Cosec and Cot Functions
We know that sec x, cosec x and cot x are the reciprocal of cos x, sin x and tan x respectively. Thus,
sec x = 1/cos x
cosec x = 1/sin x
cot x = 1/tan x
Hence, these ratios will not be defined for the following:
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12901 | https://civil.colorado.edu/~balajir/CVEN5393/lectures/chapter-08.pdf | 93 8. PIECEWISE LINEARIZATION 8.1 INTRODUCTION Most water resource planning and/or operation problems can be expressed in terms of linear con-straints. Mass balance or limits on resource use, for example, are generally linear functions.
Many objective functions, however, tend to be non-linear. Design problems for which the objective is to minimize cost or maximize benefits minus costs usually have cost functions with economies of scale. This implies a non-linear function as shown in Figure 8.1.
Cost, C X C = a Xb Figure 8.1: Typical Non-linear (Concave) Cost Function In Figure 8.1, the constant exponent b determines the degree of non-linearity and is usually between 0.4 and 1.0. For b = 1, f(x) is linear and curvature increases as b decreases. As an example, b ≈ 0.6 is a common value for pipelines.
Non-linearities may also occur in some types of constraints. For example, hydropower problems in which both flow rate (Q) and head on turbines (H) are variables require the non-linear con-straint: Power generated = f(Q • H) ...[8.1] Various approaches exist for solving non-linear problems. One of these is to divide the nonlinear functions into several linear sections (piecewise linearization). The advantage of this approach is that we then have a linear problem to which any LP algorithm, such as LINGO, can be applied.
Two approaches to this concept will be presented.
8.2 UNBOUNDED FUNCTION APPROACH This method is limited to maximizing strictly concave functions, such as that illustrated in Figure 8.1, or minimizing convex functions such as that shown in Figure 8.2.
94 f(x) X Figure 8.2: Convex Function Assume that the problem is to maximize the concave function in Figure 8.3 subject to the con-straint X ≤ 5. The problem is, of course, trivial because the solution is X = 5. However, if there were 10 variables in both the objective and the constraint we could not draw a picture of the problem, but the concept which follows would still apply.
X f(X) a 1 a 2 a 3 f(X) f1 b1 f 2 f 3 3 10 Figure 8.3: Unbounded Approach to Piecewise Linearization Instead, write a new problem: Max u ...[8.2] s.t.: u ≤ f1 = a1 + b1 X ...[8.3] u ≤ f2 = a2 + b2 X ...[8.4] u ≤ f3 = a3 + b3 X ...[8.5] X ≤ 5 ...[8.6] 95 This is an LP problem because each new fi is linear and each fi ≈ f(X) over some range of X.
The LP solution will be u = f2(X) because it is less than f1 or f3 and, therefore, closer to f(X) when 3 ≤ X ≤ 10. So the max value of u = a2 + b2 (5). Note that in the range 0 ≤ X ≤ 3, f1 is the smallest and for X ≥ 10, f3 is smallest.
Similarly we could minimize a convex function: Min f(X) ...[8.7] s.t.: g(X) ≥ b ...[8.8] by using Min u ...[8.9] s.t.: u ≥ f1 = a1 + b1 X ...[8.10] u ≥ f2 = a2 + b2 X ...[8.11] u ≥ f3 = a3 + b3 X ...[8.12] This is a very simple method that guarantees global optimum solutions, but is limited to the con-cave maximum or convex minimum restrictions given above. A more general approach (but one that guarantees only local optima without the same concave maximum/convex minimum restric-tions) follows.
8.3 BOUNDED VARIABLE APPROACH Consider the nonlinear function f(X1,X2) which has been approximated by three nonlinear seg-ments in the X1 plane of Figure 8.4. The f(X2) portion is not shown but one can imagine similar linear segments in the X2 direction which produce linear planes in three dimensions or linear hyperplanes in n dimensions.
The following notation demonstrates the method in n dimensions. The basic idea is to write the problem in terms of new artificial variables, Wji, in which i identifies which original Xi is being divided into linear pieces. Variable Wji is active between the end points j and j+1.
96 f( ) f( ) f( ) f( ) X X a 11 a 21 a 31 a 41 1 2 a 11 a 21 a 31 a 41 Figure 8.4: Bounded Variable Approach Consider the original generalized problem: Max Z = f(xi) ...[8.13] s.t.: gk(xi) ≤ bk ( k = 1, 2 ... L) ...[8.14] The piecewise linear problem can be written as follows: Max Z = i  j  f(aji) Wji ...[8.15] s.t.: j  aji Wji = Xi (i = 1,2, ... N) ...[8.16] j  Wji = 1 (i = 1,2, ... N) ...[8.17] i  j  gk(aji) Wji ≤ bk (k = 1,2, ... L) ...[8.18] Note that the last type of constraint is needed only for non-linear constraints; otherwise use the original gk(xi) ≤ bk.
97 This method guarantees a global optimum solution only for maximization problems when the function to be maximized is concave, or for minimization problems when the function to be minimized is convex. However, it may be used for other functions if restricted basis entry (i.e., only two adjacent Wji are allowed to enter the basis) software is available or if adjacent Wji in are forced into the basis by iteratively using any LP software.
For example, if we were minimizing a concave function such as shown in Figure 8.5, the solu-tion without restricted basis would be: x = w1 a1 + w4 a4 and f(x) = f1 ...[8.19] but this is incorrect because w1 and w4 are not adjacent and therefore f(w1,w4) is not a good approximation of f(x). Restricted basis entry will prevent such solutions, and x = w2 a2 + w3 a3 and f = f2 ...[8.20] will therefore be selected for the constraint shown.
a a a f f constraint a f(x) x 2 1 1 2 3 4 f(w ,w ) 2 3 f(w ,w ) 1 4 Figure 8.5: Restricted Basis Entry 98 8.4 EXAMPLE PROBLEMS 8.4.1 A Simple LP Problem Develop an LP model for solution of the following non-linear problem by the bounded variable method.
Max Z = X1 (5 - X1 2) + X2 (14 - 6 X2) ...[8.21] s.t.: X1 + 4 X2 ≤ 18 (Constraint 1) ...[8.22] 6 X1 + 2 X2 ≤ 26 (Constraint 2) ...[8.23] X2 2 ≤ 5 (Constraint 3) ...[8.24] Solution: Assume a grid interval of one unit and, for simplicity, an upper bound on X1 and X2 of 3. From this, a table of basic information about the problem can be constructed, as illustrated in Table 8.1.
Table 8.1: Calculation of Coefficients for Use in Piecewise Linearization j aji f(aj1) f(aj2) g3(aj2) 1 0 0 0 0 2 1 4 8 1 3 2 2 4 4 4 3 -12 -12 9 From the information tabulated in Table 8.1, the coefficients of an LP problem can be con-structed, as illustrated in Table 8.2.
99 Table 8.2: Piecewise Linear LP Model Coefficients Variable Constraint W11 W21 W31 W41 W12 W22 W32 W42 X1 X2 RHS Obj. Fn. Z = 0 4 2 -12 0 8 4 -12 (1) 1 4 ≤ 18 (2) 6 2 ≤ 26 (3) 0 1 4 9 ≤ 5 1 1 1 1 = 1 1 1 1 1 = 1 0 1 2 3 -1 = 0 0 1 2 3 -1 = 0 8.4.2 Two-Reservoir Example Problem Let us complete the discussion of piecewise linearization by considering the following multiple reservoir-multiple use design problem. This is a slightly modified version of a problem given by Dorfman in Maass et al. (1962).
Consider the sequential, two-reservoir problem shown in Figure 8.6. There are two uses for the water: irrigation and hydropower. Assume that optimization is to be based on average year flows of 3.3x106 acft during the wet season and 1.4x106 acft during the dry season, plus the wet and dry season inflows shown at two tributaries. The capacities of reservoirs A and B (Ya and Yb) are variables that are to be determined such that the total capacity will be filled during the wet season and completely released during the dry season. This policy will produce the river flows shown (or calculated) in various reaches. Irrigation volume (I) is a variable that must be allocated as shown during wet and dry seasons. The objective is to determine Ya, Yb, I, and E (where E is the energy generated per year in 109 KWH) such that net benefits from agriculture plus energy production are maximized. Flow through the turbine is related to energy as follows: E = 0.144 F ...[8.25] where F is flow in 106 acft. The demand for energy is uniform so that half must be generated during the wet season and half during the dry season. This means that: Fw = [6.9 - Ya - Yb - 0.275 I] ≥ (0.5 E / 0.144) = 3.47 E ...[8.26] FD = [3.9 + Ya + Yb - 0.125 I] ≥ (0.5 E / 0.144) = 3.47 E ...[8.27] Therefore, eliminating Fw and FD, we have: Ya + Yb + 0.275 I + 3.47 E ≤ 6.9 ...[8.28] - Ya - Yb + 0.125 I + 3.47 E ≤ 3.9 ...[8.29] 100 Res. Y (3.3 - Y ,1.4 + Y ) 0.425(I),0.575(I) Irrigated Area Res. Y (3.0,2.1) (F ,F ) Power Plant 0.15(I),0.45(I) Q = (3.3,1.4) t Q = (0.6,0.4) t a b a a w d Figure 8.6: Two-Reservoir Irrigation and Power Problem The objective is to maximize net benefits where: NB = B1(E) + B2(I) - C1(Ya) - C2(Yb) - C3(E) - C4(I) ...[8.30] These functions are as follows: The benefit functions, B(E) and B(I), are the present value of yearly energy and agricultural pro-duction in $106. The cost functions, Ci, are the capital costs of building the reservoirs, power plant, and irrigation systems in $106. The cost functions are as follows: C1(Ya) = 43 Y a (1 + 0.2 Ya) ...[8.31] C2(Yb) = 47 Yb (1 + 0.3 Yb) ...[8.32] 101 C3(E) = 30.6 E - E2 ...[8.33] C4(I) = 54 I ...[8.34] Note that the first three cost functions are non-linear due to economies of scale.
The benefit functions are: B1(E) = 250 E (marginal benefit is constant) ...[8.35] B2(I) = 30(I) + 1045 ln[1 + 0.2(I)] ...[8.36] The second benefit function has been derived by integrating a marginal benefit function.
Let us now proceed to set up the LP matrix for this problem using separable programming to lin-earize the objective function.
Solution: The intuitive approach is to write the usual mass balance constraints at each reservoir plus each junction. We can do this, but in a simpler form since we assume both reservoirs will fill during the wet season and empty during the dry season. This implies St is not a variable (S1 = 0, S2 = Smax, S3 = 0). Therefore, we can simply write constraints that prevent negative flows in each reach of the river. Analysis shows that several of these constraints are redundant so we use only the non-redundant set to reduce the size of the model.
Let: Ya, Yb be the yields (Smax) of reservoirs A and B, respectively I be the total annual irrigation demand (both seasons) E be the annual production of electrical energy in units of 109 KWH Table 8.3 summarizes the constraints that will be required.
102 Table 8.3: Summary of Constraints for the Two-Reservoir Problem Model Constraint in Model Constraints River Reach 1 2 3.3 - Ya ≥$ 0 1.4 + Ya ≥ 0 (this is redundant because Ya ≥ 0) The tributary adds flows, so both constraints are redundant.
1 2 3 4 3.9 Ya - 0.425 I ≥ 0 1.8 + Ya - 0.575 I ≥$ 0 Only return flows are added, so both constraints are redundant.
3 4 5 6 3.9 - Ya - Yb - 0.275 I ≥ 0 1.8 + Ya + Yb - 0.125 I ≥ 0 Only tributary flows are added, so these are redundant.
5 6 7, 8 Hydropower availability constraints, as previously given.
7 The objective function can be written as: Max Z = f1(Ya) + f2(Yb) + f3(I) + f4(E) ...[8.37] where f1(Ya) = - 43 Y a (1 + 0.2 Ya) ...[8.38] f2(Yb) = - 47 Yb (1 + 0.3 Yb) ...[8.39] f3(I) = -24 I + 1045 ln(1 + 0.2 I) ...[8.40] f4(E) = 229.4 E + E2 ...[8.41] Table 8.4 presents the calculations for the piecewise linearization of the nonlinear functions.
103 Table 8.4: Calculation of Coefficients for Piecewise Linearization of the Two-Reservoir Hydropower and Irrigation Problem j (Ya, Yb, E, I) (aji) f1(Ya) [or f1(aj1)] f2(Yb) [or f2(aj2)] f3(I) [or f3(aj3)] f4(E) [or f4(aj4)] 1 0 0 0 0 0 2 1 -35.8 -36.1 166.5 220.4 3 2 -61.0 -58.7 304 442.8 4 3 -80.6 -74.2 419 667.2 5 4 -95.5 -85.2 518 893.6 6 5 -94.0 604.3 1122 7 6 690 It should be noted that in this example, we are maximizing an objective function that is not con-cave. Therefore, an LP solution will not be correct unless restricted basis entry is used. Most commercial LP algorithms have this capability (limiting basis variables to two adjacent Wji for each i). However, if one is using an LP algorithm without this capability, restricted basis entry can be accomplished manually by solving several LP problems iteratively unless the problem is so large that the number of iterations becomes prohibitive. The procedure is to force one of the non-adjacent Wji to 0, re-solve the problem, and continue until only adjacent Wji are in the basis.
Bishop and Narayanan (1977) give a good example of the use of separable programming for a large non-linear planning problem.
8.5 SEPARABILITY Note that the piecewise linear concepts presented work only because functions were ‘separable’ into terms which are functions of only a single non-linear variable. For example, f(X1,X2) = 2 X1 2 + 3 X2 3 ...[8.42] is separable. However, a function such as g(X1,X2) = X1 X2 ...[8.43] (which is of the form of the Q•H hydropower function mentioned previously) presents difficulty because it cannot be linearized until X1 and X2 are separated. One way to do this is to do a trans-formation into two new variables, X3 and X4 as follows: Let: X3 = X1 + X2 2 ...[8.44] 104 X4 = X1 - X2 2 ...[8.45] Then X1 X2 = X3 2 - X4 2 ...[8.46] because X1 + X2 2 È Î Í ˘ ˚ ˙ 2 - X1 - X2 2 È Î Í ˘ ˚ ˙ 2 = X1 2 + 2 X1 X2 + X2 2 4 - X1 2 - 2 X1 X2 + X2 2 4 = X1 X2 ...[8.47] Products of terms with other polynomials can also separated by using the generalized transfor-mation: X1 a X2 b = X3 2a - X4 2b ...[8.48] 8.6 NON-LINEAR OPTIMIZATION SOFTWARE The need for piecewise linearization is not as great now as it was in the past because software is available to optimize problems in non-linear form. This is true particularly for problems with a non-linear objective function, but with linear constraints. However, non-linear constraints still may need to be linearized. Also, no software exists to handle an integer programming problem with a nonlinear objective.
8.7 SIMULATION VERSUS OPTIMIZATION The previous problem has only four real decision variables (until our linearization technique expanded it to 28 variables). This suggests that perhaps some simulation approach might be more appropriate since non-linearity would then present no difficulty. Two possible approaches are exhaustive enumeration and Monte Carlo simulation. The former will require a truly large number of calculations. If, for example, we wish to search over a grid with 0.1-unit increments, we need to have four nested “do-loops” where 0 ≤ Ya ≤ 3.3 requires 34 iterations, 0 ≤ Yb ≤ 3.9 requires (40)x(34) iterations, and E and I limits of 6 each require (40)x(34)x(61)x(61) = 5(106) calculations. One could alternatively search a coarser grid to locate the “good” solution vicinity and then do a finer grid search of this smaller area.
105 Consider, however, the Monte Carlo approach where a random number generator may be used to simultaneously vary the four decision variables. One could generate trial values of Ya, Yb, I, and E which are always within the possible range of each variable (but which may violate some of the six constraints). One could then test feasibility by checking the constraint set and save only those solutions which are both feasible and which improve the best previous objective function.
Solutions of this type have been shown to be within 5.5 percent of the LP solution after 5000 iterations and within 2.8 percent of the LP solution after 20,000 iterations. Note that the LP solution also includes some error due to the difference between the piecewise linear functions of the true non-linear functions. Both LP and Monte Carlo methods seem to produce good models with reasonable computational effort. In a much larger problem, however, with hundreds--rather than four--decision variables, the Monte Carlo approach will become computationally prohibi-tive, while LP models can be readily solved with thousands of variables. Relatively recent advances in optimization techniques, called genetic algorithms (that are similar to Monte Carlo methods in that they employ random but controlled search methods) show considerable promise for use in water resources planning and management. These are discussed in a later chapter.
106 8.8 PROBLEMS 1.
Solve Problem 1 from Chapter 7 as a piecewise linear problem. The problem is the same as discussed in Chapter 7, except that the capital cost function is continuous and non-linear as shown in Table 8.5, below. The reservoir may be any capacity between 0 and 900.
Table 8.5: Reservoir Capacity and Capital Cost Data Reservoir Capacity (acft) Capital Cost ($/year) 0 0 350 5,000 700 15,000 900 18,000 2.
Develop a piecewise linear model of the multiple use-two reservoir problem presented in this chapter. Then solve the model using LINGO. Note that manual iterations to achieve restricted basis entry will be necessary since you are maximizing functions that are not all concave. One can avoid the necessity of these manual operations with judicious use of integer variables to control restricted basis entry.
Solve the same problem using the non-linear programming capability of LINGO instead of piecewise linearization.
3.
Solve the same reservoir problem by using a Monte Carlo simulation approach. Compare the answers for an increasing number of trials. Also compare the answers of Exercises 2 and 3. Which is more correct? Is there any error in the Exercise 2 answer? |
12902 | https://www.wolframalpha.com/examples/Optimization.html | Examples for
Optimization
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12903 | https://www.chegg.com/homework-help/questions-and-answers/20-year-old-male-struck-side-head-baseball-bat-hours-later-becomes-disoriented-vomits-frie-q133677492 | Solved A 20-year-old male is struck on the side of the head | Chegg.com
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A 20-year-old male is struck on the side of the head by a baseball bat. A few hours later, he becomes disoriented and vomits. His friend takes him to the Emergency Department where â CT scan reveals the formation of an epidural hematoma and a skull fracture at the location of pterion. Which of the following blood vessels was most likely ruptured by the blow
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Question: A 20-year-old male is struck on the side of the head by a baseball bat. A few hours later, he becomes disoriented and vomits. His friend takes him to the Emergency Department where â CT scan reveals the formation of an epidural hematoma and a skull fracture at the location of pterion. Which of the following blood vessels was most likely ruptured by the blow
A 2 0-year-old male is struck on the side of the head by a baseball bat. A few hours later, he becomes disoriented and vomits. His friend takes him to the Emergency Department where â CT scan reveals the formation of an epidural hematoma and a skull fracture at the location of pterion. Which of the following blood vessels was most likely ruptured by the blow to his head? Vertebral artery Middle meningeal artery Internal carotid artery Internal jugular vein
Here’s the best way to solve it.Solution Share Share Share done loading Copy link Epidural Hematoma Analysis
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12904 | https://pubmed.ncbi.nlm.nih.gov/10219438/ | Endoscopic sinus surgery for rhinocerebral mucormycosis - PubMed
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. 1999 Mar-Apr;13(2):105-9.
doi: 10.2500/105065899782106751.
Endoscopic sinus surgery for rhinocerebral mucormycosis
R S Jiang1,C Y Hsu
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1 Department of Otolaryngology, Taichung Veterans General Hospital, Taiwan, Republic of China.
PMID: 10219438
DOI: 10.2500/105065899782106751
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Clinical Trial
Endoscopic sinus surgery for rhinocerebral mucormycosis
R S Jiang et al. Am J Rhinol.1999 Mar-Apr.
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. 1999 Mar-Apr;13(2):105-9.
doi: 10.2500/105065899782106751.
Authors
R S Jiang1,C Y Hsu
Affiliation
1 Department of Otolaryngology, Taichung Veterans General Hospital, Taiwan, Republic of China.
PMID: 10219438
DOI: 10.2500/105065899782106751
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Abstract
Rhinocerebral mucormycosis is a fulminant, often fatal, disease. Aggressive surgical debridement has been considered an important part of treatment. Traditionally, an external or transantral approach has been the classic method. Recently, endoscopic sinus surgery (ESS) has been tried on several occasions to reach the goal of radical resection. Since 1991, ESS has been used to treat 9 rhinocerebral mucormycosis patients in our department. Among them, ESS was the only surgical procedure in six patients. The other three patients were treated by ESS combined with a transantral procedure. As a result, eight patients (88.9%) have survived the disease. One patient died 5 days after ESS because of an internal carotid artery occlusion. We conclude that ESS can be used to treat rhinocerebral mucormycosis alone or in combination with the traditional surgical procedures. It has the advantage of less operative morbidity and greater operative accuracy.
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12905 | https://digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/lecture_05.pdf | Simplification of Expression By: Ali Mustafa How to simplify the Boolean Function • Algebric method • K-Map method (Karnaugh) • Quine – Mc Cluskey method • VEM (Variable entered mapping) Algebraic Forms ( A way to represent any function) Sum of Products (SOP) Product of Sums (POS) Sum of Products (SOP) • Switching functions formed by SUMMING (ORing) PRODUCT (ANDed) terms. Sum-of-products(SOP) • Product terms are known as minterm • Output = 1 , for SOP Sum-of-products (cont.) • ANDed product–input combination for which output is true • Each variable appears exactly once, in true or inverted form (but not both) Sum-of-products Example : Simplify the following three variable boolean expression Y = ∑ m (2,4,6) Solution: y = m2 +m4 + m6 =A’BC’+AB’C’+ABC’=A’BC’+AC’(B’+B) =A’BC’+AC’= C’(A’B + A) Hint =(A’+ A = 1) = C’(A’+ A) (B + A) = C’(B + A) How to build a circuit from the (SOP)function? F = AB'C + A'B’C + AB'C + ABC' + ABC Answer: SOP AND/OR Two-level Implementation Product of Sums (POS) • Switching functions formed by taking the PRODUCT (ANDing) of SUM (ORed) terms. Product-of-sums • Sum terms are known as Maxterm • Output = 0 , POS Product-of-sums (cont’d) • Sum term (or maxterm) • ORed sum of literals –input combination for which output is false • Each variable appears exactly once, in true or inverted form (but not both) Product-of-sums (Self Task) • Example : Simplify the following three variable boolean expression Y = ∏ M (1,3,5) ?
How to build a circuit from (POS) function? F(A, B, C) = (A + B + C) (A + B' + C) (A + B’ + C’) Answer POS: OR/AND Two-level Implementation SOP and POS Represent the same function POS vs. SOP • Any expression can be written either way • Can convert from one to another using theorems • Sometimes SOP looks simpler AB + CD = ( A + C )( B + C )( A + D )( B + D ) • Sometimes POS looks simpler (A + B)(C + D) = BD + AD + BC + AC • SOP will be most commonly used Standard or canonical SOP & POS forms Sr # Logical Expression Type of Expression 1 Y = AB + ABC’ + A’BC NON STANDARD 2 Y = AB + A’B + A’B’ STANDARD 3 Y = (A’ + B)(A + B)(A + B’) STANDARD 4 Y = (A’ + B)(A + B +C) NON STANDARD Conversion Procedure for Standard SOP 1. For Each term ,we find the missing literal 2. Then, we AND term with the term formed by ORing the missing literal and its compliment Y = AB + AC’ + BC Conversion Procedure for Standard SOP Example: Convert the expression into the standard SOP form Y = AB + AC’ + BC Solution: Given expression Y = AB + AC’ + BC Step 1: Find the missing literals Y = AB + AC’ + BC Step 2: AND each term with (Missing literal + its compliment) Y = AB(C + C’) + AC’(B + B’) + BC(A +A’) Simplify the expression to get the standard SOP form Y = ABC + ABC’ + ABC’+ AB’C’ + ABC + A’BC Y = ABC + ABC’+ AB’C’ + A’BC (Each term consists of all literal) C B A A + A = A Conversion Procedure for Standard POS • Example: Convert the expression into the standard SOP form Y = ( A+B )( A+C )( B +C’) Step 1: Find the missing literals Step 2: OR each term with (Missing literal + its compliment) Y = ( A + B + CC’ )( A+ BB’ +C )( AA’ + B +C’ ) Y =( A + B + C )( A + B + C’ ) ( A+ B +C )( A+ B’ +C ) ( A + B +C’ )( A’ + B +C’ ) Y =( A + B + C )( A + B + C’ )( A+ B’ +C )( A’ + B + C’ ) SELF TASKS • Convert the following expressions into standard SOP & POS forms 1. Y = AB + AC + BC 2. Y = (A + B) (B’ + C ) 3. Y = A + BC + ABC Boolean Expression from a Logic Circuit • To derive the Boolean expression for a given logic circuit, begin at the left-most inputs and work toward the final output, writing the expression for each gate. Example |
12906 | https://chemistry.stackexchange.com/questions/38065/do-all-conversion-factors-have-an-infinite-number-of-significant-figures | units - Do all conversion factors have an infinite number of significant figures? - Chemistry Stack Exchange
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Do all conversion factors have an infinite number of significant figures?
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I believe conversion factors have an infinite number of significant figures. They should be exact values; but not all of them are, are they?
I looked at a pdf document (via the Internet Archive) and it agreed with the fact that all conversion factors have an infinite number of significant figures.
I know this is true with a conversion factor such as 1 c a l g∘C 1 c a l g∘C, because this is how a calorie is defined. Similarly, it's the same for 1 k g 1000 g 1 k g 1000 g, 4 q u a r t s g a l 4 q u a r t s g a l, etc.. We know these are exact, and in dimensional analysis, if you multiplied by the first conversion factor, you wouldn't end up with one significant figure because of the single significant figure in 1 c a l 1 c a l.
However, what is the case with measured conversions, such as 4.184 J c a l 4.184 J c a l or 1.609 34 k m m i 1.609 34 k m m i? These were measured using significant figures, so it would make sense to round with significant figures when multiplying (dimensional analysis), but the above link says otherwise.
What should I do, especially with the below type of conversions?
units
significant-figures
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edited Jan 29, 2020 at 17:21
Martin - マーチン♦
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Jonathan LamJonathan Lam
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Your intuition is correct. There are two types of conversion factor.
One kind of conversion factor is based on definitions. These conversions are exact (infinite significant figures).
In addition to your examples of the specific heat of water (in calories), the kilogram/gram conversion, and the quart/gallon conversion, other conversions are exact, including:
The speed of light - c=299,792,458 m/s c=299,792,458 m/s the length of the meter is then derived from this constant.
The price of gasoline/petrol.
The inch is defined as exactly 25.4 m m 25.4 m m, and thus the conversions from all imperial to metric lengths (including your are exact as well.
The specific heat of water in joules - the relationship between the calorie and the joule is defined exactly as 1 c a l=4.184 J 1 c a l=4.184 J
Note that most conversions of this type are unit conversions. Many units have been defined exactly in terms of SI-units.
Other conversion factors are based on measurements. One common example is atomic mass. Atomic masses are valuable conversion factors that are based on measurements. However, some unit conversions live in this category.
The electronvolt is based on the measurement of the elementary charge of the electron (which is not exact). Thus the electronvolt is 1 e V=1.6021766208(98)×10−19 J 1 e V=1.6021766208(98)×10−19 J.
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edited Jun 10, 2020 at 14:04
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answered Sep 29, 2015 at 1:23
Ben NorrisBen Norris
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To clarify, how should I round with measurement-based conversions?Jonathan Lam –Jonathan Lam 2015-09-29 01:27:50 +00:00 Commented Sep 29, 2015 at 1:27
4 @jlam55555 Most measurement-based conversion factors have a functionally-infinite number of significant figures for the purposes of routine chemistry work. That being said, to be rigorous, one would have to track down the original source of the measurements (or a reputable secondary source such as Perry's Chem Eng Handbook or the CRC Handbook), and see how many sigfigs were reported there. That would then become the number of sigfigs to use in figuring out the final rounding, same as if it were a measurement you made yourself.hBy2Py –hBy2Py 2015-09-29 03:06:31 +00:00 Commented Sep 29, 2015 at 3:06
Thanks to the new SI, the electron volt is now exactly 1.602176634×10−19 J. Atomic weights are still empirically measured, though No Name –No Name 2023-09-05 10:59:42 +00:00 Commented Sep 5, 2023 at 10:59
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12907 | https://www.gauthmath.com/solution/1836267045002241/Determine-the-x-intercepts-of-the-graph-of-the-quadratic-y-x2-2x-8 | Solved: Determine the x-intercepts of the graph of the quadratic y=x^2+2x-8 [Math]
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Determine the x-intercepts of the graph of the quadratic y=x^2+2x-8
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The x-intercepts are -4 and 2.
Explanation
Set y=0 in the given quadratic equation y = x² + 2x - 8. This gives x² + 2x - 8 = 0.
Factor the quadratic expression: (x+4)(x-2) = 0.
Solve for x by setting each factor to zero: x+4=0 or x-2=0.
This gives the solutions x = -4 or x = 2.
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1 1号线
2 2号线
3 3号线
4 4号线
5 5号线
6 6号线
7 7号线
8 8号线
9 9号线
10 10号线
11 11号线
12 12号线
13 13号线
14 14号线
15 15号线
16 16号线
17 17号线
18 18号线
19 浦江线
20 注释
21 参考文献
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上海地铁是服务于中华人民共和国上海市的地铁系统，为上海市城市轨道交通网络的组成部分，于1993年5月28日开始运营，是中国大陸第三个投入运营的城市轨道交通系统。截至2024年11月30日，共有20条线路投入运营，510座运营车站，运营里程达837公里。
下文中，参照上海地铁在各站点的站台上设置的最新版全网图，按不同线路分别列出上海地铁的运营车站、运营中线路的在建车站及运营中线路暂未投用的车站。
1号线
[编辑]
主条目：上海轨道交通1号线
莘庄站1号线站台（大字已经拆除）
锦江乐园站
漕宝路站
人民广场站
上海火车站站
彭浦新村站
富锦路站
1993年5月28日，徐家汇站—锦江乐园站开始观光试运行。此锦江乐园站位于位于沪闵路虹漕南路附近，与锦江乐园有一定距离。试运行期间用1列车来回跑，单程12分钟。控制列车间隔采用的是最原始的电话电报闭塞。1995年4月10日，工程全线（上海火车站站—锦江乐园站）试运营，总长16.1公里，工程总概算人民币53.9亿元。1996年12月28日，南延伸段（虹梅路站—莘庄站）开始独立运营，长5.25公里，投资6.2亿元。南延伸段虹梅路站即为现锦江乐园站。1997年7月1日 莘庄站—上海火车站站贯通运营，总长20.6公里，原锦江乐园站废弃。为了提高锦江乐园的知名度，虹梅路站于2001年5月1日改名为锦江乐园站。
配合铁路上海南站改建，“地铁一号线新龙华车站改建工程”开始，这是国内地铁第一个在不中断运营条件下的车站改造项目，改建长度约2.2公里，投资7.4亿元。新设置上海南站站，取消原有新龙华站，新车站考虑将上海地铁一号线、明珠线、轻轨L1线（结构同步预留，现已不用），与上海铁路南站紧密结合，实现最佳换乘。此前漕宝路站已有预留接口，改造工程均在夜间停运后进行，并提前了拨接工作当天的末班车。施工期间，线路进行了多次拨接，于2003年9月19日和12月5日夜间分别将下、上行线拨接至临时正线。新的上海南站上海火车站站方向和莘庄站方向站台分别在2004年10月30日和12月4日投入运营。原上海南站站旧的地面站台已拆除。
北延伸段（上海火车站站—共富新村站）工程全部纳入“共和新路高架工程”内，全长12.455公里，包括地下线2.18公里、高架路10.31公里，总投资约46.2亿元。其中汶水路站—通河新村站间上海南北高架路共和新路段同体共筑，是中国第一条高架道路与轨道交通一体化线路。2004年12月28日，北延伸段投入营运。而共富新村站—富錦路站为北北延伸，建设目的是为了接入新建的富锦路停车场，故该段工程名为“1号线富锦路停车场工程”。线路长4.31公里，投资15.339亿元。2007年12月29日，北北延伸投入营运。
1号线目前共设28座车站，有4座为地面站，9座为高架站，15座为地下站，其中11座为换乘站。
| 上海轨道交通1号线车站 |
| 车站 | 最早启用日期 | 所在地 | 车站位置 | 站台类型 | 换乘线路 |
| 莘庄 | 1996年12月28日 | 闵行区 | 地面 | 侧式站台 | █ 5号线 |
| 外环路 | 地面 | 侧式站台 [註 1] | |
| 莲花路 | 地面 | 侧式站台 | |
| 锦江乐园 | 1993年5月28日（原址）1996年4月（原址） 1996年12月28日（现址） | 徐汇区 | 地面 | 侧式站台 [註 1] | |
| 上海南站 | 1993年5月28日（原址） 2004年10月30日（现址） | 地下 | 岛式站台 | █ 3号线、 █ 15号线 |
| 漕宝路 | 1993年5月28日 | 地下 | 岛式站台 | █ 12号线 |
| 上海体育馆 | 地下 | 岛式站台 | █ 4号线 |
| 徐家汇 | 地下 | 岛式站台 | █ 9号线、 █ 11号线 |
| 衡山路 | 1995年4月10日 | 地下 | 岛式站台 | |
| 常熟路 | 地下 | 岛式站台 | █ 7号线 |
| 陕西南路 | 黄浦区 | 地下 | 岛式站台 | █ 10号线、 █ 12号线 |
| 一大会址·黄陂南路 | 地下 | 岛式站台 | █ 14号线 |
| 人民广场 | 地下 | 岛式站台[註 2] | █ 2号线、 █ 8号线 |
| 新闸路 | 地下 | 岛式站台 | |
| 汉中路 | 静安区 | 地下 | 岛式站台 | █ 12号线、 █ 13号线 |
| 上海火车站 | 地下 | 岛式站台 | █ 3号线[註 3]、 █ 4号线[註 3] |
| 中山北路 | 2004年12月28日 | 地下 | 岛式站台 | |
| 延长路 | 地下 | 岛式站台 | |
| 上海马戏城 | 地下 | 岛式站台 | |
| 汶水路 | 高架 | 侧式站台 | |
| 彭浦新村 | 高架 | 侧式站台 | |
| 共康路 | 静安区/宝山区 | 高架 | 侧式站台 | |
| 通河新村 | 宝山区 | 高架 | 侧式站台 | |
| 呼兰路 | 高架 | 侧式站台 | |
| 共富新村 | 高架 | 侧式站台 | |
| 宝安公路 | 2007年12月29日 | 高架 | 侧式站台 | |
| 友谊西路 | 高架 | 侧式站台 | |
| 富锦路 | 高架 | 一岛一侧式站台 | |
2号线
[编辑]
主条目：上海轨道交通2号线
虹桥2号航站楼站
娄山关路站
静安寺站
陆家嘴站
广兰路站
浦东1号2号航站楼站
在开通初期，徐泾东站（现国家会展中心站）至广兰路站为8节编组列车，而广兰路站至浦东国际机场站（现浦东1号2号航站楼站）则为4节编组列车，乘客需要在广兰路站进行换乘。为了进一步解决广兰路站客流对冲的问题，2016年4月25日起工作日早高峰期间，部分8节编组列车驶入广兰路站后一站的唐镇站，并以唐镇站为终点站，列车清客后空驶至创新中路站存车线进行折返。2016年8月12日起，列车载客运行至唐镇站的举措覆盖工作日晚高峰。2018年12月28日起，在平峰时段部分8节编组列车载客贯通运营至浦东国际机场。2019年4月19日至2019年10月22日，2号线开行的8节编组列车贯通运营至浦东国际机场，而仅保留工作日早高峰广兰路-浦东国际机场之间4节编组列车运营。2019年10月23日起，2号线东延伸段全天运行8节编组列车，结束以往需在广兰路站换乘的历史。
2号线目前共設30座车站，有1座為为地面站，2座为高架站，其余27座为地下站，其中15座為换乘站。
| 上海轨道交通2号线车站 |
| 车站 | 最早启用日期 | 所在地 | 车站位置 | 站台类型 | 换乘线路 |
| 国家会展中心 | 2010年3月16日 | 青浦区 | 地下 | 岛式站台 | █ 17号线[註 3] |
| 虹桥火车站 | 2010年7月1日 | 闵行区 | 地下 | 与 █ 17号线组成双岛式站台 | █ 10号线、 █ 17号线 |
| 虹桥2号航站楼 | 2010年3月16日 | 地下 | 与 █ 10号线组成三岛式站台 | █ 10号线[註 4] |
| 淞虹路 | 2006年12月30日 | 长宁区 | 地下 | 岛式站台 | |
| 北新泾 | 地下 | 岛式站台 | |
| 威宁路 | 地下 | 岛式站台 | |
| 娄山关路 | 地下 | 岛式站台 | █ 15号线 |
| 中山公园 | 1999年9月20日 | 地下 | 岛式站台 | █ 3号线、 █ 4号线 |
| 江苏路 | 地下 | 岛式站台 | █ 11号线 |
| 静安寺 | 静安区 | 地下 | 岛式站台 | █ 7号线、 █ 14号线 |
| 南京西路 | 地下 | 岛式站台 | █ 12号线[註 3]、 █ 13号线[註 3] |
| 人民广场 | 黄浦区 | 地下 | 岛式站台 | █ 1号线、 █ 8号线 |
| 南京东路 | 地下 | 岛式站台 | █ 10号线 |
| 陆家嘴 | 浦东新区 | 地下 | 岛式站台 | █ 14号线 |
| 浦东南路 | 地下 | 岛式站台 | █ 14号线[註 3] |
| 世纪大道 | 地下 | 岛式站台 | █ 4号线、 █ 6号线、 █ 9号线 |
| 上海科技馆 | 地下 | 岛式站台 | |
| 世纪公园 | 地下 | 岛式站台 | |
| 龙阳路 | 地下 | 岛式站台 | █ 7号线、 █ 16号线、 █ 18号线、 █ 磁浮线[註 5] |
| 张江高科 | 2000年12月26日－2010年2月14日（原址） 2010年2月24日（现址） | 地下 | 岛式站台 | |
| 金科路 | 2010年2月24日 | 地下 | 岛式站台 | |
| 广兰路 | 地下 | 一岛一侧式站台 | |
| 唐镇 | 2010年4月8日 | 地下 | 岛式站台 | |
| 创新中路 | 地下 | 侧式站台 | |
| 华夏东路 | 地下 | 侧式站台 | |
| 川沙 | 地下 | 侧式站台 | |
| 凌空路 | 地下 | 侧式站台 | |
| 远东大道 | 高架 | 岛式站台 | |
| 海天三路 | 高架 | 岛式站台 | |
| 浦东1号2号航站楼 | 地面 | 侧式站台 | █ 磁浮线[註 5] |
3号线
[编辑]
主条目：上海轨道交通3号线
上海南站站
中山公园站
曹杨路站
宝山路站
长江南路站
江杨北路站
上海轨道交通3号线（原名“明珠线”，规划时为“明珠线”一期），是上海的第一条高架轨道交通线路，部分路段利用了原先沪杭铁路内环线（全部）和淞沪铁路（汶水东路以南）的线位。上海南站站至江湾镇站段为一期，全长24.975公里，其中高架21.516公里，地面线路3.459公里，利用原铁路线18公里，地面线路与城市道路无交叉。全线设有19座车站，其中高架车站16座，地面车站3座，平均站距1.37公里。在石龙路站出岔，设轨道交通停车场一处，在虹桥路站及东宝兴路站附近设置2座110/35kV中心变电所，11座牵引变电所，各车站均设降压变电站。工程总投资约93.78亿元，其中利用法国政府混合贷款10.8亿法郎（折合美元1.756亿元）。于1997年动工，2000年12月26日试通车。北延伸线工程起自江湾镇站，沿逸仙路高架西侧走行，跨蕴藻浜后，沿同济路西侧继续走行，过漠河路后转向西，沿富锦路南侧走行，跨北泗塘河于小游园附近下穿郊区环线富锦路至路北，继续沿富锦路西行，穿过宝钢铁路专线后继续西行并设江杨北路站后折入车辆段。本工程以高架线路为主，南起明珠一期终点站江湾镇站，北至宝山月浦镇江杨北路站，全长15.365公里，其中高架线路12.527公里，地下浅埋及敞开段1.979公里，地面正线0.859公里。于2006年12月18日通车试运营，工程概算总额为35亿元。3号线的绝大多数路段建在高架上，使得综合造价不到同样长度地下线路的三分之一，同时也让施工进度快上两到三倍。
由于3号线建设时期较早以及设计限制等历史原因，4号线在虹桥路站至宝山路站之间没有独立路轨，紫色背景站点表示该站点3号线与4号线共线。
3号线目前共設29座车站，有4座為为地面站，24座为高架站，1座为地下站，非共线段有11座為换乘站。
| 上海轨道交通3号线车站 |
| 车站 | 最早启用日期 | 所在地 | 车站位置 | 站台类型 | 换乘线路 |
| 上海南站 | 2000年12月26日 | 徐汇区 | 地面 | 岛式站台 | █ 1号线、 █ 15号线 |
| 石龙路 | 地面 | 侧式站台[註 1] | |
| 龙漕路 | 高架 | 侧式站台 | █ 12号线 |
| 漕溪路 | 高架 | 侧式站台 | |
| 宜山路 | 高架 | 侧式站台 | █ 4号线、 █ 9号线 |
| 虹桥路 | 长宁区 | 高架 | 侧式站台 | █ 4号线、 █ 10号线 |
| 延安西路 | 高架 | 侧式站台 | █ 4号线 |
| 中山公园 | 高架 | 岛式站台 | █ 2号线、 █ 4号线 |
| 金沙江路 | 普陀区 | 高架 | 侧式站台 | █ 4号线、 █ 13号线 |
| 曹杨路 | 高架 | 侧式站台 | █ 4号线、 █ 11号线、 █ 14号线[註 3] |
| 镇坪路 | 高架 | 侧式站台 | █ 4号线、 █ 7号线 |
| 中潭路 | 高架 | 侧式站台 | █ 4号线 |
| 上海火车站 | 静安区 | 地面 | 岛式站台 | █ 1号线[註 3]、 █ 4号线 |
| 宝山路 | 高架 | 侧式站台 | █ 4号线 |
| 东宝兴路 | 虹口区 | 高架 | 侧式站台 | |
| 虹口足球场 | 高架 | 侧式站台 | █ 8号线 |
| 赤峰路 | 高架 | 侧式站台 | |
| 大柏树 | 高架 | 侧式站台 | |
| 江湾镇 | 高架 | 侧式站台 | |
| 殷高西路 | 2006年12月18日 | 宝山区 | 高架 | 侧式站台 | |
| 长江南路 | 高架 | 侧式站台 | █ 18号线 |
| 淞发路 | 高架 | 侧式站台 | |
| 张华浜 | 高架 | 侧式站台 | |
| 淞滨路 | 高架 | 侧式站台 | |
| 水产路 | 高架 | 侧式站台 | |
| 宝杨路 | 高架 | 侧式站台 | |
| 友谊路 | 高架 | 侧式站台 | |
| 铁力路 | 地下 | 侧式站台 | |
| 江杨北路 | 地面 | 侧式站台 | |
4号线
[编辑]
主条目：上海轨道交通4号线
宜山路站
大连路站
世纪大道站
鲁班路站
上海体育馆站
上海轨道交通4号线（规划时称“明珠线”，通車前也被称为“明珠线”二期），是上海轨道交通现有唯一一条环状线，也是中国首条全国“工人先锋号”地铁线路。
该线路的非共线段工程北起3号线宝山路站，至虹桥路站。工程全长22.032公里，其中与3号线接轨的两端为高架线，长1.25公里，其余为地下线。总投资约126亿元人民币。工程设17座地下车站、1处停车场、2座地下式主变电站，工程管理控制中心设置于3号线东宝兴路站控制中心大楼内。其中大木桥路站至蓝村路站的「C」字路段於2005年12月31日開通，其餘部分亦於2007年12月29日啟用。此外，4号线在虹桥路站和宝山路站之间共9个站点與3号线采取城市轨道交通系统的「共线营运」方式，即两条线共用同一條轨道，虽然节约了不少投资，但同时牺牲了相互之间的运输能力也提高了自动控制的复杂程度。下表中，黄色背景站点表示该站点4号线与3号线共线。
4号线（包括3号线共线段）目前共設26座车站，有1座為为地面站，8座为高架站，其余17座为地下站，非共线段有15座為换乘站。
| 上海轨道交通4号线车站 |
| 车站 | 最早启用日期 | 所在地 | 车站位置 | 站台类型 | 换乘线路 |
| ↑↑ 环线（外圈运行） ↑↑ |
| 宜山路 | 2005年12月31日 | 徐汇区 | 地下 | 岛式站台 | █ 3号线、 █ 9号线 |
| 虹桥路 | 长宁区 | 高架 | 侧式站台 | █ 3号线、 █ 10号线 |
| 延安西路 | 高架 | 侧式站台 | █ 3号线 |
| 中山公园 | 高架 | 岛式站台 | █ 2号线、 █ 3号线 |
| 金沙江路 | 普陀区 | 高架 | 侧式站台 | █ 3号线 |
| 曹杨路 | 高架 | 侧式站台 | █ 3号线、 █ 11号线、 █ 14号线[註 3] |
| 镇坪路 | 高架 | 侧式站台 | █ 3号线、 █ 7号线 |
| 中潭路 | 高架 | 侧式站台 | █ 3号线 |
| 上海火车站 | 静安区 | 地面 | 岛式站台 | █ 1号线[註 3]、 █ 3号线 |
| 宝山路 | 高架 | 侧式站台 | █ 3号线 |
| 海伦路 | 虹口区 | 地下 | 岛式站台 | █ 10号线 |
| 临平路 | 地下 | 一岛一侧式站台 | |
| 大连路 | 虹口区/杨浦区 | 地下 | 岛式站台 | █ 12号线 |
| 杨树浦路 | 杨浦区 | 地下 | 岛式站台 | |
| 浦东大道 | 浦东新区 | 地下 | 岛式站台 | █ 14号线 |
| 世纪大道 | 地下 | 岛式站台 | █ 2号线、 █ 6号线、 █ 9号线 |
| 向城路 | 地下 | 岛式站台 | |
| 蓝村路 | 地下 | 岛式站台 | █ 6号线 |
| 塘桥 | 2007年12月29日 | 地下 | 岛式站台 | |
| 南浦大桥 | 黄浦区 | 地下 | 侧式叠式站台 | |
| 西藏南路 | 地下 | 岛式站台 | █ 8号线 |
| 鲁班路 | 地下 | 岛式站台 | |
| 大木桥路 | 2005年12月31日 | 徐汇区 | 地下 | 岛式站台 | █ 12号线 |
| 东安路 | 地下 | 岛式站台 | █ 7号线 |
| 上海体育场 | 地下 | 岛式站台 | |
| 上海体育馆 | 地下 | 岛式站台 | █ 1号线 |
| ↓↓ 环线（内圈运行） ↓↓ |
5号线
[编辑]
主条目：上海轨道交通5号线
5号线05C02型列车正进入剑川路站
东川路站
闵行开发区站
萧塘站
奉贤新城站
上海轨道交通5号线，建设时名莘闵轻轨交通线，由上海申通集团有限公司、上海闵行城市建设投资开发有限公司、上海闵行联合发展有限公司组建的上海莘闵轨道交通线发展有限公司投资。该线路最初原为规划R1线（R1a）的南段，南起现在的金山卫站，莘庄站以南沿沪杭公路、奉柘公路、江海路、南桥路、沪杭公路、沪闵路行驶，并规划沿东川路的支线，现在的5号线即为其中的沪闵路段和沪杭公路段一部分以及支线。但闵行区希望尽早建成该段线路，故以“城市高架有轨电车莘闵线工程”为名于2000年8月8日提前开工建设并独立运营。莘闵轻轨北起闵行区莘庄站，沿沪闵路往南至交通大学附近折向东川路。最初计划止于金平路站，后改为至闵行开发区站，全长17.206公里。除起点段的410米地面线外，均为高架线路。全线设11座车站和剑川路停车场1座。除莘庄站为地面站外，其余均为高架车站。工程总投资34.37亿元。线路于2002年12月28日建成贯通，并于2003年11月25日投入试运行。
5号线南延伸工程原本在2010年就应开建、2013年就应通车；但直到2014年6月30日，5号线南延伸桩基工程才开始施工；2018年3月16日，通往奉贤新城的南延伸全线轨道贯通；为配合南延伸后的六节编组新列车，配合原有四节列车的信号改造，5号线早高峰降低运能。南延伸于2018年12月30日正式开通试运营。
5号线（包括支线）目前共設19座车站，有1座為为地面站，14座为高架站，4座为地下站，仅有1座為换乘站。
| 上海轨道交通5号线车站 |
| 车站 | 最早启用日期 | 所在地 | 车站位置 | 站台类型 | 换乘线路 |
| 莘庄 | 2003年11月25日 | 闵行区 | 地面 | 侧式站台 | █ 1号线 |
| 春申路 | 高架 | 侧式站台 | |
| 银都路 | 高架 | 侧式站台 | |
| 颛桥 | 高架 | 侧式站台 | |
| 北桥 | 高架 | 侧式站台 | |
| 剑川路 | 高架 | 侧式站台 | |
| 东川路 | 高架 | 双岛式站台 | |
| 闵行开发区方向 |
| 金平路 | 2003年11月25日 | 闵行区 | 高架 | 侧式站台 | |
| 华宁路 | 高架 | 侧式站台 | |
| 文井路 | 高架 | 侧式站台 | |
| 闵行开发区 | 高架 | 侧式站台 | |
| 奉贤新城方向 |
| 江川路 | 2018年12月30日 | 闵行区 | 高架 | 侧式站台 | |
| 西渡 | 奉贤区 | 高架 | 侧式站台 | |
| 萧塘 | 高架 | 侧式站台 | |
| 奉浦大道 | 高架 | 岛式站台 | |
| 环城东路 | 地下 | 岛式站台 | |
| 望园路 | 地下 | 岛式站台 | |
| 金海湖 | 地下 | 岛式站台 | |
| 奉贤新城 | 地下 | 岛式站台 | |
6号线
[编辑]
主条目：上海轨道交通6号线
外高桥保税区北站
云山路站
世纪大道站
上海儿童医学中心站
上南路站
6号线目前共設28座车站，有9座为高架站，其余19座为地下站，其中9座為换乘站。
| 上海轨道交通6号线车站 |
| 车站 | 最早启用日期 | 所在地 | 车站位置 | 站台类型 | 换乘线路 |
| 港城路 | 2007年12月29日 | 浦东新区 | 高架 | 与 █ 10号线组成一岛两侧式站台 | █ 10号线 |
| 外高桥保税区北 | 高架 | 侧式站台 | |
| 航津路 | 高架 | 侧式站台 | |
| 外高桥保税区南 | 高架 | 侧式站台 | |
| 洲海路 | 高架 | 侧式站台 | |
| 五洲大道 | 高架 | 侧式站台 | |
| 东靖路 | 高架 | 侧式站台 | |
| 巨峰路 | 高架 | 岛式站台 | █ 12号线 |
| 五莲路 | 高架 | 侧式站台 | |
| 博兴路 | 地下 | 侧式站台 | |
| 金桥路 | 地下 | 侧式站台 | |
| 云山路 | 地下 | 侧式站台 | █ 14号线 |
| 德平路 | 地下 | 侧式站台 | |
| 北洋泾路 | 地下 | 侧式站台 | |
| 民生路 | 地下 | 侧式站台 | █ 18号线 |
| 源深体育中心 | 地下 | 侧式站台 | |
| 世纪大道 | 地下 | 侧式站台 | █ 2号线、 █ 4号线、 █ 9号线 |
| 浦电路 | 地下 | 岛式站台 | |
| 蓝村路 | 地下 | 岛式站台 | █ 4号线 |
| 上海儿童医学中心 | 地下 | 岛式站台 | |
| 临沂新村 | 地下 | 岛式站台 | |
| 高科西路 | 地下 | 岛式站台 | █ 7号线 |
| 东明路 | 地下 | 岛式站台 | █ 13号线 |
| 高青路 | 地下 | 岛式站台 | |
| 华夏西路 | 地下 | 侧式站台 | |
| 上南路 | 地下 | 侧式站台 | |
| 灵岩南路 | 地下 | 侧式站台 | |
| 东方体育中心 | 2011年4月12日 | 地下 | 与 █ 11号线组成双岛式站台 | █ 8号线、 █ 11号线 |
7号线
[编辑]
主条目：上海轨道交通7号线
美兰湖站
上海大学站
镇坪路站
常熟路站
杨高南路站
花木路站
7号线目前共設33座车站，有2座为高架站，其余31座为地下站，其中12座為换乘站。
| 上海轨道交通7号线车站 |
| 车站 | 最早启用日期 | 所在地 | 车站位置 | 站台类型 | 换乘线路 |
| 美兰湖 | 2010年12月28日 | 宝山区 | 高架 | 侧式站台 | |
| 罗南新村 | 高架 | 侧式站台 | |
| 潘广路 | 2011年6月30日 | 地下 | 岛式站台 | |
| 刘行 | 地下 | 岛式站台 | |
| 顾村公园 | 2010年12月28日 | 地下 | 岛式站台 | █ 15号线 |
| 祁华路 | 2014年7月22日 | 地下 | 岛式站台 | |
| 上海大学 | 2009年12月5日 | 地下 | 岛式站台 | |
| 南陈路 | 地下 | 岛式站台 | |
| 上大路 | 地下 | 一岛一侧式站台 | |
| 场中路 | 地下 | 岛式站台 | |
| 大场镇 | 地下 | 岛式站台 | |
| 行知路 | 地下 | 岛式站台 | |
| 大华三路 | 地下 | 岛式站台 | |
| 新村路 | 普陀区 | 地下 | 岛式站台 | |
| 岚皋路 | 地下 | 岛式站台 | |
| 镇坪路 | 地下 | 岛式站台 | █ 3号线、 █ 4号线 |
| 长寿路 | 地下 | 岛式站台 | █ 13号线 |
| 昌平路 | 静安区 | 地下 | 侧式站台 | |
| 静安寺 | 地下 | 岛式站台 | █ 2号线、 █ 14号线 |
| 常熟路 | 徐汇区 | 地下 | 岛式站台 | █ 1号线 |
| 肇嘉浜路 | 地下 | 岛式站台 | █ 9号线 |
| 东安路 | 地下 | 岛式站台 | █ 4号线 |
| 龙华中路 | 地下 | 岛式站台 | █ 12号线 |
| 后滩 | 2010年4月20日 | 浦东新区 | 地下 | 岛式站台 | |
| 长清路 | 2009年12月5日 | 地下 | 岛式站台 | █ 13号线[註 3] |
| 耀华路 | 地下 | 岛式站台 | █ 8号线 |
| 云台路 | 地下 | 岛式站台 | |
| 高科西路 | 地下 | 岛式站台 | █ 6号线 |
| 杨高南路 | 地下 | 双岛式站台 | |
| 锦绣路 | 地下 | 岛式站台 | |
| 芳华路 | 地下 | 岛式站台 | |
| 龙阳路 | 地下 | 岛式站台 | █ 2号线、 █ 16号线、 █ 18号线、 █ 磁浮线[註 5] |
| 花木路 | 地下 | 岛式站台 | |
8号线
[编辑]
主条目：上海轨道交通8号线
市光路站
江浦路站
老西门站
中华艺术宫站
东方体育中心站
在沈杜公路站折返线开出的8号线列车
8号线目前共設30座车站，有4座为高架站，其余26座为地下站，其中13座為换乘站。
| 上海轨道交通8号线车站 |
| 车站 | 最早启用日期 | 所在地 | 车站位置 | 站台类型 | 换乘线路 |
| 市光路 | 2007年12月29日 | 杨浦区 | 地下 | 侧式站台 | |
| 嫩江路 | 地下 | 侧式站台 | |
| 翔殷路 | 地下 | 侧式站台 | |
| 黄兴公园 | 地下 | 侧式站台 | |
| 延吉中路 | 地下 | 岛式站台 | |
| 黄兴路 | 地下 | 岛式站台 | |
| 江浦路 | 地下 | 岛式站台 | █ 18号线 |
| 鞍山新村 | 地下 | 岛式站台 | |
| 四平路 | 杨浦区/虹口区 | 地下 | 岛式站台 | █ 10号线 |
| 曲阳路 | 虹口区 | 地下 | 侧式站台 | |
| 虹口足球场 | 地下 | 岛式站台 | █ 3号线 |
| 西藏北路 | 静安区 | 地下 | 岛式站台 | |
| 中兴路 | 地下 | 岛式站台 | |
| 曲阜路 | 地下 | 岛式站台 | █ 12号线 |
| 人民广场 | 黄浦区 | 地下 | 一岛一侧式站台 | █ 1号线、 █ 2号线 |
| 大世界 | 地下 | 岛式站台 | █ 14号线 |
| 老西门 | 地下 | 岛式站台 | █ 10号线 |
| 陆家浜路 | 地下 | 岛式站台 | █ 9号线 |
| 西藏南路 | 地下 | 岛式站台 | █ 4号线 |
| 中华艺术宫 | 2012年9月28日 | 浦东新区 | 地下 | 岛式站台 | |
| 耀华路 | 2007年12月29日 | 地下 | 侧式站台 | █ 7号线 |
| 成山路 | 2009年7月5日 | 地下 | 岛式站台 | █ 13号线 |
| 杨思 | 地下 | 岛式站台 | |
| 东方体育中心 | 2011年4月12日 | 地下 | 岛式站台 | █ 6号线、 █ 11号线 |
| 凌兆新村 | 2009年7月5日 | 地下 | 岛式站台 | |
| 芦恒路 | 闵行区 | 地下 | 岛式站台 | |
| 浦江镇 | 高架 | 岛式站台 | |
| 江月路 | 高架 | 岛式站台 | |
| 联航路 | 高架 | 岛式站台 | |
| 沈杜公路 | 高架 | 岛式站台 | █ 浦江线 |
9号线
[编辑]
主条目：上海轨道交通9号线
9号线使用的AC04型列车，曾用于1号线运转
上海松江站站
徐家汇站
世纪大道站
蓝天路站
曹路站
上海轨道交通9号线分为四期建设。一期工程全长30.687公里，其中地下线14.782公里，敞开段0.638公里，地面线0.518公里，高架线14.748公里。全线设13座车站，地下站9座，高架站4座，平均站间距2.54公里其中松江新城站至桂林路站段全长29.14公里，共设12站，率先于2007年12月29日开通，剩余的宜山路站于2008年12月28日开通。
二期工程起于一期宜山路站，止于杨高中路站，全长14.5公里，均为地下线，沿线共设10座地下车站，1座地下主变电站，投资89.4亿元。线路于2005年12月开工，其中宜山路站至世纪大道站于2009年12月31日开通，开通时采用非高峰运营，至2010年2月20日覆盖早晚高峰。世纪大道站至杨高中路站于2010年4月7日开通，同日，徐家汇站站内换乘开通。
三期工程包括南延伸段和东延伸段两部分。南延伸段原名松江老城区公共交通配套工程，位于松江区，线路起自沪杭客专的上海松江站，终点为9号线一期工程的起点松江新城站，线路全长5.372公里，全部为地下线，投资28亿元。于2009年12月30日开工，已经于2012年12月30日开通试运营。
三期东延伸工程与9号线二期工程杨高中路站接轨，终点止于曹路站，主要位于浦东新区的金桥、曹路镇，线路起自二期工程终点杨高中路站后存车线东端，线路沿杨高中路向东前行，穿过罗山立交、金桥立交至金海路路口，转向金海路继续向东前行，穿过A20公路、浦东运河后，止于龚华路路口。正线全长13.828公里，设车站9座，均为地下线路，投资127.13亿元。本工程与9号线一、二期工程及三期南延伸共用控制中心，位于宜山路与虹梅路附近。本工程设金桥停车场一座，由金吉路站引入（与12号、14号线共址），与12号线共用一条试车线。本工程与既有线接口，曹路站预留向东延伸的条件。工程于2013年底开工建设，2017年12月30日开通试运营。
9号线目前共設35座车站，有4座为高架站，其余31座为地下站，其中11座為换乘站。
| 上海轨道交通9号线车站 |
| 车站 | 最早启用日期 | 所在地 | 车站位置 | 站台类型 | 换乘线路 |
| 上海松江站 | 2012年12月30日 | 松江区 | 地下 | 岛式站台 | |
| 醉白池 | 地下 | 岛式站台 | |
| 松江体育中心 | 地下 | 岛式站台 | |
| 松江新城 | 2007年12月29日 | 地下 | 岛式站台 | |
| 松江大学城 | 高架 | 侧式站台 | |
| 洞泾 | 高架 | 侧式站台 | |
| 佘山 | 高架 | 岛式站台 | |
| 泗泾 | 高架 | 侧式站台 | |
| 九亭 | 地下 | 岛式站台 | |
| 中春路 | 闵行区 | 地下 | 一岛一侧式站台 | |
| 七宝 | 地下 | 岛式站台 | |
| 星中路 | 地下 | 岛式站台 | |
| 合川路 | 地下 | 岛式站台 | |
| 漕河泾开发区 | 徐汇区 | 地下 | 岛式站台 | |
| 桂林路 | 地下 | 一岛一侧式站台 | █ 15号线 |
| 宜山路 | 2008年12月28日 | 地下 | 岛式站台 | █ 3号线、 █ 4号线 |
| 徐家汇 | 2009年12月31日 | 地下 | 岛式站台 | █ 1号线、 █ 11号线 |
| 肇嘉浜路 | 地下 | 岛式站台 | █ 7号线 |
| 嘉善路 | 地下 | 岛式站台 | █ 12号线 |
| 打浦桥 | 黄浦区 | 地下 | 岛式站台 | |
| 马当路 | 地下 | 侧式站台 | █ 13号线 |
| 陆家浜路 | 地下 | 岛式站台 | █ 8号线 |
| 小南门 | 地下 | 岛式站台 | |
| 商城路 | 浦东新区 | 地下 | 岛式站台 | |
| 世纪大道 | 地下 | 岛式站台 | █ 2号线、 █ 4号线、 █ 6号线 |
| 杨高中路 | 2010年4月7日 | 地下 | 岛式站台 | █ 18号线 |
| 芳甸路 | 2017年12月30日 | 地下 | 岛式站台 | |
| 蓝天路 | 地下 | 岛式站台 | █ 14号线 |
| 台儿庄路 | 地下 | 岛式站台 | |
| 金桥 | 地下 | 岛式站台 | |
| 金吉路 | 地下 | 岛式站台 | |
| 金海路 | 地下 | 岛式站台 | █ 12号线 |
| 顾唐路 | 地下 | 岛式站台 | |
| 民雷路 | 地下 | 岛式站台 | |
| 曹路 | 地下 | 岛式站台 | |
10号线
[编辑]
主条目：上海轨道交通10号线
港城路站
新江湾城站
海伦路站
豫园站
交通大学站
龙柏新村站
虹桥火车站站
上海轨道交通10号线线路总长46.301km，设站37座，其中已经投入运营的一期工程共设31站，长36.221km；在建的二期工程共设6站，长10.08km。一期由主线和支线组成，途经闵行、长宁、徐汇、黄浦、静安、虹口和杨浦7个行政区。主线起于闵行区虹桥火车站站，经虹桥机场、虹桥路、淮海西路、复兴中路、复兴东路、河南路、武进路、四平路、淞沪路，止于杨浦区新江湾城站；支线起于闵行区航中路站，沿吴中路、虹井路、虹桥路，与主线在长宁区龙溪路站汇合。线路全长36.221公里，其中主线长31.254公里，支线长4.967公里，全线均为地下线。沿线共设置地下车站31座（其中主线28座、支线3座），主变电所2座，牵引变电所13座，停车场1处，主控制中心1座，主控制中心的信息楼与吴中路停车场的停车库合设，工程总投资237.84亿元。一期主线龙溪路站以东及支线部分于2010年4月10日先期开通试运营，主线龙溪路站以西于2010年11月30日开通。
10号线二期起于一期已建成的新江湾城站，沿淞沪路跨黄浦江后经过港城路，止于浦东区基隆路站，长10.09公里，其中地下线3.383公里，过渡段0.337公里，高架段6.36公里，共设6站。在港城路北侧、江东路东侧新设与6号线共用的停车场1座，在港城路停车场内设1座主变电所，工程总投资73.85亿元。
二期工程已于2015年1月8日开工，于2020年12月26日起开通试运营。
10号线（包括支线）目前共設37座车站，有5座为高架站，其余32座为地下站，其中14座為换乘站。
| 上海轨道交通10号线车站[註 6] |
| 车站 | 最早启用日期 | 所在地 | 车站位置 | 站台类型 | 换乘线路 |
| 基隆路 | 2020年12月26日 | 浦东新区 | 高架 | 侧式站台 | |
| 港城路 | 高架 | 与 █ 6号线组成一岛两侧式站台 | █ 6号线 |
| 高桥 | 高架 | 侧式站台 | |
| 高桥西 | 高架 | 侧式站台 | |
| 双江路 | 高架 | 侧式站台 | |
| 国帆路 | 杨浦区 | 地下 | 岛式站台 | |
| 新江湾城 | 2010年4月10日 | 地下 | 双岛式站台 | |
| 殷高东路 | 地下 | 岛式站台 | |
| 三门路 | 地下 | 岛式站台 | |
| 江湾体育场 | 地下 | 岛式站台 | |
| 五角场 | 地下 | 岛式站台 | |
| 国权路 | 地下 | 岛式站台 | █ 18号线 |
| 同济大学 | 地下 | 岛式站台 | |
| 四平路 | 杨浦区/虹口区 | 地下 | 侧式站台 | █ 8号线 |
| 邮电新村 | 虹口区 | 地下 | 侧式站台 | |
| 海伦路 | 地下 | 岛式站台 | █ 4号线 |
| 四川北路 | 地下 | 侧式站台 | |
| 天潼路 | 虹口区/静安区 | 地下 | 岛式站台 | █ 12号线 |
| 南京东路 | 黄浦区 | 地下 | 岛式站台 | █ 2号线 |
| 豫园 | 地下 | 岛式站台 | █ 14号线 |
| 老西门 | 地下 | 岛式站台 | █ 8号线 |
| 一大会址·新天地 | 地下 | 侧式站台 | █ 13号线 |
| 陕西南路 | 地下 | 岛式站台 | █ 1号线、 █ 12号线 |
| 上海图书馆 | 徐汇区 | 地下 | 岛式站台 | |
| 交通大学 | 地下 | 岛式站台 | █ 11号线 |
| 虹桥路 | 长宁区 | 地下 | 岛式站台 | █ 3号线、 █ 4号线 |
| 宋园路 | 地下 | 侧式站台 | |
| 伊犁路 | 地下 | 岛式站台 | |
| 水城路 | 地下 | 岛式站台 | |
| 龙溪路 | 地下 | 一岛一侧式站台 | |
| 航中路方向 |
| 龙柏新村 | 2010年4月10日 | 闵行区 | 地下 | 岛式站台 | |
| 紫藤路 | 地下 | 岛式站台 | |
| 航中路 | 地下 | 岛式站台 | |
| 虹桥火车站方向 |
| 上海动物园 | 2010年11月30日 | 长宁区 | 地下 | 岛式站台 | |
| 虹桥1号航站楼 | 地下 | 岛式站台[註 7] | |
| 虹桥2号航站楼 | 闵行区 | 地下 | 与 █ 2号线组成三岛式站台[註 8] | █ 2号线[註 4] |
| 虹桥火车站 | 地下 | 岛式站台[註 9] | █ 2号线、 █ 17号线 |
11号线
[编辑]
主条目：上海轨道交通11号线
迪士尼站
秀沿路站
康恒路站
东方体育中心站
徐家汇站
南翔站
嘉定北站
昌吉东路站
花桥站
上海轨道交通11号线一期工程全长46.008公里，其中地下线长21.676公里，高架线长22.283公里，地面线长2.049公里。全线共设9座高架车站（其中1座预留）、11座地下车站和1座地面车站，在主线起点附近设嘉定辅助停车场1座，在支线赛车场附近设赛车场车辆段1处，另有控制中心1处，主变电站2处，工程总投资189.5亿元。工程包括主线嘉定北站—华山路中间风井、支线嘉定新城站—安亭站，位于上海市嘉定区、普陀区和长宁区。江苏路站至嘉定北站于2009年12月31日通车试运营，线路长度33.2公里。支线除昌吉东路站外于2010年3月29日开通。
二期工程全长22.886公里，工程总投资129.65亿元。江苏路站至罗山路站于2013年8月31日通车，单线里程达到65公里，成为国内单线里程最长的地铁线路。
安亭站至花桥站段，全长6公里，均为高架线；其中，上海市境内段约450米，江苏省境内段约5.55公里，投资18.49亿元。于2013年10月16日开通，共设3座高架车站，成为全国首条跨省轨道交通，单线里程达到72公里。
罗山路站至迪士尼站段，原名浦东中部地区公共交通配套工程，全长约9.15公里，其中高架线约7.39公里，敞开段约0.3公里，地下线约1.46公里，设站3座，投资43.71亿元。2015年12月19日开通秀沿路至康新公路区段。2016年4月26日10时起，迪士尼实现载客试运营。
陈翔公路站于2020年8月25日开通。康恒路站于2024年9月28日开通。
11号线现时共設40座车站，有1座為为地面站，15座为高架站，其余24座为地下站，其中12座為换乘站。
| 上海轨道交通11号线车站 |
| 车站 | 最早启用日期 | 所在地 | 车站位置 | 站台类型 | 换乘线路 |
| 迪士尼 | 2016年4月26日 | 浦东新区 | 地下 | 岛式站台 | |
| 康新公路 | 2015年12月19日 | 高架 | 侧式站台 | |
| 秀沿路 | 高架 | 侧式站台 | |
| 罗山路 | 2013年8月31日 | 高架 | 岛式叠式站台 | █ 16号线 |
| 御桥 | 地下 | 岛式站台 | █ 18号线 |
| 康恒路 | 2024年9月28日 | 地下 | 岛式站台 | |
| 浦三路 | 2013年8月31日 | 地下 | 侧式站台 | |
| 三林东 | 地下 | 岛式站台 | |
| 三林 | 地下 | 双岛式站台 | |
| 东方体育中心 | 地下 | 与 █ 6号线组成双岛式站台 | █ 6号线、 █ 8号线 |
| 龙耀路 | 徐汇区 | 地下 | 岛式站台 | |
| 云锦路 | 地下 | 侧式站台 | |
| 龙华 | 地下 | 岛式站台 | █ 12号线 |
| 上海游泳馆 | 地下 | 岛式站台 | |
| 徐家汇 | 地下 | 岛式站台 | █ 1号线、 █ 9号线 |
| 交通大学 | 地下 | 岛式站台 | █ 10号线 |
| 江苏路 | 2009年12月31日 | 长宁区 | 地下 | 岛式站台 | █ 2号线 |
| 隆德路 | 普陀区 | 地下 | 岛式站台 | █ 13号线 |
| 曹杨路 | 地下 | 侧式叠式站台 | █ 3号线、 █ 4号线、 █ 14号线[註 3] |
| 枫桥路 | 地下 | 岛式站台 | |
| 真如 | 地下 | 一岛一侧式站台 | █ 14号线 |
| 上海西站 | 地下 | 岛式站台 | █ 15号线 |
| 李子园 | 地下 | 岛式站台 | |
| 祁连山路 | 地下 | 岛式站台 | |
| 武威路 | 地下 | 岛式站台 | |
| 桃浦新村 | 地下 | 岛式站台 | |
| 南翔 | 嘉定区 | 高架 | 一岛一侧式站台 | |
| 陈翔公路 | 2020年8月25日 | 高架 | 侧式站台 | |
| 马陆 | 2009年12月31日 | 高架 | 侧式站台 | |
| 嘉定新城 | 高架 | 一岛一侧式站台 | |
| 嘉定北方向 |
| 白银路 | 2009年12月31日 | 嘉定区 | 高架 | 侧式站台 | |
| 嘉定西 | 高架 | 侧式站台 | |
| 嘉定北 | 高架 | 侧式站台[註 10] | |
| 花桥方向 |
| 上海赛车场 | 2010年3月29日 | 嘉定区 | 地下 | 侧式站台 | |
| 昌吉东路 | 2011年4月26日 | 地面 | 侧式站台 | |
| 上海汽车城 | 2010年3月29日 | 高架 | 侧式站台 | |
| 安亭 | 高架 | 岛式站台 | |
| 兆丰路 | 2013年10月16日 | 江苏省昆山市 | 高架 | 岛式站台 | |
| 光明路 | 高架 | 侧式站台 | |
| 花桥 | 高架 | 侧式站台 | █ 苏州地铁11号线 |
12号线
[编辑]
主条目：上海轨道交通12号线
金海路站
大连路站
汉中路站
陕西南路站
龙华站
桂林公园站
七莘路站
12号线全線由闵行区莘庄镇七莘路站，经徐汇区、静安区、黄浦区、虹口区、杨浦区，下穿黄浦江最终到达浦东新区金桥金海路站。线路全长40.417公里，均为地下线，工程总投资为233.99亿元。工程于2008年底动工，2013年12月29日开通天潼路站—金海路站段，2014年5月10日曲阜路站开门迎客，2015年12月19日开通汉中路站—七莘路站段，至此全线通车。12号线是上海地铁系统中可换乘线路较多的线路，全线设16个换乘站点，除了5号线，14号线，16号线，17号线及浦江线之外，其余可换乘大部分的线路，其中桂林公园站—天潼路站区间皆为换乘站（其中南京西路站为虚拟换乘）。
12号线目前共設32座车站，全线均为地下站，其中16座為换乘站。
| 上海轨道交通12号线车站 |
| 车站 | 最早启用日期 | 所在地 | 车站位置 | 站台类型 | 换乘线路 |
| 金海路 | 2013年12月29日 | 浦东新区 | 地下 | 侧式站台 | █ 9号线 |
| 申江路 | 地下 | 岛式站台 | |
| 金京路 | 地下 | 岛式站台 | |
| 杨高北路 | 地下 | 岛式站台 | |
| 巨峰路 | 地下 | 岛式站台 | █ 6号线 |
| 东陆路 | 地下 | 岛式站台 | |
| 复兴岛 | 杨浦区 | 地下 | 岛式站台 | |
| 爱国路 | 地下 | 岛式站台 | |
| 隆昌路 | 地下 | 侧式站台 | |
| 宁国路 | 地下 | 岛式站台 | |
| 江浦公园 | 地下 | 岛式站台 | █ 18号线 |
| 大连路 | 地下 | 岛式站台 | █ 4号线 |
| 提篮桥 | 虹口区 | 地下 | 岛式站台 | |
| 国际客运中心 | 地下 | 侧式站台 | |
| 天潼路 | 地下 | 侧式站台 | █ 10号线 |
| 曲阜路 | 2014年5月10日 | 静安区 | 地下 | 岛式站台 | █ 8号线 |
| 汉中路 | 2015年12月19日 | 地下 | 岛式站台 | █ 1号线、 █ 13号线 |
| 南京西路 | 地下 | 岛式站台 | █ 2号线[註 3]、 █ 13号线[註 3] |
| 陕西南路 | 黄浦区 | 地下 | 岛式站台 | █ 1号线、 █ 10号线 |
| 嘉善路 | 徐汇区 | 地下 | 岛式站台 | █ 9号线 |
| 大木桥路 | 地下 | 岛式站台 | █ 4号线 |
| 龙华中路 | 地下 | 侧式站台 | █ 7号线 |
| 龙华 | 地下 | 岛式站台 | █ 11号线 |
| 龙漕路 | 地下 | 岛式站台 | █ 3号线 |
| 漕宝路 | 地下 | 岛式站台 | █ 1号线 |
| 桂林公园 | 地下 | 岛式站台 | █ 15号线 |
| 虹漕路 | 地下 | 岛式站台 | |
| 虹梅路 | 地下 | 岛式站台 | |
| 东兰路 | 闵行区 | 地下 | 岛式站台 | |
| 顾戴路 | 地下 | 岛式站台 | |
| 虹莘路 | 地下 | 岛式站台 | |
| 七莘路 | 地下 | 岛式站台 | |
13号线
[编辑]
主条目：上海轨道交通13号线
金运路站
金沙江路站
长寿路站
淮海中路站
一大会址·新天地站
世博大道站
长清路站
张江路站
为了给上海世博会提供过江轨道交通线路，13号线世博专线即“世博园区专用交通联络线工程”先行建设，线路长度4.887公里，共设新天地站至长清路站五站四区间。其中三个车站（马当路站、卢浦大桥站、世博大道站）属于2010年世界博览会的「世博专线」（或称「世博过江段」），马当路站至世博大道站段于2010年4月20日投入运营，作为世博园内专用轨道交通。由于该段全线位于世博园区内，因此世博会期间此段线路不纳入轨道交通运营网络，而是在经过安检和检票后免费乘坐。（世博专线马当路站实际作为世博园区的一个出入口，编号为9号门，而9号线马当路站位于园区外，两线需通过地面出站换乘）。世博专线通过位于长清路站的7-13联络线，借用9号线的列车，采取双向单线运行的模式。该工程采用BT模式建设，由上海建工（集团）总公司于2007年3月以13.38亿元人民币的投得建设权。工程竣工並试运营一年后，由招标人回购。
一期工程自金运路站至南京西路站，线路全长16.468公里，设14座车站，设北翟路车辆段1座，设隆德路开闭所及控制中心与11号线共享。工程投资124.17亿元。一期工程于2008年12月28日正式开工，2012年12月30日开通金运路站至金沙江路站段（祁连山南路站和大渡河路站未开通），2014年12月28日开通隆德路站至长寿路站段，2015年12月19日开通一期工程的剩余区段。由于一期工程至南京西路站截止，而世博专线自新天地站开始，中间仍存在一站两区间（即淮海中路站及其相邻区间）未列入。因此于2009年着手开始世博园区专用交通联络线工程调整（简称“世调”）的建设。工程也包括马当路站至长清路站的改造工程（简称“世改”），含世博专线在内总投资45.21亿元。淮海中路站于2009年11月20日开工，相邻区间分别从南京西路站及新天地站推进，形成“四龙会师”的壮观景象。
二期工程自长清路站至华夏中路站，全长10.139公里，共设7站，与三期工程合计投资129.32亿元。该段线路与一期工程于2007年一并环评获批，但由于新增北蔡站，同时调整陈春东路站后，需再次向国家环保局提出变更申请。在2011年先后对变更进行了两次环境影响评价公示。2013年12月31日，二期工程陈春路站（原绿川新村站）正式开工，当初预计2017年底便可开通运营，但开工不久便进入停工状态，原因是其它车站并不能陆续动工。2014年底，其余车站陆续开工。三期工程自中科路站至张江路站，全长5.273公里，共设3站。虽然该段规划早已存在，但未于2007年与一、二期工程一并环评，故在2013-2014年补充完成了该段的环境影响评价。报告显示，工程计划2014年7月开工，2017年12月建成通车。二期、三期于2018年12月30日开通。
13号线目前正在建设西延伸与东延伸。西延伸早在2010年虹桥商务区总体规划方案中，就已存在了相关规划并画入控规中。2016年新一轮规划编制时，13号线确定西延伸至诸光路站，远期将把诸光路段拆分给25号线，已列入2017-2025近期建设规划内。
13号线目前共設31座车站，全线均为地下站，其中14座為换乘站。
| 上海轨道交通13号线车站 |
| 车站 | 最早启用日期 | 所在地 | 车站位置 | 站台类型 | 换乘线路 |
| 金运路 | 2012年12月30日 | 嘉定区 | 地下 | 岛式站台 | |
| 金沙江西路 | 地下 | 岛式站台 | |
| 丰庄 | 地下 | 岛式站台 | |
| 祁连山南路 | 2013年6月15日 | 普陀区 | 地下 | 岛式站台 | |
| 真北路 | 2012年12月30日 | 地下 | 岛式站台 | |
| 大渡河路 | 2014年11月1日 | 地下 | 侧式站台 | █ 15号线 |
| 金沙江路 | 2012年12月30日 | 地下 | 岛式站台 | █ 3号线、 █ 4号线 |
| 隆德路 | 2014年12月28日 | 地下 | 岛式站台 | █ 11号线 |
| 武宁路 | 地下 | 岛式站台 | █ 14号线 |
| 长寿路 | 地下 | 岛式站台 | █ 7号线 |
| 江宁路 | 2015年12月19日 | 地下 | 岛式站台 | |
| 汉中路 | 静安区 | 地下 | 岛式站台 | █ 1号线、 █ 12号线 |
| 自然博物馆 | 地下 | 岛式站台 | |
| 南京西路 | 地下 | 侧式站台 | █ 2号线[註 3]、 █ 12号线[註 3] |
| 淮海中路 | 黄浦区 | 地下 | 岛式站台 | |
| 一大会址·新天地 | 地下 | 岛式站台 | █ 10号线 |
| 马当路 | 2010年4月20日－2010年11月2日（世博专线）2015年12月19日（13号线） | 地下 | 岛式站台 | █ 9号线 |
| 世博会博物馆 | 地下 | 岛式站台 | |
| 世博大道 | 浦东新区 | 地下 | 一岛一侧式站台 | |
| 长清路 | 2018年12月30日 | 地下 | 岛式站台 | █ 7号线[註 3] |
| 成山路 | 地下 | 岛式站台 | █ 8号线 |
| 东明路 | 地下 | 岛式站台 | █ 6号线 |
| 华鹏路 | 地下 | 岛式站台 | |
| 下南路 | 地下 | 岛式站台 | |
| 北蔡 | 地下 | 岛式站台 | |
| 陈春路 | 地下 | 岛式站台 | |
| 莲溪路 | 地下 | 岛式站台 | █ 18号线 |
| 华夏中路 | 地下 | 岛式站台 | █ 16号线 |
| 中科路 | 地下 | 岛式站台 | |
| 学林路 | 地下 | 岛式站台 | |
| 张江路 | 地下 | 岛式站台 | |
14号线
[编辑]
主条目：上海轨道交通14号线
封浜站
曹杨路站
静安寺站
豫园站
陆家嘴站
昌邑路站
上海轨道交通14号线全线除龙居路站外的所有30座车站于2021年12月30日正式开通运营。
14号线目前现时共設30座已开通的车站，全线均为地下站，其中14座為换乘站。
| 上海轨道交通14号线车站 |
| 车站 | 最早启用日期 | 所在地 | 车站位置 | 站台类型 | 换乘线路 |
| 封浜 | 2021年12月30日 | 嘉定区 | 地下 | 岛式站台 | |
| 乐秀路 | 地下 | 岛式站台 | |
| 临洮路 | 地下 | 岛式站台 | |
| 嘉怡路 | 地下 | 岛式站台 | |
| 定边路 | 地下 | 岛式站台 | |
| 真新新村 | 地下 | 岛式站台 | |
| 真光路 | 普陀区 | 地下 | 岛式站台 | |
| 铜川路 | 地下 | 岛式站台 | █ 15号线 |
| 真如 | 地下 | 岛式站台 | █ 11号线 |
| 中宁路 | 地下 | 岛式站台 | |
| 曹杨路 | 地下 | 一岛一侧式站台 | █ 3号线[註 3]、 █ 4号线[註 3]、 █ 11号线[註 3] |
| 武宁路 | 地下 | 岛式站台 | █ 13号线 |
| 武定路 | 静安区 | 地下 | 岛式站台 | |
| 静安寺 | 地下 | 岛式站台 | █ 2号线、 █ 7号线 |
| 一大会址·黄陂南路 | 黄浦区 | 地下 | 侧式站台 | █ 1号线 |
| 大世界 | 地下 | 岛式站台 | █ 8号线 |
| 豫园 | 地下 | 岛式站台 | █ 10号线 |
| 陆家嘴 | 浦东新区 | 地下 | 岛式站台 | █ 2号线 |
| 浦东南路 | 地下 | 岛式站台 | █ 2号线[註 3] |
| 浦东大道 | 地下 | 岛式站台 | █ 4号线 |
| 源深路 | 地下 | 侧式站台 | |
| 昌邑路 | 地下 | 岛式站台 | █ 18号线 |
| 歇浦路 | 地下 | 岛式站台 | |
| 龙居路 | 尚未启用 | 地下 | 岛式站台 | |
| 云山路 | 2021年12月30日 | 地下 | 岛式站台 | █ 6号线 |
| 蓝天路 | 地下 | 岛式站台 | █ 9号线 |
| 黄杨路 | 地下 | 岛式站台 | |
| 云顺路 | 地下 | 岛式站台 | |
| 浦东足球场 | 地下 | 岛式站台 | |
| 金粤路 | 地下 | 岛式站台 | |
| 桂桥路 | 地下 | 岛式站台 | |
15号线
[编辑]
主条目：上海轨道交通15号线
长风公园站
吴中路站
桂林路站
上海南站站
上海轨道交通15号线全线除桂林路站外的所有29座车站于2021年1月23日正式开通运营，桂林路站則於2021年5月29日开始不上下客的停站测试，6月27日开通运营。
15号线目前共設30座车站，全线均为地下站，其中8座為换乘站。
| 上海轨道交通15号线车站 |
| 车站 | 最早启用日期 | 所在地 | 车站位置 | 站台类型 | 换乘线路 |
| 顾村公园 | 2021年1月23日 | 宝山区 | 地下 | 岛式站台 | █ 7号线 |
| 锦秋路 | 地下 | 岛式站台 | |
| 丰翔路 | 地下 | 岛式站台 | |
| 南大路 | 地下 | 岛式站台 | |
| 祁安路 | 普陀区 | 地下 | 岛式站台 | |
| 古浪路 | 地下 | 一岛一侧式站台 | |
| 武威东路 | 地下 | 岛式站台 | |
| 上海西站 | 地下 | 岛式站台 | █ 11号线 |
| 铜川路 | 地下 | 岛式站台 | █ 14号线 |
| 梅岭北路 | 地下 | 岛式站台 | |
| 大渡河路 | 地下 | 岛式站台 | █ 13号线 |
| 长风公园 | 地下 | 一岛一侧式站台 | |
| 娄山关路 | 长宁区 | 地下 | 岛式站台 | █ 2号线 |
| 红宝石路 | 地下 | 岛式站台 | |
| 姚虹路 | 地下 | 岛式站台 | |
| 吴中路 | 徐汇区/闵行区 | 地下 | 岛式站台 | |
| 桂林路 | 2021年6月27日 | 徐汇区 | 地下 | 岛式站台 | █ 9号线 |
| 桂林公园 | 2021年1月23日 | 地下 | 岛式站台 | █ 12号线 |
| 上海南站 | 地下 | 岛式站台 | █ 1号线、 █ 3号线 |
| 华东理工大学 | 地下 | 岛式站台 | |
| 罗秀路 | 地下 | 岛式站台 | |
| 朱梅路 | 地下 | 岛式站台 | |
| 景洪路 | 闵行区 | 地下 | 岛式站台 | |
| 虹梅南路 | 地下 | 岛式站台 | |
| 景西路 | 地下 | 岛式站台 | |
| 曙建路 | 地下 | 岛式站台 | |
| 双柏路 | 地下 | 岛式站台 | |
| 元江路 | 地下 | 岛式站台 | |
| 永德路 | 地下 | 岛式站台 | |
| 紫竹高新区 | 地下 | 岛式站台 | |
16号线
[编辑]
主条目：上海轨道交通16号线
AC19型列车驶离罗山路站准备折返
罗山路站
临港大道站
滴水湖站
上海轨道交通16号线于2013年12月29日正式开通，最初原本規劃為11號線南段（R3s）。因开通时仅使用三节编组，该线路使用的列车被东方早报称为上海地铁的最短地铁列车。该线路是上海地铁时速最快的地铁线路，其列车报站也是上海地铁第一次在报站时加入沪语。
16号线在最初设计上，安排了站站停、大站车、直达车三种运行模式。设计全程运行时间为：直达车36分钟，大站车41分钟，普通车48分钟。运行初期采取站站停和大站车结合的模式开行。自2013年12月29日起开行的大站车停靠羅山路站、新場站、惠南站以及滴水湖站。于工作日早晚高峰期间开行，每天16对。后于2014年1月30日暂时取消大站车，站站停普通列车运行间隔从原来的20分钟缩短到10分钟左右。2016年3月21日起，16号线恢复大站车运营模式，并将运行范围扩大到龙阳路站，运营时间覆盖到工作日和节假日。但值得注意的是，大站车的运营并未包括工作日早高峰开往龙阳路方向的通勤班次。
2020年6月18日起，16号线启用新运行图，逢工作日开行直达车，共四班。
下表中，粗体字加粗的站点名称为大站快车所停站点名称。
16号线目前共設13座车站，有10座为高架站，3座为地下站，其中3座為换乘站。
| 上海轨道交通16号线车站 |
| 车站[註 11] | 最早启用日期 | 所在地 | 车站位置 | 站台类型 | 换乘线路 |
| 龙阳路 | 2014年12月28日 | 浦东新区 | 高架 | 双岛式站台 | █ 2号线、 █ 7号线、 █ 18号线、 █ 磁浮线[註 5] |
| 华夏中路 | 高架 | 岛式站台 | █ 13号线 |
| 罗山路 | 2013年12月29日 | 高架 | 岛式叠式站台 | █ 11号线 |
| 周浦东 | 高架 | 侧式站台 | |
| 鹤沙航城 | 高架 | 侧式站台 | |
| 航头东 | 高架 | 双岛式站台 | |
| 新场 | 高架 | 侧式站台 | |
| 野生动物园 | 高架 | 双岛式站台 | |
| 惠南 | 地下 | 侧式站台 | |
| 惠南东 | 高架 | 双岛式站台 | |
| 书院 | 高架 | 一岛一侧式站台 | |
| 临港大道 | 地下 | 一岛一侧式站台 | |
| 滴水湖 | 地下 | 双岛式站台 | |
17号线
[编辑]
主条目：上海轨道交通17号线
17号线列车正在接近朱家角站
17A01型列车正驶离嘉松中路站
国家会展中心站
线路编号原为20号线，在2011年上海地铁规划调整中调整线路编号为17号线，又称青浦线，最初规划为2号线西延伸青浦段（R2w）。
17号线目前共設14座车站，有7座为高架站，7座为地下站，其中2座為换乘站。
| 上海轨道交通17号线车站 |
| 车站 | 最早启用日期 | 所在地 | 车站位置 | 站台类型 | 换乘线路 |
| 虹桥火车站 | 2017年12月30日 | 闵行区 | 地下 | 与 █ 2号线组成双岛式站台 | █ 2号线、 █ 10号线 |
| 国家会展中心站 | 青浦区 | 地下 | 岛式站台 | █ 2号线[註 3] |
| 蟠龙路 | 地下 | 岛式站台 | |
| 徐盈路 | 高架 | 岛式站台 | |
| 徐泾北城 | 高架 | 岛式站台 | |
| 嘉松中路 | 高架 | 侧式站台 | |
| 赵巷 | 高架 | 侧式站台 | |
| 汇金路 | 地下 | 侧式站台 | |
| 青浦新城 | 地下 | 岛式站台 | |
| 漕盈路 | 地下 | 岛式站台 | |
| 淀山湖大道 | 地下 | 岛式站台 | |
| 朱家角 | 高架 | 岛式站台 | |
| 东方绿舟 | 高架 | 岛式站台 | |
| 西岑 | 2024年11月30日 | 高架 | 岛式站台 | |
18号线
[编辑]
主条目：上海轨道交通18号线
复旦大学站
平凉路站
御桥站
上海轨道交通18号线一期南段于2020年12月26日正式开通运营，成为上海首条最高等级全自动无人驾驶的地铁线路。一期北段則於2021年12月30日开通运营。
18號線目前共設26座車站，全線均為地下站，其中10座為轉乘站。
| 上海轨道交通18号线车站 |
| 车站 | 最早启用日期 | 所在地 | 车站位置 | 站台类型 | 换乘线路 |
| 长江南路 | 2021年12月30日 | 宝山区 | 地下 | 岛式站台 | █ 3号线 |
| 殷高路 | 地下 | 岛式站台 | |
| 上海财经大学 | 杨浦区 | 地下 | 岛式站台 | |
| 复旦大学 | 地下 | 岛式站台 | |
| 国权路 | 地下 | 岛式站台 | █ 10号线 |
| 抚顺路 | 地下 | 岛式站台 | |
| 江浦路 | 地下 | 岛式站台 | █ 8号线 |
| 江浦公园 | 地下 | 岛式站台 | █ 12号线 |
| 平凉路 | 地下 | 一岛一侧式站台 | |
| 丹阳路 | 地下 | 岛式站台 | |
| 昌邑路 | 浦东新区 | 地下 | 岛式站台 | █ 14号线 |
| 民生路 | 地下 | 岛式站台 | █ 6号线 |
| 杨高中路 | 地下 | 岛式站台 | █ 9号线 |
| 迎春路 | 地下 | 岛式站台 | |
| 龙阳路 | 地下 | 岛式站台 | █ 2号线、 █ 7号线、 █ 16号线、 █ 磁浮线[註 5] |
| 芳芯路 | 地下 | 岛式站台 | |
| 北中路 | 地下 | 岛式站台 | |
| 莲溪路 | 地下 | 岛式站台 | █ 13号线 |
| 御桥 | 2020年12月26日 | 地下 | 侧式站台 | █ 11号线 |
| 康桥 | 地下 | 岛式站台 | |
| 周浦 | 地下 | 岛式站台 | |
| 繁荣路 | 地下 | 岛式站台 | |
| 沈梅路 | 地下 | 岛式站台 | |
| 鹤涛路 | 地下 | 岛式站台 | |
| 下沙 | 地下 | 岛式站台 | |
| 航头 | 地下 | 岛式站台 | |
浦江线
[编辑]
主条目：上海轨道交通浦江线
浦江线T0110号列车
沈杜公路站
浦江线为上海市第一条采用膠輪路軌系統的旅客自动输送系统线路，于2018年3月31日正式开通运营，服务浦江镇区域。
| 上海轨道交通浦江线车站 |
| 车站 | 最早启用日期 | 所在地 | 车站位置 | 站台类型 | 换乘线路 |
| 沈杜公路 | 2018年3月31日 | 闵行区 | 高架 | 西班牙式站台 | █ 8号线 |
| 三鲁公路 | 高架 | 侧式站台 | |
| 闵瑞路 | 高架 | 侧式站台[註 10] | |
| 浦航路 | 高架 | 侧式站台[註 10] | |
| 东城一路 | 高架 | 侧式站台[註 10] | |
| 汇臻路 | 高架 | 岛式站台 | |
注释
[编辑]
维基共享资源上的相关多媒体资源：上海地铁车站列表
^ 1.0 1.1 1.2 本站上下行付费区现时不连通。
^ 上海火车站方向另有一个站台，设计时可同台换乘8号线沈杜公路方向，目前该站台面未启用。
^ 3.00 3.01 3.02 3.03 3.04 3.05 3.06 3.07 3.08 3.09 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 持交通卡的乘客可在本站出站30分钟内换乘此线路，乘坐里程连续计算；单程票用户如需在本站换乘此线路，需出站另行购票，乘坐里程不连续计算。
^ 4.0 4.1 两线市区方向（2号线浦东1号2号航站楼方向、10号线基隆路方向）之间可换乘。其他方向之间，持交通卡的乘客可在出站30分钟内换乘，乘坐里程连续计算；单程票用户如需换乘，需出站另行购票，乘坐里程不连续计算，或穿过双侧开门的列车换乘。
^ 5.0 5.1 5.2 5.3 5.4 上海磁浮示范运营线不属于上海地铁系统。本站可转乘磁浮线，乘坐里程不连续计算，需出站另行购票。
^ 10号线一期曾试行轨交网络车站编号制度（见方案二），但未有其他线路跟进，后10号线二期及一期部分更换的新导视也并未使用车站编号。
^ 本站列车靠左行驶，岛式站台南侧为虹桥火车站方向，岛式站台北侧为基隆路方向。
^ 本站列车靠左行驶，南部岛式站台南侧为虹桥火车站方向，中部岛式站台南侧与南部岛式站台北侧为基隆路方向。
^ 本站列车靠左行驶，岛式站台南侧为下客站台，岛式站台北侧为基隆路方向。
^ 10.0 10.1 10.2 10.3 本站上下行付费区现时实际连通，但相关室外连通通道不对乘客开放，以至于事实上收费区不连通。
^ 粗体显示车站为大站车停靠站。16号线开通初期有“大站车”运营模式模式，2014年1月30日一度取消，2016年3月21日又恢复开行。
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^ 浦江线换乘首末班车时刻表. 上海地铁官方网站. 2018-03-31 [2018-03-31]. （原始内容存档于2018-04-01）.
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| | | | 1号线 | 莘庄 – 外环路 – 莲花路 – 锦江乐园 – 上海南站 – 漕宝路 – 上海体育馆 – 徐家汇 – 衡山路 – 常熟路 – 陕西南路 – 一大会址·黄陂南路 – 人民广场 – 新闸路 – 汉中路 – 上海火车站 – 中山北路 – 延长路 – 上海马戏城 – 汶水路 – 彭浦新村 – 共康路 – 通河新村 – 呼兰路 – 共富新村 – 宝安公路 – 友谊西路 – 富锦路 | | 2号线 | 国家会展中心 – 虹桥火车站 – 虹桥2号航站楼 – 淞虹路 – 北新泾 – 威宁路 – 娄山关路 – 中山公园 – 江苏路 – 静安寺 – 南京西路 – 人民广场 – 南京东路 – 陆家嘴 – 浦东南路 – 世纪大道 – 上海科技馆 – 世纪公园 – 龙阳路 – 张江高科 – 金科路 – 广兰路 – 唐镇 – 创新中路 – 华夏东路 – 川沙 – 凌空路 – 远东大道 – 海天三路 – 浦东1号2号航站楼 | | 3号线 | 上海南站 – 石龙路 – 龙漕路 – 漕溪路 – 宜山路 – 虹桥路 – 延安西路 – 中山公园 – 金沙江路 – 曹杨路 – 镇坪路 – 中潭路 – 上海火车站 – 宝山路 – 东宝兴路 – 虹口足球场 – 赤峰路 – 大柏树 – 江湾镇 – 殷高西路 – 长江南路 – 淞发路 – 张华浜 – 淞滨路 – 水产路 – 宝杨路 – 友谊路 – 铁力路 – 江杨北路 | | 4号线（环线） | （←外圈，上海体育馆方向）宜山路 – 虹桥路 – 延安西路 – 中山公园 – 金沙江路 – 曹杨路 – 镇坪路 – 中潭路 – 上海火车站 – 宝山路 – 海伦路 – 临平路 – 大连路 – 杨树浦路 – 浦东大道 – 世纪大道 – 向城路 – 蓝村路 – 塘桥 – 南浦大桥 – 西藏南路 – 鲁班路 – 大木桥路 – 东安路 – 上海体育场 – 上海体育馆 – 宜山路（内圈，虹桥路方向→） | | 5号线 | | | | --- | | 主线 | 莘庄 – 春申路 – 银都路 – 颛桥 – 北桥 – 剑川路 – 东川路 – 江川路 – 西渡 – 萧塘 – 奉浦大道 – 环城东路 – 望园路 – 金海湖 – 奉贤新城 | | 支线 | 东川路 – 金平路 – 华宁路 – 文井路 – 闵行开发区 | | | 6号线 | 港城路 – 外高桥保税区北 – 航津路 – 外高桥保税区南 – 洲海路 – 五洲大道 – 东靖路 – 巨峰路 – 五莲路 – 博兴路 – 金桥路 – 云山路 – 德平路 – 北洋泾路 – 民生路 – 源深体育中心 – 世纪大道 – 浦电路 – 蓝村路 – 上海儿童医学中心 – 临沂新村 – 高科西路 – 东明路 – 高青路 – 华夏西路 – 上南路 – 灵岩南路 – 东方体育中心 | | 7号线 | 美兰湖 – 罗南新村 – 潘广路 – 刘行 – 顾村公园 – 祁华路 – 上海大学 – 南陈路 – 上大路 – 场中路 – 大场镇 – 行知路 – 大华三路 – 新村路 – 岚皋路 – 镇坪路 – 长寿路 – 昌平路 – 静安寺 – 常熟路 – 肇嘉浜路 – 东安路 – 龙华中路 – 后滩 – 长清路 – 耀华路 – 云台路 – 高科西路 – 杨高南路 – 锦绣路 – 芳华路 – 龙阳路 – 花木路 | | 8号线 | 市光路 – 嫩江路 – 翔殷路 – 黄兴公园 – 延吉中路 – 黄兴路 – 江浦路 – 鞍山新村 – 四平路 – 曲阳路 – 虹口足球场 – 西藏北路 – 中兴路 – 曲阜路 – 人民广场 – 大世界 – 老西门 – 陆家浜路 – 西藏南路 – 中华艺术宫 – 耀华路 – 成山路 – 杨思 – 东方体育中心 – 凌兆新村 – 芦恒路 – 浦江镇 – 江月路 – 联航路 – 沈杜公路 | | 9号线 | 上海松江站 – 醉白池 – 松江体育中心 – 松江新城 – 松江大学城 – 洞泾 – 佘山 – 泗泾 – 九亭 – 中春路 – 七宝 – 星中路 – 合川路 – 漕河泾开发区 – 桂林路 – 宜山路 – 徐家汇 – 肇嘉浜路 – 嘉善路 – 打浦桥 – 马当路 – 陆家浜路 – 小南门 – 商城路 – 世纪大道 – 杨高中路 – 芳甸路 – 蓝天路 – 台儿庄路 – 金桥 – 金吉路 – 金海路 – 顾唐路 – 民雷路 – 曹路 | | 10号线 | | | | --- | | 主线 | 虹桥火车站 – 虹桥2号航站楼 – 虹桥1号航站楼 – 上海动物园 – 龙溪路 – 水城路 – 伊犁路 – 宋园路 – 虹桥路 – 交通大学 – 上海图书馆 – 陕西南路 – 一大会址·新天地 – 老西门 – 豫园 – 南京东路 – 天潼路 – 四川北路 – 海伦路 – 邮电新村 – 四平路 – 同济大学 – 国权路 – 五角场 – 江湾体育场 – 三门路 – 殷高东路 – 新江湾城 – 国帆路 – 双江路 – 高桥西 – 高桥 – 港城路 – 基隆路 | | 支线 | 龙溪路 – 龙柏新村 – 紫藤路 – 航中路 | | | 11号线 | | | | --- | | 主线 | 嘉定北 – 嘉定西 – 白银路 – 嘉定新城 – 马陆 – 陈翔公路 – 南翔 – 桃浦新村 – 武威路 – 祁连山路 – 李子园 – 上海西站 – 真如 – 枫桥路 – 曹杨路 – 隆德路 – 江苏路 – 交通大学 – 徐家汇 – 上海游泳馆 – 龙华 – 云锦路 – 龙耀路 – 东方体育中心 – 三林 – 三林东 – 浦三路 – 康恒路 – 御桥 – 罗山路 – 秀沿路 – 康新公路 – 迪士尼 | | 支线 | 嘉定新城 – 上海赛车场 – 昌吉东路 – 上海汽车城 – 安亭 – 兆丰路 – 光明路 – 花桥 | | | 12号线 | 七莘路 – 虹莘路 – 顾戴路 – 东兰路 – 虹梅路 – 虹漕路 – 桂林公园 – 漕宝路 – 龙漕路 – 龙华 – 龙华中路 – 大木桥路 – 嘉善路 – 陕西南路 – 南京西路 – 汉中路 – 曲阜路 – 天潼路 – 国际客运中心 – 提篮桥 – 大连路 – 江浦公园 – 宁国路 – 隆昌路 – 爱国路 – 复兴岛 – 东陆路 – 巨峰路 – 杨高北路 – 金京路 – 申江路 – 金海路 | | 13号线 | 金运路 – 金沙江西路 – 丰庄 – 祁连山南路 – 真北路 – 大渡河路 – 金沙江路 – 隆德路 – 武宁路 – 长寿路 – 江宁路 – 汉中路 – 自然博物馆 – 南京西路 – 淮海中路 – 一大会址·新天地 – 马当路 – 世博会博物馆 – 世博大道 – 长清路 – 成山路 – 东明路 – 华鹏路 – 下南路 – 北蔡 – 陈春路 – 莲溪路 – 华夏中路 – 中科路 – 学林路 – 张江路 | | 14号线 | 封浜 – 乐秀路 – 临洮路 – 嘉怡路 – 定边路 – 真新新村 – 真光路 – 铜川路 – 真如 – 中宁路 – 曹杨路 – 武宁路 – 武定路 – 静安寺 – 一大会址·黄陂南路 – 大世界 – 豫园 – 陆家嘴 – 浦东南路 – 浦东大道 – 源深路 – 昌邑路 – 歇浦路 – 龙居路 – 云山路 – 蓝天路 – 黄杨路 – 云顺路 – 浦东足球场 – 金粤路 – 桂桥路 | | 15号线 | 紫竹高新区 – 永德路 – 元江路 – 双柏路 – 曙建路 – 景西路 – 虹梅南路 – 景洪路 – 朱梅路 – 罗秀路 – 华东理工大学 – 上海南站 – 桂林公园 – 桂林路 – 吴中路 – 姚虹路 – 红宝石路 – 娄山关路 – 长风公园 – 大渡河路 – 梅岭北路 – 铜川路 – 上海西站 – 武威东路 – 古浪路 – 祁安路 – 南大路 – 丰翔路 – 锦秋路 – 顾村公园 | | 16号线 | 龙阳路 – 华夏中路 – 罗山路 – 周浦东 – 鹤沙航城 – 航头东 – 新场 – 野生动物园 – 惠南 – 惠南东 – 书院 – 临港大道 – 滴水湖 | | 17号线 | 西岑 – 东方绿舟 – 朱家角 – 淀山湖大道 – 漕盈路 – 青浦新城 – 汇金路 – 赵巷 – 嘉松中路 – 徐泾北城 – 徐盈路 – 蟠龙路 – 国家会展中心 – 虹桥火车站 | | 18号线 | 航头 – 下沙 – 鹤涛路 – 沈梅路 – 繁荣路 – 周浦 – 康桥 – 御桥 – 莲溪路 – 北中路 – 芳芯路 – 龙阳路 – 迎春路 – 杨高中路 – 民生路 – 昌邑路 – 丹阳路 – 平凉路 – 江浦公园 – 江浦路 – 抚顺路 – 国权路 – 复旦大学 – 上海财经大学 – 殷高路 – 长江南路 | | 浦江线 | 沈杜公路 – 三鲁公路 – 闵瑞路 – 浦航路 – 东城一路 – 汇臻路 | | 磁浮线 | 龙阳路 – 浦东1号2号航站楼 | |
| | | | 2号线西西西延伸段 | 蟠祥路（ – 国家会展中心） | | 12号线西延伸段 | 洞泾 – 刘五公路 – 同乐路 – 千帆路 – 场西路 – 场东路（ – 七莘路） | | 13号线西延伸段及东延伸段 | 国家会展中心 – 运乐路 – 季乐路 – 前湾公园 – 纪翟路（ – 金运路 – 已运营区段 – 张江路 – ）高科中路 – 丹桂路 | | 15号线南延伸段 | 望园路 – 东方美谷大道 – 汇丰北路 – 西闸公路 – 兰香湖南路（ – 紫竹高新区） | | 18号线二期 | （长江南路 – ）通南路 – 长江西路 – 江杨南路 – 爱辉路 – 呼兰路 – 康文路 | | 19号线 | 虹建路 – 联曹路 – 景洪路 – 华泾路 – 三林南 – 凌兆新村 – 东方体育中心 – 德州路 – 后滩 – 世博大道 – 中华艺术宫南 – 华丰路 – 南码头路 – 塘桥 – 潍坊路 – 商城路 – 浦东南路 – 提篮桥 – 周家嘴路 – 海伦路 – 甜爱路 – 上海外国语大学 – 凉城路 – 广灵四路 – 车站北路 – 虹湾路 – 一二八纪念路 – 呼玛东路 – 江杨南路 – 泰和路 – 兰岗路 – 铁山路 – 铁通路 – 上海宝山站 | | 20号线一期 | 交通路 – 上海西站 – 真华路 – 新村路 – 平利路 – 大宁公园 – 上海马戏城 – 凉城路 – 广灵二路 – 车站南路 – 万安路 – 江湾镇 – 上海财经大学 – 三门路 – 上海体育大学 – 嫩江路 – 共青森林公园 – 浦东北路 – 张杨北路 – 台南西路 – 北新园路 | | 21号线一期 | 浦东3号航站楼 – 上海东站 – 金闻路 – 闻居路 – 施新路 – 六团 – 川沙路 – 唐黄路 – 迪士尼 – 申江南路 – 康桥东 – 康南路 – 博宇路 – 学林路 – 张衡路 – 古桐南路 – 广兰路 – 龙东大道 – 浦东足球场 – 云桥路 – 金桥 – 佳林路 – 杨高北路 – 东靖路 | | 22号线 | 金吉路 – 申江路 – 高宝路 – 凌空北路 – 长兴岛 – 陈家镇 – 东滩 – 裕安 | | 23号线一期 | 闵行开发区 – 文井路 – 华宁路 – 金平路 – 东川路 – 交大紫竹路 – 紫竹高新区 – 华东师范大学 – 紫龙路 – 吴泾 – 北吴路 – 墨江路 – 澄江路 – 吴泾北 – 景联路 – 华泾路 – 徐浦大桥 – 龙吴路 – 上海植物园 – 龙启路 – 龙漕路 – 上海体育场 | |
| | | | 5号线南延伸剩余段 | （奉贤新城 – ）平庄公路 | | 9号线三期剩余段 | （曹路 – ）曹路火车站 | |
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March 22, 2018
Heilbrunn Timeline of Art History
New York:
Metropolitan Museum of Art online, 2017.
Naraelle Hohensee
CrossRef DOI: 10.3202/caa.reviews.2018.82
Few college instructors or students of art history today are likely to be unfamiliar with the Metropolitan Museum of Art’s expansive Heilbrunn Timeline of Art History. With over one thousand thematic essays written by experts in the field, as well as more than 7,600 pages featuring artworks from the Met’s collection, the timeline is a formidable and immensely popular online resource.[i] Parallel to its larger rebranding efforts in 2016, the Met launched its new edition of the timeline, featuring a brand-new, ultra-clean interface designed by the New York–based firm CHIPS. The timeline now offers a user experience that is easily navigable, attractive, and enjoyable, while providing access to the same impressive set of scholarly resources that have made the site one of the most popular destinations for art-historical research on the web.
The Heilbrunn Timeline of Art History was always more than just a timeline, and indeed the word timeline itself is now a bit of an anachronism; in this new reboot, what were formerly called “timelines” are now “chronologies.” After its initial launch in 2000, the site’s early iterations featured thematic essays alongside the timelines, which were also closely linked to maps, all interconnected through a complex set of navigational menus.
The experience of using the Heilbrunn Timeline feels much cleaner and more cohesive now. Imagine you are an undergraduate student working on the quintessential art history course assignment: the museum paper. You have wandered through the Met’s contemporary galleries and chosen Anselm Kiefer’s Bohemia Lies by the Sea as your object of study. You go home and do a quick Google search, and the first result is the Heilbrunn Timeline’s object page for the work. It displays images of the work on the left; the object’s metadata (artist, title, date, and so on) in the center column (or, on a phone or tablet, below the images); and beneath that a short essay contextualizing the work within Kiefer’s biography and oeuvre. Below the text is a link to the painting’s entry in the Met’s online catalogue. On the right-hand side of the Heilbrunn Timeline page, unobtrusive, expandable menus provide links to an essay (on Kiefer more generally), a chronology (“Germany and Switzerland, 1900 A.D.–present”), keywords (which are essentially tags that link to lists of other timeline resources with related themes, motifs, materials, styles, periods, regions, and so forth), a link to the timeline’s page on the artist, and a link (“Connections”) to additional Met-produced online content (in this case, a talk with a curator about her favorite textures in the collection). Though there is a huge amount of content here, the clean visual organization of the page, especially its hidden multiple links under expandable right-hand menus, makes it clear and digestible.
If a user comes to the Heilbrunn Timeline of Art History without a particular work in mind, the site’s home page presents a simple and organized interface, though it is so simple that a newer user may not immediately understand how its elements link together or even what they are. However, the interface is so inviting that it feels fun just to click through and discover what lies below this minimalist surface. The secondary levels of the site—“Essays,” “Works of Art,” “Chronology,” “Artists/Makers,” and “Keywords”—are also elegant, with cleanly styled menus that allow users to easily browse and discover resources.
For teachers, one of the best new features of the site is the “Works of Art” page, which displays all 7,600-plus works of art in an attractive, tiled thumbnail format of the type made popular by websites such as Pinterest. Items here are randomized on each page visit, and different works are brought to the top of the list each time the page is loaded. Through this close visual juxtaposition of works, one can see themes and styles emerge not only in particular times and places but also across multiple media. For instance, selecting the European Renaissance thematic category brings up not just painting and sculpture but also examples of armor, musical instruments, furniture, textiles, and decorative arts. The visual proximity of these objects presents a potent visual argument for expanding the canon of art history, and it could be a wonderful source of inspiration for instructors, especially, to think differently about what objects they teach and assign.
Seeing the Met’s collection displayed this way is impressive. However, herein is also one of the Heilbrunn Timeline’s limitations: with very few exceptions, the works of art featured throughout the site are solely ones in the Met’s collection. On the one hand, this restriction is beneficial, in that users have the chance to discover works in the museum’s collection that are noncanonical for art history in general; on the other hand, a student studying for a typical art history survey exam or writing a paper on a canonical piece from another institution must search elsewhere for detailed information on those works. Because one of the timeline’s clear strengths is its linking together of a huge number of resources, the adding of links to other key institutions and online resources might be a fruitful area for growth and would make the website more powerful as a learning resource.
Of the many resources on the Heilbrunn Timeline site, the essays and the works of art pages are likely the most useful for the majority of users, as they contain clear, organized, well-written, and well-researched information that can be easily used and cited by students and scholars. The inclusion of bibliographies on many of the pages is one of the site’s more important features, and this is an area where the timeline could do even more, as parallel entries in the Met’s online catalogue (which are different from the Heilbrunn Timeline’s own artwork pages) tend to have more extensive bibliographies.
Other reviewers of the Heilbrunn Timeline have noted that it has limited utility for younger learners, as its content is mainly textual and the tone is distinctly scholarly.[ii] For the informal adult learner, the organization of the content, though quite impressive in this redesign, offers little in terms of curated paths through the resources; one could imagine the addition of online curricula that would give beginning users a more guided experience.
The popularity of the Heilbrunn Timeline of Art History points not only to its quality but also to the high level of demand for open-access scholarly art history resources on the web. With the exception of Smarthistory,[iii] which also comprises a large number of learning resources and attracts a comparable number of users, the Heilbrunn Timeline is unparalleled on the web in terms of its sheer wealth of textual and visual material. Counterintuitively, for most traditional museums, sharing and linking across institutional boundaries in the digital world and providing free content for learners are positive and necessary ways to drive visitors to both their websites and their physical galleries.[iv] In these ways, the Metropolitan Museum of Art is an example to other museums of how providing digital open access increases, rather than detracts from, the public value of its collection.
[i] The timeline receives up to 1.5 million visits per month during the academic year. “The Met Launches New Edition of the Heilbrunn Timeline of Art History,” Enfilade, June 14, 2016,
[ii] “Heilbrunn Timeline of Art History,” Common Sense Education, May 7, 2013,
[iii] Full disclosure: I am currently the Andrew W. Mellon Postdoctoral Fellow at Smarthistory.
[iv] The Met’s new open-access policy is a significant step in this direction. It makes the museum’s 375,000 images and their metadata available under a Creative Commons Zero, or CC0 (public domain), license. The museum’s digital department also recently announced partnerships with Wikipedia/Wikimedia, Artstor, Digital Public Library of America, and Pinterest. Loic Tallen, “Introducing Open Access at the Met,” Metropolitan Museum of Art online, February 7, 2017,
Naraelle Hohensee
Andrew W. Mellon Postdoctoral Fellow, Smarthistory.org
Copyright © 2025 College Art Association. Reviews and essays are licensed to the public under a under a Creative Commons Attribution-NoDerivatives 4.0 International License.
By accessing and/or using caa.reviews, you accept and agree to abide by the Terms & Conditions. |
12910 | https://testbook.com/en-us/mathematics/linear-equations | SAT Linear Equations: Forms, Types, Graphs, Methods, Examples & FAQs.
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SAT Linear Equations: Forms, Graph and Algebraic Method to Solve
Let’s talk about one of the most common—and honestly, super useful—math concepts you'll come across: linear equations! At first glance, they might just look like a bunch of letters and numbers, but once you get the hang of them, they’re actually a piece of cake. A linear equation is basically any equation where the variable’s highest power is just 1. No crazy exponents, no square roots—just straightforward math. That’s why they pop up so often in exams like the SAT, ACT, PSAT/NMSQT, GED, GRE, GMAT, AP, PERT, Accuplacer, and even the MCAT. They’re foundational!
Now, when you’ve got just one variable, like Ax + B = 0, that’s called a linear equation in one variable. It’s all about figuring out the value of x that makes the equation true. Easy, right? But when things get a little more interesting—say, you’ve got two variables—it looks like Ax + By = C. Don’t worry though, because we’ve got strategies like the substitution method and the elimination method to solve these quickly and confidently.
In this article, we’ll break down everything you need to know about linear equations. We’ll cover their different forms, show you step-by-step how to solve them, and give you plenty of practice examples so you can feel 100% ready when you see them on test day!
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What is a Linear Equation?
A mathematical statement with an equal to sign in it is termed as an equation. Such equations of degree 1 are termed as linear equations. This also means that none of the exponent in a linear equation has degree greater than 1. Such an equation represents a straight line and the solution of these equations give values when substituted to the original equation make the equation true.
Also, linear equations can have more than one variable and are named as linear equations in one, two, or more variables. Some of the examples of linear equations in one, two, and three variables are mentioned below:
Linear equations in one variable: 4x + 9 = 0, 45x = 9, (3/2)x + 7 = 0.
Linear equations in two variables: 7x + y = 3, 5a + 4b = 2, 6x – 9y = 12.
Linear equations in three variables: x + y + z = 0, a – 2b = 3c, 3x + 12y = (1/2)z.
Different Form of Linear Equations
The way by which we represent any linear equation is termed as linear equation formula. It can be expressed in three ways namely standard form, slope intercept form and point slope form. Let us check these three ways of expressing the linear equation one by one:
Linear Equation in Standard form
We know that a linear equation is a combination of variables as well as constants. A linear equation in one variable in its standard form is written as:
ax + b = 0, here a cannot be zero and x is the variable.
For a linear equation in two variables, the standard form is written as:
ax + by + c = 0, here a, and b cannot be zero and x, and y are the variables.
For a linear equation in three variables, the standard form is written as:
ax + by + cz + d = 0, here a, b, and c cannot be zero and x, y, and z are the variables.
Linear Equations in Slope Intercept Form
The slope-intercept form of a linear equation is the most commonly used form of linear equation. This can be represented as:
y = mx + c
Here, m is the slope of the line and c is the y-intercept of the line.
x, and y are the x and y coordinates of the line respectively.
For instance:
Consider a line y = 3x + 9.
Here m = 3 is the slope of the line and its y-intercept is 9.
For a line that is parallel to x-axis, the x-coordinate is zero. So, the equation becomes:
y = c.
And, for the line parallel to y-axis the y-coordinate becomes zero and the equation becomes:
mx + c = 0.
Linear Equation in Point Slope Form
In this way of representing a line, the equation is formed by considering the points on the x and y plane. It is represented as:
(y−y 1)=m(x−x 1)(y−y 1)=m(x−x 1)
Here (x 1,y 1)(x 1,y 1) are the coordinates of the point on the line.
Linear Equation Representation on Graph
The graphical representation of a linear equation in one variable, say ‘x’ is represented by a vertical line parallel to the y-axis, and vice versa. However, a linear equation in two variables forms a straight line.
Let us plot a graph for linear equation in two variables using an example:
Example:
Plot a graph of linear equation in two variables x – 3y = 9.
Following are the steps to plot the graph of x – 3y = 9.
Step 1: Convert the equation in the form y = mx + c. This will give us y=1 3 x−3 y=1 3 x−3.
Step 2: Now we have to replace the value of x and find the subsequent values of y in order to create the coordinates.
Step 3: Starting with putting x = 0 in the above equation, we get y = -3. Similarly, we put values of x = 9, we get y = 0.
Step 4: We can take more values of x and find the subsequent values of y to get a set of coordinates for plotting a line.
Let us construct a table for the obtained values:
x0 3 9 12
y-3-2 0 1
The coordinates corresponding to the given values are: (0,3), (3,-2), (9,0), and (12,1).
Step 5: Plot these points on the coordinate plane and join the points to get the desired line.
Linear equations in 1 variable
As the name suggests, any linear equation is said to be a linear equation in one variable if it has only one variable in it. It can be represented as Ax + B = 0, here ‘x’ is the variable, A is the coefficient of ‘x’ and cannot be zero. B is the constant.
The variable ‘x’ in this linear equation has only one solution. The degree of such an equation is always one and it is the easiest way to represent any mathematical statement.
Solving a linear equation in one variable is even easier. We simply have to separate the variables and bring them to one side of the equation and constant to the other side to find the value of the unknown variable.
Example:
Let the linear equation in one variable be (2 x−10)2=3(x−1)(2 x−10)2=3(x−1).
Solution:
In order to solve this linear equation we have to bring together the variable ‘x’ to one side of the equation.
Let us simplify this equation to get the value of ‘x’ step by step:
x−5=3 x−3 x−5=3 x−3
3−5=3 x−x 3−5=3 x−x
−2=2 x−2=2 x
x=−1 x=−1
So, the solution for this linear equation in one variable is x = -1.
Linear equations in 2 variable
Linear equation in two variables is a type of equation that has two variables but the highest power of both the variables is always one. Such equations are of the form Ax +By + C = 0. Here, ‘x’ and ‘y’ are the variables and A, and B are the coefficients of variables ‘x’ and ‘y’ respectively, and C is the constant. Also, A and B cannot be equal to zero.
In order to get the solution for a linear equation in two variables we need to solve two such equations simultaneously, hence, such equations are also called simultaneous linear equations.
There are multiple ways to solve such pair of linear equations namely, substitution method, elimination method and cross multiplication method about which we will read in the later part of the article.
For Example:6x + 2y = -9 is a linear equation in two variables.
Linear equations in 3 variable
Such linear equations that have three variables with degree one are called linear equations in three variables. Such equations are represented as Ax + By + Cz + D = 0.
Here, x,y,and z are the three variables and A, B, and C are their coefficients. D in the above equation is the constant. In order to solve a linear equation in three variables we need three such equations.
The most common method of solving such equations is a matrix method.
For Example:x + 3y – 4z= -7 is a linear equation in three variables.
Algebraic Method of Solving Linear Equations
In order to solve linear equations in two variables many methods can be used. The most common methods among those are:
Substitution method
Elimination method
Cross multiplication method
Let us discuss each of these one by one:
Substitution Method
To solve the equation using substitution method we have to follow these steps:
Step 1: Find the value of one variable, let us suppose ‘y’ from one equation in terms of the other variable.
Step 2: Substitute the value of ‘y’ so obtained in the second equation.
In this way we shall get an equation in one variable that is ‘x’.
Step 3: Find the value of ‘x’ using the transposition method.
Step 4:Put the value of ‘x’ so obtained in equation one to get the value of ‘y’.
Let us understand this using an example:
Example: Solve the pair of equations x + 2y = 3, and 7x – 15y = 2.
Solution:The given linear equations in two variables are:
x + 2y = 3…(1)
7x – 15y = 2…(2)
Considering equation (1), let us find the value of ‘x’ in terms of ‘y’, we get:
x = 3 – 2y…(3)
Substitute the value of x from (3) in (2), we get:
7(3-2y) – 15y = 2
21 – 14y – 15 y = 2
21 – 29y = 2
-29y = 2 – 21
-29y = -19
y = 19/29…(4)
Put this value of ‘y’ from (4) in (3)
x = 3 – 2(19/29)
x = 3 – 38/29
x = (87 – 38)/29
x = 49/29
So, the value of x = 49/29, and y = 19/29.
Elimination Method
Another method used for solving linear equations in two variables is elimination method. Following are the steps involved:
Step 1:Multiply both the equations with a non-zero constant such that the coefficient of any one of the variables in both the equations becomes the same.
Step 2:Looking at the equation, check if addition or subtraction is to be performed in order to eliminate one variable completely.
Step 3: If the resultant is a true statement, we say that the pair of linear equations has infinitely many solutions.
Step 4: If the resultant is a false statement, we say that the pair of linear equations has no solution.
Step 5: If we get the resultant has an equation in one variable, solve the equation to get the value of either ‘x’ or ‘y’.
Step 6: Substitute the value of the variable so obtained in any one of the two given equations, in order to get the desired result.
Let us understand this using an example:
Example:Solve the pair of equations 2x + 5y = 20 and 3x + 6y = 12.
Solution:The given linear equations in two variables are:
2x + 5y = 20…(1)
3x + 6y = 12…(2)
In order to solve this pair of linear equations using the elimination method, we have to multiply equation (1) by 3 and equation (2) by 2. We get:
6x + 15y = 60…(3)
6x + 12y = 24…(4)
Subtract equation (4) from equation (3), we get:
3y = 36
y = 36/3 = 12
Put the value of y in equation (1), we get:
2x + 5(12) = 20
2x + 60 = 20
2x = -40
x = -20
So, the solution for the given pair of linear equations is x = -20 and y = 12.
Cross Multiplication Method
This is another convenient method to solve linear equations in two variables. Steps involved in solving the equations are as follows:
Step 1:Keep in mind that using cross multiplication method the result are obtained in the form of ratios: x?=−y?=1?x?=−y?=1?.
Step 2:To find out the term below x, strike out the column containing the coefficients of x and cross-multiply and subtract the other two columns.
Step 3: To find out the term below -y, strike out the column containing the coefficients of y and cross-multiply and subtract the other two columns.
Step 4: To find out the term below 1, strike out the column containing the constant term and cross-multiply and subtract the other two columns.
Step 5: Compare the denominator of x and -y with that of 1, we get the value of x and y.
Let us understand this using an example:
Example:Solve the pair of equations 2x + 3y – 11 = 0, and 3x + 2y – 9 = 0.
Solution: To find the solution of the given pair of linear equations, let us first write them in the form of three columns:
2 3 -11
3 2 -9
The solution equality will be of the following form:
x?=−y?=1?x?=−y?=1?
Now we have to find the values for the question mark in the above equation.
Finding the denominator of x:
Striking out the column with coefficients of x and cross multiplying and subtracting the remaining columns:
-27-(-22)=-5 ———— (i)
Finding the denominator of -y:
Striking out the column with coefficients of y and cross multiplying and subtracting the remaining columns:
-18-(-33)=15 ———— (ii)
Finding the denominator of 1:
Striking out the column with constant terms and cross multiplying and subtracting the remaining columns:
4-9=-5 ———— (iii)
The last part of the equality becomes 1−5 1−5
Combining all the equations in our equality:
x−5=−y 15=1−5 x−5=−y 15=1−5
⇒x=−5−5=1 x=−5−5=1
⇒y=−15−5=3 y=−15−5=3
⇒x=1 and y=3
So, the solution for the given pair of linear equations is x = 1 and y = 3.
Linear Equations Solved Examples:
Que 1: The sum of two numbers is 44. If one number is 10 more than the other, find the numbers by framing a linear equation.
Solution:
Let the two numbers be ‘x’ and ‘y’.
Given that the sum of two numbers is 44, so the equation becomes x + y = 44…(1)
Also, one number is 10 more than another, so the second equation is x – y = 10…(2)
Solving the equations using elimination method:
As we can see that the coefficients of x and y are the same in both equations so we need not multiply the equations with any constant.
Adding equation (1)and (2) to eliminate ‘y’, we get:
2x = 54, or x = 27.
Put this value of x in any one of the given equations, we get:
27 + y = 44,
y = 44 – 27
y = 17
So, the values of x = 27 and y = 17 is the solution of the given pair of linear equations.
Que 2: A boat running downstream covers a distance of 20 km in 2 hrs, and while running upstream it covers the same distance in 5 hrs. What is the speed of the boat in still water?
Solution:Let us suppose the speed of the boat in still water is u km/hr and that the speed of the stream is v km/hr.
Speed downstream = (u + v) km/hr…(1)
Speed upstream =(u – v)km/hr…(2)
Given that: The boat covers a distance of 20 km in 2hr while running downstream and 20 km in 5 hrs while running upstream, so the speed of the boat can be calculated as:
Speed = Distance/ Time
Speed of the boat downstream = (20/2)km/hr = 10 km/hr…(3)
Speed of the boat upstream = (20/5)km/hr = 4 km/hr…(4)
From equation (1), (2), (3), and (4), we get:
u + v = 10, and u – v = 4
Solving the two equations simultaneously, we get:
u = 7 and v = 3
So, the speed of the boat in still water is u km/hr, i.e. 7 km/hr.
Conclusion
Linear equations might seem tricky at first, but once you break them down, they’re actually super straightforward and super useful! Whether it’s one, two, or even three variables, knowing how to solve them gives you a strong edge in exams like the SAT, ACT, GRE, and more. Mastering methods like substitution, elimination, and cross multiplication makes solving linear equations a breeze. Keep practicing—you’ve totally got this math skill locked down!
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12911 | https://www.selfstudys.com/books/slloney-solution/exam/preparation/the-elements-of-coordinate-geometry/10-the-parabola/398984 | Chapter 10
THE PARABOLA 288 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 289 290 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 291
EXAMPLES XXV 292 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition)
ANSWERS CHAPTER 10: THE PARABOLA 293
SOLUTIONS/HINTS 294 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 295 296 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 297 298 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 299 300 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 301 302 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 303
EXAMPLES XXVI 304 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 305 306 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition)
ANSWERS
SOLUTIONS/HINTS CHAPTER 10: THE PARABOLA 307 308 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 309 310 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 311 312 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 313 314 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 315 316 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 317 318 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 319 320 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 321 322 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 323 324 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition)
EXAMPLES XXVII CHAPTER 10: THE PARABOLA 325
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SOLUTIONS/HINTS 326 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 327 328 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 329 330 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 331 332 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition)
EXAMPLES XXVIII CHAPTER 10: THE PARABOLA 333 334 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 335
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SOLUTIONS/HINTS 336 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 337 338 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 339 340 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 341 342 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) CHAPTER 10: THE PARABOLA 343 344 COORDINATE GEOMETRY by S. L. LONEY (Kindle Edition) |
12912 | https://jpolak.org/photo/counting-bijective-injective-and-surjective-functions/ | Counting bijective, injective, and surjective functions | Jason Polak
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Counting bijective, injective, and surjective functions
Posted onAugust 27, 2024byjpolak
updated on August 27th, 2024 at 8:22 pm
Here I derive formulas for the number of bijective, injective, and surjective functions from one finite set to another. Actually, computing the number of bijective and injective functions is easy. But computing the number of surjective functions is much harder.
I like this problem a lot because it was one of the first problems I solved as an undergrad that got me interested in recurrence relations and combinatorics.
Anyway, let’s use the notation [n]={1,2,…,n}[n] = { 1,2,\dots,n}[n]={1,2,…,n} for an n n n-element set.
Bijective Functions
The number of bijective functions [n]→[n][n]\to [n][n]→[n] is the familiar factorial:
n!=1×2×⋯×n n! = 1\times 2\times\cdots\times n n!=1×2×⋯×n Another name for a bijection [n]→[n][n]\to [n][n]→[n] is a permutation. In fact, the set all permutations [n]→[n][n]\to [n][n]→[n] form a group whose multiplication is function composition. Here is a table of some small factorials:
n n!n n!
0 1 7 5040
1 1 8 40320
2 2 9 362880
3 6 10 3628800
4 24 11 39916800
5 120 12 479001600
6 720 13 6227020800
Here’s a graph of log(n!)\log(n!)lo g(n!) so you can see the rapid growth of the factorial visually:
Quite a few functions, don’t you think?
Injective Functions
What about injective functions [k]→[n][k]\to[n][k]→[n] where 1≤k≤n 1 \leq k\leq n 1≤k≤n? In that case, the image in [n][n][n] consists of k k k elements, and the order in which they are chosen determines the function, so the number of injective functions [k]→[n][k]\to [n][k]→[n] is
k!(n k)=n!(n−k)!.k!\binom{n}{k} = \frac{n!}{(n-k)!}.k!(k n)=(n−k)!n!. Sometimes this is denoted by nPk on calculators. For example, 73P5 = 1802440080.
Surjective Functions
Calculating the number of surjective functions [n]→[k][n]\to [k][n]→[k] where n≥k≥1 n\geq k \geq 1 n≥k≥1 is the most interesting. Let’s denote by S(n,k)S(n,k)S(n,k) this number. For example, S(n,n)=n!S(n,n) = n!S(n,n)=n! and S(n,1)=1 S(n,1) = 1 S(n,1)=1. A degenerate case is S(0,0)=1 S(0,0) = 1 S(0,0)=1, though S(n,0)=0 S(n,0) = 0 S(n,0)=0. Another easy to calculate one is S(n,2)=2 n–2 S(n,2) = 2^n – 2 S(n,2)=2 n–2: this is because there are 2 n 2^n 2 n functions [n]→[n]\to [n]→, but two of them send everything to just one element. What about an explicit formula in general?
The most natural thing to do is perhaps come up with a recurrence relation. So let’s divide up the number of surjective functions into two classes: the first is where if we restrict the function to [n−1][n-1][n−1], we still get a surjective function. It’s clear there are k S(n−1,k)kS(n-1,k)k S(n−1,k) of these. On the other hand, if after restricting to [n−1][n-1][n−1] the function is no longer surjective, then there are k S(n−1,k−1)kS(n-1,k-1)k S(n−1,k−1) of these, because to make such a function you choose one element in [k][k][k] that has n n n mapping to it, and then a surjective function [n−1][n-1][n−1] onto the remaining k−1 k-1 k−1 elements. Thus, we have the recurrence relation:
S(n,k)=k S(n−1,k)+k S(n−1,k−1).S(n,k) = kS(n-1,k) + kS(n-1,k-1).S(n,k)=k S(n−1,k)+k S(n−1,k−1). At this point, it’d be easy to compute S(n,k)S(n,k)S(n,k) for small values of n n n and k k k by hand, or via a computer using a recursive function. What about an explicit formula that is not a recurrence relation?
To find one, we can use the generating function technique. Write A k(x)=∑n S(n,k)x n A_k(x) = \sum_n S(n,k)x^n A k(x)=∑nS(n,k)x n. Multiplying the recurrence relation by x n x^n x n and summing over all n n n gives the relation
A k(x)=k x 1–k x A k−1(x).A_k(x) = \frac{kx}{1 – kx}A_{k-1}(x).A k(x)=1–k x k xA k−1(x). We also have A 0(x)=1 A_0(x) = 1 A 0(x)=1 because the only nonzero term in A 0 A_0 A 0 is S(0,0)x 0 S(0,0)x^0 S(0,0)x 0. Therefore, we have an explicit formula for this generating function
A k(x)=k!x k(1−x)(1−2 x)⋯(1−k x).A_k(x) = \frac{k!x^k}{(1-x)(1-2x)\cdots(1-kx)}.A k(x)=(1−x)(1−2 x)⋯(1−k x)k!x k. Now, if we split this fraction up using partial fractions into a sum of the form
∑j=1 k a j 1−j x\sum_{j=1}^k \frac{a_j}{1-jx}j=1∑k1−j x a j we find that
a j=(−1)k−j j!(k−j)!.a_j = \frac{(-1)^{k-j}}{j!(k-j)!}.a j=j!(k−j)!(−1)k−j. Now, using the fact that (1−j x)−1=1+j x+j 2 x 2+⋯(1-jx)^{-1} = 1 + jx + j^2x^2 + \cdots(1−j x)−1=1+j x+j 2 x 2+⋯ and that S(n,k)S(n,k)S(n,k) is by definition the coefficient of x n x^n x n in this power series, we get that
S(n,k)=∑j=1 k(−1)k−j k!j n j!(k−j)!.S(n,k) = \sum_{j=1}^k \frac{(-1)^{k-j}k!j^n}{j!(k-j)!}.S(n,k)=j=1∑kj!(k−j)!(−1)k−j k!j n. For example, S(n,3)=3(3 n−1–2 n+1)S(n,3) = 3(3^{n-1} – 2^n + 1)S(n,3)=3(3 n−1–2 n+1) so that S(12,3)=519156 S(12,3) = 519156 S(12,3)=519156.
Some readers may recognize that this formula is very similar to the one for the number of partitions of [n][n][n] into k k k nonempty subsets. In fact it’s k!k!k! times this number. That makes good combinatorial sense: to make a surjective function, you first partition [n][n][n] into k k k nonempty subsets and then in one of k!k!k! ways, assign one of these subsets for each element in [k][k][k].
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12913 | https://www.wordreference.com/definicion/joya | | | | | | | | | | | | | |
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| + Ver También: + - jorobado - jorobar - jorongo - jota - jotero - joto - joule - joven - jovial - jovialidad - joya - joyel - joyería - joyero - joystick - juanete - jubilación - jubilado - jubilar - jubileo - júbilo + Búsquedas recientes: + View All | joya [links] Escuchar: ⓘ Una o más entradas de foro concuerdan exactamente con el término buscado sinónimos | definición RAE | Gramática | en inglés | en francés | conjugar | en contexto | imágenes Inflexiones de 'joya' (nf): fpl: joyas Diccionario de la lengua española © 2005 Espasa-Calpe: joya 1. f. Objeto pequeño de piedras o metales preciosos que sirve como adorno: joya de oro y brillantes; esta pulsera es una joya antiquísima.- Cosa o persona de mucha valía: este empleado es una joya. 'joya' aparece también en las siguientes entradas: alfiler - alhaja - delicadeza - colgante - dije - fantasía - joyel - labrado - medallón - pendiente - perder - pulsera - relumbrón - señor - solitario 🗣️Preguntas en los foros con la(s) palabra(s) 'joya' en el título: joya / jolla joya, o pieza en forma de aro, que se usa alrededor del tobillo ultima joya de de la abuela Visita el foro Sólo Español. Ayuda a WordReference: Pregunta tú mismo. Go to Preferences page and choose from different actions for taps or mouse clicks. En otros idiomas: Francés | Portugués | Italiano | Alemán | Holandés | Sueco | Polaco | Rumano | Checo | Griego | Turco | Chino | Japonés | Coreano | Árabe | Inglés Enlaces: ⚙️Preferencias | Abreviaturas | Privacy Policy | Términos del Servicio | Apoyar WR | Foros | Sugerencias | | | | Publicidad | | | | Publicidad | | | | Infórmanos de los anuncios inapropiados. | | WordReference.com WORD OF THE DAY GET THE DAILY EMAIL! | | | | Conviértete en un Patrocinador de WordReference para ver este sitio sin anuncios. | | Usuario de Firefox: usa accesos directos para acelerar tus búsquedas en WordReference. | |
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🗣️Preguntas en los foros con la(s) palabra(s) 'joya' en el título:
En otros idiomas: Francés | Portugués | Italiano | Alemán | Holandés | Sueco | Polaco | Rumano | Checo | Griego | Turco | Chino | Japonés | Coreano | Árabe | Inglés
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12914 | https://www.youtube.com/watch?v=H0PiI-6kPBU | Find Exact Value of Inverse Sine of Negative Square Root of 2 Divided by 2 (sin^(-1)(-sqrt(2)/2))
Wendy
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Find the exact value of the inverse sine of (-sqrt(2)/2). As the exact value is required, it is necessary to use the unit circle. The restrictions on the inverse sine function require that the answer lie in the interval [-pi/2, pi/2] radians. As the argument of the inverse sine function is negative, the answer must lie in quadrant four so coterminal angles are used to find the correct answer under the given restrictions.
Timestamps
0:00 Introduction
0:36 Unit Circle
2:26 Find Negative Angle in QIV
Transcript:
Introduction in this example i need to find the exact value of the inverse sine of negative square root of two divided by two so quickly reviewing for the inverse sine function the input values are the domain of real numbers between negative 1 and 1 including the endpoints and the output values of the range are angles in the interval negative pi over 2 to pi over 2 again including the endpoints so what this means Unit Circle is i'm looking for angles somewhere in this shaded region on my unit circle so i could have a positive angle rotating between zero and pi over two or a negative angle rotating between zero and negative pi over 2. when you first start working on these problems it's often easier to set your inverse trig function equal to a variable i'm going to pick theta since i'm looking for an angle and then just translate this what does this mean this means that the sine of theta equals negative the square root of two over two and what is the restriction theta must lie in the region negative pi over 2 to pi over 2. these restrictions on theta mean my angle must either lie in quadrant one or quadrant four i'm looking for the sine of theta being a negative number remember the sine of theta is the y-coordinate of points on the unit circle that positive in quadrant one and then negative in quadrant four so my angle must lie in quadrant four and here's the point whose y value or sine value is negative the square root of two over two so this is the angle that produces it but be careful when your answer is in quadrant four my answer cannot equal seven pi over 4 because that doesn't lie in the given range Find Negative Angle in QIV so what we have to do is find the angle that rotates in this direction a negative angle if you can look at the unit circle and figure that out in your head great but if not you take theta you set it equal to 7 pi over 4 and then you just subtract one rotation that is 2 pi from it subtracting fractions and whole numbers turn the whole number into a fraction by dividing by 1 need a common denominator that's going to be 4 so multiply numerator and denominator by four over four so i'm going to get seven pi over four minus two times four is eight pi over four so the inverse sine of negative the square root of 2 over 2 equals 7 pi over 4 minus 8 pi over 4 so that's negative 1 pi over four or just negative pi over four |
12915 | https://genent.cals.ncsu.edu/bug-bytes/thorax/ | Thorax – ENT 425 – General Entomology
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Thorax
The second (middle) tagma of an insect’s body is called the thorax. This region is almost exclusively adapted for locomotion — it contains three pairs of walking legs and, in many adult insects, one or two pairs of wings.
Structurally, the thorax is composed of three body segments:prothorax, mesothorax, and metathorax. These segments are joined together rigidly to form a “box” that houses the musculature for the legs and wings. Each segment has a dorsal sclerite, the notum (pronotum, mesonotum, and metanotum) which may be further subdivided into an anterior scutum and a posterior scutellum. The ventral sclerite of each segment is the sternum (prosternum, mesosternum, and metasternum).The side of each segment is called the pleuron — it is usually divided by a pleural suture into at least two sclerites:; an anterior episternum and a posterior epimeron.
The pleural suture marks the location of an internal ridge of exoskeleton (an apodeme) that strengthens the sides of the thorax. Ventrally, this apodeme forms a point of articulation with the basal leg segment (the coxa). In thoracic segments that bear wings, the pleural apodeme runs dorsally into the pleural wing process, a finger-like sclerite that serves as a pivot or fulcrum for the base of the wing.
A special “strut” of exoskeleton reinforces the ventral corners of each thoracic segment and provides a rigid site for attachment of leg muscles and ventral longitudinal muscles.This structure, called the furca, forms during development when a pair of sternal apophyses fuse internally with the ridge (apodeme) from each pleural suture. The points of invagination are often visible externally as furcal pits located near the midline of the sternum (and often joined by a sternacostal suture). This internal “brace” mechanism is similar in structure to the tentorium which serves a related function inside the head capsule.
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12916 | https://www.marmetal.com/wp-content/uploads/2017/12/C280-Insert.pdf | C28000 Muntz Metal, also known as 60/40, is copper alloyed with zinc. C280 Muntz is most commonly found in sheet and plate form. Cu Zn40 is considered a duplex alpha-plus-beta brass and has excellent hot-working properties but is less corrosion resistant than C464 Naval Brass. C28000 Muntz Metal is commonly used in architectural applications, including decorative panels, elevator doors, trim work and signage. Compared to the yellow tones of C260 Cartridge Brass, C280 Muntz has a warmer appearance that readily takes to polishing. Industrial applications for 280 brass include baffles and tubesheets for condensers and heat exchangers. Density @ 68o F 0.305 lb/in3 Melting Range 1645-1650o F Hot Formability Excellent Cold Formability Limited Machinability rating (C360 = 100) 45 Brazing Good Soldering Excellent Gas-shielded arc welding Fair Oxy-acetylene welding Good Carbon-arc welding Not recommended Coated metal-arc welding Not recommended Resistant welding: spot and seam Fair Resistance Welding: butt Good Temper Designation Tensile, min ksi (MPa) Tensile, max ksi (MPa) M20 As Hot-Rolled 40 (275) 55 (380) H01 Quarter-Hard 50 (345) 62 (425) H02 Half-Hard 58 (400) 70 (485) H03 Three-Quarter Hard 60 (415) 75 (515) H04 Hard 70 (485) 85 (585 H06 Extra Hard 82 (565) 95 (655) 2950 Turnpike Drive Hatboro, PA 19040 ph: 215-675-4645 fax: 215-675-9947 www.marmetal.com sales@marmetal.com UNS No. Copper Lead Iron Zinc C28000 59.0-63.0 0.09 max 0.07 max remainder C280 MUNTZ METAL ASTM B36 |
12917 | https://www.sciencedirect.com/science/article/pii/S0049384824003335 | Platelet collagen receptors and their role in modulating platelet adhesion patterns and activation on alternatively processed collagen substrates - ScienceDirect
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1. Introduction
2. Methods
3. Results
4. Discussion
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Declaration of competing interest
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Appendix A. Supplementary data
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Thrombosis Research
Volume 244, December 2024, 109201
Platelet collagen receptors and their role in modulating platelet adhesion patterns and activation on alternatively processed collagen substrates
Author links open overlay panel T.P.Lemmens a 1, Q.Luo a d e f 1, S.J.H.Wielders a, J.L.J.M.Scheijen b, S.Al-Nasiry c, R.R.Koenen a, P.Wenzel d e f, J.M.E.M.Cosemans a
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Is all collagen the same for platelet testing? Editorial on “Platelet collagen receptors and their role in modulating platelet adhesion patterns and activation on alternatively processed collagen substrates”
Thrombosis Research, Volume 247, March 2025, Pages 109255
Kristina Mott
Highlights
•Acid-soluble collagen induces more uniform platelet adhesion than HORM collagen.
•Acid-soluble collagen better investigates inhibition across multiple collagen receptors.
•Platelet activation by rat tail collagen is influenced by its form, soluble or immobilized.
•Acid-soluble collagen can substitute for HORM collagen in platelet activation studies.
Abstract
This study examines the roles of platelet collagen receptors glycoprotein VI (GPVI), α2β1, and the GPIb-IX-V complex in platelet activation and thrombus formation on various collagen sources from different species. Type I collagens standardly used in haematology testing, i.e. collagen type I derived from equine tendon (HORM) and rat tail collagen were evaluated. Moreover, acid soluble collagen from human umbilical cord was tested. To inhibit platelet-collagen interactions, combinations of monoclonal antibodies 6B4 and 6F1, targeting GPIbα and α2β1, respectively, were used, along with the therapeutic collagen receptor GPVI antibody glenzocimab. Our findings reveal distinct dependencies on these receptors: platelet aggregation of washed platelets to HORM collagen relied on both α2β1 and GPVI, to acid soluble collagen mainly on GPVI, and to rat tail collagen solely on α2β1, respectively. In whole blood perfusion assays under non-coagulating conditions, the acid soluble collagen surface triggered a more homogenous platelet adhesion when compared to the HORM collagen surface, whilst platelet adhesion on rat tail collagen varied considerably. The GPIb-IX-V complex was shown to play a key role in initial platelet adhesion and activation across all collagen surfaces at a shear rate of 1600 s−1. At 1600 s−1, inhibiting platelet α2β1 interaction with collagen by 6F1 antibody did not affect platelet thrombus formation on acid soluble collagen, while it did reduce platelet surface coverage and P-selectin expression on HORM collagen without changing the overall thrombus morphology or contraction. Inhibiting GPVI interaction with collagen significantly reduced all thrombus parameters and abolished PS exposure and P-selectin expression on all three collagen surfaces, at both 1600 s−1 and 150 s−1. Interestingly, upon investigating combined inhibition of GPIb and α2β1, an additive inhibitor effect of 6F1 was observed on P-selectin expression and PS-exposure on acid soluble collagen but not HORM collagen at 1600s−1, suggesting that the acid soluble collagen is well suited to study reinforcing functions of collagen receptors. Overall, this study highlights the potential advantages of using alternative collagen surfaces beyond the conventional HORM collagen to detect nuanced collagen receptor dependencies, which may prove valuable in evaluating anti-platelet medication.
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Keywords
Pepsin
Methylglyoxal modification
Collagen
Haematology
Thrombosis
1. Introduction
When vascular endothelial damage occurs, platelets encounter the exposed subendothelial collagen and other matrix proteins, initiating a cascade of events essential for thrombus formation and cessation of bleeding. Of the many subendothelial matrix proteins, type I collagen has been shown to be the most important structural protein that activates platelets during haemostasis . Three polypeptide chains (two α1, and one α2 chain) wind around each other and form the quaternary structure of collagen I. This spatial structuring of collagen has not only been shown to be key in initiating platelet activating by providing specific binding sites for glycoprotein VI (GPVI) but also allow for receptor clustering to enhance downstream signalling [2,3]. GPIb-IX-V complex interactions with collagen-bound von Willebrand factor (vWF) play a crucial role in initial platelet-collagen binding. At high shear or turbulent conditions vWF undergoes conformational changes that unfold its resting globular conformation and exposes binding sites . GPIb-IX-V tethers platelets to collagen-bound vWF, allowing the platelet collagen receptors, α2β1 and GPVI, to bind to the exposed collagen . In vitro studies using mostly HORM collagen and in vivo experimental thrombosis models show that platelet α2β1 and GPVI receptors each contributing uniquely to the process of platelet activation [6,7]. Studies on binding motifs of platelet collagen receptors revealed the GFOGER sequence to bind to α2β1 and the glycine-proline-hydroxyproline (GPO) sequence as the main binding site for GPVI . Collagen related peptides with specific amino acid sequences are used widely to stimulate platelets. Interestingly, for thrombus formation on human atherosclerotic plaque material, α2β1 was found not to be required . A study involving isolated collagen from the human heart suggested that the adhesion sites for α2β1 in cardiac collagen are limited, similar to the situation found in atherosclerotic plaques . It is currently unknown whether this collagen receptor dependency is similar for collagen isolated from healthy arterial tissue. Such information could be of relevance in light of anti-platelet therapy, which aims to selectively target thrombosis without interfering with haemostasis.
Our study aims to investigate the distinct roles played by the GPIb-IX-V complex as well as collagen receptors α2β1 and GPVI for platelet activation and thrombus formation on collagen substrates from different species and preparations. We assessed the responses of platelets exposed to collagen substrates routinely used in haematology testing and research in the haemostasis field, i.e. HORM collagen and rat tail collagen type I. As modification by pepsin could influence collagen structure and thus interactions with platelet receptors, we additionally included an acid soluble collagen type I substrate sourced from human arterial vessels obtained without any pepsin processing.
2. Methods
2.1. Raw materials and reagents
Human umbilical cord artery tissue was obtained with ethical approval from the local medical ethics committee (Maastricht University Medical Centre) and kept on ice until further processing. Rat tail collagen type I was from EMD Millipore (cat# 08-115), HORM collagen derived from equine tendon and chemically modified into a fibrillar structure was obtained from Takeda (Takeda GmbH, Austria). Humanised monoclonal antibody fragment glenzocimab (ACT017) was obtained from Acticor Biotech. Monoclonal antibody 6B4 against human GPIbα was a kind gift by Prof. K. Vanhoorelbeeke (KU Leuven, Leuven, Belgium). Monoclonal antibody 6F1, targeting α2β1, was from Prof. B.S. Coller (The Rockefeller University, New York, United States). Other chemical reagents if not specifically mentioned, were from Sigma-Aldrich.
2.2. Acid soluble collagen isolation
Acid soluble collagen was extracted according to the method of Miller et al. and Karami et al. with slight modifications [13,14]. All procedures were carried out at 4°C. From the umbilical cord, arteries were dissected and surrounding connective tissue was removed. To remove non-collagenous proteins, the dissected vessel tissues were soaked in 0.1 M sodium hydroxide at a sample/solution at ratio of 1:10 (w/v) with gentle stirring for 6 h, followed by water rinsing until the solution reached a neutral pH (7.0). Fat was removed by soaking in a 10% butyl alcohol solution ((1:8 (w/v)) overnight, refreshing solvent every 12 h. Samples were rinsed with distilled water, then homogenised, and extracted with 0.5 M acetic acid for 48 h. The solution was filtered and centrifuged at 48,400 g for 2 h. Collagen was precipitated by adding solid sodium chloride (NaCl) to a final concentration of 0.9 M. After 30 min of centrifugation at 48,400 g, the collagen-containing pellet was solubilized in 0.5 M acetic acid and dialyzed against 0.05 M acetic acid (refreshed every 6 h) for 2 days at 4°C, and finally, freeze-dried. Concentration collagen was determined by the amount of hydroxyproline measured using ultra-performance liquid chromatography tandem mass spectrometry (UPLC-MS/MS) . Advanced glycation end products (AGE) related lysine and arginine modifications were determined using the same technique.
2.2.1. Sodium dodecyl sulphate polyacrylamide gel electrophoresis and western blotting
Sodium dodecyl sulphate polyacrylamide gel electrophoresis (SDS-PAGE) was performed according to the method of Laemmli . Collagen samples were dissolved in 0.05 M acetic acid, mixed with sample buffer (40 mM Tris-hydrochloride[HCl] at pH 6.7, 2% SDS, 10% glycerol, 2% 2-mercaptoethanol, 0.01% bromophenol blue) and incubated at 95°C for 10 min. Denatured and reduced collagen (10 μg/lane) was loaded onto 4% stacking gel and 7% resolving gel by electrophoresis in Tris-HCl/glycine/SDS buffer (25 mM Tris, 192 mM glycine, 0.1%SDS, pH 8.3). The gel was stained with 0.1% Coomassie blue R-250 overnight and destained with distilled water: ethanol: acetic acid=8:1:1 (v/v/v) until a clear protein band was visible. Bands images were acquired by iBright Imaging system (Invitrogen).
2.2.2. Circular dichroism spectroscopy
The conformation and stability of the triple helical structure of collagen were measured by a circular dichroism (CD) spectrometer (Chirascan V100, Applied Photophysics, UK). Collagen samples were diluted in degassed 0.05 M acetic acid at 50 μg/mL and then placed into a flat quartz cell with a path length of 10 mm. The spectrum was recorded from 190 nm to 250 nm with an interval of 0.5 nm at 21°C. The spectra obtained were expressed in terms of mean residue ellipticity (deg·cm 2·dmol−1), which was calculated according to the equation:θ MRW,λ=θ λ×MRW 10×c×d where θ is the observed ellipticity (degrees) at the wavelength of λ, d is the pathlength (cm) and c is the concentration (g/mL) .
To determine the Td, [θ]221 was recorded at the fixed wavelength at 221 nm with heating from 4°C to 50°C at the rate of 1°C/min. The Td was determined as the point temperature at which the spectrums positive peak disappeared.
2.2.2.1. Blood collection and washed platelet preparation
Human blood was collected from healthy, drug-free volunteers after full informed consent was obtained, according to the Declaration of Helsinki. Procedures of human blood collection were approved from the local Medical Ethics Committee (Maastricht University). Venous blood was collected into 3.2% trisodium citrate Vacuette tubes (Greiner Bio-One, The Netherlands). Blood samples were centrifuged for 15 min at 258 g at room temperature to obtain platelet-rich plasma (PRP). Then, in the presence of 1:10 acid citrate buffer (acid-citrate-dextrose [ACD]; 80 mM trisodium citrate, 183 mM glucose, 52 mM citric acid), a platelet pellet was obtained after centrifugation of the PRP at 866 g for 15 min. The pellet was resuspended in modified 4-(2-hydroxyethyl)-1-piperazineethanesulfonic acid (HEPES) buffer pH 6.6 (136 mM NaCl, 10 mM HEPES, 2.7 mM potassium chloride, 2 mM magnesium dichloride, 0.1% glucose, and 0.1% bovine serum albumin [BSA]) supplemented with 1:1000 apyrase and 1:15 ACD. After centrifugation at 866 g for 15 min, the washed platelet pellet was acquired and resuspended in modified HEPES buffer pH 7.45. Platelet count was adjusted to 250×10 9/L for further use.
2.2.2.2. Light transmission aggregometry
Platelet aggregation of 250×10 9/L human washed platelets in response to different sources of collagen was evaluated in a light transmittance aggregometer (Chrono-log Corporation, USA) at 37°C under constant stirring. Collagens concentrations triggering suboptimal aggregation were determined for each donor, being 1 μg/mL for HORM collagen, between 0.2 and 0.3 μg/mL for acid soluble collagen, and 6.7–10 μg/mL rat tail collagen. Platelet aggregation was monitored for 10 min and amplitude in light transmission was quantified as the percentage of maximum platelet aggregation. When appropriate, blood samples were preincubated for 15 min with vehicle, anti-α2β1 antibody 6F1 (20 μg/mL) and/or anti-GPVI antibody glenzocimab (50 μg/mL).
2.2.3. Whole blood thrombus formation
Haematocrit and platelet count were within normal range for all blood donors. Platelet adhesion and thrombus formation on collagen substrates was assessed as previously described . Briefly, glass coverslips were coated with HORM collagen (50 μg/mL), acid soluble collagen (50 μg/mL), or rat tail collagen (500 μg/mL) overnight at 4°C. The coverslips were subsequently blocked with 1% BSA in HEPES buffer with 0.1% glucose (pH 7.45) for 30 min at room temperature, washed with saline, and mounted in a flow chamber. Citrated whole blood was perfused over the collagen coated coverslips for 5.5 min at shear rate of 1600 s−1 or 150 s−1 and labelled by perfusion with a buffer containing Alexa Fluor 647-Annexin-A5 (phosphatidylserine exposure [PS], 1:200) and FITC-anti-CD62P mAb (P-selectin expression, 1:40). Brightfield and multicolour fluorescence images were taken to monitor platelet adhesion and thrombus formation during whole blood perfusion by using EVOS fluorescence microscope (Olympus UPLSAPO 60× oil-immersion objective, Life Technologies, Carlsbad, CA, USA). Per donor, all control and intervention conditions were repeated in duplicates. Based on the images, thrombus formation parameters, i.e. the surface area covered with platelets (surface area coverage, SAC%), thrombus morphology (score), thrombus contraction (score), and multilayer score were determined as previously described . P-selectin expression and PS exposure were adjusted to reflect the percentage of platelets expressing the label relative to the total number of platelets covering the surface.
2.3. Statistical analysis
Experiments with washed platelets or whole blood were performed from at least three different donors. Statistical indications were made using the unpaired t-test for comparison between two groups, and one-way ANOVA test for comparison between multiple groups. Data were displayed as mean±standard deviation (SD). P<0.05 indicates statistically significance.
3. Results
3.1. Structural and amino acid modification analysis of HORM, acid soluble and rat tail collagen
To assess native structure and configuration of the acid soluble collagen, HORM collagen and rat tail collagen, SDS-PAGE and CD was performed (Fig. 1). SDS-PAGE revealed characteristic α1 and α2 bands around 100 kDa indicative of the three collagen polypeptides (Fig. 1A). UPLC-MS/MS showed that the amount of lysine and arginine was comparable between collagen sources, but glyoxal and methylglyoxal modification of lysine and arginine respectively were increased in HORM collagen compared to acid soluble- and rat tail collagen (supplemental Fig. 1). CD of the collagens was assessed at room temperature (Fig. 1B) as well from 4°C to 50°C to observe spectral changes during denaturation (Fig. 1C & D + supplemental Fig. 2). While the acid soluble collagen and the rat tail collagen showed similar spectra with a characteristic positive peak at around 220 nm and negative peak at 195 nm , HORM collagen deviated from this pattern with a more pronounced negative peak and lower positive peak. All three collagens denatured around 37°C, but HORM collagen showed an extra positive and negative peak at around 14°C (Fig. 1C & D). One difference between HORM collagen and the acid soluble human and rat collagen solutions, is that HORM collagen contains non-soluble fibrils, that may exhibit colloidal behaviour during denaturation. The altered CD spectra of HORM collagen could be indicative of such colloidal effects, where alterations in temperature affect the structure and dispersion of the particles, giving rise to temperature-dependent spectral changes. While usually more detailed information can be extracted from CD spectra from other proteins, collagens have a very distinct spectrum making it hard to use existing protein databases .
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Fig. 1. A) SDS-PAGE gel stained with Coomassie blue showing characteristic α1 and α2 bands at around 100 kDa. B) Circular dichroism spectrum at 22°C of HORM collagen, acid soluble collagen, and rat tail collagen. Denaturation spectra at their defined positive (C) and negative (D) peak of HORM collagen, acid soluble collagen and rat tail collagen. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
3.2. Collagen type I from distinct sources trigger platelet aggregation via different receptors
Acid soluble collagen was compared to conventionally used HORM collagen and rat tail collagen in platelet function assays. First, the aggregatory response of isolated washed platelets to the different collagens was measured with LTA (Fig. 2). Aggregation response to collagen was similar between HORM collagen and acid soluble collagen, while rat tail collagen showed an increase in lag time (p<0.0006) compared to HORM collagen. Inhibition of α2β1 by monoclonal antibody 6F1 had no effect on lag time or total aggregation to acid soluble collagen (Fig. 2E & H), however it did significantly increase lag time to HORM collagen (Fig. 2D), suggesting a crucial role of α2β1 in platelet shape change upon stimulation with HORM collagen but less so to acid soluble collagen. Interestingly, rat tail collagen-induced platelet aggregation was completely prevented upon incubation with 6F1 (Fig. 2C & I), suggesting total dependence on α2β1. Inhibition of the other platelet collagen receptor GPVI by glenzocimab almost completely prevented any aggregation to HORM collagen and acid soluble collagen and had no effect on the aggregation response to rat tail collagen (Fig. 2). As expected, combined inhibition of α2β1 and GPVI resulted in complete abolishment of any aggregatory response with all collagen substrates (Fig. 2G, H, I). In sum, although all types of collagens were able to induce a typical aggregation response in washed platelets, the relative contribution of different platelet receptors varied across different collagen sources.
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Fig. 2. Inhibitory effect of 6F1 and glenzocimab on collagen-induced platelet shape change and aggregation. Washed platelets were incubated with inhibitory antibodies for 15 min followed by agonist stimulation (A, D, G=HORM collagen, B, E, H=acid soluble collagen, C, F, I=rat tail collagen). Platelet aggregation was monitored by light transmission aggregometry. Shown are representative aggregation responses (A=HORM collagen, B=acid soluble collagen, C=rat tail collagen), lag time (D–F), and percentage maximal aggregation (G–I). #=lag time observed was >300 s, hence for visual representation purposes, 60 s was assigned as maximal value. Statistical analysis was done by ordinary one-way ANOVA, Mean and SD, n=4.
3.3. Acid soluble collagen well suited for resolution of the receptor dependency in thrombus formation under flow at high shear conditions
A series of in vitro flow chamber assays was conducted to assess the thrombogenicity of the different collagen sources under flow and the involvement of platelet collagen receptors. First, a dose response curve was established (supplemental Fig. 3) to find the suboptimal coating concentration of each collagen coating. Here for, human whole blood was perfused over 3 subsequent microspot coatings of collagen substrates for 5 min. The following concentrations were selected as they submaximal triggered the formation of contracting thrombi: HORM collagen 50 μg/mL, acid soluble collagen 50 μg/mL, and rat tail collagen 500 μg/mL. In contrast with the HORM collagen surface, the acid soluble collagen surface resulted in a much more homogenous platelet adhesion, whilst platelet adhesion on rat tail collagen varied considerably between experiments (supplemental Fig. 3). The rat tail collagen appeared to be less thrombogenic as the other two surfaces and did not elicit any PS-exposure on platelets and limited platelet granule release as measured with P-selectin expression (Fig. 3, Fig. 4& supplemental Figs. 6, 8). Increasing rat tail collagen coating concentration did not lead to a higher thrombogenicity (supplemental Fig. 3).
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Fig. 3. Effect of platelet collagen receptor inhibition on parameters of thrombus formation on various collagen surfaces at shear of 1600 s−1. Blood samples pre-incubated with vehicle (control) or relevant platelet receptor inhibitors (6B4=anti-GPIb, 6F1=anti-α2β1, glenzocimab=anti-GPVI), and flowed over HORM collagen (A/D), acid soluble collagen (B/E) and rat tail collagen (C/F) surfaces. Quantification of images for surface area covered by platelets (A–C). Raw values per blood donor and condition were first averaged and then per parameter univariately scaled (0−100) across surfaces. Shown are heatmaps of scaled parameters (surface area coverage [SAC], morphological score, contraction score, PS-exposure, P-selectin expression) for all three collagen surfaces (D–F).Colour code indicates scale values between 0=blue (no platelet adhesion and/or activation) and 100=yellow (highest amount of platelet adhesion and/or activation in this dataset). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 4. Effect of platelet collagen receptor inhibition on parameters of thrombus formation on various collagen surfaces at shear of 150 s−1. Blood samples pre-incubated with vehicle (control) or relevant platelet receptor inhibitors (6F1=anti-α2β1, glenzocimab=anti-GPVI), and flowed over HORM collagen (A/D), acid soluble collagen (B/E) and rat tail collagen (C/F) surfaces. Quantification of images for surface area covered by platelets (A-C). Raw values per blood donor and condition were first averaged and then per parameter univariately scaled (0–100) across surfaces. Shown are heatmaps of scaled parameters (surface area coverage [SAC], morphological score, contraction score, PS-exposure, P-selectin expression) for all three collagens (D–F)). Colour code indicates scale values between 0=blue (no platelet adhesion and/or activation) and 100=yellow (highest amount of platelet adhesion and/or activation in this dataset). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Inhibitors against GPIbα, α2β1, and GPVI were added to the whole blood prior to perfusion over the collagen-coated coverslips. As expected, inhibiting GPIb-vWF interaction by monoclonal antibody 6B4 at high shear (1600s−1) resulted in significant reduction of thrombus size (morphological score), contraction, and platelet surface area coverage (%SAC) on all collagen surfaces (Fig. 3& supplemental Figs. 4–6). These data highlight the essential role of the GPIb-IX-V complex in platelet adhesion on collagen surfaces. In line with this, 6B4 addition also resulted in a significant reduction of P-selectin expression on a HORM collagen and acid soluble collagen (Fig. 3, supplemental Fig. 6). Interestingly, a significant reduction in PS-exposure was observed on HORM collagen surface after 6B4 treatment, but not on acid soluble collagen (Fig. 3, supplemental Fig. 6).
When α2β1 interaction with collagen was inhibited with the monoclonal antibody 6F1, no change in thrombus morphology or %SAC was observed with acid soluble collagen at 1600 s−1 (Fig. 3). This contrasts with HORM collagen where a reduction in %SAC (P=0.0011) and no change in morphology was measured (Fig. 3, supplemental Fig. 6). Compared to control, no effect on granule secretion or PS-exposure was observed on any of the included collagen surfaces (Fig. 3, supplemental Fig. 6). Glenzocimab treatment of the whole blood, inhibiting GPVI interaction with collagen, resulted in a similar inhibitory pattern on HORM collagen and acid soluble collagen. While no significant reduction in %SAC was observed on acid soluble collagen following GPVI inhibition (Fig. 3), all other thrombus parameters were significantly reduced (supplementary Fig. 6). In line with previous literature , glenzocimab abolished all PS-exposure and significantly reduced P-selectin expression on all surfaces (Fig. 3, supplementary Fig. 6).
As previously reported, deficiency in GPVI can be partially overcome by the overlapping function of other collagen receptors . Therefore, we investigated the effect on thrombus formation on collagen surfaces when more than one receptor was blocked. Since 6B4 already abolished almost all platelet adhesion at 1600 s−1 on HORM collagen and rat tail collagen, no additive effect of inhibitors against α2β1 and/or GPVI could be established on platelet surface coverage (Fig. 3). On acid soluble collagen however, an additive inhibitor effect of 6F1 combined with 6B4 could be observed in P-selectin expression and PS-exposure (Fig. 3, supplementary Fig. 6). While combined inhibition of glenzocimab and 6F1 reduced surface area coverage less than when treated with 6B4, it resulted in an almost complete absence of P-selectin, PS-exposure, and contraction of thrombi on acid soluble collagen (Fig. 3). 6F1 alone did not reduce any measured parameters. However, combined with 6B4, it did result in a significant reduction of P-selectin exposure and a strong trend (p=0.0943) towards reduced %SAC (Fig. 3B) and contraction score (supplementary Fig. 6). These data suggest that acid soluble collagen allows for experimental setups to investigate these combined treatment effects, which are valuable to understanding the overlapping or reinforcing functions of these collagen receptors.
3.4. Platelet α2β1 and GPVI modulate different thrombus formation parameters on collagen surfaces at low shear conditions
Thrombus formation under flow was also measured at a lower shear (150 s−1). Thrombi formed on HORM collagen were morphologically comparable those formed on acid soluble collagen (Fig. 4, supplementary Figs. 7, 8). Similar to higher shear conditions, rat tail collagen appears to be a less thrombogenic collagen substrate for flow experiments. Rat tail collagen did not elicit any platelet PS-exposure and only minimal platelet P-selectin expression (Fig. 4, supplementary Fig. 8). Inhibitory effects of glenzocimab and 6F1 were comparable between HORM collagen and acid soluble collagen across all thrombus parameters. 6B4 was not added as GPIb-VWF interaction is most relevant at higher shear rates . Percentage surface area coverage was significantly reduced on HORM collagen and rat tail collagen upon α2β1 inhibition with 6F1 (Fig. 4). Platelet activation during thrombus formation, as measured by P-selectin expression, was significantly reduced by 6F1 on both HORM collagen and acid soluble collagen (Fig. 4). Morphological score was unaffected by single inhibition of α2β1 or GPVI (supplementary Fig. 8). Treatment with the GPVI inhibitor glenzocimab, but not the α2β1 inhibitor 6F1, resulted in significantly decreased contraction scores on HORM collagen and acid soluble collagen (supplementary Fig. 8). Similar to 1600 s−1 shear rate, glenzocimab abolished all PS-exposure on all collagen substrates (Fig. 4, supplementary Fig. 8).
The combination of glenzocimab and 6F1 also resulted in a complete reduction of P-selectin (supplementary Fig. 8). In contrast with the higher shear condition, α2β1 seems to play a more prominent role in thrombus formation on acid soluble collagen in thrombus formation at 150 s−1 shear (Fig. 3, Fig. 4). These data suggests that at a shear of 150 s−1 both collagen receptor α2β1 and GPVI are required for thrombus formation but have a less overlapping role compared to higher shear (Fig. 3, Fig. 4).
4. Discussion
In this study the involvement of platelet collagen receptors α2β1 and GPVI, and GPIb-IX-V in platelet activation and thrombus formation across three distinct collagen sources, HORM collagen, acid soluble collagen from human umbilical cord arteries, and rat tail collagen, was investigated. We found that the acid soluble collagen isolated from human umbilical cord arteries triggered more homogenous platelet adhesion than HORM collagen and rat tail collagen. The acid soluble collagen showed different platelet receptor dependencies compared to the commonly used HORM collagen and rat tail collagen. For instance, the GPVI inhibitor glenzocimab had a more pronounced inhibitory effect on thrombus formation and platelet activation on the acid soluble collagen compared to HORM collagen. In addition, the acid soluble collagen allowed for better investigation of the combined effects of inhibiting multiple collagen receptors.
Often, HORM collagen is used as a surrogate for fibrillar collagen type I. However, because it also contains collagen type III, it is uncertain whether the observed interactions are exclusively due to collagen type I . In contrast to HORM collagen and the rat tail collagen used in this study, the acid soluble collagen used was isolated from human umbilical cord arteries with minimal chemical or enzymatic modification using pepsin. Although pepsin is useful in collagen isolation due to its ability to cleave peptide bonds, it can alter collagen structure and conformation [26,27]. Small degradation of collagen could already have a large impact on its structure with potential implications for platelet function and thrombus formation. Indeed, prior studies already demonstrated the differential importance of α2β1 and GPVI in platelet shape change between pepsin-digested collagens and HORM collagen . Next to collagen degradation, there are other factors influencing collagen-platelet receptor interaction. For instance, the α2β1 integrin binds primarily to the GFOGER motif on collagen. Variations in the surrounding of this amino acid sequence and the overall conformation of the collagen molecule can influence how α2β1 binds to this motif . During hyperglycaemia, modifications of collagen residues with MG-H1 results in decreased adhesion through α2β1 integrin in endothelial cells on collagen . This modification or similar advanced glycation end products (AGEs) can alter structure or functions of various proteins in patients with hyperglycaemia. In this paper, HORM collagen contained more CML and MG-H1 modifications than acid soluble collagen and rat tail collagen, potentially due to the glucose in HORM collagen buffer. It could be hypothesised that these modifications could also alter α2β1 integrin binding to platelets. However, our data shows less dependency on α2β1 on acid soluble collagen when compared to HORM collagen. Nonetheless, it would still be interesting to investigate how modifications of collagens can change platelet receptor binding and collagen dependent platelet activation. GPVI primarily interacts with GPO repeats in collagen and the density and spacing of these repeats, can differ between collagen from different species . The density and spacing can modulate the strength of the interaction between collagen and GPVI . In addition, there is a known difference in receptor dependency between static and flow-based platelet assays. For example, immobilized collagen peptides containing a α2β1 binding motif could trigger thrombus formation under flow but did not activate platelets under static conditions . Similarly, the authenticity of in vivo GPVI binding sites in collagen will be critically dependent on the exposure of helix segments on the fibre surface at areas of the vascular lesion . Taken together, the structural differences between the tested collagens that arise from species and production differences could contribute to the here observed difference in receptor contribution for the different collagen sources.
Over the past few years, researchers gained valuable insights into the role of α2β1 in thrombus formation, but its definitive role is still not completely defined. Kuijpers et al. demonstrated that β1-deficient platelets are still able to adhere and form thrombi under high shear conditions, albeit more looser aggregates . Jarvis et al. observed a delayed collagen dependent aggregation when platelets were treated with the α2β1 inhibitor 6F1. Furthermore, they demonstrated that α2β1 inhibition abolished shape change induced by pepsin digested bovine collagen type I but only partially inhibited the response to HORM collagen . In our whole blood perfusion assay, α2β1 inhibition by mAb 6F1 on acid soluble collagen was not effective in decreasing platelet surface area coverage and platelet activation as measured by P-selectin, PS, or contraction scores at high shear of 1600s−1. On HORM collagen, platelet surface area and P-selectin expression were reduced at 1600s−1 by α2β1 inhibition. At a lower shear of 150 s−1 it is expected for α2β1 to play a more significant role in the initial adhesion to collagen surfaces, since vWF does not play a major role in bridging the gap between GPIb-IX-V complex and collagen . This was indeed observed for the collagens tested here. In contrast to a previous report , we did not find an effect of α2β1 blockage on collagen-dependent platelet procoagulant activity on any of the tested collagen surfaces at 150 s−1 or 1000 s−1. A potential explanation could be a different level of PS exposure at baseline between setups, making it sensitive to a greater or lesser extent to pick up inhibitory effects. An advantage of the acid soluble collagen is that it allowed for combined collagen receptor inhibition in contrast to HORM collagen where these subtle differences could not be distinguished in the present flow chamber setup. While 6F1 alone did not result in lower P-selectin or PS exposure on either HORM collagen or acid soluble collagen, when combined with 6B4, a significant reduction was still detectable at 1600 s−1. Our data reinforce findings that both α2β1 and GPIb-vWF interactions with collagen are responsible for initial adhesion at high shear and their functions overlap [6,35].
In line with previous studies, decreased platelet adhesion and activity on HORM collagen were observed upon GPVI inhibition with the novel humanised mAb fragment glenzocimab , at both high and low shear conditions, supporting the hypothesis that GPVI also has a central role in the context of venous thrombosis . Reduced platelet activity was also observed on acid soluble collagen. However, this did not result in fewer platelets adhering to the actual collagen surface. It should be noted that glenzocimab treatment resulted in a distinct thrombus phenotype with a looser structure and less activated platelets on acid soluble collagen compared to control. We speculate that this effect could be due to the difference in collagen fibre coating: HORM coating has large visible tangled fibres where platelets tend to form thrombi, while the acid soluble coating is more uniform without visible fibres, allowing for more homogeneous platelet adhesion. This variability highlights the importance of carefully selecting the collagen source when testing anti-platelet therapeutics, as different sources may yield divergent results.
Our experiments were conducted under non-coagulating conditions to focus solely on platelet-collagen interaction. GPVI can bind to several other matrix proteins including fibrin(ogen), which especially under coagulating conditions can contribute to the haemostatic process and thrombus formation . GPVI has been implicated in interactions with fibrin(ogen) and neutrophils, contributing to a thrombo-inflammatory state that drives venous thrombosis . Finally, substrate coating of rat tail collagen in our whole blood flow assays led to a variable adhesion of platelets, which is not ideal and could be due to an inhomogeneous coating. However, the results we could obtain were in line with a previous knockout study in mice which showed under static conditions a complete lack of adhesion of α2β1 deficient platelets . Our group previously employed a spin coating method which could potentially solve issues with rat tail collagen micro spot coating .
Overall, our findings support the use of primary human collagen I for performing translational bioassays to investigate platelet adhesion. Acid soluble collagen from human arteries allows for investigating the combined effects of inhibiting different collagen receptors on platelet adhesion and activation, which can be valuable to understanding the overlapping or reinforcing functions of these collagen receptors.
Funding
T.P. Lemmens and J.M.E.M. Cosemans are supported by the Netherlands Organization for Scientific Research (NWO; Vidi 91716421). Q. Luo, R. Koenen, P. Wenzel, J.M.E.M. Cosemans are supported by the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 813409. J.M.E.M. Cosemans and R. Koenen are funded by a BHF-DZHK-DHF International Cardiovascular Research Partnership Award (02-001-2022-0124).
CRediT authorship contribution statement
T.P. Lemmens: Writing – review & editing, Writing – original draft, Visualization, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. Q. Luo: Writing – original draft, Visualization, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. S.J.H. Wielders: Writing – review & editing, Investigation. J.L.J.M. Scheijen: Writing – review & editing, Investigation. S. Al-Nasiry: Writing – review & editing, Resources. R.R. Koenen: Writing – review & editing, Visualization, Supervision, Investigation, Conceptualization. P. Wenzel: Writing – review & editing, Supervision, Conceptualization. J.M.E.M. Cosemans: Writing – review & editing, Writing – original draft, Visualization, Supervision, Funding acquisition, Data curation, Conceptualization.
Declaration of competing interest
The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Judith Cosemans reports financial support was provided by Netherlands Organization for Health Research and Development. Judith Cosemans reports financial support was provided by European Commission Marie Sklodowska-Curie Actions. Judith Cosemans reports financial support was provided by British Heart Foundation. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
We greatly appreciate the gift of the 6B4 antibody from Prof. K. Vanhoorelbeeke and 6F1 antibody from Prof. B.S. Coller.
Appendix A. Supplementary data
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References
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12918 | https://www.reddit.com/r/math/comments/jykezg/involutions_on_%E2%84%9D_and_beyond/ | Involutions on ℝ and beyond : r/math
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Involutions on ℝ and beyond
I just watched a Dr Peyam's video on the functional equation f(f(f(x))) = x. While this video was about the functional equation f^3(x) =x, Dr Peyam was kind enough to give a few examples of involutions ( f(f(x)) = x ) on ℝ. Dr P gave the examples of Id, f(x) = a - x, f(x) = 1 / x and f(x) = -1 / x. He also gave the example log( (e^x + 1) / (e^x - 1) ). which I managed to prove is indeed an involution. This last couple of example reminded me of Möbius transformations. Are there involutive(?) Möbius transformations on ℝ, other than the trivial +- 1 / x? Are there more functions than the above mentioned that behave like involutions on ℝ?
Also, I think I have proved that f is an involution on ℝ implies both that f^-1(f(x)) and f(f^-1(x)) =x. Is this true?
Furthermore, what about involutions on ℂ?
Edit: Just found this. Looks like the involutions are dense in the set of functions from ℝ to ℝ. I don't really know what to say about that. I would probably have been surprised to find that there was a bijection from the natural numbers to involutions on ℝ, I guess. With hindsight = 20/20.
~~Edit 2: If we can find a family of Möbius transformation coefficients a(y), b(y), c(y) and d(y) for the Möbius transformation M(x): x = (a(y) + b(y)x) / (c(y) + d(y)x), and M(M(x)) = x, we are essentially home free for y ε ℝ. This would suffice to disprove that the the involutions on ℝ are countable.~~ This is wrong if all functions a(), b(), c() and d() are just constant functions. Let a(y) = 1, b(y) = 1, c(y) = -1, d(y) = 1. This gives one involution (see comment below), not uncountably many.
Edit 3: It just struck me that for the trivial Möbius transformation f(x) = a - x, a ε ℝ, we are indeed at home. There are, trivially, uncountably many involutions on ℝ. But what if we parametrize all the possible involutions?
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Applications of topology in computer science
Topology, a branch of mathematics that studies the properties of geometric objects that are preserved under continuous deformations, has several applications in computer science. Here are some notable areas where topology is utilized:
Topological Data Analysis (TDA)
TDA is a method that uses concepts from topology to analyze the shape of data. It is particularly useful for understanding complex, high-dimensional data.
Data Clustering and Feature Extraction: TDA can identify clusters and patterns in data by analyzing its topological features, such as connected components, loops, and voids. "It's a gimmicky form of clustering. There may be niche applications in things that have a flavor of fluid dynamics to them..."
Image Processing: TDA has been used to analyze images, for example, by identifying different regions in brain images. "I did a school DL project with self organizing map AEs..."
Bioinformatics: TDA can be applied to analyze biological data, such as gene expression data, to identify patterns and relationships. "For instance, this TDA paper appeared in Nature (!) two days ago..."
Robotics and Motion Planning
Topology is used to understand the configuration spaces of robotic arms and other systems, which can be complex and high-dimensional.
Configuration Space Analysis: By modeling the possible positions and orientations of a robotic arm as a topological space, one can plan paths that avoid obstacles. "In robotics, topology is also used a bit in relation to the configuration space of a robotic arm..."
Network Analysis
Topology can help in understanding the structure and properties of networks, such as social networks or communication networks.
Network Structure: TDA can be used to identify communities and important nodes in a network by analyzing its topological features. "Not really data science but when I was at NASA my work involved applying algebraic topology to fault-tolerant distributed computing..."
Cryptography
Algebraic structures, including finite fields and group theory, which are related to topology, are central to cryptography.
Encryption and Security: These algebraic structures are used to create secure encryption algorithms. "Algebra is very central to cryptography."
Theoretical Computer Science
Topology and related fields like category theory are used to formalize and understand programming languages and systems.
Domain Theory: Topology is used to define the semantics of programming languages, particularly in functional programming. "Denotational semantics was created as a topic when Dana Scott demonstrated how to interpret the lambda calculus in continuous lattices..."
Homotopy Type Theory: This theory, which combines type theory with homotopy theory from algebraic topology, is used in the development of new programming languages and automated proof verifiers. "There are also things like Homotopy type theory and automated proof verifiers."
Machine Learning
Topology is increasingly being integrated into machine learning to improve model performance and interpretability.
Topological Machine Learning: Techniques like persistent homology are used to extract features from data that can improve the performance of machine learning models. "Deep Learning with Topological Signatures"
Manifold Learning: Understanding the underlying manifolds of data can help in dimensionality reduction and feature learning. "Look up UMAP. It is a topological approach to dimension reduction and is arguably state of the art."
Practical Considerations
While topology has many potential applications, it is still a relatively new field in computer science, and its adoption can be slow.
Complexity and Scalability: Some topological methods can be computationally intensive and may not scale well to large datasets. "Efficiently computing homology groups is not something I've ever seen scale to larger datasets."
Niche Applications: Currently, TDA and other topological methods are often used in niche applications and may not be universally applicable. "They're very niche; researchers have found uses for them in very specific scientific domains..."
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12919 | https://www.who.int/teams/global-malaria-programme/case-management/drug-efficacy-and-resistance/tools-for-monitoring-antimalarial-drug-efficacy | Skip to main content
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Drug efficacy and resistance
Tools for monitoring antimalarial drug efficacy
Monitoring antimalarial drug efficacy
Antimalarial drug efficacy database
Tools for monitoring antimalarial drug efficacy
Routine monitoring of antimalarial drug efficacy is necessary for effective case management and detection of resistance. WHO has developed a series of tools to facilitate the work of national malaria programmes and other partners involved in routine testing
of antimalarial drug efficacy.
Template protocol for therapeutic efficacy studies
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This template is an updated version of the template published in Annex 1 of the Methods for surveillance of antimalarial drug efficacy. It has been reviewed by the WHO Research Ethics Review Committee and adheres to current ICH guidelines.
Polymerase chain reaction (PCR) for identifying parasite populations
TES treatment outcomes are determined based on an assessment of parasitological and clinical outcomes. Genotyping of parasite populations, or PCR-correction, allows for study investigators to distinguish between true recrudescence (i.e. when the parasites which caused the initial infection escape the treatment and resurface) and a new infection, which can occur when antimalarial drug concentrations fall below the level required to suppress a new infection. PCR-corrected treatment failure rates are now the preferred end-point in regulatory clinical trials and for monitoring antimalarial drug efficacy. An informal consultation on the methodology used to distinguish reinfection from recrudescence in high malaria transmission areas was held in May 2021 to revise the previous guidance from 2007.
11 November 2021
### Informal consultation on methodology to distinguish reinfection from recrudescence in high malaria transmission...
This report summarizes the findings and conclusions of a virtual meeting on the methodology to distinguish reinfection from recrudescence in high malaria...
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WHO data entry and analysis tool
In conjunction with the WHO standard protocol for monitoring antimalarial drug efficacy, WHO has developed an electronic data collection form to standardize data entry and analysis of therapeutic efficacy study data. Separate versions are available for
the 28-day and 42-day studies. The program can be used in areas of both high and low-to-moderate malaria transmission. Instructions are provided within the Excel program, on the reference guide worksheet.
Data entry forms 28-day studies
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Template checklists for quality control monitoring
Quality control monitoring is an essential part of the conduct of high quality therapeutic efficacy studies. By visiting study sites at the beginning, during and at the end of a study, quality control monitors help to ensure that a study site is appropriate,
that clinical and laboratory resources are adequate, that data processing is complete and that reporting is underway.
To facilitate this process, WHO has developed 3 separate template checklists for quality control monitoring. The templates can be used as provided or adapted according to local conditions and sponsor requirements.
Pre-study visit monitoring checklist
Pre visit checklist in English| Pre visit checklist in French
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Interim visit checklist in English |
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Parasite clearance estimator
Artemisinin resistance typically refers to a delay in the clearance of malaria parasites from the bloodstream following treatment with an artemisinin-based combination therapy (ACT). WHO has developed a parasite clearance estimator tool, which provides
additional information regarding the changes to the clearance rate. This Excel-based tool incorporates all phases of the parasite clearance curve: the initial lag phase, the slope, and the terminal tail phase. Each phase of the curve is considered
to represent key biological processes, and may also be useful if the genetic heritability of artemisinin resistance is confirmed. This tool remains valid even when the parasite count is low.
Parasite clearance estimator | User's guide
Chemoprevention efficacy studies (CPES)
Preventive chemotherapy is the use of medicines, either alone or in combination, to prevent malaria infection and its consequences. It requires giving a full treatment course of an antimalarial medicine, often at predefined intervals to individuals
who have not been diagnosed with malaria. By providing antimalarial medicine to vulnerable populations, existing undiagnosed malaria infections are treated, and the medicine provides a period of protection against new infections.
Information on the efficacy of the medicines as used in these malaria chemoprevention strategies is critical for ensuring that the strategies remain effective in different settings with different levels of drug resistance.
A new publication, Malaria chemoprevention efficacy study protocol, adapts some of the principles and practices underlying treatment efficacy monitoring to provide standardized approaches for monitoring and evaluating the efficacy of medicines used for malaria chemoprevention. An update of this document will be done when additional experience is gained from studies of chemoprevention efficacy.
Chemoprevention is the use of medicines, either alone or in combination, to prevent malaria infection and its consequences. This publication provides standardized...
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12920 | https://sites.millersville.edu/bikenaga/number-theory/linear-diophantine-equations/linear-diophantine-equations.html | Linear Diophantine Equations
Linear Diophantine Equations
A Diophantine problem is one in which the solutions are required to be integers. Abusing terminology, I'll refer to Diophantine equations, meaning equations which are to be solved over the integers.
For example, the equation has many solutions over the reals. Here's a solution:
However, this equation has no nonzero integer solutions. This is a special case of Fermat's Last Theorem.
On the other hand, the following equation has infinitely many integer solutions:
and are examples of solutions.
In this section, I'll look at equations like the last one. They're called linear Diophantine equations.
Theorem. Let . Consider the Diophantine equation
(a) If , there are no solutions.
(b) If , there are infinitely many solutions of the form
Here is a particular solution, and .
If you've had a course in differential equations, you may have seen something like this. and give a general solution to the homogeneous equation
is a particular solution to . Their sum gives a general solution to the given (nonhomogeneous) equation.
Before I give the proof, I'll give some examples, and also discuss the three variable equation .
Example. Solve .
Since , there are infinitely many solutions. Divide the equation by 3 to get
By inspection, and is a particular solution. Hence, the general solution is
For example, setting produces the solution , .
In general, you may not be able to see a particular solution by inspection. In that case, you can use the Extended Euclidean algorithm to generate one. We'll see how to do this in examples that follow.
Example. Solve .
Since , the equation has no solutions.
Example. Find all the solutions to the following Diophantine equation for which x and y are both positive.
, so there are solutions.
It is too hard to guess a particular solution, so I'll use the Extended Euclidean algorithm:
Matching this with the given equation , I see that is a particular solution. The general solution is
I want solutions for which x and y are both positive. So
The integers which satisfy both of these inequalities are . Here are the values of x and y:
The solutions are , , and .
The requirement that the solutions be positive can come up in real-world problems.
Example. Phoebe buys large shirts for $18 each and small shirts for $11 each. The shirts cost a total of $1188. What is the smallest total number of shirts she could have bought?
Let x be the number of large shirts and let y be the number of small shirts. Then
Since , there are solutions.
I'll use the Extended Euclidean algorithm to get a particular solution:
and is a particular solution. The general solution is
Since the number of shirts can't be negative, I have and .
Thus, .
The total number of shirts is
For , this is smallest for , which gives
She bought 66 large shirts, no small shirts, and a total of 66 shirts.
Consider a 3-variable equation
The equation has solutions if and only if . If it has solutions, there will be infinitely many, determined by two integer parameters.
You can solve a 3-variable equation by reducing it to a 2-variable equation. Group the first two terms and factor out the greatest common divisor of their coefficients. Introduce a new variable, defining it to be what is left after the greatest common divisor is factored out. The new equation is a 2-variable Diophantine equation, which you can solve using the method described earlier.
Example. Find the general solution to the following Diophantine equation.
Let .
and is a particular solution. So
Then
and is a particular solution. The general solution is
A general linear Diophantine equation has the form
There are solutions if . If there is a solution, it will in general have parameters --- exactly as you'd expect from linear algebra.
Here's the proof of the theorem for the two-variable case.
Proof. (two variable case) Consider the linear Diophantine equation
Case 1. Suppose . If x and y are solutions to the equation, then
This contradiction shows that there cannot be a solution.
Case 2. Suppose . Write for . There are integers m and n such that
Then
Hence, , , is a solution.
Suppose , , is a particular solution. Then
This proves that , is a solution for every .
Finally, I want to show that every solution has this form. Suppose then that is a solution. Then and imply
Therefore,
Now divides the left side, so it divides the right side. However, . Therefore,
Thus,
Substitute back into the last x-y equation above:
Thus,
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14.4: The Change of Concentration with Time (Integrated Rate Laws)
Last updated
: Jul 12, 2023
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14.3: Concentration and Rates (Differential Rate Laws)
14.5: Temperature and Rate
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: 25178
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Learning Objectives
To apply rate laws to zeroth, first and second order reactions.
Either the differential rate law or the integrated rate law can be used to determine the reaction order from experimental data. Often, the exponents in the rate law are the positive integers: 1 and 2 or even 0. Thus the reactions are zeroth, first, or second order in each reactant. The common patterns used to identify the reaction order are described in this section, where we focus on characteristic types of differential and integrated rate laws and how to determine the reaction order from experimental data. The learning objective of this Module is to know how to determine the reaction order from experimental data.
Zeroth-Order Reactions
A zeroth-order reaction is one whose rate is independent of concentration; its differential rate law is
rate=k.
rate=k.
We refer to these reactions as zeroth order because we could also write their rate in a form such that the exponent of the reactant in the rate law is 0:
rate=−Δ[A]Δt=k[reactant]0=k(1)=k
rate=−Δ[A]Δt=k[reactant]0=k(1)=k(14.4.1)
Because rate is independent of reactant concentration, a graph of the concentration of any reactant as a function of time is a straight line with a slope of −k−k. The value of kk is negative because the concentration of the reactant decreases with time. Conversely, a graph of the concentration of any product as a function of time is a straight line with a slope of kk, a positive value.
The integrated rate law for a zeroth-order reaction also produces a straight line and has the general form
[A]=[A]0−kt
[A]=[A]0−kt(14.4.2)
where [A]0[A]0 is the initial concentration of reactant AA. Equation 14.4.214.4.2 has the form of the algebraic equation for a straight line,
y=mx+b,
y=mx+b,
with y=[A]y=[A], mx=−ktmx=−kt, and b=[A]0b=[A]0.)
Units
In a zeroth-order reaction, the rate constant must have the same units as the reaction rate, typically moles per liter per second.
Although it may seem counterintuitive for the reaction rate to be independent of the reactant concentration(s), such reactions are rather common. They occur most often when the reaction rate is determined by available surface area. An example is the decomposition of N2O on a platinum (Pt) surface to produce N2 and O2, which occurs at temperatures ranging from 200°C to 400°C:
2N2O(g)Pt→2N2(g)+O2(g)
2N2O(g)−→Pt2N2(g)+O2(g)(14.4.3)
Without a platinum surface, the reaction requires temperatures greater than 700°C, but between 200°C and 400°C, the only factor that determines how rapidly N2O decomposes is the amount of Pt surface available (not the amount of Pt). As long as there is enough N2O to react with the entire Pt surface, doubling or quadrupling the N2O concentration will have no effect on the reaction rate. At very low concentrations of N2O, where there are not enough molecules present to occupy the entire available Pt surface, the reaction rate is dependent on the N2O concentration. The reaction rate is as follows:
rate=−12(Δ[N2O]Δt)=12(Δ[N2]Δt)=Δ[O2]Δt=k[N2O]0=k
rate=−12(Δ[N2O]Δt)=12(Δ[N2]Δt)=Δ[O2]Δt=k[N2O]0=k(14.4.4)
Thus the rate at which N2O is consumed and the rates at which N2 and O2 are produced are independent of concentration. As shown in Figure 14.4.214.4.2, the change in the concentrations of all species with time is linear. Most important, the exponent (0) corresponding to the N2O concentration in the experimentally derived rate law is not the same as the reactant’s stoichiometric coefficient in the balanced chemical equation (2). For this reaction, as for all others, the rate law must be determined experimentally.
A zeroth-order reaction that takes place in the human liver is the oxidation of ethanol (from alcoholic beverages) to acetaldehyde, catalyzed by the enzyme alcohol dehydrogenase. At high ethanol concentrations, this reaction is also a zeroth-order reaction. The overall reaction equation is
where \ce{NAD^{+}}) (nicotinamide adenine dinucleotide) and NADHNADH (reduced nicotinamide adenine dinucleotide) are the oxidized and reduced forms, respectively, of a species used by all organisms to transport electrons. When an alcoholic beverage is consumed, the ethanol is rapidly absorbed into the blood. Its concentration then decreases at a constant rate until it reaches zero (Figure 14.4.3a14.4.3a). An average 70 kg person typically takes about 2.5 h to oxidize the 15 mL of ethanol contained in a single 12 oz can of beer, a 5 oz glass of wine, or a shot of distilled spirits (such as whiskey or brandy). The actual rate, however, varies a great deal from person to person, depending on body size and the amount of alcohol dehydrogenase in the liver. The reaction rate does not increase if a greater quantity of alcohol is consumed over the same period of time because the reaction rate is determined only by the amount of enzyme present in the liver. Contrary to popular belief, the caffeine in coffee is ineffective at catalyzing the oxidation of ethanol. When the ethanol has been completely oxidized and its concentration drops to essentially zero, the rate of oxidation also drops rapidly (part (b) in Figure 14.4.314.4.3).
These examples illustrate two important points:
In a zeroth-order reaction, the reaction rate does not depend on the reactant concentration.
A linear change in concentration with time is a clear indication of a zeroth-order reaction.
First-Order Reactions
In a first-order reaction, the reaction rate is directly proportional to the concentration of one of the reactants. First-order reactions often have the general form A → products. The differential rate for a first-order reaction is as follows:
rate=−Δ[A]Δt=k[A]
rate=−Δ[A]Δt=kA
If the concentration of A is doubled, the reaction rate doubles; if the concentration of A is increased by a factor of 10, the reaction rate increases by a factor of 10, and so forth. Because the units of the reaction rate are always moles per liter per second, the units of a first-order rate constant are reciprocal seconds (s−1).
The integrated rate law for a first-order reaction can be written in two different ways: one using exponents and one using logarithms. The exponential form is as follows:
[A]=[A]0e−kt
[A]=[A]0e−kt(14.4.6)
where [A]0[A]0 is the initial concentration of reactant AA at t=0t=0; kk is the rate constant; and e is the base of the natural logarithms, which has the value 2.718 to three decimal places. Recall that an integrated rate law gives the relationship between reactant concentration and time. Equation 14.4.614.4.6 predicts that the concentration of A will decrease in a smooth exponential curve over time. By taking the natural logarithm of each side of Equation 14.4.614.4.6 and rearranging, we obtain an alternative logarithmic expression of the relationship between the concentration of AA and tt:
ln[A]=ln[A]0−kt
ln[A]=ln[A]0−kt(14.4.7)
Because Equation 14.4.714.4.7 has the form of the algebraic equation for a straight line,
y=mx+b,
y=mx+b,
with y=ln[A]y=ln[A] and b=ln[A]0b=ln[A]0, a plot of ln[A]ln[A] versus tt for a first-order reaction should give a straight line with a slope of −k−k and an intercept of ln[A]0ln[A]0. Either the differential rate law (Equation 14.4.514.4.5) or the integrated rate law (Equation 14.4.714.4.7) can be used to determine whether a particular reaction is first order.
First-order reactions are very common. One reaction that exhibits apparent first-order kinetics is the hydrolysis of the anticancer drug cisplatin. Cisplatin, the first “inorganic” anticancer drug to be discovered, is unique in its ability to cause complete remission of the relatively rare, but deadly cancers of the reproductive organs in young adults. The structures of cisplatin and its hydrolysis product are as follows:
Both platinum compounds have four groups arranged in a square plane around a Pt(II) ion. The reaction shown in Figure 14.4.514.4.5 is important because cisplatin, the form in which the drug is administered, is not the form in which the drug is active. Instead, at least one chloride ion must be replaced by water to produce a species that reacts with deoxyribonucleic acid (DNA) to prevent cell division and tumor growth. Consequently, the kinetics of the reaction in Figure 14.4.414.4.4 have been studied extensively to find ways of maximizing the concentration of the active species.
If a plot of reactant concentration versus time is not linear but a plot of the natural logarithm of reactant concentration versus time is linear, then the reaction is first order.
The rate law and reaction order of the hydrolysis of cisplatin are determined from experimental data, such as those displayed in Table 14.4.114.4.1. The table lists initial rate data for four experiments in which the reaction was run at pH 7.0 and 25°C but with different initial concentrations of cisplatin.
Table 14.4.114.4.1: Rates of Hydrolysis of Cisplatin as a Function of Concentration at pH 7.0 and 25°C
| Experiment | [Cisplatin]0 (M) | Initial Rate (M/min) |
| 1 | 0.0060 | 9.0 × 10−6 |
| 2 | 0.012 | 1.8 × 10−5 |
| 3 | 0.024 | 3.6 × 10−5 |
| 4 | 0.030 | 4.5 × 10−5 |
Because the reaction rate increases with increasing cisplatin concentration, we know this cannot be a zeroth-order reaction. Comparing Experiments 1 and 2 in Table 14.4.114.4.1 shows that the reaction rate doubles [(1.8 × 10−5 M/min) ÷ (9.0 × 10−6 M/min) = 2.0] when the concentration of cisplatin is doubled (from 0.0060 M to 0.012 M). Similarly, comparing Experiments 1 and 4 shows that the reaction rate increases by a factor of 5 [(4.5 × 10−5 M/min) ÷ (9.0 × 10−6 M/min) = 5.0] when the concentration of cisplatin is increased by a factor of 5 (from 0.0060 M to 0.030 M). Because the reaction rate is directly proportional to the concentration of the reactant, the exponent of the cisplatin concentration in the rate law must be 1, so the rate law is rate = k[cisplatin]1. Thus the reaction is first order. Knowing this, we can calculate the rate constant using the differential rate law for a first-order reaction and the data in any row of Table 14.4.114.4.1. For example, substituting the values for Experiment 3 into Equation 14.4.514.4.5,
3.6 × 10−5 M/min = k(0.024 M)
1.5 × 10−3 min−1 = k
Knowing the rate constant for the hydrolysis of cisplatin and the rate constants for subsequent reactions that produce species that are highly toxic enables hospital pharmacists to provide patients with solutions that contain only the desired form of the drug.
Example 14.4.114.4.1
At high temperatures, ethyl chloride produces HCl and ethylene by the following reaction:
CH3CH2Cl(g)Δ→HCl(g)+C2H4(g)
CH3CH2Cl(g)−→ΔHCl(g)+C2H4(g)
Using the rate data for the reaction at 650°C presented in the following table, calculate the reaction order with respect to the concentration of ethyl chloride and determine the rate constant for the reaction.
data for the reaction at 650°C
| Experiment | [CH3CH2Cl]0 (M) | Initial Rate (M/s) |
| 1 | 0.010 | 1.6 × 10−8 |
| 2 | 0.015 | 2.4 × 10−8 |
| 3 | 0.030 | 4.8 × 10−8 |
| 4 | 0.040 | 6.4 × 10−8 |
Given: balanced chemical equation, initial concentrations of reactant, and initial rates of reaction
Asked for: reaction order and rate constant
Strategy:
Compare the data from two experiments to determine the effect on the reaction rate of changing the concentration of a species.
Compare the observed effect with behaviors characteristic of zeroth- and first-order reactions to determine the reaction order. Write the rate law for the reaction.
C Use measured concentrations and rate data from any of the experiments to find the rate constant.
Solution
The reaction order with respect to ethyl chloride is determined by examining the effect of changes in the ethyl chloride concentration on the reaction rate.
A Comparing Experiments 2 and 3 shows that doubling the concentration doubles the reaction rate, so the reaction rate is proportional to [CH3CH2Cl]. Similarly, comparing Experiments 1 and 4 shows that quadrupling the concentration quadruples the reaction rate, again indicating that the reaction rate is directly proportional to [CH3CH2Cl].
B This behavior is characteristic of a first-order reaction, for which the rate law is rate = k[CH3CH2Cl].
C We can calculate the rate constant (k) using any row in the table. Selecting Experiment 1 gives the following:
1.60 × 10−8 M/s = k(0.010 M)
1.6 × 10−6 s−1 = k
Exercise 14.4.114.4.1
Sulfuryl chloride (SO2Cl2) decomposes to SO2 and Cl2 by the following reaction:
SO2Cl2(g)→SO2(g)+Cl2(g)
SO2Cl2(g)→SO2(g)+Cl2(g)
Data for the reaction at 320°C are listed in the following table. Calculate the reaction order with regard to sulfuryl chloride and determine the rate constant for the reaction.
Data for the reaction at 320°C
| Experiment | [SO2Cl2]0 (M) | Initial Rate (M/s) |
| 1 | 0.0050 | 1.10 × 10−7 |
| 2 | 0.0075 | 1.65 × 10−7 |
| 3 | 0.0100 | 2.20 × 10−7 |
| 4 | 0.0125 | 2.75 × 10−7 |
Answer
: first order; k = 2.2 × 10−5 s−1
We can also use the integrated rate law to determine the reaction rate for the hydrolysis of cisplatin. To do this, we examine the change in the concentration of the reactant or the product as a function of time at a single initial cisplatin concentration. Figure 14.4.6a14.4.6a shows plots for a solution that originally contained 0.0100 M cisplatin and was maintained at pH 7 and 25°C.
The concentration of cisplatin decreases smoothly with time, and the concentration of chloride ion increases in a similar way. When we plot the natural logarithm of the concentration of cisplatin versus time, we obtain the plot shown in part (b) in Figure 14.4.614.4.6. The straight line is consistent with the behavior of a system that obeys a first-order rate law. We can use any two points on the line to calculate the slope of the line, which gives us the rate constant for the reaction. Thus taking the points from part (a) in Figure 14.4.614.4.6 for t = 100 min ([cisplatin] = 0.0086 M) and t = 1000 min ([cisplatin] = 0.0022 M),
slope=ln[cisplatin]1000−ln[cisplatin]1001000min−100min−k=ln0.0022−ln0.00861000min−100min=−6.12−(−4.76)900min=−1.51×10−3min−1k=1.5×10−3min−1
slope−kk=ln[cisplatin]1000−ln[cisplatin]1001000min−100min=ln0.0022−ln0.00861000min−100min=−6.12−(−4.76)900min=−1.51×10−3min−1=1.5×10−3min−1
The slope is negative because we are calculating the rate of disappearance of cisplatin. Also, the rate constant has units of min−1 because the times plotted on the horizontal axes in parts (a) and (b) in Figure 14.4.614.4.6 are in minutes rather than seconds.
The reaction order and the magnitude of the rate constant we obtain using the integrated rate law are exactly the same as those we calculated earlier using the differential rate law. This must be true if the experiments were carried out under the same conditions.
Video Example Using the First-Order Integrated Rate Law Equation:
Example Using the First-Order Integrated Rate Law Equation(opens in new window) [youtu.be]
Example 14.4.214.4.2
If a sample of ethyl chloride with an initial concentration of 0.0200 M is heated at 650°C, what is the concentration of ethyl chloride after 10 h? How many hours at 650°C must elapse for the concentration to decrease to 0.0050 M (k = 1.6 × 10−6 s−1) ?
Given: initial concentration, rate constant, and time interval
Asked for: concentration at specified time and time required to obtain particular concentration
Strategy:
Substitute values for the initial concentration ([A]0) and the calculated rate constant for the reaction (k) into the integrated rate law for a first-order reaction. Calculate the concentration ([A]) at the given time t.
Given a concentration [A], solve the integrated rate law for time t.
Solution
The exponential form of the integrated rate law for a first-order reaction (Equation 14.4.614.4.6) is [A] = [A]0e−kt.
A Having been given the initial concentration of ethyl chloride ([A]0) and having the rate constant of k = 1.6 × 10−6 s−1, we can use the rate law to calculate the concentration of the reactant at a given time t. Substituting the known values into the integrated rate law,
[CH3CH2Cl]10h=[CH3CH2Cl]0e−kt=0.0200 M(e−(1.6×10−6 s−1)[(10 h)(60 min/h)(60 s/min)])=0.0189 M
[CH3CH2Cl]10h=[CH3CH2Cl]0e−kt=0.0200 M(e−(1.6×10−6 s−1)[(10 h)(60 min/h)(60 s/min)])=0.0189 M
We could also have used the logarithmic form of the integrated rate law (Equation 14.4.714.4.7):
ln[CH3CH2Cl]10 h=ln[CH3CH2Cl]0−kt=ln0.0200−(1.6×10−6 s−1)[(10 h)(60 min/h)(60 s/min)]=−3.912−0.0576=−3.970[CH3CH2Cl]10 h=e−3.970 M=0.0189 M
ln[CH3CH2Cl]10 h[CH3CH2Cl]10 h=ln[CH3CH2Cl]0−kt=ln0.0200−(1.6×10−6 s−1)[(10 h)(60 min/h)(60 s/min)]=−3.912−0.0576=−3.970=e−3.970 M=0.0189 M
B To calculate the amount of time required to reach a given concentration, we must solve the integrated rate law for tt. Equation 14.4.714.4.7 gives the following:
ln[CH3CH2Cl]t=ln[CH3CH2Cl]0−ktkt=ln[CH3CH2Cl]0−ln[CH3CH2Cl]t=ln[CH3CH2Cl]0[CH3CH2Cl]tt=1k(ln[CH3CH2Cl]0[CH3CH2Cl]t)=11.6×10−6 s−1(ln0.0200 M0.0050 M)=ln4.01.6×10−6 s−1=8.7×105 s=240 h=2.4×102 h
ln[CH3CH2Cl]tktt=ln[CH3CH2Cl]0−kt=ln[CH3CH2Cl]0−ln[CH3CH2Cl]t=ln[CH3CH2Cl]0[CH3CH2Cl]t=1k(ln[CH3CH2Cl]0[CH3CH2Cl]t)=11.6×10−6 s−1(ln0.0200 M0.0050 M)=ln4.01.6×10−6 s−1=8.7×105 s=240 h=2.4×102 h
Exercise 14.4.214.4.2
In the exercise in Example 14.4.114.4.1, you found that the decomposition of sulfuryl chloride (SO2Cl2SO2Cl2) is first order, and you calculated the rate constant at 320°C.
Use the form(s) of the integrated rate law to find the amount of SO2Cl2SO2Cl2 that remains after 20 h if a sample with an original concentration of 0.123 M is heated at 320°C.
How long would it take for 90% of the SO2Cl2 to decompose?
Answer a
: 0.0252 M
Answer b
: 29 h
Second-Order Reactions
The simplest kind of second-order reaction is one whose rate is proportional to the square of the concentration of one reactant. These generally have the form
2A→products⋅
2A→products⋅
A second kind of second-order reaction has a reaction rate that is proportional to the product of the concentrations of two reactants. Such reactions generally have the form A + B → products. An example of the former is a dimerization reaction, in which two smaller molecules, each called a monomer, combine to form a larger molecule (a dimer).
The differential rate law for the simplest second-order reaction in which 2A → products is as follows:
rate=−Δ[A]2Δt=k[A]2
rate=−Δ[A]2Δt=k[A]2(14.4.8)
Consequently, doubling the concentration of A quadruples the reaction rate. For the units of the reaction rate to be moles per liter per second (M/s), the units of a second-order rate constant must be the inverse (M−1·s−1). Because the units of molarity are expressed as mol/L, the unit of the rate constant can also be written as L(mol·s).
For the reaction 2A → products, the following integrated rate law describes the concentration of the reactant at a given time:
1[A]=1[A]0+kt
1[A]=1[A]0+kt(14.4.9)
Because Equation 14.4.914.4.9 has the form of an algebraic equation for a straight line, y = mx + b, with y = 1/[A] and b = 1/[A]0, a plot of 1/[A] versus t for a simple second-order reaction is a straight line with a slope of k and an intercept of 1/[A]0.
Second-order reactions generally have the form 2A → products or A + B → products.
Video Discussing the Second-Order Integrated Rate Law Equation: Second-Order Integrated Rate Law Equation(opens in new window) [youtu.be]
Simple second-order reactions are common. In addition to dimerization reactions, two other examples are the decomposition of NO2 to NO and O2 and the decomposition of HI to I2 and H2. Most examples involve simple inorganic molecules, but there are organic examples as well. We can follow the progress of the reaction described in the following paragraph by monitoring the decrease in the intensity of the red color of the reaction mixture.
Many cyclic organic compounds that contain two carbon–carbon double bonds undergo a dimerization reaction to give complex structures. One example is as follows:
Figure 14.4.714.4.7
For simplicity, we will refer to this reactant and product as “monomer” and “dimer,” respectively. The systematic name of the monomer is 2,5-dimethyl-3,4-diphenylcyclopentadienone. The systematic name of the dimer is the name of the monomer followed by “dimer.” Because the monomers are the same, the general equation for this reaction is 2A → product. This reaction represents an important class of organic reactions used in the pharmaceutical industry to prepare complex carbon skeletons for the synthesis of drugs. Like the first-order reactions studied previously, it can be analyzed using either the differential rate law (Equation 14.4.814.4.8) or the integrated rate law (Equation 14.4.914.4.9).
Table 14.4.214.4.2: Rates of Reaction as a Function of Monomer Concentration for an Initial Monomer Concentration of 0.0054 M
| Time (min) | [Monomer] (M) | Instantaneous Rate (M/min) |
| 10 | 0.0044 | 8.0 × 10−5 |
| 26 | 0.0034 | 5.0 × 10−5 |
| 44 | 0.0027 | 3.1 × 10−5 |
| 70 | 0.0020 | 1.8 × 10−5 |
| 120 | 0.0014 | 8.0 × 10−6 |
To determine the differential rate law for the reaction, we need data on how the reaction rate varies as a function of monomer concentrations, which are provided in Table 14.4.214.4.2. From the data, we see that the reaction rate is not independent of the monomer concentration, so this is not a zeroth-order reaction. We also see that the reaction rate is not proportional to the monomer concentration, so the reaction is not first order. Comparing the data in the second and fourth rows shows that the reaction rate decreases by a factor of 2.8 when the monomer concentration decreases by a factor of 1.7:
5.0×10−5 M/min1.8×10−5 M/min=2.8and3.4×10−3 M2.0×10−3 M=1.7
5.0×10−5 M/min1.8×10−5 M/min=2.8and3.4×10−3 M2.0×10−3 M=1.7
Because (1.7)2 = 2.9 ≈ 2.8, the reaction rate is approximately proportional to the square of the monomer concentration.
rate ∝ [monomer]2
This means that the reaction is second order in the monomer. Using Equation 14.4.814.4.8 and the data from any row in Table 14.4.214.4.2, we can calculate the rate constant. Substituting values at time 10 min, for example, gives the following:
rate=k[A]28.0×10−5 M/min=k(4.4×10−3 M)24.1 M−1⋅min−1=k
rate8.0×10−5 M/min4.1 M−1⋅min−1=k[A]2=k(4.4×10−3 M)2=k(14.4.10)(14.4.11)(14.4.12)
We can also determine the reaction order using the integrated rate law. To do so, we use the decrease in the concentration of the monomer as a function of time for a single reaction, plotted in Figure 14.4.8a14.4.8a. The measurements show that the concentration of the monomer (initially 5.4 × 10−3 M) decreases with increasing time. This graph also shows that the reaction rate decreases smoothly with increasing time. According to the integrated rate law for a second-order reaction, a plot of 1/[monomer] versus t should be a straight line, as shown in Figure 14.4.8b14.4.8b. Any pair of points on the line can be used to calculate the slope, which is the second-order rate constant. In this example, k = 4.1 M−1·min−1, which is consistent with the result obtained using the differential rate equation. Although in this example the stoichiometric coefficient is the same as the reaction order, this is not always the case. The reaction order must always be determined experimentally.
For two or more reactions of the same order, the reaction with the largest rate constant is the fastest. Because the units of the rate constants for zeroth-, first-, and second-order reactions are different, however, we cannot compare the magnitudes of rate constants for reactions that have different orders.
Example 14.4.314.4.3
At high temperatures, nitrogen dioxide decomposes to nitric oxide and oxygen.
2NO2(g)Δ→2NO(g)+O2(g)
2NO2(g)−→Δ2NO(g)+O2(g)
Experimental data for the reaction at 300°C and four initial concentrations of NO2 are listed in the following table:
Experimental data for the reaction at 300°C and four initial concentrations of NO2
| Experiment | [NO2]0 (M) | Initial Rate (M/s) |
| 1 | 0.015 | 1.22 × 10−4 |
| 2 | 0.010 | 5.40 × 10−5 |
| 3 | 0.0080 | 3.46 × 10−5 |
| 4 | 0.0050 | 1.35 × 10−5 |
Determine the reaction order and the rate constant.
Given: balanced chemical equation, initial concentrations, and initial rates
Asked for: reaction order and rate constant
Strategy:
From the experiments, compare the changes in the initial reaction rates with the corresponding changes in the initial concentrations. Determine whether the changes are characteristic of zeroth-, first-, or second-order reactions.
Determine the appropriate rate law. Using this rate law and data from any experiment, solve for the rate constant (k).
Solution
A We can determine the reaction order with respect to nitrogen dioxide by comparing the changes in NO2 concentrations with the corresponding reaction rates. Comparing Experiments 2 and 4, for example, shows that doubling the concentration quadruples the reaction rate [(5.40 × 10−5) ÷ (1.35 × 10−5) = 4.0], which means that the reaction rate is proportional to [NO2]2. Similarly, comparing Experiments 1 and 4 shows that tripling the concentration increases the reaction rate by a factor of 9, again indicating that the reaction rate is proportional to [NO2]2. This behavior is characteristic of a second-order reaction.
B We have rate = k[NO2]2. We can calculate the rate constant (k) using data from any experiment in the table. Selecting Experiment 2, for example, gives the following:
rate=k[NO2]25.40×10−5 M/s=k(0.010M)20.54M−1⋅s−1=k
rate5.40×10−5 M/s0.54M−1⋅s−1=k[NO2]2=k(0.010M)2=k
Exercise 14.4.3
When the highly reactive species HO2 forms in the atmosphere, one important reaction that then removes it from the atmosphere is as follows:
2HO2(g)→H2O2(g)+O2(g)
The kinetics of this reaction have been studied in the laboratory, and some initial rate data at 25°C are listed in the following table:
Some initial rate data at 25°C
| Experiment | [HO2]0 (M) | Initial Rate (M/s) |
| 1 | 1.1 × 10−8 | 1.7 × 10−7 |
| 2 | 2.5 × 10−8 | 8.8 × 10−7 |
| 3 | 3.4 × 10−8 | 1.6 × 10−6 |
| 4 | 5.0 × 10−8 | 3.5 × 10−6 |
Determine the reaction order and the rate constant.
Answer
: second order in HO2; k = 1.4 × 109 M−1·s−1
If a plot of reactant concentration versus time is not linear, but a plot of 1/(reactant concentration) versus time is linear, then the reaction is second order.
Example 14.4.4
If a flask that initially contains 0.056 M NO2 is heated at 300°C, what will be the concentration of NO2 after 1.0 h? How long will it take for the concentration of NO2 to decrease to 10% of the initial concentration? Use the integrated rate law for a second-order reaction (Equation 14.4.9) and the rate constant calculated above.
Given: balanced chemical equation, rate constant, time interval, and initial concentration
Asked for: final concentration and time required to reach specified concentration
Strategy:
Given k, t, and [A]0, use the integrated rate law for a second-order reaction to calculate [A].
Setting [A] equal to 1/10 of [A]0, use the same equation to solve for t.
Solution
A We know k and [NO2]0, and we are asked to determine [NO2] at t = 1 h (3600 s). Substituting the appropriate values into Equation 14.4.9,
1[NO2]3600=1[NO2]0+kt=10.056 M+[(0.54M−1⋅s−1)(3600 s)]=2.0×103 M−1
Thus [NO2]3600 = 5.1 × 10−4 M.
B In this case, we know k and [NO2]0, and we are asked to calculate at what time [NO2] = 0.1[NO2]0 = 0.1(0.056 M) = 0.0056 M. To do this, we solve Equation 14.4.9 for t, using the concentrations given.
t=(1/[NO2])−(1/[NO2]0)k=(1/0.0056 M)−(1/0.056 M)0.54M−1⋅s−1=3.0×102 s=5.0 min
NO2 decomposes very rapidly; under these conditions, the reaction is 90% complete in only 5.0 min.
Exercise 14.4.4
In the previous exercise, you calculated the rate constant for the decomposition of HO2 as k = 1.4 × 109 M−1·s−1. This high rate constant means that HO2 decomposes rapidly under the reaction conditions given in the problem. In fact, the HO2 molecule is so reactive that it is virtually impossible to obtain in high concentrations. Given a 0.0010 M sample of HO2, calculate the concentration of HO2 that remains after 1.0 h at 25°C. How long will it take for 90% of the HO2 to decompose? Use the integrated rate law for a second-order reaction (Equation 14.4.9) and the rate constant calculated in the exercise in Example 14.4.3.
Answer
: 2.0 × 10−13 M; 6.4 × 10−6 s
In addition to the simple second-order reaction and rate law we have just described, another very common second-order reaction has the general form A+B→products, in which the reaction is first order in A and first order in B. The differential rate law for this reaction is as follows:
rate=−Δ[A]Δt=−Δ[B]Δt=k[A][B]
Because the reaction is first order both in A and in B, it has an overall reaction order of 2. (The integrated rate law for this reaction is rather complex, so we will not describe it.) We can recognize second-order reactions of this sort because the reaction rate is proportional to the concentrations of each reactant.
Summary
The reaction rate of a zeroth-order reaction is independent of the concentration of the reactants. The reaction rate of a first-order reaction is directly proportional to the concentration of one reactant. The reaction rate of a simple second-order reaction is proportional to the square of the concentration of one reactant. Knowing the rate law of a reaction gives clues to the reaction mechanism.
zeroth-order reaction: rate=−Δ[A]Δt=k[A]=[A]0−kt
first-order reaction: rate=−Δ[A]Δt=k[A][A]=[A]0e−ktln[A]=ln[A]0−kt
second-order reaction: rate=−Δ[A]Δt=k[A]21[A]=1[A]0+kt
14.3: Concentration and Rates (Differential Rate Laws)
14.5: Temperature and Rate |
12922 | https://www.slideserve.com/leal/5-3-first-derivative-test | 1 / 15
5.3 - First Derivative Test
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5.3 - First Derivative Test. MCB4U - Santowski. (A) Important Terms. Recall the following terms as they were presented in a previous lesson: turning point : points where the direction of the function changes maximum : the highest point on a function minimum : the lowest point on a function
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5.3 - First Derivative Test MCB4U - Santowski
(A) Important Terms • Recall the following terms as they were presented in a previous lesson: • turning point: points where the direction of the function changes • maximum: the highest point on a function • minimum: the lowest point on a function • local vs absolute: a max can be a highest point in the entire domain (absolute) or only over a specified region within the domain (local). Likewise for a minimum. • increase: the part of the domain (the interval) where the function values are getting larger as the independent variable gets higher; if f(x1) < f(x2) when x1 < x2; the graph of the function is going up to the right (or down to the left) • decrease: the part of the domain (the interval) where the function values are getting smaller as the independent variable gets higher; if f(x1) > f(x2) when x1 < x2; the graph of the function is going up to the left (or down to the right) • "end behaviour": describing the function values (or appearance of the graph) as x values getting infinitely large positively or infinitely large negatively or approaching an asymptote
(B) Review – Graphic Analysis of a Function • We have seen functions analyzed given the criteria intervals of increase, intervals of decrease, critical points (AKA turning points or maximum or minimum points) • We have also seen graphically how the derivative function communicates the same criteria about a function these points are summarized on the next slide:
f(x) has a max. at x = -3.1 and f (x) has an x-intercept at x = -3.1 f(x) has a min. at x = -0.2 and f(x) has a root at –0.2 f(x) increases on (-, -3.1) & (-0.2, ) and on the same intervals, f (x) has positive values f(x) decreases on (-3.1, -0.2) and on the same interval, f(x) has negative values (B) Review – Graphic Analysis of a Function
(C) Analysis of Functions Using Derivatives – A Summary • If f(x) increases, then f (x) > 0 • If f(x) decreases, then f(x) < 0 • At a max/min point, f (x) = 0 • We can also state the converse of 2 of these statements: • If f(x) > 0, then f(x) is increasing • If f (x) < 0, then f(x) is decreasing • The converse of the third statement is NOT true if f(x) = 0, then the function may NOT necessarily have a max/min so for now, we will call any point that gives f `(x) = 0 (i.e. produces a horizontal tangent line) aCRITICAL POINTS or EXTREME POINTS
(D) First Derivative Test • So if f `(x) = 0, how do we decide if the point at (x, f(x)) is a maximum, minimum, or neither (especially if we have no graph?) • Since we have done some graphic analysis with functions and their derivatives, in one sense we already now the answer: see next slide
At the max (x = -3.1), the fcn changes from being an increasing fcn to a decreasing fcn the derivative changes from positive values to negative values At a the min (x = -0.2), the fcn changes from decreasing to increasing the derivative changes from negative to positive (E) First Derivative Test - Graphically
(F) First Derivative Test - Algebraically • At a maximum, the fcn changes from being an increasing fcn to a decreasing fcn the derivative changes from positive values to negative values • At the minimum, the fcn changes from decreasing to increasing the derivative changes from negative to positive • So to state the converses: • If f (x) = 0 and f the sign of if(x) changes from positive to negative, then the critical point on f(x) is a maximum point • If f (x) = 0 and f the sign of if(x) changes from negative to positive, then the critical point on f(x) is a minimum point • So therefore, if the sign on f `(x) does not change at the critical point, then the critical point is neither a maximum or minimum we will call these points STATIONARY POINTS
(G) First Derivative Test – Example #1 • Find the local max/min values of y = x3 - 3x + 1 (Show how to use inequalities to analyze for the sign change) • f (x) = 3x2 – 3 • f(x) = 0 for the critical values • 0 = 3x2 – 3 • 0 = 3(x2 – 1) • 0 = 3(x – 1)(x + 1) • x = 1 or x = -1 • Now, what happens on the function, at x = + 1? let’s set up a chart to se what happens with the signs on the derivative so that we can determine the sign on the derivative so that we can classify the critical points
(G) First Derivative Test – Example #1
(G) First Derivative Test – Example #1 • Since the derivative changes signs from +ve to –ve, the critical point at x = -1 is a maximum (the original function changing from being an increasing fcn to now being a decreasing fcn) • Since the derivative changes signs from -ve to +ve, the critical point at x = 1 is a minimum (the original function changing from being a decreasing fcn to now being an increasing fcn) • Then, going one step further, we can say that f(-1) = 3 gives us a maximum value of 3 and then f(1) = -1 gives us a minimum value of -1 • And going another step, we can test the end behaviour of f(x): • lim x-∞ f(x) = -∞ • lim x ∞ f(x) = +∞ • Therefore, the point (-1,3) represents a local maximum (as the fcn rises to infinity “at the end”) and the point (1,-1) represents a local minimum (as the fcn drops to negative infinity “at the negative end”)
(G) First Derivative Test – Example #1 – Graphic Summary
(H) In Class Examples • Ex 2. Find the local max/min values of g(x) = x4 - 4x3 - 8x2 - 1 • Ex 3. Find the absolute minimum value of f(x) = x + 1/x for x > 0 • Ex 4. Find the intervals of increase and decrease and max/min values of f(x) = cos(x) – sin(x) on (-,) • Ex 5. Find the critical numbers, intervals of increase & decrease and max/min values of y = csc(x) – cot(x) on (-/2,3/2) • Ex 6. Find the intervals of increase/decrease and max/min points of f(x) = x2e-x • Ex 7. Find the local and absolute maximum & minimum points for f(x) = x(ln(x))2
(I) Internet Links • Visual Calculus - Maxima and Minima from UTK • Visual Calculus - Mean Value Theorem and the First Derivative Test from UTK • First Derivative Test -- From MathWorld • Tutorial: Maxima and Minima from Stefan Waner at Hofstra U
(J) Homework • Handout from Stewart, 1997, text
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12923 | https://dept.swccd.edu/mcarey/M045/Lecture%20Notes/M45%202.7%20Day%202%20Uniform%20Motion.pdf | Math 45 2.7 Uniform Motion Uniform Motion: An object or person moves with the same speed (or the same average speed) for entire event. Uniform Motion formula: RT D = D = distance (units of length) R = rate (units of distance per time) T = time (units of time) Note: The units of distance in D and R must be the same, and the units of time in T and R must be the same. If they are not, convert before writing the equation. If more than one object is moving, the DRT formula applies separately to each object. A good way to keep this organized is a chart: D R T 1st object 1st distance 1st rate 1st time 2nd object 2nd distance 2nd rate 2nd time To use a chart successfully: Step 1: Fill in the numbers known from the problem. Step 2: Look at the question, identify the unknown variable, and write it in the chart. Step 3: Use the variable, numbers, and formula to fill in expressions for the rest of the chart. Three basic set-ups: • Two objects or people move in opposite directions. The distance between them is the SUM. • Two objects or people move in the same direction. The distance between them is the DIFFERENCE. • Two objects or people move the same distance in different times or directions. Set distances EQUAL. A more advanced set-up: • Total time is given. Solve formula for T to get R D T = to get two expressions for time. Add times. 1) Two boats leave port at the same time, one heading north at 35 knots (nautical miles per hour) and the other south at 47 knots. How long will it take them to be 738 nautical miles apart? 2) Sarah, driving the U-Haul, goes 50 mph. Ralph, driving the Civic, goes 70 mph. They leave at the same time. When will they be 60 miles apart? 3) Two groups on a canoe trip leave at different times. The first left at noon, the second half-an-hour later. The second group travels at an average speed which is 0.75 mph greater than the first group. At 2:30 PM, the second group caught up to the first group. How fast was each group paddling? 4) Granville and Preston are 535 miles apart. A car leaves Preston bound for Granville at 47 mph. At the same time, another car leaves for Granville bound for Preston at 60 mph. How long will it take them to meet? 5) Two crooks rob a bank and flee to the east at 66 mph. In 30 minutes, the police follow them in a helicopter, flying at 132 mph. How long will it take for the police to overtake the robbers? 6) Two cars leave a city on the same road, one driving 15 mph faster than the other. After 5 hours, the faster car stops for lunch. After 5 hours and 30 minutes, the slower car has not passed the other car and stops for lunch. The other car is still stopped for lunch. The two stopped cars are 54 miles apart. How fast was each car driving? 7) You drive from Atlanta to Durham, a distance of 390 miles. The average speed for the first part of the trip was 60 mph. During the second part of the trip, there was construction, so your average speed was only 45 mph. How long did you travel at 45 mph if you drove 3 hours longer at 60 mph than at 45 mph? 8) On the way to California, five friends drove 60 mph. On the way home, they took the same route at 70 mph. The round trip took 10 hours. How many miles is it to California? Round to the nearest tenth of a mile if necessary. |
12924 | https://math.stackexchange.com/questions/3329546/how-can-i-solve-prove-that-81-a1-b1-c-le-abc-with-the-conditions-below | inequality - How can I solve prove that $8(1-a)(1-b)(1-c)\le abc$ with the conditions below? - Mathematics Stack Exchange
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How can I solve prove that 8(1−a)(1−b)(1−c)≤a b c 8(1−a)(1−b)(1−c)≤a b c with the conditions below? [duplicate]
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How to establish this inequality: (1−a)(1−b)(1−c)≥8 a b c(1−a)(1−b)(1−c)≥8 a b c for a+b+c=1 a+b+c=1? (3 answers)
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There was a homework about inequalities (that why I ask a bunch of inequality problems). But I couldn't solve the following:
If 0<a,b,c<1 0<a,b,c<1 and a+b+c=2 a+b+c=2, prove that 8(1−a)(1−b)(1−c)≤a b c 8(1−a)(1−b)(1−c)≤a b c
I tried many times, and finally I used Muirhead, but it failed!
L.H.S.−R.H.S.=8(1−a)(1−b)(1−c)−a b c=8−8(a+b+c)+8(a b+b c+c a)−9 a b c=−(a 3+b 3+c 3)+(a 2 b+b 2 c+c 2 a+a b 2+b c 2+c a 2)−3 a b c=1 2(∑s y m a 2 b−∑s y m a 3)+1 2(∑s y m a 2 b−∑s y m a b c)
But as (3,0,0) majorizes (2,1,0) but (2,1,0) majorizes (1,1,1), so it fails.
Could someone help? Any help is appreciated!
inequality
substitution
symmetric-polynomials
muirhead-inequality
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edited Jun 12, 2020 at 10:38
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asked Aug 21, 2019 at 5:04
Culver KwanCulver Kwan
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Oh, yes, my mistake.Sarvesh Ravichandran Iyer –Sarvesh Ravichandran Iyer 2019-08-21 05:17:49 +00:00 Commented Aug 21, 2019 at 5:17
7 Possible duplicate of How to establish this inequality: (1−a)(1−b)(1−c)≥8 a b c for a+b+c=1? (Up to a small change of variables)Arnaud D. –Arnaud D. 2019-08-21 15:39:24 +00:00 Commented Aug 21, 2019 at 15:39
@Arnaud D By your way all math problems they are a duplicate. It's not a duplicate, of course.Michael Rozenberg –Michael Rozenberg 2019-08-21 18:15:17 +00:00 Commented Aug 21, 2019 at 18:15
3 @MichaelRozenberg The equation 8 x y z≤(x+y)(x+z)(y+z) that appears in one of your solutions is almost exactly the statement of the other question, and you've obtained it by doing almost exactly the change of variables I had in mind. Moreover the accepted answers are practically identical.Arnaud D. –Arnaud D. 2019-08-22 12:06:55 +00:00 Commented Aug 22, 2019 at 12:06
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By AM-GM you have (1−a)1 2(1−b)1 2≤1−a+1−b 2=c 2(1−b)1 2(1−c)1 2≤1−b+1−c 2=a 2(1−c)1 2(1−a)1 2≤1−c+1−a 2=b 2 Multiplying these three inequalities together gives the desired inequality.
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answered Aug 21, 2019 at 6:35
ZarraxZarrax
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We need to prove that ∏c y c(a+b−c)≤a b c or ∑c y c(a 3−a 2 b−a 2 c+a b c)≥0, which is true by the following reasoning.
Let a≥b≥c.
Thus, ∑c y c(a 3−a 2 b−a 2 c+a b c)=∑c y c a(a−b)(a−c)≥≥a(a−b)(a−c)+b(b−a)(b−c)=(a−b)2(a+b−c)≥0.
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answered Aug 21, 2019 at 5:07
Michael RozenbergMichael Rozenberg
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Actually, the first inequality is standard Schur's inequality.richrow –richrow 2019-08-21 10:41:35 +00:00 Commented Aug 21, 2019 at 10:41
Yes, of course. I proved Schur Michael Rozenberg –Michael Rozenberg 2019-08-21 11:44:49 +00:00 Commented Aug 21, 2019 at 11:44
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Also, we can use Muirhead here.
Let a+b−c=z, a+c−b=y and b+c−a=x.
Thus, x=a+b+c−2 a=2(1−a)>0. Similarly, y>0 and z>0 and we need to prove that 8 x y z≤(x+y)(x+z)(y+z) or ∑c y c(x 2 y+x 2 z−2 x y z)≥0, which is true by Muirhead.
Also, we can use AM-GM: ∏c y c(x+y)≥∏c y c 2√x y=8 x y z.
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answered Aug 21, 2019 at 5:19
Michael RozenbergMichael Rozenberg
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Denote: 1−a=x,1−b=y,1−c=z. Then: x+y+z=1 A M−G M⇒1=x+y+z≥3 3√x y z Hence: 8(1−a)(1−b)(1−c)≤a b c⇒8 x y z≤(1−x)(1−y)(1−z)⟺8 x y z≤1−(x+y+z)+(x y+y z+z x)−x y z⟺9 x y z≤x y+y z+z x⟺9 x y z≤3 3√(x y z)2 A M−G M≤x y+y z+z x⟺3 3√x y z≤1. equality occurs for x=y=z=1 3, consequently, for a=b=c=2 3.
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edited Aug 23, 2019 at 3:43
answered Aug 21, 2019 at 6:21
farruhotafarruhota
32.3k 2 2 gold badges 20 20 silver badges 54 54 bronze badges
4
What is the role of AM-GM here? Isn't is better to write 8 x y z=(2 x)(2 y)(2 z)=(x+x)(y+y)(z+z)≤(1+x)(1+y)(1+z)?C.F.G –C.F.G 2019-08-21 07:13:11 +00:00 Commented Aug 21, 2019 at 7:13
@c.f.g He is using that 1+x≥2√x which is true by AM-GM.asdf –asdf 2019-08-21 07:29:14 +00:00 Commented Aug 21, 2019 at 7:29
But equality occurs when a=b=c=2 3!Culver Kwan –Culver Kwan 2019-08-22 13:13:34 +00:00 Commented Aug 22, 2019 at 13:13
@CulverKwan, there was a typo, fixed, thank you.farruhota –farruhota 2019-08-23 03:43:48 +00:00 Commented Aug 23, 2019 at 3:43
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12925 | http://mathcentral.uregina.ca/QQ/database/QQ.09.04/kate1.html | Three tangent circles
Quandaries and Queries
Two circles, C1 and C2, touch each other externally; and the line l is a common tangent. The line m is parallel to l and touches the two circls C1 and C3. The three circles are mutually tangent. If the radius of C2 is 9 and if the radius of C3 is 4, what is the radius of C1?
Hi Kate,
I drew a diagram and labeled some points. P, R and T are the centers of the circles. PQ and ST are parallel to the common tangent lines m and l.
I then redrew part of the diagram, larger so that I can include some dimensions. I let r be the radius of C 1.
Since PQ and ST are parallel to the tangent lines. PQR and RST are right triangles and hence Pythagoras' Theorem can be used to find the lengths of PQ ans ST. Now draw a line trough T, parallel to SQ and meeting PQ at W.
PWT is a right triangle and hence Pythagoras' Theorem applied to this triangle gives a quadratic you can solve for r.
This is one configuration for the circles you describe in your problem but there is at least one other configuration.
Chris
Go to Math Central |
12926 | https://www.tiger-algebra.com/drill/-2m%3C=1/ | Copyright Ⓒ 2013-2025
tiger-algebra.com
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Solution - Linear inequalities
Other Ways to Solve
Step by Step Solution
Rearrange:
Rearrange the equation by subtracting what is to the right of the less equal sign from both sides of the inequality :
-2m-(1)≤0
Step by step solution :
Step 1 :
Pulling out like terms :
1.1 Pull out like factors :
-2m - 1 = -1 • (2m + 1)
Equation at the end of step 1 :
Step 2 :
2.1 Multiply both sides by (-1)
Flip the inequality sign since you are multiplying by a negative number
2m+1 ≥ 0
2.2 Divide both sides by 2
m+(1/2) ≥ 0
Solve Basic Inequality :
Inequality Plot :
One solution was found :
How did we do?
Why learn this
Life is not binary (no matter how badly Tiger wishes it was) and we are often faced with questions with more than one answer. This is why we need inequalities. How much of a product should be produced to maximize a company's profit? What is the number of tickets that you need to sell for your band's show to be profitable? How much money do you need to make during summer break to book a ski trip in the winter? By helping explain the relationships between what we know and what we want to know, linear inequalities can help us answer these questions, and many more!
Terms and topics
Related links
Latest Related Drills Solved
Copyright Ⓒ 2013-2025
tiger-algebra.com |
12927 | https://periodictable.chemicalaid.com/element.php/K?lang=it | Potassio (K)
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Copyright © 2023 - 2025 ChemicalAid
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⚗️🧪⚛️🧑🏻🔬
Argo ‹Ar ‹
Potassio
› Calcio› Ca
Potassio (K)
elemento chimico con numero atomico 19
| Numero Atomico | 19 |
| Massa Atomica | 39.0983 |
| numero di massa | 39 |
| Gruppo | 1 |
| Periodo | 4 |
| Blocco | s |
| protone | 19 p+ |
| neutrone | 20 n 0 |
| elettrone | 19 e- |
Discover more
გოგირდი
s
Proprietà Fisica
| Raggio Atomico | 220 pm 234 pm |
| volume molare | 45,3 cm³/mol |
| Raggio covalente | 196 pm 193 pm 207 pm 203 pm |
| Metallic Radius | 203 pm 235 pm |
| raggio ionico | 137 pm 138 pm 146 pm 151 pm 155 pm 159 pm 164 pm |
| Crystal Radius | 151 pm 152 pm 160 pm 165 pm 169 pm 173 pm 178 pm |
| Raggio di van der Waals | 280 pm 275 pm 275 pm 309 pm 381,2 pm |
| massa volumica | 0,89 g/cm³ 828 kg/m³ |
Proprietà Chimica
{ERROR}
| energia |
| Discover more s გოგირდი |
| Affinità protonica | |
| affinità elettronica | 0,50147 eV/particle |
| energia di ionizzazione | 4,34066354 eV/particle 31,625 eV/particle 45,8031 eV/particle 60,917 eV/particle 82,66 eV/particle 99,4 eV/particle 117,56 eV/particle 154,87 eV/particle 175,8174 eV/particle 503,67 eV/particle 565,6 eV/particle 631 eV/particle 714,71 eV/particle 786,26 eV/particle 860,92 eV/particle 967,66 eV/particle 1.034,541 eV/particle 4.610,86998 eV/particle 4.934,0482 eV/particle |
| |
| entalpia di vaporizzazione | 2,33 kJ/mol |
| entalpia di fusione | 102,5 kJ/mol |
| entalpia standard di formazione | 89 kJ/mol |
| elettrone |
| Discover more გოგირდი s |
| electron shell | 2, 8, 8, 1 |
| |
| elettrone di valenza | 1 ⓘ |
| |
| configurazione elettronica | [Ar] 4s 1ⓘ 1s 2 2s 2 2p 6 3s 2 3p 6 4s 1 |
| |
| numero di ossidazione | -1, 1 |
| elettronegatività | 4,34 eV/particle 0.0981896 0.82 |
| Electrophilicity Index | 0,7633848415985606 eV/particle |
| stato fondamentale della materia |
| |
| fase della materia |
| gaseous state of matter | |
| Punto di Ebollizione | 1.032,15 K |
| Punto di Fusione | 336,65 K |
| pressione critica | 16 MPa |
| Temperatura critica | 2.223,15 K |
| punto triplo | |
| apparenza |
| colore | Nero #ff1493 #8f40d4 #8f40d4 |
| apparenza | silvery gray |
| indice di rifrazione | |
| proprietà materiale |
| Conduttività Termica | 79 W/(m K) |
| dilatazione termica | |
| molar heat capacity | 29,6 J/(mol K) |
| Capacità Termica Specifica | 0,757 J/(g⋅K) |
| Coefficiente di dilatazione adiabatica | |
| electrical properties |
| type | Conductor |
| conduttività elettrica | 14 MS/m |
| resistività elettrica | 0,00000007000000000002 Ωm |
| superconduttività | |
| magnetismo |
| type | paramagnetic |
| suscettività magnetica (Mass) | 0,0000000067 m³/Kg |
| suscettività magnetica (Molar) | 0,000000000262 m³/mol |
| suscettività magnetica (Volume) | 0,00000574 |
| magnetic ordering | |
| punto di Curie | |
| Temperatura di Néel | |
| struttura |
| Struttura Cristallina | {ERROR} |
| Costante di reticolo | 5,23 Å |
| Lattice Angles | π/2, π/2, π/2 |
| proprietà meccanica dei materiali |
| durezza | 0,363 MPa 0,4 MPa |
| Modulo di compressibilità | 3,1 GPa |
| Modulo di taglio | 1,3 GPa |
| Young's modulus | |
| modulo di Poisson | |
| velocità del suono | 2.000 m/s |
| classificazione |
| Categoria | Attinidi, Alkali metals |
| CAS Group | IA |
| IUPAC Group | IA |
| Glawe Number | 10 |
| Mendeleev Number | 3 |
| Pettifor Number | 10 |
| Geochemical Class | alkali metal |
| Classificazione Goldschmidt | litophile |
other
| Gas Basicity | |
| Polarizzabilità | 289,7 ± 0,3 a₀ |
| C6 Dispersion Coefficient | 3.923 a₀ 3.910 a₀ |
| allotrope | |
| cross section | 2,1 |
| Neutron Mass Absorption | 0,0018 |
| numero quantico | 2S1/2 |
| gruppo spaziale | 229 (Im_3m) |
Isotopi del potassio
| Isotopi Stabili | 2 |
| Isotopi Instabili | 27 |
| Natural Isotopes | 3 |
31 K☢️
32 K☢️
33 K☢️
34 K☢️
35 K☢️
36 K☢️
37 K☢️
38 K☢️
39 K
40 K☢️
41 K
42 K☢️
43 K☢️
44 K☢️
45 K☢️
46 K☢️
47 K☢️
48 K☢️
49 K☢️
50 K☢️
51 K☢️
52 K☢️
53 K☢️
54 K☢️
55 K☢️
56 K☢️
57 K☢️
58 K☢️
59 K☢️
31 K
| numero di massa | 31 |
| numero neutronico | 12 |
| massa atomica relativa | 31,03678 ± 0,000322 Da |
| Fattore-g | |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | |
| spin | 3/2 |
| nuclear quadrupole moment | |
| data della scoperta o invenzione | 2019 |
| parità | + |
| tipo di decadimento | intensità |
--- |
| 3p | 100% |
32 K
| numero di massa | 32 |
| numero neutronico | 13 |
| massa atomica relativa | 32,023607 ± 0,000429 Da |
| Fattore-g | |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | |
| spin | 1 |
| nuclear quadrupole moment | |
| data della scoperta o invenzione | |
| parità | + |
| tipo di decadimento | intensità |
--- |
| p (proton emission) | |
33 K
| numero di massa | 33 |
| numero neutronico | 14 |
| massa atomica relativa | 33,008095 ± 0,000215 Da |
| Fattore-g | |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | |
| spin | 3/2 |
| nuclear quadrupole moment | |
| data della scoperta o invenzione | |
| parità | + |
| tipo di decadimento | intensità |
--- |
| p (proton emission) | |
34 K
| numero di massa | 34 |
| numero neutronico | 15 |
| massa atomica relativa | 33,99869 ± 0,00021 Da |
| Fattore-g | |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | |
| spin | 1 |
| nuclear quadrupole moment | |
| data della scoperta o invenzione | |
| parità | + |
| tipo di decadimento | intensità |
--- |
| p (proton emission) | |
35 K
| numero di massa | 35 |
| numero neutronico | 16 |
| massa atomica relativa | 34,988005406 ± 0,00000055 Da |
| Fattore-g | 0,26133333333333 ± 0,0046666666666667 |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | 175,2 ± 1,9 ms |
| spin | 3/2 |
| nuclear quadrupole moment | |
| data della scoperta o invenzione | 1976 |
| parità | + |
| tipo di decadimento | intensità |
--- |
| β+ (β+ decay; β+ = ϵ + e+) | 100% |
| β+ p (β+-delayed proton emission) | 0.37% |
36 K
| numero di massa | 36 |
| numero neutronico | 17 |
| massa atomica relativa | 35,981301887 ± 0,000000349 Da |
| Fattore-g | 0,274 ± 0,0005 |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | 341 ± 3 ms |
| spin | 2 |
| nuclear quadrupole moment | |
| data della scoperta o invenzione | 1967 |
| parità | + |
| tipo di decadimento | intensità |
--- |
| β+ (β+ decay; β+ = ϵ + e+) | 100% |
| β+ p (β+-delayed proton emission) | 0.048% |
| β+α (β+-delayed α emission) | 0.0034% |
37 K
| numero di massa | 37 |
| numero neutronico | 18 |
| massa atomica relativa | 36,97337589 ± 0,0000001 Da |
| Fattore-g | 0,13547333333333 ± 0,00004 |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | 1,23651 ± 0,00094 s |
| spin | 3/2 |
| nuclear quadrupole moment | 0,109 ± 0,004 |
| data della scoperta o invenzione | 1958 |
| parità | + |
| tipo di decadimento | intensità |
--- |
| β+ (β+ decay; β+ = ϵ + e+) | 100% |
38 K
| numero di massa | 38 |
| numero neutronico | 19 |
| massa atomica relativa | 37,969081114 ± 0,000000209 Da |
| Fattore-g | 0,457 ± 0,002 |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | 7,651 ± 0,019 m |
| spin | 3 |
| nuclear quadrupole moment | |
| data della scoperta o invenzione | 1937 |
| parità | + |
| tipo di decadimento | intensità |
--- |
| β+ (β+ decay; β+ = ϵ + e+) | 100% |
39 K
| numero di massa | 39 |
| numero neutronico | 20 |
| massa atomica relativa | 38,96370648482 ± 0,00000000489 Da |
| Fattore-g | 0,26098 ± 0,0000053333333333333 |
| natural abundance | 93,2581 ± 0,0044 |
| radioattività | isotopo stabile |
| emivita | Not Radioactive ☢️ |
| spin | 3/2 |
| nuclear quadrupole moment | 0,0603 ± 0,0006 |
| data della scoperta o invenzione | 1921 |
| parità | + |
40 K
| numero di massa | 40 |
| numero neutronico | 21 |
| massa atomica relativa | 39,963998165 ± 0,00000006 Da |
| Fattore-g | |
| natural abundance | 0,0117 ± 0,0001 |
| radioattività | ☢️ radioactive element |
| emivita | 1,248 ± 0,003 Gy |
| spin | 4 |
| nuclear quadrupole moment | -0,075 ± 0,0008 |
| data della scoperta o invenzione | 1935 |
| parità
| tipo di decadimento | intensità |
--- |
| β− (β− decay) | 89.28% |
| β+ (β+ decay; β+ = ϵ + e+) | 10.72% |
41 K
| numero di massa | 41 |
| numero neutronico | 22 |
| massa atomica relativa | 40,96182525611 ± 0,00000000403 Da |
| Fattore-g | 0,143248 ± 0,0000033333333333333 |
| natural abundance | 6,7302 ± 0,0044 |
| radioattività | isotopo stabile |
| emivita | Not Radioactive ☢️ |
| spin | 3/2 |
| nuclear quadrupole moment | 0,0734 ± 0,0007 |
| data della scoperta o invenzione | 1921 |
| parità | + |
42 K
| numero di massa | 42 |
| numero neutronico | 23 |
| massa atomica relativa | 41,962402305 ± 0,000000113 Da |
| Fattore-g | -0,57125 ± 0,0003 |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | 12,355 ± 0,007 h |
| spin | 2 |
| nuclear quadrupole moment | |
| data della scoperta o invenzione | 1935 |
| parità
| tipo di decadimento | intensità |
--- |
| β− (β− decay) | 100% |
43 K
| numero di massa | 43 |
| numero neutronico | 24 |
| massa atomica relativa | 42,960734701 ± 0,00000044 Da |
| Fattore-g | 0,10886666666667 ± 0,00053333333333333 |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | 22,3 ± 0,1 h |
| spin | 3/2 |
| nuclear quadrupole moment | |
| data della scoperta o invenzione | 1949 |
| parità | + |
| tipo di decadimento | intensità |
--- |
| β− (β− decay) | 100% |
44 K
| numero di massa | 44 |
| numero neutronico | 25 |
| massa atomica relativa | 43,961586984 ± 0,00000045 Da |
| Fattore-g | -0,428 ± 0,002 |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | 22,13 ± 0,19 m |
| spin | 2 |
| nuclear quadrupole moment | |
| data della scoperta o invenzione | 1954 |
| parità
| tipo di decadimento | intensità |
--- |
| β− (β− decay) | 100% |
45 K
| numero di massa | 45 |
| numero neutronico | 26 |
| massa atomica relativa | 44,960691491 ± 0,00000056 Da |
| Fattore-g | 0,1156 ± 0,00053333333333333 |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | 17,8 ± 0,6 m |
| spin | 3/2 |
| nuclear quadrupole moment | |
| data della scoperta o invenzione | 1964 |
| parità | + |
| tipo di decadimento | intensità |
--- |
| β− (β− decay) | 100% |
46 K
| numero di massa | 46 |
| numero neutronico | 27 |
| massa atomica relativa | 45,961981584 ± 0,00000078 Da |
| Fattore-g | -0,5255 ± 0,003 |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | 96,3 ± 0,08 s |
| spin | 2 |
| nuclear quadrupole moment | |
| data della scoperta o invenzione | 1965 |
| parità
| tipo di decadimento | intensità |
--- |
| β− (β− decay) | 100% |
47 K
| numero di massa | 47 |
| numero neutronico | 28 |
| massa atomica relativa | 46,961661612 ± 0,0000015 Da |
| Fattore-g | 3,866 ± 0,018 |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | 17,38 ± 0,03 s |
| spin | 1/2 |
| nuclear quadrupole moment | 0 |
| data della scoperta o invenzione | 1964 |
| parità | + |
| tipo di decadimento | intensità |
--- |
| β− (β− decay) | 100% |
48 K
| numero di massa | 48 |
| numero neutronico | 29 |
| massa atomica relativa | 47,965341184 ± 0,00000083 Da |
| Fattore-g | |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | 6,83 ± 0,14 s |
| spin | 1 |
| nuclear quadrupole moment | |
| data della scoperta o invenzione | 1972 |
| parità
| tipo di decadimento | intensità |
--- |
| β− (β− decay) | 100% |
| β− n (β−-delayed neutron emission) | 1.14% |
49 K
| numero di massa | 49 |
| numero neutronico | 30 |
| massa atomica relativa | 48,968210753 ± 0,00000086 Da |
| Fattore-g | 2,6772 ± 0,0016 |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | 1,26 ± 0,05 s |
| spin | 1/2 |
| nuclear quadrupole moment | 0 |
| data della scoperta o invenzione | 1972 |
| parità | + |
| tipo di decadimento | intensità |
--- |
| β− (β− decay) | 100% |
| β− n (β−-delayed neutron emission) | 86% |
50 K
| numero di massa | 50 |
| numero neutronico | 31 |
| massa atomica relativa | 49,972380015 ± 0,0000083 Da |
| Fattore-g | 0 |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | 472 ± 4 ms |
| spin | 0 |
| nuclear quadrupole moment | 0 |
| data della scoperta o invenzione | 1972 |
| parità
| tipo di decadimento | intensità |
--- |
| β− (β− decay) | 100% |
| β− n (β−-delayed neutron emission) | 28.6% |
| 2n (2-neutron emission) | |
51 K
| numero di massa | 51 |
| numero neutronico | 32 |
| massa atomica relativa | 50,975828664 ± 0,000014 Da |
| Fattore-g | 0,342 ± 0,0013333333333333 |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | 365 ± 5 ms |
| spin | 3/2 |
| nuclear quadrupole moment | |
| data della scoperta o invenzione | 1983 |
| parità | + |
| tipo di decadimento | intensità |
--- |
| β− (β− decay) | 100% |
| β− n (β−-delayed neutron emission) | 65% |
| 2n (2-neutron emission) | |
52 K
| numero di massa | 52 |
| numero neutronico | 33 |
| massa atomica relativa | 51,981602 ± 0,000036 Da |
| Fattore-g | |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | 110 ± 4 ms |
| spin | 2 |
| nuclear quadrupole moment | |
| data della scoperta o invenzione | 1983 |
| parità
| tipo di decadimento | intensità |
--- |
| β− (β− decay) | 100% |
| β− n (β−-delayed neutron emission) | 72.2% |
| 2n (2-neutron emission) | 2.3% |
53 K
| numero di massa | 53 |
| numero neutronico | 34 |
| massa atomica relativa | 52,9868 ± 0,00012 Da |
| Fattore-g | |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | 30 ± 5 ms |
| spin | 3/2 |
| nuclear quadrupole moment | |
| data della scoperta o invenzione | 1983 |
| parità | + |
| tipo di decadimento | intensità |
--- |
| β− (β− decay) | 100% |
| β− n (β−-delayed neutron emission) | 64% |
| 2n (2-neutron emission) | 10% |
54 K
| numero di massa | 54 |
| numero neutronico | 35 |
| massa atomica relativa | 53,994471 ± 0,000429 Da |
| Fattore-g | |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | 10 ± 5 ms |
| spin | 2 |
| nuclear quadrupole moment | |
| data della scoperta o invenzione | 1983 |
| parità
| tipo di decadimento | intensità |
--- |
| β− (β− decay) | 100% |
| β− n (β−-delayed neutron emission) | |
| 2n (2-neutron emission) | |
55 K
| numero di massa | 55 |
| numero neutronico | 36 |
| massa atomica relativa | 55,000505 ± 0,000537 Da |
| Fattore-g | |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | |
| spin | 3/2 |
| nuclear quadrupole moment | |
| data della scoperta o invenzione | 2009 |
| parità | + |
| tipo di decadimento | intensità |
--- |
| β− (β− decay) | |
| β− n (β−-delayed neutron emission) | |
| 2n (2-neutron emission) | |
56 K
| numero di massa | 56 |
| numero neutronico | 37 |
| massa atomica relativa | 56,008567 ± 0,000644 Da |
| Fattore-g | |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | |
| spin | 2 |
| nuclear quadrupole moment | |
| data della scoperta o invenzione | 2009 |
| parità
| tipo di decadimento | intensità |
--- |
| β− (β− decay) | |
| β− n (β−-delayed neutron emission) | |
| 2n (2-neutron emission) | |
57 K
| numero di massa | 57 |
| numero neutronico | 38 |
| massa atomica relativa | 57,015169 ± 0,000644 Da |
| Fattore-g | |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | |
| spin | 3/2 |
| nuclear quadrupole moment | |
| data della scoperta o invenzione | 2018 |
| parità | + |
| tipo di decadimento | intensità |
--- |
| β− (β− decay) | |
| β− n (β−-delayed neutron emission) | |
| 2n (2-neutron emission) | |
58 K
| numero di massa | 58 |
| numero neutronico | 39 |
| massa atomica relativa | 58,023543 ± 0,000751 Da |
| Fattore-g | |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | |
| spin | 2 |
| nuclear quadrupole moment | |
| data della scoperta o invenzione | 2019 |
| parità
| tipo di decadimento | intensità |
--- |
| β− (β− decay) | |
| β− n (β−-delayed neutron emission) | |
| 2n (2-neutron emission) | |
59 K
| numero di massa | 59 |
| numero neutronico | 40 |
| massa atomica relativa | 59,030864 ± 0,000859 Da |
| Fattore-g | |
| natural abundance | |
| radioattività | ☢️ radioactive element |
| emivita | |
| spin | 3/2 |
| nuclear quadrupole moment | |
| data della scoperta o invenzione | 2018 |
| parità | + |
| tipo di decadimento | intensità |
--- |
| β− (β− decay) | |
| β− n (β−-delayed neutron emission) | |
| 2n (2-neutron emission) | |
Previous Next
Previous Next
storia
| scoperto o inventato da | Sir Humphrey Davy |
| luogo di scoperta | England |
| data della scoperta o invenzione | 1807 |
| etimologia | English: pot ash; symbol from Latin: kalium, (alkali). |
| pronuncia | pe-TASS-i-em (inglese) |
sorgente
| Abbondanza |
| Abbondanza sulla crosta terrestre | 20.900 mg/kg |
| natural abundance (oceano) | 399 mg/L |
| natural abundance (corpo umano) | 0,2% |
| natural abundance (meteoroide) | 0,07% |
| natural abundance (Sole) | 0,0004% |
| Abbondanza nell'universo | 0,0003% |
Nuclear Screening Constants
1 s 0.5105
2 p 3.9728
2 s 5.9938
3 p 11.2744
3 s 10.3201
4 s 15.5048
Potassio on ChemicalAid
Potassio
potassio
identificatore |
12928 | https://chemguide.co.uk/14to16/equations/formulaeionic.html | | |
| Chemguide: Core Chemistry 14 - 16 How to write formulae for simple ionic compounds This page explains how to work out the formulae of the simple ionic compounds that you will meet at this level. It is essential that you take our time over this, and don't leave the topic until you feel reasonably competent at writing these formulae. This is a key bit of chemistry, and the truth is that if you can't be bothered to do it properly, you might as well give up chemistry here and now. You can't succeed without this most basic chemistry tool. You will need to know about ionic bonding and have access to a Periodic Table such as the one you can download from this site. The download button is at the beginning of the second paragraph under the table. How writing formulae for ionic compounds works There are potentially thousands of ionic compounds whose formulae you could possibly be asked to write or recognise - although a relatively small number will turn up again and again. Trying to learn all of them would be a ridiculous waste of time, and really difficult. You will need to do a tiny bit of learning, but once that is done, the process is easy. First of all, of course, you need to know that the compound is ionic. There is a simple bit of guidance to help you there, and I will give you that in a minute. Secondly, all these ionic compounds are overall electrically neutral. There have to be equal numbers of positive and negative charges in the compound. Knowing what those charges are is going to take up quite a lot of the rest of this page. There are a few simple generalisations, and a small amount of learning. Suppose, then, that you wanted to write the formula for the ionic compound, magnesium chloride. Suppose you knew that magnesium ions had a 2+ charge, Mg2+, and chloride ions had a 1- charge, Cl-. The magnesium chloride has to be electrically neutral overall. To balance the charges you would need 2 chloride ions for every magnesium ion: 2 negative charges to balance the 2 positive charges. So the formula for magnesium chloride is MgCl2. As another example, if you knew that the charge on a sodium ion was +1, Na+, and the charge on an oxide ion was 2-, O2-, then it is easy to see that the formula for sodium oxide is Na2O. You need to have two sodium ions to balance the charges on the oxide ion. Notice the little numbers in these formulae. A number written smaller and subscripted (set lower) in a formula counts the number of atoms or ions immediately before it. So in the first formula, MgCl2, there are 2 chlorines, but only 1 magnesium - the 2 doesn't apply to the magnesium as well. In the second formula, Na2O, the 2 only applies to the sodium. Incidentally, the number 1 is never written into a formula - if there is no number after a symbol, then there is one of it. Once you know the charges on the ions making up the compound, working out the formula is easy. Setting the rules Some of these rules have exceptions, but you won't meet them at this introductory level - so we will ignore the exceptions for now. How do you know if a compound is ionic? A compound will be ionic if . . . it contains a metal, it is an acid in solution (which will contain H+ ions), it is an ammonium salt (containing NH4+ ions - you will need to learn that one). How do you know if an ion is positive or negative? All metals form positive ions. Two other positive ions you will meet are H+ and NH4+. Simple non-metal ions (containing only one type of atom) that you will meet are negative. How do you know if something is a metal or non-metal? There is a simple pattern from the Periodic Table. Here is a simplified version of the Table missing out the bits that aren't relevant to this level. The non-metals are shown in green. You can see that the non-metals are all found on the right-hand side of the Periodic Table. All the rest of the elements are metals. |
| Note:The dividing line between metals and non-metals isn't quite as clear-cut as this. Some of the elements just to the left of the green bit tend to have properties which are a mixture of metal and non-metal. This isn't something you need to worry about. The elements you will come across during a course at this level are clearly one or the other. |
| How do you name non-metal ions? Simple ions from non-metals have their endings changed to "ide". So the ions are called nitride, oxide, sulfide, fluoride, chloride, bromide, and iodide. Not all of the non-metals form simple ions. The ones mentioned above are the only simple ions you are likely to come across. Phosphorus does form a phosphide ion, but it isn't common. |
| Note:Are there non-simple ions? Yes! "Complex ions" are also common in the non-metals. These contain the non-metal plus other things as well - often oxygen. We will have more to say about this below. |
| How do you know how many charges an ion has? Where the name of the compound tells you the number of charges on a metal ion Quite a lot of compound names have a Roman numeral as a part of the name. (The proper term for this number is the oxidation state or oxidation number of the element. That won't concern you until you go on to do chemistry at a higher level.) For example . . . copper(II) oxide iron(III) fluoride lead(II) bromide This tells you how many positive charges the ion has. We are talking about the charges on a metal ion, and so they will always be positive. So . . . In copper(II) oxide, the copper ion has 2+ charges. In iron(III) fluoride, the iron ion has 3+ charges. In lead(II) bromide, the lead ion has 2+ charges. Where you can work out the number of charges on a metal ion from a Periodic Table Let's look again at the Periodic Table from earlier on. The numbers at the top of the Groups (the vertical columns in the table) are the old group numbers - from 1 to 7 and then 0. These ignore the transition metals. This has the big advantage for students at this level that it counts the number of electrons in the outer energy level of the atoms of each element. |
| Note:The last Group in the Periodic Table (the Noble Gases) is usually called Group 0, but was sometimes called Group 8. Helium, of course, only has room in its outer level for 2 electrons. |
| Modern numbering includes the transition metals, and the numbers go from 1 to 18. You will find this on the Periodic Table I suggested at the top of the page. To convert the new numbering to the old one, just subtract 10 from groups 13 to 17. Why does this matter? For the metals in Groups,1 2 and 3 (on the old numbering) the number of charges on the ions is the same as the Group number. That is because they have 1, 2 or 3 electrons in their outer levels to give away to something else. So. . . Potassium is in Group 1, and so its ion carries a 1+ charge, K+. Barium is in Group 2, and so its ion carries a 2+ charge, Ba2+. Aluminium is in Group 3, and so its ion carries a 3+ charge, Al3+. The positive ions you will need to learn Group 1, 2 and 3 metals are easy, and so are all those with the Roman numerals in their name. There are a couple of metals, though, where the Roman numerals are left out more often than not. You will need to learn . . . Silver ions are Ag+. Zinc ions are Zn2+. Don't forget that you will also need to know about these positive ions which don't contain metals . . . Hydrogen ions are H+. Ammonium ions are NH4+. Working out the charges on non-metal ions using the Periodic Table We are talking here about simple non-metal ions. The ones you need to know are oxide, sulfide, fluoride, chloride, bromide, and iodide in Groups 6 and 7. Elements at the top of Group 5 (nitrogen and phosphorus) also form negative ions in rare cases - so you should know how to work out the charges on nitride and phosphide ions as well just in case. Elements in Groups 5, 6 and 7 have 5, 6 and 7 electrons in their outer energy levels. That means they have room to gain 3, 2 or 1 electrons to form ions with 8 electrons in the outer level. Elements in Group 7 can gain 1 electron and so their ions have a 1- charge, e.g. Cl-. Elements in Group 6 can gain 2 electrons and so their ions have a 2- charge, e.g. S2-. Elements in Group 5 can gain 3 electrons and so their ions have a 3- charge, e.g. N3-. |
| Note:For Group 5, this only really applies to nitrogen and to an even smaller extent phosphorus. Most of time, nitrogen and phosphorus form covalent bonds, or are a part of a complex ion (see below for one with nitrogen in). |
| Awkward complex negative ions A complex ion is one made up of more than one sort of atom. | | | --- | | hydroxide ions | OH- | | nitrate ions | NO3- | | carbonate ions | CO32- | | sulfate ions | SO42- | You just have to learn these. Sorry! You may come across one or two more during the course, but these are enough to get started with. Some worked examples You need to have a copy of the Periodic Table available which has the names of the elements as well as their symbols. As you read through these examples, try to work them out for yourself before you read on. The more you pactise this, the easier it will become. What is the formula for chromium(III) chloride? First, you need the symbol for chromium. If you don't know it, find it! The name tells you that it has 3+ charges. By this time, you will probably know that the symbol for chlorine is Cl. It is in Group 7 and so has 7 outer electrons. There is room for 1 more in the outer level, and so a chloride ion has one extra negative electron. The ions are therefore Cr3+ and Cl-. To balance the charges on the chromium you will need 3 chloride ions. The formula is CrCl3. What is the formula for potassium sulfate? First, you need the symbol for potassium. If you don't know it, find it! Potassium is in Group 1 and so forms a 1+ ion. Sulfate is an ion you can't work out - you have to remember it. Go back up this page and look it up for now. The ions are therefore K+ and SO42-. To balance the charges on the sulfate ion you will need 2 potassium ions. The formula is K2SO4. What is the formula for iron(III) oxide? You need the symbol for iron. The name tells you that it has 3+ charges. The symbol for oxygen is O. It is in Group 6 and so has room for 2 more electrons in the outer level - so an oxide ion has 2 negative charges. The ions are therefore Fe3+ and O2-. This is slightly more tricky! The only way to balance the charges is to have 2 iron(III) ions and 3 oxide ions: 6 pluses and 6 negatives. The formula is Fe2O3. What is the formula for magnesium nitrate? Notice this is magnesium nitrate, NOT magnesium nitride - the ends of the words really matter. The nitrate ion is an ion which you will have to learn. The nitrate ion is NO3-. Magnesium is in Group 2 and so has 2+ charges, Mg2+. To balance the charges, you will need two nitrate ions for every magnesium ion. The formula is Mg(NO3)2. This is important! Notice the brackets around the nitrate group. If you need to have more than one complex ion, you must enclose it in brackets before you write any number after it. The "2" in the magnesium nitrate formula has to apply to the whole of the nitrate group. The only way of showing that is to enclose it in brackets. However, if you only need one of the group, you must not put brackets around it - that would be wrong. See, for example, the formula for potassium sulfate above. There are no brackets in this, because there is only one sulfate group. Some problems for you to do You will need a copy of the Periodic Table to refer to which must include the names of the elements. I suggest that you also make a list of those ions that need to be learnt on a piece of paper so that you can refer to them quickly. But do the problems first, and worry about learning them later. The fact that you keep will keep writing them down will mean that by the time you have finished this, you will probably have remembered most of them anyway. I have listed the problems in sets, and after each set, you will find a link to the answers for that set. Do one set at a time, and don't go on until you are sure that you completely understand any formula that you might have got wrong. Don't worry if it takes you a long time to start with - that is entirely to be expected. You will find it speeds up considerably once you get to know your way around the Periodic Table and the ions get more familiar. Once you have got to the end of this successfully, you will have cracked one of the big problems students find in the early stages of chemistry courses. To save you searching the whole Periodic Table for symbols, the metals used in the examples will mainly be from Groups 1, 2 and 3 (only aluminium in Group 3) or from the top row of the transition elements - but there will be a couple of other examples as well. Set 1 potassium iodide, iron(II) sulfide, calcium chloride, magnesium carbonate, lithium oxide set 1 answers Set 2 sodium bromide, copper(II) sulfate, barium hydroxide, nickel(II) chloride, lithium nitrate set 2 answers Set 3 aluminium chloride, lithium nitride, sodium nitrate, copper(II) nitrate, zinc sulfide set 3 answers Set 4 potassium carbonate, aluminium oxide, silver iodide, lead(II) nitrate, aluminium hydroxide set 4 answers Set 5 barium sulfate, zinc nitrate, ammonium chloride, sulfuric acid ("hydrogen sulfate"), lead(II) sulfide set 5 answers Set 6 ammonium sulfate, lead(II) bromide, nitric acid ("hydrogen nitrate"), chromium(III) sulfate, iron(III) hydroxide set 6 answers What now? If you made more than the occasional mistake, wait a day, re-read this page, and then do the problems again. It is essential that you don't leave this page until you can write formulae successfully. It doesn't matter if you haven't learnt all the awkward ions like nitrate or ammonium or silver at the moment. What matters is that you can put together a formula, even if you need to look some things up. If you have been successful most or all of the time, learn any of the ions which you don't know properly. You will find a complete list of the awkward ones below. | | | --- | | ammonium ions | NH4+ | | hydrogen ions | H+ | | silver ions | Ag+ | | zinc ions | Zn2+ | | hydroxide ions | OH- | | nitrate ions | NO3- | | carbonate ions | CO32- | | sulfate ions | SO42- | You will get more practice at writing formulae when we go on to look at writing equations. Where would you like to go now? To the formulae and equations menu . . . To the Chemistry 14-16 menu . . . To Chemguide Main Menu . . . © Jim Clark 2019 | |
12929 | https://espressomd.github.io/tutorials/lennard_jones/lennard_jones.html | Introductory Tutorial: Lennard-Jones Liquid¶
Table of Contents¶
Introduction
Background
The Lennard-Jones Potential
Units
First steps
Overview of a simulation script
System setup
Placing and accessing particles
Setting up non-bonded interactions
Energy minimization
Choosing the thermodynamic ensemble, thermostat
Integrating equations of motion and taking manual measurements
Automated data collection
Further Exercises
Binary Lennard-Jones Liquid
References
Introduction¶
Welcome to the basic ESPResSo tutorial!
In this tutorial, you will learn, how to use the ESPResSo package for your research. We will cover the basics of ESPResSo, i.e., how to set up and modify a physical system, how to run a simulation, and how to load, save and analyze the produced simulation data.
More advanced features and algorithms available in the ESPResSo package are described in additional tutorials.
Background¶
Research on Soft Condensed Matter has brought the needs for having a flexible, extensible, reliable, and efficient (parallel) molecular simulation package. For this reason ESPResSo (Extensible Simulation Package for Research on Soft Matter Systems) has been developed at the Max Planck Institute for Polymer Research, Mainz, and at the Institute for Computational Physics at the University of Stuttgart in the group of Prof. Dr. Christian Holm [2,3]. The ESPResSo package is a flexible and extensible simulation package. It is specifically developed for coarse-grained molecular dynamics (MD) simulation of polyelectrolytes but is not necessarily limited to this. For example, it could also be used to simulate granular media. ESPResSo has been nominated for the Heinz-Billing-Preis for Scientific Computing in 2003 .
Prior knowledge¶
Basics on thermodynamics and statistical mechanics.
Basics on coarse-grained simulation methods: molecular dynamics and Langevin dynamics.
Correlation functions.
Pair structure and radial distribution function.
The Lennard-Jones Potential¶
A pair of neutral atoms or molecules is subject to two distinct forces in the limit of large separation and small separation: an attractive force at long ranges (van der Waals force, or dispersion force) and a repulsive force at short ranges (the result of overlapping electron orbitals, referred to as Pauli repulsion from the Pauli exclusion principle). The Lennard-Jones potential (also referred to as the LJ potential, 6-12 potential or, less commonly, 12-6 potential) is a simple mathematical model that represents this behavior. It was proposed in 1924 by John Lennard-Jones. The LJ potential is of the form
\begin{equation} V(r) = 4\epsilon \left[ \left( \dfrac{\sigma}{r} \right)^{12} - \left( \dfrac{\sigma}{r} \right)^{6} \right] \end{equation}
where $\epsilon$ is the depth of the potential well, providing a measure of the attraction strength, $\sigma$ is the (finite) distance at which the inter-particle potential is zero and $r$ is the distance between the particles. The $\left(\frac{1}{r}\right)^{12}$ term describes repulsion and the $\left(\frac{1}{r}\right)^{6}$ term describes attraction. The Lennard-Jones potential is an approximation. The form of the repulsion term has no theoretical justification; the repulsion force should depend exponentially on the distance, but the inverse power form of the repulsion term of the LJ formula is more convenient due to the ease and efficiency of computing $r^{12}$ as the square of $r^6$.
In simulations, the LJ potential is typically cut off beyond a specified distance $r_{\mathrm{cut}}$, i.e. the potential is zero for distances larger than the cutoff distance. Also an additional potential shift $c_{\mathrm{shift}}$ can be added:
\begin{equation} V(r) = \begin{cases}4\epsilon \left[ \left( \dfrac{\sigma}{r} \right)^{12} - \left( \dfrac{\sigma}{r} \right)^{6} \right] + c_{\mathrm{shift}} && 0 r_{\mathrm{cut}} \end{cases} \end{equation}
For example, the LJ potential could be cut off at $r_{\mathrm{cut}}=2^{1/6}\sigma$, which corresponds to the location of the potential minimum. One can then shift the potential up by $c_{\mathrm{shift}}=\epsilon$ to make it continuous, resulting in a purely repulsive version of the LJ potential, commonly known as the Weeks–Chandler–Andersen potential. Another common choice for the cutoff is $r_{\mathrm{cut}}=2.5\sigma$, beyond which the potential is consistently $|V(r>r_{\mathrm{cut}})| < 0.02 \epsilon$. In this case, $c_{\mathrm{shift}}$ is chosen such that the potential is continuous at $r_{\mathrm{cut}}$. The main advantage of this truncated version of the LJ potential is that it is significantly cheaper computationally, while the error due to the missing tail is negligible.
Originally LJ has been introduced to describe Van der Waals interactions in molecular systems, but nowadays it is often used to represent generic particles in coarse-grained simulations, such as polymer systems.
In :
import numpy as np % matplotlib importmatplotlib.pyplot as plt plt. rcParams. update({'font.size': 18}) def lj_pot(x, epsilon, sigma, r_cut, c_shift =0.0): pot =4.0 epsilon((sigma/ x) 12 -(sigma/ x) 6) + c_shift pot[x> r_cut] =0. return pot epsilon =1.0 # energy in units of k_BT sigma =1.0 # distance in units of sigma xs = np. linspace(0.5, 3, 100) ys_lj = lj_pot(xs, epsilon, sigma, r_cut = xs[- 1]) ys_WCA = lj_pot(xs, epsilon, sigma, r_cut = 2(1/ 6) sigma, c_shift = epsilon) fig = plt. figure(figsize =(10, 6)) plt. plot(xs, ys_lj, label = 'LJ') plt. plot(xs, ys_WCA, label = 'WCA') plt. axhline(y = 0, color = 'grey') plt. xlabel(r"$r/\sigma$") plt. ylabel(r"$V(r)/(k_{\mathrm{B}}T)$") plt. legend() plt. ylim(-1.5,2.5) plt. show()
Units¶
Novice users must understand that ESPResSo has no fixed unit system. The unit system is set by the user. Conventionally, reduced units are employed. For instance, the potential can be expressed in units of $k_{\mathrm{B}}T$: $$V^(r^)=\beta V(r^) = 4\epsilon^ \left[ \left( \dfrac{1}{r^} \right)^{12} - \left( \dfrac{1}{r^} \right)^{6} \right]$$ with $\beta=(k_{\mathrm{B}}T)^{-1}$, $\epsilon^=\beta\epsilon$ and $r^=r/\sigma$. We have used $\sigma$ as unit of distance.
For more information on units, please check the documentation on units.
Lennard-Jones fluid¶
In the following simulations, we are going to model a single-species Lennard-Jones fluid at constant number of particles $N$, density $\rho=\frac{N}{V}$ and temperature $T$. Depending on the chosen parameters, the system might present different phase behavior, leading to solid, liquid and gas phases. Simulations will allow us to characterize these different phases structurally by means of the radial distribution function.
For a more detailed discussion on simulation results about phase diagrams of Lennard-Jones fluids, see .
First steps¶
What is ESPResSo? It is an extensible, efficient Molecular Dynamics package specially powerful on simulating charged systems. In depth information about the package can be found in the relevant sources [1,4,2,3].
ESPResSo consists of two components. The simulation engine is written in C++ for the sake of computational efficiency. The steering or control level is interfaced to the kernel via an interpreter of the Python scripting languages.
The kernel performs all computationally demanding tasks. Before all, integration of Newton's equations of motion, including calculation of energies and forces. It also takes care of internal organization of data, storing the data about particles, communication between different processors or cells of the cell-system.
The scripting interface (Python) is used to setup the system (particles, boundary conditions, interactions etc.), control the simulation, run analysis, and store and load results. The user has at hand the full reliability and functionality of the scripting language. For instance, it is possible to use the SciPy package for analysis and PyPlot for plotting. With a certain overhead in efficiency, it can also be used to reject/accept new configurations in combined MD/MC schemes. In principle, any parameter which is accessible from the scripting level can be changed at any moment of runtime. In this way methods like thermodynamic integration become readily accessible.
Note: This tutorial assumes that you already have a working ESPResSo installation on your system. If this is not the case, please consult the first chapters of the user's guide for installation instructions.
Overview of a simulation script¶
Typically, a simulation script consists of the following parts:
System setup (box geometry, thermodynamic ensemble, integrator parameters)
Placing the particles
Setup of interactions between particles
Equilibration (bringing the system into a state suitable for measurements)
Integration loop (propagate the system in time and record measurements)
System setup¶
The functionality of ESPResSo for python is provided via a python module called espressomd. At the beginning of the simulation script, it has to be imported.
In :
import espressomd importespressomd.observables importespressomd.accumulators importespressomd.analyze importespressomd.zn required_features =["LENNARD_JONES"] espressomd. assert_features(required_features)
The function espressomd.assert_features() expects a list of features as argument and checks they are available in the ESPResSo executable. If a required feature is missing, the program will print an error message and halt. To compile ESPResSo with a different set of features, see the documentation on features.
In :
importscipy.optimize np. random. seed(42) # System parameters N_PART = 200 DENSITY =0.75 BOX_L = np. power(N_PART/ DENSITY,1.0/3.0) np. ones(3)
The next step would be to create an instance of the System class. This instance is used as a handle to the simulation system. At any time, only one instance of the System class can exist.
Exercise:
Create an instance of an espresso system and store it in a variable called system; use BOX_L as box length.
See ESPResSo documentation and module documentation.
In :
```
SOLUTION CELL system = espressomd. System(box_l = BOX_L)
```
In :
```
Test solution of Exercise 1 assert isinstance(system, espressomd. System)
```
It can be used to store and manipulate the crucial system parameters like the time step and the size of the simulation box (time_step, and box_l).
In :
SKIN =0.4 TIME_STEP =0.01 system. time_step = TIME_STEP system. cell_system. skin = SKIN
The parameter SKIN affects how often the Verlet lists will be updated. This parameter does not influence the physics of the simulation. It can however have a significant impact on the performance of the simulation. Depending on system parameters such as density and temperature, the optimal SKIN parameter can vary considerably. Please be aware that ESPResSo implements the function tune_skin() that automatically tunes SKIN for optimal performance.
Placing and accessing particles¶
Particles in the simulation can be added and accessed via the part property of the System class. Individual particles should be referred to by the particle handle returned upon creation. You can also retrieve them by an integer id, e.g. system.part.by_id(0). If id is unspecified when creating a particle, an unused particle id is automatically assigned. It is also possible to use common python iterators and slicing operations to add or access several particles at once.
Particles can be grouped into several types, so that, e.g., a binary fluid can be simulated. Particle types are identified by integer ids, which are set via the particles' type attribute. If it is not specified, zero is implied.
Create N_PART particles at random positions, store their handles in a variable called particles.
Use system.part.add().
Use an (N_PART x 3) array for positions. Use np.random.random() to generate random numbers.
In :
```
SOLUTION CELL particles = system. part. add(type = N_PART, pos = np. random. random((N_PART, 3)) system. box_l)
```
In :
```
Test that now we have indeed N_PART particles in the system assert len(system. part) == N_PART
```
The particle properties can be accessed using standard numpy slicing syntax:
In :
```
Access position of a single particle print(f"position of particle with id 0: {system. part. by_id(0). pos} ")# Iterate over the first five particles for the purpose of demonstration. first_five = system. part. by_ids(range(5)) for p in first_five: print(f "id {p. id} position: {p. pos} ") print(f "id {p. id} velocity: {p. v} ") # Obtain particle positions for the particles created until now cur_pos = particles. pos
```
position of particle with id 0: [2.41076339 6.1193638 4.7115492 ] id 0 position: [2.41076339 6.1193638 4.7115492 ] id 0 velocity: [0. 0. 0.] id 1 position: [3.85332274 1.00422894 1.00407369] id 1 velocity: [0. 0. 0.] id 2 position: [0.37386074 5.57522583 3.86913442] id 2 velocity: [0. 0. 0.] id 3 position: [4.55757705 0.13249407 6.24291778] id 3 velocity: [0. 0. 0.] id 4 position: [5.35809689 1.36674105 1.17033384] id 4 velocity: [0. 0. 0.]
You can also get all particles using system.part.all(), but particles already contains all particles that are in the simulation so far.
Many objects in ESPResSo have a string representation, and thus can be displayed via python's print function:
In :
print(system. part. by_id(0))
ParticleHandle({'id': 0, 'pos': (2.4107633923737293, 6.119363804209853, 4.711549202763617), 'ext_torque': (0.0, 0.0, 0.0), 'omega_body': (0.0, 0.0, 0.0), 'swimming': {'is_engine_force_on_fluid': False, 'f_swim': 0.0}, 'lees_edwards_flag': 0, 'vs_relative': [-1, 0.0, array([1., 0., 0., 0.])], 'dipm': 0.0, 'gamma': (-1.0, -1.0, -1.0), 'v': (0.0, 0.0, 0.0), 'propagation': , 'mass': 1.0, 'gamma_rot': (-1.0, -1.0, -1.0), 'bonds': (), 'exclusions': (), 'f': (0.0, 0.0, 0.0), 'torque_lab': (0.0, 0.0, 0.0), 'pos_folded': (2.4107633923737293, 6.119363804209853, 4.711549202763617), 'rotation': (False, False, False), 'omega_lab': (0.0, 0.0, 0.0), 'vs_quat': (1.0, 0.0, 0.0, 0.0), 'ext_force': (0.0, 0.0, 0.0), 'image_box': (0, 0, 0), 'q': 0.0, 'fix': (False, False, False), 'lees_edwards_offset': 0.0, 'director': (0.0, 0.0, 1.0), 'quat': (1.0, 0.0, 0.0, 0.0), 'mol_id': 0, 'type': 0, 'dip': (0.0, 0.0, 0.0), 'rinertia': (1.0, 1.0, 1.0), 'node': 0, 'dip_fld': (0.0, 0.0, 0.0), 'mu_E': (0.0, 0.0, 0.0)})
Setting up non-bonded interactions¶
Non-bonded interactions act between all particles of a given combination of particle types. In this tutorial, we use the Lennard-Jones non-bonded interaction. First we define the LJ parameters
In :
```
use LJ units: EPS=SIG=1 LJ_EPS =1.0 LJ_SIG =1.0 LJ_CUT =2.5 LJ_SIG
```
In a periodic system, it is in general not straightforward to calculate all non-bonded interactions. As mentioned earlier in the text, usually a cutoff distance $r_{\mathrm{cut}}$ is applied for infinite-range potentials like Lennard-Jones, such that $V(r>r_{\mathrm{cut}}) = 0$. The potential can be shifted to zero at the cutoff value to ensure continuity using the shift='auto' option of espressomd.interactions.LennardJonesInteraction. For comparison with the fundamental work on MD simulations of LJ systems , we do NOT shift the potential to zero at the cutoff in this tutorial and instead correct for the long-range error $V_{\mathrm{lr}}$ later. However, it is strongly recommended to implement shift='auto' in any other case to ensure continuity of the potential!
To avoid spurious self-interactions of particles with their periodic images one usually forces that the shortest box length is at least twice the cutoff distance:
In :
assert(BOX_L - 2 SKIN> LJ_CUT). all()
Exercise:
Setup a Lennard-Jones interaction with $\epsilon=$LJ_EPS and $\sigma=$LJ_SIG that is cut at $r_{\mathrm{cut}}=$LJ_CUT$\times\sigma$ and not shifted.
Hint:
Have a look at the documentation on non-bonded interactions
In :
```
SOLUTION CELL system. non_bonded_inter[0, 0]. lennard_jones. set_params(epsilon = LJ_EPS, sigma = LJ_SIG, cutoff = LJ_CUT, shift = 0)
```
Energy minimization¶
In many cases, including this tutorial, particles are initially placed randomly in the simulation box. It is therefore possible that particles overlap, resulting in a huge repulsive force between them. In this case, integrating the equations of motion would not be numerically stable. Hence, it is necessary to remove possible overlaps. This is typically done by performing a steepest descent minimization of the potential energy until a maximal force criterion is reached.
Note: Making sure a system is properly equilibrated highly depends on the system details. In most cases a relative convergence criterion on the forces and/or energies works well but you might have to make sure that the total force is smaller than a threshold value f_max at the end of the minimization. Depending on the simulated system, other strategies might be necessary, such as simulating with a small time step or with capped forces.
In :
```
suitable minimization parameters for this LJ system F_TOL =1e-2 DAMPING = 30 MAX_STEPS = 10000 MAX_DISPLACEMENT =0.01 LJ_SIG EM_STEP = 10
```
We will use espressomd.integrate.set_steepest_descent() to relax the initial configuration. The particle displacement is related to the particle force via a damping constant $\gamma$, such that: $$\vec{x}_i(t + \Delta t) = \vec{x}_i(t) + \min\left(|\gamma\vec{F}_i(t)\Delta t|, r_{\text{max}}\right) \cdot \vec{F}_i(t)/|\vec{F}_i(t)|$$
with $r_{\text{max}}$ the maximal displacement, $\gamma$ the friction coefficient, $\vec{x}$ the particle position, $\vec{F}$ the force on the particle, $\Delta t$ the time step, and $i$ the vector index. We will integrate until the largest particle force in the system falls below a specific threshold value FMAX, chosen in such a way that integrating the system with that force would lead to a displacement inferior or equal to 1% of the particle diameter.
In :
MASS =1.0 FMAX =0.01 LJ_SIG MASS/ system. time_step 2 system. integrator. set_steepest_descent(f_max = FMAX, gamma = 10, max_displacement =0.01) system. integrator. run(200) assert np. all(np. abs(system. part. all(). f)< FMAX),"Overlap removal did not converge!"
In :
```
reset clock system. time =0.
```
Choosing the thermodynamic ensemble, thermostat¶
Simulations can be carried out in different thermodynamic ensembles such as NVE (particle Number, Volume, Energy), NVT (particle Number, Volume, Temperature) or isotropic NpT (particle Number, pressure, Temperature).
In this tutorial, we use a NVT ensemble with the Langevin thermostat.
In :
```
Parameters for the Langevin thermostat# reduced temperature T = k_B T / LJ_EPS TEMPERATURE =0.827# value from Tab. 1 in GAMMA =1.0
```
Exercise:
Use system.integrator.set_vv() to use a Velocity Verlet integration scheme and system.thermostat.set_langevin() to turn on the Langevin thermostat. Set the reduced temperature to TEMPERATURE and the solvent friction coefficient to GAMMA. Initialize the random number generator of the thermostat with an integer seed.
The Langevin thermostat maintains the system temperature at a constant value by mimicking the effect of an implicit solvent with large heat capacity that acts as a heat reservoir. This is achieved by introducing both a friction term and a stochastic term in the Newton's equations of motion (for details see the documentation on thermostats).
In :
```
SOLUTION CELL system. integrator. set_vv() system. thermostat. set_langevin(kT = TEMPERATURE, gamma = GAMMA, seed = 42)
```
System visualization¶
We will use ZnDraw to visualize and interact with the simulation box. On the server has started, a new frame will appear in the notebook with a menu bar and various buttons. The bottom numbers represent the number of frames and the current position in the animation. We will soon integrate the system and update the visualizer with the new system configurations.
In :
```
Visualizer Parameters color ={0: "#00f0f0"} # Particle color by type radii ={0: LJ_SIG/2.0} # Particle size by type # Initializing Visualizer vis = espressomd. zn. Visualizer(system, colors = color, radii = radii) vis. update()
```
Out:
Integrating equations of motion and taking manual measurements¶
Now, we integrate the equations of motion and take measurements of relevant quantities.
In :
```
Integration parameters STEPS_PER_SAMPLE = 20 N_SAMPLES = 1000 times = np. zeros(N_SAMPLES) e_total = np. zeros_like(times) e_kin = np. zeros_like(times) T_inst = np. zeros_like(times)
```
Exercise:
Integrate the system and measure the total and kinetic energy. Take N_SAMPLES measurements every STEPS_PER_SAMPLE integration steps. Notice that the total simulated time in LJ units is given by the product N_SAMPLES $\times$ STEPS_PER_SAMPLE $\times$ TIME_STEP.
Calculate the total and kinetic energies using the analysis method system.analysis.energy().
Use the containers times, e_total and e_kin from the cell above to store the time series.
From the simulation results, calculate the kinetic temperature $T_{\mathrm{inst}} = 2/3 \times E_{\mathrm{kin}}$ / N_PART.
Call vis.update() at the end of each loop to update the visualizer. Click inside the ZnDraw frame and press the space bar to animate the frames.
In :
```
SOLUTION CELL for i in range(N_SAMPLES): times[i] = system. time energy = system. analysis. energy() e_total[i] = energy['total'] e_kin[i] = energy['kinetic'] system. integrator. run(STEPS_PER_SAMPLE) vis. update() T_inst =2./3. e_kin/ N_PART
```
In :
plt. figure(figsize =(10, 6)) plt. plot(times, T_inst, label = r"$T_{\mathrm{inst}}$") plt. plot(times,[TEMPERATURE] len(times), label = r"$T_{\mathrm{thermostat}}$") plt. legend() plt. xlabel("Simulation time") plt. ylabel("Temperature") plt. show()
Since the ensemble average $\langle E_{\mathrm{kin}}\rangle=3/2 N k_B T$ is related to the temperature, we may compute the actual temperature of the system via $k_B T= 2/(3N) \langle E_{\mathrm{kin}}\rangle$. Note the thermodynamic temperature $T$ is fixed and does not fluctuate in the NVT ensemble! The kinetic temperature calculated via $T_{\mathrm{inst}} = 2/(3N) E_{\mathrm{kin}}$ (without ensemble averaging) is allowed to fluctuate, but it is not the temperature of the system ($T = \langle T_{\mathrm{inst}} \rangle$).
In the first simulation run, we picked STEPS_PER_SAMPLE arbitrarily. To ensure proper statistics, we will subsample the time series to reduce correlation between consecutive measurements. In order to do this, we first have to remove the beginning of the time series, because the system was out of equilibrium. The time to reach equilibrium depends on the thermostat friction coefficient gamma and can be determined visually from the plot.
In :
```
Use only the data where the system is at equilibrium equilibration_time = 15 e_total = e_total[times> equilibration_time] e_kin = e_kin[times> equilibration_time] times = times[times> equilibration_time] times -= times
```
Notice that equilibration_time is not known a priori. It has to be found by trial and error for the given set of input parameters.
Autocorrelation function and correlation time¶
In simulations, the knowledge of the time evolution of the particle positions and velocities allows for the calculation of autocorrelation functions and autocorrelation time.
Correlation functions are useful to quantify how microscopic observables at different positions or times of a system are related one another. One particular example is the equilibrium velocity autocorrelation function $$C(\tau)=\langle {\bf v}(0)\cdot{\bf v}(\tau)\rangle,$$ that relates the velocities at different times along an equilibrium trajectory . Time correlation functions are typically decaying functions in fluid systems, whose decay is characterized by the so-called correlation time $\xi$. We say that two samples of a given system taken at time $t'$ and $t''$, respectively, are uncorrelated for a certain observable $X$, if $t'-t''\gg \xi_X $.
The ESPResSo module espressomd.analyze provides a function to compute the autocorrelation of time series. By means of espressomd.analyze.autocorrelation(X), it is possible to calculate the unnormalized autocorrelation function of an observable $X$ measured at time $t$ with constant time step for lag times $\tau$. This method neither subtracts the mean value nor normalizes by the variance of the provided time series.
In :
def autocor(x): x = np. asarray(x) mean = np. mean(x) var = np. var(x) xp = x - mean corr = espressomd. analyze. autocorrelation(xp)/ var return corr def fit_correlation_time(data, ts): data = np. asarray(data) data/= data def fitfn(t, t_corr): return np. exp(- t/ t_corr) popt, pcov = scipy. optimize. curve_fit(fitfn, ts, data) return popt
Exercise
Calculate the autocorrelation of the total energy (store in e_total_autocor). Calculate the correlation time $\xi$ (corr_time). Calculate a quantity steps_per_subsample that represents the number of integration steps necessary to advance the simulation time by $3\xi$. This is a conservative quantity that will help us subsample the time series in such a way that the correlation between two consecutive samples is small ($e^{-3} \simeq 5\%$).
The value steps_per_subsample is somewhat arbitrary and was chosen as a trade-off between accuracy and simulation runtime for the purpose of this tutorial. In a research project, you would run simulations for a lot longer and increase the time between subsamples to reduce the residual correlation even further. Please refer to the Error Analysis tutorial for an in-depth discussion of time series autocorrelation.
In :
```
SOLUTION CELL e_total_autocor = autocor(e_total) corr_time = fit_correlation_time(e_total_autocor[: 100], times[: 100]) steps_per_subsample = int(np. ceil(3 corr_time/ system. time_step))
```
In :
print(f'steps_per_subsample = {steps_per_subsample} ')
steps_per_subsample = 233
We plot the autocorrelation function and the fit to visually confirm a roughly exponential decay
In :
plt. figure(figsize =(10, 6)) plt. plot(times[1:], e_total_autocor[1:], label = 'data') plt. plot(times[1:], np. exp(- times[1:]/ corr_time), label = 'exponential fit') plt. plot(2[steps_per_subsample system. time_step],[min(e_total_autocor), 1], label = 'subsampling interval') plt. ylim(top =1.2, bottom =-0.15) plt. legend() plt. xscale('log') plt. xlabel('Simulation time') plt. ylabel('Total energy autocorrelation') plt. show()
In order to obtain equilibrium properties, we need to consider ensemble-averaged quantities. Assuming that the simulation is an ergodic process, averaging over uncorrelated samples would provide equilibrium results.
Exercise:
Calculate the mean and standard error of the mean potential energy per particle using the formula for uncorrelated samples (define mean_pot_energy and SEM_pot_energy).
Hint
You know how many steps are between samples in e_total and how many steps are between subsamples. So you have to figure out how many samples to skip.
In :
```
SOLUTION CELL subsample_step = int(np. ceil(steps_per_subsample/ STEPS_PER_SAMPLE)) pot_energies =(e_total - e_kin)[:: subsample_step]/ N_PART mean_pot_energy = np. mean(pot_energies) SEM_pot_energy = np. std(pot_energies)/ np. sqrt(len(pot_energies))
```
In :
print(f'mean potential energy = {mean_pot_energy:.2f} +/- {SEM_pot_energy:.2f} ')
mean potential energy = -4.97 +/- 0.01
For comparison to literature values we need to account for the error made by the LJ truncation. For an isotropic system one can assume that the density is homogeneous behind the cutoff, which allows to calculate the so-called long-range corrections to the energy and pressure: $$V_{\mathrm{lr}} = \frac{1}{2} \rho \int_{r_{\mathrm{cut}}}^\infty 4 \pi r^2 g(r) V(r) \,\mathrm{d}r$$ Using that the radial distribution function $g(r)=1$ for $r>r_{\mathrm{cut}}$ one obtains $$V_{\mathrm{lr}} = -\frac{8}{3}\pi \rho \epsilon \sigma^3 \left[\frac{1}{3} \left(\frac{\sigma}{r_{\mathrm{cut}}}\right)^9 - \left(\frac{\sigma}{r_{\mathrm{cut}}}\right)^3 \right].$$ Similarly, a long-range contribution to the pressure can be derived .
In :
tail_energy_per_particle =8./3. np. pi DENSITY LJ_EPS LJ_SIG 3(1./3.(LJ_SIG/ LJ_CUT) 9 -(LJ_SIG/ LJ_CUT) 3) mean_pot_energy_corrected = mean_pot_energy + tail_energy_per_particle print(f'corrected mean potential energy = {mean_pot_energy_corrected:.2f} ')
corrected mean potential energy = -5.37
This value differs quite strongly from the uncorrected one but agrees well with the literature value $U^i = -5.38$ given in Table 1 of Ref. .
Automated data collection¶
As we have seen, it is easy to manually extract information from an ESPResSo simulation, but it can get quite tedious. Therefore, ESPResSo provides a number of data collection tools to make life easier (and less error-prone). We will now demonstrate those with the calculation of the radial distribution function.
Observables extract properties from the particles and calculate some quantity with it, e.g. the center of mass, the total energy or a histogram.
Accumulators allow the calculation of observables while running the system and then doing further analysis. Examples are a simple time series or more advanced methods like correlators. For our purposes we need an accumulator that calculates the average of the RDF samples.
In :
```
Parameters for the radial distribution function BIN_WIDTH =0.05 R_MIN =0.0 R_MAX = system. box_l/2.0 N_BINS = int((R_MAX - R_MIN)/ BIN_WIDTH)
```
Exercise
Instantiate a RDF observable
Instantiate a MeanVarianceCalculator accumulator to track the RDF over time. Samples should be taken every steps_per_subsample steps.
Add the accumulator to the auto_update_accumulators of the system for automatic updates
In :
```
SOLUTION CELL rdf_obs = espressomd. observables. RDF(ids1 = system. part. all(). id, min_r = R_MIN, max_r = R_MAX, n_r_bins = N_BINS) rdf_acc = espressomd. accumulators. MeanVarianceCalculator(obs = rdf_obs, delta_N = steps_per_subsample) system. auto_update_accumulators. add(rdf_acc)
```
Now we don't need an elaborate integration loop anymore, instead the RDFs are calculated and accumulated automatically.
In :
system. integrator. run(N_SAMPLES steps_per_subsample)
Exercise
Get the mean RDF (define rdf) from the accumulator
Get the histogram bin centers (define rs) from the observable
In :
```
SOLUTION CELL rdf = rdf_acc. mean() rs = rdf_obs. bin_centers()
```
In :
fig, ax = plt. subplots(figsize =(10, 7)) ax. plot(rs, rdf, label = 'simulated') plt. legend() plt. xlabel('r') plt. ylabel('RDF') plt. show()
We now plot the experimental radial distribution. Empirical radial distribution functions have been determined for pure fluids , mixtures and confined fluids . We will compare our distribution $g(r)$ to the theoretical distribution $g(r^, \rho^, T^)$ of a pure fluid .
In :
```
comparison to literature def calc_literature_rdf(rs, temperature, density, LJ_eps, LJ_sig): T_star = temperature/ LJ_eps rho_star = density LJ_sig 3# expression of the factors Pi from Equations 2-8 with coefficients qi from Table 1# expression for a,g def P(q1, q2, q3, q4, q5, q6, q7, q8, q9): return q1 + q2 np. exp(- q3 T_star) + q4 np. exp(- q5 T_star) + q6/ rho_star + q7/ rho_star 2 + q8 np. exp(- q3 T_star)/ rho_star 3 + q9 np. exp(- q5 T_star)/ rho_star 4 a = P(9.24792, -2.64281,0.133386, -1.35932,1.25338,0.45602, -0.326422,0.045708, -0.0287681) g = P(0.663161, -0.243089,1.24749, -2.059,0.04261,1.65041, -0.343652, -0.037698,0.008899)# expression for c,k def P(q1, q2, q3, q4, q5, q6, q7, q8): return q1 + q2 np. exp(- q3 T_star) + q4 rho_star + q5 rho_star 2 + q6 rho_star 3 + q7 rho_star 4 + q8 rho_star 5 c = P(-0.0677912, -1.39505,0.512625,36.9323, -36.8061,21.7353, -7.76671,1.36342) k = P(16.4821, -0.300612,0.0937844, -61.744,145.285, -168.087,98.2181, -23.0583)# expression for b,h def P(q1, q2, q3): return q1 + q2 np. exp(- q3 rho_star) b = P(-8.33289,2.1714,1.00063) h = P(0.0325039, -1.28792,2.5487)# expression for d,l def P(q1, q2, q3, q4): return q1 + q2 np. exp(- q3 rho_star) + q4 rho_star d = P(-26.1615,27.4846,1.68124,6.74296) l = P(-6.7293, -59.5002,10.2466, -0.43596) # expression for s def P(q1, q2, q3, q4, q5, q6, q7, q8): return(q1 + q2 rho_star + q3/ T_star + q4/ T_star 2 + q5/ T_star 3)/(q6 + q7 rho_star + q8 rho_star 2) s = P(1.25225, -1.0179,0.358564, -0.18533,0.0482119,1.27592, -1.78785,0.634741) # expression for m def P(q1, q2, q3, q4, q5, q6): return q1 + q2 np. exp(- q3 T_star) + q4/ T_star + q5 rho_star + q6 rho_star 2 m = P(-5.668, -3.62671,0.680654,0.294481,0.186395, -0.286954) # expression for n def P(q1, q2, q3): return q1 + q2 np. exp(- q3 T_star) n = P(6.01325,3.84098,0.60793)# fitted expression (=theoretical curve)# slightly more than 1 to smooth out the discontinuity in the range [1.0, 1.02] theo_rdf_cutoff =1.02 theo_rdf = 1 + 1/ rs 2(np. exp(-(a rs + b)) np. sin(c rs + d) + np. exp(-(g rs + h)) np. cos(k rs + l)) theo_rdf[np. nonzero(rs<= theo_rdf_cutoff)] = s np. exp(-(m rs + n) 4)[np. nonzero(rs<= theo_rdf_cutoff)] return theo_rdf
```
In :
theo_rdf = calc_literature_rdf(rs, TEMPERATURE, DENSITY, LJ_EPS, LJ_SIG)
In :
ax. plot(rs, theo_rdf, label = 'literature') ax. legend() fig
Further Exercises¶
Binary Lennard-Jones Liquid¶
A two-component Lennard-Jones liquid can be simulated by placing particles of two types (0 and 1) into the system. Depending on the Lennard-Jones parameters, the two components either mix or separate.
Modify the code such that half of the particles are of type=1. Type 0 is implied for the remaining particles.
Specify Lennard-Jones interactions between type 0 particles with other type 0 particles, type 1 particles with other type 1 particles, and type 0 particles with type 1 particles (set parameters for system.non_bonded_inter[i,j].lennard_jones where {i,j} can be {0,0}, {1,1}, and {0,1}. Use the same Lennard-Jones parameters for interactions within a component, but use a different lj_cut_mixed parameter for the cutoff of the Lennard-Jones interaction between particles of type 0 and particles of type 1. Set this parameter to $2^{\frac{1}{6}}\sigma$ to get de-mixing or to $2.5\sigma$ to get mixing between the two components.
Record the radial distribution functions separately for particles of type 0 around particles of type 0, type 1 around particles of type 1, and type 0 around particles of type 1. This can be done by changing the ids1/ids2 arguments of the espressomd.observables.RDF command. You can record all three radial distribution functions in a single simulation. It is also possible to write them as several columns into a single file.
Plot the radial distribution functions for all three combinations of particle types. The mixed case will differ significantly, depending on your choice of lj_cut_mixed. Explain these differences.
References¶
H. J. Limbach, A. Arnold, B. Mann and C. Holm. ESPResSo: An extensible simulation package for research on soft matter systems. Computer Physics Communications, 174(9):704–727, 2006. DOI:10.1016/j.cpc.2005.10.005 A. Arnold, O. Lenz, S. Kesselheim, R. Weeber, F. Fahrenberger, D. Rohm, P. Košovan, and C. Holm. ESPResSo 3.1 — molecular dynamics software for coarse-grained models. In M. Griebel and M. A. Schweitzer, editors, Meshfree Methods for Partial Differential Equations VI, volume 89 of Lecture Notes in Computational Science and Engineering, pages 1–23. Springer Berlin Heidelberg, 2013. DOI:10.1007/978-3-642-32979-1_1 A. Arnold, BA Mann, HJ Limbach, and C. Holm. ESPResSo–An Extensible Simulation Package for Research on Soft Matter Systems. In K. Kremer and V. Macho, editors, Forschung und wissenschaftliches Rechnen, Beiträge zum Heinz-Billing-Preis 2003, volume 63, pages 43–59. Gesellschaft für wissenschaftliche Datenverarbeitung mbH Göttingen, 2004. B. Smit. Phase diagrams of Lennard‐Jones fluids. Journal of Chemical Physics, 96:8639, 1992. DOI:10.1063/1.462271 W. P. Allen and D. J. Tildesley. Computer Simulation of Liquids. Oxford University Press, 2017. L. Verlet, Computer “Experiments” on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules, Physical Review, 159(1):98–103, 1967. DOI:10.1103/PhysRev.159.98 D. Frenkel and B. Smit. Understanding Molecular Simulation: From Algorithms to Applications. Second edition. Academic Press, 2002. Morsali, Goharshadi, Mansoori, Abbaspour. An accurate expression for radial distribution function of the Lennard-Jones fluid. Chemical Physics, 310(1–3):11–15, 2005. DOI:10.1016/j.chemphys.2004.09.027 Matteoli. A simple expression for radial distribution functions of pure fluids and mixtures. The Journal of Chemical Physics, 103(11):4672, 1995. DOI:10.1063/1.470654 Abbaspour, Akbarzadeha, Abroodia. A new and accurate expression for the radial distribution function of confined Lennard-Jones fluid in carbon nanotubes. RSC Advances, 5(116): 95781–95787, 2015. DOI:10.1039/C5RA16151G |
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Order of Operations
Order of Operations Worksheets
Rich with scads of practice, our printable order of operations worksheets get learners in grade 4 through grade 7 acquainted with the rule of performing the operations in the right order. The pdfs help grasp a procedural understanding of how to apply the order of operations using mnemonics like PEMDAS, DMAS, BEDMAS, or BODMAS in some countries, and the latest addition being GEMS to solve arithmetic expressions involving whole numbers, integers, fractions and decimals. Go ahead, use our free order of operations worksheets and solve arithmetic expressions in a jiffy!
»Evaluating Expressions with Exponents
»Evaluating Expressions with Parentheses
Order of Operations | 4 Basic Operations - Level 1
Apply the order of operations on expressions involving three whole numbers or integers. Direct grade 4 and grade 5 children to multiply or divide first, then add or subtract to solve the arithmetic expressions.
Order of Operations | DMAS - Level 2
Offering three levels of difficulty, the printable order of operations worksheets provide practice in using DMAS on expressions with 4 integers and 3 operators; You miss the order, you miss the answer!
Solving using DMAS - Level 3
Get this collection of worksheets to practice applying the DMAS rule. Follow the correct order of operations to solve each problem accurately.
Parentheses, Brackets, and Braces
Demonstrate your knowledge of the order of operations by simplifying what's within the innermost layer of the parentheses () first, moving outward simplifying the square brackets [], and curly braces {}.
Order of Operations with Fractions and Decimals
The time is just ripe to blend the order of operations with fractions and decimals. Simplify the expressions in the parentheses first, followed by multiplication or division, then addition or subtraction.
Comparing Two Quantities
Revisit order of operations as you solve the expressions on either side and compare the quantities using = or ≠ in Part A, < > or = in Part B and match expressions in Part C in these printable PEMDAS worksheets.
Finding the Missing Number
Keeping in line with the order of operations, solve the known part, perform inverse operations on either side to isolate and find the unknown part that makes the equation true.
Finding the Missing Operator
The twist in this section of order of operation worksheets is finding the missing operator. Figure out if it is an addition, subtraction, multiplication or division operation that will make the equation true.
Filling in the Numbers
Refine the problem-solving skills of your grade 6 and grade 7 learners as they fill the specified operands in the right places to keep the equation balanced, using the order of operations.
Filling in the Operators
Go pondering, puzzling, and scratching your head as you try to fix the right operator in the right place to balance the two sides of the equation in these order of operation worksheet pdfs.
GEMS
GEMS is a foolproof order of operations strategy, where G stands for Groupings: parentheses, brackets, braces, E for Exponents, M for Multiply/Divide, and S for Subtract/Add whichever comes first to solve the expressions.
Evaluating Numerical Expressions with Exponents
With the positive and negative exponents taking the lead, the expressions may seem a little tricky. The order of operation worksheets perfectly fit the gap and propel 6th grade and 7th grade students to spades of practice.
(59 Worksheets)
Evaluating Numerical Expressions with Parentheses
Cruise through this batch of pdf worksheets where students are expected to evaluate numerical expressions with parentheses, and nested parentheses in accordance with the order of operations.
(72 Worksheets)
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12931 | https://books.google.com/books/about/Antenna_Theory_and_Design.html?id=xhZRA1K57wIC | Antenna Theory and Design - Warren L. Stutzman, Gary A. Thiele - Google Books
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Antenna Theory and Design ========================= Warren L. Stutzman, Gary A. Thiele John Wiley & Sons, May 22, 2012 - Technology & Engineering - 848 pages This introduction to antenna theory and design is suitable for senior undergraduate and graduate courses on the subject. Its emphasis on both principles and design makes it perfect both as a college text and as a reference to the practicing engineer. The final three chapters on computational electromagnetics for antennas are suitable for graduate work. Stutzman provides more of a pedagogical approach than its competitors, placing a greater emphasis on a concise easily understandable presentation of fundamentals and applications as well as computational methods. This third edition has been completely revised. New topics have been added on antennas for personal and mobile communications and base station antennas. Coverage of systems applications of antennas, arrays, microstrip and low-profile antennas, and antenna measurements has been updated and expanded, including more examples applied to modern applications. More » Preview this book »
Selected pages
Title Page
Table of Contents
Index
References
Contents
Antenna Fundamentals 23
The Ideal Dipole 32
Chapter 3 70
System Applications for Antennas 100
Line Sources 128
Wire Antennas 151
Broadband Antennas 218
Array Antennas 271
John Wiley Sons 509
Terminal and Base Station Antennas for Wireless Applications 536
Chapter Antenna Measurements 559
The Method of Moments 587
Finite Difference Time Domain Method 652
HighFrequency Methods 700
Appendix A Frequency Bands 781
Appendix 783
More
Aperture Antennas 344
Antenna Synthesis 433
LowProfile Antennas and Personal Communication Antennas 465
Trigonometric Relations 789
Index811
Copyright
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Oct 16, 2012
1998
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Apr 21, 1981
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Common terms and phrases
amplitudeangleantennaapertureapplicationsapproximatearray factorbandbandwidthbeamwidthbroadsidecalculatedcircularcommunicationscomparedComputeconstantcurrent distributioncurveDerivedetermineddipoledirectivitydiscusseddistancedistributioneffectiveefficiencyelectric fieldelementendfireequalequationEvaluateexampleexcitationexpressionfeedFigurefrequencyfunctiongaingeometrygivengivesground planehalf-wavehornimpedanceincreasesinput impedanceintegralleadslengthline sourcelinearlooplossmain beammatchingmaximummeasuredmethodmodemonopolenormalizedNoteobtainedoperatingpatchpatternperformancephaseplotpolarizationproduceradiation patternradiation resistanceratioreceivingreducedreflectorregionresistanceresonantscanshapeshortshown in Figshowsside lobespacingsurfaceTabletransmission lineuniformusuallyvaluesvectorwavewavelengthwireyieldszero
About the author(2012)
WARREN L. STUTZMAN received his BS in electrical engineering and AB in mathematics degrees from the University of Illinois in 1964 and received MS and Ph.D. degrees in electrical engineering from Ohio State University in 1965 and 1969, respectively. Dr. Stutzman has been on the electrical engineering faculty of Virginia Polytechnic Institute and State University since 1969 and has served as the director of the Antenna Group from its beginning in 1983 until 2001. He served two terms as Interim Department Head. He is currently Principal Investigator for the AWINN (Advanced Wireless Integrated Navy Network) research program sponsored by the Office of Naval Research. He is co-author of the textbook Antenna Theory and Design, John Wiley, 1981 and 1998, and author of Polarization in Electromagnetic Systems, Artech House, 1993. He is a Fellow of the IEEE and served as President of the IEEE Antennas and Propagation Society in 1992.
Bibliographic information
Title Antenna Theory and Design
Antenna Theory and Design, Warren L. Stutzman
AuthorsWarren L. Stutzman, Gary A. Thiele
Edition 3, illustrated
Publisher John Wiley & Sons, 2012
ISBN 0470576642, 9780470576649
Length 848 pages
SubjectsTechnology & Engineering
›
Electronics
›
General
Technology & Engineering / Electrical
Technology & Engineering / Electronics / General
Export CitationBiBTeXEndNoteRefMan
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12932 | https://artofproblemsolving.com/wiki/index.php/Directed_angles?srsltid=AfmBOoocmMZZM-9jwzz5QzeCJjXb0b0lRzFcu2krO7YqnAiPh1ijy2bS | Art of Problem Solving
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Directed angles
Directed Angles is a method to express angles that can be very useful in angle chasing problems where there are configuration issues.
Contents
[hide]
1 Definition
2 Warning
3 Important Properties
4 Application
5 See Also
Definition
Given any two non-parallel lines and , the directed angle is defined as the measure of the angle starting from and ending at , measured counterclockwise and modulo (or say it is modulo ). With this definition in place, we can define , where and are lines (rather than segments).
An equivalent statement for is that, is positive if the vertices , , appear in clockwise order, and negative otherwise, then we take the angles modulo (or modulo ).
Figure 1: The directed angle , while the directed angle
Figure 2: Here, and
Note that in some other places, regular notation is also used for directed angles. Some writers will also use sign instead of a regular equal sign to indicate this modulo nature of a directed angle.
Warning
The notation introduced in this page for directed angles is still not very well known and standard. It is recommended by many educators that in a solution, it is needed to explicitly state the usage of directed angles.
Never take a half of a directed angle. Since directed angles are modulo , taking half of a directed angle may cause unexpected problems.
Do not use directed angles when the problem only works for a certain configuration.
Important Properties
Oblivion: .
Anti-Reflexivity: .
Replacement: if and only if , , are collinear.
Right Angles: If , then .
Addition: .
Triangle Sum: .
Isosceles Triangles: if and only if .
Inscribed Angle Theorem: If points , , is on a circle with center, then .
Parallel Lines: If , then .
Cyclic Quadrilateral: Points , , , lie on a circle if and only if .
Application
The slope of a line in a coordinate system can be given as the tangent of the directed angle between -axis and this line. (Remember the tangent function has a period , so we have our "modulo " part in tangent function)
Other than that, direct angles can be very useful when a geometric (usually angle chasing) problem have a lot of configuration issues. We can avoid solving the same problem twice (sometimes even multiple times) by applying direct angles.
Here are some examples with directed angles:
Proof of the Miquel's Point
Proof of the Orthic Triangle
Proof of the Pascal's Theorem
2002 IMO Shortlist Problems G4
2010 IMO Shortlist Problems G1
1998 APMO Problem 4
See Also
Handout on Evan Chen's Website
An article on a Romanian Mathematical Journal, DIDACTICA MATHEMATICA
Retrieved from "
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12933 | https://www.cuemath.com/amplitude-formula/ | Amplitude Formula
Before going to learn what is amplitude formula, let us recall what is amplitude. The maximum displacement of any particle of a medium, from its state or a position of equilibrium, is called the amplitude. Amplitude is represented with 'A'. In a periodic function with a bounded range, the amplitude is half the distance between the minimum and maximum values. The amplitude is the height from the centerline to the peak or to the trough. Let us learn the amplitude formula along with a few solved examples.
What is Amplitude Formula?
Amplitude refers to the maximum change of a variable from its mean value. The amplitude formula helps in determining the sine and cosine functions. Amplitude is represented by A. The sine function (or) cosine function can be expressed as,
x = A sin (ωt + ϕ) or x = A cos (ωt + ϕ)
Here,
x = displacement of wave (meter)
A = amplitude
ω = angular frequency (rad/s)
t = time period
ϕ = phase angle
The amplitude formula is also expressed as the average of the maximum and minimum values of the sine or cosine function. We always take the absolute value of the amplitude.
Amplitude = (max + min) / 2
Want to find complex math solutions within seconds?
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Examples Using Amplitude Formula
Example 1: y = 2sin(4t) is a wave. Find its amplitude.
Solution:
Given: equation of wave y = 2sin(4t)
Using amplitude formula,
x = A sin(ωt + ϕ)
On comparing it with the wave equation:
A = 2
ω = 4
ϕ = 0
Therefore, theamplitude of the wave = 2 units.
Example 2: The equation of a wave is given by x = 10sin(5πt+π) is a wave. Find its amplitude.
Solution:
Given: equation of wave y = 10sin(5πt + π)
Now, using amplitude formula,
x=A sin(ωt + ϕ)
On comparing it with the wave equation:
A = 10
ω = 5π
ϕ = π
Therefore, the amplitude of the wave = 10 units.
Example 3: If y = 6 cos (7t + 1) is a wave. Find its amplitude.
Solution:
Given: equation of wave y = 6cos(7t + 1)
Using amplitude formula,
x= A cos (ωt + ϕ)
On comparing it with the wave equation:
A = 6
ω = 7
ϕ = 1
Therefore, the amplitude of the wave = 6 units.
FAQs on Amplitude Formula
What is Meant by Amplitude Formula?
Amplitude refers to the maximum change of a variable from its mean value. The amplitude formula helps in determining the sine and cosine functions. Amplitude is represented by A. In a periodic function with a bounded range, the amplitude is half the distance between the minimum and maximum values. The amplitude is the height from the centerline to the peak or to the trough. The formula is x = A sin (ωt + ϕ) or x = A cos (ωt + ϕ)
What is the Formula to Find the Amplitude?
The sine function (or) cosine function can be expressed as,
x = A sin (ωt + ϕ) or x = A cos (ωt + ϕ)
Here,
x = displacement of wave (meter)
A = amplitude
ω = angular frequency (rad/s)
t = time period
ϕ = phase angle
What is the Amplitude Formula in Maximum and Minimum Form?
The amplitude formula is also expressed as the average of the maximum and minimum values of the sine or cosine function. i.e.,
Amplitude = (max + min) / 2
What are the Units of an Amplitude Formula?
The unit in an amplitude formula is the meter (m). The amplitude of a wave is the maximum disturbance or displacement of the medium from the equilibrium position.
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12934 | https://www.dummies.com/article/academics-the-arts/math/geometry/find-locus-points-equidistant-two-points-230066/ | Opens in a new window Opens an external website Opens an external website in a new window
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Find the Locus of Points Equidistant from Two Points
By
Mark Ryan
Updated
2016-12-08 4:41:58
From the book
Geometry For Dummies
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If you're given two points, and you're asked to find the locus of points equidistant from these two points, you'll always find the same thing: that the locus of points is actually the perpendicular bisector of the segment that joins the two points.
If that sounds a little technical, don't worry—the following example will make everything clear!
To find the locus of all points equidistant from two given points, follow these steps:
Identify a pattern.
The figure shows the two given points, A and B, along with four new points that are each equidistant from the given points.
Identifying points that work.
Do you see the pattern? You got it—it's a vertical line that goes through the midpoint of the segment that connects the two given points. In other words, it's that segment's perpendicular bisector.
2. Look outside the pattern.
You come up empty in Step 2. Check any point not on the perpendicular bisector of line AB, and you see that it's not equidistant from A and B. Thus, you have no points to add.
3. Look inside the pattern.
Nothing noteworthy here, either. Every point on the perpendicular bisector of line AB is, in fact, equidistant from A and B. Thus, no points should be excluded. (Warning: Don't allow yourself to get a bit lazy and skip Steps 2 and 3!)
4. Draw the locus and describe it in words.
This figure shows the locus, and the caption gives its description.
The locus of points equidistant from two given points is the perpendicular bisector of the segment that joins the two points.
About This Article
This article is from the book:
Geometry For Dummies
About the book author:
Mark Ryan has more than three decades’ experience as a calculus teacher and tutor. He has a gift for mathematics and a gift for explaining it in plain English. He tutors students in all junior high and high school math courses as well as math test prep, and he’s the founder of The Math Center on Chicago’s North Shore. Ryan is the author of Calculus For Dummies, Calculus Essentials For Dummies, Geometry For Dummies, and several other math books.
This article can be found in the category:
Geometry |
12935 | https://clinicalgate.com/disorders-of-malabsorption/ | Disorders of Malabsorption | Clinical Gate
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Disorders of Malabsorption
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Chapter 330 Disorders of Malabsorption
David Branski
All disorders of malabsorption are associated with diminished intestinal absorption of one or more dietary nutrients. Malabsorption can result from a defect in the nutrient digestion in the intestinal lumen or from defective mucosal absorption. Malabsorption disorders can be categorized into generalized mucosal abnormalities usually resulting in malabsorption of multiple nutrients (Table 330-1) or malabsorption of specific nutrients (carbohydrate, fat, protein, vitamins, minerals, and trace elements) (Table 330-2). Almost all the malabsorption disorders are accompanied by chronic diarrhea (Chapter 333).
Table 330-1MALABSORPTION DISORDERS AND CHRONIC DIARRHEA ASSOCIATED WITH GENERALIZED MUCOSAL DEFECT
Mucosal disorders
Gluten-sensitive enteropathy (celiac disease)
Cow’s milk and other protein-sensitive enteropathies
Eosinophilic enteropathy
Protein-losing enteropathy
Lymphangiectasia (congenital and acquired)
Disorders causing bowel mucosal inflammation, Crohn disease
Congenital bowel mucosal defects
Microvillous inclusion disease
Tufting enteropathy
Carbohydrate-deficient glycoprotein syndrome
Enterocyte heparan sulfate deficiency
Enteric anendocrinosis (NEUROG 3 mutation)
Immunodeficiency disorders
Congenital immunodeficiency disorders
Selective IgA deficiency (can be associated with celiac disease)
Severe combined immunodeficiency
Agammaglobulinemia
X-linked hypogammaglobulinemia
Wiskott-Aldrich syndrome
Common variable immunodeficiency disease
Chronic granulomatous disease
Acquired immune deficiency
HIV infection
Immunosuppressive therapy and post-bone marrow transplantation
Autoimmune enteropathy
IPEX (i mmune dysregulation, p olyendocrinopathy, e nteropathy, X-linked inheritance)
Miscellaneous
Immunoproliferative small intestinal disease
Short bowel syndrome
Chronic malnutrition
Radiation enteritis
IgA, immunoglobulin A.
Table 330-2CLASSIFICATION OF MALABSORPTION DISORDERS AND CHRONIC DIARRHEA BASED ON THE PREDOMINANT NUTRIENT MALABSORBED
CARBOHYDRATE MALABSORPTION
Lactose malabsorption
Congenital lactase deficiency
Hypolactasia (adult type)
Secondary lactase deficiency
Congenital sucrase-isomaltase deficiency
Glucose galactose malabsorption
FAT MALABSORPTION
Abetalipoproteinemia
Lymphangiectasia
Homozygous hypobetalipoproteinemia
Chylomicron retention disease (Anderson disease)
Cystic fibrosis
Shwachman-Diamond syndrome
Blizzard Johanson syndrome
Pearson syndrome
Secondary exocrine pancreatic insufficiency
Isolated enzyme deficiency
Enterokinase deficiency
Trypsinogen deficiency
Lipase/co-lipase deficiency
Chronic pancreatitis
Protein-calorie malnutrition
Decreased pancreozymin/cholecystokinin secretion
Disrupted enterohepatic circulation of bile salts
Cholestatic liver disease
Bile acid synthetic defects
Bile acid malabsorption (terminal ileal disease)
AMINO ACID MALABSORPTION
Lysinuric protein intolerance (defect in dibasic amino acid transport,Chapter 79.13)
Hartnup disease (defect in free neutral amino acids)
Blue diaper syndrome (isolated tryptophan malabsorption)
Oast-house urine disease (defect in methionine absorption)
Lowe syndrome (lysine and arginine malabsorption)
MINERAL AND VITAMIN MALABSORPTION
Congenital chloride diarrhea
Congenital sodium absorption defect
Acrodermatitis enteropathica (zinc malabsorption)
Menkes disease (copper malabsorption)
Vitamin D–dependent rickets
Folate malabsorption
Congenital
Secondary to mucosal damage (celiac disease)
Vitamin B 12 malabsorption
Autoimmune pernicious anemia
Decreased gastric acid (H 2 blockers or proton pump inhibitors)
Terminal ileal disease (e.g., Crohn disease) or resection
Inborn errors of vitamin B 12 transport and metabolism
Primary hypomagnesemia
DRUG INDUCED
Sulfasalazine: folic acid malabsorption
Cholestyramine: calcium and fat malabsorption
Anticonvulsant drugs such as phenytoin (causing vitamin D deficiency and calcium malabsorption)
Clinical Approach
The clinical features depend on the extent and type of the malabsorbed nutrient. The common presenting features, especially in toddlers with malabsorption, are diarrhea, abdominal distention, and failure to gain weight, with a fall in growth chart percentiles. Physical findings include muscle wasting and the disappearance of the subcutaneous fat, with subsequent loose skinfolds (Fig. 330-1). The nutritional consequences of malabsorption are more dramatic in toddlers because of the limited energy reserves and higher proportion of calorie intake being used for weight gain and linear growth. In older children, malnutrition can result in growth retardation, as is commonly seen in children with late diagnosis of celiac disease. If malabsorption is untreated, linear growth slows, and with prolonged malnutrition, death can follow (Chapter 43). This extreme outcome is usually restricted to children living in the developing world, where resources to provide enteral and parenteral nutrition support may be limited. Specific findings on examination can guide toward a specific disorder; edema is usually associated with protein-losing enteropathy, digital clubbing with cystic fibrosis and celiac disease, perianal excoriation and gaseous abdominal distention with carbohydrate malabsorption, perianal and circumoral rash with acrodermatitis enteropathica, abnormal hair with Menkes syndrome, and the typical facial features diagnostic of the Johanson-Blizzard syndrome.
Figure 330-1An 18 mo old boy with active celiac disease. Note the loose skinfolds, marked proximal muscle wasting, and distended abdomen. The child looks ill.
Many children with malabsorption disorders have very good appetites as they try to compensate for the fecal protein and energy losses. In exocrine pancreatic insufficiency, fecal losses of up to 40% of ingested protein and energy do not lead to malnutrition, as long as they are compensated by an increased appetite. In conditions associated with villous atrophy or inflammation (celiac disease, postinfectious enteropathy), fecal protein and energy losses are usually modest, but associated anorexia and reduced food intake results in malnutrition.
The nutritional assessment is an important part of clinical evaluation in children with malabsorptive disorders (Chapter 41). Long-term calcium and vitamin D malabsorption can lead to reduced bone mineral density and metabolic bone disease, with increased risk of bone fractures. Vitamin K malabsorption, irrespective of the underlying mechanism (fat malabsorption, mucosal atrophy), can result in coagulopathy. Severe protein-losing enteropathy is often associated with malabsorption syndromes (celiac disease, intestinal lymphangiectasia) and causes hypoalbuminemia and edema. Other nutrient deficiencies include iron malabsorption causing microcytic anemia and low reticulocyte count, low serum folate levels in conditions associated with mucosal atrophy, and low serum vitamin A and vitamin E concentrations in fat malabsorption.
Clinical history alone might not be sufficient to make a specific diagnosis, but it can direct the pediatrician toward a more structured and rational investigative approach. Diarrhea is the main clinical expression of malabsorption. Onset of diarrhea in early infancy suggests a congenital defect (Table 330-3). In secretory diarrhea due to disorders such as congenital chloride diarrhea and microvillus inclusion disease, the stool is watery and voluminous and can be mistaken for urine. Onset of symptoms after introduction of a particular food into a child’s diet can provide diagnostic clues, such as with sucrose in sucrase-isomaltase deficiency. The nature of the diarrhea may be helpful: explosive watery diarrhea suggests carbohydrate malabsorption; loose, bulky stools are associated with celiac disease; and pasty and yellowish offensive stools suggest an exocrine pancreatic insufficiency. Stool color is usually not helpful; green stool with undigested “peas and carrots” can suggest rapid intestinal transit in toddler’s diarrhea, which is a self-limiting condition unassociated with failure to thrive.
Table 330-3DIARRHEAL DISEASES APPEARING IN THE NEONATAL PERIOD
| CONDITION | CLINICAL FEATURES |
:---: |
| Microvillus inclusion disease | Secretory watery diarrhea |
| Tufting enteropathy | Secretory watery diarrhea |
| Congenital glucose-galactose malabsorption | Acidic diarrhea |
| Congenital lactase deficiency | Acidic diarrhea |
| Congenital chloride diarrhea | Hydramnion, secretory watery diarrheaMetabolic alkalosis |
| Congenital defective jejunal Na+/H+ exchange | Hydramnion, secretory watery diarrhea |
| Congenital bile acid malabsorption | Steatorrhea |
| Congenital enterokinase deficiency | Failure to thrive, edema |
| Congenital trypsinogen deficiency | Failure to thrive, edema |
| Congenital lipase and/or co-lipase deficiency | Failure to thrive, oily stool |
| Enteric anendocrinosis (NEUROG 3 mutation) | Hyperchloremic acidosis, failure to thrive |
Adapted from Schmitz J: Maldigestion and malabsorption. In Walker WA, Durie PR, Hamilton JR, et al, editors: Pediatric gastrointestinal disease, ed 3, Hamilton, Ontario, 2000, BC Decker, p 55.
330.1 Evaluation of Children with Suspected Intestinal Malabsorption
Michael J. Lentze and David Branski
The investigation is guided by the history and physical examination. In a child presenting with chronic or recurrent diarrhea, the initial work-up should include stool cultures and antibody tests for parasites, stool microscopy for ova and parasites such as Giardia, and stool occult blood and leukocytes to exclude inflammatory disorders. Stool pH and reducing substances for carbohydrate malabsorption, and quantitative stool fat examination and α 1-antitrypsin to demonstrate fat and protein malabsorption, respectively, should also be determined. Fecal stool elastase-1 can determine exocrine pancreatic insufficiency.
A complete blood count including peripheral smear for microcytic anemia, lymphopenia (lymphangiectasia), neutropenia (Shwachman syndrome), and acanthocytosis (abetalipoproteinemia) is useful. If celiac disease is suspected, serum immunoglobulin A (IgA) and tissue transglutaminase (TG2) antibody levels should be determined. Depending on the initial test results, more-specific investigations can be planned.
Investigations for Carbohydrate Malabsorption
Measurement of carbohydrate in the stool, using a Clinitest reagent that identifies reducing substances, is a simple screening test. An acidic stool with >2+ reducing substance suggests carbohydrate malabsorption. Sucrose or starch in the stool is not recognized as a reducing sugar until after hydrolysis with hydrochloric acid, which converts them to reducing sugars.
Breath hydrogen test is used to identify the specific carbohydrate that is malabsorbed. After an overnight fast, the suspected sugar (lactose, sucrose, fructose, or glucose) is administered as an oral solution (carbohydrate load 1-2 g/kg, maximum 50 g). In malabsorption, the sugar is not digested or absorbed in the small bowel, passes on to the colon, and is metabolized by the normal bacteria flora. One of the products of this process is hydrogen gas, which is absorbed through the colon mucosa and excreted in the breath. Increased hydrogen concentration in the breath samples suggests carbohydrate malabsorption. A rise in breath hydrogen of 20 ppm above the baseline is considered a positive test. The child should not be on antibiotics at the time of the test, because colonic flora is essential for fermenting the sugar.
Small bowel mucosal biopsies can measure mucosal disaccharidase (lactase, sucrase, maltase, palatinase) concentrations directly. In primary enzyme deficiencies the mucosal enzyme levels are low and small bowel mucosal morphology is normal. Partial or total villous atrophy due to disorders such as celiac disease, or following rotavirus gastroenteritis can result in secondary disaccharidase deficiency and transient lactose intolerance. The disaccharidase levels revert to normal after mucosal healing.
Investigations for Fat Malabsorption
The presence of fat globules in the stool suggests fat malabsorption. The ability to assimilate fat varies with age; a premature infant can absorb only 65-75% of dietary fat, a full-term infant absorbs almost 90%, and an older child absorbs >95% of fat while on a regular diet. Quantitative determination of fat malabsorption requires a 3-day stool collection for evaluation of fat excretion and determination of the coefficient of fat absorption:
where fat intake and fat losses are in grams. Because fecal fat balance studies are cumbersome, expensive, and unpleasant to perform, simpler tests are often preferred. Among these stool tests, the acid steatocrit test is the most reliable. When bile acid deficiency is suspected of being the cause of fat malabsorption, the evaluation of bile acid levels in duodenal fluid aspirate may be useful.
Fat malabsorption and exocrine pancreatic insufficiency are usually associated with deficiencies of fat-soluble vitamins A, D, E, and K. Serum concentrations of vitamins A, D, and E can be measured. A prolonged prothrombin time is an indirect test to assess vitamin K deficiency.
Investigations for Protein Losing Enteropathy
Dietary and endogenous proteins secreted into the bowel are almost completely absorbed; <1 g of protein from these sources passes into the colon. The majority of the stool nitrogen is derived from gut bacterial proteins. Excessive bowel protein loss usually manifests as hypoalbuminemia. However, the most common cause of hypoalbuminemia in children is a renal disorder; therefore, urinary protein excretion must be determined. Other potential causes of hypoalbuminemia include liver disease (reduced production) and inadequate protein intake. Very rarely hypoalbuminemia can result from an extensive skin disorder causing protein loss via the skin. Measurement of stool α 1-antitrypsin is a useful screening test for protein-losing enteropathy. This serum protein has a molecular weight similar to albumin’s; however, unlike albumin it is resistant to digestion in the gastrointestinal (GI) tract. Excessive α 1-antitrypsin excretion in the stool should prompt further investigations to identify the specific cause of gut or stomach (Menetrier disease) protein loss.
Investigations for Exocrine Pancreatic Function (Fig. 330-2)
Cystic fibrosis is the most common cause of exocrine pancreatic insufficiency in children; therefore, a sweat chloride test must be performed before embarking on invasive tests to investigate possible exocrine pancreatic insufficiency. Many cases of cystic fibrosis are detected by neonatal genetic screening programs; occasional rare mutations are undetected.
Figure 330-2Algorithm for assessment of exocrine pancreatic function. If not available, use other test. Perform appropriate imaging studies of the pancreas. In case of borderline values, consider repeating the test with three independent samples. Consider differential diagnosis (especially consider mucosal villous atrophy and dilution effect of watery stool). GI, gastrointestinal.
(Adapted from Walkowiak J, Nousia-Arvanitakis S, Henker J, et al: Indirect pancreatic function tests in children, J Pediatr Gastroenterol Nutr 40:107–114, 2005.)
Fecal elastase-1 estimation is a sensitive test to assess exocrine pancreatic function in chronic cystic fibrosis and pancreatitis. Elastase-1 is a stable endoprotease unaffected by exogenous pancreatic enzymes. One disadvantage of the fecal elastase-1 test is the lack of full differentiation between primary exocrine pancreatic insufficiency and exocrine pancreatic dysfunction secondary to intestinal villous atrophy. The proximal small bowel is the site for pancreozymin/cholecystokinin production; the latter is the hormone that stimulates enzyme secretion from the exocrine pancreas. Mucosal atrophy can lead to diminished pancreozymin/cholecystokinin secretion and subsequently to exocrine pancreatic insufficiency. Fecal elastase-1 can also give a false-positive result during acute episodes of diarrhea.
Serum trypsinogen concentration can also be used as a screening test for exocrine pancreatic insufficiency. In cystic fibrosis, the levels are greatly elevated early in life, and then they gradually fall, so that by 5-7 yr of age, most patients with cystic fibrosis with pancreatic insufficiency have subnormal levels. Patients with cystic fibrosis and adequate exocrine pancreatic function tend to have normal or elevated levels. In such patients, observing the trend in serial serum trypsinogen estimation may be useful in monitoring exocrine pancreatic function. In Shwachman syndrome, another condition associated with exocrine pancreatic insufficiency, the serum trypsinogen level is low.
Other tests for pancreatic insufficiency (NBT-PABA test and pancreolauryl test) measure urine or breath concentrations of substances released and absorbed across the mucosal surface following pancreatic digestion. These tests lack specificity and are rarely used in clinical practice.
The gold standard test for exocrine pancreatic function is direct analysis of duodenal aspirate for volume, bicarbonate, trypsin and lipase upon secretin and pancreozymin/cholecysto-kinin stimulation. This involves duodenal intubation, and only a few centers perform this test (Chapter 340).
Investigations for Intestinal Mucosal Disorders
Establishing a specific diagnosis for malabsorption often requires histologic examination of small bowel mucosal biopsies. These are obtained during endoscopy, which allows multiple biopsies to be performed, because mucosal involvement can be patchy, especially in celiac disease. Periodic acid–Schiff (PAS) staining of mucosal biopsies and electron microscopy are necessary in congenital diarrhea to assess congenital microvillus atrophy. Bowel mucosal lesions can also be segmental in cases of intestinal lymphangiectasia. In these situations radiographic small bowel series or repeated ultrasonographies can identify a region of thickened bowel responsible for protein loss. During endoscopy, mucosal biopsies can be obtained to measure mucosal disaccharidase activities. Duodenal aspirates can be performed to measure pancreatic enzyme concentration as well as quantitative bacterial cultures. Aspirates to demonstrate other infections and infestations such as Giardia may be useful.
Imaging Procedures
Plain radiographs and barium contrast studies might suggest a site and cause of intestinal motility disorders. Although flocculations of barium and dilated bowel with thickened mucosal folds have been attributed to diffuse malabsorptive lesions such as celiac disease, these abnormalities are nonspecific. Diffuse fluid-filled bowel loops during sonography also suggest malabsorption.
330.2 Gluten-Sensitive Enteropathy (Celiac Disease)
David Branski,Riccardo Troncone
Etiology and Epidemiology
Celiac disease is an immune-mediated disorder elicited by the ingestion of gluten in genetically susceptible persons and characterized by chronic inflammation of the small intestine. It is considered an autoimmune condition because of the presence of anti–TG2 antibodies and the association with other autoimmune diseases (thyroid, liver, diabetes, adrenal).
Celiac disease is triggered by the ingestion of wheat gluten and related prolamines from rye and barley. In most studies oats proved to be safe; however, a few celiac patients have oats prolamine–reactive mucosal T cells that can cause mucosal inflammation.
Celiac disease is a common disorder (1% prevalence of biopsy-proven disease). It is thought to be rare in Central Africa and East Asia. Environmental factors might affect the risk of developing celiac disease or the timing of its presentation. Prolonged breastfeeding has been associated with a reduced incidence of symptomatic disease. Less clear is the effect of the time of gluten introduction in the infant diet; the ingestion of increased amounts of gluten in the 1st year of life can increase the incidence. Infectious agents have been hypothesized to play a role because frequent rotavirus infections are associated with an increased risk. It is plausible that the contact with gliadin at a time when there is ongoing intestinal inflammation, altered intestinal permeability, and enhanced antigen presentation can increase the risk of developing the disease, at least in a subset of persons (Fig. 330-3).
Figure 330-3Causative factors in celiac disease. HLA, human leukocyte antigen.
(From Di Sabatino A, Corazza GR: Coeliac disease, Lancet 373:1480-1490, 2009.)
Genetics and Pathogenesis
A genetic predisposition is suggested by the family aggregation and the concordance in monozygotic twins, which approaches 100%. It is suggested that the primary association of CD is with the DQ αβ heterodimer encoded by the DQA105_ and the _DQB102 genes. Such a DQ molecule is present in ≥95% of celiac patients compared with 20-30% of controls. DQ2-negative celiac patients are invariably HLA DQ8 positive (DQA10301/DQB10302). A gene dosage effect has been suggested, and a molecular hypothesis for such a phenomenon has been proposed, based on the impact of the number and quality of the HLA DQ2 molecules on gluten peptide presentation to T cells. Other non-HLA genes confer susceptibility to celiac disease. Genome-wide association studies have shown risk variants in genes controlling the immune response, some being shared with type 1 diabetes.
Celiac disease is a T cell–mediated chronic inflammatory disorder with an autoimmune component. Altered processing by intraluminal enzymes, changes in intestinal permeability, and activation of innate immunity mechanisms may be involved and precede the activation of the adaptive immune response. Immunodominant epitopes from gliadin are highly resistant to intraluminal and mucosal digestion; incomplete degradation favors the immunostimulatory and toxic effects. Some gliadin peptides (p31-43) can activate innate immunity, and in particular they induce interleukin 15 (IL-15). Others activate lamina propria T cells in the context of HLA-DQ2 or DQ8 molecules. Gliadin-specific T-cell responses are enhanced by the action of TG2; the enzyme converts particular glutamine residues into glutamic acid, which results in higher affinity of these gliadin peptides for HLA-DQ2 or HLA-DQ8. The pattern of cytokines produced following gliadin activation is dominated by interferon-γ (IFN-γ) (Th1 skewed); IFN-α, IL-18, and IL-21 are also upregulated. A complex remodeling of the mucosa then takes place, involving increased levels of metalloproteinases and growth factors, which leads to the classic flat mucosa. Increased density of CD8+ cytotoxic intraepithelial lymphocytes are a hallmark of celiac disease. IL-15 is implicated in the expression of natural killer receptors CD94 and NKG2D, as well as in epithelial expression of stress molecules, thus enhancing cytotoxicity, cell apoptosis, and villous atrophy.
The most evident expression of autoimmunity is the presence of serum antibodies to TG2. However, the mechanisms leading to autoimmunity are largely unknown. The finding of IgA deposits on extracellular TG2 in the liver, lymph nodes, and muscles indicates that TG2 is accessible to the gut-derived autoantibodies. Several extraintestinal clinical manifestations of celiac disease (e.g., liver, heart, nervous system) are possibly related to the presence of autoantibodies.
Clinical Presentation and Associated Disorders
Clinical features of celiac disease vary considerably (Table 330-4). Intestinal symptoms are common in children whose disease is diagnosed within the 1st 2 years of life; failure to thrive, chronic diarrhea, vomiting, abdominal distention, muscle wasting, anorexia, and irritability are present in most cases (see Fig. 330-1). Occasionally there is constipation, rectal prolapse, or intussusception. As the age at presentation of the disease shifts to later in childhood, and with the more liberal use of serologic screening tests, extraintestinal manifestations and associated disorders, without any accompanying digestive symptoms, have increasingly become recognized, affecting almost all organs (Table 330-5).
Table 330-4SOME CLINICAL MANIFESTATIONS OF CELIAC DISEASE IN CHILDREN AND ADOLESCENTS
| SYSTEM | MANIFESTATION | (POSSIBLE) CAUSE |
:---:
| Gastrointestinal | DiarrheaDistended abdomenVomitingAnorexiaWeight lossFailure to thriveAphthous stomatitis | Atrophy of the small bowel mucosaMalabsorption |
| Hematologic | Anemia | Iron malabsorption |
| Skeletal | RicketsOsteoporosisEnamel hypoplasia of the teeth | Calcium/vitamin D malabsorption |
| Muscular | Atrophy | Malnutrition |
| Neurologic | Peripheral neuropathyEpilepsyIrritability | Thiamine/vitamin B 12 deficiency |
| Endocrinologic | Short staturePubertas tardaSecondary hyperparathyroidism | MalnutritionCalcium/vitamin D malabsorption |
| Dermatologic | Dermatitis herpetiformisAlopecia areataErythema nodosum | Autoimmunity |
| Respiratory | Idiopathic pulmonary hemosiderosis | |
Adapted from Mearin ML: Celiac disease among children and adolescents, Curr Prob Pediatr Adolesc Health Care 37:81–112, 2007.
Table 330-5RISK GROUPS FOR CELIAC DISEASE CASE-FINDING
1st-degree relatives
Dermatitis herpetiformis
Unexplained iron deficiency anaemia
Autoimmune thyroiditis
Type 1 diabetes
Unexplained infertility
Recurrent abortion
Dental enamel hypoplasia
Cryptic hypertransaminasemia
Autoimmune liver disease
Short stature
Delayed puberty
Down, Williams, and Turner syndromes
Irritable bowel syndrome
Unexplained osteoporosis
Sjögren syndrome
Epilepsy with occipital calcifications
Selective IgA deficiency
Addison disease
IgA, immunoglobulin A.
Modified from Di Sabatino A, Corazza GR: Coeliac disease, Lancet 373:1480–1490, 2009.
The most common extraintestinal manifestation of celiac disease is iron-deficiency anemia, unresponsive to iron therapy. Osteoporosis may be present; in contrast to the situation in adults, it can be reversed by a gluten-free diet, with restoration of normal peak bone densitometric values. Other extraintestinal manifestations include short stature, endocrinopathies, arthritis and arthralgia, epilepsy with bilateral occipital calcifications, peripheral neuropathies, cardiomyopathy, chronic lung disease, isolated hypertransaminasemia, dental enamel hypoplasia, aphthous stomatitis, and alopecia. The mechanisms responsible for the severity and the variety of clinical presentations remain obscure. Nutritional deficiencies or abnormal immune responses have been advocated.
Silent celiac disease is being increasingly recognized, mainly in asymptomatic 1st-degree relatives of celiac patients investigated during screening studies. However, small bowel biopsy in these people reveals severe mucosal damage consistent with celiac disease. Potential celiac disease is defined when patients are identified by positive screening studies but without documented celiac disease on small bowel biopsy. It is important to follow these patients because they can develop established celiac disease in the future (Table 330-6).
Table 330-6CLINICAL SPECTRUM OF CELIAC DISEASE
SYMPTOMATIC
Frank malabsorption symptoms: chronic diarrhea, failure to thrive, weight loss
Extraintestinal manifestations: anemia, fatigue, hypertransaminasemia, neurologic disorders, short stature, dental enamel defects, arthralgia, aphthous stomatitis
SILENT
No apparent symptoms in spite of histologic evidence of villous atrophy
In most cases identified by serologic screening in at-risk groups (see Table 330-1)
LATENT
Subjects who have a normal histology, but at some other time, before or after, have shown a gluten-dependent enteropathy
POTENTIAL
Subjects with positive celiac disease serology but without evidence of altered jejunal histology
It might or might not be symptomatic
Some diseases, many with an autoimmune pathogenesis, are found with a higher than normal incidence in celiac patients. Among these are type 1 diabetes, autoimmune thyroid disease, Addison disease, Sjögren syndrome, autoimmune cholangitis, autoimmune hepatitis, primary biliary cirrhosis, IgA nephropathy, alopecia, and dilated cardiomyopathy. Such associations have been interpreted as a consequence of the sharing of identical HLA haplotypes. The relation between celiac disease and other autoimmune diseases is poorly defined; once those diseases are established, they are not influenced by a gluten-free diet. Other associated conditions include selective IgA deficiency, Down syndrome, Turner syndrome, and Williams syndrome.
Patients with celiac disease show increased long-term mortality, the risk rising with delayed diagnosis and/or poor dietary compliance. Non-Hodgkin lymphoma is the main cause of death. Adult patients can develop complications such a refractory celiac disease, ulcerative jejunoileitis, or enteropathy-associated T-cell lymphoma.
Diagnosis
Serologic tests have a crucial role in the diagnosis of celiac disease; sensitivity of the IgA anti-TG2 is 61-100% (mean, 87%), and specificity is 86-100% (mean, 95%). Some 10% of patients whose disease is diagnosed earlier than 2 yr of age show absence of IgA anti-TG2. For them, the measurement of serum antigliadin antibodies is generally advised. Antibodies against gliadin-derived deamidated peptides (D-AGA) have been assessed. Compared with conventional AGA, the peptide antibodies (IgG and IgA) have a greater sensitivity and specificity. A problem with serology is represented by the association of celiac disease with IgA deficiency (10-fold increase compared to the general population). Serum IgA should always be checked, and in the case of IgA deficiency, D-AGA, IgG anti-endomysium, or TG2 should be sought. Negative serology should not preclude a biopsy examination when the clinical suspicion is strong.
Genetic tests have an increasing role in the diagnosis. Less than 2% of celiac patients lack both HLA specificities; at the same time, approximately one third of the “normal” population has one or the other marker; that means that the measurement of HLA DQ2 and/or DQ8 has a strong negative predictive value but a very weak positive predictive value for the diagnosis of celiac disease. With these limitations the test can prove useful to exclude celiac disease when the genetic studies are negative in subjects on a gluten-free diet or in subjects belonging to an at-risk group (e.g., 1st-degree relatives, insulin-dependent diabetics, patients with Down syndrome) to avoid long-term follow-up.
The ultimate diagnosis of celiac disease relies on the demonstration of specific, though not pathognomonic, histopathologic abnormalities in the small bowel mucosa (Table 330-7). According to The European Society for Pediatric Gastroenterology, Hepatology and Nutrition (ESPGHAN) current criteria, the 2 requirements mandatory for the diagnosis of celiac disease are the finding of villous atrophy with hyperplasia of the crypts and abnormal surface epithelium, while the patient is eating adequate amounts of gluten, and a full clinical remission after withdrawal of gluten from the diet. The finding of circulating IgA celiac disease–associated antibodies at the time of diagnosis and their disappearance on a gluten-free diet adds weight to the diagnosis. A control biopsy to verify the consequences of the gluten-free diet on the mucosal architecture is considered mandatory only in patients with an equivocal clinical response to the diet. Gluten challenge is not considered mandatory except in situations where there is doubt about the initial diagnosis, for example, when an initial biopsy was not performed or when the biopsy specimen was inadequate or atypical of celiac disease.
Table 330-7OTHER CAUSES OF FLAT MUCOSA
Autoimmune enteropathy
Tropical sprue
Giardiasis
HIV enteropathy
Bacterial overgrowth
Crohn disease
Eosinophilic gastroenteritis
Cow’s milk enteropathy
Soy protein enteropathy
Primary immunodeficiency
Graft-versus-host disease
Chemotherapy and radiation
Protein energy malnutrition
Tuberculosis
Lymphoma
Non-gluten food intolerances
Modified from Di Sabatino A, Corazza GR: Coeliac disease, Lancet 373:1480–1490, 2009.
It is now accepted that the spectrum of histologic abnormalities in the celiac small intestine is wider than previously recognized. In some celiac disease patients, only subtle changes of crypt elongation with an increase in intraepithelial lymphocytes may be present. In those cases, it is very important to also evaluate the serology and the HLA typing so as to reach the correct diagnosis. Analysis of multiple biopsies is also very important.
Nevertheless, many cases of celiac disease are undiagnosed, and the ratio between patients with diagnosed and with undiagnosed disease may be as high as 1:7. Case finding by liberal use of anti-endomysium or anti-TG2 antibodies, followed by confirmatory jejunal biopsy, is more cost effective in primary care than mass screening is. Patients with symptoms or diseases known to be associated with celiac disease should undergo serologic evaluation.
Treatment
The only treatment for celiac disease is lifelong strict adherence to a gluten-free diet (Fig. 330-4). This requires a wheat-, barley-, and rye-free diet . Despite evidence that oats are safe for most patients with celiac disease, there is concern regarding the possibility of contamination of oats with gluten during harvesting, milling, and shipping. Nevertheless, it seems wise to add oats to the gluten-free diet only when the latter is well established, so that possible adverse reactions can be readily identified. There is a consensus that all celiac disease patients should be treated with a gluten-free diet regardless of the presence of symptoms. However, whereas it is relatively easy to assess the health improvement after treatment of celiac disease in patients with clinical symptoms of the disease, it proves difficult in persons with asymptomatic celiac disease. The nutritional risks, particularly osteopenia, are those mainly feared for subjects who have silent celiac disease and continue on a gluten-containing diet. Little is known about the health risks in untreated patients with minor enteropathy, which may be clinically silent. There are no guidelines concerning the need for a gluten-free diet in subjects with “potential” celiac disease (patients with positive celiac disease–associated serology but without enteropathy).
Figure 330-4Gluten-sensitive enteropathy. Growth curve demonstrates initial normal growth from 0 to 9 mo, followed by onset of poor appetite with intermittent vomiting and diarrhea after initiation of gluten-containing diet (single arrow). After biopsy confirmed diagnosis and treatment with gluten-free diet (double arrow), growth improves.
The Codex Alimentarius Guidelines define gluten-free as <20 ppm, but, although analytical methods for gluten detection have already reached a satisfactory degree of sensitivity, more information is needed on the daily gluten amount that may be tolerated by celiac disease patients. The data available so far seem to suggest that the threshold should be set to <50 mg/day, although individual variability makes it difficult to set a universal threshold.
It is important that an experienced dietician with specific expertise in celiac disease counseling educates the family and the child about dietary restriction. Compliance with a gluten-free diet can be difficult, especially in adolescents. It is recommended that children with celiac disease be monitored with periodic visits for assessment of symptoms, growth, physical examination, and adherence to the gluten-free diet. Periodic measurements of TG2 antibody levels to document reduction in antibody titers can be helpful as indirect evidence of adherence to a gluten-free diet, although they are inaccurate in detecting slight dietary transgressions.
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330.3 Other Malabsorptive Syndromes
Philip M. Sherman,David Branski,Olivier Goulet
Congenital Intestinal Mucosal Defects
Microvillus Inclusion Disease (Congenital Microvillus Atrophy)
Microvillus inclusion disease is an autosomal recessive disorder, which manifests at birth with profuse watery secretory diarrhea. It is the most commonly recognized cause of congenital diarrhea. Light microscopy of the small bowel mucosa demonstrates diffuse thinning of the mucosa, with hypoplastic villus atrophy and no inflammatory infiltrate. Diagnosis may be easily performed with light microscopy using PAS and CD10 staining, which shows a very thin or absent brush border, together with positive PAS and CD10 intracellular inclusions. Electron microscopy shows enterocytes with absent or sparse microvilli. The apical cytoplasm of the enterocytes contains electron-dense secretory granules; the hallmark is presence of microvilli within involutions of the apical membrane. Polyhydramnios is not a classic presentation of MID. Neonates usually present with dehydration and failure to thrive. Despite parenteral nutrition, diarrhea continues and initial fluid management is difficult. The disease is fatal without long-term parenteral nutrition support. Some infants present with rapid onset of liver disease, which is associated with pruritus. Most children die in infancy or early childhood. The long-acting somatostatin analog octreotide has been used as treatment and can reduce the volume of stool in some infants (Chapter 331). Intestinal transplantation is the only definitive treatment for this rare disease. Rarely, in milder forms of the disease, the patient can reach young adulthood and enjoy partially oral feeding. The underlying gene defect is a mutation in MYO5B, which encodes a protein involved in subcellular protein trafficking. Several types of mutations are involved.
Tufting Enteropathy
Tufting enteropathy (intestinal epithelial dysplasia) manifests in the 1st weeks of life with persistent watery diarrhea and accounts for a small fraction of infants with protracted diarrhea of infancy. Symptoms typically do not begin immediately after birth but occur in early infancy. The distinctive feature on small intestinal mucosal biopsy is focal epithelial “tufts” (teardrop-shaped groups of closely packed enterocytes with apical rounding of the plasma membrane) involving 80-90% of the epithelial surface. However, the typical pathology does not appear immediately after birth, and in other known enteropathies tufts are seen on ≤15% of the epithelial surface. Colonic epithelium shows abnormalities that are more difficult to identify. Electron microscopy does not help in the diagnosis.
The pathogenesis of this disorder may be due to a disorder of cell-cell and cell-matrix interactions, because there is an abnormal distribution of α 2 β 1-integrin along the crypt-villus axis, increased expression of desmoglein, and ultrastructural changes of desmosomes. Tufting enteropathy is often associated with punctiform keratitis and conjunctival dysplasia resembling typical pictures of tufts. The genetic basis of tufting enteropathy supports this speculation, because a single amino acid substitution in exon 4 of the EPCAM gene encoding an epithelial cell adhesion molecule protein has been described.
No treatment has been effective, so management requires permanent parenteral nutrition with possible intestinal transplantation (Chapter 331). Several types of mutations are involved, opening the door to genotype-phenotype analysis.
Enteric Anendocrinosis
Mutations of the NEUROG3 gene produce generalized mucosal malabsorption, vomiting, diarrhea, failure to thrive, dehydration, and a hyperchloremic metabolic acidosis. Oral alimentation with anything other than water produces diarrhea. Villus-crypt architecture in small bowel biopsies is normal, but staining for neuroendocrine cells (e.g., employing anti-chromogranin antibodies) demonstrates a complete absence of this secretory cell lineage with preservation of goblet cells and Paneth cells. Treatment is with total parenteral nutrition and small bowel transplantation.
Carbohydrate-Deficient Glycoprotein Syndrome and Enterocyte Heparan Sulfate Deficiency
Congenital disorders of glycosylation (also carbohydrate-deficient glycoprotein, CDG) are genetic disorders of assembly of N-glycans in the cytosol and endoplasmic reticulum, resulting in a variety of manifestations (Chapter 81.6). The subtypes of CDG I are all associated with protein-losing enteropathy. Diagnosis can be established by isoelectric focusing of serum transferrin, enzyme analysis, and/or DNA analysis. Oral mannose can provide effective therapy in CDG Ib, so early identification of children presenting with hypoglycemia, hypothyroidism, and/or thyroid binding globulin deficiency is beneficial.
Congenital enterocyte heparan deficiency (CEHD) is a rare cause of intractable diarrhea with protein-losing enteropathy, which may be an unusual presentation of the carbohydrate-deficient glycoprotein syndrome (CDGS) type 1 (also known as Jaeken syndrome) (Chapter 81.6). Heparan sulfate is a glycosaminoglycan with multiple roles in the intestine, including restriction of charged macromolecules, such as albumin, in the vascular lumen.
Intestinal Lymphangiectasia
Obstruction of the lymphatic drainage of the intestine can be due to either congenital defects in lymphatic duct formation or to secondary causes (Table 330-8). The congenital form is often associated with lymphatic abnormalities elsewhere in the body, as occur with Turner, Noonan, and Klippel-Trenaunay-Weber syndromes. Causes of secondary lymphangiectasia include constrictive pericarditis, heart failure, retroperitoneal fibrosis, abdominal tuberculosis, and retroperitoneal malignancies. Lymph rich in proteins, lipids, and lymphocytes leaks into the bowel lumen, resulting in protein-losing enteropathy, steatorrhea, and lymphocyte depletion. Hypoalbuminemia, hypogammaglobulinemia, edema, lymphopenia, malabsorption of fat and fat-soluble vitamins, and chylous ascites often occur. Intestinal lymphangiectasia can also manifest with ascites, peripheral edema, and a low serum albumin.
Table 330-8CAUSES OF PROTEIN-LOSING ENTEROPATHY
Mucosal inflammation
Infection
Cytomegalovirus (CMV)
Bacterial overgrowth
Invasive bacterial infection
Gastric inflammation
Menetrier disease
Eosinophilic gastroenteropathy
Intestinal inflammation
Celiac disease
Crohn disease
Eosinophilic gastroenteropathy
Tropical sprue
Radiation enteritis
Primary intestinal lymphangiectasia
Secondary intestinal lymphangiectasia
Constrictive pericarditis
Congestive heart failure
Post Fontan procedure
Malrotation
Lymphoma
Sarcoidosis
Radiation therapy
Colonic inflammation
Inflammatory bowel diseases
Necrotizing enterocolitis
Congenital disorders of glycosylation
The diagnosis is suggested by the typical findings in association with an elevated fecal α 1-antitrypsin clearance. Radiologic findings of uniform, symmetric thickening of mucosal folds throughout the small intestine are characteristic but nonspecific. Small bowel mucosal biopsy can show dilated lacteals with distortion of villi and no inflammatory infiltrate. A patchy distribution and deeper mucosal involvement on occasion causes false-negative results on small bowel histology. Treatment of lymphangiectasia includes restricting the amount of long-chain fat ingested and administering a formula containing protein and medium-chain triglycerides (MCTs). Supplementing a low-fat diet with MCT oil in cooking is used in the management of older children with lymphangiectasia. Rarely, parenteral nutrition is required. If only a portion of the intestine is involved, surgical resection may be considered.
Syndromic Diarrhea
Syndromic diarrhea (SD), also known as phenotypic diarrhea (PD) or tricho-hepato-enteric (THE) syndrome, is a congenital enteropathy manifesting with early onset of severe diarrhea requiring parenteral nutrition. The estimated prevalence is approximately 1/300,000-400,000 live births in Western Europe. Patients born small for gestational age present with diarrhea starting in the 1st 6 mo of life (<1 mo of age in most cases). They have an abnormal phenotype, including facial dysmorphism with prominent forehead, broad nose, and hypertelorism and a distinct abnormality of hair, trichorrhexis nodosa. Hairs are woolly, easily removed, and poorly pigmented. Liver disease affects about half of the patients with extensive fibrosis or cirrhosis. The patients have defective antibody responses despite normal serum immunoglobulin levels and defective antigen-specific skin tests despite positive proliferative responses in vitro. Microscopic analysis shows twisted hair (pili torti), aniso- and poilkilotrichosis, and trichorrhexis nodosa. Histopathologic analysis shows nonspecific villus atrophy with or without mononuclear cell infiltration of the lamina propria, without specific histologic abnormalities involving the epithelium. Recently mutations in the TTC37 gene were found as the cause of THE syndrome. The common association of the disorder with parental consanguinity and/or affected siblings suggests a genetic origin with an autosomal recessive transmission. Prognosis of this type of intractable diarrhea of infancy is poor because most patients have died between the ages of 2 and 5 yr, some of them with early-onset liver disease.
Autoimmune Enteropathy
Symptoms of autoimmune enteropathy usually occur after the 1st 6 mo of life with chronic diarrhea, malabsorption, and failure to thrive. Histologic findings in the small bowel include partial or complete villous atrophy, crypt hyperplasia, and an increase in chronic inflammatory cells in the lamina propria. In contrast to gluten-sensitive enteropathy, an increase in intra-epithelial lymphocytes is not a prominent feature of autoimmune enteropathy. Specific serum anti-enterocyte antibodies can be identified in ≥50% of patients by indirect immunofluorescent staining of normal small bowel mucosa and kidney. In some patients anti-goblet cell antibodies can be demonstrated as well. The colon is also often involved, with inflammation and clinical features of colitis.
Extraintestinal autoimmune disorders are usual and include arthritis, membranous glomerulonephritis, insulin-dependent diabetes, thrombocytopenia, autoimmune hepatitis, hypothyroidism, and hemolytic anemia. It is essential to exclude underlying primary immune deficiency, particularly in boys with other autoimmune features (diabetes mellitus), because a proportion have underlying I mmune dysregulation, P olyendocrinopathy, E nteropathy, X linked (IPEX) syndrome (Chapter 120.5). This systemic autoimmune disorder is due to mutations in FOXP3, a transcriptional regulator essential for the normal development of regulatory T cells (T regs). Autoimmune enteropathy is reported in cases of Schimke immunoosseous dysplasia.
Treatment for autoimmune enteropathy includes immune suppression drugs including prednisone, azathioprine, cyclophosphamide, cyclosporine, and tacrolimus. Bone marrow transplantation is curative for children with IPEX syndrome.
Proprotein Convertase 1/3 Deficiency
Chronic watery, neonatal onset diarrhea is described in infants with hyperinsulinism, hypoglycemia, hypogonadism, and hypoadrenalism. Small bowel biopsy reveals a nonspecific enteropathy. A clue to the autosomal recessive condition is subsequent onset of marked obesity with hyperphagia in the toddler years in both affected probands and symptomatic siblings.
Bile Acid Malabsorption
In primary bile acid malabsorption, mutation of the ileal sodium–bile acid cotransporter gene, SLC10A2, results in congenital diarrhea, steatorrhea, interruption of enterohepatic circulation of bile acids, and reduced plasma cholesterol levels. Bile acids are normally synthesized from cholesterol in the liver and secreted into the small intestine, where they facilitate absorption of fat, fat-soluble vitamins, and cholesterol. Bile acids are reabsorbed in the distal ileum, return to the liver via the portal venous circulation, and are resecreted into the bile. Normally, the enterohepatic circulation of bile acids is an extremely efficient process; only 10% of the intestinal bile acids escape reabsorption and are eliminated in feces. Bile acid secretion is largely autoregulated, but there is only a limited capacity to increase bile acid secretion. Reduction in the bile acid pool due to bile acid malabsorption causes steatorrhea, which requires restriction of dietary fat. Unabsorbed bile acids stimulate chloride excretion in the colon, resulting in diarrhea, which responds to cholestyramine, an anion-binding resin. Secondary bile acid malabsorption can result from ileal disease, such as in Crohn disease, and following an ileal resection.
Chronic neonatal-onset diarrhea has also been described in autosomal recessive cerebrotendinous xanthomatosis, which is caused by an inborn error of bile acid synthesis due to 27-hydroxylase deficiency. These children also present with juvenile-onset cataracts and developmental delay. Neonatal cholestasis has also been described as a presenting feature. Tendon xanthomas develop in the second and third decades of life. The diagnosis is important to establish, because treatment is effective when employing oral chenodeoxycholic acid.
Abetalipoproteinemia
Abetalipoproteinemia is a rare autosomal recessive disorder of lipoprotein metabolism (Bassen Kornsweig syndrome) (Chapter 80). It is associated with severe fat malabsorption from birth. Children fail to thrive during the 1st year of life, with stools that are pale, foul smelling, and bulky. The abdomen is distended and deep tendon reflexes are absent as a result of peripheral neuropathy, which is secondary to vitamin E (fat-soluble vitamin) deficiency. Intellectual development tends to be slow. After 10 yr of age, intestinal symptoms are less severe, ataxia develops, and there is a loss of position and vibration sensation with the onset of intention tremors unless vitamin E levels are maintained in the normal range. These latter symptoms reflect involvement of the posterior columns, cerebellum, and basal ganglia. In adolescence, atypical retinitis pigmentosa develops without adequate supplemental of vitamin E; for instance, using a TPGS formulation of the vitamin.
Diagnosis rests on the presence of acanthocytes in the peripheral blood smear and extremely low plasma levels of cholesterol (<50 mg/dL); triglycerides are also very low (<20 mg/dL). Chylomicrons and very low density lipoproteins are not detectable, and the low-density lipoprotein (LDL) fraction is virtually absent from the circulation. Marked triglyceride accumulation in villus enterocytes occurs in the duodenal mucosa. Steatorrhea occurs in younger patients, but other processes of nutrient assimilation are intact. Rickets may be an unusual initial manifestation of abetalipoproteinemia and hypobetalipoproteinemia. Rickets is caused by steatorrhea-induced calcium losses and vitamin D deficiency. Patients have mutations of the microsomal triglyceride transfer protein (MTP) gene, resulting in absence of MTP function in the small bowel. This protein is required for normal assembly and secretion of very low density lipoproteins and chylomicrons.
Specific treatment is not available. Large supplements of the fat-soluble vitamins A, D, E, and K should be given. Vitamin E (100-200 mg/kg/24 hr) appears to arrest neurologic and retinal degeneration. Limiting long-chain fat intake can alleviate intestinal symptoms; medium-chain triglycerides can be used to supplement the fat intake.
Homozygous Hypobetalipoproteinemia
Homozygous hypobetalipoproteinemia (Chapter 80) is transmitted as an autosomal dominant trait. The homozygous form is indistinguishable from abetalipoproteinemia. The parents of these patients, as heterozygotes, have reduced plasma LDL and apoprotein-β concentrations, whereas the parents of patients with abetalipoproteinemia have normal levels. On transmission electron microscopy of small bowel biopsies, the size of lipid vacuoles in enterocytes differentiates between abetalipoproteinemia and hypobetalipoproteinemia: Many small vacuoles are present in hypobetalipoproteinemia, and larger vacuoles are seen in abetalipoproteinemia.
Chylomicron Retention Disease (Anderson Disease)
In chylomicron retention disease, a rare recessive disorder, there is a defect in chylomicron exocytosis from enterocytes. Sar1-GTP promotes the formation of endoplasmic reticulum to Golgi transport carriers, and Sar1b is defective in Anderson disease. These patients have severe intestinal symptoms with steatorrhea, chronic diarrhea, and failure to thrive. Acanthocytosis is rare, and neurologic manifestations are less severe than those observed in abetalipoproteinemia. Plasma cholesterol levels are moderately reduced (<75 mg/dL) and fasting triglycerides are normal, but the fat-soluble vitamins, particularly A and E, are very low. Treatment is early aggressive therapy with fat-soluble vitamins and modification of dietary fat intake, as in the treatment of abetalipoproteinemia.
Wolman Disease
Wolman disease is a rare, lethal lipid storage disease that leads to lipid accumulation in multiple organs, including the small intestine. In addition to vomiting, severe diarrhea, and hepatosplenomegaly, patients have steatorrhea as a result of lymphatic obstruction. Deficiency of lysosomal acid lipase is the underlying cause of disease (Chapter 80). Successful long-term bone marrow engraftment has resulted in normalization of peripheral blood leukocyte lysosomal enzyme acid lipase activity, with subsequent resolution of diarrhea and the restoration of developmental milestones.
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330.4 Intestinal Infections and Infestations Associated with Malabsorption
David Branski,Raanan Shamir
Malabsorption is a rare consequence of primary intestinal infection and infestation in immunocompetent children, but it can occur after infection with Campylobacter, Shigella, Salmonella, Giardia, cryptosporidium, coccidioidosis, and rotavirus. These infectious causes of malabsorption are more common in immunocompromised children.
Postinfectious Diarrhea
In infants and very young toddlers chronic diarrhea can appear following infectious enteritis, regardless of the nature of the pathogen. The pathogenesis of the diarrhea is not always clear and may be related to secondary lactase deficiency, food protein allergy, antibiotic-associated colitis (including pseudomembranous colitis due to Clostridium difficile toxin), or a combination of these.
Treatment is supportive and may include a lactose-free diet in the presence of secondary lactase deficiency; infants might require a semi-elemental diet. The beneficial effect of probiotic should await well-controlled clinical trials.
Bacterial Overgrowth
Bacteria are normally present in large numbers in the colon (10 11-10 13 colony-forming units [CFU]/gram of feces) and have a symbiotic relationship with the host, providing nutrients and protecting the host from pathogenic organisms. Excessive numbers of bacteria in the small bowel or stomach are harmful. Bacteria are usually present only in a small number in the stomach and small bowel. Gastric acid pH prevents the ingested organisms from colonizing the small bowel. Small bowel motility and the migrating motor complex cleanse the small bowel between meals and at night; the ileocecal valve prevents colonic bacteria from refluxing into the ileum. Mucosal defenses such as mucin and immunoglobulins prevent bacterial overgrowth in the small bowel. Bacterial overgrowth can result from clinical conditions that alter the gastric pH or small bowel motility, including disorders such as partial bowel obstruction, diverticula, short bowel, intestinal duplications, diabetes mellitus, idiopathic intestinal pseudo-obstruction syndrome, and scleroderma. Prematurity, immunodeficiency, and malnutrition are other factors associated with bacterial overgrowth of the small bowel.
Diagnosis of bacterial overgrowth can be made by culturing small bowel aspirate or by lactulose hydrogen breath test. Lactulose is a synthetic disaccharide, which is not digested by mucosal brush border enzymes but can be fermented by bacteria. High baseline hydrogen and a quick rise in hydrogen in expired breath samples support the diagnosis of bacterial overgrowth, but false-positive tests are common.
Bacterial overgrowth leads to inefficient intraluminal processing of dietary fat and to steatorrhea due to bacterial deconjugation of bile salts, vitamin B 12 malabsorption, and microvillus brush border damage with malabsorption. Bacterial consumption of vitamin B 12 and enhanced synthesis of folate result in decreased vitamin B 12 and increased folate serum levels. Overproduction of D-lactate (the isomer of L-lactate) can cause stupor, neurologic dysfunction, and shock from D-lactic acidosis. Lactic acidosis should be suspected in children at risk of bacterial overgrowth, who show signs of neurologic deterioration and a high anion gap metabolic acidosis not explained by measurable acids such as L-lactate. Measurement of D-lactate is required because standard lactate assay only measures the L-isomer.
Treatment of bacterial overgrowth focuses on correction of underlying causes such as partial obstruction. The oral administration of antibiotics is the mainstay of therapy. Initial treatment with 2-4 wk of metronidazole can provide relief for many months. Cycling of antibiotics including azithromycin, trimethoprim-sulfamethoxazole, ciprofloxacin, and metronidazole is required. Other alternatives are oral nonabsorbable antibiotics such as aminoglycosides. Occasionally, antifungal therapy is required to control fungal overgrowth of the bowel.
Tropical Sprue
Natives and expatriates of certain tropical regions can present with a diffuse lesion of the small intestinal mucosa—tropical sprue, even long after emigration. The endemic regions include South India, the Philippines, and some islands in the Caribbean. It is uncommon in Africa, Jamaica, and Southeast Asia. The etiology of this disorder is unclear; because it follows outbreaks of acute diarrheal disease and improves with antibiotic therapy, an infectious etiology is suspected. The incidence is decreasing worldwide, possibly due to common use of antibiotics for gastroenteritis in developing countries.
Clinical symptoms include fever and malaise followed by watery diarrhea. After about a week the acute features subside, and anorexia, intermittent diarrhea, and chronic malabsorption result in severe malnutrition characterized by glossitis, stomatitis, cheilosis, night blindness, hyperpigmentation, and edema. Muscle wasting is often marked, and the abdomen is often distended. Megaloblastic anemia results from folate and vitamin B 12 deficiencies.
Diagnosis is made by small bowel biopsy, which shows villous flattening, crypt hyperplasia, and a chronic inflammatory cell infiltrate of the lamina propria with adjacent lipid accumulation in the surface epithelium.
Treatment requires nutritional supplementation, including supplementation of folate and vitamin B 12. To prevent recurrence, 6 mo of therapy with oral folic acid (5 mg) and tetracycline or sulfonamides is recommended. Relapses occur in 10-20% of patients who continue to reside in an endemic tropical region; additional courses of antibiotics may be necessary.
Whipple Disease
Whipple disease is a chronic multisystem disorder. It is a rare disease, especially in childhood. The disease is caused by an infectious agent, Tropheryma whipplei, which can be cultured from a lymph node in the involved tissue. The syndrome encompasses weight loss, diarrhea, abdominal pain, occult bleeding from the bowel mucosa, hepatosplenomegaly, hepatitis, and ascites. Other organs and systems such as the joints, eyes, heart, and kidneys can also be affected and there may be neurologic and psychiatric manifestations as well.
Diagnosis is made upon demonstration of PAS-positive macrophages in the biopsy material and later by positive identification of polymerase chain reaction (PCR) for T. whipplei.
Treatment requires antibiotics such as cotrimoxazole for 1-2 yr. Recently a 2-wk course of intravenous ceftriaxone or meropenem, followed by cotrimoxazole for 1 yr, has been recommended.
Bibliography
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330.5 Immunodeficiency Disorders
Ernest G. Seidman,David Branski
Malabsorption can occur with congenital immunodeficiency disorders, and chronic diarrhea with failure to thrive is often the mode of presentation. Defects of humoral and or cellular immunity may be involved, including selective IgA deficiency, agammaglobulinemia, common variable immunodeficiency disease (CVID), severe combined immunodeficiency, Wiskott-Aldrich syndrome, or chronic granulomatous disease. Although most patients with selective IgA deficiency are asymptomatic, malabsorption due to giardiasis or nonspecific enteropathy with bacterial overgrowth can occur. Malabsorption syndrome or chronic noninfectious diarrhea has been reported in 60% of children with CVID, most often in the subgroup with low memory B cell counts. Malabsorption has also been reported in ∼10% of patients with late-onset CVID, often secondary to giardiasis. Celiac disease is more common in patients with IgA deficiency and CVID. Paradoxically, it is more difficult to exclude the diagnosis of celiac disease because of the lack of reliability of IgA- and IgG-based serologic tests. Malabsorption due to chronic rotavirus, giardiasis, bacterial overgrowth, and protein-losing enteropathy are well-recognized complications of X-linked agammaglobulinemia. Malabsorption associated with immunodeficiency is exacerbated by villus atrophy and secondary disaccharidase deficiency. In chronic granulomatous disease, phagocytic function is impaired and granulomas develop throughout the GI tract, mimicking Crohn disease. In addition to failure to thrive, it is important to consider that malabsorption associated with immunodeficiency is often complicated by micronutrient deficiencies, including vitamins A, E, and B 12 and calcium, zinc, and iron.
Overall, immunodeficiencies such as hypogammaglobulinemia in the pediatric age group are more often secondary to other conditions such as cancer and chemotherapy, chronic infections, malabsorption, nephrotic syndrome, or cardiac disease. Malnutrition, diarrhea, and failure to thrive are common in untreated children with HIV infection. The risk of GI infection is related to the depression of the CD4 count. Opportunistic infections include Cryptosporidium parvum, cytomegalovirus, Mycobacterium avium-intracellulare, Isospora belli, Enterocytozoon bieneusi, Candida albicans, astrovirus, calicivirus, adenovirus, and the usual bacterial enteropathogens. In these patients, Cryptosporidium can cause a chronic secretory diarrhea.
Cancer chemotherapy can damage the bowel mucosa, leading to secondary malabsorption of disaccharides such as lactose. After bone marrow transplantation, mucosal damage from graft vs host disease can cause diarrhea and malabsorption. Small bowel biopsies show nonspecific villus atrophy, mixed inflammatory cell infiltrates, and increased apoptosis. Cancer chemotherapy and bone marrow transplantation have been associated with pancreatic damage leading to exocrine pancreatic insufficiency.
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330.6 Immunoproliferative Small Intestinal Disease
Ernest G. Seidman,David Branski
Malignant lymphomas of the small intestine are categorized into 3 subtypes: Burkitt lymphoma, non-Hodgkin lymphomas, and Mediterranean lymphoma. Burkitt lymphoma, the most common form in children, characteristically involves the terminal ileum with extensive abdominal involvement. The relatively uncommon “Western” type of non-Hodgkin lymphomas (usually large B-cell type), can involve various parts of the small intestine. Mediterranean lymphoma predominantly involves the proximal small intestine. The World Health Organization (WHO) recommended the term immunoproliferative small intestinal disease (IPSID) for the syndrome associated with Mediterranean lymphoma, because in its early stages it does not appear to be a truly malignant lymphoma. Many of the patients with “secretory” IPSID syndrome have variable levels of abnormal immunoglobulin in serum or other body fluids, identified as truncated α heavy chain. The WHO classification lists IPSID with heavy chain diseases as a special variant of extranodal marginal zone B-cell small intestinal mucosa associated lymphoid tissue (MALT) lymphoma.
IPSID occurs most often in the proximal small intestine in older children and young adults in the Mediterranean basin, Middle East, Asia, and Africa. Poverty and frequent episodes of gastroenteritis during infancy are antecedent risk factors. The initial clinical presentation is intermittent diarrhea and abdominal pain. Later, chronic diarrhea with malabsorption (60-80%), protein-losing enteropathy, weight loss, digital clubbing, and growth failure ensue. Intestinal obstruction, abdominal masses, and ascites are common in advanced stages.
In contrast to primary non-immunoproliferative small intestinal lymphomas, in which the pathology in the intestine is usually focal, involving specific segments of the intestine and leaving the segments between the involved areas free of disease, the pathology in IPSID is diffuse, with a mucosal cellular infiltrate involving large segments of the intestine and sometimes the entire length of the intestine, thus producing malabsorption. Molecular and immunohistochemical studies demonstrated an association with Campylobacter jejuni infection . The differential diagnosis includes chronic enteric infections (parasites, tropical sprue), celiac disease, and other lymphomas. Radiologic findings include multiple filling defects, ulcerations, strictures, and enlarged mesenteric lymph nodes on CT scan.
The diagnosis is usually established by endoscopic biopsies and/or laparotomy. Upper endoscopy shows thickening, erythema, and nodularity of the mucosal folds in the duodenum and proximal jejunum. As the disease progresses, tumors usually appear in the proximal small intestine and rarely in the stomach. The diagnosis requires multiple duodenal and jejunal mucosal biopsies showing dense mucosal infiltrates, consisting of centrocyte-like and plasma cells. Progression to higher-grade large-cell lymphoplasmacytic and immunoblastic lymphoma is characterized by increased plasmocytic atypia with formation of aggregates and later sheets of dystrophic plasma cells and immunoblasts invading the submucosa and muscularis propria. A serum marker of IgA, α heavy-chain paraprotein, is present in most cases.
Treatment of early-stage IPSID with antibiotics results in complete remission in 30-70% of cases. However, the majority of untreated IPSID cases progress to lymphoplasmacytic and immunoblastic lymphoma invading the intestinal wall and mesenteric lymph nodes and can metastasize to distant organs, requiring chemotherapy.
Bibliography
Al-Saleem T, Al-Mondhiry H. Immunoproliferative small intestinal disease (IPSID): a model for mature B-cell neoplasms. Blood. 2005;105:2274-2280.
Economidou I, Manousos ON, Triantafillidis JK, et al. Immunoproliferative small intestinal disease in Greece: presentation of 13 cases including two from Albania. Eur J Gastroenterol Hepatol. 2006;18:1029-1038.
Lecuit M, Abachin E, Martin A, et al. Immunoproliferative small intestinal disease associated with C ampylobacter jejuni. N Engl J Med. 2004;350:239-248.
Salem PA, Estephan HH. Immunoproliferative small intestinal disease, Curr Concepts. Cancer J. 2005;11:374-382.
Witzig TE, Wahner-Roedler DL. Heavy chain disease. Curr Treat Options Oncol. 2002;3:247-254.
Zamir A, Parasher G, Moukarzel A, et al. Immunoproliferative small intestinal disease in a 16-year-old boy presenting as severe malabsorption with excellent response to tetracycline treatment. J Clin Gastroenterol. 1998;27:85-89.
330.7 Short Bowel Syndrome
Jon A. Vanderhoof and David Branski
Short bowel syndrome results from congenital malformations or resection of the small bowel. Causes of short bowel syndrome are listed inTable 330-9. Loss of >50% of the small bowel, with or without a portion of the large intestine, can result in symptoms of generalized malabsorption disorder or in specific nutrient deficiencies, depending on the region of the bowel resected. At birth, the length of small bowel is 200-250 cm; by adulthood, it grows to 300-800 cm. Bowel resection in an infant has a better prognosis than in an adult because of the potential for intestinal growth. An infant with as little as 15 cm of bowel with an ileocecal valve, or 20 cm without, has the potential to survive and be eventually weaned from total parenteral nutrition (TPN).
Table 330-9CAUSES OF SHORT BOWEL SYNDROME
CONGENITAL
Congenital short bowel syndrome
Multiple atresias
Gastroschisis
BOWEL RESECTION
Necrotizing enterocolitis
Volvulus with or without malrotation
Long segment Hirschsprung disease
Meconium peritonitis
Crohn disease
Trauma
In addition to the length of the bowel, the anatomic location of the resection is also important. The jejunum has more circular folds and longer villi. The proximal 100-200 cm of jejunum is the main site for carbohydrate, protein, iron, and water-soluble vitamin absorption, whereas fat absorption occurs over a longer length of the small bowel. Depending on the region of the bowel resected, specific nutrient malabsorption can result. Vitamin B 12 and bile salts are only absorbed in the distal ileum (Fig. 330-5). Jejunal resections are generally tolerated better than ileal resections because the ileum can adapt to absorb nutrients and fluids. Net sodium and water absorption is relatively much higher in the ileum. Ileal resection has a profound effect on fluid and electrolyte absorption due to malabsorption of sodium and water by the remaining ileum; ileal malabsorption of bile salts stimulates increased colonic secretion of fluid and electrolytes.
Figure 330-5Absorption of nutrients in the small bowel varies with the region.
Treatment
After bowel resection, treatment of short bowel syndrome is initially focused on repletion of the massive fluid and electrolyte losses while the bowel initially accommodates to absorb these losses. Nutritional support is often provided via parenteral nutrition. A central venous catheter should be inserted to provide parenteral fluid and nutrition support. The ostomy or stool output should be measured and fluid and electrolyte losses adequately replaced. Measurement of urinary Na+ to assess body Na+ stores is useful to prevent Na+ depletion. Maintaining urinary Na+ higher than K+ ensures that Na+ intake is adequate. Use of oral glucose electrolyte solutions improves intestinal sodium absorption, particularly in patients without a colon.
After the initial few weeks following resection, fluid and electrolyte losses stabilize, and the focus of therapy shifts to bowel rehabilitation with the gradual reintroduction of enteral feeds. Continuous small-volume trophic enteral feeding should be initiated with a protein hydrolysate and medium-chain triglyceride–enriched formula to stimulate gut hormones and promote mucosal growth. Enteral feeding also increases pancreatobiliary flow and reduces parenteral nutrition-induced hepatotoxicity. As soon as possible, the infant should be given a small amount of water, and then formula by mouth to maintain an interest in oral feeding and minimize or avoid the development of oral aversion. As intestinal adaptation occurs, enteral feeding increases and parenteral supplementation decreases. The bowel mucosa proliferates and bowel lengthens with growth.
After achieving the maximal increase in bowel absorptive capacity, management of specific micronutrient and vitamin deficiencies and treatment of transient problems such as postinfectious mucosal malabsorption are required. GI infections such as rotavirus or small bowel bacterial overgrowth can cause setbacks in the progression to full enteral feeding in patients with marginal absorptive function. A marked increase in stool output or evidence of carbohydrate malabsorption (stool pH <5.5 and positive test for reducing substances) contraindicate further increases in enteral feeds. Slow advancement of continuous enteral feeding rates continues until all nutrients are provided enterally. Then the feeds can be altered to include increased oral or bolus feeding volumes.
In patients with large stool outputs, the addition of soluble fiber and antidiarrheal agents such as loperamide and anticholinergics can be beneficial, although these drugs can increase the risk of bacterial overgrowth. Cholestyramine can be beneficial for patients with distal ileal resection, but its potential depletion of the bile acid pool can increase steatorrhea. Bacterial overgrowth is common in infants with a short bowel and can delay progression of enteral feedings. Empirical treatment with metronidazole or other antibiotics is often useful. Diets high in fat and lower in carbohydrate may be helpful in reducing bacterial overgrowth as well as enhancing adaptation.
Complications
Long-term complications of short bowel syndrome include those of parenteral nutrition: central catheter infection, thrombosis, hepatic cholestasis and cirrhosis, and gallstones. Appropriate care of the central line to prevent infection and catheter-related thrombosis is extremely important. Some patients need long-term parenteral nutritional support, and lack of central line access is potentially life-threatening; inappropriate removal or changes of central lines in the neonatal period should be avoided. Other complications of terminal ileal resection include vitamin B 12 deficiency, which might not appear until 1-2 yr after parenteral nutrition is withdrawn. Long-term monitoring for deficiencies of vitamin B 12, folate, iron, fat-soluble vitamins, and trace minerals such as zinc and copper is important. Renal stones can occur as a result of hyperoxaluria secondary to steatorrhea (calcium binds to the excess fat and not to oxalate, so more oxalate is reabsorbed and excreted in the urine). Venous thrombosis and vitamin deficiency have been associated with hyperhomocystinemia in short bowel syndrome. Bloody diarrhea secondary to patchy, mild colitis can develop during the progression of enteral feedings. The pathogenesis of this “feeding colitis” is unknown, but it is usually benign and can improve with a hypoallergenic diet or treatment with sulfasalazine.
In patients who are unable to achieve full enteral feeding after several years of nutritional rehabilitation, surgical bowel lengthening procedures may be considered. In some children with complications of parenteral nutrition, especially impending liver failure, small intestinal and liver transplantation may be considered (Chapter 331).
330.8 Chronic Malnutrition
Raanan Shamir,David Branski
Primary malnutrition (i.e., undernutrition) is very common in developing countries and is directly related to increased disease burden and mortality (Chapter 43). In developed countries, chronic malnutrition occurs mainly as a result of decreased food intake, malabsorption syndromes, and increased nutritional needs in children with chronic diseases, and it affects 11-50% of hospitalized children. Child neglect and improper preparation of formula can result in severe malnutrition. Malnutrition can be identified by evaluating dietary intake, by medical history (anorexia, vomiting, dysphagia, mood and behavioral changes, abdominal pain, diarrhea), by anthropometric measurements (e.g., reduced weight per age and weight per height, BMI <5th percentile) by clinical signs of nutrient deficiencies (atrophic tongue in iron deficiency anemia or alopecia in zinc deficiency).
Malnourished children suffer from impaired immunity, poor wound healing, muscle weakness, and diminished psychologic drive. Malnutrition has short-term consequences (increased disability, morbidity, and mortalitiy) and long-term consequences (final adult size, lower IQ, economic productivity). Undernutrition in hospitalized children is related to increased infectious complications, delayed recovery, increased length of stay and costs, increased readmission rate, and increased mortality.
Nutritional rehabilitation in malnourished children is discussed inChapter 43.
Chronic malnutrition complicated by diarrheal dehydration is a commonly observed phenomenon. Infectious diarrhea is common in tropical and subtropical countries, in the setting of poor hygiene practices, in immunocompromised hosts (HIV, congenital immunodeficiency), and when impairment of the immune response is due to chronic malnutrition itself. In children with chronic disorders, diarrhea may be related to the underlying disease and should be sought. Examples include noncompliance with a gluten-free diet in celiac disease, noncompliance with pancreatic enzyme treatment in cystic fibrosis, and disease relapse in inflammatory bowel disease (IBD). In the case of IBD, relapse should be diagnosed only after infectious diarrhea and C. difficile infection have been ruled out. Malnutrition per se can lead to exocrine pancreatic insufficiency, which, in turn, aggravates malabsorption and diarrhea.
In infants and children with severe malnutrition, many of the signs normally used to assess the state of hydration or shock are unreliable. Severe malnutrition might be accompanied by sepsis; thus, children with septic shock might not have diarrhea, thirst, or sunken eyes, but may be hypothermic, hypoglycemic, or febrile. The electrocardiogram (ECG) often shows tachycardia, low amplitude, and flat or inverted T waves. Cardiac reserve seems lowered, and heart failure is a common complication.
Despite clinical signs of dehydration, urinary osmolality may be low in the chronically malnourished child. Renal acidifying ability is also limited in patients with malnutrition.
Management of the diarrhea in chronically malnourished children is based on 3 principles: oral rehydration to correct dehydration, rapid resumption of regular feeds with avoidance of periods of nothing by mouth, and treating the etiology of the diarrhea.
When treating the dehydration, it must be remembered that in dehydrated and malnourished infants there appears to be overexpansion of the extracellular space accompanied by extracellular and presumably intracellular hypo-osmolality. Thus, reduced or hypotonic osmolarity oral rehydration solutions are indicated in this setting. When oral rehydration is not possible, the route of choice is nasogastric, and intravenous therapy should be avoided if possible.
Initial intravenous therapy in profound dehydration is designed to improve the circulation and expand extracellular volume. For patients with edema, the quality of fluid and the rate of administration might need to be readjusted from recommended levels to avoid overhydration and pulmonary edema. Blood should be given if the patient is in shock or severely anemic. Potassium salts can be given early if urine output is good. Clinical and ECG improvement may be more rapid with magnesium therapy.
Children with chronic malnutrition are at risk for the refeeding syndrome. Therefore, initial calorie provision should not exceed the previous daily intake and is usually begun at 50-75% of estimated resting energy expenditure, with rapid increase to caloric goals once there are no severe abnormalities in sodium, potassium, phosphorus, calcium, or magnesium. Correction of malnutrition and catch-up growth are not part of the primary treatment of these children, but a nutrition rehabilitation plan is necessary.
Bibliography
Black RE, Allen LH, Bhutta ZA, et al. Maternal and child undernutrition: global and regional exposures and health consequences. Lancet. 2008;371:243-260.
Brown KH. Diarrhea and malnutrition. J Nutr. 2003;133:328S-332S.
Caulfield LE, de Onis M, Blossner M, et al. Undernutrition as an underlying cause of child deaths associated with diarrhea, pneumonia, malaria, and measles. Am J Clin Nutr. 2004;80:193-198.
Fonseca BK, Holdgate A, Craig JC. Enteral vs. intravenous rehydration therapy for children with gastroenteritis: a meta-analysis of randomized controlled trials. Arch Pediatr Adolesc Med. 2004;158:483-490.
Guarino A, Albano F, Ashkenazi S, et al. ESPGHAN/ESPID guidelines for the management of acute gastroenteritis in children in Europe. J Pediatr Gastroenterol Nutr. 2008;46:S81-S122.
World Health Organization. Management of severe malnutrition: a manual for physicians and other senior health workers. Geneva: World Health Organization; 1999.
330.9 Enzyme Deficiencies
Michael J. Lentze and David Branski
Carbohydrate Malabsorption
Symptoms of carbohydrate malabsorption include loose watery diarrhea, flatulence, abdominal distention, and pain. Some children are asymptomatic unless the malabsorbed carbohydrate is consumed in large amounts. Disaccharidases are present on the brush border membrane of the small bowel. Disaccharidase deficiency can be due to a genetic defect or secondarily due to damage to the small bowel epithelium, as occurs with infection or inflammatory disorders.
Unabsorbed carbohydrates enter the large bowel and are fermented by intestinal bacteria, producing organic acids and gases such as methane and hydrogen. The gases can cause discomfort and the unabsorbed carbohydrate and the organic acids cause osmotic diarrhea characterized by an acidic pH and presence of either reducing or nonreducing sugars in the stool. Hydrogen gas can be detected in the breath as a sign of fermentation of unabsorbed carbohydrates (H 2-breath test).
Lactase Deficiency
Congenital lactase deficiency is rare and is associated with symptoms occurring on exposure to lactose in milk. Fewer than 50 cases have been reported worldwide. In patients with congenital lactase deficiency, 5 distinct mutations in the coding region of the LCT gene were found. In most patients (84%), homozygosity for a nonsense mutation, 4170T-A (Y1390X; OMIM 223000), designated Fin (major), was found.
Primary adult type-hypolactasia is caused by a physiologic decline in lactase actively that occurs following weaning in most mammals. The brush border lactase is expressed at low levels during fetal life; activity increases in late fetal life and peaks from term to 3 yr, after which levels gradually decrease with age. This decline in lactase levels varies between ethnic groups. Lactase deficiency occurs in ∼15% of white adults, 40% of Asian adults, and 85% of black adults in the United States. Lactase is encoded by a single gene (LCT) of ∼50 kb located on chromosome 2q21. C/T (−13910) polymorphisms of the MCM6 gene were found to be related to adult-type hypolactasia in most European populations. In 3 African populations—Tanzanians, Kenyans, and Sudanese—3 SNPs, G/C (−14010), T/G(−13915), and C/G(−13907) were identified with lactase persistence and have derived alleles that significantly enhance transcription from the lactase gene promoter in vitro.
Secondary lactose intolerance follows small bowel mucosal damage (celiac disease, rotavirus infection) and is usually transient, improving with mucosal healing.
Lactase deficiency can be diagnosed by H 2-breath test or by measurement of lactase activity in mucosal tissue retrieved by small bowel biopsy. Diagnostic testing is not mandatory, and often simple dietary changes that reduce or eliminate lactose from the diet relieve symptoms.
Treatment of lactase deficiency consists of a milk-free diet. A lactose-free formula (based on either soy or cow’s milk) can be used in infants. In older children, low-lactose milk can be consumed. Addition of lactase to dairy products usually abbreviates the symptoms.
Live-culture yogurt contains bacteria that produce lactase enzymes and is therefore tolerated in most patients with lactase deficiency. Hard cheeses have a small amount of lactose and are generally well tolerated.
Fructose Malabsorption
Children consuming a large quantity of juice rich in fructose, corn syrup, or natural fructose in fruit juices can present with diarrhea, abdominal distention, and slow weight gain. Restricting the amount of juice in the diet resolves the symptoms and helps avoid unnecessary investigations. Fructose H 2 breath test can be helpful in the diagnosis of fructose malabsorption. The reason for fructose malabsorption is the reduced abundance of GLUT-5 transporter on the surface of the intestinal brush border membrane, which occurs in about 5% of the population.
Sucrase-Isomaltase Deficiency
Sucrase-isomaltase deficiency is a rare autosomal recessive disorder with a complete absence of sucrase and reduced maltase digestive activity. The sucrase-isomaltase complex is composed of 1,927 amino acids encoded by a 3,364 bp mRNA. The gene locus on chromosome 3 has 30 exons spanning 106.6 kb. The majority of sucrase-isomaltase mutations result in a lack of enzyme protein synthesis (null mutation). Post-translational processing defects are also identified.
Approximately 2% of Europeans and Americans are mutant heterozygote. Sucrase deficiency is especially common in indigenous Greenlanders (estimated 5%) in whom it is often accompanied by lactase deficiency.
Symptoms of sucrase-isomaltase deficiency usually begin when the infant is exposed to sucrose or a glucose polymer diet. This can occur with ingestion of non–lactose based infant formula or on the introduction of pureed food, especially fruits and sweets. Diarrhea, abdominal pain, and poor growth are observed. Occasional patients present with symptoms in late childhood or even adult life, but careful history often indicates that symptoms appeared earlier. Diagnosis of sucrase-isomaltase malabsorption requires acid hydrolysis of stool for reducing substances because sucrase is a nonreducing sugar. Alternatively, diagnosis can be achieved with hydrogen breath test or direct enzyme assay of small bowel biopsy.
The mainstay of treatment is lifelong dietary restriction of sucrose-containing foods. Enzyme replacement with a purified yeast enzyme, sacrosidase (Sucraid), is a highly effective adjunct to dietary restriction.
Glucose-Galactose Malabsorption
More than 30 different mutations of the sodium/glucose co-transporter gene (SGLT1) are identified. These mutations cause a rare autosomal recessive disorder of intestinal glucose and galactose/Na+ co-transport system that leads to osmotic diarrhea. Because most dietary sugars are polysaccharides or disaccharides with glucose or galactose moieties, diarrhea follows the ingestion of glucose, breast milk, or conventional lactose-containing formulas. Dehydration and acidosis can be severe, resulting in death.
The stools are acidic and contain sugar. Patients with the defect have normal absorption of fructose, and their small bowel function and structure are normal in all other aspects. Intermittent or permanent glycosuria after fasting or after a glucose load is a common finding due to the transport defect also being present in the kidney. The presence of reducing substances in watery stools and slight glycosuria despite low blood sugar levels is highly suggestive of glucose-galactose malabsorption. Malabsorption of glucose and galactose is easily identified using the breath hydrogen test. It is safe to perform the 1st test with a dose of 0.5 g/kg of glucose; if necessary, a second test can be performed using 2 g/kg. Breath H 2 will rise more than 20 ppm. The small intestinal biopsy is useful to document a normal villous architecture and normal disaccharidase activities. The identification of mutations of SGLT1 makes it possible to perform prenatal screening in families at risk for the disease.
Treatment consists of rigorous restriction of glucose and galactose. Fructose, the only carbohydrate that can be given safely, should be added to a carbohydrate-free formula at a concentration of 6-8%. Diarrhea immediately ceases when infants are given such a formula. Although the defect is permanent, later in life, limited amounts of glucose, such as starches or sucrose may be tolerated.
Exocrine Pancreatic Insufficiency
Disorders of exocrine pancreatic insufficiency are discussed inChapter 341. Cystic fibrosis is the most common congenital disorder associated with exocrine pancreatic insufficiency. Although rare, the next most common cause of pancreatic insufficiency in children is Shwachman-Diamond syndrome. Other rare disorders causing exocrine pancreatic insufficiency are Blizzard-Johanson syndrome (severe steatorrhea, aplasia of alae nasi, deafness, hypothyroidism, scalp defects), Pearson bone marrow syndrome (sideroblastic anemia, variable degree of neutropenia, thrombocytopenia), and isolated pancreatic enzyme deficiency (lipase, colipase and lipase-colipase, trypsinogen, amylase). Deficiency of enterokinase—a key enzyme that is produced in the proximal small bowel and is responsible for the activation of trypsinogen to trypsin—manifests clinically as exocrine pancreatic insufficiency.
Autoimmune polyendocrinopathy syndrome type 1, a rare autosomal recessive disorder, is caused by mutation in the autoimmune regulator gene (AIRE). Chronic mucocutaneous candidiasis is associated with failure of parathyroid gland, adrenal cortex, pancreatic β-cells, gonads, gastric parietal cells, and thyroid gland. Pancreatic insufficiency and steatorrhea have been associated with this condition.
Enterokinase (Enteropeptidase) Deficiency
Enterokinase (enteropeptidase) is a brush border enzyme of the small intestine. It is responsible for the activation of trypsinogen into trypsin. Deficiency of this enzyme results in severe diarrhea, malabsorption, failure to thrive, and hypoproteinemic edema after birth.
Enterokinase deficiency is caused by mutation in the serine protease-7 gene (PRSS7) on chromosome 21q21. The diagnosis can be established by measuring the enzyme level in intestinal tissue. Treatment of this rare autosomal recessive disorder consists of replacement with pancreatic enzymes and administration of a protein hydrolyzed formula with added MCT oil in infancy.
Trypsinogen Deficiency
Trypsinogen deficiency is a rare syndrome with symptomatology similar to that of enterokinase deficiency. Enterokinase catalyzes the conversion of trypsinogen to trypsin, which, in turn, activates the various pancreatic proenzymes such as chymothrypsin, procaboxypeptidase, and proelastase for their active forms. Deficiency of trypsinogen results in severe diarrhea, malabsorption, failure to thrive and hypoproteinemic edema soon after birth.
The trypsinogen gene is encoded on chromosome 7q35. Treatment is the same as for enterokinase deficiency, with pancreatic enzymes and protein hydrolysate formula with added MCT oil in infancy.
330.10 Liver and Biliary Disorders Causing Malabsorption
Anil Dhawan and David Branski
Absorption of fats and fat-soluble vitamins depends to a great extent on adequate bile flow providing bile acids to the small intestine. Most of the liver and biliary disorders lead to impairment of the bile flow, contributing to malabsorption of long-chain fatty acids and vitamins such as A, D, E, and K. In addition, severe portal hypertension can lead to portal hypertensive enteropathy, resulting in poor absorption of the nutrients. Decompensated liver disease leads to anorexia and increased energy expenditures, further widening the gap between calorie intake and net absorption, leading to severe malnutrition. Adequate management of nutrition is essential to improve the outcome with or without liver transplantation. This is usually achieved by using medium-chain triglyceride-rich milk formula, supplemental vitamins, and continuous or bolus enteral feed where oral intake is poor.
Vitamin D deficiency is commonly observed on biochemical tests, and children rarely present with pathologic fractures. Simultaneous administration of vitamin D with the water-soluble vitamin E preparation (d-α-tocopherol polyethylene glycol 1,000 succinate [TPGS]) enhances absorption of vitamin D. In young infants, oral vitamin D 3 is given at a dose of 1,000 IU/kg/24 hr. After 1 mo, if the serum 25-hydroxyvitamin D level is low, the same dose of oral vitamin D is mixed with TPGS. 25-hydroxyvitamin D is then monitored every 3 mo, with adjustment of doses as necessary.
Vitamin E deficiency in patients with chronic cholestasis is not usually symptomatic, but it can manifest as a progressive neurologic syndrome, which includes peripheral neuropathy (manifesting as loss of deep tendon reflexes and ophthalmoplegia), cerebellar ataxia, and posterior column dysfunction. Early in the course, findings are partially reversible with treatment; late features might not be reversible. It may be difficult to identify vitamin E deficiency because the elevated blood lipid levels in cholestatic liver disease can falsely elevate the serum vitamin E level. Therefore, it is important to measure the ratio of serum vitamin E to total serum lipids; the normal level for patients <12 yr of age is >0.6, and for patients >12 yr it is >0.8. The neurologic disease can be prevented with the use of an oral water-soluble vitamin E preparation (TPGS, Liqui-E) at a dose of 25-50 IU/day in neonates and 1 IU/kg/day in children.
Vitamin K deficiency can occur as a result of cholestasis and poor fat absorption. In children with liver disease it is very important to differentiate between the coagulopathy related to vitamin K deficiency and one secondary to the synthetic failure of the liver. A single dose of vitamin K administered intravenously does not correct the prolonged prothrombin time in liver failure, but the deficiency state responds within a few hours. Easy bruising may be the 1st sign. In neonatal cholestasis, coagulopathy due to vitamin K deficiency can manifest with intracranial bleeds with devastating consequences, and prothrombin time should be routinely measured to monitor for deficiency in children with cholestasis. All children with cholestasis should receive vitamin K supplements.
Vitamin A deficiency is rare and is associated with night blindness, xerophthalmia, and increased mortality if patients contract measles. Serum vitamin A levels should be monitored and adequate supplementation considered.
In practice, children with cholestasis are prescribed twice the recommended daily allowance of the commonly available multivitamin preparations while awaiting blood levels.
330.11 Rare Inborn Defects Causing Malabsorption
Peter Zimmer,David Branski
Some congenital (primary) malabsorption disorders originate from a defect of integral membrane proteins, which fulfill a transport function as receptor or channel across the apical or basolateral membrane of enterocytes for nutritional components. Histologic examination of the small and large bowel is typically normal. Most of these disorders are inherited in an autosomal recessive pattern. Most are rare, and patients present with a broad phenotypic heterogeneity due to modifier genes and nutritional and other secondary factors.
Disorders of Carbohydrate Absorption
Patients with Fanconi-Bickel syndrome (FBS) present with tubular nephropathy; rickets; hepatomegaly; glycogen accumulation in liver, kidney, and small bowel; failure to thrive; and fasting hypoglycemia. The disorder is caused by homozygous mutations of GLUT2, the facilitative glucose (and galactose) transporter at the basolateral membrane of enterocytes hepatocytes, renal tubules, pancreatic islet cells, and cerebral neurons. Because severe osmotic diarrhea is not a feature of FBS, a GLUT2-independent basolateral transport for glucose is suggested. GLUT2 seems to modulate insulin secretion, renal reabsorption, and glucose uptake from the apical membrane of enterocytes in response to the (postprandial) sugar environment. Diagnostic signs are elevated galactose levels in the blood (found in the neonatal screening program), neonatal bilateral cataracts, marked glycosuria, generalized aminoaciduria, and excessive renal losses of phosphate and calcium. Liver and kidneys are enlarged. Therapy includes substitution of electrolyte losses and vitamin D and supplying uncooked cornstarch to prevent hypoglycemia. Patients who present in the neonatal period need frequent small meals and galactose-free milk.
Disorders of Amino Acid and Peptide Absorption
Owing to their ontogenic origins, enterocytes and renal tubules express amino acid transporter in common. Their highest intestinal transporter activity is found in the jejunum. The transporters causing Hartnup disease, cystinuria, iminoglycinuria, and dicarboxylic aminoaciduria are located in the apical membrane, and those causing lysinuric protein intolerance (LPI) and blue diaper syndrome are anchored in the basolateral membrane of the intestinal epithelium.
Dibasic amino acids, including cystine, ornithine, lysine, and arginine are taken up by the Na-independent SLC3A1/SLC7A9, which is defective in cystinuria. The overall prevalence of the disease is 1 in 7,000 newborns. This disorder is not associated with any GI or nutritional consequences because of compensation by alternative transporter. However, hypersecretion of cystine in the urine leads to recurrent cystine stones, which account for up to 1% of all urinary tract stones. Ample hydration, urine alkalinization, and cystine-binding thiol drugs can increase the solubility of cystine. Cystinuria type I is inherited as an autosomal recessive trait, and the transmission of type II is autosomal dominant with incomplete penetrance. Cystinuria type I has been described in association with 2p21 deletion syndrome and hypotonia-cystinuria syndrome.
Hartnup disease is characterized by malabsorption of neutral amino acids, including the essential amino acid tryptophan, with aminoaciduria, photosensitive pellagra-like rash, headaches, cerebellar ataxia, delayed intellectual development, and diarrhea. The clinical spectrum ranges from asymptomatic patients to severely affected patients with progressive neurodegeneration leading to death by adolescence. SLC6A19, which is the major luminal sodium-dependent neutral amino acid transporter of small intestine and renal tubules, has been identified as the defective protein. Its association with collectrin and angiotensin-converting enzyme (ACE) II is likely to be involved in the phenotypic heterogeneity of Hartnup disorder. Tryptophan is a precursor of NAD(P)H biosynthesis; therefore the disorder can be treated by nicotinamide in addition to a diet of 4 g protein/kg. The use of lipid-soluble esters of amino acids and tryptophan ethylester has also been reported.
In the blue diaper syndrome (indicanuria, Drummond syndrome) tryptophan is specifically malabsorbed and the defect is expressed only in the intestine and not in the kidney, in contrast to Hartnup disease. Intestinal bacteria convert the unabsorbed tryptophan to indican, which is responsible for the bluish discoloration of the urine after its hydrolysis and oxidation. Symptoms can include digestive disturbances such as vomiting, constipation, poor appetite, failure to thrive, hypercalcemia, nephrocalcinosis, fever, irritability, and ocular abnormalities. The molecular genetic defect of this disorder has not yet been characterized.
The underlying defect of iminoglycinuria is the malabsorption of proline, hydroxyproline, and glycine due to the proton amino acid transporter SLC36A2 defect, with a possible participation of modifier genes, one of which (SLC6A20), is present in the intestinal epithelium. This disorder is usually benign, but sporadic cases with encephalopathy, mental retardation, deafness, blindness, kidney stones, hypertension, and gyrate atrophy have been described.
The excitatory amino acid carrier SLC1A1 is affected in dicarboxylic aminoaciduria. This carrier is present in the small intestine, kidney, and brain, and transports the anionic acids L-glutamate, L– and D-aspartate, and L-cysteine. There are single case reports indicating that this disorder could be associated with hyperprolinemia and neurologic symptoms such as POLIP (polyneuropathy, ophthalmoplegia, leukoencephalopathy, intestinal pseudo-obstruction).
A histidine-specific transport system has also been proposed. A few patients have been reported with an intestinal and renal defect of this carrier. It has not been confirmed that patients with histidinuria, who have low plasma histidine levels, in contrast to histidinemia, develop neurologic symptoms (e.g., hearing loss, myoclonic seizures).
A methionine-preferring transporter in the small intestine was suggested to be affected in Smith-Strang disease (oasthouse urine disease), which is characterized by purple, red-brown-colored urine with a cabbage-like odor, containing 2-hydroxybutyric acid, valine, and leucine. The potential symptoms of methionine malabsorption include neurologic signs, white hair, and diarrhea. Large amounts of methionine and branched-chain amino acids are present in the feces but not in the urine. A low-methionine diet is recommended to alleviate the symptoms.
Among the diseases (see the earlier discussion of cystinuria) with a membrane transport defect of cationic amino acids (lysine, arginine, ornithine), lysinuric protein intolerance (LPI) is the 2nd most common, with a prevalence in Finland of 1 in 60,000. The y+LAT-1 (SLC7A7) carrier at the basolateral membrane of the intestinal and renal epithelium is affected, with failure to deliver cytosolic dibasic cationic amino acids into the paracellular space in exchange for Na+ and neutral amino. This defect is not compensated by the SLC3A1/SLC7A9 transporter (at the apical membrane), the latter being affected in cystinuria. The symptoms of LPI, which appear after weaning, include diarrhea, failure to thrive, hepatosplenomegaly, nephritis, respiratory insufficiency, alveolar proteinosis, pulmonary fibrosis, and osteoporosis. Abnormalities of bone marrow have also been described in a subgroup of LPI patients. The disorder is characterized by low plasma concentrations of dibasic amino acids (in contrast to high levels of citrulline, glutamine, and alanine) and massive excretion of lysine (as well as orotic acid, ornithine, and arginine in moderate excess) in the urine. Hyperammonemia and coma usually develop after episodic attacks of vomiting, after fasting, or following administration of large amounts of protein (or alanine load), possibly due to a deficiency of intramitochondrial ornithine. Some patients show moderate retardation. Cutaneous manifestations can include alopecia, perianal dermatitis, and sparse hair. Some patients avoid protein-containing food. Treatment includes orally administered citrulline (200 mg/kg/day), which is well absorbed from the intestine; dietary protein restriction (<1.5 g/kg/day); and carnitine supplementation. One patient with isolated lysinuria has been reported with growth failure, seizures, and mental retardation.
Disorders of Fat Transport
Abetalipoproteinemia, hypobetalipoproteinemia, and chylomicron retention disease are described inChapter 80. The long-chain fatty acid (FATP4) and cholesterol transporters, the latter being called Niemann-Pick C1-like protein (NPC1L1), have been characterized at the intestinal brush border in knock-out mice models showing a hyperproliferative hyperkeratosis and an impaired fatty acid and cholesterol uptake. NPC1L1 is inhibited by ezetimibe, which is used to restrict the absorption of dietary cholesterol.
Tangier disease is characterized by the absence of high-density lipoprotein cholesterol (HDL-C), which is caused by mutations in the adenosine triphosphate (ATP)-binding cassette transporter A1 (ABCA1) gene. The failure of intracellular phospholipids and cholesterol efflux to lipid-poor apolipoprotein acceptors such as HDL predisposes to premature coronary heart disease and accumulation of cholesterol in liver, spleen, lymph nodes (tonsils), and small intestine.
Features of Tangier disease include orange tonsils, hepatosplenomegaly, relapsing neuropathy, orange-brown spots on the colon and ileum, diarrhea in association with decreased plasma cholesterol levels (apo A-I, apo A-II), and normal or elevated triglyceride levels. Specific therapy for Tangier disease has not yet been established.
In sitosterolemia defective efflux of sterol leads to increased absorption of dietary sterols; normally, <5% are retained by the GI tract. Patients carry mutations of the ABCG5 (sterolin-1) and ABCG8 (sterolin-2) transporters. The disorder is associated with tendon xanthomas, increased atherosclerosis, and hemolysis. Plasma levels of phytosterols (mainly sitosterol) are typically >10 mg/dL.
Disorders of Vitamin Absorption
Transporters and receptors of the intestinal epithelium have been described for water-soluble but not fat-soluble vitamins, the latter being absorbed primarily by enterocytes, by passive diffusion after emulsification of fats by bile salts. Transfer proteins (retinol-binding protein, RBP4 and α-tocopherol transfer protein, TTP1) have been involved in deficiency states of vitamins E (spinocerebellar ataxia) and A (ophthalmologic signs), respectively.
Vitamin B 12 (cobalamin) is used exclusively by microorganisms and is acquired mostly from meat and milk. Its absorption starts with the removal of cobalamin from dietary protein by gastric acidity and its binding to haptocorrin. In the duodenum, pancreatic proteases hydrolyze the cobalamin-haptocorrin complex, allowing the binding of cobalamin to intrinsic factor (IF), which originates from parietal cells. The receptor of the cobalamin-IF complex is located at the apical membrane of the ileal enterocytes and represents a heterodimer consisting of cubulin and amnionless, with endocytic uptake of this ligand into endosomes, where it binds to megalin and forms a cobalamin–transcobalamin-2 complex (after cleavage of IF) for further transcytosis. As a cofactor for methionine synthase, cobalamin converts homocysteine to methionine. Cobalamin deficiency can be caused by inadequate intake of the vitamin (e.g., breast-feeding by mothers on a vegetarian diet), primary or secondary achlorhydria including autoimmune gastritis, exocrine pancreatic insufficiency, bacterial overgrowth (Chapter 330.4), ileal disease (Crohn disease, Chapter 328), ileal (or gastric) resection, infections (fish tapeworm), and Whipple disease (Chapter 333).
Clinical signs of congenital cobalamin malabsorption, which usually appear from a few months to 14 yr of age, are pancytopenia including megaloblastic anemia, fatigue, failure to thrive, and neurologic symptoms including developmental delay. Recurrent infections and bruising may be present. Laboratory evaluation indicates low serum cobalamin, hyperhomocysteinemia, methylmalonicacidemia, and mild proteinuria. The Schilling test is useful to differentiate between lack of IF and malabsorption of cobalamin. Three rare autosomal recessive disorders of congenital cobalamin deficiency affect absorption and transport of cobalamin (in addition to 7 other inherited defects of cobalamin metabolism). These include mutations of the gastric IF (GIF) gene with absence of IF (but normal acid secretion and lack of autoantibodies against IF or parietal cells), mutations of the amnionless (AMN) and cubilin (CUBN) genes (Imerslund-Grasbeck syndrome), and mutations in the transcobalamin 2 cDNA. These disorders require long-term parenteral cobalamin treatment: intramuscular injections of hydroxycobalamin 1 mg daily for 10 days and then once a month. High-dose substitution with oral cyanocobalamin (1 mg biweekly) does not seem to be sufficient for all patients with congenital cobalamin deficiency.
Folate is an essential vitamin required to synthesize methionine from homocysteine. It is found mainly in green leafy vegetables, legumes, and oranges. It is converted to 5-methyltetrahydrofolate (5MTHF) after its uptake by enterocytes. Secondary folate deficiency is caused by insufficient folate intake, villous atrophy (e.g., celiac disease, IBD), treatment with phenytoin, and trimethoprim among others (Chapter 448.1). Several inherited disorders of folate metabolism and transport have been described.
Hereditary folate malabsorption is characterized by a defect of the proton-coupled folate transporter (PCFT, formerly reported to be HCP1, a heme carrier) of the brush border, leading to impaired absorption of folate in the upper small intestine as well as impaired transport of folate into the central nervous system. Mutations of the reduced folate carrier (RFC1, SLC19A1) have not been found in this entity. Sulfasalazine and methotrexate are potent inhibitors of PCFT. Symptoms of congenital folate malabsorption are diarrhea, failure to thrive, megaloblastic anemia (in the 1st few months of life), glossitis, infections (Pneumocystis jirovecii) with hypoimmunoglobulinemia, and neurologic abnormalities (seizures, mental retardation, and basal ganglia calcifications). Macrocytosis, with or without neutropenia, multilobulated polymorphonuclear cells, increased LDH and bilirubin, increased saturation of transferrin, and decreased cholesterol can be found. Low levels of folate are present in serum and cerebrospinal fluid. Plasma homocysteine concentrations as well as urine excretion of formiminoglutamic acid and orotic acid are elevated. Long-lasting deficiency is best documented using red cell folate. Therapy involves large doses of oral (up to 100 mg/day) or systemic (intrathecal) folate.
The molecular basis of intestinal transport of other water-soluble vitamins such as vitamin C (Na+-dependent vitamin C transporter, SVCT1 and SVCT2), pyridoxine/vitamin B 6, and biotin/vitamin B 5 (Na+-dependent multivitamin transporter, SMVT) have been described; however congenital defects of these transporter systems have not yet been found in humans. A thiamine/vitamin B 1-responsive megaloblastic anemia (TRMA) syndrome, which is associated with early-onset type 1 diabetes mellitus and sensorineural deafness, is caused by mutations of the thiamine transporter protein, THTR-1 (SLC19A2), present in the brush border.
Disorders of Electrolyte and Mineral Absorption
Congenital chloride diarrhea belongs to the more common causes of severe congenital diarrhea, with prevalence in Finland of 1: 20,000. It is caused by a defect of the SLC26A3 gene, which encodes a Na+-independent Cl−/HCO 3− exchanger within the apical membrane of ileal and colonic epithelium. Founder mutations have been described in Finnish, Polish, and Arab patients: V317del, I675-676ins, and G187X, respectively. The Cl−/HCO 3− exchanger absorbs chloride originating from gastric acid and the cystic fibrosis transmembrane conductance regulator (CFTR) and secretes bicarbonate into the lumen, neutralizing the acidity of gastric secretion.
Prenatal clinical signs of this disorder are a dilated small bowel that can mislead to a diagnosis of intestinal obstruction. Newborns with congenital chloride diarrhea present with severe life-threatening secretory diarrhea during the 1st weeks of life. Laboratory findings are metabolic alkalosis, hypochloremia, hypokalemia, and hyponatremia (with high plasma renin and aldosterone activities). Fecal chloride concentrations are >90 mmol/L and exceed the sum of fecal sodium and potassium. Early diagnosis and aggressive lifelong enteral substitution of KCl in combination with NaCl (chloride doses of 6-8 mmol/kg/day for infants and 3-4 mmol/kg/day for older patients) prevent mortality and long-term complications (such as urinary infections, hyperuricemia with renal calcifications, renal insufficiency, and hypertension) and allow normal growth and development. Orally administered proton pump inhibitors, cholestyramine, and butyrate can reduce the severity of diarrhea. The diarrheal symptoms usually tend to regress with age. However, febrile diseases are likely to exacerbate symptoms as a consequence of severe dehydration and electrolyte imbalances. (SeeChapter 52 for fluid and electrolyte management.)
The classic form of congenital sodium diarrhea manifests with polyhydramnios, massive secretory diarrhea, severe metabolic acidosis, alkaline stools (fecal pH >7.5) and hyponatremia as a result of fecal losses of Na+ (fecal Na+>70 mmol/L). Urinary secretion of sodium is low to normal. There is partial villous atrophy. The molecular genetic defect could not be located in the Na+-H+ exchangers (NHEs), which were thought to be impaired because they seem to be mainly responsible for Na+ absorption in the small intestine. In addition, a syndromic form of congenital sodium diarrhea with choanal or anal atresia, hypertelorism, and corneal erosions has been related to mutations of SPINT2, encoding a serine-protease inhibitor, whose pathophysiologic action on intestinal Na+ absorption is unclear. Some patients can be weaned from parenteral nutrition later in childhood but depend on oral sodium citrate supplementation.
The congenital form of acrodermatitis enteropathica manifests with severe deficiency of body zinc soon after birth in bottle-fed children or after weaning from breastfeeding. Clinical signs of this disorder are anorexia, diarrhea, failure to thrive, humoral and cell-mediated immunodeficiency (poor wound healing, recurrent infections), male hypogonadism, skin lesions (vesicobullous dermatitis on the extremities and perirectal, perigenital, and perioral regions, and alopecia), and neurologic abnormalities (tremor, apathy, depression, irritability, nystagmus, photophobia, night blindness, and hypogeusia). The genetic defect of acrodermatitis enteropathica is caused by a mutation in the Zrt-Irt-lik protein 4 (ZIP4, SLC39A4), normally expressed on the apical membrane, which enables the uptake of zinc into the cytosol of enterocytes. The zinc-dependent alkaline phosphatase and plasma zinc levels are low. Paneth cells in the crypt of the small intestinal mucosa show inclusion bodies. Acrodermatitis enteropathica requires long-term treatment with elemental zinc 1 mg/kg/day. Maternal zinc deficiency impairs embryonic, fetal, and postnatal development. Acquired forms of zinc deficiency are described inChapter 51.
Menkes disease and occipital horn syndrome are both caused by mutations in the gene encoding Cu 2+ transporting ATPase, alpha polypeptide (ATP7A), also called Menkes or MNK protein. ATP7A is mainly expressed by enterocytes, placental cells, and CNS and is localized in the trans-Golgi network for copper transfer to enzymes in the secretory pathway or to endosomes to facilitate copper efflux. Copper values in liver and brain are low in contrast to an increase in mucosal cells, including enterocytes and fibroblasts. Plasma copper and ceruloplasmin levels decline postnatally. Clinical features of Menkes disease are progressive cerebral degeneration (convulsions), feeding difficulties, failure to thrive, hypothermia, apnea, infections (urinary tract), peculiar facies, hair abnormalities (kinky hair), hypopigmentation, bone changes, and cutis laxa. Patients with the classic form of Menkes disease usually die before the age of 3 yr. A therapeutic trial with copper-histidinate should start before the age of 6 wk. In contrast to Menkes disease, occipital horn syndrome usually manifests during adolescence with borderline intelligence, craniofacial abnormalities, skeletal dysplasia (short clavicles, pectus excavatum, genu valgum), connective tissue abnormalities, chronic diarrhea, orthostatic hypotension, obstructive uropathy, and osteoporosis. It should be differentiated from Ehlers-Danlos syndrome type V.
Active calcium absorption is mediated by the transient receptor potential channel 6 (TRPV6) at the brush border membrane, calbindin, and the CaATPase, or the Na+-Ca++ exchanger for calcium efflux at the basolateral membrane within the proximal small bowel. A congenital defect of these transporters has not yet been described.
Intestinal absorption of dietary magnesium, which occurs via the transient receptor potential channel TRPM6 at the apical membrane, is impaired in familial hypomagnesemia with secondary hypocalcemia, which manifests with neonatal seizures and tetany.
Intestinal iron absorption consists of several complex regulated processes starting with the uptake of heme-containing iron by heme carrier protein 1 (HCP1) and Fe 2+ (after luminal reduction of oxidized Fe 3+) by the divalent metal transporter 1 (DMT1) at the apical membrane, followed by the efflux of Fe 2+ by ferroportin 1 (also called iron-regulated transporter[ IREG1]) at the basolateral membrane of duodenal enterocytes. Mutations of the ferroportin 1 gene have been found in the autosomal dominant form of hemochromatosis type 4. Mutations of HFE (Cys282Tyr, His63Asn, Ser65Cys) of classic hemochromatosis reduce the endocytic uptake of diferric transferrin by the transferrin receptor-1 (TfR1) at the basolateral membrane of the intestinal epithelium. Hepcidin antimicrobial peptide (HAMP) encodes hepcidin, a hepatic peptide hormone, which inhibits the efflux of iron through ferroportin and can be induced by IL-6. It is the defective gene of juvenile hemochromatosis (type 2, subtype B).
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Heinz-Erian P, Muller T, Krabichler B, et al. Mutations in SPINT2 cause a syndromic form of congenital sodium diarrhea. Am J Hum Genet. 2009;84:188-196.
Hihnala S, Hoglund P, Lammi L, et al. Long-term clinical outcome in patients with congenital chloride diarrhea. J Pediatr Gastroenterol Nutr. 2006;42:369-375.
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330.12 Malabsorption in Eosinophilic Gastroenteritis
Ernest G. Seidman,David Branski
The diagnosis of eosinophilic gastroenteritis is based on GI symptoms, GI eosinophilic infiltrates, and no demonstrable cause of the eosinophilia such as parasitic infection (most commonly Enterobius vermicularis in children ) or a specific allergic response. Peripheral eosinophilia is variable and not uniformly considered a criterion for diagnosis. The majority (50-70%) of patients have a history of other allergic disorders, and others might have associated connective tissue diseases. Approximately 10% of patients with this disorder have an immediate family member with this disorder as well, suggesting that eosinophilic GI disorders stem from a genetic predisposition, common environmental factors, or, most likely, a combination. Hypersensitivity to specific food allergens has been postulated as an etiologic factor. Symptoms depend on the severity and location of eosinophilic inflammation. Any region or layer (mucosa, submucosa, and serosa) of the gut may be involved, alone or in combination.
Diagnosis requires panendoscopy and biopsies in combination with other imaging diagnostic procedures. Eosinophilic infiltrates dominate the histologic findings, and signs of other inflammatory diseases are absent; in particular, the crypt architecture remains normal, no parasites are identified, and no eggs or larvae are seen. An increase in mast cells and IgE-containing plasma cells may be observed. The mucosal layer might be involved focally, or it might not be involved at all, in which case mucosal biopsy will fail to establish the diagnosis. Small bowel capsule endoscopy is useful in that it characteristically reveals mucosal erythema with marked focal villous atrophy. The most common sites of involvement are the stomach and small intestine.
Diarrhea and subsequent malabsorption can occur if small bowel involvement with villous blunting is extensive. Eosinophilic gastroenteritis can cause abdominal pain and excessive gas, weight loss, and failure to thrive. Other than peripheral eosinophilia, hypoalbuminemia due to protein-losing enteropathy and iron deficiency anemia are the more common laboratory findings.
Nutritional exclusion (or elemental) diets and corticosteroids are the mainstay of treatment. Less well documented treatments such as mast cell stabilizers and leukotriene antagonists have been used in small, uncontrolled trials. Clinical trials using biological modalities such as monoclonal anti-IgE (omalizumab) and anti-IL-5 (SCH55700/reslizumab and mepolizumab) are anticipated for severe cases.
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330.13 Malabsorption in Inflammatory Bowel Disease
Ernest G. Seidman,David Branski
Crohn disease and ulcerative colitis represent the 2 forms of chronic, immune-mediated IBD that commonly affect pediatric patients (Chapter 328). Because the small bowel is involved in the majority of pediatric Crohn disease patients, malabsorption of nutrients is far more of a problem than in ulcerative colitis. At the time of diagnosis, significant weight loss is observed in up to 85% of pediatric patients with Crohn disease and in about 65% with ulcerative colitis, due to inadequate intake of energy and micronutrients as well as diarrhea and malabsorption. Consequently, growth failure due to chronic undernutrition is far more common in Crohn disease than in ulcerative colitis, affecting up to 40% of cases.
In addition to malabsorption, energy intake is lower in patients with Crohn disease compared to healthy controls, in part due to lesser appetite. Excessive levels of proinflammatory cytokines have been implicated in causing the anorexia as well as in mediating impaired growth. Affected children often have a lower desire to eat, because symptoms, including abdominal pain, nausea, vomiting, and diarrhea, can lead to reduced food intake. This can, in turn negatively affect nutritional status during a child’s critical period of growth and development. Patients with IBD are also at risk of developing nutritional deficiencies because of restrictive diets imposed by caregivers or by the patients themselves.
In children with active disease, inadequate intakes of energy and of a number of micronutrients have been observed. Reduced energy intake during active disease can contribute to poor weight gain and impaired growth. Patients with IBD, particularly Crohn disease, often have multiple nutritional deficiencies and are in negative nitrogen balance, due to decreased intake and malabsorption of macro- and micronutrients. Quantifying nutrient intake, determining micronutrient deficiencies, and ascertaining requirements for nutritional supplementation are essential components of successful management in pediatric IBD.
Optimizing nutritional status and growth are key priorities in the management of IBD in children and adolescents. Energy intake should meet the added costs of catch-up growth and are usually in the range of 40-70 kcal/kg ideal body weight per day. Protein requirements are higher in Crohn disease (1-1.5 g/kg/day). Bone mineral density deficit is common, even in pediatric patients who have not been exposed to systemic corticosteroid therapy. Osteoporosis or osteopenia is best assessed by bone densitometry, and levels of vitamin 25-hydroxyvitamin D should be monitored. Other micronutrient deficiencies that result from inadequate intake, malabsorption, and gut losses are shown inTable 330-10.
Enteral nutrition support is favored over parenteral for all but Crohn disease patients with extreme short gut. Patients requiring hospitalization for a severe relapse should receive nutrition support if they are already malnourished or their intake is likely to be severely curtailed for ≥1 wk. Preoperative nutrition support is essential to the prevention of morbidity and mortality. However, clinicians must be aware of the risk of the refeeding syndrome in patients with severe malnutrition. In ulcerative colitis, nutrition support is adjunctive therapy; there is no evidence that bowel rest or TPN influences the outcome of severe ulcerative colitis.
Table 330-10COMMON MICRONUTRIENT DEFICIENCIES IN INFLAMMATORY BOWEL DISEASE
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12936 | https://www.ime.usp.br/~oliveira/POSTOWARDLAW.pdf | Row Rank Equals Column Rank Author(s): William P. Wardlaw Reviewed work(s): Source: Mathematics Magazine, Vol. 78, No. 4 (Oct., 2005), pp. 316-318 Published by: Mathematical Association of America Stable URL: .
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316 MATHEMATICS MAGAZINE Proof Refer to FIGURE 3. We find that there are twelve pairs of triangles, each pair of which has a common side or a common angle (or two congruent angles one from each triangle of the pair). Applying P1 and P2 on these triangles we have AM Thus, if we let IA = a, IC = c, IM = m, IN = n, we get a-m which simplifies to 1 Hence a butterfly inscribed in a quadrilateral satisfies the same relation (1) as a butterfly inscribed in a circle. Equivalently, the conclusion of the theorem indicates that the ratio of the ratios, (AM/IM)/(CN/IN), is the same as the ratio IA/IC, or that the harmonic mean of IC and IM equals the harmonic mean of IA and IN. In either case, if IC = IA, we have IM = IN thereby the analog of the usual butterfly theorem for quadrilaterals. Acknowledgment. The author would like to thank the referees for their helpful suggestions, and Joseph Kung for preparation of the article. REFERENCE 1. Leon Bankoff, The metamorphosis of the butterfly problem, this MAGAZINE 60 (1987), 195-210. Row Rank Equals Column Rank WILLIAM P. WARDLAW U. S. Naval Academy Annapolis, MD 21402-5000 wpw@usna.edu Dedicated to George Mackiw, a good friend and an excellent mathematical expositor The purpose of this note is to present a short (perhaps shortest?) proof that the row rank of a matrix is equal to its column rank. The proof is elementary and accessi- ble to students in a beginning linear algebra course. It requires only the definition of VOL. 78, NO. 4, OCTOBER 2005 317 matrix multiplication and the fact that a minimal spanning set is a basis. It differs in approach from proofs given in textbooks as well as from some interesting proofs in MAA journals [1, 2]. And, unlike the latter, this proof is valid over any field of scalars. But first, recall that if the m x n matrix A = BC is a product of the m x r matrix B and the r x n matrix C, then it follows from the definition of matrix multiplication that the ith row of A is a linear combination of the r rows of C with coefficients from the ith row of B, and the jth column of A is a linear combination of the r columns of B with coefficients from the jth column of C. (If you have trouble understanding this or the next paragraph, you should construct several examples of small matrix products, say, a 3 x 2 times a 2 x 3 matrix, etc., with small integer as well as symbolic entries.) On the other hand, if any collection of r row vectors c1, spans the row space of A, an r x n matrix C can be formed by taking these vectors as its rows. Then the ith row of A is a linear combination of the rows of C, say bil1- birCr. This means A = BC, where B = (bij) is the m x r matrix whose ith row, bir), is formed from the coefficients giving the ith row of A as a linear combination of the r rows of C. Similarly, if any r column vectors span the column space of A, and B is the m x r matrix formed by these columns, then the r x n matrix C formed from the appropriate coefficients satisfies A = BC. Now the four sentence proof. THEOREM. If A is an m x n matrix, then the row rank of A is equal to the column rank of A. Proof If A = 0, then the row and column rank of A are both 0; otherwise, let r be the smallest positive integer such that there is an m x r matrix B and an r x n matrix C satisfying A = BC. Thus the r rows of C form a minimal spanning set of the row space of A and the r columns of B form a minimal spanning set of the column space of A. Hence, row and column ranks are both r. E Several other properties of the rank of a matrix over a field are also very easy to ob- tain. The factorizations A = ImA = AIn show that r < m and r < n, which proves that the rank of A is less than or equal to the number of rows and the number of columns of A. Since A = BC implies AT = CT BT, the transpose clearly has the same rank as the original matrix. Since A = BC and D - EF implies AD = B(CD) = (AE)F, the rank of AD must be less than or equal to the rank of A and to the rank of D. These concepts suggest the following definition [5, p. 123]: DEFINITION. Let R be a commutative ring with identity and let A be an m x n matrix over R. Then the spanning rank of A is 0 if A = 0 and otherwise is the smallest positive integer r such that there is an m x r matrix B and an r x n matrix C satisfying A = BC. This definition is one way of extending the notion of rank to matrices over commu- tative rings. Even if the ring has no identity, it can be embedded in a ring with identity so that the definition can be used. Care must be taken in considering rank over commu- tative rings, because several different extensions of the definitions over a field can give different results over rings, even though they all give the standard concept of rank over a field. Nonetheless, if the above definition is used, matrices over rings automatically have row rank equal to column rank, have rank less than or equal to the number of rows and the number of columns, the rank of the transpose is equal to the rank of the matrix, and the rank of a product is less than or equal to the rank of either factor. Another application of the spanning rank, first used by the author in a problem and later a Note in the MAGAZINE, is the proof that a matrix over a commutative 318 MATHEMATICS MAGAZINE ring with spanning rank r satisfies a polynomial equation of degree at most r + 1. For if A = BC is an n x n matrix of spanning rank r, then D = CB is an r x r matrix with characteristic polynomial fD (x) = det(xI - D) of degree r and fD(D) = 0 fol- lows from the Cayley-Hamilton Theorem. (The author has shown that the Cayley- Hamilton Theorem holds for matrices over commutative rings.) Thus there is a poly- nomial m(x) of smallest positive degree such that m(D) = 0. Then p(x) = xm(x) is a polynomial of degree 1 such that p(A) = Am(A) = BCm(BC) = Bm(CB)C = Bm(D)C = 0. REFERENCES 1. H. Liebeck, A proof of the equality of column and row rank of a matrix, Amer Math. Monthly 73 (1966), 1114. 2. G. Mackiw, A note on the equality of column and row rank of a matrix, this MAGAZINE 68 (1995), 285-286. 3. W. Wardlaw, problem 1179, this MAGAZINE 56 (1983), 326, and solution 1179, this MAGAZINE 57 (1984), 303. 4. - A transfer device for matrix theorems, this MAGAZINE 59 (1986), 30-33. 5. - Minimum and characteristic polynomials of low-rank matrices, this MAGAZINE 68 (1995), 122-127. A Modern Approach to a Medieval Problem AWANI KUMA R, Director Lucknow Zoological Garden Lucknow 226001 INDIA awanieva@eth.net The following problem from Lilavati , a mathematical treatise written by Bhaskaracharya, a 12th-century Indian mathematician and astronomer, deserves a modem approach: A snake's hole is at the foot of pillar, nine cubits high, and a peacock is perched on its summit. Seeing a snake at the distance of thrice the pillar gliding towards his hole, he pounces obliquely upon him. Say quickly at how many cubits from the snake's hole they meet, both proceeding an equal distance. Since both proceed an equal distance, it is reasonable to assume that their speeds are equal. Readers are invited to solve this problem before proceeding. Assuming that the peacock flies along the hypotenuse of a right-angled triangle and knows the Pythagorean Theorem, it will grab the snake at a distance of 12 cubits from the pillar. Practically, however, such a thing does not happen. Why should a peacock know-a priori-to fly along the hypotenuse of a right-angled triangle having a base of 12 cubits? A more peacock-like behavior would be to keep an eagle eye on the snake and change its direction at every instant, always aiming toward the snake. This type of pursuit problem has a history of over five hundred years. However, this particular problem is a bit different from most. Instead of the prey running away from the predator, here prey and the predator are moving toward each other. Even so, the results are startling. Although the reader may have seen similar problems, I offer a general analysis. We assume that the snake and the peacock move at different, but constant, speeds: the snake in a straight line toward its hole and the peacock along a curve, changing its direction at every instant so as to be flying directly toward the snake. |
12937 | https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-6a/v/derivative-properties-example | Basic derivative rules: table (video) | Khan Academy
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AP®︎/College Calculus AB
Course: AP®︎/College Calculus AB>Unit 2
Lesson 6: Derivative rules: constant, sum, difference, and constant multiple: introduction
Basic derivative rules
Basic derivative rules: find the error
Basic derivative rules: find the error
Basic derivative rules: table
Basic derivative rules: table
Justifying the basic derivative rules
Math>
AP®︎/College Calculus AB>
Differentiation: definition and basic derivative rules>
Derivative rules: constant, sum, difference, and constant multiple: introduction
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Basic derivative rules: table
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Let's explore a problem involving two functions, f and g, and their derivatives at specific points. Our goal is to find the derivative of a new function, h(x), which is a combination of these functions: 3f(x)+2g(x). By applying basic derivative rules, we determine the derivative—and thus the slope of the tangent line—of h(x) at x = 9.Created by Sal Khan.
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Chirag Jindal 9 years ago Posted 9 years ago. Direct link to Chirag Jindal's post “We are given that h(x)=3f...” more We are given that h(x)=3f(x)+2g(x) also x=9. Now if i calculate h(9). It comes out to be 3f(9)+2g(9) which is equal to 3+18=21.
Then differentiating h(x) we get h'(x) as 0 because differentiation of a constant is 0.
Please help me out. :)
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Arnab Chowdhury 10 years ago Posted 10 years ago. Direct link to Arnab Chowdhury's post “How did Sal estimate that...” more How did Sal estimate that the slope of 2-x is -1?
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robshowsides 10 years ago Posted 10 years ago. Direct link to robshowsides's post “y = 2 - x is a straight l...” more y = 2 - x is a straight line. Any straight line with equation y = mx + b has a slope m. That is, the slope is the coefficient of x. So in y = 2 - x, the coefficient of x is -1, so that's the slope of that graph.
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Hasan 11 years ago Posted 11 years ago. Direct link to Hasan's post “How else could we find wh...” more How else could we find what is g'(9) except using a graph and with using knowledge that we have learned so far?
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Joshua LeClerg 6 years ago Posted 6 years ago. Direct link to Joshua LeClerg's post “Is there a video covering...” more Is there a video covering how to derive an absolute value function on here that I'm just not seeing? Like, I see this and I think I understand, but my professor expects us to show all of our derivatives algebraically, so I'm trying to find a resource to help me learn how to do it (the course is online so I don't have a lecture, but exams are in person). Thanks!
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kubleeka 6 years ago Posted 6 years ago. Direct link to kubleeka's post “The absolute value functi...” more The absolute value function is just f(x)=x for positive inputs, and f(x)=-x for negative inputs. You know how to differentiate both of those functions, so differentiating and absolute value function is just a matter of finding the positive/negative regions of the input.
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handsomemoosekip a year ago Posted a year ago. Direct link to handsomemoosekip's post “In previous videos, Sal s...” more In previous videos, Sal says that equations with sharp turns aren't differentiable as they have opposite slopes when approaching zero, but here he takes an absolute value equation and breaks it into pieces like the first thing I said isn't true. Where is the disconnect?
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Venkata a year ago Posted a year ago. Direct link to Venkata's post “Being able to break up a ...” more Being able to break up a function into two parts doesn't disturb the fact that it is not differentiable at a point. We broke the function up because it is piecewise. However, when we consider the derivative, we still agree that it isn't defined at x = 0.
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Sudarshan 11 years ago Posted 11 years ago. Direct link to Sudarshan's post “What is dy/dx? Is I encou...” more What is dy/dx? Is I encountered this in a physics textbook in a section for acceleration and was just wondering...
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Barrett Southworth 11 years ago Posted 11 years ago. Direct link to Barrett Southworth's post “dy/dx is the derivative o...” more dy/dx is the derivative of y with respect to x.
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lance.redpath 9 months ago Posted 9 months ago. Direct link to lance.redpath's post “My understanding is that ...” more My understanding is that d/dx is Leibniz's notation and f'(x) is Lagrange's notation. Now on a pure understanding level, it appears fine to use both interchangeably. However, for exams etc is there an issue using both interchangeably, or should you stick to one?
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Venkata 9 months ago Posted 9 months ago. Direct link to Venkata's post “You can use them intercha...” more You can use them interchangeably. One notation might be better than the other in some cases, but there's no harm in using either.
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jo123 2 years ago Posted 2 years ago. Direct link to jo123's post “i thought you can't find ...” more i thought you can't find the derivative of an absolute value function. sal still did it anyways. what's the explanation for that?
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Tyler Thibeau 2 years ago Posted 2 years ago. Direct link to Tyler Thibeau's post “The absolute value functi...” more The absolute value function is not differentiable at the vertex, but has a definite slope everywhere else.
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Yellow Shiƒt» 9 years ago Posted 9 years ago. Direct link to Yellow Shiƒt»'s post “At 6:40, why is the deriv...” more At 6:40 , why is the derivative of g(x) x - 1 plus one? Isn't the derivative of a constant just 0?
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cherry a year ago Posted a year ago. Direct link to cherry's post “Isn’t the derivative of a...” more Isn’t the derivative of a constant always 0? So how come g’(9) = 1?
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Jerry Nilsson a year ago Posted a year ago. Direct link to Jerry Nilsson's post “𝑔ʹ(9) is not the derivat...” more 𝑔ʹ(9) is not the derivative of 𝑔(9).
𝑔ʹ(𝑥) is the derivative of 𝑔(𝑥) and 𝑔ʹ(9) is that derivative evaluated at 𝑥 = 9.
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Video transcript
Voiceover: We've been given some interesting information here about the functions f, g, and h. For f, they tell us for given values of x what f of x is equal to and what f prime of x is equal to. Then they defined g of x for us in terms of this kind of absolute value expression. Then they define h of x for us, in terms of both f of x and g of x. What we're curious about is what is the derivative with respect to x, of h of x at x is equal to nine. I encourage you to pause this video and think about it on your own before I work through it. Let's think about it a little bit. Another way just to get familiar with the notation of writing this, the derivative of h of x with respect to x at x equals nine. This is equivalent to h, we need that blue color, it is equivalent to h prime and the prime signifies that we're taking the derivative. H prime of x, when x equals nine so h prime of nine is what this really is. Actually I'm going to do this in a different color. This is h prime of nine. Let's think about what that is. Let's take the derivative of both sides of this expression to figure out what the derivative with respect to x of h is. We get a derivative, I'll do that same white color. A derivative with respect to x of h of x is going to be equal to the derivative with respect to x of all of this business. I could actually just, well I'll just rewrite it. Three times f of x, plus two times g of x. Now this right over here, the derivative of the sum of two terms that's going to be the same thing as the sum of the derivatives of each of the terms. This is going to be the same thing as the derivative with respect to x of three times, I'll write that a little bit neater. Three times f of x, plus the derivative with respect to x of two times g of x. Now the derivative of a number or I guess you could say a scaling factor times a function. The derivative of a scalar times the function is the same thing as a scalar times the derivative of the function. What does that mean? Well that just means that this first term right over here that's going to be equivalent to three times the derivative with respect to x of f, of our f of x, plus this part over here is the same thing as two. Okay, make sure I don't run out of space here, plus two times the derivative with respect to x. The derivative with respect to x of g of x. Derivative of h with respect to x is equal to three times the derivative of f with respect to x, plus two times the derivative of g with respect to x. If we want to write it in this kind of prime notation here, we could rewrite it as h prime of x is equal to three times f prime of x, so this part right over here that is the same thing as f prime of x. It's three times f prime of x, plus two times g prime of x. Once you are more fluent with this property, the derivative of the sum of two things is the sum of the derivatives. The derivative of a scalar times something is the same thing as a scalar times the derivative of that something. You really could have gone straight from here to here, pretty quickly. Now why is this interesting, well now we can evaluate this function when x is equal to nine. H prime of nine is the same thing as three times f prime of nine, plus two times g prime of nine. Now what is f prime of nine? The derivative of our function f when x is equal to nine. Well they tell us, when x is equal to nine, f of nine is one but more importantly f prime of nine is three. This part right over here evaluates that part's three. What's g prime of nine? Let's look at this function a little bit more closely. There's a couple of ways we could think about it. Actually let's try to graph it, now I think that could be interesting. Just to visualize what's going on here. Let's say that's our y-axis and do this right over here is our x-axis. Now when does an absolute value function like this, when is this going to hit a minimum point? Well the absolute value of something is always going to be non-negative. It hits a minimum point when this thing is equal to zero. Well when is this thing equal to zero? When x equals one, this thing is equal to zero. We hit a minimum point when x is equal to one, and when x equals one, this term is zero absolute value of zero, zero. G of one is one. We have this point right over there. Now what happens after that? What happens for x greater than one? Actually let me write this down. G of x is equal to, and in general whenever you have an absolute value, a relatively simple absolute value function like this you could think of it, you could break it up into two function or you could think about this function over different intervals when the absolute value is non-negative and when the absolute value is negative. When the absolute value is non-negative that's when x is greater than or equal to zero. When the absolute value is non-negative, if you're taking the absolute value of a non-negative number that is just going to be itself. The absolute value of zero, zero. Absolute value of one is one. The absolute value of a hundred is a hundred. Then you could ignore the absolute value for x is greater than or equal to, not greater than or equal to zero, for x is greater than or equal to one. X is greater than or equal to one, this thing right over here is non-negative. It will just evaluate to x minus one. This is going to be x minus one plus one. Which is the same thing as just x, minus one plus one, they just cancel out. Now when this term right over here is negative and that's going to happen for x is less than one. Well then the absolute value is going to be the opposite of it. You give me the absolute value of a negative number that's going to be the opposite. Absolute value of negative eight is positive eight. It's going to be that the negative of x minus one is one minus x, plus one. Or we could say two minus x. For x is greater or equal to one, we would look at this expression, now what's the slope of that? Well the slope of that is one. We're going to have a curve that looks like or a line I guess we could say that looks like this. For all x is greater than or equal to one. The important thing, remember, we're going to think about the slope of the tangent line when we think about the derivative of g. Slope is equal to one. For x less than one or our slope now, if we look right over here our slope is negative one. It's going to look like this. It's going to look like that. For the pointing question, if we're thinking about g prime of nine so nine is some place out here, so what is g prime of nine? G prime of nine, let me make it clear, this graph right over here, this is the graph of g of x or we could say y, this is the graph y equals g of x. Y is equal to g of x. What is g prime of nine? Well that's the slope when x is equal to nine. The slope is going to be equal to one. G prime of nine is one. What does this evaluate to? This is going to be three times three, so this part right over here is nine plus two times one, plus two, which is equal to 11. The slope of the tangent line of h when x is equal to nine is 11.
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12938 | https://www.themeasureofthings.com/correction.php?comp=distance&unit=in&amt=30&i=412 | Correction for the length of a Red Blood Cell | The Measure of Things
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Correction for the length of a Red Blood Cell
Thanks for your input! Let us know more about what's wrong in the form below.
How long is 30 inches?
It's about 100,000 times as long as a Red Blood Cell
The length of a Red Blood Cell is about 0.0003 inches.
(a.k.a. RBCs, a.k.a. haematids, a.k.a. erythrocyte, a.k.a. erythroid cells, a.k.a. red blood corpuscles)
An average Red Blood Cell measures 7.5 µm (micrometers) in diameter. An average human has 20 - 30 trillion red blood cells in their body and each cell completes a circulatory lap in about 20 seconds.
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12939 | https://tentzeris.ece.gatech.edu/appphys23_lynch.pdf | 1 Super Realized Gain Antenna Array Donal Patrick Lynch, Student Member, IEEE Manos M. Tentzeris, Fellow, IEEE Vincent Fusco, Fellow, IEEE and Stylianos D. Assimonis Member, IEEE Abstract In this study, we investigate and fabricate a superdirective antenna array composed of strip dipole elements operating at a frequency of 3.5 GHz. The spacing, dimensions, and phase difference of the elements are optimized to achieve a super realized gain antenna with a theoretical efficiency of 98.8% and computed efficiency of 99.3%. By employing an element spacing of 0.2λ, the end-fire antenna array demonstrates a maximum theoretical realized gain of 6.4 dBi, and a maximum computed realized gain of 6.3 dBi. Significantly, our proposed superdirective antenna array distinguishes itself from existing approaches by achieving high directivity, high radiation efficiency, and impedance matching to 50 Ωsolely through careful adjustments in the strip dimensions and the inter-element phase. This eliminates the need for additional impedance matching networks, amplifiers, or attenuators.
Index Terms Antennas, Antenna arrays, Microstrip antenna arrays, Dipole antennas, Directive antennas, Superdirective antenna arrays I. INTRODUCTION T HE advent of modern mobile communication networks, including fifth-generation (5G), sixth-generation (6G), and beyond, has been driven by the ever-increasing demand for faster download speeds and low latency, enabling seamless connectivity to work and social digital platforms. With the exponential growth of data-intensive applications and the need for reliable connectivity, these systems have become a necessary response to meet the evolving requirements of today’s digital society.
One of the critical aspects of these advanced communication technologies is the deployment of efficient and advanced antenna systems , to support the enhanced capabilities of these networks.
In the context of 5G , which is currently the most prevalent mobile communication technology, the frequency band most widely used for applications falls within the range of 3.3 to 4.2 GHz. The selection of the sub-6 GHz range for 5G deployment is driven by the desire to strike a balance between coverage and capacity. However, a notable challenge faced by 5G antenna systems is their relatively large size, which gives rise to intricate and complex geometries. These larger and more complex antenna systems pose difficulties in the fabrication process, making their practical implementation more challenging. The increased complexity adds to the manufacturing complexity and costs, thereby necessitating innovative design and fabrication techniques to overcome these practical implementation challenges in deploying 5G antenna systems effectively.
To address this challenge, researchers and engineers have been exploring alternative solutions, and one promising approach is the use of superdirective antenna arrays (SDAs). SDAs offer compact geometries and higher directivity compared to traditional uniform antenna arrays. This increased directivity is achieved through the close placement of antenna elements, which results in strong coupling between them. Several studies, including the work of Uskov and others –, have demonstrated that a typical linear antenna array with N elements yields a maximum directivity of N 2 + 2N as the inter-element distance tends to zero. In general, this superdirectivity is higher than that of a corresponding uniform antenna array with the same number of elements. Furthermore, a superdirective antenna array is end-fire, meaning that the radiation pattern of the array is directed predominantly along the axis perpendicular to the array’s elements.
By leveraging SDAs in 5G systems, it becomes possible to enhance the received power and extend the communication distance. This, in turn, leads to improved power efficiency and ensures reliable connectivity, even in challenging environments.
The compact nature of SDAs also helps address the issue of large antenna size, enabling easier integration into various devices and infrastructure.
However, it is important to note that SDAs also face certain disadvantages and impediments to practical implementation.
One significant drawback is the presence of high ohmic losses, which reduce radiation efficiency and antenna gain. These losses limit the overall performance of the antenna system and affect its ability to efficiently transmit and receive signals.
Additionally, the impedance characteristics of SDA elements often exhibit high reactance, making it challenging to achieve impedance matching at the standard 50 Ω, thereby complicating the practical implementation of superdirective arrays.
In this paper, our first objective is to conduct a theoretical study on an antenna array consisting of two dipoles, with a focus on superdirectivity. This study can be readily extended to an antenna array consisting of N elements. In contrast to existing research, we prioritize the analysis of the realized gain rather than solely examining directivity or gain. We make this choice because the realized gain factor takes into account both ohmic and return losses. Additionally, we propose implementing Donal Patrick Lynch, Vincent Fusco and Stylianos D. Assimonis are with the Institute of Electronics Communications and Information Technology, Queen’s University Belfast, BT3 9DT Belfast, U.K. (e-mail: dlynch27@qub.ac.uk; v.fusco@ecit.qub.ac.uk; s.assimonis@qub.ac.uk).
Manos M. Tentzeris is with the Georgia Institute of Technology, School of Electrical and Computer Engineering, Atlanta, GA 30332 USA (e-mail: etentze@ece.gatech.edu) arXiv:2309.09889v1 [physics.app-ph] 18 Sep 2023 2 dipoles with slightly different lengths and radii (or widths), excited by signals of equal magnitude but different phases. This innovative approach allows us to achieve impedance matching of the antenna array elements to 50 Ω, eliminating the need for additional impedance matching networks, hence, this technique reduces ohmic losses and enhances radiation efficiency.
Moreover, by adopting this design concept, there is no need for active amplifiers or attenuators to regulate the magnitude of the excitation signal, resulting in a significant reduction in the antenna design process. Additionally, the power efficiency of the superdirective antenna system is enhanced by minimizing power consumption. As the next step, we conduct a comprehensive numerical analysis using full-electromagnetic simulation. Finally, we proceed to fabricate and measure the proposed antenna array with super realized gain. The measured results closely align with the simulated and theoretical findings.
II. ANTENNA DESIGN A. Theoretical Analysis When antennas are located in close proximity, the effect of mutual coupling between them cannot be ignored. The mutual impedance serves as an indicator of the degree to which antenna cross interaction proximity effects are present. This section presents the theoretical analysis of a two-element antenna array, as depicted in Fig. 1. The array consists of linear wire dipoles arranged in a parallel, side by side format along the ρ-axis at a distance d from each other, with centers at positions (xi, yi), lengths Li, radii ai, input voltages and currents Vi and Ii, respectively, where i = 1, 2. In order to evaluate the mutual and self impedances Zij of the antenna array, where i, j = 1, 2, we consider it as a two-port network. Therefore, in general, when both antennas are excited, the relationship between the driving voltages and input currents are expressed as follows : V1 V2 = Z11 Z12 Z21 Z22 I1 I2 ⇔v = Zn in (1) The impact of the first dipole on the second dipole is represented by the mutual impedance Z21, which is defined as : Z21 = V oc 2 I1 (2) Thus, the mutual impedance Z21 is defined as the ratio of the induced open-circuited voltage at the terminals of the second dipole when only the first dipole is driven, and vice versa for Z12. Please note that according to reciprocity, Z21 = Z12. The induced open-circuited voltage is given by : V oc 2 = −1 I2 Z l2 −l2 Ez (z) I2 (z) dz, (3) where l2 = L2/2 and Ez(z) is the electric field caused by the driven first dipole on the second dipole. To calculate Ez(z), we need to define the currents that flow through the dipoles. We assume that the dipoles have lengths close to half wavelength, so we can use a sinusoidal current distribution in this analysis. Thus, we have: I2 (z) = I2 sin [k (l2 −|z|)] sin [k l2] , |z| ≤l2, (4) where I2 ∈C is the input current of the second dipole, k = 2π/λ is the wavenumber, and λ is the wavelength. Please note that I1(z) can also be given by (4) by setting I2 →I1 and l2 →l1 = L1/2. The electric field along the second antenna is given by : (5) Ez (z) = −j η0I1 4π sin [k l1] e −jkR(21) a R(21) a + e −jkR(21) b R(21) b −2 cos [k l1] e −jkR(21) c R(21) c !
, where η0 is the characteristic impedance of free space and z ∈[−l2, l2], and R(21) a = q d2 + (z −l1)2 R(21) b = q d2 + (z + l1)2 (6) R(21) c = p d2 + z2 By substituting (3)-(6) into (2): Z21 = j η0 4π sin [k l1] sin [k l2] Z l2 −l2 A21 (z) dz (7) where, (8) A21 (z) = e −jkR(21) a R(21) a + e −jkR(21) b R(21) b −2 cos [k l1] e −jkR(21) c R(21) c !
sin [k (l2 −|z|)] .
3 ρ z R(21) c E(z) R(21) b R(21) a I2(z) I1(z) L1 L2 2a1 2a2 d Fig. 1. An array of wire dipole antennas with an inter-element distance of d, lengths of elements L1 and L2, and radii of elements a1 and a2.
Note that the integral in (7) does not have an analytical solution. Therefore, we utilized numerical integration techniques, specifically global adaptive quadrature, to accurately evaluate the integral .
To obtain the near-field on the surface of the first dipole, we set d →a1 and l2 →l1 in (6) because the integral is now estimated on the first dipole, and not on the second. The resulting expression is: Z11 = j η0 4π sin2 [k l1] Z l1 −l1 A11 (z) dz, (9) where now, (10) A11 (z) = e −jkR(11) a R(11) a + e −jkR(11) b R(11) b −2 cos [k l1] e −jkR(11) c R(11) c !
sin [k (l1 −|z|)] and R(11) a = q a2 1 + (z −l1)2 R(11) b = q a2 1 + (z + l1)2 (11) R(11) c = q a2 1 + z2.
Similar analysis can be applied to estimate Z22.
With the given driven voltages v, solving equation (1) provides the input currents in, which are used to define the sinusoidal currents Ii(z) based on (4). Thus, in = Zn −1 v.
(12) where, ( )−1 denotes the inverse matrix.
By determining the currents in, the radiation pattern of the array can be obtained, and the radiation intensity can be expressed as: (13) U (in, θ, ϕ) = η0 8 π2 N X i=1 Ii cos [k li cos θ] −cos [k li] sin [k li] sin θ e j⃗ k· ⃗ di 2 where N is the number of antenna array elements (in this example N = 2 since we have two dipoles), ⃗ k = kˆ r is the wavevector, where ˆ r = sin θ cos ϕ ˆ x + sin θ sin ϕ ˆ y + cos θ ˆ z, (14) is the unit vector in spherical coordinates, and ⃗ di = xi, ˆ x + yi, ˆ y + zi, ˆ z is the vector that indicates the position of the dipoles.
Hence, the directivity is given by D ≜4π U (in, θ, ϕ) Pr , (15) where Pr ≜ Z 2π ϕ=0 Z π θ=0 U (in, θ, ϕ) sin θ dθdϕ, (16) 4 is the total radiated power. The radiated power represents a portion of the input power to the two-port system and is defined as: Pin ≜Pr + Pl, (17) where Pl represents the ohmic losses on the dipoles. In the calculation of directivity, it is assumed that there are no ohmic losses on the antenna array, and therefore all the input power is radiated. In this scenario (i.e., Pl →0), it can be presumed that Pin = Pr, as stated in : Pr = 1 2Re n in HZn in o = 1 2Re n Zn −1 v H v o , (18) where, ( )H denotes the Hermitian transpose.
On the other hand, when calculating the gain of an antenna array, it is essential to consider the ohmic losses associated with the wire dipoles. In general, the gain of an antenna is given by: G ≜4π U (il, θ, ϕ) Pin = 4π U (il, θ, ϕ) Pr + Pl , (19) where now Pl ̸= 0. Ohmic losses are a result of the skin effect . Based on this effect, we can derive the loss resistance per unit length on the i-th conductive wire dipole as: rl,i = 1 2ai r fµ0 πσ , (20) where f, µ0 = 4π ×10−7 H/m, and σ are the operating frequency, magnetic permeability of free space, and wire conductivity, respectively. Thus, for the current distribution of (4), Rl,i = rl,i Z li −li Ii (z) Ii 2 dz = kLi −sin [kLi] 4kai sin2 k Li 2 r fµ0 πσ .
(21) Additionally, the relationship between the driving voltages and the input currents is now expressed as follows: v = Zl il = (Zn + Rl) il, (22) where Rl = diag(Rl,1, . . . , Rl,N), and il is the matrix of the input currents of the lossy network. Hence, similarly to (12): il = (Zn + Rl)−1 v.
(23) The input power is now given by: Pin = 1 2Re n il H (Zn + Rl) il o = 1 2Re (Zn + Rl)−1 v H v .
(24) The definition of the directivity and gain of an antenna array incorporates the corresponding power density. The power density is determined by the input currents, which do not consider the ohmic losses on the radiating elements when estimating directivity, but do take into account the ohmic losses when estimating gain. Consequently, the power density for directivity (i.e., U(in, θo, ϕo)) differs from the power density for gain (i.e., U(il, θo, ϕo)). Additionally, the radiation efficiency, defined as the maximum gain divided by the maximum directivity, is represented by the following equation: η ≜G D = Umax (il, θo, ϕo) Pr Umax (in, θo, ϕo) Pin , (25) which incorporates the maximum radiation density for the lossless case (i.e., Umax(in, θo, ϕo)) and the lossy case (i.e., Umax(il, θo, ϕo)).
It is noted that in the literature, the radiation efficiency is often defined as the ratio of the radiated power to the input power, expressed as: η = Pr Pin .
(26) Equation (26) follows from equation (25), assuming an identical power density in both directivity and gain estimation.
However, it is crucial to note that power density depends on the input currents, which vary when estimating directivity and gain. The input currents account for ohmic losses in the gain estimation, but this consideration is absent in the directivity estimation. Consequently, this disparity leads to inaccuracies when using (26) to calculate radiation efficiency. For instance, 5 when considering the scenario where copper wires with lengths L1 = L2 = λ/2, radii a1 = a2 = λ/1001, are placed side-by-side at a distance d = λ/2, and driven by voltages V1 = V2 = 1 V, the calculated value of η based on (25) is 99%. On the other hand, applying (26) yields a different value of 100.94%, clearly demonstrating the inaccuracies caused by the application of this simplified formula.
Another important parameter is the realized gain, which is defined as the product of the port efficiency ηport and the gain, and thus: GR ≜ηport G, (27) where, ηport = 1 −|Γa|2 = vH I −SHS v vHv , (28) where Γa is the total active reflection coefficient , , S is the S-parameter matrix of the two-port network, calculated at the reference impedance of Z0 = 50 Ω, and I is an identity matrix with the same dimension as S.
The antenna array was optimized to achieve superdirectivity. The goal was to maximize the directivity, gain (which considers ohmic losses, equivalently radiation efficiency), and realized gain (which accounts for both radiation efficiency and return losses at 50 Ωin our case) by varying the inter-element distance from 0.1λ to 0.5λ, where λ is the operating frequency wavelength (assumed to be 3.5 GHz for sub-6 GHz 5G systems). The analysis is based on theoretical calculations and formulas, i.e., on (15), (19), and (27). Design parameters include the lengths (L1, L2), radii (a1, a2), and inter-element phase difference (∆ϕ).
In contrast to the state-of-the-art approach , –, we fixed the driven voltages’ magnitude at 1 V/m to avoid using additional components like amplifiers or attenuators.
Particle Swarm Optimization (PSO) was employed as the optimization method for the solution of the problem: Maximize {L1,L2,a1,a2,∆ϕ} f (L1, L2, a1, a2, ∆ϕ) subj. to: L1, L2 ∈[0.4λ, 0.6λ] , a1, a2 ∈[λ/2001, λ/201] , ∆ϕ ∈[0◦, 360◦] , (29) where, function f represents either the directivity (15), the gain (19) or the realised gain (27). Each inter-element distance had an optimal set of design parameters for maximum directivity, gain, and realized gain. The results in Fig. 2 reveal a significant increase in directivity as the inter-element distance approaches zero, indicating superdirectivity. The gain reaches its maximum when the inter-element distance is approximately 0.1λ, accounting for ohmic losses. Similarly, the realized gain, which considers return losses at 50 Ω, peaks at an inter-element distance of around 0.2λ (specifically at 0.17λ). In contrast, the directivity trend alone suggests enhancement as d tends to zero with appropriate excitation signals, this finding does not account for ohmic losses or return losses. Moreover, the directivity calculated at d = 0.2λ is 7.3 dBi. Additionally, the directivity estimated in for two isotropic elements is 3.5 when using linear scaling. In our specific scenario, considering dipoles of length close to half-wavelength and a theoretical maximum directivity of 1.67, the predicted directivity is 10 log10(1.67 · 3.5) ≈7.7 dBi, which closely aligns with our findings.
Fig. 3 depicts the optimal inter-element phase difference (∆ϕ) for achieving maximum directivity, gain, and realized gain as a function of inter-element distance. While ∆ϕ displays significant variation for realized gain, it remains approximately 200◦ for directivity and gain, even up to an inter-element distance of 0.4λ. This observation suggests that the directivity and gain are relatively insensitive to the phase setting.
TABLE I OPTIMUM DESIGN PARAMETERS: THEORETICAL ANALYSIS AT 3.5 GHZ d/λ L1/λ L2/λ a1/λ a2/λ ∆ϕ (deg.) ηport η (%) GR (dBi) 0.05 0.480 0.482 0.0050 0.005 345.8◦ 0.763 94.2 6.1 0.1 0.473 0.467 0.0050 0.005 320.8◦ 0.807 98.4 6.3 0.2 0.479 0.452 0.0015 0.002 239.3◦ 0.925 98.8 6.4 0.3 0.474 0.437 0.0009 0.005 208.5◦ 0.978 99.2 6.1 0.4 0.466 0.440 0.0012 0.005 193.5◦ 0.997 99.5 5.3 0.5 0.448 0.448 0.0050 0.005 180◦ 0.987 99.8 4.3 Finally, Table I provides insight into the lengths (L1, L2) and radii (a1, a2) yielding the maximum realized gain as a function of the inter-element distance. Notably, the lengths of the wire-dipoles consistently remain below half-wavelength for inter-element distances up to 0.5λ. This suggests that optimizing the lengths within this range is crucial for achieving high realized gain. Additionally, the radii of the wire-dipoles appear to approach the upper limit of λ/201 ≈0.005λ in most cases.
This observation implies that increasing the radius of the wires may lead to even higher realized gain. However, to maintain 6 0.1 0.2 0.3 0.4 0.5 d=6 2 3 4 5 6 7 8 Maximum Magnitude (dBi) D, theor.
G, theor.
GR, theor.
D, numer.
G, numer.
GR, numer.
Fig. 2. The end-fire directivity, gain, and realized gain of the optimal antenna array in terms of dB-scaling are analyzed in relation to the inter-element distance (analytical and numerical results).
0.1 0.2 0.3 0.4 0.5 d=6 150 200 250 300 350 "? (deg.) D G GR Fig. 3. Inter-element phase difference for maximum end-fire directivity, gain, and realized gain of a two-wire dipole antenna array versus inter-element distance. The remaining design parameters (i.e., L1, L2, a1, and a2) are held constant at their optimal values.
the assumption of a valid sinusoidal current distribution over the dipoles, we adhered to the empirical rule of keeping the wires as thin as possible.
According to Harrington’s study , the maximum directivity, Dmax, of a lossless antenna that completely fills a sphere with radius R is given by the equation: Dmax = (kR)2 + 2kR (30) In our specific case, with an inter-element distance of 0.2λ, the resulting radius R is approximately 22 mm. Applying Harrington’s findings, the maximum directivity of such an antenna is 7.7 dBi, which aligns with our expectations. Additionally, the achieved realized gain reaches a maximum of 6.4 dBi, indicating close proximity to this theoretical upper limit.
These analytical findings suggest that it is feasible to implement a practical super-realized gain antenna by carefully designing the dipoles in the array.
B. Numerical Analysis Concluding the theoretical analysis presented in the previous Section, we validated our findings through numerical analysis.
Specifically, we first applied the method of moments using the Antenna Toolbox of MATLAB . We modeled the dipoles using the dipoleCylindrical function and the array using the linearArray function. The conductor was defined using the metal function, with a conductivity of 5.8 × 107 S/m and a thickness of 35 µm. We used the same optimum design parameters as in the theoretical study. The results are depicted in Fig. 2. Regarding the directivity, there is perfect agreement between the theoretical and numerical results from 0.05λ to 0.5λ. A good agreement is also observed for the gain case. However, as the dipoles get closer to each other or become thicker, the agreement for the realized gain decreases. This has an impact on the estimation of (21), which in turn affects the resulting impedance matrix Zl in (22). The impedance matrix is used to calculate the S-parameters in (28). Please note that the numerical method estimates the current distribution on the surface of a cylinder with radius ai, which differs from the linear distribution predicted in the theoretical analysis based on (4). Therefore, as the radius increases, the assumption of (4) becomes less accurate.
One of the main objectives of this work is to construct a superdirective antenna array. Therefore, although wire dipoles offer analytical expressions and their theoretical study is feasible, we have chosen to focus on strip dipoles instead due to the advantages offered by the fabrication process. For example, fabricating wire dipoles with accurate inter-element distance is challenging compared to the fabrication of strip dipoles with accurate widths. Strip dipoles can be easily manufactured by etching metal traces on PCB substrates, resulting in structurally robust configurations, and accurate geometries. Additionally, strip dipoles often exhibit a wider bandwidth compared to wire dipoles .
For comprehensive electromagnetic numerical simulations, we utilized the commercial solver CST Studio Suite 2022 , specifically the time domain solver. This software allowed accurate modeling of the antenna array, analysis of its radiation pattern characteristics, and examination of the array’s impedance. The simulated array is depicted in Fig. 4. Copper material with a conductivity of 5.96 × 107 S/m and a thickness of 35 µm was used for the metallic components. The strip dipoles were defined by their respective lengths, L1 and L2, widths, w1 and w2, and positioned at a distance of d from each other. The dipoles were driven by input voltages V1 and V2. Discrete ports with 50 Ωwere used to model the excitation of the array.
For practical reasons, the strips were modelled on a substrate based on RO4003C, with a thickness of 0.813 mm, a dielectric constant of ϵr = 3.55, and a dissipation factor of tan δ = 0.0027. The modeled substrate was assumed to have dimensions of 50 × 50 mm.
7 y x z − + V1 − + V2 L1 L2 w1 w2 d Fig. 4. Strip dipole layout with inter-element distance d, lengths of elements L1, L2, widths of elements w1, w2, excitation signal applied to elements V1, V2.
Fig. 5.
The antenna array’s realized gain for the optimized and uniform cases versus inter-element distance at 3.5 GHz (simulated results). The 3D radiation pattern for the optimized result when d = 0.2λ is also shown.
0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° -10 0 Fig. 6. Simulated realized gain radiation pattern, horizontal plane (H-plane).
Maximum occurs at ϕ = 0◦(i.e., end-fire antenna array). The front-to-back ratio at 3.5 GHz is 8.44 dB, and the angular width (3 dB) is 127◦.
TABLE II OPTIMUM DESIGN PARAMETERS: NUMERICAL ANALYSIS AT 3.5 GHZ (STRIP DIPOLES ON SUBSTRATE) d/λ L1 (mm) L2 (mm) w1 (mm) w2 (mm) ∆ϕ (deg.) η (%) GR (dBi) 0.05 36.23 28.89 2.05 3.18 256◦ 99.2 3.4 0.1 42.39 30.39 4.85 3.19 210◦ 99.5 5.0 0.2 34.27 29.48 4.40 3.19 214◦ 99.3 6.3 0.3 33.33 29.62 5.14 2.57 197◦ 99.5 6.0 0.4 31.47 29.83 5.31 3.28 196◦ 99.6 5.2 0.5 30.74 29.77 4.14 3.24 172◦ 99.7 4.1 The antenna array was optimized to achieve superdirectivity by maximizing the realized gain while varying the inter-element distance from 0.05λ to 0.5λ (at 3.5 GHz). The design parameters included the lengths L1, L2, the widths w1, w2, and the inter-element phase difference ∆ϕ ∈[0, 360◦] of the elements. To keep implementation complexity low, the magnitude of the driven voltages was again fixed at 1 V/m.
PSO method was employed once again. The outcomes are shown in Fig. 5, which displays the maximum achieved antenna array realized gain as a function of the inter-element distance d, maximized to 6.3 dBi at d = 0.2λ. It is evident that each d value has an optimal set of design parameters that yield the highest realized gain (Table II).
When considering the uniform case for d = 0.2λ, the array exhibits a realized gain of 1.1 dBi and is broadside. This indicates an improvement of approximately 5.2 dBi for the superdirective array.
Also, for d ≥λ/2, the antenna array becomes broadside, and the improvement is marginal, as the optimized antenna realized gain only slightly differs from the uniform case. This aligns with expectations, as the superdirectivity phenomenon does not 8 Fig. 7.
Simulated surface current distribution for simultaneous optimal excitation at 3.5 GHz, when the inter-element distance is 0.2λ. Ports are numbered, and strip dipoles lie on an RO4003C substrate.
160 175 215 253 268 "?(deg.) 4.8 5.3 5.8 6.3 Realized Gain (dBi) Fig. 8. The effect of the phase difference ∆ϕ on the realized gain (simulated results): when ∆ϕ ranges from 175◦to 253◦, the realized gain experiences a decrease of merely 0.5 dB.
occur under these conditions (i.e., when d ≥λ/2).
The radiation efficiency η as a function of d/λ is also listed in Table II for inter-element distances up to 0.5λ. It can be observed that the antenna array exhibits extremely high radiation efficiency, exceeding 99.2% for all inter-element cases computed.
Fig. 6 illustrates the simulated realized gain in the horizontal plane (H-plane) for the optimal case with an element spacing of d = 0.2λ. The maximum gain achieved is 6.3 dBi, observed at ϕ = 0◦, indicating an end-fire antenna array configuration.
The angular width, estimated at the 3 dB drop-off points, is 126◦.
In superdirective antenna arrays, the surface current distribution plays a crucial role in achieving high directivity and gain , and it is typically non-uniform. It is characterized by strong currents flowing in specific regions of the array elements while minimizing currents in other areas. This non-uniform current distribution helps in shaping the radiation pattern and achieving high directivity. The specific current distribution pattern depends on the design and geometry of the array elements.
The spacing, size, and arrangement of the elements, as well as the excitation amplitudes and phases, all contribute to the desired surface current distribution. In this work, the simulated surface current distribution is depicted in Fig. 7. It is evident that distribution is not uniform, as expected.
The impact of the phase difference ∆ϕ on the realized gain, with all other design parameters at their optimal values, is depicted in Fig. 8. The graph demonstrates that varying the phase difference to 175◦or 253◦from the maximum at 215◦leads to a reduction in realized gain of 0.5 dB. This observation is of significance in fabrication, as minor deviations in the phase difference do not exert a substantial influence on the maximum realized gain.
C. Implementation and Measurements After completing the theoretical study and conducting a comprehensive numerical analysis using full-electromagnetic simu-lation, we proceed to the implementation phase. The antenna array is fabricated based on the design concept discussed earlier, which involves strip dipoles with slightly different lengths and radii (or widths). These dipoles are excited by signals of equal magnitude but different phases, enabling impedance matching to 50 Ω.
For validation purposes, the antenna array with an inter-element distance of d = 0.2λ was selected. To achieve a precise phase difference, coaxial cables (RG405) of different lengths were used to feed the strip dipoles. The strips were etched onto an RO4003C substrate with a thickness of 0.813 mm, a dielectric constant of ϵr = 3.55, and a dissipation factor of tan δ = 0.0027.
The substrate had dimensions of 50 × 50 mm. A balun (balanced-to-unbalanced) of length λ/4 was incorporated to match the balanced structure of the dipole to the unbalanced structure of the coaxial cable. SMA connectors were employed for the feeding. Before the fabrication and measurement, the antenna array was subjected to numerical simulation to determine the optimal design parameters. Similar to the previous design, the lengths L1, L2, w1, and w2 were considered. However, this time, to achieve the optimal phase difference, the length of the first coaxial cable (Lc1) was fixed at Lc1 = λ/2, and the optimal length of the second cable (Lc2) was determined. Following numerical optimization, the final design parameters were obtained: L1 = 33.7 mm, L2 = 29.1 mm, w1 = 4.8 mm, w2 = 3.5 mm, and Lc2 = 81.3 mm.
To measure the realized gain, we employed the method of three antennas . This method involves the use of a transmitting antenna, the antenna under test, and a reference antenna. In our setup, we opted to use identical transmitting and reference antennas to simplify calculations. A signal generator was utilized at the transmitter, emitting at a frequency of 3.5 GHz with a power of 0 dBm, while a spectrum analyzer was employed at the receiver. Additionally, we took into account any losses incurred by the cables used in our setup. Instead of using a power divider, we performed our measurement in two steps and 9 Fig. 9. (left) Prior to fabrication, the antenna array underwent simulation to obtain optimal values for the design parameters. In this scenario, the inter-element phase difference was achieved by utilizing coaxial cables of different lengths: Lc1 was fixed at λ/2, while Lc2 was estimated as a design parameter instead of ∆ϕ. Additionally, a pawsey balun with a length of λ/4 was implemented on both dipoles to ensure a smooth transition from the coaxial cables to the strip dipoles. (right) Fabricated antenna array.
0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° -10 0 10 Fig. 10. Measured and simulated results of the realized gain in the horizontal plane (H-plane): indicating that the proposed antenna is behaving as a super realized gain antenna array, with a maximum realized gain of 6.3 dBi, at 3.5 GHz.
Fig. 11.
Measured and simulated S-parameters of the fabricated antenna array. It is evident that the reflection coefficient at both dipoles is less than −10 dB, indicating a well impedance-matched to 50 Ω.
then combined the results algebraically. Specifically, in the first step, we measured the received power at the first dipole while terminating the second dipole at 50 Ω. In the second step, we repeated the process but with the roles of the first and second dipoles reversed. The antenna under test was rotated to obtain the received power in the horizontal plane (H-plane). Due to the antenna’s symmetry, we measured the rotation angles from 0◦to 180◦.
The measured results, along with the simulated results, are shown in Fig. 10. Good agreement between the measured and simulated results is observed. At 3.5 GHz, the proposed antenna exhibits a measured realized gain of 6.3 dBi. The observed ripples could be due to balun manufacturing tolerances, the finite size of the substrate, or attenuation.
Fig. 11 depicts the measured and simulated S-parameters of the fabricated antenna array, demonstrating a high level of agreement between them. Additionally, at 3.5 GHz, both S11 and S22 exhibit values below −10 dB, indicating low return losses at 50 Ω. By analyzing the S-parameters, we can estimate the impedance of each antenna element . Specifically, at 3.5 GHz, the first and second dipoles have impedance values of 53.5 + j1.2 and 57.9 + j2, respectively, the measured system resulting in reflection coefficients of −14.5 dB and −11.2 dB for dipole 1 and dipole 2, respectively. Based on Fig. 11, the antenna array operates (i.e., the reflection coefficient is below −10 dB for both dipoles at a 50 Ωimpedance), within the frequency range of 3.44 GHz to 3.62 GHz, resulting in a measured fractional bandwidth of 5.1%.
The measurements demonstrate a close alignment with the simulated and theoretical results, confirming the successful realization of super realized gain. This validates the efficacy of our approach in reducing losses, increasing radiation efficiency, and enhancing the power efficiency of the superdirective antenna system.
III. CONCLUSION This study undertook a thorough investigation of a superdirective dipole antenna array specifically designed to cater to the requirements of directional 5G wireless communication applications. The research focused on careful design considerations, including the selection of appropriate radiating elements, to achieve the desired superdirective performance. Impedance matching was effectively addressed through adjustments in strip dimensions, leveraging innovative techniques not previously introduced in the literature. Phase matching was ensured by employing unequal coaxial cable lengths to achieve the desired inter-element phase difference. Furthermore, the antenna array’s efficiency was optimized by minimizing both ohmic and return losses through 10 appropriate low-profile antenna design, resulting in high radiation efficiency. The measured and simulated results exhibited exceptional agreement, confirming the effectiveness of the proposed design. This work contributes to the advancement of high-directivity antennas and provides valuable insights for future research and development in the field of superdirective antenna arrays.
REFERENCES S. D. Assimonis, M. A. B. Abbasi, and V. Fusco, “Millimeter-wave multi-mode circular antenna array for uni-cast multi-cast and OAM communication,” Scientific Reports, vol. 11, no. 1, p. 4928, 2021.
K. Hu, T. W. Callis, and M. M. Tentzeris, “Additively manufactured flexible on-package phased antenna arrays with integrated microfluidic cooling channels for 5G/mmwave system-on-package designs,” IEEE Microwave and Wireless Technology Letters, vol. 33, no. 6, pp. 899–902, 2023.
A. Eid, J. G. D. Hester, and M. M. Tentzeris, “5G as a wireless power grid,” Scientific Reports, vol. 11, no. 1, p. 636, Jan 2021.
A. I. Uzkov, “An approach to the problem of optimum directive antenna design,” Comptes Rendus (Doklady) de l’Academie des Sciences de l’URSS, vol. 53, p. 35–38, 1946.
N. Yaru, “A note on super-gain antenna arrays,” Proceedings of the IRE, vol. 39, no. 9, pp. 1081–1085, 1951.
M. Uzsoky and L. Solym´ ar, “Theory of super-directive linear arrays,” Acta Physica Academiae Scientiarum Hungaricae, vol. 6, no. 2, pp. 185–205, 1956.
R. Harrington, “On the gain and beamwidth of directional antennas,” IRE Transactions on Antennas and Propagation, vol. 6, no. 3, pp. 219–225, 1958.
E. Altshuler, T. O’Donnell, A. Yaghjian, and S. Best, “A monopole superdirective array,” IEEE Transactions on Antennas and Propagation, vol. 53, no. 8, pp. 2653–2661, 2005.
M. Morris, M. Jensen, and J. Wallace, “Superdirectivity in mimo systems,” IEEE Transactions on Antennas and Propagation, vol. 53, no. 9, pp.
2850–2857, 2005.
M. T. Ivrlaˇ c and J. A. Nossek, “High-efficiency super-gain antenna arrays,” in 2010 International ITG Workshop on Smart Antennas (WSA), 2010, pp.
369–374.
O. S. Kim, S. Pivnenko, and O. Breinbjerg, “Superdirective magnetic dipole array as a first-order probe for spherical near-field antenna measurements,” IEEE Transactions on Antennas and Propagation, vol. 60, no. 10, pp. 4670–4676, 2012.
T. L. Marzetta, “Super-directive antenna arrays: Fundamentals and new perspectives,” in 2019 53rd Asilomar Conference on Signals, Systems, and Computers, 2019, pp. 1–4.
K. Dovelos, S. D. Assimonis, H. Q. Ngo, and M. Matthaiou, “Superdirective arrays with finite-length dipoles: Modeling and new perspectives,” in GLOBECOM 2022 - 2022 IEEE Global Communications Conference, 2022, pp. 6517–6522.
L. Han, H. Yin, and T. L. Marzetta, “Coupling matrix-based beamforming for superdirective antenna arrays,” in ICC 2022 - IEEE International Conference on Communications, 2022, pp. 5159–5164.
K. Dovelos, S. D. Assimonis, H. Q. Ngo, and M. Matthaiou, “Superdirective antenna pairs for energy-efficient terahertz massive mimo,” arXiv preprint arXiv:2207.00697, 2022.
A. Tornese, A. Clemente, and C. Delaveaud, “A new method for gain prediction of superdirective end-fire arrays,” in 2022 16th European Conference on Antennas and Propagation (EuCAP), 2022, pp. 1–4.
L. Han, H. Yin, M. Gao, and J. Xie, “A superdirective beamforming approach with impedance coupling and field coupling for compact antenna arrays,” arXiv preprint arXiv:2302.08203, 2023.
R. W. Ziolkowski, “Superdirective unidirectional mixed-multipole antennas: Designs, analysis, and simulations,” IEEE Transactions on Antennas and Propagation, vol. 71, no. 7, pp. 5566–5581, 2023.
B. Couraud, R. Vauche, S. N. Daskalakis, D. Flynn, T. Deleruyelle, E. Kussener, and S. Assimonis, “Internet of things: A review on theory based impedance matching techniques for energy efficient rf systems,” Journal of Low Power Electronics and Applications, vol. 11, no. 2, p. 16, Mar 2021.
[Online]. Available: D. M. Pozar, Microwave Engineering, 4th ed.
Hoboken, NJ: Wiley, 2011.
S. J. Orfanidis. (2016) Electromagnetic waves and antennas. [Online]. Available: L. F. Shampine, “Vectorized adaptive quadrature in MATLAB,” Journal of Computational and Applied Mathematics, vol. 211, pp. 131–140, 2008.
C. A. Balanis, Antenna theory: Analysis and Design.
John Wiley & Sons, 2005.
M. Manteghi and Y. Rahmat-Samii, “Broadband characterization of the total active reflection coefficient of multiport antennas,” in IEEE Antennas and Propagation Society International Symposium. Digest. Held in conjunction with: USNC/CNC/URSI North American Radio Sci. Meeting (Cat.
No.03CH37450), vol. 3, 2003, pp. 20–23 vol.3.
S. Moradi, B. Honarbakhsh, and T. F. Eibert, “Some excitation independent bounds for the total active reflection coefficient of antenna arrays,” IEEE Transactions on Antennas and Propagation, vol. 70, no. 12, pp. 11 743–11 751, 2022.
The MathWorks, Inc., “MATLAB,” Version R2022b, Natick, Massachusetts, 2022.
Dassault Systemes, “CST Studio Suite,” Computer software, 2022, version 2022. [Online]. Available: products/cst-studio-suite/ SunAR RF Motion, “DRH Series Broadband Horn Antennas,” 2023, datasheet. [Online]. Available: |
12940 | https://mathworld.wolfram.com/ExponentialDistribution.html | Exponential Distribution -- from Wolfram MathWorld
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Given a Poisson distribution with rate of change , the distribution of waiting times between successive changes (with ) is
(1)
(2)
(3)
and the probability distribution function is
(4)
It is implemented in the Wolfram Language as ExponentialDistribution[lambda].
The exponential distribution is the only continuousmemorylessrandom distribution. It is a continuous analog of the geometric distribution.
This distribution is properly normalized since
(5)
The raw moments are given by
(6)
the first few of which are therefore 1, , , , , .... Similarly, the central moments are
(7)
(8)
where is an incomplete gamma function and is a subfactorial, giving the first few as 1, 0, , , , , ... (OEIS A000166).
The mean, variance, skewness, and kurtosis excess are therefore
(9)
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The characteristic function is
(13)
(14)
where is the Heaviside step function and is the Fourier transform with parameters .
If a generalized exponential probability function is defined by
(15)
for , then the characteristic function is
(16)
The central moments are
(17)
and the raw moments are
(18)
(19)
and the mean, variance, skewness, and kurtosis excess are
(20)
(21)
(22)
(23)
See also
Extreme Value Distribution, Geometric Distribution, Poisson Distribution
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More things to try:
exponential distribution
{25, 35, 10, 17, 29, 14, 21, 31}
Champernowne constant
References
Balakrishnan, N. and Basu, A.P. The Exponential Distribution: Theory, Methods, and Applications. New York: Gordon and Breach, 1996.Beyer, W.H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp.534-535, 1987.Sloane, N.J.A. Sequence A000166/M1937 in "The On-Line Encyclopedia of Integer Sequences."Spiegel, M.R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, p.119, 1992.
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Weisstein, Eric W. "Exponential Distribution." From MathWorld--A Wolfram Resource.
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12941 | https://www.youtube.com/watch?v=q4wDcrCkkfQ | Example of Symmetric Equations of a Line
MathDoctorBob
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Multivariable Calculus: Find the symmetric equations of the line through the point (1,0,3) and perpendicular to the plane x+2y-z=6.
For more videos like this one, please visit the Multivariable Calculus playlist at my channel.
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Transcript:
Find the symmetrical equations of the line through the point 103 and perpendicular to the plane x + 2 y - z = 6. Now what do I need to make a line? I need a point and a direction. So what are we given? I'm given my point 103. So, I need to find a direction. Well, we're told perpendicular to the plane x + 2 y - z = 6. So, what do we get from that equation? Well, if I'm given the equation of a plane, I can get the perpendicular direction, the normal direction by peeling the coefficients off of x, y, and z. So, in this case, we have a perpendicular direction given by 1 2 minus one. So, that's what we'll use for the direction of our line. Now having a point and a direction we can get the vector function for our line by taking our point adding our direction time t. So in vector function form we have r of t = 1 + t 2 t 3 - t. That's our line. Now next step symmetrical equations. So the reason we can't stop here. So while this does give us the equation of our line, we're taking an outside parameter to be able to define it. So that's what t is doing. So the idea is t is sort of outside of the line. I don't need t to get a handle on our line. So we want to get rid of t in our equations. The way we do that, we're to set each coordinate to x, y, and z. Solve for t. And then I can get rid of the t by just letting what we solve for be equal to each other. And that's what we call our symmetrical equations. Now doing that, what do we have? I have x= 1 + t, y = 2t, z = 3 - t. So we solve for each of those gives me t = x - 1, t = y / 2, t = 3 - z. So we don't have a degenerate case here where we can't solve for one of our variables in terms of t. So the idea then is just let these three items be equal to each other and that's symmetric equations. Okay. The only thing I have to check is just to see that my point actually satisfies those equations. So we put one3 in. You know we'll get 0als 0 equals 0. |
12942 | https://mathoverflow.net/questions/219003/greatest-number-of-coprime-numbers-between-two-numbers | Stack Exchange Network
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Greatest number of coprime numbers between two numbers
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We know that from prime number theorem that the number of primes below $n$ is approximately $$\frac{n}{\log_en}.$$
$\star$ Given $n,m$, what is the largest list of pairwise coprime numbers that one can come up above $n$ and below $m$? Will it be asymptotically same as number of primes or something different? In particular can we replace $\log_en$ by something smaller? If so, how much smaller?
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edited Sep 23, 2015 at 5:48
GH from MO
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user76479user76479
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$\begingroup$ Related: mathoverflow.net/questions/56099/… $\endgroup$
joro
– joro
2015-09-23 05:31:27 +00:00
Commented Sep 23, 2015 at 5:31
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For a given $n$, let $P\subset{2,3\dots,n}$ be the set of primes up to $n$, and let $S\subset{2,3\dots,n}$ be any subset with pairwise coprime elements. Consider the function $f:S\to P$ that assigns to any $s\in S$ its smallest prime factor $p\in P$. It is clear that $f$ is an injection, whence $|S|\leq|P|$.
In short, the set of primes up to $n$ is the largest subset of ${2,3,\dots,n}$ with the given property.
Added 1. Péter Komjáth kindly called my attention to a 1962 survey by Paul ErdÅs (in Hungarian), which discusses some related problems. In particular, Problem 4 can be solved by the argument above: if $1\leq a_1<\dots1938 paper of ErdÅs and some later developments. Most relevant is Problem 18 that discusses the OP's question in general intervals. He denotes by $F(n,k)$ the maximal size of a subset $S\subset{n+1,\dots,n+k}$ with pairwise coprime elements, and he mentions that estimating this quantity is a difficult problem. In particular, he says that he is far from solving completely the problem of determining or estimating $\max_n F(n,k)$. I am sure that digesting the vast ErdÅs archive would bring up many related questions and results, in particular estimates for $F(n,k)$.
Added 2. The problem of estimating $F(n,k)$ (see previous section) is discussed in more detail in this 1971 paper of ErdÅs. See also the relevant OEIS entry.
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answered Sep 23, 2015 at 5:37
GH from MOGH from MO
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$\begingroup$ oops I meant an interval. I was looking at joro's post and forgot to update. $\endgroup$
user76479
– user76479
2015-09-23 05:38:32 +00:00
Commented Sep 23, 2015 at 5:38
$\begingroup$ Using PNT we can count number of primes between $m$ and $n$. Do we have roughly the same statistics for number of pairwise coprime numbers between $m$ and $n$ (this was what was in my mind)? $\endgroup$
user76479
– user76479
2015-09-23 05:40:36 +00:00
Commented Sep 23, 2015 at 5:40
$\begingroup$ I think from your answer it should be same. $\endgroup$
user76479
– user76479
2015-09-23 05:42:46 +00:00
Commented Sep 23, 2015 at 5:42
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$\begingroup$ @GH from MO: renyi.hu/~p_erdos/1962-23.pdf $\endgroup$
Péter Komjáth
– Péter Komjáth
2015-09-23 06:53:59 +00:00
Commented Sep 23, 2015 at 6:53
1
$\begingroup$ @Arul: See my added sections. $\endgroup$
GH from MO
– GH from MO
2015-09-23 20:06:07 +00:00
Commented Sep 23, 2015 at 20:06
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12943 | https://www.statisticshowto.com/geometric-mean/ | Skip to content
Geometric Mean: Definition, Examples, Formula, Uses
Statistics Definitions > Geometric Mean
Contents.
Basics:
How to Find the Geometric Mean.
Technology Options.
Real Life Uses.
More Advanced Info:
Equal Ratios and a Geometric Explanation.
Logarithmic Values and Dealing with Negative Numbers.
Arithmetic Mean-Geometric Mean Inequality
Antilog of a Geometric Mean
Geometric Mean Definition and Formula
Watch the video for three examples of how to find the geometric mean:
Can’t see the video? Click here to watch it on YouTube.
The geometric mean is a type of average , usually used for growth rates, like population growth or interest rates. While the arithmetic mean adds items, the geometric mean multiplies items. Also, you can only get the geometric mean for positive numbers.
Like most things in math, there’s an easy explanation, and there’s a more, ahem, mathematical way of stating the same thing. Formally, the geometric mean is defined as “…the nth root of the product of n numbers.” In other words, for a set of numbers {xi}Ni=1, the geometric mean is:
What this formula is saying in English is: multiply your items together and then take the nth root (where n is the number of items).
The π symbol in the formula is product notation: It’s the mathematical notation for “product”, similar to the (probably more familiar) Σ in summation notation.
How to Find the Geometric Mean (Examples)
Need help with a homework question? Check out our tutoring page!
Example 1: What is the geometric mean of 2, 3, and 6? First, multiply the numbers together and then take the cubed root (because there are three numbers) = (236)1/3 = 3.30
Note: The power of (1/3) is the same as the cubed root 3√. To convert a nth root to this notation, just change the denominator in the fraction to whatever “n” you have. So:
5th root = to the (1/5) power
12th root = to the (1/12) power
99th root = to the (1/99) power.
Example 2: What is the geometric mean of 4,8.3,9 and 17? First, multiply the numbers together and then take the 5th root (because there are 5 numbers) = (4 8 3 9 17)(1/5) = 6.81
Example 3: What is the geometric mean of 1/2, 1/4, 1/5, 9/72 and 7/4? First, multiply the numbers together and then take the 5th root: (1/21/41/59/727/4)(1/5) = 0.35.
Example 4: The average person’s monthly salary in a certain town jumped from $2,500 to $5,000 over the course of ten years. Using the geometric mean, what is the average yearly increase?
Solution:
Step 1: Find the geometric mean. (25005000)^(1/2) = 3535.53390593.
Step 2: Divide by 10 (to get the average increase over ten years). 3535.53390593 / 10 = 353.53.
The average increase (according to the GM) is 353.53.
Tip: The geometric mean of a set of data is always less than the arithmetic mean with one exception: if all members of the data set are the same (i.e. 2, 2, 2, 2, 2, 2, then the two means are equal.
A More Detailed Example
Let’s say you own a piece of art that increases in value by 50% the first year after you buy it, 20% the second year, and 90% the third year. What these numbers tell you is that at the end of the first year the value was multiplied by 150% or 1.5, the second year the value at the end of year 1 was multiplied by 120% or 1.2 and at the end of the third year the value at the end of year 2 was multiplied by 190% or 1.9. As these are multiplied, what you are looking for is the geometric mean which can be calculated in the following way: (1.51.21.9)(1/3) = 1.50663725458… or about 1.51
What the answer of 1.51 is telling you is that if you multiplied your initial investment by 1.51 each year, you would get the same amount as if you had multiplied it by 1.5, 1.2 and 1.9.
Art work value year 0: $90,000.
Art work value year 1: $90,000 1.5 = $135,000
Art work value year 2: $135,000 1.2 = $162,000
Art work value year 3: $90,000 1.9 = $307,000
or, using the geometric mean:$90,000 1.506637254583 = $307,000
If you do this calculation, it’s slightly different because of the number of decimal places I wrote here. In other words, you should get the exact result on a calculator.
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Why not use the Arithmetic Mean Instead?
The arithmetic mean is the sum of the data items divided by the number of items in the set:(1.5+1.2+1.9)/3 = 1.53
As you can probably tell, adding 1.53 to your initial price won’t get you anywhere, and multiplying it will give you the wrong result.
$90,000 1.53 1.53 1.53 = $322,343.91
Technology Options for Calculating the GM
JMP does not have a formula, but you can create one with the formula editor.
SAS: If you use SAS/STAT 12.1 or later, specify the ALLGEO statistic keyword in the PROC SURVEYMEANS statement. Earlier editions lack this capability.
Excel: use the GEOMEAN function for any range of positive data. The syntax is GEOMEAN(number1, [number2], …)
SPSS: Use the MEANS command. In the SPSS menus, choose Analyze>Compare Means>Means, then click on the “Options” button and select from the list of available statistics on the left.
MINITAB: use the GMEAN function. The syntax is GMEAN(number), where “number” is the column number. All numbers must be positive.
MAPLE: The calling sequence GeometricMean has a variety of options for calculating the mean from a data sample (A), a matrix data set(M) or a random variable or distribution(X). See this article for the full parameters.
TI83: there is no built in function. As a workaround, enter your data into a list and then enter the geometric mean formula on the home screen.
TI89: like the TI83, there isn’t a built in function. You could install an app, like the “Statistics and Probability Made Easy” app, which I recommend as I used it in grad school :).
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Real Life Uses
Aspect Ratios
The geometric mean has been used in film and video to choose aspect ratios (the proportion of the width to the height of a screen or image). It’s used to find a compromise between two aspect ratios, distorting or cropping both ratios equally.
Computer Science
Computers use mind-boggling amounts of data which often has to be summarized using statistics. One study compared the precision of several statistics (arithmetic means, geometric means, and percentage in the top x%) for a mind-boggling 97 trillion pieces of citation data. The study found that the geometric mean was the most precise (see this Cornell University Library article).
Geometry
1. Mean Proportional The geometric mean is used as a proportion in geometry (and is sometimes called the “mean proportional”). The mean proportional of two positive numbers a and b, is the positive number x, so that:
In this image, triangles ADC, ADB, and CAB are similar.If you have similar triangles, you can use the proportion to find missing sides. For example, the leg rule states that: hypotenuse/leg = leg/projection. These types of problems appear in high school geometry classes.
2. Golden Ratio The golden mean has a value of about 1.618 and can be derived from the geometric mean and similar rectangles. Let’s say you have a rectangle with width “a” and length “b”. Create a square within the rectangle with sides “a”: The smaller rectangle to the left is similar to the larger rectangle. Both rectangles contain the golden ratio, which is the ratio of the rectangle’s length to width. You could write a statement about the relationship between the two rectangles: b : a = a : b – a This statement, a proportional statement about the rectangles, also defines the geometric mean as “a”.
Medicine
The Geometric Mean has many applications in medicine. It has been called the “gold standard” for some measurements, including for the calculation of gastric emptying timesJNM.
Proportional Growth
As the examples at the beginning of this article demonstrate, the geometric mean is useful for calculating proportional growth, like the growth seen in long term investments using the Treasury bill rate as your riskfree rate (the riskfree rate is the theoretical return rate on a risk free investment). According to NYU corporate finance and valuation professor Aswath Damodoran, the geometric mean is appropriate for estimating expected returns over long term horizons. For short terms, the arithmetic mean is more appropriate.
It’s use isn’t limited to financial markets—it can be applied anywhere there is some type of proportional growth. For example, let’s say the amount of cells in a culture are 100, 180, 210, and 300 over a four day period. This gives a growth of 1.8 for day 2, 1.167 for day 3, and 1.42 for day 4. The geometric mean is (1.8 1.167 1.42)(1/3) = 1.44, meaning a daily growth of .44 or 44%.
United Nations Human Development Index
The Human Development Index (HDI) is an index that takes into account factors other than economic development when reporting a country’s growth. It is “…a summary measure of average achievement in key dimensions of human development: a long and healthy life, being knowledgeable and have a decent standard of living. The HDI is the geometric mean of normalized indices for each of the three [categories].”UNDP The “normalized” indexes refer to the fact that the geometric mean isn’t affected by differences in scoring indices. For example, if standard of living is scored on a scale of 1 to 5 and longevity on a scale of 1 to 100, a country that scores better in longevity would score better overall if the arithmetic mean were used. The geometric mean isn’t affected by those factors.
Water Quality Standards
Test results for water quality (specifically, fecal coliform bacteria concentrations) are sometimes reported as geometric means. Water authorities identify a threshold geometric mean where beaches or shellfish beds must be closed. According to CA.GOV, the dampening effect of the geometric mean is especially useful in water quality calculations, as bacteria levels can vary from 10 to 10,000 fold over a period of time. Back to Top
Equal Ratios
When you look at the geometric mean and the numbers you put into the calculation, an interesting thing happens. Let’s say you wanted to find the geometric mean of 4 and 9. The calculation would be √(4 9) = 6.
The ratio of the first number (4) and the geometric mean (6) is 4/6, which reduces to 2/3.
The ratio of the second number (9) and the geometric mean (6) is 6/9, which reduces to 2/3.
As you can see, the ratios are the same. This tells you that the geometric mean is a sort of “average” of all of the multipliers you are putting into the equation. Take the numbers 2 and 18. What number could you put in the center so that the ratio of 2 (to this number) is the same as the ratio of this number to 18?
2 (?) 18
If you guessed 6, you’re right, because 2 3 = 6 and 6 3 = 18. For more complex numbers, the ratios would be difficult to work out, which is why the formula is used.
You can get the same result (6) by using the formula, so if you’re ever presented the above type of problem in a math class, just find the square root of the numbers multiplied together:√(2 18) = 6
A Geometric Explanation
Let’s say you had a rectangle with sides of 2″ and 18″. The perimeter of this rectangle has the same perimeter as a square with four sides of 10″ each. 10 is what you would get if you worked out the arithmetic mean: (2 + 18) / 2 = 20 / 2 = 10.
The geometric mean can also be equated to the rectangle-square scenario: the square root of the sides (i.e. √ 218) is the length of the sides of a square with the same area as the rectangle. A rectangle 2″ x 18″ = 36 square inches and 6″ x 6″ is also 36 square inches. 6 is what you would get if you worked out the geometric mean: √(218) = 6.
Similarly, the geometric mean of three numbers a, b, and c is a cuboid with sides a, b, and c equal to the three numbers.
Back to Top
Logarithmic Values and the Geometric Mean
One way to think of the geometric mean is that it’s the average of logarithmic values converted back to base 10. If you’re familiar with logarithms, this can be a very intuitive way to look at it. For example, let’s say you wanted to calculate the geometric mean of 2 and 32.
Step 1: Convert the numbers to base 2 logs (you can theoretically use any base):
2 = 21
32 = 25
Step 2:Find the (arithmetic) average of the exponents in Step 1. The average of 1 and 5 is 3. We’re still working in base 2 here, so our average gives us 23, which gives us the geometric mean of 2 2 2 = 8.
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Dealing with Negative Numbers
Generally, you can only find the geometric mean of positive numbers. If you have negative numbers (common with investments), it’s possible to find a geometric mean, but you have to do a little math beforehand (which is not always easy!).
Example: What is the geometric mean for an investment that shows a growth in year 1 of 10 percent and a decrease the next year of 15 percent?
Step 1: Figure out the total amount of growth for the investment for each year. At the end of the first year you have 110% (or 1.1) of what you started with. The following year, you have 90% (or 0.9) of what you started year 2 with.
Step 2: Calculate the geometric mean based on the figures in Step 1: GM = √(1.1 0.9) = 0.99.
The geometric mean for this scenario is 0.99. Your investment is slowly losing money at a rate of about 1% per year.
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Arithmetic Mean-Geometric Mean Inequality
The Arithmetic Mean-Geometric Mean Inequality (AM-GM inequality) states that for a list of non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean. Taking the formulas for both types of mean, we get the inequality: For example, for the sequence of numbers {9, 12, 54}, the arithmetic mean, 25, is greater than the geometric mean, 18. The multiple proofs of this inequality are beyond the scope of this statistics site, but Bjorn Poonen offers this simple one line proof for the AM-GM inequality with two variables:
Back to Top
Antilog of a Geometric Mean
The antilog of a number is raising 10 to its power. Let’s say your geometric mean is 8. Raise base 10 to 8: 108 The general formula is: antilog(g) = 10g = 10√(ab)
Back to Top
References
Agresti A. (1990) Categorical Data Analysis. John Wiley and Sons, New York. Klein, G. (2013). The Cartoon Introduction to Statistics. Hill & Wamg. Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences, Wiley. Poonen, B. Retrieved March 3, 2023 from: Vogt, W.P. (2005). Dictionary of Statistics & Methodology: A Nontechnical Guide for the Social Sciences. SAGE.
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12944 | https://www.cuemath.com/algebra/cramers-rule/ | Cramers Rule
Cramer's rule is invented by mathematician Gabriel Cramer in 1750s. This rule is used to find the solution of a system of equations with any number of variables and the same number of equations. Sometimes, when we are solving a system of equations in 3 variables, say x, y, and z, we may need to solve for two variables x and y to solve for variable z. But using Cramer's rule, we can find the value of any variable without finding the values of the other variables.
But this rule has some limitations with respect to the solutions. This rule can be applied only when the system has unique solutions. But how do we know when a system has unique solution? Let us learn more about this along with the definition and formula of Cramer's Rule.
| | |
--- |
| 1. | What is Cramer's Rule? |
| 2. | Cramer's Rule Formula |
| 3. | Cramer's Rule For 2 x 2 |
| 4. | Cramer's Rule For 3 x 3 |
| 5. | Cramer's Rule Chart |
| 6. | Cramer's Rule Condition |
| 7. | FAQs on Cramer's Rule |
What is Cramer's Rule?
Cramer's rule is one of the methods used to solve a system of equations. This rule involves determinants. i.e., the values of the variables in the system are found with the help of determinants. Let us consider a system of equations in n variables x₁, x₂, x₃, ..., xₙ written in the matrix form AX = B, where
A = the coefficient matrix which is a square matrix
X = the column matrix with variables
B = the column matrix with the constants (which are on the right side of the equations)
Cramer's Rule Formula
Here is the Cramer's rule formula to solve the system AX = B (or) to find the values of the variables x₁, x₂, x₃, ..., xₙ. To solve the system of equations:
Find det |A| and represent it by D.
Find the determinants Dₓ₁, Dₓ₂, Dₓ₃, ..., Dₓₙ, where Dₓᵢ is the determinant of matrix A where the ith column is replaced by the column matrix B.
We divide each of these determinants by D to find the value of the corresponding variables. i.e.,
x₁ = Dₓ₁/D, x₂ = Dₓ₂/D, ...., xₙ = Dₓₙ/D.
Note that the system of equations has a unique solution only when D ≠ 0.
Are you getting confused with this general formula of Cramer's rule? Let us see this rule for 2 x 2 and 3 x 3 system of equations for clarification.
Cramer's Rule For 2 x 2
Using the above formula, let us see how to solve a system of 2 equations in 2 variables using Cramer's rule. Here are the steps to solve this system of 2x2 equations in two unknowns x and y using Cramer's rule.
Step-1: Write this system in matrix form is AX = B.
Step-2: Find D which is the determinant of A. Also, find the determinants Dₓ and Dᵧ where
Dₓ = det (A) where the first column is replaced with B
Dᵧ = det (A) where the second column is replaced with B
Step-3: Find the values of the variables x and y by dividing each of Dₓ and Dᵧ by D respectively.
Consider a system of two equations in two variables x and y.
a₁x + b₁y = c₁ and
a₂x + b₂y = c₂
Let us apply the above steps to solve the above system.
Step-1: Write this system in matrix form is AX = B, where
A = ⎡⎢⎣a1b1a2b2⎤⎥⎦ = the coefficient matrix
X = ⎡⎢⎣xy⎤⎥⎦ = the variable matrix
B = ⎡⎢⎣c1c2⎤⎥⎦ = the constant matrix
Step-2: Calculate the determinants D, Dₓ, and Dᵧ, where
D = det (A) = ∣∣
∣∣a1b1a2b2∣∣
∣∣
Dₓ = det (A) where the first column is replaced with B = ∣∣
∣∣c1b1c2b2∣∣
∣∣
Dᵧ = det (A) where the second column is replaced with B = ∣∣
∣∣a1c1a2c2∣∣
∣∣
Step-3: Find x and y (when D ≠ 0) using
x = Dₓ/D
y = Dᵧ/D
Cramer's Rule For 3 x 3
We will just extend the same process of Cramer's rule for 2 equations for a 3x3 system of equations as well. Here are the steps to solve this system of 3x3 equations in three variables x, y, and z by applying Cramer's rule.
Step-1: Write this system in matrix form is AX = B.
Step-2: Find D which is the determinant of A. i.e., D = det (A). Also, find the determinants Dₓ, Dᵧ, and Dz where
Dₓ = det (A) where the first column is replaced with B
Dᵧ = det (A) where the second column is replaced with B
Dz = det (A) where the third column is replaced with B
Step-3: Find the values of the variables x, y, and z by dividing each of Dₓ, Dᵧ, and Dz by D respectively.
Consider a system of three equations in three variables x, y, and z.
a₁x + b₁y + c₁z = d₁
a₂x + b₂y + c₂z = d₂ and
a₃x + b₃y + c₃z = d₃
Let us apply the above steps to solve 3x3 equations.
Step-1: We will write the system in matrix form is AX = B, where
A = ⎡⎢⎣a1b1c1a2b2c2a3b3c3⎤⎥⎦ = the coefficient matrix
X = ⎡⎢⎣xyz⎤⎥⎦ = the variable matrix
B = ⎡⎢⎣d1d2d3⎤⎥⎦ = the constant matrix
Step-2: Compute the determinants D, Dₓ, Dᵧ, and Dz where
D = det (A) = ∣∣
∣∣a1b1c1a2b2c2a3b3c3∣∣
∣∣
Dₓ = det (A) where the first column is replaced with B = ∣∣
∣∣d1b1c1d2b2c2d3b3c3∣∣
∣∣
Dᵧ = det (A) where the second column is replaced with B = ∣∣
∣∣a1d1c1a2d2c2a3d3c3∣∣
∣∣
Dz = det (A) where the third column is replaced with B = ∣∣
∣∣a1b1d1a2b2d2a3b3d3∣∣
∣∣
Step-3: Find the values of the variables x, y, and z (when D ≠ 0) using
x = Dₓ/D
y = Dᵧ/D
z = Dz/D
Cramer's Rule Chart
If we observe the formula of Cramer's rule in all the above three sections, we have mentioned that D ≠ 0 everywhere. This is because while finding the values of the variables, D is in the denominator and if D = 0, the fraction (the value of the variable) goes undefined. So this rule is applicable only when D ≠ 0. But what about the system of equations when D = 0? Then there are two possibilities.
The system may have no solution.
The system may have an infinite number of solutions.
Though Cramer's rule doesn't help in finding the infinite number of solutions, we can determine whether the system has "no solution" or "infinite number of solutions" using the determinants which we compute as the process of applying the rule.
If D ≠ 0, we say that the system AX = B has unique solution.
If D = 0 and atleast one of the numerator determinants is a 0, then the system has infinitely many solutions.
If D = 0 and none of the numerator determinants is 0, then the system has no solution.
You can visualize this from the following Cramer's rule chart.
Cramer's Rule Condition
From the above chart and explanation, it is very clear that Cramer's rule is NOT applicable when D = 0. i.e., when the determinant of the coefficient matrix is 0, we cannot find the solution of the system of equations using Cramer's rule. In this case, we can find the solution (if any) by using Gauss Jordan Method.
Thus, Cramer's rule is used to find the solution of a system only when the system has a unique solution.
Important Notes on Cramer's Rule:
Here are some important notes related to the application of Cramer's rule:
If there are n variables and n equations, we have to compute (n + 1) determinants.
This rule can give the solutions only when D ≠ 0.
If D = 0, the system has either an infinite number of solutions or no solutions.
We cannot find solutions by using this rule when the system has an infinite number of solutions.
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Cramer's Rule Examples
Example 1: Solve the following system of 2x2 equations: x + y = 5 and 2x - 3y = -4.
Solution:
The given system can be written in the matrix form AX = B where, A = ⎡⎢⎣112−3⎤⎥⎦, X = ⎡⎢⎣xy⎤⎥⎦, and B = ⎡⎢⎣5−4⎤⎥⎦.
Now, we will find the determinants.
D = det(A) = ∣∣
∣∣112−3∣∣
∣∣ = 1(-3) - 1(2) = -3 - 2 = -5.
Dₓ = ∣∣
∣∣51−4−3∣∣
∣∣ = 5(-3) - 1(-4) = -15 + 4 = -11.
Dᵧ = ⎡⎢⎣152−4⎤⎥⎦ = 1(-4) - 5(2) = -4-10 = -14.
Now, by Cramer's rule for 2 equations,
x = Dₓ/D = (-11) / (-5) = 11/5
y = Dᵧ/D = (-14) / (-5) = 14/5
Answer: The solution of the given system is, x = 11/5 and y = 14/5.
2. Example 2: Solve the following system of 3 equations in 3 variables using Cramer's rule: x + y + z = 2, 2x + y + 3z = 9, and x - 3y + z = 10.
Solution:
The given system can be written in the matrix form AX = B where, A = ⎡⎢⎣1112131−31⎤⎥⎦, X = ⎡⎢⎣xyz⎤⎥⎦, and B = ⎡⎢⎣2910⎤⎥⎦.
We will compute the determinants.
D = det(A) = ∣∣
∣∣1112131−31∣∣
∣∣
= 1(1 + 9) - 1(2 - 3) + 1(-6 - 1)
= 10 + 1 - 7
= 4
Dₓ = ∣∣
∣∣21191310−31∣∣
∣∣
= 2(1 + 9) - 1(9 - 30) + 1(-27 - 10)
= 20 + 21 - 37
= 4
Dᵧ = ∣∣
∣∣1212931101∣∣
∣∣
= 1(9 - 30) - 2(2 - 3) + 1(20 - 9)
= -21 + 2 + 11
= - 8
Dz = ∣∣
∣∣1122191−310∣∣
∣∣
= 1(10 + 27) - 1(20 - 9) + 2(-6 - 1)
= 37 - 11 - 14
= 12
Now, we apply the formulas:
x = Dₓ/D = 4/4 = 1
y = Dᵧ/D = -8/4 = -2
z = Dz = 12/4 = 3
Answer: The solution of the given system is x = 1, y = -2, and z = 3.
3. Example 3: Determine whether the following system has unique solution, infinite number of solutions, or no solution: x - 2y + 3z = 17, 2x + y + 2z = 6, and 2x - 4y + 6z = 34.
Solution:
The given system can be written in the matrix form AX = B where, A = ⎡⎢⎣1−232122−46⎤⎥⎦, X = ⎡⎢⎣xyz⎤⎥⎦, and B = ⎡⎢⎣17634⎤⎥⎦.
Let us calculate the determinants.
D = det(A) = ∣∣
∣∣1−232122−46∣∣
∣∣
= 1(6 + 8) + 2 (12 - 4) + 3(-8 - 2)
= 14 + 16 - 30
= 0
Dₓ = ∣∣
∣∣17−2361234−46∣∣
∣∣
= 17(6 + 8) + 2(36 - 68) + 3(-24 - 34)
= 238 - 64 - 174
= 0
Here, D = 0, and one of Dₓ, Dᵧ, and Dz is 0.
By Cramer's rule, the system has infinitely many solutions.
Answer: The given system has infinitely many solutions.
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Practice Questions on Cramer's Rule
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FAQs on Cramer's Rule
How Does Cramer's Rule Work?
Cramer's rule is used to find the solution of the system of equations with a unique solution. It is also used to find whether the system has a unique solution, no solution, or an infinite number of solutions.
What is Cramer's Rule Definition?
Cramer's rule says the solution of the system of equations written in the matrix form AX = B (where A is the matrix of coefficients, X is the column matrix of variables, and B is the column matrix of coefficients) is obtained by dividing det (A) by the same determinant where the respective columns are replaced by the matrix B.
Who Invented Cramer's Rule of Matrices?
Cramer's rule for matrices is invented by a mathematician called Gabriel Cramer in 1750. He is a Swiss mathematician. This rule is very helpful in finding any variable right away without needing of finding any other variables.
What is Cramer's Rule 2x2?
First, write the given system of 2x2 equations as AX = B, where X is a column matrix of the variables x and y. Then find the determinants D, Dₓ, and Dᵧ, where D = det(A) and Dₓ and Dᵧ are same as det(A) where the first and second columns are respectively replaced by the matrix B. Then use the following to find the variables x and y.
x = Dₓ/D
y = Dᵧ/D
How Do You Use Cramer's Rule for 2x3 Equations?
Cramer's rule deals with the determinants and determinants can be found only for square matrices. But if we write 2x3 equations in the form of AX = B, then A is NOT a square matrix (it is a rectangular matrix) and hence this rule cannot be applied in this case.
What is Dₓ in Cramer's Rule?
To solve a system of equations using Cramer's Rule, first, we write it in the form AX = B. Then Dₓ is a Cramer's rule determinant of the coefficient matrix where the first column is replaced with the column matrix B.
What are the Advantages of Cramer's Rule?
Cramer's rule is used to solve the system of equations where the number of variables is equal to the number of equations. Also, using this rule, we can find the value of finding any variable right away without finding the other variables.
How to Apply Cramer's Rule When the Determinant is Zero?
While solving a system AX = B using Cramer's rule, if det A = 0, then the system either has an infinite number of solutions or no solution. In either case, we cannot conclude/find anything using Cramer's rule. This is because while finding every variable using Cramer's rule formula, we have to divide the determinants by det A and a fraction is undefined when its denominator is 0. Hence, when det A = 0, the Cramer's rule cannot be used.
What is Cramer's Rule 3x3?
First, write the given system of 3x3 equations as AX = B, where X is a column matrix of the variables x, y, and z. Then find the determinants D, Dₓ, Dᵧ, and Dz, where D = det(A) and Dₓ, Dᵧ, and Dz are same as det(A) where the first, second, and third columns are respectively replaced by the matrix B. Then use the following to find the variables x, y, and z.
x = Dₓ/D
y = Dᵧ/D
z = Dz/D
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domain of convergence: Laplace transform of sin b t sinb t
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To find Laplace transform of sin b t sinb t, I get
How to find the domain of convergence.
Attempt:
It seems to me that it will be for all s−i b>0 s−i b>0 (the domain for L(e b t)L(e b t) is ∀s−a>0∀s−a>0) but the actual answer I know s>0 s>0.
Also L(e a t sin b t)=b(s−a)2+b 2 L(e a t sinb t)=b(s−a)2+b 2 then what will be domain of convergencce of L(e a t sin b t)L(e a t sinb t)?
calculus
laplace-transform
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edited May 11, 2017 at 18:23
user1942348user1942348
asked May 11, 2017 at 18:14
user1942348user1942348
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Note that the Laplace Transform of sin(b t)sin(b t) is given by
L{sin(b t)}(s)=∫∞0 sin(b t)e−s t d t(1)(1)L{sin(b t)}(s)=∫0∞sin(b t)e−s t d t
We have the estimate
∣∣∣∫L 0 sin(b t)e−s t d t∣∣∣≤∫L 0 e−Re(s)t d t=1−e−Re(s)L Re(s)|∫0 L sin(b t)e−s t d t|≤∫0 L e−Re(s)t d t=1−e−Re(s)L Re(s)
Evidently, the integral in (1)(1) converges whenever Re(s)>0 Re(s)>0.
If Re(s)≤0 Re(s)≤0, then it is easy to see that the integral diverges.
Therefore, the domain of convergence is Re(s)>0 Re(s)>0 as was to be shown!
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answered May 11, 2017 at 19:16
Mark ViolaMark Viola
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then what will be the domain of convergence of L(e a t sin b t)=b(s−a)2+b 2 L(e a t sinb t)=b(s−a)2+b 2?user1942348 –user1942348 2017-05-12 03:20:02 +00:00 Commented May 12, 2017 at 3:20
1 Re(s−a)>0 Re(s−a)>0 Mark Viola –Mark Viola 2017-05-12 03:22:16 +00:00 Commented May 12, 2017 at 3:22
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12946 | https://www.nagwa.com/en/videos/950149357985/ | Question Video: Determining the Relations between Two Vectors | Nagwa
Question Video: Determining the Relations between Two Vectors | Nagwa
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Question Video: Determining the Relations between Two Vectors Mathematics • Third Year of Secondary School
Given the two vectors 𝐀 = (8𝐢 − 7𝐣 + 𝐤) and 𝐁 = (64𝐢 − 56𝐣 + 8𝐤), determine whether these two vectors are parallel, perpendicular, or otherwise.
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Given the two vectors 𝐀 is equal to eight 𝐢 minus seven 𝐣 plus 𝐤 and 𝐁 is equal to 64𝐢 minus 56𝐣 plus eight 𝐤, determine whether these two vectors are parallel, perpendicular, or otherwise.
We know that two vectors are parallel if they are scalar multiples of each other. Vector 𝐀 must be equal to 𝑘 multiplied by vector 𝐁. Two vectors 𝐀 and 𝐁 are perpendicular on the other hand, if their scalar or dot product is equal to zero. Let’s firstly consider whether our two vectors 𝐀 and 𝐁 are parallel. If one vector is a scalar multiple of another vector, then the ratio of their individual components must be equal. In this case, 64 over eight must be equal to negative 56 over negative seven, which must be equal to eight over one. 64 divided by eight is equal to eight, and eight divided by one is equal to eight.
Dividing a negative number by a negative number gives a positive answer. Therefore, negative 56 divided by negative seven is also equal to eight. We can therefore conclude that vector 𝐁 is equal to eight multiplied by vector 𝐀 or vector 𝐀 is equal to one-eighth of vector 𝐁. The two vectors 𝐀 and 𝐁 are therefore parallel. Whilst they cannot be parallel and perpendicular, let’s just check that the scalar product is not equal to zero. The 𝐢-components of our vector are eight and 64. The 𝐣-components are negative seven and negative 56. The 𝐤-components are one and eight. Eight multiplied by 64 is 512. Negative seven multiplied by negative 56 is 392. One multiplied by eight is equal to eight. The scalar product of vectors 𝐀 and 𝐁 is therefore equal to 912. As this is not equal to zero, the vectors 𝐀 and 𝐁 are not perpendicular.
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12947 | https://ocw.mit.edu/courses/8-20-introduction-to-special-relativity-january-iap-2021/6f52a05ddc356a017d2b33a50ecd9908_MIT8_20iap21_pset4.pdf | Massachusetts Institute of Technology Department of Physics Course: 8.20 —Special Relativity Term: IAP 2021 Instructor: Markus Klute Problem Set 4 handed out January 23rd, 2021 Note: In most problems, I will work with the convention that a 3-vector is denoted by an arrow (p ~), while a 4-vector would just be denoted as p (I have dropped the subscript term, pµ, as this is the convention). In particle physics, we often set the speed of light to the unit-less value of 1 (c = 1). Masses are often referred to in terms of energy (i.e. 511 keV, 938 MeV, etc...) Problem 1: Acceleration in Special Relativity [25 pts] In class we determined that the momentum of a particle traveling at velocity ~ u with respect to an observer is given by p ~ = √ m~ u . 1−u2/c2 ~ (a) Find the force F by taking the derivative with respect to ordinary time. (b) It is possible to also define a 4-vector for acceleration, just like we did for 4 velocity, by taking again the time derivative with respect to proper time µ dηµ d2x αµ = = dτ dτ 2 Find the components of αµ. (c) Express those components in terms of the force term you found in part (a). 1 Problem 2: π0 Decay [25 pts] The π0 is a heavy meson with a mass of 135 MeV/c2 that decays almost immediately to two back-to-back photons (with a lifetime of τ = 8.4 × 10−17 s). (a) What are the energies of the two photons emitted in the center-of-mass frame of the π0 when it decays? (b) Suppose one of the two photons makes an angle θ with respect to the x-axis in the center of mass frame. What is the minimum energy the π0 must have in order for both photons to be boosted in the forward direction (i.e. make an angle less than 90◦ from the positive x-axis)? This is convenient if your detector doesn’t fully encompass the region surrounding your pion. (c) Suppose with your detector (read as, lab frame) you measure both photons and each makes a ±45◦ angle with respect to the beam axis. From this information, tell me how far the π0 moved from when it was created to when it decayed. Problem 3: Review: Mendelstam Variables [25 pts] High energy physicists try as best they can to express various quantities (energy, momentum, cross-sections, etc.) in terms of invariant quantities. This is not mere aesthetics; it is far easier to make calculations if those calculations are independent of what frame one is working in. One such tool are Mendelstam variables, which describe the energy-momentum exchange when 2 particles collide with one another. Consider the (inelastic) collision of two particles (1 and 2) to yield two different particles (3 and 4), each with a different mass mi=1,2,3,4. The Mendelstam variables are defined as follows: s ≡ (p1 + p2)2/c2 t ≡ (p1 − p3)2/c2 u ≡ (p1 − p4)2/c2 (a) Calculate the quantity s + t + u. (b) Find the lab-frame energy of particle 1 in terms of Mendelstam variables (Suppose we are working with a fixed-target experiment, where lab frame implies particle 2 is at rest.) (c) Finally, find the total center-of-mass energy (E1 + E2) in terms of Mendelstam variables. 2 Problem 4: Collider versus Linac [25 pts] Suppose you were determined to discover a new particle (say, the Higgs) that required √ a very high energy center-of-mass energy, s to create1 . You decide you will create this elusive particle by slamming two identical particles of mass m (say, two protons) against each other. You have two choices on how to build your machine. You can build either (a) a collider, which slams the two particles in a head-to-head collision or (b) a linac (linear accelerator) which slams one particle against a fixed stationary target of material. (a) For a given kinetic energy K (which is either given solely to the proton in the linac or split evenly among the two protons in the collider) which option is the better choice to reach your targeted center-of-mass energy? (b) For what other reasons might you chose the other, less energetic, option? Figure 1: Photograph of the Fermi National Accelerator Facility. One can see both fixed target beamlines (linacs) and the main proton-anti-proton collider. Image courtesy of DOE. √ 1If you are confused as to what s is, look at Task 3 in this problem set. 3 MIT OpenCourseWare 8.20 Introduction to Special Relativity IAP 2021 For information about citing these materials or our Terms of Use, visit: |
12948 | https://www.finra.org/rules-guidance/guidance/interpretations-financial-operational-rules/sea-rule-15c3-1-and-related-interpretations | Cookie Policy
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SEA Rule 15c3-1 and Related Interpretations
Publication Date: August 11, 2025
Interpretations are marked in blue background beneath the rule text to which they relate.
15c3-1 Net capital requirements for brokers or dealers.
15c3-1(a) Every broker or dealer must at all times have and maintain net capital no less than the greater of the highest minimum requirement applicable to its ratio requirement under paragraph (a)(1) of this section, or to any of its activities under paragraph (a)(2) of this section, and must otherwise not be “insolvent” as that term is defined in paragraph (c)(16) of this section. In lieu of applying paragraphs (a)(1) and (a)(2) of this section, an OTC derivatives dealer shall maintain net capital pursuant to paragraph (a)(5) of this section. Each broker or dealer also shall comply with the supplemental requirements of paragraphs (a)(4) and (a)(9) of this section, to the extent either paragraph is applicable to its activities. In addition, a broker or dealer shall maintain net capital of not less than its own net capital requirement plus the sum of each broker's or dealer's subsidiary or affiliate minimum net capital requirements, which is consolidated pursuant to appendix C, § 240.15c3-1c.
15c3-1(a)/001 Moment to Moment Net Capital
Broker-dealers must maintain sufficient net capital at all times prior to, during and after purchasing or selling proprietary securities. Broker-dealers must have at all times (including intraday) sufficient net capital to meet the haircut requirements of the Capital Rule before taking on any new proprietary positions, even if the intention of the firm is to liquidate or cover the positions before the end of the same day. Broker-dealers are expected to be able to demonstrate moment to moment compliance with the Capital Rule.
(SEC Staff to NYSE) (NYSE Interpretation Memo 99-8, August 1999)
15c3-1(a)/01 Additional Net Capital Requirement
The net capital requirement is increased by one percent of accrued liabilities that are excluded from aggregate indebtedness under the provisions specified in interpretation 15c3-1(c)(2)(iv)(C)/095.
(SEC Letter to NASD, July 24, 1984)
(NYSE Interpretation Memo 87-6, May 1987)
(SEC Staff to FINRA) (FINRA Regulatory Notice 21-27)
15c3-1(a)/02 Consolidations, Minimum Net Capital Requirement
The minimum net capital requirement of the consolidated entity is determined by adding the amount of net capital required for compliance by each consolidated subsidiary subject to the Rule to the minimum dollar net capital requirement of the parent broker-dealer.
See interpretation 15c3-1c(c)/022 (Appendix C).
(SEC Staff to NYSE)
15c3-1(a)/03 Inactive Exchange Members
An inactive Exchange member is not subject to a net capital requirement so long as he is not conducting or engaged in the securities business.
(SEC Staff to NYSE)
15c3-1(a)/04 Registered Traders
A registered Trader (other than a registered trader in options) is subject to a net capital requirement if he trades for his own account. This is so even if he is associated with another broker-dealer. If he trades only for the account of the broker-dealer he is associated with, he is not individually subject to a requirement.
(SEC Staff to NYSE)
Ratio Requirements
Aggregate Indebtedness Standard
15c3-1(a)(1)(i) No broker or dealer, other than one that elects the provisions of paragraph (a)(1)(ii) of this section, shall permit its aggregate indebtedness to all other persons to exceed 1500 percent of its net capital (or 800 percent of its net capital for 12 months after commencing business as a broker or dealer).
15c3-1(a)(1)(i)/01 New Broker-Dealers
A new broker-dealer is considered to have commenced doing a business on the date it becomes effectively registered with the Commission. If a firm remains inactive for all or a portion of its first year of existence to circumvent the 8 to 1 ratio requirement, the SEC has the authority to cancel its registration pursuant to Section 15b-5 of the Securities Exchange Act of 1934, as amended.
(SEC Staff to NASD)
Alternative Standard
A broker or dealer may elect not to be subject to the Aggregate Indebtedness Standard of paragraph (a)(1)(i) of this section. That broker or dealer shall not permit its net capital to be less than the greater of $250,000 or 2 percent of aggregate debit items computed in accordance with the Formula for Determination of Reserve Requirements for Brokers and Dealers (Exhibit A to Rule 15c3-3, § 240.15c3-3a). Such broker or dealer shall notify its Examining Authority, in writing, of its election to operate under this paragraph (a)(1)(ii). Once a broker or dealer has notified its Examining Authority, it shall continue to operate under this paragraph unless a change is approved upon application to the Commission. A broker or dealer that elects this standard and is not exempt from Rule 15c3-3 shall:
15c3-1(a)(1)(ii)(A) Make the computation required by §240.15c3-3(e) and set forth in Exhibit A, §240.15c3-3a, on a weekly basis and, in lieu of the 1% reduction of certain debit items required by Note E (3) in the computation of its Exhibit A requirement, reduce aggregate debit items in such computation by 3%; provided, however, that, if a broker or dealer is required to make the computation required by §240.15c3-3(e) and set forth in Exhibit A, §240.15c3-3a, on a daily basis, the broker or dealer may reduce aggregate debit items in such computation by 2%;
15c3-1(a)(1)(ii)(B) Include in Items 7 and 8 of Exhibit A, § 240.15c3-3a, the market value of items specified therein more than 7 business days old;
15c3-1(a)(1)(ii)(C) Exclude credit balances in accounts representing amounts payable for securities not yet received from the issuer or its agent which securities are specified in paragraphs (c)(2)(vi)(A) and (E) of this section and any related debit items from the Exhibit A requirement for 3 business days; and
15c3-1(a)(1)(ii)(D) Deduct from net worth in computing net capital 1 percent of the contract value of all failed to deliver contracts or securities borrowed that were allocated to failed to receive contracts of the same issue and which thereby were excluded from Items 11 or 12 of Exhibit A, § 240.15c3-3a.
15c3-1(a)(1)(ii)/01 Minimum Capital Requirement
The 2% minimum net capital requirement is based on the aggregate SEA Rule 15c3-3 Reserve Formula debit items before the 3% reduction required by SEA Rule 15c3-1(a)(1)(ii)(A).
The net capital requirement as of a given moment in time is based on the aggregate Reserve Formula debits then existing just as if a Formula computation had been prepared. The moment-to-moment requirement is not based on the most recent formal weekly Reserve Formula computation.
(SEC Staff to NYSE) (NYSE Interpretation Memo 76-2, February 1976)
15c3-1(a)(1)(ii)/02 Approximation and Netting
A broker electing the alternative may not use the approximation and netting procedures outlined in SEC Release 34-9922 (under SEA Rule 15c3-3(e) Special Reserve Bank Account for the Exclusive Benefit of Customers) in making the weekly computation. A complete and accurate calculation must be made every week.
(SEC Staff to NYSE)
Futures Commission Merchants
No broker or dealer registered as a futures commission merchant shall permit its net capital to be less than the greater of its requirement under paragraph (a)(1) (i) or (ii) of this section, or 4 percent of the funds required to be segregated pursuant to the Commodity Exchange Act and the regulations thereunder (less the market value of commodity options purchased by option customers on or subject to the rules of a contract market, each such deduction not to exceed the amount of funds in the customer's account).
15c3-1(a)(2) Minimum Requirements
See Appendix E (§ 240.15c3-1e) for temporary minimum requirements.
Brokers or Dealers That Carry Customer Accounts
15c3-1(a)(2)(i) A broker or dealer (other than one described in paragraphs (a)(2)(ii) or (a)(8) of this section) shall maintain net capital of not less than $250,000 if it carries customer or broker or dealer accounts and receives or holds funds or securities for those persons. A broker or dealer shall be deemed to receive funds, or to carry customer or broker or dealer accounts and to receive funds from those persons if, in connection with its activities as a broker or dealer, it receives checks, drafts, or other evidences of indebtedness made payable to itself or persons other than the requisite registered broker or dealer carrying the account of a customer, escrow agent, issuer, underwriter, sponsor, or other distributor of securities. A broker or dealer shall be deemed to hold securities for, or to carry customer or broker or dealer accounts, and hold securities of, those persons if it does not promptly forward or promptly deliver all of the securities of customers or of other brokers or dealers received by the firm in connection with its activities as a broker or dealer. A broker or dealer, without complying with this paragraph (a)(2)(i), may receive securities only if its activities conform with the provisions of paragraphs (a)(2) (iv) or (v) of this section, and may receive funds only in connection with the activities described in paragraph (a)(2)(v) of this section.
15c3-1(a)(2)(i)/01 Introducing Brokers
Introducing brokers who do not meet the requirements outlined in interpretation 15c3-1 (a)(2)(iv)/01 shall be subject to the requirements of brokers that carry customer accounts.
(SEC Staff to NYSE) (NYSE Interpretation Memo 93-6, November 1993)
15c3-1(a)(2)(i)/02 Non-Carrying Brokers
A broker who clears and carries only accounts of “non-customers” is subject to the minimum net capital requirement under SEA Rule 15c3-1(a)(2)(i).
(SEC Staff to NYSE)
15c3-1(a)(2)(i)/03 Prime Broker Capital Requirements
A broker-dealer that acts as a prime broker must maintain net capital of not less than $1,500,000.
A broker-dealer acting as an executing broker in a prime broker relationship who self clears or a broker-dealer clearing prime broker transactions on behalf of an introducing executing broker must have minimum net capital of at least $1,000,000.
A broker-dealer must notify its DEA that it intends to act as a prime broker.
(SEC Letter to SIA, January 24, 1994)
(NYSE Interpretation Memo 94-5, May 1994)
15c3-1(a)(2)(ii) A broker or dealer that is exempt from the provisions of § 240.15c3-3 pursuant to paragraph (k)(2)(i) thereof shall maintain net capital of not less than $100,000.
15c3-1(a)(2)(ii)/01 Permitted to Elect Alternative Standard
Broker-dealers who are exempt from SEA Rule 15c3-3 pursuant to a (k)(2)(i) exemption may elect the alternative standard to determine their ratio requirements. See other requirements for alternative election under paragraph (a)(1)(ii).
(SEC Release 34-31511, December 2, 1992)
(NYSE Interpretation Memo 93-6, November 1993)
15c3-1(a)(2)(iii) Dealers
A dealer shall maintain net capital of not less than $100,000. For the purposes of this section, the term “dealer” includes:
15c3-1(a)(2)(iii)(A) Any broker or dealer that endorses or writes options otherwise than on a registered national securities exchange or a facility of a registered national securities association; and
15c3-1(a)(2)(iii)(B) Any broker or dealer that effects more than ten transactions in any one calendar year for its own investment account. This section shall not apply to those persons engaging in activities described in paragraphs (a)(2)(v), (a)(2)(vi) or (a)(8) of this section, or to those persons whose underwriting activities are limited solely to acting as underwriters in best efforts or all or none underwritings in conformity with paragraph (b)(2) of § 240.15c2-4, so long as those persons engage in no other dealer activities.
15c3-1(a)(2)(iii)(B)/01 Commodities Transactions - Not Counted as Dealer Transactions
Commodities transactions made by an introducing broker-dealer shall not be counted as dealer transactions pursuant to SEA Rule 15c3-1(a)(2)(iii).
(SEC Staff to NYSE) (NYSE Interpretation Memo 90-11, December 1990)
15c3-1(a)(2)(iii)(B)/011 Dealers That Operate Pursuant to 15c3-1(a)(6)
A dealer that elects to operate under SEA Rule 15c3-1(a)(6) will be subject to the minimum net capital requirements of a dealer pursuant to SEA Rule 15c3-1(a)(2)(iii).
(SEC Staff to NYSE) (NYSE Interpretation Memo 95-3, May 1995)
15c3-1(a)(2)(iii)(B)/012 Dealer/Ten Proprietary Transactions, Buys and Sells are Individually Included
Buy and sell transactions are counted as individual transactions even if the buy and sell transactions are for the same security.
(SEC Staff of DMR to NASD, May 1993) (NASD Notice to Members 93-30, May 1993)
15c3-1(a)(2)(iii)(B)/013 Dealer/Ten Proprietary Transactions, Money Market Fund Transactions are Excluded
Transactions in a money market fund registered as a fund under the Investment Company Act of 1940 and defined as a money market fund under Rule 2(a)7 of the Investment Company Act of 1940 are excluded from the ten-transaction limitation explained under “dealer” in SEA Rule 15c3-1(a)(2)(iii)(B).
(SEC Staff of DMR to NASD, May 1993) (NASD Notice to Members 93-30, May 1993)
15c3-1(a)(2)(iii)(B)/014 Dealer/Ten Proprietary Transactions, Mutual Fund Transactions within the Same Family of Funds are Included
Transactions between mutual funds (excluding money market mutual funds) within the same family of funds count toward the ten-transaction limitation explained under “dealer” in SEA Rule 15c3-1(a)(2)(iii)(B).
(SEC Staff of DMR to NASD, May 1993)
15c3-1(a)(2)(iii)(B)/015 Dealer/Ten Proprietary Transactions, Single Monthly Investments of $1,000 or Less into a Mutual Fund are Excluded
A single monthly investment of $1,000 or less into an established mutual fund account for the firm would not be considered as a transaction for the purpose of the ten-transaction limitation explained under “dealer” in SEA Rule 15c3-1(a)(2)(iii)(B).
(NASD Notice to Members 93-46, July 1993)
15c3-1(a)(2)(iii)(B)/02 Sole Proprietor Joint Securities Account With Spouse
See interpretation 15c3-1(a)(2)(vi)(B)/04.
15c3-1(a)(2)(iii)(B)/03 Sole Proprietor IRA, Keogh or ERISA Accounts
See interpretation 15c3-1(a)(2)(vi)(B)/05.
15c3-1(a)(2)(iv) Brokers or Dealers That Introduce Customer Accounts And Receive Securities
A broker or dealer shall maintain net capital of not less than $50,000 if it introduces transactions and accounts of customers or other brokers or dealers to another registered broker or dealer that carries such accounts on a fully disclosed basis, and if the broker or dealer receives but does not hold customer or other broker or dealer securities. A broker or dealer operating under this paragraph (a)(2)(iv) of this section may participate in a firm commitment underwriting without being subject to the provisions of paragraph (a)(2)(iii) of this section, but may not enter into a commitment for the purchase of shares related to that underwriting.
15c3-1(a)(2)(iv)/01 Requirements for Broker-Dealers Who Introduce Accounts on a Fully Disclosed Basis
Firms who introduce their accounts on a fully disclosed basis and wish to maintain their minimum Net Capital requirement pursuant to this paragraph (a)(2)(iv), must meet the following requirements:
The introducing firm must maintain a written clearing agreement (signed by the clearing broker-dealer) which states that for purposes of SIPA and SEA Rules 15c3-3, and 15c3-1, the customers are customers of the clearing firm and not the introducing firm;
The clearing firm must issue all account statements directly to customers;
Account statements must disclose the fact that all customer funds and/or securities are located at the clearing broker-dealer; and
Account statements must provide a contact person or department at the clearing firm who can address customer inquiries regarding their account(s).
If the introducing firm fails to meet any of the above requirements, it would be required to comply with the greater minimum net capital requirements of a broker-dealer that carries customer accounts.
The introducing firm should also maintain procedures to prevent their customers from transmitting funds (other than checks made out to appropriate third parties) to the firm (except by error). Procedures should address the actions the broker-dealer will take to advise its customers (in writing) should they send funds to the firm by error.
(SEC Release 34-31511, December 2, 1992)
(NYSE Interpretation Memo 93-6, November 1993)
15c3-1(a)(2)(iv)/02 Introducing Brokers - Receiving Funds
Any introducing broker that receives customer funds (checks made payable to itself and or cash), except by error, will be subject to the minimum net capital requirements of a broker-dealer that carries customer accounts (See SEA Rule 15c3-1(a)(2)(i).)
(SEC Release 34-31511, December 2, 1992)
(NYSE Interpretation Memo 93-6, November 1993)
15c3-1(a)(2)(iv)/03 Commission Recapture/Commission Rebate Program of Introducing Brokers
Any introducing broker who rebates a portion of its commission back to its customers either as a cash payment or to a creditor of the customer is required to maintain a minimum net capital requirement of at least $250,000. It is also considered a carrying firm for purposes of SEA Rule 15c3-3 unless it elects the following method for the handling of the customers’ rebates:
The introducing broker deposits money into a separate 15c3-3 bank account similar to those accounts established under a SEA Rule 15c3-3(k)(2)(i) exemption and the balance in the bank account at all times must equal or exceed the payables to customers. The firm issues checks from this bank account to pay the customer or the creditor of the customer.
(SEC Staff to NYSE) (NYSE Interpretation Memo 02-3, February 2002)
15c3-1(a)(2)(v) Brokers or Dealers Engaged in the Sale of Redeemable Shares of Registered Investment Companies and Certain Other Share Accounts
A broker or dealer shall maintain net capital of not less than $25,000 if it acts as a broker or dealer with respect to the purchase, sale and redemption of redeemable shares of registered investment companies or of interests or participations in an insurance company separate account directly from or to the issuer on other than a subscription way basis. A broker or dealer operating under this section may sell securities for the account of a customer to obtain funds for the immediate reinvestment in redeemable securities of registered investment companies. A broker or dealer operating under this paragraph (a)(2)(v) must promptly transmit all funds and promptly deliver all securities received in connection with its activities as a broker or dealer, and may not otherwise hold funds or securities for, or owe money or securities to, customers.
15c3-1(a)(2)(vi) Other Brokers or Dealers
A broker or dealer that does not receive, directly or indirectly, or hold funds or securities for, or owe funds or securities to, customers and does not carry accounts of, or for, customers and does not engage in any of the activities described in paragraphs (a)(2) (i) through (v) of this section shall maintain net capital of not less than $5,000. A broker or dealer operating under this paragraph may engage in the following dealer activities without being subject to the requirements of paragraph (a)(2)(iii) of this section:
15c3-1(a)(2)(vi)(A) In the case of a buy order, prior to executing such customer's order, it purchases as principal the same number of shares or purchases shares to accumulate the number of shares necessary to complete the order, which shall be cleared through another registered broker or dealer or
15c3-1(a)(2)(vi)(B) In the case of a sell order, prior to executing such customer's order, it sells as principal the same number of shares or a portion thereof, which shall be cleared through another registered broker or dealer.
15c3-1(a)(2)(vi)/01 Riskless Principal Transactions
A broker who does riskless principal transactions in effectuating customer trades may be subject to a $5,000 minimum requirement, provided these transactions are made on a fully disclosed basis.
(SEC Staff to NYSE)
15c3-1(a)(2)(vi)/02 Requirements for Broker-Dealers Who Introduce Accounts on a Fully Disclosed Basis and Do Not Receive Securities
To be subject to the minimum requirements of paragraph (a)(2)(vi), introducing brokers must meet the requirements outlined in interpretation 15c3-1(a)(2)(iv)/01.
Introducing brokers should also maintain procedures to prevent their customers from transmitting securities and/or funds (other than checks made out to appropriate third parties) to the firm (except by error). Procedures should address the actions the broker will take to advise the customer (in writing) should they send securities and/or funds to the firm by error.
(SEC Release 34-31511, December 2, 1992)
(NYSE Interpretation Memo 93-6, November 1993)
15c3-1(a)(2)(vi)/021 Requirement to use a Qualified Escrow Agent
A broker-dealer operating pursuant to the $5,000 minimum net capital requirement of SEA Rule 15c3-1(a)(2)(vi) must comply with the provisions of SEA Rule 15c2-4(b)(2) when participating in a contingent best efforts underwriting or offering. This provision of the rule requires that customer funds in a contingent offering must be deposited in an escrow account with a qualified escrow agent, that has agreed in writing to hold such funds in escrow for the persons who have the beneficial interests therein and to transmit or return such funds directly to the persons entitled thereto, when the appropriate event or contingency has occurred. A qualified escrow agent must be a bank that is unaffiliated with either the issuer; general partner of the issuer; or the broker-dealer. A bank is defined in Section 3(a)(6) of the Securities Exchange Act and does not include a Savings and Loan Association or Credit Union. Failure to comply with the escrow requirements of SEA Rule 15c2-4(b)(2) subjects a $5,000 broker-dealer to a $250,000 minimum net capital requirement and nullifies its SEA Rule 15c3-3 exemption.
(SEC Staff of DMR to NASD Notice to Members 84-7)
15c3-1(a)(2)(vi)/03 Introducing Brokers - Receiving Funds
Any introducing broker that receives customer funds (checks made payable to itself and or cash), except by error, will be subject to the minimum net capital requirements of a broker-dealer that carries customer accounts (SEA Rule 15c3-1(a)(2)(i)).
(SEC Release 34-31511, December 2, 1992)
(NYSE Interpretation Memo 93-6, November 1993)
15c3-1(a)(2)(vi)/031 Error Transactions of Floor Brokers – (Rescinded)
(NYSE Interpretation Memo 02-7, August 2002)
15c3-1(a)(2)(vi)/032 Error Transactions of Floor Brokers
When a broker-dealer, which is primarily in the business of acting as a floor broker, makes an error in executing a transaction, which is done as a floor broker for another broker, no haircut need be taken on the resulting error position provided the security position is immediately liquidated upon discovery, but no later than the closing of the business day after the day the error occurred.
A broker-dealer is considered to be primarily in the business of acting as a floor broker when 75% of its gross revenue is derived from floor brokerage commissions.
This interpretation is applicable for a floor broker which either owns its seat or leases its seat.
(SEC Staff to NYSE) (NYSE Interpretation Memo 02-7, August 2002)
15c3-1(a)(2)(vi)/033 Introducing Broker-Dealers that Receive Only Customer Dividends or Capital Gains
An introducing broker-dealer that receives checks payable to itself, from a mutual fund, which result from dividends or capital gains in a customer’s account, will have a net capital requirement of $250,000 pursuant to SEA Rule 15c3-1(a)(2)(i), regardless of whether the customer requested this arrangement.
(SEC Staff of DMR to NASD, May 1993) (NASD Notice to Members 93-30, May 1993)
15c3-1(a)(2)(vi)/04 Sole Proprietor Joint Securities Account With Spouse
A sole proprietor broker-dealer’s joint securities account with a spouse should be reported on the Focus Balance Sheet. The transactions in this account should be counted in determining whether the broker-dealer effected more than ten (10) transactions in any one calendar year and subject to SEA Rule 15c3-1(a)(2)(iii).
Note: The account would be subject to PAB account requirements.
(SEC Staff to NYSE) (NYSE Interpretation Memo 01-3, March 2001)
(SEC Staff to FINRA) (FINRA Regulatory Notice 15-25)
15c3-1(a)(2)(vi)/05 Sole Proprietor IRA, Keogh or ERISA Accounts
Securities positions and money balances in IRA, Keogh or ERISA accounts of a sole proprietor broker-dealer do not need to be reported on the Focus Balance Sheet. The transactions in these accounts are also not counted in determining whether the broker-dealer effected more than ten (10) transactions in any one calendar year pursuant to SEA Rule 15c3-1(a)(2)(iii).
Note: These accounts would not be subject to PAB account requirements.
(SEC Staff to NYSE) (NYSE Interpretation Memo 01-3, March 2001)
(SEC Staff to FINRA) (FINRA Regulatory Notice 15-25)
15c3-1(a)(2)(vi)/06 Certificates of Deposit Issued by Bank
In order to not be considered as carrying customer accounts for purposes of SEA Rule 15c3-1 and SEA Rule 15c3-3, a broker-dealer that acts as agent for both the purchaser and seller in effecting transactions in bank certificates of deposit must have these clients sign a written agreement that explicitly acknowledges the clients’ understanding that the broker-dealer will have no obligation to the clients for the value of any bank certificates of deposit, any purchase price, or failure of any party with whom a transaction has been arranged to complete the transaction in accordance with its terms. The certificate of deposit must be issued by the bank in the name of the customer. If these conditions are met, this activity falls within the $5,000 minimum net capital requirement of SEA Rule 15c3-1(a)(2)(vi)
(SEC Letter to NASD, November 1993)
(SEC Letter to NASD, January 2004)
15c3-1(a)(3) [Reserved]
15c3-1(a)(4) Capital Requirements for Market Makers
A broker or dealer engaged in activities as a market maker as defined in paragraph (c)(8) of this section shall maintain net capital in an amount not less than $2,500 for each security in which it makes a market (unless a security in which it makes a market has a market value of $5 or less, in which event the amount of net capital shall be not less than $1,000 for each such security) based on the average number of such markets made by such broker or dealer during the 30 days immediately preceding the computation date. Under no circumstances shall it have net capital less than that required by the provisions of paragraph (a) of this section, or be required to maintain net capital of more than $1,000,000 unless required by paragraph (a) of this section.
15c3-1(a)(4)/01 Debt Securities
Market makers’ minimum capital requirements do not apply to bonds or other debt securities.
(SEC Staff to NYSE)
15c3-1(a)(4)/02 Average Number of Markets
The average number of markets made by a broker-dealer during the 30 calendar days immediately preceding the computation date is determined as follows:
Add the total number of markets made each day during the 30 calendar day period for securities with a representative ask price over $5 per share and divide the sum by the total number of business days in the same 30 calendar day period. Round quotient to the next highest number to determine the average number of markets made.
The same procedure should be used when determining the average number of markets made in securities with a representative ask price of $5 or less.
(SEC Staff to NASD)
15c3-1(a)(4)/03 Specialist Net Capital Requirements
The minimum net capital dollar amount requirement based on the number of securities in which the broker-dealer makes a market does not apply for securities in which the broker-dealer is registered as a specialist on a national securities exchange.
(SEC Staff to NYSE) (NYSE Interpretation Memo 76-4, April 1976)
15c3-1(a)(4)/04 Warrant and Underlying Common Stock
When a broker-dealer makes a market in a warrant and the underlying common stock which trade as separate issues, they must be treated as separate securities in determining the appropriate amount of net capital required to be maintained under SEA Rule 15c3-1(a)(4).
A registration statement describing the issue as a unit consisting of common stock and warrants has no effect on this determination.
(SEC Letter to Corna and Co. Inc., Investment Securities, March 17, 1989)
(NYSE Interpretation Memo 89-9, July 1989)
15c3-1(a)(5)(i) In accordance with appendix F to this section (§ 240.15c3-1f), the Commission may grant an application by an OTC derivatives dealer when calculating net capital to use the market risk standards of appendix F as to some or all of its positions in lieu of the provisions of paragraph (c)(2)(vi) of this section and the credit risk standards of appendix F to its receivables (including counterparty net exposure) arising from transactions in eligible OTC derivative instruments in lieu of the requirements of paragraph (c)(2)(iv) of this section. An OTC derivatives dealer shall at all times maintain tentative net capital of not less than $100 million and net capital of not less than $20 million.
15c3-1(a)(5)(ii) An OTC derivatives dealer that is also registered as a security-based swap dealer under section 15F of the Act (15 U.S.C. 78o-10) is subject to the capital requirements in §§ 240.18a-1, 240.18a-1a, 240.18a-1b, 240.18a-1c and 240.18a-1d instead of the capital requirements of this section and its appendices.
15c3-1(a)(6) Market Makers, Specialists and Certain Other Dealers
15c3-1(a)(6)(i) A dealer who meets the conditions of paragraph (a)(6)(ii) of this section may elect to operate under this paragraph (a)(6) and thereby not apply, except to the extent required by this paragraph (a)(6), the provisions of paragraphs (c)(2)(vi) or appendix A (§ 240.15c3-1a) of this section to market maker and specialist transactions and, in lieu thereof, apply thereto the provisions of paragraph (a)(6)(iii) of this section.
15c3-1(a)(6)(i)/01 Optional Financial Responsibility Standard
The optional standard can be used for any securities in which the broker-dealer makes a market; it is not restricted to specialist options or stocks.
The broker-dealer may elect to treat certain dealer securities under the optional standard, leaving other dealer securities subject to regular haircuts. For example, he might choose to clear his specialist options positions through another broker-dealer pursuant to SEA Rule 15c3-1(a)(6), yet clear his specialist stock positions himself. In this case the options positions are exempt from haircuts, while the stock positions are not.
(SEC Staff to NYSE) (NYSE Interpretation Memo 76-2, February 1976)
15c3-1(a)(6)(ii) This paragraph (a)(6) shall be available to a dealer who does not effect transactions with other than brokers or dealers, who does not carry customer accounts, who does not effect transactions in options not listed on a registered national securities exchange or facility of a registered national securities association, and whose market maker or specialist transactions are effected through and carried in a market maker or specialist account cleared by another broker or dealer as provided in paragraph (a)(6)(iv) of this section.
15c3-1(a)(6)(ii)/01 Introducing Options Specialist or Market Maker
An options specialist or market maker who introduces customers’ accounts to another broker-dealer on a fully disclosed basis may elect to operate under SEA Rule 15c3-1(a)(6) in regard to its specialist or market maker activities provided that the conditions of subparagraph (a)(6)(ii) are met in every respect.
(SEC Letter to Invemed Associates Inc., April 10, 1979)
(NYSE Interpretation Memo 79-10, December 1979)
15c3-1(a)(6)(ii)/02 Commodity Futures Trading
An options specialist or market maker will not lose the election to meet capital requirements under SEA Rule 15c3-1(a)(6) solely through trading in commodity futures.
(ASE Information Circular NYSE Interpretation Memo 78-22, October 26, 1978)
(NYSE Interpretation Memo 83-5, November 1983)
15c3-1(a)(6)(ii)/03 Specialist’s Personal Account – Non-Customer
When a specialist’s personal account is carried by the same broker-dealer that carries the specialist account, this personal account must be treated as a non-customer account. (See interpretation 15c3-3(a)(1)/011.)
(SEC Staff to NYSE) (NYSE Interpretation Memo 87-6, May 1987)
15c3-1(a)(6)(iii) A dealer who elects to operate pursuant to this paragraph (a)(6) shall at all times maintain a liquidating equity in respect of securities positions in his market maker or specialist account at least equal to:
15c3-1(a)(6)(iii)(A) An amount equal to 25 percent (5 percent in the case of exempted securities) of the market value of the long positions and 30 percent of the market value of the short positions; provided, however, in the case of long or short positions in options and long or short positions in securities other than options which relate to a bona fide hedged position as defined in paragraph (c)(2)(x)(C) of this section, such amount shall equal the deductions in respect of such positions specified by appendix A (§ 240.15c3-1a).
15c3-1(a)(6)(iii)(B) Such lesser requirement as may be approved by the Commission under specified terms and conditions upon written application of the dealer and the carrying broker or dealer.
15c3-1(a)(6)(iii)(C) For purposes of this paragraph (a)(6)(iii), equity in such specialist or market maker account shall be computed by
15c3-1(a)(6)(iii)(C)(1) marking all securities positions long or short in the account to their respective current market values,
15c3-1(a)(6)(iii)(C)(2) adding (deducting in the case of a debit balance) the credit balance carried in such specialist or market maker account, and
15c3-1(a)(6)(iii)(C)(3) adding (deducting in the case of short positions) the market value of positions long in such account.
15c3-1(a)(6)(iii)/01 Equity On Deposit With Carrying Broker
The equity on deposit with the carrying broker to meet the requirements specified in this subparagraph is not deducted from net worth in computing net capital. The requirements may be met by depositing in the account any cash or securities that may be used by the market maker or specialist, including all cash and securities contributed as subordinated liabilities or capital, whether pursuant to conforming agreements or not, as well as trading and investment account securities in which the computing broker-dealer is not a market maker or specialist.
(SEC Staff to NYSE) (NYSE Interpretation Memo 76-2, February 1976)
15c3-1(a)(6)(iv) The dealer shall obtain from the broker or dealer carrying the market maker or specialist account a written undertaking which shall be designated “Notice Pursuant to § 240.15c3-1(a)(6) of Intention to Carry Specialist or Market Maker Account.” Said undertaking shall contain the representations required by paragraph (a)(6) of this section and shall be filed with the Commission's Washington, DC, Office, the regional office of the Commission for the region in which the broker or dealer has its principal place of business and the Designated Examining Authorities of both firms prior to effecting any transactions in said account. The broker or dealer carrying such account:
15c3-1(a)(6)(iv)(A) Shall mark the account to the market not less than daily and shall issue appropriate calls for additional equity which shall be met by noon of the following business day;
15c3-1(a)(6)(iv)(B) Shall notify by telegraph the Commission and the Designated Examining Authorities pursuant to 17 CFR 240.17a-11, if the market maker or specialist fails to deposit any required equity within the time prescribed in paragraph (a)(6)(iv)(A) of this section; said telegraphic notice shall be received by the Commission and the Designated Examining Authorities not later than the close of business on the day said call is not met;
15c3-1(a)(6)(iv)(C) Shall not extend further credit in the account if the equity in the account falls below that prescribed in paragraph (a)(6)(iii) of this section, and
15c3-1(a)(6)(iv)(D) Shall take steps to liquidate promptly existing positions in the account in the event of a failure to meet a call for equity.
15c3-1(a)(6)(v) No such carrying broker or dealer shall permit the sum of
15c3-1(a)(6)(v)(A) the deductions required by paragraph (c)(2)(x)(A) of this section in respect of all transactions in market maker accounts guaranteed, indorsed or carried by such broker or dealer pursuant to paragraph (c)(2)(x) of this section and
15c3-1(a)(6)(v)(B) the equity required by paragraph (iii) of this paragraph (a)(6) in respect of all transactions in the accounts of specialists of market makers in options carried by such broker or dealer pursuant to this paragraph (a)(6) to exceed 1,000 percent of such broker's or dealer's net capital as defined in paragraph (c)(2) of this section for any period exceeding five business days; Provided, That solely for purposes of this paragraph (a)(6)(v), deductions or equity required in a specialist or market maker account in respect of positions in fully paid securities (other than options), which do not underlie options listed on the national securities exchange or facility of a national securities association of which the specialist or market marker is a member, need not be recognized. Provided further, That if at any time such sum exceeds 1,000 percent of such broker's or dealer's net capital, then the broker or dealer shall immediately transmit telegraphic notice of such event to the principal office of the Commission in Washington, DC, the regional office of the Commission for the region in which the broker or dealer maintains its principal place of business, and such broker's or dealer's Designated Examining Authority. Provided further, That if at any time such sum exceeds 1,000 percent of such broker's or dealer's net capital, then such broker or dealer shall be subject to the prohibitions against withdrawal of equity capital set forth in paragraph (e) of this section, and to the prohibitions against reduction, prepayment and repayment of subordination agreements set forth in paragraph (b)(11) of § 240.15c3-1d, as if such broker or dealer's net capital were below the minimum standards specified by each of the aforementioned paragraphs.
15c3-1(a)(7) Alternative Net Capital Computation for Broker-Dealers Authorized to Use Models
In accordance with appendix E to this section (§ 240.15c3-1e), the Commission may approve, in whole or in part, an application or an amendment to an application by a broker or dealer to calculate net capital using the market risk standards of appendix E to compute a deduction for market risk on some or all of its positions, instead of the provisions of paragraphs (c)(2)(vi) and (c)(2)(vii) of this section, and using the credit risk standards of appendix E to compute a deduction for credit risk on certain credit exposures arising from transactions in derivatives instruments, instead of the provisions of paragraph (c)(2)(iv) of this section, subject to any conditions or limitations on the broker or dealer the Commission may require as necessary or appropriate in the public interest or for the protection of investors. A broker or dealer that has been approved to calculate its net capital under appendix E must:
15c3-1(a)(7)(i)(A) At all times maintain tentative net capital of not less than $5 billion and net capital of not less than the greater of $1 billion or the sum of the ratio requirement under paragraph (a)(1) of this section and:
15c3-1(a)(7)(i)(A)(1) Two percent of the risk margin amount; or
15c3-1(a)(7)(i)(A)(2) Four percent or less of the risk margin amount if the Commission issues an order raising the requirement to four percent or less on or after the third anniversary of this section's compliance date; or
15c3-1(a)(7)(i)(A)(3) Eight percent or less of the risk margin amount if the Commission issues an order raising the requirement to eight percent or less on or after the fifth anniversary of this section's compliance date and the Commission had previously issued an order raising the requirement under paragraph (a)(7)(i)(B) of this section;
15c3-1(a)(7)(i)(B) If, after considering the capital and leverage levels of brokers or dealers subject to paragraph (a)(7) of this section, as well as the risks of their security-based swap positions, the Commission determines that it may be appropriate to change the percentage pursuant to paragraph (a)(7)(i)(A)(2) or (3) of this section, the Commission will publish a notice of the potential change and subsequently will issue an order regarding any such change.
15c3-1(a)(7)(ii) Provide notice that same day in accordance with § 240.17a-11(g) if the broker's or dealer's tentative net capital is less than $6 billion. The Commission may, upon written application, lower the threshold at which notification is necessary under this paragraph (a)(7)(ii), either unconditionally or on specified terms and conditions, if a broker or dealer satisfies the Commission that notification at the $6 billion threshold is unnecessary because of, among other factors, the special nature of its business, its financial position, its internal risk management system, or its compliance history; and
15c3-1(a)(7)(iii) Comply with § 240.15c3-4 as though it were an OTC derivatives dealer with respect to all of its business activities, except that paragraphs (c)(5)(xiii), (c)(5)(xiv), (d)(8), and (d)(9) of § 240.15c3-4 shall not apply.
15c3-1(a)(8) Municipal securities brokers' brokers.
15c3-1(a)(8)(i) A municipal securities brokers' brokers, as defined in subsection (ii) of this paragraph (a)(8), may elect not to be subject to the limitations of paragraph (c)(2)(ix) of this section provided that such brokers' broker complies with the requirements set out in paragraphs (a)(8) (iii), (iv) and (v) of this section.
15c3-1(a)(8)(i)/01 SEC Approval Required
A broker that elects to operate under this paragraph may not switch back to the paragraph (a) method of computing net capital without SEC approval.
(SEC Letter to NASD, October 24, 1983)
(NYSE Interpretation Memo 87-6, May 1987)
15c3-1(a)(8)(ii) The term municipal securities brokers' broker shall mean a municipal securities broker or dealer who acts exclusively as an undisclosed agent in the purchase or sale of municipal securities for a registered broker or dealer or registered municipal securities dealer, who has no “customers” as defined in this rule and who does not have or maintain any municipal securities in its proprietary or other accounts.
15c3-1(a)(8)(ii)/01 Acceptable Security Investments
Idle cash may be invested in short term investments in government securities falling within subparagraph (c)(2)(vi)(A)(1) Category 1 or securities qualifying under subparagraph (c)(2)(vi)(E)(1). No other proprietary positions are permitted.
Municipal securities are prohibited as SDN collateral.
(SEC Letter to NASD, October 24, 1983)
(NYSE Interpretation Memo 87-6, May 1987)
15c3-1(a)(8)(iii) In order to qualify to operate under this paragraph (a)(8), a brokers' broker shall at all times have and maintain net capital of not less than $150,000.
15c3-1(a)(8)(iv) For purposes of this paragraph (a)(8), a brokers' broker shall deduct from net worth 1% of the contract value of each municipal failed to deliver contract which is outstanding 21 business days or longer. Such deduction shall be increased by any excess of the contract price of the fail to deliver over the market value of the underlying security.
15c3-1(a)(8)(iv)/01 Fail to Deliver Extensions Prohibited
The extension provision contained in subparagraph (c)(2)(ix) is not available. Twenty one business days is deemed sufficient.
(SEC Letter to NASD, October 24, 1983)
(NYSE Interpretation Memo 87-6, May 1987)
15c3-1(a)(8)(v) For purposes of this paragraph (a)(8), a brokers' broker may exclude from its aggregate indebtedness computation indebtedness adequately collateralized by municipal securities outstanding for not more than one business day and offset by municipal securities failed to deliver of the same issue and quantity. In no event may a brokers' broker exclude any overnight bank loan attributable to the same municipal securities failed to deliver contract for more than one business day. A brokers' broker need not deduct from net worth the amount by which the market value of securities failed to receive outstanding longer than thirty (30) calendar days exceeds the contract value of those failed to receive as required by Rule 15c3-1(c)(2)(iv)(E).
15c3-1(a)(8)(v)/01 Corporate Bond Brokers' Broker
A corporate bond brokers' broker has been permitted to operate according to this paragraph (a)(8), with certain modifications.
(SEC Letter to Wolfe & Drizos Corporates, Inc., August 19, 1986)
(NYSE Interpretation Memo 87-6, May 1987)
15c3-1(a)(9) Certain Additional Capital Requirements for Brokers or Dealers Engaging in Reverse Repurchase Agreements
A broker or dealer shall maintain net capital in addition to the amounts required under paragraph (a) of this section in an amount equal to 10 percent of:
15c3-1(a)(9)(i) The excess of the market value of United States Treasury Bills, Bonds and Notes subject to reverse repurchase agreements with any one party over 105 percent of the contract prices (including accrued interest) for reverse repurchase agreements with that party;
15c3-1(a)(9)(ii) The excess of the market value of securities issued or guaranteed as to principal or interest by an agency of the United States or mortgage related securities as defined in section 3(a)(41) of the Act subject to reverse repurchase agreements with any one party over 110 percent of the contract prices (including accrued interest) for reverse repurchase agreements with that party; and
15c3-1(a)(9)(iii) The excess of the market value of other securities subject to reverse repurchase agreements with any one party over 120 percent of the contract prices (including accrued interest) for reverse repurchase agreements with that party.
15c3-1(a)(10) Broker-Dealers Registered as Security-Based Swap Dealers
A broker or dealer registered with the Commission as a security-based swap dealer, other than a broker or dealer subject to the provisions of paragraph (a)(7) of this section, must:
15c3-1(a)(10)(i)
15c3-1(a)(10)(i)(A) At all times maintain net capital of not less than the greater of $20 million or the sum of the ratio requirement under paragraph (a)(1) of this section and:
15c3-1(a)(10)(i)(A)(1) Two percent of the risk margin amount; or
15c3-1(a)(10)(i)(A)(2) Four percent or less of the risk margin amount if the Commission issues an order raising the requirement to four percent or less on or after the third anniversary of this section's compliance date; or
15c3-1(a)(10)(i)(A)(3) Eight percent or less of the risk margin amount if the Commission issues an order raising the requirement to eight percent or less on or after the fifth anniversary of this section's compliance date and the Commission had previously issued an order raising the requirement under paragraph (a)(10)(i)(B) of this section;
15c3-1(a)(10)(i)(B) If, after considering the capital and leverage levels of brokers or dealers subject to paragraph (a)(10) of this section, as well as the risks of their security-based swap positions, the Commission determines that it may be appropriate to change the percentage pursuant to paragraph (a)(10)(i)(A)(2) or (3) of this section, the Commission will publish a notice of the potential change and subsequently will issue an order regarding any such change; and
15c3-1(a)(10)(ii) Comply with § 240.15c3-4 as though it were an OTC derivatives dealer with respect to all of its business activities, except that paragraphs (c)(5)(xiii) and (xiv), and (d)(8) and (9) of § 240.15c3-4 shall not apply.
15c3-1(b) Exemptions:
15c3-1(b)(1) The provisions of this section shall not apply to any specialist:
15c3-1(b)(1)(i) Whose securities business, except for an occasional non-specialist related securities transaction for its own account, is limited to that of acting as an options market maker on a national securities exchange;
15c3-1(b)(1)(i)/01 Occasional Transactions
Excess funds may be invested in reverse repurchase agreement transactions as often as necessary, and not be counted as occasional investment transactions.
(SEC Staff to NYSE)
15c3-1(b)(1)(ii) That is a member in good standing and subject to the capital requirements of a national securities exchange;
15c3-1(b)(1)(iii) That does not transact a business in securities with other than a broker or dealer registered with the Commission under section 15 or section 15C of the Act or a member of a national securities exchange; and
15c3-1(b)(1)(iv) That is not a clearing member of The Options Clearing Corporation and whose securities transactions are effected through and carried in an account cleared by another broker or dealer registered with the Commission under section 15 of the Act.
15c3-1(b)(1)/01 Non-Specialist Transactions
A specialist, market maker or a competitive options trader operating under this paragraph (b)(1) exemption may not engage in trading non-specialist securities. However, they may engage in hedging transactions if they are directly related to their market making or specialist activity. They may also make occasional investment account transactions in non-specialist securities (not more than 10 per year).
Excess funds may be invested in reverse repurchase agreement transactions as often as necessary, and not be counted as occasional.
(SEC Staff to NYSE) (NYSE Interpretation Memo 90-11, December 1990)
15c3-1(b)(1)/02 Servicing Family Accounts
A specialist member organization may not service the customer accounts of members of its partners' (or stockholders') families without losing its exemption from the rule.
(SEC Staff to NYSE)
15c3-1(b)(1)/021 Servicing Partners Accounts
A specialist member organization may not service the individual accounts of its partners' or stockholders' without losing its exemption from the rule.
(SEC Staff to NYSE) (NYSE Interpretation Memo 89-6, June 1989)
15c3-1(b)(1)/03 Specialist Trading in Futures
A broker who is exempt from SEA Rule 15c3-1 under this section will not lose the exemption solely through trading in commodity futures.
(ASE Circular NYSE Interpretation Memo 78-72, October 26, 1978)
(SEC Staff to NYSE) (NYSE Interpretation Memo 83-5, November 1983)
15c3-1(b)(2) A member in good standing of a national securities exchange who acts as a floor broker (and whose activities do not require compliance with other provisions of this rule), may elect to comply, in lieu of the other provisions of this section, with the following financial responsibility standard: The value of the exchange membership of the member (based on the lesser of the most recent sale price or current bid price for an exchange membership) is not less than $15,000, or an amount equal to the excess of $15,000 over the value of the exchange membership is held by an independent agent in escrow: Provided, That the rules of such exchange require that the proceeds from the sale of the exchange membership of the member and the amount held in escrow pursuant to this paragraph shall be subject to the prior claims of the exchange and its clearing corporation and those arising directly from the closing out of contracts entered into on the floor of such exchanges.
15c3-1(b)(2)/01 Floor Brokers Elective
Exchange floor brokers currently satisfying capital requirements under this section will not lose their elective solely through trading in commodity futures.
(ASE Circular NYSE Interpretation Memo 78-72, October 26, 1978)
(SEC Staff to NYSE) (NYSE Interpretation Memo 83-5, November 1983)
15c3-1(b)(2)/011 Non-Exchange Member Transactions
An exempt floor broker who executes transactions for a broker-dealer who is not a member of the same exchange is subject to the minimum requirements of SEA Rule 15c3-1(a).
(SEC Staff to NYSE) (NYSE Interpretation Memo 93-6, November 1993)
15c3-1(b)(2)/02 Availability of Exemption
The exemption is available even though the rules of a national securities exchange do not require that the proceeds from the sale be held in escrow and be subject to prior claims, provided that the escrow agreement provides for such treatment.
(SEC Staff to NYSE) (NYSE Interpretation Memo 87-6, May 1987)
15c3-1(b)(2)/03 Floor Broker Error Transactions – (Rescinded)
(NYSE Interpretation Memo 99-5, May 1999)
15c3-1(b)(2)/031 Error Transactions of Floor Brokers – (Rescinded)
(NYSE Interpretation Memo 03-2, March 2003)
15c3-1(b)(2)/032 Error Transactions of Floor Brokers
When a broker-dealer, which is primarily in the business of acting as a floor broker, makes an error in executing a transaction, which is done as a floor broker for another broker, no haircut need be taken on the resulting error position provided the security position is immediately liquidated upon discovery, but no later than the closing of the business day after the day the error occurred.
A broker-dealer is considered to be primarily in the business of acting as a floor broker when 75% of its gross revenue is derived from floor brokerage commissions.
This interpretation is applicable for a floor broker which either owns its seat or leases its seat.
(SEC Staff to NYSE) (NYSE Interpretation Memo 03-2, March 2003)
15c3-1(b)(2)/04 Floor Brokers Activities
This elective is available to a floor broker that does not receive, directly or indirectly, or hold, securities for, or owe funds or securities to, customers and does not carry accounts of, or for, customers and does not engage in any of the activities described in paragraphs (a)(2)(i) through (v) of SEA Rule 15c3-1.
(SEC Staff to NYSE) (NYSE Interpretation Memo 93-6, November 1993)
15c3-1(b)(3) The Commission may, upon written application, exempt from the provisions of this section, either unconditionally or on specified terms and conditions, any broker or dealer who satisfies the Commission that, because of the special nature of its business, its financial position, and the safeguards it has established for the protection of customers' funds and securities, it is not necessary in the public interest or for the protection of investors to subject the particular broker or dealer to the provisions of this section.
15c3-1(c) Definitions. For the purpose of this section:
15c3-1(c)(1) Aggregate Indebtedness
The term aggregate indebtedness shall be deemed to mean the total money liabilities of a broker or dealer arising in connection with any transaction whatsoever and includes, among other things, money borrowed, money payable against securities loaned and securities “failed to receive,” the market value of securities borrowed to the extent to which no equivalent value is paid or credited (other than the market value of margin securities borrowed from customers in accordance with the provisions of 17 CFR 240.15c3-3 and margin securities borrowed from non-customers), customers' and non-customers' free credit balances, credit balances in customers' and non-customers' accounts having short positions in securities, equities in customers' and non-customers' future commodities accounts and credit balances in customers' and non-customers' commodities accounts, but excluding:
15c3-1(c)(1)/01 Guarantees in Aggregate Indebtedness
Guarantees are generally included.
(SEC Staff to NYSE)
15c3-1(c)(1)/02 Subordinations - Accrued Interest
Accrued interest payable to subordinated lenders is excluded.
(SEC Staff to NYSE)
15c3-1(c)(1)/03 Unsold Long Securities
The phrase "... and which have not been sold ..." in exclusions (i), (ii) and (iii) prevents a broker from excluding from aggregate indebtedness liabilities relating to securities sold to a customer on a COD basis which have not yet been delivered by a broker-dealer using settlement date accounting. This phrase should not be interpreted as referring to trade date sales.
(SEC Staff to NYSE)
15c3-1(c)(1)/031 Securities Pledged Against Securities Borrowed
Liabilities collateralized by; (1) securities borrowed from other broker-dealers obtained by pledging proprietary securities, or (2) securities pledged by other broker-dealers upon the lending of proprietary securities, need not be included in aggregate indebtedness.
(SEC Staff to NYSE) (NYSE Interpretation Memo 77-2, June 1977)
15c3-1(c)(1)/04 Deferred Income
Some deferred income items are excluded from aggregate indebtedness. For example, unearned plan fees are excludable to the extent they will not have to be refunded.
(SEC Staff to NYSE)
15c3-1(c)(1)/05 Letters of Credit
Letters of credit unsecured, or secured by proprietary securities and/or spot commodities, are excludable from aggregate indebtedness. Letters of credit secured by customers' securities and/or spot commodities are includable in aggregate indebtedness but only to the extent needed to satisfy margin or other obligations.
(SEC Staff to NYSE)
15c3-1(c)(1)/06 Continuous Net Settlement (CNS) Balances
The net allocated "customer" balance in the continuous net settlement (CNS) account, if a credit, is included in aggregate indebtedness. The net allocated "customer" balance (debit or credit) is also included in the SEA Rule 15c3-3 Reserve Formula.
(SEC Staff to NYSE)
15c3-1(c)(1)/07 Bank Overdrafts
Bank overdrafts and balances in drafts payable accounts representing checks or drafts drawn in excess of book balances may be reduced to the extent of debit balances in accounts representing cash balances carried by banks provided that:
both the debit and the credit represent balances carried at the same bank, and
the bank has the absolute right of offsetting the balance in one account against the balance in the other account.
(SEC Staff to NYSE) (NYSE Interpretation Memo 76-4, April 1976)
15c3-1(c)(1)/08 Drafts for Immediate Credit
The immediate credit proceeds advanced on drafts with securities attached deposited for collection are included in aggregate indebtedness unless they are collateralized by securities owned (and which have not been sold) by the broker-dealer or by securities collateralizing a secured demand note pursuant to SEA Rule 15c3-1d (Appendix D). (See interpretation 15c3-1(c)(1)/03.)
(SEC Staff to NYSE) (NYSE Interpretation Memo 76-4, April 1976)
15c3-1(c)(1)/09 Intercompany Accounts with Subsidiaries
An unsubordinated amount payable to a subsidiary is Aggregate Indebtedness of the parent.
An unsubordinated amount payable to a parent is Aggregate Indebtedness of the subsidiary.
The above apply when there is no consolidation of assets and liabilities for Net Capital purposes as stipulated in SEA Rule 15c3-1c. (Appendix C)
(Also, see Certain Receivables under interpretation 15c3-1(c)(2)(iv)(C)/07.)
(SEC Staff to NYSE) (NYSE Interpretation Memo 79-4, March 1979)
15c3-1(c)(1)/10 Unsecured Receivables and Payable With the Same Party
See other deductions under interpretation 15c3-1(c)(2)(iv)(E)/04.
(SEC Staff to NYSE) (NYSE Interpretation Memo 82-2, April 1982)
15c3-1(c)(1)/11 Accrued Liability for Concessions or Commissions Payable
That portion of such accrued liabilities that are payable more than twelve months from the computation date may be excluded from aggregate indebtedness under the provisions specified at interpretation 15c3-1(c)(2)(iv)(C)/095.
(SEC Letter to NASD, July 24, 1984)
(NYSE Interpretation Memo 87-6, May 1987)
(SEC Staff to FINRA) (FINRA Regulatory Notice 21-27)
15c3-1(c)(1)/12 Liability for Law Suit Damages, Penalties, etc.
Where long term liabilities, such as damages in a lawsuit, penalties, etc., are payable in installments or a lump sum over a long term, the full amount of the liability must be recorded and included as Aggregate Indebtedness.
In the event the liability is recorded on the books of account at present value under GAAP, the full amount of the liability (not the present value amount) must be included in Aggregate Indebtedness. (Also, see interpretation 15c3-1(c)(2)/012.)
(SEC Staff to NYSE) (NYSE Interpretation Memo 90-11, December 1990)
15c3-1(c)(1)/13 Accrued "Soft Dollar" Research Liabilities
Section 28(e) of the Securities Exchange Act of 1934 provides a safe harbor to investment managers who use the commission dollars of their advised accounts to obtain investment research and brokerage services ("soft dollar" research) by permitting such persons, under certain circumstances, to "pay up" for such services received from broker-dealers.
Accrued liabilities under "soft dollar" arrangements resulting from the receipt of commission revenues from customers before the third party research is provided is includable in Aggregate Indebtedness. The accrued liabilities are to be accounted for on a customer-by-customer basis and may not be netted with deferred research expenses unless both are with the same customer. These liabilities may not be added back to net worth for Net Capital computation purposes.
Accruals for "soft dollar" research liabilities are to be determined in accordance with generally accepted accounting principles including the proper matching of revenues and expenses during each accounting period. The possibility of inflated revenues, accelerated expenses recognition and the propriety of accruals associated with "soft dollar" arrangements should be of particular concern. (See interpretation 15c3-1(c)(2)(iv)(E)/15.)
Appropriate accounting records which show the research obligations and expenses should be maintained for each customer having a "soft dollar" research arrangement covered by this interpretation
(SEC Staff to NYSE) (NYSE Interpretation Memo 91-13, August 1991)
15c3-1(c)(1)/14 Adverse Award in an Arbitration Proceeding
A broker-dealer that is the subject of an adverse award in an arbitration proceeding should book said award as an actual liability at the time the award is made, even though the appeal process has not been exhausted and no judgment has been rendered, because grounds for revision on appeal are very limited. In addition, the award would be included in Aggregate Indebtedness as there is no exclusion available under SEA Rule 15c3-1(c)(1).
(SEC Staff of DMR to NASD, September 1988)
15c3-1(c)(1)/15 Commission, Registration, or Wrap Fee Refund
A broker-dealer that offers customers fee-related services, and that permits a partial refund of fees if the customer terminates the arrangement before the entire fee is earned, must reflect the unused portion of the fee subject to a refund as a liability and a component of aggregate indebtedness.
(SEC Staff of DMR to NASD, June 16, 1993)
15c3-1(c)(1)/16 Court Judgment Rendered against a Broker-Dealer
A court judgment adverse to a broker-dealer is, at a minimum, a contingent liability of the firm and included in the calculation of aggregate indebtedness unless an opinion of counsel indicates otherwise. If the broker-dealer has exhausted its remedies, the liability must be booked. Each situation must be analyzed on the particular facts present in the matter.
(SEC Letter to NASD, February 8, 1978)
15c3-1(c)(1)/17 Deposit in a Special Agency or Trust Account
Funds related to a best efforts underwriting that are on deposit in a special trust or agency account must be included in the calculation of aggregate indebtedness under SEA Rule 15c3-1(c)(1).
(SEC Staff of DMR to NASD, September 1983)
15c3-1(c)(1)/18 Distributor or Underwriter of Mutual Funds and Other Investment Companies Includes Payables in Aggregate Indebtedness
A broker-dealer that is an underwriter of investment company shares may not net amounts payable to the investment company against related receivables from other broker-dealers that had purchased shares of the investment company. Such a broker-dealer must include the amounts it owes to the investment company in its aggregate indebtedness under SEA Rule 15c3-1(c)(1). A distributor of mutual fund shares is covered by this interpretation.
(SEC Letter to NASD, May 1, 1979)
(SEC Letter to NASD, July 16, 1981)
(SEC Letter to F. Eberstadt & Co., May 21, 1979)
15c3-1(c)(1)/19 Good Faith Deposits Placed with the Managing Underwriter
Good faith deposits, placed with the managing underwriter by joint account participants in connection with municipal underwritings, represent payables of the managing underwriter to the joint account members and are included in the managing underwriter’s calculation of aggregate indebtedness under SEA Rule 15c3-1(c)(1).
(SEC Letter to Seasongood & Mayer, June 7, 1977)
15c3-1(c)(1)/20 Temporary Capital Infusions in a Broker-dealer – (Rescinded)
(SEC Staff to FINRA) (FINRA Regulatory Notice 14-06)
15c3-1(c)(1)/21 One-Day Bank Loan for DVP Transactions
A broker-dealer uses a local bank as agent for delivery of securities and collection of payment for out-of-town delivery versus payment (DVP) transactions. At the same time, the firm obtains a loan from the same bank collateralized by the securities to be delivered. The loan remains outstanding for one additional day after the DVP transaction has been completed. Said loan must be classified as aggregate indebtedness pursuant to SEA Rule 15c3-1(c)(1) until the funds are actually received by the bank and the loan is eliminated.
(SEC Staff of DMR to NASD, December 1982)
15c3-1(c)(1)/22 Partner’s Securities that are used to Collateralize Firm Loans
If securities owned by a general partner are left with the broker-dealer to use in the ordinary course of its business and such securities are used to collateralize firm loans, such indebtedness is included in the broker-dealer’s calculation of aggregate indebtedness as required by SEA Rule 15c3-1(c)(1), absent a related approved secured demand note or a capital contribution of said securities pursuant to the partnership agreement.
(SEC Letter to Stone and Youngberg Investment Securities, January 6, 1978)
15c3-1(c)(1)/23 Partnership “Capital” that was Contributed by Non-Partners
In a partnership, persons other than partners may contribute capital to a broker-dealer only by means of an approved subordination agreement or secured demand note. Any other contributions by such persons are loans made to the partnership and constitute liabilities of the broker-dealer. These liabilities must be included in the calculation of aggregate indebtedness under SEA Rule 15c3-1(c)(1).
(SEC Staff of DMR to NASD, September 1988)
15c3-1(c)(1)/24 Deferred Prepaid Investment Advisory Fees
Deferred prepaid investment advisory fees, that have not yet been earned, must be included in the broker-dealer’s aggregate indebtedness pursuant to SEA Rule 15c3-1.
(SEC Letter to A.R. Schmeidler & Co., Inc., December 4, 1976)
15c3-1(c)(1)/25 Rent Payments that are Not Required during Part of Lease Term (Free Rent)
The following sets forth the appropriate treatment of free rent. By way of illustration, assume that a broker-dealer enters into an office lease agreement for a five-year term with an annual rent of $12,000. The lease contains a provision that there is no payment of rent required for the first year of the lease; that is, free rent for one year.
The total rent payment over the life of the lease ($48,000) should be divided by the total number of months in the lease period (60 months). The result will be an $800 per month prorated rent expense that should be recorded monthly on the firm’s books and records, even though actual rent payments under the lease do not begin until the second year.
(SEC Staff of DMR to NASD, June 1985)
15c3-1(c)(1)/26 Fines and Other Monetary Penalties Assessed by a Governmental Agency or Self-Regulatory Organization
A fine, an order to pay restitution or similar penalty imposed by a governmental agency or self-regulatory organization (“fine”), at a minimum, shall be treated as a contingent liability of the broker-dealer and included in the computation of aggregate indebtedness at the time such fine is imposed.
In addition, under Generally Accepted Accounting Principles (GAAP), broker-dealers have an ongoing obligation to assess the specific facts applicable to each pending or decided matter that may result or has resulted in the imposition of a fine and to make a determination as to whether an actual liability must be recorded in the financial statements.
In any event, once all available appeals or other remedies have been exhausted, the broker-dealer must record the full amount of the fine as a liability in its financial statements.
Note: This interpretation does not apply to adverse awards resulting from arbitration proceedings or adverse court judgments. See interpretations 15c3-1(c)(1)/14 (Adverse Award in an Arbitration Proceeding) and 15c3-1(c)(1)/16 (Court Judgment Rendered Against a Broker-Dealer) for the applicable Net Capital treatment in such instances.
(SEC Staff to FINRA) (FINRA Regulatory Notice 13-44)
15c3-1(c)(1)(i) Exclusions From Aggregate Indebtedness
Indebtedness adequately collateralized by securities which are carried long by the broker or dealer and which have not been sold or by securities which collateralize a secured demand note pursuant to appendix D to this section 17 CFR 240.15c3-1d; indebtedness adequately collateralized by spot commodities which are carried long by the broker or dealer and which have not been sold; or, until October 1, 1976, indebtedness adequately collateralized by municipal securities outstanding for not more than one business day and offset by municipal securities failed to deliver of the same issue and quantity, where such indebtedness is incurred by a broker or dealer effecting transactions solely in municipal securities who is either registered with the Commission or temporarily exempt from such registration pursuant to 17 CFR 240.15a-1(T) or 17 CFR 240.15Ba2-3(T);
15c3-1(c)(1)(i)/01 Commingled and Inadequately Secured Loans
Commingled bank loans and inadequately secured firm bank loans are treated as follows:
If a bank loan which has been arranged as a customer loan contains securities other than customers' securities, the entire bank loan should be treated as aggregate indebtedness.
If the bank loan contains any non-exempted securities (until June 1, 1976, other than municipals), the 4% option set forth in subparagraph (c)(2)(xiii) is not available.
If a bank loan which has been arranged as a firm loan contains non-firm (customers' and/or "non-customers'") securities, the amount of the bank loan should be prorated based upon the loan value of the collateral and the prorated amounts treated in accordance with the pertinent provisions of SEA Rule 15c3-1, as either:
excluded from aggregate indebtedness;
included in aggregate indebtedness; or
if appropriate, as eligible for the 4% option set forth in subparagraph (c)(2)(xiii).
The following formula is used for pro-ration:
| | | | | | |
--- --- --- |
| Amount includable in aggregate indebtedness (or eligible for 4% option) | = | Amount Borrowed | | x | loan value of non-firm collateral loan value of all collateral |
| | |
For example, assume that the amount borrowed is $1,000,000; that firm collateral market value is $2,000,000; that non-firm collateral market value is $1,000,000; and that the broker can borrow 50% on firm collateral and 80% on non-firm collateral.
| | | |
---
| Amount includable in aggregate indebtedness | = | $1,000,000 x 80% of $1,000,000 80% of $1,000,000 + 50% of $2,000,000 |
| | = | $444,444 |
If a bank loan which has been arranged as a firm loan contains firm securities only but is not "adequately secured", the amount of the bank loan should be prorated between adequately secured and unsecured based upon the loan value of the collateral and the prorated amounts treated in accordance with the pertinent provisions of SEA Rule 15c3-1, as either excluded from, or included in, aggregate indebtedness.
Notes: The terms "non-customer" and "adequately secured" are defined in the rule.
Securities that are the subject of satisfactory subordination agreements are considered firm securities for purposes of these interpretations.
(SEC Staff to NYSE)
15c3-1(c)(1)(i)/02 Indebtedness in the Proprietary Trading Account of a Broker-Dealer
When a broker-dealer has a proprietary trading account carried by another broker-dealer, indebtedness in such account is considered “adequately collateralized” if the broker-dealer’s equity in such account is at least equal to the haircut requirements specified in SEA Rules 15c3-1(c)(2)(vi) and (vii) on the positions in such account. If the proprietary trading account is not adequately collateralized (i.e., the equity in the account is less than the haircuts required on the positions), the excess of the haircut amount over the equity in the account, up to the amount of the indebtedness, shall be included as aggregate indebtedness. The remainder of the indebtedness may be considered adequately collateralized and may be excluded from aggregate indebtedness.
Example: A broker-dealer has incurred $100,000 of indebtedness in a proprietary trading account carried by another broker. The account also contains $100,000 of marketable equity securities and $50,000 of non-marketable securities. In this case, the firm’s equity in the account is $50,000, and the SEA Rule 15c3-1(c)(2)(vi) and (vii) haircut requirements on the positions in the account are $65,000 (15% of $100,000 and 100% of $50,000). Since the firm’s equity is less than the haircut requirements, the indebtedness is not adequately collateralized. The firm must include the amount by which the haircuts on the positions exceed the equity in the account (i.e., $15,000) in aggregate indebtedness. The remainder of the indebtedness (i.e., $85,000) may be excluded from aggregate indebtedness.
(SEC Staff to FINRA) (FINRA Regulatory Notice 21-27)
15c3-1(c)(1)(ii) Amounts payable against securities loaned, which securities are carried long by the broker or dealer and which have not been sold or which securities collateralize a secured demand note pursuant to Appendix (D) (17 CFR 240.15c)
15c3-1(c)(1)(ii)/01 Continuous Net Settlement (CNS) Balances Allocation
Securities loaned allocating to long CNS positions (securities of the same issue and quantity) are excluded from aggregate indebtedness.
(SEC Staff to NYSE) (NYSE Interpretation Memo 06-5, June 2006)
15c3-1(c)(1)(iii) Amounts payable against securities failed to receive which securities are carried long by the broker or dealer and which have not been sold or which securities collateralize a secured demand note pursuant to Appendix (D) (17 CFR 240.15c3-1d) or amounts payable against securities failed to receive for which the broker or dealer also has a receivable related to securities of the same issue and quantity thereof which are either fails to deliver or securities borrowed by the broker or dealer;
15c3-1(c)(1)(iii)/01 Continuous Net Settlement (CNS) Balances Allocation
Fails to receive allocated to long CNS positions (securities of the same issue and quantity) are excluded from aggregate indebtedness. (See interpretation 15c3-1(c)(1)/06.)
(SEC Staff to NYSE) (NYSE Interpretation Memo 79-4, March 1979)
15c3-1(c)(1)(iii)/02 Institutional Short Positions
Credit balances in the account of an institutional customer which results from the non-delivery of securities sold by the institution by settlement date:
Shall not be excluded from Aggregate indebtedness as if it were Fail to Receive.
The use of the “fail to receive” concept should be limited to transactions between brokers and dealers.
(SEC Letter to M. A. Schapiro & Co., Inc., June 7, 1983)
(NYSE Interpretation Memo 87-6, May 1987)
15c3-1(c)(1)(iv) Credit balances in accounts representing amounts payable for securities or money market instruments not yet received from the issuer or its agent which securities are specified in paragraph (c)(2)(vi)(E) and which amounts are outstanding in such accounts not more than three (3) business days;
15c3-1(c)(1)(v) Equities in customers' and non-customers' accounts segregated in accordance with the provisions of the Commodity Exchange Act and the rules and regulations thereunder;
15c3-1(c)(1)(vi) Liability reserves established and maintained for refunds of charges required by section 27(d) of the Investment Company Act of 1940, but only to the extent of amounts on deposit in a segregated trust account in accordance with 17 CFR 270.27d-1 under the Investment Company Act of 1940;
15c3-1(c)(1)(vii) Amounts payable to the extent funds and qualified securities are required to be on deposit and are deposited in a “Special Reserve Bank Account for the Exclusive Benefit of Customers” pursuant to 17 CFR 240.15c3-3 under the Securities Exchange Act of 1934;
15c3-1(c)(1)(vii)/01 Funds on Deposit in a “Special Bank Account For The Exclusive Benefit Of Customers”
Broker-dealers that use a “Special Account for the Exclusive Benefit of Customers” pursuant to SEA Rule 15c3-3(k)(2)(i) can not exclude the amounts payable to customers pursuant to SEA Rule 15c3-1(c)(1)(vii). Amounts payable to customers, to the extent funds are on deposit in a “Special Account for the Exclusive Benefit of Customers”, must be included in the calculation of aggregate indebtedness.
(SEC Letter to MSE, June 2, 1977)
15c3-1(c)(1)(vii)/02 Reserve Bank Account Deposit/Requirement
Pursuant to SEA Rule 15c3-1(c)(1)(vii), a broker/dealer may exclude from aggregate indebtedness customer-related liabilities to the extent funds are on deposit and required to be on deposit in a Special Reserve Bank Account for the Exclusive Benefit of Customers established pursuant to SEA Rule 15c3-3(e). At its option, prior to the next required reserve deposit timeframe, a firm may exclude the amount that was required to be and was on deposit as of the previous reserve formula calculation date from the current calculation of aggregate indebtedness, even if the current reserve formula calculation has a lower requirement. Should the current reserve formula calculation have a higher deposit requirement than the previous reserve formula calculation, the firm may elect to use the current reserve formula calculation amount to reduce aggregate indebtedness, provided the necessary funds were on deposit in the Special Reserve Bank Account for the Exclusive Benefit of Customers as of that date.
(SEC Staff of DMR to NASD, June 1980)
15c3-1(c)(1)(viii) Fixed liabilities adequately secured by assets acquired for use in the ordinary course of the trade or business of a broker or dealer but no other fixed liabilities secured by assets of the broker or dealer shall be so excluded unless the sole recourse of the creditor for nonpayment of such liability is to such asset;
15c3-1(c)(1)(viii)/01 Assets Acquired in the Ordinary Course of Business
“Assets acquired for use in the ordinary course of the trade or business of a broker or dealer” are those which generally are necessary for the conduct of a brokerage business. This is determined on a case-by-case basis. An office building acquired for the purpose of rental income would not qualify. A truck primarily used to make deliveries of securities would qualify.
(SEC Staff to NYSE)
15c3-1(c)(1)(viii)/011 Automobiles Do Not Qualify
Fixed liabilities secured by automobiles cannot be excluded from A. I. as the automobiles are not necessary for the conduct of a brokerage business.
(SEC Letter to Investors Diversified Services, Inc., January 18, 1983)
(NYSE Interpretation Memo 87-6, May 1987)
15c3-1(c)(1)(viii)/012 Demonstrated Collateral Value
To receive favorable treatment, the broker-dealer must demonstrate to its designated examining authority that the asset adequately secures the indebtedness within the meaning of subparagraph (c)(5) of SEA Rule 15c3-1.
(SEC Letter to NASD, April 17, 1986)
(NYSE Interpretation Memo 87-6, May 1987)
15c3-1(c)(1)(viii)/013 Partially Collateralized Loan
A partially collateralized loan may receive favorable treatment up to the amount that the loan is adequately collateralized.
(SEC Staff to NYSE) (NYSE Interpretation Memo 87-6, May 1987)
15c3-1(c)(1)(viii)/02 Capitalized Leases
When the lease of computer or telephone equipment from a vendor-lessor or bank or other financial institution is capitalized, 50% of the capitalized lease liability may be treated as adequately secured for a period of two years after the lease was entered into without demonstration of the adequacy of the collateral.
(SEC Letter to NASD, April 17, 1986)
(NYSE Interpretation Memo 87-6, May 1987)
15c3-1(c)(1)(viii)/021 Capitalized Leases - Sole Recourse
To be treated as adequately secured for more than 50% of the liability, sole recourse must be to the asset pledged or the broker-dealer must demonstrate that the asset adequately secures the liability.
(SEC Staff to NYSE) (NYSE Interpretation Memo 87-6, May 1987)
15c3-1(c)(1)(viii)/03 Fixed Liabilities - Remaining Maturity of One Year or More
The term "fixed liabilities" includes only those liabilities with remaining maturity of one year or more. The portion of such liability which matures in less than one year is considered current and may not be excluded from aggregate indebtedness whether or not sole recourse of the creditor for nonpayment of such liability is to assets securing the liability.
(SEC Staff to NYSE) (NYSE Interpretation Memo 91-6, July 1991)
15c3-1(c)(1)(ix) Liabilities on open contractual commitments;
15c3-1(c)(1)(x) Indebtedness subordinated to the claims of creditors pursuant to a satisfactory subordination agreement, as defined in Appendix (D) (17 CFR 240.15c3-1d);
15c3-1(c)(1)(x)/01 Cash SDN Collateral
Money deposited as collateral for a secured demand note is not included in aggregate indebtedness except to the extent that it exceeds the principal amount of the note.
(SEC Staff to NYSE) (NYSE Interpretation Memo 76-4, April 1976)
15c3-1(c)(1)(x)/02 Payment Prohibited on a Subordination, which Continues to be Capital and Excluded from Aggregate Indebtedness
Subordinated agreements that have reached their stated or accelerated maturity date and for which payment is prohibited pursuant to subparagraph (b)(8) of SEA Rule 15c3-1d (Appendix D) shall continue to be excluded from aggregate indebtedness. Such subordinated agreements are treated as capital until the broker-dealer is able to repay the loan.
(SEC Staff of DMR to NASD, September 1975)
(SEC Letter to Philadelphia Stock Exchange, July 11, 1977)
15c3-1(c)(1)(xi) Liabilities which are effectively subordinated to the claims of creditors (but which are not subject to a satisfactory subordination agreement as defined in Appendix (D) (17 CFR 240.15c3-1d)) by non-customers of the broker or dealer prior to such subordination, except such subordinations by customers as may be approved by the Examining Authority for such broker or dealer;
15c3-1(c)(1)(xii) Credit balances in accounts of general partners;
15c3-1(c)(1)(xiii) Deferred tax liabilities;
15c3-1(c)(1)(xiii)/01 Deferred Tax Exclusion
All deferred tax liabilities are excluded from aggregate indebtedness, including those that might be added back to net worth as described in SEA Rule 15c3-1(c)(2)(i)(C).
(SEC Staff to NYSE)
15c3-1(c)(1)(xiv) Eighty-five percent of amounts payable to a registered investment company related to fail to deliver receivables of the same quantity arising out of purchases of shares of those registered investment companies; and
15c3-1(c)(1)(xv) Eighty-five percent of amounts payable against securities loaned for which the broker or dealer has receivables related to securities of the same class and issue and quantity that are securities borrowed by the broker or dealer.
15c3-1(c)(2) Net Capital
The term net capital shall be deemed to mean the net worth of a broker or dealer, adjusted by:
15c3-1(c)(2)/01 Generally Accepted Accounting Principles (GAAP)
Net worth is to be computed in accordance with generally accepted accounting principles.
Some exceptions to this are:
Unlisted options are carried at their intrinsic values (or “in the money” amounts) - under GAAP they are carried at the unamortized premium amounts.
Leases on unoccupied premises such as closed branch offices continue to be accounted for as leases until their expiration dates - under GAAP the discounted present value of the lease payments is generally accounted for as a money liability.
Flow through capital from affiliated entities pursuant to SEA Rule 15c3-1c (Appendix C) is required to be reported on a one line basis for unaudited FOCUS Reports - under GAAP flow through capital from affiliated entities is generally reported on a line-by-line basis.
Broker-dealers that use the trade date basis of record keeping are in compliance with the AICPA Guide for Audits of Brokers and Dealers in Securities while broker-dealers that use the settlement date basis of recordkeeping are in compliance with the Guide only if the difference between trade date and settlement date accounting is not material.
A broker-dealer must have a consistent policy of reflecting all transactions either on a trade date or a settlement date basis and must compute its net capital on the same basis as it uses in recording its transactions. However, if settlement date accounting is used, and
there is a “material difference” between trade date accounting, the net capital computation must reflect the trade date position for proprietary positions, and
if there is a material difference on more than an occasional basis (i.e., twice in a six month period), trade date accounting should probably be used on a consistent basis.
The broker-dealer must continue to use the method chosen unless it advises its designated examining authority of a change in method or a change is required because of a “material difference”.
The securities record required by SEA Rule 17a-3(a)(5) and the formula computation set forth in SEA Rule 15c3-3a (Exhibit A) must be maintained and computed, respectively, on a settlement date basis.
Any net receivables or payables resulting from the recording of proprietary positions on a trade date basis are not deducted from net worth nor included in aggregate indebtedness.
(SEC Staff to NYSE) (NYSE Interpretation Memo 76-4, April 1976)
(SEC Letter to A.I.C.P.A., April 23, 1986) (NYSE Interpretation Memo 86-10, December 1986)
(SEC Staff to NYSE) (NYSE Interpretation Memo 02-7, August 2002)
15c3-1(c)(2)/011 Guarantees and Contingencies
Losses which could result from guarantees or contingencies should be accrued and deducted in computing net worth when occurrence of a loss is probable and the amount can be reasonably estimated.
(SEC Staff to NYSE) (NYSE Interpretation Memo 81-10, February 1981)
15c3-1(c)(2)/0110 Lawsuits as Contingent Liabilities
A broker-dealer that is the subject of a lawsuit that could have a material impact on its net capital must obtain an opinion of outside counsel regarding the potential effect of such a suit on the firm’s financial condition. Absent such opinion, the item must be considered, at a minimum, a contingent liability, and be included in the calculation of aggregate indebtedness that is required by SEA Rule 15c3-1(c)(1).
(SEC Staff of DMR to NASD, September 1988)
15c3-1(c)(2)/012 Liabilities Recorded on a Present Value Basis
Broker-dealers may, in some instances, record long term liabilities on the books of account on a present value basis. In that event the full amount of the liability, and not the present value amount, must be treated as a liability in the net worth computation and the full amount must be deducted in the net capital computation. (Also, see interpretation 15c3-1(c)(1)/12.)
(SEC Staff to NYSE) (NYSE Interpretation Memo 90-11, December 1990)
15c3-1(c)(2)/013 Accrual Method of Accounting
All registered broker-dealers are required to use the accrual method of accounting in order to ensure a proper matching of revenues and expenses and to provide an accurate reflection of a broker-dealer's financial condition.
(SEC Release 34-18737, May 13, 1982)
(NYSE Interpretation Memo 93-6, November 1993)
15c3-1(c)(2)/014 Reporting Customers’ Balances on FOCUS Report
The reporting of customers’ balances on a trade date basis only on the FOCUS Balance Sheet is acceptable provided this does not conflict with Generally Accepted Accounting Practices (GAAP). Customers’ balances must still be reported on a settlement date basis in preparing the Reserve Formula.
(SEC Staff to NYSE) (NYSE Interpretation Memo 99-5, May 1999)
15c3-1(c)(2)/015 Broker-Dealer As Qualified Lender
When a broker-dealer is a Qualified Lender for a revolving subordinated loan agreement (See interpretation 15c3-1d(a)(2)(v)(F)/01 (Appendix D)) the following applies:
it takes a charge to net capital for the full amount of the loan commitment whether or not drawn down by the borrower, from the effective date of the revolving subordinated loan agreement through the maturity date of the loans thereunder; and
at the time it enters into the revolving subordinated loan agreement, after taking the charge to net capital set forth above, its aggregate indebtedness does not exceed 1000% of its net capital nor is its net capital less than 120% of the minimum dollar amount required by SEA Rule 15c3-1 or, in the case of a broker-dealer operating pursuant to paragraph (a)(1)(ii) of SEA Rule 15c3-1, its net capital would not be less than 5 percent of its aggregate debit items computed in accordance with SEA Rule 15c3-3a (Exhibit A), or if registered as a futures commission merchant, its net capital would not be less than 7 percent of the funds required to be segregated pursuant to the Commodity Exchange Act and the regulations thereunder, if greater.
The charge to net capital for the portion of the loan commitment that is not drawn down by the borrower should be deducted by the Qualified Lender on the FOCUS Computation of Net Capital, line item number 3610.
(SEC Staff to NYSE) (NYSE Interpretation Memo 01-4, April 2001)
15c3-1(c)(2)/016 Federal Tax Claim on a Broker-Dealer
The Internal Revenue Service has the power to attach assets of the appellant to insure payment of the disputed tax in certain situations. A stay of tax payment is not automatic in the Tax Court forum; rather, the appellant must post a bond equal to the amount of the claim and subsequently file an appeal. Consequently, although all legal remedies have not been exhausted, for purposes of computing net capital under SEA Rule 15c3-1(c)(2), the tax claim should usually be booked as an actual liability at the time the administrative procedures are exhausted with the IRS.
(SEC Staff of DMR to NASD, September 1992)
15c3-1(c)(2)/02 Discretionary Liabilities
In determining net capital, accrued amounts that are payable solely at the discretion of the broker-dealer for bonuses, profit-sharing, etc., may be added back to net worth net of any related tax benefit. Such amounts must still be reported as proposed or scheduled capital withdrawals on reports pursuant to SEA Rule 17a-5, and while excludable from Aggregate Indebtedness are still considered as liabilities for purposes of paragraphs (d) and (e) and SEA Rule 15c3-1d (Appendix D). (See interpretation 15c3-1(c)(2)/07 for treatment of Employee Stock Ownership Plans.)
(SEC Letter to Becker Securities Corporation, April 14, 1976)
(SEC Staff to NYSE) (NYSE Interpretation Memo 79-4, March 1979)
15c3-1(c)(2)/03 Partners’ Securities Capital
Securities may be contributed by partners as capital with any related income or loss accruing to the partner individually. Generally accepted accounting principles prescribe that ordinarily such securities be valued at market prices and included in net worth. However, since for the purpose of this rule such securities are not to be given any value in computing net capital and the debt-equity total, they are treated by interpretation as not being included in net worth.
A partner may contribute securities as capital and have them included in net worth for net capital purposes, provided the securities are then owned by the firm for all purposes (i.e., the firm has all of the incidents of ownership), are thus recorded in the firm trading and investment accounts and are subject to appropriate haircuts. Of course partners wishing to make contributions without giving up the incidences of ownership may still do so through the use of a “secured demand note” as provided under SEA Rule 15c3-1d (Appendix D).
(SEC Staff to NYSE)
15c3-1(c)(2)/031 Ownership of Proprietary Securities
Securities in the proprietary account of a broker-dealer that were contributed to capital must be owned by the firm (that is, the firm has all of the incidents of ownership) and therefore must be held in bearer, nominee, or firm name in order to have value for net capital purposes. An individual may not serve as a nominee.
(SEC Letter to NASD, June 28, 1978)
(SEC Staff of DMR to NASD, July 1978)
15c3-1(c)(2)/04 Forward Commodity Contracts
In determining net worth, net positions in forward commodity contracts may be valued at the price quoted on a recognized commodity exchange for the future contract month closest to the forward delivery date or by interpolation, provided that the terms of the contract are comparable to the contract traded on the commodity exchange.
(SEC Staff to NYSE)
15c3-1(c)(2)/05 Deferred Taxes in Accordance with GAAP
Unincorporated entities, as well as corporations, that are subject to taxes on income must accrue deferred taxes in accordance with GAAP.
(SEC Staff to NYSE) (NYSE Interpretation Memo 76-4, April 1976)
15c3-1(c)(2)/06 Options
See SEA Rule 15c3-1a (Appendix A).
15c3-1(c)(2)/07 Employee Stock Ownership Plans (ESOPS)
So long as a broker-dealer does not have reason to believe that an ESOP will be disapproved by the Internal Revenue Service, an ESOP that has been adopted subject to such approval may be treated for net capital purposes as if the approval had been received. However, FOCUS reports and net capital computations should be footnoted to indicate that such approval is pending. Any changes to the plan in order to secure approval should also be taken into consideration.
To the extent that contributions to an ESOP are at the broker-dealer's discretion, provisions for anticipated contributions should be treated as described in interpretation 15c3-1(c)(2)/02. However, if the investing activities of the ESOP are restricted to buying stock directly from the broker-dealer (as opposed to its existing shareholders) the amounts accrued need not be considered as scheduled capital withdrawals for FOCUS report purposes.
Contributions that have been declared as payable by the broker-dealer to the ESOP are liabilities that generally are includable in aggregate indebtedness. If the activities of the ESOP are limited to purchasing the broker-dealer's stock directly from the broker-dealer, the liability of the broker-dealer to the ESOP is not included in aggregate indebtedness.
Leveraged ESOP's should be treated as follows:
An obligation of an ESOP should be recorded in the employer's financial statements as a liability when the obligation is covered by the employer's guarantee or his commitment to make future contributions to the ESOP sufficient to meet the debt service requirements.
The offsetting debit to the liability should be shown as a reduction of shareholders' equity.
The debit should be reduced and the offset to shareholders' equity should be restored only as the ESOP makes payments on the debt.
The employer's compensation expense is the amount contributed or committed to be contributed for a given year.
Exchange member organizations proposing to establish an ESOP should submit drafts of the proposals to Surveillance Director.
(SEC Staff to NYSE) (NYSE Interpretation Memo 77-2, June 1977)
15c3-1(c)(2)/08 Capital Contributions from Parent, Using Borrowed Funds
The parent of a broker-dealer may borrow funds and infuse those funds as additional paid-in capital into the firm without adverse net capital consequences provided the broker-dealer:
Is not, in any way, a party to the lending arrangement;
Has no assets, directly or indirectly, pledged to secure the loan; and
Is not subject to any recourse of any kind to the lender for collection of the loan against the parent.
(SEC Staff of DMR to NASD)
15c3-1(c)(2)(i) Adjustments to net worth related to unrealized profit or loss, deferred tax provisions, and certain liabilities.
15c3-1(c)(2)(i)(A) Adding unrealized profits (or deducting unrealized losses) in the accounts of the broker or dealer;
15c3-1(c)(2)(i)(B)
15c3-1(c)(2)(i)(B)(1) In determining net worth, all long and all short positions in listed options shall be marked to their market value and all long and all short securities and commodities positions shall be marked to their market value.
15c3-1(c)(2)(i)(B)(1)/01 Non-Marketable Securities, Mark to Market
Securities, long or short, with no ready market should be valued at fair value as determined by the management of the broker-dealer.
Valuation procedures for securities that are not readily marketable should be designed to approximate the value that would have been established by market forces and “are generally a good faith estimate by management to determine the value of non-marketable securities”. Among other things, consideration should be given to prices at which recent sales (purchases) were made with clients, customers or others.
(SEC Letter to Power Securities Corporation, October 3, 1988)
(NYSE Interpretation Memo 89-6, June 1989)
15c3-1(c)(2)(i)(B)(2) In determining net worth, the value attributed to any unlisted option shall be the difference between the option's exercise value and the market value of the underlying security. In the case of an unlisted call, if the market value of the underlying security is less than the exercise value of such call it shall be given no value and in the case of an unlisted put if the market value of the underlying security is more than the exercise value of the unlisted put it shall be given no value.
15c3-1(c)(2)(i)(B)(2)/01 Long and Short Unlisted Options
In determining net worth, both long and short unlisted options are valued at their “in the money” amounts. The time value portion of the premium is ignored. (See interpretation 15c3-1(c)(2)/01.)
(SEC Staff to NYSE)
15c3-1(c)(2)(i)(C) Adding to net worth the lesser of any deferred income tax liability related to the items in (1), (2), and (3) below, or the sum of (1), (2) and (3) below;
15c3-1(c)(2)(i)(C)(1) The aggregate amount resulting from applying to the amount of the deductions computed in accordance with paragraph (c)(2)(vi) of this section and Appendices A and B, § 240.15c3-1a and 240.15c3-1b, the appropriate Federal and State tax rate(s) applicable to any unrealized gain on the asset on which the deduction was computed;
15c3-1(c)(2)(i)(C)(2) Any deferred tax liability related to income accrued which is directly related to an asset otherwise deducted pursuant to this section;
15c3-1(c)(2)(i)(C)(3) Any deferred tax liability related to unrealized appreciation in value of any asset(s) which has been otherwise deducted from net worth in accordance with the provisions of this section; and,
15c3-1(c)(2)(i)(C)/01 Local Taxes (City)
Local (city) income taxes also qualify to be added back.
(SEC Staff to NYSE)
15c3-1(c)(2)(i)(C)/02 Haircuts and Undue Concentration Charges
The potential addback is computed by applying the tax rates to the total of all the haircuts and undue concentration charges. It is not computed only on positions which have an unrealized gain.
Example:
| | | |
---
| 30% Securities | Cost | Market Value |
| A | 100 | 200 |
| B | 100 | 100 |
| C | 100 | 80 |
| Total | 300 | 380 |
The tax rates are applied to 30% of $380, or $114.
This amount is still effectively limited to the actual deferred tax credit attributable to specified items such as these. For example, assume that the broker had no timing differences other than the net unrealized appreciation in these three securities. His deferred tax credit account would reflect the tax rate applied to the $80. His addback would thus be limited to the amount in the deferred tax credit account.
(SEC Staff to NYSE)
15c3-1(c)(2)(i)(C)/03 Short Positions (Related Taxes)
This subparagraph also applies to taxes related to short positions.
(SEC Staff to NYSE)
15c3-1(c)(2)(i)(D) Adding, in the case of future income tax benefits arising as a result of unrealized losses, the amount of such benefits not to exceed the amount of income tax liabilities accrued on the books and records of the broker or dealer, but only to the extent such benefits could have been applied to reduce accrued tax liabilities on the date of the capital computation, had the related unrealized losses been realized on that date.
15c3-1(c)(2)(i)(D)/01 Deferred Tax Debit
The rule provides that tax benefits can be added to net worth to the extent that income tax liabilities could have been reduced on the date of the capital computation, if the related unrealized losses had been realized on that date. This means that, for example, a calendar year company that has realized gains but unrealized losses at December 3lst can offset the unrealized losses to the extent of the realized gains on that date but immediately at the beginning of the next year will not have the right of offset for the unrealized losses, since no realized gains as yet exist.
Certain deferred tax debits relating to deferred compensation payable may be treated in a like manner. Where a deferred compensation plan is structured so that the broker-dealer may at any time at its option make payment, the related deferred income tax receivable may be applied to reduce the tax liability on the broker-dealer's books for other items.
(SEC Staff to NYSE) (NYSE Interpretation Memo 77-2, June 1977)
15c3-1(c)(2)(i)(E) Adding to net worth any actual tax liability related to income accrued which is directly related to an asset otherwise deducted pursuant to this section.
15c3-1(c)(2)(i)(E)/01 Add-Back to Net Worth of Deferred Tax Liabilities Directly Related to Certain Non-Allowable Assets
A broker-dealer may add-back to net worth the amount of its deferred tax liabilities which are directly related to the following non-allowable assets:
Software assets capitalizing certain costs associated with internal software development;
Prepaid advertising; and
Distribution network and prepaid selling commission.
(SEC Letter to Charles Schwab & Co., Inc., October 25, 1999)
(SEC Letter to Advanced Clearing, Inc., October 25, 1999)
(NYSE Interpretation Memo 01-3, March 2001)
(SEC Letter to John Hancock Funds, LLC, December 12, 2006)
(NYSE Interpretation Memo 07-4, April 2007)
15c3-1(c)(2)(i)(F) Subtracting from net worth any liability or expense relating to the business of the broker or dealer for which a third party has assumed the responsibility, unless the broker or dealer can demonstrate that the third party has adequate resources independent of the broker or dealer to pay the liability or expense.
15c3-1(c)(2)(i)(G) Subtracting from net worth any contribution of capital to the broker or dealer:
15c3-1(c)(2)(i)(G)/01 Services Arrangement with a Parent or an Affiliate
Any payment made by a broker-dealer, directly or indirectly, to its parent or an affiliate, in connection with an arrangement whereby the parent or affiliate provides services to the broker-dealer (“services arrangement”) shall be considered a capital withdrawal for purposes of SEA Rules 15c3-1(c)(2)(i)(G) and 15c3-1(e), unless the broker-dealer demonstrates that the following conditions are met:
at the time the service(s) were provided to the broker-dealer, the services arrangement was in writing and specified such service(s) with a reasonable and consistent basis for determining the cost of each service (e.g., utilizing a percentage of the broker-dealer’s net income to determine the cost to be charged by a parent or affiliate for technology services provided by a parent or affiliate, for example, may not be deemed “a reasonable basis” because the cost of obtaining such services generally does not fluctuate based on the level of a broker-dealer’s net income);
the service(s) provided were related to the broker-dealer’s business;
the parent or affiliate had the ability to provide such service(s); and
the parent or affiliate provided the service(s).
For purposes of this interpretation, “payment” shall include any reduction or forgiveness of a receivable from the broker-dealer’s parent or affiliate.
(SEC Staff to FINRA) (FINRA Regulatory Notice 21-27)
15c3-1(c)(2)(i)(G)(1) Under an agreement that provides the investor with the option to withdraw the capital; or
15c3-1(c)(2)(i)(G)(2) That is intended to be withdrawn within a period of one year of contribution. Any withdrawal of capital made within one year of its contribution is deemed to have been intended to be withdrawn within a period of one year, unless the withdrawal has been approved in writing by the Examining Authority for the broker or dealer.
15c3-1(c)(2)(ii) Subordinated liabilities. Excluding liabilities of the broker or dealer which are subordinated to the claims of creditors pursuant to a satisfactory subordination agreement, as defined in appendix (D) (17 CFR 240.15c3-1d).
15c3-1(c)(2)(iii) Sole proprietors. Deducting, in the case of a broker or dealer who is a sole proprietor, the excess of liabilities which have not been incurred in the course of business as a broker or dealer over assets not used in the business.
15c3-1(c)(2)(iv) Assets not readily convertible into cash. Deducting fixed assets and assets which cannot be readily converted into cash (less any indebtedness excluded in accordance with subdivision (c)(1)(viii) of this section) including, among other things:
15c3-1(c)(2)(iv)/01 Liabilities Secured by Non Allowable Assets
For a fixed liability to be allowed as an offset against assets, as described in subparagraph (c)(1)(viii) of SEA Rule 15c3-1, the loan agreement must be submitted to and acceptable to the Exchange prior to such reduction becoming effective.
(NYSE Information Memo 80-66) (NYSE Interpretation Memo 83-2, April 1983)
15c3-1(c)(2)(iv)/011 Fixed Liabilities-Definition
Fixed liabilities are liabilities with remaining maturity of one year or more. The portion of such liabilities which matures in less than one year is considered current and cannot be used as an offset against assets, as described in subparagraph (c)(1)(viii) of SEA Rule 15c3-1.
(SEC Staff to NYSE) (NYSE Interpretation Memo 91-6, July 1991)
15c3-1(c)(2)(iv)/02 Suspense Accounts
The net capital treatment of security positions and money balances whose ultimate disposition is not known, e.g. DKs and suspense items which remain unresolved seven (7) business days after discovery (see interpretation 15c3-1(c)(2)(iv)/022), is as follows:
Long position and related debit balance, and short position and related credit balance – proprietary commitments that cannot operate to increase net capital;
Long positions only – ignored;
Short position only – deduct appropriate % of current market value (see SEA Rule 15c3-1(c)(2)(v)(A));
Credit balance only – included in aggregate indebtedness;
Debit balance only – charged;
DKs and other suspense items which as of the capital computation date are not yet seven business days old need not be deducted from net worth or included in Aggregate Indebtedness in the current capital computation. If resolved, such items may be treated properly (as appropriate) upon reclassification or resolution. However, all items that are seven business days or older as of the computation date described would be reported as suspense items, even if resolved before the FOCUS is filed, and are treated as above.
(SEC Staff to NYSE) (NYSE Interpretation Memo 79-4, March 1979)
15c3-1(c)(2)(iv)/021 Reconciliation Differences
Net overall unfavorable bank account, correspondent account, clearing corporation and securities depository reconciliation differences disclosed by reconciliations other than those required under SEA Rule 17a-13 (Quarterly Security Counts) shall be deducted in computing net capital if not resolved within the below named number of business days from the date of receipt of the statement of account from the carrying entity. Each broker-dealer having any such differences shall maintain a record of the date of receipt of the pertinent statement of account or in the absence of such record, shall compute the elapsed days from the date of the statement. The treatment of differences disclosed by reconciliations required under SEA Rule 17a-13 shall be governed by the requirements of that Rule.
| | |
--- |
| | Not Resolved Within |
| Bank accounts | Seventeen (17 business days |
| Correspondents' accounts | Seventeen (17) business days |
| Clearing corporations | Seven (7) business days |
| Securities depositories | Seven (7) business days |
Differences which have been resolved but which have not yet been appropriately corrected in the records shall be so identified on the reconciliations and may be considered resolved for purposes of this interpretation. It is expected that such differences will not carry over to the following reconciliation. Unresolved favorable and unfavorable differences with the same carrying entity may be netted for purposes of determining the impact, if any, upon the capital computation. The net overall amount or value of an unresolved favorable difference shall be disregarded for capital purposes. The net overall amount or value of an unresolved unfavorable difference within a carrying entity shall be deducted in computing net capital.
For example, the following represent deductions because differences were not resolved with the carrying entity within 17 business days from the date of receipt of the statements of accounts:
| | | | |
--- --- |
| A Bank Account | Item One | Item Two | Totals |
| Deposits per books | $15,000 | $16,000 | $31,000 |
| Deposits per bank statement | 10,000 | 18,000 | 28,000 |
| Differences | $ 5,000 | $-2,000 | $ 3,000 |
A deduction of $3,000 is required in computing net capital.
Statement of Account Received from a Correspondent
| | |
--- |
| Long 100 XYC - current value | $10,000 |
| Per books 100 XYB- current value | 12,000 |
| Deduction required in computing net capital | $ 2,000 |
NYSE Rule 440.10 calls for receipt of position statements as frequently as good business practice requires, but not less than once per month with respect to securities held by clearing corporations, the Depository Trust Company, other organizations, or correspondents. The rule also requires the reconciliation at least once per month of all such securities and money balances by comparison of the clearing corporation’s or correspondent’s position statements to the broker-dealer’s books and records. It also requires that differences be promptly reported to the contra organization and that such differences be promptly resolved by both.
(SEC Staff to NYSE) (NYSE Interpretation Memo 79-4, March 1979)
15c3-1(c)(2)(iv)/022 Security Suspense and Differences
In computing net capital, the deductions for long and short security suspense and difference positions with related money amounts and short positions without related money noted in interpretations 15c3-1(c)(2)(iv)/02 and /021 above may be graduated over 28 business days in accordance with the schedule described under subparagraph (c)(2)(v) of SEA Rule 15c3-1.
(SEC Staff to NYSE) (NYSE Interpretation Memo 83-2, April 1983)
15c3-1(c)(2)(iv)/023 Cash Suspense and Securities Difference Accounts Are Not to be Netted
Amounts in cash suspense accounts are to be treated broadly (that is, the debits, as of the net capital computation date, are to be treated as non-allowable assets, and the total credits are to be included in aggregate indebtedness).
Securities difference accounts may not be netted for any reason. Long securities differences may not increase net capital.
(SEC Staff of DMR to NASD, April 1981)
15c3-1(c)(2)(iv)/03 Subordinations - Non-Conforming
Both the asset and the liability relating to a non-conforming subordination shall be ignored in computing net capital if the claim of the lender to the related asset is subordinate to the claims of general creditors.
(SEC Staff to NYSE)
15c3-1(c)(2)(iv)/04 Cash Surrender Value
The cash surrender value of life insurance policies is not a charge if it is owned by the broker-dealer and the face amount of the policies are payable to the broker-dealer.
(SEC Staff to NYSE) (NYSE Interpretation Memo 76-4, April 1976)
15c3-1(c)(2)(iv)/05 Cash Surrender Value – “Split Dollar” Life Insurance Policy
The cash surrender value of a “split dollar” life insurance policy recorded as an asset on the books of a broker-dealer is not required to be deducted from net worth in determining net capital provided such asset is readily convertible into cash and is payable directly to the broker-dealer in the event of the death of the insured or the termination of the policy by the broker-dealer.
Under a “split dollar” life insurance policy, the cash surrender value is payable to the broker-dealer and any excess of face amount of the policy over the cash surrender value upon the death of the insured is paid to a designated beneficiary.
(SEC Letter to Smith Barney, Harris Upham & Co., Inc., August 10, 1978) (NYSE Interpretation Memo 79-4, March 1979)
15c3-1(c)(2)(iv)/06 Assets Secured by a Letter of Credit
Assets not otherwise readily convertible into cash, secured only by a letter of credit (whether collateralized or not) must be deducted from net worth in computing net capital.
(SEC Letter to NASD, August 17, 1981)
(SEC Release 34-18737, May 12, 1982)
(NYSE Interpretation Memo 83-2, April 1983)
15c3-1(c)(2)(iv)/07 Whole Loan Mortgages
Broker-dealer's investments in whole loan mortgages or whole loan mortgage pools (e.g., mortgage loans that have not been converted into securitized form) are not allowable for purposes of computing net capital.
(SEC Staff to NYSE) (NYSE Interpretation Memo 92-4, January 1992)
15c3-1(c)(2)(iv)/08 DTC Preferred Stock
The par value ($100 per share) of the new variable rate non-cumulative non-voting Series “A” Preferred Stock issued by the Depository Trust Corporation (DTC), for which a broker-dealer is required to purchase under DTC’s clearing fund formula can be considered an allowable asset. This amount should also be reported on the Focus Balance Sheet under Clearing Organizations – Other on line item number 290.
(SEC Letter to The Depository Trust Company, August 21, 2000)
(NYSE Interpretation Memo 01-3, March 2001)
15c3-1(c)(2)(iv)/09 Assets Pledged as Collateral for a Surety Bond
A broker-dealer that is required to maintain a surety bond by the state in which it is headquartered and pledges its interest in a savings account as collateral for the bond must treat the pledged asset as non-allowable for net capital purposes.
(SEC Staff of DMR to NASD, April 3, 1980)
15c3-1(c)(2)(iv)(A) Fixed assets and prepaid items. Real estate; furniture and fixtures; exchange memberships; prepaid rent, insurance and other expenses; goodwill, organization expenses;
15c3-1(c)(2)(iv)(A)/01 Capitalized Leases
Capitalized leases are treated as purchases of assets which collateralize a financing loan. The liability, representing the present value of the future lease payments, is treated as indebtedness collateralized by the asset. To receive favorable treatment, the assets must either be “acquired for use in the ordinary course of the trade or business of a broker or dealer”, or the lessor’s sole recourse must be limited to the leased property.
These arrangements require Exchange approval under NYSE Rule 328 and may be deductions for computations under NYSE Rule 326 if the agreements contain acceleration clauses.
When the lease of computer or telephone equipment from a vendor-lessor or bank or other financial institution is capitalized, 50% of the capitalized lease liability may be treated as adequately secured for a period of two years after the lease was entered into without demonstration of the adequacy of the collateral.
The portion of the lease liability which matures in less than one year, is not to be treated as indebtedness collateralized by the asset. Favorable treatment is allowed for the assets only to the extent of that portion of the liability which matures in more than one year.
(SEC Staff to NYSE) (NYSE Information Memo 80-66)
(SEC Staff to NYSE) (NYSE Interpretation Memo 83-2, April 1983)
(SEC Letter to NASD, April 17, 1986)
(NYSE Interpretation Memo 88-14, August 1988)
(SEC Staff to NYSE) (NYSE Interpretation Memo 91-6, July 1991)
15c3-1(c)(2)(iv)(A)/02 Prepaid Non-Allowable Assets – Add-Back of Tax Liability
See interpretation 15c3-1(c)(2)(E)(i)/01.
15c3-1(c)(2)(iv)(B) Certain Unsecured and Partly Secured Receivables
All unsecured advances and loans; deficits in customers' and non-customers' unsecured and partly secured notes; deficits in omnibus credit accounts maintained in compliance with the requirements of 12 CFR 220.7(f) of Regulation T under the Act, or similar accounts carried on behalf of another broker or dealer, after application of calls for margin, marks to the market or other required deposits that are outstanding 5 business days or less; deficits in customers' and non-customers' unsecured and partly secured accounts after application of calls for margin, marks to market or other required deposits that are outstanding 5 business days or less, except deficits in cash accounts as defined in 12 CFR 220.8 of Regulation T under the Act for which not more than one extension respecting a specified securities transaction has been requested and granted, and deducting for securities carried in any of such accounts the percentages specified in paragraph (c)(2)(vi) of this section or appendix A, § 240.15c3-1a; the market value of stock loaned in excess of the value of any collateral received therefor; receivables arising out of free shipments of securities (other than mutual fund redemptions) in excess of $5,000 per shipment and all free shipments (other than mutual fund redemptions) outstanding more than 7 business days, and mutual fund redemptions outstanding more than 16 business days; and any collateral deficiencies in secured demand notes as defined in appendix D, § 240.15c3-1d; a broker or dealer that participates in a loan of securities by one party to another party will be deemed a principal for the purpose of the deductions required under this section, unless the broker or dealer has fully disclosed the identity of each party to the other and each party has expressly agreed in writing that the obligations of the broker or dealer do not include a guarantee of performance by the other party and that such party's remedies in the event of a default by the other party do not include a right of setoff against obligations, if any, of the broker or dealer.
15c3-1(c)(2)(iv)(B)/01 Free Shipments
On other than mutual fund redemptions, the charge for a free shipment outstanding seven business days or less applies only to the excess over $5,000, which is computed by aggregating together all shipments made on a particular day to a particular broker.
Securities sent to a transfer agent for tender or exchange are not considered to be related to free shipments.
A shipment by mail of securities to a bank, which serves as the seller’s agent for the collection of proceeds, is not a free shipment.
To the extent a broker-dealer used the facilities of a bank, a clearing agency such as Correspondent Delivery and Collection Service, or a common carrier such as Brink’s to act as the seller’s agent in effectuating securities deliveries, the deliveries are not considered to be free shipments. Once the agent releases control of the securities to the purchaser, payment should be promptly receive by the agent or the broker-dealer within a few hours in accordance with trade custom.
(SEC Staff to NYSE)
(SEC Release 34-11854, November 20, 1975)
(SEC Staff to NYSE) (NYSE Interpretation Memo 76-4, April 1976)
15c3-1(c)(2)(iv)(B)/02 Cash Accounts
Haircut and liquidating deficit charges on a deficit cash account do not apply to the entire account. Rather, they apply only to cash transactions that are:
In deficit; and
The subject of more than one extension of time or for which an extension should have been, but was not, obtained; and
Part of an account that liquidates to a deficit.
If all three conditions do not exist, there would be no charge with respect to the subject transaction.
For municipal securities, since Regulation T does not apply, the charges are imposed on transactions that are unpaid after five business days beyond settlement date (for C.O.D. transactions, 42 calendar days after trade date).
For sales transactions, the time frames of SEA Rule 15c3-3(m) apply subject however to extensions of time that have been obtained. For sales transactions in securities exempt from SEA Rule 15c3-3(m), charges shall be applied if the securities are not received within ten business days after settlement date.
(SEC Staff to NYSE) (NYSE Interpretation Memo 76-4, April 1976)
(NYSE Interpretation Memo 93-6, November 1993)
15c3-1(c)(2)(iv)(B)/021 Advances to Customers Collateralized by Uncertificated Mutual Funds
Where a customer pledges uncertificated mutual fund shares carried in an account by the fund in the customer’s name, the broker-dealer may have a perfected security interest under one of the following conditions:
The fund upon the customer’s instruction issues certificates for the pledged shares to the broker-dealer.
The pledged shares are transferred to an account carried by the fund under the conditions described in interpretation 15c3-3(c)(1)/04 relating to control of Uncertificated Mutual Fund Shares.
The fund is notified and agrees that subject to the broker-dealer’s lien, the customer may not redeem, transfer, exchange or otherwise effect transactions involving their fund shares pledged to the broker-dealer. Under this condition the shares pledged may not be used to satisfy possession or control requirements related to other customers.
(SEC Staff to NYSE) (NYSE Interpretation Memo 86-8, August 1986)
15c3-1(c)(2)(iv)(B)/03 Joint Trading and Investment Accounts
See interpretation 15c3-1(c)(2)(vi)/02.
15c3-1(c)(2)(iv)(B)/04 Cash Transactions in GNMA's Including TBAs and Standbys
See interpretation 15c3-1(c)(2)(xii)/02.
15c3-1(c)(2)(iv)(B)/05 (Reserved)
15c3-1(c)(2)(iv)(B)/06 Unendorsed Stock Certificates
A broker-dealer holding unendorsed stock certificates registered in the names of its customers, which are no longer in such customers’ accounts, should deduct the market value of the securities from net worth in computing net capital when the certificates have been held by the firm for ten business days past settlement date of the sales transaction and the customer has been paid for the sale.
(SEC Letter to NASD, October 27, 1983)
(NYSE Interpretation Memo 92-13, December 1992)
15c3-1(c)(2)(iv)(B)/061 Partial Payment on Unendorsed Stock Certificate
When a customer sells a security and has been paid for only a portion of their sale and the broker-dealer is holding an unendorsed stock certificate registered in the name of the customer, the market value of the securities less the free credit balance remaining in the customer's account would be deducted from net worth in computing net capital.
(SEC Staff to NYSE) (NYSE Interpretation Memo 96-4, November 1996)
15c3-1(c)(2)(iv)(B)/07 Stock Loaned
The market value of stock loaned in excess of the value of any collateral received may be computed on a broker-by-broker basis.
(SEC Letter to Oppenheim, Appel, Dixon & Co., May 7, 1979)
(NYSE Interpretation Memo 79-10, December 1979)
15c3-1(c)(2)(iv)(B)/071 Stock Loan Collateralized by a Letter of Credit
A letter of credit received as collateral to a stock loan has no value in the computation of stock loan deficit charges. The position would be considered an unsecured short position.
(SEC Staff to NYSE) (NYSE Interpretation Memo 92-4, January 1992)
15c3-1(c)(2)(iv)(B)/072 Securities Loaned through Euroclear
Securities loaned through Euroclear against an irrevocable guarantee, which constitutes the legal and functional equivalent of an irrevocable letter of credit, shall have no value in computing securities loaned deficit charges. The position would be considered an unsecured short position similar to interpretation 15c3-1(c)(2)(iv)(B)/071.
(SEC Staff to NYSE) (NYSE Interpretation Memo 93-6, November 1993)
15c3-1(c)(2)(iv)(B)/08 Stock Loan Deficits
Stock loan deficits need not be deducted in computing net capital for one business day from the day the deficit arises provided the broker-dealer issues a mark-to-market call and collects payment on that day.
Stock loan deficits need not be deducted in computing net capital, to the extent the deficit can be offset with equity in securities borrowed items with the same broker-dealer, provided that such securities loan contracts can be legally offset.
Stock loan deficits need not be deducted in computing net capital for one business day from the day the deficit arises provided the broker-dealer returns the stock loan on that day. This provision can be applied on an overnight stock loan contract only if at the time of origination the contract was properly collateralized.
The broker-dealer will be subject to the deficit capital charge if the stock loan contract that is in deficit is rolled over without additional funds or securities received from the counterparty.
(SEC Letter to Bear, Stearns & Co., September 16, 1980)
(SEC Staff to NYSE) (NYSE Interpretation Memo 81-3, July 1981)
(SEC Staff to NYSE) (NYSE Interpretation Memo 83-2, April 1983)
(SEC Staff to NYSE) (NYSE Interpretation Memo 05-8, April 2005)
15c3-1(c)(2)(iv)(B)/081 Issuing Mark to Market Call and Collecting Payment on Stock Loan Deficits Focusing Around Bank Holidays
A broker-dealer can reduce or eliminate a stock loan deficit by issuing a mark to market call and collecting payment from the lender by the close of the next business day following the date of such determination.
In cases where the broker-dealer is unable to process the mark because the next business day is a domestic or foreign bank holiday, but the securities exchanges are open, the bank holiday will not count as a business day. As such, a broker-dealer will be allowed to reduce or eliminate a stock loan deficit by issuing a mark to market call and collecting payment from the lender by the close of the next business day after the bank holiday.
(SEC Staff to NYSE) (NYSE Interpretation Memo 02-7, August 2002)
15c3-1(c)(2)(iv)(B)/09 Securities Borrowed Deficits
A broker-dealer which has borrowed securities (borrower) must mark the borrowed securities to the market each business day, as of the close of the prior day’s business, and determine the amount of collateral held by any securities lender (lender) which exceeds the current market value of the securities borrowed from that lender (excess collateral). The borrower must deduct from its net worth in computing its net capital:
the amount of collateral held by any one lender which exceeds one hundred and five percent (105%) of the current market value of the securities borrowed from that lender; or
if greater, the amount of excess collateral held by any one lender to the extent the excess collateral is greater than twenty percent (20%) of the borrower’s excess net capital (net capital greater than the minimum required); plus
the total amount of excess collateral held by all lenders in aggregate which exceeds three hundred percent (300%) of the borrower’s excess net capital reduced by the charge that the broker-dealer has already incurred under the above standards.
Securities borrowed deficits need not be deducted in computing net capital for one business day from the day the deficit arises provided the broker-dealer issues a mark-to-market call and collects payment on that day.
In computing deficits excess collateral related to other securities borrowed or securities loaned due to the same lender may be considered provided that such contracts may be legally offset.
Securities borrowed deficits need not be deducted in computing net capital for one business day from the day the deficit arises provided the broker-dealer returns the securities borrowed on that day. This provision can be applied on an overnight securities borrowed contract only if at the time of origination the contract was properly collateralized.
The broker-dealer will be subject to the deficit capital charge if the securities borrowed contract that is in deficit is rolled over without additional funds or securities received from the counterparty.
(SEC Letter to CBOE, December 7, 1983) (NYSE Interpretation Memo 84-1, January 1984)
(SEC Staff to NYSE) (NYSE Interpretation Memo 05-8, April 2005)
15c3-1(c)(2)(iv)(B)/091 Issuing Mark to Market Call and Collecting Payment on Securities Borrowed Deficits Focusing Around Bank Holidays
A broker-dealer can reduce or eliminate a securities borrowed deficit by issuing a mark to market call and collecting payment from the lender by the close of the next business day following the date of such determination.
In cases where the broker-dealer is unable to process the mark because the next business day is a domestic or foreign bank holiday, but the securities exchanges are open, the bank holiday will not count as a business day. As such, a broker-dealer will be allowed to reduce or eliminate a securities borrowed deficit by issuing a mark to market call and collecting payment from the lender by the close of the next business day after the bank holiday.
(SEC Staff to NYSE) (NYSE Interpretation Memo 02-7, August 2002)
15c3-1(c)(2)(iv)(B)/092 Non-Marketable Securities Collateralizing Purpose Borrow Transactions
Securities that are non-marketable, as defined in SEA Rule 15c3-1, and which have been received as collateral to a securities borrowed transaction that was originally transacted for a permitted purpose pursuant to FRB Reg. T Section 220.10(a) (“Purpose Borrow”), where cash or other marketable securities, as defined in SEA Rule 15c3-1, have been pledged, shall be subject to a 100% net capital charge if they allocate to a box location for more than five (5) business days in the case of a pre-borrow transaction and for more than two (2) business days in all other circumstances.
Where the securities received as collateral to a Purpose Borrow transaction are non-marketable, as defined in SEA Rule 15c3-1, but margin has been collected by the broker-dealer, the applicable net capital charge would be the cash receivable or market value of the securities pledged, less the margin collected.
See SEA Rule 15c3-1(c)(2)(iv)(F)(3)(ii)/06 (Non-Marketable Securities Collateralizing Reverse Repurchase Transactions).
(SEC Staff to NYSE) (NYSE Interpretation Memo 03-3, April 2003)
(SEC Staff to NYSE) (NYSE Interpretation Memo 06-5, June 2006)
15c3-1(c)(2)(iv)(B)/093 Non-Purpose Equity Securities Borrowed Transactions
A broker-dealer, including an “exempted borrower” as defined pursuant to FRB Regulation T Section 220.2, that engages in equity securities borrowed transactions (versus cash collateral) solely for financing the positions of another broker-dealer, where the equity securities were initially borrowed without a “permitted purpose” pursuant to FRB Regulation T Section 220.10(a) and placed in a box location, should maintain equity in such “non-purpose” securities borrowed account with each counterparty at least equal to the haircut deduction on the market value of the equity securities as required under SEA Rule 15c3-1 subparagraphs (c)(2)(vi) and (c)(2)(vii) (excluding undue concentration charges pursuant to subparagraph (c)(2)(vi)(M)).
A separate identifiable “non-purpose” securities borrowed account with each counterparty should be set up to record the activity.
If the equity in the account with the counterparty is not equal to or greater than the total haircut deduction computed for the positions carried in the “non-purpose” securities borrowed account, the broker-dealer must charge its own capital for the deficiency. If the broker-dealer’s account with the counterparty liquidates to a deficit, the charge to the broker-dealer will be for the sum of the deficit and the applicable haircut deduction.
The computed charges are applicable even if the “non-purpose” securities borrowed positions are subsequently used for a permitted purpose pursuant to FRB Regulation T Section 220.10(a).
Example:
A broker-dealer that engages in a non-purpose equity securities borrowed transaction and provides to the counterparty $1,700 in cash collateral should receive from the counterparty an equity security with a market value of at least $2,000.
The equity of $300 ($2,000 market value - $1,700 cash collateral) maintained on the non-purpose equity securities borrowed transaction is equal to the 15% haircut deduction on the equity security with a market value of $2,000.
(SEC Staff to NYSE) (NYSE Interpretation Memo 05-2, January 2005)
(SEC Staff to NYSE) (NYSE Interpretation Memo 05-8, April 2005)
15c3-1(c)(2)(iv)(B)/10 Non-Purpose Loans Collateralized by Certificates of Deposit
Non-Purpose Loans collateralized by negotiable certificates of deposit need not be deducted provided the following conditions are satisfied:
The negotiable CDs are issued by U.S. Banks and are denominated in U.S. Dollars.
Negotiable CDs with maturity of one year or less securing a non-purpose loan shall have an initial and maintenance margin requirement of 10%. Negotiable CDs with longer maturities securing a non-purpose loan account shall have an initial and maintenance margin requirement of 25%. Any cash margin deficiencies must be charged to capital.
CDs held as collateral for non-purpose loans to customers may not be deposited in the Special Reserve Bank Account to satisfy a SEA Rule 15c3-3 reserve deposit requirement.
Two percent of the total amount of non-purpose loan debits must be deducted in the broker-dealer’s computation of net capital under NYSE Rule 325.
Broker-dealers are prohibited from selling CDs short to customers.
Brokered CDs with any one bank, which are used for collateral, may not exceed 100 percent of the broker-dealer’s excess net capital.
(SEC Letter to NYSE, August 13, 1984)
(SEC Letter to NYSE, November 5, 1987)
(NYSE Interpretation Memo 88-14, August 1988)
15c3-1(c)(2)(iv)(B)/11 Deficits in Introduced Accounts
Deficits in unsecured and partly secured introduced accounts shall be deducted by the carrying broker-dealer and the introducing broker-dealer when the clearing agreement states that such deficits are the liability of the introducing broker-dealer.
If the carrying broker-dealer subordinates its receivable for the deficit amount to the claims of creditors of the introducing broker-dealer, the subordinated receivable shall be deducted as an unallowable asset by the carrying broker-dealer. The introducing broker-dealer may exclude the subordinated liability from Aggregate Indebtedness; however, it shall be considered as a liability in the determination of net worth since it is not subject to a satisfactory subordination agreement as defined in SEA Rule 15c3-1d (Appendix D). (See paragraph (c)(1)(xi) of SEA Rule 15c3-1.)
If the carrying broker-dealer subordinates capital to the introducing broker-dealer to offset the deduction, the carrying broker-dealer has a double deduction, one for the deficit and one for the subordinated amount.
The amount is deductible by the carrying broker-dealer upon occurrence after application of timely calls for margin, marks to market or other required deposits which are not outstanding for more than five (5) business days unless there is reason to believe payment will not be made. The introducing broker-dealer must deduct the charge on the day after it becomes a charge to the carrying broker and the carrying broker-dealer must advise the introducing broker-dealer in writing on a daily basis of all such deficits to be charged.
(SEC Staff to NYSE) (NYSE Interpretation Memo 88-14, August 1988)
(SEC Staff to NYSE) (NYSE Interpretation Memo 89-12, October 1989)
15c3-1(c)(2)(iv)(B)/111 Customers’ Unsecured/Partly Secured Deficits Offset by Correspondent’s Deposits
Deficits in customers’ unsecured and partly secured accounts of an introducing broker-dealer do not have to be deducted from net capital by the carrying broker-dealer provided sufficient deposits were received from the introducing broker-dealer which can be legally applied to cover (fully secure) the applicable deficits. The introducing broker-dealer must still take the customer’s deficits as a deduction to net capital when the clearing agreements state that such deficits are its liability (see interpretation 15c3-1(c)(2)(iv)(B)/11). The amount of the introducing broker-dealer’s deposits must also be included in the carrying broker-dealer’s PAB reserve formula computation.
(SEC Staff to NYSE) (NYSE Interpretation Memo 02-3, February 2002)
(SEC Staff to FINRA) (FINRA Regulatory Notice 15-25)
15c3-1(c)(2)(iv)(B)/12 Credit Extended Upon Exercise of Employee Stock Option
When a broker-dealer exercises an employee stock option for a customer, it must have acknowledgement from the issuer that a freely transferable, readily salable and marketable security in negotiable form will be promptly delivered to the broker-dealer within 13 business days after exercise notice is given to the issuer (when acknowledgement is given by telephone, the condition should be restated in the transmittal to the issuer). The exercise shall be subject to the following:
When the security to be received from the exercise has not been sold and is not received within 13 business days after notice of exercise has been given, any related debit balance shall be treated as an unsecured debit for net capital purposes;
When the security to be received from the exercise has been sold and is not receivable from the issuer within 13 business days after notice of exercise has been given, the position shall be subject to a cash margin deficiency charge computed without allowing any value for the security not received (and is subject to the buy-in provision under SEA Rule 15c3-3(m) unless an extension of time is requested and approved under paragraph (n) of that rule). See interpretation 15c3-1(c)(2)(xii)/05.
(SEC Staff to NYSE) (NYSE Interpretation Memo 88-14, August 1988) (NYSE Interpretation Memo 97-6, October 1997)
15c3-1(c)(2)(iv)(B)/13 Customers’ Debits Secured by Control or Restricted Stock
Only securities which can be publicly sold can be recognized in determining whether a customer’s account is partially secured or unsecured. Where the value of publicly saleable securities is not sufficient to fully secure the customer debit, such debit is to be treated as an unsecured debit or partially secured debit, as applicable.
(SEC Staff to NYSE) (NYSE Interpretation Memo 92-1, January 1992)
15c3-1(c)(2)(iv)(B)/14 Customer Deficits in Listed Currency Options Secured by Letters of Credit
A letter of credit received from a customer shall be recognized to the extent it covers the margin required by the Philadelphia Stock Exchange for listed currency options. A liquidating deficit on the basis of not allowing value for the letter of credit must be deducted in computing net capital and the account treated as a partly secured account.
(SEC Staff to NYSE) (NYSE Interpretation Memo 93-6, November 1993)
15c3-1(c)(2)(iv)(B)/15 Ownership of Collateral
In order for collateral to secure an otherwise non-allowable asset in conformity with the net capital rule, the collateral itself, among other things, must be an allowable asset under the rule. The securities collateralizing the receivable held by a firm must be in the bearer, nominee, or firm name to have value for net capital purposes. Securities that are registered in the name of a custodian, such as Depository Trust Company, or securities that have been forwarded to a transfer agent for transfer into the name of the firm, can be considered acceptable for purposes of securing a receivable, provided all required documentation for transfer have been forwarded with the securities.
(SEC Staff of DMR to NASD, December 1983)
15c3-1(c)(2)(iv)(B)/16 Deficits or Unsecured Balances in Securities Transactions with a Federal Reserve Bank
Deficits or unsecured balances in securities transactions with a Federal Reserve Bank need not be deducted in computing net capital under SEA Rule 15c3-1(c)(2)(iv)(B).
(SEC Staff to FINRA) (FINRA Regulatory Notice 21-27)
15c3-1(c)(2)(iv)(C) Interest receivable, floor brokerage receivable, commissions receivable from other brokers or dealers (other than syndicate profits which shall be treated as required in paragraph (c)(2)(iv)(E) of this section), mutual fund concessions receivable and management fees receivable from registered investment companies, all of which receivables are outstanding longer than thirty (30) days from the date they arise; dividends receivable outstanding longer than thirty (30) days from the payable date; good faith deposits arising in connection with a non-municipal securities underwriting, outstanding longer than eleven (11) business days from the settlement of the underwriting with the issuer; receivables due from participation in municipal securities underwriting syndicates and municipal securities joint underwriting accounts which are outstanding longer than sixty (60) days from settlement of the underwriting with the issuer and good faith deposits arising in connection with an underwriting of municipal securities, outstanding longer than sixty (60) days from settlement of the underwriting with the issuer; and receivables due from participation in municipal securities secondary trading joint accounts, which are outstanding longer than sixty (60) days from the date all securities have been delivered by the account manager to the account members;
15c3-1(c)(2)(iv)(C)/01 Good Faith Deposits
A good faith deposit made in anticipation of an underwriting may be outstanding indefinitely without its being required to be deducted in computing net capital, so long as there is no indication that the broker has been eliminated from participation in the underwriting.
Good faith deposits relating to instances where there is no underwriting, such as a rejection of all bids under a competitive bid procedure, are required to be deducted subsequent to eleven business days after it is clear that the underwriting will not occur. Rejection of a bid would be indicative of such an occurrence.
(SEC Staff to NYSE)
15c3-1(c)(2)(iv)(C)/02 Income Taxes Receivable
Income taxes receivable are deducted in computing net capital even if acknowledged for payment by the taxing authority. Note however, that under certain circumstances the effect of deducting deferred tax debits might be offset by the addback to net worth of part or all of those amounts. (See interpretation 15c3-1(c)(2)(i)(D)/01.)
(SEC Staff to NYSE)
15c3-1(c)(2)(iv)(C)/03 Receivables, Deductible Fees
The following receivables are deducted in computing net capital:
Fees receivable from banks for federal funds placement;
Tender fees receivable from offerors;
Commercial paper fees receivable from issuers.
(SEC Staff to NYSE)
15c3-1(c)(2)(iv)(C)/04 Exchange Membership - Proceeds of Sale
The net proceeds receivable from the sale of an exchange seat are not deducted in computing net capital provided they will be received within 30 days of the date of sale.
(SEC Staff to NYSE)
15c3-1(c)(2)(iv)(C)/05 Investment Company Management Fees Receivable
If investment company management fees are payable on a quarterly basis, are billed within eleven business days after the close of the quarter and are not outstanding more than eleven business days thereafter such receivables would not be deducted from net worth when computing net capital.
(SEC Staff to NYSE) (NYSE Interpretation Memo 79-7, August 1979)
15c3-1(c)(2)(iv)(C)/06 Trade Date Commissions
Commissions receivable from customers on unsettled regular way transactions need not be deducted even if they are currently unsecured provided that appropriate liabilities such as registered representative's compensation and taxes have been accrued on the related income.
(SEC Staff to NYSE) (NYSE Interpretation Memo 77-2, June 1977)
15c3-1(c)(2)(iv)(C)/07 Intercompany Accounts With Guaranteed Subsidiaries – (Rescinded)
(SEC Staff to FINRA) (FINRA Regulatory Notice 13-44)
15c3-1(c)(2)(iv)(C)/071 Intercompany Accounts With Non-Guaranteed Subsidiaries
Unsecured amounts due from a parent to a non-guaranteed broker-dealer subsidiary are not allowable assets in computing the subsidiary’s net capital. However, where the parent is clearing transactions of the subsidiary, unsecured receivables represent funds transferred to the parent for bona-fide securities purchases or proceeds from the sale of securities held for the account of the subsidiary are an allowable asset for one business day pending settlement of a purchase or transfer of proceeds to the subsidiary or to a bank account in the name of the subsidiary over which the parent has no direct control.
(SEC Letter to Moore & Schley Cameron & Co., December 5, 1979)
(NYSE Interpretation Memo 81-10, December 1981)
15c3-1(c)(2)(iv)(C)/072 Intercompany Accounts with a Parent Bank
Cash deposits with a parent or affiliated bank will only have net capital value to the extent that the deposit represents normal day-to-day operating balances.
(SEC Staff to NYSE) (NYSE Interpretation Memo 84-9, November 1984)
15c3-1(c)(2)(iv)(C)/073 Netting of Intercompany Receivables and Payables with Affiliates
An account with an affiliate or a parent which could be classified as a customer under SEA Rule 15c3-3 (securities account) may not be netted with accounts of other affiliates. For this purpose they would be treated as cash accounts, margin accounts, etc.
Other accounts (non-securities accounts) of the same affiliate may be netted. However, cross netting of such accounts with different affiliated entities will not be permitted.
Netting can be accomplished by recording all non-securities transactions with affiliates in a single account carried for a parent organization. Transactions (payments and receipts) should be made according to instructions from the parent and details can be maintained in sub accounts. However, it must be clear that any creditor or other claimant against any of the entities can only proceed against the net account carried for the parent.
Accounts with parent organizations or affiliates which are not classifiable as securities accounts should not be carried in a securities account.
(SEC Staff to NYSE) (NYSE Interpretation Memo 91-9, July 1991)
15c3-1(c)(2)(iv)(C)/074 Reduction of Intercompany Accounts Receivable
The SEC has barred a Chief Financial Officer of a broker-dealer for violations of SEA Rule 15c3-1 that involved reducing unsecured intercompany accounts receivables by clearing house checks received by the broker-dealer from its parent when the parent did not have sufficient cash or liquid assets to cover the presented checks. This violation involved a “seg-offset” banking arrangement which was improperly used.
The SEC staff has advised that it is a violation to increase regulatory net capital through the improper recognition of uncleared checks received from an affiliated entity and that it may be regarded as an intentional circumvention of the rule.
Also, see interpretations 15c3-3a(Item 1) /24 (Exhibit A), /25 and /26.
(SEC Release 34-30444, March 4, 1992)
(NYSE Interpretation Memo 92-8, June 1992)
15c3-1(c)(2)(iv)(C)/075 Treatment of an Unsecured Receivable Due From a Guaranteed Subsidiary
An unsecured receivable due from a guaranteed subsidiary shall be treated as a non-allowable asset.
(SEC Staff to FINRA) (FINRA Regulatory Notice 13-44)
15c3-1(c)(2)(iv)(C)/08 Rebates or Interest Receivable Resulting From Securities Borrowed or Securities Loaned
Rebates or interest receivable in connection with securities borrowed or securities loaned from brokers or dealers need not be deducted from net worth when computing net capital provided the rebates or interest receivable are billed promptly and not aged over 30 days from the date they arise.
(SEC Staff to NYSE) (NYSE Interpretation Memo 81-10, December 1981)
(SEC Staff to NYSE) (NYSE Interpretation Memo 06-5, June 2006)
15c3-1(c)(2)(iv)(C)/081 Rebates or Interest Receivable From Institutions Pursuant to Securities Borrowed Transactions
Rebates or interest receivable from non-broker-dealer institutions resulting from securities borrowed transactions shall be deducted in computing net capital unless:
The written agreement between the broker-dealer and the non-broker-dealer institution required by SEA Rule 15c3-3(b)(3), provides for prompt payment by the institution of the rebates or interest owed to the broker-dealer.
For purposes of securities borrows transacted pursuant to agency securities lending arrangements, a broker-dealer may maintain the written agreement required by SEA Rule 15c3-3(b)(3) with the agent lender in lieu of obtaining individual written agreements with each underlying non-broker-dealer institution principal counter party; and
The amount or value of the collateral held by the lender does not exceed 105% of the market value of the securities borrowed.
For purposes of securities borrows transacted pursuant to agency securities lending arrangements, a broker-dealer may establish compliance of this provision with the agent lender in lieu of establishing compliance with each underlying non-broker-dealer institution principal counter party; and
The rebates or interest receivable are billed promptly and not aged more than thirty (30) calendar days from the billing date, which should be at least monthly.
For purposes of securities borrows transacted pursuant to agency securities lending arrangements, a broker-dealer may establish compliance of this provision with the agent lender in lieu of establishing compliance with each underlying non-broker-dealer institution principal counter party, provided that the written agreement between the broker-dealer and the agent lender states that each underlying principal counter party has duly authorized the agent lender to perform this function on its behalf.
For purposes of this interpretation, agency securities lending does not refer to situations where the principal counter parties negotiate directly with the broker-dealer.
(SEC Staff to NYSE) (NYSE Interpretation Memo 90-11, December 1990)
(NYSE Interpretation Memo 06-5, June 2006) (NYSE Interpretation Memo 07-4, April 2007)
15c3-1(c)(2)(iv)(C)/09 Commissions or Concessions Receivable versus Commissions or Concessions Payable - (Rescinded)
(SEC Staff to FINRA) (FINRA Regulatory Notice 21-27)
15c3-1(c)(2)(iv)(C)/091 Concessions Receivable from Individual Variable Annuities are Allowable for 30 Days; from Group Variable Annuities an Offset is Permitted
The staff of the SEC’s Division of Market Regulation has concluded that since individual variable annuity products are registered under the Investment Company Act of 1940 (’40 Act), any concessions receivable therefrom come within the “...mutual fund concessions receivable and management fees receivable from registered investment companies...” language of SEA Rule 15c3-1(c)(2)(iv)(C). Therefore, a broker-dealer may give allowable asset treatment to concessions receivable from the sale of individual variable annuities for 30 days from the date they arise.
In contrast, group variable annuities are exempt from registration under the ’40 Act; therefore, concessions receivable from the sale of these products are not allowable assets under SEA Rule 15c3-1(c)(2)(iv)(C). However, the staff of the SEC’s Division of Market Regulation has no objection to member firms considering concessions receivable from the sale of group variable annuities as allowable assets if the member complies with interpretation 15c3-1(c)(2)(iv)(C)/095.
(SEC Staff to NASD, May 1999)
(SEC Staff to FINRA) (FINRA Regulatory Notice 21-27)
15c3-1(c)(2)(iv)(C)/092 Concessions Receivable, Offset Not Permitted with Payables of a Particular Program or Syndication
The permissible offset between concessions receivable and related commissions payable pursuant to NASD Notices to Members 84-48 and 85-5 may not be extended to payables related to sales of a particular program or syndication. For example, a broker may not offset concessions receivable against legal fees payable.
(SEC Staff of DMR to NASD, April 1985)
15c3-1(c)(2)(iv)(C)/093 Distinction between Commission and Concession Receivable
A commission is generally considered to be a charge added to the gross price of securities purchased or deducted from the gross price of securities sold. A concession is an amount included in the gross price of the transaction that is retained by or payable to the broker-dealer executing the transaction. While a commission may occur in a retail transaction between a broker-dealer and a customer or between broker-dealers, a concession is more likely to occur between a broker-dealer and an issuer or another member of an underwriting group.
(SEC Staff to DMR to NASD, 1985)
15c3-1(c)(2)(iv)(C)/094 Concessions Receivable from a “Best Efforts” Offering, from Bank Escrow Agent
A broker-dealer shall receive allowable asset treatment for concessions receivable arising from a “best efforts” offering of securities in which an escrow account is utilized as required by SEA Rule 15c2-4, provided there is a written agreement between the bank escrow agent and the broker-dealer specifying that the bank escrow agent accepts responsibility for direct disbursement of the sales concessions due the broker-dealer.
(NASD Notice to Members 80-49, September 24, 1980)
15c3-1(c)(2)(iv)(C)/095 Unsecured Receivables and Related Payables
Instances arise where a broker-dealer accrues revenue in connection with the sale of products or for services it has provided and at the same time accrues expenses incurred as a result of selling such products or providing such services.
An unsecured amount receivable that results from the recording of such revenue need not be deducted from net worth to the extent the broker-dealer has also accrued a payable as the direct result of the expenses incurred in connection with such revenue, and:
A written contract exists between the broker-dealer and the payee, in which:
The broker-dealer’s liability for the amount payable is limited solely to the proceeds of the receivable; and
The payee waives payment of the amount payable until the broker-dealer has received payment of the related amount receivable; and
If the broker-dealer is subject to the Aggregate Indebtedness Standard of paragraph (a)(1)(i),
The portion of the payable due within twelve months is included in aggregate indebtedness; and
The broker-dealer’s net capital requirement shall be increased by an amount equal to one percent of the portion of the payable that was not included in aggregate indebtedness
(SEC Staff to FINRA) (FINRA Regulatory Notice 21-27)
15c3-1(c)(2)(iv)(C)/10 Due Bills Receivable
Due bills receivable from broker-dealers or banks covering dividends, stock splits or similar distributions, need not be deducted in computing net capital provided the broker-dealer promptly presented the due bill for payment and it is not older than 30 days from its payable date.
(SEC Staff to NYSE) (NYSE Interpretation Memo 81-10, December 1981)
15c3-1(c)(2)(iv)(C)/11 Commissions Receivable From a Broker-Dealer Parent
Commissions receivable from a broker-dealer parent are good assets for 30 days.
(SEC Staff to NYSE) (NYSE Interpretation Memo 88-14, August 1988)
15c3-1(c)(2)(iv)(C)/12 Commissions Receivable From a Foreign Broker-Dealer Parent
Commissions receivable from a foreign broker-dealer parent are considered unsecured and are not allowable for net capital purposes.
(SEC Staff to NYSE) (NYSE Interpretation Memo 88-14, August 1988)
15c3-1(c)(2)(iv)(C)/13 Commissions Receivable From a Savings and Loan
Commissions receivable from savings and loan associations and/or other thrift institutions must be deducted in the computation of net capital.
(SEC Letter to First United Fund, Ltd., August 15, 1984)
(NYSE Interpretation Memo 88-14, August 1988)
15c3-1(c)(2)(iv)(C)/14 Floor Brokerage Receivable
Floor brokerage commissions receivable need not be deducted from net worth for a period of 30 days from the month end accrual date provided they are billed promptly after the close of the month.
(SEC Letter to Philadelphia Stock Exchange, July 15, 1983)
(NYSE Interpretation Memo 88-14, August 1988)
15c3-1(c)(2)(iv)(C)/15 Introduced Commissions/Fees Receivable
Commissions or fees receivable from a broker-dealer on introduced account activity need not be deducted from net worth for a period of 30 days from the month end accrual date provided they are billed promptly after the close of the month.
(SEC Letter to East/West Securities Co., March 27, 1989)
(NYSE Interpretation Memo 89-9, July 1989)
15c3-1(c)(2)(iv)(C)/16 Federal Funds Sales & Swap Transactions
Federal Funds Sales - a broker-dealer may enter into loans of Federal funds without deducting the value of the transaction from net worth if:
The loan is with a depository institution;
The loan cannot exceed one business day, and
The Federal funds held by the broker-dealer resulted from the clearance of securities on the day the loan is made.
Federal Funds Swaps - A broker-dealer may engage in Federal funds swap transactions in which excess funds contained in their clearance account are swapped with registered broker-dealers or counterparties other than registered broker-dealers without deducting the value of the transaction from net worth if:
The swap does not exceed one business day, and
The broker-dealer must receive a certified check (in an amount equal to the swap) drawn on a bank as defined in Section 3(a)(6) of the Securities Exchange Act of 1934 simultaneously with the release of the funds.
(SEC Letters to Securities Industry Association, November 19, 1991)
(NYSE Interpretation Memo 92-7, May 1992)
(NYSE Interpretation Memo 93-6, November 1993)
15c3-1(c)(2)(iv)(C)/17 Investment Advisory Fees
Investment advisory fees receivable from customers that have been earned and recorded as receivable but not yet deducted from the customer’s account, may be treated as an allowable asset in computing net capital if all of the following conditions are met:
The broker-dealer has entered into a written agreement with each customer that permits the broker-dealer to deduct from the customer’s securities account the full amount of the fees earned and accrued as receivable from such customer immediately upon the occurrence of either of the following events:
the termination of the advisory relationship or the initiation by the customer of a transfer of the account, whichever occurs first; or
the broker-dealer’s filing for protection under applicable bankruptcy laws and/or the issuance of a protective decree under SIPA;
The broker-dealer has implemented controls to enable it to monitor and collect outstanding investment advisory fees due from a customer prior to the completion of any request to transfer the customer’s account to another financial institution; including, where applicable, arranging for its carrying broker-dealer to provide notification of outgoing customer account transfer requests received by the carrying broker-dealer and to collect the outstanding fees from the customer’s account;
The broker-dealer has procedures and controls reasonably designed to assure that the net liquid assets in the customer’s account, equal or exceed the investment advisory fees earned and recorded as receivable by the broker-dealer from each such customer, but which have not yet been deducted from the customer’s account; and
The fees are collected by the broker-dealer no later than six months from the date they are recorded as receivable.
(SEC Staff to FINRA) (FINRA Regulatory Notice 13-44)
15c3-1(c)(2)(iv)(D) Insurance claims. Insurance claims which, after seven (7) business days from the date the loss giving rise to the claim is discovered, are not covered by an opinion of outside counsel that the claim is valid and is covered by insurance policies presently in effect; insurance claims which after twenty (20) business days from the date the loss giving rise to the claim is discovered and which are accompanied by an opinion of outside counsel described above, have not been acknowledged in writing by the insurance carrier as due and payable; and insurance claims acknowledged in writing by the carrier as due and payable outstanding longer than twenty (20) business days from the date they are so acknowledged by the carrier; and,
15c3-1(c)(2)(iv)(D)/01 Insurance Claim Extensions
Extensions of time beyond the twenty day time frames specified in the Rule might possibly be granted by the SEC, but only on a case-by-case basis.
(SEC Staff to NYSE)
15c3-1(c)(2)(iv)(D)/02 Counsel’s Opinion for Lost Certificates
Generally, there is no applicable charge for lost security certificates even though opinion of outside counsel has not been obtained provided the broker-dealer takes prompt steps for replacement of the loss. These steps should include placing a stop transfer order with the transfer agent, obtaining a replacement bond from an insurance company and transmitting it to the transfer agent, and whatever other steps would be necessary to expedite the replacement.
(SEC Staff to NYSE) (NYSE Interpretation Memo 77-2, June 1977)
15c3-1(c)(2)(iv)(E) Other deductions. All other unsecured receivables; all assets doubtful of collection less any reserves established therefor; the amount by which the market value of securities failed to receive outstanding longer than thirty (30) calendar days exceeds the contract value of such fails to receive; and the funds on deposit in a “segregated trust account” in accordance with 17 CFR 270.27d-1 under the Investment Company Act of 1940, but only to the extent that the amount on deposit in such segregated trust account exceeds the amount of liability reserves established and maintained for refunds of charges required by sections 27(d) and 27(f) of the Investment Company Act of 1940; Provided, That the following need not be deducted:
15c3-1(c)(2)(iv)(E)(1) Any amounts deposited in a Customer Reserve Bank Account or PAB Reserve Bank Account pursuant to § 240.15c3-3(e) or in the “special reserve account for the exclusive benefit of security-based swap customers” established pursuant to § 240.15c3-3(p)(3),
15c3-1(c)(2)(iv)(E)(2) Cash and securities held in a securities account at a carrying broker or dealer (except where the account has been subordinated to the claims of creditors of the carrying broker or dealer), and
15c3-1(c)(2)(iv)(E)(3) Clearing deposits.
15c3-1(c)(2)(iv)(E)/01 Fails to Receive Outstanding More Than 30 Calendar Days
The amount by which the market value of fails to receive outstanding longer than 30 calendar days exceeds the contract value is computed on a contract-by-contract basis.
(SEC Letter to Oppenheim, Appel, Dixon & Co., May 7, 1979)
(NYSE Interpretation Memo 79-10, December 1979)
15c3-1(c)(2)(iv)(E)/011 Syndicate Receivables
Syndicate profits receivable must be deducted (see SEA Rule 15c3-1(c)(2)(iv)(C)) unless the asset:
Adequately secures (see definition at SEA Rule 15c3-1(c)(5)) a fixed liability and are the sole recourse of the creditor for nonpayment of the liability, and
The loan agreement has been submitted to and found acceptable by the Exchange.
(SEC Staff to NYSE) (NYSE Interpretation Memo 88-14, August 1988)
15c3-1(c)(2)(iv)(E)/012 Emerging Markets Clearing Corporation (EMCC)
When computing net capital member firms of the Emerging Markets Clearing Corporation (EMCC) do not take deductions on open fails to receive outstanding longer than thirty calendar days and on open fails to deliver outstanding five business days or longer on Brady Bonds as defined by EMCC, if EMCC continues to:
guarantee settlement of all open fail positions; and
compute and collect daily net debit marks on all open fail positions.
(SEC Staff to NYSE) (NYSE Interpretation Memo 00-6, September 2000)
15c3-1(c)(2)(iv)(E)/02 Clearing Deposits Maintained With Broker-Dealers – (Rescinded)
(NYSE Interpretation Memo 99-6, May 1999)
15c3-1(c)(2)(iv)(E)/021 Clearing Deposits of Introducing Brokers
Clearing deposits of introducing brokers must be maintained with a registered broker or dealer pursuant to a written clearing agreement and the clearing agreement must:
Permit the return of the deposit within 30 calendar days after cancellation of the agreement; and
State that the deposit does not represent an ownership interest in the clearing broker.
The 30 calendar day period referred to above shall commence 5 business days after the date of the initial transfer of customer accounts and not on the date that notice of termination is given by either party to the clearing agreement. The amount of any clearing deposit held under the terms of the clearing agreement that is not returned to the introducing broker-dealer within 30 calendar days after the aforementioned 5 business day grace period, shall be treated as a non-allowable asset in the computation of the introducing broker-dealer’s net capital commencing on the 31st calendar day after such grace period.
Note: If there is a monetary penalty against the introducing broker-dealer resulting from the voluntary termination of a clearing agreement, see interpretation 15c3-1(c)(2)(iv)(E)/0211.
(SEC Staff to NYSE) (NYSE Interpretation Memo 99-6, May 1999)
(SEC Staff to FINRA) (FINRA Regulatory Notice 08-46)
(SEC Staff to FINRA) (FINRA Regulatory Notice 13-44)
(SEC Staff to FINRA) (FINRA Regulatory Notice 15-25)
15c3-1(c)(2)(iv)(E)/0211 Monetary Penalty Resulting From the Voluntary Termination of a Clearing Agreement
If a monetary penalty against an introducing broker-dealer results from its voluntary termination of a clearing agreement (termination penalty), the introducing broker-dealer must apply a net capital charge for the lesser of the amount of the termination penalty or the amount of its clearing deposit held by the clearing broker-dealer. The net capital charge must be applied on the date that the introducing broker-dealer provides notice of the termination to the clearing broker-dealer and continue until such date as the clearing broker-dealer returns the clearing deposit to the introducing broker-dealer.
The introducing broker-dealer must also make a determination, under generally accepted accounting principles, whether it must accrue a liability on its financial statements to reflect the effect of the voluntary termination of its clearing agreement. An introducing broker-dealer that accrues a liability for the full amount of the termination penalty may reduce the aforementioned net capital charge by the amount of such accrued liability. Introducing broker-dealers that use the basic method of computing their net capital requirements pursuant to SEA Rule 15c3-1 must also include the amount of any accrued liability in their computation of aggregate indebtedness.
(SEC Staff to FINRA) (FINRA Regulatory Notice 08-46)
(SEC Staff to FINRA) (FINRA Regulatory Notice 13-44)
15c3-1(c)(2)(iv)(E)/0212 Clearing Agreements Containing a Termination Penalty Clause
Due to the potential lien by the clearing broker-dealer on an introducing broker-dealer’s clearing deposit while a termination penalty clause remains in effect, if such introducing broker-dealer becomes the subject of a protective decree issued pursuant to the Securities Investor Protection Act of 1970, the introducing broker-dealer must treat any clearing deposit assets held by its clearing broker-dealer, up to the amount of the termination penalty, as non-allowable in computing its net capital. To avoid such a deduction, the clearing agreement must contain the following clause:
In the event that [the Introducing Broker-Dealer] is the subject of the issuance of a protective decree pursuant to the Securities Investor Protection Act of 1970 (15 USC 78aaa-lll), [the Clearing Firm’s] claim for payment of a termination fee under this Agreement shall be subordinate to claims of [the Introducing Broker’s] customers that have been approved by the Trustee appointed by the Securities Investor Protection Corporation pursuant to the issuance of such protective decree.
Further, when the termination penalty exceeds the amount of the clearing deposit, absent inclusion of the foregoing clause in the clearing agreement, a broker-dealer must determine, in accordance with generally accepted accounting principles, if any additional expense should be recorded or liability accrued.
The foregoing clause is voluntary for clearing broker-dealers. However, in order for an introducing broker-dealer that is party to a clearing agreement that contains a termination penalty clause to treat its clearing deposit at the clearing broker-dealer as an allowable asset for net capital purposes, its clearing agreement must include such clause. The clearing broker-dealer may provide for the inclusion of such clause either through an amended clearing agreement or an addendum to an existing clearing agreement.
The clearing agreement must also include the provisions of interpretation 15c3-1(c)(2)(iv)(E)/021. See also interpretation 15c3-1(c)(2)(iv)(E)/0211 for the treatment of a monetary penalty resulting from the voluntary termination of a clearing agreement.
(SEC Staff to FINRA) (FINRA Regulatory Notice 08-46)
(SEC Staff to FINRA) (FINRA Regulatory Notice 13-44)
15c3-1(c)(2)(iv)(E)/022 Introducing Firms with No Proprietary Trading Accounts – (Rescinded)
(SEC Staff to FINRA) (FINRA Regulatory Notice 15-25)
15c3-1(c)(2)(iv)(E)/023 Introducing Firm’s Net Equity – (Rescinded)
(SEC Staff to FINRA) (FINRA Regulatory Notice 15-25)
15c3-1(c)(2)(iv)(E)/024 Proprietary Accounts of Other Broker-Dealers – (Rescinded)
(SEC Staff to FINRA) (FINRA Regulatory Notice 15-25)
15c3-1(c)(2)(iv)(E)/025 U.S. Broker-Dealer’s Deposit at Foreign Entity
A U.S. broker-dealer’s deposit held by a foreign entity is not subject to PAB account requirements. However, the deposit would be subject to the net capital treatment as is normally accorded to such deposits.
(SEC Staff to NYSE) (NYSE Interpretation Memo 99-6, May 1999)
(SEC Staff to FINRA) (FINRA Regulatory Notice 15-25)
15c3-1(c)(2)(iv)(E)/026 DVP/RVP Accounts – (Rescinded)
(SEC Staff to FINRA) (FINRA Regulatory Notice 15-25)
15c3-1(c)(2)(iv)(E)/027 Piggyback Carrying Arrangements
Piggyback carrying arrangements generally involve two introducing firms and one carrying and clearing firm. In these arrangements, an introducing firm acts as intermediary to another introducing firm (the “piggyback” firm) in obtaining clearing services from the carrying and clearing firm, pursuant to an agreement that meets all applicable requirements of FINRA Rule 4311.
As part of this arrangement, the clearing deposit of the piggyback firm must be clearly identified in a separate proprietary account in the name of the piggyback firm, by the intermediary firm to the carrying and clearing firm. The proprietary accounts, including the clearing deposit accounts, of both introducing firms maintained at the carrying and clearing firm are subject to PAB account requirements.
Further, the clearing deposit of both introducing firms can be treated as an allowable asset for net capital purposes, only if the carrying agreement meets the applicable requirements of interpretations 15c3-1(c)(2)(iv)(E)/021 (Clearing Deposits of Introducing Brokers) and 15c3-1(c)(2)(iv)(E)/0212 (Clearing Agreements Containing a Termination Penalty Clause).
(SEC Staff to NYSE) (NYSE Interpretation Memo 00-6, September 2000)
(SEC Staff to FINRA) (FINRA Regulatory Notice 15-25)
15c3-1(c)(2)(iv)(E)/028 Aged Commissions Receivables
An introducing broker’s commissions’ receivables, which are over 30 days old, may be considered allowable assets for net capital purposes, provided:
The carrying and clearing firm credits these commissions to a proprietary securities account of the introducing broker that is included in the PAB reserve formula computation; and
The carrying and clearing firm notifies the introducing broker in writing that the commissions were credited to the proprietary account.
(SEC Staff to NYSE) (NYSE Interpretation Memo 00-6, September 2000)
(SEC Staff to FINRA) (FINRA Regulatory Notice 15-25)
15c3-1(c)(2)(iv)(E)/029 Sole Proprietor Joint Securities Account With Spouse
See interpretation 15c3-1(a)(2)(vi)(B)/04.
15c3-1(c)(2)(iv)(E)/030 Sole Proprietor IRA, Keogh or ERISA Accounts
See interpretation 15c3-1(a)(2)(vi)(B)/05.
15c3-1(c)(2)(iv)(E)/03 Prepaid Fails To Receive
Where advance payments are made for securities which a broker-dealer is failing to receive, the securities are considered unsecured short positions and accordingly their market value is deducted immediately. However, no deduction is required if the selling broker-dealer carries the securities in an account titled “Special Custody Account for Accommodation Transfers for the Exclusive Benefit of Customers of (name of purchasing broker-dealer)”. Also, see interpretation 15c3-3a(Item 4)/02 (Exhibit A).
(SEC Letter to Midwest Stock Exchange, Inc., August 3, 1976)
(NYSE Interpretation Memo 77-2, June 1977)
15c3-1(c)(2)(iv)(E)/04 Unsecured Receivables and Payables With the Same Party
Undisputed unsecured monies receivable from and payable to the same party in connection with underwritings, syndicates, private placements, joint trading accounts, investment banking and similar services, as well as good faith deposits related to unsuccessful syndicate bids may be netted to determine the net credit balance includable in aggregate indebtedness subparagraph (c)(l) or net debit balance treated as unallowable under subparagraph (c)(2)(iv)(E).
(SEC Staff to NYSE) (NYSE Interpretation Memo 82-2, April 1982)
15c3-1(c)(2)(iv)(E)/05 Correspondent Account Balances - (Rescinded)
(NYSE Interpretation Memo 99-6, May 1999)
15c3-1(c)(2)(iv)(E)/06 Eurodollar and Other Offshore Deposits – (Rescinded)
(NYSE Interpretation Memo 06-5, June 2006)
15c3-1(c)(2)(iv)(E)/061 Offshore Demand Deposits and Time Deposits
An offshore demand deposit or time deposit will have no value for net capital purposes unless it is deposited with a non-affiliated financial institution that meets the following conditions:
The non-affiliated financial institution issues certificates of deposit, bank deposit notes, bankers acceptances or bills of exchange that are rated investment grade by at least two NRSROs, has shareholders’ equity of at least US $1 billion and its capital is subject to supervision by an authority of a sovereign national government where a major money market is located; or
The non-affiliated financial institution has shareholders’ equity of at least US $1.5 billion and its capital is subject to supervision by an authority of a sovereign national government where a major money market is located; and
The deposit must be redeemable, without restrictions, at a major money market.
See interpretation 15c3-1(c)(2)(vii)/09 for a list of major money markets.
(SEC Staff to NYSE) (NYSE Interpretation Memo 06-5, June 2006)
15c3-1(c)(2)(iv)(E)/062 Cayman Island Branches of Foreign Banks
Negotiable certificates of deposits (“CDs”) that are issued by the branches of foreign banks located in the Cayman Island may be treated as allowable assets for the purpose of SEA Rule 15c3-1 if the following conditions are satisfied:
The CDs are rated in one of the three highest categories for CDs by at least two nationally recognized statistical ratings organizations;
The CDs are issued by Banks with shareholder equity in excess of $500 million which are subject to supervision by authorities of a sovereign national government other than the Cayman Islands;
The CDs are unconditionally guaranteed by a Bank’s New York branch or agency. Each Guarantor is regulated and supervised by both the Superintendent of Banks of the State of New York and the Board of Governors of the Federal Reserve System. The CD and the Guarantee may not be transferred independently of each other. The Guarantee unconditionally guarantees payment of all amounts payable under the CD, without any requirement that the holder, first proceed against the Cayman Branch. The guarantee by the New York branch of the bank must be in writing, signed by legal counsel of the bank and maintained in the broker-dealer’s records.
The CDs are payable (a) to bearer upon presentation by the Bank in Tokyo, Japan or London, England, as well as in the Cayman Islands; or (b) payable under the Guarantee by the New York branch or agency in New York; and
The CDs are in denominations of at least $100,000, and are issued on an interest bearing or discount basis with maturities of seven days to one year.
(SEC Letter to Seward & Kissel, February 24, 1992)
(NYSE Interpretation Memo 92-7, May 1992)
(NYSE Interpretation Memo 06-5, June 2006)
15c3-1(c)(2)(iv)(E)/07 Custodial Fees Receivable for IRA Custody Accounts
Accrued custodial fees receivable for IRA custody accounts need not be deducted provided the individual fees are secured by and chargeable to each related individual IRA custody account.
(SEC Staff to NYSE) (NYSE Interpretation Memo 90-11, December 1990)
15c3-1(c)(2)(iv)(E)/08 Unendorsed Stock Certificates
See interpretation 15c3-1(c)(2)(iv)(B)/06.
15c3-1(c)(2)(iv)(E)/09 Collateral to Receivable Must be in Physical Possession or Control
Under the law of a particular state, it may be possible to establish a “security interest” in collateral for a receivable without obtaining physical possession or control of the collateral and be “secured” under the state law.
This subparagraph requires that the receivable be considered unsecured and deducted due to the lack of physical possession or control of the collateral.
(SEC Letter to CBOE, August 21, 1981)
(NYSE Interpretation Memo 88-14, August 1988)
15c3-1(c)(2)(iv)(E)/10 Letter of Credit Not Acceptable as Collateral
Any asset that relies upon a letter of credit as its collateral must be considered unsecured and deducted under the provisions of this subparagraph.
(SEC Letter to NASD, August 17, 1981)
(NYSE Interpretation Memo 88-14, August 1988)
15c3-1(c)(2)(iv)(E)/11 Foreign Issued, Foreign Settled Securities - Haircut Alternative to Buy-In for Aged Items
Broker-dealers may, in lieu of the buy-in requirements of paragraph (d)(2) (fail to receives over 30 calendar days old) and of paragraph (m) (securities due from customer on long sales over 10 days old) of SEA Rule 15c3-3, apply alternative procedures regarding foreign issued, foreign settled securities.
In the event such alternative procedures are used (see interpretation 15c3-3(d)(2)/01), the following treatment pursuant to SEA Rule 15c3-1 shall apply:
Thirty days after settlement date, a proprietary haircut charge shall be taken for foreign issued, foreign settled securities failed to receive or on those due from a customer, reduced by the equity (or increased by the deficit) in the transaction on a mark-to-market basis. In those countries where settlement is on a seller’ option basis rather than on a customary settlement cycle, the settlement date for purposes of this alternative will be considered to be a day not more than 30 days from the trade date.
Broker-dealers electing to use this alternative procedure should be aware that the deficit deductions provided under SEA Rule 15c3-1 paragraph (c)(2)(iv)(E) and (c)(2)(ix) (i.e., fails to receive aged over 30 calendar days and haircuts on fails to deliver outstanding 5 business days or longer) will continue to be based on the settlement date of the transaction as determined by the settlement cycle specified under SEA Rule 15c6-1(a).
(SEC Letter to SIA, June 16, 1988) (NYSE Interpretation Memo 88-14, August 1988)
(SEC Staff to FINRA) (FINRA Regulatory Notice 18-03)
15c3-1(c)(2)(iv)(E)/12 NSCC’s RECAPS Program
Broker-dealers participating in the NSCC’s Reconfirmation and Pricing Service (RECAPS) Program may treat the RECAPS settlement date and price as the date of the fail for aging and contract price purposes.
(SEC Letter to NSCC, June 11, 1987)
(NYSE Interpretation Memo 89-6, June 1989)
15c3-1(c)(2)(iv)(E)/13 Foreign Issued, Foreign Settled Securities Fail to Receive - Deficit Deduction Alternative
Broker-dealers may, in lieu of the treatment required by this provision (c)(2)(iv)(E) to deduct deficits for aged fail to receive of foreign issued, foreign settled securities, apply alternative procedures.
In the event such alternative procedures are elected, the following treatment shall apply:
Thirty calendar days after settlement date (in accordance with the current foreign settlement cycle) or forty-five calendar days after trade date, whichever comes first, the broker-dealer shall take a charge equal to the amount by which the market value of the foreign issued, foreign settled securities failed to receive exceeds the contract value of such securities failed to receive (the “deficit”). In those countries where settlement is on a seller's option basis, the settlement date for purposes of this computation will be considered to be a day not more than thirty calendar days from trade date;
During the period from settlement date until the aged failed to receive charge is required to be taken, the broker-dealer will take a concentration charge on a mark-to-market basis equal to 100 percent of the excess of all failed to receive deficits with a single counterparty in excess of ten percent of the broker-dealer's tentative net capital;
In determining a required deduction, the broker-dealer may reduce such deficit by any margin or other deposit held by the broker-dealer in connection with such transaction with the same party and any net equity in failed to deliver transactions not older than five business days past settlement date and/or the net equity in all other failed to receive transactions with the same party;
In determining a required deduction, the broker-dealer may reduce such deficit by any margin calls issued by the broker-dealer, outstanding not more than two business days. A broker-dealer may take advantage of this provision regarding margin calls only if it has a written agreement with the customer regarding the issuance and satisfaction of margin calls;
The broker-dealer shall file a written notice with the national securities exchange or registered national securities association which is its designated examining authority of its intention to apply this alternative treatment in lieu of the requirements of subparagraph (c)(2)(iv)(E) of SEA Rule 15c3-1;
The broker-dealer will maintain in its records a schedule of the current settlement cycle of each country in which it trades; and
The broker-dealer shall maintain and preserve separate records, in whatever form appropriate, detailing, by country, the total number of failed to receive and failed to deliver contracts, and the total contractual value of those contracts and transactions.
(SEC Letter to Securities Industry Association, June 5, 1989)
(NYSE Interpretation Memo 89-9, July 1989)
15c3-1(c)(2)(iv)(E)/14 CNS System Fails to Receive - Not Aged
Fails to receive from a registered clearing agency or a registered securities depository operating under a continuous net settlement system which is marked to market daily shall be considered as continually current and is not subject to the aging provision of subparagraph (c)(2)(iv)(E).
(SEC Staff to NYSE) (NYSE Interpretation Memo 90-11, December 1990)
15c3-1(c)(2)(iv)(E)/15 Deferred “Soft Dollar” Research Expenses
Section 28(e) of the Securities Exchange Act of 1934 provides a safe harbor to investment managers who use the commission dollars of their advised accounts to obtain investment research and brokerage services (“soft dollar” research) by permitting such persons, under certain circumstances, to “pay up” for such services received from broker-dealers.
Deferred expenses relating to “soft dollar” research arrangements and resulting from the payment of third party research expenses for customers prior to receiving compensating commission revenues, must be deducted as a non-allowable asset in computing net capital on a customer-by-customer basis and may not be netted with accrued “soft dollar” liabilities unless both are with the same customer.
Deferred “soft dollar” research expenses are to be determined in accordance with generally accepted accounting principles including the proper matching of revenues and expenses during each accounting period. The possibility of inflated revenues, accelerated expense recognition and the propriety of accruals associated with “soft dollar” arrangements should be of particular concern. (Also see interpretation 15c3-1(c)(1)/13.)
Appropriate accounting records which show the research obligations and expenses should be maintained for each customer having a “soft dollar” research arrangement and covered by this interpretation.
(SEC Staff to NYSE, August 8, 1991) (NYSE Interpretation Memo 91-13, August 1991)
15c3-1(c)(2)(iv)(E)/16 Compensating Balances Deposited With Others
Compensating balances or other assets (e.g. Cash, CDs, time deposits, commercial paper, etc.) deposited with or held by subordinated lenders, or lenders under a NYSE Rule 328 fixed asset agreement, should be treated as assets not readily convertible to cash. These balances should be treated as non-allowable assets up to the value of the liability to the related party.
(SEC Staff to NYSE) (NYSE Interpretation Memo 92-4, January 1992)
15c3-1(c)(2)(iv)(E)/160 Proceeds of a Subordination Used to Purchase Certificates of Deposit of Lending Bank
When a broker-dealer enters into a subordinated loan with a bank and uses the proceeds of the loan to purchase certificates of deposit issued by the lender or an affiliate of the lender, the lender is in possession (although for the account of the broker-dealer) of the subordinated funds. Since the funds that were the subject of the loan have been returned to the lender, the loan is not a satisfactory subordination under SEA Rule 15c3-1d (Appendix D).
(SEC Staff of DMR to NASD, June 1983)
15c3-1(c)(2)(iv)(E)/17 Petty Cash
Petty cash shall be considered a non-allowable asset in the computation of net capital.
(SEC Staff to NYSE) (NYSE Interpretation Memo 04-3, June 2004)
15c3-1(c)(2)(iv)(E)/18 Fixed Income Clearing Corporation (FICC) – Fails to Receive Not Aged
Government securities broker-dealers that are FICC Netting Members need not age open fails to receive and comprehend deductions under the provisions of SEA Rule 15c3-1(c)(2)(iv)(E), for trades processed through FICC’s Netting System which operates on a continuous settlement basis that marks to the market daily.
(Department of the Treasury Letter to GSCC, November 22, 1989)
(SEC Staff to NYSE) (NYSE Interpretation Memo 05-8, April 2005)
15c3-1(c)(2)(iv)(E)/19 Financial Advisory Fees Receivable from a Municipality
Financial advisory fees receivable, which represent income earned for providing assistance to municipalities, in connection with the issuance of bonds, are non-allowable assets pursuant to SEA Rule 15c3-1(c)(2)(iv)(E) for net capital purposes.
(SEC Letter to M.E. Allison & Co., Inc. June 11, 1987)
15c3-1(c)(2)(iv)(F)
15c3-1(c)(2)(iv)(F)(1) For purposes of this paragraph:
15c3-1(c)(2)(iv)(F)(1)(i) The term reverse repurchase agreement deficit shall mean the difference between the contract price for resale of the securities under a reverse repurchase agreement and the market value of those securities (if less than the contract price).
15c3-1(c)(2)(iv)(F)(1)(ii) The term repurchase agreement deficit shall mean the difference between the market value of securities subject to the repurchase agreement and the contract price for repurchase of the securities (if less than the market value of the securities).
15c3-1(c)(2)(iv)(F)(1)(iii) As used in paragraph (c)(2)(iv)(F)(1) of this section, the term contract price shall include accrued interest.
15c3-1(c)(2)(iv)(F)(1)(iv) Reverse repurchase agreement deficits and the repurchase agreement deficits where the counterparty is the Federal Reserve Bank of New York shall be disregarded.
15c3-1(c)(2)(iv)(F)(2)
15c3-1(c)(2)(iv)(F)(2)(i) In the case of a reverse repurchase agreement, the deduction shall be equal to the reverse repurchase agreement deficit.
15c3-1(c)(2)(iv)(F)(2)(i)/01 Reverse Repurchase Agreements Collateral
When a broker-dealer enters into reverse repurchase agreement transaction for its own account, the securities subject to the agreement must be in the possession or control of the broker-dealer and outside of the control of the counterparty in order to treat the contract as an allowable asset for net capital purposes. Securities held by the counterparty in a Special Reserve Bank Account to satisfy a reserve requirement under SEA Rule 15c3-3 would be considered acceptable collateral in treating the reverse repurchase contact as an allowable asset. (Also, see interpretation 15c3-1(c)(2)(iv)(E)/09.)
(SEC Letter to CBOE, August 21, 1981)
(NYSE Interpretation Memo 92-4, January 1992)
15c3-1(c)(2)(iv)(F)(2)(i)/011 Collateral Consisting of Book-Entry Securities Segregated in Affiliate’s Customer Account at the Federal Reserve Bank
If a broker-dealer enters into a reverse repurchase agreement transaction with a bank affiliate, and if the affiliate maintains possession of the book-entry exempt securities collateralizing the transaction, even though segregated in the affiliate's customer account at the Federal Reserve Bank and identified as belonging to the broker-dealer on the books of the affiliate, it is a non-allowable asset for purposes of SEA Rule 15c3-1.
(SEC Staff of DMR to NASD, June 1987)
15c3-1(c)(2)(iv)(F)(2)(ii) In determining the required deductions under paragraph (c)(2)(iv)(F)(2)(i) of this section, the broker or dealer may reduce the reverse repurchase agreement deficit by:
15c3-1(c)(2)(iv)(F)(2)(ii)(A) Any margin or other deposits held by the broker or dealer on account of the reverse repurchase agreement;
15c3-1(c)(2)(iv)(F)(2)(ii)(B) Any excess market value of the securities over the contract price for resale of those securities under any other reverse repurchase agreement with the same party;
15c3-1(c)(2)(iv)(F)(2)(ii)(C) The difference between the contract price for resale and the market value of securities subject to repurchase agreements with the same party (if the market value of those securities is less than the contract price); and
15c3-1(c)(2)(iv)(F)(2)(ii)(D) Calls for margin, marks to the market, or other required deposits which are outstanding one business day or less.
15c3-1(c)(2)(iv)(F)(3)
15c3-1(c)(2)(iv)(F)(3)(i) In the case of repurchase agreements, the deduction shall be:
15c3-1(c)(2)(iv)(F)(3)(i)(A) The excess of the repurchase agreement deficit over 5 percent of the contract price for resale of United States Treasury Bills, Notes and Bonds, 10 percent of the contract price for the resale of securities issued or guaranteed as to principal or interest by an agency of the United States or mortgage related securities as defined in section 3(a)(41) of the Act and 20 percent of the contract price for the resale of other securities and;
15c3-1(c)(2)(iv)(F)(3)(i)(B) The excess of the aggregate repurchase agreement deficits with any one party over 25 percent of the broker or dealer's net capital before the application of paragraph (c)(2)(vi) of this section (less any deduction taken with respect to repurchase agreements with that party under paragraph (c)(2)(iv)(F)(3)(i)(A) of this section) or, if greater;
15c3-1(c)(2)(iv)(F)(3)(i)(C) The excess of the aggregate repurchase agreement deficits over 300 percent of the broker's or dealer's net capital before the application of paragraph (c)(2)(vi) of this section.
15c3-1(c)(2)(iv)(F)(3)(ii) In determining the required deduction under paragraph (c)(2)(iv)(F)(3)(i) of this section, the broker or dealer may reduce a repurchase agreement deficit by:
15c3-1(c)(2)(iv)(F)(3)(ii)(A) Any margin or other deposits held by the broker or dealer on account of a reverse repurchase agreement with the same party to the extent not otherwise used to reduce a reverse repurchase deficit;
15c3-1(c)(2)(iv)(F)(3)(ii)(B) The difference between the contract price and the market value of securities subject to other repurchase agreements with the same party (if the market value of those securities is less than the contract price) not otherwise used to reduce a reverse repurchase agreement deficit; and
15c3-1(c)(2)(iv)(F)(3)(ii)(C) Calls for margin, marks to the market, or other required deposits which are outstanding one business day or less to the extent not otherwise used to reduce a reverse repurchase agreement deficit.
15c3-1(c)(2)(iv)(F)/01 Reverse Repurchase and Repurchase Agreements - Application
Net capital treatments applicable to government securities subject to reverse repurchase, repurchase and matched repurchase agreements are interpreted to apply also to money market instruments not subject to Regulation T.
On a reverse repurchase agreement, the greater of any cash margin deficiency (based on the margin requirement of the examining authority) or the charge required by this paragraph (c)(2)(iv)(F) shall be deducted. (See interpretation 15c3-1(c)(2)(xii)/01)
Interest receivable on a reverse repurchase agreement and interest payable on a repurchase agreement are to be treated as part of the contracted receivable or payable.
The reverse repurchase side of a matched repurchase agreement is treated as described in paragraph (c)(2)(iv)(F)(2). The repurchase side is a financing arrangement comparable to a bank loan.
The deficit, if any, applicable to securities subject to repurchase agreements is treated as described in paragraph (c)(2)(F)(3).
If a broker-dealer has reason to believe a repurchase agreement transaction or a reverse repurchase agreement transaction may not be honored by the party on the other side of such transaction, the transaction will be treated as a proprietary commitment for purposes of computing net capital but such treatment cannot operate to increase net capital.
(SEC Letter to Stuart Brothers, March 29, 1976)
(SEC Letter to Stuart Brothers, August 25, 1976)
(SEC Letter to Stuart Brothers, November 4, 1976)
(SEC Staff to NYSE) (NYSE Interpretation Memo 83-2, April 1983)
(SEC Staff to NYSE) (NYSE Interpretation Memo 88-11, June 1988)
15c3-1(c)(2)(iv)(F)/011 Overnight Reverse Repurchase and Repurchase Deficit Charges
Broker-dealers shall not be subject to capital charges on overnight reverse repurchase or repurchase agreement contracts that are in deficit, provided that at the time of origination the contract was properly collateralized.
If the contract that is in deficit is rolled over without additional funds or securities received from the counterparty, the firm would be subject to a capital charge under paragraph (c)(2)(iv)(F)(2)(i).
(SEC Staff to NYSE) (NYSE Interpretation Memo 96-4, November 1996)
15c3-1(c)(2)(iv)(F)/02 Accrued Coupon Interest
Accrued coupon interest on reverse repurchase and repurchase securities can be added to the market values in determining the deficit charges and additional capital requirements if applicable.
(SEC Letter to NYSE, July, 1992) (NYSE Interpretation Memo 92-12, December 1992)
15c3-1(c)(2)(iv)(F)/03 Buy-Sell Transactions
Buy-sell (sell-buy) transactions are to be considered as reverse repurchase and repurchase positions and all charges for reverse repurchase and repurchase positions under paragraph (c)(2)(iv)(F) shall apply. All open buy-sell balances must be recorded to the broker-dealer’s books and records as required by SEA Rules 17a-3 and 17a-4. A broker-dealer entering into a sell-buy (repurchase) hold in custody transaction must obtain a written agreement from the counterparty of the transaction as required by SEA Rule 15c3-3(b)(4).
(SEC Staff to NYSE) (NYSE Interpretation Memo 96-4, November 1996)
15c3-1(c)(2)(iv)(F)/04 Repurchase Transactions to Maturity – (Rescinded)
(SEC Staff to FINRA) (FINRA Regulatory Notice 13-44)
15c3-1(c)(2)(iv)(F)/05 GSCC’s Netting System and Repo and Reverse Repo Deficits
GSCC Netting Members need not deduct from their net worth repo and reverse repo deficits, outstanding one business day or less, arising from repo and reverse repo agreements that are netted and guaranteed by GSCC as part of GSCC’s netting system.
(SEC Letter to GSCC, April 1, 1998)
(NYSE Interpretation Memo 98-9, July 1998)
15c3-1(c)(2)(iv)(F)/06 Non-Marketable Securities Collateralizing Reverse Repurchase Transactions
Securities that are non-marketable, as defined in SEA Rule 15c3-1, and which have been received as collateral to a reverse repurchase transaction, where cash or other marketable securities, as defined in SEA Rule 15c3-1, have been pledged, shall be subject to a 100% net capital charge if they allocate to a box location for more than two (2) business days.
Where the securities received as collateral to a reverse repurchase transaction are non-marketable, as defined in SEA Rule 15c3-1, but margin has been collected by the broker-dealer, the applicable net capital charge would be the cash receivable or market value of the securities pledged, less the margin collected.
Also, see SEA Rule 15c3-1(c)(2)(iv)(B)/092 (Non-Marketable Securities Collateralizing Purpose Borrow Transactions).
(SEC Staff to NYSE) (NYSE Interpretation Memo 03-3, April 2003)
(SEC Staff to NYSE) (NYSE Interpretation Memo 06-5, June 2006)
15c3-1(c)(2)(iv)(G) Securities borrowed. 1 percent of the market value of securities borrowed collateralized by an irrevocable letter of credit.
15c3-1(c)(2)(iv)(G)/01 Deduction 1 percent charge
The deduction of 1% of the market value of securities borrowed is applied regardless of whether the letter of credit is secured or unsecured.
(SEC Staff to NYSE) (NYSE Interpretation Memo 83-2, April 1983)
15c3-1(c)(2)(iv)(G)/02 Deduction of 1 percent for Securities Borrowed through Euroclear System
The 1% deduction applies to the market value of securities borrowed through the Euroclear System which are collateralized by an irrevocable guarantee constituting the legal and functional equivalent of an irrevocable letter of credit.
(SEC Staff to NYSE) (NYSE Interpretation Memo 93-6, November 1993)
15c3-1(c)(2)(iv)(H) Any receivable from an affiliate of the broker or dealer (not otherwise deducted from net worth) and the market value of any collateral given to an affiliate (not otherwise deducted from net worth) to secure a liability over the amount of the liability of the broker or dealer unless the books and records of the affiliate are made available for examination when requested by the representatives of the Commission or the Examining Authority for the broker or dealer in order to demonstrate the validity of the receivable or payable. The provisions of this subsection shall not apply where the affiliate is a registered broker or dealer, registered government securities broker or dealer or bank as defined in section 3(a)(6) of the Act or insurance company as defined in section 3(a)(19) of the Act or investment company registered under the Investment Company Act of 1940 or federally insured savings and loan association or futures commission merchant registered pursuant to the Commodity Exchange Act.
15c3-1(c)(2)(v)
15c3-1(c)(2)(v)(A) Deducting the market value of all short securities differences (which shall include securities positions reflected on the securities record which are not susceptible to either count or confirmation) unresolved after discovery in accordance with the following schedule:
| | |
--- |
| Differences 1 | Numbers of business days after discovery |
| 25 percent | 7 |
| 50 percent | 14 |
| 75 percent | 21 |
| 100 percent | 28 |
| 1 Percentage of market value of short securities differences. | |
15c3-1(c)(2)(v)(A)/01 Quarterly Security Counts Under SEA Rule 17a-13
This rule requires quarterly security counts to be made by most broker-dealers. In a letter to the Exchange, the SEC highlighted its position that where outside auditors have agreed to perform a security count for a broker-dealer, it does not relieve the broker-dealer of his obligation to inquire promptly about the results of the count because auditors perform this service for their clients only in satisfaction of the broker-dealer’s obligation to comply. The letter went on to state that all unresolved security count differences should be entered in the broker-dealer’s books of account and records within seven business days of the date of the security count. The graduated deductions (per subparagraph (c)(2)(v)) in computing Net Capital commence on the seventh day. Obviously any delay in inquiring about the results of the count reduces the amount of time available to resolve differences before they impact Net Capital.
(SEC Letter to NYSE, July 16, 1979)
(SEC Release 34-18417, January 13, 1982)
(NYSE Interpretation Memo 83-2, April 1983)
15c3-1(c)(2)(v)(B) Deducting the market value of any long securities differences, where such securities have been sold by the broker or dealer before they are adequately resolved, less any reserves established therefor;
15c3-1(c)(2)(v)(B)/01 Long Differences - Resolution
Once funds or securities are forwarded to an escheat (abandoned property) fund, the long difference is deemed to be resolved and its market value need not be deducted from net worth, even if the item has been sold.
(SEC Staff to NYSE)
15c3-1(c)(2)(v)(C) The designated examining authority for a broker or dealer may extend the periods in (v)(A) of this section for up to 10 business days if it finds that exceptional circumstances warrant an extension.
15c3-1(c)(2)(vi) Securities Haircuts
Deducting the percentages specified in paragraphs (c)(2)(vi) (A) through (M) of this section (or the deductions prescribed for securities positions set forth in Appendix A (§ 240.15c3-1a) of the market value of all securities, money market instruments or options in the proprietary or other accounts of the broker or dealer.
15c3-1(c)(2)(vi)/01 Haircuts Application
Where a broker-dealer has both regular-way settled positions and open contractual commitment positions, the total in each category for actual and contractual are added together to determine the combined amounts to which haircuts are to be applied.
Example:
| | | | | | | |
--- --- ---
| | | Contractual | | | | |
| Common Stocks | Actual | Market Value | Contact Value |
| Long | $1,600,000 | $ 400,000 | $ 400,000 |
| Short | $ 500,000 | 100,000 | 110,000 |
| | Combined | | |
| Long | $2,000,000 | | |
| Short | $ 600,000 | | |
Charge
| | | |
---
| | Longs (Actual) - 15% | $240,000 |
| | Longs (Contractual) - 30% | $120,000 |
| | Shorts - 15% of (short value less 25% of long value) | $ 15,000 |
| | reduced by $10,000 profit on contractuals | $(10,000) |
| | Total Charge | $365,000 |
A loss on a contractual commitment should be deducted in arriving at net worth.
Contractual commitment charges vary for certain securities, see SEA Rule 15c3-1 (c)(2)(viii)(A) to (c) for further information.
When-issued and delayed delivery transactions that will be consummated in 30 days or less are treated as actual without any restrictions.
(SEC Staff to NYSE) (NYSE Interpretation Memo 77-2, June 1977)
15c3-1(c)(2)(vi)/02 Joint Trading and Investment Accounts
A broker-dealer is required to include as a proprietary commitment its portion of a joint account in which it is a participant, whether or not it carries the account.
In the event the broker-dealer is carrying the entire joint account, the other participants are to be considered as “non-customers” or “customers” as appropriate, since they are dealing for their own accounts. In the event such an account is in deficit, in effect it is to be considered as a proprietary account in the computation of net capital. If there is an equity in the account, the other participants’ portion must be sufficient to meet the margin requirements of the designated examining authority. If not, the deficiency is charged pursuant to paragraph (c)(2)(xii).
(SEC Staff to NYSE)
15c3-1(c)(2)(vi)/03 Accrued Interest
The accrued interest on proprietary positions is not subject to haircuts.
(SEC Staff to NYSE)
15c3-1(c)(2)(vi)/031 Accrued Interest Receivable on Proprietary Municipal Bonds
A broker-dealer that accrues interest receivable on a monthly basis for municipal bonds with scheduled coupon payable dates may include the receivable as an allowable asset for purposes of SEA Rule 15c3-1(c)(2)(iv)(C), provided that:
No interest payments are in default;
The issue does not trade flat; and
The interest receivable is not outstanding more than 30 days from the end of the calendar month in which the subsequent coupon payable date occurred.
(SEC Staff of DMR to NASD, November 1982)
15c3-1(c)(2)(vi)/032 Book-Entry Securities Held by an Affiliate
A broker-dealer may treat proprietary exempt securities as allowable assets in the computation of net capital when such securities are issued in book-entry form and are held by an affiliate in the affiliates account at the Federal Reserve Bank, provided:
The securities are fully paid for by the broker-dealer.
The securities are not subject to any liens by the affiliate.
The securities are segregated in the affiliates customer account at the Federal Reserve Bank.
The affiliate maintains books and records clearly showing that the broker-dealer is the owner of said securities and is entitled to all principal and interest on demand.
(SEC Staff of DMR to NASD, June 1987)
15c3-1(c)(2)(vi)/04 Rating Organizations; Nationally Recognized - (Rescinded)
(NYSE Interpretation Memo 04-3, June 2004)
15c3-1(c)(2)(vi)/041 Nationally Recognized Statistical Rating Organizations (“NRSROs”) - (Rescinded)
(SEC Staff to FINRA) (FINRA Regulatory Notice 14-38)
15c3-1(c)(2)(vi)/05 Stripped Bonds and Coupons
Where there is a ready market for bonds from which the coupons have been stripped the market value may be included in computing net capital with haircuts taken under the appropriate subsections of SEA Rule 15c3-1(c)(2)(iv)(A), (F) or (J) provided the stripped bonds meet all the qualification requirements in the respective categories.
Where there is a ready market for the individual coupons or sheets of coupons which have been stripped from bonds, the market value of such coupons may be included in computing net capital with haircuts taken under the appropriate subsection of SEA Rule 15c3-1(c)(2)(vi)(A), (F) or (J) wherein the underlying bond would be appropriately classified provided all the qualification requirements in the respective categories are met. Maturities can be based on coupon due date for individual coupons or the due date of the last coupon for sheets of coupons. Ready market is determined as under SEA Rule 15c3-1(c)(11).
(SEC Staff to NYSE) (NYSE Interpretation Memo 83-2, April 1983)
15c3-1(c)(2)(vi)/06 Intercompany Securities Holding – (Rescinded)
(NYSE Interpretation Memo 05-2, January 2005)
15c3-1(c)(2)(vi)/061 Intercompany Securities Holding – Redeemable Debt Instruments
A broker-dealer holding proprietary positions in Certificates of Deposit, Bankers Acceptances, Commercial Paper, Corporate Non-Convertible Bonds, Trust Preferred Securities or similar instruments (“debt instruments”), which are redeemable debt and issued by a parent or an affiliated entity, will receive no net capital value for such positions unless the following conditions are met:
The broker-dealer must maintain a liability or financing loan payable to the parent or affiliated entity on its books equal to the market value of the debt instruments that can be turned into equity capital (it will also be acceptable for a broker-dealer to have a liability or financing loan with the parent offset by debt instruments issued by an affiliated entity); and
The parent or affiliated entity agrees in writing that it will forgive its liability or financing loan by making an equity contribution in an amount equal to the market value of any debt instruments which are not redeemed when presented for payment; and
The ownership of the debt instruments must be transferable to the parent or affiliated entity in complete satisfaction of the liability or financing loan should the parent or affiliated entity become insolvent; and
The agreement must provide that the liability or loans will not be paid or mature prior to the earlier of either:
The sale of the debt instruments by the affiliated broker-dealer, or
The maturity or early redemption of the debt instruments.
However, if the parent or affiliated entity has not provided a contingent equity contribution to the broker-dealer as outlined above, net capital value will be allowed for proprietary positions in redeemable debt instruments for no more than two business days, only if the securities purchased or received were related to the underwriting activity of the broker-dealer.
Proprietary positions in redeemable debt instruments that meet the conditions of this interpretation must still be subject to the haircut deduction required under SEA Rule 15c3-1 subparagraphs (c)(2)(vi) and (c)(2)(vii) for net capital purposes.
Proprietary positions in asset-backed securities and trust assets issued by a parent or an affiliated entity are not subject to the provisions of this interpretation and should be treated in accordance with the marketability requirements of SEA Rule 15c3-1 for net capital purposes.
Proprietary positions in common stock, preferred stock and convertible bonds issued by a parent or an affiliated entity do not fall within the provisions of this interpretation and should be treated as non-marketable securities for net capital purposes.
(SEC Letter to White & Case, April 14, 1989) (NYSE Interpretation Memo 89-9, July 1989)
(SEC Staff to NYSE) (NYSE Interpretation Memo 05-2, January 2005)
(SEC Staff to NYSE) (NYSE Interpretation Memo 05-8, April 2005)
15c3-1(c)(2)(vi)/07 Foreign Securities
See interpretations 15c3-1(c)(2)(vii)/08 and /09 for marketability of certain foreign and domestic debt, banker's acceptances and money market instruments.
15c3-1(c)(2)(vi)/08 Haircut Deduction on a Foreign Currency Balance
A foreign currency balance shall be treated as “inventory” and subject to the applicable haircut deduction to cover any currency risk that has not been eliminated by an offsetting balance, security position, futures contract or contractual commitment in the same foreign currency.
The haircut deduction applicable on a foreign currency balance is as follows:
A 6% haircut deduction shall be applied on the US dollar equivalent amount of a foreign currency net debit or credit balance in any of the five major foreign currencies (Euro, British pound, Swiss franc, Canadian dollar and Japanese yen).
A 20% haircut deduction shall be applied on the US dollar equivalent amount of a foreign currency net debit or credit balance in all other foreign currencies.
(SEC Staff to NYSE) (NYSE Interpretation Memo 90-11, December 1990)
(SEC Staff to FINRA) (FINRA Regulatory Notice 13-44)
15c3-1(c)(2)(vi)/09 Haircut Deductions on Inventory Positions Denominated in a Foreign Currency
An additional currency risk haircut deduction of 6% must be applied on inventory positions denominated in any of the five major foreign currencies (Euro, British pound, Swiss franc, Canadian dollar and Japanese yen). An additional currency risk haircut deduction of 20% must be applied on inventory positions denominated in all other foreign currencies.
In determining the additional currency risk haircut deductions to be applied, inventory positions denominated in a foreign currency may be offset by a balance, security position, futures contract or contractual commitment in the same foreign currency.
However, no offsetting is permitted for purposes of computing the currency risk haircut deductions required hereunder on any inventory positions denominated in a foreign currency that receive hedging treatment under paragraph (c)(2)(vi)(A) or (c)(2)(vi)(F) of SEA Rule 15c3-1.
Note: The above currency risk haircut deductions are additive to any haircut deductions applicable to the underlying inventory positions denominated in a foreign currency, as provided in paragraph (c)(2)(vi) of SEA Rule 15c3-1.
(SEC Staff to NYSE) (No.92-12, December 1992)
(SEC Staff to FINRA) (FINRA Regulatory Notice 13-44)
15c3-1(c)(2)(vi)/10 Securities Deposited by U.S. Subsidiaries with Foreign Parent
Securities deposited with a foreign parent company are deemed "not readily convertible to cash" and subject to a 100% deduction from net worth.
A broker-dealer who is a subsidiary of a foreign broker-dealer parent and deposits securities with its parent, may allow net capital value for those securities under the following conditions.
The proprietary securities must be registered in the subsidiary's name;
The proprietary securities must be physically segregated in the foreign parent's vault abroad;
The foreign parent must submit a letter to the subsidiary which is provided to its Designated Examining Authority which will assure that such proprietary securities will not be subject to any encumbrances or liens by the foreign parent;
Each firm will provide a letter to its Designated Examining Authority from the subsidiary's fidelity bond company which verifies that coverage extends to the proprietary securities in the custody of the foreign parent, or the foreign parent's insurance/bonding company submits a letter which provides equivalent coverage;
The amount of the subsidiary's proprietary securities in the custody of the foreign parent does not exceed the subsidiary's tentative net capital for more than three (3) consecutive business days;
The subsidiary must be treated by the parent the same as any other customer of the foreign parent for such purposes as bankruptcy of the parent under the laws of the foreign parents country;
In complying with SEA Rule 17a-5, the subsidiary's deposited proprietary securities must be inspected quarterly by parent company employees and the results of those inspections must be reported within 15 days of completion of the inspections to the independent public accountant for the parent for review.
The foreign parent must remain in compliance with the foreign regulatory net capital provisions; and
The independent public accountant for the subsidiary considers items 1, 2, 5 and 7 above, in connection with the supplemental schedule on net capital requirement by SEA Rule 17a-5(d).
(SEC Letter to NYSE, July 30, 1986) (NYSE Interpretation Memo 86-9, August 1986)
(SEC Staff to NYSE) (NYSE Interpretation Memo 93-6, November 1993)
15c3-1(c)(2)(vi)/101 Securities Deposited by U.S. Subsidiaries with Foreign Parent for Two Business Days or Less
Proprietary securities on deposit at a foreign parent of a U.S. broker-dealer subsidiary that results from clearance activities and that represents assets utilized for regulatory capital of the subsidiary broker dealer would not be subject to the provisions of interpretation 15c3-1(c)(2)(vi)/10 (Securities Deposited by U.S. Subsidiaries with Foreign Parent), if maintained at the foreign parent for two business days or less. A daily determination of the aging of the securities needs to be recorded by the U.S. broker-dealer subsidiary to avail themselves of the two-day relief period.
(SEC Staff to NYSE) (NYSE Interpretation Memo 98-5, May 1998)
15c3-1(c)(2)(vi)/11 (Rescinded)
(NYSE Interpretation Memo 97-5, September 1997)
15c3-1(c)(2)(vi)/12 Bona Fide Arbitrage - paragraphs (c)(2)(vi)(A) through (H)
A bona fide arbitrage treatment can also be applied to any security position in paragraphs (c)(2)(vi) that is convertible into or exchangeable within 90 days subject to no conditions other than the payment of money for the same security that the firm is short. Securities which cannot be publicly offered or sold because of statutory, regulatory or contractual agreements or other restrictions are not considered the same as unrestricted securities and cannot be used for this treatment.
(SEC Staff to NYSE) (NYSE Interpretation Memo 95-3, May 1995)
15c3-1(c)(2)(vi)/13 Error Transactions of Floor Brokers – (Rescinded)
(NYSE Interpretation Memo 03-2, March 2003)
15c3-1(c)(2)(vi)/14 Error Transactions of Floor Brokers
When a broker-dealer, which is primarily in the business of acting as a floor broker, makes an error in executing a transaction, which is done as a floor broker for another broker, no haircut need be taken on the resulting error position provided the security position is immediately liquidated upon discovery, but no later than the closing of the business day after the day the error occurred.
A broker-dealer is considered to be primarily in the business of acting as a floor broker when 75% of its gross revenue is derived from floor brokerage commissions.
This interpretation is applicable for a floor broker which either owns its seat or leases its seat.
(SEC Staff to NYSE) (NYSE Interpretation Memo 03-2, March 2003)
15c3-1(c)(2)(vi)(A)
15c3-1(c)(2)(vi)(A)(1) In the case of a security issued or guaranteed as to principal or interest by the United States or any agency thereof, the applicable percentages of the market value of the net long or short position in each of the categories specified below are:
Category 1
(i) Less than 3 months to maturity—0 percent.
(ii) 3 months but less than 6 months to maturity—1⁄2 of 1 percent.
(iii) 6 months but less than 9 months to maturity—3⁄4 of 1 percent.
(iv) 9 months but less than 12 months to maturity—1 percent.
Category 2
(i) 1 year but less than 2 years to maturity—1 1⁄2 percent.
(ii) 2 years but less than 3 years to maturity—2 percent.
Category 3
(i) 3 years but less than 5 years to maturity—3%.
(ii) 5 years but less than 10 years to maturity—4%.
Category 4
(i) 10 years but less than 15 years to maturity—4 1⁄2%.
(ii) 15 years but less than 20 years to maturity—5%.
(iii) 20 years but less than 25 years to maturity—5 1⁄2%.
(iv) 25 years or more to maturity—6%.
Brokers or dealers shall compute a deduction for each category above as follows: Compute the deductions for the net long or short positions in each subcategory above. The deduction for the category shall be the net of the aggregate deductions on the long positions and the aggregate deductions on the short positions in each category plus 50% of the lesser of the aggregate deductions on the long or short positions.
15c3-1(c)(2)(vi)(A)(2) A broker or dealer may elect to deduct, in lieu of the computation required under paragraph (c)(2)(vi)(A)(1) of this section, the applicable percentages of the market value of the net long or short positions in each of the subcategories specified in paragraph (c)(2)(vi)(A)(1) of this section.
15c3-1(c)(2)(vi)(A)(3) In computing deductions under paragraph (c)(2)(vi)(A)(1) of this section, a broker or dealer may elect to exclude the market value of a long or short security from one category and a security from another category, Provided, That:
15c3-1(c)(2)(vi)(A)(3)(i) Such securities have maturity dates:
15c3-1(c)(2)(vi)(A)(3)(i)(A) Between 9 months and 15 months and within 3 months of one another.
15c3-1(c)(2)(vi)(A)(3)(i)(B) Between 2 years and 4 years and within 1 year of one another; or
15c3-1(c)(2)(vi)(A)(3)(i)(C) Between 8 years and 12 years and within 2 years of one another.
15c3-1(c)(2)(vi)(A)(3)(ii) The net market value of the two excluded securities shall remain in the category of the security with the higher market value.
15c3-1(c)(2)(vi)(A)(4) In computing deductions under paragraph (c)(2)(vi)(A)(1) of this section, a broker or dealer may include in the categories specified in paragraph (c)(2)(vi)(A)(1) of this section, long or short positions in securities issued by the United States or any agency thereof that are deliverable against long or short positions in futures contracts relating to Government securities, traded on a recognized contract market approved by the Commodity Futures Trading Commission, which are held in the proprietary or other accounts of the broker or dealer. The value of the long or short positions included in the categories shall be determined by the contract value of the futures contract held in the account. The provisions of Appendix B to Rule 15c3-1 (17 CFR 240.15c3-1b) will in any event apply to the positions in futures contracts.
15c3-1(c)(2)(vi)(A)(5) In the case of a Government securities dealer that reports to the Federal Reserve System, that transacts business directly with the Federal Reserve System, and that maintains at all times a minimum net capital of at least $50,000,000, before application of the deductions provided for in paragraph (c)(2)(vi) of this section, the deduction for a security issued or guaranteed as to principal or interest by the United States or any agency thereof shall be 75 percent of the deduction otherwise computed under paragraph (c)(2)(vi)(A) of this section.
15c3-1(c)(2)(vi)(A)/01 Quasi-agency Securities
Government securities haircuts apply also to securities issued under the sponsorship of the United States or any agency thereof, where the securities are traded in the market on a basis similar to securities issued or guaranteed as to principal or interest by the United States or any agency thereof.
Some examples of debt securities to which government securities haircuts apply are:
Bank for Cooperatives
District of Columbia
Export-Import Banks
Farm Credit Bank
Federal Agricultural Mortgage Corporation
Federal Home Loan Banks
Federal Home Loan Mortgage Corp.
Federal Intermediate Credit Banks
Federal Land Banks
Federal National Mortgage Assn.
General Services Administration
General National Mortgage Assn.
International Bank for Reconstruction and Development
Tennessee Valley Authority
Washington Metropolitan Area Transit Authority
(SEC Staff to NYSE) (NYSE Interpretation Memo 79-2, January 1979)
(SEC Staff to NYSE) (NYSE Interpretation Memo 00-6, September 2000)
15c3-1(c)(2)(vi)(A)/011 Government Agency Securities that are Direct Obligations of or Guaranteed as to Principal or Interest by the United States
Securities that are direct obligations of, or obligations guaranteed as to principal or interest by the United States, such as Government National Mortgage Association (GNMA) securities, Public Housing Authority bonds guaranteed as to principal or interest by the United States, and the portion of loans guaranteed as to principal and interest by the Small Business Administration are subject to the haircuts specified in SEA Rule 15c3-1(c)(2)(vi)(A) (Government Securities).
(SEC Letter to Robison, Curphey & O'Connell, August 2, 1977)
(SEC Letter to Rogers & Lamb, August 30, 1977)
15c3-1(c)(2)(vi)(A)/02 GNMA's (Temporary Interpretation)
All transactions in GNMA's maturing in twenty-five years or more (including issued positions, commodity exchange futures, and TBA's ) are treated as a special separate category subject to haircut percentages as follows:
| | | |
---
| Issued positions or TBA’s unhedged | | 6% of the net long or short market value |
| Commodity exchange futures unhedged and spreads in commodity exchange futures | | In accordance with SEA Rule 15c3-1b (Appendix B) subparagraph (a)(3)(xiv) |
| Issued positions hedged by commodity exchange futures or TBA’s | | No haircut |
| TBA’s hedged by commodity exchange futures | | No haircut |
| Issued positions hedging issued positions | | No haircut |
| TBA’s hedging TBA’s | | No haircut |
(See interpretation 15c3-1a/021 (Appendix A) for treatment of GNMA standbys.)
Subject to modification
TBA's (to be announced) are delayed delivery and "when issued" type transactions in GNMA's. Generally, GNMA pool numbers are not announced or assigned to these transactions on trade date.
(SEC Staff to NYSE) (NYSE Interpretation Memo 79-10, December 1979)
15c3-1(c)(2)(vi)(A)/021 FHLMC’s (Federal Home Loan Mortgage Corporation -Temporary Interpretation)
All transactions in FHLMC’s maturing in twenty-five years or more (including issued positions, and TBA’s) are treated as a special separate category subject to haircut percentages as follows:
| | | |
---
| Issued positions or TBA’s unhedged | | 6% of the net long or short market value |
| Issued positions hedged by TBA’s | | No haircut |
| Issued positions hedging issued positions | | No haircut |
| TBA’s hedging TBA’s | | No haircut |
Subject to modification
TBA’s (to be announced) are delayed delivery and “when issued” type transactions in FHLMC’s. Generally, FHLMC group numbers are not announced or assigned to these transactions on trade date.
(SEC Staff to NYSE) (NYSE Interpretation Memo 79-10, December 1979)
(NYSE Interpretation Memo 82-3, August 1982)
15c3-1(c)(2)(vi)(A)/03 GNMA’s
If a broker-dealer’s net position in GNMA’s is comprised of either issued or futures positions, the broker-dealer may elect to include the net position in the government securities haircut computation as specified in paragraph (c)(2)(vi)(A).
(SEC Staff to NYSE) (NYSE Interpretation Memo 82-3, December 1982)
15c3-1(c)(2)(vi)(A)/04 Alternative for Pass -Through mortgage Securities
Haircuts on pass -through mortgage securities sponsored by the Unites States government or any agency thereof, may be calculated using the treatment outlined in the SEC No-Action Letter to the NYSE, dated December 30, 1996. (See NYSE Interpretation Memo 97-2 for a copy of the letter.)
(SEC Letter to NYSE, December 30, 1996) (No.97-2, February 1997)
15c3-1(c)(2)(vi)(B)
15c3-1(c)(2)(vi)(B)(1) In the case of any municipal security which has a scheduled maturity at date of issue of 731 days or less and which is issued at par value and pays interest at maturity, or which is issued at a discount, and which is not traded flat or in default as to principal or interest, the applicable percentages of the market value on the greater of the long or short position in each of the categories specified below are:
15c3-1(c)(2)(vi)(B)(1)(i) Less than 30 days to maturity—0%.
15c3-1(c)(2)(vi)(B)(1)(ii) 30 days but less than 91 days to maturity—1⁄8 of 1%.
15c3-1(c)(2)(vi)(B)(1)(iii) 91 days but less than 181 days to maturity—1⁄4 of 1%.
15c3-1(c)(2)(vi)(B)(1)(iv) 181 days but less than 271 days to maturity—3⁄8 of 1%.
15c3-1(c)(2)(vi)(B)(1)(v) 271 days but less than 366 days to maturity—1⁄2 of 1%.
15c3-1(c)(2)(vi)(B)(1)(vi) 366 days but less than 456 days to maturity—3⁄4 of 1%.
15c3-1(c)(2)(vi)(B)(1)(vii) 456 days but less than 732 days to maturity—1%.
15c3-1(c)(2)(vi)(B)(2) In the case of any municipal security, other than those specified in paragraph (c)(2)(vi)(B)(1), which is not traded flat or in default as to principal or interest, the applicable percentages of the market value of the greater of the long or short position in each of the categories specified below are:
15c3-1(c)(2)(vi)(B)(2)(i) Less than 1 year to maturity—1%.
15c3-1(c)(2)(vi)(B)(2)(ii) 1 year but less than 2 years to maturity—2%.
15c3-1(c)(2)(vi)(B)(2)(iii) 2 years but less than 3 1⁄2 years to maturity—3%.
15c3-1(c)(2)(vi)(B)(2)(iv) 3 1⁄2 years but less than 5 years to maturity—4%.
15c3-1(c)(2)(vi)(B)(2)(v) 5 years but less than 7 years to maturity—5%.
15c3-1(c)(2)(vi)(B)(2)(vi) 7 years but less than 10 years to maturity—5 1⁄2%.
15c3-1(c)(2)(vi)(B)(2)(vii) 10 years but less than 15 years to maturity—6%.
15c3-1(c)(2)(vi)(B)(2)(viii) 15 years but less than 20 years to maturity—6 1⁄2%.
15c3-1(c)(2)(vi)(B)(2)(ix) 20 years or more to maturity—7%.
15c3-1(c)(2)(vi)(B)/01 Zero Coupon Issues - Application
Zero coupon bonds may be treated under the haircut provisions of paragraph (c)(2)(vi)(B)(2) providing that other securities issued by the municipality are not in default or trading flat as to principal and interest. If other securities of the municipality are in default or are trading flat as to principal or interest, the haircut will be the percentage as specified in SEA Rule 15c3-1(c)(2)(vi)(J).
(SEC Staff to NYSE) (NYSE Interpretation Memo 82-3, December 1982)
15c3-1(c)(2)(vi)(B)/02 Municipal Securities - Presumed Marketability
Where municipal securities are valued under the presumed marketability method as under interpretation 15c3-1(c)(2)(vii)/02 the haircut to be taken is the greater of the deduction specified above or the deduction taken in arriving at the presumed value.
(SEC Staff to NYSE) (NYSE Interpretation Memo 86-8, August 1986)
15c3-1(c)(2)(vi)(B)/03 Municipal Put Bonds - Endorsed by Issuer
A one percent (1%) haircut can be taken on the aggregate principal amount of proprietary municipal put bonds when:
The issue is a variable or floating rate municipal security which should normally trade at or near par with a non-severable periodic demand feature which entitles the holder to put the underlying security to the issuer through the re-marketing agent at its par value at designated times.
The issuer supports its ability to satisfy the holder's demand with an irrevocable letter of credit, a standby bond purchase agreement or other liquidity feature which provides third-party credit support to ensure the availability of sufficient funds to allow a holder to recover the principal amount of the instrument upon the exercise of the demand feature.
(SEC Letter to NYSE and NASD, August 18, 1988)
(NYSE Interpretation Memo 88-19, October 1988)
(NYSE Interpretation Memo 93-6, November 1993)
15c3-1(c)(2)(vi)(B)/04 Municipal Put Bonds - Endorsed by Broker-dealers
A municipal bond that has been sold with a put endorsed or guaranteed by the broker-dealer where the bond can be put to the broker-dealer with no obligation of the issuer to purchase it back, should be treated as an uncovered OTC put under SEA Rule 15c3-1a(c)(2) (Appendix A).
(SEC Staff to NYSE) (NYSE Interpretation Memo 86-8, August 1986) (NYSE Interpretation Memo 93-6, November 1993)
15c3-1(c)(2)(vi)(C) Canadian Debt Obligations. In the case of any security issued or unconditionally guaranteed as to principal and interest by the Government of Canada, the percentages of market value to be deducted shall be the same as in paragraph (A) of this section.
15c3-1(c)(2)(vi)(D)
15c3-1(c)(2)(vi)(D)(1) In the case of redeemable securities of an investment company registered under the Investment Company Act of 1940, which assets consist of cash or money market instruments and which is described in § 270.2a-7 of this chapter, the deduction will be 2% of the market value of the greater of the long or short position.
15c3-1(c)(2)(vi)(D)(2) In the case of redeemable securities of an investment company registered under the Investment Company Act of 1940, which assets are in the form of cash or securities or money market instruments of any maturity which are described in paragraph (c)(2)(vi) (A) through (C) or (E) of this section, the deduction shall be 7% of the market value of the greater of the long or short positions.
15c3-1(c)(2)(vi)(D)(3) In the case of redeemable securities of an investment company registered under the Investment Company Act of 1940, which assets are in the form of cash or securities or money market instruments which are described in paragraphs (c)(2)(vi) (A) through (C) or (E) and (F) of this section, the deduction shall be 9% of the market value of the long or short position.
15c3-1(c)(2)(vi)(D)/01 Money Market Funds – (Rescinded)
(SEC Staff to FINRA) (FINRA Regulatory Notice 14-06)
15c3-1(c)(2)(vi)(D)/02 Registered Investment Companies with Repurchase Agreements in the Portfolio
Repurchase agreements in the portfolio of registered investment companies do not alter the haircuts applicable to the registered investment company issue pursuant to SEA Rule 15c3-1(c)(2)(vi)(D)(1), (2), and (3).
(SEC Staff of DMR to NASD, April 1981)
15c3-1(c)(2)(vi)(D)/03 Redeemable Securities of an Investment Company Registered Under the Investment Company Act of 1940
If the prospectus issued by an investment company registered under the Investment Company Act of 1940 indicates that the investment company may invest in, or its assets may consist of securities or money market instruments that are not specified in paragraphs (D)(2) and (D)(3) of this Rule, the haircut deduction to be applied shall be 15% of the market value of the greater of the long or short positions held by the broker-dealer in such redeemable securities.
(SEC Staff to FINRA) (FINRA Regulatory Notice 13-44)
15c3-1(c)(2)(vi)(E) Commercial paper, bankers' acceptances and certificates of deposit. In the case of any short term promissory note or evidence of indebtedness which has a fixed rate of interest or is sold at a discount, which has a maturity date at date of issuance not exceeding nine months exclusive of days of grace, or any renewal thereof, the maturity of which is likewise limited and has only a minimal amount of credit risk, or in the case of any negotiable certificates of deposit or bankers' acceptance or similar type of instrument issued or guaranteed by any bank as defined in section 3(a)(6) of the Securities Exchange Act of 1934 (15 U.S.C. 78c(a)(6)), the applicable percentage of the market value of the greater of the long or short position in each of the categories specified below are:
15c3-1(c)(2)(vi)(E)(1) Less than 30 days to maturity—0 percent.
15c3-1(c)(2)(vi)(E)(2) 30 days but less than 91 days to maturity 1⁄8 of 1 percent.
15c3-1(c)(2)(vi)(E)(3) 91 days but less than 181 days to maturity 1⁄4 of 1 percent.
15c3-1(c)(2)(vi)(E)(4) 181 days but less than 271 days to maturity 3⁄8 of 1 percent.
15c3-1(c)(2)(vi)(E)(5) 271 days but less than 1 year to maturity 1⁄2 of 1 percent; and
15c3-1(c)(2)(vi)(E)(6) With respect to any negotiable certificate of deposit or bankers acceptance or similar type of instrument issued or guaranteed by any bank, as defined above, having 1 year or more to maturity, the deduction shall be on the greater of the long or short position and shall be the same percentage as that prescribed in paragraph (c)(2)(vi)(A) of this section.
15c3-1(c)(2)(vi)(E)/01 Commercial Paper Rating - (Rescinded)
(SEC Staff to FINRA) (FINRA Regulatory Notice 14-38)
15c3-1(c)(2)(vi)(E)/011 Commercial Paper Marketability
See interpretation 15c3-1(c)(2)(vii)/06 for ready market criteria.
15c3-1(c)(2)(vi)(E)/012 Commercial Paper Marketability under Section 936 Market
See interpretation 15c3-1(c)(2)(vii)/07 for ready market criteria.
15c3-1(c)(2)(vi)(E)/013 Concentration Charge on Money Market Instruments - Greater than 30% of Tentative Net Capital
See interpretation 15c3-1(c)(2)(vii)/09 for concentration criteria.
15c3-1(c)(2)(vi)(E)/02 Certificates of Deposit - Banks
The haircuts in this subparagraph apply to certificates of deposit issued by Federal Savings and Loan Associations and certain state chartered insured institutions, as authorized by the Federal Home Loan Bank Board.
(SEC Letter to A.G. Becker & Co., Inc., March 10, 1976)
(NYSE Interpretation Memo 76-4, April 1976)
15c3-1(c)(2)(vi)(E)/03 Federally Chartered Savings and Loan Association Short-Term Promissory Notes
Short-term promissory notes issued by federally chartered savings and loan associations should be treated as if commercial paper, provided the notes have a maturity date at issuance not exceeding nine months exclusive of date of grace, or any renewal thereof (the maturity of which is likewise limited) and have only a minimal amount of credit risk (see paragraphs (c)(2)(vi)(E) and (c)(2)(vi)(I) of SEA Rule 15c3-1).
(SEC Letter to A.G. Becker Inc., August 1, 1979)
(NYSE Interpretation Memo 79-10, December 1979)
(SEC Staff to FINRA) (FINRA Regulatory Notice 14-38)
15c3-1(c)(2)(vi)(E)/04 Certificates of Deposit-Non-Negotiable
Non-negotiable certificates of deposit subject to immediate withdrawal at any time pursuant to the requirements of Regulation Q of the Federal Reserve System are allowable assets for net capital purposes, less the early withdrawal penalties, if any, and less the haircut charges stipulated in this subparagraph (c)(2)(vi)(E).
No capital value is allowable if not subject to immediate withdrawal.
(SEC Staff to NYSE) (NYSE Interpretation Memo 88-15, September 1988)
15c3-1(c)(2)(vi)(E)/05 Foreign Bankers Acceptances and Bankers Deposit Notes
See interpretation 15c3-1(c)(2)(vii)/09 (Marketability of Money Market Instruments).
15c3-1(c)(2)(vi)(F)
15c3-1(c)(2)(vi)(F)(1) Nonconvertible debt securities. In the case of nonconvertible debt securities having a fixed interest rate and a fixed maturity date, which are not traded flat or in default as to principal or interest and which have only a minimal amount of credit risk, the applicable percentages of the market value of the greater of the long or short position in each of the categories specified below are:
15c3-1(c)(2)(vi)(F)(1)(i) Less than 1 year to maturity—2%
15c3-1(c)(2)(vi)(F)(1)(ii) 1 year but less than 2 years to maturity—3%
15c3-1(c)(2)(vi)(F)(1)(iii) 2 years but less than 3 years to maturity—5%
15c3-1(c)(2)(vi)(F)(1)(iv) 3 years but less than 5 years to maturity—6%
15c3-1(c)(2)(vi)(F)(1)(v) 5 years but less than 10 years to maturity—7%
15c3-1(c)(2)(vi)(F)(1)(vi) 10 years but less than 15 years to maturity—7 1⁄2%
15c3-1(c)(2)(vi)(F)(1)(vii) 15 years but less than 20 years to maturity—8%
15c3-1(c)(2)(vi)(F)(1)(viii) 20 years but less than 25 years to maturity—8 1⁄2%
15c3-1(c)(2)(vi)(F)(1)(ix) 25 years or more to maturity—9%
15c3-1(c)(2)(vi)(F)(2) A broker or dealer may elect to exclude from the above categories long or short positions that are hedged with short or long positions in securities issued by the United States or any agency thereof or nonconvertible debt securities having a fixed interest rate and a fixed maturity date and which are not traded flat or in default as to principal or interest, and which have only a minimal amount of credit risk if such securities have maturity dates:
15c3-1(c)(2)(vi)(F)(2)(i) Less than five years and within 6 months of each other;
15c3-1(c)(2)(vi)(F)(2)(ii) Between 5 years and 10 years and within 9 months of each other;
15c3-1(c)(2)(vi)(F)(2)(iii) Between 10 years and 15 years and within 2 years of each other; or
15c3-1(c)(2)(vi)(F)(2)(iv) 15 years or more and within 10 years of each other.
The broker-dealer shall deduct the amounts specified in paragraphs (c)(2)(vi)(F) (3) and (4) of this section.
15c3-1(c)(2)(vi)(F)(3) With respect to those positions described in paragraph (c)(2)(vi)(F)(2) of this section that include a long or short position in securities issued by the United States or any agency thereof, the broker or dealer shall exclude the hedging short or long United States or agency securities position from the applicable haircut category under paragraph (c)(2)(vi)(A) of this section. The broker or dealer shall deduct the percentage of the market value of the hedged long or short position in nonconvertible debt securities as specified in each of the categories below:
15c3-1(c)(2)(vi)(F)(3)(i) Less than 5 years to maturity—1 1⁄2%
15c3-1(c)(2)(vi)(F)(3)(ii) 5 years but less than 10 years to maturity—2 1⁄2%
15c3-1(c)(2)(vi)(F)(3)(iii) 10 years but less than 15 years to maturity—2 3⁄4%
15c3-1(c)(2)(vi)(F)(3)(iv) 15 years or more to maturity—3%
15c3-1(c)(2)(vi)(F)(4) With respect to those positions described in paragraph (c)(2)(vi)(F)(2) of this section that include offsetting long and short positions in nonconvertible debt securities, the broker or dealer shall deduct a percentage of the market value of the hedged long or short position in nonconvertible debt securities as specified in each of the categories below:
15c3-1(c)(2)(vi)(F)(4)(i) Less than 5 years to maturity—1 3⁄4%
15c3-1(c)(2)(vi)(F)(4)(ii) 5 years but less than 10 years to maturity—3%
15c3-1(c)(2)(vi)(F)(4)(iii) 10 years but less than 15 years to maturity—3 1⁄4%
15c3-1(c)(2)(vi)(F)(4)(iv) 15 years or more to maturity—3 1⁄2%
15c3-1(c)(2)(vi)(F)(5) In computing deductions under paragraph (c)(2)(vi)(F)(3) of this section, a broker or dealer may include in the categories specified in paragraph (c)(2)(vi)(F)(3) of this section, long or short positions in securities issued by the United States or any agency thereof that are deliverable against long or short positions in futures contracts relating to Government securities, traded on a recognized contract market approved by the Commodity Futures Trading Commission, which are held in the proprietary or other accounts of the broker or dealer. The value of the long or short positions included in the categories shall be determined by the contract value of the futures contract held in the account.
15c3-1(c)(2)(vi)(F)(5)/01 Hedging of Debt Securities
When determining haircut charges under Rule 15c3-1(c)(2)(vi)(F)(3) through (5), debt securities can only be hedged with other debt securities that are denominated in the same currency.
(SEC Staff to NYSE) (NYSE Interpretation Memo 97-6, September 1997)
15c3-1(c)(2)(vi)(F)(6) The provisions of Appendix B to Rule 15c3-1 (17 CFR 240.15c3-1b) will in any event apply to the positions in futures contracts.
15c3-1(c)(2)(vi)(F)/01 Zero Coupon Bonds - Application
Corporate zero coupon bonds having a fixed maturity date may be treated under the haircut provisions of paragraph (c)(2)(vi)(F) providing:
That other securities issued by the entity are not in default as to principal or interest, and
The bonds have only a minimal amount of credit risk (see paragraphs (c)(2)(vi)(F) and (c)(2)(vi)(I) of SEA Rule 15c3-1).
(SEC Staff to NYSE) (NYSE Interpretation Memo 82-3, December 1982)
(SEC Staff to FINRA) (FINRA Regulatory Notice 14-38)
15c3-1(c)(2)(vi)(F)/02 Taxable Securities Issued By Church And Health Care Institutions
For the above securities to be treated under this subparagraph, only one rating in one of the four highest categories by a nationally recognized statistical rating organization is required.
(SEC Letter to O’Connor & Hannan, June 29, 1993)
(NYSE Interpretation Memo 93-6, November 1993)
(SEC Staff to NYSE) (NYSE Interpretation Memo 97-6, October 1997)
15c3-1(c)(2)(vi)(F)/03 Repurchase Agreements and Reverse Repurchase Agreements in Non-Marginable Corporate Bonds
See subparagraph (c)(2)(iv)(F) of SEA Rule 15c3-1.
15c3-1(c)(2)(vi)(F)/04 Foreign Sovereign National Government Debt Securities
See interpretation15c3-1(c)(2)(vii)/08 (Marketability of Certain Foreign and Domestic Securities).
15c3-1(c)(2)(vi)(F)/05 Canadian Province or Municipal Debt Securities
See interpretation 15c3-1(c)(2)(vii)/08 (Marketability of Certain Foreign and Domestic Securities).
15c3-1(c)(2)(vi)(F)/06 Foreign Non-Convertible Debt Securities
See interpretation 15c3-1(c)(2)(vii)/08 (Marketability of Certain Foreign and Domestic Securities).
15c3-1(c)(2)(vi)(F)/07 Secondary Mortgage Market Enhancement Act (SMMEA) Debt Securities
Debt securities issued under the Secondary Mortgage Market Enhancement Act (SMMEA) which are rated in one of the two highest rating categories by one of the nationally recognized rating organizations may be included and combined as non-convertible debt securities under this subparagraph (c)(2)(vi)(F)(1) through (6) of SEA Rule 15c3-1 for haircut purposes.
(SEC Letter to SIA, June 12, 1992) (NYSE Interpretation Memo 92-12, December 1992)
15c3-1(c)(2)(vi)(F)/08 Defense Security Assistance Agency Guaranteed Debt Securities
Debt securities which are 90% guaranteed by the Defense Security Assistance Agency pursuant to Public Law No. 100.202 and 31 C.F.R. Part 25 (Foreign Operations Export Financing and Related Programs Appropriations Act of 1988 - Foreign Military Sales Debt Reform), may be included and combined under this subparagraph (c)(2)(vi)(F)(1) through (6) of SEA Rule 15c3-1 for haircut purposes.
(SEC Staff to NYSE) (NYSE Interpretation Memo 90-11, December 1990)
15c3-1(c)(2)(vi)(F)/09 Corporate Put Bond Haircuts
Variable interest rate corporate debt securities with a non-severable periodic demand feature are subject to a two percent (2%) deduction from the aggregate principal amount provided the securities:
Are supported by a Credit Facility provided by a credit-worthy institution that covers the payment of principal, interest and premium on any bonds tendered to the issuer under the put;
Trade generally at par;
Have a Put Date no greater than six months from the previous Put Date; and
Otherwise meet the criteria set forth in paragraph (c)(2)(vi)(F) of SEA Rule 15c3-1.
(SEC Letter to NYSE and NASD, November 26, 1990)
(NYSE Interpretation Memo 90-11, December 1990)
(NYSE Interpretation Memo 93-6, November 1993)
(SEC Staff to FINRA) (FINRA Regulatory Notice 14-38)
15c3-1(c)(2)(vi)(F)/10 Nonconvertible Debt Securities Not Highly Rated
See interpretation 15c3-1(c)(2)(vii)/10 (Marketability of Nonconvertible Debt Securities Which Are Not Highly Rated).
15c3-1(c)(2)(vi)(F)/11 Government Stripped Bonds and Coupons
U.S. government bonds and coupons which have been stripped by the U.S. government, shall be subject to haircuts under the appropriate subsections of SEA Rule 15c3-1(c)(2)(vi)(A).
U.S. Treasury notes, bonds and coupons which have been stripped by an entity other than the U.S. government, shall be subject to haircuts under the appropriate subsections of SEA Rule 15c3-1(c)(2)(vi)(F).
(SEC Staff to NYSE) (No.97-6, October 1997)
(SEC Staff to FINRA) (FINRA Regulatory Notice 14-38)
15c3-1(c)(2)(vi)(F)/12 Nonconvertible Debt Securities with Variable Interest Rate
For nonconvertible debt securities having a variable interest rate and a fixed maturity date and which are not traded flat or in default as to principal or interest and which have only a minimal amount of credit risk (see paragraphs (c)(2)(vi)(F) and (c)(2)(vi)(I) of SEA Rule 15c3-1), the applicable haircut percentages on the market value of the greater of the long or short position in each of the categories specified under SEA Rule 15c3-1(c)(2)(vi)(F)(1) shall be applied.
When computing the haircut charge under SEA Rule 15c3-1(c)(2)(vi)(F)(1), a broker-dealer may combine the variable interest rate nonconvertible debt securities that meet the aforementioned requirements with the fixed interest rate nonconvertible debt securities that meet the requirements of SEA Rule 15c3-1(c)(2)(vi)(F)(1).
The haircut on variable interest rate nonconvertible debt securities under SEA Rule 15c3-1(c)(2)(vi)(F)(1) is to be determined based upon the maturity date of the instrument and not the “next interest rate reset date”.
(SEC Staff to NYSE) (NYSE Interpretation Memo 05-8, April 2005)
(SEC Staff to FINRA) (FINRA Regulatory Notice 14-38)
15c3-1(c)(2)(vi)(G) Convertible debt securities. In the case of a debt security not in default which has a fixed rate of interest and a fixed maturity date and which is convertible into an equity security, the deductions shall be as follows: If the market value is 100 percent or more of the principal amount, the deduction shall be determined as specified in paragraph (c)(2)(vi)(J) of this section; if the market value is less than the principal amount, the deduction shall be determined as specified in paragraph (F) of this section; if such securities are rated as required of paragraph (F) of this section;
15c3-1(c)(2)(vi)(G)/01 High Coupon Interest
Convertible debt securities having market values in excess of 100% of the principal amount may still be subject to the lower haircuts specified under SEA Rule 15c3-1(c)(2)(vi)(F) if:
The coupon or specified interest rate of the convertible bond is greater than the current interest market rate for similar (nonconvertible) bonds of the same grade or quality and the market price represents an equivalent current yield;
The current market value of the equity security to be received upon conversion is not such that the conversion would be other than a loss conversion; and
The securities are rated as required under subdivision (F).
(SEC Staff to NYSE) (NYSE Interpretation Memo 83-5, November 1983)
15c3-1(c)(2)(vi)(G)/02 Foreign Convertible Debt Securities
See interpretation 15c3-1(c)(2)(vii)/08 (Marketability of Certain Foreign and Domestic Securities).
15c3-1(c)(2)(vi)(H) In the case of cumulative, non-convertible preferred stock ranking prior to all other classes of stock of the same issuer, which has only a minimal amount of credit risk and which are not in arrears as to dividends, the deduction shall be 10% of the market value of the greater of the long or short position.
15c3-1(c)(2)(vi)(H)/01 Foreign Debt and Preferred Equity Securities
See interpretation 15c3-1(c)(2)(vii)/08 (Marketability of Certain Foreign and Domestic Securities).
15c3-1(c)(2)(vi)(H)/02 Municipal Auction Rate Cumulative Preferred Stock
Municipal Auction Rate Cumulative Preferred Stocks are subject to the requirements of paragraphs (c)(2)(vi)(H) and (c)(2)(vi)(I) of SEA Rule 15c3-1.
(SEC Staff to NYSE) (NYSE Interpretation Memo 93-6, November 1993)
(SEC Staff to FINRA) (FINRA Regulatory Notice 14-38)
15c3-1(c)(2)(vi)(I) In order to apply a deduction under paragraphs (c)(2)(vi)(E), (c)(2)(vi)(F)(1), (c)(2)(vi)(F)(2), or (c)(2)(vi)(H) of this section, the broker or dealer must assess the creditworthiness of the security or money market instrument pursuant to policies and procedures for assessing and monitoring creditworthiness that the broker or dealer establishes, documents, maintains, and enforces. The policies and procedures must be reasonably designed for the purpose of determining whether a security or money market instrument has only a minimal amount of credit risk. Policies and procedures that are reasonably designed for this purpose should result in assessments of creditworthiness that typically are consistent with market data. A broker-dealer that opts not to make an assessment of creditworthiness under this paragraph may not apply the deductions under paragraphs (c)(2)(vi)(E), (c)(2)(vi)(F)(1), (c)(2)(vi)(F)(2), or (c)(2)(vi)(H) of this section.
Note to paragraph (c)(2)(vi)(I):
For a discussion of the “minimal amount of credit risk” standard, see Removal of Certain References to Credit Ratings Under the Securities Exchange Act of 1934, Exchange Act Release No. 34-71194 (Dec. 27, 2013), at
15c3-1(c)(2)(vi)(J) All Other Securities
In the case of all securities or evidences of indebtedness, except those described in appendix A, § 240.15c3-1a, which are not included in any of the percentage categories enumerated in paragraphs (c)(2)(vi) (A) through (H) of this section or paragraph (c)(2)(vi)(K)(ii) of this section, the deduction shall be 15 percent of the market value of the greater of the long or short positions and to the extent the market value of the lesser of the long or short positions exceeds 25 percent of the market value of the greater of the long or short positions, the percentage deduction on such excess shall be 15 percent of the market value of such excess. No deduction need be made in the case of:
15c3-1(c)(2)(vi)(J)(1) A security that is convertible into or exchangeable for another security within a period of 90 days, subject to no conditions other than the payment of money, and the other securities into which such security is convertible or for which it is exchangeable, are short in the accounts of such broker or dealer; or
15c3-1(c)(2)(vi)(J)(2) A security that has been called for redemption and that is redeemable within 90 days.
15c3-1(c)(2)(vi)(J)/01 Bona Fide Arbitrage
A bona fide arbitrage exists when a long security is convertible into or exchangeable for a short security, subject to the conditions stated in this subparagraph (J) text, and may, in lieu of the treatment prescribed in other sections of SEA Rule 15c3-1 be treated as if the exchange or conversion had been effected, i.e., at realizable values.
For net capital purposes, any cash to be received or paid is credited or charged to net worth, respectively, in lieu of long or short market values. An example follows.
XYZ preferred is convertible into XYZ common on a share for share basis. XYZ preferred sells for $120; XYZ common sells for $100. The $20 per share loss, resulting had the preferred been converted, is charged as a haircut in the computation of net capital.
(SEC Staff to NYSE) (NYSE Interpretation Memo 88-15, September 1988)
15c3-1(c)(2)(vi)(J)/011 Settlement - Different Periods
Exchangeable securities or the same security purchased and sold simultaneously in different markets but with different settlement dates are considered offsetting positions, provided that settled positions will result in equivalent long and short positions.
(SEC Staff to NYSE)
15c3-1(c)(2)(vi)(J)/012 Bona Fide Arbitrage - Same Securities
Bona fide arbitrage treatment can be applied to a security position that is convertible into or exchangeable for the same security that the firm is short. For example, a long restricted preferred security that is convertible into a restricted equity security cannot be offset by a short unrestricted equity position to obtain bona fide arbitrage treatment. Securities which cannot be publicly offered or sold because of statutory, regulatory or contractual agreements or other restrictions, are not considered the same as unrestricted securities.
(SEC Staff to NYSE) (NYSE Interpretation Memo 93-6, November 1993)
15c3-1(c)(2)(vi)(J)/013 Exchange Traded Funds - Unhedged
Exchange traded funds such as SPDRs, DIAMONDS, HOLDRS, WEBS, Index Shares, etc, on high-capitalization broad-based indexes will be subject to a haircut charge of 10% and on non-high-capitalization broad-based and narrow-based or sector indexes will be subject to a haircut charge of 15%.
(SEC Staff to NYSE) (NYSE Interpretation Memo 01-5, August 2001)
15c3-1(c)(2)(vi)(J)/014 Exchange Traded Funds Hedged With Underlying Securities
Exchange traded funds (ETFs) such as SPDRs, DIAMONDS, HOLDRS, WEBS, Index Shares etc., offset by qualified stock baskets of the underlying securities to these products, will be subject to a minimum haircut charge of 5% on the market value of the qualified stock baskets for high-capitalization broad-based and narrow-based or sector indexes including the U.S. NASD Market index and 7 ½% for non-high-capitalization broad-based indexes. The market value of underlying securities in excess of the hedged ETF amount should be treated as other securities and subject to paragraph (J) while the market value of ETFs in excess of the hedged underlying securities amount should be subject to interpretation 15c3-1(c)(2)(vi)(J)/013. (Also, see SEA Rule 15c3-1a(b)(1)(i)(D) (Appendix A).)
(SEC Staff to NYSE) (NYSE Interpretation Memo 01-5, August 2001)
15c3-1(c)(2)(vi)(J)/015 Exchange Traded Funds With Underlying Commodity Products - Unhedged
An Exchange Traded Fund where the underlying instruments are commodity products is subject to a haircut charge of 15% when unhedged.
(SEC Staff to NYSE) (NYSE Interpretation Memo 05-8, April 2005)
15c3-1(c)(2)(vi)(J)/016 Exchange Traded Funds With Underlying Commodity Products - Hedged
An Exchange Traded Fund where the underlying instruments are commodity products, when fully hedged with either a commodity futures contract, a commodity future options contract traded on an exchange, the underlying spot commodity or a commodity forward contract (which go out for 30 days or less), is subject to a haircut charge of 5%.
(SEC Staff to NYSE) (NYSE Interpretation Memo 05-8, April 2005)
15c3-1(c)(2)(vi)(J)/02 Tenders
There is no deduction for securities that are the subject of an irrevocable tender offer or which have been officially accepted provided that the transaction can be consummated within 90 days.
Note that most offering circulars contain boiler plate contingency clauses which by their terms negate irrevocability. In these cases a haircut is charged until the securities have been officially accepted and/or paid for.
(SEC Staff to NYSE)
15c3-1(c)(2)(vi)(J)/03 Foreign Equity Securities – FTSE World Index
Equity securities of a foreign issuer that are listed on the FTSE World Index may be included under this subparagraph (c)(2)(vi)(J) for haircut purposes. All other foreign equities will have to meet the ready market criteria outlined in subparagraph (c)(11). For cumulative, nonconvertible preferred stock ranking prior to all other classes of stock of the same issuer, see interpretation 15c3-1(c)(2)(vi)(H)/01.
(SEC Letter to SIA, August 13, 1993)
(NYSE Interpretation Memo 93-5, September 1993)
(NYSE Interpretation Memo 93-6, November 1993)
(SEC Staff to NYSE) (NYSE Interpretation Memo 01-3, March 2001)
15c3-1(c)(2)(vi)(K) Securities with a limited market. In the case of securities (other than exempted securities, nonconvertible debt securities, and cumulative nonconvertible preferred stock) which are not:
15c3-1(c)(2)(vi)(K)(1) Traded on a national securities exchange;
15c3-1(c)(2)(vi)(K)(2) designated as "OTC Margin Stock" pursuant to Regulation T under the Securities Exchange Act of 1934;
15c3-1(c)(2)(vi)(K)(3) quoted on "NASDAQ"; or
15c3-1(c)(2)(vi)(K)(4) redeemable shares of investment companies registered under the Investment Company Act of 1940, the deduction shall be as follows:
15c3-1(c)(2)(vi)(K)(4)(i) In the case where there are regular quotations in an inter-dealer quotations system for the securities by three or more independent market-makers (exclusive of the computing broker or dealer) and where each such quotation represents a bona fide offer to brokers or dealers to both buy and sell in reasonable quantities at stated prices, or where a ready market as defined in paragraph (c)(11) (ii) is deemed to exist, the deduction shall be determined in accordance with paragraph (c)(2)(vi)(J) of this section;
15c3-1(c)(2)(vi)(K)(4)(i)/01 Ohio Dealer Data Service, Inc., Does Not Qualify
The Ohio Dealer Data Service, Inc. does not qualify as an inter-dealer quotation system. It is not recognized as an established securities market as required by the "ready market" definition in SEA Rule 15c3-1(c)(11)(i).
(SEC Letter to Pierre R. Smith & Co., August 19, 1986)
(NYSE Interpretation Memo 88-15, September 1988)
15c3-1(c)(2)(vi)(K)(4)(ii) In the case where there are regular quotations in an inter-dealer quotations system for the securities by only one or two independent market-makers (exclusive of the computing broker or dealer) and where each such quotation represents a bona fide offer to brokers or dealers both to buy and sell in reasonable quantities, at stated prices, the deduction on both the long and short position shall be 40 percent.
15c3-1(c)(2)(vi)(K)(4)(ii)/01 Convertible Preferred Stock
A preferred stock having a limited market, that is freely convertible without restriction into common stock that is readily marketable, may be considered to have a market value equal to the common stock into which it is convertible.
(SEC Staff to NYSE) (NYSE Interpretation Memo 79-4, March 1979)
15c3-1(c)(2)(vi)(L) Where a broker or dealer demonstrates that there is sufficient liquidity for any securities long or short in the proprietary or other accounts of the broker or dealer which are subject to a deduction required by paragraph (c)(2)(vi)(K) of this section, such deduction, upon a proper showing to the Examining Authority for the broker or dealer, may be appropriately decreased, but in no case shall such deduction be less than that prescribed in paragraph (c)(2)(vi)(J) of this section.
15c3-1(c)(2)(vi)(M) Undue Concentration
15c3-1(c)(2)(vi)(M)(1) In the case of money market instruments, or securities of a single class or series of an issuer, including any option written, endorsed or held to purchase or sell securities of such a single class or series of an issuer (other than “exempted securities” and redeemable securities of an investment company registered pursuant to the Investment Company Act of 1940), and securities underwritten (in which case the deduction provided for herein shall be applied after 11 business days), which are long or short in the proprietary or other accounts of a broker or dealer, including securities that are collateral to secured demand notes defined in appendix D, § 240.15c3-1d, and that have a market value of more than 10 percent of the “net capital” of a broker or dealer before the application of paragraph (c)(2)(vi) of this section or appendix A, § 240.15c3-1a, there shall be an additional deduction from net worth and/or the Collateral Value for securities collateralizing a secured demand note defined in appendix D, § 240.15c3-1d, equal to 50 percent of the percentage deduction otherwise provided by this paragraph (c)(2)(vi) of this section or appendix A, § 240.15c3-1a, on that portion of the securities position in excess of 10 percent of the “net capital” of the broker or dealer before the application of paragraph (c)(2)(vi) of this section and appendix A, § 240.15c3-1a. In the case of securities described in paragraph (c)(2)(vi)(J), the additional deduction required by this paragraph (c)(2)(vi)(M) shall be 15 percent.
15c3-1(c)(2)(vi)(M)(1)/01 Municipal Securities - Undue Concentration Charge
For purposes of SEA Rule 15c3-1(c)(2)(vi)(M)(1), municipal securities are not considered “exempted securities” and are subject to undue concentration charges.
See SEA Rule 15c3-1(c)(2)(vi)(M)(4) and interpretation 15c3-1(c)(2)(vi)(M)(4)/01.
(SEC Staff to NYSE) (NYSE Interpretation Memo 06-5, June 2006)
15c3-1(c)(2)(vi)(M)(2) This paragraph (c)(2)(vi)(M) shall apply notwithstanding any long or short position exemption provided for in paragraph (c)(2)(vi)(J) of this section (except for long or short position exemptions arising out of the first proviso to paragraph (c)(2)(vi)(J)) and the deduction on any such exempted position shall be 15 percent of that portion of the securities position in excess of 10 percent of the broker or dealer's net capital before the application of paragraph (c)(2)(vi) of this section and appendix A, § 240.15c3-1a.
15c3-1(c)(2)(vi)(M)(3) This paragraph (c)(2)(vi)(M) shall be applied to an issue of equity securities only on the market value of such securities in excess of $10,000 or the market value of 500 shares, whichever is greater, or $25,000 in the case of a debt security.
15c3-1(c)(2)(vi)(M)(4) This paragraph (c)(2)(vi)(M) will be applied to an issue of municipal securities having the same security provisions, date of issue, interest rate, day, month and year of maturity only if such securities have a market value in excess of $500,000 in bonds ($5,000,000 in notes) or 10 percent of tentative net capital, whichever is greater, and are held in position longer than 20 business days from the date the securities are received by the syndicate manager from the issuer.
15c3-1(c)(2)(vi)(M)(4)/01 Municipal Securities – Undue Concentration Charge
Initial and Secondary Offerings
A municipal security held in inventory by a syndicate manager for a period longer than 20 business days from the date such security is received from the issuer, is subject to an undue concentration charge. The undue concentration charge is applicable to the greater of the market value of the municipal security in excess of $500,000 in bonds ($5,000,000 in notes) or 10 percent of tentative net capital.
Secondary Market
Municipal securities acquired in the secondary market are subject to undue concentration charges immediately upon being acquired. The undue concentration charge is applicable to the greater of the market value of the municipal security in excess of $500,000 in bonds ($5,000,000 in notes) or 10 percent of tentative net capital.
(SEC Staff to NYSE) (NYSE Interpretation Memo 06-5, June 2006)
15c3-1(c)(2)(vi)(M)(5) Any specialist that is subject to a deduction required by this paragraph (c)(2)(vi)(M), respecting its specialty stock, that can demonstrate to the satisfaction of the Examining Authority for such broker or dealer that there is sufficient liquidity for such specialist's specialty stock and that such deduction need not be applied in the public interest for the protection of investors, may upon a proper showing to such Examining Authority have such undue concentration deduction appropriately decreased, but in no case shall the deduction prescribed in paragraph (c)(2)(vi)(J) of this section above be reduced. Each such Examining Authority shall make and preserve for a period of not less than 3 years a record of each application granted pursuant to this paragraph (c)(2)(vi)(M)(5), which shall contain a summary of the justification for the granting of the application.
15c3-1(c)(2)(vi)(M)/01 (Rescinded)
(NYSE Interpretation Memo 97-5, September 1997)
15c3-1(c)(2)(vi)(M)/011 Theoretical Options Pricing Models - Concentration Charges – (Rescinded)
(NYSE Interpretation Memo 04-3, June 2004)
15c3-1(c)(2)(vi)(M)/012 Theoretical Options Pricing Models - Concentration Charges
Equity securities that are included in a Theoretical Options Pricing Model computation but not fully offset by options and/or futures positions shall remain subject to this section of the rule.
(SEC Staff to NYSE) (NYSE Interpretation Memo 04-3, June 2004)
15c3-1(c)(2)(vi)(M)/021 (Rescinded)
(NYSE Interpretation Memo 97-5, September 1997)
15c3-1(c)(2)(vi)(M)/03 Securities Concentration - Positions Included
Undue concentration charges apply to securities positions as follows:
| | | | |
--- --- |
| | POSITION | | UNDUE CONCENTRATION |
| | Trading and investment accounts | | Applies |
| | Customers’ and non-customers’ partly secured accounts (including partly secured cash accounts with more than one Regulation T extension) | | Applies |
| | Secured demand note collateral necessary for proper collateralization | | Applies |
| | Aged fails to deliver | | Does not apply |
15c3-1(c)(2)(vi)(M)/04 Tentative Net Capital
In computing tentative net capital (or net capital before the application of haircuts and undue concentration charges on securities and option positions), a broker-dealer must comply with these procedures:
The part of the adjustment to net worth for the deferred tax credit add back which relates to the undue concentration deduction (SEA Rule 15c3-1(c)(2)(i)(C)(1)) shall be ignored for purposes of determining the undue concentration deduction.
The following must be added back to net capital to arrive at tentative net capital:
Haircuts (and undue concentration charges) on customers’ and non-customers’ partly secured securities accounts (including partly secured cash accounts containing transactions that are the subject of more than one Regulation T extension);
Haircuts (and undue concentration charges) on open contractual commitments net of unrealized profits used to reduce such charges;
Haircuts (and undue concentration charges) on regular and proprietary positions;
Haircuts (and undue concentration charges) on options and related underlying securities positions, reduced by the related profits and losses that would result upon the exercise of the options, and plus or net of adjustments to net worth relating to listed options;
Deductions (net where applicable) relating to listed option spread positions;
Deductions representing the excess of the long market value of exchangeable securities over the short market value of securities into which such long securities are convertible (see interpretation 15c3-1(c)(2)(vi)(J)/01); and
Deduction for the deficit in a single customer’s account or accounts controlled by such persons exceeding the specified percentage of tentative net capital (see interpretation 15c3-1(c)(2)(xii)/02).
The following are not added back to net capital to arrive at tentative net capital:
100% deduction for non-marketable securities.
Charges for aged fails to deliver.
(SEC Staff to NYSE)
15c3-1(c)(2)(vi)(M)/05 Single Class or Series
Securities of the same issue with the same maturity but different coupon rates, or the same coupon rate but different maturities are treated as separate positions for undue concentration purposes.
(SEC Staff to NASD and NYSE) (NYSE Interpretation Memo 76-4, April 1976)
15c3-1(c)(2)(vi)(M)/06 Non-Marketable Short Securities - Undue Concentration
Undue concentration charges apply, as appropriate, to the fair value (see interpretation 15c3-1(c)(2)(i)(B)(1)/01) of short securities which are not readily marketable.
(SEC Letter to Power Securities Corporation, October 3, 1988)
(NYSE Interpretation Memo 89-6, June 1989)
15c3-1(c)(2)(vi)(M)/07 Hedged Positions
An undue concentration charge is not applicable to the hedged portion of a long (short) position which is convertible or exchangeable for short (long) positions as permitted under paragraph (c)(2)(vi)(J). Only the unhedged portion of the position is subject.
(SEC Letter to Kelly Drye & Warren, January 25, 1989)
(NYSE Interpretation Memo 89-9, July 1989)
15c3-1(c)(2)(vi)(M)/08 (Rescinded)
(NYSE Interpretation Memo 97-5, September 1997)
15c3-1(c)(2)(vi)(M)/09 Nonconvertible Debt Securities Which Are Not Highly Rated
Undue concentration charges on nonconvertible debt securities which are not highly rated and subject to haircut percentages outlined in interpretation 15c3-1(c)(2)(vii)/10 (Marketability of Nonconvertible Debt Securities Which Are Not Highly Rated) shall be 15%. This undue concentration charge may be used as an offset to the portfolio concentration charge under interpretation 15c3-1(c)(2)(vii)/10.
(SEC Staff to NYSE) (NYSE Interpretation Memo 96-3, April 1996)
15c3-1(c)(2)(vi)(M)/10 Concentration Charge on Money Market Instruments - Greater than 30% of Tentative Net Capital
See interpretation 15c3-1(c)(2)(vii)/09 for concentration criteria.
15c3-1(c)(2)(vi)(M)/11 Exchange Traded Funds – Undue Concentration Charge
An Exchange Traded Fund where the underlying instruments are securities is not subject to an undue concentration charge.
An Exchange Traded Fund where the underlying instruments are commodity products is subject to an undue concentration charge of 15%.
(SEC Staff to NYSE) (NYSE Interpretation Memo 05-8, April 2005)
15c3-1(c)(2)(vi)(N) Any specialist that limits its securities business to that of a specialist (except for an occasional non-specialist related securities transaction for its own account), that does not transact a business in securities with other than a broker or dealer registered with the Commission under section 15 or 15C of the Act or a member of a national securities exchange, and that is not a clearing member of The Options Clearing Corporation need not deduct from net worth in computing net capital those deductions, as to its specialty securities, set forth in paragraph (c)(2)(vi) of this section or appendix A to this section, except for paragraph (e) of this section limiting withdrawals of equity capital and appendix D to this section relating to satisfactory subordination agreements. As to a specialist that is solely an options specialist, in paragraph (e) the term “net capital” shall be deemed to mean “net capital before the application of paragraph (c)(2)(vi) of this section or appendix A to this section” and “excess net capital” shall be deemed to be the amount of net capital before the application of paragraph (c)(2)(vi) of this section or appendix A to this section in excess of the amount of net capital required under paragraph (a) of this section. In reports filed pursuant to § 240.17a-5 and in making the record required by § 240.17a-3(a)(11) each specialists shall include the deductions that would otherwise have been required by paragraph (c)(2)(vi) of this section or appendix A to this section in the absence of this paragraph (c)(2)(vi)(N).
15c3-1(c)(2)(vi)(N)/01 Non-Specialist Securities Transactions
A specialist may not engage in any non-specialist related securities transactions except for investments made on an occasional basis. However, he may engage in hedging transactions including options transactions directly related to his specialist securities.
A specialist operating under this paragraph may not engage in trading non-specialist securities. However, they may make occasional investment account transactions in non-specialist securities (not more than 10 per year).
Excess funds may be invested in reverse repurchase agreement transactions as often as necessary, and not be counted as occasional investment transactions.
(SEC Staff to NYSE) (NYSE Interpretation Memo 90-11, December 1990)
15c3-1(c)(2)(vi)(N)/02 Servicing Family Accounts
A specialist member organization who services the customer accounts of members of its partners’ (or stockholders’) families shall not remain subject to this paragraph.
(SEC Staff to NYSE)
15c3-1(c)(2)(vi)(N)/021 Servicing Partners Accounts
A specialist member organization who services the individual accounts of its partners’ or stockholders’ shall not remain subject to this paragraph.
(SEC Staff to NYSE) (NYSE Interpretation Memo 89-6, June 1989)
15c3-1(c)(2)(vi)(N)/03 Joint Trading and Investment Account
A specialist in stocks may carry a joint specialist trading and investment account in which he participates and remain subject to this paragraph.
(SEC Staff to NYSE)
15c3-1(c)(2)(vi)(N)/04 Exchange Specialist Trading in Futures
A specialist under this paragraph may trade in commodity futures.
(ASE Circular NYSE Interpretation Memo 78-72, October 26, 1978)
(SEC Staff to NYSE) (NYSE Interpretation Memo 83-5, November 1983)
15c3-1(c)(2)(vi)(N)/05 Exchange Specialist Trading in Options
An exchange specialist trading in listed options transactions that are directly related to the specialist activities, shall remain subject to this paragraph.
(SEC Staff to NYSE) (NYSE Interpretation Memo 83-5, November 1983)
15c3-1(c)(2)(vi)(O) Cleared security-based swaps. In the case of a cleared security-based swap held in a proprietary account of the broker or dealer, deducting the amount of the applicable margin requirement of the clearing agency or, if the security-based swap references an equity security, the broker or dealer may take a deduction using the method specified in § 240.15c3-1a.
15c3-1(c)(2)(vi)(P) Non-cleared security-based swaps —
15c3-1(c)(2)(vi)(P)(1) Credit default swaps —
15c3-1(c)(2)(vi)(P)(1)(i) Short positions (selling protection). In the case of a non-cleared security-based swap that is a short credit default swap, deducting the percentage of the notional amount based upon the current basis point spread of the credit default swap and the maturity of the credit default swap in accordance with table 1 to § 240.15c3-1(c)(2)(vi)(P)(1)(i):
| | | | | | | |
--- --- ---
| Table 1 to § 240.15c3-1(c)(2)(vi)(P)(1)(i) | | | | | | |
| Length of time to maturity of credit default swap contract | Basis point spread | | | | | |
| 100 or less % | 101-300 % | 301-400 % | 401-500 % | 501-699 % | 700 or more % |
| Less than 12 months | 1.00 | 2.00 | 5.00 | 7.50 | 10.00 | 15.00 |
| 12 months but less than 24 months | 1.50 | 3.50 | 7.50 | 10.00 | 12.50 | 17.50 |
| 24 months but less than 36 months | 2.00 | 5.00 | 10.00 | 12.50 | 15.00 | 20.00 |
| 36 months but less than 48 months | 3.00 | 6.00 | 12.50 | 15.00 | 17.50 | 22.50 |
| 48 months but less than 60 months | 4.00 | 7.00 | 15.00 | 17.50 | 20.00 | 25.00 |
| 60 months but less than 72 months | 5.50 | 8.50 | 17.50 | 20.00 | 22.50 | 27.50 |
| 72 months but less than 84 months | 7.00 | 10.00 | 20.00 | 22.50 | 25.00 | 30.00 |
| 84 months but less than 120 months | 8.50 | 15.00 | 22.50 | 25.00 | 27.50 | 40.00 |
| 120 months and longer | 10.00 | 20.00 | 25.00 | 27.50 | 30.00 | 50.00 |
15c3-1(c)(2)(vi)(P)(1)(ii) Long positions (purchasing protection). In the case of a non-cleared security-based swap that is a long credit default swap, deducting 50 percent of the deduction that would be required by paragraph (c)(2)(vi)(P)(1)(i) of this section if the non-cleared security-based swap was a short credit default swap, each such deduction not to exceed the current market value of the long position.
15c3-1(c)(2)(vi)(P)(1)(iii) Long and short credit default swaps. In the case of non-cleared security-based swaps that are long and short credit default swaps referencing the same entity (in the case of non-cleared credit default swap security-based swaps referencing a corporate entity) or obligation (in the case of non-cleared credit default swap security-based swaps referencing an asset-backed security), that have the same credit events which would trigger payment by the seller of protection, that have the same basket of obligations which would determine the amount of payment by the seller of protection upon the occurrence of a credit event, that are in the same or adjacent spread category, and that are in the same or adjacent maturity category and have a maturity date within three months of the other maturity category, deducting the percentage of the notional amount specified in the higher maturity category under paragraph (c)(2)(vi)(P)(1)(i) or (ii) on the excess of the long or short position. In the case of non-cleared security-based swaps that are long and short credit default swaps referencing corporate entities in the same industry sector and the same spread and maturity categories prescribed in paragraph (c)(2)(vi)(P)(1)(i) of this section, deducting 50 percent of the amount required by paragraph (c)(2)(vi)(P)(1)(i) of this section on the short position plus the deduction required by paragraph (c)(2)(vi)(P)(1)(ii) of this section on the excess long position, if any. For the purposes of this section, the broker or dealer must use an industry sector classification system that is reasonable in terms of grouping types of companies with similar business activities and risk characteristics and the broker or dealer must document the industry sector classification system used pursuant to this section.
15c3-1(c)(2)(vi)(P)(1)(iv) Long security and long credit default swap. In the case of a non-cleared security-based swap that is a long credit default swap referencing a debt security and the broker or dealer is long the same debt security, deducting 50 percent of the amount specified in paragraph (c)(2)(vi) or (vii) of this section for the debt security, provided that the broker or dealer can deliver the debt security to satisfy the obligation of the broker or dealer on the credit default swap.
15c3-1(c)(2)(vi)(P)(1)(v) Short security and short credit default swap. In the case of a non-cleared security-based swap that is a short credit default swap referencing a debt security or a corporate entity, and the broker or dealer is short the debt security or a debt security issued by the corporate entity, deducting the amount specified in paragraph (c)(2)(vi) or (vii) of this section for the debt security. In the case of a non-cleared security-based swap that is a short credit default swap referencing an asset-backed security and the broker or dealer is short the asset-backed security, deducting the amount specified in paragraph (c)(2)(vi) or (vii) of this section for the asset-backed security.
15c3-1(c)(2)(vi)(P)(2) Non-cleared security-based swaps that are not credit default swaps. In the case of a non-cleared security-based swap that is not a credit default swap, deducting the amount calculated by multiplying the notional amount of the security-based swap and the percentage specified in paragraph (c)(2)(vi) of this section applicable to the reference security. A broker or dealer may reduce the deduction under this paragraph (c)(2)(vi)(P)(2) by an amount equal to any reduction recognized for a comparable long or short position in the reference security under paragraph (c)(2)(vi) of this section and, in the case of a security-based swap referencing an equity security, the method specified in § 240.15c3-1a.
15c3-1(c)(2)(vii) Non-marketable securities. Deducting 100 percent of the carrying value in the case of securities or evidence of indebtedness in the proprietary or other accounts of the broker or dealer, for which there is no ready market, as defined in paragraph (c)(11) of this section, and securities, in the proprietary or other accounts of the broker or dealer, which cannot be publicly offered or sold because of statutory, regulatory or contractual arrangements or other restrictions.
15c3-1(c)(2)(vii)/001 FOCUS Reporting of Non-Marketable Inventory Positions
An inventory long position that is determined to be non-marketable and subject to a 100% deduction under the requirements of SEA Rule 15c3-1 should be reported as a non-allowable asset on line 610 (Securities owned not readily marketable) in the Statement of Financial Condition section of the FOCUS Report.
The amount of the deduction for an inventory short position that is determined to be non-marketable and which is subject to a 40% deduction under the requirements of interpretation 15c3-1(c)(2)(vii)/05 should be reported on line 3736 (Haircuts on securities – Other) in the Computation of Net Capital section of the FOCUS Report.
(SEC Staff to FINRA) (FINRA Regulatory Notice 13-44)
15c3-1(c)(2)(vii)/01 Marketplace Blockage
When it can be established that the marketplace can absorb only a limited number of shares of a security for which a ready market seemingly exists, the non-marketable portion of that position is subject to a 100% deduction (and treated as a non-allowable asset).
For shares of common stock or preferred stock not covered by paragraph (c)(2)(vi)(H) of SEA Rule 15c3-1 (highly rated preferred stock), the Division will raise no question nor recommend any action to the Commission if a broker-dealer, when faced with a blockage in securities, treats as readily marketable securities that portion of the block which equals the aggregate of the most recent four-week, inter-dealer trading volume. The number of shares exceeding the aggregate of the most recent four-week inter-dealer trading volume should be considered non-marketable and subject to a 100% deduction (and treated as a non-allowable asset) unless the broker-dealer demonstrates to the satisfaction of its Designated Examining Authority that a ready market exists for these shares.
Those securities purchased by the computing broker-dealer during the most recent four-week period shall be excluded from the determination of trading volume.
Subsequent sale of securities deemed non-marketable due to marketplace blockage will be considered as demonstration that a ready market exists, provided the sale takes place within a reasonable period of time after the net capital computation date. The reasonable period of time will be determined on a case-by-case basis, but will generally be within about 5 business days.
(SEC Letter to NYSE, October 5, 1987) (NYSE Interpretation Memo 87-11, December 1987)
(SEC Staff to NYSE) (NYSE Interpretation Memo 90 11, December 1990)
(SEC Staff to NYSE and NASD) (NYSE Interpretation Memo 96-4, November 1996)
15c3-1(c)(2)(vii)/011 Marketplace Blockage - 1% Exemption
The long market value of an inventory position that is subject to marketplace blockage requirements as defined under interpretation 15c3-1(c)(2)(vii)/01 and which is less than 1% of the broker-dealer’s total long inventory market value, shall be exempt from the marketplace blockage charge.
For purposes of this interpretation, the total long inventory market value shall include only securities ordinarily subject to marketplace blockage charge (Index Arbitrage positions would therefore be excluded).
This exemption does not apply to securities that were underwritten or distributed by the broker-dealer.
(SEC Staff to NYSE) (NYSE Interpretation Memo 97-6, October 1997)
(SEC Staff to NYSE) (NYSE Interpretation Memo 07-4, April 2007)
15c3-1(c)(2)(vii)/02 Municipal Securities Valuation
Municipal securities dealers should value their municipal securities inventories at market or, if such values are unavailable, at the lower of cost or estimated fair value for a period of 30 calendar days following settlement date. Thereafter, in the absence of further price or transaction data, the dealer should mark down or reduce the value of such positions by 5% of the original cost per month until these capital values decline to zero. At that point, the position should be considered a non-marketable security for net capital purposes. This valuation is for net capital purposes only. (Note: This interpretation replaces previous interpretation which exempted municipal securities from marketability provisions.)
A ready market may be established if the securities are collateral for a bank loan. (See interpretation 15c3-1(c)(11)(ii)/01, NYSE Rule 328(c) and NYSE Information Memo No. 80-66, dated December 31, 1980.)
(SEC Release 34-18737, May 12, 1982)
15c3-1(c)(2)(vii)/03 Non-Recourse Indebtedness
No deduction need be taken on non-marketable securities up to the amount of fixed term non-recourse, indebtedness collateralized by these securities.
(SEC Staff to NYSE)
15c3-1(c)(2)(vii)/04 Non-Transferable, Restricted or Unregistered Securities as Collateral to a Sole Recourse Fixed Term Loan
Securities that do not have a ready market or cannot be publicly offered or sold because of statutory, regulatory or contractual arrangements or as a result of other restrictions must be deducted from net capital, unless:
Such securities are collateral to a fixed term loan and are the sole recourse of the creditor for nonpayment of the liability;
The broker-dealer and the creditor have entered into a written loan agreement which has a minimum term of three years and identifies the specific securities (which may not be substituted) that collateralize the loan;
The remaining life to maturity of the fixed term loan must be in excess of one year at the time of the net capital computation; a fixed term loan with a lesser remaining life to maturity cannot be used to obtain the net capital relief provided in this interpretation;
The portion of a fixed term loan where the remaining life to maturity is less than one year must be included in aggregate indebtedness; and
The loan agreement has been submitted to and has been found acceptable by the Designated Examining Authority before the broker-dealer may rely on this interpretation.
A broker-dealer is not required to apply a haircut charge on the value of the non-marketable securities pledged up to the amount of the fixed term loan proceeds received from the lender. However, the value of the non-marketable securities pledged in excess of the fixed term loan must be treated as a non-allowable asset for net capital purposes.
In accordance with interpretation 15c3-1(c)(11)(ii)/03 (Non-Transferable or Restricted Securities), a broker-dealer may not rely on this interpretation to establish a “ready market” as defined in SEA Rule 15c3-1(c)(11)(ii).
(SEC Letter to Warburg Paribus Becker, Inc., March 16, 1982)
(NYSE Interpretation Memo 88-16, October 1988)
(SEC Staff to NYSE) (NYSE Interpretation Memo 07-4, April 2007)
15c3-1(c)(2)(vii)/05 Non-Marketable Short Securities - Haircuts
The SEC staff has issued a no-action letter which states that the fair value (see interpretation 15c3-1(c)(2)(i)(B)(1)/01) of short securities that are not readily marketable shall be subject to a 40% haircut.
(SEC Letter to Power Securities Corporation, October 3, 1988) (NYSE Interpretation Memo 89-6, June 1989)
15c3-1(c)(2)(vii)/06 Marketability of Commercial Paper
Commercial paper, whether or not exempted from the registration requirement under section 3(a)(3) of the Securities Act, may be deemed to have a ready market under subparagraph (c)(2)(vii) of SEA Rule 15c3-1 and not be subject to a deduction of 100% of its carrying value, if the following conditions are met:
The commercial paper is not traded flat or in default as to principal or interest.
The commercial paper is not issued by a parent or an affiliated company of the broker-dealer.
The commercial paper is rated in one of the “two” highest categories by at least two of the nationally recognized statistical rating organizations (“NRSROs”).
If at any time, any of the two ratings is reduced below the two highest categories the broker-dealer will deduct from net worth, when computing net capital, 15% of the carrying value of the commercial paper. Any time after the thirtieth day subsequent to the date when any of the two ratings is reduced below the two highest categories, there shall be a deduction from net worth equal to 100% of the carrying value of the position.
The commercial paper is the subject of a commercial paper program which:
is administered by an issuing and paying agent bank and there exists a dealer willing to make a market in said commercial paper, or
is administered by a direct issuer pursuant to a direct placement program.
(SEC Letter to SIA, March 10, 1992)
(NYSE Interpretation Memo 92-7, May 1992)
15c3-1(c)(2)(vii)/07 Marketability of Commercial Paper under Section 936 Market
Commercial paper, whether or not exempted from the registration requirement under section 3(a)(3) of the Securities Act, which is sold in Puerto Rico under the section 936 market, may be deemed to have a ready market under subparagraph (c)(2)(vii) of SEA Rule 15c3-1 and not be subject to a deduction of 100% of its carrying value if the following conditions are met:
The commercial paper is not traded flat or in default as to principal or interest.
The commercial paper is not issued by a parent or an affiliated company of the broker-dealer.
The purchase of the commercial paper by a corporation that is allowed a tax credit pursuant to section 936, constitutes, either (i) an investment under section 936(d)(2) of the Internal Revenue Code for the purpose of deriving “Qualified Possession Source Investment Income”, or (ii) an investment in an “eligible activity” under section 6.2.4 of Regulation Number 3582 of the Commissioner of Financial Institutions of the Commonwealth of Puerto Rico and, in either instance, the principal and interest owed as a result of the commercial paper obligation are payable in the Commonwealth of Puerto Rico.
The commercial paper is rated investment grade by at least two NRSROs, or is absolutely and without condition guaranteed as to principal and interest by an institution whose commercial paper is rated in one of the two highest grades by at least two NRSROs.
If at any time, any of the two ratings is reduced below the two highest categories the broker-dealer will deduct from net worth, when computing net capital, 15% of the carrying value of the commercial paper. Any time after the thirtieth day subsequent to the date when any of the two ratings is reduced below the two highest categories, there shall be a deduction from net worth equal to 100% of the carrying value of the position.
The commercial paper is the subject of a commercial paper program which:
is administered by an issuing and paying agent bank and there exists a dealer willing to make a market in said commercial paper, or
is administered by a direct issuer pursuant to a direct placement program.
(SEC Letter to SIA, March 10, 1992)
(NYSE Interpretation Memo 92-7, May 1992)
15c3-1(c)(2)(vii)/08 Marketability of Certain Foreign and Domestic Securities
The SEC Division of Market Regulation will not recommend enforcement action to the Commission if broker-dealers, in computing their net capital apply the haircuts described below to the securities held in their proprietary and other accounts. For the purposes of this letter it is irrelevant whether the securities may be publicly offered or sold without registration under Section 5 of the Securities Act.
Foreign Securities - Sovereign Issued Debt
A debt security that: (1) is issued as a general obligation of a sovereign government; (2) has a fixed maturity date; (3) is not traded flat or in default as to principal or interest; and (4) is rated (implicitly or explicitly) in one of the four highest rating categories by at least two NRSROs may be treated in accordance with the haircut provisions set forth in paragraph (c)(2)(vi)(F) of SEA Rule 15c3-1.
Nonconvertible Debt Securities Issued by a Supranational Organization or a Domestic or Non-Domestic Issuer
A nonconvertible debt security that: (1) is issued by a supranational organization or a domestic or non-domestic issuer; (2) has a fixed rate of interest and fixed maturity date; (3) is not traded flat or in default as to principal or interest; and (4) is rated in one of the four highest rating categories by at least two NRSROs may be treated in accordance with the haircut provisions set forth in paragraph (c)(2)(vi)(F) of SEA Rule 15c3-1.
Convertible Debt Securities
A debt security that: (1) is issued by a domestic or non-domestic issuer; (2) has a fixed rate of interest and a fixed maturity date; (3) is not traded flat or in default as to principal or interest; (4) is convertible into an equity security; and (5) is rated in one of the four highest rating categories by at least two NRSROs or readily convertible within ninety days into a security that is deemed to have a ready market may be treated in accordance with the haircut provisions set forth in paragraph (c)(2)(vi)(G) of SEA Rule 15c3-1.
Preferred Stock
Cumulative, nonconvertible preferred stock ranking prior to all other classes of stock of the same issuer that is: (1) issued by a domestic or non-domestic issuer; (2) rated in one of the four highest categories by at least two NRSROs; and (3) not in arrears as to dividends may be treated in accordance with the haircut provisions set forth in paragraph (c)(2)(vi)(H) of SEA Rule 15c3-1. Convertible preferred stock that is: (1) issued by a domestic or non-domestic issuer; (2) rated in one of the four highest rating categories by at least two NRSROs or readily convertible within ninety days into a security that is deemed to have a ready market; and (3) not in arrears as to dividends may be treated in accordance with the haircut provisions set forth in paragraph (c)(2)(vi)(J) of SEA Rule 15c3-1.
Securities Issued Under the Secondary Mortgage Enhancement Act of 1984
Debt securities that are issued under the Secondary Mortgage Enhancement Act of 1984 and rated in one of the two highest rating categories by at least one NRSRO may be treated in accordance with the haircut provisions set forth in paragraph (c)(2)(vi)(F) of SEA Rule 15c3-1.
(SEC Letter to SIA, June 12, 1992)
(NYSE Interpretation Memo 92-12, December 1992)
15c3-1(c)(2)(vii)/09 Marketability of Money Market Instruments
The SEC Division of Market Regulation will not recommend enforcement action to the Commission if broker-dealers, in computing their net capital apply the haircuts described below to money market instruments held in their proprietary and other accounts under the circumstances described below.
MONEY MARKET INSTRUMENTS
The following proprietary positions may be deemed to have a ready market under subparagraph (c)(2)(vii) of SEA Rule 15c3-1 and not subject to a deduction of 100% of its carrying value, if one or more of the conditions set forth in paragraphs (A), (B) or (C), below are met.
negotiable certificates of deposit and bank deposit notes,
that are not issued by a parent or an affiliated company of the broker-dealer, and
where the funds are deposited and payable in a major money market. (For list of major money markets, see note at end of this interpretation)
negotiable bankers acceptances and bills of exchange that,
are not issued or accepted by a parent or an affiliated company of the broker-dealer, and
are issued or accepted by a bank when the obligation is booked and payable in a major money market.
Conditions:
The certificate of deposit or bank deposit note is issued or unconditionally guaranteed as to principal and interest, or the bankers acceptance or bill of exchange is issued or accepted as to principal and interest by:
a bank as defined in section 3(a)(6) of the Exchange Act or a building and loan or savings and loan institution, and such bank, building and loan or savings and loan institution is:
subject to supervision by a federal banking authority, and
rated investment grade by at least two of the nationally recognized statistical rating organizations ("NRSROs").
a bank is defined in section 3(a)(6) of the Exchange Act that:
is not rated,
has shareholders' equity of at least $400 million, and
is subject to supervision by a federal banking authority.
building and loan or savings and loan institution that:
is not rated,
that is subject to supervision by a federal banking authority, and
has shareholders' equity of at least $500 million.
The certificate of deposit, bank deposit note, bankers acceptance or bill of exchange is rated investment grade by at least two NRSROs and is issued or accepted as to principal and interest by a foreign commercial bank,
that has shareholders' equity of at least US$1 billion, and
whose capital is subject to supervision by an authority of a sovereign national government where a major money market is located.
The certificate of deposit, bank deposit note, bankers acceptance or bill of exchange is issued or accepted as to principal and interest by a foreign commercial bank,
that has shareholders' equity of at least US$1.5 billion,
whose capital is subject to supervision by an authority of a sovereign national government where a major money market is located, and
that is not rated.
If a broker-dealer holds a combined position consisting of negotiable or non-negotiable certificates of deposit, bank deposit notes, bankers acceptances or bills of exchange issued or unconditionally guaranteed as to principal and interest, or accepted as to principal and interest by (i) a single bank as defined in section 3(a)(6) of the Exchange Act, or (ii) a single building and loan or savings and loan institution, or (iii) a single foreign commercial bank, which combined position is in the proprietary or other account of a broker-dealer for more than five business days, there shall be a deduction from net worth equal to 100% of the carrying value of that position exceeding 30% of the broker-dealer's net capital before the application of the adjustment set forth in subparagraph (c)(2)(vi) and Appendices A and B of SEA Rule 15c3-1.
2. SECTION 936 MARKET
Negotiable certificates of deposit and bank deposit notes ("obligations") which are sold in Puerto Rico for the section 936 market, may be deemed to have a ready market under subparagraph (c)(2)(vii) of SEA Rule 15c3-1 and not subject to a deduction of 100% of its carrying value, provided that the following conditions are met:
the obligations are not issued by a parent or an affiliated company of the broker-dealer, and
the purchase of the obligations by a corporation that is allowed a tax credit pursuant to section 936 or that is an "eligible institution" under Regulation Number 3582 of the Commissioner of Financial Institutions of the Commonwealth of Puerto Rico ("Reg. 3582"), constitutes, either
an investment under section 936(d)(2) of the Internal Revenue Code for the purpose of deriving "Qualified Possession Source Investment Income," or
an investment in an "eligible activity" under section 6.2.4 of Reg. 3582, and
in either instance, the funds securing such obligations are deposited and payable in the Commonwealth of Puerto Rico, provided that one or more of the conditions set forth in paragraphs (A), (B) or (C) below are met.
Conditions:
The obligations are issued by, or backed, absolutely and without condition (as to principal and interest), by an irrevocable letter of credit issued by:
a bank as defined in section 3(a)(6) of the Exchange Act or a building and loan or savings and loan institution which is:
subject to supervision by a federal banking authority, and
rated in one of the two highest grades by at least two NRSROs
a bank as defined in Section 3(a)(6) of the Exchange Act that:
is not rated,
has stockholders' equity of at least $400 million, and
is subject to supervision by a federal banking authority
a building and loan or savings and loan institution that:
is not rated,
has stockholders' equity of at least $500 million, and
is subject to supervision by a federal banking authority
The obligations are backed absolutely and without condition (as to principal and interest) by an irrevocable letter of credit issued by a foreign commercial bank rated in one of the two highest grades by a least two NRSROs,
that has shareholders' equity of at least US$1 billion, and
whose capital is subject to supervision by an authority of a sovereign national government where a major money market is located
The obligations are backed (as to principal and interest) by an irrevocable letter of credit issued by a Federal Home Loan Bank or an agency of the Federal Government of the U.S.
Non-negotiable certificates of deposit that would otherwise qualify for treatment under one or more of the provisions set forth in either sections I or II above, for which the only restriction relative to early withdrawal at any time prior to maturity is the forfeiture of interest may be included under subparagraph (c)(2)(vi)(E) of SEA Rule 15c3-1 if the broker-dealer takes an additional deduction for the amount of interest subject to forfeiture.
With regard to money market instruments that are deemed to have a ready market under Section A and B above: if either of the ratings required to qualify a money market instrument as having a ready market is reduced below the two highest ratings categories, the broker-dealer holding such money market instruments will deduct from net worth, when computing net capital, 15% of the carrying value of the money market instruments. At the end of 30 days subsequent to the date when any of the two ratings is reduced below the two highest categories, if the broker-dealer continues to hold the position it must prove otherwise that the money market instruments have a ready market in order to include such position under subparagraph (c)(2)(vi)(E) of SEA Rule 15c3-1.
Note: For purpose of this interpretation 15c3-1(c)(2)(vii)/09 only, major money markets include: Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Hong Kong, Ireland, Italy, Japan, Luxembourg, Mexico, Netherlands, New Zealand, Norway, Portugal, Singapore, Spain, Sweden, Switzerland, United States and United Kingdom.
(SEC Letter to SIA, August 21, 1992)
(NYSE Interpretation Memo 92-12, December 1992)
15c3-1(c)(2)(vii)/10 Marketability of Nonconvertible Debt Securities Which Are Not Highly Rated
The SEC Division of Market Regulation will not recommend enforcement action to the Commission if broker-dealers apply the haircuts described below to nonconvertible debt securities held in their proprietary accounts which are not rated in one of the four highest rating categories by at least two NRSROs provided the following conditions are met:
The securities can be publicly sold without registration with the Commission under Section 5 of the Securities Act of 1933, and
Current information concerning the issuer is available to the public. Current information is deemed to be available to the public if:
The issuer filed with the Commission public reports consisting of the most recently required periodic financial report, or
The issuer (provided it is the subject of a bankruptcy proceeding) filed with a bankruptcy court, public information that is sufficient to value the assets and liabilities of the issuer and such information is dated not more than six months prior to the date of the broker-dealer's capital computation.
The current information requirement will also be deem to be satisfied by the existence of current ratings by two NRSROs on any issuance of the issuer. A rating by an NRSRO is considered to be current if the NRSRO itself continues to hold out its rating for the issuance of the issuer. Non-investment grade nonconvertible debt securities shall be treated as follows:
The broker-dealer shall deduct from its net worth the following percentages applied to the greater of the gross long or the gross short market value of non-investment grade, nonconvertible debt securities' positions in each of the categories specified below:
| | |
--- |
| a. An initial issuance of at least $100 million | 15% |
| b. An initial issuance of at least $75 million and less than $100 million | 20% |
| c. An initial issuance of at least $50 million and less than $75 million | 25% |
| d. An initial issuance of at least $20 million and less than $50 million | 50% |
| e. An initial issuance of less than $20 million or have been held in inventory for more than 90 days as the result of the failure to complete an underwriting | 100% |
Broker-dealers may not include the value of non-investment grade, nonconvertible debt securities, subject to the haircut percentages set forth above, in paragraph (c)(2)(vi)(J) of SEA Rule 15c3-1 for the purposes of netting long or short securities positions under paragraph (c)(2)(vi)(J).
2. The broker-dealer shall take an additional portfolio concentration charge on the securities in categories (b), (c) and (d) above, to the extent the market value of the greater of the gross total long or gross total short positions in categories (b), (c) and (d) combined exceeds 25 percent of the broker-dealer's tentative net capital. The portfolio concentration charge shall be 50 percent of the haircuts otherwise taken on that portion of the total market value of the securities in categories (b), (c) and (d) in excess of 25 percent of tentative net capital. This portfolio concentration charge may be reduced by any undue concentration charge computed in accordance with paragraph (c)(2)(vi)(M) of SEA Rule 15c3-1.
Securities with an initial issuance of less than $20 million will be deemed to be included in category (d) above if the issuer has another outstanding issue of non-investment grade, nonconvertible debt securities, which has an initial issuance of $50 million or more.
Rule 144A Debt securities are not deemed to have a ready market pursuant to this interpretation since they cannot be publicly sold.
(SEC Letter to SIA, February 14, 1994)
(NYSE Interpretation Memo 94-5, May 1994)
(NYSE Interpretation Memo 97-6, October 1997)
15c3-1(c)(2)(vii)/101 Portfolio Concentration Charge - Sample Computations
EXAMPLE 1:
| | | | |
--- --- |
| HAIRCUT CATEGORY | LONG VALUE | SHORT VALUE | CAPITAL CHARGE |
| (b) 20% | $300,000,000 | $55,000,000 | $ 60,000,000 |
| (c) 25% | 150,000,000 | 27,000,000 | 37,500,000 |
| (d) 50% | 75,000,000 | 13,750,000 | 37,500,000 |
| TOTAL | $525,000,000 | $95,750,000 | $135,000,000 |
| 1. Total LV (b) + (c) + (d) = | | | $525,000,000 |
| 2. Tentative Net Capital | | | $2,000,000,000 |
| 3. Portfolio concentration | | | x25% |
| 4. Concentration threshold | | | $500,000,000 |
| 5. Amount subject to concentration | | | $25,000,000 |
| Concentration charge | | | 12.86% |
| CONCENTRATION CHARGE | | | $3,214,286 |
Formula for computing portfolio concentration haircut is 50% of total haircuts charges on categories (b), (c), and (d) divided by the greater of the gross total long or gross total short market value of the positions in these three categories. In the above example the total haircut charges 135m / 525m x .5 = 12.86%.
EXAMPLE 2:
| | | | |
--- --- |
| HAIRCUT | LONG | SHORT | CAPITAL |
| CATEGORY | VALUE | VALUE | CHARGE |
| (b) 20% | $300,000,000 | $ 55,000,000 | $ 60,000,000 |
| (c) 25% | 27,500,000 | 150,000,000 | 37,500,000 |
| (d) 50% | 75,000,000 | 13,750,000 | 37,500,000 |
| TOTAL | $402,500,000 | $218,750,000 | $135,000,000 |
| 1. Total LMV (b) + (c) + (d) = | | | $402,500,000 |
| 2. Tentative Net Capital | | | $2,000,000,000 |
| 3. Portfolio concentration | | | x25% |
| 4. Concentration threshold | | | $500,000,000 |
| 5. Amount subject to concentration | | | |
| $402,500,000 - $500,000,000 = | | | -0- |
| Concentration charge | | | 16.77% |
| CONCENTRATION CHARGE | | | -0- |
Formula for computing portfolio concentration haircut is 50% of total haircuts charges on categories (b), (c), and (d) divided by the greater of the gross total long or short market values market value of the positions in these three categories. In the above example the total haircut charges 135m / 402.5m x .5 = 16.8%.
(SEC Staff to NYSE) (NYSE Interpretation Memo 95-3, May 1995)
15c3-1(c)(2)(vii)/11 Marketability of Restricted Securities
The SEC Division of Market Regulation will not recommend enforcement action to the Commission if broker-dealers, in computing their net capital, apply the haircuts described below to their proprietary restricted securities (as defined in Rule 144(a)(3) of the Securities Act of 1933) that cannot be publicly offered or sold.
Nonconvertible debt securities, convertible debt securities, preferred stock or convertible preferred stock:
rated in one of the two highest rating categories by at least one NRSRO, would be subject to a 15% haircut, if:
the issuer, whether domestic or foreign, has long or short term ratings in one of the four highest rating categories by two NRSRO's on any debt issue, and
the issue's single rating is at least equal to or higher than the above referenced investment grade rating on the domestic or foreign debt, and
the NRSRO rating the issue is one of the organizations that rated the debt.
that are not rated by a NRSRO, or
that have a below-investment grade rating by a NRSRO, would be subject to a haircut of between 15% and 100%, based on initial issuance size as outlined in interpretation 15c3-1(c)(2)(vii)/10 for Nonconvertible Debt Securities Which Are Not Highly Rated, if:
the issuer, whether domestic or foreign, has long or short term ratings in one of the four highest rating categories by two NRSROs on any debt issue that is pari passu with the issue described in (b) or (c) above, or
the issuer has issued common stock included in the S&P 500 or in the FTSE World Index.
These securities would be subject to portfolio concentration charges as described in the Non Highly Rated Debt interpretation.
Commercial paper issues with one NRSRO rating in one of the two highest rating categories would be subject to a 15% haircut if:
the issuer, whether domestic or foreign, has long term or short term ratings in one of the four highest rating categories by two NRSROs on any debt issue, and
the NRSRO rating the commercial paper is one of the organizations that rated the debt.
Securities which can be sold pursuant to an exemption from registration, regardless of rating, that are freely convertible into publicly traded securities meeting the ready market provisions of the net capital rule, would be subject to the haircut charges pursuant to SEA Rule 15c3-1(c)(2)(vi) on the security to which it is convertible.
In addition to this haircut, an additional charge should be taken for the cost of conversion or the loss upon conversion.
These securities would be subject to the portfolio concentration charges described in the Non Highly Rated Debt interpretation.
Securities which can be sold pursuant to an exemption from registration and which have registration rights that provide for an exchange offer of the securities for registered securities are to be treated as readily marketable and subject to the respective capital charges set forth in the Non Highly Rated Debt interpretation, if:
the registered securities meet the other criteria of that interpretation, and
the company issuing the securities is listed in the S&P 500 or has an equity issue that is included in the FTSE World Index.
If these securities are not registered within 90 days from the settlement date with the issuer, the haircuts shall increase by 25% each month until such time as the security is subject to a 100% deduction.
These securities would be subject to the portfolio concentration charges described in the Non Highly Rated Debt interpretation.
Securities which can be sold pursuant to an exemption from registration would be subject to a contractual commitment deduction under SEA Rule 15c3-1(c)(2)(viii) of 30%. Any positions remaining in a firm's inventory after the date of the contractual commitment would be subject to the haircuts outlined above.
Consistent with the Non Highly Rated Debt interpretation, securities held in inventory longer than 90 days that were unsuccessfully offered pursuant to Rule 144A would be subject to a 100% capital charge.
The securities described above shall not include any security the transfer of which is subject to any contractual restriction or limitation imposed by the issuer or the seller of the security, if such restriction or limitation is not for the purpose of assuring compliance with the Federal securities laws or the securities laws of any state regarding the offer and sale of such securities in a transaction not involving a public offering. Such securities would be deemed non-marketable and subject to 100% deduction.
(SEC Letter to SIA, March 15, 1996)
(NYSE Interpretation Memo 96-3, April 1996)
15c3-1(c)(2)(vii)/12 Foreign Sovereign Debt Not Highly Rated
A nonconvertible foreign sovereign debt security held in a proprietary account that: 1) is issued as a general obligation of a sovereign government; 2) has a fixed maturity date; 3) is not traded flat or in default as to principal or interest; and 4) is not rated in one of the four highest categories (“investment grade rating”) by at least two NRSROs shall be subject to the charges stated below if at least one of the following conditions is met:
The foreign sovereign debt (denominated in either local currency or another currency, including collateralized and non-collateralized Brady bonds) is rated in one of the four highest ratings categories by one NRSRO and some satisfactory transaction volume can be demonstrated, or
The foreign sovereign debt is rated in the next highest category below investment grade by one NRSRO and both substantial volume and transactions can be demonstrated to indicate liquidity exists.
The charges shall be:
Haircut Charge
The broker-dealer shall deduct 15 percent of the market value of the greater of the long or short positions in nonconvertible Foreign Sovereign Debt by country; and
2. Country Concentration
The broker-dealer shall take an additional 7 ½ percent country concentration charge on the market value of the net long or net short position of an individual country that is in excess of 10 percent of the broker-dealer’s tentative net capital. Those securities for which a 100 percent capital deduction has already been taken should not be included in the country concentration charge; and
3. Portfolio Concentration
The broker-dealer shall take an additional 15 percent portfolio concentration charge on the sum of each country’s net long or net short securities market value to the extent that this total exceeds 50 percent of the broker-dealer's tentative net capital. Those securities for which a 100 percent capital deduction has already been taken should not be included in the portfolio charge. In addition, this portfolio concentration charge may be reduced by any undue concentration charge computed in accordance with paragraph (c)(2)(vi)(M) of SEA Rule 15c3-1.
(SEC Staff to NYSE) (NYSE Interpretation Memo 99-5, May 1999)
15c3-1(c)(2)(viii) Open Contractual Commitments
Deducting, in the case of a broker or dealer that has open contractual commitments (other than those option positions subject to appendix A, § 240.15c3-1a), the respective deductions as specified in paragraph (c)(2)(vi) of this section or appendix B, § 240.15c3-1b, from the value (which shall be the market value whenever there is a market) of each net long and each net short position contemplated by any open contractual commitment in the proprietary or other accounts of the broker or dealer.
15c3-1(c)(2)(viii)(A) The deduction for contractual commitments in those securities that are treated in paragraph (c)(2)(vi)(J) of this section shall be 30 percent unless the class and issue of the securities subject to the open contractual commitment deduction are listed for trading on a national securities exchange or are designated as NASDAQ National Market System Securities.
15c3-1(c)(2)(viii)(A)/01 Deductions for Exchange Listed and NASDAQ NMS Securities
The decreased charge (15%, pursuant to paragraph (c)(2)(vi)(J)) for securities that are listed for trading on a national securities exchange or are designated as NASDAQ National Market System Securities does not apply to initial public offerings of securities.
(SEC Staff to NYSE) (NYSE Interpretation Memo 93-6, November 1993)
15c3-1(c)(2)(viii)(A)/02 Offerings in Debt Convertible into Exchange Listed or NASDAQ NMS Securities
Initial public offerings of debt securities which are immediately convertible into equity securities that are listed for trading on a national securities exchange or are designated NASDAQ National Market System securities may be treated as if they have been converted into the equity security and subject to the reduced 15% open contractual commitment charge. If the conversion results in a loss (the market value of the equity securities is less than the market value of the debt securities), an additional charge must be taken for the amount of the loss.
(SEC to NASD) (NYSE Interpretation Memo 93-6, November 1993)
15c3-1(c)(2)(viii)(B) A broker or dealer that maintains in excess of $250,000 of net capital may add back to net worth up to $150,000 of any deduction computed under this paragraph (c)(2)(viii)(B).
15c3-1(c)(2)(viii)(B)/01 Application of $150,000 Addback
When a broker or dealer engages in more than one underwriting, the aggregate amount added back under SEA Rule 15c3-1(c)(2)(viii)(B) shall be the lessor of $150,000 or the contractual commitment charge taken.
(SEC Staff to NYSE) (NYSE Interpretation Memo 97-6, October 1997)
15c3-1(c)(2)(viii)(C) The deduction with respect to any single commitment shall be reduced by the unrealized profit in such commitment, in an amount not greater than the deduction provided for by this paragraph (or increased by the unrealized loss), in such commitment, and in no event shall an unrealized profit on any closed transactions operate to increase net capital.
15c3-1(c)(2)(viii)(C)/01 Losses on Open Contractual Commitments
Unrealized losses on open contractual commitments are treated as charges in arriving at net worth and the debt/equity total. (See SEA Rule 15c3-1(c)(2)(i)(A).)
(SEC Staff to NYSE)
15c3-1(c)(2)(viii)(C)/02 Profits on Open Contractual Commitments
Unrealized profits on open contractual commitments are allowed to reduce haircuts but not to otherwise increase net worth or net capital.
(SEC Staff to NYSE)
15c3-1(c)(2)(viii)(C)/021 GNMA's - Unrealized Profits and Losses
The aggregate of all unrealized profits reduced by the aggregate of all unrealized losses on open GNMA transactions (including issued positions, commodity exchange futures, and TBA's) may reduce the aggregate of all haircuts related to such transactions. Any excess net unrealized profits may not operate to increase net capital.
(See interpretation 15c3-1(c)(2)(vi)(A)/02 for haircuts and interpretation 15c3-1a/02 (Appendix A) for standbys in GNMA transactions.)
(SEC Staff to NYSE) (NYSE Interpretation Memo 76-3, February 1976)
15c3-1(c)(2)(viii)(C)/03 Haircuts on Contractual Commitments
Open contractual commitments are subject to the haircut percentage set forth in SEA Rule 15c3-1(c)(2)(vi). (See interpretation 15c3-1(c)(2)(vi)/01 for application of haircuts to combinations of open contractual commitment and actual positions.)
(SEC Staff to NYSE) (NYSE Interpretation Memo 77-2, June 1977)
15c3-1(c)(2)(viii)(C)/031 Underwriting Commitments
Where a firm commitment is contingent upon the effectiveness of a registration statement the appropriate deductions will apply when the registration becomes effective.
Where securities underwritten are not subject to registration requirements deductions will be applied when the broker or dealer is irrevocably committed to the underwriting.
(SEC Staff to NYSE) (NYSE Interpretation Memo 84-9, November 1984)
15c3-1(c)(2)(viii)(C)/032 Offsetting Sale Commitments in a Registered Offering
Underwriting commitments in a registered offering may be reduced by binding contracts for sale.
So called "indications of interest" received prior to the effectiveness of a registration statement are not binding contracts for sale. The Securities Act prohibits the sale of a security in a registered offering until the registration statement has become effective.
(SEC Letter to NYSE, November 18, 1983) (NYSE Interpretation Memo 84-9, November 1984)
(SEC Staff to FINRA) (FINRA Regulatory Notice 21-27)
15c3-1(c)(2)(viii)(C)/033 Offsetting Sale Commitments in an Unregistered Offering
Underwriting commitments in an unregistered offering may be reduced by binding contracts for sale, including bona fide presales.
A bona fide presale is a binding purchase commitment that is:
received by the broker-dealer prior to the effectiveness of its underwriting commitment; and
not subject to any conditions (other than the issuance of the securities) at the time the underwriting commitment becomes effective.
A bona fide presale may be evidenced in a variety of ways, including a written schedule of the bona fide presales, maintained by the broker-dealer as a record pursuant to SEA Rules 17a-3 and 17a-4.
Under the Securities Act, the sale of a security, which includes entry into a binding contract for sale, may be made without registration if the security is an exempted security or the transaction is exempt from registration.
(SEC Staff to FINRA) (FINRA Regulatory Notice 21-27)
15c3-1(c)(2)(viii)(C)/04 Selling Group Participations
Haircuts need not be applied to best efforts selling group participations in firm commitment underwritings to the extent that the selling group member has an unconditional right evidenced by a written agreement with the underwriting participants to return any unsold securities.
Once the issue trades regular way, haircuts do not apply to unsold shares returned to the underwriter or participant no later than the settlement date of the issue.
$5,000 broker-dealers may not participate in firm commitment underwritings even on a best efforts basis.
(SEC Letter to NASD, December 15, 1976)
(NYSE Interpretation Memo 83-2, April 1983)
15c3-1(c)(2)(viii)(C)/05 (Rescinded)
(NYSE Interpretation Memo 97-5, September 1997)
15c3-1(c)(2)(viii)(C)/06 Underwriting Backstop Agreement
A member of an underwriting syndicate in a firm commitment underwriting (the “backstop recipient”) is not required to take an open contractual commitment charge arising from its underwriting commitment if the backstop recipient enters into a written agreement with another syndicate member (the “backstop provider”) that:
is executed and effective at or before the time the backstop recipient becomes obligated to the underwriting commitment;
requires the backstop provider to deduct in its net capital computation any applicable open contractual commitment charge that the backstop recipient would otherwise be required to take into account in its net capital computation; and
unequivocally requires the backstop provider to purchase any unsold securities allocated to the backstop recipient under the underwriting agreement.
The backstop provider and backstop recipient must comply with all other applicable laws, rules, and regulations of the Commission and any self-regulatory organization of which they are members.
(SEC Staff to FINRA) (FINRA Regulatory Notice 19-11)
15c3-1(c)(2)(ix) Deducting from the contract value of each failed to deliver contract that is outstanding five business days or longer (21 business days or longer in the case of municipal securities) the percentages of the market value of the underlying security that would be required by application of the deduction required by paragraph (c)(2)(vi) of this section. Such deduction, however, shall be increased by any excess of the contract price of the failed to deliver contract over the market value of the underlying security or reduced by any excess of the market value of the underlying security over the contract value of the failed to deliver contract, but not to exceed the amount of such deduction. The designated examining authority for the broker or dealer may, upon application of the broker or dealer, extend for a period up to 5 business days, any period herein specified when it is satisfied that the extension is warranted. The designated examining authority upon expiration of the extension may extend for one additional period of up to 5 business days, any period herein specified when it is satisfied that the extension is warranted.
15c3-1(c)(2)(ix)/01 Undue Concentration Charges
Undue concentration charges do not apply against aged fails to deliver.
15c3-1(c)(2)(ix)/02 Short Positions - Related to Fails to Deliver
If an aged fail to deliver relates to a proprietary short sale, the percentage deduction is 15% of the market value of the underlying security regardless of whether the proprietary short position is less than the long position or the computing broker-dealer is subject to the capital requirements of paragraph (a) of Rule 15c3-1. However, if the fail to deliver and related short positions are in securities other than those subject to subparagraph (c)(2)(vi)(J) of SEA Rule 15c3-1, the aged fail to deliver is subject to the haircut provisions of subparagraphs (c)(2)(vi)(A) - (I).
(SEC Letter to Mauney Company, December 11, 1975)
(NYSE Interpretation Memo 76-4, April 1976)
15c3-1(c)(2)(ix)/03 Securities Drafted to Another Broker-Dealer
Securities drafted to another broker-dealer to satisfy a fail to deliver are not subject to the aged fail to deliver deduction unless not accepted, in which case it would be aged from the original settlement date.
(SEC Letter to Prince, Langheinrich & Greer Inc., April 22, 1977)
(NYSE Interpretation Memo 79-4, March 1979)
15c3-1(c)(2)(ix)/04 Continuous Net Settlement Systems
Where a clearing organization operates on a continuous settlement system and marks to market daily, open fails to deliver to the organization need not be aged and no deductions need be made by the broker-dealer under this provision of SEA Rule 15c3-1.
(SEC Staff to NYSE) (NYSE Interpretation Memo 81-10, December 1981)
15c3-1(c)(2)(ix)/041 NSCC’s RECAPS Program
Broker-dealers participating in the NSCC’s Reconfirmation and Pricing Service (RECAPS) Program may treat the RECAPS settlement date and price as the date of the fail for aging and contract price purposes.
(SEC Letter to NSCC, June 11, 1987)
(NYSE Interpretation Memo 89-6, June 1989)
15c3-1(c)(2)(ix)/05 Municipal Securities Brokers’ Broker
The extension provision is not available. Twenty-one business days as provided in subparagraph (a)(8) is deemed sufficient.
(SEC Letter to NASD, October 24, 1983)
(NYSE Interpretation Memo 88-16, October 1988)
15c3-1(c)(2)(ix)/06 Foreign Issued and Settled Securities Fail to Deliver - Haircut Alternative
Broker-dealers may, in lieu of the treatment required by this provision (c)(2)(ix) for aged fail to deliver of foreign issued, foreign settled securities, apply alternative procedures. In the event such alternative procedures are elected, the following treatment shall apply:
Five business days after settlement date (in accordance with the customary foreign settlement cycle), the broker-dealer shall take a proprietary haircut charge for the foreign issued, foreign settled securities failed to deliver pursuant to SEA Rule 15c3-1, reduced by the equity (or increased by the deficit) in the transaction on a mark-to-market basis. In those countries where settlement is on a seller’s option basis, the settlement date for purposes of this computation will be considered to be a day not more than thirty calendar days from the trade date;
During the period from trade date until the aged failed to deliver charge is required to be taken, the broker-dealer shall take a concentration charge on a mark-to-market basis equal to 100 percent of the excess of all trade date based deficits with a single counterparty in excess of 10 percent of the broker-dealer’s tentative net capital;
In determining a required deduction, the broker-dealer may reduce such deficit by any margin or other deposit held by the broker-dealer in connection with such transaction with the same party and any net equity in all failed to receive transactions, on a trade date basis, with the same party;
In determining a required deduction, the broker-dealer may reduce such deficit by any margin calls issued by the broker-dealer, outstanding not more than two business days. A broker-dealer may take advantage of this provision regarding margin calls only if it has a written agreement with the customer regarding the issuance and satisfaction of margin calls;
The broker-dealer shall file a written notice with the national securities exchange or registered national securities association which is its designated examining authority of its intention to apply this alternative treatment instead of the requirements of subparagraph (c)(2)(ix) of SEA Rule 15c3-1;
The broker-dealer will maintain in its records a schedule of the current settlement cycle of each country in which it trades; and
The broker-dealer shall maintain and preserve separate records, in whatever form appropriate, detailing, by country, the total number of failed to receive and failed to deliver contracts, and the total contractual value of those contracts and transactions.
A “ready market” is deemed to exist with respect to certain foreign securities that satisfy the criteria discussed in interpretation 15c3-1(c)(11)(i)/02, which specifically includes (but is not limited to) securities listed on any of the principal exchanges in the major money markets outside the United State, i.e. –
| | | | | | |
--- --- --- |
| Amsterdam | Frankfurt | London | Montreal | Sydney | Toronto |
| Brussels | Johannesburg | Luxenbourg | Paris | Tokyo | Zurich |
When foreign issued, foreign settled securities fail to deliver do not meet the SEC’s criteria as readily marketable securities under SEA Rule 15c3-1(c)(11) (see interpretation 15c3-1(c)(11)(i)/02)) and such securities are traded on exchanges in the countries shown on the next page, the haircut charge to be applied shall be a multiple of the haircut charge for securities which meets the criteria for readily marketable securities. This treatment has no effect on the ready market criteria set forth at subparagraph (c)(11) of SEA Rule 15c3-1.
| | | | | |
--- ---
| Australia | Federal Republic of Germany | Italy | Netherlands | South Africa |
| Austria | | Japan | Spain | New Zealand |
| Belgium | Finland | Norway | Sweden | Luxembourg |
| Canada | France | Malaysia | Portugal | Switzerland |
| Denmark | Hong Kong | Mexico | Singapore | United Kingdom |
| | |
--- |
| Outstanding From Settlement Date | Deduction |
| 5-29 calendar days | Standard proprietary haircut charge |
| 30-59 calendar days | Twice standard proprietary haircut charge (but not greater than 30%) |
| 60-89 calendar days | Four times standard proprietary haircut charge (but not greater than 60%) |
| 90 or more calendar days | 100% market value |
The market value of all other fails to deliver of foreign issued, foreign settled securities shall be deducted, in full, 5 business days after settlement date.
Settlement date for these purposes determined in accordance with Item a. of this Interpretation.
(SEC Letter to SIA, June 5, 1989)
(NYSE Interpretation Memo 89-9, July 1989)
(SEC Staff to NYSE) (NYSE Interpretation Memo 90-7, September 1990)
15c3-1(c)(2)(ix)/07 Failed to Deliver Resulting from NYSE Rule 412(e) (ACATS) Customer Securities Account Transfers
Failed to deliver contracts resulting from the Automated Customer Account Transfer System (ACATS) which are identified as NYSE Rule “412 Fails” aged 5 business days or longer (21 business days or longer in the case of municipal securities) need not be subject to the net capital charges specified herein so long as they are carried in compliance with subparagraph (c), i.e., close outs within 10 days, and (f) i.e., exemptive provisions, of NYSE Rule 412. (See NYSE Rule 412 and interpretations in the NYSE Interpretation Handbook.)
(SEC Staff to NYSE) (NYSE Interpretation Memo 90-1, February 1990)
15c3-1(c)(2)(ix)/08 Failed to Deliver Charges on Non-Marketable Securities
Broker-dealers can apply the following percentages on foreign and domestic non-marketable securities (as defined under SEA Rule 15c3-1(c)(2)(vii)) when calculating the haircut portion of the aged fail to deliver charges:
| | | |
---
| Days Failed to Deliver Contract is Outstanding | | Percentage Deduction |
| 5 to 14 business days | | 15 percent |
| 15 to 21 business days | | 50 percent |
| 22 to 28 business days | | 75 percent |
| 29 business days or more | | 100 percent |
In addition, broker-dealers must increase the amount of the deduction set forth above by any excess of the contract price of the failed to deliver contract over the market value of the underlying security or reduce the deduction by any excess of the market value of the underlying security over the contract value of the failed to deliver contract, but not to exceed the amount of such deduction. Further, broker-dealers relying on the schedule must have the ability to demonstrate to appropriate staff of the Commission or the broker-dealer’s designated examining authority that the firm has an adequate process for monitoring risk associated with aged fails.
(SEC Letter to SIA Capital Committee, July 13, 2001)
(NYSE Interpretation Memo 02-3, February 2002)
15c3-1(c)(2)(ix)/09 Fixed Income Clearing Corporation (FICC) – Fails to Deliver Not Aged
Government securities broker-dealers that are FICC Netting Members need not age open fails to deliver and comprehend deductions under the provisions of SEA Rule 15c3-1(c)(2)(ix), for trades processed through FICC’s Netting System which operates on a continuous settlement basis that marks to the market daily.
(Department of the Treasury Letter to GSCC, November 22, 1989))
(SEC Staff to NYSE) (NYSE Interpretation Memo 05-8, April 2005)
15c3-1(c)(2)(x) Brokers or Dealers Carrying Accounts of Listed Options Specialists
15c3-1(c)(2)(x)(A) With respect to any transaction of a specialist in listed options, who is either not otherwise subject to the provisions of this section or is described in paragraph (c)(2)(vi)(N) of this section, for whose specialist account a broker or dealer acts as a guarantor, endorser, or carrying broker or dealer, such broker or dealer shall adjust its net worth by deducting as of noon of each business day the amounts computed as of the prior business day pursuant to § 240.15c3-1a. The required deductions may be reduced by any liquidating equity that exists in such specialist's market-maker account as of that time and shall be increased to the extent of any liquidating deficit in such account. Noon shall be determined according to the local time where the broker or dealer is headquartered. In no event shall excess equity in the specialist's market-maker account result in an increase of the net capital of any such guarantor, endorser, or carrying broker or dealer.
15c3-1(c)(2)(x)(B) Definitions.
15c3-1(c)(2)(x)(B)(1) The term listed option shall mean any option traded on a registered national securities exchange or automated facility of a registered national securities association.
15c3-1(c)(2)(x)(B)(2) For purposes of this section, the equity in an individual specialist's market-maker account shall be computed by:
15c3-1(c)(2)(x)(B)(2)(i) Marking all securities positions long or short in the account to their respective current market values;
15c3-1(c)(2)(x)(B)(2)(ii) Adding (deducting in the case of a debit balance) the credit balance carried in such specialist's market-maker account; and
15c3-1(c)(2)(x)(B)(2)(iii) Adding (deducting in the case of short positions) the market value of positions long in such account.
15c3-1(c)(2)(x)(C) No guarantor, endorser, or carrying broker or dealer shall permit the sum of the deductions required pursuant to § 240.15c3-1a in respect of all transactions in specialists' market-maker accounts guaranteed, endorsed, or carried by such broker or dealer to exceed 1,000 percent of such broker's or dealer's net capital as defined in § 240.15c3-1(c)(2) for any period exceeding three business days. If at any time such sum exceeds 1,000 percent of such broker's or dealer's net capital, then the broker or dealer shall:
15c3-1(c)(2)(x)(C)(1) Immediately transmit telegraphic or facsimile notice of such event to the Division of Market Regulation in the headquarters office of the Commission in Washington, DC, to the regional office of the Commission for the region in which the broker or dealer maintains its principal place of business, and to its examining authority designated pursuant to section 17(d) of the Act (15 U.S.C. 78q(d)) (“Designated Examining Authority”); and
15c3-1(c)(2)(x)(C)(2) Be subject to the prohibitions against withdrawal of equity capital set forth in § 240.15c3-1(e) and to the prohibitions against reduction, prepayment, and repayment of subordination agreements set forth in paragraph (b)(11) of § 240.15c3-1d, as if such broker or dealer's net capital were below the minimum standards specified by each of those paragraphs.
15c3-1(c)(2)(x)(D) If at any time there is a liquidating deficit in a specialist's market-maker account, then the broker or dealer guaranteeing, endorsing, or carrying listed options transactions in such specialist's market-maker account may not extend any further credit in that account, and shall take steps to liquidate promptly existing positions in the account. This paragraph shall not prevent the broker or dealer from, upon approval by the broker's or dealer's Designated Examining Authority, entering into hedging positions in the specialist's market-maker account. The broker or dealer also shall transmit telegraphic or facsimile notice of the deficit and its amount by the close of business of the following business day to its Designated Examining Authority and the Designated Examining Authority of the specialist, if different from its own.
15c3-1(c)(2)(x)(E) Upon written application to the Commission by the specialist and the broker or dealer guaranteeing, endorsing, or carrying options transactions in such specialist's market-maker account, the Commission may approve upon specified terms and conditions lesser adjustments to net worth than those specified in § 240.15c3-1a.
15c3-1(c)(2)(xi) Brokers or dealers carrying specialists or market makers accounts. With respect to a broker or dealer who carries a market maker or specialist account, or with respect to any transaction in options listed on a registered national securities exchange for which a broker or dealer acts as a guarantor or endorser of options written by a specialist in a specialist account, the broker or dealer shall deduct, for each account carried or for each class or series of options guaranteed or endorsed, any deficiency in collateral required by paragraph (a)(6) of this section.
15c3-1(c)(2)(xii)(A) Deduction from net worth for certain undermargined accounts. Deducting the amount of cash required in each customer's or non-customer's account to meet the maintenance margin requirements of the Examining Authority for the broker or dealer, after application of calls for margin, marks to the market or other required deposits which are outstanding 5 business days or less.
15c3-1(c)(2)(xii)(A)/01 Reverse Repurchase Agreements – (Rescinded)
(NYSE Interpretation Memo 06-5, June 2006)
15c3-1(c)(2)(xii)(A)/011 Reverse Repurchase Agreements
On reverse repurchase agreement transactions, the greater of the cash margin deficiency based on the margin requirements of the Designated Examining Authority described herein or the amount required by subparagraph (c)(2)(iv)(F) of SEA Rule 15c3-1 shall be deducted.
Cash reverse repurchase transactions in “exempted securities” as defined in Section 3(a)(12) of the Securities and Exchange Act of 1934, and cash reverse repurchase transactions in mortgage related securities as defined in Section 3(a)(41) of the Securities and Exchange Act of 1934, as well as certain non-equity securities described in NYSE Rule 431(a)(9) through 431(a)(11), may be maintained in a special account subject to the provisions of NYSE Rule 431(e)(2)(F), which provides that broker-dealers entering into transactions with “exempt accounts”, as defined in NYSE Rule 431(a)(13), need not collect margin from such “exempt accounts”, however are subject to the Exchange’s capital requirements described in NYSE Rule 431(e)(2)(F), (G) and (H).
All other non-exempt accounts entering into cash reverse repurchase transactions are subject to the following minimum margin requirements:
1% to 6% of the current market value of U. S. Government obligations (See NYSE Rule 431(e)(2)(A));
7% of the current market value of all other exempted securities other than obligations of the United States (See NYSE Rule 431(e)(2)(B));
10% of the current market value in the case of investment grade debt securities (See NYSE Rule 431(e)(2)(C)(i)); and
20% of the current market value or 7% of the principal amount, whichever amount is greater, in the case of all other listed non-equity securities, and all other marginable non-equity securities as defined in NYSE Rule 431(a)(16) (See NYSE Rule 431(e)(2)(C)(ii)).
(SEC Staff to NYSE) (NYSE Interpretation Memo 06-5, June 2006)
15c3-1(c)(2)(xii)(A)/02 Government National Mortgage Association (GNMA)
See Interpretation Handbook for treatment under NYSE Rule 431(e)(2)(F)/04 through /047.
(NYSE Interpretation Memo 88-15, September 1988)
15c3-1(c)(2)(xii)(A)/03 Regulation T Calls for Margin
Only calls for margin, which are outstanding five business days or less, may be applied in computing cash margin deficiencies under this provision.
(SEC Staff to NYSE) (NYSE Interpretation Memo 81-10, December 1981)
(SEC Staff to NYSE) (NYSE Interpretation Memo 06-5, June 2006)
15c3-1(c)(2)(xii)(A)/04 Non-Purpose Loans Collateralized by Certificates of Deposit
See interpretation 15c3-1(c)(2)(iv)(B)/10.
15c3-1(c)(2)(xii)(A)/05 Credit Extended Upon Exercise of Employee Stock Option
When a broker-dealer exercises an employee stock option for a customer, it must have acknowledgement from the issuer that a freely transferable, readily salable and marketable security in negotiable form will be promptly delivered to the broker-dealer within 13 business days after exercise notice is given to the issuer (when acknowledgment is given by telephone, the condition should be restated in the transmittal to the issuer). The exercise shall be subject to the following:
When the security to be received from the exercise has been sold and is not received from the issuer within 13 business days after notice of exercise has been given, the position shall be subject to a cash margin deficiency charge computed without allowing any value for the security not received (and is subject to the buy-in provisions under SEA Rule 15c3-3(m) unless an extension of time is requested and approved under paragraph (n) of that rule);
When the security to be received from the exercise has not been sold and is not received within 13 business days after notice of exercise has been given, any related debit balance shall be treated as an unsecured debit for net capital purposes. (See interpretation 15c3-1(c)(2)(iv)(B)/12.)
(SEC Staff to NYSE) (NYSE Interpretation Memo 88-15, September 1988)
(NYSE Interpretation Memo 97-6, October 1997)
15c3-1(c)(2)(xii)(A)/06 Credit Extended to Customers on Control or Restricted Stock
Credit extended to customers on control or restricted stock shall be subject to maintenance margin requirements of NYSE Rule 431(e)(8). Any resulting cash margin deficiencies should be charged pursuant this provision.
(SEC Staff to NYSE) (NYSE Interpretation Memo 92-1, January 1992)
15c3-1(c)(2)(xii)(A)/061 Customers Foreign Currency Options Collateralized by Letters of Credit
See interpretation 15c3-1(c)(2)(iv)(B)/013.
15c3-1(c)(2)(xii)(A)/07 Maintenance Requirement for Proprietary Accounts Carried for Joint Back Office Broker-Dealers
Broker-dealers operating joint back offices and carrying proprietary accounts of other broker-dealers that are participants in the joint back office must maintain equity in such accounts at least equal to the haircut percentages provided under SEA Rule 15c3-1 subparagraphs (c)(2)(vi) (excluding subparagraph (c)(2)(vi)(M)) or 15c3-1a (Appendix A) as appropriate. If the equity in the account is not equal to or greater than the total haircuts computed for the positions carried in the participant's account, the carrying broker-dealer must obtain additional allowable collateral or charge its own capital for the deficiency. No benefit may be taken by the carrying broker-dealer if equity in such accounts exceeds the required haircuts. If the participant's account liquidates to a deficit, the charge to the carrying broker will be for the sum of the deficit and the applicable haircuts.
(SEC Staff to NYSE) (NYSE Interpretation Memo 92-12, December 1992)
(NYSE Interpretation Memo 96-4, November 1996) (NYSE Interpretation Memo 97-5, September 1997)
15c3-1(c)(2)(xii)(B) Deducting the amount of cash required in the account of each security-based swap and swap customer to meet the margin requirements of a clearing agency, Examining Authority, the Commission, derivatives clearing organization, or the Commodity Futures Trading Commission, as applicable, after application of calls for margin, marks to the market, or other required deposits which are outstanding within the required time frame to collect the margin, mark to the market, or other required deposits.
15c3-1(c)(2)(xiii) Deduction from net worth for indebtedness collateralized by exempted securities. Deducting, at the option of the broker or dealer, in lieu of including such amounts in aggregate indebtedness, 4 percent of the amount of any indebtedness secured by exempted securities or municipal securities if such indebtedness would otherwise be includable in aggregate indebtedness.
15c3-1(c)(2)(xiii)/01 Optional Treatment of Liabilities vs Municipal Collateral
The optional deduction applies to bank loans, fail to receive, securities loaned or other such liabilities includable in aggregate indebtedness which are collateralized by exempted or municipal securities.
(SEC Staff to NYSE) (NYSE Interpretation Memo 77-4, November 1977)
15c3-1(c)(2)(xiv) Deduction from net worth for excess deductible amounts related to fidelity bond coverage. Deducting the amount specified by rule of the Examining Authority for the broker or dealer with respect to a requirement to maintain fidelity bond coverage.
15c3-1(c)(2)(xv) Deduction from net worth in lieu of collecting collateral for non-cleared security-based swap and swap transactions — -
15c3-1(c)(2)(xv)(A) Security-based swaps. Deducting the initial margin amount calculated pursuant to § 240.18a-3(c)(1)(i)(B) for the account of a counterparty at the broker or dealer that is subject to a margin exception set forth in § 240.18a-3(c)(1)(iii), less the margin value of collateral held in the account.
15c3-1(c)(2)(xv)(B) Swaps. Deducting the initial margin amount calculated pursuant to the margin rules of the Commodity Futures Trading Commission in the account of a counterparty at the broker or dealer that is subject to a margin exception in those rules, less the margin value of collateral held in the account.
15c3-1(c)(2)(xv)(C) Treatment of collateral held at a third-party custodian. For the purposes of the deductions required pursuant to paragraphs (c)(2)(xv)(A) and (B) of this section, collateral held by an independent third-party custodian as initial margin may be treated as collateral held in the account of the counterparty at the broker or dealer if:
15c3-1(c)(2)(xv)(C)(1) The independent third-party custodian is a bank as defined in section 3(a)(6) of the Act or a registered U.S. clearing organization or depository that is not affiliated with the counterparty or, if the collateral consists of foreign securities or currencies, a supervised foreign bank, clearing organization, or depository that is not affiliated with the counterparty and that customarily maintains custody of such foreign securities or currencies;
15c3-1(c)(2)(xv)(C)(2) The broker or dealer, the independent third-party custodian, and the counterparty that delivered the collateral to the custodian have executed an account control agreement governing the terms under which the custodian holds and releases collateral pledged by the counterparty as initial margin that is a legal, valid, binding, and enforceable agreement under the laws of all relevant jurisdictions, including in the event of bankruptcy, insolvency, or a similar proceeding of any of the parties to the agreement, and that provides the broker or dealer with the right to access the collateral to satisfy the counterparty's obligations to the broker or dealer arising from transactions in the account of the counterparty; and
15c3-1(c)(2)(xv)(C)(3) The broker or dealer maintains written documentation of its analysis that in the event of a legal challenge the relevant court or administrative authorities would find the account control agreement to be legal, valid, binding, and enforceable under the applicable law, including in the event of the receivership, conservatorship, insolvency, liquidation, or a similar proceeding of any of the parties to the agreement.
15c3-1(c)(3) Exempted Securities
The term exempted securities shall mean those securities deemed exempted securities by section 3(a)(12) of the Securities Exchange Act of 1934 and rules thereunder.
15c3-1(c)(4) Contractual Commitments
The term contractual commitments shall include underwriting, when issued, when distributed and delayed delivery contracts, the writing or endorsement of puts and calls and combinations thereof, commitments in foreign currencies, and spot (cash) commodities contracts, but shall not include uncleared regular way purchases and sales of securities and contracts in commodities futures. A series of contracts of purchase or sale of the same security conditioned, if at all, only upon issuance may be treated as an individual commitment.
15c3-1(c)(5) Adequately Secured
Indebtedness shall be deemed to be adequately secured within the meaning of this section when the excess of the market value of the collateral over the amount of the indebtedness is sufficient to make the loan acceptable as a fully secured loan to banks regularly making secured loans to brokers or dealers.
15c3-1(c)(6) Customer
The term customer shall mean any person from whom, or on whose behalf, a broker or dealer has received, acquired or holds funds or securities for the account of such person, but shall not include a broker or dealer or a registered municipal securities dealer, or a general, special or limited partner or director or officer of the broker or dealer, or any person to the extent that such person has a claim for property or funds which by contract, agreement, or understanding, or by operation of law, is part of the capital of the broker or dealer. Provided, however, That the term “customer” shall also include a broker or dealer, but only insofar as such broker or dealer maintains a special omnibus account carried with another broker or dealer in compliance with 12 CFR 220.4(b) of Regulation T under the Securities Exchange Act of 1934.
15c3-1(c)(7) Non-Customer
The term non-customer means a broker or dealer, registered municipal securities dealer, general partner, limited partner, officer, director and persons to the extent their claims are subordinated to the claims of creditors of the broker or dealer.
15c3-1(c)(7)/01 Municipal Securities Dealers
A bank municipal securities dealer that does not transact its municipals business through a separately identifiable department or division and accordingly registers as an undivided entity, is a non-customer only with respect to its transactions as a municipal securities dealer.
(SEC Release 34-11969, January 2, 1976)
(NYSE Interpretation Memo 76-2, February 1976)
15c3-1(c)(7)/02 Foreign Banks as Brokers or Dealers
To be treated as a broker, the foreign bank must be engaged in the business of effecting transactions in securities for the account of others within the meaning of Section 3(a)(4) of the '34 Act.
To be treated as a dealer, the foreign bank must be engaged in the business of buying and selling securities for its own account, through a broker or otherwise, within the meaning of Section 3(a)(5) of the '34 Act.
In addition, to be treated as either a non-customer broker or a non-customer dealer, the foreign bank must not fall within the definition of "bank" set forth in Section 3(a)(6) of the '34 Act, which provides as follows:
The term "bank" means (A) a banking institution organized under the laws of the United States, (B) a member bank of the Federal Reserve System, (C) any other banking institution, whether incorporated or not, doing business under the laws of any State or of the United States, a substantial portion of the business of which consists of receiving deposits or exercising a fiduciary power similar to those permitted to national banks under Section 11(k) of the Federal Reserve Act, as amended, and which is supervised and examined by State or Federal authority having supervision over banks, and which is not operated for the purpose of evading the provisions of this title, and (D) a receiver, conservator, or other liquidating agent of any institution or firm included in clauses (A), (B) or (C) of this paragraph.
If the foreign bank falls within the above definition of a bank, it is to be treated as a customer for purposes of SEA Rules 15c3-1 and 15c3-3.
There are at least three forms of foreign banking operations that you may be doing business with (1) representative offices, (2) agencies and (3) branches. Agencies and branches are subject to certain reporting requirements of the Federal Reserve Board and some states have specific regulations concerning foreign bank entry and operation, including examination and supervision and may be required to be treated as customers. Representative offices generally do not conduct normal banking operations but merely act as liaison offices between the head office and its customers. Generally speaking, there are no state regulations as to examination and supervision of representative offices other than simple registration with the state in which business is being conducted. Representative offices may be eligible for treatment as a non-customer.
(SEC Letter to UBS-DB Corporation, March 5, 1977)
(SEC Staff to NYSE) (NYSE Interpretation Memo 78-2, May 1978)
15c3-1(c)(8) Market Maker
The term market maker shall mean a dealer who, with respect to a particular security,
15c3-1(c)(8)(i) regularly publishes bona fide, competitive bid and offer quotations in a recognized interdealer quotation system; or
15c3-1(c)(8)(ii) furnishes bona fide competitive bid and offer quotations on request; and,
15c3-1(c)(8)(iii) is ready, willing and able to effect transactions in reasonable quantities at his quoted prices with other brokers or dealers.
15c3-1(c)(8)/01 Broker-Dealer as Market Maker
A broker-dealer is not considered a market maker when entering a "bid" or an "offer" quotation in an inter-dealer quotation service in response to a customer's order.
A broker-dealer is considered a market maker when entering an inter-dealer quotation service with either "OW - BW" or its name, if it furnishes bona fide competitive quotations on request. Other factors to be considered are trading activity and its position in the security in question. The burden of proof is on the broker-dealer to demonstrate that it is not making a market.
In addition to "NASDAQ" and the "NQB" Pink Sheets, other quotations in an inter-dealer quotation service may constitute a bona fide market. Discretion should be used in those instances where the market appears to be limited or regional. A determination should be made regarding depth of the market and the source of the quotation.
(SEC Staff to NASD)
15c3-1(c)(9) Promptly Transmit and Deliver
A broker or dealer is deemed to “promptly transmit” all funds and to “promptly deliver” all securities within the meaning of paragraphs (a)(2)(i) and (a)(2)(v) of this section where such transmission or delivery is made no later than noon of the next business day after the receipt of such funds or securities; provided, however, that such prompt transmission or delivery shall not be required to be effected prior to the settlement date for such transaction.
15c3-1(c)(10) Promptly Forward
A broker or dealer is deemed to “promptly forward” funds or securities within the meaning of paragraph (a)(2)(i) of this section only when such forwarding occurs no later than noon of the next business day following receipt of such funds or securities.
15c3-1(c)(11) Ready Market
15c3-1(c)(11)(i) The term ready market shall include a recognized established securities market in which there exists independent bona fide offers to buy and sell so that a price reasonably related to the last sales price or current bona fide competitive bid and offer quotations can be determined for a particular security almost instantaneously and where payment will be received in settlement of a sale at such price within a relatively short time conforming to trade custom.
15c3-1(c)(11)(i)/01 Initial Distribution Period
A ready market is deemed to exist during the initial distribution period of a taxable corporate debt offering. The securities should be valued at their public offering price less the dealer discount. If a market for the securities does not develop shortly after the initial distribution period, a ready market would be deemed to exist only under the conditions described in SEA Rule 15c3-1(c)(11)(ii).
(SEC Letter to the Ohio Company, December 15, 1975)
15c3-1(c)(11)(i)/02 Foreign Securities
A ready market will be deemed to exist with respect to foreign securities if they are either:
Equity securities of a foreign issuer that are listed on the FTSE World Index;
Securities that meet the marketability criteria of either interpretations 15c3-1(c)(2)(vii)/08 (Marketability of Certain Foreign and Domestic Securities), or 15c3-1(c)(2)(vii)/09 (Marketability of Money Market Instruments), or
Securities, bankers acceptances and bankers deposit notes of a foreign issuer that have been accepted as collateral for a loan by any major financial institution (including foreign banks) where the broker or dealer is able to demonstrate to the satisfaction of the Designated Examining Authority and the Commission that such securities adequately secure such loans. (See NYSE Rule 328(c) for further requirements.)
(SEC Staff to NYSE) (NYSE Interpretation Memo 92-12, December 1992)
(SEC Letter to SIA, August 13, 1993) (NYSE Interpretation Memo 93-5, September 1993)
(NYSE Interpretation Memo 93-6, November 1993)
(SEC Staff to NYSE) (NYSE Interpretation Memo 01-3, March 2001)
15c3-1(c)(11)(i)/021 Mexican Stock Exchange Securities (MSE)
Broker-dealers may treat equities listed on the Mexican Stock Exchange (MSE) that are included in the FT-Actuaries Mexico Index ("FT-A Mexico Index") as having a ready market.
(SEC Letter to Comision Nacional de Valores, August 28, 1992)
(NYSE Interpretation Memo 92-12, December 1992)
15c3-1(c)(11)(i)/03 National Daily Quotation Service, BW-OW Quotes
Where Bid Wanted - Offer indications published in National Daily Quotation Service are the only available source of market value, limited market provisions under subparagraphs (K)(i) or (K)(ii) may apply under the following circumstances:
The broker-dealer can show that two or three or more market makers (other than the computing broker-dealer) in the quotation sheets, even if the market makers do not display actual quotations in the sheets; and
The broker-dealer can show the existence of bona fide inter-dealer trades within five business days before or after the date of valuation. The trades must be of sufficient volume to justify a reasonable belief that the price used would support the liquidation of the entire position at or near that price.
(SEC Letter to NASD, June 21, 1985)
15c3-1(c)(11)(i)/04 Ohio Dealer Data Service, Inc. - Not a Ready Market
The Ohio Dealer Data Service, Inc. is not recognized as an established securities market nor does it qualify as an inter-dealer quotation system under (c)(2)(vi)(K).
(SEC Letter to Pierre R. Smith & Co., August 19, 1986)
(NYSE Interpretation Memo 88-20, December 1988)
15c3-1(c)(11)(i)/05 Ready Market of Commercial Paper
See interpretation 15c3-1(c)(2)(vii)/06 for ready market criteria.
15c3-1(c)(11)(i)/06 Ready Market of Commercial Paper under Section 936 Market
See interpretation 15c3-1(c)(2)(vii)/07 for ready market criteria.
15c3-1(c)(11)(ii) A ready market shall also be deemed to exist where securities have been accepted as collateral for a loan by a bank as defined in section 3(a)(6) of the Securities Exchange Act of 1934 and where the broker or dealer demonstrates to its Examining Authority that such securities adequately secure such loans as that term is defined in paragraph (c)(5) of this section.
15c3-1(c)(11)(ii)/001 See requirements under NYSE Rule 328(c) and NYSE Information Memo No. 80-66, dated December 31, 1980.
(NYSE Information Memo 80-66, December 31, 1980)
15c3-1(c)(11)(ii)/01 Securities Collateralizing Bank Loan
Bank loan collateral may be treated as having a ready market for purposes of this subsection only if the securities are actually adequately collateralizing an outstanding bank loan.
(SEC Letter to Lex Jolly & Co. Inc., May 15, 1976)
(NYSE Interpretation Memo 77-2, June 1977)
15c3-1(c)(11)(ii)/011 Value to be Included
Where “ready market” is based on collateral value of securities collateralizing a bank loan, the amount to be included will be the lower of the amount of the loan or fair market value of the security less the applicable haircut.
(SEC Staff to NYSE) (NYSE Interpretation Memo 86-8, August 1986)
15c3-1(c)(11)(ii)/02 Repurchase Agreements
Sale of securities under a repurchase agreement will not serve to establish a “ready market” under this provision.
(SEC Staff to NYSE) (NYSE Interpretation Memo 81-3, July 1981)
15c3-1(c)(11)(ii)/03 Non-Transferable or Restricted Securities
Securities which cannot be publicly offered or sold because of statutory, regulatory or contractual arrangements or other restrictions may not be considered as having a “ready market” under this provision. The securities must be fully transferable.
(SEC Staff to NYSE) (NYSE Interpretation Memo 81-3, July 1981)
15c3-1(c)(11)(ii)/04 Net Capital Treatment of Securities Positions In Suspended Securities
The net capital treatment for suspended securities positions, transactions, and obligations are as follows:
Long proprietary positions and collateral held for Secured Demand Note(s) are non-allowable assets.
Short proprietary positions are to be valued at the last sale prior price prior to suspension. The broker-dealer shall reduce its net capital by the applicable haircut deduction (the haircut percentage that would have been applied prior to the suspension) on the presumed market value of the short proprietary position.
Fails to Receive and uncompleted customer's sale transactions shall be valued at the original contract price.
Fails to Deliver are non-allowable assets, until collected or the suspension is lifted.
(SEC Letter to NASD, June 8, 1973) (SEC Release 10209)
15c3-1(c)(12) Examining Authority
The term Examining Authority of a broker or dealer shall mean for the purposes of 17 CFR 240.15c3-1 and 240.15c3-1a-d the national securities exchange or national securities association of which the broker or dealer is a member or, if the broker or dealer is a member of more than one such self-regulatory organization, the organization designated by the Commission as the Examining Authority for such broker or dealer, or if the broker or dealer is not a member of any such self-regulatory organization, the Regional Office of the Commission where such broker or dealer has its principal place of business.
15c3-1(c)(13) Entities That Have a Principal Regulator
15c3-1(c)(13)(i) For purposes of § 240.15c3-1e and § 240.15c3-1g, the term entity that has a principal regulator shall mean a person (other than a natural person) that is not a registered broker or dealer (other than a broker or dealer registered under section 15(b)(11) of the Act (15 U.S.C. 78o(b)(11)), provided that the person is:
15c3-1(c)(13)(i)(A) An insured depository institution as defined in section 3(c)(2) of the Federal Deposit Insurance Act (12 U.S.C. 1813(c)(2));
15c3-1(c)(13)(i)(B) Registered as a futures commission merchant or an introducing broker with the Commodity Futures Trading Commission;
15c3-1(c)(13)(i)(C) Registered with or licensed by a State insurance regulator and issues any insurance, endowment, or annuity policy or contract;
15c3-1(c)(13)(i)(D) A foreign bank as defined in section 1(b)(7) of the International Banking Act of 1978 (12 U.S.C. 3101(7)) that has its headquarters in a jurisdiction for which any foreign bank has been approved by the Board of Governors of the Federal Reserve System to conduct business pursuant to the standards set forth in 12 CFR 211.24(c), provided such foreign bank represents to the Commission that it is subject to the same supervisory regime as the foreign bank previously approved by the Board of Governors of the Federal Reserve System;
15c3-1(c)(13)(i)(E) Not primarily in the securities business, and the person is:
15c3-1(c)(13)(i)(E)(1) A corporation organized under section 25A of the Federal Reserve Act (12 U.S.C. 611 through 633); or
15c3-1(c)(13)(i)(E)(2) A corporation having an agreement or undertaking with the Board of Governors of the Federal Reserve System under section 25 of the Federal Reserve Act (12 U.S.C. 601 through 604a); or
15c3-1(c)(13)(i)(F) A person that the Commission finds is another entity that is subject to comprehensive supervision, has in place appropriate arrangements so that information that the person provides to the Commission is sufficiently reliable for the purposes of determining compliance with § 240.15c3-1e and § 240.15c3-1g, and it is appropriate to consider the person to be an entity that has a principal regulator considering all relevant circumstances, including the person's mix of business.
15c3-1(c)(13)(ii) For purposes of §§ 240.15c3-1e, 240.15c3-1g, 240.17h-1T, and 240.17h2T, the term ultimate holding company that has a principal regulator shall mean a person (other than a natural person) that:
15c3-1(c)(13)(ii)(A) Is a financial holding company or a company that is treated as a financial holding company under the Bank Holding Company Act of 1956 (12 U.S.C. 1840 et seq.), or
15c3-1(c)(13)(ii)(B) The Commission determines to be an ultimate holding company that has a principal regulator, if that person is subject to consolidated, comprehensive supervision; there are in place appropriate arrangements so that information that the person provides to the Commission is sufficiently reliable for the purposes of determining compliance with § 240.15c3-1e and § 240.15c3-1g; and it is appropriate to consider the person to be an ultimate holding company that has a principal regulator in view of all relevant circumstances, including the person's mix of business.
15c3-1(c)(14) The term municipal securities shall mean those securities included within the definition of “municipal securities” in section 3(a)(29) of the Securities Exchange Act of 1934.
15c3-1(c)(15) The term tentative net capital shall mean the net capital of a broker or dealer before deducting the securities haircuts computed pursuant to paragraph (c)(2)(vi) of this section and the charges on inventory computed pursuant to appendix B to this section (§ 240.15c3-1b). However, for purposes of paragraph (a)(5) of this section, the term tentative net capital means the net capital of an OTC derivatives dealer before deducting the charges for market and credit risk as computed pursuant to appendix F to this section (§ 240.15c3-1f) or paragraph (c)(2)(vi) of this section, if applicable, and increased by the balance sheet value (including counterparty net exposure) resulting from transactions in eligible OTC derivative instruments which would otherwise be deducted by virtue of paragraph (c)(2)(iv) of this section. For purposes of paragraph (a)(7) of this section, the term tentative net capital means the net capital of the broker or dealer before deductions for market and credit risk computed pursuant to § 240.15c3-1e or paragraph (c)(2)(vi) of this section, if applicable, and increased by the balance sheet value (including counterparty net exposure) resulting from transactions in derivative instruments which would otherwise be deducted by virtue of paragraph (c)(2)(iv) of this section. Tentative net capital shall include securities for which there is no ready market, as defined in paragraph (c)(11) of this section, if the use of mathematical models has been approved for purposes of calculating deductions from net capital for those securities pursuant to § 240.15c3-1e.
15c3-1(c)(16) Insolvent
For the purposes of this section, a broker or dealer is insolvent if the broker or dealer:
15c3-1(c)(16)(i) Is the subject of any bankruptcy, equity receivership proceeding or any other proceeding to reorganize, conserve, or liquidate such broker or dealer or its property or is applying for the appointment or election of a receiver, trustee, or liquidator or similar official for such broker or dealer or its property;
15c3-1(c)(16)(ii) Has made a general assignment for the benefit of creditors;
15c3-1(c)(16)(iii) Is insolvent within the meaning of section 101 of title 11 of the United States Code, or is unable to meet its obligations as they mature, and has made an admission to such effect in writing or in any court or before any agency of the United States or any State; or
15c3-1(c)(16)(iv) Is unable to make such computations as may be necessary to establish compliance with this section or with § 240.15c3-3.
15c3-1(c)(17) The term risk margin amount means the sum of:
15c3-1(c)(17)(i) The total initial margin required to be maintained by the broker or dealer at each clearing agency with respect to security-based swap transactions cleared for security-based swap customers; and
15c3-1(c)(17)(ii) The total initial margin amount calculated by the broker or dealer with respect to non-cleared security-based swaps pursuant to § 240.18a-3(c)(1)(i)(B).
15c3-1(d) Debt-equity requirements. No broker or dealer shall permit the total of outstanding principal amounts of its satisfactory subordination agreements (other than such agreements which qualify under this paragraph (d) as equity capital) to exceed 70 percent of its debt-equity total, as hereinafter defined, for a period in excess of 90 days or for such longer period which the Commission may, upon application of the broker or dealer, grant in the public interest or for the protection of investors. In the case of a corporation, the debt-equity total shall be the sum of its outstanding principal amounts of satisfactory subordination agreements, par or stated value of capital stock, paid in capital in excess of par, retained earnings, unrealized profit and loss or other capital accounts. In the case of a partnership, the debt-equity total shall be the sum of its outstanding principal amounts of satisfactory subordination agreements, capital accounts of partners (exclusive of such partners' securities accounts) subject to the provisions of paragraph (e) of this section, and unrealized profit and loss. In the case of a sole proprietorship, the debt-equity total shall include the sum of its outstanding principal amounts of satisfactory subordination agreements, capital accounts of the sole proprietorship and unrealized profit and loss. Provided, however, That a satisfactory subordination agreement entered into by a partner or stockholder which has an initial term of at least three years and has a remaining term of not less than 12 months shall be considered equity for the purposes of this paragraph (d) if:
15c3-1(d)(1) It does not have any of the provisions for accelerated maturity provided for by paragraphs (b)(9)(i), (10)(i) or (10)(ii) of Appendix (D) (17 CFR 240.15c3-1d) and is maintained as capital subject to the provisions restricting the withdrawal thereof required by paragraph (e) of this section or
15c3-1(d)(2) The partnership agreement provides that capital contributed pursuant to a satisfactory subordination agreement as defined in Appendix (D) (17 CFR 240.15c3-1d) shall in all respects be partnership capital subject to the provisions restricting the withdrawal thereof required by paragraph (e) of this section.
15c3-1(d)/01 Debt-Equity Subordination Agreements
The debt-equity total includes satisfactory subordination agreements as defined in SEA Rule 15c3-1d (Appendix D), and net worth computed under generally accepted accounting principles adjusted by unrealized profits and losses (if this adjustment has not already been made), and by the write-off of the time value of any unlisted option, long or short. (The time value is the excess of the unamortized cost or proceeds of the option over the in the money amount.)
(SEC Staff to NYSE)
15c3-1(d)/02 Partners' Subordinations
A contribution under a satisfactory subordination agreement which is to be considered as capital under a partnership agreement does not have to have an initial term of at least three years and a remaining term of not less than one year for it to be considered equity. However, if the subordination agreement is separate from the partnership agreement, the time limits will apply and may not be satisfied by automatic rollover provisions.
(SEC Staff to NYSE) (NYSE Interpretation Memo 79-4, March 1979)
15c3-1(d)/03 Equity Subordination
The following are all the conditions which must be met for a subordinated liability to be considered as equity in the debt-equity total:
Lender is a partner or stockholder,
Subordination agreement has initial term of 3 years,
Subordination agreement has remaining term of l year,
Lender cannot accelerate the maturity of the liability as described in SEA Rule 15c3-1d(b)(9)(i) and (b)(10) (Appendix D),
Liability is subject to the withdrawal restrictions of Rule 15c3-1(e), and
It is in all other respects a satisfactory subordination as defined in SEA Rule 15c3-1d (Appendix D).
For satisfactory subordination agreements contributed as capital under a partnership agreement see interpretation 15c3-1(d)/02 above.
A subordination agreement entered into by an officer of a corporation may not be considered as equity irrespective of maturity, unless the officer is a stockholder (regardless of size of holding).
Satisfactory subordination agreements entered into by banks, institutions and vendors are not considered equity for the purposes of the debt-equity requirement. They are considered debt.
(SEC Staff to NASD) (NYSE Interpretation Memo 77-2, June 1977)
15c3-1(d)/031 Limited Partner Subordination
Limited partner subordinations may be included as equity in computing debt to equity ratio provided the conditions specified for such treatment in subparagraph (d) are satisfied. However, there should be assurance that the limited partner is adequately informed as to the operational and financial condition of the business.
(SEC Staff to NYSE) (NYSE Interpretation Memo 88-20, November 1988)
15c3-1(d)/04 Death or Retirement
A subordinated loan contributed by a partner or stockholder for an initial term of three years and a remaining term of more than one year should, upon his death or retirement, continue to be treated as equity. It may be treated as equity even in the last year of its term provided it is to be renewed. Once it is renewed, it is treated as debt.
15c3-1(d)/05 Non-conforming Subordinations
Liabilities which are effectively subordinated to the claims of creditors but which are not subject to a satisfactory subordination agreement are disregarded for purposes of the debt-equity ratio.
(SEC Staff to NYSE) (NYSE Interpretation Memo 76-4, April 1976)
15c3-1(d)/06 Discretionary Liabilities
See interpretation 15c3-1(c)(2)/02.
15c3-1(e)(1) Notice provisions relating to limitations on the withdrawal of equity capital. No equity capital of the broker or dealer or a subsidiary or affiliate consolidated pursuant to appendix C (17 CFR 240.15c3-1c) may be withdrawn by action of a stockholder or a partner or by redemption or repurchase of shares of stock by any of the consolidated entities or through the payment of dividends or any similar distribution, nor may any unsecured advance or loan be made to a stockholder, partner, sole proprietor, employee or affiliate without written notice given in accordance with paragraph (e)(1)(iv) of this section:
15c3-1(e)(1)(i) Two business days prior to any withdrawals, advances or loans if those withdrawals, advances or loans on a net basis exceed in the aggregate in any 30 calendar day period, 30 percent of the broker or dealer's excess net capital. A broker or dealer, in an emergency situation, may make withdrawals, advances or loans that on a net basis exceed 30 percent of the broker or dealer's excess net capital in any 30 calendar day period without giving the advance notice required by this paragraph, with the prior approval of its Examining Authority. Where a broker or dealer makes a withdrawal with the consent of its Examining Authority, it shall in any event comply with paragraph (e)(1)(ii) of this section; or
15c3-1(e)(1)(ii) Two business days after any withdrawals, advances or loans if those withdrawals, advances or loans on a net basis exceed in the aggregate in any 30 calendar day period, 20 percent of the broker or dealer's excess net capital.
15c3-1(e)(1)(iii) This paragraph (e)(1) does not apply to:
15c3-1(e)(1)(iii)(A) Securities or commodities transactions in the ordinary course of business between a broker or dealer and an affiliate where the broker or dealer makes payment to or on behalf of such affiliate for such transaction and then receives payment from such affiliate for the securities or commodities transaction within two business days from the date of the transaction; or
15c3-1(e)(1)(iii)(B) Withdrawals, advances or loans which in the aggregate in any thirty calendar day period, on a net basis, equal $500,000 or less.
15c3-1(e)(1)(iv) Each required notice shall be effective when received by the Commission in Washington, DC, the regional office of the Commission for the region in which the broker or dealer has its principal place of business, the broker or dealer's Examining Authority and the Commodity Futures Trading Commission if such broker or dealer is registered with that Commission.
15c3-1(e)(2) Limitations on Withdrawal of equity capital. No equity capital of the broker or dealer or a subsidiary or affiliate consolidated pursuant to appendix C (17 CFR 240.15c3-1c) may be withdrawn by action of a stockholder or a partner or by redemption or repurchase of shares of stock by any of the consolidated entities or through the payment of dividends or any similar distribution, nor may any unsecured advance or loan be made to a stockholder, partner, sole proprietor, employee or affiliate, if after giving effect thereto and to any other such withdrawals, advances or loans and any Payments of Payment Obligations (as defined in appendix D (17 CFR 240.15c3-1d)) under satisfactory subordination agreements which are scheduled to occur within 180 days following such withdrawal, advance or loan if:
15c3-1(e)(2)(i) The broker or dealer's net capital would be less than 120 percent of the minimum dollar amount required by paragraph (a) of this section;
15c3-1(e)(2)(ii) The broker-dealer is registered as a futures commission merchant, its net capital would be less than 7 percent of the funds required to be segregated pursuant to the Commodity Exchange Act and the regulations thereunder (less the market value of commodity options purchased by option customers on or subject to the rules of a contract market, each such deduction not to exceed the amount of funds in the option customer's account);
15c3-1(e)(2)(iii) The broker-dealer's net capital would be less than 25 percent of deductions from net worth in computing net capital required by paragraphs (c)(2)(vi), (f) and appendix A, of this section, unless the broker or dealer has the prior approval of the Commission to make such withdrawal;
15c3-1(e)(2)(iv) The total outstanding principal amounts of satisfactory subordination agreements of the broker or dealer and any subsidiaries or affiliates consolidated pursuant to appendix C (17 CFR 240.15c3-1c) (other than such agreements which qualify as equity under paragraph (d) of this section) would exceed 70% of the debt-equity total as defined in paragraph (d) of this section;
15c3-1(e)(2)(v) The broker or dealer is subject to the aggregate indebtedness limitations of paragraph (a) of this section, the aggregate indebtedness of any of the consolidated entities exceeds 1000 percent of its net capital; or
15c3-1(e)(2)(vi) The broker or dealer is subject to the alternative net capital requirement of paragraph (f) of this section, its net capital would be less than 5 percent of aggregate debit items computed in accordance with 17 CFR 240.15c3-3a.
15c3-1(e)(3)(i) Temporary restrictions on withdrawal of net capital. The Commission may by order restrict, for a period of up to twenty business days, any withdrawal by the broker or dealer of equity capital or unsecured loan or advance to a stockholder, partner, sole proprietor, member, employee or affiliate under such terms and conditions as the Commission deems necessary or appropriate in the public interest or consistent with the protection of investors if the Commission, based on the information available, concludes that such withdrawal, advance or loan may be detrimental to the financial integrity of the broker or dealer, or may unduly jeopardize the broker or dealer's ability to repay its customer claims or other liabilities which may cause a significant impact on the markets or expose the customers or creditors of the broker or dealer to loss without taking into account the application of the Securities Investor Protection Act of 1970.
15c3-1(e)(3)(ii) An order temporarily prohibiting the withdrawal of capital shall be rescinded if the Commission determines that the restriction on capital withdrawal should not remain in effect. A hearing on an order temporarily prohibiting the withdrawal of capital will be held within two business days from the date of the request in writing by the broker or dealer.
15c3-1(e)(4)(i) Miscellaneous provisions. Excess net capital is that amount in excess of the amount required under paragraph (a) of this section. For the purposes of paragraphs (e)(1) and (e)(2) of this section, a broker or dealer may use the amount of excess net capital and deductions required under paragraphs (c)(2)(vi), (f) and appendix A of this section reported in its most recently required filed Form X-17A-5 for the purposes of calculating the effect of a projected withdrawal, advance or loan relative to excess net capital or deductions. The broker or dealer must assure itself that the excess net capital or the deductions reported on the most recently required filed Form X-17A-5 have not materially changed since the time such report was filed.
15c3-1(e)(4)(ii) The term equity capital includes capital contributions by partners, par or stated value of capital stock, paid-in capital in excess of par, retained earnings or other capital accounts. The term equity capital does not include securities in the securities accounts of partners and balances in limited partners' capital accounts in excess of their stated capital contributions.
15c3-1(e)(4)(iii) Paragraphs (e)(1) and (e)(2) of this section shall not preclude a broker or dealer from making required tax payments or preclude the payment to partners of reasonable compensation, and such payments shall not be included in the calculation of withdrawals, advances, or loans for purposes of paragraphs (e)(1) and (e)(2) of this section.
15c3-1(e)(4)(iv) For the purpose of this paragraph (e) of this section, any transaction between a broker or dealer and a stockholder, partner, sole proprietor, employee or affiliate that results in a diminution of the broker or dealer's net capital shall be deemed to be an advance or loan of net capital.
15c3-1(e)/01 Services Arrangement with a Parent or an Affiliate
Any payment made by a broker-dealer, directly or indirectly, to its parent or an affiliate, in connection with an arrangement whereby the parent or affiliate provides services to the broker-dealer (“services arrangement”) shall be considered a capital withdrawal for purposes of SEA Rules 15c3-1(c)(2)(i)(G) and 15c3-1(e), unless the broker-dealer demonstrates that the following conditions are met:
at the time the service(s) were provided to the broker-dealer, the services arrangement was in writing and specified such service(s) with a reasonable and consistent basis for determining the cost of each service (e.g., utilizing a percentage of the broker-dealer’s net income to determine the cost to be charged by a parent or affiliate for technology services provided by a parent or affiliate, for example, may not be deemed “a reasonable basis” because the cost of obtaining such services generally does not fluctuate based on the level of a broker-dealer’s net income);
the service(s) provided were related to the broker-dealer’s business;
the parent or affiliate had the ability to provide such service(s); and
the parent or affiliate provided the service(s).
For purposes of this interpretation, “payment” shall include any reduction or forgiveness of a receivable from the broker-dealer’s parent or affiliate.
(SEC Staff to FINRA) (FINRA Regulatory Notice 21-27) |
12949 | https://stackoverflow.com/questions/58370577/seating-arrangement-problem-in-a-circular-table | python 3.x - Seating arrangement problem in a circular table - Stack Overflow
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Seating arrangement problem in a circular table [closed]
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Asked 5 years, 11 months ago
Modified2 years, 11 months ago
Viewed 4k times
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N people sit around a circular table. You have to find the probability that two particular people won't be sitting together.
The input will have the number N and the output should have the probability printed as a float type number rounded off to four decimal places.
python-3.x
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asked Oct 14, 2019 at 5:29
Arjun MurthyArjun Murthy
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2 What have you tried so far? meta.stackoverflow.com/questions/334822/…blueenvelope –blueenvelope 2019-10-14 05:38:07 +00:00 Commented Oct 14, 2019 at 5:38
Have understood the theoretical logic and the mathematical derivation to land upto this solution. However, I am unable to conclude with a program stub.Arjun Murthy –Arjun Murthy 2019-10-14 05:44:02 +00:00 Commented Oct 14, 2019 at 5:44
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Here's the link for the derived formula
You can find the step by step derivation over there
Here's the simple python implementation as per the thread
python
n = 5
result = (n-3)/(n-1)
print(result)
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answered Oct 14, 2019 at 6:42
saintlyzerosaintlyzero
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python
n= int(input())
import math
print(round(1-math.factorial(n-2)math.factorial(2)/math.factorial(n-1),4))
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edited Oct 16, 2022 at 10:44
Adrian Mole
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answered Oct 11, 2022 at 7:43
KaranKaran
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Adrian Mole
Adrian MoleOver a year ago
While this code may solve the question, including an explanation of how and why this solves the problem would really help to improve the quality of your post, and probably result in more up-votes. Remember that you are answering the question for readers in the future, not just the person asking now. Please edit your answer to add explanations and give an indication of what limitations and assumptions apply.
2022-10-16T10:45:13.42Z+00:00
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12950 | https://www.effortlessmath.com/wp-content/uploads/2020/04/Simplifying-Ration.pdf?srsltid=AfmBOooXj11x4RXUNcWgZcmN92srkbdVEBCpKAJ6Th0FbDcfsZVMNjiP | Math Worksheets Name: ____
Date: _____
… So Much More Online! Please visit: www.EffortlessMath.com
Simplifying Ratios
Reduce each ratio.
1) 12 : 8 = _: _
2) 2: 20 = _: _
3) 3: 36 = _: _
4) 8: 16 = _: _
5) 6: 100 = _: _
6) 10 : 60 = _: _
7) 21 : 49 = _: _
8) 20 : 40 = _: _
9) 10 : 50 = _: _
10) 14 : 18 = _: _
11) 45 : 27 = _: _
12) 49 : 21 = _: _
13) 100 : 10 = _: _
14) 35 : 45 = _: _
15) 8: 20 = _: _
16) 25 : 35 = _: _
17) 21 : 27 = _: _
18) 52 : 82 = _: _
19) 12 : 36 = _: _
20) 24 : 3 = _: _
21) 15 : 30 = _: _
22) 14 : 63 = _: _
23) 68 : 80 = _: _
24) 8: 80 = _: _
Write each ratio as a fraction in simplest form.
25) 2: 4 =
26) 6: 20 =
27) 5: 35 =
28) 10 : 55 =
29) 8: 24 =
30) 9: 42 =
31) 12 : 48 =
32) 6: 40 =
33) 15 : 36 =
34) 18 : 82 =
35) 22 : 26 =
36) 8: 36 =
37) 16 : 128 =
38) 14 : 77 =
39) 12 : 180 =
40) 36 : 108 =
41) 24 : 42 =
42) 18 : 120 =
43) 44 : 82 =
44) 60 : 240 =
45) 36 : 180 =Math Worksheets Name: ____
Date: _____
… So Much More Online! Please visit: www.EffortlessMath.com
Answers
1) 3 ∶ 2
2) 1 ∶ 10
3) 1 ∶ 12
4) 1 ∶ 2
5) 3 ∶ 50
6) 1 ∶ 6
7) 3 ∶ 7
8) 1 ∶ 2
9) 1 ∶ 5
10) 7 ∶ 9
11) 5 ∶ 3
12) 7 ∶ 3
13) 10 ∶ 1
14) 7 ∶ 9
15) 2 ∶ 5
16) 5 ∶ 7
17) 7 ∶ 9
18) 26 ∶ 41
19) 1 ∶ 3
20) 8 ∶ 1
21) 1 ∶ 2
22) 2 ∶ 9
23) 17 ∶ 20
24) 1 ∶ 10
25) 1
2
26) 3
10
27) 1
7
28) 2
11
29) 1
3
30) 3
14
31) 1
4
32) 3
20
33) 5
12
34) 9
41
35) 11
13
36) 2
9
37) 1
8
38) 2
11
39) 1
15
40) 1
3
41) 4
7
42) 3
20
43) 22
41
44) 1
4
45) 1 |
12951 | https://www.chemteam.info/AcidBase/Buffer-Probs21-to-30.html | Buffers and the Henderson-Hasselbalch EquationProblems #21 - 30
| | | | |
--- --- |
| Fifteen Buffer Examples | Buffer Problems 1-10 | Buffer Problems 11-20 | Buffer Problems 31-40 |
| Intro. to the Henderson-Hasselbalch Equation | Return to the Acid Base menu |
Problem #21: What volume of 6.00 M NaOH must be added to 0.250 L of 0.300 M HNO2 to prepare a pH = 4.00 buffer?
1) The first thing to do is look up the Ka for nitrous acid, to find:
4.0 x 10¯4
Several different values can be found. I selected the one above since it seemed more common than the others.
2) Some comments on the chemistry involved:
We know that this reaction will take place:
HNO2 + OH¯ ---> H2O + NO2¯
The hydroxide MUST be the limiting reagent. Why? If some NaOH was left over, the solution would be a mixture of a strong base (the NaOH) and some NaNO2 (the salt of a weak acid). This is NOT a buffer and the pH would be calculated using the concept of a strong base.
So the final solution will be a mixture of HNO2 and NO2¯. Since this solution is a buffer, the Henderson-Hasselbalch equation will be employed.
3) Filling in the H-H, step 1:
pH = pKa + log [base / acid]
We know we must have a buffer of pH = 4.00:
4.00 = pKa + log [base / acid]
4) Filling in the H-H, step 2:
4.00 = 3.40 + log [base / acid]
3.40 is the pKa for HNO2.
5) Filling in the H-H, step 3:
I'm going to give you the log portion and then comment on it:
| | |
--- |
| | x |
| 4.00 = 3.40 + log | –––––––––– |
| | 0.0750 − x |
The unknown 'x' is the amount of NO2¯ that got produced by the HNO2 reacting with the OH¯.
The 0.0750 part of (0.0750 − x) comes from this calculation:
(0.300 mol/L) (0.250 L) = 0.0750 mol
and the 'x' is the moles of HNO2 that reacted. The amount of HNO2 lost is equal to the amount of NO2¯ gained.
6) Algebra!
4.00 = 3.40 + log [x / (0.0750 − x)]
log [x / (0.0750 − x)] = 0.60
x / (0.0750 − x) = 3.98
x = 0.2985 − 3.98x
4.98x = 0.2985
x = 0.05994 moles <--- don't round off yet
By the 1:1 molar ratio of the overall reaction, that's the moles of hydroxide that need to be added.
7) Determine the volume of sodium hydroxide solution required:
0.05994 mol / 6.00 mol/L = 0.00999 L
Based on the Ka value, 2 sig figs seems best ---> 10. mL
Problem #22: If an acetate buffer solution was going to be prepared by neutralizing HC2H3O2 with 0.10 M NaOH, what volume (in mL) of 0.10 M NaOH would need to be added to 10.0 mL of 0.10 M HC2H3O2 to prepare a solution with pH = 5.50?
Solution:
Comment: In doing the salt (sodium acetate) and the acid (acetic acid), I'm going to use moles rather than molarity. Since everything occurs in the same volume of solution, the ratio of salt moles to acid moles is the same as the ratio of molarities. Besides, we don't know the final molarities since we are adding an unknown volume of NaOH solution.
1) We need to know the initial moles of acetic acid in the solution:
(0.010 L) (0.1 mol/L) = 0.001 mol
2) Let us insert values into the H-H equation:
pH = pKa + log [base / acid]
5.50 = 4.752 + log [x / (0.001 − x)]
4.752 is the pKa of acetic acid x is the moles of sodium acetate produced by the NaOH reacting 0.001 − x is the amount of acetic acid remaining in solution.
The moles of acetate will give us moles of NaOH since there is a 1:1 molar ratio between the two.
3) Continue solving:
log (x / 0.001 − x) = 0.748
4) Antilog both sides
(x / 0.001 − x) = 5.598
5) Cross multiply & simplify to get:
6.598x = 5.598 x 10¯3
x = 8.5 x 10¯4 moles
6) Let us determine the volume of NaOH required:
8.5 x 10¯4 mol divided by 0.1 mol/L = 8.5 x 10¯3 L = 8.5 mL
Problem #23: A beaker with 175 mL of an acetic acid buffer with a pH of 5.000 is sitting on a benchtop. The total molarity of acid and conjugate base in this buffer is 0.100 M. A student adds 8.40 mL of a 0.300 M HCl solution to the beaker. What is the new pH? The pKa of acetic acid is 4.752.
Solution:
1) Use the Henderson-Hasselbalch to get the molarities of the base and the acid:
pH = pKa + log [base / acid]
5.000 = 4.752 + log [x / (0.1 − x)]
Note that x and 0.1 − x add up to 0.1, which is the total molarity
log [x / (0.1 − x)] = 0.248
[x / (0.1 − x)] = 1.7701
x = 0.17701 − 1.7701x
2.7701x = 0.17701
x = 0.0639 M <--- that's the acetate concentration in the pH 5 buffer
0.1 − 0.0639 = 0.0361 <--- that's the acetic acid concentration
2) Now, we need to know the moles of acetic acid, acetate and HCl:
acetate ---> (0.0639 mol/L) (0.175 L) = 0.0111825 mol acetic acid ---> (0.0361 mol/L) (0.175 L) = 0.0063175 mol HCl ---> (0.300 mol/L) (0.00840 L) = 0.00252 mol
3) The HCl will react with the acetate and turn it into acetic acid. The acetate amount goes down and the acetic acid goes up. Note that all the reaction stoichiometries are 1:1.
acetate ---> 0.0111825 mol − 0.00252 mol = 0.0086625 mol acetic aid ---> 0.0063175 mol + 0.00252 mol = 0.0088375 mol
4) Now, for the HH equation again:
pH = 4.752 + log (0.0086625 / 0.0088375)
pH = 4.752 + log 0.98019802
pH = 4.752 + (−0.009)
pH = 4.743
Problem #24: 200.0 mL of an acetate/acetic acid buffer is 0.100 M in total molarity and has a pH of 5.000. After 6.30 mL of 0.490 M HCl is added, what is the new pH?
Solution:
1) We ned to know the amount of acetic acid (HAc) and acetate ion (Ac¯) in the pH 5 buffer:
pH = pKa + log [base / acid]
5.000 = 4.752 + log [x / (0.02 − x)]
0.02 comes from:
MV = (0.100 mol/L) (0.2000 L) = 0.0200 mol
log [x / (0.02 − x)] = 0.248
x / (0.02 − x) = 1.770109
x = 0.03540218 − 1.770109x
2.770109x = 0.03540218
x = 0.01278 mol <--- that's the acetate moles
0.02 − 0.01278 = 0.00722 mol <--- that's the acetic acid moles
2) Determine the moles of HCl to be added:
MV = (0.490 mol/L) (0.00630 L) = 0.003087 mol
3) The HCl will protonate the Ac¯, causing its amount to decrease and the HAc amount to increase.
Ac¯ ---> 0.01278 − 0.003087 = 0.009693 mol HAc ---> 0.00722 + 0.003087 = 0.010307 mol
4) Use the Henderson-Hasselbalch Equation to calculate the new pH:
pH = pKa + log [base / acid]
pH = 4.752 + log [0.009693 / 0.010307]
pH = 4.752 + [−0.027]
pH = 4.725
Problem #25a: We desire to make a pH 5.000 buffer and we choose a weak acid (let's call it HA) with a pKa of 4.700. Starting with 0.100 M each HA and NaA, we desire to make 100. mL buffer solution.
Solution:
1) Use the Henderson-Hasselbalch equation:
5.000 = 4.700 + log [A¯] / [HA]
[A¯] / [HA] = 100.300
[A¯] / [HA] = 2.00
2) Use the definition of molarity:
M = moles / volume
moles = MV
moles of A¯ = (0.100 mol / L) (LA¯) moles of HA = (0.100 mol / L) (LHA)
3) Our ratio now becomes:
[(0.100 mol/L) (LA¯)] / [(0.100 mol/L) (LHA)] = 2.00
(LA¯) / (LHA) = 2.00
4) Set variables and substitute into above ratio:
let LA¯ = x therefore LHA = 0.1 − x
Comment: I used 0.1 because total volume = 100. mL or 0.1 L.
x / (0.1 − x) = 2.00
x = 0.2 − 2x
3x = 0.2
x = 0.667 L
We require 66.7 mL of NaA and 33.3 mL of HA to make our pH 5 buffer.
Problem #25b: Determine how you would prepare 1.00 L of this buffer starting with 0.100 M HA, 0.100 M NaOH and water where the total concentration of HA plus NaA is 0.010 M.
Solution:
pH = pKa + log [base / acid]
The total moles of HA and NaA will be 0.010 mol. This comes from the total molarity (0.0100 M) times the final volume of the solution (1.00 L). I will use moles in the log portion of the Henderson-Hasselbalch Equation.
5.000 = 4.700 + log [x / (0.01 − x)]
0.300 = log [x / (0.01 − x)]
x / (0.01 − x) = 2
x = 0.02 − 2x
x = 0.00667 mole of the base (the A¯)
Comments:
(a) x represents the moles of the NaA (the salt) in the 1.00 L of solution (b) 0.010 − x = 0.00333; this is the moles of HA in 1.0 L of solution.
How to prepare the buffer:
| | |
--- |
| (i) | Take 100. mL of 0.100 M HA. This represents 0.0100 mole of HA. |
| (ii) | Add 66.7 mL of 0.100 M NaOH solution. This is 0.0067 mole of NaOH. The NaOH reacts with the HA to form NaA, the salt of HA. |
| (iii) | Dilute to 1.00 L with water. |
Problem #26: Calculate the volume (in mL) of 0.170 M NaOH that must be added to 311 mL of 0.0485 M HA (a generic weak acid) to give the solution a pH of 7.55. The pKa of HA = 7.18.
Solution:
1) Since the moles of HA and the salt formed from HA/NaOH (for which I will use A¯) are in the same volume of solution, we can use moles in the log portion of the Henderson-Hasselbalch equation:
7.55 = 7.18 + log (A¯) / (HA − A¯)
2) The moles of A¯ will be the unknown:
moles HA ---> (0.0485 mol/L) (0.311 L) = 0.0150835 mol
3) I'll use the unrounded off number.
7.55 = 7.18 + log [(x) / (0.0150835 − x)]
0.37 = log (x) / (0.0150835 − x)
4) antilog both sides
(x) / (0.0150835 − x) = 2.34423
5) cross multiply
x = 0.035359 − 2.34423x
3.34423x = 0.035359
x = 0.0105732 moles of A¯ required
6) mL of NaOH required:
0.0105732 mol / 0.170 mol/L = 0.0621953 L = 62.2 mL (to three sig figs)
Problem #27: What mass of HCl would need to be added to a 250. mL solution containing 0.500 M NaC2H3O2 and 0.500 M HC2H3O2, to make the pH = 4.25? Ka of HC2H3O2 is 1.77 x 10-5.
Solution with moles:
pH = pKa + log [base / acid]
4.25 = 4.752 + log [base / acid]
−0.502 = log [base / acid]
[base/acid] = 0.314775 <--- this is the base/acid ratio needed to create a pH of 4.25
(0.125 − x) / (0.125 + x) = 0.314775 <--- note use of total moles in solution as opposed to molarity
0.039346875 + 0.314775x = 0.125 − x
1.314775x = 0.085653125
x = 0.0651466 mol
(0.0651466 mol) (36.5 g/mol) = 2.38 g
Solution with molarities:
(0.5 − x) / (0.5 + x) = 0.314775
0.1573875 + 0.314775x = 0.5 − x
1.314775x = 0.3426125
x = 0.2605864 M <--- the molarity of the HCl that must be achieved to create the desired pH of 4.25
MV = g/molar mass
(0.2605864) (0.25) = x / 36.5
x = 2.38 g
Problem #28: What mass of HCl would need to be added to a 250. mL solution containing 0.500 M NaC2H3O2 and 0.500 M HC2H3O2, to make the pH = 4.25? Ka of HC2H3O2 is 1.77 x 10¯5
Solution:
pH = pKa + log [base / acid]
4.25 = 4.752 + log [base / acid]
−0.502 = log [base / acid]
[base / acid] = 0.314775 <--- we need to create a solution with this base/acid ratio to get our pH of 4.25
(0.125 − x) / (0.125 + x) = 0.314775 <--- the 0.125 is the moles of each solute [from (0.5 mol/L) (0.25 L)]
The HCl converts the acetate (the base) into the acid (the acetic acid)
0.04001375 + 0.314775x = 0.125 − x
1.314775x = 0.08498625
x = 0.0646394 mol
(0.0646394 mol) (36.4609 g/mol) = 2.36 g
Problem #29: How many mL of 0.75 M HCl must be added to 120 mL of 0.90 M sodium formate to make a buffer of pH = 4.00? pKa of formic acid = 3.75
Solution:
1) I'll use the Henderson-Hasselbalch equation to solve the problem, but the amounts of acid and base will be expressed in terms of moles, not molarities.
pH = pKa + log [base / acid]
2) Determine moles formate:
(0.90 mol/L) (0.12 L) = 0.108 mol <--- formate is the base in this problem.
3) The HCl will protonate some of the formate, making formic acid. So, the formate amount will go down from 0.108 mol and the formic acid amount will go up from zero.
4.00 = 3.75 + log (0.108 − x) / x
log (0.108 − x) / x = 0.25
(0.108 − x) / x = 1.77828
1.77828x = 0.108 − x
2.77828x = 0.108
x = 0.038873 mol <--- this is the amount of formic acid formed
4) Since HCl reacted and formic acid formed are in a 1:1 molar ratio, we can determine the volume of HCl required:
0.038873 mol / 0.75 mol/L = 0.05183 L = 51.8 mL
5) As a check, let's insert the moles back into the H-H equation as follows:
pH = 3.75 + log (0.069127 / 0.038873)
pH = 3.75 + 0.25 = 4.00
You can also use molarities (by dividing moles by the total volume of 0.1718 L).
Problem #30a: You need to prepare a buffer solution of pH 4.178 from 25.0 mL of 0.282 M solution of a sodium salt of a weak acid, NaA where the pKa of the weak acid HA is 4.270. What volume of 0.329 M HCl would you need to add?
Solution:
1) The Henderson-Hasselbalch Equation is this:
| | |
--- |
| | [base] |
| pH = pKa + log | ––––– |
| | [acid] |
2) We already know two values:
| | |
--- |
| | [base] |
| 4.178 = 4.270 + log | ––––– |
| | [acid] |
3) The anion of the weak acid (the A¯) is the base and HA will be the acid. We know that adding some HCl will turn some of the A¯ into HA. Let's determine how much A¯ we have present:
(0.282 mol/L) (0.025 L) = 0.00705 mol
4) That amount of A¯ is going to go down by some unknown amount when it reacts with the HCl. But, here's the key: the HA amount will go up by the exact same amount. This allows me to fill in the H-H equation:
| | |
--- |
| | 0.00705 − x |
| 4.178 = 4.270 + log | –––––––––– |
| | x |
0.00705 − x ---> that's the amount of A¯ remaining (after all the HCl is used up) x ---> that's the amount of HA produced
5) This whole thing works because of the 1:1 molar ratio between A¯ used up and HA produced. Now, for some algebra:
| | | |
---
| | 0.00705 − x | |
| log | –––––––––– | = −0.092 |
| | x | |
| | |
--- |
| 0.00705 − x | |
| –––––––––– | = 0.809096 |
| x | |
0.00705 − x = 0.809096x
1.809096x = 0.00705
x = 0.00389697 mol <--- the moles of HA needed in the base/acid ratio
6) Because of the 1:1 molar ratio in the chemical reaction, it's also the moles of HCl we need. On to the volume of HCl:
0.00389697 mol / 0.329 mol/L = 0.0118449 L = 11.8449 mL
To three sig figs, the answer is 11.8 mL
Problem #30b: You need to prepare an acetate buffer of pH 5.83 from a 0.642 M acetic acid solution and a 2.31 M KOH solution. If you have 975 mL of the acetic acid solution, how many milliliters of the KOH solution do you need to add to make a buffer of pH 5.830? The pKa of acetic acid is 4.752.
Solution:
1) We will use the Henderson-Hasselbalch equation:
pH = pKa + log [base / acid]
2) Here is the H-H, set up with what we know:
5.830 = 4.752 + log [base / acid]
3) The base and the acid amounts will be expressed in moles. The base, by the way, is the acetate anion, not the KOH.
We do not know how much acetate is required, so we will call it x.
The acetic acid amount is this:
0.62595 − x
The 0.62595 mol comes from this:
(0.642 mol/L) (0.975 L) = 0.62595 mol
The minus x come from the fact that some of the acetic acid will be converted to acetate in a 1:1 molar ratio.
4) We are now ready for the completed H-H:
5.830 = 4.752 + log [x / (0.62595 − x)]
5) Now, for a bit of algebra:
log [x / (0.62595 − x)] = 1.078
x / (0.62595 − x) = 11.9674
x = 7.490994 − 11.9674x
12.9674x = 7.490994
x = 0.57768 mol of acetate required
6) Because of the 1:1 molar ratio between OH¯ consumed and acetate produced, the moles of acetate equals the moles of KOH required.
0.57768 mol divided by 2.31 mol/L = 0.250078 L
0.250078 L = 250. mL (to three sig figs)
Bonus Problem: pKa for phenophthalein is 9.3 at room temp.
(a) Calculate ratio of its anionic form to acid form at pH 8.2 and at pH 10. (b) Using these values, explain the colour change within this pH range.
Solution to part (a):
1) At pH = 8.2:
8.2 = 9.3 + log (base form / acid form)
log (base form / acid form) = −1.1
ratio of base form to acid form = 0.0794 to 1 (call it 8 to 100)
2) At pH = 10.:
10 = 9.3 + log (base form / acid form)
log (base form / acid form) = 0.7
ratio of base form to acid form = 5.01 to 1 (call it 500 to 100)
Solution to part (b):
1) Key fact:
it's the anionic (or base) form that is colored pink. The acid form is colorless.
2) At pH = 8.3:
the pink form is in the minority at this pH. For every 100 acid (colorless) forms, there are only 8 base (pink) forms.
3) At pH = 10.:
the colorless form is in the minority. For every 100 acid (colorless) forms present, there are now 500 base (pink) forms present.
4) This means:
From pH = 8.3 to pH = 10., there has been a 6250% increase in pink forms (from 8:100 to 500:100).
While there might be a slight pink color (viewed against a white background) at pH 8.3, the population of pink forms has greatly increased by pH = 10., to the point where the pink color is now easily seen, even without the aid of a white background.
| | | | |
--- --- |
| Fifteen Buffer Examples | Buffer Problems 1-10 | Buffer Problems 11-20 | Buffer Problems 31-40 |
| Intro. to the Henderson-Hasselbalch Equation | Return to the Acid Base menu | |
12952 | https://math.stackexchange.com/questions/3056563/whats-the-degree-of-a-multivariate-polynomial-in-artin-algebra | algebraic geometry - What's the degree of a multivariate polynomial in Artin Algebra? - Mathematics Stack Exchange
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What's the degree of a multivariate polynomial in Artin Algebra?
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According to Degree of a polynomial, it is highest degree among the monomials. Where is this in Artin Algebra?
In Chapter 11.9, an exercise gives a degree to an irreducible complex polynomial of two variables
In Chapter 11.9, there's a brief discussion of degrees of complex polynomials of two variables in connection with something called Bézout's bound or Bézout's theorem
In Chapter 11.2, I don't think there's an explicit definition of degree of a polynomial in several variables
I thought the exercise in Chapter 11.9 refers to f f's degree in x because I think that f f has positive degree in x x is part of the definition of irreducible in Chapter 11.9, but apparently that may not be the case. There's a line:
Let's assume that the polynomial f is irreducible - that it is not a product of two nonconstant polynomials, and also that it has positive degree in the variable x.
I think it is unclear whether or not "it has positive degree in the variable x." is part of the definition of irreducible.
Here is the context:
Guesses:
We infer it's highest degree among the monomials because we can write the monomials in a polynomial in several variables as x i=x i 1 1…x i n n x i=x 1 i 1…x n i n
I think I read in Cox, O'Shea and Little something about monomial orders like choosing which of the i i's is highest varies by choice of order. Therefore, Artin does not have a definition (at least thus far) for the degree of a multivariate polynomial and thus:
d d in the Chapter 11.9 exercise refers to f f's degree in x and
the discussion after the statement of Theorem 11.9.10 is part of Artin's usual method of discussing terms that are not yet defined, much similar to how he defines a subring before a ring and a subfield before a field.
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asked Dec 30, 2018 at 6:43
user198044 user198044
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The degree of a multivariable polynomial always refers to the highest degree of its monomials, unless indicated otherwise. This is what Artin always means when he refers to degree with no further qualification. This includes every single usage that you have quoted except for the one that says "degree in the variable x x".
When Artin says "degree in the variable x x", that means instead the degree when you pretend all the other variables are coefficients and you just have a polynomial in x x. In other words, the "degree in x x" is the largest power of x x appearing in the polynomial.
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answered Dec 30, 2018 at 7:33
Eric WofseyEric Wofsey
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What is the justification for your first sentence?user198044 –user198044 2018-12-30 07:43:25 +00:00 Commented Dec 30, 2018 at 7:43
2 Um, my personal knowledge? This is a standard definition that is used everywhere.Eric Wofsey –Eric Wofsey 2018-12-30 08:04:12 +00:00 Commented Dec 30, 2018 at 8:04
Okay thank you.user198044 –user198044 2018-12-30 12:18:01 +00:00 Commented Dec 30, 2018 at 12:18
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12953 | https://goldbook.iupac.org/terms/view/D01533 | IUPAC - debye (D01533)
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Non-SI unit of electric dipole moment. It is equal to the electric dipole moment for two charges of 10−10 f r a n k l i n separated by 1 å n g s t r ö m, D=10−18 Fr cm≈3.335 64×10−30 C m.
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12954 | https://en.wikipedia.org/wiki/Stagnation_temperature | Published Time: 2005-07-11T21:58:55Z
Stagnation temperature - Wikipedia
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1 DerivationToggle Derivation subsection
1.1 Adiabatic
1.2 Flow with heat addition
2 Solar thermal collectors
3 See also
4 References
Stagnation temperature
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Concept in thermodynamics and fluid mechanics
In thermodynamics and fluid mechanics, stagnation temperature is the temperature at a stagnation point in a fluid flow. At a stagnation point, the speed of the fluid is zero and all of the kinetic energy has been converted to internal energy and is added to the local static enthalpy. In both compressible and incompressible fluid flow, the stagnation temperature equals the total temperature at all points on the streamline leading to the stagnation point.: 657–659, §14.1 See gas dynamics.
Derivation
[edit]
Adiabatic
[edit]
Stagnation temperature can be derived from the first law of thermodynamics. Applying the steady flow energy equation: Eq (5.50) and ignoring the work, heat and gravitational potential energy terms, we have:
h 0=h+V 2 2{\displaystyle h_{0}=h+{\frac {V^{2}}{2}}\,}
where:
h 0={\displaystyle h_{0}=\,} mass-specific stagnation (or total) enthalpy at a stagnation point h={\displaystyle h=\,} mass-specific static enthalpy at the point of interest along the stagnation streamline V={\displaystyle V=\,} velocity at the point of interest along the stagnation streamline
Substituting for enthalpy by assuming a constant specific heat capacity at constant pressure (h=C p T{\displaystyle h=C_{p}T}) we have:
T 0=T+V 2 2 C p{\displaystyle T_{0}=T+{\frac {V^{2}}{2C_{p}}}\,}
or
T 0 T=1+γ−1 2 M 2{\displaystyle {\frac {T_{0}}{T}}=1+{\frac {\gamma -1}{2}}M^{2}\,}
where:
C p={\displaystyle C_{p}=\,}specific heat capacity at constant pressure T 0={\displaystyle T_{0}=\,} stagnation (or total) temperature at a stagnation point T={\displaystyle T=\,} temperature (or static temperature) at the point of interest along the stagnation streamline V={\displaystyle V=\,} velocity at the point of interest along the stagnation streamline M={\displaystyle M=\,} Mach number at the point of interest along the stagnation streamline γ={\displaystyle \gamma =\,}Ratio of Specific Heats (C p/C v{\displaystyle C_{p}/C_{v}}), ~1.4 for air at ~300 K
Flow with heat addition
[edit]
h 02=h 01+q{\displaystyle h_{02}=h_{01}+q}T 02=T 01+q C p{\displaystyle T_{02}=T_{01}+{\frac {q}{C_{p}}}}q = Heat per unit mass added into the system
Strictly speaking, enthalpy is a function of both temperature and density. However, invoking the common assumption of a calorically perfect gas, enthalpy can be converted directly into temperature as given above, which enables one to define a stagnation temperature in terms of the more fundamental property, stagnation enthalpy.
Stagnation properties (e.g., stagnation temperature, stagnation pressure) are useful in jet engine performance calculations. In engine operations, stagnation temperature is often called total air temperature. A bimetallic thermocouple is frequently used to measure stagnation temperature, but allowances for thermal radiation must be made.
Solar thermal collectors
[edit]
Performance testing of solar thermal collectors utilizes the term stagnation temperature to indicate the maximum achievable collector temperature with a stagnant fluid (no motion), an ambient temperature of 30C, and incident solar radiation of 1000W/m 2. The aforementioned figures are 'worst case scenario values' that allow collector designers to plan for potential overheat scenarios in the event of collector system malfunctions.
See also
[edit]
Stagnation point
Stagnation pressure
Total air temperature
References
[edit]
^ Jump up to: abVan Wylen, Gordon J.; Sonntag, Richard Edwin; Borgnakke, Claus (1994). Fundamentals of classical thermodynamics (4th ed.). New York: Wiley. ISBN9780471593959.
^Planning and Installing Solar Thermal Systems: A Guide for Installers, Architects and Engineers. German Solar Energy Society (DGS). 2005. ISBN978-1844071258.
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12955 | https://emis.dsd.sztaki.hu/journals/INTEGERS/papers/h44/h44.pdf | INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A44 A CONGRUENCE FOR PRODUCTS OF BINOMIAL COEFFICIENTS MODULO A COMPOSITE Andrew D. Loveless Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195-4350, USA aloveles@math.washington.edu Received: 8/28/07, Revised: 9/17/07, Accepted: 9/24/07, Published: 10/1/07 Abstract For a positive composite integer n, we investigate the residues of !mn k " for positive integers m and k. First, we discuss divisibility of such coefficients. Then we study congruence identities relating products of binomial coefficients modulo n. Certainly the Chinese Remainder The-orem can be used in combination with prime power results to evaluate binomial coefficients modulo a composite. However, in this study we investigate residues modulo a composite di-rectly without appealing to the Chinese Remainder Theorem. And as a result we get closed form identities modulo a composite.
One of the many consequences of the main result (Theorem 8) is the following: If p, q and r are primes and m is a positive integer, then mr !mpqr pq " ≡ !mqr q "!mpr p " (mod pqr). Several numerical examples of these results are included.
Introduction Several areas of number theory and discrete mathematics make use of arithmetical properties of binomial coefficients. Congruence relations for binomial coefficients have been studied by many prominent mathematicians including Gauss, Kummer, Legendre, and Lucas. Andrew Granville gives a wonderful survey of many of the elegant results pertaining to binomial coefficients modulo prime powers. In addition, Prof. Granville maintains an on-line ‘dynamic survey’ devoted to these results. In this study, we extend some of these result to composite moduli.
We will focus mainly on binomial coefficients !a b " modulo n, where n is a divisor of a.
This focus is motivated by their extensive use in the study of probabilistic primality testing (also called compositeness testing). For the remainder of this introduction, we motivate this statement.
The subject of probabilistic primality testing is full of congruence relations that always INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A44 2 hold for a prime and rarely hold for a composite. A large portion of these congruences are proved, at least in part, by making use of the following well-known property.
Theorem 1. The positive integer n is a prime if and only if # n k $ ≡0 (mod n) for all k with 1 ≤k ≤n −1.
This characterization of primes is not an efficient form of primality testing since it requires the direct computation of binomial coefficients. However, this theorem is used indirectly to give effective probabilistic primality tests. The main goal in researching such tests is to study when and why a composite integer may satisfy a congruence that is usually only satisfied by primes.
In this study, we let n be a composite number and we examine the generalizations of Theorem 1. The results naturally extend to the case when the modulus divides the ‘top’ values in the binomial coefficient so we include this case as well. We believe that these results are interesting and elegant in their own right, but we ultimately hope that they will help in the study of primality testing.
To summarize, we are motivated by the following question: Question: For a composite integer n, what can be said about the residue of !mn k " modulo n for 1 ≤k ≤n −1?
We will show that such binomial coefficients can be expressed in closed form in terms of products of binomial coefficients in which the divisors of (n, k) are removed in a certain way, where (n, k) is the greatest common divisor of n and k. The techniques are elementary, but the results seem interesting in their own right. The only results of a similar nature that this author could find was in a chapter of the book “Computational Recreations in Mathematica” . On pages 65-66 of this book, the following result is given.
Theorem 2. If f(n) := # n −1 (n −1)/2 $ is defined for odd integers n, then f(pq) ≡f(p)f(q) (mod pq), and f(pqr)f(p)f(q)f(r) ≡f(pq)f(pr)f(qr) (mod pqr).
Although this is not the same class of binomial coefficients that we consider in this study, the nature of our results is similar to Theorem 2.
At the end of this study, we also give some minor extensions to binomial coefficients !a b " modulo n, for which n does not divide a, by using well-known binomial coefficient identities.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A44 3 1. Binomial Coefficients Divisible By n In this section we characterize all values k, 1 ≤k ≤n −1, such that !n k " ≡0 (mod n).
Various results are known concerning the divisibility of binomial coefficients by a positive integer. One of the most useful of these is the so-called Kummer’s Theorem.
Theorem 3. Kummer’s Theorem.
If n and k are integers and p is a prime, then the largest power of p dividing !n k " is given by the number of borrows required when subtracting the base p representations of k from n.
The last part of the theorem is often equivalently stated as the number of carries when k and n −k are added in the base p. For convenience we will use the notation pa||n to mean that a is the largest power of the prime p dividing n. A simple application of Kummer’s Theorem gives the following.
Theorem 4. If m, n, and k are positive integers with (n, k) = 1, then !mn k " ≡0 (mod n).
Proof. Let p be an arbitrary prime divisor of n with pa||n. Write k = krpr +· · ·+k1p+k0 in the base p and note that k0 ̸= 0 because (n, k) = 1. Since mn = mn0pa for some integer n0, subtracting the base p representation of k from mn requires at least a borrows. Kummer’s Theorem gives !mn k " ≡0 (mod pa). The prime p was an arbitrary divisor of n. Thus, !mn k " ≡0 (mod n).
The converse of this theorem is not true. In fact, !63 48 " ≡0 (mod 63) and (63, 48) = 3 ̸= 1.
A true characterization of binomial coefficients congruent to zero is given by the following corollary to Kummer’s result.
Corollary 1. For a positive integer n, !n k " ≡0 (mod n) if and only if for each pa||n the subtraction of the base p representation of k from n requires at least a borrows.
Proof. Note that !n k " ≡0 (mod n) if and only if !n k " ≡0 (mod pa) for all primes p dividing n, where pa||n. By Kummer’s Theorem, !n k " ≡0 (mod pa) if and only if the number of borrows, when subtracting the base p representation of k from n, is at least a.
In the example !63 48 " ≡0 (mod 63), we have n = 63 = 32 · 7 and k = 48. Taking pa = 32, we have n = 2 · 33 + 32 and k = 33 + 2 · 32 + 3. So n −k requires 2 ≥2 = a borrows in the base p = 3. Taking pa = 71, we have n = 72 + 2 · 7 and k = 6 · 7 + 6. So n −k requires 2 ≥1 = a borrows in the base p = 7. Thus, the binomial coefficient has to be congruent to 0 as predicted by the characterization.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A44 4 2. Binomial Coefficient Residues Modulo n In this section we are concerned with finding a way to simplify binomial coefficients modulo n. For a prime p, the following theorem of Lucas is well-known.
Theorem 5. Lucas Theorem.
If p is a prime and n and k are positive integers with base p representations n = nrpr +· · ·+ n1p + n0 and k = krpr + · · · + k1p + k0, respectively, then # n k $ = # nrpr + · · · + n1p + n0 krpr + · · · + k1p + k0 $ ≡ # nr kr $ · · · # n1 k1 $ # n0 k0 $ (mod p).
The case with a prime power modulus is considered by Davis and Webb .
They determined that binomial coefficients modulo pa depends on blocks of a consecutive digits in the base p representation of n and k.
In addition, we will make use of the following result which is attributed to Jacobsthal in and .
Theorem 6. If n and k are positive integers and p is prime p > 3, then # np kp $ ≡ # n k $ (mod pr), where r is the largest power of p dividing p3nk(n −k) !n k " .
In addition, we will need the following theorem for the cases p = 2 and p = 3 which are consequences of Proposition 5 of .
Theorem 7. If n and k are positive integers and p = 2 or p = 3, then # pn pk $ ≡ # n k $ (mod pr), where r is the largest power of 3 dividing 18nk(n −k).
Thus, in all cases we have the following general result.
Corollary 2. If n and k are positive integers and p is a prime, then # pn pk $ ≡ # n k $ (mod pr), where r is the largest power of p dividing pnk(n −k).
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A44 5 The remainder of this study focuses on the simplification of binomial coefficients modulo composites. Certainly, one could apply Lucas’ Theorem and prime power generalizations for each prime power dividing n then combine these results using the Chinese Remainder Theorem. However, this approach does not easily lend itself to closed form evaluations. Here we focus on closed form methods for evaluating these binomial coefficients in hopes to build on the general theory.
By combining the prime power results we get the main result. But first, we need some definitions.
Definition 1. For positive integers n and k, define An,k = {p prime : p|(n, k)}.
For any set A, define O(A) = {B ⊆A : |B| is odd}, and E(A) = {B ⊆A : |B| is even}.
Finally, for a set of primes B, define dB = % p∈B p In addition, define d∅= 1.
Note that if (n, k) = 1, then E(An,k) contain one element (the empty set), while O(An,k) contains no elements, and the product on the right below is defined to be zero. Thus, when (n, k) = 1, Theorem 8 reduces to Theorem 1.
Theorem 8. If n, m, and k are positive integers, then % B∈E(An,k) # mn/dB k/dB $ ≡ % B∈O(An,k) # mn/dB k/dB $ (mod n).
Proof. Let pa||n, where a > 0. Consider the two cases where p ∤(n, k) and p|(n, k).
If p does not divide (n, k), then p does not divide k, so (pa, k/dB) = 1 for all B ⊆An,k.
Thus, # mn/dB k/dB $ ≡0 (mod pa) by Theorem 4. Thus, the congruence holds modulo pa.
If pb||(n, k) for some b, 0 < b ≤a, then we can use Corollary 2 to give a correspondence between each binomial coefficient on the left-hand side of the congruence and a binomial coefficient on the right-hand side of the congruence modulo pa.
If B ⊆An,k and p ̸∈B, then, letting B′ = B ∪{p}, we obtain # mn/dB k/dB $ ≡ # mn/dB′ k/dB′ $ (mod pa). To justify this last congruence, write mn/dB = αp and k/dB = βp INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A44 6 and note that pa divides pαβ (in particular pa−1 divides α) so Corollary 2 ensures that the congruence holds. Finally, note that one of the cardinalities of B and B′ is odd and the other is even.
Similarly, if B ⊆An,k and p ∈B, then, letting B′ = B{p}, we obtain # mn/dB k/dB $ ≡ # mn/dB′ k/dB′ $ (mod pa) in the same manner. Thus, there is a one-to-one correspondence for each binomial coefficient from a set B on the left-hand side yielding a congruent binomial coefficient from a set B′ on the right-hand side.
Therefore, for all pa||n, we have & B∈E(A) # mn/dB k/dB $ ≡& B∈O(A) # mn/dB k/dB $ (mod pa).
Hence, the congruence holds modulo n.
Note that dB is always square-free as defined. That is, in the theorem above we only ‘cancel’ one factor of each prime p when pa|(n, k). We may be able to prove more in special cases by using generalizations of Jacobsthal’s result. We leave such an investigation for a different study. Such results would likely lack the elegance and generality of Theorem 8.
As a quick illustration of the usefulness of Theorem 8, consider the following.
Corollary 3. If p, q and r are primes and m is a positive integer, then mr # mpqr pq $ ≡ # mqr q $ # mpr p $ (mod pqr).
Proof. Note that % B∈E(Apqr,pq) # mpqr/dB pq/dB $ = # mr 1 $ # mpqr pq $ and % B∈O(Apqr,pq) # mpqr/dB pq/dB $ = # mpr p $ # mqr q $ .
Letting n = pqr and k = pq in Theorem 8, we have the result.
3. Special Cases and Examples Here we consider special forms for the prime factorization of n. For these special forms we simplify the corresponding binomial coefficients for all k values.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A44 7 Corollary 4. If n = paqb is the product of two odd prime powers and m is a positive integer, then # mn k $ ≡ 0 (mod n) , if (k, n) = 1; # mn/p k/p $ (mod n) , if (n, k) = pj, 1 ≤j ≤a; # mn/q k/q $ (mod n) , if (n, k) = qj, 1 ≤j ≤b; mn/p k/p mn/q k/q mn/(pq) k/(pq) (mod n) , if pq|(n, k).
Proof. The first case follows from Theorem 4, the second and third cases follow from Corol-lary 2, and the fourth case follows from Theorem 8.
We must be careful when using formulas such as the fourth case of Corollary 4. It is understood that the evaluation and simplification of the binomial coefficients occurs before the expression is evaluated modulo n. This comment is essential since # mn/(pq) k/(pq) $ is not guaranteed to have an inverse modulo n. Similar evaluations can be given when n has more than two prime factors.
To illustrate these ideas we give the following examples: (1) Consider the following binomial coefficient modulo 323. Here, m = 1, n = 323, k = 85, p = 17.
323 85 $ = # 17 · 19 5 · 17 $ ≡ # 19 5 $ (mod 323).
(2) Consider the following binomial coefficient modulo 30.
30 15 $ = # 2 · 3 · 5 3 · 5 $ ≡ # 2 · 5 5 $ # 2 · 3 3 $ # 2 1 $ = 1 2 # 10 5 $ # 6 3 $ (mod 30).
In general, Theorem 8 allows for the evaluation of !mn k " in terms of binomial coefficients involving smaller numbers modulo n as is given below.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A44 8 Theorem 9. If n, m and k are positive integers, then # mn k $ ≡ 0 (mod n) , if (n, k) = 1 % B∈O(A) mn/nB k/nB % B∈E(A){∅} mn/nB k/nB (mod n) , if (n, k) > 1.
4. Composite Moduli and General Binomial Coefficients The study so far has only focused on binomial coefficients of the form !mn k " modulo n. Now we show how these results can be extended to general coefficients of the form !a b " modulo n.
By using the following lemma, we will be able to attack binomial coefficients in general.
Lemma 1. If a, b, and k are positive integers, then # a b $ = +k j=0 # k j $ # a −k b −j $ .
Proof. This result is given by repeated application of the identity !a b " = !a−1 b " + !a−1 b−1 " .
Thus, given a binomial coefficient !a b " with a ≥n we can use Lemma 1 in combination with our previous results by writing a = mn + k.
Corollary 5. If n, a and b are positive integers such that a = mn + k, then # a b $ = # mn + k b $ = k , j=0 # k j $ # mn b −j $ .
When evaluating this sum modulo n, Theorem 4 guarantees that the only nonzero terms are those where (b −j, n) > 1.
Theorem 10. If n, a and b are positive integers such that a = mn + k, then # a b $ = k , j = 0 (b −j, n) > 1 # k j $ # mn b −j $ (mod n).
We can draw several conclusions from this. Here is one such result.
Corollary 6. If n, a and b are positive integer such that a = mn+k, b ≥k, and ( b!
(b−k)!, n) = 1, then !a b " ≡ !mn b−k " (mod n).
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A44 9 Proof. Since ( b!
(b−k)!, n) = 1, we have (b −j, n) = 1 for j = 0, 1, . . . k −1. Hence, !a b " = !mn b−k " + +k−1 j=0 !k j "!mn b−j " ≡ !mn b−k " +0 (mod n), by Theorem 4.
As a summary example, letting n = 35 we evaluate the binomial coefficient: # 38 13 $ ≡ # 35 10 $ , by Corollary 5, since (35, 13 · 12 · 11) = 1.
= # 5 · 7 2 · 5 $ ≡ # 7 2 $ , by Theorem 8 or Corollary 2.
= 21 (mod 35).
5. Conclusions Hopefully, these results give added insight into the properties of these important numbers.
Thank you to my advisor Prof. William Webb who introduced me to the interesting divis-ibility properties of binomial coefficients. I also sincerely thank the referee who corrected several small typos and greatly improved the presentation and clarity of this article.
References A. Granville: Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers. Organic mathematics (Burnaby, BC, 1995), 253-276. CMS Conf. Proc. 20, Amer. Math. Soc.
Providence, RI, 1997.
I. Vardi: Computational Recreations in Mathematica. Addison-Wesley Publishing Company, 1991 K.S. Davis and W.A. Webb: Lucas’ Theorem for prime powers. Europ. J. Combinatorics, 11 (1990), 229-233. |
12956 | https://tches.iacr.org/index.php/TCHES/article/view/12219 | All You Need is XOR-Convolution: A Generalized Higher-Order Side-Channel Attack with Application to XEX/XE-based Encryptions | IACR Transactions on Cryptographic Hardware and Embedded Systems
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Vol. 2025 No. 3/
Articles
All You Need is XOR-Convolution: A Generalized Higher-Order Side-Channel Attack with Application to XEX/XE-based Encryptions
Authors
Rei Ueno Kyoto University, Yoshidahommachi, Sakyo-ku, Kyoto-shi, Kyoto 606–8501, Japan
Akira Ito NTT Social Informatics Laboratories, Nippon Telegraph and Telephone Corporation, 3–9–11 Midori-cho, Musashino-shi, Tokyo, 180-8585, Japan
Yosuke Todo NTT Social Informatics Laboratories, Nippon Telegraph and Telephone Corporation, 3–9–11 Midori-cho, Musashino-shi, Tokyo, 180-8585, Japan
Akiko Inoue NEC Secure System Platform Laboratories, 1753 Shimonumabe, Nakahara, Kawasaki, Kanagawa 211–8666, Japan
Kazuhiko Minematsu NEC Secure System Platform Laboratories, 1753 Shimonumabe, Nakahara, Kawasaki, Kanagawa 211–8666, Japan
Hibiki Ishikawa Tohoku University, 2–1–1 Katahira, Aoba-ku, Sendai-shi, Miyagi, 980-8577, Japan
Naofumi Homma Tohoku University, 2–1–1 Katahira, Aoba-ku, Sendai-shi, Miyagi, 980-8577, Japan
DOI:
Keywords:
Higher-order side-channel attack, Collision analysis, Masking, Noise amplification, Template attack, DL-SCA, XEX/XE, OCB, PMAC, XTS, Tweakable block cipher
Abstract
The XEX/XE scheme has been widely used to realize authenticated encryptions (AEs), message authentication codes (MACs), and storage encryptions, such as OCB, PMAC, and XTS. Although these schemes have been extensively deployed in the real world, limited studies have evaluated side-channel attacks (SCAs) on them. In this study, we propose an efficient SCA that can be applied to the XEX/XE scheme. Despite the fact that the offset generated in these modes is guaranteed to have no full offset collision with an overwhelming probability, we analyze their offset-generating routines to exploit the partial offset collisions. Then, we propose a new profiled SCA named XOR-convoluting collision analysis (XCCA), which estimates the sum of keys from two leakages by XOR-convoluting probability distributions that model the leakages. The proposed collision SCA effectively erases the effect of random offsets by using XOR-convolution, whereas conventional collision SCAs are ineffective in this scenario. We validated the proposed SCA through simulations and experimental attacks using real traces. The results confirmed that the proposed SCA reduces the number of traces by up to 90% to achieve a success rate identical to that of a state-of-the-art SCA on OCB in TCHES 2022. Furthermore, we show that the proposed SCA distinguisher (XCCA distinguisher) is a generalization of higher-order SCAs, including non-collision SCAs on masked implementations. The profiled higher-order SCAs on masked implementations can be written in the form of an XCCA distinguisher using XOR-convolution with the new concept of leaking and target selection functions. The generalized representation clarifies how and why a higher-order SCA has better or worse performance from the theoretical viewpoint of noise amplification, which is also demonstrated through experiments and a spectrum analysis based on Walsh–Hadamard transform (WHT). Our analysis reveals that the random offsets of XEX/XE would work as masking from an SCA perspective, and XEX/XE-based encryption would have an inherent first-order SCA resilience under certain conditions.
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2025-06-05
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Copyright (c) 2025 Rei Ueno, Akira Ito, Yosuke Todo, Akiko Inoue, Kazuhiko Minematsu, Hibiki Ishikawa, Naofumi Homma
This work is licensed under a Creative Commons Attribution 4.0 International License.
How to Cite
Ueno, R., Ito, A., Todo, Y., Inoue, A., Minematsu, K., Ishikawa, H., & Homma, N. (2025). All You Need is XOR-Convolution: A Generalized Higher-Order Side-Channel Attack with Application to XEX/XE-based Encryptions. IACR Transactions on Cryptographic Hardware and Embedded Systems, 2025(3), 317-360.
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12957 | https://en.wikipedia.org/wiki/Continuously_compounded_nominal_and_real_returns | Jump to content
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2 Real return
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Continuously compounded nominal and real returns
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| This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) | | | This article is an orphan, as no other articles link to it. Please introduce links to this page from related articles. (July 2016) | | | | This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: "Continuously compounded nominal and real returns" – news · newspapers · books · scholar · JSTOR (March 2021) (Learn how and when to remove this message) | | | | This article needs attention from an expert in Accounting. The specific problem is: Article needs reliable source citations to establish notability. WikiProject Accounting may be able to help recruit an expert. (January 2025) | (Learn how and when to remove this message) |
Return rate is a corporate finance and accounting tool which calculates the gain and loss of investment over a certain period of time.
Nominal return
[edit]
Let Pt be the price of a security at time t, including any cash dividends or interest, and let Pt − 1 be its price at t − 1. Let RSt be the simple rate of return on the security from t − 1 to t. Then
The continuously compounded rate of return or instantaneous rate of return RCt obtained during that period is
If this instantaneous return is received continuously for one period, then the initial value Pt-1 will grow to during that period. See also continuous compounding.
Since this analysis did not adjust for the effects of inflation on the purchasing power of Pt, RS and RC are referred to as nominal rates of return.
Real return
[edit]
Let be the purchasing power of a dollar at time t (the number of bundles of consumption that can be purchased for $1). Then , where PLt is the price level at t (the dollar price of a bundle of consumption goods). The simple inflation rate ISt from t –1 to t is . Thus, continuing the above nominal example, the final value of the investment expressed in real terms is
Then the continuously compounded real rate of return is
The continuously compounded real rate of return is just the continuously compounded nominal rate of return minus the continuously compounded inflation rate.
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12958 | https://rheumnow.com/news/updated-eular-recommendations-treatment-systemic-sclerosis | Search
Updated EULAR Recommendations on the Treatment of Systemic Sclerosis Save
Medscape
Jul 16, 2024 7:03 pm
Medscape has published an overview to the 2024 updated recommendations for the treatment of systemic sclerosis (SSc) presented in Vienna at EULAR 2024 by Professor Francesco Del Galdo, MD, PhD on behalf of a 27 member task force.
The new recommendations (n=20) supercede the 2017 (n=16) EULAR recommendations, providing specific evidence based suggestions for 8 specific domains of SSc (Raynauds, digital ulcers, Pulmonary hypertension, skin fibrosis, interstitial lung disease [ILD], musculoskeletal involvement [MSK], gastrointestinal [GI], and renal crisis).
"The most impactful new recommendation relates to the evidence for immunosuppressive agents and antifibrotics for the treatment of skin fibrosis and lung fibrosis," said Francesco Del Galdo, MD, PhD, professor of experimental medicine, consultant rheumatologist, and scleroderma and connective tissue diseases specialist at Leeds Teaching Hospitals NHS Trust, Leeds, England. Del Galdo presented the update at the EULAR 2024 Annual Meeting.
"But there are also new recommendations, including a redefined target population for hematopoietic stem cell transplantation following cyclophosphamide, the upfront combination treatment at the time of diagnosis of pulmonary arterial hypertension [PAH], and a negative recommendation for the use of anticoagulants for pulmonary arterial hypertension," noted Del Galdo, highlighting key updates in the 2024 recommendations.
Six new recommendations appear, covering drugs like mycophenolate mofetil, nintedanib, rituximab, and tocilizumab. None of these therapies were present in the 2017 recommendations.
Del Galdo highlighted the new immunosuppression continuum and associated treatments for skin and lung fibrosis. "For skin involvement, the task force recommended mycophenolate, methotrexate, and rituximab, with tocilizumab having a lower level of evidence and lower recommendation strength; similarly, in interstitial lung disease, we have rituximab, mycophenolate, cyclophosphamide, and nintedanib, and these all have the highest strength of evidence. Tocilizumab is assigned one strength of evidence below the other drugs."
He also cited the phosphodiesterase 5 inhibitor (PDE5i) drugs that are used across Raynaud's phenomenon, digital ulcers, and pulmonary arterial hypertension, which together form a vascular therapeutic continuum.
The complexity of systemic sclerosis and multiple manifestations was a major determinant of the recommendations, Del Galdo pointed out. "The task force realized that since this is such a complex disease, we cannot recommend one treatment unconditionally. For example, with mycophenolate mofetil, what works for most patients for the skin and lung manifestations might not for someone who experiences severe diarrhea, in which mycophenolate is contraindicated. So, the highest degree of recommendation that the task force felt comfortable with was 'should be considered.'"
Turning to new evidence around drug use, Del Galdo said that rituximab has the highest level of evidence across skin and lung manifestations, nintedanib is new in lung, and tocilizumab is new across both skin and lung.
To treat systemic sclerosis-pulmonary arterial hypertension (SSc-PAH), as long as there are no contraindications, the task force recommends using PDE5i and endothelin receptor antagonists (ERAs) at diagnosis. Data from phase 3 trials show a better outcome when the combination is established early.
The task force suggests avoiding the use of warfarin in PAH. "This is supported by a signal from two trials showing an increase in morbidity and mortality in these patients," noted Del Galdo.
He also pointed out that selexipag and riociguat were new and important second-line additions for the treatment of PAH, and —consistent with the ERA approach — the EULAR recommendation supports frequent follow-up to establish a treat-to-target approach to maximizing clinical outcomes in SSc-PAH and SSc-ILD. "Specifically, for the first time, we recommend monitoring the effect of any chosen intervention selected within 3-6 months of starting. The evidence suggests there is a group of patients who respond and some who respond less well and who might benefit from a second-line intervention."
For example, results of one trial support the approach of adding an antifibrotic agent to reduce progression in people with progressive lung fibrosis. "Similarly, for pulmonary hypertension, we recommend putting patients on dual treatment, and if this fails, place them on selexipag or switch the PDE5i to riociguat," Del Galdo said.
Recommendations by SSc domains are summarized in the table below.
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Updated Systemic Sclerosis Recommendations from EULAR Use 'Therapeutic Continuums'
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12959 | https://www.organicchemistrytutor.com/topic/nomenclature-of-alkynes/ | Nomenclature of Alkynes — Organic Chemistry Tutor
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Nomenclature of Alkynes
Organic Chemistry
Nomenclature of Organic Compounds
Nomenclature of Alkynes
In this tutorial, I wanna talk about the nomenclature of alkynes.
Alkynes are unsaturated hydrocarbons with at least one carbon-to-carbon triple bond. They’re a fundamental part of organic chemistry and show up all over the place in health sciences and pre-med studies. That’s because they’re biologically relevant, highly versatile in synthetic chemistry, and just generally useful. And yes, you’ll definitely need to know how to name them for your test.
That’s exactly what we’re going to cover here: how to name alkynes and cycloalkynes, along with the edge cases that tend to trip students up on exams. So grab your coffee, open up your notebook to work through the examples with me, hit that LIKE button for good luck on the test, and let’s dive in.
Fundamentals of the Alkyne Nomenclature
As a functional group, alkynes are named with the suffix “-yne.” Just like with any other hydrocarbon, we start by finding the longest continuous carbon chain that includes the triple bond. For instance, in the first example, we count nine carbons in the main chain. The next step is to number that chain in a way that gives the lowest possible number to the first carbon of the triple bond.
In this case, numbering from the left puts the triple bond starting at carbon 3. If we tried it from the right, we’d hit carbon 6 first, which is obviously higher. And if there’s a tie in numbering the triple bond, we go to the usual tiebreakers—first checking the positions of substituent groups, and if still tied, we use alphabetical order.
Once the chain is numbered, we identify and alphabetize the substituents. Here, we have two methyl groups on carbons 2 and 7. That makes it 2,7-dimethyl. A quick reminder: prefixes like “di-” don’t count in the alphabetical order—only the first letter of the group name does, so “methyl” gets ranked by “m.”
We also have a chlorine on carbon 8, so that gives us 8-chloro. In the final name, substituents are always listed alphabetically, regardless of their positions. I know it doesn’t always feel logical, but it’s what the rules say, so we roll with it.
Next, we mark the position of the triple bond within the parent chain. Since it starts at carbon 3, we name the parent chain as non-3-yne.
Putting it all together, the full name is 8-chloro-2,7-dimethylnon-3-yne.
Now, if we had any stereochemistry involved—say, if there are chiral centers—we would include stereodescriptors at the beginning of the name, just like for other organic molecules. So if we add stereochemistry at the chiral carbons here and assign R/S descriptors, let’s say the methyl-bearing carbon is R and the chlorine-bearing carbon is S. That gives us (7R,8S)-8-chloro-2,7-dimethylnon-3-yne.
And listen, make sure you know how to assign these descriptors. I can’t tell you how many students come out of exams telling me they lost points just because they forgot about stereochemistry. I swear, if I had a quarter for every time that happened, I’d have my mortgage paid off by now. But anyway…
Examples
Let’s take a look at another example.
In this one, we’ve got eight carbons in the longest chain. When numbering, you’ll notice that whether we go from the left or right, the triple bond ends up starting at carbon 4 either way. So, we go to our tiebreakers. Numbering from the left gives us two substituents at carbon 3, while from the right we get one on carbon 3 and two on carbon 6. Since we want the lowest positions for our groups, left-side numbering wins.
The substituents here are two ethyl groups on carbons 3 and 6, and one methyl group also on carbon 3. That makes it 3,6-diethyl-3-methyloct-4-yne. This molecule doesn’t have any chiral centers, so no need for stereochemistry here.
Now, the next example does include stereochemistry. We’ve got six carbons in the longest chain, and since the aromatic ring is attached at position 1, we’re going to start numbering there—you can’t get any lower than 1, after all.
We also have an isopropyl and a chlorine substituent. Just a heads-up: the “iso” in “isopropyl” does count for alphabetical purposes.
So, we put the name together as 5-chloro-3-isopropyl-1-phenylhex-1-yne. Since we have chiral centers at positions 3 and 5, and both are R, the full name becomes (3R,5R)-5-chloro-3-isopropyl-1-phenylhex-1-yne.
Now, what about cyclic alkynes?
Just like with any other cyclic molecule, we number the ring starting from the functional group—in this case, the triple bond—so that it gets the lowest possible number. You always, and I mean always, number through the triple bond, not around it. From there, we keep numbering to give the lowest locants to any substituents. In our case, we number clockwise to make that happen.
So, the final name for this molecule is 4,4-dichloro-7,7-dimethylundecyne. Notice that we don’t specify where the triple bond is—because in cyclic molecules, numbering starts from it by default. Also, since there’s no stereochemistry, we don’t need to mention any descriptors.
One last note before we move on: locants always go directly before the functional group. This is the current IUPAC Preferred IUPAC Name format (PIN). You might still see older names where the locant comes before the parent chain name, but that’s outdated and no longer recommended.
Alkenes vs Alkynes (When Both Are Present)
Now, here comes the tricky part: what if there’s both a double and a triple bond in the same molecule?
Good news: IUPAC has clear rules for that. Bad news: those rules have changed a few times over the last few decades, and not all instructors are caught up. So I’ll tell you what the 2013 IUPAC guidelines say, but you should double check with your instructor. After all, they’re the ones grading you. That said, standardized exams like the ACS and MCAT do use the rules I’m about to explain.
If the double and triple bonds end up with the same number, the double bond takes priority. So, for example, the correct name would be non-2-en-7-yne, not non-7-en-2-yne.
If numbering from one end gives the double bond a lower number and the triple bond a higher one—or vice versa—then we go with the option that gives the lowest number to either.
So you’d get names like hept-4-en-1-yne or dec-3-en-6-yne, depending on the structure.
And that wraps it up.
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12960 | https://www.youtube.com/watch?v=OyZ6Gog5prY | 2.6.2 Find Limit at Infinity (Indeterminate Forms)
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Posted: 9 Sep 2020
Lecture series for Calculus 1 (Differential Calculus). Textbook used: James Stewart. Calculus - Early Transcendentals, 8th edition. Cengage. This video introduces general methods of finding limits at infinity for indeterminate forms. Thank you for watching! For more videos, please subscribe (right below the video) or check out the playlist
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The objective of this lecture is to introduce the general methods of evaluating limits at infinity for indeterminate forms. Different types of functions and indeterminate forms are discussed among multiple examples.
Transcript:
hello this is paul from foster tech the last lecture we defined the image at infinity right and then we know how to guess or estimated the limits using a table list and in this lecture we're going to find that the exact limit okay so this is important we will find the exact limit at the infinitive okay the major the mutual um results we're going to use is this one okay we only have one theorem and that the theorems stays in this if r greater than zero is a rational number okay here is a rational number then this limit equals zero um first let me go into a look at it is this a minute the result meaningful yes why because this is intuitive we think either just intuitive and the r is a greater than zero a number you can imagine i guess raised to the other power the denominator goes to water goes to infinity right so one out of infinity so one can match one out of infinity should we go down to zero okay and if we don't going to prove it this should be true true intuitively but how to exactly prove because later and a lot of examples going to use the theory so we need the proof okay exactly and of course so firstly we look at it these are is a rational number okay we can write the rational number in this way right and then uh integers okay and then so we look at it we can directly define that the limits go to fuse depends and then the final result is zero just to do calculation okay and then so the first step look at it the first step and the r is uh a rational axopolin rational at this lender we can write to write into uh this is a reticle right so we can write the rational power into radical so the radical is a and theoretical and throat and there is the power of n to remember this is the definition and if we write it in this way and then another step is going to move the position of the limit by water by 30 meter loss we learned that before okay see so first uh the base and the residual power and then we have water we have power law okay the limiter can go into the base right and then we have a roll the law so the limiter can go into the road okay under the road therefore finally we go into this way right we just use the limited law we learned and now we have these results do you remember the last lecture we exactly proved and then what is the limit of y of x that's zero right okay therefore the result is zero definitely okay and uh a little bit more okay so r is a rational can be extended into any real number in the future we were going to introduce okay r is any real number greater than zero the result is they are true and the general result is just as this we used this right is one out of infinite always zero okay keep in mind well out of infinity we always know it's zero let me show you a few examples so the first example and we need to find okay the limit or if the limit does not exist sure the limit does not exist or if the limited are not existed by the infinity okay so should the infinity okay so how about this so the first step we look at is a rational right a polynomial over polynomial if it directly belongs to infinity to the bottom infinity to the bar to the bottom infinity to the top what do we get it we get a infinity over infinity so let me rod and there's water infinite over infinite if someone like it you can put that like this okay because the infinite is not a number okay it's just a formal so this is what is a form this formula is same as like a similar as zero over zero we call the water zero over zero we got it in determinate form it's called a yin determinate form uh deter form y is indeterminate let me show uh one example for you and this is exactly indeterminate we cannot determine just by the form okay we need to look at the detail what can you see this i'll give you two examples uh the first example is this if it's a two aggressor overaccus can you see is the water infinity over infinity right this is the infinite over infinity okay because this is infinite is equal to infinity however we know the result okay because the water and the these are good and that these are goods are the same right so common factor we factored out divided the out so 2 the limit over 2 is 2. see the first cell we get at the limit there is a 2 i give you another and if i change it two to three and therefore why not this cancer accuracy what do we get we get the three but this is also the form of infinity over infinity see both the form are the same infinity over infinity however the results are different which means water this is indeterminable okay so indeterminate form you have to look at the detail and for these questions we cannot affect it out and then divided out of something okay however we have another way so i will show you another way let's do this divide it let me just write it down and this is a type of formal to find the limit the major way are the same can you see is do this manipulate it okay manipulate this function by dividing let me see write down when you do the why did the uh the numerator numerator numerator and the denominator by x square divided by x squared okay why are we divided by a square so what is the arc square aggregate square is what it is the largest determine the bottom this is the largest keep in mind always these are the the largest the largest of the term in the denominator in denominator okay we simply say we always do this okay manipulator find it the the largest determinant the denominator then divided by the numerator and the denominator by the largest term in the denominator okay and the in this example divided by square see what do we get i will get this let me write what do we get we get i guess -2 divided by square that's the top the border i guess square plus one divided by a square okay why divide it because another step see the numerator is fraction we can expand this fraction expanded every expander you will find something okay well we're going to do this it could span all the fractions in the top and the inter-bottom okay after we expanded see we will get something believe me that i got supposed to infinity so aggressive over x square is a 1 out of x minus 2 over i guess square okay see and the original is one fraction now we have a two why we wrote this do you remember what the limit of this zero what the limit of zero by the theory okay so that's the reason why we're going to divide it by the highest or the largest term and the why we need to expand okay because we are going to use the theory so the button we can spend x square over square is one right plus one over i guess square okay now all we need to do is use the limited law okay and the quotient the law and the sum and law and the difference law whatever and we use the limited law we learned that before the limit the loss and then this limiter can go to the first go to the second the occurred the third because here is one and they go to the fourth which means uh uh after we use this limit you know the limit of the first step is zero this is zero this is zero and then you can use this limit the law and the theorem we just discussed okay see 1 over x raised to the earth power and then what do we have we will get this uh so the limit we don't have all right so we don't have limit and now the limit goes to the top but this is zero because to here is zero because to here is a one goes to here is zero okay all the zero is from the theorem and then do calculation is zero that's the result okay here's the first example and then i have a few other examples to show you okay keep in mind that so we have the basic idea is to do this manipulate the fractions but divided by get the largest determinant in the denominator let me show you one more so here is the at the same okay so the other similar what are we going to do is a fraction uh if we directly plot infinity and this is infinity so it's di infinity over infinity okay so we said yeah how this infinite over infinite so the indeterminate what are we going to do divide it by the largest determinant which is x right okay divided let me see divided by the uh why the numerator denominator by the same by x and someone would say y is not divided by 4x 4 does not matter okay so we only divided by x is good after we double divided the backers to see what do we get we will get this we will get that the limit i guess approaches infinity is here and this is the bigger term um let me write it this way i guess plus three i guess square this is the original but it divided by x right okay the bottom is uh four i guess minus one divided by x okay now what are we going to do and then we need to put these accuracy inside the square root let me show you and if we put a number inside the square root we need to raise to the power of two okay and the opposite if we take a square outer from the square root we take a two away can you imagine that okay so which means that divided by x or divided by x square into is divided by x squared does that make sense divided by x squared in the square root okay let me direct the reward well we simply say now is just put i put this accuracy in put it into a square root okay put it into square root see what do we have we will have this and that's the limit there i guess support g infinite okay so here is a big square root and the aggregates plus three x squared this is the original is divided by what divided divided by x okay outside these axis so inside there is i guess a square you can imagine okay square root that goes to the top is the original square root equal to the bottom square root of like square which is x okay keep in mind that don't make a mistake uh i guess go inside that goes to x square that's all and of course the button let me write the others i'll just copy it for i guess minus one over x okay and then what do we do um we just expand okay just expand expand the first what do we get can you see this i will get the this weight i will get the limit i guess supposed to infinity i just expanded okay you can spend there is a square root right so it's a square root i guess over x square is a one over x plus three x squared over x squared is three this gets much simpler okay so how about the bottom so the bottom goes to this four i guess over x is a go to four right okay minus one out of over x okay so this is just expand why would you expand it because we're going to use the theory well out of that get supported zero one other factors which is zero okay and then what are we going to do you can use the um it's the other same our condo you give me the law okay you deliver the laws and the theory we discussed okay and then what do we have we directly have the results so this is a which means the limited go inside is a square root of zero plus three right is four the limit goes to here is minus zero and the do calculations so you will get a square root of 3 over 4. these are the results the idea is they are same right it's they are similar all you need to pay addition is a number going inside the square root that goes to square okay so you need to put the origin to the square okay let's do the second power one more can we look at this question um they are the same idea okay idea is the same first step we look at the form and i guess approach infinity e raised to the x power imaging okay goes to infinity so the top is infinite okay probably you say negative good so let me put a is negative infinity because the negative infinite is the top right what is the problem the body is infinite okay oh don't care about the negative infinity over infinite is already indeterminable okay so this is the indeterminate so they are indeterminate and then now what are we going to do the same right the same idea the y divided the numerator denominator by the highest determinant in the bottom of course it is a year raised to the x power e raised to the x power you find the determinant divided and then you will have the similar results easy to get it now what do we do is the limit i guess approximate infinity okay so can you see and one minus the e raised to the x is at the top divided by e raised to the x and the bottom is a 1 plus 2 e x and also divided by e raised to the x power and then what to do expand the right okay expand the fraction expand the fraction then we can use the theory okay the fractions what do we do and then we will have this okay we have the limit i guess supposed to infinite okay now one divided by this is what i just wrote one over i guess raised to the x power e raised to the x power this over this is a one right so minus one one over this is d r one over e raised to the x power two times this over this is together two okay so now you need one more result and the limit of the law is the same right the limit goes to here the limit okay here what kind of we need to know this is a one over infinity okay so let me show you here is a one over infinite we need to know this is zero okay keep reminder this is just a kind of a form if that this means what if the denominator goes to infinity and then flip is going to zero okay and then what's the result and that is also should that be equal to here and because it to fraction this goes to zero right so this is zero minus one this is zero plus two we get a negative one out of two so this is the results we get okay the results we get so they are similar all these three questions are very similar the idea but this question is different okay what to do so this dummy is not a fraction it's water it's just a minus okay uh first let me look let us check the formula and if a prologue is good so here's the infinitive right uh i say this is a infinite i guess it goes to infinity okay so keep in mind that and this is also in determinator in determination and why this is infinite mind infinity is also indeterminate let me give you example okay okay look at this oh we take a limit of a negative so it goes to infinity can you see i guess plus one this goes to infinity right and then minus x this axis also goes to infinity so what it is is the infinite minus infinite of course the result is what i guess minus is always one one the limit of one is what the limit of 1 is 1 right there's always one the same you can imagine okay i just change i guess plus 2 minus x and then we work out that the limit that goes approach infinity 2 the limit of 2 is 2 this is also infinity minus infinity just give yourself an idea okay infinite minus infinite is a formal we cannot determine the results we need to look at the detail okay so this is a form which is the aquatic indeterminate form i show you a method how to do this we needed the square root okay what are we going to do rationalize remember this well we made a square form rationalized and here we know denominator so we can rationalize the numerator okay so we can do this we can uh re channelize numerator okay and we can rationalize numerate okay so let me go into rot we're going to uh rationalize which means what i have to times the conjugated right so this is a little bit longer okay and the original let me put it here again i guess square plus one minus x that's the original what are we going to do our country is the conjugate remember the conjugator is what is a plastic is right okay it's still not that long enough okay logic is manageable because the plots this is the conjugate so your time is the conjugated to the bottom should be i guess square plus one uh plus x see and these end these are the same when we cancel it as a nothing right okay but now what we can do we can simplify the numerator by the difference of squares okay do you remember the difference of squares the difference of squares okay which means the sum time at the sum time the difference is the square of the the difference of squares okay so square of this minus square of this square of the first you know is like a square plus one right let me write the one one more step and this if we wrote one more step by the difference of square and this square is uh i guess square plus one minus i guess square does that this square root of square i'll take the square root of away right so this is the difference of square and at the bottom i just copy and i can square plus one plus x okay of course the numerator we can simplify right okay this is the time can you see we will get something new this time the top will go to one the bottom goes to something but the this time is good okay why the denominator goes to infinity this is the infinity polished infinity is water okay a infinite number okay infinite polarity infinite the definitely does the infinity right there's no problem okay so what do we get we got it here we get this is a one over infinity one over infinitely remember we have the result is zero right so that's the result zero okay and this one and uh look at another one so here what this okay so let's say you could something i guess square these are supposed to negative infinity no matter what it is it goes to infinite what is negative number goes to uh up the power is a negative so this goes to negative infinity right infinite plus a negative infinity which is water which is a uh infinite minus do you remember we just said this is a indeterminate right okay this is the indeterminate um indeterminate but this time we cannot rationalize how to do okay can you see x square x square is common factor you can factor out it okay so this way is just as a trick okay let's just do this you factored out i guess squared why we do this because later you will see if we factor out let's see what do we get we will get a another form i guess approaching negative infinity after the factor there is x squared and one left for the two times the accuracy raised to the fifth power left okay why this works why can you see uh this is the product right if someone you say i use the product of the law and the product the law okay the limited to the first okay someone said the first stickers reward is not a number divergence but it diverges to positive infinity after the second c1 is okay so this number goes to negatively infinite okay so if we write it here i just use the form okay whatever so we will get this this goes to the positive infinity this goes into negatively infinite uh sorry for the negative and this is times you can imagine positive infinity times negative infinity what do we get what do we get we get a negative infinite we get a negative infinite this is the result okay of course you know negative infinity is not a number this means water the limit does not exist however it doubles to negative infinity to remember the questions at the beginning if the limit does not exist but equal to infinity we have to write the infinity right okay we have the last questions to show how about the sign what is sign sign is what kind of you learned before okay in trigonometry sine is uh uh like a wave like okay it's a goes r because down goes up goes down we call that isolate okay so we can oscillate between negative one and the one right okay then you look at it the infinity can the function point to a thickest number no never because it's oscillate so uh what are we going to see because of the oscillation so the limit does not exist i write the lead up the more because of sine function okay this is a oscillate okay between what as between neck at the wall and the what right always like this so if you look at the right tail they never approach can never approach never gets stable so what the limiter does not exist so it does not exist off the limit okay so this is the reason that's all thank you |
12961 | https://www.khanacademy.org/math/ka-math-class-11/x0419e5b3b578592a:probability-ncert-new/x0419e5b3b578592a:probability-using-combinatorics/v/probability-of-a-given-selection | Probability of a given selection (video) | Khan Academy
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KA Math Class 11
Course: KA Math Class 11>Unit 14
Lesson 5: Probability using Combinatorics
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Probability of a given selection
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This video focuses on calculating probabilities when making selections from a larger group, using the example of drawing 5 marbles from a box containing 10 red, 20 blue, and 30 green marbles. We'll calculate the total number of ways to draw 5 marbles. Then, we'll find the probability of specific outcomes, such as all marbles drawn being blue, at least one marble being green, and at most 2 marbles being blue.Created by Ashish Gupta.
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12962 | https://brainly.com/question/38362976 | [FREE] Why might Palestinians and Kurds be considered stateless nations? - brainly.com
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Why might Palestinians and Kurds be considered stateless nations?
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Palestinians and Kurds are considered stateless nations because they are ethnic groups that do not possess their own sovereign state and are not the majority population in any sovereign state. Palestinians are native to the region of Palestine and Kurds to the region of Kurdistan, but both lack independent states of their own.
Explanation
The terms Palestinians and Kurds often come up in discussions about stateless nations. A stateless nation refers to an ethnic or cultural group that does not possess its own sovereign state and is not the majority population in any sovereign state. Palestinians, native to the region of Palestine (currently divided between Israel, the West Bank, and the Gaza Strip), do not have a recognized, independent state of their own. Similarly, Kurds, an ethnic group originating in a region known as Kurdistan (encompassing parts of Turkey, Iran, Iraq, and Syria), also lack an officially recognized, independent state of their own. Both groups are engaged in long-standing conflicts over self-determination and sovereignty with the states currently controlling their traditional homelands.
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Palestinians and Kurds are considered stateless nations because they lack their own recognized, independent sovereign states despite having a shared cultural identity. Their historical struggles for sovereignty and self-determination illustrate the ongoing conflicts and challenges they face. Both groups remain divided geographically among multiple existing states, which complicates their pursuit of autonomy.
Explanation
Palestinians and Kurds are often referred to as stateless nations because they are ethnic groups that do not have their own independent, recognized sovereign states.
Who: The term stateless nations applies to various ethnic groups, with Palestinians and Kurds being two prominent examples.
What: A stateless nation is defined as a group of people sharing a common identity, culture, or history but lacking their own state or political sovereignty.
When: The modern context of these groups being considered stateless nations arises from historical events - for Palestinians, this includes the establishment of Israel in 1948, and for Kurds, from the division of their homelands among Turkey, Iraq, Iran, and Syria after World War I.
Where:
Palestinians: They are primarily located in the region of Palestine, which includes areas currently known as Israel, the West Bank, and the Gaza Strip.
Kurds: They inhabit the region called Kurdistan, which stretches across parts of Turkey, Iran, Iraq, and Syria.
Why: Both groups have experienced displacement and conflict over their right to self-determination. For Palestinians, their land was significantly affected by the creation of the state of Israel, leading to ongoing disputes. The Kurds have faced oppression and have sought autonomy in their regions without achieving full statehood.
How: Both groups engage in political movements striving for recognition and independence. Palestinian efforts include the pursuit of statehood recognition in international organizations like the United Nations, while Kurdish groups have sought greater autonomy through political and sometimes armed resistance, particularly in Iraq and Turkey.
In summary, Palestinians and Kurds are considered stateless nations primarily due to their shared cultural identity without a recognized, independent state, and their historical struggles for sovereignty. Their situations highlight the complexities of national identity and geopolitical boundaries in modern times.
Examples & Evidence
Examples of stateless nations include:
Palestinians, who are primarily located in the West Bank and Gaza Strip, where they are striving for statehood but are currently contained within Israel's borders.
Kurds, who inhabit regions across Turkey, Iran, Iraq, and Syria, and have sought recognition and autonomy, particularly in Iraq where they have established an autonomous region.
Research shows that the Palestinians have been recognized as a state by multiple countries and organizations, yet they do not exercise full control over their territories. Additionally, the Kurds represent the largest stateless ethnic group in the world, existing as a significant population across several countries without a recognized independent nation.
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12963 | https://www.geeksforgeeks.org/maths/a-union-b-formula/ | AUB Formula
Last Updated :
23 Jul, 2025
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AUB is defined by the elements that belong to either set A or set B. The union operation i.e., A∪B is one of the important operations in sets. This article explores the A∪B formula, the number of elements in A∪B formula, A∪B formula in probability, and the complement of the A union B formula.
We will also solve some examples and answer some FAQs related to the A∪B formula. Let's start our learning on the topic "A∪B formula".
Table of Content
What is A Union B?
What is A∪ B Formula?
Number of Elements in A∪B Formula
A∪B Formula in Probability
Complement of AUB Formula
What is A Union B?
A union B represented as A∪B is the set with elements that belong to either of the sets. The union is represented by the symbol ∪. If A and B are two sets the union of A and B is denoted as A∪B. Union is a set operation that unites the elements of two or more sets.
Union of two sets is performed by combining the elements of the two sets and taking common sets only once.
A∪B Formula
A∪B formula is defined as the elements that either belong to set A or set B. The combination of the elements of A or B gives the A∪B formula. The set builder form representation of the A∪B formula is:
A ∪ B = {x : x ∈ A or x ∈ B}
Venn Diagram for AUB Formula
The Venn diagram for AUB formula is given below. In the below diagram, the pink region shows the union of two sets A and B.
Number of Elements in A∪B Formula
Number of elements in the A∪B is defined as the sum of the number of elements in A and B minus number of elements in A∩B. The formula for the number of elements in A∪B formula is given by:
n(A∪B) = n(A) + n(B) - n(A∩B)
where,
n(A∪B) is number of elements in A∪B
n(A) is number of elements in A
n(B) is number of elements in B
n(A∩B) is number of elements in A∩B
A∪B Formula in Probability
A∪B formula in Probability is used to find the probability of A∪B with the help of the probability of the events A, B and A∩B respectively. There are two A∪B formula in probability one for the mutually exclusive events and other for non-mutually exclusive events.
A∪B Formula for Non-Mutually Exclusive Events
The A∪B formula for non-mutually exclusive events uses the probabilities of A, B and A∩B.
P(AUB) = P(A) + P(B) - P(A∩B)
where,
P(A∪B) is probability of A∪B
P(A) is probability of A
P(B) is probability of B
P(A∩B) is probability of A∩B
A∪B Formula for Mutually Exclusive Events
The A∪B formula for mutually exclusive events uses the probabilities of A and B as for mutually exclusive events A∩B =0.
P(A∪B) = P(A) + P(B)
where,
P(A∪B) is probability of A∪B
P(A) is probability of A
P(B) is probability of B
Complement of AUB Formula
The complement of A∪B formula is defined as the intersection of complement of A and complement of B. This is called De Morgan's law.
(A ∪ B)' = A' ∩ B'
(A ∪ B)' = {x: x ∉ (A∪B)'}
Venn Diagram for Complement of A∪B Formula
Complement of A union B is the region except the region of A union B. The Venn diagram for complement of A∪B is given below. The pink region shows the region of compliment of A union B.
What is A Union B Union C?
The A union B union C is defined as the set of elements that belongs to set A or set B or set C. It is represented as A ∪ B ∪ C. The set builder form of A ∪ B ∪ C is:
A ∪ B ∪ C = {x: x∈ A or x ∈ B or x ∈ C}
A Union B Union C Formula
The formula for finding number of elements in A U B U C set is given by:
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A∩B) - n(B∩C) - n(A∩C) + n(A∩B∩C)
where,
n(A∪B∪C) is number of elements in A∪B∪C
n(A) is number of elements in A
n(B) is number of elements in B
n(A∩B) is number of elements in A∩B
n(B∩C) is number of elements in B∩C
n(A∩C) is number of elements in A∩C
n(A∩B∩C) is number of elements in A∩B∩C
A Union B Union C Venn Diagram
The Venn diagram for A Union B Union C is given below. All the three circles consisting of white region shows A union B union C.
What is A union B intersection C?
The A union B intersection C is defined as the set of elements that belongs to either A or B∩C. The set builder form of A union B intersection C is:
A ∪ (B ∩ C) = {x: x ∈ A or x ∈ (B ∩C)}
A union B intersection C Formula
The formula for the A union B intersection C is given by:
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A union B intersection C Venn Diagram
The Venn diagram for A union B intersection C is given below. The region colored by blue colour shows A union B intersection C.
Also Check
A intersection B
Intersection of Sets
Set Operations
Solved Examples on AUB Formula
Example 1. Find A∪B using A∪B formula where, A = {9, 22} and B = {10, 16}.
Solution:
Given
A = {9, 22} and B = {10, 16}
A∪B = {9, 10, 16, 22}
Example 2. Find the cardinality of X∪Y where, X = {x, y} and Y = {x, z}.
Solution:
Given
X = {x, y} and Y = {x, z}
X ∩ Y = {x}
n(X) = 2 and n(Y) = 2
n(X ∩ Y) = 1
By number of elements in AUB formula
n(X∪Y) = n(X) + n(Y) - n(X ∩ Y)
n(X∪Y) = 2 + 2 - 1
n(X∪Y) = 3
Example 3. If the number of elements in P, Q and P∩Q is 10, 12 and 5 respectively, then find the number of elements in P∪Q.
Solution:
Given
n(P) = 10 and n(Q) = 12
n(P ∩ Q) = 5
By number of elements in AUB formula
n(P∪Q) = n(P) + n(Q) - n(P ∩ Q)
n(P∪Q) = 10 + 12 - 5
n(P∪Q) = 17
Example 4. If the probability of event A is 0.7, probability of event B is 0.9 and the probability of event A∩B is 0.8 then find probability of A∪B.
Solution:
Given
Probability of event A = 0.7
Probability of event B = 0.9
Probability of event A∩B = 0.8
P(A∪B) = P(A) + P(B) - P(A∩B)
P(A∪B) = 0.7 + 0.9 - 0.8
P(A∪B) = 0.8
Practice Questions on AUB Formula
Q1. Find A∪B using A∪B formula where, A = {1, 2} and B = {5, 6}.
Q2. Find the cardinality of P∪Q where, P = {a, b, c} and Q = {e, f}
Q3. If the number of elements in A, B and A∩B is 3, 4 and 1 respectively, then find the number of elements in A∪B.
Q4. If the probability of event A is 0.5, probability of event B is 0.7 and the probability of event A∩B is 0.2 then find probability of A∪B.
A
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12964 | https://physics.stackexchange.com/questions/734702/why-is-an-equilateral-triangle-not-a-2d-unit-cell | condensed matter - Why is an equilateral triangle not a 2d unit cell? - Physics Stack Exchange
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Why is an equilateral triangle not a 2d unit cell?
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An equilateral triangle obeys the crystallographic restriction theorem, but it is not a part of 2d crystal structure. What symmetry does it lack? Why can't it be a Bravais lattice?
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edited Nov 1, 2022 at 11:12
Thomas Fritsch
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It is impossible to make a Bravais lattice of triangular cells (equilateral or not). It is the essence of the Bravais lattice concept that it is based on translational symmetry (in 2D, along two independent directions). In order to build a Bravais lattice with an equilateral triangular cell, one would need to ensure that, by rigidly displacing a triangle along two of its sides, it is possible to have a tessellation (i.e., a covering) of the plane without overlapping and without holes. This is not possible with triangles. One needs parallelograms or other 2D figures that do not need rotations after displacement to tassell the plane.
What could be misleading is that, in 2D, there is the so-called triangular lattice. This is a misname. The elementary cell is a rhombus with 60∘60∘ acute angles. The Wigner-Seitz cell is instead a regular hexagon made of six equilateral triangles. However, no cell is a triangle. If we see it as a crystalline structure, it can be thought as a triangular Bravais lattice with a basis made by two equilateral triangles sharing one side. But that it is not a Bravais lattice of triangles.
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edited Nov 1, 2022 at 13:08
answered Nov 1, 2022 at 11:55
GiorgioP-DoomsdayClockIsAt-89GiorgioP-DoomsdayClockIsAt-89
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It is a possible 2d lattice structure, it just so happens that in the case of equilateral triangles we can observe them to tile together into hexagons. A hexagonal lattice is sometimes called a triangular lattice.
We can see clearly how the lattice vectors of a hexagon can form the lattice vector of an equilateral in the following figure
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edited Nov 1, 2022 at 11:24
answered Nov 1, 2022 at 11:19
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Thanks for clarification LEO PHYSICS –LEO PHYSICS 2022-11-01 11:23:16 +00:00 Commented Nov 1, 2022 at 11:23
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12965 | https://www.youtube.com/watch?v=3JzpcVMXbhg | 2.3.1 (Part 3) | Harmonic Oscillator | Introduction to Quantum Mechanics (Griffiths)
Hayashi Manabu
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In my upcoming uploads I will derive the properties of the quantum harmonic oscillator using the ingenious method of ladder operators as developed by Paul Dirac.
In the third video we will establish an important claim: if a function ψ satisfies the time-independent Schrodinger Equation, then applying the a+ operator to the same function will give us another function that would also satisfy the Schrodinger Equation. This important result will allow us to derive all the energy states of the harmonic oscillator later on.
This is going to correspond to section 2.3.1 of Griffiths's textbook.
(I'm using the 2nd Edition textbook. I don't have the 3rd Edition textbook but the 2nd Edition is widely available online, so I assume more people would be using it. If you're using the 3rd Edition, the problem numbering might not match sometimes. Credits go entirely to Griffiths for writing this excellent textbook.)
Transcript:
in the last video we proved a very important result that the hamiltonian operator can be expressed in one of two ways in terms of the a plus a minus operators so as usual what we proved in the last video will come in handy in the next step of our derivation so moving on with our derivation the next thing that we're going to do is that I'm going to make a claim I'm going to claim that if there is a function is I that satisfies the time independent Schrodinger equation with energy level e the know I'm going to claim that if this is true then this function or this function over here a plus sign so na plus is a operator applied to the functions I so this gives us this new function a plus sign I'm going to claim that a plus I is also satisfies this Kronecker equation with an energy level of e + H Omega so when I say that something satisfies this Kronecker equation with energy level e that means this relationship is true so this is exactly what this code of equation is so when I'm saying that this also satisfy this growing equation I'm saying that this expression here can also satisfy this expert this equality over here so in order to prove this let's try to let's try to work our way through this sconedogger equation to see if whether both sides matches so I'm going to apply this the Hamiltonian operator to the function a plus is I over here and then if what I claimed is true I'm going to end up with an expression that is going to be at energy level e + H Omega times the function itself so let's see if this is true so applying the Hamiltonian operator to this function here I'm going to invoke the result that we had last time so I'm going to use this definition of the Hamiltonian operator and then I'm using this one instead of this one for a special reason so you'll see later on so let's just stick to using this definition for that so this is just apply directly applying the definition of this scoring this Hamiltonian operator and then the next step I'm going to absorb this a plus operator inside the brackets so I can do this it's valid to put this inside the brackets as long as I retain the order so I can't put this a plus in the middle at the front I have to make sure that this a plus is going to be applied to this function at first so I have to put this on the right-hand side and then I'm going to pull this left a plus out of the brackets and then because this is just a constant I can move it around so I can pull the a plus outside as well so now you arrived at something that looks pretty similar so this expression here looks pretty similar to this Hamiltonian operator right but the only difference is that instead of a minus we have the plus over here so so let's try to use this definition of the Hamiltonian operator as well let's try to apply it to what we have over here so in order to do that I'm going to have to change this plus sign into a negative so if I change this to a negative side I need to overcompensate by adding one so as you can see one plus one minus one half I'll just get back there one half over here and so moving on I'm going to group up some of these terms together so I'm going to pull out the A+ to the very front going to move these constants in and I have a minus a plus minus 1/2 sorry and then we also have this plus one that it's going to be multiplied by these constants so we're also going to ever H Omega is I so obviously I've done this so that I could use this definition over here so recall that this is also the Hamiltonian operator so right now I have the Hamiltonian operator applied to the function inside so here I'm assuming that zai is a function that satisfies the square root equation so this relationship is true so I can apply this relationship directly for our next step so these Hamiltonian operator applied to this function it's going to give us the energy level times the function itself so grouping up the constants together and then because these are constants I can shift the operator a plus operator over to the right and so there we have it this is exactly what we wanted to find so going back what we've proved that was just that just now is that the Hamiltonian operator applied to a plus is I is equal to this expression which is exactly this code nuclear equation so the function is n plus is I and the energy level is e plus h or beta so we've proven this claim that if Z is a function that satisfies the sconedogger equation with energy level e then a plus si is also another function that satisfied this correlation with the energy level of B plus H Omega and then you can do something you can do the exact same thing for any - as I as well so I'm not going to prove what this leads to is you can try yeah you can you need to go through a similar process so you can try that out yourself but after you go through the same process again you will get e minus H Omega anybody's sorry so here we've shown two things so if so I satisfies the scroll to that equation with energy level e then both a plus sign and a minus i solutions to the scrolling third equation with these corresponding energy levels |
12966 | https://www.ncbi.nlm.nih.gov/books/NBK604196/figure/article-150900.image.f1/ | [Figure, Central Poststroke Pain Syndrome, Potential...] - StatPearls - NCBI Bookshelf
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StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2025 Jan-.
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Central Poststroke Pain Syndrome, Potential Sites of Involvement. Lesions at various levels of the spinothalamic tract, including the thalamus, can contribute to central poststroke pain syndrome. Thalamic lesions were initially thought to be solely responsible, but later research identified the involvement of the lateral medulla, pons, lenticulocapsular area, and cortex. The condition may result from the loss of somatosensory integration and changes in cortical plasticity. Betancur DFA, da Graça Lopes Tarragó M, da Silva Torres IL, Fregni F, Caumo W. Central post-stroke pain: an integrative review of somatotopic damage, clinical symptoms, and neurophysiological measures. F ront Neurol. 2021;12:678198. doi: 10.3389/fneur.2021.678198.
From: Central Post-Stroke Pain Syndrome
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Hyperbaric oxygen therapy for thalamic pain syndrome: case report.[Front Neurol. 2024]Hyperbaric oxygen therapy for thalamic pain syndrome: case report.Slade JB, Kwan N, Lennox P, Gray R. Front Neurol. 2024; 15:1364716. Epub 2024 Mar 13.
Allodynia in patients with post-stroke central pain (CPSP) studied by statistical quantitative sensory testing within individuals.[Pain. 2004]Allodynia in patients with post-stroke central pain (CPSP) studied by statistical quantitative sensory testing within individuals.Greenspan DJ, Ohara S, Sarlani E, Lenz AF. Pain. 2004 Jun; 109(3):357-366.
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Anosike KC, Rajaram Manoharan SVR. Central Post-Stroke Pain Syndrome. [Updated 2024 Jun 7]. In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2025 Jan-. [Figure, Central Poststroke Pain Syndrome, Potential...] Available from:
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12967 | https://www.rapidtables.com/convert/energy/ev-to-joule.html | Home›Conversion›Energy conversion› eV to joules
Electron-volts to joules conversion
Electron-volts (eV) to joules (J) conversion calculator and how to convert.
eV to joules conversion calculator
How to convert eV to joules
eV to joules conversion table
eV to joules conversion calculator
Enter the energy in electron-volts and press the Convert button:
Joules to eV conversion ►
How to convert eV to joules
One electron-volt is equal to 1.602176565⋅10-19 joules:
1eV = 1.602176565e-19 J = 1.602176565⋅10-19 J
So the energy in joules E(J) is equal to the energy in electron-volts E(eV) times 1.602176565⋅10-19:
E(J) = E(eV) × 1.602176565⋅10-19
eV to joules conversion table
| Energy (eV) | Energy (J) |
--- |
| 1 eV | 1.602177⋅10-19 J |
| 2 eV | 3.204353⋅10-19 J |
| 3 eV | 4.806530⋅10-19 J |
| 4 eV | 6.408706⋅10-19 J |
| 5 eV | 8.010883⋅10-19 J |
| 6 eV | 9.613059⋅10-19 J |
| 7 eV | 1.121524⋅10-18 J |
| 8 eV | 1.281741⋅10-18 J |
| 9 eV | 1.441959⋅10-18 J |
| 10 eV | 1.602677⋅10-18 J |
| 20 eV | 3.204353⋅10-18 J |
| 30 eV | 4.806530⋅10-18 J |
| 40 eV | 6.408706⋅10-18 J |
| 50 eV | 8.010883⋅10-18 J |
| 60 eV | 9.613059⋅10-18 J |
| 70 eV | 1.121524⋅10-17 J |
| 80 eV | 1.281741⋅10-17 J |
| 90 eV | 1.441959⋅10-17 J |
| 100 eV | 1.602677⋅10-17 J |
| 200 eV | 3.204353⋅10-17 J |
| 300 eV | 4.806530⋅10-17 J |
| 400 eV | 6.408706⋅10-17 J |
| 500 eV | 8.010883⋅10-17 J |
| 600 eV | 9.613059⋅10-17 J |
| 700 eV | 1.121524⋅10-16 J |
| 800 eV | 1.281741⋅10-16 J |
| 900 eV | 1.441959⋅10-16 J |
| 1000 eV | 1.602677⋅10-16 J |
Joules to eV conversion ►
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Chapter 4.3 - Rational Inequalities
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Pre-Calculus Course Content
6. Polynomial and rational inequalities
1 Functions & Relations
2 Linear Functions
3. Quadratic Functions
4. Polynomial Functions
5. Rational Functions
6. Polynomial and rational inequalities
7. Inverse functions
8. Exponential & Logarithmic Functions
9. Introduction to Trigonometry
10. Trigonometric Functions
11. Analytical Trigonometry
12. Complex Numbers
13. Vectors
14. Systems of Equations
15. Conic Sections
Rational Inequalities
Polynomial Inequalities
Rational Inequalities
Absolute Value Equations and Inequalities
Rational Inequalities
Rational Inequalities
It is commonly observed in a precalculus class that, despite many warnings and demonstrations, students “cross multiply out” denominators. The fact that denominators provide valuable information should be emphasized. As before, technology helps, but students are better served if it is not used.
In this module we extend the notions introduced in polynomial inequalities to rational inequalities. In addition to zeros of the function we must also include singular points to study the sign of a rational function.
Review
zeros and singular points of rational functions
addition and subtraction of rational expressions
Learning Objectives:
Solve a rational inequality
Write the solution in terms of interval notation |
12969 | https://www.brainkart.com/article/Rational-Roots_39121/ | | | |
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Definition, Solved Example Problems | Theory of Equations - Rational Roots | 12th Mathematics : UNIT 3 : Theory of Equations
Posted On : 12.05.2019 12:37 pm
Chapter: 12th Mathematics : UNIT 3 : Theory of Equations
Rational Roots
Nature of Roots and Nature of Coefficients of Polynomial Equations
Rational Roots
If all the coefficients of a quadratic equation are integers, then Δ is an integer, and when it is positive, we have, √Δ is rational if, and only if, Δ is a perfect square. In other words, the equation ax2 + bx+ c= 0 with integer coefficients has rational roots, if, and only if, Δ is a perfect square.
What we discussed so far on polynomial equations of rational coefficients holds for polynomial equations with integer coefficients as well. In fact, multiplying the polynomial equation with rational coefficients, by a common multiple of the denominators of the coefficients, we get a polynomial equation of integer coefficients having the same roots. Of course, we have to handle this situation carefully. For instance, there is a monic polynomial equation of degree 1 with rational coefficients having 1/2 as a root, whereas there is no monic polynomial equation of any degree with integer coefficients having 1/2 as a root.
Example 3.11
Show that the equation 2x2 - 6x+ 7 = 0 cannot be satisfied by any real values of x.
Solution
∆= b2 − 4ac= −20 < 0 . The roots are imaginary numbers.
Example 3.12
If x2 + 2 (k+ 2)x+ 9k= 0 has equal roots, find k.
Solution
Here Δ = b2 - 4ac= 0 for equal roots. This implies 4 (k+ 2)2 = 4 (9) k.This implies k= 4 or 1.
Example 3.13
Show that, if p, q, r are rational, the roots of the equation x2 - 2 px+ p2 - q2 + 2qr- r2 = 0 are rational.
Solution
The roots are rational if Δ = b2 - 4ac= (-2 p)2 - 4 ( p2 - q2 + 2qr- r2 ) .
But this expression reduces to 4 (q2 - 2qr+ r2 ) or 4 (q- r)2 which is a perfect square. Hence the roots are rational.
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Tags : Definition, Solved Example Problems | Theory of Equations , 12th Mathematics : UNIT 3 : Theory of Equations
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12th Mathematics : UNIT 3 : Theory of Equations : Rational Roots | Definition, Solved Example Problems | Theory of Equations
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Exercise 3.2: Polynomial Equation in Geometry - Problem Questions with Answer, Solution
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Exercise 3.6: Descartes Rule - Problem Questions with Answer, Solution
Choose the correct Answers - Theory of Equations
Summary - Theory of Equations
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12970 | https://www.reddit.com/r/learnmath/comments/189pohw/what_is_the_method_of_finding_the_slope_of/ | what is the method of finding the slope of intersection of parabola : r/learnmath
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12971 | https://www.youtube.com/watch?v=Zze36spqjF0 | How to find the asymptotes of the tangent function
Brian McLogan
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Posted: 15 Dec 2018
👉 Learn how to graph a tangent function. To graph a tangent function, we first determine the period (the distance/time for a complete oscillation), the phase shift (the horizontal shift from the parent function), the vertical shift (the vertical shift from the parent function) and the asymptotes (the two vertical lines which the graph does not touch). After we have obtained these features, we plot the points on the graph and graph accordingly.
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Transcript:
it's called apples and bananas all right um let's just go and review this because the main thing i want to do is to talk to you guys about you know let's i just work on identifying the parts and this is a review on koten or on tangent but obviously we can relate this to what we're going to be doing today so first of all if i asked you to find the important parts which we covered in your notes the first thing we'd want to figure out is now is this in our form of b times x minus c no so i want to be careful here so i'm going to rewrite this as 3 halves tangent if i factor out a 2 that's like dividing out a 2 i would get a 2 times x plus pi over 4. we talked about this last class period but is everybody okay with my factoring skills there again check your work 2 times x is 2x 2 times pi over 4 is pi halves when you factor out the 2 it's like dividing out of 2. pi halves divided by 2. dividing by 2 is the same thing as multiplying by the reciprocal right okay so now let's go ahead and discuss what is some of the topics or information we know the period remember guys for tangent as well as for cotangent the period is going to be 2 pi divided by b so our b so here's our a here's our b and here's our c so b is pi divided by 2. not really anything else we need to do there we got it um you take period divided by b okay for sine and cosine it's two pi divided by b um the phase shift remember phase shift is your c but it's only your c when you factored out that b it is not pi halves to the left it is pi over 4 to the left and then and then what about this three halves is that the amplitude don't fall for the trap guys amplitude is the half distance from the max to the min tangent cotangent cosecant and cosecant do not have a max or a min right so there is no amplitude amplitude's only for sine and cosine so this is everything we can we have here it's basically everything we're talking about now what can we understand about that three halves what is actually that doing to the graph it's a stretch it's a fraction so stretching or compression it's a fraction that's larger than one so it's vertically stretching the graph okay and then obviously the two we know is shrinking because the period is two now it's pi half so obviously we know the period is getting compressed so you could also think of the two as like a horizontal compression now let's go and figure out the asymptotes because the range is negative infinity to infinity so if you guys remember from your notes the asymptote of tangent of x is x equals pi halves plus pi n you don't believe me that's fine go and look at the unit circle right here when is tangent equal to zero or i'm sorry when is tangent going to be undefined when is y over x undefined one over zero right at pi halves and then to get to the next undefined value which is three pi f how far do i have to travel pi and if i travel pi again i get to the next undefined value right so pi half plus pi n is not something you need to memorize you can just kind of refer back to that every time yes and eventually you'll do it so many times you're like i already have this memorized now thank you i didn't need to get out my flash cards right okay now we have changed what is happening inside this function though right think about it so here's here's the tangent function this tangent function has now been compressed and then shifted that just moved the asymptotes right compressing it changes the asymptotes and then shifting it left or right changes the asymptotes would you guys agree so we need a way to represent both of those um or we don't need a way those both of those changes are represented inside of the function the three halves does not change anything to the asymptotes right if i have my graph and i have asymptotes if i move this up or down that's not changing the asymptotes right if i stretch it up or down that's not changing where the asymptotes occur only horizontal changes so if i want to know what my new asymptotes are based on these transformations inside of the function just set whatever's inside the function equal to your asymptote or to the original asymptote now we just got to solve for x so these are the changes that happen to x if x had no changes if there was an x there your asymptotes would be pi halves plus pi pi n right but now we multiply by two and add pi f's so divide by 2 and then we have x plus pi over 4 equals pi over 4 plus pi halves n and then to get rid of the pi over 4 i subtract pi over 4. now should i subtract the pi over 4 to both of them like i divided because i divided to both of them should i subtract both of them why not why can you do that from division but you can't do that for subtraction should we just do it anyways well guys i know it's kind of confusing the pies and the twos and the fractions but can you subtract like x minus five they're not like terms right you can't do three x minus three or three x minus five it's not like terms pi halves n and pi over four are not like terms you can't subtract them right and then plus that's distributed this is like the distributive property of division that we're doing so that's okay so anyways you get x equals pi over 4 minus pi over 4 is 0. do i really need to write 0 or can i just kind of forget about it so your asymptotes occurs at pi half's n now some of you might say miss mclogan |
12972 | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-spring-2020/7d7d5c35490f41b7b037cafbda7019ad_MIT6_006S20_lec14.pdf | Restrictions SSSP Algorithm Graph Weights Name Running Time O(·) General Unweighted BFS |V | + |E| DAG Any DAG Relaxation |V | + |E| General Non-negative Dijkstra Bellman-Ford |V | log |V | + |E| General Any Introduction to Algorithms: 6.006 Massachusetts Institute of Technology Instructors: Erik Demaine, Jason Ku, and Justin Solomon Lecture 14: Johnson’s Algorithm Lecture 14: Johnson’s Algorithm Previously |V | · |E| All-Pairs Shortest Paths (APSP) • Input: directed graph G = (V, E) with weights w : E → Z • Output: δ(u, v) for all u, v ∈ V , or abort if G contains negative-weight cycle • Useful when understanding whole network, e.g., transportation, circuit layout, supply chains... • Just doing a SSSP algorithm |V | times is actually pretty good, since output has size O(|V |2) – |V | · O(|V | + |E|) with BFS if weights positive and bounded by O(|V | + |E|) – |V | · O(|V | + |E|) with DAG Relaxation if acyclic – |V | · O(|V | log |V | + |E|) with Dijkstra if weights non-negative or graph undirected – |V | · O(|V | · |E|) with Bellman-Ford (general) • Today: Solve APSP in any weighted graph in |V | · O(|V | log |V | + |E|) time 2 Lecture 14: Johnson’s Algorithm Approach • Idea: Make all edge weights non-negative while preserving shortest paths! • i.e., reweight G to G0 with no negative weights, where a shortest path in G is shortest in G0 • If non-negative, then just run Dijkstra |V | times to solve APSP • Claim: Can compute distances in G from distances in G0 in O(|V |(|V | + |E|)) time – Compute shortest-path tree from distances, for each s ∈ V 0 in O(|V | + |E|) time (L11) – Also shortest-paths tree in G, so traverse tree with DFS while also computing distances – Takes O(|V | · (|V | + |E|)) time (which is less time than |V | times Dijkstra) • But how to make G0 with non-negative edge weights? Is this even possible?? • Claim: Not possible if G contains a negative-weight cycle • Proof: Shortest paths are simple if no negative weights, but not if negative-weight cycle • Given graph G with negative weights but no negative-weight cycles, can we make edge weights non-negative while preserving shortest paths? Making Weights Non-negative • Idea! Add negative of smallest weight in G to every edge! All weights non-negative! :) • FAIL: Does not preserve shortest paths! Biases toward paths traversing fewer edges :( • Idea! Given vertex v, add h to all outgoing edges and subtract h from all incoming edges • Claim: Shortest paths are preserved under the above reweighting • Proof: – Weight of every path starting at v changes by h – Weight of every path ending at v changes by −h – Weight of a path passing through v does not change (locally) • This is a very general and useful trick to transform a graph while preserving shortest paths! 3 Lecture 14: Johnson’s Algorithm • Even works with multiple vertices! • Define a potential function h : V → Z mapping each vertex v ∈ V to a potential h(v) • Make graph G0: same as G but edge (u, v) ∈ E has weight w0(u, v) = w(u, v)+h(u)−h(v) • Claim: Shortest paths in G are also shortest paths in G0 • Proof: Pk – Weight of path π = (v0, . . . , vk) in G is w(π) = i=1 w(vi−1, vi) Pk – Weight of π in G0 is: i=1 w(vi−1, vi) + h(vi−1) − h(vi) = w(π) + h(v0) − h(vk) – (Sum of h’s telescope, since there is a positive and negative h(vi) for each interior i) – Every path from v0 to vk changes by the same amount – So any shortest path will still be shortest Making Weights Non-negative • Can we find a potential function such that G0 has no negative edge weights? • i.e., is there an h such that w(u, v) + h(u) − h(v) ≥ 0 for every (u, v) ∈ E? • Re-arrange this condition to h(v) ≤ h(u) + w(u, v), looks like triangle inequality! • Idea! Condition would be satisfied if h(v) = δ(s, v) and δ(s, v) is finite for some s • But graph may be disconnected, so may not exist any such vertex s... :( • Idea! Add a new vertex s with a directed 0-weight edge to every v ∈ V ! :) • δ(s, v) ≤ 0 for all v ∈ V , since path exists a path of weight 0 • Claim: If δ(s, v) = −∞ for any v ∈ V , then the original graph has a negative-weight cycle • Proof: – Adding s does not introduce new cycles (s has no incoming edges) – So if reweighted graph has a negative-weight cycle, so does the original graph • Alternatively, if δ(s, v) is finite for all v ∈ V : – w0(u, v) = w(u, v) + h(u) − h(v) ≥ 0 for every (u, v) ∈ E by triangle inequality! – New weights in G0 are non-negative while preserving shortest paths! 4 Lecture 14: Johnson’s Algorithm Johnson’s Algorithm • Construct Gx from G by adding vertex x connected to each vertex v ∈ V with 0-weight edge • Compute δx(x, v) for every v ∈ V (using Bellman-Ford) • If δx(x, v) = −∞ for any v ∈ V : – Abort (since there is a negative-weight cycle in G) • Else: – Reweight each edge w0(u, v) = w(u, v) + δx(x, u) − δx(x, v) to form graph G0 – For each u ∈ V : ∗ Compute shortest-path distances δ0(u, v) to all v in G0 (using Dijkstra) ∗ Compute δ(u, v) = δ0(u, v) − δx(x, u) + δx(x, v) for all v ∈ V Correctness • Already proved that transformation from G to G0 preserves shortest paths • Rest reduces to correctness of Bellman-Ford and Dijkstra • Reducing from Signed APSP to Non-negative APSP • Reductions save time! No induction today! :) Running Time • O(|V | + |E|) time to construct Gx • O(|V ||E|) time for Bellman-Ford • O(|V | + |E|) time to construct G0 • O(|V | · (|V | log |V | + |E|)) time for |V | runs of Dijkstra • O(|V |2) time to compute distances in G from distances in G0 • O(|V |2 log |V | + |V ||E|) time in total MIT OpenCourseWare 6.006 Introduction to Algorithms Spring 2020 For information about citing these materials or our Terms of Use, visit: |
12973 | https://www.khanacademy.org/python-program/calculator/4888260022222848 | Khan Academy | Khan Academy
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12974 | https://math.stackexchange.com/questions/2045559/is-x2-strictly-increasing-on-0-infty | Stack Exchange Network
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Is x^2 strictly increasing on $[0, \infty]$
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$\begingroup$
I'm new here so please let me know if I'm doing something wrong.
Me and my brother are arguing since few hours on something, and we can really not figure it out, so I'm asking you some advice.
Let assume that we have a simple function $x^2$, this function is defined, continuous and derivable on the entire Real domain.
Of course we can prove that $f'(0) = 0$. Therefore as from my university books function should not be strictly increasing on $[0, \infty]$ (as $f'(x)$ is not always greather than 0).
But we know so only because we know the function on the entire domain and therefore we have negative and positive limits arround x=0. If we assume that the function is defined only on $[0, \infty]$ and here we know just that each single value of
$$f(x) > f(x - \epsilon)$$
and therefore should logically be strictly increasing.
What are your thoughts on this?
derivatives
inequality
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edited Dec 5, 2016 at 22:59
Sophie
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asked Dec 5, 2016 at 22:15
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$\begingroup$ You may see at every x in the domain other than 0 ,f'(x)>0 so it's strictly increasing on (0,inf),also f(0)=0 $\endgroup$
Sathasivam K
– Sathasivam K
2016-12-05 22:31:07 +00:00
Commented Dec 5, 2016 at 22:31
1
$\begingroup$ You can try proving: a differentiable function $f$ with $f'>0$ except for some finite set where $f'=0$ is strictly increasing. $\endgroup$
A.Γ.
– A.Γ.
2016-12-05 23:03:14 +00:00
Commented Dec 5, 2016 at 23:03
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5 Answers 5
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The usual definition of a strictly increasing function on a domain $D$ is that for all $x,y\in D$ with $x0$ for $0\leq x
On the other hand, $f'(x)>0$ implies $f(x)$ is increasing on $D$. In your case, you have $f'(0)=0$. However, $f'(x)=0$ does not imply $f$ is not strictly increasing. For example, take $f(x)=x^3$, so that $f'(0)=0$, yet $f(x)$ is strictly increasing. In the case of these boundary situations, $f(x)$ would not be strictly increasing if $f'(x)=0$ for all $x$ in an interval $I$
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answered Dec 5, 2016 at 22:20
Alex R.Alex R.
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That's not a theorem. It is true that if a function $f:\mathbb{R} \to \mathbb{R}$ has $f'(x)>0$ for all $x\in [a,b]$, then $f$ is strictly increasing on $[a,b]$. The proof is an application of the mean value theorem.
However, it is not true that if a function is strictly increasing, then it must have $f'(x)>0$ for all $x$. You have provided a counterexample.
To answer the title: yes, the function $f(x)=x^{2}$ is strictly increasing on $[0,\infty)$
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answered Dec 5, 2016 at 22:19
preferred_anonpreferred_anon
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$\begingroup$ Can u say any other function with f'(x)=0 but strictly increasing $\endgroup$
Sathasivam K
– Sathasivam K
2016-12-05 22:34:10 +00:00
Commented Dec 5, 2016 at 22:34
$\begingroup$ In fact if every point in domain other than some point (say x1)has f'(x)>0. Then the point is maximum or minimum or point of inflection.in this case it's a minimum point. $\endgroup$
Sathasivam K
– Sathasivam K
2016-12-05 22:37:48 +00:00
Commented Dec 5, 2016 at 22:37
$\begingroup$ How you say f²(x)>0 for all x[a,b]? It must be an open interval (a,b). because if you say it's a closed interval ,then,at end points how you find whether the f'(x) is positive or negative?in fact the derivative may or may not exists $\endgroup$
Sathasivam K
– Sathasivam K
2016-12-05 22:39:06 +00:00
Commented Dec 5, 2016 at 22:39
$\begingroup$ $x^{n}$ for any $n>1$ (in fact if you take $n$ to be odd you can take the domain to be all of $\mathbb{R}$ $\endgroup$
preferred_anon
– preferred_anon
2016-12-05 22:39:22 +00:00
Commented Dec 5, 2016 at 22:39
$\begingroup$ @SathasivamK Are you sure it makes a difference? $\endgroup$
preferred_anon
– preferred_anon
2016-12-05 22:40:53 +00:00
Commented Dec 5, 2016 at 22:40
| Show 2 more comments
1
$\begingroup$
Go back to the definition of a "strictly increasing" function. $f(x)$ is strictly increasing (by the definition) if whenever $x_1 > x_2$ and $x_1$ and $x_2$ are in the domain of $f$, $f(x_1) > f(x_2)$. Note that we said nothing at all about derivatives, or even that $f(x)$ has to have a derivative, or even that $f(x)$ needs to be continuous!
So your function $f(x)=x^2$ defined on the domain $[0,\infty)$ is strictly increasing.
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answered Dec 5, 2016 at 22:22
Mark FischlerMark Fischler
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$\begingroup$ Yes ,you are correct and I agree with u,but op is asked this question because he has problems with it while he use derivative.it would be nice if you explain it using Derivatives too $\endgroup$
Sathasivam K
– Sathasivam K
2016-12-05 22:47:59 +00:00
Commented Dec 5, 2016 at 22:47
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$\begingroup$
Sounds like an argument over definitions. Many authors give slightly different definitions of things such as "strictly increasing", all of which may not be equivalent. Personally I prefer: $f$ is increasing iff $a>b \implies f(a)>f(b)$. Under this definition, $x^2$ is indeed strictly increasing on $[0, \infty)$.
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answered Dec 5, 2016 at 22:24
OviOvi
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In fact,result that you use is the consequence of mean value theorem and it requires f(x) is differentiable in (a,b) and not [a,b).
This is where you make mistake and you show f'(x)=0 but the result you use won't work if the interval is closed interval or semi closed interval
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answered Dec 5, 2016 at 22:53
Sathasivam KSathasivam K
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12975 | https://www.youtube.com/watch?v=M9stogwMZDw | Learn to solve a proportion with the variable on the denominator 9/g = 15/11
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👉 Learn how to solve proportions. Two ratios are said to be proportional when the two ratios are equal. Thus, proportion problems are problems involving the equality of two ratios. When given a proportion problem with an unknown, we usually cross-multiply the two ratios and then solve for the unknown using the applicable operations.
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Transcript:
all right in this example what we have is 9 divided by g equals 15 over 10. so again i'll show you guys two different methods now automatically you could say hey we have a proportion right so i know once we have a proportion we can solve by cross multiplication right so you can say oh well this is not bad just cross multiply so you say 9 times 10 is going to be 90 and then 15 times g is 15 g divided by 15 divided by 15 and therefore you get g equals 6. however ladies gentlemen we could also do this another way if you guys look at this if you see g's on the bottom we can't solve when g is in the denominator so another way that you guys can do this is multiply by g on both sides so if you had 9 over g and this works whenever you have a variable on the denominator we need to get that variable off the denominator so to do that what i can do is multiply by g on both sides and what that does is now that eliminates the g on the left side i have 9 equals 15 over 10 g now the same way remember what happens when we have a a fraction multiplied by our variable what did we have to do to get rid of that fraction we had to multiply by the reciprocal so you can just multiply by the circle ten fifteenths those multiply to one ten fifteens over one so therefore now you have to do ten fifteenths times nine over one well ten times nine is ninety divided by five is six so there's two different ways to solve that i know you guys probably prefer cross multiplication is not a problem just want you to understand the other way as well okay um so ladies and gentlemen that is your next |
12976 | https://www.cuemath.com/numbers/multiplicative-inverse/ | LearnPracticeDownload
Multiplicative Inverse
The multiplicative inverse is defined as the reciprocal of a given number. It is used to simplify mathematical expressions. The word 'inverse' implies something opposite/contrary in effect, order, position, or direction. A number when multiplied to its multiplicative inverse results in 1.
| | |
--- |
| 1. | What is Multiplicative Inverse? |
| 2. | Multiplicative Inverse Property |
| 3. | How to Find Multiplicative Inverse? |
| 4. | Multiplicative Inverse of Complex Numbers |
| 5. | Modular Multiplicative Inverse |
| 6. | FAQs on Multiplicative Inverse |
What is Multiplicative Inverse?
The multiplicative inverse of a number is defined as a number that when multiplied by the original number gives the product as 1. The multiplicative inverse of 'a' is denoted by a-1 or 1/a. In other words, when the product of two numbers is 1, they are said to be multiplicative inverses of each other. The multiplicative inverse of a number is defined as the division of 1 by that number. It is also called the reciprocal of the number. The multiplicative inverse formula says that the product of a number and its reciprocal is 1.
There are different types of numbers like natural numbers, fractions, unit fractions, negative numbers, etc. Let us understand the multiplicative inverse formula for each type of number.
Natural numbers are counting numbers starting from 1. The multiplicative inverse of a natural number a is 1/a.For example, 3 is a natural number. If we multiply 3 by 1/3, the product is 1. Therefore, the multiplicative inverse of 3 is 1/3. Similarly, the multiplicative inverse of 110 is 1/110.
Multiplicative Inverse of Integers
Finding the multiplicative inverse of positive integers is the same as natural numbers (explained above). Just like positive integers, the product of a negative number and its reciprocal must be equal to 1. Thus, the multiplicative inverse of any negative number is its reciprocal. For example, (-6) × (-1/6) = 1, therefore, the multiplicative inverse of -6 is -1/6. Note that the multiplicative inverse of a negative number is always negative. And, in the multiplicative inverse of a negative integer, the negative sign will attach to the numerator, and not with the denominator.
Multiplicative Inverse of a Fraction
The multiplicative inverse of a fraction a/b is b/a because a/b × b/a = 1 when (a,b ≠ 0). For example, the multiplicative inverse of 2/7 is 7/2. If we multiply 2/7 by 7/2, the product is 1 (2/7 × 7/2 = 1). The multiplicative inverse of 76/43 is 43/76. If we multiply 76/43 by 43/76, the product is 1 (76/43 × 43/76 = 1).
A unit fraction is a fraction with the numerator 1. If we multiply a unit fraction 1/x by x, the product is 1. Thus, the multiplicative inverse of a unit fraction 1/x is x.
Examples:
The multiplicative inverse of the unit fraction 1/7 is 7. If we multiply 1/7 by 7, the product is 1 (1/7 × 7 = 1).
The multiplicative inverse of the unit fraction 1/50 is 50. If we multiply 1/50 by 50, the product is 1 (1/50 × 50 = 1).
Multiplicative Inverse of a Mixed Fraction
To find the multiplicative inverse of a mixed fraction, convert the mixed fraction into an improper fraction, then determine its reciprocal. For example, let us find the multiplicative inverse of (3\dfrac{1}{2}).
Step 1: Convert (3\dfrac{1}{2}) to an improper fraction, that is 7/2.
Step 2: Find the reciprocal of 7/2, that is 2/7. Thus, the multiplicative inverse of (3\dfrac{1}{2}) is 2/7.
It is interesting to note that the multiplicative inverse of a mixed number is always a proper fraction whose value is less than 1.
Multiplicative Inverse of 0
As per the definition of multiplicative inverse, it is the number that when multiplied to the original number results in 1 as the product. But with 0, we know that the product of 0 with any number is always 0. So, the multiplicative inverse of 0 does not exist.
We can also understand this using the properties of division which states that the division of any number by 0 is not defined. The multiplicative inverse of 0 can be written as 1/0, but its value is not defined. So, it does not exist.
Multiplicative Inverse Property
The multiplicative inverse property states that the product of a number with its reciprocal is always equal to 1. Look at the image given below where 1/n is the multiplicative inverse of the number n and 1 is the product.
For example, let us consider 5 apples. Now, divide the apples into five groups of 1 each. To make them into groups of 1 each, we need to divide them by 5. Dividing a number by itself is equivalent to multiplying it by its multiplicative inverse. Hence, 5 ÷ 5 = 5 × 1/5 = 1. Here, 1/5 is the multiplicative inverse of 5.
How to Find Multiplicative Inverse?
The multiplicative inverse of a number is also known as its reciprocal. It is very easy to find the multiplicative inverse of a number using the following steps:
Step 1: Divide the given number by 1.
Step 2: Write it in the form of a fraction. Say, the reciprocal of a is 1/a.
Step 3: Simplify and get the answer.
Let us find the multiplicative inverse of 2/3. The first step is to divide it by 1, which will result in 1/(2/3) = 3/2. Therefore, the reciprocal of 2/3 is 3/2.
Multiplicative Inverse of Complex Numbers
Complex numbers of the form Z = a + ib, such as Z = 3+i√2, where 3 is the real number and i√2 is the imaginary number. The multiplicative inverse of a complex number Z is 1/Z. The reciprocal of this complex number is 1/3+i√2. It can be simplified by multiplying and dividing it by 3-i√2, such that: (3-i√2)/(3+i√2)(3-i√2) = (3-i√2)/(9-i22) = (3-i√2)/(9+2) = (3-i√2)/11. Therefore, 3/11 - i√2/11 is the multiplicative inverse of 3+i√2.
Follow the steps given below to find the multiplicative inverse of a complex number a + ib:
Step 1: Write the reciprocal in the form of 1/(a+ib).
Step 2: Multiply and divide this number by the conjugate of (a+ib).
Step 3: Apply the following formulae: (a + b)(a - b) = a2 - b2, and i2 = -1.
Step 4: Simplify to the lowest form.
Modular Multiplicative Inverse
The modular multiplicative inverse of an integer p is another integer x such that the product px is congruent to 1 with respect to the modulus m. It can be represented as: px (\equiv ) 1 (mod m). In other words, m divides px - 1 completely. Also, the modular multiplicative inverse of an integer p can exist with respect to the modulus m only if gcd(p, m) = 1.
In a nutshell, the multiplicative inverses are as follows:
| Type | Multiplicative Inverse | Example |
---
| Natural Number x | 1/x | Multiplicative Inverse of 4 is 1/4 |
| Integer x, x ≠ 0 | 1/x | Multiplicative Inverse of -4 is -1/4 |
| Fraction x/y; x,y ≠ 0 | y/x | Multiplicative Inverse of 2/7 is 7/2 |
| Unit Fraction 1/x, x ≠ 0 | x | Multiplicative Inverse of 1/20 is 20 |
Tips and Tricks:
The multiplicative inverse of a fraction can be obtained by flipping the numerator and denominator.
The multiplicative inverse of 1 is 1.
The multiplicative inverse of 0 is not defined.
The multiplicative inverse of a number x is written as 1/x or x-1.
The multiplicative inverse of a mixed fraction can be obtained by converting the mixed fraction into an improper fraction and determining its reciprocal.
Important Notes
The multiplicative inverse of a number is also called it's reciprocal.
The product of a number and its multiplicative inverse is equal to 1.
☛ Related Topics:
Multiplicative Inverse Calculator
Additive Inverse Calculator
Inverse Operations
Reciprocal Function
Multiplicative Inverse Examples
Example 1: A pizza is sliced into 8 pieces. Tom keeps 3 slices of the pizza at the counter and leaves the rest on the table for his 3 friends to share. What is the portion that each of his friends gets? Do we apply multiplicative inverse here?
Solution:
Since Tom ate 3 slices out of 8, it implies he ate 3/8th part of the pizza.
The pizza left out = 1 - 3/8 = 5/8
5/8 to be shared among 3 friends ⇒ 5/8 ÷ 3.
We take the multiplicative inverse of the divisor to simplify the division.
5/8 ÷ 3/1
= 5/8 × 1/3
= 5/24
Answer: Each of Tom's friends will be getting a 5/24 portion of the left-over pizza.
2. Example 2: The total distance from Mark's home to school is 3/4 of a kilometer. He can ride his cycle 1/3 kilometer in a minute. In how many minutes will he reach his school from home?
Solution:
Total distance from home to school = 3/4 km
Distance covered in a minute = 1/3 km
The time taken to cover the total distance = total distance/distance covered in a minute
= 3/4 ÷ 1/3
The multiplicative inverse of 1/3 is 3.
3/4 × 3 = 9/4 = 2.25 minutes
Answer: Time taken to cover the total distance by Mark is 2.25 minutes.
3. Example 3: Find the multiplicative inverse of -9/10. Also, verify your answer.
Solution:
The multiplicative inverse of -9/10 is -10/9.
To verify the answer, we will multiply -9/10 with its reciprocal and check if the product is 1.
(-9/10) × (-10/9) = 1.
Answer: The multiplicative inverse of -9/10 is -10/9.
View Answer >
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FAQs on Multiplicative Inverse
What is the Meaning of Multiplicative Inverse?
The multiplicative inverse of any number is another number that when multiplied by the original number gives the product as 1. For example, the multiplicative inverse of 2 is 1/2. It is also known as the reciprocal of a number.
What is the Difference between Reciprocal and Multiplicative Inverse?
Reciprocal and multiplicative inverse mean the same in mathematics. When the product of two numbers is 1, then the numbers are said to be reciprocals or multiplicative inverses of each other.
How to Calculate Multiplicative Inverse?
To find the multiplicative inverse of a number, we divide it by 1. So, the multiplicative inverse of x is 1/x.
What is the Multiplicative Inverse of 9?
If we multiply 9 by 1/9, the product is 1. Therefore, the multiplicative inverse of 9 is 1/9.
What is the Multiplicative Inverse of 1?
If we multiply 1 by 1, the product is 1. Therefore, the multiplicative inverse of 1 is 1 itself.
What is the Multiplicative Inverse of -20?
If we multiply -20 by -1/20, the product is 1. Therefore, the multiplicative inverse of -20 is -1/20.
What is the Multiplicative Inverse of a Rational Number?
The multiplicative inverse of a rational number is its reciprocal. The multiplicative inverse of any rational number, x/y, where x,y ≠ 0 is y/x. For example, the multiplicative inverse of -2/3 is -3/2. We just flip the numerator and denominator to find the multiplicative inverse.
What is the Multiplicative Inverse Property?
The multiplicative inverse property states that the product of a number and its multiplicative inverse is always one. For example, 9 × 1/9 = 1.
Why do we Use Multiplicative Inverse?
In math, the multiplicative inverse is used to simplify expressions. One major application of multiplicative inverse is while solving division problems. While dividing two numbers, we multiply the reciprocal of the divisor to the dividend. For example, 2 ÷ 4 = 2 × 1/4 = 1/2.
What is the Multiplicative Inverse of 0?
The division by zero is not defined, therefore, the multiplicative inverse of 0 is undefined.
How to Find Multiplicative Inverse Modulo?
The modular multiplicative inverse of an integer a is another integer x such that the product ax is congruent to 1 with respect to the modulus m. It can be represented as: ax (\equiv ) 1 (mod m). The multiplicative inverse of “a modulo m” exists if and only if a and m are relatively prime, i.e. gcd(a, m) = 1.
Q1: The reciprocal of a number is also known as its _____.
Q2: Choose the number whose multiplicative inverse is equal to itself.
Q3: The multiplicative inverse of _____ does not exist.
Q4: The multiplicative inverse of (2/3)⁻3 is:
Q5: The multiplicative inverse of $$4 - 3i$$ is:
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12977 | https://math.stackexchange.com/questions/1604316/abc-is-an-isosceles-triangle-prove-ad-bc | geometry - ABC is an isosceles triangle -prove AD=BC - Mathematics Stack Exchange
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ABC is an isosceles triangle -prove AD=BC
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ABC is an isosceles triangle having ∠B=∠C=2∗∠A∠B=∠C=2∗∠A. If BD bisecting ∠B∠B meets AC in D,prove that AD=BC.
I know congruent triangles would help but am not able to figure out how to use them. ADB can not be congruent to CBD. I am trying to figure out which triangles might be congruent(triangles with AD as side and BC as side).
geometry
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asked Jan 8, 2016 at 12:59
GayatriGayatri
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See angles are 72,72,36 72,72,36 of triangle ABC . So BD is bisector of B implies each angle=36. So A D=B D...(1)A D=B D...(1)isoceles triangle theorem now in triangle BDC angle B D C=72 B D C=72 but also D C B=72 D C B=72 so B D=B C B D=B C..(2) isoceles triangle theorem thus from 1,2 A D=B C A D=B C thas all.
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answered Jan 8, 2016 at 13:07
Archis WelankarArchis Welankar
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Instead of looking for congruent triangles, you should instead find the angle of triangles themselves.
We know that :
2×∠A=∠B=∠C 2×∠A=∠B=∠C
Since ∠A,∠B,∠C∠A,∠B,∠C are parts of single triangle then :
∠A+∠B+∠C=π∠A+∠B+∠C=π
solve for these angles we would find:
∠A=π 5,∠B=∠C=2 π 5∠A=π 5,∠B=∠C=2 π 5
B D B D bisect ∠B∠B means that
∠C B D=∠A B D=π 5∠C B D=∠A B D=π 5
Which tells us that triangle A B D A B D is isosceles thus
A D=B D A D=B D
We can also solve for ∠B D C∠B D C to find that
∠B D C=2 π 5∠B D C=2 π 5
thus triangle B D C B D C is also isosceles and
B D=B C B D=B C
finally we proved that
A D=B C A D=B C
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answered Jan 8, 2016 at 13:18
KavinkulKavinkul
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By construction we have:
∠D B A=∠D C B=1 2∠A B C=1 2∠A C B=∠C A B∠D B A=∠D C B=1 2∠A B C=1 2∠A C B=∠C A B
so, for the triangle A D B A D B:
∠D A B=∠C A B=1 2∠A B C=∠D B A⇒D A=D B∠D A B=∠C A B=1 2∠A B C=∠D B A⇒D A=D B
and for the triangle C D B C D B:
∠C D B=180°−(∠D C B+∠D B C)=(180°−∠D B C)−∠D C B=∠C D B=180°−(∠D C B+∠D B C)=(180°−∠D B C)−∠D C B=
=(180°−∠C A B)−∠D C B=2×∠D C B−∠D C B=∠D C B⇒D B=C B=(180°−∠C A B)−∠D C B=2×∠D C B−∠D C B=∠D C B⇒D B=C B
So: D A=D B D A=D B
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answered Jan 8, 2016 at 14:00
Emilio NovatiEmilio Novati
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12979 | https://math.tools/calculator/unit/mass/gr-to-t-oz | Grains to Troy Ounces Converter
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Grains to Troy Ounces Converter
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Conversion Table
Grains to Troy Ounces Conversion Table
| Grains | Troy Ounces |
--- |
| 1 gr | 0.002083333118995 t oz |
| 2 gr | 0.00416666623799 t oz |
| 3 gr | 0.006249999356985 t oz |
| 4 gr | 0.00833333247598 t oz |
| 5 gr | 0.010416665594975 t oz |
| 6 gr | 0.01249999871397 t oz |
| 7 gr | 0.014583331832965 t oz |
| 8 gr | 0.01666666495196 t oz |
| 9 gr | 0.018749998070955 t oz |
| 10 gr | 0.02083333118995 t oz |
| 11 gr | 0.022916664308945 t oz |
| 12 gr | 0.02499999742794 t oz |
| 13 gr | 0.027083330546935 t oz |
| 14 gr | 0.02916666366593 t oz |
| 15 gr | 0.031249996784925 t oz |
| 16 gr | 0.03333332990392 t oz |
| 17 gr | 0.035416663022915 t oz |
| 18 gr | 0.03749999614191 t oz |
| 19 gr | 0.039583329260905 t oz |
| 20 gr | 0.0416666623799 t oz |
Troy Ounces to Grains Conversion Table
| Troy Ounces | Grains |
--- |
| 1 t oz | 480.00004938356 gr |
| 2 t oz | 960.00009876711 gr |
| 3 t oz | 1440.0001481507 gr |
| 4 t oz | 1920.0001975342 gr |
| 5 t oz | 2400.0002469178 gr |
| 6 t oz | 2880.0002963013 gr |
| 7 t oz | 3360.0003456849 gr |
| 8 t oz | 3840.0003950685 gr |
| 9 t oz | 4320.000444452 gr |
| 10 t oz | 4800.0004938356 gr |
| 11 t oz | 5280.0005432191 gr |
| 12 t oz | 5760.0005926027 gr |
| 13 t oz | 6240.0006419862 gr |
| 14 t oz | 6720.0006913698 gr |
| 15 t oz | 7200.0007407534 gr |
| 16 t oz | 7680.0007901369 gr |
| 17 t oz | 8160.0008395205 gr |
| 18 t oz | 8640.000888904 gr |
| 19 t oz | 9120.0009382876 gr |
| 20 t oz | 9600.0009876711 gr |
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12980 | https://www.researchgate.net/publication/230725473_The_Schur_Complement_and_Its_Applications | Published Time: 2005-01-01
The Schur Complement and Its Applications
Book
The Schur Complement and Its Applications
January 2005
DOI:10.1007/b105056
Publisher: Springer-Verlag
ISBN: 0-387-24271-6
Authors:
Zhang F. Z.
Zhang F. Z.
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Citations (1,433)References (4)
Abstract
Historical Introduction: Issai Schur and the Early Development of the Schur Complement.- Basic Properties of the Schur Complement.- Eigenvalue and Singular Value Inequalities of Schur Complements.- Block Matrix Techniques.- Closure Properties.- Schur Complements and Matrix Inequalities: Operator-Theoretic Approach.- Schur complements in statistics and probability.- Schur Complements and Applications in Numerical Analysis.
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Citations (1,433)
References (4)
... (D.35)] be positive definite. By Schur complement (Theorem 1.12 in Ref. ), we know that ...
... (D.61)] be positive definite. By Schur complement (Theorem 1.12 in Ref. ), we have ...
... (E.39)] be positive definite. By Schur complement (Theorem 1.12 in Ref. ), we have ...
Thermodynamic criteria for signaling in quantum channels
Preprint
Full-text available
Jun 2025
Yutong Luo
Simon Milz
Felix Binder
Signaling quantum channels are fundamental to quantum communication, enabling the transfer of information from input to output states. In contrast, thermalisation erases information about the initial state. This raises a crucial question: How does the thermalising tendency of a quantum channel constrain its signaling power and vice versa? In this work, we address this question by considering three thermodynamic tasks associated with a quantum channel: the generation, preservation, and transmission of athermality. We provide faithful measures for athermality generation and athermality preservation of quantum channels, and prove that their difference quantifies athermality transmission. Analysing these thermodynamic tasks, we find that the signaling ability of a quantum channel is upper-bounded by its athermality preservation and lower-bounded by its athermality transmission, thereby establishing a fundamental relationship between signaling and thermodynamic properties of channels for quantum communication. We demonstrate this interplay for the example of the quantum switch, revealing an explicit trade-off between the signaling ability and athermality of the quantum channels it can implement.
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... Remark 2. The method in Lemma 5 is called the Schur complement method [49,50], which is important for solving optimization issues using linear matrix inequalities (LMIs). There is an LMI toolbox and extended toolbox in MATLAB for LMI-based design and synthesis . ...
... Proof. Substituting H in (27a) and other symbols defined in (15a)-(15i) and (30a)-(30d) into (31) and (46), one can obtain (50) and (51), respectively. Moreover, (32) is equivalent to (52). ...
... □ Remark 7. Equality (52) can be solved by using inequality (14). By solving (14), (50), and (51), one can obtain the matrices P i , M i , and Y i . For i ∈ {1, 2, · · · , N}, one can then calculate the observer gains as follows: ...
Actuator Fault Estimation for Distributed Interconnected Lipschitz Nonlinear Systems with Direct Feedthrough Inputs
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Apr 2025
Ling Fang
Zhi-Wei Gao
Yuanhong Liu
Distributed interconnected systems are complex dynamic systems where every single subsystem has an impact on other subsystems. Actuators are key components in interconnected dynamic systems, which are prone to faults due to age and unexpected conditions. Therefore, there is motivation to develop an effective diagnosis algorithm for distributed interconnected systems, which is a starting point for predictive maintenance. In this study, an actuator fault estimation approach is proposed for a class of nonlinear interconnected systems with direct feedthrough inputs. Specifically, the original interconnected system is transformed into an augmented system by setting an extended state vector composed of an original state vector and actuator fault vector. An additional control term is used to eliminate the impact from unknown disturbances on the estimator error dynamics. Regional pole constraints are considered in the design of the distributed robust observer so that the poles are placed into a desired stable region. The observer gains are obtained by solving simultaneous linear matrix inequalities. Finally, the effectiveness of the proposed method is demonstrated by simulation studies, and a comparison is also provided.
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... Lastly, we verify det N = det V AB det V A . Note that N coincides with the Schur complement of block matrix V A of the whole matrix V AB , denoted by V AB /V A [47,48]. Following Schur determinant formula (see Eq. (0.3.2) in Ref. ), the deter- ...
... Note that N coincides with the Schur complement of block matrix V A of the whole matrix V AB , denoted by V AB /V A [47,48]. Following Schur determinant formula (see Eq. (0.3.2) in Ref. ), the deter- ...
Gaussian Atemporality: When Gaussian Quantum Correlations Imply Common Cause
Preprint
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Aug 2025
Minjeong Song
Jayne Thompson
Matthew Scott Winnel
Mile Gu
Conventionally, covariances do not distinguish between spatial and temporal correlations. The same covariance matrix could equally describe temporal correlations between observations of the same system at two different times or correlations made on two spatially separated systems that arose from some common cause. Here, we demonstrate Gaussian quantum correlations that are atemporal, such that the covariances governing their quadrature measurements are unphysical without postulating some common cause. We introduce Gaussian atemporality robustness as a measure of atemporality, illustrating its efficient computability and operational meaning as the minimal noise needed to remove this uniquely quantum phenomenon. We illustrate that (i) specific spatiotemporal Gaussian correlations possess an intrinsic arrow of time, such that Gaussian atemporality robustness is zero in one temporal direction and not the other and (ii) that it measures quantum correlations beyond entanglement.
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... Furthermore, according to the Schur complement , and considering condition (15d) with P = Π −1 , we know that ...
... We stress the fact that both the dt-GNS (28) and A-dt-GNS (31) are assumed to be unknown (i.e., matrices and A), and we just have the dictionary (29) which results in constructing (32). We now proceed with building the sets of the A-dt-GNS (31 . ...
Data-Driven Dynamic Controller Synthesis for Discrete-Time General Nonlinear Systems
Conference Paper
May 2025
Behrad Samari
Abolfazl Lavaei
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... This can be further simplified using the Schur complement. 83 This decomposition allows us to rewrite the integral with the constrained coordinate x Nτ as The integral over the constrained coordinate x Nτ can be solved analytically in terms of the error function. ...
... Rate constants k (which have not been normalized by the reactant partition function) for different couplings ∆ for the system of two coupled harmonic oscillators. xα kNA-instZR kBO-InstZR kGR-InstZR kexactZR kEyringZR kMarcusZR 10 −4 0.83 1.77 ×10 −22 -1.86 ×10 −22 1.95 ×10 −22 -1.82 ×10 −30 10 −3 0.63 1.77 ×10 −20 -1.86 ×10 −20 1.95 ×10 −20 -1.82 ×10 −28 10 −2 0.37 1.77 ×10 −18 -1.86 ×10 −18 1.95 ×10 −18 -1.82 ×10 −26 0.1 0.20 1.79 ×10 −16 -1.86 ×10 −16 1.96 ×10 −16 -1.82 ×10 −24 0.25 0.03 1.16 ×10 −15 1.06 ×10 −13 1.16 ×10 −15 1.25 ×10 −15 1.69 ×10 −22 1.14 ×10 −23 0.5 0.0 5.07 ×10 −15 1.05 ×10 −13 4.65 ×10 −15 5.48 ×10 −15 7.59 ×10 −22 4.56 ×10 −23 1.5 0.0 1.16 ×10 −13 3.99 ×10 −13 4.19 ×10 −14 1.27 ×10 −13 3.06 ×10 −22 4.10 ×10 −22 3 0.0 1.01 ×10 −11 1.72 ×10 −11 1.67 ×10 −13 1.12 ×10 −11 2.48 ×10 −22 1.64 ×10 −21 Appendix B: Nonstandard steepest-descent integrals ...
Nonadiabatic ring-polymer instanton rate theory: a generalised dividing-surface approach
Preprint
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May 2025
Rhiannon A. Zarotiadis
Joseph E. Lawrence
Jeremy O Richardson
Constructing an accurate approximation to nonadiabatic rate theory which is valid for arbitrary values of the electronic coupling has been a long-standing challenge in theoretical chemistry. Ring-polymer instanton theories offer a very promising approach to solve this problem, since they can be rigorously derived using semiclassical approximations and can capture nuclear quantum effects such as tunnelling and zero-point energy at a cost similar to that of a classical calculation. A successful instanton rate theory already exists within the Born--Oppenheimer approximation, for which the optimal tunnelling pathway is located on a single adiabatic surface. A related instanton theory has also been developed for nonadiabatic reactions using two weakly-coupled diabatic surfaces within the framework of Fermi's golden rule. However, many chemical reactions do not satisfy the conditions of either limit. By employing a tunable dividing surface which measures the flux both along nuclear coordinates as well as between electronic states, we develop a generalised nonadiabatic instanton rate theory that bridges between these two limits. The resulting theory approximates the quantum-mechanically exact rates well for the systems studied and, in addition, offers a novel mechanistic perspective on nonadiabatic reactions.
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... Specifically, the DNN is designed to predict a reduced stiffness matrix, representing a global stiffness matrix condensed to selected degrees of freedom using the Schur complement approach. Condensing global stiffness matrices is a well-established strategy for analyzing modular systems and is widely employed in disciplines such as domain decomposition (Zhang, 2006), where deep learning has been applied to subdomain solving (Klawonn et al., 2024). While approximating the Schur complement is commonly used to construct multilevel preconditioners (Notay, 1998;Benzi and DeLong, 2000;Axelsson et al., 2009), Kraus (2012 demonstrated that sparse approximations can be derived locally from overlapping subdomains. ...
... The Schur complement method represents the foundational form of domain decomposition (Zhang, 2006). This method usually divides a finite element problem into separate, non-overlapping subdomains, eliminating the unknown variables located within the interior of these subdomains. ...
Symmetric positive definite convolutional network for surrogate modeling and optimization of modular structures
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Apr 2025
ENG APPL ARTIF INTEL
Liya Gaynutdinova
Martin Doškář
Ondřej Rokoš
Ivana Pultarova
While modular structures offer great construction efficiency, scalability, safety, and reusability in engineering and architectural applications, their wide-spread adoption is hindered by the perceived material inefficiency and low design flexibility. Finding an optimal design within a modular system is a significant challenge, mostly because of associated computational complexity. Existing methods of accelerating combinatorial optimization with machine learning rely on heuristics and are often not transferrable between varying domain shapes, boundary conditions, and external loads. In this work, we present two key contributions to address this issue: (i) a deep neural network (DNN)-based surrogate model that accelerates the evaluation of mechanical responses by predicting reduced-order stiffness matrices, and (ii) a stochastic gradient optimization method that leverages the surrogate’s capability to compute sensitivities of the structure’s response to changes in module types. Our model combines convolutional layers with a physics-guided approach, ensuring that the output stiffness matrices are symmetric positive definite, consistent with the structure’s reduced-order representation via Schur’s complement. A distinguishing feature of our approach is its intrinsic independence from the specific domain shape, boundary conditions, and applied loads, allowing for broader applicability once the DNN-based surrogate is trained on a specific module set. We validate our method by optimizing multiple modular layout plans differing in size and loading conditions and demonstrate its efficacy by comparing its performance against the standard density-based topology optimization method. We achieve a computational speed-up of up to 1000x compared to the full-scale simulation, with a fast converging optimization for different domain sizes. This work lays the foundation for more flexible, efficient, and scalable modular design processes.
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... Furthermore, according to the Schur complement , and considering condition (16d) with P = Π −1 , we know that ...
... Moreover, C 1 = − 1 15 and C 2 = 1 15 in (2), which implies that the control input is constrained by −15 ≤ ≤ 15. We assume that the dt-GNS (31) is unknown, and just the dictionary ( , ) = 1 2 ln(1 + 2 1 ) ln(1 + 2 2 ) ln(1 + 2 ) cos( 1 ) cos( 2 ) cos( ) sin( ) ⊤ (32) with irrelevant terms is available based on the insight into the system. The dt-GNS in (31) can be reformulated in the form of (3) using the dictionary (32) and the matrix as ...
Data-Driven Dynamic Controller Synthesis for Discrete-Time General Nonlinear Systems
Preprint
Mar 2025
Behrad Samari
Abolfazl Lavaei
Synthesizing safety controllers for general nonlinear systems is a highly challenging task, particularly when the system models are unknown, and input constraints are present. While some recent efforts have explored data-driven safety controller design for nonlinear systems, these approaches are primarily limited to specific classes of nonlinear dynamics (e.g., polynomials) and are not applicable to general nonlinear systems. This paper develops a direct data-driven approach for discrete-time general nonlinear systems, facilitating the simultaneous learning of control barrier certificates (CBCs) and dynamic controllers to ensure safety properties under input constraints. Specifically, by leveraging the adding-one-integrator approach, we incorporate the controller's dynamics into the system dynamics to synthesize a virtual static-feedback controller for the augmented system, resulting in a dynamic safety controller for the actual dynamics. We collect input-state data from the augmented system during a finite-time experiment, referred to as a single trajectory. Using this data, we learn augmented CBCs and the corresponding virtual safety controllers, ensuring the safety of the actual system and adherence to input constraints over a finite time horizon. We demonstrate that our proposed conditions boil down to some data-dependent linear matrix inequalities (LMIs), which are easy to satisfy. We showcase the effectiveness of our data-driven approach through two case studies: one exhibiting significant nonlinearity and the other featuring high dimensionality.
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... Similar index results are also proved for the cyclic sums of the second kind (see Theorem 4.4). In particular, if det w(Y 1 , Y m ) ̸ = 0, cyclic sums (1.7) coincide with the positive and negative inertia of the Schur complement ...
... and the inertia of block symmetric matrices with the blocks A = A T ∈ C k×k , D = D T ∈ C l×l , and B ∈ C k×l (see [30,Theorem 2.3]) . Firstly we recall the definition of the comparative index in more details, see . ...
Cyclic sums of comparative indices and oscillation theory of symplectic difference systems
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Feb 2025
Julia V. Elyseeva
In this paper we generalize the notion of the comparative index which has fundamental applications in oscillation theory of symplectic difference systems and linear differential Hamiltonian systems. We introduce cyclic sums μ c(Y 1,Y 2,…,Y m)\mu_c(Y_1,Y_2,\dots,Y_m) and μ c∗(Y 1,Y 2,…,Y m),m≥2\mu_c^{}(Y_1,Y_2,\dots,Y_m),\,m\ge 2 of the comparative indices for the set of n−n- dimensional Lagrangian subspaces. We formulate and prove main properties of the cyclic sums, in particular, we connect the cyclic sums of the comparative indices with the number of positive and negative eigenvalues of m n×m n mn\times mn symmetric matrices defined in terms of the Wronskians Y_i^T\cJ Y_j,i,j=1,…,m.i,j=1,\dots,m. We present first applications of the cyclic sums in the oscillation theory of the discrete symplectic systems connecting the number of focal points of their conjoined bases with the positive and negative inertia of symmetric matrices.
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... The Schur complement is a useful tool in many fields such as control theory, numerical algebra, polynomial optimization, magnetic resonance imaging and simulation [5,13,14,23,31]. In 2005, Liu and Zhang provided the following properties of the Schur complement related to SDD matrices. ...
On Schur complements of locally doubly strictly diagonally dominant matrices and its applications
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Aug 2025
NUMER ALGORITHMS
Yang Hu
Jianzhou Liu
Yebo Xiong
Wenlong Zeng
In this paper we propose a new scaling method to study the Schur complements of locally doubly strictly diagonally dominant (for shortly, LDSDD) matrices. Compared with the previous methods, the proposed method can obtain a more accurate numerical estimate in a more succinctly manner. Based on the Schur complement, a new upper bound of the infinity norm for the inverse of LDSDD matrices is presented. We apply the new bound to derive an error bound for linear complementarity problems of LB-matrices. Many numerical experiments with lots of random matrices are presented to show the efficiency and superiority of our results.
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... Here B D and B C are also the Schur complements (Ref. ) of S D and S C , respectively. Our strategy starts to establish a quantitative estimate on B −1 D − B −1 C . ...
Homogenization rates of beam lattices to micropolar continua
Preprint
Aug 2025
Eric T. Chung
Kuang Huang
Changqing Ye
As the size of a mechanical lattice with beam-modeled edges approaches zero, it undergoes homogenization into a continuum model, which exhibits unusual mechanical properties that deviate from classical Cauchy elasticity, named micropolar elasticity. Typically, the homogenization process is qualitative in the engineering community, lacking quantitative homogenization error estimates. In this paper, we rigorously analyze the homogenization process of a beam lattice to a continuum. Our approach is initiated from an engineered mechanical problem defined on a triangular lattice with periodic boundary conditions. By applying Fourier transformations, we reduce the problem to a series of equations in the frequency domain. As the lattice size approaches zero, this yields a homogenized model in the form of a partial differential equation with periodic boundary conditions. This process can be easily justified if the external conditions in the frequency domain are nonzero only at low-frequency modes. However, through numerical experiments, we discover that beyond the low-frequency regime, the homogenization of the beam lattice differs from classical periodic homogenization theory due to the additional rotational degrees of freedom in the beams. A crucial technique in our analysis is the decoupling of displacement and rotation fields, achieved through a linear algebraic manipulation known as the Schur complement. Through dedicated analysis, we establish the coercivity of the Schur complements in both lattice and continuum models, which enables us to derive convergence rate estimates for homogenization errors. Numerical experiments validate the optimality of the homogenization rate estimates.
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... The first expression (22) (for z = (x, y)) follows from Schur's complement formula (see, for example, [47,48]), and (23) and (24) follow from a relation, given in ), between cofactors of the matrices B = (b i j ) 1≤i, j≤n (12) and B = (b i j ) 1≤i, j≤n (21). Below, we present a more general solution, parametrized by an arbitrary function h(x, y; p, q), which reduces to Vein's solution and, thus, to the Yamazaki-Hori solution in a specific limit. ...
On the Yamazaki-Hori solution of the Ernst equation
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A. Melikyan
A family of solutions to the Ernst equation is presented, which, in a certain limit, recovers the Yamazaki-Hori solution - an extension of the Tomimatsu-Sato solutions for all integer values of the deformation parameter δ. Our solution also recovers, as a special case, the formulation given by Vein, which is equivalent to the Yamazaki-Hori solution. Furthermore, our construction establishes a connection to a nonlinear differential equation proposed by Cosgrove, which is also associated with the Ernst equation.
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... Lemma 2.5 (SLFG, [19,27,30] Proof We provide a short proof of (2.4). Let ...
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We compute asymptotic expansions for the negative eigenvalues of the Pauli operator in two dimensions perturbed by a weakly coupled potential with definite sign, whereas previous results were limited to the case of radial magnetic fields and potentials, we are able to drop the assumption of radial symmetry entirely.
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... D † ) stands for the Moore-Penrose inverse of A (resp. D) . For a matrix M we denote by C(M ) its column space, i.e., the linear span of the columns of M . ...
Matrix Fejér-Riesz type theorem for a union of an interval and a point
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Jul 2025
Shengding Sun
A. Zalar
The matrix Fejér-Riesz theorem characterizes positive semidefinite matrix polynomials on the real line. In the previous work of the second-named author this was extended to the characterization on arbitrary closed semialgebraic sets K in R\mathbb{R} by using matrix quadratic modules from real algebraic geometry. In the compact case there is a denominator-free characterization, while in the non-compact case denominators are needed except when K is the whole line, an unbounded interval, a union of two unbounded intervals, and it was conjectured also when K is a union of an unbounded interval and a point or a union of two unbounded intervals and a point. In this paper, we confirm this conjecture by solving the truncated matrix-valued moment problem (TMMP) on a union of a bounded interval and a point. The presented technique for solving the corresponding TMMP can potentially be used to determine degree bounds in the positivity certificates for matrix polynomials on compact sets K.
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... Using the Schur complement , , this marginalization results in: ...
Performance Analysis of FGO Windowing Strategies for PDR+GNSS Fusion Architecture
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Amjad Hussain Magsi
Luis Enrique Díez Blanco
Smartphone-based pedestrian navigation has become increasingly popular due to the widespread availability of low-cost Android devices. However, these smartphones often suffer from low positional accuracy caused by inexpensive GNSS antennas and receivers. To address these limitations, fusion mechanisms are employed, with Factor Graph Optimization (FGO) emerging as a prominent method in navigation solutions, particularly for INS+GNSS integration. While FGO has also gained popularity in Simultaneous Localization and Mapping (SLAM) applications, its reliance on batch data processing, referred to as Batch Factor Graph Optimization (BFGO) and which is used as the baseline for this study, results in significant computational overhead, making it impractical for real-time or resource-constrained scenarios. To address these challenges, two windowing-based FGO approaches are proposed: the Naive Windowing FGO (NW-FGO) and the Marginalized Windowing FGO (MW-FGO). By incorporating marginalization, MWFGO retains critical information while reducing computational costs. Both methods are evaluated under two error conditions GNSS multipath and PDR errors to assess their impact on positional accuracy and efficiency. Our findings reveal that optimal window sizes, such as 50s for GNSS errors and 30s for PDR errors, achieve accuracy comparable to BFGO while significantly reducing computational time from 2500 seconds to 38 seconds with MWFGO. Furthermore, MWFGO outperforms NW-FGO, improving mean positional accuracy by 2.18% across varying window sizes, underscoring the effectiveness of marginalization. This study demonstrates the feasibility of windowing-based FGO, particularly MWFGO, in balancing computational efficiency and accuracy. By optimizing window sizes and leveraging marginalization, we provide a robust alternative to FGO, making real-time pedestrian navigation viable in resource-constrained environments.
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... According to the Schur complement , it follows that ...
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Jun 2025
Meng Kaitao
Kawon Han
Christos Masouros
L. Hanzo
Integrated sensing and communication (ISAC) networks strive to deliver both high-precision target localization and high-throughput data services across the entire coverage area. In this work, we examine the fundamental trade-off between sensing and communication from the perspective of base station (BS) deployment. Furthermore, we conceive a design that simultaneously maximizes the target localization coverage, while guaranteeing the desired communication performance. In contrast to existing schemes optimized for a single target, an effective network-level approach has to ensure consistent localization accuracy throughout the entire service area. While employing time-of-flight (ToF) based localization, we first analyze the deployment problem from a localization-performance coverage perspective, aiming for minimizing the area Cramer-Rao Lower Bound (A-CRLB) to ensure uniformly high positioning accuracy across the service area. We prove that for a fixed number of BSs, uniformly scaling the service area by a factor κincreases the optimal A-CRLB in proportion to κ^{2β}, where βis the BS-to-target pathloss exponent. Based on this, we derive an approximate scaling law that links the achievable A-CRLB across the area of interest to the dimensionality of the sensing area. We also show that cooperative BSs extends the coverage but yields marginal A-CRLB improvement as the dimensionality of the sensing area grows.
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... Based on the Schur complementary theorem , an equivalent SDP problem is ...
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Rui Zhang
Fuwei Wu
Bing Gao
Jiawei Zhang
To enhance the signature of an extended target in a SAR image, a robust waveform design method is presented for spectrally dense environments. First, the problem is formulated by maximizing the worst-case signal-to-clutter ratio (SCR) over the uncertainty set of statistics for both target and background scattering characteristics, subject to energy, similarity, and spectrum constraints. Second, the closed-form solutions for the uncertain statistics are derived. The problem of maximizing worst-case SCR is boiled down to a nonconvex fractional quadratically constrained quadratic problem (QCQP). Resorting to the Dinkelbach’s algorithm and Lagrange duality, the QCQP is split into a series of solvable semidefinite programming problems. A convergence analysis is conducted, where a sufficient condition for global convergence is derived. Finally, numerical examples are presented to demonstrate the performance of the proposed scheme.
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... It follows from Page 20 of that the inverse of K σ (x(u), π(u), Y(u)) can be expressed as ...
Characterization of the Convergence Rate of the Augmented Lagrange for the Nonlinear Semidefinite Optimization Problem
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Yule Zhang
Jia Wu
Jihong Zhang
Haoyang Liu
The convergence rate of the augmented Lagrangian method (ALM) for solving the nonlinear semidefinite optimization problem is studied. Under the Jacobian uniqueness conditions, when a multiplier vector (π,Y) and the penalty parameter σ are chosen such that σ is larger than a threshold σ>0 and the ratio ∥(π,Y)−(π,Y)∥/σ is small enough, it is demonstrated that the convergence rate of the augmented Lagrange method is linear with respect to ∥(π,Y)−(π,Y)∥ and the ratio constant is proportional to 1/σ, where (π,Y) is the multiplier corresponding to a local minimizer. Furthermore, by analyzing the second-order derivative of the perturbation function of the nonlinear semidefinite optimization problem, we characterize the rate constant of local linear convergence of the sequence of Lagrange multiplier vectors produced by the augmented Lagrange method. This characterization shows that the sequence of Lagrange multiplier vectors has a Q-linear convergence rate when the sequence of penalty parameters {σk} has an upper bound and the convergence rate is superlinear when {σk} is increasing to infinity.
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... Applying the Schur complement to γ ⊤ γ ≤ β 2 (which is equivalent to ∥γ∥≤β in (14c)), we get the equivalent linear matrix inequality (LMI) βI s γ γ ⊤ β ⪰ 0 and subsequently optimization problem (14) can be recast as an optimization problem with a convex objective function and a set of linear matrix/vector inequalities along with a matrix equation with bi-linear terms as follows: ...
Improving Power Systems Controllability Via Edge Centrality Measures
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Jan 2025
MirSaleh Bahavarnia
Muhammad Nadeem
Ahmad F. Taha
Improving the controllability of power networks is crucial as they are highly complex networks operating in synchrony; even minor perturbations can cause desynchronization and instability. To that end, one needs to assess the criticality of key network components (buses and lines) in terms of their impact on system performance. Traditional methods to identify the key nodes/edges in power networks often rely on static centrality measures based on the network's topological structure ignoring the network's dynamic behavior. In this paper, using multimachine power network models and a new control-theoretic edge centrality matrix (ECM) approach, we: (i) quantify the influence of edges (i.e., the line susceptances) in terms of controllability performance metrics, (ii) identify the most influential lines, and (iii) compute near-optimal edge modifications that improve the power network controllability. Employing various IEEE power network benchmarks, we validate the effectiveness of the ECM-based algorithm and demonstrate improvements in system reachability, control, and damping performance.
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... Applying the Schur complement to γ ⊤ γ ≤ β 2 (which is equivalent to ∥γ∥ ≤ β in (14c)), we get the equivalent linear matrix inequality (LMI) βI s γ γ ⊤ β ⪰ 0 and subsequently optimization problem (14) can be recast as an optimization problem with a convex objective function and a set of linear matrix/vector inequalities along with a matrix equation with bi-linear terms as follows: ...
Improving Power Systems Controllability via Edge Centrality Measures
Preprint
Full-text available
May 2025
MirSaleh Bahavarnia
Muhammad Nadeem
Ahmad F. Taha
Improving the controllability of power networks is crucial as they are highly complex networks operating in synchrony; even minor perturbations can cause desynchronization and instability. To that end, one needs to assess the criticality of key network components (buses and lines) in terms of their impact on system performance. Traditional methods to identify the key nodes/edges in power networks often rely on static centrality measures based on the network's topological structure ignoring the network's dynamic behavior. In this paper, using multi-machine power network models and a new control-theoretic edge centrality matrix (ECM) approach, we: (i) quantify the influence of edges (i.e., the line susceptances) in terms of controllability performance metrics, (ii) identify the most influential lines, and (iii) compute near-optimal edge modifications that improve the power network controllability. Employing various IEEE power network benchmarks, we validate the effectiveness of the ECM-based algorithm and demonstrate improvements in system reachability, control, and damping performance.
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... The first level of restriction is physics based, and is primarily motivated by the multi-physics nature of certain problems such as the interaction between velocity and pressure in fluid dynamics. This approach is based on two key insights: (1) in many cases, one field can be effectively reduced in terms of another, which is rigorously justified in linear problems through the computation of the Schur complement matrix [Zha06]. This suggests that it is possible to solve the least squares problem using only one of the sub-physics without loss of information. ...
Two-Level Sketching Alternating Anderson acceleration for Complex Physics Applications
Preprint
Full-text available
May 2025
Nicolas Barnafi
Massimiliano Lupo Pasini
We present a novel two-level sketching extension of the Alternating Anderson-Picard (AAP) method for accelerating fixed-point iterations in challenging single- and multi-physics simulations governed by discretized partial differential equations. Our approach combines a static, physics-based projection that reduces the least-squares problem to the most informative field (e.g., via Schur-complement insight) with a dynamic, algebraic sketching stage driven by a backward stability analysis under Lipschitz continuity. We introduce inexpensive estimators for stability thresholds and cache-aware randomized selection strategies to balance computational cost against memory-access overhead. The resulting algorithm solves reduced least-squares systems in place, minimizes memory footprints, and seamlessly alternates between low-cost Picard updates and Anderson mixing. Implemented in Julia, our two-level sketching AAP achieves up to 50% time-to-solution reductions compared to standard Anderson acceleration-without degrading convergence rates-on benchmark problems including Stokes, p-Laplacian, Bidomain, and Navier-Stokes formulations at varying problem sizes. These results demonstrate the method's robustness, scalability, and potential for integration into high-performance scientific computing frameworks. Our implementation is available open-source in the AAP.jl library.
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... In essence, the S-procedure lemma facilitates the conversion of a quadratic inequality into a more manageable form, such as a set of linear matrix inequalities (LMIs), which can then be employed in optimization problems. • Schur Complement : The Schur complement is a technique used in linear algebra to reduce a block matrix to a simpler form and analyze positive definiteness, invertibility, or solve systems of equations. The Schur complement is frequently employed to transform nonlinear matrix inequalities into linear matrix inequalities, thereby rendering them manageable in the context of convex optimization. ...
Physical Layer Security for Integrated Sensing and Communication: A Survey
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Apr 2025
Toshiki Matsumine
Hideki Ochiai
Junji Shikata
Integrated sensing and communication (ISAC) has become a crucial technology in the development of next-generation wireless communication systems. The integration of communication and sensing functionalities on a unified spectrum and infrastructure is expected to enable a variety of emerging use cases. The introduction of ISAC has led to various new challenges and opportunities related to the security of wireless communications, resulting in significant research focused on ISAC system design in relation to physical layer security (PLS). The shared use of spectrum presents a risk where confidential messages embedded in probing ISAC signals may be exposed to potentially malicious sensing targets. This situation creates a tradeoff between sensing performance and security performance. The sensing functionality of ISAC offers a unique opportunity for PLS by utilizing sensing information regarding potential eavesdroppers to design secure PLS schemes. This study examines PLS methodologies to tackle the specified security challenge associated with ISAC. The study begins with a brief overview of performance metrics related to PLS and sensing, as well as the optimization techniques commonly utilized in the existing literature. A thorough examination of existing literature on PLS for ISAC is subsequently presented, with the objective of emphasizing the current state of research. The study concludes by outlining potential avenues for future research pertaining to secure ISAC systems.
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... It remains to be shown that R is positive semi-definite. We use [45,Th. 1.20] which states that this is equivalent to (i) the matrix M σ is positive semi-definite, (ii) the column spaces match, i.e., C(X foil ) ⊆ C(M σ ), (iii) the Schur complement G foil − X ⊤ foil M + σ X foil is positive semi-definite. ...
A generalized energy-based modeling framework with application to field/circuit coupled problems
Preprint
Full-text available
Apr 2025
Robert Altmann
Idoia Cortes Garcia
Elias Paakkunainen
Sebastian Schöps
This paper presents a generalized energy-based modeling framework extending recent formulations tailored for differential-algebraic equations. The proposed structure, inspired by the port-Hamiltonian formalism, ensures passivity, preserves the power balance, and facilitates the consistent interconnection of subsystems. A particular focus is put on low-frequency power applications in electrical engineering. Stranded, solid, and foil conductor models are investigated in the context of the eddy current problem. Each conductor model is shown to fit into the generalized energy-based structure, which allows their structure-preserving coupling with electrical circuits described by modified nodal analysis. Theoretical developments are validated through a numerical simulation of an oscillator circuit, demonstrating energy conservation in lossless scenarios and controlled dissipation when eddy currents are present.
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... Thus, Q(γ) k is positive definite and so has only positive eigenvalues. Applying either [35, Definition 1.59, Property a] or Schur determinant formula to the characteristic polynomial of Q(γ) we get: ...
Mathematical programs with complementarity constraints and application to hyperparameter tuning for nonlinear support vector machines
Preprint
Full-text available
Apr 2025
Samuel Ward
Alain B. Zemkoho
Selin Damla AHIPASAOGLU
We consider the Mathematical Program with Complementarity Constraints (MPCC). One of the main challenges in solving this problem is the systematic failure of standard Constraint Qualifications (CQs). Carefully accounting for the combinatorial nature of the complementarity constraints, tractable versions of the Mangasarian Fromovitz Constraint Qualification (MFCQ) have been designed and widely studied in the literature. This paper looks closely at two such MPCC-MFCQs and their influence on MPCC algorithms. As a key contribution, we prove the convergence of the sequential penalisation and Scholtes relaxation algorithms under a relaxed MPCC-MFCQ that is much weaker than the CQs currently used in the literature. We then form the problem of tuning hyperparameters of a nonlinear Support Vector Machine (SVM), a fundamental machine learning problem for classification, as a MPCC. For this application, we establish that the aforementioned relaxed MPCC-MFCQ holds under a very mild assumption. Moreover, we program robust implementations and comprehensive numerical experimentation on real-world data sets, where we show that the sequential penalisation method applied to the MPCC formulation for tuning SVM hyperparameters can outperform both the Scholtes relaxation technique and the state-of-the-art derivative-free methods from the machine learning literature.
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... Finally, we integrate these innovations into SCHURCFCM, proving its complexity and approximation guarantee (Theorem 4.7). Then L −1 −S can be represented as : ...
Fast Maximization of Current Flow Group Closeness Centrality
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Full-text available
Apr 2025
Haisong Xia
Zhongzhi Zhang
Derived from effective resistances, the current flow closeness centrality (CFCC) for a group of nodes measures the importance of node groups in an undirected graph with n nodes. Given the widespread applications of identifying crucial nodes, we investigate the problem of maximizing CFCC for a node group S subject to the cardinality constraint ∣S∣=k≪n|S|=k\ll n. Despite the proven NP-hardness of this problem, we propose two novel greedy algorithms for its solution. Our algorithms are based on spanning forest sampling and Schur complement, which exhibit nearly linear time complexities and achieve an approximation factor of 1−k k−1 1 e−ϵ 1-\frac{k}{k-1}\frac{1}{\mathrm{e}}-\epsilon for any 0<ϵ<1 0<\epsilon<1. Extensive experiments on real-world graphs illustrate that our algorithms outperform the state-of-the-art method in terms of efficiency and effectiveness, scaling to graphs with millions of nodes.
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... Moreover, to attain an LMI, we employ the Schur complement with symmetric positive-definite matrix to transform (55) into ...
Robust Joint Active and Passive Beamforming for Reconfigurable Intelligent Surface Assisted Full-Duplex Transmissions Under Imperfect Channels
Article
Full-text available
Jan 2025
Li-Hsiang Shen
Chia-Jou Ku
Kai-Ten Feng
The sixth-generation (6G) wireless technology recognizes the potential of reconfigurable intelligent surfaces (RIS) as an effective technique for intelligently manipulating channel paths through reflection to serve desired users. Full-duplex (FD) systems, enabling simultaneous transmission and reception from a base station (BS), offer the theoretical advantage of doubled spectrum efficiency. However, the presence of strong self-interference (SI) in FD systems significantly degrades performance, which can be mitigated by leveraging the capabilities of RIS. Moreover, accurately obtaining channel state information (CSI) from RIS poses a critical challenge. Our objective is to maximize downlink (DL) user data rates while ensuring quality-of-service (QoS) for uplink (UL) users under imperfect CSI from reflected channels. To address this, we propose a robust active BS and passive RIS beamforming (RAPB) scheme for RIS-FD, accounting for both SI and imperfect CSI. RAPB incorporates distributionally robust design, conditional value-at-risk (CVaR), and penalty convex-concave programming (PCCP) techniques. Simulation results demonstrate the UL/DL rate improvement are achieved by considering different levels of imperfect CSI. The proposed RAPB schemes validate their effectiveness across different RIS deployments and RIS/BS configurations. Benefited from robust beamforming, RAPB outperforms the existing methods in terms of non-robustness, deployment without RIS, conventional approximation, and half-duplex systems.
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... Results on the Schur complement of special H-matrices including spectra localizations and closure properties can be found in [9,11,12,27,39,48]. ...
New scaling criteria for H -matrices and applications
Article
Full-text available
Jan 2025
Maja Nedović
Dunja Arsić
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... and the Schur complement theorem guarantees that ...
Gaussian Blahut-Arimoto Algorithm for Capacity Region Calculation of Gaussian Vector Broadcast Channels
Preprint
Mar 2025
Tian Jiao
Yanlin Geng
Yonghui Chu
Zai Yang
This paper is concerned with the computation of the capacity region of a continuous, Gaussian vector broadcast channel (BC) with covariance matrix constraints. Since the decision variables of the corresponding optimization problem are Gaussian distributed, they can be characterized by a finite number of parameters. Consequently, we develop new Blahut-Arimoto (BA)-type algorithms that can compute the capacity without discretizing the channel. First, by exploiting projection and an approximation of the Lagrange multiplier, which are introduced to handle certain positive semidefinite constraints in the optimization formulation, we develop the Gaussian BA algorithm with projection (GBA-P). Then, we demonstrate that one of the subproblems arising from the alternating updates admits a closed-form solution. Based on this result, we propose the Gaussian BA algorithm with alternating updates (GBA-A) and establish its convergence guarantee. Furthermore, we extend the GBA-P algorithm to compute the capacity region of the Gaussian vector BC with both private and common messages. All the proposed algorithms are parameter-free. Lastly, we present numerical results to demonstrate the effectiveness of the proposed algorithms.
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... This can further help us to analyze the relationship of the two conditional information gains. Since the covariance matrix K pmt , K \m and K m ,m are both positive semi-definite (PSD), then according to the Schur complement theorem , we know K m ,m −B T (K \m +σ −2 ϵ I) −1 B and K \m + σ −2 ϵ I are also PSD. Then, considering the Minkowski determinant inequality that for PSD matrices C and D, we have |C+D| ≥ |C|+|D| ≥ |C|. ...
(θ l,θ u\boldsymbol{\theta}_l, \boldsymbol{\theta}_u)-Parametric Multi-Task Optimization: Joint Search in Solution and Infinite Task Spaces
Preprint
Full-text available
Mar 2025
Tingyang Wei
Jiao Liu
Abhishek Gupta
Yew Soon Ong
Multi-task optimization is typically characterized by a fixed and finite set of optimization tasks. The present paper relaxes this condition by considering a non-fixed and potentially infinite set of optimization tasks defined in a parameterized, continuous and bounded task space. We refer to this unique problem setting as parametric multi-task optimization (PMTO). Assuming the bounds of the task parameters to be (θ l\boldsymbol{\theta}_l, θ u\boldsymbol{\theta}_u), a novel (θ l\boldsymbol{\theta}_l, θ u\boldsymbol{\theta}_u)-PMTO algorithm is crafted to enable joint search over tasks and their solutions. This joint search is supported by two approximation models: (1) for mapping solutions to the objective spaces of all tasks, which provably accelerates convergence by acting as a conduit for inter-task knowledge transfers, and (2) for probabilistically mapping tasks to the solution space, which facilitates evolutionary exploration of under-explored regions of the task space. At the end of a full (θ l\boldsymbol{\theta}_l, θ u\boldsymbol{\theta}_u)-PMTO run, the acquired models enable rapid identification of optimized solutions for any task lying within the specified bounds. This outcome is validated on both synthetic test problems and practical case studies, with the significant real-world applicability of PMTO shown towards fast reconfiguration of robot controllers under changing task conditions. The potential of PMTO to vastly speedup the search for solutions to minimax optimization problems is also demonstrated through an example in robust engineering design.
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... A similar structure can be observed in the two-block Gauss-Newton normal equations. The Schur complement [11,16] offers a means of reducing the system dimension by eliminating one of the block variables. For example, if A is invertible, we can write ...
A Nonlinear Extension of the Variable Projection (VarPro) Method for NURBS-based Conformal Surface Flattening
Preprint
Full-text available
Feb 2025
Masaaki Miki
In the field of computer graphics, conformal surface flattening has been widely studied for tasks such as texture mapping, geometry processing, and mesh generation. Typically, existing methods aim to flatten a given input geometry while preserving conformality as much as possible, meaning the result is only as conformal as possible. By contrast, this study focuses on surfaces that can be flattened conformally without singularities, making the process a coupled problem: the input (or target) surface must be recursively refined while its flattening is computed. Although the uniformization theorem or the Riemann mapping theorem guarantees the existence of a conformal flattening for any simply connected, orientable surface, those theorems permit singularities in the flattening. If singularities are not allowed, only a special class of surfaces can be conformally flattened-though many practical surfaces do fall into this class. To address this, we develop a NURBS-based approach in which both the input surface and its flattening are refined in tandem, ensuring mutual conformality. Because NURBS surfaces cannot represent singularities, the resulting pair of surfaces is naturally singularity-free. Our work is inspired by the form-finding method by [Miki and Mitchell 2022, 2024], which solves bilinear PDEs by iteratively refining two surfaces together. Building on their demonstration of the effectiveness of variable projection (VarPro), we adopt a similar strategy: VarPro alternates between a linear projection and a nonlinear iteration, leveraging a partially linear (separable) problem structure. However, since our conformal condition separates into two nonlinear subproblems, we introduce a nonlinear extension of VarPro. Although this significantly increases computational cost, the quality of the results is noteworthy.
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A Convex Optimization Framework for Computing Robustness Margins of Kalman Filters
Conference Paper
Jul 2025
Himanshu Prabhat
Raktim Bhattacharya
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Location Optimization for Fluid Antenna-Assisted Near-Field System
Conference Paper
Dec 2024
Jingxuan Zhou
Yinchao Yang
Zhaohui Yang
M. Shikh-Bahaei
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A Learned Approach to Adaptive Sampling for Seabed Identification with Autonomous Vehicles
Conference Paper
Jun 2025
John Lipor
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Interlacing Polynomial Method for Matrix Approximation via Generalized Column and Row Selection
Article
Jul 2025
FOUND COMPUT MATH
Jian-Feng Cai
Zhiqiang Xu
Zili Xu
This paper delves into the spectral norm aspect of the Generalized Column and Row Subset Selection (GCRSS) problem. Given a target matrix A∈R n×d\textbf{A}\in \mathbb {R}^{n\times d}, the objective of GCRSS is to select a column submatrix B:,S∈R n×k\textbf{B}{:,S}\in \mathbb {R}^{n\times k} from the source matrix B∈R n×d B\textbf{B}\in \mathbb {R}^{n\times d_B} and a row submatrix C R,:∈R r×d\textbf{C}{R,:}\in \mathbb {R}^{r\times d} from the source matrix C∈R n C×d\textbf{C}\in \mathbb {R}^{n_C\times d}, such that the residual matrix (I n−B:,S B:,S†)A(I d−C R,:†C R,:)(\textbf{I}n-\textbf{B}{:,S}\textbf{B}{:,S}^{\dagger })\textbf{A}(\textbf{I}_d-\textbf{C}{R,:}^{\dagger } \textbf{C}{R,:}) has a small spectral norm. By employing the method of interlacing polynomials, we show that the smallest possible spectral norm of a residual matrix can be bounded by the largest root of a related expected characteristic polynomial. A deterministic polynomial time algorithm is provided for the spectral norm case of the GCRSS problem. We next apply our results to two specific GCRSS scenarios, one where r=0, simplifying the problem to the Generalized Column Subset Selection (GCSS) problem, and the other where B=C=I d\textbf{B}=\textbf{C}=\textbf{I}_d, reducing the problem to the submatrix selection problem. In the GCSS scenario, we connect the expected characteristic polynomials to the convolution of multi-affine polynomials, leading to the derivation of the first provable reconstruction bound on the spectral norm of a residual matrix. In the submatrix selection scenario, we show that for any sufficiently small ε>0\varepsilon >0 and any square matrix A∈R d×d\textbf{A}\in \mathbb {R}^{d\times d}, there exist two subsets S⊂[d]S\subset [d] and R⊂[d]R\subset [d] of sizes O(d⋅ε 2)O(d\cdot \varepsilon ^2) such that ∥A S,R∥2≤ε⋅∥A∥2\Vert \textbf{A}{S,R}\Vert _2\le \varepsilon \cdot \Vert \textbf{A}\Vert _2. Unlike previous studies that have produced comparable results for very special cases where the matrix is either a zero-diagonal or a positive semidefinite matrix, our results apply universally to any square matrix A\textbf{A}.
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Complementable operators and their Schur complements
Article
Jul 2025
INDIAN J PURE AP MAT
Sachin Naik
P. Sam Johnson
In this paper, we characterize complementable operators and provide more precise expressions for the Schur complement of these operators using a single Douglas solution. We demonstrate the existence of subspaces where the given operator is invariably complementable. Additionally, we investigate the range-Hermitian property of these operators.
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Applications to Power Systems with Inclusion of Wide-Area Delays
Chapter
Jul 2025
Hua Ye
Yutian Liu
Xiaofan Jia
The operation and control of modern power systems always depend on a reliable information system. Based on computer, communication and sensor technologies, the primary power system is closely and organically integrated with the information system, leading to a cyber-physical power system (CPPS) [1,2,3]. With the consideration of time delays introduced by the information system, the CPPS becomes a time-delay CPPS. Accordingly, there is a demand for establishing a rounded system of small-signal stability modeling, analysis and control for time-delay CPPSs [4, 5]. To this end, this chapter illustrates applications of PSD-based eigenvalue computation methods elaborated in previous sections to time-delay CPPSs.
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Triangle, Sphere & Cuboid: A Unified Projector-Based Framework for Fractional Operators and Arithmetic Space-Forms
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Full-text available
Jul 2025
Oussama Basta
We present a compact unification of three seemingly distant objects: 1. the cyclic 3-sign system on the unit circle, 2. the OPi-BAZ projector sphere for vector-fractional calculus, and 3. the long-standing perfect-cuboid (Euler-brick) problem viewed as a "universe in a box". First, we show that the equatorial triangle of projector operators {T 0 , T 1 , T 2 } ⊂ S 2 is a faithful geometric realisation of the 3-sign group, generating a 3-dimensional 2-step, nilpotent Lie algebra whose BCH polygons encode every closed-form identity catalogued in From Circle to Sphere: A Unified Operator Projector for Fractional Calculus, Topology, and Gauge-Symmetric Physics. Next, we prove a precise longitude-latitude dictionary that identifies this triangle-sphere with the OPi-BAZ kernel sphere, establishing operator-level equivalence of the two formalisms. Turning to Cartesian geometry, we map the projector construction to the 3-torus that parametrises Euler bricks; imposing three quadratic "face-diagonal" constraints slices the torus by an elliptic curve. A hypothetical perfect cuboid is then the missing rational north-pole of that curve, mirroring the north-pole horizon of the projector sphere. Hence the same projector algebra simultaneously governs fractional-calculus identities and the arithmetic of cuboid space-time. 1 We conclude with a three-column "Theory-of-Everything" table: S 1 for signs, S 2 for operators, and T 3 /E for arithmetic boxes-one projector, three avatars.
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Efficient NVH Trim Representation with Compressed Reduced Impedance Matrices
Article
Jun 2025
André Antonio Andrade Paiva
Benoit Van den Nieuwenhof
Gregory Lielens
Efficient finite element simulations are crucial for tackling the computational demands of large-scale models and high-frequency ranges in noise, vibration, and harshness (NVH) modeling. This paper introduces novel methods for compressing reduced impedance matrices (RIM) to optimize computational resources while preserving simulation accuracy. The proposed methodology employs diverse reduction techniques, including Singular Value Decomposition (SVD), pellicular modes, and thin shell modes, to identify relevant vibration modes and optimize modal projection of impedance matrices. By eliminating less relevant modes or generating pertinent modal information of the coupling surfaces, the method aims to significantly compress RIMs without sacrificing accuracy. A comparative analysis on a car model demonstrates the effectiveness of the compression technique. The numerical results with the developed approach show significant reduction in storage and computational time, highlighting the method's ability to reduce complex NVH models with precision. This proposed approach presents a promising solution to the computational challenges inherent in NVH modeling, offering opportunities for enhanced efficiency and reliability in NVH simulations. With its capacity to reduce computational time and storage requirements, the method is well-suited for large-scale NVH models across distinct engineering industries.
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Robust Scheduling of a Charging Station Considering Uncertain Charging Demand and Demand Response
Conference Paper
Nov 2024
Yuling Ren
Mao Tan
Zibin Li
Wei Tan
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Connected and Automated Vehicle Trajectory Control in Stochastic Heterogeneous Traffic Flow with Human-Driven Vehicles Under Communication Delay and Disturbances
Article
Full-text available
May 2025
Meiqi Liu
Yang Chen
Ruochen Hao
In this paper, we study the stability of the stochastically heterogeneous traffic flow involving connected and automated vehicles (CAVs) and human-driven vehicles (HDVs). Taking the stochasticity of vehicle arrivals and behaviors into account, a general robust H∞ platoon controller is proposed to address the communication delay and unexpected disturbances such as prediction or perception errors on HDV motions. To simplify the problem complexity from a stochastically heterogeneous traffic flow to multiple long vehicle control problems, three types of sub-platoons are identified according to the CAV arrivals, and each sub-platoon can be treated as a long vehicle. The car-following behaviors of HDVs and CAVs are simulated using the optimal velocity model (OVM) and the cooperative adaptive cruise control (CACC) system, respectively. Later, the robust H∞ platoon controller is designed for a pair of a CAV long vehicle and an HDV long vehicle. The time-lagged system and the closed-loop system are formulated and the H∞ state feedback controller is designed. The robust stability and string stability of the heterogeneous platoon system are analyzed using the H∞ norm of the closed-loop transfer function and the time-lagged bounded real lemma, respectively. Simulation experiments are conducted considering various settings of platoon sizes, communication delays, disturbances, and CAV penetration rates. The results show that the proposed H∞ controller is robust and effective in stabilizing disturbances in the stochastically heterogeneous traffic flow and is scalable to arbitrary sub-platoons in various CAV penetration rates in the heterogeneous traffic flow of road vehicles. The advantages of the proposed method in stabilizing heterogeneous traffic flow are verified in comparison with a typical car-following model and the linear quadratic regulator.
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On generalized Schur complement of matrices and its applications to real and integer matrix factorizations
Article
Full-text available
Jan 2022
Effat Golpar-Raboky
We provide a general finite iterative approach for constructing factorizations of a matrix A under a common framework of a general decomposition A = BC based on the generalized Schur complement. The approach applies a zeroing process using two index sets. Different choices of the index sets lead to different real and integer matrix factorizations. We also provide the conditions under which this approach is well-defined.
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QoS-Based Beamforming and Compression Design for Cooperative Cellular Networks via Lagrangian Duality
Article
Jan 2025
IEEE T SIGNAL PROCES
Xilai Fan
Ya-Feng Liu
Liang Liu
Tsung-Hui Chang
This paper considers the quality-of-service (QoS)-based joint beamforming and compression design problem in the downlink cooperative cellular network, where multiple relay-like base stations (BSs), connected to the central processor via ratelimited fronthaul links, cooperatively transmit messages to the users. The problem of interest is formulated as the minimization of the total transmit power of the BSs, subject to all users’ signal-to-interference-plus-noise ratio (SINR) constraints and all BSs’ fronthaul rate constraints. In this paper, we first show that there is no duality gap between the considered joint optimization problem and its Lagrangian dual by showing the tightness of its semidefinite relaxation (SDR). Then, we propose an efficient algorithm based on the above duality result for solving the considered problem. The proposed algorithm judiciously exploits the special structure of an enhanced Karush-Kuhn-Tucker (KKT) conditions of the considered problem and approaches the solution that satisfies the enhanced KKT conditions via two fixed point iterations. Two key features of the proposed algorithm are: (1) it is able to detect whether the considered problem is feasible or not and find its globally optimal solution when it is feasible; (2) it is highly efficient because both of the fixed point iterations in the proposed algorithm are linearly convergent and function evaluations in the fixed point iterations are computationally cheap. Numerical results show the global optimality and efficiency of the proposed algorithm.
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Convergence of Complementable Operators
Article
Apr 2025
LINEAR ALGEBRA APPL
Sachin Naik
P. Sam Johnson
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High-speed computation method for condition numbers in the range restricted general minimum residual method
Article
Full-text available
Apr 2025
J SUPERCOMPUT
Miho Chiyonobu
Masami Takata
Kinji Kimura
Yoshimasa Nakamura
In this study, we propose a method for fast computation of the stopping condition of the range restricted general minimum residual (RRGMRES) method. The RRGMRES method is iterative, causing the matrix size to increase with the number of iterations. The stopping criterion for the iterations is based on the condition number. The condition number was expressed as the ratio of the largest singular value to the smallest singular value. In the RRGMRES method, the computation of the condition number of the upper triangular matrix is a bottleneck. Conventional methods compute the smallest and largest singular values required for the condition number in each iteration. Conversely, the proposed method leverages the increasing matrix size with each iteration and utilizes the results from the previous iteration. In the proposed method, the smallest singular value can be obtained with high speed and accuracy using the inverse iteration method, and the largest singular value can be converted into a problem of computing the smallest singular value: using the inverse matrix. However, the matrix in this study may contain clusters of singular values around the largest singular value. Therefore, the largest singular values had to be separated from the clusters by using the Cholesky LR method. The accuracy of RRGMRES using the proposed method was nearly equal to that of the conventional method. By comparing the experimental results with those obtained using conventional methods, the proposed method was 10 times faster than the conventional method.
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Bias-corrected estimation for G I 0\mathcal{G}^0_I regression with applications
Article
Mar 2025
ASTA-ADV STAT ANAL
M. F. S. S. Sousa
Josimar Mendes de Vasconcelos
Abraão D. C. Nascimento
Synthetic aperture radar (SAR) systems are highly efficient tools for addressing remote sensing challenges. They offer several advantages, such as operating independently of atmospheric conditions and producing high spatial resolution images. However, SAR images are often contaminated by a type of interference called speckle noise, which complicates their analysis and processing. Therefore, proposing statistical methods, such as regression models, that account for speckle behavior is an important step for users of SAR systems. In the work [ISPRS J. Photogramm. Remote Sens., 213, 1–13, 2024], the G I 0{\mathcal{G}^{0}{I}} regression model (short for R G I 0\mathcal{R} {\mathcal{G}^{0}{I}}) was proposed as an interpretable tool to relate SAR intensity features to other physical properties. The authors employed maximum likelihood estimators (MLEs), known for their good asymptotic properties but prone to considerable bias in small and medium sample sizes. In this paper, we propose a matrix expression for the second-order bias of MLEs for R G I 0\mathcal{R} {\mathcal{G}^{0}_{I}} parameters, based on the Cox and Snell method. This proposal is justified by the necessity of using small and moderate windows when processing SAR images, such as for classification and filtering purposes. We compare bias-corrected MLEs with their counterparts using both Monte Carlo experiments and an application to SAR data from a Brazilian region. Numerical evidence demonstrates the effectiveness of our proposal.
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Resistance distance and Kirchhoff index of the splitting-joins of two graphs
Article
Jan 2024
Yanan Li
Xiaoling Ma
Shian Deng
Dandan Chen
Let G be agraph. The splitting graph SP(G) of G is the graph received from G by putting a new vertex w? for each w ? VG and joining w? to all vertices of G adjacent to w. Let SG be the set of such new vertices of the splitting graph SP(G). Let G1 and G2 be two simple connected graphs, the splitting V-vertex join graph is obtained by taking one copy of SP(G1) and joining each vertex in VG1 to each vertex in VG2, denoted by G1 ? G2 . The splitting S-vertex join of G1 and G2, denoted by G1 ? G2, is a graph obtained from SP(G1) and G2 by joining each vertex in SG1 to each vertex in VG2. In this paper, we calculate the resistance distance and Kirchhoff index of1 G ?2 G and1 G ?2 G for regular graphs1 G and2 G, respectively.
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Fast Computation Method for Stopping Condition of Range Restricted GMRES Method
Chapter
Mar 2025
Miho Chiyonobu
Masami Takata
Kinji Kimura
Yoshimasa Nakamura
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Finite-time model-based event-triggered H∞H∞{\mathcal{H}}_{\infty } control for fuzzy T–S system
Article
Full-text available
Mar 2025
Davood Nazari Maryam Abadi
Ali Moarefianpour
Nima Mahdian Dehkordi
In this paper, finite-time H∞{\mathcal{H}}{\infty } control is studied for discrete-time Takagi–Sugeno (T–S) fuzzy systems. A dynamic model-based event-triggered mechanism (DMBETM) is utilized to monitor the data transmission from the plant to the controller, which reduces the amount of transmitted data more efficiently than conventional static mechanisms. First, it is demonstrated that the adopted DMBETM can avoid Zeno behavior and yield a larger minimal inter-execution time compared to the static model-based mechanism. Subsequently, an enhanced finite-time H∞{\mathcal{H}}{\infty } performance criterion is proposed, by employing a new Lyapunov-like function that incorporates an internal dynamic variable. Finally, the finite-time H∞{\mathcal{H}}_{\infty } control of the closed-loop system is being investigated for a DC–DC boost converter.
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Robust Beamforming Design for Monostatic ISAC Systems Based on Minimum Mean-Square Error Estimation
Conference Paper
Dec 2024
Chao Chen
Cui Yuanhao
Ronghui Zhang
Xiaojun Jing
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Safety Barrier Certificates for Stochastic Control Systems with Wireless Communication Networks
Conference Paper
Dec 2024
Omid Akbarzadeh
Sadegh Soudjani
Abolfazl Lavaei
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Retrofit Control of DC Microgrids: A Reliability-oriented Control Approach
Conference Paper
Dec 2024
Mahdieh S. Sadabadi
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Extremal Characterizations of the Schur Complement and Resulting Inequalities
Article
Full-text available
Aug 2006
SIAM REV
Chi-Kwong Li
Roy Mathias
Let H=\pmatrix{H_{11}&H_{12}\cr H_{12}^&H_{22}} be an n×n n\times n positive semidefinite matrix, where H 11 H_{11} is k×k k\times k with 1≤k<n 1 \le k < n. The {\it generalized Schur complement} of H 11 H_{11} in H is defined as S(H)=H 22−H 12∗H 11†H 12, S(H) = H_{22} - H_{12}^H_{11}^{\dag}H_{12}, where H 11†H_{11}^{\dag} is the Moore\,--Penrose generalized inverse of H 11 H_{11}. It has theextremal characterizations S(H) = \max\left{W: H - \pmatrix{0_k & 0 \cr 0 & W\cr} \ge 0, W \hbox{ is } (n-k) \times (n-k) \hbox{ Hermitian} \right} and\rule[-4.75pt]{0pt}{0pt} S(H)=min{[Z∣I n−k]H[Z∣I n−k]∗:Z is(n−k)×k}. S(H) = \min \left{[Z|I_{n-k}] H [Z|I_{n-k}]^: Z \ \hbox{ is } \ (n-k) \times k \right}. \noindent These characterizations are used to deduce many old and new inequalities for Schur complements of positive semidefinite matrices. In many cases, stronger statements and shorter proofs can be obtained using the extremal characterizations.
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Epsilon algorithm and related topics
Article
Oct 2000
J COMPUT APPL MATH
P. R. Graves-Morris
D.E. Roberts
A. Salam
The epsilon algorithm is recommended as the best all-purpose acceleration method for slowly converging sequences. It exploits the numerical precision of the data to extrapolate the sequence to its limit. We explain its connections with Padé approximation and continued fractions which underpin its theoretical base. Then we review the most recent extensions of these principles to treat application of the epsilon algorithm to vector-valued sequences, and some related topics. In this paper, we consider the class of methods based on using generalised inverses of vectors, and the formulation specifically includes the complex case wherever possible.
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What are Schur complements, anyway?
Article
Feb 1986
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B Partial Fraction Decomposition - "Telescoping sum"
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The discussion revolves around using partial fraction decomposition to simplify the sum of fractions of the form 1/(k(k+1)). The decomposition reveals a telescoping nature, where most terms cancel out, leaving only the first and last terms of the series. Participants explore how to identify when to apply partial fraction decomposition, emphasizing the importance of recognizing patterns through experience and experimentation. They also discuss the general concept of telescoping sums and the method of summation by anti-differencing, which can be useful in deriving closed-form formulas for series. Understanding these techniques is crucial for future applications, especially in calculus.
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There is a problem in a PreCalculus book that I'm going over that states:Express the sum ##\frac{1}{2⋅3}+\frac{1}{3⋅4}+\frac{1}{4⋅5}+...+\frac{1}{2019⋅2020}## as a fraction of whole numbers in lowest terms.It goes on to state that each term in the sum is of the form ##\frac{1}{k\left(k+1\right)}## which is obvious.The partial fraction decomposition of this is ##\frac{1}{k\left(k+1\right)}=\frac{1}{k}-\frac{1}{k+1}## Again, no problems here.Now it follows that by using this decomposition for each term, we get:##\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\left(\frac{1}{4}-\frac{1}{5}\right)+...+\left(\frac{1}{2019}-\frac{1}{2020}\right)##After removing the parenthesis, the only terms that are left, after adding them all up, is the first and last term.My question is, in being presented with this problem, what "flags" if you will, should we be looking that tells us to use partial fraction decomposition? In trying this problem myself, I got to just the basic setup of ##\frac{1}{k\left(k+1\right)}##, but from here, I would never guess that it needed to be decomposed.
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And an additional question- we can clearly see that after the parentheses are removed, that terms start cancelling, but what's to say that 20 terms down the line, they don't cancel? How can we be sure?
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That's a matter of experience or persistence. If you try long enough without any efforts, you finally try everything and the one working path will be among them. On the other hand does experience a similar job. There are a handful of tricks which by experience migrate into your standard repertoire and partial fractions are among them.The usual way, however, is to check small numbers and see if you can recognize a pattern. Latest at ##10## the pattern should be obvious.
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opus said:
And an additional question- we can clearly see that after the parentheses are removed, that terms start cancelling, but what's to say that 20 terms down the line, they don't cancel? How can we be sure?
Write the sum with more terms. In the expression below, where you have the ellipsis (...), fill in three or four general terms (terms involving n), and you can see which terms cancel with which other terms. ##\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\left(\frac{1}{4}-\frac{1}{5}\right)+...+\left(\frac{1}{2019}-\frac{1}{2020}\right)##
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opus said:
And an additional question- we can clearly see that after the parentheses are removed, that terms start cancelling, but what's to say that 20 terms down the line, they don't cancel? How can we be sure?
Write it with sums: $$ \sum_{k=2}^{2019}\dfrac{1}{k(k+1)}=\sum_{k=2}^{2019}\left( \dfrac{1}{k}-\dfrac{1}{k+1} \right)=\sum_{k=2}^{2019}\dfrac{1}{k}-\sum_{k=3}^{2020}\dfrac{1}{k}=\sum_{k=2}^{2}\dfrac{1}{k}-\sum_{k=2020}^{2020}\dfrac{1}{k}+\sum_{k=3}^{2019}\left(\dfrac{1}{k}-\dfrac{1}{k} \right)=\dfrac{1}{2}-\dfrac{1}{2020} $$
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fresh_42 said:
That's a matter of experience or persistence. If you try long enough you without any efforts, you finally try everything and the one working path will be among them. On the other hand does experience a similar job. There are a handful of tricks which by experience migrate into your standard repertoire and partial fractions are among them.The usual way, however, is to check small numbers and see if you can recognize a pattern. Latest at ##10## the pattern should be obvious.
That sounds fair. So this isn't a cover-all case, and there can be similar cases which wouldn't necessarily have these same steps and reasoning?
Mark44 said:
Write the sum with more terms. In the expression below, where you have the ellipsis (...), fill in three or four general terms (terms involving n), and you can see which terms cancel with which other terms.##\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\left(\frac{1}{4}-\frac{1}{5}\right)+...+\left(\frac{1}{2019}-\frac{1}{2020}\right)##
Ok so once we can start to see a pattern of reasonable length, we can safely assume that the pattern will continue?
fresh_42 said:
Write it with sums:$$\sum_{k=2}^{2019}\dfrac{1}{k(k+1)}=\sum_{k=2}^{2019}\left( \dfrac{1}{k}-\dfrac{1}{k+1} \right)=\sum_{k=2}^{2019}\dfrac{1}{k}-\sum_{k=3}^{2020}\dfrac{1}{k}=\sum_{k=2}^{2}\dfrac{1}{k}-\sum_{k=2020}^{2020}\dfrac{1}{k}+\sum_{k=3}^{2019}\left(\dfrac{1}{k}-\dfrac{1}{k} \right)=\dfrac{1}{2}-\dfrac{1}{2020}$$
What do those big E symbols mean? Is that Calculus? I haven't been introduced to those yet.
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That sounds fair. So this isn't a cover-all case, and there can be similar cases which wouldn't necessarily have these same steps and reasoning?
Yes. After a few dozens of (different) cases, in which partial fractions helped, you automatically get used to check this possibility. Latest if it comes to integration, you should remember partial fraction decomposition.
What do those big E symbols mean? Is that Calculus? I haven't been introduced to those yet.
That is the big Greek letter ##S## which stands for "sum". It's an abbreviation to avoid those dots. It simply means $$ \sum_{i=1}^{N} a_i = a_1 +a_2 + a_3 + a_4 + ... \text{ many } a_i \text{ later} \ldots +a_{N-2} +a_{N-1}+a_N $$ It is a method to handle such sums by administrating the indices instead.
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Ok that makes sense. So it's safe to say that partial fraction decomposition is a very important skill and I'll be using it later?
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Ok that makes sense. So it's safe to say that partial fraction decomposition is a very important skill and I'll be using it later?
Probably. For integration it is an important tool. See in your example: ##\dfrac{1}{k^2+k}## is much more inconvenient than ##\dfrac{1}{k}## or ##\dfrac{1}{k+1}## is.
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Ok great. Thanks guys.
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opus said:
My question is, in being presented with this problem, what "flags" if you will, should we be looking that tells us to use partial fraction decomposition?
The aspect of partial fractions is a special case for some series, however the idea of "telescoping sum" is quite general. In general, if you want to find a "closed form" formula for the series ##\sum_{i=1}^n f_i = f_1 + f_2 + ... f_n## you try to find a function ##F(k)## such that ##F(k+1) - F(k) = f_k##. This technique is called "summation by anti-differencing". ##\sum_{i=1}^n f_i = f_1 + f_2 + ...f_n = (F(2)-F(1) + (F(3) - F(2)) + ...(F(n+1)-F(n)) ## ## = F(n+1) - F(1)##. Results like this are studied in the "Calculus of Finite Differences", which is often taught after people have studied calculus, but can be profitably studied before taking calculus. For example, the result ##\sum_{i=1}^n = 1 + 2 + ...n = n(n+2)/2 ## is often taught by telling how the young Gauss derived the formula, but a more general way is to look for a second degree polynomial ##F(k) = Ak^2 + Bk + C## such that ##F(k+1) - F(k) =k ##. This leads to ##A = 1/2, B= -1/2## and ##C## an arbitrary constant. That approach can be applied to deriving formulae for sums like ##\sum_{i=1}^n i^3##. In that case, we'd be looking for a fourth degree polynomial.
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And an additional question- we can clearly see that after the parentheses are removed, that terms start cancelling, but what's to say that 20 terms down the line, they don't cancel? How can we be sure?
You use the final Peano axiom for "natural numbers": The principle of induction. From ( "The intuitive notion that each natural number can be obtained by applying successor sufficiently often to zero requires an additional axiom, which is sometimes called the axiom of induction.
If K is a set such that:
0 is in K, and
for every natural number n, n being in K implies that S(n) is in K,then K contains every natural number."
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12982 | https://web.evanchen.cc/exams/USAMO-2015-notes.pdf | USAMO 2015 Solution Notes Evan Chen《陳誼廷》 23 July 2025 This is a compilation of solutions for the 2015 USAMO. The ideas of the solution are a mix of my own work, the solutions provided by the competition organizers, and solutions found by the community. However, all the writing is maintained by me.
These notes will tend to be a bit more advanced and terse than the “official” solutions from the organizers. In particular, if a theorem or technique is not known to beginners but is still considered “standard”, then I often prefer to use this theory anyways, rather than try to work around or conceal it. For example, in geometry problems I typically use directed angles without further comment, rather than awkwardly work around configuration issues. Similarly, sentences like “let R denote the set of real numbers” are typically omitted entirely.
Corrections and comments are welcome!
Contents 0 Problems 2 1 Solutions to Day 1 3 1.1 USAMO 2015/1, proposed by Titu Andreescu . . . . . . . . . . . . . . . .
3 1.2 USAMO 2015/2, proposed by Zuming Feng, Jacek Fabrykowski . . . . . .
4 1.3 USAMO 2015/3, proposed by Gabriel Carroll . . . . . . . . . . . . . . . .
7 2 Solutions to Day 2 9 2.1 USAMO 2015/4, proposed by Maria Monks Gillespie . . . . . . . . . . . .
9 2.2 USAMO 2015/5, proposed by Mohsen Jamaali . . . . . . . . . . . . . . .
11 2.3 USAMO 2015/6, proposed by Iurie Boreico . . . . . . . . . . . . . . . . .
12 1 USAMO 2015 Solution Notes web.evanchen.cc, updated 23 July 2025 §0 Problems 1. Solve in integers the equation x2 + xy + y2 = x + y 3 + 1 3 .
2. Quadrilateral APBQ is inscribed in circle ω with ∠P = ∠Q = 90◦and AP = AQ < BP. Let X be a variable point on segment PQ. Line AX meets ω again at S (other than A). Point T lies on arc AQB of ω such that XT is perpendicular to AX. Let M denote the midpoint of chord ST.
As X varies on segment PQ, show that M moves along a circle.
3. Let S = {1, 2, . . . , n}, where n ≥1. Each of the 2n subsets of S is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set T ⊆S, we then write f(T) for the number of subsets of T that are blue.
Determine the number of colorings that satisfy the following condition: for any subsets T1 and T2 of S, f(T1)f(T2) = f(T1 ∪T2)f(T1 ∩T2).
4. Steve is piling m ≥1 indistinguishable stones on the squares of an n × n grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions (i, k), (i, l), (j, k), (j, l) for some 1 ≤i, j, k, l ≤n, such that i < j and k < l. A stone move consists of either removing one stone from each of (i, k) and (j, l) and moving them to (i, l) and (j, k) respectively, or removing one stone from each of (i, l) and (j, k) and moving them to (i, k) and (j, l) respectively.
Two ways of piling the stones are equivalent if they can be obtained from one another by a sequence of stone moves. How many different non-equivalent ways can Steve pile the stones on the grid?
5. Let a, b, c, d, e be distinct positive integers such that a4 + b4 = c4 + d4 = e5. Show that ac + bd is a composite number.
6. Fix 0 < λ < 1, and let A be a multiset of positive integers. Let An = {a ∈A : a ≤ n}. Assume that for every n ∈N, the multiset An contains at most nλ numbers.
Show that there are infinitely many n ∈N for which the sum of the elements in An is at most n(n+1) 2 λ.
2 USAMO 2015 Solution Notes web.evanchen.cc, updated 23 July 2025 §1 Solutions to Day 1 §1.1 USAMO 2015/1, proposed by Titu Andreescu Available online at
Problem statement Solve in integers the equation x2 + xy + y2 = x + y 3 + 1 3 .
We do the trick of setting a = x + y and b = x −y. This rewrites the equation as 1 4 (a + b)2 + (a + b)(a −b) + (a −b)2 = a 3 + 1 3 where a, b ∈Z have the same parity. This becomes 3a2 + b2 = 4 a 3 + 1 3 which is enough to imply 3 | a, so let a = 3c. Miraculously, this becomes b2 = (c −2)2(4c + 1).
So a solution must have 4c + 1 = m2, with m odd. This gives x = 1 8 3(m2 −1) ± (m3 −9m) and y = 1 8 3(m2 −1) ∓(m3 −9m) .
For mod 8 reasons, this always generates a valid integer solution, so this is the complete curve of solutions. Actually, putting m = 2n + 1 gives the much nicer curve x = n3 + 3n2 −1 and y = −n3 + 3n + 1 and permutations.
For n = 0, 1, 2, 3 this gives the first few solutions are (−1, 1), (3, 3), (19, −1), (53, −17), (and permutations).
3 USAMO 2015 Solution Notes web.evanchen.cc, updated 23 July 2025 §1.2 USAMO 2015/2, proposed by Zuming Feng, Jacek Fabrykowski Available online at
Problem statement Quadrilateral APBQ is inscribed in circle ω with ∠P = ∠Q = 90◦and AP = AQ < BP. Let X be a variable point on segment PQ. Line AX meets ω again at S (other than A). Point T lies on arc AQB of ω such that XT is perpendicular to AX. Let M denote the midpoint of chord ST.
As X varies on segment PQ, show that M moves along a circle.
We present three solutions, one by complex numbers, two more synthetic. (A fourth solution using median formulas is also possible.) Most solutions will prove that the center of the fixed circle is the midpoint of AO (with O the center of ω); this can be recovered empirically by letting • X approach P (giving the midpoint of BP) • X approach Q (giving the point Q), and • X at the midpoint of PQ (giving the midpoint of BQ) which determines the circle; this circle then passes through P by symmetry and we can find the center by taking the intersection of two perpendicular bisectors (which two?).
¶ Complex solution (Evan Chen).
Toss on the complex unit circle with a = −1, b = 1, z = −1 2. Let s and t be on the unit circle. We claim Z is the center.
It follows from standard formulas that x = 1 2 (s + t −1 + s/t) thus 4 Re x + 2 = 2 x + 1 x + 2 = s + t + 1 s + 1 t + s t + t s which depends only on P and Q, and not on X. Thus 4 z −s + t 2 2 = |s + t + 1|2 = 3 + (4 Re x + 2) does not depend on X, done.
¶ Homothety solution (Alex Whatley).
Let G, N, O denote the centroid, nine-point center, and circumcenter of triangle AST, respectively. Let Y denote the midpoint of AS. Then the three points X, Y , M lie on the nine-point circle of triangle AST, which is centered at N and has radius 1 2AO.
4 USAMO 2015 Solution Notes web.evanchen.cc, updated 23 July 2025 A B S T O X P Q M G N Y Let R denote the radius of ω. Note that the nine-point circle of △AST has radius equal to 1 2R, and hence is independent of S and T. Then the power of A with respect to the nine-point circle equals AN2 − 1 2R 2 = AX · AY = 1 2AX · AS = 1 2AQ2 and hence AN2 = 1 2R 2 + 1 2AQ2 which does not depend on the choice of X. So N moves along a circle centered at A.
Since the points O, G, N are collinear on the Euler line of △AST with GO = 2 3NO it follows by homothety that G moves along a circle as well, whose center is situated one-third of the way from A to O. Finally, since A, G, M are collinear with AM = 3 2AG it follows that M moves along a circle centered at the midpoint of AO.
¶ Power of a point solution (Zuming Feng, official solution).
Let Y be the foot of the altitude from S to AT. Then XY ⊥AO, so Y lies on line PQ too. We then complete the picture by letting K be the foot of A to ST.
5 USAMO 2015 Solution Notes web.evanchen.cc, updated 23 July 2025 A B S T O X M Y K P Q V The main claim is: Claim — Quadrilateral PQKM is cyclic.
Proof. To see this, we use power of a point: let V = QXY P ∩SKMT. One approach is that since (V K; ST) = −1 we have V Q · V P = V S · V T = V K · V M. A longer approach is more elementary: V Q · V P = V S · V T = V X · V Y = V K · V M using the nine-point circle, and the circle with diameter ST.
But the circumcenter of PQKM, is the midpoint of AO, since it lies on the perpendicular bisectors of KM and PQ. So it is fixed, the end.
6 USAMO 2015 Solution Notes web.evanchen.cc, updated 23 July 2025 §1.3 USAMO 2015/3, proposed by Gabriel Carroll Available online at
Problem statement Let S = {1, 2, . . . , n}, where n ≥1. Each of the 2n subsets of S is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set T ⊆S, we then write f(T) for the number of subsets of T that are blue.
Determine the number of colorings that satisfy the following condition: for any subsets T1 and T2 of S, f(T1)f(T2) = f(T1 ∪T2)f(T1 ∩T2).
For an n-coloring C (by which we mean a coloring of the subsets of {1, . . . , n}), define the support of C as supp(C) = {T | f(T) ̸= 0} .
Call a coloring nontrivial if supp(C) ̸= ∅(equivalently, the coloring is not all red). In that case, notice that • the support is closed under unions and intersections: since if f(T1)f(T2) ̸= 0 then necessarily both f(T1 ∩T2) and f(T1 ∪T2) are nonzero; and • the support is obviously upwards closed.
Thus, the support must take the form supp(C) = [X, S] def = {T | X ⊆T ⊆S} for some set X (for example by letting X be the minimal (by size) element of the support).
Say C has full support if X = ∅(equivalently, ∅is blue).
Lemma For a given n and B ⊆{1, . . . , n}, there is exactly one n-coloring with full support in which the singletons colored blue are exactly those in B. Therefore there are exactly 2n n-colorings with full support.
Proof. To see there is at least one coloring, color only the subsets of B blue. In that case f(T) = 2|T∩B| which satisfies the condition by Inclusion-Exclusion. To see this extension is unique, note that f({b}) is determined for each b and we can then determine f(T) inductively on |T|; hence there is at most one way to complete a coloring of the singletons, which completes the proof.
For a general nontrivial n-coloring C, note that if supp(C) = [X, S], then we can think of it as an (n−|X|)-coloring with full support. For |X| = k, there are n k possible choices of X ⊆S. Adding back in the trivial coloring, the final answer is 1 + n X k=0 n k 2k = 1 + 3n .
7 USAMO 2015 Solution Notes web.evanchen.cc, updated 23 July 2025 Remark. To be more explicit, the possible nontrivial colorings are exactly described by specifying two sets X and Y with X ⊆Y , and coloring blue only the sets T with X ⊆T ⊆Y .
In particular, one deduces that in a working coloring, f(T) is always either zero or a power of two. If one manages to notice this while working on the problem, it is quite helpful for motivating the solution, as it leads one to suspect that the working colorings have good structure.
8 USAMO 2015 Solution Notes web.evanchen.cc, updated 23 July 2025 §2 Solutions to Day 2 §2.1 USAMO 2015/4, proposed by Maria Monks Gillespie Available online at
Problem statement Steve is piling m ≥1 indistinguishable stones on the squares of an n × n grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions (i, k), (i, l), (j, k), (j, l) for some 1 ≤i, j, k, l ≤n, such that i < j and k < l. A stone move consists of either removing one stone from each of (i, k) and (j, l) and moving them to (i, l) and (j, k) respectively, or removing one stone from each of (i, l) and (j, k) and moving them to (i, k) and (j, l) respectively.
Two ways of piling the stones are equivalent if they can be obtained from one another by a sequence of stone moves. How many different non-equivalent ways can Steve pile the stones on the grid?
The answer is m+n−1 n−1 2. The main observation is that the ordered sequence of column counts (i.e. the number of stones in the first, second, etc. column) is invariant under stone moves, as does the analogous sequence of row counts.
¶ Definitions.
Call these numbers (c1, c2, . . . , cn) and (r1, r2, . . . , rn) respectively, with P ci = P ri = m. We say that the sequence (c1, . . . , cn, r1, . . . , rn) is the signature of the configuration. These are the 2m blue and red numbers shown in the example below (in this example we have m = 8 and n = 3).
c1 = 5 c2 = 2 c3 = 1 r1 = 3 r2 = 3 r3 = 2 Signature: (5, 2, 1; 3, 3, 2) By stars-and-bars, the number of possible values (c1, . . . , cn) is m+n−1 n−1 . The same is true for (r1, . . . , rn). So if we’re just counting signatures, the total number of possible signatures is m+n−1 n−1 2.
¶ Outline and setup.
We are far from done. To show that the number of non-equivalent ways is also this number, we need to show that signatures correspond to pilings. In other words, we need to prove: 1. Check that signatures are invariant around moves (trivial; we did this already); 9 USAMO 2015 Solution Notes web.evanchen.cc, updated 23 July 2025 2. Check conversely that two configurations are equivalent if they have the same signatures (the hard part of the problem); and 3. Show that each signature is realized by at least one configuration (not immediate, but pretty easy).
Most procedures to the second step are algorithmic in nature, but Ankan Bhattacharya gives the following far cleaner approach. Rather than having a grid of stones, we simply consider the multiset of ordered pairs (x, y) corresponding to the stones. Then: • a stone move corresponds to switching two y-coordinates in two different pairs.
• we redefine the signature to be the multiset (X, Y ) of x and y coordinates which appear. Explicitly, X is the multiset that contains ci copies of the number i for each i.
For example, consider the earlier example which had • Two stones each at (1, 1), (1, 2).
• One stone each at (1, 3), (2, 1), (2, 3), (3, 2).
Its signature can then be reinterpreted as (5, 2, 1; 3, 3, 2) ← → ( X = {1, 1, 1, 1, 1, 2, 2, 3} Y = {1, 1, 1, 2, 2, 2, 3, 3}.
In that sense, the entire grid is quite misleading!
¶ Proof that two configurations with the same signature are equivalent.
The second part is completed just because transpositions generate any permutation. To be explicit, given two sets of stones, we can permute the labels so that the first set is (x1, y1), . . . , (xm, ym) and the second set of stones is (x1, y′ 1), . . . , (xm, y′ m). Then we just induce the correct permutation on (yi) to get (y′ i).
¶ Proof that any signature has at least one configuration.
Sort the elements of X and Y arbitrarily (say, in non-decreasing order). Put a stone whose x-coordinate is the ith element of X, and whose y-coordinate is the ith element of Y , for each i = 1, 2, . . . , m.
Then this gives a stone placement of m stones with signature (X, Y ).
For example, if X = {1, 1, 1, 1, 1, 2, 2, 3} Y = {1, 1, 1, 2, 2, 2, 3, 3} then placing stones at (1, 1), (1, 1), (1, 1), (1, 2), (1, 2), (2, 2), (2, 3), (3, 3) gives a valid piling with this signature.
10 USAMO 2015 Solution Notes web.evanchen.cc, updated 23 July 2025 §2.2 USAMO 2015/5, proposed by Mohsen Jamaali Available online at
Problem statement Let a, b, c, d, e be distinct positive integers such that a4 + b4 = c4 + d4 = e5. Show that ac + bd is a composite number.
Assume to the contrary that p = ac + bd, so that ac ≡−bd (mod p) = ⇒a4c4 ≡b4d4 (mod p) = ⇒a4(e5 −d4) ≡(e5 −a4)d4 (mod p) = ⇒a4e5 ≡d4e5 (mod p) = ⇒e5(a4 −d4) ≡0 (mod p) and hence p | e5(a −d)(a + d)(a2 + d2).
Claim — We should have p > e.
Proof. We have e5 = a4 + b4 ≤a5 + b5 < (ac + bd)5 = p5.
Thus the above equation implies p ≤max(a −d, a + d, a2 + d2) = a2 + d2. Similarly, p ≤b2 + c2. So ac + bd = p ≤min a2 + d2, b2 + c2 or by subtraction 0 ≤min {a(a −c) + d(d −b), b(b −d) + c(c −a)} .
But since a4 + b4 = c4 + d4 the numbers a −c and d −b should have the same sign, and so this is an obvious contradiction.
11 USAMO 2015 Solution Notes web.evanchen.cc, updated 23 July 2025 §2.3 USAMO 2015/6, proposed by Iurie Boreico Available online at
Problem statement Fix 0 < λ < 1, and let A be a multiset of positive integers. Let An = {a ∈A : a ≤n}.
Assume that for every n ∈N, the multiset An contains at most nλ numbers. Show that there are infinitely many n ∈N for which the sum of the elements in An is at most n(n+1) 2 λ.
For brevity, #S denotes |S|. Let xn = nλ −#An ≥0. We now proceed by contradiction by assuming the conclusion fails for n large enough; that is, n(n + 1) 2 λ < X a∈An a = 1(#A1 −#A0) + 2(#A2 −#A1) + · · · + n(#An −#An−1) = n#An −(#A1 + · · · + #An−1) = n(nλ −xn) −[(λ −x1) + (2λ −x2) + · · · + ((n −1)λ −xn−1)] = n(n + 1) 2 λ −nxn + (x1 + · · · + xn−1).
This means that for all sufficiently large n, say n ≥N0, we have xn < x1 + · · · + xn−1 n ∀n ≥N0.
In particular, each xn is the less than the average of all preceding terms. Intuitively this means xn should become close to each other, since they are also nonnegative.
However, we have a second condition we haven’t used yet: the “integer” condition implies |xn+1 −xn| = |λ −#{n ∈A}| > ε for some fixed ε > 0, namely ε = min {λ, 1 −λ}. Using the fact that consecutive terms differ by some fixed ε, we will derive a contradiction.
If we let M be the average of x1, . . . , xN0, then we ought to have xn < M ∀n > N0.
Hence for n > N0 we have xn + xn+1 < 2M −ε, and so for large enough n the average must drop to just above M −1 2ε. Thus for some large N1 > N0, we will have xn < M −1 3ε ∀n > N1.
If we repeat this argument then with a large N2 > N1, we obtain xn < M −2 3ε ∀n > N2 and so on and so forth. This is a clear contradiction.
12 USAMO 2015 Solution Notes web.evanchen.cc, updated 23 July 2025 Remark. Note that if A = {2, 2, 3, 4, 5, . . . } and λ = 1 then contradiction. So the condition that 0 < λ < 1 cannot be dropped, and (by scaling) neither can the condition that A ⊆Z.
Remark (Suggested by Zhao Ting-wei). Despite the relation xn < x1 + · · · + xn−1 n ∀n ≥N0 implying that xn is bounded, it does not alone imply that xn converges, not even to some nonzero value. Zhao Ting-Wei showed me that one can have a sequence which is zero “every so often” yet where the average is nonzero.
A counterexample is given explicitly by xn = 1000 n = 1 0 n is a power of 10 1 + 1 n otherwise which does not have a limit. For completeness, let’s check this — let Hn denote the n’th harmonic number, and compute n−1 X 1 xn = 1000 + (n −1) + Hn−1 − ⌊log10 n⌋ X k=1 1 + 1 10k > n + 999 + Hn−1 −log10 n − 1 + 1 10 + . . .
> n + 997 + Hn−1 −log10 n > n + 1 so 1 + 1 n < 1 n Pn−1 1 xn as needed.
13 |
12983 | https://en.wikipedia.org/wiki/Umbilical_artery | Jump to content
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Contents
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1 Structure
1.1 Development
1.2 After development
2 Clinical significance
3 Additional images
4 See also
5 References
6 External links
Umbilical artery
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From Wikipedia, the free encyclopedia
Artery in the abdominal and pelvic regions
Blood vessel
| Umbilical artery |
| Fetal circulation; the umbilical vein is the large, red vessel at the far left. The umbilical arteries are purple and wrap around the umbilical vein. |
| Scheme of placental circulation. |
| Details |
| Source | Internal iliac artery |
| Branches | Superior vesical arteryartery of the ductus deferens |
| Vein | Umbilical vein |
| Identifiers |
| Latin | arteria umbilicalis |
| MeSH | D014469 |
| TA98 | A12.2.15.020 |
| TA2 | 4316 |
| TE | artery_by_E6.0.1.3.0.0.4 E6.0.1.3.0.0.4 |
| FMA | 18820 |
| Anatomical terminology [edit on Wikidata] |
The umbilical artery is a paired artery (with one for each half of the body) that is found in the abdominal and pelvic regions. In the fetus, it extends into the umbilical cord.
Structure
[edit]
Development
[edit]
The umbilical arteries supply systemic arterial blood from the fetus to the placenta. Although this blood is sometimes referred to as deoxygenated blood it is not, and has the same oxygen saturation and nutrients as blood distributed to the other fetal tissues. There are usually two umbilical arteries present together with one umbilical vein in the umbilical cord. The umbilical arteries surround the urinary bladder and then carry all the deoxygenated blood out of the fetus through the umbilical cord. Inside the placenta, the umbilical arteries connect with each other at a distance of approximately 5 mm from the cord insertion in what is called the Hyrtl anastomosis. Subsequently, they branch into chorionic arteries or intraplacental fetal arteries.
The umbilical arteries are actually the anterior division of the internal iliac arteries, and retain part of this function after birth.
The umbilical arteries are one of two arteries in the human body that carry deoxygenated blood, the other being the pulmonary arteries.
The pressure inside the umbilical artery is approximately 50 mmHg. Resistance to blood flow decreases during development as the artery grows wider.
After development
[edit]
The umbilical artery regresses after birth. A portion obliterates to become the medial umbilical ligament (not to be confused with the median umbilical ligament, a different structure that represents the remnant of the embryonic urachus). A portion remains open as a branch of the anterior division of the internal iliac artery. The umbilical artery is found in the pelvis, and gives rise to the superior vesical arteries, which in males usually supplies the artery to the ductus deferens. Alternately, the latter artery can be supplied by the inferior vesical artery in some individuals.
Clinical significance
[edit]
A catheter may be inserted into one of the umbilical arteries of critically ill babies for drawing blood for testing. This is a common procedure in neonatal intensive care, and can often be performed until 2 weeks after birth (when the arteries start to decay too much). The umbilical arteries are typically not suitable for infusions.
Additional images
[edit]
Model of human embryo, 1.3 mm. long.
Transverse section of human embryo, eight and a half to nine weeks old.
Tail end of human embryo, twenty-five to twenty-nine days old.
Inguinal fossae
Umbilical artery. Deep dissection. Anterior view.
Umbilical artery. Deep dissection. Serial cross-section.
See also
[edit]
Single umbilical artery
References
[edit]
^ Gordon, Z.; Elad, D.; Almog, R.; Hazan, Y.; Jaffa, A. J.; Eytan, O. (2007). "Anthropometry of fetal vasculature in the chorionic plate". Journal of Anatomy. 211 (6): 698–706. doi:10.1111/j.1469-7580.2007.00819.x. PMC 2375851. PMID 17973911.
^ Hsieh, FJ; Kuo, PL; Ko, TM; Chang, FM; Chen, HY (1991). "Doppler velocimetry of intraplacental fetal arteries". Obstetrics and Gynecology. 77 (3): 478–82. PMID 1992421.
^ Adamson, S. Lee; Myatt, Leslie; Byrne, Bridgette M. P. (2004-01-01), Polin, Richard A.; Fox, William W.; Abman, Steven H. (eds.), "Chapter 72 - Regulation of Umbilical Blood Flow", Fetal and Neonatal Physiology (Third Edition), W.B. Saunders, pp. 748–758, doi:10.1016/b978-0-7216-9654-6.50075-8, ISBN 978-0-7216-9654-6, retrieved 2020-11-16
^ Fetal and maternal blood circulation systems From Online course in embryology for medicine students. Universities of Fribourg, Lausanne and Bern (Switzerland). Retrieved on 6 April 2009
^ Geipel, Annegret; Gembruch, Ulrich (2009-01-01), Wladimiroff, Juriy W; Eik-Nes, Sturla H (eds.), "Chapter 11 - Evaluation of fetal and uteroplacental blood flow", Ultrasound in Obstetrics and Gynaecology, Edinburgh: Elsevier, pp. 209–227, doi:10.1016/b978-0-444-51829-3.00011-8, ISBN 978-0-444-51829-3, retrieved 2020-11-16
^ a b Bell, Edward F. (2011-01-01), Goldsmith, Jay P.; Karotkin, Edward H. (eds.), "27 - Nutritional Support", Assisted Ventilation of the Neonate (Fifth Edition), Philadelphia: W.B. Saunders, pp. 466–483, doi:10.1016/b978-1-4160-5624-9.00027-5, ISBN 978-1-4160-5624-9, retrieved 2020-11-16
^ Durand, DAVID J.; Phillips, BARRY; Boloker, JUDD (2003-01-01), Goldsmith, Jay P.; Karotkin, Edward H.; Siede, Barbara L. (eds.), "Chapter 17 - BLOOD GASES: Technical Aspects and Interpretation", Assisted Ventilation of the Neonate (Fourth Edition), W.B. Saunders, pp. 279–292, doi:10.1016/b978-0-7216-9296-8.50022-2, ISBN 978-0-7216-9296-8, retrieved 2020-11-16
^ Wald, Samuel H.; Mendoza, Julianne; Mihm, Frederick G.; Coté, Charles J. (2019-01-01), Coté, Charles J.; Lerman, Jerrold; Anderson, Brian J. (eds.), "49 - Procedures for Vascular Access", A Practice of Anesthesia for Infants and Children (Sixth Edition), Philadelphia: Elsevier, pp. 1129–1145.e5, doi:10.1016/b978-0-323-42974-0.00049-5, ISBN 978-0-323-42974-0, S2CID 81592410, retrieved 2020-11-16
External links
[edit]
Anatomy photo:43:13-0203 at the SUNY Downstate Medical Center - "The Female Pelvis: Branches of Internal Iliac Artery"
| v t e Arteries of the abdomen and pelvis |
| Abdominalaorta | | | | --- | | Inferior phrenic | Superior suprarenal | | Celiac | | | | --- | | Left gastric | Esophageal branches | | Common hepatic | Proper hepatic + cystic Right gastric Gastroduodenal + right gastroepiploic + superior pancreaticoduodenal + supraduodenal | | Splenic | Pancreatic branches + greater + dorsal Short gastrics Left gastroepiploic | | | Superior mesenteric | Inferior pancreaticoduodenal Intestinal + jejunal + ileal + arcades + vasa recta Ileocolic + colic + anterior cecal + posterior cecal + ileal branch + appendicular Right colic Middle colic + Marginal | | Suprarenal | Middle suprarenal | | Renal | Inferior suprarenal Ureteral | | Gonadal | Testicular artery Ovarian artery | | Lumbar | Lumbar arteries | | Inferior mesenteric | Left colic + Marginal Sigmoid Superior rectal | | Common iliac | | | | | | | | | | | | | | | | | | | | | | | --- --- --- --- --- --- --- --- --- --- | Internal iliac | | | | | | --- --- | | Posterior surface | | | | --- | | Iliolumbar | Lumbar branch Iliac branch | | | Anterior surface | | | | --- | | Superior vesical artery | Umbilical artery + Medial umbilical ligament to ductus deferens | | Obturator | Anterior branch + Pubic branch Posterior branch + Acetabular branch + Cruciate anastomosis Corona mortis | | Middle rectal | Vaginal branch ♀ / Prostatic branch ♂ | | Uterine ♀ | Arcuate Vaginal branches Ovarian branches Tubal branches Spiral | | Vaginal ♀ / Inferior vesical ♂ | | Inferior gluteal | Accompanying of sciatic nerve Cruciate anastomosis | | Internal pudendal | Inferior rectal Perineal + posterior scrotal + posterior labial Bulb of penis/vestibule Urethral Deep artery of the penis + helicine Deep artery of clitoris Dorsal of the penis Dorsal of the clitoris | | | | External iliac | Inferior epigastric + Corona mortis Deep circumflex iliac Femoral + see arteries of lower limbs | | | Median sacral | Coccygeal glomus | |
| v t e Membranes of the fetus and embryo |
| Embryo | Trophoblast + Cytotrophoblast + Syncytiotrophoblast + Intermediate trophoblast Allantois Decidua + Decidual cells Chorionic villi/Intervillous space Amnion + sac + cavity |
| Fetus | Umbilical cord + Umbilical artery + Umbilical vein + Wharton's jelly |
| Circulatory | Placenta Chorion |
| Other | Blastocoel Heuser's membrane Reichert's membrane Vitelline duct Gestational sac |
| v t e Development of the circulatory system |
| Heart | | | | --- | | Tubular heart | Truncus arteriosus Bulbus cordis Primitive ventricle Primitive atrium Sinus venosus | | Chamber formation | Atrioventricular + Primary interventricular foramen + Endocardial cushions + Septum intermedium + Atrioventricular canal Atrial + Septum primum + Foramen secundum + Primary interatrial foramen + Septum secundum + Foramen ovale | | Other | Aorticopulmonary septum Protein signalling in heart development | |
| Vessels | | | | --- | | Arteries | Dorsal aorta Aortic arches Aortic sac | | Veins | Anterior cardinal vein Posterior cardinal vein Common cardinal veins | | Lymph vessels | Lymph sacs | | Other | Vascular remodelling in the embryo | |
| Extraembryonichemangiogenesis | Blood islands Chorion Connecting stalk Yolk sac Placenta |
| Fetal circulation | umbilical cord: Umbilical vein → Ductus venosus → Inferior vena cava → Heart → Pulmonary artery → Ductus arteriosus → Aorta → Umbilical artery yolk sac: Vitelline veins Vitelline arteries |
| | |
--- |
| Authority control databases | Terminologia Anatomica |
Retrieved from "
Categories:
Arteries of the abdomen
Embryology of cardiovascular system
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Time Constant Calculator
Calculate and visualize the time constant (τ) for RC and RL circuits.
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Understanding the Time Constant
What is a Time Constant?
The time constant, denoted by the Greek letter tau (τ), is a measure of how quickly a circuit responds to a change in voltage or current. In simple terms, it represents the time required for the voltage or current in a charging or discharging circuit to reach approximately 63.2% of the difference between its initial and final values.
The Formulas
The time constant is calculated differently for RC and RL circuits:
For a Resistor-Capacitor (RC) circuit: τ = R × C
For a Resistor-Inductor (RL) circuit: τ = L / R
Where R is resistance in Ohms (Ω), C is capacitance in Farads (F), and L is inductance in Henrys (H).
Why 5τ for Full Charge/Discharge?
While theoretically the circuit never reaches 100% of its final value, it gets very close. After five time constants (5τ), the circuit has reached over 99.3% of its final value. This is considered fully charged or discharged for most practical engineering purposes. Here's a quick breakdown:
After 1τ: ~63.2% complete
After 2τ: ~86.5% complete
After 3τ: ~95.0% complete
After 4τ: ~98.2% complete
After 5τ: ~99.3% complete
Frequently Asked Questions
What does a small time constant mean?
A small time constant (τ) indicates a fast-responding circuit. This means the capacitor or inductor will charge and discharge very quickly. This is desirable in high-frequency applications.
What does a large time constant mean?
A large time constant (τ) indicates a slow-responding circuit. The capacitor or inductor takes a longer time to charge or discharge. This is useful for creating timing delays or in filtering out high-frequency noise.
Where are time constants used in real life?
Time constants are a fundamental concept in electronics. They are used in timing circuits (like in a blinking LED circuit), filters in audio equipment to separate different frequencies, and in power supplies to smooth out voltage ripples.
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12985 | https://byjus.com/physics/nuclear-binding-energy/ | During the 20th century, the popular Albert Einstein came up with the revolutionary theory, “the theory of relativity”. The theory explained that mass and energy are interconvertible; the mass can be converted into energy and vice-versa. This new dimension to physics helped to resolve plenty of unsolved problems and formed a forum for lot of new theories. One among them is the existence of Nuclear Binding energy. Thereby giving a clear insight into nuclear mass and inter-nuclei interactions.
Mass Defect and Binding Energy
An atom comprises a nucleus at the center and electrons revolving around it in an orbital fashion. Nuclei constitute of Protons and Neutrons, Combined called nucleons. Thus, we expect that mass of the nucleus will be the same as the sum of individual masses of neutrons and protons. But it is not true. The total mass of the nucleus(mnuc) is less than the sum of individual masses of neutrons and protons which in fact constitutes it. This difference in the mass is called mass defect given by,
Zmp is the total mass of the protons.
(A-Z)mn is the total mass of the neutrons.
mnuc is the mass of the nucleus.
According to Einstein’s theory of relativity, The mass-energy is equivalent. That is the mass of a system measures the total energy of the system. Given by the famous equation E=mc2. Hence, it is implied that the total energy of the nucleus is less than the sum of the energies of individual protons and neutrons(nucleons). This implies that when the nucleus disintegrates into constituent nucleons releasing some energy in the form of heat energy. (Exothermic reaction). The energy emitted here is mathematically expressed using
Now imagine the situation of breaking the nucleus. To attain this, a certain amount of energy is put into the system. The amount of energy required to achieve this is called nuclear binding energy. Thus, we can define nuclear binding energy as,
“The minimum energy required to separate nucleons into its constituent protons and neutrons.” and is given by-
Thus, the difference in the mass is converted into Nuclear binding energy.
How to calculate binding energy?
The nuclear binding energy can be calculated following the below-given steps:
Once the mass defect is known, the nuclear binding energy can be calculated by converting that mass to energy using the formula.
Make sure that the mass is in the units of kgs
Once the energy obtained is known, it can be scaled into per-nucleon and per- mole quantities.
You may want to check out the following articles for a better understanding of nuclear binding energy.
| |
| Radioactive Decay |
| Radioactivity and Alpha decay |
| Electrons and Protons |
Binding Energy Calculation
1)Calculate the Binding Energy of the Deuteron. Give data: a mass of the deuteron is 1875.61MeV/c2 or 3.34359✕10-27 Kg.
Solution:
Given:
Mass of deuteron mD = 1875.61MeV/c2 or 3.34359✕10-27 Kg.
Atomic Mass number of Deuteron; A=2
The atomic number of Deuteron; Z=1
Mass defect Δm =?
Binding energy Eb = ?
Formula used:
Mass defect is given by
Binding energy is given by
Calculation:
Mass defect is given by
. Here Z=1 and (A-Z)=1, mn =939.57MeV/c2 and mp =938.28 MeV/c2. Substituting the values we get-
Δm = mp + mn– mD
= 938.28 MeV/c2 + 939.57MeV/c2 – 1875.61MeV/c2
=2.24 MeV/c2
Thus, the mass defect is 2.24 MeV/c2
The binding energy of the Deuteron is thus given by
= (2.24 MeV/c2)(c2)
=2.24 MeV
A minimum of 2.24 million electron volt energy is required to break Deuteron into Proton and Neutron. This is a very large value. However, the energy required to separate an electron from a hydrogen atom by overcoming electromagnetic force(Coulomb force) is approximately 10eV. This comparison clearly indicates the strength of the nuclear force. Although its range lies in multiple Femto meters, It is one of the strongest forces in nature.
Stay tuned with BYJU’S for more such interesting topics. Register to “BYJU’S-The Learning App” for the latest set of interactive engaging physics videos.
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12986 | http://library.rumsfeld.com/doclib/sp/347/2003-07-23%20to%20Steve%20Bucci%20re%20DoD%20Dictionary-%20Memo%20Attachment.pdf | TO: Col. Bucci FROM: Donald Rumsfeld PA SUBJECT: DoD Dictionary Please get me the complete Department of Defense Dictionary of Military and Associated Terms.
Second, I would like the definitions for: insurgency, guerrilla war, and belligerency, each on a single piece of paper.
Thanks.
Attach.
7/10/03 Note from USD(I) DHR:dh 072103-4 Please respond by U20913 /03 TO: Col. Bucci FROM: Donald Rumsfeld P A SUBJECT: DoD Dictionary Please get me the complete Department of Defense Dictionary of Military and Associated Terms. Second, I would like the definitions for: insurgency, guerrilla war, and belligerency, each on a single piece of paper. Thanks. Attach. 7/10/03 Note from USD(1) DHR:dh 072103-4 Please respond by 7/25 Id3 FROM: TO: Col. Bucci Donald Rumsfeld V/\ SUBJECT: DoD Dictionary z..; July)t; 200 Piease get me the complete Department of Defense Dictionary of Military and Associated Terms.
Second, I would like the definitions for: insurgency, guerrilla war, and belligerency, each on a single piece of paper.
Thanks.
Attach.
7110/03 Note from USD(I) DHR:dh 072103-4 ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• Please respond by /+-I_2-;{-L/_o 3 _ U2091S 103 Certified As Unclassified January 9 2009 IAW EO 12958, as amended Chief, RDD, ESD, WHS OFFICE OF THE SECRETARY OF DEFENSE /i'o7o3 MEMO FOR ¡:-J a AJ. ,2&r ,ìZ a ,L ¿s ' P a Y Le.4 11Lf -cI a.c .' e ¿A, £4i Q KI4'c fe1 /4ea 'D 4iarI / c,4 c L I i (4r/"J C.4)cPfAI J:/ /D C(('.iv; f 4 e 7 et4s /'1C/14t j,yh/ ,'.oT'.
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C tA ~. IJ;" q y J.oe. / Jl.J I /e~ Ir-"''''Ti IZJ- ;,~/IA-"/' R o Certified As Unclassified January 9 2009 IAW EO 12958, as amended Chief, RDD, ESD, WHS Regional Notes NEAR EAST ¿-J Iraq: / Insurgency Sustj.fab1e Despite Shortcomings (SI/NF) SECDEF HAS SEEN JUL 8 2003 Titi-Coalition groups in Iraq have begun to link their military tactics to political efforts, but they still seem to lack broad public support. The insurgentsnon-Ba'thist nationalists, former regime loyalists, Sunni Islamic extremists, and foreign fightersare moving beyond defensive resistance against Coalition forces to insurgency tactics, including guerrilla warfare, propaganda, intimidation, and subversion.
The insurgents believe daily losses will)ead US politicians to withdraw US troops, according to foreign government service sensitive reporting. The insurgents may see the current level of attacks, which has grown since May, as sufficient to do this.
Former Ba 'th Party militants probably have the resources to sustain the current level of attacks indefinitely. Iraq is awash in weapons after the collapse of the Iraqi Army, and Saddam or other former regime officials could still have a war chest of more than $100 million even after Coalition seizures in the past three months. (SI/NF) Effective insurgencies historically have political as well as military expressions and developed political agendas. The former Ba'th militants are reorganizing under the name Hizb al- 'Awda (The Party of the Return) and other names in the Sunni heartland and in some cities in Shia-dominated southern Iraq. 'Awda uses printed fliers and word of mouth to spread its goals in towns in northern and central Iraq, according to sensitive reporting and US military reporting.
A CIA field assessment indicates elements of the old regime are coalescing and trying to recruit former senior military officers, unemployed former Ba'th Party officials, and middle-class professionals who think the Coalition Provisional Authority (CPA) is disenfranchising them. (S//NF) Insurgent groups would have to overcome a host of weaknesses, however, to pose a greater threat to the Coalition. The majority of Iraq's population fears and despises former Ba'th Party leaders, according to press reports, constraining the insurgents' support base. The propaganda of 'Awda and other anti-Coalition groups has limited appeal because it fails to offer a compelling political alternative.
8 July 2003 .L R egio na 1 NEAR EAST Iraq: nti-Coalition groups in Iraq have begun to link their military tactics to political efforts, but they still seem to lack broad public support. The insurgents-non-Ba’thist nationalists, former regime loyalists, Sunni Islamic extremists, and foreign fighters-are moving beyond defensive resistance against Coalition forces to insurgency tactics, including guerrilla warfare, propaganda, intimidation, and subversion. - The insurgents believe daily losses willlead US politicians to withdraw US troops, according to foreign government service sensitive reporting. The insurgents may see the current level of attacks, which has grown since May, as sufficient to do this. - Former Ba ’th Party militants probably have the resources to sustain the current level o f attacks indefinitely. Iraq is awash in weapons after the collapse of the Iraqi Army, and Saddam or other former regime officials could still have a war chest of more than $100 million even after Coalition seizures in the past three months. (S//NF) Effective insurgencies historically have political as well as military expressions and developed political agendas. The former Ba’ th militants are reorganizing under the name Hizb al-‘Awda (The Party of the Return) and other names in the Sunni heartland and in some cities in Shia-dominated southern Iraq. ‘Awda uses printed fliers and word of mouth to spread its goals in towns in northern and central Iraq, according to sensitive reporting and US military reporting. - A CIA field assessment indicates elements of the old regime are coalescing and trying to recruit former senior military officers, unemployed former Ba’th Party officials, and middle-class professionals who think the Coalition Provisional Authority (CPA) is disenfranchising them. (S//NF) Insurgent groups would have to overcome a host o f weaknesses, however, to pose a greater threat to the Coalition. The majority of Iraq’s population fears and despises former Ba’th Party leaders, according to press reports, constraining the insurgents’ support base. The propaganda of ‘Awda and other anti-Coalition groups has limited appeal because it fails to offer a compelling political alternative. I 8 July 2003 • • • ...
Regional Notes NEAREAST Iraq: nti-Coalition groups in Iraq have begun to link their military tactics to political efforts, but they still seem to lack broad public support. The insurgents-non-Ba'thist nationalists, former regime loyalists, Sunni Islamie extremists, and foreign fighters-are moving beyond defensive resistanee against Coalition forees 10 insurgeney taetics, including guerrilla warfare, propaganda, intimidation, and subversion.
-The insurgents believe daily losses will)ead US politieians to withdraw US troops, aceording to foreign government service sensitive reporting. The insurgents may see the current level of attacks, which has grown since May, as sufficient to do this.
-Fonner Ba'th Party militants probably have the resources to sustain the current level ofattacks indefinitely. Iraq is awash in weapons after the eollapse of the Iraqi Army, and Saddam or other former regime offieials could still have a war ehest of more than $100 million even after Coalition seizures in the past three months. (S//NF) Effective insurgencies historieally have political as weIl as military expressions and developed politieal agendas. The former Ba'th militants are reorganizing under the name Hizb al-'Awda (The Party ofthe Return) and other names in the Sunni heartland and in some eities in Shia-dominated southern Iraq. 'Awda uses printed fliers and word of mouth to spread its goals in towns in northern and eentral Iraq, aeeording to sensitive reporting and US military reporting.
-A CIA field assessment indicates elements of the old regime are eoalescing and trying to recruit former senior military officers, unemployed former Ba'th Party officials, and middle-class professionals who think the Coalition Provisional Authority (CPA) is disenfranchising them. (S//NF) Insurgent groups would have to overcome a host ofweaknesses, however, to pose a greater threat to the Coalition. The majority of Iraq's population fears and despises former Ba'th Party leaders, aeeording to press reports, eonstraining the insurgents' support base. The propaganda of 'Awda and other anti-Coalition groups has limited appeal beeause it fails to offer a compelling politieal alternative.
8 July 2003 Certified As Unclassified January 9 2009 IAW EO 12958, as amended Chief, RDD, ESD, WHS Department of Defense Dictionary of Military and Associated Terms Joint Publication 1-02 12 April 2001 (As Amended Through S June 2003) a a Certified As Unclassified January 9 2009 IAW EO 12958, as amended Chief, RDD, ESD, WHS AftJ f:r-ir'd PATO A Ameiided Through 5 June 2003 v instrument flight () Flight in which the path and attitude of the aircraft are controlled solely by reference to instruments.
instrument landing system () A system of radio navigation intended to assist aircraft in landing which provides lateral and vertical guidance, which may include indications of distance from the optimum point of landing. Also called ILS.
instrument meteorological conditions - Meteorological conditions expressed in terms of visibility, distance from cloud, and ceiling; less than minimums specified for visual meteorological conditions. Also called ¡MC. See also visual meteorological conditions.
(JP 3-04.1) in support () An expression used to denote the task of providing artillery supporting fire to a formation or unit. Liaison and observation are not normally provided. See also at priority call; direct support.
in support of Assisting or protecting another formation, unit, or organization while remaining under original control.
insurgency () An organized movement aimed at the overthrow ola constituted government through use of subversion and armed conflict.
insurgent - Member of a political party who rebels against established leadership. See also antiterrorism; counterinsurgency; insurgency. (W 3-07.2) Integrated Consumable Item Support - A decision support system that takes time-phased force and deployment data (i.e., Department of Defense deployment plans) and calculates the ability of the Defense Logistics Agency, the warehousing unit of the Department of Defense, to support those plans Integrated Consumable Item Support can calculate for the planned deployment supply/demand curves for over two million individual items stocked by the Defense Logistics Agency in support of deployment. Integrated Consumable Item Support allows planners to identify critical end items and anticipated shortfalls in the Defense Logistics Agency inventories.
Integrated Consumable Item Support provides materiel readiness information for Defense Logistics Agency managed items to Defense Logistics Agency management, to all Services, and to the Joint Staff, to be used as a piece of the larger wartime logistic picture, which ultimately is used to assess total readiness and sustainability for deliberately planned contingencies. The goals and objectives of Integrated Consumable Item Support are to know the "war stoppers," know the weapons systems affected, and know when the Defense Logistics Agency will run out of stock. Also called ICIS. (JP 4-03) integrated fire control system A system that performs the functions of target acquisition, tracking, data computation, and engagement control, primarily using electronic means and assisted by electromechanical devices.
JP I-02 As Amended ‘I’hrough 5 lune 2003 instrument flight - () Flight in which the path and attitude of the aircraft are controlled solely by reference to instruments. instrument landing system - () A system of radio navigation intended to assist aircraft in landing which provides lateral and vertical guidance, which may include indications of distance from the optimum point of landing. Also called ILS. instrument meteorological conditions - Meteorological conditions expressed in terms of visibility, distance from cloud, and ceiling; less than minimums specified for visual meteorological conditions. Also called JMC. See also visual meteorological conditions. (JP 3-04.1) in support - () An expression used to denote the task of providing artillery supporting fire to a formation or unit. Liaison and observation are not normally provided. See also at priority call; direct support. in support of - Assisting or protecting another formation, unit, or organization while remaining under original control. n organized movement aimed at the overthrow of a constituted government ubversion and armed conflict. +z -- w~p UATO insurgent - Member of a political party who rebels against established leadership. See also U p % d bf- bD antiterrorism; counterinsurgency; insurgency. (JP 3-07.2) Integrated Consumable Item Support - A decision support system that takes time-phased force and deployment data (Le., Department of Defense deployment plans) and calculates the ability of the Defense Logistics Agency, the warehousing unit of the Department of Defense, to support those plans. Integrated Consumable Item Support can calculate for the planned deployment supply/demand curves for over two million individual items stocked by the Defense Logistics Agency in support of deployment. Integrated Consumable Item Support allows planners to identifl critical end items and anticipated shortfalls in the Defense Logistics Agency inventories. Integrated Consumable Item Support provides materiel readiness information for Defense Logistics Agency managed items to Defense Logistics Agency management, to all Services, and to the Joint Staff, to be used as a piece of the larger wartime logistic picture, which ultimately is used to assess total readiness and sustainability for deliberately planned contingencies. The goals and objectives of Integrated Consumable Item Support are to know the “war stoppers,” know the weapons systems affected, and know when the Defense Logistics Agency will run out of stock. Also called ICIS. (JP 4-03) integrated fire control system - A system that performs the functions of target acquisition, tracking, data computation, and engagement control, primarily using electronic means and assisted by electromechanical devices. A~ Amenueu Throllgh 5 .JlIne 2003 ~1J'i-Al I instrument flight -() Flight in which the path and attitude of the aircraft are controlled solely by reference to instruments.
instrument Ianding system -() A system of radio navigation intended to assist aircraft in landing which provides lateral and vertical guidance, which may include indications of distance from the optimum point oflanding. Also called ILS.
instrument meteorological conditions -Meteorological conditions expressed in terms of visibility, distance from cloud, and ceiling; less than minimums specified for visual meteorological conditions. Also called IMC. See also visual meteorological conditions.
(JP 3-04.1) in support -() An expression used to denote the task ofproviding artillery supporting fire to a formation orunit. Liaison and observation are not normally provided. See alsoat priority call; direct support.
in supportof-Assisting orproteeting anotherformation, unit, ororganization whileremaining under original contro!.
insurgency >() An organized movement aimed at the overthrow ofa constituted government through use of subversion and armed conflict.
insurgent -Member of a political party who rebels against established leadership. See also antiterrorism; counterinsurgency; insurgency. (JP 3-07.2) Integrated Consumable Item Support -A decision support system that takes time-phased force and deployment data (Le., Department of Defense deployment plans) and calculates the ability of the Defense Logistics Agency, the warehousing unit of the Department of Defense, to support those plans. Integrated Consumable Item Support can calculate for the planned deployment supply/demand curves for over two million individual items stocked by the Defense Logistics Agency in support of deployment. Integrated Consumable ltem Supportallows planners to identify critical end iterns andanticipated shortfalls in the Defense Logistics Agency inventories.
Integrated Consumable Item Support provides materiel readiness information for Defense Logistics Agency managed items to Defense Logistics Agency management, to all Services, and to the Joint Staff, to be used as a piece of the larger wartime logistic picture, which ultimately is used to assess total readiness and sustainability for deliberately plannedcontingencies. The goals and objectives ofIntegrated Consumable Item Support are to know the "war stoppers," know the weapons systems affected, and know when the Defense Logistics Agency will ron out of stock. Also called ICIS. (JP 4-03) integrated lire control system -A system that performs the functions of target acquisition, tracking, data computation, and engagement control, prirnarily using electronic means and assisted by electromechanical devices.
260 lP 1-02 Certified As Unclassified January 9 2009 IAW EO 12958, as amended Chief, RDD, ESD, WHS |
12987 | https://nrich.maths.org/problems/quad-solve | Quad solve | NRICH
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Quad solve
Can you solve this problem involving powers and quadratics?
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Problem Getting Started Student Solutions
Problem
Find all real solutions to this equation:
(2−x 2)x 2−3 2 x+4=1
Extension:What if x is permitted to be a complex number?
Did you know ... ?
Quadratic equations and powers are commonly used throughout school and university mathematics and beyond. It is also important to remember that algebraic manipulations might not necessarily find all solutions to a problem; you always need to reason carefully that all possibilities have been considered. Moreover, in complicated situations it is necessary to check that all proposed solutions unearthed by algebra are in fact valid solutions. Powers, roots and quadratics all link together very nicely when complex numbers are considered.
Getting Started
To do this problem you will need to know that a 0=1 when a≠0 and 1 b=1 for any b along with other rules of indices.
Note that 0 0 is not defined as a number.
You will also need to know the formula for the solution of a quadratic equation.
If you are studying complex numbers then you can also bring those ideas into the problem if desired.
Student Solutions
By applying the quadratic equation formula we can factorise this expression
(2−x 2)x 2−3 2 x+4=1
into
−(x−2)(x+2)(x−2 2)=1
For real solutions we can make use of the fact that for any real a≠0 we have
a 0=1 1 a=1
First look at the base. This equals 1 if and only if x=±1.
Next look at the exponent. This equals zero if and only if x=2 or 2 2. However, when x=2 both the base and exponent are zero. Therefore, three valid real solutions to the equation are
x=±1,x=2 2
Is this all? Not necessarily. We might also have (−1)2 n=1 for any positive whole number n. Are there any solutions to this? 2−x 2=−1 has solutions x=±3. In this case the exponent becomes 7±3 6, which is not of the form 2 n. There are therefore no more real solutions.
Note: If we extend to complex numbers then we also have to take into account the fact that there are multiple complex roots of 1. For example:
(−1±3 i 2)3=1
EXTENSION: READ ON ONLY IF VERY, VERY KEEN!!
Any complex number z can be written in modulus argument form z=r e i θ and the logarithm of a complex number becomes
lnz=ln(r e i θ)=lnr+ln(e i θ)=lnr+i θ
Using these facts we can attack our problem in the complex numbers. Taking logs of the equations gives
(z 2−3 2 z+4)ln(2−z 2)=2 n π i n∈Z(†)
To progress with the complex logarithm of (2−z 2) it is natural to use modulus argument form of (2−z 2). If we suppose that z=r e i θ then
2−z 2=2−r 2 e 2 θ i=2−r 2 cos2 θ−i r 2 sin2 θ=(2−r 2 cos2 θ)2+(r 2 sin2 θ)2 e i tan−1(r 2 sin2 θ r 2 cos2 θ−2)
Our equation (†) then becomes
A(r,θ)×B(r,θ)=2 n π i
where
A(r,θ)=(4+r 2 cos2 θ−3 2 r cosθ)+i(r 2 sin2 θ−3 2 r sinθ)
and
B(r,θ,m)=1 2 ln|4−4 r 2 cos 22 θ+r 4|+i[2 m π+tan−1(r 2 sin2 θ r 2 cos2 θ−2)]m∈Z
There are no non-real solutions for n=m=0 as this excellent piece of mathematics by Stephen Lynch shows (I include his full solution).
However, there might well be solutions for other values of n and m. In principle we can attempt to solve this by equating real and imaginary parts for various choices of n and m.
Editor's note: further analysis appears to be time consuming, so I looked for numerical answers as follows :
Finding an exact solution would appear to be, well, complex. I simply attempted a numerical solution. To find a numerical solution I looked for small values of the expression
X(r,θ,n,m)=|A(r,θ)B(r,θ,m)−2 n π i|2
I looked at the case n=1,m=0.
I found a solution
r=3.3858990007,θ=0.1902641501
This solves the equation in the sense that
X(3.3858990007,0.1902641502,1,0)<10−17
From this numerical exploration is seems likely that other solutions exist, although proof that the iterative scheme used does indeed converge on a genuine solution would require more work. (As an aside it is worth noting that complex numbers enter into all sorts of applied mathematics, such as fluid dynamics and air flow. In such situations numerical solutions of complex number equations are crucial.)
You might wish to take a look at the a spreadsheet which computes the values X(r,θ,n,m) for various values.
With numerics there is always the chance of hard to spot error, but here is the input used if you wish to check the logic for yourself (this is a good habit to get into)
Image
Fortunately X(2 2,0,0,0)=X(1,0,0,0)=X(1,π,0,0)=0 as we would expect from our analysis of the real case. This sort of check is essential when performing numerics.
Needless to say, if you spot an error please let us know!
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12989 | https://www.cuemath.com/ncert-solutions/prove-that-the-tangent-drawn-at-the-mid-point-of-an-arc-of-a-circle-is-parallel-to-the-chord-joining-the-end-points-of-the-arc/ | Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc
Solution:
Given, a tangent is drawn at the midpoint of an arc of a circle.
We have to prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.
From the figure,
Centre of the circle is O.
P is the midpoint of arc APB
PT is a tangent to a circle at P
We know that the radius of the circle is perpendicular to the tangent at the point of contact.
So, OP ⟂ PT
∠OPT = 90°
Since P is the midpoint of arc APB,
arc AP = arc BP
∠AOP = ∠BOP
∠AOM = ∠BOM
Considering triangles AOM and BOM,
OA = OB = radius of circle
OM = OM = common side
Also, ∠AOM = ∠BOM
By SAS criterion, the triangles AOM and BOM are similar.
Corresponding parts of congruent triangles or cpct is used to denote the relation between the sides and the angles of two congruent triangles.
By cpct,
∠AMO = ∠BMO —------------------ (1)
The linear pair angles ∠AMO + ∠BMO = 180°
From (1), ∠AMO + ∠AMO = 180°
2∠AMO = 180°
∠AMO = 90°
So, ∠AMO = ∠BMO = 90°
We know that the corresponding angles are equal.
From the figure,
∠BMO = ∠OPT = 90°
This implies AB || PT
Therefore, the tangent at midpoint of the arc is parallel to the chord joining the end points of the arc.
✦ Try This: Prove that the line joining the midpoint of a chord to the centre of the circle passes through the midpoint of the corresponding minor arc.
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 10
NCERT Exemplar Class 10 Maths Exercise 9.4 Problem 9
Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc
Summary:
It is proven that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc
☛ Related Questions: |
12990 | https://www.reddit.com/r/learnmath/comments/1jyl085/how_to_calculate_points_on_a_small_circle_on_a/ | How to calculate points on a small circle on a sphere? : r/learnmath
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r/learnmath•6 mo. ago
azroscoe
How to calculate points on a small circle on a sphere?
If I have a small circle on a sphere with center point of the circle denoted (long,lat) and an angular radius R, how can I calculate points along the circle's circumference? I am looking for a spherical analog to the 2D formula:
x = h + r cos(angle), y = k + r sin(angle)
I am reasonably familiar with spherical trig, but this one eludes me.
Thanks!
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There are different possible answers to this question, depending on the standard of proof one needs and the background knowledge one brings to the question.
Mathematical consistency and patterns
Try solving each of these problems, paying attention to the previous set of problems as you do so. Look for patterns to make solving the problems easier.
3 × 3 = ?
3 × 2 = ?
3 × 1 = ?
3 × 0 = ?
3 × -1 = ?
3 × -2 = ?
3 × -3 = ?
2 × -3 = ?
1 × -3 = ?
0 × -3 = ?
-1 × -3 = ?
-2 × -3 = ?
-3 × -3 = ?
The answers to these problems are below but I really do recommend taking the time to solve the problems above on your own first, so you get the sense of how students might think through this set of problems.
3 × 3 = 9
3 × 2 = 6
3 × 1 = 3
3 × 0 = 0
At this stage, many people will notice the answers are 3 smaller each time and the number being multiplied by 3 is one smaller each time, so they continue that pattern to answer the following questions.
3 × -1 = -3
3 × -2 = -6
3 × -3 = -9
Now, we decrease the first number in the pattern by 3 and one has to make some deductions about what the answer should be.
2 × -3 = -6
1 × -3 = -3
0 × -3 = 0
One might now notice that the answers are going up by 3 each time as we increase the first number, and so it is reasonable to continue this pattern.
-1 × -3 = 3
-2 × -3 = 6
-3 × -3 = 9
While to some this pattern may seem obvious, when someone is still in the middle of learning this concept, they have less cognitive capacity available to accomplish the task at hand (multiplying numbers together) and accomplish the additional task of looking for patterns in their answers, so this is where someone else prompting them to stop and look for patterns in their work so far will be very useful.
Prerequisite knowledge: One has to know what these symbols mean, what is meant by finding one number times another, and how negative numbers work in terms of counting down and subtraction.
Mathematical consistency and mathematical properties
Let’s look at a problem that we can do in more than one way, borrowed from the Khan Academy.
5 × (3 + -3) = ?
If we add the numbers inside the parenthesis first, then this is 5 times 0 which is 0, since 3 + -3 = 0.
5 × (3 + -3) = 0
But what if we distribute 5 through both terms first?
5 × 3 + 5 × -3 = ?
Since distributing the 5 across the addition does not change the value of the expression, we know this is still equal to 0.
5 × 3 + 5 × -3 = 0
But this means that 5 × 3 and 5 × -3 are opposite signs, so since 5 × 3 = 15, then 5 × -3 is -15. Let’s look at another example.
-5 × (3 + -3) = ?
We know that this is the same as -5 times 0, so this has a value of 0.
-5 × (3 + -3) = 0
Similar to before, we distribute -5 through both terms.
-5 × 3 + -5 × -3 = ?
Again, the distribution of terms does not change the value of the expression on the left-hand side of the equation, so the result is still 0.
-5 × 3 + -5 × -3 = 0
We know from before that -5 × 3 is -15 so we can substitute that value for -5 × 3 in the left-hand side of the equation.
-15 + -5 × -3 = 0
Therefore -15 and -5 × -3 are opposites since they add to 0, so -5 × -3 must be positive.
Nothing in what we did for the two examples above is specific to the value of 5 × 3, so we can repeat this argument for every other multiplication fact we want to derive, so these two ideas can be generalized.
Prerequisite knowledge: One has to know what these symbols mean, what is meant by finding one number times another, how the distributive property works, and how negative numbers can be defined as the opposites of positive numbers.
Representation on a number line
Imagine we represent multiplication as jumps on a number line.
For 3 × 3, we draw 3 groups of 3 moving to the right. Both the number of groups and the direction of each group are to the right.
But what about 3 × -3? Now we have 3 groups of the number still, but the number is negative.
If we find -3 × 3, the size and direction of the number we multiply are the same, but now we are finding -3 groups of that number. One way to think of this is to think of taking 3 groups of the number away. Another is to think of -3 times a number as being a reflection of 3 times the same number.
So -3 × -3 is, therefore, a reflection of 3 × -3 across the number line.
In one sense though, this visual argument is just mathematical consistency represented using a number line. If multiplication by a negative is a reflection across 0 on the number line, and we think of negative numbers as being reflections across 0 of the number line, then multiplication of a negative number times a negative number is a double-reflection.
Context
Karen Lew has this analogy.
Multiplying by a negative is repeated subtraction. When we multiply a negative number times a negative number, we are getting less negative.
This analogy between multiplication and addition and subtraction helps students nicely connect the two concepts.
Joseph Rourke shared this context.
A gambler loses $10 per day. How much more money did they have 5 days ago?
Here, the loss per day is one negative and going backwards in time is another.
@M_Teacher_w_T shared this analogy:
“An enemy of my enemy is my friend.”
This aims not at the algebraic or arithmetic properties of numbers but more at the oppositeness of negative numbers.
Prerequisite knowledge: All contexts that build new understanding require students to understand the pieces of the context fairly well, so it is especially important to probe how students understand an idea when it is presented contextually.
Algebraic proof from first principles
From Dr. Alex Eustis, we have this algebraic proof that a negative times a negative is a positive.
First, he states a set of axioms that apply to any ring with unity. A ring is basically a number system with two operations. Each operation is closed, which means that using these operations (such as addition and multiplication on the real numbers) leads to another number within the number system. Each operation also has an identity element or an element that does not change another element in the system when applied to it. For example, under addition, 0 is the additive identity. Under multiplication, 1 is the multiplicative identity. The full set of axioms required is below.
| | |
--- |
| Axiom 1: a + b = b + a | (Additive commutivity) |
| Axiom 2: (a + b) + c = a + (b + c) | (Additive associativity) |
| Axiom 3: 0 + a = a | (Additive identity) |
| Axiom 4: There exists −a satisfying a + (−a) = 0 | (Additive inverse) |
| Axiom 5: 1 × a = a × 1 = a | (Multiplicative identity) |
| Axiom 6: (a × b) × c = a × (b × c) | (Multiplicative associativity) |
| Axiom 7: a × (b + c) = a × b + a × c | (Left multiplicative distribution) |
| Axiom 8: (b + c) × a = b × a + c × a | (Right multiplication distribution) |
From these axioms, we can prove that a negative times a negative is a positive. I’ll reproduce Dr. Eustis’s proof below and include the reference to the axioms used. First, we prove that a = −(−a).
Corrolary 1
| | |
--- |
| a = a + 0 | (Axiom 3 and Axiom 1) |
| a = a + (−a + −(−a)) | (Axiom 4 applied to −a ) |
| a = (a + (−a)) + (−(−a)) | (Axiom 2 – the associative property) |
| a = 0 + (−(−a)) | (Axiom 4) |
| a = −(−a) | (Axiom 3) |
So now we know that if we introduce negative numbers a is equal to −(−?).
Corrolary 2
| | |
--- |
| 0 = a + (−a) | (Axiom 4) |
| 0 = (0 + 1) × a + (−a) | (Axiom 3 and Axiom 5) |
| 0 = 0 × a + 1 × a + (−a) | (Axiom 8) |
| 0 = 0 × a + (a + (−a)) | (Axiom 5 and Axiom 2) |
| 0 = 0 × a + 0 | (Axiom 4) |
| 0 = 0 × a | (Axiom 3 and Axiom 1) |
Proving that 0 = 0 × a is the kind of painfully obvious idea that hardly requires proof but it establishes a relationship between multiplication and the additive identity in the real numbers, which is not yet included in the axioms above.
Next, we prove that (−1) × a = −a.
Corrollary 3
| | |
--- |
| −a = −a + 0 × a | (Corrolary 2 and Axiom 3) |
| −a = −a + (1 + (−1)) × a | (Axiom 4) |
| −a = −a + 1 × a + (−1) × a | (Axiom 8) |
| −a = (−a + a) + (−1) × a | (Axiom 5 and Axiom 2) |
| −a = 0 + (−1) × a | (Axiom 4) |
| −a = 0 + (−1) × a | (Axiom 3) |
Now, finally, we can prove that (−a) × (−b) = ab.
| | |
--- |
| (−a) × (−b) = (a × (−1)) × (−b) | (Corrolary 3) |
| (−a) × (−b) = a × ((−1) × (−b)) | (Axiom 6) |
| (−a) × (−b) = a × (−(−b)) | (Corrolary 3) |
| (−a) × (−b) = a × b | (Corrolary 1) |
This last “proof” though is unlikely to justify that a negative times a negative is a positive for any students though. It’s the kind of thing which is a required level of justification for a mathematician interested in rigorous proof who would likely consider the other justifications “patterning” and not sufficient.
A critical idea of proof though is that the intended audience of a proof is left convinced that an idea is true, and so I posit that the algebraic “proof” presented here is no proof at all for almost everyone.
Prerequisite knowledge: While I went through and added the justification for each step of the proof that was missing, I needed a fair bit of fluency with the original set of axioms. I also needed to not lose sight of the overall goal and to be able to recognize the structure of each part of the argument and match that structure to the axioms.
A simpler algebraic proof
This algebraic proof from Benjamin Dickman is much simpler than going back to a proof based on the axioms of arithmetic.
a + (−a) = 0
a × b + (−a) × b = 0 × b
ab + (−ab) = 0
From this, we can show that ab and –ab have opposite signs and therefore that a positive times a negative is a negative. Using the fact multiplication is commutative, a negative times a positive is also negative.
Similarly, we can prove that a negative times a negative is a positive.
a + (−a) = 0
a × (−b) + (−a) × (−b) = 0 × (−b)
−ab + (−a) × (−b) = 0
Since we know that −ab is negative, and the sum of these two terms is 0, therefore (−a) × (−b) is positive.
Prerequisite knowledge: The prerequisite knowledge for this proof is much less than the other one, but it does assume a fair bit of fluency with manipulation of algebraic structures.
Conclusion:
Given that the goal of an argument that something is true is to leave the other person convinced of the truth of the argument, whenever anyone uses any justification, representation, or proof, it behooves one to check that one’s audience is left convinced.
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kentilton says:
Using the number line again, and considering just -1 as a multiplier and p as some positive number:
-1 times p produces -p, a value p units in the opposite direction on the negative range of the number line, and equidistant from zero;
if we multiply by -1 again, we get p, a value in the original position;
on the complex plane, multiplying by -1 is a 180 degree rotation;
I like to say, if I turn around twice I am going in the same direction.
Left as an exercise is generalizing from -1 to -n.
Next fun question: we tell students to reverse an inequality when multiplying both sides by a negative. Justify _that_!
March 15, 2020 — 5:19 pm
Reply
mike says:
If 0 < a a > b.
If a < 0 < b, flipping signs flips which sides of zero they're on, so b < 0 < a.
January 6, 2024 — 3:10 pm
2. #### Emma says:
I also like to show students that 9 x 9 (which they know is 81) is the same as (10-1)(10-1). Expanding to give the first 3 terms using the distributive law or the grid or box method yields 100 – 10 – 10 which gives 80. So -1 x -1 must be +1 to give 81 in total. Not a proof though!
March 25, 2020 — 3:14 pm
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Comment by post author
#### David Wees says:
That’s a really nice argument though that relies on consistency and student knowledge of the distributive property. “Hey look, we know these two things are true, therefore this third thing must also be true.”
March 25, 2020 — 3:47 pm
#### Craig Cyr says:
Awesome observation!!
November 20, 2023 — 6:45 pm
#### senan says:
That’s a great one to show it to primary kids while explaining negative times negative is positive.
December 28, 2023 — 10:44 am
3. #### Robert says:
Not a proof, but for an example in realia from the physical world of levers:
If you add weight (pos.) on the right side of the lever (pos.), the moment (pos. weight times pos. distance) is clockwise (pos.). –> + times + = +
If you remove weight (neg.) on the right side of the lever (pos.), the moment (neg. weight times pos. distance) is counterclockwise (neg.) –> – times + = –
If you add weight (pos.) on the left side of the lever (neg.), the moment (pos. weight times neg. distance) is counterclockwise (neg.). –> + times – = –
If you remove weight (neg.) on the left side of the lever (neg.), the moment (neg. weight times neg. distance) is clockwise (pos.) –> – times – = +
June 1, 2021 — 1:22 pm
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Sugarbrenda@yahoo.com says:
That is absurd. If I owe my friend $1000 every month, I am in the negative by 1000 each month. -1000. If I have to pay for 10 months, -10×-1000 = -10,000. Now if by mental masterbation I can just say that two negatives multiplied equals a positive, then my friend owes me $10,000! Great, let’s tell my mortgage company!!!
October 7, 2023 — 3:07 am
Comment by post author
#### David Wees says:
Paying for ten months is +10 × -1000 = -10,000 since the number of months you pay is a positive quantity. Using your analogy, what if your friend took away 10 months that you had to owe them 10,000? This is like -10 × -1000. How much money is this the equivalent of them giving you?
October 18, 2023 — 4:03 am
4. #### Warren Jones says:
so a couple years ago my son asked me why a negative times a negative is a positive, i didn’t have a good answer so we looked online for a proof, and i didn’t find anything i found super compelling, so i started trying to think about what we are literally saying
(-3)5 is easier to understand because we are saying “5 groups of -3” or (-3)+(-3)+(-3)+(-3)+(-3), which tracks, but I think some of the cognitive dissonance is that subtraction is just adding the inverse. So what are we saying when we say 3(-5)? 5 groups of the inverse of 3? if that is the case, then this also tracks ya? the inverse of 3 is -3 so 5 groups of (-3) which is (-15) ok so what are we saying when it’s (-3)(-5)?
5 inverse groups of (-3)? which is 15.
does that sound right?
July 14, 2021 — 7:32 pm
Reply
Warren Jones says:
also thanks for this post, it would have been so helpful when we were initially looking, but maybe i wouldn’t have tried to reason it out like this if i had seen this.
July 14, 2021 — 7:33 pm
Comment by post author
#### David Wees says:
Yeah, this is a good explanation, mostly because it makes sense to me and to you! That’s all that is really needed for an explanation to be useful — at least one person who understands it but of course two is ideal.
July 20, 2021 — 1:08 am
#### Krishna says:
But I am still surprised, what if a bag contains 5 rotten apples and we put 3 times of 5 rotten apples? Are they good apples 🍎? Contradictory of negative times negative is positive.
April 5, 2023 — 3:06 pm
Comment by post author
#### David Wees says:
Putting is “a positive”, so putting 3 times of 5 rotten apples is adding 3 times -5 or +3 times -5. Instead, what if we had a bag of 15 rotten apples and we removed (eg. subtracted) 3 groups of rotten apples. Now the bag improves in quality by 15 rotten apples, eg. we added 15 good apples to cancel out each of the rotten apples in the bag.
April 6, 2023 — 9:41 pm
5. #### Denise says:
Oh wow. Lol. None of these explanations helped me. #WalksAwayInADaze
December 23, 2021 — 12:23 am
Reply
6. #### Eric says:
The real problem is trying to explain what a group of negatives is. People can visualize groups of things but a negative number of things has no physical meaning for them. Ultimately a negative sign just positions a number relative to zero. That is why in chemistry we have to use the absolute temperature scale for any mathematical calculations involving temperature. Zero degrees Celsius and zero degrees Fahrenheit are both arbitrary assignments.
April 19, 2022 — 11:52 am
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Petr says:
Distances are more useful than “number of things” in this case.
September 4, 2022 — 12:39 pm
7. #### Hannah says:
I’m reading Neil de Grasse’ Tyson’s Origens. In it he states -3 -3 = 9.
And this site agrees. I get it that a negative a positive = 0.
For now I’ll just memorize a negative a negative = a positive.
I’ll ask my school teacher daughter to explain it,
and thank you for trying to make it clear,
first time I used this good site! Hannah
October 13, 2022 — 12:33 am
Reply
Comment by post author
#### David Wees says:
Yeah, sorry. I’ll see if I can find other explanations that may land for people. I focused on more algebraic ideas here but there are some visuals that might help.
November 3, 2022 — 4:15 pm
8. #### Ralph Cobo says:
Oh wow. You guys need to consider that numbers are directional vectors. The opposite of a vector is the negative (-) of that vector. Therefore -3 x -3 is the opposite of the direction sum -9 which is +9.
December 9, 2022 — 11:29 pm
Reply
9. #### lamarr says:
i use direction on a rail. you can only go one way, or another. a positive will tell you to proceed in the direction you’re going. A negative will tell you to turn around, no matter what direction you’re going. I don’t expect men to get this because who asks for directions? lol
December 12, 2022 — 6:15 pm
Reply
Denise says:
This one helps . Ha ha ha
March 2, 2024 — 9:37 am
10. #### BK says:
I really want to provide this response to my credit card debt! Spending $20 means my balance equals -20.00.
So, do that 3 times -3, and viola, -20×-3=60.
Hooray! They owe me!
Makes about as much sense as all the junk I had to memorize to get thru algebra and calculus. Mental masterbation.
July 29, 2023 — 9:26 pm
Reply
HD says:
Charging your credit card 3 times is a positive instance. So 3 x -20 = -60. It still checks out.
August 28, 2024 — 2:33 pm
11. #### james says:
I’m afraid Benjamin Dickman’s approach has one fundamental flaw.
It goes without showing that the distributive property can be used with a negative number composition at all.
When doing that it’s automatically assuming the legality of negative times negative…
August 7, 2023 — 9:52 am
Reply
12. #### howard says:
Another explanation about the number line diagram about negative times negative equal positive is that the point zero is the initial point,first negative,-a means you having decreasing values,times second negative,-b means you losing the amount by b times. -3 x -3 In this situation,you having the decreasing value,-3 when you at zero initial point,you losing it by 3 time,which you left the total decreasing values of -9,which means you gaining back your values,9.
October 8, 2023 — 12:36 pm
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13. #### Zach Fairchild says:
This article showed proof of absolutely nothing. Every “proof” is assuming a negative x negative is a positive. It’s hard for people to comprehend, but you have to imagine a negative object existing. A negative apple sitting there, taking up space negatively. If I multiply 3 negative apples by 5 negative apples, they wouldn’t magically turn positive. They wouldn’t know what a positive apple looks like. What if I’m multiplying cells. Would all the negative cells I multiply together magically turn positive? No, and that’s why everything looks good on paper.
October 17, 2023 — 9:55 am
Reply
14. #### John Harding says:
Let (x-a)(x-a)=y
Then x^2 – 2xa + (-a)^2 = y
Now if y=0 then x=a
So a^2 – 2a^2 + (-a)^2 = 0
Then -a^2 + (-a)^2 = 0
Or (-a)^2 = +a^2
Ie. (-a) x (-a) = + (a x a)
So minus x minus = plus
December 6, 2023 — 8:58 am
Reply
Satish Shah says:
Superb proof.🙏🏽
January 30, 2024 — 8:13 am
15. #### Rajesh says:
How to write 2×0 & 0×2 both in a successive “addition” way separately in mathematics?
December 22, 2023 — 8:26 pm
Reply
16. #### mike says:
Apparently angle brackets destroy text.
To sum up : Flipping signs of all numbers (multiplying entire equality by a negative) swaps which side of zero a number is on while preserving relative magnitude, so the relation between them is inverted.
January 6, 2024 — 3:13 pm
Reply
Neena says:
How do you measure the magnitude of a literal number?
July 29, 2024 — 7:49 am
17. #### Dustin Robert Kengott says:
How does this debate differ from people who claim that 0.9(repeating) is equal to one?
…
A nine-fold absence of something that is thrice absent already is not a miracle made.
Like this: I overdrafted $100.00 from my bank account so now I am only eating food that I can buy 2-for-1 until my debt is washed.
I owe you a dollar? No problem, here is a cheeseburger.
Wait just a second, the sale paper says pickles are buy one get one free, and if you can’t find anywhere that cheeseburger is accepted, I will consider taking it off your hands for $0.50.
Yah I bet 27-more of anything is looking pretty good now that you sat on someone’s discarded chewing gum…
Oh man, I love seagulls.
January 22, 2024 — 6:33 am
Reply
18. #### Eritheus says:
Hello, why does the -3 x 3 example move from 1 to -8 on the graph?
April 6, 2024 — 2:17 am
Reply
19. #### Scott says:
Easy peasy, 2 negatives are positive. This is Truth. Galatians 3:13-“Christ hath redeemed us from the curse of the law, being made a curse for us: for it is written, Cursed is every one that hangeth on a tree:” The curse is negative, Christ was cursed, the result is POSITIVE! That settles it for me! glory to God!
September 7, 2024 — 2:44 pm
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Give examples of oxides that are neutral and acidic oxides.
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Hint: Oxides are chemical compounds with one or more oxygen atoms combined with another element (example-$L{i_2}O$) . Oxides are binary compounds of oxygen with another element, example-$C{O_2},S{O_2},CaO,CO,ZnO,Ba{O_2},{H_2}O$ etc. Oxides can be generated with multiple reactions. Complete answer:Oxides are called because here, oxygen is in combination with only one element. Based on their acid-base characteristics oxides are classified as acidic, basic, amphoteric and neutral:[1.] An oxide that combines with water to give an acid is termed as an acid oxide.$2.$ The oxide that gives a base in water is known as basic oxide.$3.$ An amphoteric solution is a substance that can chemically react as either acid or base.$4.$ However, it is also possible for an oxide to be neither acidic nor basic, but is a neutral oxide.There are different properties which help distinguish between the three types of oxide. The term anhydride (without water) refers to compounds that assimilate${H_2}O$ to form either an acid or a base upon the addition of water.Acidic oxide: Acidic oxide are the oxides of non-metals and these acid anhydride form acid with water.$ \bullet $ Sulfurous acid- $S{O_2} + {H_2}O \to {H_2}S{O_3}$$ \bullet $ Sulfuric acid- $S{O_3} + {H_2}O \to {H_2}S{O_4}$$ \bullet $ Carbonic acid- $C{O_2} + {H_2}O \to {H_2}C{O_3}$Acidic oxides are known as acid anhydrides (example- sulfur dioxide is sulfurous anhydride and sulfur trioxide is sulfuric anhydride) and when combined with bases, they produce salts, example-$Cu{(OH)_2}\xrightarrow{\Delta }CuO + {H_2}O$$4.$$4.$$S{O_2} + 2NaOH \to N{a_2}S{O_3} + {H_2}O$Neutral oxide- Neutral oxide show neither basic nor acidic properties and hence do not form salts when reacted with acids and bases, example- carbon monoxide $(CO)$, nitrous oxide $({N_2}O)$, nitric oxide $(NO)$, etc. are neutral oxides.Preparation of Oxides:$ \bullet $ By direct heating of an element with oxygen:$2Mg + {O_2}\xrightarrow{{heat}}2MgO$$2Ca + {O_2}\xrightarrow{{heat}}2CaO$$S + {O_2}\xrightarrow{{heat}}S{O_2}$${P_4} + {O_2}\xrightarrow{{heat}}2{P_2}{O_5}$$ \bullet $ By reaction of oxygen with compounds at higher temperature:$1.$ sulfides are usually oxidized when heated with oxygen.$2PbS + 3{O_2}\xrightarrow{\Delta }2PbO + 2S{O_2}$$2ZnS + 3{O_2}\xrightarrow{\Delta }2ZnO + 2S{O_2}$$2.$ When heated with oxygen, compounds contained carbon and hydrogen are oxidized.${C_2}{H_5}OH + 3{O_2} \to 2C{O_2} + 3{H_2}O$$3.$ By thermal decomposition of certain compounds like hydroxides, carbonates. And nitrates$CaC{O_3}\xrightarrow{\Delta }CaO + C{O_2}$$2Cu{(N{O_3})_2}\xrightarrow{\Delta }2CuO + 4N{O_2} + {O_2}$$Cu{(OH)_2}\xrightarrow{\Delta }CuO + {H_2}O$[4.] By oxidation of some metals with nitric acid[2Cu + 8HN{O_3}\xrightarrow{{heat}}2CuO + 8N{O_2} + 4{H_2}O + {O_2}][Sn + 4HN{O_3}\xrightarrow{{heat}}Sn{O_2} + 4N{O_2} + 2{H_2}O][5.] By oxidation of some nonmetallic with nitric acid.[C + 4HN{O_3} \to C{O_2} + 4N{O_2} + 2{H_2}O]Note:The oxide of elements in a period becomes progressively more acidic as one goes from left to right in a period of the periodic table. Metal oxide on the left side of the periodic table produces a basic solution in water (example- $N{a_2}O$). Nonmetallic oxides on the right side of the table produce an acidic solution (example- $C{l_2}O$).
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12994 | https://apjcn.nhri.org.tw/server/apjcn/21/3/464.pdf | 464 Asia Pac J Clin Nutr 2012;21 (3):464-469 Case Study High output enterocutaneous fistula: a literature review and a case study Chung Yan Tong RD, CNSC1, Li Lin Lim MRCP2, Rebecca A Brody PHD, RD, CNSC3 1Dieteitcs Department, National University Hospital, Singapore 2Department of Gastroenterology and Hepatology, National University Hospital, Singapore 3Department of Nutritional Sciences, School of Health Related Professions, University of Medicine and Dentistry of New Jersey, United States An enterocutaneous (EC) fistula is referred to as a channel between the gut and the skin. Effluent of an EC fistula of more than 500 ml per day is considered as high output. Patients with high output EC fistulae have a high mor-bidity and mortality rate. No evidence-based guidelines are available for this condition and more research is re-quired to evaluate the effectiveness of treatment. Nevertheless, patients with fistulae should be managed based on the available evidence, detailed clinical and nutrition assessment, and close monitoring. Management of high out-put EC fistula is complex and challenging. It involves nutrition, medical, skin care and psychological treatment, which is best managed by a multidisciplinary team. It requires an individualized nutrition and clinical treatment plan to maximize patient outcomes. Up to 70% of patients with fistulae have malnutrition and it is a significant prognostic factor of spontaneous fistula closure. Nutrition therapies including macronutrient and micronutrient delivery, enteral nutrition and parenteral nutrition are discussed in this review. A case study of a patient with mul-tiple EC fistulae is presented to illustrate the management of high output EC fistulae. Key Words: intestinal fistula, cutaneous fistula, parenteral nutrition, enteral nutrition, intestinal secretions INTRODUCTION Fistula is defined as an abnormal communication of an organ to another organ, skin or wound.1,2 If a fistula is between the gut and another part of the body, it is gener-ally called a gastrointestinal (GI) fistula.1 An enterocuta-neous (EC) fistula is referred to as a channel between the gut and the skin.2 The pathophysiology of fistulae can be explored through various techniques, including the me-thylene blue test, fistulography, computed tomography scan, ultrasonography, magnetic resonance imaging and endoscopy.2 Surgery accounts for 75% to 85% of gastrointestinal fistula development and the fistula usually arises five to ten days after an operation.3 It is likely to happen in emergency surgery when pre-operative preparation is usually poor.1 For spontaneous fistula development, Crohn’s disease is the major cause and 40% of this group develops fistula.2 Effluent of an EC fistula of more than 500ml per day is considered as high output. Patients with high output EC fistulae have a high morbidity and mortal-ity rate.1 This review focuses on high output EC fistulae. LITERATURE REVIEW ON FISTULA MANAGE-MENT Prognosis of a GI fistula depends on the patient character-istics, nutritional status, fistula characteristics and other co-morbidities.4-6 The endpoint of GI fistula is fistula closure, either spontaneously or surgically. Table 1 sum-marizes the factors that may cause delay in spontaneous fistula closure and Table 2 summarizes the management of GI fistula based on literature review. Overall, management of high output EC fistula is complex and challenging. It involves nutrition, medical, skin care and psychological treatment. A case study of a patient with multiple EC fistulae is presented to illustrate the management of high output EC fistulae. CASE STUDY PRESENTATION JY, is a 23-year-old Chinese female, admitted to the hos-pital due to a serious motorcycle accident. She suffered haemoperitoneum with a mesenteric tear at the ileum, sigmoid colon and caecum. JY had resection of ischemic parts of the gut in the first two days of admission. On hospital day 25, a serosal tear of small bowel was noted and surgical repair was done (Table 3). The initial defect had increased in size and a new defect was found 3 days later. Her small bowel was extremely frail to any suturing. On hospital day 37, a third small bowel defect had devel-oped, and small bowel endoscopic stenting was attempted to repair the fistulae but this was unsuccessful. On hospi-tal day 40, the most proximal fistula had increased to 4 Corresponding Author: Ms. Chung Yan Tong, National Uni-versity Hospital, Dietetics Department, Main Building, Level 1, 5 Lower Kent Ridge Road, Singapore 119074. Tel: (65)67725166; Fax: (65)67791938 Email: cherie_tong@nuhs.edu.sg Manuscript received 22 September 2011. Initial review com-pleted 17 January 2012. Revision accepted 5 March 2012. High output enterocutaneous fistula 465 cm. Dense adhesions were found in the intestines and other organs described by the surgeons as like a ‘cocoon.’ JY was on parenteral nutrition (PN) and antibiotic treatment for persistent intrabdominal sepsis for the fol-lowing 6 months (Tables 3 and 4). The output from the proximal fistula remained very high, ranging from 1.6-4.6 L/day. When she developed line sepsis eight months after admission, the surgeon operated on her to close the three fistulae. One week after the operation, a new small jeju-num perforation was found and repaired. However, this new defect formed a small fistula with moderate output of 200 – 400ml per day. She was started on an oral diet and she remained an inpatient for another three months for antibiotic treatment and rehabilitation. She was dis-charged ten months after admission. She had her stoma dressing done at a nearby clinic on a weekly basis and had monthly follow-up appointments with the surgeon, the dietitian and the wound ostomy nurse in the hospital. NUTRITION MANAGEMENT Baseline nutrition assessment was performed using the Subjective Global Assessment (SGA) according to weight history, dietary intake change, GI symptoms, functional capacity, disease state and physical exams.35 The SGA rating was A which implied JY was well nourished at admission. SGA was performed every 3 months and JY’s SGA rating dropped to B (moderately malnourished) in the third to sixth month due to multiple infections, reiter-ated surgery and persistent high fistulae output despite ongoing PN support. JY remained moderately malnour-Table 1. Factors that may cause delay in spontaneous fistula closure 1,2,4-6 Old age >65 Malnutrition Organ involved: stomach, duodenum, ileum. Fistula duration >4-6 weeks. Fistula output >500 ml/day. Etiology of fistula: malignancy, inflammatory bowel disease, radiation enteritis Co-morbidities: sepsis, diabetes or renal failure, under chemotherapy, radiation or corticosteroid treatment. Fistula presentation: External, complex, multiple or end fistula. Fistula tract <2 cm or defect >1 cm. Eversion of mucosa or distal occlusion. Poor or diseased adjacent bowel. Presence of abscess or foreign body. Presence of abdominal wall defect. Management errors: failure to diagnose an anastamotic leak, a delay in surgical exploration, attempt to restore intestinal continuity too early and failure to initiate nutrition support. Table 2. Management of gastrointestinal (GI) fistula based on literature review 1-34 Goals To prevent complications, promote spontaneous closure and minimize morbidity and mortality.1-4 Multidisciplinary approach A gastroenterologist, surgeon, radiologist, nutrition support dietitian, wound ostomy nurse, pharmacist and psychologist.7,8 Fluid & electrolyte management Active replacement to prevent dehydration, electrolyte imbalance (eg, hyponatremia, hypokalemia) and metabolic acidosis or alkalosis.2,9,10 Nutrition assessment Up to 70% of patients with fistulae have malnutrition and it is a significant prognostic factor for sponta-neous fistula closure. Baseline and regular nutrition assessment is fundamental.1,2,9,19-21 Enteral nutrition Oral diet Fistuloclysis Preferred route of nutrition support, unless it causes significantly increased fistula output, abdominal pain or diarrhea.9,22,23 High sodium, low residue diet and use of oral rehydration solutions to replace fistula losses.1,15,33 1) delivering feed via the fistula site by inserting a tube under radiology into the fistula; 2) feeding the fistula effluent from the proximal fistula to the distal fistula.16,34 Parenteral nutrition Individualized parenteral nutrition regimen should be planned to meet nutrition, fluid and electrolyte re-quirements and to minimize PN-related complications (eg, hyperglycemia, bacterial translocation, cathe-ter sepsis, vein thrombosis, cholestasis, steatosis and metabolic bone disease).10,23-30 Nutritional Requirements Low output GI fistula:1 Resting energy expenditure (REE) or 25 kcal/kg body weight/day, 1.0-1.5 g protein/kg body weight/day. High output GI fistula:1 1.5 x REE or 30 kcal/kg body weight/day,1.5-2.0 g protein/kg body weight/day, 2 x daily recommended intake (DRI) of vitamins and trace minerals, 5 x DRI of vitamin C and zinc. At risk of vitamin B12, zinc, magnesium and selenium deficiency.15,33 Medications To reduce gut motility and digestive secretion: somatostatin and its analog (Octeotide), loperamide, di-phenoxylate, codeine, opium tincture, stress ulceration prophylaxis (eg, proton pump inhibitors and H2 receptor antagonist).1,4,11-16 Artificial fistula closure Using fibrin glue via fistuloscopy, stenting via endoscopy, surgical fistula closure (at least three months after the patient has been hemodynamically stable with adequate nutritional support to maximize the chance of a successful surgery).1,2,4-6,9,11-18 466 CY Tong, LL Lim and RA Brody ished at discharge but her nutritional status gradually im-proved with increased oral intake, reduced fistula output and the absence of infection. Electrolyte levels and fluid balance were monitored closely. The PN regime was adjusted and intravenous electrolyte replacement was given accordingly (Table 4). Her serum sodium, potassium and magnesium levels were depleted initially when the fistula output was more than 2L/day. The maximum compatible dose of magnesium and zinc were prescribed in the PN solution and intrave-nous zinc supplementation was given to JY daily while she was on PN to compensate for her losses through the high output EC fistulae. The amount of fistula output was associated with her oral diet consumption; her fistula output increased up to 4L/day when she had good oral intake in the third, sixth and seventh month post injury (Table 3). The output re-duced to 1.2-1.8 L/day when her appetite was poor in the fourth and fifth months of hospitalization. She refused oral rehydration solution, semi-elemental and elemental supplements. Her poor dietary compliance was likely due to depression caused by the long hospital stay, chronic forearm wound, high output fistulae and financial concern. JY did not tolerate fistuloclysis with fistula effluent. Somatostatin was prescribed twice for a duration of 3 weeks each and it reduced the fistula output to approxi-mately 1L per day. Folate, vitamin B-12, zinc and an iron panel were monitored in the third month. Her folate, vitamin B-12 and zinc levels were normal. Her iron, total iron binding capacity, and transferrin were low with elevated ferritin indicating chronic disease and repeated operation-related acute blood loss. JY received regular blood transfusions when her hemoglobin was low after each operation. Parenteral nutrition was the major source of nutrition support for JY with a total of 229 days of PN during her hospital stay. She suffered cholestasis and steatosis three months after PN started. Cyclic PN of 16 hours was Table 3. JY’s weight trend, laboratory data, fistula output and nutrition support regime Hospital Stay (month) Weight (kg) Pertinent Laboratory Data† Key Medial Events Fistula Output (L/day) Nutrition Support Regime‡ 1 55 BUN: 50-120 mg/dL Cr: 22 mg/dL Glu: 150-200 mg/dL Alb: 15-25 g/L LFT: normal TG: normal Major operations at right forearm and ab-domen. Formation of 3 EC fistulae. Stayed in intensive care unit. 0-0.2 Oral: Nill by mouth. PN: initial regime in Table 4 with standard MVI and trace element. Enteral feeding was initiated in ICU and discontinued when small bowel perforated. 3 52 Na: 132-137 mmol/L K: 3-4.5 mmol/L Mg: 0.79-0.94 mmol/L Glu: 85-175 mg/dL Alb: 26-31 g/L ALT: 34-129 U/L ALP: 272-549 U/L GGT: 362 U/L Bil: 11-22 mol/L TG: 237-356 mg/dL Started somatostatin for 3 weeks in view of very high fistula out-put. JY was very depressed and she was allowed to have oral diet. 1.6-4.0 Cyclic PN: 16 hours. Reduce lipid infusion to x 2/week. Addition IV zinc: 5 mg/day. Trial of SMOF lipid. Oral: Refused ORS, semi-elemental and elemental feed, educated on high protein, low residue diet but with poor diet compliance. Electrolyte replacement:IV K and Mg. 6 53-55 Glu: 126-128 mg/dL K: 3.6-4.6 mmol/L Mg: 0.45-0.93 mmol/L ALT: 14-93U/L ALP: 167-498 U/L GGT: 89-306 U/L Bil: 2-8 mol/L TG: 90-182 mg/dL Discontinued soma-tostatin due to high cost and unlikely spontaneous fistula closure. Line sepsis occurred at the end of 7th month. JY was encouraged to increase oral intake. 2.1-4.6 PN: discontinued at the end of 7th months due to line sepsis. Oral: Normal diet with increased intake, poor diet compliance, refused ORS, small amount of semi-elemental feed and some standard oral supplements. JY lost 2 kg in 2 weeks when PN was off. 9 53 Glu: 104-135 mg/dL K: 3.0-4.6 mmol/L Mg: 0.60-1.00 mmol/L ALT: 29-93U/L ALP: 199-585 U/L GGT: 123-400 U/L Bil: 10-58 mol/L TG: 192-276 mg/dL Small bowel fistulae closure, adhesiolysis and took down colos-tomy. New small jejunal fistula formed with moderate output. 0.2-1.7 PN: restarted after fistula closure with same previous regime, weaned off in middle of 9th month when oral intake increased with moderate fis-tula output. Electrolyte replacement: IV K and Mg Oral: as above. † Abbreviation: full text (normal range): BUN: blood urea nitrogen (12-39 mg/dL), Cr: serum creatinine (5.5-9.9 mg/dL), Na: serum sodium (135-150 mmol/L), K: serum potas-sium (3.5-5.0 mmol/L), Glu: blood glucose (72-140 mg/dL), Mg: magnesium (0.75-1.07 mmol/L), Alb: serum albumin (38-48 g/L), LFT: liver function test, ALT: alanine transaminase (10-70 U/L), ALP: alkaline phosphatase (40-130 U/L), GGT: gamma-glutamyltransferase (10-80 U/L), Bil: total bilirubin (5-30 mol/L) , TG: triglyceride (<150 mg/dL) ‡ MVI: multivitamin infusion, SMOF: soybean oil, medium chain triglycerides, olive oil and fish oil, ORS: oral rehydration solution. IV: intravenous infusion High output enterocutaneous fistula 467 arranged and lipid infusion was decreased to twice a week. Metronidazole was prescribed to treat possible bacterial overgrowth which may have compromised her liver func- tion. Her total bilirubin level improved but the liver func-tion test (LFT), gamma-glutamyltransferase (GGT) and triglyceride (TG) levels remained high (Table 3). Soy-bean oil, medium chain triglycerides, olive oil and fish oil (SMOF) lipid (SMOFlipid®, Fresenius Kabi) was used in her PN formulation in the fourth month and reductions of LFT, GGT and TG were observed. When PN was stopped due to line sepsis, the LFT and GGT returned to normal ranges. Parenteral nutrition-associated liver disease (PNALD) was evident because LFT, GGT, TG and total bilirubin levels were elevated again when PN was re-started after the fistulae closure operation. Upon discharge, JY was educated about the optimal oral diet to maintain her weight and nutritional status. She was advised on sufficient fluid intake and how to monitor her hydration status. She was also discharged with oral potassium, magnesium and iron supplements. CONCLUSIONS This case study illustrates that high output EC fistula is a complex, demanding condition which is best managed by a multidisciplinary team. Not only does it have a high morbidity and mortality rate, it also has a significant fi-nancial and psychological impact. It requires an individu-alized nutrition and clinical treatment plan to maximize patient outcomes. More research is required to evaluate the effectiveness of treatment and to develop evidence-based guidelines. Meanwhile, patients with fistulae should be managed based on the available evidence, de-tailed clinical and nutrition assessment, and close moni-toring. AUTHOR DISCLOSURES The authors do not have any financial support or relationships that may pose a conflict of interest. The authors also do not have any industrial links and affiliations. REFERENCES 1. Cozzaglio L, Farinella E, Bagnoli P, Sciannameo F, Doci R. Gastrointestinal fistulas. Nutr Ther Metabol. 2007;25:113-34. 2. Lloyd DAJ, Gabe SM, Windsor ACJ. Nutrition and management of enterocutaneous fistula. Br J Surg. 2006;93: 1045-55. 3. Berry SM, Fischer JE. Classification and pathophysiology of enterocutaneous fistulas. Surg Clin North Am. 1996;76: 1009-18. 4. Campos AC, Andrade DF, Campos GM, Matias JE, Coelho JC. A multivariate model to determine prognostic factors in gastrointestinal fistulas. J Am Coll Surg. 1999;188:483-90. 5. Hollington P, Mawdsley J, Lim W, Gabe SM, Forbes A, Windsor AJ. An 11-year experience of enterocutaneous fistula. Br J Surg. 2004;91:1646-51. 6. Haffejee AA. Surgical management of high output enterocutaneous fistulae: a 24-year experience. Curr Opin Clin Nutr Metab Care. 2004;7:309-16. 7. Datta V, Engledow A, Chan S, Forbes A, Cohen CR, Windsor A. The management of enterocutaneous fistula in a regional unit in the United kingdom: a prospective study. Dis Colon Rectum. 2010;53:192-9. 8. McNaughton V, Canadian Association for Enterostomal Therapy ECF Best Practice Recommendations Panel, Brown J, Hoeflok J, Martins L, McNaughton V, Nielsen EM, Thompson G, Westendorp C. Summary of best practice recommendations for management of enterocutaneous fistulae from the Canadian Association for Enterostomal Therapy ECF Best Practice Recommendations Panel. J Wound Ostomy Continence Nurs. 2010;37:173-84. 9. Willcutts K, Eddins C. Ostomies and fisulas: a collaborative approach. Pract Gastroenterol. 2005;33:65-79. 10. Gonzalez-Pinto I, Gonzalez EM. Optimising the treatment of upper gastrointestinal fistulae. Gut. 2001;49(S4):iv22-31. 11. Martineau P, Shwed JA, Denis R. Is octreotide a new hope for enterocutaneous and external pancreatic fistulas closure? Am J Surg. 1996;172:386-95. 12. Alivizatos V, Felekis D, Zorbalas A. Evaluation of the effectiveness of octreotide in the conservative treatment of postoperative enterocutaneous fistulas. Hepatogastro-enterology. 2002;49:1010-2. 13. Leandros E, Antonakis PT, Albanopoulos K, Dervenis C, Konstadoulakis MM. Somatostatin versus octreotide in the treatment of patients with gastrointestinal and pancreatic fistulas. Can J Gastroenterol. 2004;18:303-6. 14. Hesse U, Ysebaert D, de Hemptinne B. Role of somatostatin-14 and its analogues in the management of gastrointestinal fistulae: clinical data. Gut. 2001;49(S4): iv11-21. 15. Buchman AL. Short-bowel syndrome. Clin Gastroenterol Hepatol. 2005;3:1066-70. Table 4. JY’s nutrition management plan Nutrition Goals: 1. maintain nutritional status by delivering adequate energy and nutrients 2. provide sufficient protein for skin integrity and replacing losses from fistula output 3. maintain fluid balance and normalize electrolytes 4. facilitate wound healing and fistula closure 5. prevent and promote recovery from infection and sepsis 6. prevent vitamin and mineral deficiencies 7. prevent parenteral nutrition related complications. Estimated nutrition requirements: Admission weight: 55 kg Height: 1.6 m BMI: 22 kg/m2 Energy: 30 kcal/kg/day = 1650 kcal/day Protein: 1.5 g/kg body weight/day = 83 g/day Aim for fluid balance: urine output, fistula output, drainage, blood loss etc. Vitamin and minerals: DRI levels Parenteral nutrition (PN) prescription (per day) Type of PN Calories kcal (kcal/kg) Protein g (g/kg) Dextrose g (g/kg) Lipid g (g/kg) Magnesium mmol (mmol/kg) Zinc mg (mg/kg) Initial 1678 (31) 90 (1.64) 270 (4.9) 40 (0.73) 13 (0.24) 11.5 (0.21) Cyclic 1532 (28) 125 (2.27) 270 (4.9) 40x2/wk (0.73) 15 (0.27) 11.5+5 (IV) = 16.5(0.3) 468 CY Tong, LL Lim and RA Brody 16. Willcutts K. The Art of Fistuloclysis: Nutritional Management of Enterocutaneous Fistulas. Pract Gastro-enterol. 2010;87:47-56. 17. Lynch AC, Delaney CP, Senagore AJ, Connor JT, Remzi FH, Fazio VW. Clinical outcome and factors predictive of recurrence after enterocutaneous fistula surgery. Ann Surg. 2004;240:825-31. 18. Conter RL, Roof L, Roslyn JJ. Delayed reconstructive surgery for complex enterocutaneous fistulae. Am Surg. 1988;54:589-93. 19. Windsor JA, Hill GL. Protein depletion and surgical risk. Aust N Z J Surg. 1988;58:711-5. 20. Windsor JA, Hill GL. Risk factors for postoperative pneumonia. The importance of protein depletion. Ann Surg. 1988;208:209-14. 21. Reilly JJ, Jr., Hull SF, Albert N, Waller A, Bringardener S. Economic impact of malnutrition: a model system for hospitalized patients. JPEN J Parenter Enteral Nutr. 1988; 12:371-6. 22. Meguid MM, Campos AC. Nutritional management of patients with gastrointestinal fistulas. Surg Clin North Am. 1996;76:1035-80. 23. Levy E, Frileux P, Cugnenc PH, Honiger J, Ollivier JM, Parc R. High-output external fistulae of the small bowel: management with continuous enteral nutrition. Br J Surg. 1989;76:676-9. 24. Campos AC, Meguid MM, Coelho JC. Factors influencing outcome in patients with gastrointestinal fistula. Surg Clin North Am. 1996;76:1191-8. 25. Dudrick SJ, Maharaj AR, McKelvey AA. Artificial nutritional support in patients with gastrointestinal fistulas. World J Surg. 1999;23:570-6. 26. Rombeau JL, Rolandelli RH. Enteral and parenteral nutrition in patients with enteric fistulas and short bowel syndrome. Surg Clin North Am. 1987;67:551-71. 27. Alexander JW. Bacterial translocation during enteral and parenteral nutrition. Proc Nutr Soc. 1998;57:389-93. 28. Sedman PC, MacFie J, Palmer MD, Mitchell CJ, Sagar PM. Preoperative total parenteral nutrition is not associated with mucosal atrophy or bacterial translocation in humans. Br J Surg. 1995;82:1663-7. 29. Zera RT, Bubrick MP, Sternquist JC, Hitchcock CR. Enterocutaneous fistulas. Effects of total parenteral nutrition and surgery. Dis Colon Rectum. 1983;26:109-12. 30. Deitel M. Elemental diet and enterocutaneous fistula. World J Surg. 1983;7:451-4. 31. American Dietetic Association. American Dietetic Association Evidence Analysis Library. [cited 2011/7/15]; Available from: 32. Essential Evidence Plus. [cited 2011/7/15]; Available from: content/ 33. Buchman AL. Etiology and initial management of short bowel syndrome. Gastroenterology. 2006;130(2 Suppl 1): S5-S15. 34. Teubner A, Morrison K, Ravishankar HR, Anderson ID, Scott NA, Carlson GL. Fistuloclysis can successfully replace parenteral feeding in the nutritional support of patients with enterocutaneous fistula. Br J Surg. 2004;91: 625-31. 35. Detsky AS, McLaughlin JR, Baker JP, Johnston N, Whittaker S, Mendelson RA, Jeejeebhoy KN. What is subjective global assessment of nutritional status? JPEN J Parenter Enteral Nutr. 1987;11:8-13. High output enterocutaneous fistula 469 Case Study High output enterocutaneous fistula: A literature review and a case study Chung Yan Tong RD, CNSC1, Li Lin Lim MRCP2, Rebecca A Brody PHD, RD, CNSC3 1National University Hospital, Singapore 2Department of Gastroenterology and Hepatology, National University Hospital, Singapore 3Department of Nutritional Sciences, School of Health Related Professions, University of Medicine and Dentistry of New Jersey, United States 高流量肠外瘘：文獻回顧及病例研究 肠外瘘是指肠道和皮肤之间的通道。如果肠外瘘每日排出超过五百毫升便被视 为高流量肠外瘘。高流量肠外瘘患者的患病和死亡的机率很高。目前没有以科 研证据为基础的肠外瘘医疗指引，而需要更多的研究以评估治疗的有效性。然 而，肠外瘘患者的治疗应根据现有的科研证据，详细的临床和营养评估，并密 切监测管理来拟定。高流量肠外瘘的治疗是复杂和具有挑战性的。它涉及到营 养、医学、皮肤护理和心理治疗。这需由不同专业医疗小组依据个别状况来拟 定一个适合病患的营养和临床治疗方案，才能达到最佳的治疗效果。高达七十 百分比的肠外瘘患者患有营养不良，而营养不良是预测肠外瘘自发愈合的一个 重要因素。本篇回顧所要讨论的营养治疗包括：主要营养素及微量营养素的供 給，肠内营养和肠外营养辅助。本文将以一个患有多重肠外瘘病人的个案来说 明高流量肠外瘘的管理。 关键词：肠瘘、皮肤瘘、肠外营养、肠内营养、肠分泌 |
12995 | https://www.ixl.com/math/grade-6/find-what-percent-one-number-is-of-another | IXL | Find what percent one number is of another | 6th grade math
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Complete the following statement. Write your answer as a decimal or whole number.
% of $4 = $3
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% of 80 = 60
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A percent is 1 part out of 100.
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Proportions can be used to solve percent problems.
part whole=percent 100
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The part is 60.
The whole is 80.
Let n represent the percent.
Write a proportion to solve for n.
60 80=n 100
60 · 100=80 n
6,000=80 n
6,000 ÷ 80=80 n ÷ 80
75=n
75% of 80 = 60
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60=0.8 n
60÷0.8=0.8 n÷0.8
75=n
75% of 80 = 60
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12996 | https://www.youtube.com/watch?v=OxLRhmvuKVM | Chapter 7 - Cell Membrane & Transport (Active & Passive Transport, Osmosis, Diffusion, Bulk)
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⏰ Timestamps:
1:34 Intro to the Cell Membrane
2:22: Phospholipids
4:24 Fluid Mosaic Model and factors of membrane fluidity
9:03 Membrane proteins and function
11:36 Functions of surface proteins
15:44 Selective permeability
19:01 Transport Proteins
20:58 Types of Transport (Active vs. Passive)
21:50 Diffusion & concentration gradients
25:08 Passive Transport (Simple Diffusion, Osmosis, Facilitated Diffusion)
27:16 Osmosis
29:31 Tonicity (hypotonic, hypertonic, isotonic)
35:19 Facilitated Diffusion
36:52 Channel Proteins
39:12 Active Transport (Electrogenic Pumps, Cotransport, and Bulk transport)
51:15 Exocytosis
51:57 Endocytosis (phagocytosis, pinocytosis, receptor-mediated endocytosis)
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Transcript:
hello and welcome back to this online lecture series in the last lecture you were taken through a tour of the cell where you were introduced to the organelles and the network of cytoskeleton components that gives organization and structure to the cells knowing the organelles is important to understanding how the cell functions being able to distinguish between prokaryotes and eukaryotes and it will definitely help in future physiology courses you also learned about the selective barrier surrounding every cell called the plasma membrane this lecture will dive further into the structure and function of the plasma membrane and the implications for cellular functions we'll cover the plasma membrane membrane proteins and functions osmosis and tonicity passive and active transport including bulk transport across the plasma membrane after this lecture you will be able to demonstrate an understanding of the structure and functions of cell membranes and the implications for cellular processes in particular you'll be able to describe the structure of biological membranes including the role of lipids and proteins describe the function of biological membranes and what it means that the cell membrane is selectively permeable and you will be able to describe how various substances move across the cell membrane and distinguish between passive and active transport as you should now know the plasma membrane is the boundary that separates the living cell from its surroundings every single cell has a plasma membrane whether they are prokaryotic or eukaryotic its most crucial role is as in a selectively permeable barrier which allows some substances to cross more easily than others this selectivity is so important to a cell's ability to regulate homeostasis and maintain controlled interactions with its environment we will be learning all about how the cell accomplishes this including how transport proteins are often responsible for controlling passage across cellular membranes but first let's review the structure of the plasma membrane knowing the structure of the membrane is very important to understanding how it functions remember back to lecture one the structure function relationships plasma membranes are comprised of a special type of lipid called a phospholipid recall from lecture 5 that a phospholipid is two fatty acids and a phosphate group attached to a glycerol the two fatty acids are hydrophobic hydrocarbons but the phosphate group forms a hydrophilic head this makes the phospholipid both hydrophilic and hydrophobic and this type of molecule is called amphipathic part of the molecule is water loving while part of it is water fearing because of this the phospholipids orient themselves so that the hydrophobic tails are sheltered inside the membrane while the hydrophilic heads are exposed to water and fluids on either side given the opportunity phospholipids will automatically orient themselves like this and you can see the lipid bilayer sphere structure in this image notice how it's different from the single layer lipid sphere which would not allow water to come in contact with the exposed hydrophobic tails that are on the inside of that sphere our membranes are phospholipid bilayers there's two layers of phospholipids so that the intracellular environment can be water-based because after all we are 70 water this special orientation of the phospholipids is what gives the membrane selective permeability and more on that in a little bit remember cells are the smallest units of life and must therefore perform all cellular operations that are required of life to achieve this successfully the plasma membrane possesses a unique property of fluidity whereby the phospholipids and a variety of protein components of the membrane can shift and move laterally through the bilayer this is termed the fluid mosaic model the image on the right is a tile mosaic which is a pattern that's made of small regular or irregular pieces of colored stone or glass or ceramic the membrane is a type of mosaic but it's made of proteins bobbing in a fluid bilayer of phospholipids however don't mistake this to mean that the cell has no control of the fluidity or the mosaic of proteins proteins are not randomly distributed in the membrane the cell is of course a finely tuned piece of organic machinery and everything is highly orchestrated to maintain homeostasis in the support of life movement is free sideways or laterally within the membrane but there are rare proteins that can actually flip across the membrane from one side of the phospholipid layer to another and scientists were very creative in naming these proteins they're called flip base and flop ace let's take a look at how the cell monitors the fluidity of the membranes recall from lecture five that the main type of bond holding phospholipids together in the membrane are weak hydrophobic interactions because these bonds are weak the phospholipids have a lot of flexibility to move around within the phospholipid bilayer and recall that the three types of lipids are fats made of saturated or unsaturated fatty acids phospholipids and steroids just like fatty oils such as olive oil are slick and fluid so too is the phospholipid bilayer in fact membranes are about as fluid as salad oil there are a few ways that the cell can tinker with the plasma membrane to affect its fluidity this is important for the cell's ability to effectively respond to varying environmental conditions to maintain an optimal physiological environment the first is temperature as temperatures cool membranes switch from a fluid state to a solid state you experience this at a macroscopic level when you heat butter to melt it the temperature at which a membrane solidifies depends on the types of lipids that are present for example membranes that are rich in unsaturated fatty acids are more fluid than those that are rich in saturated fatty acids and just look at the structure of an unsaturated fatty acid it is kinked it's bent which means that they can't pack in as closely which leaves for more fluidity and movement whereas saturated fatty acids are usually straight and they can pack in more closely they would be more dense there'd be less movement and fluidity membranes must be fluid to work properly again think of salad oil the last thing affecting fluidity is the steroid cholesterol which has different effects on the membrane of animal cells at different temperatures at warm temperatures such as 37 celsius which is your body temperature cholesterol actually restrains movement of phospholipids at cooler temperatures it has the opposite effect it maintains fluidity by preventing tight packing of the phospholipids plant cells use a different system than animal cells for controlling their fluidity although cholesterol is present in in plant cells they use a different set of related steroid lipids to buffer their membrane fluidity let's have a look at membrane proteins and their functions somewhat like the tile mosaic we saw earlier a membrane is a collage of different proteins that are often clustered in groups and embedded in the fluid matrix of the lipid bilayer phospholipids form the main fabric of the membrane while proteins determine most of the membrane's functions there are two major categories of proteins within the plasma membrane peripheral proteins and integral proteins peripheral proteins are bound to the surface of the membrane periphery means outside the boundary of something integral proteins are different integral proteins penetrate the hydrophobic core of the membrane some integral proteins span the entire membrane and these are called transmembrane proteins transmeaning across this is an image of a transmembrane protein notice how it spans across the entire membrane from the extracellular side to the cytoplasmic side extracellular means outside the cell and of course cytoplasmic is the cytoplasm within the cell intracellular in the bottom image we see there are many different forms that transmembrane proteins can take oftentimes there are multiple regions winding through the membrane as seen in example two and three the parts of the integral and transmembrane proteins that penetrate the plasma membranes are hydrophobic regions consisting of one or more stretches of non-polar hydrophobic amino acids recall from lecture five that there are 20 amino acids which are either polar nonpolar acidic or basic any parts of the proteins that are interacting with the hydrophobic phospholipids in the membrane must also be hydrophobic the nonpolar stretches of amino acids are often coiled into alpha helises just like the one you see here and this protein has seven transmembrane domains example three is showing domains that are actually made of beta pleated sheets cell surface proteins carry out several functions important for life including transport of molecules across the membrane enzymatic activity to speed up our bio chemical reactions signal transduction to catalyze an intracellular response to an extracellular stimulus cell cell recognition for immuno immunological responses intercellular joining for stability and structure and attachment to the cytoskeleton and the extracellular matrix carbohydrates play a very important role in cell cell recognition cells are able to recognize each other by binding to molecules which often contain carbohydrates that are on the extracellular surface of the plasma membrane they sort of act like tags membrane carbohydrates may be covalently bonded to lipids which form glycolipids or more commonly the carbohydrates are covalently bonded to proteins and these are called glycoproteins notice the root words here glyco meaning sugar a glycolipid is a fat with a sugar on it and a glycoprotein is a protein with a sugar on it carbohydrates on the extracellular side of the plasma membrane vary among species individuals and even cell types in an individual cell surface proteins are also important in the medical field glycoprotein and glycolipid patterns that are on a cell's surface can give many viruses an opportunity for infection in the example that is shown here hiv must bind to the immune cell surface protein cd4 which is shown in purple and a co-receptor ccr5 in order to infect a cell hiv cannot enter the cells of resistant individuals who lack the ccr5 co-receptor so drugs are being developed to mask the ccr5 protein these viruses are able to invade these cells because the cells have binding sites on their surfaces that are specific to and compatible with certain viruses as you now know the phospholipids orient themselves into the phospholipid bilayer forming the plasma membrane but the plasma membrane isn't the only part of the cell that's made of phospholipids all of the organelles in the cell use phospholipid bilayer as their membrane this includes the nucleus mitochondria endoplasmic reticulum golgi apparatus and vesicles used for transport within the cell in the last lecture we learned about the functions of each of these organelles take a look at the picture showing the flow of material through the cell's organelles the ability of the er to bud off into a vesicle for that vesicle to fuse with the golgi and then for vesicles to be released from the golgi and then fuse with the plasma membrane for secretion this is all crucial for the endomembrane system the fact that all membranes are made of the same phospholipids allow for easy movement of materials through the cell you'll also notice from this picture that there's a sightedness to the membranes meaning they have distinct inside and outside faces each side has an asymmetrical distribution of proteins specific to each compartment's specialized role in the cell we have already discussed the structure of the membranes and the resulting selective permeability selective permeability means the membrane allows some substances to cross more easily than others it's very important that you understand this concept before we continue a cell must exchange materials with its surroundings which is controlled by the plasma membrane recall that plasma membranes are amphipathic they have hydrophilic and hydrophobic regions in the phospholipids this characteristic is what helps move some materials through the membrane and hinders the movement of others most molecules in your body are not able to cross this phospholipid bilayer so what molecules are able to move freely through the plasma membrane remember the phrase like dissolves like small hydrophobic nonpolar molecules can dissolve in the lipid bilayer and pass through the membrane rapidly hydrophilic molecules including ions and polar molecules do not cross the membrane easily or at all large molecules of any type will have an issue crossing the membrane and require special transport which we will be discussing later in this lecture so to answer the question what can move freely through the plasma membrane small nonpolar and lipid soluble material with a low molecular weight like oxygen and carbon dioxide can easily slip through the membrane's hydrophobic lipid core substances like fat soluble vitamins which would be vitamin a vitamin d vitamin e and vitamin k also can rapidly pass through the plasma membrane in the digestive tract and other tissues fat soluble drugs and hormones also gain pretty easy entry into cells and readily transport themselves into the body's tissues and organs water is also able to pass through though pretty slowly since it's a polar molecule and we'll talk about another way that water can get through the membrane this selective permeability allows for tight regulation of molecules that are trafficking into and out of the cell and this is crucial to the cell's ability to regulate its internal environment and maintain a favorable homeostasis which means steady state nearly all cells in our body are in contact with the bloodstream which acts as this highway that's carrying nutrients and gases and signaling molecules and even toxins like alcohol throughout the body so it's necessary for the cell to be able to regulate what can or can't enter from the bloodstream and the cell membrane is the first and most important barrier of defense in this process and part of that defense are the transport proteins which we will focus on in this lecture transport proteins are responsible for controlling passage across cellular membranes specifically for hydrophilic polar substances polar substances present problems for the membrane while some polar molecules can connect easily with the cells outside they cannot readily pass through the plasma membrane's lipid core and additionally while small ions could easily slip through the spaces of the membrane's mosaic their charge prevents them from doing so ions such as sodium and potassium calcium and chloride need special means of penetrating the plasma membrane because of their charge simple sugars and amino acids also need the help of various protein channels to transport them across the plasma membrane because of their polarity we have two major types of transport proteins these are channel proteins and carrier proteins channel proteins have a hydrophilic channel that certain molecules or ions can use as a tunnel one very specialized type of channel protein is called an aquaporin which facilitates the passage of water molecules hence the word aqua carrier proteins bind to molecules and change shape to shuttle them across that membrane and we call this a conformational change that shape change but whether it is a channel or a carrier protein all transport proteins are very specific for the substance that they're moving across the membrane now that we have the basics of the proteins let's talk about the transport we have two main types of transport in our cells active transport and passive transport to put it very simply active transport uses energy which is usually atp and passive transport does not use energy passive transport works by moving substances down their concentration gradient while active transport is using the energy to move things up or against their concentration gradient there are examples of active and passive transport listed here and we will discuss each in detail before we talk further about the proteins involved in protein transport we have to learn the concepts of diffusion and concentration gradients as this is what propels passive transport diffusion is defined as the tendency for molecules to spread out evenly into an available space you encounter diffusion daily though you may not even notice it pouring milk into coffee or spraying perfume into the air demonstrates how the substances will move from where they are more concentrated to where they are less concentrated they're spreading out evenly in their spaces over time this diffusion requires no energy and as we'll see in a moment the gradients are a special form of energy called potential energy look at the image on the right which uses blue lines to follow the motion of a few molecules this motion is called brownian motion which is the random motion of particles suspended in a liquid or a gas caused by the collision of those particles with water molecules at the atomic level molecules are vibrating and moving due to thermal and kinetic energy and therefore they're randomly bumping into one another however although each molecule moves randomly diffusion of a population of molecules may be directional for example movement tends to be away from where things are more concentrated at dynamic equilibrium just as many molecules are crossing the membrane in one direction as in the other and we'll encounter the concept of equilibrium a few times in this lecture concentration gradients play a big role in the movement of these molecules a concentration gradient is the gradual difference in concentration over the distance between the regions of high and low concentration concentration is a mass or number of something per unit of volume see how the image with the red dots has more dots per area in the high concentration than in the low concentration area over time the molecules will move due to brownian motion in the direction of high to low and we call this moving down the concentration gradient again this requires no energy input concentration gradients are a form of potential energy which will dissipate as the gradient is eliminated the substance will move from a high concentration to a low concentration area until the concentration is equal across a space this is a spontaneous and irreversible process which increases entropy of a system because particles will not spontaneously reorder themselves and we'll learn more about entropy in the next lecture now that we have diffusion and concentration gradients defined we'll cover passive transport and the various forms of passive transport let's start with a few rules for passive transport for you to remember as we move through the rest of the lecture passive transport uses no energy molecules always move down the concentration gradient and both channel proteins and carrier proteins may be used in passive transport the first type of passive transport is simple diffusion in simple diffusion the molecules are able to diffuse across the plasma membrane of course they are diffusing down their concentration gradients from high concentration to low concentration and no energy is required we already talked about what type of molecules can or can't move across the plasma membrane small electrically neutral atoms are able to diffuse through by slipping between the spaces of the lipid molecules examples are oxygen gas and carbon dioxide gas water is also able to move through via simple diffusion although water is polar so it doesn't move very rapidly in this image we see two sides that are separated by a permeable membrane we begin with a concentration gradient of blue molecules with the higher concentration in the extracellular space in the middle panel the molecules have begun to diffuse and they'll continue until equilibrium is reached in the rightmost panel we see the solution has reached equilibrium and equilibrium does not mean equal concentration equilibrium means equal rates of diffusion in other words just as many blue molecules are moving into the cell as are moving out as i mentioned water can also diffuse across a selectively permeable membrane and this has a special term called osmosis pay close attention to this explanation water diffuses across a membrane from a region of lower solute concentration to a region of higher solute concentration until the solute concentrations equal on both sides take a look at the image on the right these beakers are filled with water and solute at different concentrations separated by a selectively permeable membrane there are two components here water and solute each with a different concentration gradient the membrane is only permeable to water yet everything wants to reach equilibrium because water is the only thing that can cross it's going to want to diffuse down its own concentration gradient which means it will move from an area of higher water concentration to lower water concentration this means it's actually moving from lower solute concentration to higher solute concentration the left and right sides of the tube begin with equal amounts of water but in equal solute concentrations which means in equal water concentrations the left has a higher water concentration than the right so water will move from the left towards the right which is from lower solute concentration to higher solute concentration the osmotic pressure is so great that the water level on the right exceeds the water level on the left osmosis allowed for equilibrium with similar concentration of solute on both sides of the membrane and equal rates moving across the membrane if you've ever heard of reverse osmosis it's a water purification process that uses a partially permeable membrane to separate out ions and unwanted molecules and larger particles from your drinking water now that you know the basics of diffusion in osmosis we can apply this to our cells as i keep mentioning homeostatic balance is crucial to all aspects of cellular physiology the proper balance of water in the cell and in the body is included in that tonicity is the ability of a surrounding solution to cause a cell to gain or lose water the tonicity of that solution depends on its concentration of solutes that cannot cross the membrane relative to the inside of the cell what this does is create a solute concentration gradient which will drive osmosis water diffusion from or into the cell organisms have highly adept mechanisms for controlling osmosis and we call this osmoregulation osmoregulation is the control of solute concentrations and water balance as examples unicellular eukaryotes like paramecium are hypertonic relative to their pond water environment so it has a contractile vacuole that can pump water bacteria and archaea that live in excessively salty environments have their cellular mechanisms to ensure that they don't lose too much water to their environment there are three types of tonic solutions isotonic hypertonic and hypotonic let's start with isotonic as it's the easiest one to remember isotonic solutions have a solute concentration that is the same as inside the cell therefore there's no net movement across the plasma membrane the center panel shows an isotonic red blood cell which is able to maintain its proper shape and therefore function notice there is water diffusion into and out of the cell osmosis is therefore in equilibrium here again with no net change in movement if you've ever had a saline drip intravenously at a hospital that saline iv would have to be a concentration of salts and nutrients that are isotonic to your blood otherwise the saline iv would create a net diffusion of water either into or out of your cells which would either dehydrate or burst your cells the animal cell likes isotonic environments the next type of tonicity is a hypertonic solution in a hypertonic solution the site concentration is greater than inside the cell so the cell loses water the top panel shows cells in hypertonic solutions when the solution has a site concentration that's greater than that within the cell water wants to create equal concentrations therefore water will diffuse from the area of the lower solute concentration inside the cell to the area of the higher site concentration outside the cell this causes the red blood cell to shrivel a process that's termed crenation the last type of tonicity is hypotonic in a hypotonic solution the concentration of solutes is less than that inside the cell therefore the cell will gain water the bottom panel shows cells in hypotonic solutions the concentration of solute in this solution is less than the concentration within the cell once again water wants to move from an area of lower solute concentration to higher solute concentration therefore water will rush into the cell and will cause it to swell until it bursts or lyses plant cells are a bit different because they have sturdy cell walls to help maintain water balance remember plants have both a plasma membrane and a cell wall and they like when the membrane is pressed firmly against the cell wall this is seen in a hypotonic environment the plant cell is thriving in the hypotonic solution water rushes into the cell but it does not burst because it has the strong cell wall to maintain its structure this cell is said to be turgid and is the optimal form for plant cells this is why plants love water water has far less concentration of solutes and nutrients than plant cells and therefore osmotic pressure will cause the diffusion of water into plant cells and keep them turgid the plant cell in the middle panel is in an isotonic solution where there is no net movement into the cell this cell becomes flaccid or lymph which is not ideal in the hypertonic solution the plant has lost water and the membrane pulls away from the cell wall causing the plant to wilt the cell walls do not buckle but the plasma membranes have shriveled and pulled away from the cell walls here this is called plasmolysis which is potentially lethal to plants hypertonic solutions are detrimental to both animal and plant cells now you've had a complete review of simple diffusion the movement of molecules in diffusion is driven solely by concentration gradients simple diffusion is also unique in that it uses no transport proteins it only has to do with movement of water or solutes across that selectively permeable membrane now we will learn the second type of passive transport called facilitated diffusion this type of diffusion does use proteins to aid in transport but still does not require energy it is still a type of diffusion after all in facilitated diffusion transport proteins speed the process of passive movement of molecules across the plasma membrane transport proteins are the two types we discussed earlier in the lecture channel proteins and carrier proteins channel proteins have a hydrophilic channel that allows charged molecules or ions to use as a tunnel one special type of channel protein is the aquaporin which facilitates diffusion of water there are also basic ion channels which to facilitate the transport of ions carrier proteins are different when they bind to their molecules they change shape to shuttle them across the membrane transport proteins are always always always highly specific for the substance that they move let's look at channel proteins specifically the aquaporin aquaporins provide corridors for water to rapidly cross the membrane notice how it's a transmembrane protein with the hydrophilic domains exposed to the intracellular and extracellular fluids and a hydrophilic channel through the membrane core that provides a hydrated opening through the hydrophobic membrane layers passage through the channel allows polar compounds to avoid the plasma membrane's non-polar inner layer that would otherwise slow down or prevent their entry into the cell here's a model of an aquaporin transporting water molecules across that membrane the yellow molecule is a water molecule that's highlighted to help you follow the flow of molecules this movement is driven by diffusion and is passive transport aquaporins are most prevalent in the kidneys for water reabsorption they help to maintain fluid homeostasis in several different tissues including the kidney lung the gi tract and brain here's another type of channel protein the gated ion channel ion channels facilitate the transport of ions across the membrane some of these ion channels are gated meaning they open or close in response to a particular stimulus examples of stimuli are voltage and ligand binding prime example is in nerve cells where sodium ion channels open in response to electrical stimulus to conduct an action potential and you'll learn more about that in physiology or in your psychology courses check your understanding and retention match the term in the left column to the correct answer in the right column pause the video then play to check your answers now we will cover active transport let's set a few rules for active transport for you to remember as we move through the lecture active transport requires energy from atp hydrolysis molecules always move up or against a concentration gradient all proteins involved in active transport are carrier proteins and active transport creates important electrochemical gradients this is a good time to review atp hydrolysis as this is how our cells harness energy atp hydrolysis is the metabolic reaction to release chemical energy in high energy bonds atp is adenosine triphosphate which is an adenine attached to a ribose sugar attached to three phosphate groups the bonds between the phosphate groups are the phospho-anhydride bonds when atp is broken down we create adp which is adenosine diphosphate and an inorganic phosphate inorganic simply means it is not attached to a molecule that contains carbon it was an organic phosphate when attached to the atp but once cleaved it is an inorganic phosphate hydrolysis of the terminal phospho-anhydride bond is highly exergonic meaning it releases a lot of energy the gibbs free energy is negative 34 kilojoules per mole and we'll learn what this is in the next lecture chemical energy present within the bonds of that molecule is released by splitting the bonds and harnessed for mechanical energy to do work energy is neither created nor destroyed so we are simply transforming one form of energy into another chemical energy into mechanical energy adp can be further hydrolyzed to give energy amp is created here which is adenosine monophosphate and we get another inorganic phosphate we're about to get into the first type of active transport which is electrogenic pumps but first i think it's helpful to explain what electrogenic means very simply electrogenic pumps create electrochemical gradients so what's an electrochemical gradient first we have to understand the concept of membrane potential remember earlier i mentioned that gradients are a form of potential energy the membrane potential is the voltage across a membrane you know what voltage is maybe related to energy or a battery voltage is created by difference in the distribution of positive and negative ions across a membrane look at these cell membrane image to illustrate this in this image the yellow circles are positive potassium ions the purple hexagons are positive sodium ions and the green circles are negative chloride ions all of these ions are distributed differently across the two sides of the membrane and overall the intracellular space is more negative than the extracellular space and this is true of all cells they are more negative on the inside and this is voltage when we measure neurons they are negative 70 millivolts there's an imbalance in both charge and the individual ion concentrations these two forces make up the electrochemical gradient which drives the diffusion of ions across the membrane the two forces are called the chemical force and the electrical force the chemical force is the ion concentration gradient so for example there is a higher concentration of sodium outside the cell than inside the cell that is the chemical force the electrical force is the effect of the membrane potential the charge on the ions movement so there is a higher concentration of negative ions inside the cell so chloride is a negatively charged ion and it would want to move down that concentration gradient if we were looking just at the electrical force but in reality both of these forces the chemical force and the electrical force are factored into how ions will move across the membrane and look these two forces make up the gradient name electro chemical gradient so back to electrogenic pumps electrogenic pumps create these electrochemical gradients they are transport proteins that generate voltage across the membrane they do this by using energy atp to pump substances against their concentration gradient this ability for active transport proteins to move solute up their concentration gradients allows cells to maintain concentration gradients that differ from their surroundings the sodium potassium pump is the major electrogenic pump in animal cells and the proton pump is prevalent in plants fungi and bacteria electrogenic pumps big job is to help store energy that can be used for cellular work again this would be potential energy so the first type of active transport is the electrogenic pump here we have the sodium potassium pump the sodium potassium pump uses atp and carrier proteins to move substances against their gradients animal cell has a higher potassium k plus and a much lower sodium which is n a plus concentration compared to its surroundings this is controlled by the sodium potassium pump let's walk through the basic steps of how the sodium potassium pump functions the overall goal is to move three sodium ions out of the cell and two potassium ions into the cell make sure you note that three sodium out two potassium in the carrier proteins begin open facing the interior of the cell three orange sodium ions bind in step two the energy molecule atp is hydrolyzed resulting in the transfer of that phosphate group onto the carrier protein this activates the protein for movement the carrier protein has a conformational change shape change and releases these sodium ions to the outside then it's open for the two potassium ions to bind and the phosphate group is removed resulting in the final conformational shape change as seen in step four the process then repeats and each cycle requires the hydrolysis of one atp molecule next we have another electrogenic pump the proton pump here an integral membrane protein pump builds up a proton gradient across a membrane in the image we see atp being used to drive hydrogen ions to the outside where there is a higher proton concentration already and a corresponding low ph proton pumps are prevalent in the stomach to create an acidic environment for our digestion and you may also find proton pumps as a part of co-transport co-transport is another type of active transport and it's also called secondary active transport secondary active transport does not require direct atp usage instead it is the movement of material due to the electrochemical gradient that's established by primary active transport using atp of another substance in this process the diffusion of an actively transported solute down its concentration gradient is coupled with the transport of a second substance that will be going against its own concentration gradient there are two types of proteins that are involved in code transport we have anti-porters and symporters anti-porters move two substances across the membrane in opposite directions and symporters move two substances across in the same direction the sodium potassium pump is actually an antiporter and many amino acids and glucose actually enter the cell via cotransport in the center image the yellow symporter is moving both sodium and glucose into the cell in the same direction notice sodium's concentration gradient remember the sodium gradient was created by the sodium potassium pump that we just learned about which utilized atp energy it's favorable for sodium to then move down its concentration gradient into the cell via diffusion without any energy however glucose is more concentrated in the cell so moving from outside to inside is against its gradient and would need energy the symporter allows glucose to take advantage of sodium's gradient and hitch a ride into the cell so in this way glucose is indirectly using atp to move into the cell which is why this is called secondary active transport and that is co-transport the final type of active transport is bulk transport through this lecture we've discussed how small molecules and water enter or leave the cell through the lipid bilayer or via transport proteins however large molecules present a problem because they don't fit through the membrane or through transport proteins large molecules such as polysaccharides and polypeptide proteins cross the membrane in bulk via vesicles we have two basic forms exocytosis and endocytosis exo means outside so exocytosis is things exiting the cell while endo means inside so endocytosis is things entering the cell of course moving large things across a membrane is an arduous endeavor so bulk transport requires atp energy in exocytosis transport vesicles migrate to the membrane fuse with it and release their contents outside the cell remember vesicles are usually coming off from the golgi apparatus and with some finished products after having gone through the endomembrane system vesicles are of course surrounded by phospholipid bilayer so they easily fuse with the plasma membrane many secretory cells use exocytosis to export their products to the extracellular matrix or nearby cells or through ducts in endocytosis the cell takes in macromolecules by forming vesicles from the plasma membrane endocytosis is a reversal of exocytosis involving different proteins there are three types of endocytosis phagocytosis meaning cellular eating pinocytosis meaning cellular drinking and receptor-mediated endocytosis we learned of phagocytosis in the previous lecture when we learned about vacuoles in phagocytosis or cellular eating a cell engulfs a particle or another cell in a special vacuole called a food vacuole in pinocytosis or cellular drinking molecules that are dissolved in droplets are taken up when extracellular fluid is gulped into tiny vesicles and in receptor-mediated endocytosis binding of a specific solute to its receptor which is sitting on the outside of the cell will trigger vesicle formation and taking in of that that triggering molecule human cells use receptor-mediated endocytosis to take in cholesterol which is carried through our bloodstream in particles called ldls or low-density lipoproteins individuals with the disease familial hypercholesterolemia have missing or defective ldl receptor proteins this leaves individuals with a higher risk of heart disease and a greater risk of early heart attack and that will conclude this lecture on membrane transport i know that was a lot so i've created this schematic to help you mentally map out all that we've covered use it as inspiration for your own notes and add details where needed to help you learn the material and i will see you next lecture [Music] you |
12997 | https://pmc.ncbi.nlm.nih.gov/articles/PMC3023181/ | Molecular determinants of renal glucose reabsorption. Focus on “Glucose transport by human renal Na+/d-glucose cotransporters SGLT1 and SGLT2” - PMC
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Am J Physiol Cell Physiol
. 2011 Jan;300(1):C6–C8. doi: 10.1152/ajpcell.00444.2010
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Molecular determinants of renal glucose reabsorption. Focus on “Glucose transport by human renal Na+/d-glucose cotransporters SGLT1 and SGLT2”
Volker Vallon
Volker Vallon
1 Departments of Medicine and Pharmacology, University of California San Diego and Veterans Affairs San Diego Healthcare System, San Diego, California
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1 Departments of Medicine and Pharmacology, University of California San Diego and Veterans Affairs San Diego Healthcare System, San Diego, California
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Address for reprint requests and other correspondence: V. Vallon, Depts. of Medicine and Pharmacology, Univ. of California San Diego and VA San Diego Healthcare System, 3350 La Jolla Village Dr., San Diego, CA 92161 (e-mail: vvallon@ucsd.edu).
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PMCID: PMC3023181 PMID: 21048164
about 180 g of glucose are filtered daily in the glomeruli of the kidneys in a healthy normoglycemic subject, which is equivalent to approximately one third of the total energy consumed by the human body in a day. Most of the glucose entering the tubular system is reabsorbed along the nephron segments, primarily in the proximal tubule, such that urine is almost free of glucose. This is different in diabetes, where the filtered glucose exceeds the transport capacity of the tubular system for glucose and glucosuria occurs. On the basis of transport studies in membrane vesicles and analyses of mRNA expression in isolated nephron segments of rat and rabbit kidneys, performed largely between 1981 and 1995, the concept has been established that the bulk of tubular glucose uptake across the apical membrane occurs in the early proximal tubule and is mediated by the low-affinity/high-capacity Na+-glucose cotransporter SGLT2 (SLC5A2); in comparison, the high-affinity/low-capacity SGLT1 (SLC5A1) is thought to “clean up” most of the remaining luminal glucose in further distal parts of the proximal tubule (2, 4, 6, 7, 9, 11, 18–21)(see Fig. 1). With newly available specific antibodies, recent studies directly document specific expression of brush border membrane SGLT2 in early proximal tubule and SGLT1 in later sections of the proximal tubule (1, 14). Much of the evidence for the relative quantitative contribution of these proteins to renal glucose reabsorption in human derives from the phenotype of subjects carrying gene mutations. Whereas mutations in SGLT1 are associated with intestinal glucose malabsorption with little or no glucosuria, individuals with gene mutations in SGLT2 have persistent renal glucosuria (10).
Fig. 1.
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Glucose transport in the kidney. Under normoglycemia, ∼97% of filtered glucose is reabsorbed via the Na+-glucose cotransporter SGLT2 primarily in the early segments of the proximal tubule. A significant capacity of SGLT1 to reabsorb glucose in later segments of the proximal tubule is unmasked by SGLT2 inhibition (∼40% of filtered glucose under normoglycemia; see numbers in parentheses), on the basis of our previous work (14) and the assumption that apical tubular glucose uptake in the kidney is primarily mediated by SGLT2 and SGLT1. The glucose transporters GLUT2 and GLUT1 mediate glucose transport across the basolateral membrane. Na+-glucose cotransport is electrogenic, and luminal K+ channels serve to stabilize the membrane potential (12, 13): KCNE1/unknown α-subunit and KCNE1/KCNQ1 in early and late proximal tubule, respectively.
Interest in SGLTs has recently been sparked by the development of a novel antidiabetic therapeutic approach, namely, the selective pharmacological inhibition of SGLT2, which inhibits renal reabsorption of glucose, thereby increasing its urinary excretion and reducing plasma glucose levels (17). During diabetes, excess glucose uptake via SGLTs may contribute to the glucose toxicity in the diabetic kidney. Moreover, an increase in SGLT-mediated sodium/glucose reabsorption has been implicated in the enhanced proximal tubular sodium reabsorption in the diabetic kidney which lowers luminal NaCl concentration at the macula densa and, because of the normal physiology of tubuloglomerular feedback, can contribute to glomerular hyperfiltration observed in early diabetes (15). Thus, SGLT2 inhibitors may also have the potential to reduce the glucose toxicity and hyperfiltration observed in the diabetic kidney. Despite this new clinical interest in renal glucose handling, surprisingly little information has been available on the specific characteristics of human SGLT2 and SGLT1.
The timely study by Hummel and colleagues (5) in this issue of American Journal of Physiology-Cell Physiology builds on the previous pioneering studies of Wright's group in the field of SGLTs, which included the cloning of SGLT1 (4), showing that defects in SGLT1 trafficking and function cause intestinal glucose-galactose malabsorption (8), and their contributions to cloning of SGLT2 (18) and delineating the crystal structure of a sodium galactose transporter to reveal mechanistic insights into Na+/sugar symport (3). The new studies aimed to provide relevant insights on the characteristics of hSGLT2 and hSGLT1 in an experimental setting close to physiological conditions. To this end hSGLT2 and hSGLT1 were expressed in mammalian epithelial cells (HEK293T) and studied using whole cell patch-clamp electrophysiology at 37°C. The studies show that, under these conditions, the Na+:glucose coupling ratio equals a value of 1 for hSGLT2 and 2 for hSGLT1 (see Fig. 1). Na+-glucose uptake is electrogenic, and previous studies indicated that the ensuing depolarization is partly offset by luminal K+ exit (12, 13). Hummel and colleagues further found that hSGLT2 transports glucose with similar affinity (5 versus 2 mM) and has lower concentrative power than hSGLT1. The studies confirm that unlike in hSGLT1, d-galactose is a poor substrate for hSGLT2. The results provide further information about how inhibitors block Na+/d-glucose cotransport. This includes data showing that the 10-fold lower affinity of phlorizin for hSGLT1 versus hSGLT2 is mainly due to differences in the rate of inhibitor release. Notably, the phlorizin-sensitive maximal transport rate, which depends on both protein copy number and on turnover rates, was 20-fold greater for hSGLT1 over hSGLT2 in this experimental setup. Whereas the hSGLT1 turnover and copy number can be determined from the measured maximum rate of transport and the magnitude of pre-steady-state charge movements, this analysis could not be performed for hSGLT2 since no pre-steady-state currents were detectable. Therefore, the reason behind the observed difference in the maximal transport rate between hSGLT1 and hSGLT2 and whether this is unique to this expression system remain unclear. Since diabetes and SGLT2 inhibitors will enhance the delivery of glucose to the later sections of the proximal tubule, the capacity of SGLT1 to transport glucose is of clinical relevance.
The present results are consistent with a dominant role of SGLT2 in renal glucose reabsorption complemented by a significant capacity of SGLT1 to reabsorb filtered glucose in the late proximal tubule of the human kidney. Hummel and colleagues inferred, on the basis of the current and previous data, that SGLT2 works at only 50% capacity in fasted human subjects. Recent micropuncture studies in knockout mice provided direct evidence that SGLT2 is responsible for all glucose reabsorption in the early proximal tubule and, overall, is the major pathway of glucose reabsorption in the kidney, whereas mice heterozygous for SGLT2 showed no urinary glucose loss (14). Surprisingly, the lack of SGLT2 suppressed renal SGLT1 mRNA and protein expression by ∼40%. This may reflect a mechanism to blunt the increase in glucose reabsorption in the late proximal segments under conditions of increased luminal glucose delivery and uptake. Despite lacking SGLT2 and having suppressed SGLT1 expression, the SGLT2 knockout mice have increased absolute glucose reabsorption along the late proximal tubule and maintain a mean fractional renal glucose reabsorption of ∼36% (between 10 and 60%, varying inversely with the amount of glucose filtered) (14). Preliminary studies in mice lacking SGLT1 indicated normal renal SGLT2 protein expression and a significant but minor reduction in fractional renal glucose reabsorption from 99.8% to 96.9% (16). If SGLT1 is the major pathway for renal glucose uptake in mice lacking SGLT2, then its contribution to glucose uptake is significantly enhanced by inhibition of SGLT2 (see Fig. 1). The physiological relevance of shifting glucose reabsorption to later segments of the proximal tubule deserves further investigation. Owing to differences in the Na+:glucose coupling ratio, shifting glucose reabsorption from SGLT2 to SGLT1 is expected to attenuate the renal sodium loss in response to SGLT2 inhibition.
The in vivo capacity of SGLT1 may be reflected by the observed high maximal glucose transport rate in the current expression studies by Hummel and colleagues (5). Overall, their studies provide new insights into the molecular physiology of hSGLT2 and hSGLT1 and a novel method for evaluating hSGLT inhibitors in a mammalian cell system. This will help to further elucidate the molecular mechanisms involved in the regulation of these cotransporters and their relevance in the physiology and pathophysiology of renal glucose transport and the development of new therapeutic strategies.
GRANTS
The author's work was supported by the National Institutes of Health (Grants DK-56248, DK-28602, HL-094728, GM-66232, and P30 DK-079337) and the Department of Veterans Affairs.
DISCLOSURES
Author's research is supported by Bristol-Myers Squibb, Astra-Zeneca, and Takeda Pharmaceuticals America.
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12998 | https://www.youtube.com/watch?v=j0ORPea9ICA | Restrict domains of functions to make them invertible : Khan Academy
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this will be for the con exercise restrict domains of functions to make them invertible all right so if you recall the inverse of a function is when you exchange the X and Y values and also if you recall the function test the vertical line if something is a function each x value has only one y value to test this on the graph you draw a vertical line anywhere on the graph and it should not intersect more than once all right if it intersects more than once it's not a function now for this exercise we are going to be testing if the function is invertible using the horizontal line test for the um to prove that the function is invertible each y value has to have only one x value and the way to test that on the graph you draw a horizontal line anywhere on the graph and it should not intersect more than once so that's what we're going to be doing on this exercise if it intersects more than once then it is not invertible all right function f graph below is not an invertible function so before we even start we can see it's not invertible by the horizontal line test if I drew a horizontal line right here we can clearly see it intersects the graph more than once so that's what makes it not invertible however on this exercise we are going to restrict the domains to make them invertible so the domains are basically the possible X values all right so that's the definition of domain and if we restrict some of those X values it may be possible that the function is going to be invertible and have it pass the horizontal line test all right so we're going to have to do the test for each of the multiple choices all right to which intervals can we restrict the domain to make it invertible choose all answers that apply so it may be more than one all right so for each of these responses we're going to do the horizontal line test on each one in the first case we can see that X is in between 0 and 8 and if it has the bar on the bottom of the inequality symbol it also includes 0 and 8 it also includes those numbers so let's test X between 0 and 8 and we have to include both 0 and 8 cuz it's less than or equal to or greater than or equal to all right we go to the x axis this is xal 0 right here so we um make this so it's divided into this one section here this is X = 0 and if we go on the xais 4 6 8 right here is x = 8 so we are restricting the domain to this section only okay and we are going to do the horizontal or vertical no horizontal line test all right so we're looking at the graph here we're only in between this section right here and if I draw a horizontal line anywhere you can see it intersects the graph only once it does not intersect the graph more than once so in this section the answer is yes it is invertible in just this section right here because it does not intersect more than once so answer a is good moving on to answer B all right X is going to be between -8 and -4 and it includes 8 and4 because of the bar here so let's go to these boundaries -4 and8 and test it out all right on the x axis here is -4 this whole thing is right here x = -4 on the xaxis -4 -6 -8 x = 8 is right there if we were to draw the line for that that that's 8 okay and we're doing the horizontal line test only in this section so you can clearly see it passes if I draw a horizontal line it intersects only once right here only once if it does it twice the answer is no but if it's only once the answer is yes answer B is good finally answer c x is between 1 and six and it has the equal sign so it includes one it includes six let's go to the graph and run the test all right x axis here's x = 1 and let me fix that x axis here's x = 6 all right so this is x = 6 this is x = 1 we are only testing in this section here horizontal line test only once intersects the graph only once okay there is nowhere that it intersects more than once so this section is also a yes all right so answer C is also correct choose all answers that apply all right f is not invertible and we can see that to which intervals can we restrict the domain all right let's test it out X is between -4 and pos4 it also equals those numbers all right x axis here's -4 xais here is pos4 all right so this one is -4 pos4 we are testing in this section only and if we test it out down here horizontal line we can see it intersects more than once so in this section the answer is no so part A is no good part b x is less than or equal to zero all right let's go to x = 0 draw the line in this is zero and it said X is less than or equal to zero less than is to the left if we test it out with the horizontal the line hits only once hits only once or even all the way down here it's only once so in this section the answer is yes and I saw for part C it said greater than or equal to zero so we might as well do that side as well here's zero this side to the right is greater than or equal to zero horizontal line hits only once hits once on the bottom there is nowhere intersects more than once so for the greater than or equal to zero it's also going to be a yes this is for less than this is for greater than but both of them are yeses all right so B is good and C is good as well all right last one this is not invertible so we are going to run the test on each of these we can see the first one answer a is between -7 and -3 it includes both of those numbers go to the xaxis here is -7 x axis here is 1 2 -3 all right let's run the test in this area alone and I can see already the answer is no if I draw the line here it intersects right there and right there it intersects more than once so for this section the answer is no it is not invertible so leave a alone answer b x is bound by -3 and -1 it includes both all right so go to x = -3 and go to x = -1 we are running the test in this section only horizontal line if you draw one it's going to intersect only once all right so in this section it only intersects more than one it only intersects once so the answer is yes just for this section only the function is invertible so answer B is good answer C is between one and five all right so let's try x = 1 and five go to the x-axis here's one go to the x-axis here's five so this right here is x = 1 this right here is x = 5 we're going to run the test in between this area right here and I can see if I run the test horizontal line right there it's going to intersect twice okay so for this section the answer is no it intersects more than once so C is no good just answer B |
12999 | https://cstheory.stackexchange.com/questions/48917/what-does-x-y-notation-mean | pl.programming languages - What does x.y notation mean? - Theoretical Computer Science Stack Exchange
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What does x.y notation mean?
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In Harper's PFPL (Ed. 2, top of page 8), this notation is used but I don't see a definition. What does x.y x.y mean?
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asked May 8, 2021 at 19:49
andiandi
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This is the notation for Harper's "abstract binding structures": x.t represents the binding site of a variable x and the term t the variable scopes over.
Apparently you are in the parts that define variable bindings. B[X]s B[X]s appears to be the set of terms, or binding structures at sort s s whose free variables are among X X. So I would expect (but I don't have the book) that there is in fact an explanation for this notation close by.
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answered May 9, 2021 at 5:15
gaschegasche
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