The practical issues involved in parameter estimation via weighted least-squares minimization are... more The practical issues involved in parameter estimation via weighted least-squares minimization are reviewed, with emphasis on fitting compartmental models. An example is given and solved using the MLAB mathematical and statistical modeling package. The problems of weight generation and of guessing an initial starting point are discussed and various heuristic methods for dealing with these problems are presented. Finally, the pitfalls of round-off error, algorithm instability, algorithm ineffectiveness, and ill-conditioning are described, together with some amelioration devices.
I wrote this book to help teach LISP to students in a course on data structures. Consequently it ... more I wrote this book to help teach LISP to students in a course on data structures. Consequently it contains a careful description of the data structures manipulated by LISP functions. These data structures and others, notably hash tables, are also used in constructing a LISP interpreter. I wish to acknowledge the help of readers in shaping and debugging this material. The study of LISP, coupled with the study of a LISP interpreter intended for exhibition, is of special interest to students in the areas of programming languages and computer architecture as well as data structures. Indeed, I hope this book will be useful to students in all areas of computer science, and I also hope it will be useful to autodidacts, professional programmers, and computer enthusiasts in a wide variety of fields. This book is intended to be accessible to a wide range of interested readers from high school students through professional programmers. I would very much like to see students use this book to hel...
1≤i≤m yibi. Now let P = {x ∈ (R) | Ax ≤ b and x ≥ 0}. Suppose x is a feasible point for our prima... more 1≤i≤m yibi. Now let P = {x ∈ (R) | Ax ≤ b and x ≥ 0}. Suppose x is a feasible point for our primal linear programming problem, so x ∈ P , i.e., Ax ≤ b and x ≥ 0. Also suppose we can choose y1, . . . , ym defining an m-covector y such that (yA)i ≥ ci and yi ≥ 0 for i = 1, . . . , n. Then c x ≤ yAx, and we have yAx ≤ yb, so altogether we have cx ≤ yAx ≤ yb. We see that yAx and yb are both upper-bounds of our primal objective function ζ(x) = cx for admissible choices of x and y, and yb is independent of x, so for any admissible choice of y, yb is an upper-bound of cx for all feasible points x ∈ P .
The MLAB advanced mathematical and statistical modeling system is a useful tool for mathematical ... more The MLAB advanced mathematical and statistical modeling system is a useful tool for mathematical modeling as is demonstrated by the following example. MLAB is a general-purpose system which can be applied to model enzyme kinetics, multiple site binding equilibrium, or any of a wide variety of other situations. Many such examples are given at the web-site www.civilized.com. In this paper, we review some of MLAB’s capabilities in a practical context of independent interest. Spaces between the cells that make-up the human body are filled with molecules that instigate a myriad of actions and reactions that form the biochemical activity of life. Some of these molecules, such as calcium and potassium ions, flow through channels within cell membranes to control muscle and heart cell contraction, or to act as the stuff of nerve impulses. Other molecules, such as estrogen and testosterone, have a more global role in many physiological sub-systems of the body. Mathematical modeling, either gr...
Given an analog signal f(t), for s0 ≤ t ≤ s1, we may wish to transmit this curve to a remote site... more Given an analog signal f(t), for s0 ≤ t ≤ s1, we may wish to transmit this curve to a remote site. For example, f may be a voice signal, or a continuous time series measurement reading of some transducer. Digital encoding generally allows data compression, permitting fewer bits per second to be transmitted, and error-correcting codes can be used as desired to further improve accuracy. We may transmit a train of rectangular pulses, using repeaters as needed, or we may use any of the various amplitude, frequency, or phase modulation methods for transmission as a band-limited oscillating signal. Digital encoding for data compression is also a useful device in storing digitally-encoded curves. One common scheme is so-called delta-modulation encoding. This method can be used for sound and/or video digital radio transmissions and recordings as well as other types of signals. If the signal, f , is to be digitized for computational purposes, then delta-modulation encoding serves as a cheap ...
The permeability, PC , of a membrane for a particular chemical species, C, is a measure of the ea... more The permeability, PC , of a membrane for a particular chemical species, C, is a measure of the ease of diffusion of C across the membrane in the presence of a concentration difference on either side of the membrane. In particular, upon entering the membrane, one C-molecule will travel, on the average, (PCaF/(βRT ))Em cm/sec transversely across the membrane, where PC = uβRT/(aF ), and Em, u, β, R, T , a, and F are defined as follows.
A constructive proof is given for the fact that a suitably-nice self-intersecting curve can alway... more A constructive proof is given for the fact that a suitably-nice self-intersecting curve can always be traversed without crossing itself. I recently discovered that you can traverse a “scribble” like that shown below, without crossing your path. (You might enjoy imagining an algorithm by which you could generate such “random” scribbles at will.) You may need to touch the previously-traced path at a point, or even retrace a section of it, but you need not cross it. It seems likely to me that this fact must have been observed before, perhaps many times, but I don’t know of any prior report. It is convenient to restrict our attention to scribbles that are closed planar continuous curves, and moreover we want the curves we consider to self-intersect at only a finite number of points, or to overlap themselves along a finite number of disjoint connected segments (or both.) Such curves are suitably nice for our purposes. The existence of non-crossing traversal paths for such curves is linke...
The study of this reaction is common in chemistry and biochemistry. For example, F could be a hor... more The study of this reaction is common in chemistry and biochemistry. For example, F could be a hormone or drug and G the associated receptor sites. The symbol B represents the bound complex. Let us also allow F (t), G(t), and B(t) to be functions which specify the concentrations of F , G, and B respectively at time t. Then we have the differential equation model: dB dt (t) = k 1 F (t)G(t) − k 2 B(t), B(0) = B 0 , F (t) = F 0 − (B(t) − B 0), G(t) = G 0 − (B(t) − B 0), where F 0 , G 0 , and B 0 are the initial concentrations of F , G, and B respectively, and the molar association and dissociation rate constants k 1 and k 2 appear as the proportionality constants for the terms which occur in dB/dt. Note that k 1 = (dB dt (0) + k 2 B 0)/(F 0 G 0), so when B 0 = 0 and F 0 and G 0 are known, k 1 may be estimated from the initial velocity dB/dt(0), which can, in turn, be estimated from a few points with t near 0. The solution to our differential equation is: B(t) = (S(B 0 − R) − R(B 0 − S)e d•k 1 •t)/(B 0 − R − (B 0 − S)e d•k 1 •t),
Drug-combination analysis is discussed and demonstrated. Suppose we have a set of drugs (or other... more Drug-combination analysis is discussed and demonstrated. Suppose we have a set of drugs (or other treatments which we misclassify as “drugs” for the purpose of our analysis.) These drugs are supposed to be treatments for some disease, e.g., HIV infection. Our goal is to assess whether, and to what extent, treatment with a particular combination of these drugs is more or less effective than some other such combination treatment. Such combination treatments include the case of treatment with a single drug, so we can also compare the efficacy of two distinct single drugs with our method. For each treatment with a combination of drugs D1, . . . , Dk, the data we have is a set of k + 1 dimensional points of the form (d1, . . . , dk, y), where dj is the dose of drug Dj (measured in any desirable units) and y is the response observed in the subject represented by the point at hand. The observed response is always non-negative, and is generally a value in the interval [0, 1]. Such a value y...
The practical issues involved in parameter estimation via weighted least-squares minimization are... more The practical issues involved in parameter estimation via weighted least-squares minimization are reviewed, with emphasis on fitting compartmental models. An example is given and solved using the MLAB mathematical and statistical modeling package. The problems of weight generation and of guessing an initial starting point are discussed and various heuristic methods for dealing with these problems are presented. Finally, the pitfalls of round-off error, algorithm instability, algorithm ineffectiveness, and ill-conditioning are described, together with some amelioration devices.
The MLAB advanced mathematical and statistical modeling system is a useful tool for mathematical ... more The MLAB advanced mathematical and statistical modeling system is a useful tool for mathematical modeling as is demonstrated by the following example. MLAB is a general-purpose system which can be applied to model enzyme kinetics, multiple site binding equilibrium, or any of a wide variety of other situations. Many such examples are given at the web-site www.civilized.com. In this paper, we review some of MLAB's capabilities in a practical context of independent interest. Spaces between the cells that make-up the human body are filled with molecules that instigate a myriad of actions and reactions that form the biochemical activity of life. Some of these molecules, such as calcium and potassium ions, flow through channels within cell membranes to control muscle and heart cell contraction, or to act as the stuff of nerve impulses. Other molecules, such as estrogen and testosterone, have a more global role in many physiological subsystems of the body.
Thousands of genes are expressed at such very low levels (Յ1 copy per cell) that global gene expr... more Thousands of genes are expressed at such very low levels (Յ1 copy per cell) that global gene expression analysis of rarer transcripts remains problematic. Ambiguity in identification of rarer transcripts creates considerable uncertainty in fundamental questions such as the total number of genes expressed in an organism and the biological significance of rarer transcripts. Knowing the distribution of the true number of genes expressed at each level and the corresponding gene expression level probability function (GELPF) could help resolve these uncertainties. We found that all observed large-scale gene expression data sets in yeast, mouse, and human cells follow a Pareto-like distribution model skewed by many low-abundance transcripts. A novel stochastic model of the gene expression process predicts the universality of the GELPF both across different cell types within a multicellular organism and across different organisms. This model allows us to predict the frequency distribution of all gene expression levels within a single cell and to estimate the number of expressed genes in a single cell and in a population of cells. A random "basal" transcription mechanism for protein-coding genes in all or almost all eukaryotic cell types is predicted. This fundamental mechanism might enhance the expression of rarely expressed genes and, thus, provide a basic level of phenotypic diversity, adaptability, and random monoallelic expression in cell populations.
Proceedings of the 1971 Acm Sigfidet Workshop, Nov 11, 1971
Page 1. Expandable Open Addressing Hash Table Storage and Retrieval By Gary D. Knott Heuristics L... more Page 1. Expandable Open Addressing Hash Table Storage and Retrieval By Gary D. Knott Heuristics Laboratory Division of Computer Research and Technology National Institutes of Health Department of Health, Education and Welfare Bethesda, Maryland 20014 187 Page 2. ...
It is often necessary to generate the sequence of combinations of s things chosen from among n th... more It is often necessary to generate the sequence of combinations of s things chosen from among n things, or to generate a random member of this sequence. Various algorithms have been given for these tasks. Algorithm 94 [1] is a good combination generator, and methods for generating a random combination are discussed in [2, pp. 121-125].
The kinetics of the reaction between human chorionic gonadotropin (hCG) and specific gonadotropin... more The kinetics of the reaction between human chorionic gonadotropin (hCG) and specific gonadotropin receptors in the rat testis were determined at 24 and 37O, over a wide range of hormone concentrations. Hormone concentrations were corrected for the binding activity of the [1251] hCG tracer preparations. Analysis of the experimental data was performed with an interactive nonlinear curve fitting program, based upon the second-order chemical kinetic differential equation. The mean values for the association rate constant (k l) were 4.7 X lo7 M-' min-' at 24O, and 11.0 X lo7 M-' min-l at 37O. At both temperatures,
The practical issues involved in parameter estimation via weighted least-squares minimization are... more The practical issues involved in parameter estimation via weighted least-squares minimization are reviewed, with emphasis on fitting compartmental models. An example is given and solved using the MLAB mathematical and statistical modeling package. The problems of weight generation and of guessing an initial starting point are discussed and various heuristic methods for dealing with these problems are presented. Finally, the pitfalls of round-off error, algorithm instability, algorithm ineffectiveness, and ill-conditioning are described, together with some amelioration devices.
I wrote this book to help teach LISP to students in a course on data structures. Consequently it ... more I wrote this book to help teach LISP to students in a course on data structures. Consequently it contains a careful description of the data structures manipulated by LISP functions. These data structures and others, notably hash tables, are also used in constructing a LISP interpreter. I wish to acknowledge the help of readers in shaping and debugging this material. The study of LISP, coupled with the study of a LISP interpreter intended for exhibition, is of special interest to students in the areas of programming languages and computer architecture as well as data structures. Indeed, I hope this book will be useful to students in all areas of computer science, and I also hope it will be useful to autodidacts, professional programmers, and computer enthusiasts in a wide variety of fields. This book is intended to be accessible to a wide range of interested readers from high school students through professional programmers. I would very much like to see students use this book to hel...
1≤i≤m yibi. Now let P = {x ∈ (R) | Ax ≤ b and x ≥ 0}. Suppose x is a feasible point for our prima... more 1≤i≤m yibi. Now let P = {x ∈ (R) | Ax ≤ b and x ≥ 0}. Suppose x is a feasible point for our primal linear programming problem, so x ∈ P , i.e., Ax ≤ b and x ≥ 0. Also suppose we can choose y1, . . . , ym defining an m-covector y such that (yA)i ≥ ci and yi ≥ 0 for i = 1, . . . , n. Then c x ≤ yAx, and we have yAx ≤ yb, so altogether we have cx ≤ yAx ≤ yb. We see that yAx and yb are both upper-bounds of our primal objective function ζ(x) = cx for admissible choices of x and y, and yb is independent of x, so for any admissible choice of y, yb is an upper-bound of cx for all feasible points x ∈ P .
The MLAB advanced mathematical and statistical modeling system is a useful tool for mathematical ... more The MLAB advanced mathematical and statistical modeling system is a useful tool for mathematical modeling as is demonstrated by the following example. MLAB is a general-purpose system which can be applied to model enzyme kinetics, multiple site binding equilibrium, or any of a wide variety of other situations. Many such examples are given at the web-site www.civilized.com. In this paper, we review some of MLAB’s capabilities in a practical context of independent interest. Spaces between the cells that make-up the human body are filled with molecules that instigate a myriad of actions and reactions that form the biochemical activity of life. Some of these molecules, such as calcium and potassium ions, flow through channels within cell membranes to control muscle and heart cell contraction, or to act as the stuff of nerve impulses. Other molecules, such as estrogen and testosterone, have a more global role in many physiological sub-systems of the body. Mathematical modeling, either gr...
Given an analog signal f(t), for s0 ≤ t ≤ s1, we may wish to transmit this curve to a remote site... more Given an analog signal f(t), for s0 ≤ t ≤ s1, we may wish to transmit this curve to a remote site. For example, f may be a voice signal, or a continuous time series measurement reading of some transducer. Digital encoding generally allows data compression, permitting fewer bits per second to be transmitted, and error-correcting codes can be used as desired to further improve accuracy. We may transmit a train of rectangular pulses, using repeaters as needed, or we may use any of the various amplitude, frequency, or phase modulation methods for transmission as a band-limited oscillating signal. Digital encoding for data compression is also a useful device in storing digitally-encoded curves. One common scheme is so-called delta-modulation encoding. This method can be used for sound and/or video digital radio transmissions and recordings as well as other types of signals. If the signal, f , is to be digitized for computational purposes, then delta-modulation encoding serves as a cheap ...
The permeability, PC , of a membrane for a particular chemical species, C, is a measure of the ea... more The permeability, PC , of a membrane for a particular chemical species, C, is a measure of the ease of diffusion of C across the membrane in the presence of a concentration difference on either side of the membrane. In particular, upon entering the membrane, one C-molecule will travel, on the average, (PCaF/(βRT ))Em cm/sec transversely across the membrane, where PC = uβRT/(aF ), and Em, u, β, R, T , a, and F are defined as follows.
A constructive proof is given for the fact that a suitably-nice self-intersecting curve can alway... more A constructive proof is given for the fact that a suitably-nice self-intersecting curve can always be traversed without crossing itself. I recently discovered that you can traverse a “scribble” like that shown below, without crossing your path. (You might enjoy imagining an algorithm by which you could generate such “random” scribbles at will.) You may need to touch the previously-traced path at a point, or even retrace a section of it, but you need not cross it. It seems likely to me that this fact must have been observed before, perhaps many times, but I don’t know of any prior report. It is convenient to restrict our attention to scribbles that are closed planar continuous curves, and moreover we want the curves we consider to self-intersect at only a finite number of points, or to overlap themselves along a finite number of disjoint connected segments (or both.) Such curves are suitably nice for our purposes. The existence of non-crossing traversal paths for such curves is linke...
The study of this reaction is common in chemistry and biochemistry. For example, F could be a hor... more The study of this reaction is common in chemistry and biochemistry. For example, F could be a hormone or drug and G the associated receptor sites. The symbol B represents the bound complex. Let us also allow F (t), G(t), and B(t) to be functions which specify the concentrations of F , G, and B respectively at time t. Then we have the differential equation model: dB dt (t) = k 1 F (t)G(t) − k 2 B(t), B(0) = B 0 , F (t) = F 0 − (B(t) − B 0), G(t) = G 0 − (B(t) − B 0), where F 0 , G 0 , and B 0 are the initial concentrations of F , G, and B respectively, and the molar association and dissociation rate constants k 1 and k 2 appear as the proportionality constants for the terms which occur in dB/dt. Note that k 1 = (dB dt (0) + k 2 B 0)/(F 0 G 0), so when B 0 = 0 and F 0 and G 0 are known, k 1 may be estimated from the initial velocity dB/dt(0), which can, in turn, be estimated from a few points with t near 0. The solution to our differential equation is: B(t) = (S(B 0 − R) − R(B 0 − S)e d•k 1 •t)/(B 0 − R − (B 0 − S)e d•k 1 •t),
Drug-combination analysis is discussed and demonstrated. Suppose we have a set of drugs (or other... more Drug-combination analysis is discussed and demonstrated. Suppose we have a set of drugs (or other treatments which we misclassify as “drugs” for the purpose of our analysis.) These drugs are supposed to be treatments for some disease, e.g., HIV infection. Our goal is to assess whether, and to what extent, treatment with a particular combination of these drugs is more or less effective than some other such combination treatment. Such combination treatments include the case of treatment with a single drug, so we can also compare the efficacy of two distinct single drugs with our method. For each treatment with a combination of drugs D1, . . . , Dk, the data we have is a set of k + 1 dimensional points of the form (d1, . . . , dk, y), where dj is the dose of drug Dj (measured in any desirable units) and y is the response observed in the subject represented by the point at hand. The observed response is always non-negative, and is generally a value in the interval [0, 1]. Such a value y...
The practical issues involved in parameter estimation via weighted least-squares minimization are... more The practical issues involved in parameter estimation via weighted least-squares minimization are reviewed, with emphasis on fitting compartmental models. An example is given and solved using the MLAB mathematical and statistical modeling package. The problems of weight generation and of guessing an initial starting point are discussed and various heuristic methods for dealing with these problems are presented. Finally, the pitfalls of round-off error, algorithm instability, algorithm ineffectiveness, and ill-conditioning are described, together with some amelioration devices.
The MLAB advanced mathematical and statistical modeling system is a useful tool for mathematical ... more The MLAB advanced mathematical and statistical modeling system is a useful tool for mathematical modeling as is demonstrated by the following example. MLAB is a general-purpose system which can be applied to model enzyme kinetics, multiple site binding equilibrium, or any of a wide variety of other situations. Many such examples are given at the web-site www.civilized.com. In this paper, we review some of MLAB's capabilities in a practical context of independent interest. Spaces between the cells that make-up the human body are filled with molecules that instigate a myriad of actions and reactions that form the biochemical activity of life. Some of these molecules, such as calcium and potassium ions, flow through channels within cell membranes to control muscle and heart cell contraction, or to act as the stuff of nerve impulses. Other molecules, such as estrogen and testosterone, have a more global role in many physiological subsystems of the body.
Thousands of genes are expressed at such very low levels (Յ1 copy per cell) that global gene expr... more Thousands of genes are expressed at such very low levels (Յ1 copy per cell) that global gene expression analysis of rarer transcripts remains problematic. Ambiguity in identification of rarer transcripts creates considerable uncertainty in fundamental questions such as the total number of genes expressed in an organism and the biological significance of rarer transcripts. Knowing the distribution of the true number of genes expressed at each level and the corresponding gene expression level probability function (GELPF) could help resolve these uncertainties. We found that all observed large-scale gene expression data sets in yeast, mouse, and human cells follow a Pareto-like distribution model skewed by many low-abundance transcripts. A novel stochastic model of the gene expression process predicts the universality of the GELPF both across different cell types within a multicellular organism and across different organisms. This model allows us to predict the frequency distribution of all gene expression levels within a single cell and to estimate the number of expressed genes in a single cell and in a population of cells. A random "basal" transcription mechanism for protein-coding genes in all or almost all eukaryotic cell types is predicted. This fundamental mechanism might enhance the expression of rarely expressed genes and, thus, provide a basic level of phenotypic diversity, adaptability, and random monoallelic expression in cell populations.
Proceedings of the 1971 Acm Sigfidet Workshop, Nov 11, 1971
Page 1. Expandable Open Addressing Hash Table Storage and Retrieval By Gary D. Knott Heuristics L... more Page 1. Expandable Open Addressing Hash Table Storage and Retrieval By Gary D. Knott Heuristics Laboratory Division of Computer Research and Technology National Institutes of Health Department of Health, Education and Welfare Bethesda, Maryland 20014 187 Page 2. ...
It is often necessary to generate the sequence of combinations of s things chosen from among n th... more It is often necessary to generate the sequence of combinations of s things chosen from among n things, or to generate a random member of this sequence. Various algorithms have been given for these tasks. Algorithm 94 [1] is a good combination generator, and methods for generating a random combination are discussed in [2, pp. 121-125].
The kinetics of the reaction between human chorionic gonadotropin (hCG) and specific gonadotropin... more The kinetics of the reaction between human chorionic gonadotropin (hCG) and specific gonadotropin receptors in the rat testis were determined at 24 and 37O, over a wide range of hormone concentrations. Hormone concentrations were corrected for the binding activity of the [1251] hCG tracer preparations. Analysis of the experimental data was performed with an interactive nonlinear curve fitting program, based upon the second-order chemical kinetic differential equation. The mean values for the association rate constant (k l) were 4.7 X lo7 M-' min-' at 24O, and 11.0 X lo7 M-' min-l at 37O. At both temperatures,
Let Z denote the set of integers {.. . , −2, −1, 0, 1, 2,. . .} and let Z + denote the set of pos... more Let Z denote the set of integers {.. . , −2, −1, 0, 1, 2,. . .} and let Z + denote the set of positive integers {1, 2,. . .}. Also let R denote the set of real numbers (also specified as all the numbers in the open interval (−∞, ∞)). For a ∈ Z + ∪ {0}, let N a = {1, 2,. .. , a}; when a = 0, N a is empty, i.e., N a = ∅ where ∅ denotes the empty set. Let S a be the set of all length-a sequences that are composed of distinct elements of N a ; these sequences are permutations of the elements of N a , thus S a is the set of (length-a) permutations of 1, 2,. .. , a. For example, S 3 = {123, 132, 213, 231, 312, 321}. Note we sometimes, as in this example, write sequences without commas separating the sequence elements. We have #S a = a! where a! = a(a − 1)(a − 2) · · · 2 · 1. (In general, we write #S to denote the number of elements in the set S, thus for example, #N a = a.) The symbol ! is called the factorial operator. We define 0! = 1. To construct an element of S a , we may choose the first element in a ways, and then the remaining sequence of a − 1 elements may be chosen in (a − 1)! ways, yielding a! ways overall. This is essentially a proof by induction, where the empty sequence comprising the set S 0 can be chosen in just one way. Note there is just 0! = 1 permutation of the empty set.
Uploads
Papers by Gary Knott