#paperoftheday — Public Fediverse posts

The #paperOfTheDay is a book by Gauss: "Theoria motus corporum coelestium in sectionibus conicis solem ambientium" (theory of the motion of the heavenly bodies about the sun in conic sections) from 1809. Gauss wrote this in German, but had to translate it to Latin to get it published, and a few decades later various translations (including back to German) of this Latin text appeared.
The bulk of the 300 page book is a detailed practical guide how to compute orbits of celestial bodies from observations, how the various parameters of the orbits are related, etc. Many of these calculations are straightforward for us thanks to electronic computers, but back then, a major consideration was how to solve equations with the least amount of work, given only tables of logarithms and trigonometric functions.
Besides practical algorithms, the book introduces major innovations in pure #mathematics . In particular, in the process of approximately solving a certain transcendental equation, Gauss arrives for the first time at the power series of what is now known as Gaussian hypergeometric functions (whose mathematical theory he developed in a separate paper that I mentioned a few weeks ago). Also, he argues that independent experimental observations should give rise to values distributed according to what we now know as Gaussian normal distribution. Using this distribution, he develops the method of least squares to solve an over-determined system of equations in the best possible way (this method had been known to other authors previously, but apparently not widely used and systematically understood). #astronomy #physics
https://openlibrary.org/books/OL19572834M/Theoria_motus_corporum_coelestium

The #paperOfTheDay is a book by Gauss: "Theoria motus corporum coelestium in sectionibus conicis solem ambientium" (theory of the motion of the heavenly bodies about the sun in conic sections) from 1809. Gauss wrote this in German, but had to translate it to Latin to get it published, and a few decades later various translations (including back to German) of this Latin text appeared.
The bulk of the 300 page book is a detailed practical guide how to compute orbits of celestial bodies from observations, how the various parameters of the orbits are related, etc. Many of these calculations are straightforward for us thanks to electronic computers, but back then, a major consideration was how to solve equations with the least amount of work, given only tables of logarithms and trigonometric functions.
Besides practical algorithms, the book introduces major innovations in pure #mathematics . In particular, in the process of approximately solving a certain transcendental equation, Gauss arrives for the first time at the power series of what is now known as Gaussian hypergeometric functions (whose mathematical theory he developed in a separate paper that I mentioned a few weeks ago). Also, he argues that independent experimental observations should give rise to values distributed according to what we now know as Gaussian normal distribution. Using this distribution, he develops the method of least squares to solve an over-determined system of equations in the best possible way (this method had been known to other authors previously, but apparently not widely used and systematically understood). #astronomy #physics
https://openlibrary.org/books/OL19572834M/Theoria_motus_corporum_coelestium

The #paperOfTheDay is a book by Gauss: "Theoria motus corporum coelestium in sectionibus conicis solem ambientium" (theory of the motion of the heavenly bodies about the sun in conic sections) from 1809. Gauss wrote this in German, but had to translate it to Latin to get it published, and a few decades later various translations (including back to German) of this Latin text appeared.
The bulk of the 300 page book is a detailed practical guide how to compute orbits of celestial bodies from observations, how the various parameters of the orbits are related, etc. Many of these calculations are straightforward for us thanks to electronic computers, but back then, a major consideration was how to solve equations with the least amount of work, given only tables of logarithms and trigonometric functions.
Besides practical algorithms, the book introduces major innovations in pure #mathematics . In particular, in the process of approximately solving a certain transcendental equation, Gauss arrives for the first time at the power series of what is now known as Gaussian hypergeometric functions (whose mathematical theory he developed in a separate paper that I mentioned a few weeks ago). Also, he argues that independent experimental observations should give rise to values distributed according to what we now know as Gaussian normal distribution. Using this distribution, he develops the method of least squares to solve an over-determined system of equations in the best possible way (this method had been known to other authors previously, but apparently not widely used and systematically understood). #astronomy #physics
https://openlibrary.org/books/OL19572834M/Theoria_motus_corporum_coelestium

The #paperOfTheDay is a book by Gauss: "Theoria motus corporum coelestium in sectionibus conicis solem ambientium" (theory of the motion of the heavenly bodies about the sun in conic sections) from 1809. Gauss wrote this in German, but had to translate it to Latin to get it published, and a few decades later various translations (including back to German) of this Latin text appeared.
The bulk of the 300 page book is a detailed practical guide how to compute orbits of celestial bodies from observations, how the various parameters of the orbits are related, etc. Many of these calculations are straightforward for us thanks to electronic computers, but back then, a major consideration was how to solve equations with the least amount of work, given only tables of logarithms and trigonometric functions.
Besides practical algorithms, the book introduces major innovations in pure #mathematics . In particular, in the process of approximately solving a certain transcendental equation, Gauss arrives for the first time at the power series of what is now known as Gaussian hypergeometric functions (whose mathematical theory he developed in a separate paper that I mentioned a few weeks ago). Also, he argues that independent experimental observations should give rise to values distributed according to what we now know as Gaussian normal distribution. Using this distribution, he develops the method of least squares to solve an over-determined system of equations in the best possible way (this method had been known to other authors previously, but apparently not widely used and systematically understood). #astronomy #physics
https://openlibrary.org/books/OL19572834M/Theoria_motus_corporum_coelestium

The #paperOfTheDay is a book by Gauss: "Theoria motus corporum coelestium in sectionibus conicis solem ambientium" (theory of the motion of the heavenly bodies about the sun in conic sections) from 1809. Gauss wrote this in German, but had to translate it to Latin to get it published, and a few decades later various translations (including back to German) of this Latin text appeared.
The bulk of the 300 page book is a detailed practical guide how to compute orbits of celestial bodies from observations, how the various parameters of the orbits are related, etc. Many of these calculations are straightforward for us thanks to electronic computers, but back then, a major consideration was how to solve equations with the least amount of work, given only tables of logarithms and trigonometric functions.
Besides practical algorithms, the book introduces major innovations in pure #mathematics . In particular, in the process of approximately solving a certain transcendental equation, Gauss arrives for the first time at the power series of what is now known as Gaussian hypergeometric functions (whose mathematical theory he developed in a separate paper that I mentioned a few weeks ago). Also, he argues that independent experimental observations should give rise to values distributed according to what we now know as Gaussian normal distribution. Using this distribution, he develops the method of least squares to solve an over-determined system of equations in the best possible way (this method had been known to other authors previously, but apparently not widely used and systematically understood). #astronomy #physics
https://openlibrary.org/books/OL19572834M/Theoria_motus_corporum_coelestium

#paperOfTheDay is another one by Euler: "Elementa doctrinae solidorum" from 1758. The English title would be "Elements of the doctrine of solids". This is the article that establishes what's now known as Euler's polyhedra formula. Namely, consider a finite 3-dimensional solid bounded by planes, where H is the number of these plane surfaces, and A is the number of edges (=1-dimensional intersection between two planes), and S is the number of corners (=0-dimensional intersections between three or more planes). Then
S + H - A = 2.
This is easily checked for typical examples, e.g. a cube has S=8 corners and H=6 surfaces and A=12 edges, and 8+6-12=2 indeed. This theorem holds for all 3-dimensional polyhedra, it is not required that their sides are any particular regular n-gons. The proof is surprisingly long and "un-geometrical", Euler complains several times that although 2-dimensional #geometry has been studied in great detail, even such elementary properties of 3-dimensional geometry are completely obscure. From his theorem, and simple observations relating the various counts (e.g. if one of the side surfaces has n corners, then it is adjacent to n edges, etc), Euler derives numerous corollaries and bounds regarding the minimum or maximum allowed values of various counts. For example, he shows that there is no polyhedron with 7 edges, but all integer values >=8 are possible.
Today, we know that the polyhedral theorem holds much more generally, e.g. for graphs or for triangulations of surfaces, but the coefficient 2 on the right sometimes is another integer.
This is #Euler works no 230. #mathematics
https://scholarlycommons.pacific.edu/euler-works/230/
Of course, various translations exist, e.g. German https://www.agtz.mathematik.uni-mainz.de/algebraische-geometrie/van-straten/euler-kreis-mainz/

#paperOfTheDay is another one by Euler: "Elementa doctrinae solidorum" from 1758. The English title would be "Elements of the doctrine of solids". This is the article that establishes what's now known as Euler's polyhedra formula. Namely, consider a finite 3-dimensional solid bounded by planes, where H is the number of these plane surfaces, and A is the number of edges (=1-dimensional intersection between two planes), and S is the number of corners (=0-dimensional intersections between three or more planes). Then
S + H - A = 2.
This is easily checked for typical examples, e.g. a cube has S=8 corners and H=6 surfaces and A=12 edges, and 8+6-12=2 indeed. This theorem holds for all 3-dimensional polyhedra, it is not required that their sides are any particular regular n-gons. The proof is surprisingly long and "un-geometrical", Euler complains several times that although 2-dimensional #geometry has been studied in great detail, even such elementary properties of 3-dimensional geometry are completely obscure. From his theorem, and simple observations relating the various counts (e.g. if one of the side surfaces has n corners, then it is adjacent to n edges, etc), Euler derives numerous corollaries and bounds regarding the minimum or maximum allowed values of various counts. For example, he shows that there is no polyhedron with 7 edges, but all integer values >=8 are possible.
Today, we know that the polyhedral theorem holds much more generally, e.g. for graphs or for triangulations of surfaces, but the coefficient 2 on the right sometimes is another integer.
This is #Euler works no 230. #mathematics
https://scholarlycommons.pacific.edu/euler-works/230/
Of course, various translations exist, e.g. German https://www.agtz.mathematik.uni-mainz.de/algebraische-geometrie/van-straten/euler-kreis-mainz/

#paperOfTheDay is another one by Euler: "Elementa doctrinae solidorum" from 1758. The English title would be "Elements of the doctrine of solids". This is the article that establishes what's now known as Euler's polyhedra formula. Namely, consider a finite 3-dimensional solid bounded by planes, where H is the number of these plane surfaces, and A is the number of edges (=1-dimensional intersection between two planes), and S is the number of corners (=0-dimensional intersections between three or more planes). Then
S + H - A = 2.
This is easily checked for typical examples, e.g. a cube has S=8 corners and H=6 surfaces and A=12 edges, and 8+6-12=2 indeed. This theorem holds for all 3-dimensional polyhedra, it is not required that their sides are any particular regular n-gons. The proof is surprisingly long and "un-geometrical", Euler complains several times that although 2-dimensional #geometry has been studied in great detail, even such elementary properties of 3-dimensional geometry are completely obscure. From his theorem, and simple observations relating the various counts (e.g. if one of the side surfaces has n corners, then it is adjacent to n edges, etc), Euler derives numerous corollaries and bounds regarding the minimum or maximum allowed values of various counts. For example, he shows that there is no polyhedron with 7 edges, but all integer values >=8 are possible.
Today, we know that the polyhedral theorem holds much more generally, e.g. for graphs or for triangulations of surfaces, but the coefficient 2 on the right sometimes is another integer.
This is #Euler works no 230. #mathematics
https://scholarlycommons.pacific.edu/euler-works/230/
Of course, various translations exist, e.g. German https://www.agtz.mathematik.uni-mainz.de/algebraische-geometrie/van-straten/euler-kreis-mainz/

#paperOfTheDay is another one by Euler: "Elementa doctrinae solidorum" from 1758. The English title would be "Elements of the doctrine of solids". This is the article that establishes what's now known as Euler's polyhedra formula. Namely, consider a finite 3-dimensional solid bounded by planes, where H is the number of these plane surfaces, and A is the number of edges (=1-dimensional intersection between two planes), and S is the number of corners (=0-dimensional intersections between three or more planes). Then
S + H - A = 2.
This is easily checked for typical examples, e.g. a cube has S=8 corners and H=6 surfaces and A=12 edges, and 8+6-12=2 indeed. This theorem holds for all 3-dimensional polyhedra, it is not required that their sides are any particular regular n-gons. The proof is surprisingly long and "un-geometrical", Euler complains several times that although 2-dimensional #geometry has been studied in great detail, even such elementary properties of 3-dimensional geometry are completely obscure. From his theorem, and simple observations relating the various counts (e.g. if one of the side surfaces has n corners, then it is adjacent to n edges, etc), Euler derives numerous corollaries and bounds regarding the minimum or maximum allowed values of various counts. For example, he shows that there is no polyhedron with 7 edges, but all integer values >=8 are possible.
Today, we know that the polyhedral theorem holds much more generally, e.g. for graphs or for triangulations of surfaces, but the coefficient 2 on the right sometimes is another integer.
This is #Euler works no 230. #mathematics
https://scholarlycommons.pacific.edu/euler-works/230/
Of course, various translations exist, e.g. German https://www.agtz.mathematik.uni-mainz.de/algebraische-geometrie/van-straten/euler-kreis-mainz/

#paperOfTheDay is another one by Euler: "Elementa doctrinae solidorum" from 1758. The English title would be "Elements of the doctrine of solids". This is the article that establishes what's now known as Euler's polyhedra formula. Namely, consider a finite 3-dimensional solid bounded by planes, where H is the number of these plane surfaces, and A is the number of edges (=1-dimensional intersection between two planes), and S is the number of corners (=0-dimensional intersections between three or more planes). Then
S + H - A = 2.
This is easily checked for typical examples, e.g. a cube has S=8 corners and H=6 surfaces and A=12 edges, and 8+6-12=2 indeed. This theorem holds for all 3-dimensional polyhedra, it is not required that their sides are any particular regular n-gons. The proof is surprisingly long and "un-geometrical", Euler complains several times that although 2-dimensional #geometry has been studied in great detail, even such elementary properties of 3-dimensional geometry are completely obscure. From his theorem, and simple observations relating the various counts (e.g. if one of the side surfaces has n corners, then it is adjacent to n edges, etc), Euler derives numerous corollaries and bounds regarding the minimum or maximum allowed values of various counts. For example, he shows that there is no polyhedron with 7 edges, but all integer values >=8 are possible.
Today, we know that the polyhedral theorem holds much more generally, e.g. for graphs or for triangulations of surfaces, but the coefficient 2 on the right sometimes is another integer.
This is #Euler works no 230. #mathematics
https://scholarlycommons.pacific.edu/euler-works/230/
Of course, various translations exist, e.g. German https://www.agtz.mathematik.uni-mainz.de/algebraische-geometrie/van-straten/euler-kreis-mainz/

#paperOfTheDay is Euler's "De summis serierum reciprocarum" from 1740. This article concerns the sum of inverse squares, like
1+ 1/4 + 1/9 + 1/16 + ...
Euler starts with the remark that such series had been studied intensively by various scholars, including himself, where nobody had been able to find an exact value for their sum. Now, however, he found a new approach. The key idea is to examine the power series representations of sine and cosine, such as
sin(x) = x - x^3/(1*2*3) + x^5/(1*2*3*4*5) - ...
One then studies the inverse relation, i.e. for which x is sin(x)=0? Clearly x=0, but what are the others? Geometrically, we know that these are the multiples of pi, but it must also follow from the series, by studying the solutions of
0 = 1-x^2/(1*2*3) + x^4/(1*2*3*4*5) - ....
If the right and side vanishes at some x=A, this means that it can be written with this linear factor, i.e. (x-A)*(something else). Using a sequence of clever transformations, one eventually isolates the sought-after sum of inverse squares, and finds its value to be pi^2/6. Likewise, the sums of other inverse even powers can be found, and evaluated to powers of pi, but not the sums of inverse odd powers.
From today's perspective, this is reflected in the fact that the Riemann zeta function, for even arguments, evaluates to a power of pi, whereas it gives complicated numbers for odd arguments.
The only non-trivial input needed for Euler's paper is the series representation of sine and cosine, and the geometric knowledge how these functions are periodic, and related to pi. Everything else is elementary algebra, used in an ingenious way.
#mathematics
https://scholarlycommons.pacific.edu/euler-works/41/

#paperOfTheDay is Euler's "De summis serierum reciprocarum" from 1740. This article concerns the sum of inverse squares, like
1+ 1/4 + 1/9 + 1/16 + ...
Euler starts with the remark that such series had been studied intensively by various scholars, including himself, where nobody had been able to find an exact value for their sum. Now, however, he found a new approach. The key idea is to examine the power series representations of sine and cosine, such as
sin(x) = x - x^3/(1*2*3) + x^5/(1*2*3*4*5) - ...
One then studies the inverse relation, i.e. for which x is sin(x)=0? Clearly x=0, but what are the others? Geometrically, we know that these are the multiples of pi, but it must also follow from the series, by studying the solutions of
0 = 1-x^2/(1*2*3) + x^4/(1*2*3*4*5) - ....
If the right and side vanishes at some x=A, this means that it can be written with this linear factor, i.e. (x-A)*(something else). Using a sequence of clever transformations, one eventually isolates the sought-after sum of inverse squares, and finds its value to be pi^2/6. Likewise, the sums of other inverse even powers can be found, and evaluated to powers of pi, but not the sums of inverse odd powers.
From today's perspective, this is reflected in the fact that the Riemann zeta function, for even arguments, evaluates to a power of pi, whereas it gives complicated numbers for odd arguments.
The only non-trivial input needed for Euler's paper is the series representation of sine and cosine, and the geometric knowledge how these functions are periodic, and related to pi. Everything else is elementary algebra, used in an ingenious way.
#mathematics
https://scholarlycommons.pacific.edu/euler-works/41/

#paperOfTheDay is Euler's "De summis serierum reciprocarum" from 1740. This article concerns the sum of inverse squares, like
1+ 1/4 + 1/9 + 1/16 + ...
Euler starts with the remark that such series had been studied intensively by various scholars, including himself, where nobody had been able to find an exact value for their sum. Now, however, he found a new approach. The key idea is to examine the power series representations of sine and cosine, such as
sin(x) = x - x^3/(1*2*3) + x^5/(1*2*3*4*5) - ...
One then studies the inverse relation, i.e. for which x is sin(x)=0? Clearly x=0, but what are the others? Geometrically, we know that these are the multiples of pi, but it must also follow from the series, by studying the solutions of
0 = 1-x^2/(1*2*3) + x^4/(1*2*3*4*5) - ....
If the right and side vanishes at some x=A, this means that it can be written with this linear factor, i.e. (x-A)*(something else). Using a sequence of clever transformations, one eventually isolates the sought-after sum of inverse squares, and finds its value to be pi^2/6. Likewise, the sums of other inverse even powers can be found, and evaluated to powers of pi, but not the sums of inverse odd powers.
From today's perspective, this is reflected in the fact that the Riemann zeta function, for even arguments, evaluates to a power of pi, whereas it gives complicated numbers for odd arguments.
The only non-trivial input needed for Euler's paper is the series representation of sine and cosine, and the geometric knowledge how these functions are periodic, and related to pi. Everything else is elementary algebra, used in an ingenious way.
#mathematics
https://scholarlycommons.pacific.edu/euler-works/41/

#paperOfTheDay is Euler's "De summis serierum reciprocarum" from 1740. This article concerns the sum of inverse squares, like
1+ 1/4 + 1/9 + 1/16 + ...
Euler starts with the remark that such series had been studied intensively by various scholars, including himself, where nobody had been able to find an exact value for their sum. Now, however, he found a new approach. The key idea is to examine the power series representations of sine and cosine, such as
sin(x) = x - x^3/(1*2*3) + x^5/(1*2*3*4*5) - ...
One then studies the inverse relation, i.e. for which x is sin(x)=0? Clearly x=0, but what are the others? Geometrically, we know that these are the multiples of pi, but it must also follow from the series, by studying the solutions of
0 = 1-x^2/(1*2*3) + x^4/(1*2*3*4*5) - ....
If the right and side vanishes at some x=A, this means that it can be written with this linear factor, i.e. (x-A)*(something else). Using a sequence of clever transformations, one eventually isolates the sought-after sum of inverse squares, and finds its value to be pi^2/6. Likewise, the sums of other inverse even powers can be found, and evaluated to powers of pi, but not the sums of inverse odd powers.
From today's perspective, this is reflected in the fact that the Riemann zeta function, for even arguments, evaluates to a power of pi, whereas it gives complicated numbers for odd arguments.
The only non-trivial input needed for Euler's paper is the series representation of sine and cosine, and the geometric knowledge how these functions are periodic, and related to pi. Everything else is elementary algebra, used in an ingenious way.
#mathematics
https://scholarlycommons.pacific.edu/euler-works/41/

#paperOfTheDay is Euler's "De summis serierum reciprocarum" from 1740. This article concerns the sum of inverse squares, like
1+ 1/4 + 1/9 + 1/16 + ...
Euler starts with the remark that such series had been studied intensively by various scholars, including himself, where nobody had been able to find an exact value for their sum. Now, however, he found a new approach. The key idea is to examine the power series representations of sine and cosine, such as
sin(x) = x - x^3/(1*2*3) + x^5/(1*2*3*4*5) - ...
One then studies the inverse relation, i.e. for which x is sin(x)=0? Clearly x=0, but what are the others? Geometrically, we know that these are the multiples of pi, but it must also follow from the series, by studying the solutions of
0 = 1-x^2/(1*2*3) + x^4/(1*2*3*4*5) - ....
If the right and side vanishes at some x=A, this means that it can be written with this linear factor, i.e. (x-A)*(something else). Using a sequence of clever transformations, one eventually isolates the sought-after sum of inverse squares, and finds its value to be pi^2/6. Likewise, the sums of other inverse even powers can be found, and evaluated to powers of pi, but not the sums of inverse odd powers.
From today's perspective, this is reflected in the fact that the Riemann zeta function, for even arguments, evaluates to a power of pi, whereas it gives complicated numbers for odd arguments.
The only non-trivial input needed for Euler's paper is the series representation of sine and cosine, and the geometric knowledge how these functions are periodic, and related to pi. Everything else is elementary algebra, used in an ingenious way.
#mathematics
https://scholarlycommons.pacific.edu/euler-works/41/

The #paperOfTheDay is "The Ehrhart polynomial of a matroid specializes to the beta invariant" from 2025.
A #matroid is an abstract generalization of a set of vectors in a vector space: Consider some set of k vectors in R^n (allowing k>n). Some of these k vectors could be linearly independent, but assume that not all of them are. If some subset is linearly independent, then so is every subset of that subset. This and a few other properties jointly give the definition of a matroid: Basically, consider the same logical relations arising from linear independence, but without demanding that the objects under consideration are actually vectors in R^n.
If one declares the elements of a matroid to be unit vectors in some abstract vector space (these basis vectors are all linearly independent in that abstract space, whereass not all elements are independent in the matroid), then the maximum set of independent vectors in the matroid amounts to some subset of these abstract vectors, whose convex hull is a polytope. This polytope contains a finite number of points of Z^n. If one rescales the polytope by an integer t, the number of lattice points changes. It turns out that the number of lattice points included is a polynomial of t (not a more complicated function), this defines the Ehrhart polynomial.
Every graph gives rise to a matroid (but not every matroid can be represented as a graph). For graphs, the Tutte polynomial is a classical quantity. The linear term of the Tutte polynomial is the Crapo-beta invariant (and can also be defined for non-graphical matroids). The present paper proves that a linear term of the Ehrhart polynomial coincides with beta. https://arxiv.org/abs/2504.15518 #mathematics #graphTheory

The #paperOfTheDay is "The Ehrhart polynomial of a matroid specializes to the beta invariant" from 2025.
A #matroid is an abstract generalization of a set of vectors in a vector space: Consider some set of k vectors in R^n (allowing k>n). Some of these k vectors could be linearly independent, but assume that not all of them are. If some subset is linearly independent, then so is every subset of that subset. This and a few other properties jointly give the definition of a matroid: Basically, consider the same logical relations arising from linear independence, but without demanding that the objects under consideration are actually vectors in R^n.
If one declares the elements of a matroid to be unit vectors in some abstract vector space (these basis vectors are all linearly independent in that abstract space, whereass not all elements are independent in the matroid), then the maximum set of independent vectors in the matroid amounts to some subset of these abstract vectors, whose convex hull is a polytope. This polytope contains a finite number of points of Z^n. If one rescales the polytope by an integer t, the number of lattice points changes. It turns out that the number of lattice points included is a polynomial of t (not a more complicated function), this defines the Ehrhart polynomial.
Every graph gives rise to a matroid (but not every matroid can be represented as a graph). For graphs, the Tutte polynomial is a classical quantity. The linear term of the Tutte polynomial is the Crapo-beta invariant (and can also be defined for non-graphical matroids). The present paper proves that a linear term of the Ehrhart polynomial coincides with beta. https://arxiv.org/abs/2504.15518 #mathematics #graphTheory

The #paperOfTheDay is "The Ehrhart polynomial of a matroid specializes to the beta invariant" from 2025.
A #matroid is an abstract generalization of a set of vectors in a vector space: Consider some set of k vectors in R^n (allowing k>n). Some of these k vectors could be linearly independent, but assume that not all of them are. If some subset is linearly independent, then so is every subset of that subset. This and a few other properties jointly give the definition of a matroid: Basically, consider the same logical relations arising from linear independence, but without demanding that the objects under consideration are actually vectors in R^n.
If one declares the elements of a matroid to be unit vectors in some abstract vector space (these basis vectors are all linearly independent in that abstract space, whereass not all elements are independent in the matroid), then the maximum set of independent vectors in the matroid amounts to some subset of these abstract vectors, whose convex hull is a polytope. This polytope contains a finite number of points of Z^n. If one rescales the polytope by an integer t, the number of lattice points changes. It turns out that the number of lattice points included is a polynomial of t (not a more complicated function), this defines the Ehrhart polynomial.
Every graph gives rise to a matroid (but not every matroid can be represented as a graph). For graphs, the Tutte polynomial is a classical quantity. The linear term of the Tutte polynomial is the Crapo-beta invariant (and can also be defined for non-graphical matroids). The present paper proves that a linear term of the Ehrhart polynomial coincides with beta. https://arxiv.org/abs/2504.15518 #mathematics #graphTheory

The #paperOfTheDay is "The Ehrhart polynomial of a matroid specializes to the beta invariant" from 2025.
A #matroid is an abstract generalization of a set of vectors in a vector space: Consider some set of k vectors in R^n (allowing k>n). Some of these k vectors could be linearly independent, but assume that not all of them are. If some subset is linearly independent, then so is every subset of that subset. This and a few other properties jointly give the definition of a matroid: Basically, consider the same logical relations arising from linear independence, but without demanding that the objects under consideration are actually vectors in R^n.
If one declares the elements of a matroid to be unit vectors in some abstract vector space (these basis vectors are all linearly independent in that abstract space, whereass not all elements are independent in the matroid), then the maximum set of independent vectors in the matroid amounts to some subset of these abstract vectors, whose convex hull is a polytope. This polytope contains a finite number of points of Z^n. If one rescales the polytope by an integer t, the number of lattice points changes. It turns out that the number of lattice points included is a polynomial of t (not a more complicated function), this defines the Ehrhart polynomial.
Every graph gives rise to a matroid (but not every matroid can be represented as a graph). For graphs, the Tutte polynomial is a classical quantity. The linear term of the Tutte polynomial is the Crapo-beta invariant (and can also be defined for non-graphical matroids). The present paper proves that a linear term of the Ehrhart polynomial coincides with beta. https://arxiv.org/abs/2504.15518 #mathematics #graphTheory

The #paperOfTheDay is "The Ehrhart polynomial of a matroid specializes to the beta invariant" from 2025.
A #matroid is an abstract generalization of a set of vectors in a vector space: Consider some set of k vectors in R^n (allowing k>n). Some of these k vectors could be linearly independent, but assume that not all of them are. If some subset is linearly independent, then so is every subset of that subset. This and a few other properties jointly give the definition of a matroid: Basically, consider the same logical relations arising from linear independence, but without demanding that the objects under consideration are actually vectors in R^n.
If one declares the elements of a matroid to be unit vectors in some abstract vector space (these basis vectors are all linearly independent in that abstract space, whereass not all elements are independent in the matroid), then the maximum set of independent vectors in the matroid amounts to some subset of these abstract vectors, whose convex hull is a polytope. This polytope contains a finite number of points of Z^n. If one rescales the polytope by an integer t, the number of lattice points changes. It turns out that the number of lattice points included is a polynomial of t (not a more complicated function), this defines the Ehrhart polynomial.
Every graph gives rise to a matroid (but not every matroid can be represented as a graph). For graphs, the Tutte polynomial is a classical quantity. The linear term of the Tutte polynomial is the Crapo-beta invariant (and can also be defined for non-graphical matroids). The present paper proves that a linear term of the Ehrhart polynomial coincides with beta. https://arxiv.org/abs/2504.15518 #mathematics #graphTheory

The #paperOfTheDay is "Duality of O(N) and Sp(N) random tensor models: tensors with symmetries" from 2023.
In previous weeks I have often mentioned the "vector model", which is the field-theoretic limit of a random distribution on a lattice where the elementary quantities do not just have a scalar numerical value (such as a temperature), but are a N-dimensional vector. Tensor models are a generalization of these, the "field" quantity is a tensor, i.e. an array of numbers of higher dimension (a vector is a 1-tensor and a matrix is a 2-tensor in this sense). Still, N is the size of these objects, so a 3-dimensional tensor would be a N x N x N array of numbers.
What is interesting is that these models often depend on the parameter N in a smooth way. Physically it makes no sense to have a "vector with 1.5 components", but mathematically the dependence on N is some polynomial, and one can insert any value. In particular, one can let N be negative. From a perspective of #Feynmandiagram s , the N-dependent polynomial counts all ways to find closed circuits in the graph, and assigns a factor N to each of them. If N is negative, each circuit gets a negative sign, which reminds of the behaviour of fermions. In fact, it had long been known that the O(N) vector model for negative N is equivalent to a vector model with positive N, but based upon the symplectic group Sp(N). The present article proves the analogous duality, O(-N)= Sp(N), for more general tensor models. For the moment this is a #mathematics statement, but in the long run such dualities have often given rise to unexpected insights about the #physics of such theories. https://iopscience.iop.org/article/10.1088/1751-8121/ad0af4

The #paperOfTheDay is "Duality of O(N) and Sp(N) random tensor models: tensors with symmetries" from 2023.
In previous weeks I have often mentioned the "vector model", which is the field-theoretic limit of a random distribution on a lattice where the elementary quantities do not just have a scalar numerical value (such as a temperature), but are a N-dimensional vector. Tensor models are a generalization of these, the "field" quantity is a tensor, i.e. an array of numbers of higher dimension (a vector is a 1-tensor and a matrix is a 2-tensor in this sense). Still, N is the size of these objects, so a 3-dimensional tensor would be a N x N x N array of numbers.
What is interesting is that these models often depend on the parameter N in a smooth way. Physically it makes no sense to have a "vector with 1.5 components", but mathematically the dependence on N is some polynomial, and one can insert any value. In particular, one can let N be negative. From a perspective of #Feynmandiagram s , the N-dependent polynomial counts all ways to find closed circuits in the graph, and assigns a factor N to each of them. If N is negative, each circuit gets a negative sign, which reminds of the behaviour of fermions. In fact, it had long been known that the O(N) vector model for negative N is equivalent to a vector model with positive N, but based upon the symplectic group Sp(N). The present article proves the analogous duality, O(-N)= Sp(N), for more general tensor models. For the moment this is a #mathematics statement, but in the long run such dualities have often given rise to unexpected insights about the #physics of such theories. https://iopscience.iop.org/article/10.1088/1751-8121/ad0af4

The #paperOfTheDay is "Duality of O(N) and Sp(N) random tensor models: tensors with symmetries" from 2023.
In previous weeks I have often mentioned the "vector model", which is the field-theoretic limit of a random distribution on a lattice where the elementary quantities do not just have a scalar numerical value (such as a temperature), but are a N-dimensional vector. Tensor models are a generalization of these, the "field" quantity is a tensor, i.e. an array of numbers of higher dimension (a vector is a 1-tensor and a matrix is a 2-tensor in this sense). Still, N is the size of these objects, so a 3-dimensional tensor would be a N x N x N array of numbers.
What is interesting is that these models often depend on the parameter N in a smooth way. Physically it makes no sense to have a "vector with 1.5 components", but mathematically the dependence on N is some polynomial, and one can insert any value. In particular, one can let N be negative. From a perspective of #Feynmandiagram s , the N-dependent polynomial counts all ways to find closed circuits in the graph, and assigns a factor N to each of them. If N is negative, each circuit gets a negative sign, which reminds of the behaviour of fermions. In fact, it had long been known that the O(N) vector model for negative N is equivalent to a vector model with positive N, but based upon the symplectic group Sp(N). The present article proves the analogous duality, O(-N)= Sp(N), for more general tensor models. For the moment this is a #mathematics statement, but in the long run such dualities have often given rise to unexpected insights about the #physics of such theories. https://iopscience.iop.org/article/10.1088/1751-8121/ad0af4

The #paperOfTheDay is "Duality of O(N) and Sp(N) random tensor models: tensors with symmetries" from 2023.
In previous weeks I have often mentioned the "vector model", which is the field-theoretic limit of a random distribution on a lattice where the elementary quantities do not just have a scalar numerical value (such as a temperature), but are a N-dimensional vector. Tensor models are a generalization of these, the "field" quantity is a tensor, i.e. an array of numbers of higher dimension (a vector is a 1-tensor and a matrix is a 2-tensor in this sense). Still, N is the size of these objects, so a 3-dimensional tensor would be a N x N x N array of numbers.
What is interesting is that these models often depend on the parameter N in a smooth way. Physically it makes no sense to have a "vector with 1.5 components", but mathematically the dependence on N is some polynomial, and one can insert any value. In particular, one can let N be negative. From a perspective of #Feynmandiagram s , the N-dependent polynomial counts all ways to find closed circuits in the graph, and assigns a factor N to each of them. If N is negative, each circuit gets a negative sign, which reminds of the behaviour of fermions. In fact, it had long been known that the O(N) vector model for negative N is equivalent to a vector model with positive N, but based upon the symplectic group Sp(N). The present article proves the analogous duality, O(-N)= Sp(N), for more general tensor models. For the moment this is a #mathematics statement, but in the long run such dualities have often given rise to unexpected insights about the #physics of such theories. https://iopscience.iop.org/article/10.1088/1751-8121/ad0af4

The #paperOfTheDay is "Duality of O(N) and Sp(N) random tensor models: tensors with symmetries" from 2023.
In previous weeks I have often mentioned the "vector model", which is the field-theoretic limit of a random distribution on a lattice where the elementary quantities do not just have a scalar numerical value (such as a temperature), but are a N-dimensional vector. Tensor models are a generalization of these, the "field" quantity is a tensor, i.e. an array of numbers of higher dimension (a vector is a 1-tensor and a matrix is a 2-tensor in this sense). Still, N is the size of these objects, so a 3-dimensional tensor would be a N x N x N array of numbers.
What is interesting is that these models often depend on the parameter N in a smooth way. Physically it makes no sense to have a "vector with 1.5 components", but mathematically the dependence on N is some polynomial, and one can insert any value. In particular, one can let N be negative. From a perspective of #Feynmandiagram s , the N-dependent polynomial counts all ways to find closed circuits in the graph, and assigns a factor N to each of them. If N is negative, each circuit gets a negative sign, which reminds of the behaviour of fermions. In fact, it had long been known that the O(N) vector model for negative N is equivalent to a vector model with positive N, but based upon the symplectic group Sp(N). The present article proves the analogous duality, O(-N)= Sp(N), for more general tensor models. For the moment this is a #mathematics statement, but in the long run such dualities have often given rise to unexpected insights about the #physics of such theories. https://iopscience.iop.org/article/10.1088/1751-8121/ad0af4

#paperOfTheDay is "Comparison between large-order estimates and perturbation series in a scalar field theory with Gaussian propagator" from 1978.
The perturbation series for small coupling in #quantumFieldTheory is usually divergent (because the theory with negative coupling describes different #physics and is not "smoothly" connected with the case of positive coupling). To obtain finite physical predictions for the positive coupling case, a #resummation is needed. In the 1970s and 1980s, the computational techniques were for the first time sufficient to reach, say, order 4 in perturbation series computed with #FeynmanDiagram s, and the question arose how to actually do the resummation in practice.
The present paper studies a model quantum field theory, where the propagator is exp(-p^2). This is not physically accurate, but it allows to compute perturbation series to very high order (order 8 here) because the #FeynmanIntegral s are easier. Secondly, one can carry out an #instanton calculation to determine the leading large-order growth. This allows to study whether or not the series coefficients match the expected growth rate, and how to use various different methods of resummation. In particular, the instanton analysis predicts the location and algebraic exponent for the leading singularity in the Borel plane, and this can be used as an additional input: Either, compute the Borel transform directly with a shifted-gamma function to account for the singularity, or make an ansatz of the type (known singularity)x(Pade approximant) for the Borel transformed function. The numerical data shows that this gives substantially higher accuracy for the resummation. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.17.2144

#paperOfTheDay is "Comparison between large-order estimates and perturbation series in a scalar field theory with Gaussian propagator" from 1978.
The perturbation series for small coupling in #quantumFieldTheory is usually divergent (because the theory with negative coupling describes different #physics and is not "smoothly" connected with the case of positive coupling). To obtain finite physical predictions for the positive coupling case, a #resummation is needed. In the 1970s and 1980s, the computational techniques were for the first time sufficient to reach, say, order 4 in perturbation series computed with #FeynmanDiagram s, and the question arose how to actually do the resummation in practice.
The present paper studies a model quantum field theory, where the propagator is exp(-p^2). This is not physically accurate, but it allows to compute perturbation series to very high order (order 8 here) because the #FeynmanIntegral s are easier. Secondly, one can carry out an #instanton calculation to determine the leading large-order growth. This allows to study whether or not the series coefficients match the expected growth rate, and how to use various different methods of resummation. In particular, the instanton analysis predicts the location and algebraic exponent for the leading singularity in the Borel plane, and this can be used as an additional input: Either, compute the Borel transform directly with a shifted-gamma function to account for the singularity, or make an ansatz of the type (known singularity)x(Pade approximant) for the Borel transformed function. The numerical data shows that this gives substantially higher accuracy for the resummation. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.17.2144

#paperOfTheDay is "Comparison between large-order estimates and perturbation series in a scalar field theory with Gaussian propagator" from 1978.
The perturbation series for small coupling in #quantumFieldTheory is usually divergent (because the theory with negative coupling describes different #physics and is not "smoothly" connected with the case of positive coupling). To obtain finite physical predictions for the positive coupling case, a #resummation is needed. In the 1970s and 1980s, the computational techniques were for the first time sufficient to reach, say, order 4 in perturbation series computed with #FeynmanDiagram s, and the question arose how to actually do the resummation in practice.
The present paper studies a model quantum field theory, where the propagator is exp(-p^2). This is not physically accurate, but it allows to compute perturbation series to very high order (order 8 here) because the #FeynmanIntegral s are easier. Secondly, one can carry out an #instanton calculation to determine the leading large-order growth. This allows to study whether or not the series coefficients match the expected growth rate, and how to use various different methods of resummation. In particular, the instanton analysis predicts the location and algebraic exponent for the leading singularity in the Borel plane, and this can be used as an additional input: Either, compute the Borel transform directly with a shifted-gamma function to account for the singularity, or make an ansatz of the type (known singularity)x(Pade approximant) for the Borel transformed function. The numerical data shows that this gives substantially higher accuracy for the resummation. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.17.2144

#paperOfTheDay is "Comparison between large-order estimates and perturbation series in a scalar field theory with Gaussian propagator" from 1978.
The perturbation series for small coupling in #quantumFieldTheory is usually divergent (because the theory with negative coupling describes different #physics and is not "smoothly" connected with the case of positive coupling). To obtain finite physical predictions for the positive coupling case, a #resummation is needed. In the 1970s and 1980s, the computational techniques were for the first time sufficient to reach, say, order 4 in perturbation series computed with #FeynmanDiagram s, and the question arose how to actually do the resummation in practice.
The present paper studies a model quantum field theory, where the propagator is exp(-p^2). This is not physically accurate, but it allows to compute perturbation series to very high order (order 8 here) because the #FeynmanIntegral s are easier. Secondly, one can carry out an #instanton calculation to determine the leading large-order growth. This allows to study whether or not the series coefficients match the expected growth rate, and how to use various different methods of resummation. In particular, the instanton analysis predicts the location and algebraic exponent for the leading singularity in the Borel plane, and this can be used as an additional input: Either, compute the Borel transform directly with a shifted-gamma function to account for the singularity, or make an ansatz of the type (known singularity)x(Pade approximant) for the Borel transformed function. The numerical data shows that this gives substantially higher accuracy for the resummation. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.17.2144

#paperOfTheDay is "Comparison between large-order estimates and perturbation series in a scalar field theory with Gaussian propagator" from 1978.
The perturbation series for small coupling in #quantumFieldTheory is usually divergent (because the theory with negative coupling describes different #physics and is not "smoothly" connected with the case of positive coupling). To obtain finite physical predictions for the positive coupling case, a #resummation is needed. In the 1970s and 1980s, the computational techniques were for the first time sufficient to reach, say, order 4 in perturbation series computed with #FeynmanDiagram s, and the question arose how to actually do the resummation in practice.
The present paper studies a model quantum field theory, where the propagator is exp(-p^2). This is not physically accurate, but it allows to compute perturbation series to very high order (order 8 here) because the #FeynmanIntegral s are easier. Secondly, one can carry out an #instanton calculation to determine the leading large-order growth. This allows to study whether or not the series coefficients match the expected growth rate, and how to use various different methods of resummation. In particular, the instanton analysis predicts the location and algebraic exponent for the leading singularity in the Borel plane, and this can be used as an additional input: Either, compute the Borel transform directly with a shifted-gamma function to account for the singularity, or make an ansatz of the type (known singularity)x(Pade approximant) for the Borel transformed function. The numerical data shows that this gives substantially higher accuracy for the resummation. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.17.2144

The #paperOfTheDay is today a little book on pure #mathematics : "On Applications and Theory of Functional Equations" from 1969. Today, one would perhaps call it a proceeding, it consists of the notes of two different talks given by the author at various meetings in the 1960s. Basically, they are a discussion of classical examples of functional equations. These are, roughly speaking, equations for one or more unknown functions, expressed as relations between these functions for different argument values. Maybe the most famous one is the functional equation
f(x+y) = f(x) + f(y).
Such equations are treated by clever manipulations of the arguments. One can for example set y=0 to find f(x) = f(x) + f(0) and therefore f(0)=0. For rational values of x, the unique solution is f(x) = c*x, with some constant c. If one allows arbitrary real x, an additional condition is required, e.g. that f(x) should be bounded in bounded intervals, or differentiable.
Of course, this book is outdated, and by now the state of affairs might have changed. But at that time, apparently there were only relatively few general theorems about whether a functional equation has solutions at all, or whether it is unique, or how to construct or approximate it. Apparently, sometimes a single functional equation that involves multiple functions can determine all of these functions at once. Compare this to ordinary algebraic equations, or even differential equations, where one often knows at least whether a solution exists and how many free parameters it has, and in many cases there are even efficient numerical procedures for finding approximate solutions.
https://openlibrary.org/books/OL5512535M/On_applications_and_theory_of_functional_equations.

The #paperOfTheDay is today a little book on pure #mathematics : "On Applications and Theory of Functional Equations" from 1969. Today, one would perhaps call it a proceeding, it consists of the notes of two different talks given by the author at various meetings in the 1960s. Basically, they are a discussion of classical examples of functional equations. These are, roughly speaking, equations for one or more unknown functions, expressed as relations between these functions for different argument values. Maybe the most famous one is the functional equation
f(x+y) = f(x) + f(y).
Such equations are treated by clever manipulations of the arguments. One can for example set y=0 to find f(x) = f(x) + f(0) and therefore f(0)=0. For rational values of x, the unique solution is f(x) = c*x, with some constant c. If one allows arbitrary real x, an additional condition is required, e.g. that f(x) should be bounded in bounded intervals, or differentiable.
Of course, this book is outdated, and by now the state of affairs might have changed. But at that time, apparently there were only relatively few general theorems about whether a functional equation has solutions at all, or whether it is unique, or how to construct or approximate it. Apparently, sometimes a single functional equation that involves multiple functions can determine all of these functions at once. Compare this to ordinary algebraic equations, or even differential equations, where one often knows at least whether a solution exists and how many free parameters it has, and in many cases there are even efficient numerical procedures for finding approximate solutions.
https://openlibrary.org/books/OL5512535M/On_applications_and_theory_of_functional_equations.

The #paperOfTheDay is today a little book on pure #mathematics : "On Applications and Theory of Functional Equations" from 1969. Today, one would perhaps call it a proceeding, it consists of the notes of two different talks given by the author at various meetings in the 1960s. Basically, they are a discussion of classical examples of functional equations. These are, roughly speaking, equations for one or more unknown functions, expressed as relations between these functions for different argument values. Maybe the most famous one is the functional equation
f(x+y) = f(x) + f(y).
Such equations are treated by clever manipulations of the arguments. One can for example set y=0 to find f(x) = f(x) + f(0) and therefore f(0)=0. For rational values of x, the unique solution is f(x) = c*x, with some constant c. If one allows arbitrary real x, an additional condition is required, e.g. that f(x) should be bounded in bounded intervals, or differentiable.
Of course, this book is outdated, and by now the state of affairs might have changed. But at that time, apparently there were only relatively few general theorems about whether a functional equation has solutions at all, or whether it is unique, or how to construct or approximate it. Apparently, sometimes a single functional equation that involves multiple functions can determine all of these functions at once. Compare this to ordinary algebraic equations, or even differential equations, where one often knows at least whether a solution exists and how many free parameters it has, and in many cases there are even efficient numerical procedures for finding approximate solutions.
https://openlibrary.org/books/OL5512535M/On_applications_and_theory_of_functional_equations.

The #paperOfTheDay is today a little book on pure #mathematics : "On Applications and Theory of Functional Equations" from 1969. Today, one would perhaps call it a proceeding, it consists of the notes of two different talks given by the author at various meetings in the 1960s. Basically, they are a discussion of classical examples of functional equations. These are, roughly speaking, equations for one or more unknown functions, expressed as relations between these functions for different argument values. Maybe the most famous one is the functional equation
f(x+y) = f(x) + f(y).
Such equations are treated by clever manipulations of the arguments. One can for example set y=0 to find f(x) = f(x) + f(0) and therefore f(0)=0. For rational values of x, the unique solution is f(x) = c*x, with some constant c. If one allows arbitrary real x, an additional condition is required, e.g. that f(x) should be bounded in bounded intervals, or differentiable.
Of course, this book is outdated, and by now the state of affairs might have changed. But at that time, apparently there were only relatively few general theorems about whether a functional equation has solutions at all, or whether it is unique, or how to construct or approximate it. Apparently, sometimes a single functional equation that involves multiple functions can determine all of these functions at once. Compare this to ordinary algebraic equations, or even differential equations, where one often knows at least whether a solution exists and how many free parameters it has, and in many cases there are even efficient numerical procedures for finding approximate solutions.
https://openlibrary.org/books/OL5512535M/On_applications_and_theory_of_functional_equations.

The #paperOfTheDay is today a little book on pure #mathematics : "On Applications and Theory of Functional Equations" from 1969. Today, one would perhaps call it a proceeding, it consists of the notes of two different talks given by the author at various meetings in the 1960s. Basically, they are a discussion of classical examples of functional equations. These are, roughly speaking, equations for one or more unknown functions, expressed as relations between these functions for different argument values. Maybe the most famous one is the functional equation
f(x+y) = f(x) + f(y).
Such equations are treated by clever manipulations of the arguments. One can for example set y=0 to find f(x) = f(x) + f(0) and therefore f(0)=0. For rational values of x, the unique solution is f(x) = c*x, with some constant c. If one allows arbitrary real x, an additional condition is required, e.g. that f(x) should be bounded in bounded intervals, or differentiable.
Of course, this book is outdated, and by now the state of affairs might have changed. But at that time, apparently there were only relatively few general theorems about whether a functional equation has solutions at all, or whether it is unique, or how to construct or approximate it. Apparently, sometimes a single functional equation that involves multiple functions can determine all of these functions at once. Compare this to ordinary algebraic equations, or even differential equations, where one often knows at least whether a solution exists and how many free parameters it has, and in many cases there are even efficient numerical procedures for finding approximate solutions.
https://openlibrary.org/books/OL5512535M/On_applications_and_theory_of_functional_equations.

#paperOfTheDay is "Effective five-wave Hamiltonian for surface water waves" from 1997. This article concerns (a certain idealization of) waves on the surface of deep water in one spacial dimension. These waves have a non-linear equation of motion, therefore they can "scatter" when they meet. This has nothing to do with #quantumFieldTheory , but still, the scattering amplitudes can be computed with #FeynmanDiagram s: The sum of all tree-level diagrams reproduces the non-quantum classical field theory. Such computation has the advantage that it is more structured, and ultimately simpler, than "manually" modifying and expanding the interaction Hamiltonian in superpositions of plane waves.
Since 1-dimensional water waves are not the most fashionable area of theoretical #physics right now, this work was maybe not widely known. However, it has resurfaced now as part of an ongoing effort trying to compute, or re-interpret, sums of tree-level Feynman diagrams (=non-quantum scattering amplitudes) in more geometric terms. While these endeavors in quantum field theory always have the shortcoming of leaving out the actual quantum part, for a classical field theory such as water waves, the sum of tree-level diagrams is a genuine physically relevant observable.
https://www.sciencedirect.com/science/article/pii/S0375960197002107?__cf_chl_f_tk=rgYKJNN8nI6263mmLkztzvasv.4l8IQQrTh0qPjse9Y-1783072682-1.0.1.1-BKQujlGYbuT1Onn9XcARc8fj7OS2.QuI8_CKS90K3dA

#paperOfTheDay is "Effective five-wave Hamiltonian for surface water waves" from 1997. This article concerns (a certain idealization of) waves on the surface of deep water in one spacial dimension. These waves have a non-linear equation of motion, therefore they can "scatter" when they meet. This has nothing to do with #quantumFieldTheory , but still, the scattering amplitudes can be computed with #FeynmanDiagram s: The sum of all tree-level diagrams reproduces the non-quantum classical field theory. Such computation has the advantage that it is more structured, and ultimately simpler, than "manually" modifying and expanding the interaction Hamiltonian in superpositions of plane waves.
Since 1-dimensional water waves are not the most fashionable area of theoretical #physics right now, this work was maybe not widely known. However, it has resurfaced now as part of an ongoing effort trying to compute, or re-interpret, sums of tree-level Feynman diagrams (=non-quantum scattering amplitudes) in more geometric terms. While these endeavors in quantum field theory always have the shortcoming of leaving out the actual quantum part, for a classical field theory such as water waves, the sum of tree-level diagrams is a genuine physically relevant observable.
https://www.sciencedirect.com/science/article/pii/S0375960197002107?__cf_chl_f_tk=rgYKJNN8nI6263mmLkztzvasv.4l8IQQrTh0qPjse9Y-1783072682-1.0.1.1-BKQujlGYbuT1Onn9XcARc8fj7OS2.QuI8_CKS90K3dA

#paperOfTheDay is "Effective five-wave Hamiltonian for surface water waves" from 1997. This article concerns (a certain idealization of) waves on the surface of deep water in one spacial dimension. These waves have a non-linear equation of motion, therefore they can "scatter" when they meet. This has nothing to do with #quantumFieldTheory , but still, the scattering amplitudes can be computed with #FeynmanDiagram s: The sum of all tree-level diagrams reproduces the non-quantum classical field theory. Such computation has the advantage that it is more structured, and ultimately simpler, than "manually" modifying and expanding the interaction Hamiltonian in superpositions of plane waves.
Since 1-dimensional water waves are not the most fashionable area of theoretical #physics right now, this work was maybe not widely known. However, it has resurfaced now as part of an ongoing effort trying to compute, or re-interpret, sums of tree-level Feynman diagrams (=non-quantum scattering amplitudes) in more geometric terms. While these endeavors in quantum field theory always have the shortcoming of leaving out the actual quantum part, for a classical field theory such as water waves, the sum of tree-level diagrams is a genuine physically relevant observable.
https://www.sciencedirect.com/science/article/pii/S0375960197002107?__cf_chl_f_tk=rgYKJNN8nI6263mmLkztzvasv.4l8IQQrTh0qPjse9Y-1783072682-1.0.1.1-BKQujlGYbuT1Onn9XcARc8fj7OS2.QuI8_CKS90K3dA

#paperOfTheDay is "Effective five-wave Hamiltonian for surface water waves" from 1997. This article concerns (a certain idealization of) waves on the surface of deep water in one spacial dimension. These waves have a non-linear equation of motion, therefore they can "scatter" when they meet. This has nothing to do with #quantumFieldTheory , but still, the scattering amplitudes can be computed with #FeynmanDiagram s: The sum of all tree-level diagrams reproduces the non-quantum classical field theory. Such computation has the advantage that it is more structured, and ultimately simpler, than "manually" modifying and expanding the interaction Hamiltonian in superpositions of plane waves.
Since 1-dimensional water waves are not the most fashionable area of theoretical #physics right now, this work was maybe not widely known. However, it has resurfaced now as part of an ongoing effort trying to compute, or re-interpret, sums of tree-level Feynman diagrams (=non-quantum scattering amplitudes) in more geometric terms. While these endeavors in quantum field theory always have the shortcoming of leaving out the actual quantum part, for a classical field theory such as water waves, the sum of tree-level diagrams is a genuine physically relevant observable.
https://www.sciencedirect.com/science/article/pii/S0375960197002107?__cf_chl_f_tk=rgYKJNN8nI6263mmLkztzvasv.4l8IQQrTh0qPjse9Y-1783072682-1.0.1.1-BKQujlGYbuT1Onn9XcARc8fj7OS2.QuI8_CKS90K3dA

#paperOfTheDay is "Effective five-wave Hamiltonian for surface water waves" from 1997. This article concerns (a certain idealization of) waves on the surface of deep water in one spacial dimension. These waves have a non-linear equation of motion, therefore they can "scatter" when they meet. This has nothing to do with #quantumFieldTheory , but still, the scattering amplitudes can be computed with #FeynmanDiagram s: The sum of all tree-level diagrams reproduces the non-quantum classical field theory. Such computation has the advantage that it is more structured, and ultimately simpler, than "manually" modifying and expanding the interaction Hamiltonian in superpositions of plane waves.
Since 1-dimensional water waves are not the most fashionable area of theoretical #physics right now, this work was maybe not widely known. However, it has resurfaced now as part of an ongoing effort trying to compute, or re-interpret, sums of tree-level Feynman diagrams (=non-quantum scattering amplitudes) in more geometric terms. While these endeavors in quantum field theory always have the shortcoming of leaving out the actual quantum part, for a classical field theory such as water waves, the sum of tree-level diagrams is a genuine physically relevant observable.
https://www.sciencedirect.com/science/article/pii/S0375960197002107?__cf_chl_f_tk=rgYKJNN8nI6263mmLkztzvasv.4l8IQQrTh0qPjse9Y-1783072682-1.0.1.1-BKQujlGYbuT1Onn9XcARc8fj7OS2.QuI8_CKS90K3dA

#paperOfTheDay is "The Diagrammar of Quantum Magnusian" from 2026.
The Magnusian is, morally, the logarithm of the S-matrix in #quantumFieldTheory . It has a perturbative expansion similarly to #FeynmanDiagrams , but these are not exactly the same diagrams, and they come with different combinatorial factors. The Magnus expansion has long been well understood for the case of differential equations, but the application to full quantum field theory is relatively new and so far had been rather ad-hoc, by expanding the corresponding series and explicitly matching terms. The present paper provides a systematic description of which diagrams, and with which combinatorial factors, appear in the Magnus expansion in QFT. Even for a scalar QFT, this is more complicated than ordinary Feynman diagrams, e.g. because the Magnus diagrams have directed edges (representing retarded propagators). https://arxiv.org/abs/2605.25473

#paperOfTheDay is "The Diagrammar of Quantum Magnusian" from 2026.
The Magnusian is, morally, the logarithm of the S-matrix in #quantumFieldTheory . It has a perturbative expansion similarly to #FeynmanDiagrams , but these are not exactly the same diagrams, and they come with different combinatorial factors. The Magnus expansion has long been well understood for the case of differential equations, but the application to full quantum field theory is relatively new and so far had been rather ad-hoc, by expanding the corresponding series and explicitly matching terms. The present paper provides a systematic description of which diagrams, and with which combinatorial factors, appear in the Magnus expansion in QFT. Even for a scalar QFT, this is more complicated than ordinary Feynman diagrams, e.g. because the Magnus diagrams have directed edges (representing retarded propagators). https://arxiv.org/abs/2605.25473

#paperOfTheDay is "The Diagrammar of Quantum Magnusian" from 2026.
The Magnusian is, morally, the logarithm of the S-matrix in #quantumFieldTheory . It has a perturbative expansion similarly to #FeynmanDiagrams , but these are not exactly the same diagrams, and they come with different combinatorial factors. The Magnus expansion has long been well understood for the case of differential equations, but the application to full quantum field theory is relatively new and so far had been rather ad-hoc, by expanding the corresponding series and explicitly matching terms. The present paper provides a systematic description of which diagrams, and with which combinatorial factors, appear in the Magnus expansion in QFT. Even for a scalar QFT, this is more complicated than ordinary Feynman diagrams, e.g. because the Magnus diagrams have directed edges (representing retarded propagators). https://arxiv.org/abs/2605.25473

#paperOfTheDay is "The Diagrammar of Quantum Magnusian" from 2026.
The Magnusian is, morally, the logarithm of the S-matrix in #quantumFieldTheory . It has a perturbative expansion similarly to #FeynmanDiagrams , but these are not exactly the same diagrams, and they come with different combinatorial factors. The Magnus expansion has long been well understood for the case of differential equations, but the application to full quantum field theory is relatively new and so far had been rather ad-hoc, by expanding the corresponding series and explicitly matching terms. The present paper provides a systematic description of which diagrams, and with which combinatorial factors, appear in the Magnus expansion in QFT. Even for a scalar QFT, this is more complicated than ordinary Feynman diagrams, e.g. because the Magnus diagrams have directed edges (representing retarded propagators). https://arxiv.org/abs/2605.25473

#paperOfTheDay is "The Magnus expansion in relativistic quantum field theory" from 2025.
One of the well-established ways to do computations in #quantumFieldTheory is perturbation theory in terms of #FeynmanIntegral s. From an analytic perspective, these integrals are the coefficients of a Dyson series, which is a power-series expansion of the time-evolution operator U (especially the S-matrix is a special case of U). This operator U is supposed to be unitary, so it seems plausible to write it as U=exp(i N) with some operator N, and to directly compute the quantity N. This is known as the #Magnus expansion, and has been rather well studied in the theory of differential equations in #mathematics . It has nice properties, in particular that even an inaccurate truncation of N still guarantees U to be unitary.
In recent years, the Magnus expansion has also received new attention in the field of scattering amplitudes in theoretical #physics . The present preprint concerns using Magnus expansions instead of Dyson series in perturbative quantum field theory. The series coefficients are distinct from Feynman integrals, but they have an analogous graphical representation, using retarded propagators in place of Feynman propagators, and slightly different #combinatorics on the corresponding #graphs . The paper contains excessively many examples which clearly indicate the general formulas, which are then stated (but not proved). Besides the actual application to future physics calculations, I think it could be an interesting project in #combinatorics to prove all these formulas, and relate them to the existing mathematical literature. https://arxiv.org/abs/2512.05017

#paperOfTheDay is "The Magnus expansion in relativistic quantum field theory" from 2025.
One of the well-established ways to do computations in #quantumFieldTheory is perturbation theory in terms of #FeynmanIntegral s. From an analytic perspective, these integrals are the coefficients of a Dyson series, which is a power-series expansion of the time-evolution operator U (especially the S-matrix is a special case of U). This operator U is supposed to be unitary, so it seems plausible to write it as U=exp(i N) with some operator N, and to directly compute the quantity N. This is known as the #Magnus expansion, and has been rather well studied in the theory of differential equations in #mathematics . It has nice properties, in particular that even an inaccurate truncation of N still guarantees U to be unitary.
In recent years, the Magnus expansion has also received new attention in the field of scattering amplitudes in theoretical #physics . The present preprint concerns using Magnus expansions instead of Dyson series in perturbative quantum field theory. The series coefficients are distinct from Feynman integrals, but they have an analogous graphical representation, using retarded propagators in place of Feynman propagators, and slightly different #combinatorics on the corresponding #graphs . The paper contains excessively many examples which clearly indicate the general formulas, which are then stated (but not proved). Besides the actual application to future physics calculations, I think it could be an interesting project in #combinatorics to prove all these formulas, and relate them to the existing mathematical literature. https://arxiv.org/abs/2512.05017

#paperOfTheDay is "The Magnus expansion in relativistic quantum field theory" from 2025.
One of the well-established ways to do computations in #quantumFieldTheory is perturbation theory in terms of #FeynmanIntegral s. From an analytic perspective, these integrals are the coefficients of a Dyson series, which is a power-series expansion of the time-evolution operator U (especially the S-matrix is a special case of U). This operator U is supposed to be unitary, so it seems plausible to write it as U=exp(i N) with some operator N, and to directly compute the quantity N. This is known as the #Magnus expansion, and has been rather well studied in the theory of differential equations in #mathematics . It has nice properties, in particular that even an inaccurate truncation of N still guarantees U to be unitary.
In recent years, the Magnus expansion has also received new attention in the field of scattering amplitudes in theoretical #physics . The present preprint concerns using Magnus expansions instead of Dyson series in perturbative quantum field theory. The series coefficients are distinct from Feynman integrals, but they have an analogous graphical representation, using retarded propagators in place of Feynman propagators, and slightly different #combinatorics on the corresponding #graphs . The paper contains excessively many examples which clearly indicate the general formulas, which are then stated (but not proved). Besides the actual application to future physics calculations, I think it could be an interesting project in #combinatorics to prove all these formulas, and relate them to the existing mathematical literature. https://arxiv.org/abs/2512.05017

#paperOfTheDay is "The Magnus expansion in relativistic quantum field theory" from 2025.
One of the well-established ways to do computations in #quantumFieldTheory is perturbation theory in terms of #FeynmanIntegral s. From an analytic perspective, these integrals are the coefficients of a Dyson series, which is a power-series expansion of the time-evolution operator U (especially the S-matrix is a special case of U). This operator U is supposed to be unitary, so it seems plausible to write it as U=exp(i N) with some operator N, and to directly compute the quantity N. This is known as the #Magnus expansion, and has been rather well studied in the theory of differential equations in #mathematics . It has nice properties, in particular that even an inaccurate truncation of N still guarantees U to be unitary.
In recent years, the Magnus expansion has also received new attention in the field of scattering amplitudes in theoretical #physics . The present preprint concerns using Magnus expansions instead of Dyson series in perturbative quantum field theory. The series coefficients are distinct from Feynman integrals, but they have an analogous graphical representation, using retarded propagators in place of Feynman propagators, and slightly different #combinatorics on the corresponding #graphs . The paper contains excessively many examples which clearly indicate the general formulas, which are then stated (but not proved). Besides the actual application to future physics calculations, I think it could be an interesting project in #combinatorics to prove all these formulas, and relate them to the existing mathematical literature. https://arxiv.org/abs/2512.05017

#paperOfTheDay is "The Magnus expansion in relativistic quantum field theory" from 2025.
One of the well-established ways to do computations in #quantumFieldTheory is perturbation theory in terms of #FeynmanIntegral s. From an analytic perspective, these integrals are the coefficients of a Dyson series, which is a power-series expansion of the time-evolution operator U (especially the S-matrix is a special case of U). This operator U is supposed to be unitary, so it seems plausible to write it as U=exp(i N) with some operator N, and to directly compute the quantity N. This is known as the #Magnus expansion, and has been rather well studied in the theory of differential equations in #mathematics . It has nice properties, in particular that even an inaccurate truncation of N still guarantees U to be unitary.
In recent years, the Magnus expansion has also received new attention in the field of scattering amplitudes in theoretical #physics . The present preprint concerns using Magnus expansions instead of Dyson series in perturbative quantum field theory. The series coefficients are distinct from Feynman integrals, but they have an analogous graphical representation, using retarded propagators in place of Feynman propagators, and slightly different #combinatorics on the corresponding #graphs . The paper contains excessively many examples which clearly indicate the general formulas, which are then stated (but not proved). Besides the actual application to future physics calculations, I think it could be an interesting project in #combinatorics to prove all these formulas, and relate them to the existing mathematical literature. https://arxiv.org/abs/2512.05017

#paperOfTheDay is "On the Local Borel transform of perturbation theory" from 2010. This article is about perturbative #QuantumFieldTheory , where one computes the terms of the perturbation series in terms of #FeynmanIntegral s. It is well known that the number of Feynman diagrams grows factorially with the loop order, and also there are individual classes of diagrams whose integral grows factorially. Since this is a growth rate faster than exponentially, it implies that the perturbation series does not converge.
For scalar quantum field theories in 2 or even 3 dimensions, this situation is relatively well under control because the second of the two effects is not present. In 4 dimensions, it is more tricky. It was established in the 1970s that at least the two effects do not conspire to give factorial-squared growth, but still, bounds were relatively crude. The present paper aims to give numerically more tight bounds of the magnitude of correlation functions depending on loop order, number of legs, or external momenta.
Notice that this statement is different from "Borel resummability": The latter asks whether the Borel transform is finite and not too quickly growing on the entire positive line, whereas the present article asks whether it has non-zero radius of convergence in the complex plane (e.g. bounded by singularities on the negative line).

Curiously, none of the existing articles, not even this present one, establish a factorial *lower* bound, i.e. from what is proven, it would be conceivable that the perturbation series is actually convergent. However, all numerical data we have is suggestive of factorial growth.
#physics
https://link.springer.com/article/10.1007/s00220-009-0979-x