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On The Rank Of Generalized Order-Preserving Transformation Semigroups, Haytham Darweesh Mustafa Abusarri̇s, Gonca Ayik Jan 2023

On The Rank Of Generalized Order-Preserving Transformation Semigroups, Haytham Darweesh Mustafa Abusarri̇s, Gonca Ayik

Turkish Journal of Mathematics

For any two non-empty (disjoint) chains $X$ and $Y$, and for a fixed order-preserving transformation $\theta : Y \rightarrow X$, let $\mathcal{GO} (X,Y; \theta )$ be the generalized order-preserving transformation semigroup. Let $\mathcal{O}(Z)$ be the order-preserving transformation semigroup on the set $Z=X\cup Y$ with a defined order. In this paper, we show that $\mathcal{GO}(X,Y;\theta)$ can be embedded in $O(Z,Y)=\{\, \alpha\in \mathcal{O}(Z)\, :\, Z\alpha \subseteq Y\,\}$, the semigroup of order-preserving transformations with restricted range. If $\theta \in \mathcal{GO}(Y,X)$ is one-to-one, then we show that $\mathcal{GO}(X,Y; \theta)$ and $O(X, im (\theta))$ are isomorphic semigroups. If we suppose that $\left X \right =m$,\, …

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Szabos Algorithm And Applications, Michel Bertrand Djiadeu Ngaha, Salomon Joseph Mbatakou, Romain Nimpa Pefoukeu Jan 2023

Szabos Algorithm And Applications, Michel Bertrand Djiadeu Ngaha, Salomon Joseph Mbatakou, Romain Nimpa Pefoukeu

Turkish Journal of Mathematics

In this paper, Szabos algorithm is used as the main tool to find locally symmetric left invariant Riemannian metrics on some $4$-dimensional Lie groups. Locally symmetric left invariant Riemannian Lie groups constitute an important subclass of Riemannian Lie groups with zero-divergence Weyl-tensor the so-called C-manifolds. Some properties of the curvature operator of these 4-dimensional C-manifolds are studied.

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Perfect Fluid Spacetimes And $K$-Almost Yamabe Solitons, Krishnendu De, Uday Chand De, Aydin Gezer Jan 2023

Perfect Fluid Spacetimes And $K$-Almost Yamabe Solitons, Krishnendu De, Uday Chand De, Aydin Gezer

Turkish Journal of Mathematics

In this article, we presumed that a perfect fluid is the source of the gravitational field while analyzing the solutions to the Einstein field equations. With this new and creative approach, here we study $k$-almost Yamabe solitons and gradient $k$-almost Yamabe solitons. First, two examples are constructed to ensure the existence of gradient $k$-almost Yamabe solitons. Then we show that if a perfect fluid spacetime admits a $k$-almost Yamabe soliton, then its potential vector field is Killing if and only if the divergence of the potential vector field vanishes. Besides, we prove that if a perfect fluid spacetime permits a …

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Spin Structures On Generalized Real Bott Manifolds, Asli Güçlükan İlhan, Sabri̇ Kaan Gürbüzer, Semra Pamuk Jan 2023

Spin Structures On Generalized Real Bott Manifolds, Asli Güçlükan İlhan, Sabri̇ Kaan Gürbüzer, Semra Pamuk

Turkish Journal of Mathematics

In this paper, we give a necessary and sufficient condition for a generalized real Bott manifold to have a spin structure in terms of column vectors of the associated matrix. We also give an interpretation of this result to the associated acyclic $\omega$-weighted digraphs. Using this, we obtain a family of real Bott manifolds that does not admit spin structure.

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Struwe Compactness Results For A Critical $P-$Laplacian Equation Involving Critical And Subcritical Hardy Potential On Compact Riemannian Manifolds, Tewfik Ghomari, Youssef Maliki Jan 2023

Struwe Compactness Results For A Critical $P-$Laplacian Equation Involving Critical And Subcritical Hardy Potential On Compact Riemannian Manifolds, Tewfik Ghomari, Youssef Maliki

Turkish Journal of Mathematics

Let $(M,g)$ be a compact Riemannian manifold. In this paper, we prove Struwe-type decomposition formulas for Palais-Smale sequences of functional energies corresponding to the equation: \begin{equation*} \Delta_{g,p}u-\frac{h(x)}{(\rho_{x_{o}}(x))^{s}}\left u\right ^{p-2}u =f(x)\left u\right ^{p^{\ast}-2}u, \end{equation*} where $\Delta_{g,p} $ is the $p-$Laplacian operator, $p^*=\frac{np}{n-p}$, $0

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On Orthogonally Additive Band Operators And Orthogonally Additive Disjointness Preserving Operators, Bahri̇ Turan, Demet Tülü Jan 2023

On Orthogonally Additive Band Operators And Orthogonally Additive Disjointness Preserving Operators, Bahri̇ Turan, Demet Tülü

Turkish Journal of Mathematics

Let $M$ and $N$ be Archimedean vector lattices. We introduce orthogonally additive band operators and orthogonally additive inverse band operators from $M$ to $N$ and examine their properties. We investigate the relationship between orthogonally additive band operators and orthogonally additive disjointness preserving operators and show that under some assumptions on vector lattices $M$ or $N$, these two classes are the same. By using this relation, we show that if ${\mu }$ is a bijective orthogonally additive band operator (resp. orthogonally additive disjointness preserving operator) from $M$ into $N$ then ${\mu }^{-1}$:$N$${\rightarrow}$$M$ is an orthogonally additive band operator (resp. orthogonally additive …

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Spectral And Topological Properties Of Linear Operators On A Hilbert Space, Salah Mecheri, Aissa Nasli Bakir Jan 2023

Spectral And Topological Properties Of Linear Operators On A Hilbert Space, Salah Mecheri, Aissa Nasli Bakir

Turkish Journal of Mathematics

We introduce the class of $(M, k)$-quasi-$*$-paranormal operators on a Hilbert space $H$. This class extends the classes of $*$-paranormal and $k$-quasi-$*$-paranormal operators. An operator $T$ on a complex Hilbert space is called $(M, k)$-quasi-$*$-paranormal if there exists $M>0$ such that \begin{equation*} \sqrt{M}\left\Vert T^{k+2}x\right\Vert \left\Vert T^{k}x\right\Vert \geq \left\Vert T^{\ast }T^{k}x\right\Vert ^{2} \end{equation*} for all $x\in H.$ In the present article, we give operator matrix representation of a $(M, k)$-quasi-$*$-paranormal operator. The compactness, the invariant subspace, and some topological properties of this class of operators are studied. Several properties of this class of operators are also presented.

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Left-Definite Hamiltonian Systems And Corresponding Nested Circles, Eki̇n Uğurlu Jan 2023

Left-Definite Hamiltonian Systems And Corresponding Nested Circles, Eki̇n Uğurlu

Turkish Journal of Mathematics

This work aims to construct the Titchmarsh-Weyl $M(\lambda )-$theory for an even-dimensional left-definite Hamiltonian system. For this purpose, we introduce a suitable Lagrange formula and selfadjoint boundary conditions including the spectral parameter $\lambda $. Then we obtain circle equations having nesting properties. Using the intersection point belonging to all the circles we share a lower bound for the number of Dirichlet-integrable solutions of the system.

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Energy Decay And Blow-Up Of Solutions For A Class Of System Of Generalized Nonlinear Klein-Gordon Equations With Source And Damping Terms, Zeynep Sümeyye Çeli̇k, Şevket Gür, Erhan Pi̇şki̇n Jan 2023

Energy Decay And Blow-Up Of Solutions For A Class Of System Of Generalized Nonlinear Klein-Gordon Equations With Source And Damping Terms, Zeynep Sümeyye Çeli̇k, Şevket Gür, Erhan Pi̇şki̇n

Turkish Journal of Mathematics

In this work, we investigate generalized coupled nonlinear Klein-Gordon equations with nonlinear damping and source terms and initial-boundary value conditions, in a bounded domain. We obtain decay of solutions by use of Nakao inequality. The blow up of solutions with negative initial energy is also established.

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Σ-Ary, Minnesota State University Moorhead, Mathematics Department Jan 2023

Σ-Ary, Minnesota State University Moorhead, Mathematics Department

Math Department Newsletters

No abstract provided.

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On 2-Primitive Triangle Decompositions Of Cocktail Party Graphs, Ian P. Waddell Jan 2023

On 2-Primitive Triangle Decompositions Of Cocktail Party Graphs, Ian P. Waddell

Theses, Dissertations and Capstones

A decomposition of a graph Γ is a collection C of subgraphs, perhaps nonisomorphic, that partition the edges of Γ. Analogously, consider a group of truck drivers whose non-overlapping routes jointly cover all of the roads between a set of cities; that is, each road is traversed by precisely one driver. In this scenario, the cities are the vertices of the graph, the roads are the edges between vertices, and the drivers’ routes are the subgraphs in the decomposition. Given a graph H, we call C an H-decomposition of Γ if each subgraph in C is isomorphic to …

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On A New Subclass Of Biunivalent Functions Associated With The $(P,Q)$-Lucas Polynomials And Bi-Bazilevic Type Functions Of Order $\Rho+I\Xi$, Hali̇t Orhan, İbrahi̇m Aktaş, Hava Arikan Jan 2023

On A New Subclass Of Biunivalent Functions Associated With The $(P,Q)$-Lucas Polynomials And Bi-Bazilevic Type Functions Of Order $\Rho+I\Xi$, Hali̇t Orhan, İbrahi̇m Aktaş, Hava Arikan

Turkish Journal of Mathematics

Using $ (p, q) $-Lucas polynomials and bi-Bazilevic type functions of order $\rho +i\xi,$ we defined a new subclass of biunivalent functions. We obtained coefficient inequalities for functions belonging to the new subclass. In addition to these results, the upper bound for the Fekete-Szegö functional was obtained. Finally, for some special values of parameters, several corollaries were presented.

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Global Bifurcation Of Positive Solutions For A Class Of Superlinear First-Order Differential Systems, Lijuan Yang, Ruyun Ma Jan 2023

Global Bifurcation Of Positive Solutions For A Class Of Superlinear First-Order Differential Systems, Lijuan Yang, Ruyun Ma

Turkish Journal of Mathematics

We are concerned with the first-order differential system of the form $$\left\{ \begin{array}{ll} u'(t)+a(t)u(t)=\lambda b(t) f(v(t-\tau(t))), &t\in\mathbb{R},\\ v'(t)+a(t)v(t)=\lambda b(t)g(u(t-\tau(t))), &t\in\mathbb{R},\\ \end{array} \right. $$ where $\lambda\in\mathbb{R}$~is a parameter. $a,b\in C(\mathbb{R},[0,+\infty))$ are two $\omega$-periodic functions such that $\int_0^\omega a(t)\text{d}t>0$,~$\int_0^\omega b(t)\text{d}t>0$,~$\tau\in C(\mathbb{R},\mathbb{R})$ is an $\omega$-periodic function. The nonlinearities~$f,g\in C(\mathbb{R},(0,+\infty))$~are two nondecreasing continuous functions and satisfy superlinear conditions at infinity.~By using the bifurcation theory,~we will show the existence of an unbounded component of positive solutions, which is bounded in positive $\lambda$-direction.

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On The Properties Of Solutions For Nonautonomous Third-Order Stochastic Differential Equation With A Constant Delay, Ayman Mohammed Mahmoud, Doaa Ali Mohamed Bakhit Jan 2023

On The Properties Of Solutions For Nonautonomous Third-Order Stochastic Differential Equation With A Constant Delay, Ayman Mohammed Mahmoud, Doaa Ali Mohamed Bakhit

Turkish Journal of Mathematics

In this work, complete Lyapunov functionals (LFs) are constructed and used for the established conditions on the nonlinear functions appearing in the main equation, to guarantee stochastically asymptotically stable (SAS), uniformly stochastically bounded (USB) and uniformly exponentially asymptotically stable (UEAS) in probability of solutions to the nonautonomous third-order stochastic differential equation (SDE) with a constant delay as \begin{align*} \begin{split} \dddot{x}(t)&+a(t)f(x(t),\dot{x}(t))\ddot{x}(t)+b(t)\phi(x(t))\dot{x}(t) +c(t)\psi(x(t-r))\\&+g(t,x)\dot{\omega}(t)=p(t,x(t),\dot{x}(t),\ddot{x}(t)). \end{split} \end{align*} In Section 4, we give two numerical examples as an application to illustrate the results.

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Jackson-Type Theorem In The Weak $L_{1}$-Space, Rashid Aliev, Eldost Ismayilov Jan 2023

Jackson-Type Theorem In The Weak $L_{1}$-Space, Rashid Aliev, Eldost Ismayilov

Turkish Journal of Mathematics

The weak $L_{1}$-space meets in many areas of mathematics. For example, the conjugate functions of Lebesgue integrable functions belong to the weak $L_{1}$-space. The difficulty of working with the weak $L_{1}$-space is that the weak $L_{1}$-space is not a normed space. Moreover, infinitely differentiable (even continuous) functions are not dense in this space. Due to this, the theory of approximation was not produced in this space. In the present paper, we introduced the concept of the modulus of continuity of the functions from the weak $L_{1}$-space, studied its properties, found a criterion for convergence to zero of the modulus of …

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On The Measure Of Noncompactness In $L_P(\Mathbb{R}^+)$ And Applications To A Product Of $N$-Integral Equations, Mohamed M. A. Metwali, Vishnu Narayan Mishra Jan 2023

On The Measure Of Noncompactness In $L_P(\Mathbb{R}^+)$ And Applications To A Product Of $N$-Integral Equations, Mohamed M. A. Metwali, Vishnu Narayan Mishra

Turkish Journal of Mathematics

In this article, we prove a new compactness criterion in the Lebesgue spaces $L_p({\mathbb{R}}^+), 1 \leq p < \infty$ and use such criteria to construct a measure of noncompactness in the mentioned spaces. The conjunction of that measure with the Hausdroff measure of noncompactness is proved on sets that are compact in finite measure. We apply such measure with a modified version of Darbo fixed point theorem in proving the existence of monotonic integrable solutions for a product of $n$-Hammerstein integral equations $n\geq 2$.

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Global Regularity For The 3d Axisymmetric Incompressible Hall-Mhd System With Partial Dissipation And Diffusion, Meilin Jin, Quansen Jiu, Huan Yu Jan 2023

Global Regularity For The 3d Axisymmetric Incompressible Hall-Mhd System With Partial Dissipation And Diffusion, Meilin Jin, Quansen Jiu, Huan Yu

Turkish Journal of Mathematics

In this paper, we study the Cauchy problem for the 3D incompressible axisymmetric Hall-MHD system with horizontal velocity dissipation and vertical magnetic diffusion.We obtain a unique global smooth solution of which in the cylindrical coordinate system the swirl velocity fields, the radial and the vertical components of the magnetic fields are trivial. This type of solution has been studied for the MHD system in [17][16] and [15] and for the Hall-MHD system with total dissipation and diffusion in [11]. Some new and fine estimates are obtained in this paper to overcome the difficulties raised from the Hall term and the …

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Einstein's Model Of "The Movement Of Small Particles In A Stationary Liquid" Revisited: Finite Propagation Speed, Akif Ibragimov, Zeev Sobol, Isanka Hevage Jan 2023

Einstein's Model Of "The Movement Of Small Particles In A Stationary Liquid" Revisited: Finite Propagation Speed, Akif Ibragimov, Zeev Sobol, Isanka Hevage

Turkish Journal of Mathematics

The aforementioned celebrated model, though a breakthrough in Stochastic processes and a great step toward the construction of the Brownian motion, leads to a paradox: infinite propagation speed and violation of the 2nd law of thermodynamics. We adapt the model by assuming the diffusion matrix is dependent on the concentration of particles, rather than constant it was up to Einstein, and prove a finite propagation speed under the assumption of a qualified decrease of the diffusion for small concentrations. The method involves a nonlinear degenerated parabolic PDE in divergent form, a parabolic Sobolev-type inequality, and the Ladyzhenskaya-Ural'tseva iteration lemma.

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On $\Gamma$-Hypersemigroups, Niovi Kehayopulu Jan 2023

On $\Gamma$-Hypersemigroups, Niovi Kehayopulu

Turkish Journal of Mathematics

The results on $\Gamma$-hypersemigroups are obtained either as corollaries of corresponding results on $\vee e$ or $poe$-semigroups or on the line of the corresponding results on $le$-semigroups. It has come to our attention that Theorem 3.22 in [4] cannot be obtained as corollary to Theorem 2.2 of the same paper as for a $\Gamma$-hypersemigroup, $({\cal P}^*(M),\Gamma,\subseteq)$ is a $\vee e$-semigroup and not an $le$-semigroup. Also on p. 1850, l. 12 in [4], the "$le$-semigroup" should be changed to "$\vee e$-semigroup". In the present paper we prove Theorems 3.26 and 3.28 stated without proof in [4]. On this occasion, some further …

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On Bell Based Appell Polynomials, Zeynep Özat, Mehmet Ali̇ Özarslan, Bayram Çeki̇m Jan 2023

On Bell Based Appell Polynomials, Zeynep Özat, Mehmet Ali̇ Özarslan, Bayram Çeki̇m

Turkish Journal of Mathematics

Recently, several Bell based polynomials such as Bernoulli, Euler, Genocchi and Apostol versions were defined and investigated. The main aim of this paper is to introduce the general family of Bell based Appell polynomials, which includes many new members in addition to the existing ones, and to investigate their properties including determinantal representation, recurrence relation, derivative formula, addition formula, shift operators and differential equation. Furthermore, we introduce 2-iterated Bell-Appell polynomials and investigate their similar properties. With the help of this 2-iterated family, we also obtain the closed form summation formulae between the usual and the generalized versions of the Bell …

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On $K$-Generalized Lucas Sequence With Its Triangle, Abdullah Açikel, Amrouche Said, Hacene Belbachir, Nuretti̇n Irmak Jan 2023

On $K$-Generalized Lucas Sequence With Its Triangle, Abdullah Açikel, Amrouche Said, Hacene Belbachir, Nuretti̇n Irmak

Turkish Journal of Mathematics

In this paper, we investigate several identities of $k$-generalized Lucas numbers with $k$-generalized Fibonacci numbers. We also establish a link between generalized $s$-Lucas triangle and bi$^{s}$nomial coefficients given by the coefficients of the development of a power of $(1+x+x^{2}+\cdots+x^{s}),$ with $s \in \mathbb{N}.$

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On The Geometry Of Nearly Trans-Sasakian Manifolds, Aligadzhi R. Rustanov, Tatiana L. Melekhina, Svetlana V. Kharitonova Jan 2023

On The Geometry Of Nearly Trans-Sasakian Manifolds, Aligadzhi R. Rustanov, Tatiana L. Melekhina, Svetlana V. Kharitonova

Turkish Journal of Mathematics

The geometry of nearly trans-Sasakian manifolds is researched in this paper. The complete group of structural equations and the components of the Lee vector on the space of the associated $G$-structure are obtained for such manifolds. Conditions are found under which a nearly trans-Sasakian structure is a trans-Sasakian, a cosymplectic, a closely cosymplectic, a Sasakian structure or a Kenmotsu structure. The conditions are obtained when the nearly trans-Sasakian structure is a special generalized Kenmotsu structure of the second kind. A complete classification of nearly trans-Sasakian manifolds is obtained, i.e. it is proved that a nearly trans-Sasakian manifold is either a …

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A Generalization Of The Notion Of Helix, Pascual Lucas, Jose Antonio Ortega-Yagues Jan 2023

A Generalization Of The Notion Of Helix, Pascual Lucas, Jose Antonio Ortega-Yagues

Turkish Journal of Mathematics

In this paper we generalize the notion of helix in the three-dimensional Euclidean space, which we define as that curve $\alpha$ for which there is an $F$-constant vector field $W$ along $\alpha$ that forms a constant angle with a fixed direction $V$ (called an axis of the helix). We find the natural equation and the geometric integration of helices $\alpha$ where the $F$-constant vector field $W$ is orthogonal to its axis.

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Polynomials Taking Integer Values On Primes In A Function Field, Tuangrat Chaichana, Vichian Laohakosol, Rattiya Meesa, Boonrod Yuttanan Jan 2023

Polynomials Taking Integer Values On Primes In A Function Field, Tuangrat Chaichana, Vichian Laohakosol, Rattiya Meesa, Boonrod Yuttanan

Turkish Journal of Mathematics

Let $\mathbb{F}_q[x]$ be the ring of polynomials over a finite field $\mathbb{F}_q$ and $\mathbb{F}_q(x)$ its quotient field. Let $\mathbb{P}$ be the set of primes in $\mathbb{F}_q[x]$, and let $\mathcal{I}$ be the set of all polynomials $f$ over $\mathbb{F}_q(x)$ for which $f(\mathbb{P})\subseteq\mathbb{F}_q[x]$. The existence of a basis for $\mathcal{I}$ is established using the notion of characteristic ideal; this shows that $\mathcal{I}$ is a free $\mathbb{F}_q[x]$-module. Through localization, explicit shapes of certain bases for the localization of $\mathcal{I}$ are derived, and a well-known procedure is described as to how to obtain explicit forms of some bases of $\mathcal{I}$.

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The Generalized Lyapunov Function As Ao’S Potential Function: Existence In Dimensions 1 And 2, Haoyu Wang, Wenqing Hu, Xiaoliang Gan, Ping Ao Jan 2023

The Generalized Lyapunov Function As Ao’S Potential Function: Existence In Dimensions 1 And 2, Haoyu Wang, Wenqing Hu, Xiaoliang Gan, Ping Ao

Mathematics and Statistics Faculty Research & Creative Works

By using Ao's decomposition for stochastic dynamical systems, a new notion of potential function has been introduced by Ao and his collabora-tors recently. We show that this potential function agrees with the generalized Lyapunov function of the deterministic part of the stochastic dynamical sys-tem. We further prove the existence of Ao's potential function in dimensions 1 and 2 via the solution theory of first-order partial differential equations. Our framework reveals the equivalence between Ao's potential function and Lyapunov function, the latter being one of the most significant central notions in dynamical systems. Using this equivalence, our existence proof can also …

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Three Solutions For Discrete Anisotropic Kirchhoff-Type Problems, Martin Bohner, Giuseppe Caristi, Ahmad Ghobadi, Shapour Heidarkhani Jan 2023

Three Solutions For Discrete Anisotropic Kirchhoff-Type Problems, Martin Bohner, Giuseppe Caristi, Ahmad Ghobadi, Shapour Heidarkhani

Mathematics and Statistics Faculty Research & Creative Works

In this article, using critical point theory and variational methods, we investigate the existence of at least three solutions for a class of double eigenvalue discrete anisotropic Kirchhoff-type problems. An example is presented to demonstrate the applicability of our main theoretical findings.

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Nilary Group Rings And Algebras, Omar Al-Mallah, Gary Birkenmeier, Hafedh Alnogashi Jan 2023

Nilary Group Rings And Algebras, Omar Al-Mallah, Gary Birkenmeier, Hafedh Alnogashi

Turkish Journal of Mathematics

A ring $A$ is (principally) nilary, denoted (pr-)nilary, if whenever $XY=0,$ then there exists a positive integer $n$ such that either $X^n=0$ or $Y^n=0$ for all (principal) ideals $X$, $Y$ of $A$. We determine necessary and/or sufficient conditions for the group ring $A[G]$ to be (principally) nilary in terms of conditions on the ring $A$ or the group $G$. For example, we show that: (1) If $A[G]$ is (pr-)nilary, then $A$ is (pr-)nilary and either $G$ is prime or the order of each finite nontrivial normal subgroup of $G$ is nilpotent in $A$. (2) Assume that $G$ is finite. Then …

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On The Cohomology Of Solvable Leibniz Algebras, Jorg Feldvoss, Friedrich Wagemann Jan 2023

On The Cohomology Of Solvable Leibniz Algebras, Jorg Feldvoss, Friedrich Wagemann

University Faculty and Staff Publications

This paper is a sequel to J. Feldvoss and F. Wagemann: On Leibniz cohomology, (2021), where we mainly consider semisimple Leibniz algebras. It turns out that the analogue of the Hochschild-Serre spectral sequence for Leibniz cohomology cannot be applied to many ideals, and therefore this spectral sequence seems not to be applicable for computing the cohomology of non-semi-simple Leibniz algebras. The main idea of the present paper is to use similar tools as developed by Farnsteiner for Hochschild cohomology (see R. Farnsteiner: On the cohomology of associative algebras and Lie algebras (1987) and R. Farnsteiner: On the vanishing of homology …

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Calculus I Companion, Beth Campbell Hetrick, Kimberly R. Spayd Jan 2023

Calculus I Companion, Beth Campbell Hetrick, Kimberly R. Spayd

Open Educational Resources

A course pack for supplementing Calculus 1 with algebra, geometry, trigonometry, and precalculus topics, including reading material, activities, and practice problems assembled from various OER texts.

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Part 3: James Clerk Maxwell's Zoetrope, Daniel Silver Jan 2023

Part 3: James Clerk Maxwell's Zoetrope, Daniel Silver

A Five-Minute History of Knot Theory

James Clerk Maxwell attempted to understand how three rings of ether would pass through each other. On October 6, 1868, he wrote to Kelvin: "[Helmholtz's] 3 rings do as 2 rings in his own paper that is those in front expand and go slower those behind contract and when small go faster and thread through the others. I drew 3 to make the motion more slow and visible, not that I have solved the case of 3 rings more than to get a rough notion about this case and to make the sum of the three areas [constant] I have …

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