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Subjects -> MATHEMATICS (Total: 1013 journals)     - APPLIED MATHEMATICS (92 journals)    - GEOMETRY AND TOPOLOGY (23 journals)    - MATHEMATICS (714 journals)    - MATHEMATICS (GENERAL) (45 journals)    - NUMERICAL ANALYSIS (26 journals)    - PROBABILITIES AND MATH STATISTICS (113 journals) MATHEMATICS (714 journals)            First | 1 2 3 4
 Showing 601 - 538 of 538 Journals sorted alphabetically Research in the Mathematical Sciences Research Journal of Pure Algebra       (Followers: 1) Researches in Mathematics Results in Control and Optimization Results in Mathematics Results in Nonlinear Analysis Review of Symbolic Logic       (Followers: 2) Reviews in Mathematical Physics       (Followers: 1) Revista Baiana de Educação Matemática Revista Bases de la Ciencia Revista BoEM - Boletim online de Educação Matemática Revista Colombiana de Matemáticas       (Followers: 1) Revista de Ciencias Revista de Educación Matemática Revista de la Escuela de Perfeccionamiento en Investigación Operativa Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas Revista de Matemática : Teoría y Aplicaciones       (Followers: 1) Revista Digital: Matemática, Educación e Internet Revista Electrónica de Conocimientos, Saberes y Prácticas Revista Integración : Temas de Matemáticas Revista Internacional de Sistemas Revista Latinoamericana de Etnomatemática Revista Latinoamericana de Investigación en Matemática Educativa Revista Matemática Complutense Revista REAMEC : Rede Amazônica de Educação em Ciências e Matemática Revista SIGMA Ricerche di Matematica RMS : Research in Mathematics & Statistics Royal Society Open Science       (Followers: 7) Russian Journal of Mathematical Physics Russian Mathematics Sahand Communications in Mathematical Analysis Sampling Theory, Signal Processing, and Data Analysis São Paulo Journal of Mathematical Sciences Science China Mathematics       (Followers: 1) Science Progress       (Followers: 1) Sciences & Technologie A : sciences exactes Selecta Mathematica       (Followers: 1) SeMA Journal Semigroup Forum       (Followers: 1) Set-Valued and Variational Analysis SIAM Journal on Applied Mathematics       (Followers: 11) SIAM Journal on Computing       (Followers: 11) SIAM Journal on Control and Optimization       (Followers: 18) SIAM Journal on Discrete Mathematics       (Followers: 8) SIAM Journal on Financial Mathematics       (Followers: 3) SIAM Journal on Mathematics of Data Science       (Followers: 1) SIAM Journal on Matrix Analysis and Applications       (Followers: 3) SIAM Journal on Optimization       (Followers: 12) Siberian Advances in Mathematics Siberian Mathematical Journal Sigmae SILICON SN Partial Differential Equations and Applications Soft Computing       (Followers: 7) Statistics and Computing       (Followers: 14) Stochastic Analysis and Applications       (Followers: 3) Stochastic Partial Differential Equations : Analysis and Computations       (Followers: 2) Stochastic Processes and their Applications       (Followers: 6) Stochastics and Dynamics       (Followers: 2) Studia Scientiarum Mathematicarum Hungarica       (Followers: 1) Studia Universitatis Babeș-Bolyai Informatica Studies In Applied Mathematics       (Followers: 1) Studies in Mathematical Sciences       (Followers: 1) Superficies y vacio Suska Journal of Mathematics Education       (Followers: 1) Swiss Journal of Geosciences       (Followers: 1) Synthesis Lectures on Algorithms and Software in Engineering       (Followers: 2) Synthesis Lectures on Mathematics and Statistics       (Followers: 1) Tamkang Journal of Mathematics Tatra Mountains Mathematical Publications Teaching Mathematics       (Followers: 10) Teaching Mathematics and its Applications: An International Journal of the IMA       (Followers: 4) Teaching Statistics       (Followers: 8) Technometrics       (Followers: 8) The Journal of Supercomputing       (Followers: 1) The Mathematica journal The Mathematical Gazette       (Followers: 1) The Mathematical Intelligencer The Ramanujan Journal The VLDB Journal       (Followers: 2) Theoretical and Mathematical Physics       (Followers: 7) Theory and Applications of Graphs Topological Methods in Nonlinear Analysis Transactions of the London Mathematical Society       (Followers: 1) Transformation Groups Turkish Journal of Mathematics Ukrainian Mathematical Journal Uniciencia Uniform Distribution Theory Unisda Journal of Mathematics and Computer Science Unnes Journal of Mathematics       (Followers: 1) Unnes Journal of Mathematics Education       (Followers: 2) Unnes Journal of Mathematics Education Research       (Followers: 1) Ural Mathematical Journal Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki Vestnik St. Petersburg University: Mathematics VFAST Transactions on Mathematics       (Followers: 1) Vietnam Journal of Mathematics Vinculum Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics       (Followers: 2) Water SA       (Followers: 1) Water Waves Zamm-Zeitschrift Fuer Angewandte Mathematik Und Mechanik       (Followers: 1) ZDM       (Followers: 2) Zeitschrift für angewandte Mathematik und Physik       (Followers: 2) Zeitschrift fur Energiewirtschaft Zetetike

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Similar Journals
 Russian Journal of Mathematical PhysicsJournal Prestige (SJR): 0.696 Citation Impact (citeScore): 1Number of Followers: 0      Subscription journal ISSN (Print) 1061-9208 - ISSN (Online) 1555-6638 Published by Springer-Verlag  [2469 journals]
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PubDate: 2022-09-01

• Triviality Theorem for Quasirepresentations in Reflexive Banach Spaces

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• Bifurcations of Essential Spectra Generated by a Small Non-Hermitian Hole.
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Abstract: The paper focuses on bifurcations that occur in the essential spectrum of certain non-Hermitian operators. We consider an eigenvalue problem for an elliptic differential operator in a multidimensional tube-like domain which is infinite along one dimension and can be bounded or unbounded in other dimensions. This self-adjoint eigenvalue problem is perturbed by a small hole cut out of the domain. The boundary of the hole is described by a non-Hermitian Robin-type boundary condition. Our main result consists in sufficient conditions ensuring that this singular and non-Hermitian perturbation results in discrete eigenvalues or resonances bifurcating either from the edge or from certain internal points of the essential spectrum of the unperturbed problem. The location of the bifurcating eigenvalues and resonances is described in terms of asymptotic expansions with respect to the small linear size of the hole. Several illustrative examples are discussed.
PubDate: 2022-09-01

• Hamiltonian Systems with a Functional Parameter in the Form of a Potential

Abstract: Let $$T=T(h)$$ be a real-analytic at $$0\in \mathbb{R}$$ function, $$T(0)=\pi$$ . Let $$H(x,y) = x^2+y^2 + {\cal O} _4(x,y)$$ be a real-analytic at $$0\in \mathbb{R} ^2$$ even function. We prove that there exists a real-analytic $$\varphi = \varphi (x) = {\cal O} _4(x)$$ even function such that any solution of the Hamiltonian equations $$\dot x = \partial_y(H + \varphi ), \quad \dot y = - \partial_x(H + \varphi )$$ near the origin on the energy level $$h>0$$ has the period $$T(h)$$ . We discuss motivations and possible generalizations of this result.
PubDate: 2022-09-01

• On the Instability of Monotone Traveling-Wave Solutions for a Generalized
Korteweg–de Vries–Burgers Equation

Abstract: For a generalized Korteweg–de Vries–Burgers equation with a flux function having two inflection points, we construct an example in which three different monotone structures of nonclassical (special) discontinuities, or undercompressive shocks, propagate at the same velocity. The linear stability of these structures is analyzed by applying the Evans function method. It is shown that some of them are neither linearly nor globally stable. The evolution of monotone unstable structures of undercompressive shocks is studied. The asymptotic behavior of the Evans function for large increments of perturbations is examined when the dissipation coefficient depends on the coordinate and time. A general approach to the construction of asymptotic expressions for the Evans function is described.
PubDate: 2022-09-01

• Degenerate Whitney Numbers of First and Second Kind of Dowling Lattices

Abstract: Dowling constructed the Dowling lattice $$Q_{n}(G)$$ , for any finite set with $$n$$ elements and any finite multiplicative group $$G$$ of order $$m$$ , which is a finite geometric lattice. He also defined the Whitney numbers of the first and second kind for any finite geometric lattice. These numbers for the Dowling lattice $$Q_{n}(G)$$ are the Whitney numbers of the first kind $$V_{m}(n,k)$$ and those of the second kind $$W_{m}(n,k)$$ , which are given by Stirling number-like relations. In this paper, by ‘degenerating’ such relations we introduce the degenerate Whitney numbers of the first kind and those of the second kind and investigate, among other things, generating functions, recurrence relations and various explicit expressions for them. As further generalizations of the degenerate Whitney numbers of both kinds, we also consider the degenerate $$r$$ -Whitney numbers of both kinds.
PubDate: 2022-09-01

• Solarity and Proximinality in Generalized Rational Approximation in Spaces
$$C(Q)$$ and $$L^p$$

Abstract: The present paper is concerned with problems of solarity, proximinality, approximative compactness, stability, and monotone path-connectedness in generalized rational approximation in spaces $$L^p$$ and $$C(Q)$$ . Efficiency of the new approaches and concepts used in the paper is illustrated by a number of examples. Solar properties of sets of generalized rational functions in the space $$C(Q)$$ are proved with the help of the new concept of $$\mathring B$$ -complete sets: a closed set $$M$$ is called $$\mathring B$$ -complete if, for each $$x\in X$$ and $$r>0$$ , the condition $$M_0:= \mathring B (x,r)\cap M)\ne\varnothing$$ implies that $${}\kern.2em\overline{\kern-.2em M}{} _0 \supset M\cap B(x,r)$$ . Existence and generalized approximative compactness in problems of generalized rational approximations in spaces $$C(Q)$$ and $$L^p$$ are proved using the new definition of algebraic completeness with the machinery of regular Deutsch convergence.
PubDate: 2022-09-01

• Functional-Difference Equations and Their Link with Perturbations of the
Mehler Operator

Abstract: This work deals with the spectral properties of the functional-difference equations that arise in a number of applications in the diffraction of waves and quantum scattering. Their link with some of the spectral properties of perturbations of the Mehler operator is addressed. The latter naturally arise in studies of functional-difference equations of the second order with a meromorphic potential which depend on a characteristic parameter. In particular, this kind of equations is frequently encountered with in the asymptotic treatment of eigenfunctions of the Robin Laplacians in wedge- or cone-shaped domains. The unperturbed selfadjoint Mehler operator is studied by means of the modified Mehler–Fock transform. Its resolvent and spectral measure are described. These results are obtained by use of some additional analysis applied to the known Mehler formulas. For a class of compact perturbations of this operator, sufficient conditions of existence and finiteness of the discrete spectrum are then discussed. Applications to the functional-difference equations are also addressed. An example of a problem leading to the study of the spectral properties for a functional-difference equation is considered. The corresponding eigenfunctions and characteristic values are found explicitly in this case.
PubDate: 2022-09-01

• Continuity Criterion for Locally Bounded Finite-Dimensional
Representations of Simply Connected Solvable Lie Groups

Abstract: We prove a continuity criterion for locally bounded finite-dimensional representations of simply connected solvable Lie groups.
PubDate: 2022-06-01
DOI: 10.1134/S1061920822020066

• Topological and Frame Properties of Certain Pathological $$C^*$$ -Algebras

Abstract: We introduce a classification of locally compact Hausdorff topological spaces with respect to the behavior of $$\sigma$$ -compact subsets and, relying on this classification, we study properties of the corresponding $$C^*$$ -algebras in terms of frame theory and the theory of $${\mathscr A}$$ -compact operators in Hilbert $$C^*$$ -modules; some pathological examples are constructed.
PubDate: 2022-06-01
DOI: 10.1134/S1061920822020029

• Bounds for the Kolmogorov Widths of the Sobolev Weighted Classes with
Conditions on the Zero and Highest Derivatives

Abstract: In this paper, order estimates for the Kolmogorov widths of sets defined by restrictions on the norm in the Sobolev weighted space $$W^r_{p_1}$$ and the weighted space $$L_{p_0}$$ are obtained.
PubDate: 2022-06-01
DOI: 10.1134/S106192082202008X

• Geometry of the Singular Set of Hypersurfaces and the Eikonal Equation

Abstract: The paper is concerned with singularities of classical solutions to the eikonal equation. With this aim in view, we study the relation between the geometry of hypersurfaces and the set of singular points of its distance function from both sides of this hypersurface. We also give an algorithm for the construction of reflection caustics and compare the results and illustrations of mathematical simulation with real-world images.
PubDate: 2022-06-01
DOI: 10.1134/S1061920822020078

• New Integral Representations for the Maslov Canonical Operator on an
Isotropic Manifold with a Complex Germ

Abstract: The objective of our paper is to generalize the representation obtained in 2017 by S.Yu. Dobrokhotov, V.E. Nazaikinskii, and A.I. Shafarevich to the general case of the canonical operator on an isotropic manifold with complex germ.
PubDate: 2022-06-01
DOI: 10.1134/S1061920822020030

• Existence of Contrast Structures in a Problem with Discontinuous Reaction

Abstract: In the paper, a boundary value problem for a singularly perturbed reaction-diffusion-advection equation is considered in a two-dimensional domain in the case of discontinuous coefficients of reaction and advection, whose discontinuity occurs on a predetermined curve lying in the domain. It is shown that this problem has a solution with a sharp internal transition layer localized near the discontinuity curve. For this solution, an asymptotic expansion in a small parameter is constructed, and also sufficient conditions are obtained for the input data of the problem under which the solution exists. The proof of the existence theorem is based on the asymptotic method of differential inequalities. It is also shown that a solution of this kind is Lyapunov asymptotically stable and locally unique. The results of the paper can be used to create mathematical models of physical phenomena at the interface between two media with different characteristics, as well as for the development of numerical-analytical methods for solving singularly perturbed problems.
PubDate: 2022-06-01
DOI: 10.1134/S1061920822020042

• Index of Differential-Difference Operators on an Infinite Cylinder

Abstract: Differential-difference operators are considered on an infinite cylinder. The objective of the paper is to present an index formula for the operators in question. We define the operator symbol as a triple consisting of an internal symbol and conormal symbols on plus and minus infinity. The conormal symbols are families of operators with a parameter and periodic coefficients. Our index formula contains three terms: the contribution of the internal symbol on the base manifold, expressed by an analog of the Atiyah–Singer integral, the contributions of the conormal symbols at infinity, described in terms of the $$\eta$$ -invariant, and also the third term, which also depends on the conormal symbol. The result thus obtained generalizes the Fedosov–Schulze–Tarkhanov formula.
PubDate: 2022-06-01
DOI: 10.1134/S1061920822020091

• Inverse Spectral Problem for Band Operators and Their Sparsity Criterion
In Terms of Inverse Problem Data

Abstract: We consider certain classes of operators generated by infinite band matrices (called band operators). They can be applied to the integration via the Lax pair formalism of nonlinear dynamical systems (e.g., Volterra type lattices) by using the inverse problem theory, in particular, by the inverse spectral problem method. A key role in this method is played by the moments of the Weyl matrix of a given band operator, which are used for unique reconstruction of the latter. These band operators have a special sparse structure, namely, they contain only two nonzero diagonals. We find a description of their sparsity in terms of these moments.
PubDate: 2022-06-01
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• Asymptotics of the Solution of the Cauchy Problem with Localized Initial
Conditions for a Wave Type Equation with Time Dispersion. I. Basic
Structures

Abstract: In many physical problems describing the propagation of waves, there are dispersion effects, both spatial and temporal. In these cases, the equations describing these problems and relating the temporal frequency and spatial momenta turn out to be inhomogeneous with respect to these variables. At the same time, these relations are even not polynomial. An example with time dispersion is given by Maxwell’s equations in a situation with rapidly changing electric and magnetic fields; then the second time derivative $$t$$ is replaced by a function of this derivative (a pseudodifferential operator). In the paper, asymptotic formulas are constructed for the solution of the Cauchy problem with localized initial data for the equation $$g^2\bigl(-ih\frac{\partial}{\partial t}\bigr)u=-h^2\langle \nabla,\, c^2(x)\nabla\rangle u$$ with variable speed and a small parameter characterizing fast oscillations of the propagating waves. One of the main considerations in use is that the constructive asymptotics for the solution of problems of this type are represented by functions given parametrically, and the corresponding parameters are the natural coordinates on the Lagrangian manifolds defining these asymptotics. Here, in contrast to problems with spatial dispersion, the corresponding Lagrangian manifold is defined in the phase space containing the time $$t$$ and the corresponding conjugate momentum coordinate, the frequency $$\omega$$ .
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DOI: 10.1134/S1061920822020017

• Continuity Criterion for Locally Bounded Automorphisms of Central
Extensions of Perfect Lie Groups with Discrete Center

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• On a Nonlinear Nonlocal Parabolic Problem

Abstract: We consider the Cauchy problem for a parabolic equation with a Ё-Laplacian or a general second-order quasilinear equation with boundary conditions of the Bitsadze–Samarskii type. We prove that at least one generalized solution of such problem exists.
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• Integrable Hamiltonian Systems on the Symplectic Realizations of
$$\textbf{e}(3)^*$$

Abstract: The phase space of a gyrostat with a fixed point and a heavy top is the Lie–Poisson space $$\textbf{e}(3)^*\cong \mathbb{R}^3\times \mathbb{R}^3$$ dual to the Lie algebra $$\textbf{e}(3)$$ of the Euclidean group $$E(3)$$ . One has three naturally distinguished Poisson submanifolds of $$\textbf{e}(3)^*$$ : (i) the dense open submanifold $$\mathbb{R}^3\times \dot{\mathbb{R}}^3\subset \textbf{e}(3)^*$$ which consists of all $$4$$ -dimensional symplectic leaves ( $$\vec{\Gamma}^2>0$$ ); (ii) the $$5$$ -dimensional Poisson submanifold of $$\mathbb{R}^3\times \dot{\mathbb{R}}^3$$ defined by $$\vec{J}\cdot \vec{\Gamma} = \mu \vec{\Gamma}$$ ; (iii) the $$5$$ -dimensional Poisson submanifold of $$\mathbb{R}^3\times \dot{\mathbb{R}}^3$$ defined by $$\vec{\Gamma}^2 = \nu^2$$ , where $$\dot{\mathbb{R}}^3:= \mathbb{R}^3\backslash \{0\}$$ , $$(\vec{J}, \vec{\Gamma})\in \mathbb{R}^3\times \mathbb{R}^3\cong \textbf{e}(3)^*$$ and $$\nu < 0$$ , $$\mu$$ are some fixed real parameters. Using the $$U(2,2)$$ -invariant symplectic structure of Penrose twistor space we find full and complete $$E(3)$$ -equivariant symplectic realizations of these Poisson submanifolds which are $$8$$ -dimensional for (i) and $$6$$ -dimensional for (ii) and (iii). As a consequence of the above, Hamiltonian systems on $$\textbf{e}(3)^*$$ lift to Hamiltonian systems on the above symplectic realizations. In this way, after lifting the integrable cases of a gyrostat with a fixed point and of a heavy top, we obtain a large family of integrable Hamiltonian systems on the phase spaces defined by these symplectic realizations.
PubDate: 2022-03-01
DOI: 10.1134/S1061920822010095

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Email: journaltocs@hw.ac.uk
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# Execution Operators

Domains:

PHP supports one execution operator: backticks (). Note that these are not single-quotes! PHP will attempt to execute the contents of the backticks as a shell command; the output will be returned (i.e., it won't simply be dumped to output; it can be assigned to a variable). Use of the backtick operator is identical to shell_exec().

<?php
$output = ls -al; echo "<pre>$output</pre>";
?>

Note:

The backtick operator is disabled when safe mode is enabled or shell_exec() is disabled.

Note:

Unlike some other languages, backticks have no special meaning within double-quoted strings.

See also the manual section on Program Execution functions, popen() proc_open(), and Using PHP from the commandline.

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# Lecture Series

These lecture series are coordinated by Missing in Architecture, a platform for research and collaboration between architects, educators, students, local communities, councils and everyone in between. We want to promote creativity, action and equality within the profession and we are interested in filling the gaps in architecture that we care about.

The lectures are through which architecture is examined through a wider lens with themes relating to gender, disability, equality + diversity, education, the environment and the future for practice,


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# Interesting Properties of Matrix Norms and Singular Values

$\DeclareMathOperator*{\argmax}{arg\,max}$

Matrix norms and singular values have special relationships. Before I forget about them, I’ll summarized them in this post.

### Definitions

• Schatten p-Norm

The Schatten p-Norm is defined as the following.1

• Nuclear Norm

The nuclear norm of a matrix is defined as a special case of the Schatten p-norm where $p=1$.

• Frobenius Norm

• Matrix p-Norm

Matrix p-norm is defined as

In another word, matrix p-Norm is defined as the largest scalar that you can get for a unit vector $e$.

• Spectral Norm

Largest singular value of a matrix $\sigma_1(X)$.

Special case of the matrix p-norm where $p=2$ when the matrix $X$ is positive semi-definite. For negative definite matrix, the matrix 2-norm is not necessarily the largest norm.

### Lemmas

1. $A \in \mathbf{S}^n \; tr(A) = \sum_i^n \lambda_i =\lVert A \rVert_{S_1}$

Trace of a symmetric matrix $A$ is equal to the sum of eigen values. Let A be a symmetric matrix $A \in \mathcal{S}^{n}$. Then there exists a orthogonal matrix $U$ and diagonal matrix $\Lambda$ such that $A = U \Lambda U^T$.

\begin{align} tr(A) & = tr(A^T)\\ & = \sum_i^n e_i^T A^T e_i \\ & = \sum_i^n e_i^T U^T \Lambda U e_i \\ & = \sum_i^n \sum_j^n u_{ji}^T \lambda_j u_{ji} \\ & = \sum_j^n \lambda_j \sum_i u{ji}^2\\ & = \sum_j^n \lambda_j \end{align}

Where $U = [ u_1, u_2, u_3, \dots u_n]$. We used the fact that $u_i^T u_i = 1$.

2. $A \in \mathbf{S}_+^n \; tr(A) =\sum_i^n \lvert \sigma_i \rvert = |A|_{S_1}$

Trace of a positive semi-definite matrix $A$ is equal to the L1 norm of singular values, or is equal to the Schatten 1-Norm (Nuclear Norm).

This is the direct extension of Lemma 1.

\begin{align} tr(A) & = \sum_i^n \lambda_i\\ & = \sum_i^n \sigma_i\\ & = \sum_i^n \| \sigma_i \|\\ & = \|A\|_{S_1} \end{align}

Since the L1 norm of singular values enforce sparsity on the matrix rank, yhe result is used in many application such as low-rank matrix completion and matrix approximation.

3. $\lVert X\rVert_F = \sqrt{ \sum_i^n \sigma_i^2 } = \lVert X\rVert_{S_2}$

Frobenius norm of a matrix is equal to L2 norm of singular values, or is equal to the Schatten 2 norm.

\begin{align} \|X\|_F & = \sqrt{ tr(X^T X) }\\ & = \sqrt{ tr(V \Sigma U^T U \Sigma V^T) }\\ & = \sqrt{ tr(V\Sigma^2 V^T)}\\ & = \sqrt{ \sum_i \sigma_i^2 }\\ & = \|X\|_{S_2} \end{align}
4. $\lVert A \rVert_1 = \max_j \sum_i^n \lvert a_{ij} \rvert$

L1 matrix norm of a matrix is equal to the maximum of L1 norm of a column of the matrix.

To begin with, the solution of L1 optimization usually occurs at the corner. If the function of interest is piece-wise linear, the extrema always occur at the corners. One can show this by showing that $\frac{|Ax|_1}{|x|_1}$ is Lipschitz except $x = 0$ and differentiate $\frac{|Ax|_1}{|x|_1}$ w.r.t. $x_i$. Then show that they are non-zero. Use the fact on line 1 and 2.

Finally, we can just compute the L1 norms of each column. Let’s reformulate the problem.

\begin{align} \|A\|_1 & = \sup_{x \neq 0} \frac{\|Ax\|_1}{\|x\|_1}\\ & = \max_{\|x\|_1 = 1} \sum_j \sum_i^n \lvert a_{ij}\rvert \lvert x_j \rvert\\ & = \max_j \sum_i \lvert a_{ij} \rvert \end{align}
5. $\lVert A \rVert_\infty = \max_i \sum_j^n \lvert a_{ij} \rvert$

The supremum occurs at the corner of the hypercube since infinity nomr of a vector is the absolute value of the largest element in it. Thus,

\begin{align} \|A\|_\infty & = \max \frac{\|Ax\|_\infty}{\|x\|_\infty}\\ & = \max_{\|x_j\| = 1} \|Ax\|_\infty \\ & = \max_i \|a_i^T\| \end{align}

Where $a_i^T$ is the $i$ th row of the matrix $A$.

1. http://en.wikipedia.org/wiki/Schatten_norm

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# Minimum vertex-weight directed spanning tree where the weight function depends on the tree

Given a directed graph $G=(V,E)$ and a node $r\in V$, I need to grow a tree $T$ rooted at $r$ that has a minimum weight and spans all reachable nodes in $G$.

The weight function assigns a non-negative weight to each node, which depends on the node's ancestors in $T$. Specifically, for some fixed sets of nodes $S_1, S_2, \dots, S_k \subseteq V$, the weight of node $v$ is the number of sets $S_i$ that contain $v$ and all its ancestors in $T$.

Any suggestion how to approach this problem?

• Do you mean ancestors?
– Joe
Sep 10, 2012 at 8:39
• Edited for clarity. Sep 12, 2012 at 22:32
• Interesting. Can you give some information about the background/motivation? Maybe there is another way to solve your problem. What are your constraints, e.g. runtime? The naive algorithm just tries all possible trees. The new formulation seems to imply that $T$ has to have directed paths from $r$ to all leaves (i.e. you can only use edges of $G$ in their original orientation); is that intended? Also, what have you tried?
– Raphael
Sep 13, 2012 at 6:24
• Are there any guarantees on how the function changes depending on the ancestors? In other words, do you know that it can only increase with more ancestors or some such? If so, a variant of Djikstra's might work or possibly dynamic programming.
– SamM
Sep 13, 2012 at 19:34


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### numbertheorist17's blog

By numbertheorist17, 4 years ago, ,

# Problem A

It is a necessary and sufficient condition that we have exactly 2 distinct values for x and y. If we have less than 2 distinct values for any variable, then there is no way to know the length of that dimension. If there are at least 3 distinct values for any variable, then that means more than 3 vertices lie on that dimension, which cannot happen since there can be at most 2 vertices in a line segment. The area, if it can be found, is just the difference of values of the x coordinates times the difference of values of the y coordinates.

Complexity: O(1)

Code: Solution

# Problem B

No matter what, we make |b1| operations to make a1 equal to b1. Once this is done, a2, a3, ... an = b1. Then no matter what, we must make |b2 - b1| operations to make a2 equal to b2. In general, to make ai = bi we need to make |bi - bi - 1| operations, so in total we make |b1| + |b2 - b1| + |b3 - b2| + ... + |bn - bn - 1| operations.

Complexity: O(n)

Code: Solution

# Problem C

Note that if there is an integer d so that the number of wi equal to d differs from the number of the given squares whose weight equals d, then the answer is automatically "NO". This can be easily checked by using a map for the wi and the weights of the squares and checking if the maps are the same. This step takes time.

Let d be an integer, and let D be the set of all i so that wi = d. Let W be the set of all special points (x, y) so that the weight of (x, y) is d. Note that W and D have the same number of elements. Suppose that i1 < i2 < ... < ik are the elements of D. Let (a, b) < (c, d) if a < c or a = c and b < d. Suppose that (x1, y1) < (x2, y2) < ... < (xk, yk) are the elements of W. Note that the point (xj, yj) has to be labeled by ij for 1 ≤ j ≤ k.

Now, each special point is labeled. It remains to check if this is a valid labeling. This can be done by taking an array of vectors. The vector arr[i] will denote the points with x-coordinate i. This vector can be easily made from the points given in O(n) time, and since the points are already labeled, arr[i][j] will denote the label for the point (i, j). Now, for all points (i, j), the point (i, j + 1) (if it is special) and the point (i + 1, j) (if it is special) must have a greater number than (i, j). This step takes a total of O(n) time.

Complexity:

Code: Solution

Bonus: Can you do this problem in O(n) time?

Comments: This problem was inspired by the representation theory of the group of permutations Sn (Representation theory of the Symmetric Group). Essential objects in the study of Sn are Young diagrams and standard Young tableau (Young Tableau). The weight of a point as defined by the problem is basically the same thing as the content of a square in a standard Young tableaux. If you have questions, feel free to message me.

# Problem D

Let us solve this problem using dynamic programming.

First let us reindex the trees by sorting them by x-coordinate. Let f(i, j, b1, b2) where we would like to consider the problem of if we only have trees i... j standing where b1 = 1 indicates that tree i - 1 falls right and b1 = 0 if it falls left and b2 = 1 indicates that tree j + 1 falls right and b2 = 0 if it falls left.

We start with the case that Wilbur chooses the left tree and it falls right. The plan is to calculate the expected length in this scenario and multiply by the chance of this case occurring, which is . We can easily calculate what is the farthest right tree that falls as a result of this and call it wi.

Then if wi >  = j this means the entire segment falls, from which the length of the ground covered by trees in i... j can be calculated. However, be careful when b2 = 0, as there may be overlapping covered regions when the tree j falls right but the tree j + 1 falls left.

If only wi < j, then we just consider adding the length of ground covered by trees i... wi falling right and add to the value of the subproblem f(wi + 1, j, 1, b2).

There is another interesting case where Wilbur chooses the left tree and it falls left. In this case we calculate the expected length and multiply by the chance of this occurring, which is . The expected length of ground covered by the trees here is just the length contributed by tree i falling left, which we must be careful calculating as there might be overlapping covered regions with the ith tree falling left and the i - 1th tree falling right. Then we also add the value of subproblem f(i + 1, j, 0, b2).

Doing this naively would take O(n3) time, but this can be lowered to O(n2) by precalculating what happens when tree i falls left or right.

We should also consider the cases that Wilbur chooses the right tree, but these cases are analogous by symmetry.

Complexity: O(n2)

Code: Solution

# Problem E

Solution 1: Suppose that s is a string in the query. Reverse s and the direction of all the moves that can be made on the table. Note that starting at any point that is part of a cycle, there is a loop and then edges that go out of the loop. So, for every point, it can be checked by dfs whether the s can be made by starting at that point by storing what is in the cycle.

Moreover, note that in the reversed graph, each point can only be a part of one cycle. Therefore, the total time for the dfs in a query is O(nm·SIGMA + |s|). This is good enough for q queries to run in time.

Complexity: where SIGMA = 10 is the number of distinct characters in the table, and si is the query string for the i th query.

Code: Solution

Solution 2 (Actually too slow, see comment by waterfalls below for more details): For each string s, dfs from every node that has in degree equal to 0 in the original graph. There will be a path which leads into a cycle after which anything in the cycle can be used any number of times in s. Only every node with in degree equal to 0 has to be checked because every path which leads to a cycle is part of a larger path which starts with a vertex of in degree 0 that leads into a cycle.

This solution is slower, but it works in practice since it is really hard for a string to match so many times in the table. Each query will take O(n2·m2 + si) time, but it is much faster in practice.

Complexity: where SIGMA = 10 is the number of distinct characters in the table, and si is the query string of the i th query.

• +113

 » 4 years ago, # |   0 Fast editorial!
 » 4 years ago, # |   0 Editorial was published before contest end.
•  » » 4 years ago, # ^ |   +61 It wasn't publish, it was saved as draft.
•  » » » 4 years ago, # ^ |   0 Great work :)
 » 4 years ago, # |   0 can someone plz explain problem statement of C..
•  » » 4 years ago, # ^ | ← Rev. 2 →   +21
•  » » 4 years ago, # ^ |   +5
 » 4 years ago, # | ← Rev. 2 →   +5 There's a typo I think in E's part, O(n2m2 + si)should be O(n2m2 + si)am I rigt? Anyway thanks for very-very fast editorial :) A good practice it is.
•  » » 4 years ago, # ^ |   0 Yeah. I saw that and fixed it.
 » 4 years ago, # |   +6 Bonus: Can you do this problem in O(n) time?Yes, yes I do :)http://codeforces.com/contest/596/submission/14283371
•  » » 4 years ago, # ^ |   0 omg o_O 363 lines
•  » » » 4 years ago, # ^ |   0 Well, the solution is only in the method solve() which is only 57 lines :)
•  » » 4 years ago, # ^ | ← Rev. 2 →   0 14282779 is mine O(n) solution. It's a bit shorter. :)
 » 4 years ago, # | ← Rev. 2 →   +3 How can the obtained labelling in Problem C be checked in O(n) time? (in O(nlogn) solution)
•  » » 4 years ago, # ^ |   0 check out my solution 14282779
•  » » 4 years ago, # ^ |   +5 Updated the editorial.
•  » » 4 years ago, # ^ |   0 We can do this without sorting. Observe that we can greedily place a certain w[i] to the "earliest slot" in a diagonal (y — x).There are only 200000 diagonals of the form y — x, so we can just store these "earliest slots" in an array. We can use either x-coordinate or y-coordinate to represent the earliest slot, and just use the fact that w = y — x to get the other coordinate.Therefore, to do this in O(n), just iterate array w, and if the earliest possible slot in the diagonal w[i] is available (i.e. its left and bottom adjacent positions are already filled), then we can proceed to fill the others. Otherwise, if the slot is not a valid point or its adjacents are not yet filled, there is no more solution and we print "NO".Here's my solution :)
 » 4 years ago, # |   0 I solved D problem at the end of contest. i forgot x == y , the left index == the right index. I am so sad.
 » 4 years ago, # |   +4 Test 76 on C is a winner. Who's idea was to add it?
•  » » 4 years ago, # ^ |   0 :( For 0,0 I've put position 1 instead of the first position in array w of 0...
•  » » 4 years ago, # ^ |   0 Could be someone's great hack added later to the main suit. Killed my solution for C :(
•  » » 3 years ago, # ^ |   0 seeing this conversation, i made a personal note to myself to specially not get caught on that test case. well, i did....
 » 4 years ago, # |   0 When we can see results?
•  » » 4 years ago, # ^ | ← Rev. 2 →   0 Results have been posted on blog
 » 4 years ago, # |   -6 For Problem B you can simulate the operations optimally using a BIT and you get a O(n * log(n)) time complexity. Here is my solution http://codeforces.com/contest/596/submission/14287599. (Unfortunately I didn't manage to get Accepted during contest due to integer overflow.)
•  » » 4 years ago, # ^ |   0 O(n) faster
 » 4 years ago, # |   +6 'Now, each special point is labeled. It remains to check if this is a valid labeling, which can easily be done in O(n) time.' — Can someone explane me checking it in O(n)? My solution consists BIT — O(n log n).
•  » » 4 years ago, # ^ |   0 I will update this soon.
•  » » » 4 years ago, # ^ |   0 Thanks. The tasks were great and interesting to me. Only I expected a little easier third and fourth, but this is also ok :)
•  » » 4 years ago, # ^ |   0 Updated the editorial.
•  » » » 3 years ago, # ^ | ← Rev. 2 →   -10 okay
 » 4 years ago, # |   0 This contest made me Newbie. I couldn't make a good result.
 » 4 years ago, # |   +28 I think the second solution to E is too slow. I don't think "it works in practice since it is really hard for a string to match so many times in the table" really matters, since we can just make all queries a single character that doesn't appear in the table at all (it then looks through the whole path). If there are a lot of degree 0 nodes (even O(n)), the complexity can be O(n2mq) which shouldn't work for n = m = q = 200. For example see this test: http://pastebin.com/qbRypHZC (basically a winding path with inputs in the first and last column).
•  » » 4 years ago, # ^ |   0 Wow. Well, I stand corrected.
 » 4 years ago, # |   0 For Problem C:This nlogn solution gots TLE, why ?solution: 14286507
•  » » 4 years ago, # ^ |   +27 The lower_bound function on sets takes O(n) time. You should do s.lower_bound(something) which takes O(log(n)) time not lower_bound(all(s),something) which would take O(n) time.
•  » » » 4 years ago, # ^ |   0 First time to know this, thank you :)
 » 4 years ago, # |   0 It was also possible to do C with a 2D binary indexed tree, by only allocating memory for parts of the tree that were nonzero. It took O(nlog^2 n) time, and the same amount of memory. It just barely fits in the time limit though. Here is the code.
 » 4 years ago, # |   0 Here's another idea for E. For each vertex in the graph, build jump hashes in powers of two. Then for any query, we can binary search from each vertex to see if the query string matches. The complexity should be something like O(qn2log(Σi|si|)).
•  » » 4 years ago, # ^ |   +5 This is an interesting idea. Thanks for sharing that!
 » 4 years ago, # |   0 Excuse me, I don't know why my solution was always runtime error on test 3, I have tried serveral times,but I can't find why it is wrong, this code can be run smoothly in my own computer, if you can help me, thx very much . my code 14310580 thx very much
•  » » 4 years ago, # ^ |   0 I solved it , thx~
 » 4 years ago, # |   +5 In problem C solution, I have problem in this part of the code. for (int i=0; i
•  » » 4 years ago, # ^ |   +5 No, it doesn't become like this because each point is added exactly once making a total of n points.
 » 4 years ago, # |   0 May I ask what is meaning of some variable in solution of Problem D? like int sz[2][MAX_N]; , I guess it used to save the preprocess value of sth , but i couldn't understand how it use.int nl=min(rig,lef+sz[0][lef]-1);, I also don't understand what's nl mean sorry for my poor English
•  » » 4 years ago, # ^ |   +6 So, the sz array is used to store the preprocessing. The value sz[0][i] says how many trees fall down if the i th tree falls down. The value sz[1][i] is the same thing if the i th tree falls right. The variables nl and nr are for the last tree that falls if the lef falls right and the last tree that falls if rig falls left, respectively. Note that these are bounded by rig and lef, respectively. So, you have to take the minimums.
•  » » » 4 years ago, # ^ |   0 Thank you so much!!! Fully understand
 » 4 years ago, # |   0 In problem C, many submissions cannot pass this test case: 5 0 0 0 1 0 2 1 0 1 1 0 1 0 2 -1
•  » » 2 years ago, # ^ |   0 It should be "NO", right?
 » 3 years ago, # |   0 in solution C, isn'it TLE when inputs are 100000 1 100000 2 100000 3 100000 .... 100000 100000 ....while (diag[a].size()<=b) diag[a].push_back(0); because of this line?
•  » » 3 years ago, # ^ |   0 Read the problem carefully again: Also, it is guaranteed that if some point (x, y) is present in the input, then all points (x', y'), such that 0  ≤  x' ≤ x and 0  ≤  y'  ≤  y, are also present in the input. So the test you constructed is invalid.Also read this comment.
•  » » » 3 years ago, # ^ |   0 Thanks for your explanation :)
 » 3 years ago, # |   0 can somebody explain problem E editorial


text:
## dimanche 1 juillet 2018

### Bac

Among the 4 documents that you have prepared, the jury makes you pick 2 subjects out of 4, without looking, then you choose 1 out of 2, after reading them.

For the exercise, they'll make you choose 1 out of 2 subjects, on different themes, among probabilities, sequences, statistics and functions.

1) You'll have 20 min to prepare the exercise that they'll give you, (on probabilities, sequences, functions or statistics) and you'll have 10 min to present it on the white board. (The subjects of the exercises of Baccalauréat DNL of the last 6 years : ).

2) When you present the second part of the oral (the document), introduce your subject make a plan and announce it (1st part, 2nd part), then develop itand as a conclusion give your personal opinion about the document. You'll have 5 min to present your document and then 5 min of questions about it (or about more general questions).

Do not bring any Powerpoint or any documents from home.
Do it all from memory, without learning by heart.
Talk about it freely, but with a structure (introduction, development, conclusion).

Baccalaureat : how are the marks given during the oral exam ?

(T. S or T. ES)
2) Wealth inequality (T. S or T. ES)
3) Fractals (T. S or T. ES)
4) Four physics riddles (T. S.)
5) Sex and marriage theorems (T. S. or T. ES)
6) Kakeya's needle problem (T. S.)
7) Blob Infinity (T. S.)
8) Hilbert hotel (T. S. or T. ES)
9) Futurama theorem (T. S. or T. ES)
10) Rubik's cube and the Enigma machine (Alan Turing) (T. S.)
11) Magnus effect (Physics) (T. S.)
12) Anti-gravity wheel (Physics) (T. S.)
13) Computer Science Unplugged. (T. S. or T. ES)
14) Conditional probability (T. S. or T. ES)
15) Monty Hall (T. S. or T. ES)

## vendredi 8 septembre 2017

### Wealth inequality

The content of this video is explained here :

# Why Do Americans Tolerate Extreme Wealth Inequality?

New data shows Americans haven’t a clue how stunningly massive the wealth gap in their country really is. An excellent video blowing up on YouTube depicts this disparity and explains how badly Americans tend to underestimate it. No matter how many statistics you’ve read on the subject, the video deserves six minutes of your time.

To summarize the video's data:
• The richest 20 percent of Americans hold 84 percent of the nation’s wealth, while the bottom 40 percent have less than 1 percent.
• Americans believe their country is much less inegalitarian than it really is.
• The top 1 percent have a bigger slice of the pie (40 percent) than nine of out 10 Americans believe the top 20 percent should have.
• While the top 1 percent command 40 percent of the wealth, the bottom 80 percent have a measly 7 percent.
• The top 1 percent “own half of the country’s stocks, bonds and mutual funds. The bottom 50 percent of Americans own only half a percent of these investments.”
• “The average worker needs to work more than a month to earn what the CEO makes in one hour.”
Putting it all together, the wealth distribution in the U.S. looks like this(note that the top 1 percent is, literally, off the chart) :

The video concludes with this call to arms:
We certainly don’t have to go all the way to socialism to find something that is fair for hard-working Americans. We don’t even have to achieve what most of us consider might be ideal. All we need to do is wake up and realize that the reality in this country is not at all what we think it is.
There’s the rub. If the 2.2 million+ viewers of the video were to expand, Gangnam style, to all 311 million Americans and everyone finally saw with clear eyes just how vast wealth inequality truly is in their country, things would change, right?

Maybe, but I seriously doubt it. Even leaving aside Republican resistance to measures that could dent the wealth gap, many Americans would likely remain opposed to policies that would be necessary to seriously address the problem. The apparently most efficient mechanism for redistributing the pie is a tax on wealth like the “solidarity tax” (impôt de solidarité sur la fortune) in France: citizens with a net worth surpassing 1 million Euros are asked to fork over 0.25 percent of the value of their property every year; the annual levy rises to 0.5 percent for fortunes over 3 million. Such a measure cannot be implemented in the United States: Article 1, section 9, clause 4 of the U.S. Constitution explicitly prohibits “direct” taxation. Should we try to amend the Constitution to permit direct taxes? Well, we could, but if the American experience is anything like that of France, the wealth tax will not actually have a significant redistributive effect, and the 1 percenters would move to Canada to keep their assets intact. That would smooth out the curve at the top without improving the lives of poor and middle class Americans a bit.

Short of a wealth tax, then, how could we bring the American distribution more in line with what Americans would like? Judging by the tenor of his State of the Union address, President Obama seems committed to chipping away at the inequalities in American society by expanding opportunities for preschool and raising the minimum wage, among other proposals. Policies like these may help reduce wealth inequality to some limited extent, but the problem is much too big and too entrenched in American society to resolve with small-bore solutions. Without fundamental changes to the income tax code eliminating “stealth” subsidies for corporations, further raising rates on the wealthy and hiking the estate tax (something Obama has given up on), there is little on the horizon that has a good chance of bridging the rich-poor rift.

Maybe we really do need a socialist revolution.

## mercredi 17 mai 2017

### Fractals

1º) Introduce the subject with a summary of what you are going to present.
2º) Present it with a few steps (What is it ? What are they good for ?)
3º) Conclude and give your personal opinion

# Fractal

A Sierpinski triangle, after 7 iterations.
The Mandelbrot set is a famous example of a fractal.
fractal is any pattern, that when seen as an image, produces a picture, which when zoomed into will still make the same picture. It can be cut into parts which look like a smaller version of the picture that was started with. The word fractal was made by Benoît Mandelbrot in 1975 from the Latin word fractus, which means "broken" or "fractured". A simple example is a tree that branches into smaller branches, and those branches into smaller branches and so on. Fractals are not only beautiful, but also have many practical applications.

## Contents

• 1Examples
• 1.1The Koch Curve
• 1.2Similarity dimension
• 1.3Koch snowflake
• 2Uses

## Examples

There are many types of fractals, made in a large variety of ways. One example is the Sierpinski triangle, where there are an infinite number of small triangles inside the large one. Another example is the Mandelbrot set, named for Benoît Mandelbrot. The Sierpinksi triangle is constructed using patterns, but the Mandelbrot set is based on an equation.
There are also many natural examples of fractals in nature including trees, snowflakes, some vegetables and coastlines.

### The Koch Curve

How to make the Koch Curve
The Koch Curve is a simple example of a fractal. First, start with part of a straight line - called a straight line segment. Cut the line into 3 same-sized pieces. Get rid of the middle of those pieces, and put in the top part of a triangle with sides which are the same length as the bit to cut out. We now have 4 line segments which are touching at the ends. We can now do what we just did to the first segment to each of the 4 bits. We can now do the same thing again and again to all the bits we end up with. We now do this forever and look at what we end up with.
The length of the Koch Curve is infinity, and the area of the Koch Curve is zero. This is quite strange. A line segment (with dimension 1) could have a length of 1, but it has an area of 0. A square of length 1 and width 1 (with dimension 2) will have area 1 and length of infinity.

### Similarity dimension

So, the Koch Curve seems to be bigger than something of dimension 1, and smaller than something of dimension 2. The idea of the similarity dimension is to give a dimension which gives a better idea of length or area for fractals. So, for a Koch Curve, we want a dimension between 1 and 2.
The Koch Curve can be cut into 4 pieces, each of which are ${\displaystyle {\frac {1}{3}}}$ of the size of the original.
The Koch Curve is one of the simplest fractal shapes, and so its dimension is easy to work out. Its similarity dimension and Hausdorff dimension are both the same. This is not true for more complex fractals.

### Koch snowflake

The Koch snowflake (or Koch star) is the same as the Koch curve, except it starts with an equilateral triangle instead of a line segment.

## Uses

Fractals have many applications e.g. in biology (lung, kidneys, heart rate variability, etc...), in earthquakes, in finance where it is related to the so called heavy tail distributions and in physics. This indicates that fractals should be studied to understand why fractals are so frequent in nature.
Some fractals exist only for artistic reasons, but others are very useful. Fractals are very efficient shapes for radio antennas and are used in computer chips to efficiently connect all the components. Also, coastlines can be thought of as fractals

## lundi 24 avril 2017

### Four physics riddles

1) Introduce the subject.
2) Summarize it shortly in 4 minutes (only what you can explain).
3) Conclude and give your personal opinion on the subject of open problems.

## lundi 20 mars 2017

### Hilbert hotel

1) Introduce the subject.
2) Summarize it shortly in 4 minutes. (Only what you can explain).

The sequence u(n) = n + 1 proves you can always find space for a new guest.
The sequence v(n) = 2n proves that there are as many even numbers than integers.

3) Conclude for example, by explaining why infinity is a surprising subject full of paradoxes. (Part of an infinite set can have the same number of elements than the whole set).

David Hilbert (1862-1943)

Imagine an ordinary hotel of 500 rooms, all of them occupied by some clients. You arrive there one evening and the receptionist tells you that there's no vacancy. You walk away sadly. There's no paradox about that.

Now, imagine a hotel in which there are infinitely many rooms, all of them occupied. Although the hotel is full, the receptionist is able to give you a room.How is it possible ? Later, on the same day, infinitely many persons arrive, and the receptionist is able to make room for all of them, (earning a fortune in doing so).

The german mathematician David Hilbert invented these paradoxes in the 1920s to illustrate the mysterious properties of infinity.

Lets see how we can obtain a room in Hilbert's hotel infinity. When only you arrive and the hotel is full, the receptionist is able to give you a room by asking the client who was in room 1 to go to room 2, the one who was in room 2 to go to room 3, and so on forever, each client moving to the next room. Thus, room 1 is now empty for you.

To be able to receive an infinity of new clients, all the previous ones move to the rooms of even numbers, the one who was in room 1 moving to room 2, the one who was in room 2 moving to room 4, the one who was in room 3 moving to room 6, and so on, every client moving to the room having the number double from his previous room. The receptionist is now able to move all the new clients to the rooms of odd numbers.

Hilbert's hotel infinity paradox may be understood with Cantor's theory of transfinite numbers. Thus, while in a normal hotel, the number of odd numbered rooms is inferior to the total number of rooms, in an infinite hotel, the "number" of odd numbered rooms is not inferior to the total "number" of rooms. (The mathematicians use the term of "cardinality" to refer themselves to the size of these sets of rooms).

The receptionist can even receive infinitely many buses containing infinitely many travelers each, by using all the powers of 2 for the previous clients, all the powers of 3 for the first bus, and all the powers of each different prime numbers for each different bus.

See also Zeno's paradox (445 B.C.), Cantor's transfinite numbers (1874), Peano's axioms (1889) and Hilbert's twenty three problems (1900).