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Abstract: Abstract This paper explores the concept of approximate Birkhoff–James orthogonality in the context of operators on semi-Hilbert spaces. These spaces are generated by positive semi-definite sesquilinear forms. We delve into the fundamental properties of this concept and provide several characterizations of it. Using innovative arguments, we extend a widely known result initially proposed by Magajna [17]. Additionally, we improve a recent result by Sen and Paul [24] regarding a characterization of approximate numerical radius orthogonality of two semi-Hilbert space operators, such that one of them is \(A\) -positive. Here, \(A\) is assumed to be a positive semi-definite operator. PubDate: 2024-06-25

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Abstract: Abstract Let \(S\) be a set of four points chosen independently, uniformly at random from a square. Join every pair of points of \(S\) with a straight line segment. Color these edges red if they have positive slope and blue, otherwise. We show that the probability that \(S\) defines a pair of crossing edges of the same color is equal to \(1/4\) . This is connected to a recent result of Aichholzer et al. [1] who showed that by 2-colouring the edges of a geometric graph and counting monochromatic crossings instead of crossings, the number of crossings can be more than halved. Our result shows that for the described random drawings, there is a coloring of the edges such that the number of monochromatic crossings is in expectation \(\frac{1}{2}-\frac{7}{50}\) of the total number of crossings. PubDate: 2024-06-18

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Abstract: Abstract Let \(H\) be a real or complex Hilbert space with the dimension greater than one and \(B(H)\) the algebra of all bounded linear operators on \(H\) . Assume that \(\delta\) is a linear mapping from \(B(H)\) into itself which is Jordan derivable at a given element \(\Omega\in B(H)\) , in the sense that \(\delta(A\circ B)=\delta(A)\circ B+A\circ\delta (B)\) holds for all \(A,B\in B(H)\) with \(A\circ B = \Omega\) , where \(\circ\) denotes the Jordan product \( {A\circ B } =AB+BA\) . In this paper, we show that if \(\Omega\) is an arbitrary but fixed nonzero operator, then \(\delta\) is a derivation; if \(\Omega\) is a zero operator, then \(\delta\) is a generalized derivation. PubDate: 2024-06-15

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Abstract: Abstract We show that the category of X-generated E-unitary inverse monoids with greatest group image G is equivalent to the category of G-invariant, finitary closure operators on the set of connected subgraphs of the Cayley graph of G. Analogously, we study F-inverse monoids in the extended signature \((\cdot, 1, ^{-1}, ^\mathfrak m)\) , and show that the category of X-generated F-inverse monoids with greatest group image G is equivalent to the category of G-invariant, finitary closure operators on the set of all subgraphs of the Cayley graph of G. As an application, we show that presentations of F-inverse monoids in the extended signature can be studied by tools analogous to Stephen’s procedure in inverse monoids, in particular, we introduce the notions of F-Schützenberger graphs and P-expansions. PubDate: 2024-06-14

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Abstract: Abstract We study Markov population processes on large graphs, with the local state transition rates of a single vertex being a linear function of its neighborhood. A simple way to approximate such processes is by a system of ODEs called the homogeneous mean-field approximation (HMFA). Our main result is showing that HMFA is guaranteed to be the large graph limit of the stochastic dynamics on a finite time horizon if and only if the graph-sequence is quasi-random. An explicit error bound is given and it is \(\frac{1}{\sqrt{N}}\) plus the largest discrepancy of the graph. For Erdős–Rényi and random regular graphs we show an error bound of order the inverse square root of the average degree. In general, diverging average degrees is shown to be a necessary condition for the HMFA to be accurate. Under special conditions, some of these results also apply to more detailed type of approximations like the inhomogenous mean field approximation (IHMFA). We pay special attention to epidemic applications such as the SIS process. PubDate: 2024-06-11

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Abstract: Abstract Suppose K is a knot in a 3-manifold Y, and that Y admits a pair of distinct contact structures. Assume that K has Legendrian representatives in each of these contact structures, such that the corresponding Thurston-Bennequin framings are equivalent. This paper provides a method to prove that the contact structures resulting from Legendrian surgery along these two representatives remain distinct. Applying this method to the situation where the starting manifold is \(-\Sigma(2,3,6m+1)\) and the knot is a singular fiber, together with convex surface theory we can classify the tight contact structures on certain families of Seifert fiber spaces. PubDate: 2024-06-10

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Abstract: Abstract On the class of typically real odd polynomials of degree \(2N-1\) $$F(z)=z+\sum_{j=2}^Na_jz^{2j-1}$$ we consider two problems: 1) stretching the central unit disc under the above polynomial mappings and 2) estimating the coefficient \(a_2.\) It is shown that $$\begin{gathered} {F(z)} \le \frac12\csc^2\left({\frac{\pi}{2N+2}}\right),\\-1+4\sin^2\left({\frac{\pi}{2N+4}}\right)\le a_2\le-1+4\cos^2\left({\frac{\pi}{N+2}}\right) \quad \text{for odd $N$,}\end{gathered} $$ and $$-1+4(\nu_N)^2\le a_2\le -1+4\cos^2\left({\frac{\pi}{N+2}}\right) \quad \text{for even $N$,}$$ where \(\nu_N\) is a minimal positive root of the equation \(U'_{N+1}(x) = 0\) with \(U'_{N + 1}(x)\) being the derivative of the Chebyshev polynomial of the second kind of the corresponding order. The above boundaries are sharp, the corresponding estremizers are unique and the coefficients are determined. PubDate: 2024-06-08

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Abstract: Abstract A major open problem of AF-embedding is whether every separable exact quasidiagonal \(C^*\) -algebra can be embedded into an AF-algebra. In this paper we characterize AF-embeddable \(C^*\) -algebras by representations to observe their similarity to the separable exact quasidiagonal \(C^*\) -algebras. As an application, we show that every separable exact quasidiagonal \(C^*\) -algebra is AF-embeddable if and only if every faithful essential representation of a separable exact quasidiagonal \(C^*\) -algebra is a certain kind of \(*\) -representation. We also show that a separable \(C^*\) -algebra is AF-embeddable if and only if it can be embedded into a particular \(C^*\) -algebra. PubDate: 2024-06-08

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Abstract: Abstract We introduce a new potential characterization of Osserman algebraic curvature tensors. An algebraic curvature tensor is Jacobi-orthogonal if \(\mathcal{J}_XY\perp\mathcal{J}_YX\) holds for all \(X\perp Y\) , where \(\mathcal{J}\) denotes the Jacobi operator. We prove that any Jacobi-orthogonal tensor is Osserman, while all known Osserman tensors are Jacobi-orthogonal. PubDate: 2024-06-05

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Abstract: Abstract Let \((a(n) : n \in \mathbb{N})\) denote a sequence of nonnegative integers. Let \(0.a(1)a(2) \ldots \) denote the real number obtained by concatenating the digit expansions, in a fixed base, of consecutive entries of \((a(n) : n \in \mathbb{N})\) . Research on digit expansions of this form has mainly to do with the normality of \(0.a(1)a(2) \ldots \) for a given base. Famously, the Copeland-Erdős constant \(0.2357111317 \ldots {}\) , for the case whereby \(a(n)\) equals the \(n^{\text{th}}\) prime number \(p_{n}\) , is normal in base 10. However, it seems that the “inverse” construction given by concatenating the decimal digits of \((\pi(n) : n \in \mathbb{N})\) , where \(\pi\) denotes the prime-counting function, has not previously been considered. Exploring the distribution of sequences of digits in this new constant \(0.0122 \ldots 9101011 \ldots \) would be comparatively difficult, since the number of times a fixed \(m \in \mathbb{N} \) appears in \((\pi(n) : n \in \mathbb{N})\) is equal to the prime gap \(g_{m} = p_{m+1} - p_{m}\) , with the behaviour of prime gaps notoriously elusive. Using a combinatorial method due to Szüsz and Volkmann, we prove that Cramér’s conjecture on prime gaps implies the normality of \(0.a(1)a(2) \ldots \) in a given base \(g \geq 2\) , for \(a(n) = \pi(n)\) . PubDate: 2024-06-05

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Abstract: Abstract The mixed area of a Reuleaux polygon and its symmetric with respect to the origin is expressed in terms of the mixed area of two explicit polygons. This gives a geometric explanation of a classical proof due to Chakerian. Mixed areas and volumes are also used to reformulate the minimization of the volume under constant width constraint as isoperimetric problems. In the two dimensional case, the equivalent formulation is solved, providing another proof of the Blaschke–Lebesgue theorem. In the three dimensional case the proposed relaxed formulation involves the mean width, the area and inclusion constraints. PubDate: 2024-06-05

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Abstract: Abstract Let P be a set of n points in general position in the plane. The Second Selection Lemma states that for any family of \(\Theta(n^3)\) triangles spanned by P, there exists a point of the plane that lies in a constant fraction of them. For families of \(\Theta(n^{3-\alpha})\) triangles, with \(0\le \alpha \le 1\) , there might not be a point in more than \(\Theta(n^{3-2\alpha})\) of those triangles. An empty triangle of P is a triangle spanned by P not containing any point of P in its interior. Bárány conjectured that there exists an edge spanned by P that is incident to a super-constant number of empty triangles of P. The number of empty triangles of P might be as low as \(\Theta(n^2)\) ; in such a case, on average, every edge spanned by P is incident to a constant number of empty triangles. The conjecture of Bárány suggests that for the class of empty triangles the above upper bound might not hold. In this paper we show that, somewhat surprisingly, the above upper bound does in fact hold for empty triangles. Specifically, we show that for any integer n and real number \(0\leq \alpha \leq 1\) there exists a point set of size n with \(\Theta(n^{3-\alpha})\) empty triangles such that any point of the plane is only in \(O(n^{3-2\alpha})\) empty triangles. PubDate: 2024-05-23

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Abstract: Abstract We consider the Lipschitz class functions on [0, 1] and special series of their Fourier coefficients with respect to general orthonormal systems (ONS). The convergence of classical Fourier series (trigonometric, Haar, Walsh systems) of Lip 1 class functions is a trivial problem and is well known. But general Fourier series, as it is known, even for the function f (x) = 1 does not converge. On the other hand, we show that such series do not converge with respect to general ONSs. In the paper we find the special conditions on the functions \(\varphi_{n}\) of the system \((\varphi_{n})\) such that the above-mentioned series are convergent for any Lipschitz class function. The obtained result is the best possible. PubDate: 2024-05-10

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Abstract: Abstract We provide a non-vanishing region for an infinite sum of weight zero Hecke–Maass L-functions for the full modular group inside the critical strip. For given positive parameters T and \(1 \leq M \ll \frac{T}{\log T}\) , T large, we also count the number of Hecke–Maass cusp forms whose L-values are non-zero at any point s in this region and whose spectral parameters \(t_j\) lie in short intervals. PubDate: 2024-05-09

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Abstract: Abstract We extend the concept of a G-Drazin inverse from the set \(M_n\) of all \(n\times n\) complex matrices to the set \(\mathcal{R}^{D}\) of all Drazin invertible elements in a ring \(\mathcal{R}\) with identity. We also generalize a partial order induced by G-Drazin inverses from \(M_n\) to the set of all regular elements in \(\mathcal{R}^{D}\) , study its properties, compare it to known partial orders, and generalize some known results. PubDate: 2024-05-08 DOI: 10.1007/s10474-024-01429-8

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Abstract: Abstract It is consistent that the continuum be arbitrary large and no absolute \(\kappa\) -Borel set X of density \(\kappa\) , \(\aleph_1<\kappa<\mathfrak{c}\) ,condenses onto a compactum. It is consistent that the continuum be arbitrary large and any absolute \(\kappa\) -Borel set X of density \(\kappa\) , \(\kappa\leq\mathfrak{c}\) , containing a closed subspace of the Baire space of weight \(\kappa\) , condenses onto a compactum. In particular, applying Brian's results in model theory, we get the following unexpected result. Given any \(A\subseteq \mathbb{N}\) with \(1\in A\) , there is a forcing extension in which every absolute \(\aleph_n\) -Borel set, containing a closed subspace of the Baire space of weight \(\aleph_n\) , condenses onto a compactum if and only if \(n\in A\) . PubDate: 2024-05-06 DOI: 10.1007/s10474-024-01428-9

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Abstract: Abstract Let \(S\) be a semigroup, \(Z(S)\) the center of \(S\) and \(\sigma \colon S \rightarrow S\) is an involutive automorphism. Our main results is that we describe the solutions of the Kannappan-Wilson functional equation \(\int_{S} f(xyt)\, d\mu(t) + \int_{S} f(\sigma(y)xt)\, d\mu(t)= 2f(x)g(y),\ \ x,y\in S,\) and the Van Vleck-Wilson functional equation \(\int_{S} f(xyt)\, d\mu(t) - \int_{S} f(\sigma(y)xt)\, d\mu(t)= 2f(x)g(y),\ \ x,y\in S,\) where \(\mu\) is a measure that is a linear combination of Dirac measures \((\delta_{z_i})_{i\in I}\) , such that \(z_i\in Z(S)\) for all \(i\in I\) . Interesting consequences of these results are presented. PubDate: 2024-05-06 DOI: 10.1007/s10474-024-01433-y

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Abstract: Abstract Let D be a division ring. The first aim of this paper is to describe all unipotent matrices of index 2 in the general linear group \(\mathrm {GL}_n(D)\) of degree n and in the Vershik–Kerov group \(\mathrm{GL} _{\rm VK}(D)\) . As a corollary, the subgroups generated by such matrices are investigated. The next aim is to seek a positive integer d such that every matrix in these groups is a product of at most d unipotent matrices of index 2. For example, we show that if every element in the derived subgroup \(D'\) of \(D^*=D\backslash \{0\}\) is a product of at most c commutators in \(D^*\) , then every matrix in \(\mathrm{GL}_n(D)\) (resp., \(\mathrm{GL} _{\rm VK}(D)\) , which is a product of some unipotent matrices of index 2, can be written as a product of at most 4+3c (resp.,5 + 3c) of unipotent matrices of index 2 in \(\mathrm{GL}_n(D)\) (resp., \(\mathrm{GL}_{\rm VK}(D))\) . PubDate: 2024-05-06 DOI: 10.1007/s10474-024-01427-w

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Abstract: Abstract A topological space is called submetrizable if it can be mapped onto a metrizable topological space by a continuous one-to-one map. In this paper we answer two questions concerning sequence-covering maps on submetrizable spaces. PubDate: 2024-04-10 DOI: 10.1007/s10474-024-01426-x

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Abstract: Abstract Let \((t(m))_{m \ge0}\) be Thue-Morse sequence and \(b>2\) be an integer. In this paper, we prove that the real numbers \(1\) , \(\sum_{m=0}^\infty {\frac{t(m^2)}{{b}^{m+1}}}\) and \(\sum_{m=0}^\infty {\frac{t(m^3)}{{b}^{m+1}}}\) are linearly independent over \(\mathbb{Q}\) . PubDate: 2024-02-29 DOI: 10.1007/s10474-024-01417-y