Acta Mathematica Hungarica最新文献

We characterize the approximate Cohen–Macaulayness of amodule in terms of the length function and the Hilbert coefficient of the modulewith respect to an almost p-standard system of parameters (a strict subclass ofd-sequences). As applications, we characterize the approximate Cohen–Macaulayproperty of Stanley–Reisner rings, localizations, idealizations, and power seriesrings. Furthermore, for power series rings, we construct almost p-standard systemsof parameters of them. From this result, we give a class of Cohen–MacaulayRees algebras and give precise formulas computing all Hilbert coefficients of theformal power series ring with respect to an almost p-standard system of parameters.

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.

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.

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 (Omegain B(H)), in the sense that (delta(Acirc B)=delta(A)circ B+Acircdelta (B)) holds for all (A,Bin B(H)) with (Acirc B = Omega), where (circ) denotes the Jordan product ( {Acirc 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.

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.

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.

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.

On the class of typically real odd polynomials of degree (2N-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

and

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.

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.

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.