Scientific Reports
The preprint collection of the Department of Mathematics.
Full-text generally not available for preprints prior to may 2006.
Found 868 products
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QDD235 - 03/02/2023
Colombo, F; Krausshar R.S.; Sabadini, I
Octonionic monogenic and slice monogenic Hardy and Bergman spaces | Abstract | | In this paper we discuss some basic properties of octonionic Bergman and Hardy spaces. In the first part we review some fundamental concepts of the general theory of octonionic Hardy and Bergman spaces together with related reproducing kernel functions in the monogenic setting.
We explain how some of the fundamental problems in well-defining a reproducing kernel can be overcome in the non-associative setting by looking at the real part of an appropriately defined para-linear octonion-valued inner product. The presence of a weight factor of norm 1 in the definition of the inner product is an intrinsic new ingredient in the octonionic setting.
Then we look at the slice monogenic octonionic setting using the classical complex book structure. We present explicit formulas for the slice monogenic reproducing kernels for the unit ball, the right octonionic half-space and strip domains bounded in the real direction. In the setting of the unit ball we present an explicit sequential characterization which can be obtained by applying the special Taylor series representation of the slice monogenic setting together with particular octonionic calculation rules that reflect the property of octonionic para-linearity.
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QDD234 - 08/01/2022
Colombo F.; Krausshar R.S.; Sabadini I.
Slice monogenic theta series | Abstract | | In this paper we introduce a generalization of theta series in the context of the slice monogenic function theory in the n+1-dimensional Euclidean space where me make use of the so-called *-exponential function in a hypercomplex variable. Together with the Eisenstein and Poincaré series that we introduced in a previous paper, the theta series construction in this paper completes the fundament of the basic theory of modular forms in the slice monogenic setting. We introduce a suitable generalized Poisson summation formula in this framework and we apply an properly adapted Fourier transform. As a direct application we prove a transformation formula for slice monogenic theta series. Then we introduce a family of conjugated theta functions. These are used to construct a slice monogenic generalization of the third power of the Dedekind eta function and of the modular discriminant. We also investigate their transformation behavior. Finally, we show that these theta series are special solutions to a generalization of the heat equation associated with the slice derivative. We round off by discussing the monogenic case.
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QDD233 - 11/23/2020
Bertacchi D.; Braunsteins, P.; Hautphenne, S.; Zucca, F.
Extinction probabilities in branching processes with countably many types: a general framework | Abstract | | We consider Galton-Watson branching processes with countable typeset X. We study the vectors q(A)=(q_x(A))x?X recording the conditional probabilities of extinction in subsets of types A?X, given that the type of the initial individual is x. We first investigate the location of the vectors q(A) in the set of fixed points of the progeny generating vector and prove that q_x({x}) is larger than or equal to the xth entry of any fixed point, whenever it is different from 1. Next, we present equivalent conditions for q_x(A) |
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QDD232 - 09/24/2019
Pavani, R.
Challenging mathematical insights into masonry domes over the centuries | Abstract | | From a mathematical point of view, an approximation of a dome is provided by a rotation solid whose cross-section gives the generating curve. Here we want to highlight the role of parabola and catenary used as generating curves. Actually, catenary is the curve of a hanging chain, which exhibits a tension strength only. When it is "frozen" and inverted it exhibits a compression strength only, which means that it supports itself. Parabola does not exhibit such structural property, but catenary may differ from a convenient parabola very slightly. Here, we investigate the mathematical connection between catenary and parabola in masonry dome structure, referring to historical domes. |
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QDD231 - 11/13/2018
Andrà, C.; Brunetto, D.; Pini, A.
A contribution to understand STEM students' difficulties with mathematics | Abstract | | Drop out during the first year at university STEM courses is a plague spreading all around the world: it has been estimated that, on average, 40% of freshmen abandon their studies before the end of the first academic year.
Research in Mathematics Education has revealed that mathematics is one of the main causes for drop out: not only the students' mathematical knowledge, but also affective issues such as attitudes towards learning mathematics, views about mathematics itself, as well as emotions determine the students' success or failure in university career. On the one's hand, thus, it is important to develop suitable and reliable means for investigating both cognitive and affective dimensions, and on the other hand it becomes necessary to reflect on the kind of information the researcher can get from these means of investigation. One of the central issues is the private versus public dimension of learning mathematics. This is connected to the public and private nature of telling about one's emotions and views. We understand ``public'' versus ``private'' as identifiable versus anonymous questionnaires and tests, respectively. In this paper, we discuss gains and drawbacks of either approach. In doing so, we also investigate the intertwining of cognitive and affective dimensions in freshmen Engineering students attending a bridge course in mathematics at the beginning of the first semester at the Politecnico di Milano. |
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QDD230 - 04/10/2018
Sabadini, I; Sommen, F.
Radon type transforms for holomorphic functions in the Lie ball | Abstract | | In this paper we consider holomorphic functions on the $m$-dimensional Lie ball $LB(0,1)$ which admit a square integrable extension on the Lie sphere. We then define orthogonal projections of this set onto suitable subsets of functions defined in lower dimensional spaces to obtain several Radon-type transforms. For all these transforms we provide the kernel and an integral representation, besides other properties. In particular, we introduce and study a generalization to the case of the Lie ball of the Szego-Radon transform, and various types of Hua-Radon transforms.
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QDD229 - 01/09/2018
Alpay, D.; Colombo, F.; Sabadini, I.
de Branges spaces and characteristic operator function: the quaternionic case | Abstract | | This work inserts in the very fruitful study of quaternionic linear operators. This study is a generalization of the complex case, but the noncommutative setting of quaternions shows several interesting new features, see e.g. the so-called $S$-spectrum and $S$-resolvent operators. In this work, we study de Branges spaces, namely the quaternionic counterparts of spaces of analytic functions (in a suitable sense) with some specific reproducing kernels, in the unit ball of quaternions or in the half space of quaternions with positive real parts. The spaces under consideration will be Hilbert or Pontryagin or Krein spaces. These spaces are closely related to operator models that are also discussed. We also introduce a notion of the characteristic operator function of a bounded linear operator $A$ with finite real part and we address several questions like the study of $J$-contractive functions, where $J$ is self-adjoint and unitary, and we also treat the inverse problem namely to characterize which $J$-contractive functions are characteristic operator functions of an operator. In particular, we prove the counterpart of Potapov's factorization theorem in
this framework. Besides other topics, we also consider canonical differential equations in the setting of slice hyperholomorphic functions.
We define the lossless inverse scattering problem in the present setting. We also consider the inverse scattering problem associated to canonical differential equations. These equations provide a convenient unifying framework to discuss a number of
questions pertaining, for example, to inverse scattering, non-linear partial differential equations and are studied in the last section of this paper.
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QDD228 - 07/06/2017
Bucur, D.; Fragalà, I.; Velichkov, B.; Verzini, G.
On the honeycomb conjecture for a class of minimal convex partitions | Abstract | | We prove that the planar hexagonal honeycomb is asymptotically optimal for a large class of optimal partition problems, in which the cells are assumed to be convex, and the criterion is to minimize either the sum or the maximum among the energies of the cells, the cost being a shape functional F which satisfies a few assumptions. They are: monotonicity under inclusions; homogeneity under dilations; a Faber-Krahn inequality for convex hexagons; a convexity-type inequality for the map which associates with every integer n the minimizers of F among convex n-gons with given area. In particular, our result allows to obtain the honeycomb conjecture for the Cheeger constant and for the logarithmic capacity (still assuming the cells to be convex). Moreover we show that, in order to get the conjecture also for the first Dirichlet eigenvalue of the Laplacian, it is sufficient to establish some facts about its behaviour among convex pentagons, hexagons, and heptagons with prescribed area.
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