Geometric representations of linear codes

We say that a linear code C over a field F is triangular representable if there exists a two dimensional simplicial complex $\Delta$ such that C is a punctured code of the kernel ker $\Delta$ of the incidence matrix of $\Delta$ over F and there is a linear mapping between C and ker $\Delta$ which is a bijection and maps minimal codewords to minimal codewords. We show that the linear codes over rationals and over GF(p), where p is a prime, are triangular representable. In the case of finite fields, we show that this representation determines the weight enumerator of C. We present one application of this result to the partition function of the Potts model. On the other hand, we show that there exist linear codes over any field different from rationals and GF(p), p prime, that are not triangular representable. We show that every construction of triangular representation fails on a very weak condition that a linear code and its triangular representation have to have the same dimension.Comment: 20 pages, 8 figures, v3 major change

Implied Calibration of Stochastic Volatility Jump Diffusion Models

In the context of arbitrage-free modelling of financial derivatives, we introduce a novel calibration technique for models in the affine- quadratic class for the purpose of contingent claims pricing and risk- management. In particular, we aim at calibrating a stochastic volatility jump diffusion model to the whole market volatility surface at any given time. We numerically implement the algorithm and show that the proposed approach is both stable and accurate.Affine-quadratic models, Option pricing, Model Calibration

Geometric representation of binary codes and computation of weight enumerators

For every linear binary code $C$, we construct a geometric triangular configuration $\Delta$ so that the weight enumerator of $C$ is obtained by a simple formula from the weight enumerator of the cycle space of $\Delta$. The triangular configuration $\Delta$ thus provides a geometric representation of $C$ which carries its weight enumerator. This is the initial step in the suggestion by M. Loebl, to extend the theory of Pfaffian orientations from graphs to general linear binary codes. Then we carry out also the second step by constructing, for every triangular configuration $\Delta$, a triangular configuration $\Delta'$ and a bijection between the cycle space of $\Delta$ and the set of the perfect matchings of $\Delta'$.Comment: 16 pages, 11 figures, submitted to Advances in Applied Mathematics, v2: major conceptual change

Critical thermodynamics of the two-dimensional +/-J Ising spin glass

We compute the exact partition function of 2d Ising spin glasses with binary couplings. In these systems, the ground state is highly degenerate and is separated from the first excited state by a gap of size 4J. Nevertheless, we find that the low temperature specific heat density scales as exp(-2J/T), corresponding to an ``effective'' gap of size 2J; in addition, an associated cross-over length scale grows as exp(J/T). We justify these scalings via the degeneracy of the low-lying excitations and by the way low energy domain walls proliferate in this model

The association between Ponticulus Posticus and Dental Agenesis: a retrospective study

OBJECTIVE: Neural tube defects may increase the risk of an abnormal development of skull, vertebral column and teeth formation, including dental agenesis in non syndromic patients. The association between the presence of a congenital Dental Agenesis (DA) and the Atlantooccipital Ligament (AOL) calcification, known as "Ponticulus Posticus" (PP), as possible links can be investigated. DESIGN: After a systematic review of the scientific literature on this topic, two independent examiners assessed the AOL calcification in lateral cephalograms of 350 non syndromic patients(7-21 years old). The results were compared with a control group (non syndromic patients, without congenital missing teeth). RESULTS: The 16.3% of the population studied by cephalometric analysis revealed a prevalence rate of PP (both complete and partial) with a slight male predominance is seen, not statistically significant (χ square test = 0.09; p= 0.76). In both sexes complete PP is more observed. In the patients affected by DA the frequency of PP is the 66.6% (both complete than partial). The χ square test with Yates correction showed a significative difference(χ= 66.20; p value= 0.00) between PP in patients with DA compared to not affected by DA. CONCLUSIONS: PP is not an uncommon anomaly. Since orofacial pain like migraine and other symptoms are often associated to PP, during routine radiographic examination, if detected, it should be documented in patients' health record and with symptoms, further investigation should be sought for. These findings encourage to think there's an association between DA in non syndromic patients and neuro-crestal cells defects

Video Augmentation in Education: in-context support for learners through prerequisite graphs

The field of education is experiencing a massive digitisation process that has been ongoing for the past decade. The role played by distance learning and Video-Based Learning, which is even more reinforced by the pandemic crisis, has become an established reality. However, the typical features of video consumption, such as sequential viewing and viewing time proportional to duration, often lead to sub-optimal conditions for the use of video lessons in the process of acquisition, retrieval and consolidation of learning contents. Video augmentation can prove to be an effective support to learners, allowing a more flexible exploration of contents, a better understanding of concepts and relationships between concepts and an optimization of time required for video consumption at different stages of the learning process. This thesis focuses therefore on the study of methods for: 1) enhancing video capabilities through video augmentation features; 2) extracting concept and relationships from video materials; 3) developing intelligent user interfaces based on the knowledge extracted. The main research goal is to understand to what extent video augmentation can improve the learning experience. This research goal inspired the design of EDURELL Framework, within which two applications were developed to enable the testing of augmented methods and their provision. The novelty of this work lies in using the knowledge within the video, without exploiting external materials, to exploit its educational potential. The enhancement of the user interface takes place through various support features among which in particular a map that progressively highlights the prerequisite relationships between the concepts as they are explained, i.e., following the advancement of the video. The proposed approach has been designed following a user-centered iterative approach and the results in terms of effect and impact on video comprehension and learning experience make a contribution to the research in this field

Ground State Wave Function of the Schr\"odinger Equation in a Time-Periodic Potential

Using a generalized transfer matrix method we exactly solve the Schr\"odinger equation in a time periodic potential, with discretized Euclidean space-time. The ground state wave function propagates in space and time with an oscillating soliton-like wave packet and the wave front is wedge shaped. In a statistical mechanics framework our solution represents the partition sum of a directed polymer subjected to a potential layer with alternating (attractive and repulsive) pinning centers.Comment: 11 Pages in LaTeX. A set of 2 PostScript figures available upon request at [email protected] . Physical Review Letter

Noisy Covariance Matrices and Portfolio Optimization

According to recent findings [1,2], empirical covariance matrices deduced from financial return series contain such a high amount of noise that, apart from a few large eigenvalues and the corresponding eigenvectors, their structure can essentially be regarded as random. In [1], e.g., it is reported that about 94% of the spectrum of these matrices can be fitted by that of a random matrix drawn from an appropriately chosen ensemble. In view of the fundamental role of covariance matrices in the theory of portfolio optimization as well as in industry-wide risk management practices, we analyze the possible implications of this effect. Simulation experiments with matrices having a structure such as described in [1,2] lead us to the conclusion that in the context of the classical portfolio problem (minimizing the portfolio variance under linear constraints) noise has relatively little effect. To leading order the solutions are determined by the stable, large eigenvalues, and the displacement of the solution (measured in variance) due to noise is rather small: depending on the size of the portfolio and on the length of the time series, it is of the order of 5 to 15%. The picture is completely different, however, if we attempt to minimize the variance under non-linear constraints, like those that arise e.g. in the problem of margin accounts or in international capital adequacy regulation. In these problems the presence of noise leads to a serious instability and a high degree of degeneracy of the solutions.Comment: 7 pages, 3 figure