The Structure of a Graph Inverse Semigroup

Given any directed graph E one can construct a graph inverse semigroup G(E), where, roughly speaking, elements correspond to paths in the graph. In this paper we study the semigroup-theoretic structure of G(E). Specifically, we describe the non-Rees congruences on G(E), show that the quotient of G(E) by any Rees congruence is another graph inverse semigroup, and classify the G(E) that have only Rees congruences. We also find the minimum possible degree of a faithful representation by partial transformations of any countable G(E), and we show that a homomorphism of directed graphs can be extended to a homomorphism (that preserves zero) of the corresponding graph inverse semigroups if and only if it is injective.Comment: 19 pages; corrected errors, improved organization, strengthened a result (Theorem 20), added reference

Belgian corporate finance in a European perspective

When analysing corporate finance, it is necessary to take account of various factors which may cause significant disparities between firms, such as their size and their sector of activity. Taking account of the size aspect, by neutralising sectoral disparities, there are few differences between the debt levels of small, medium-sized and large firms. Conversely, the debt structure appears to depend on the firm’s size : small firms are more dependent on bank loans. That is confirmed by the high degree to which they make use of credit facilities. Nonetheless, surveys indicate that access to finance is not a major constraint for SMEs, be they Belgian or European : they perceive access to finance, and more specifically access to bank finance, as relatively easy. In contrast, the financial structure of firms differs widely between sectors, and depends to a great extent on the associated intrinsic activity and the scale of the investments. Sectors with high investment ratios, such as the transport and communication sector or the energy sector, mainly use long-term finance. Ample equity capital enables them to maintain a balanced financial situation. Conversely, highly labour-intensive sectors, such as construction or trade, display much higher debt-to-equity ratios ; their debts are mainly short term and they make extensive use of trade credit. A more detailed analysis of the manufacturing sector also reveals differences of financial structure between firms which are classed as innovative and those which are not. In particular, if the chemical industry is excluded, the firms in the innovative sectors make less use of bank loans and record more short-term debt than firms in non-innovative sectors. That may reflect the lenders’ desire to limit the risk incurred, particularly by using the threat of non-renewal of the loan to encourage the manager to behave efficiently. The qualitative surveys appear to indicate that the financial constraint is felt more by innovative firms than by SMEs in general. That expresses a financing need specific to innovative SMEs. At the early stages in their development, they depend almost exclusively on the entrepreneur’s personal resources and those of his friends and family, and venture capital only takes over in the later stages. Finally, as regards the financing structure, a comparison between Belgian firms and their European counterparts, after neutralising the specific effects of size and sector, indicates that the former issue larger amounts of capital. Abundant intra-group financial flows and a favourable institutional context are conducive to that situation.bank lending, venture capital, corporate finance.

The Belgian deposit guarantee scheme in a European perspective

During the recent financial crisis, the deposit guarantee scheme in Belgium – as in other European countries – played a role in preventing bank runs and restoring confidence : to that end, the intervention ceilings were raised substantially and the scope of the scheme was extended to include certain life insurance policies. Finally, the expansion of the system’s coverage had to be financed by a sharp increase in the contributions from financial institutions. First of all, that measure had a positive impact on the budget ; secondly, increased contributions may also boost the credibility of the deposit guarantee system. A recent European initiative proposes further ambitious reforms. Besides a better consumer protection, the European deposit guarantee schemes would be largely harmonised, thus also promoting European financial integration. Risk-weighted financing of the schemes should counteract moral hazard, benefiting financial stability. However, this proposal has yet to be approved by the European Parliament and the Council. Its impact ought to be assessed in the light of the broader package of measures aimed at making the financial system more resilient, such as the new prudential supervision structure, the Basel III proposal for stricter capital and liquidity requirements, and the possible new levies on the financial sector.deposit assurance, financial institutions, financial crisis, moral hazard

Iterated Function System Models in Data Analysis: Detection and Separation

We investigate the use of iterated function system (IFS) models for data analysis. An IFS is a discrete dynamical system in which each time step corresponds to the application of one of a finite collection of maps. The maps, which represent distinct dynamical regimes, may act in some pre-determined sequence or may be applied in random order. An algorithm is developed to detect the sequence of regime switches under the assumption of continuity. This method is tested on a simple IFS and applied to an experimental computer performance data set. This methodology has a wide range of potential uses: from change-point detection in time-series data to the field of digital communications

Simplicial Multivalued Maps and the Witness Complex for Dynamical Analysis of Time Series

Topology based analysis of time-series data from dynamical systems is powerful: it potentially allows for computer-based proofs of the existence of various classes of regular and chaotic invariant sets for high-dimensional dynamics. Standard methods are based on a cubical discretization of the dynamics and use the time series to construct an outer approximation of the underlying dynamical system. The resulting multivalued map can be used to compute the Conley index of isolated invariant sets of cubes. In this paper we introduce a discretization that uses instead a simplicial complex constructed from a witness-landmark relationship. The goal is to obtain a natural discretization that is more tightly connected with the invariant density of the time series itself. The time-ordering of the data also directly leads to a map on this simplicial complex that we call the witness map. We obtain conditions under which this witness map gives an outer approximation of the dynamics, and thus can be used to compute the Conley index of isolated invariant sets. The method is illustrated by a simple example using data from the classical H\'enon map.Comment: laTeX, 9 figures, 32 page

Configurations of Rank-40r Extremal Even Unimodular Lattices (r=1,2,3)

We show that if L is an extremal even unimodular lattice of rank 40r with r=1,2,3 then L is generated by its vectors of norms 4r and 4r+2. Our result is an extension of Ozeki's result for the case r=1.Comment: 5 pages. To appear, Journal de Theorie des Nombres de Bordeau

Universal Reconfiguration of (Hyper-)cubic Robots

We study a simple reconfigurable robot model which has not been previously examined: cubic robots comprised of three-dimensional cubic modules which can slide across each other and rotate about each others' edges. We demonstrate that the cubic robot model is universal, i.e., that an n-module cubic robot can reconfigure itself into any specified n-module configuration. Additionally, we provide an algorithm that efficiently plans and executes cubic robot motion. Our results directly extend to a d-dimensional model.Comment: 5 pages, 2 figure

Foreign financial transactions of Belgian non-financial sectors

The open character of the Belgian economy is also reflected in its financial relations with other countries. At the end of 2007, Belgium’s net foreign assets totalled around 44.4 p.c. of GDP. In the past decade, Belgium’s net financial assets have shown a marked increase, against the backdrop of the introduction of the common currency and the progressive financial integration and globalisation. The degree to which the various non-financial sectors have responded to these developments varies greatly between sectors. The public sector is the one which has done most to adjust its financing and seen the biggest rise in the share of the rest of the world in its total debt, namely from 20 p.c. at the end of 1997 to around 46 p.c. at the end of 2007. When the euro was introduced on the financial markets, it was imperative for the Treasury to widen its investment base : appropriate diversification of the corps of Primary Dealers and Recognised Dealers was one of the ways in which it achieved that. Non-financial corporations, which traditionally maintain very close international financial contacts, still saw a steady increase in the share of the rest of the world in their financial transactions between 1997 and 2007 : during that period, in the case of financial liabilities, the figure was up from 24 p.c. at the end of 1997 to over 37 p.c. at the end of 2007 ; for financial assets, the share was up from 28 p.c. to 39 p.c. However, these orders of magnitude are subject to a strong upward influence exerted by the coordination centres based in Belgium, which perform the function of a financial intermediary for the multinational group to which they are attached. In the past decade, there have been frequent exchanges of shareholdings between Belgian and foreign companies, reflecting the process of mergers and acquisitions, and these have also contributed to the growth of direct investment between Belgium and the rest of the world. Another point worth mentioning is that the past decade has brought a strong rise in loans granted by foreign financial institutions as a percentage of total bank lending to Belgian firms : that figure increased from 12 p.c. at the end of 1997 to 35 p.c. at the end of 2007. The share of foreign assets in the household portfolio dropped from 30 p.c. at the end of 1997 to 17 p.c. at the end of 2007. Naturally, that is due partly to the introduction on 1 July 2005 of the European directive on the taxation of savings ; it has now ceased to be possible for individuals to avoid the tax on income from interest-bearing assets held in other countries, and that has ultimately led to the repatriation of those assets. In contrast, in the case of non interest-bearing assets, foreign investment flows increased between 1997 and 2007. Finally, it should be pointed out that the share of the rest of the world in the liabilities of households is still negligible.flow of funds, financial flow