## Computer simulations and experimental testing of nonliner rod model by measuring actual deformations

This is the continuation of my dissertation. Starting from the mathematical model that I developed earlier computer algorithm and a program should be developed. Then the experiment should be set up in order to explore in what conditions the model actually agrees with the actual 3D body behaviour. It seems that this has not been done yet.I would like to connect with the researchers from the field of metrology.

In the mathematical model of rod structure, the question on existence and stability for general structures is still open (solved only for stents). Professor Igor Velčić gave some suggestions here (new kind of a domain for stent), and I suppose new mathematical structures and algorithms should be developed.

## Maritime connection of island Ugljan with the mainland

We conducted a survey which measured the demand for the transportation on the ferry line Zadar-Preko. Data we collected is now publicly available. We also analyzed the results. Together with some collaborators from Universiti of Zadar we intend to analyze the data with respect to optimization of transportation supply (number of connections, etc.) to best suit the demand. We hope to find ways to make the savings and better fulfill the needs of travelers.

## Eigenvalues for general complex matrix

From what I have seen in this area, I have an impression that all iterative procedures rest on an assumption that a starting matrix is close to a certain form which is then exploited to finish with a matrix that is even closer to this form. Iterations mostly even stagnate at first and this is where most of the time is spent, before the (quadratic) convergence catches on. For an any such algorithm there are often quite a few decisions that are arbitrarily set: for example it can happen that the direction of the sweep, which was overlooked for four decades, turns out to be very important (see Christian Mehl, On asymptotic convergence of non-symmetric Jacobi algorithms, 2008).

As I see it, one of the reasons for the success of the QR algorithm is the introduction of the first step: transition to the Hessenberg form. Then, the QR iterations exploit this form and converge quickly. As I understand it, QR algorithm alone is not spectacular at all when applied directly to the general matrix. It seems as the other iterative methods lack this first step, to efficiently transform general matrix close to that form which the algorithm exploits, for example, close to normal matrices or close to Hermitian matrices.

The third observation is that algorithms most often look to the matrix as a table of numbers, and try to cancel certain entries, one by one, each cancellation taking one step. Then, from step to step there is nothing to stop the ruining of the work already done. This seems like a Heuristics that in the end after many steps miraculously starts to work. What lacks, it seems to me, is the geometric viewpoint --- or what happens with the matrix as an operator in a whole. For example, one iterative procedure that works well for matrices close to normal, concentrates on a certain entry in each step while neglecting whether that step actually brings the whole matrix (in a certain norm) closer to normal. I would like to dedicate 6 months or so exclusively to the topic and try to see if deepening of these observations can bring something new to the subject.

## Topology of distance in time and geography of traffic

How long does it take to transfer from point A to point B? This is their distance in time. It depends on the vehicles that can be used, budget, time of the day. Set mentioned parameters and then build a map that best show what are the distances in time between points. On such a map, two points are close if they can be traveled between very fast. It can happen that the physically close points end up far away on such a map, for example, two nearby island which are only connected by ferry to the mainland each.

In which cases does such map exist? If not can we use third dimension to "escape" from the constrains of 2D (colour, height). Is 3D enough? If not, can we make some kind approximation with some kind of error that can be quantified? I would try to use optimization theory to build such a map. I hope that the results could unlock better understanding of traffic connections. Afterwards, we could visually see how the introduction of a certain connection, for example road or a flight, changes the traffic map.

## Optimal maritime routes between islands of Zadar archipelago and the mainland

This is a classical optimization problem. I need some data which I hope some students would be able to provide. What should be optimized: travel time, total expense of the connections, etc.? I would guess that the solution is more or less the same for a number of such functions that are minimized. I hope some of the results of the project on topology of distances in time could be used to facilitate the presentation of results. I have became interested on this subject when I heard prof. Ratko Radulić speak at the public library in Zadar.

## Course as the axis of rotation for travel on main circles on Earth

When traveling on Earth today (sea and air travel) the direction is given by course with regard to Earth's magnetic plovidbe axis. This direction is not the strait line on the ball, and so during the travel the course has to be changed in order to traverse the shortest distance possible. This planning uses some time and crew activitiy. I hope that the course could be changed by some other parameter that corresponds with the main circles. This would shorten the trip planning activities and ease holding the "course" during the travel. It is possible that this is not a significant step forward now that computers solve most of these tasks automatically.

## Make an estimate of the commitments that Republic of Croatia has taken on trough the pension system

This is a big commitment that is probably higher than the official public debt. Having established the method and the estimate we could better determine the change of this obligation from the introduction of new laws, change in demography, etc.

## Optimization of vignette (sticker) prices for Croatian highways that maximizes the public good

Poll on which prices the drivers are prepared to pay should be taken and other benefits from vignettes quantified. If the tax payer (citizen) is ready to pay X for yearly sticker, and we set price Y under X, then such policy brings Y as the highway income and X-Y as the citizen's benefit. If that driver is foreigner only Y enhances the public good. We certainly have constrains, for example maybe the policy-makers expect vignette highway income higher than the current income.

This above is oversimplification as there highways can be substituted with the other roads, so this effect must be accounted for.

## Other projects

Take a look at my current and past research.