CiteExport$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_upper_j_idt149",{id:"formSmash:upper:j_idt149",widgetVar:"widget_formSmash_upper_j_idt149",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:upper:exportLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_upper_j_idt150_j_idt152",{id:"formSmash:upper:j_idt150:j_idt152",widgetVar:"widget_formSmash_upper_j_idt150_j_idt152",target:"formSmash:upper:j_idt150:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});

Digital Geometry, Combinatorics, and Discrete OptimizationPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
function selectAll()
{
var panelSome = $(PrimeFaces.escapeClientId("formSmash:some"));
var panelAll = $(PrimeFaces.escapeClientId("formSmash:all"));
panelAll.toggle();
toggleList(panelSome.get(0).childNodes, panelAll);
toggleList(panelAll.get(0).childNodes, panelAll);
}
/*Toggling the list of authorPanel nodes according to the toggling of the closeable second panel */
function toggleList(childList, panel)
{
var panelWasOpen = (panel.get(0).style.display == 'none');
// console.log('panel was open ' + panelWasOpen);
for (var c = 0; c < childList.length; c++) {
if (childList[c].classList.contains('authorPanel')) {
clickNode(panelWasOpen, childList[c]);
}
}
}
/*nodes have styleClass ui-corner-top if they are expanded and ui-corner-all if they are collapsed */
function clickNode(collapse, child)
{
if (collapse && child.classList.contains('ui-corner-top')) {
// console.log('collapse');
child.click();
}
if (!collapse && child.classList.contains('ui-corner-all')) {
// console.log('expand');
child.click();
}
}
2010 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Stockholm: Department of Mathematics, Stockholm University , 2010. , p. 42
##### Keywords [en]

Digital geometry, Khalimsky topology, Khalimsky plane, Khalimsky-continuous function, digital straight line segments, discrete optimization, discrete convexity, integral convexity, lateral convexity, marginal function
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:su:diva-47399ISBN: 978-91-7447-196-0 (print)OAI: oai:DiVA.org:su-47399DiVA, id: diva2:373896
##### Public defence

2011-01-21, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
##### Opponent

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt459",{id:"formSmash:j_idt459",widgetVar:"widget_formSmash_j_idt459",multiple:true});
##### Supervisors

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt470",{id:"formSmash:j_idt470",widgetVar:"widget_formSmash_j_idt470",multiple:true});
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt476",{id:"formSmash:j_idt476",widgetVar:"widget_formSmash_j_idt476",multiple:true});
##### Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 4: Submitted. Paper 5: Manuscript. Paper 6: Manuscript.Available from: 2010-12-29 Created: 2010-12-01 Last updated: 2011-01-12Bibliographically approved
##### List of papers

This thesis consists of two parts: digital geometry and discrete optimization.

In the first part we study the structure of digital straight line segments. We also study digital curves from a combinatorial point of view.

In Paper I we study the straightness in the 8-connected plane and in the Khalimsky plane by considering vertical distances and unions of two segments. We show that we can investigate the straightness of Khalimsky arcs by using our knowledge from the 8-connected plane.

In Paper II we determine the number of Khalimsky-continuous functions with 2, 3 and 4 points in their codomain. These enumerations yield examples of known sequences as well as new ones. We also study the asymptotic behavior of each of them.

In Paper III we study the number of Khalimsky-continuous functions with codomain Z and N. This gives us examples of Schröder and Delannoy numbers. As a byproduct we get some relations between these numbers.

In Paper IV we study the number of Khalimsky-continuous functions between two points in a rectangle. Using a generating function we get a recurrence formula yielding this numbers.

In the second part we study an analogue of discrete convexity, namely lateral convexity.

In Paper V we define by means of difference operators the class of lateral convexity. The functions have plus infinity in their codomain. For the real-valued functions we need to check the difference operators for a smaller number of points. We study the relation between this class and integral convexity.

In Paper VI we study the marginal function of real-valued functions in this class and its generalization. We show that for two points with a certain distance we have a Lipschitz property for the points where the infimum is attained. We show that if a function is in this class, the marginal function is also in the same class.

1. Chord properties of digital straight linesegments$(function(){PrimeFaces.cw("OverlayPanel","overlay386153",{id:"formSmash:j_idt538:0:j_idt542",widgetVar:"overlay386153",target:"formSmash:j_idt538:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. The number of Khalimsky-continuous functions on intervals$(function(){PrimeFaces.cw("OverlayPanel","overlay374230",{id:"formSmash:j_idt538:1:j_idt542",widgetVar:"overlay374230",target:"formSmash:j_idt538:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. The number of continuous curves in digitalgeometry$(function(){PrimeFaces.cw("OverlayPanel","overlay386167",{id:"formSmash:j_idt538:2:j_idt542",widgetVar:"overlay386167",target:"formSmash:j_idt538:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. The number of Khalimsky-continuous functions between two points$(function(){PrimeFaces.cw("OverlayPanel","overlay374166",{id:"formSmash:j_idt538:3:j_idt542",widgetVar:"overlay374166",target:"formSmash:j_idt538:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. Discrete convexity built on differences$(function(){PrimeFaces.cw("OverlayPanel","overlay386177",{id:"formSmash:j_idt538:4:j_idt542",widgetVar:"overlay386177",target:"formSmash:j_idt538:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

6. Convexity of marginal functions in the discrete case$(function(){PrimeFaces.cw("OverlayPanel","overlay386183",{id:"formSmash:j_idt538:5:j_idt542",widgetVar:"overlay386183",target:"formSmash:j_idt538:5:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

isbn
urn-nbn$(function(){PrimeFaces.cw("Tooltip","widget_formSmash_j_idt1295",{id:"formSmash:j_idt1295",widgetVar:"widget_formSmash_j_idt1295",showEffect:"fade",hideEffect:"fade",showDelay:500,hideDelay:300,target:"formSmash:altmetricDiv"});});

CiteExport$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_lower_j_idt1348",{id:"formSmash:lower:j_idt1348",widgetVar:"widget_formSmash_lower_j_idt1348",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:lower:exportLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_lower_j_idt1349_j_idt1351",{id:"formSmash:lower:j_idt1349:j_idt1351",widgetVar:"widget_formSmash_lower_j_idt1349_j_idt1351",target:"formSmash:lower:j_idt1349:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});