I n this paper we show how mathematics can illuminate the study of cake-cutting in ways that have practical implications. Specifically, we analyze cake-cutting algorithms that use a minimal number of cuts (n − 1 if there are n people), where a cake is a metaphor for a heterogeneous, divisible good, whose parts may be valued differently by different people. These algorithms not only establish th...

Effects of the suspended titanium dioxide nanoparticles (TiO2 NPs, 50 mg/L) on the cake layer formation in a submerged MBR were systematically investigated. With nanometer sizes, TiO2 NPs were found to aggravate membrane pore blocking but postpone cake layer fouling. TiO2 NPs showed obvious effects on the structure and the distribution of the organic and the inorganic compounds in cake layer. C...

The classic fair cake-cutting problem [Steinhaus, 1948] is extended by introducing geometric constraints on the allocated pieces. Specifically, agents may demand to get their share as a square or a rectangle with a bounded length/width ratio. This is a plausible constraint in realistic cake-cutting applications, notably in urban and agricultural economics where the “cake” is land. Geometric con...

An unceasing problem of our prevailing society is the fair division of goods. The problem of fair cake cutting is dividing a heterogeneous and divisible resource, the cake, among n players who value pieces according to their own measure function. The goal is to assign each player a not necessarily connected part of the cake that the player evaluates at least as much as her proportional share. I...

The aim of this study was to determine the rheological properties of sponge cake batters andphysical (volume, density, moisture content, weight after baking and color) and sensory properties of spongecake formulated with four different levels of wild sage seed gum (0, 0.5, 0.75 and 1.0 %). Sponge cake battersformulated with gums showed pseudoplastic (shear-thinning) and thixotropic (time-depend...

Resistant starch type III (RS III) derived from enzymatically debranched high amylose rice starch was prepared and used to make butter cake at different levels (0, 5, 10, 15 and 20%) in place of wheat flour. Physico-chemical properties, sensory evaluation, and in vitro starch hydrolysis rate of the developed butter cake were investigated. This study showed that the content of resistant starch i...

Imagine a cake that must be divided between a group of gluttonous children. To complicate matters, the cake is heterogeneous: two pieces of cake may differ in terms of their toppings, so the children have different preferences over the pieces (one may weakly prefer a larger proportion of chocolate curls, while another may single-mindedly desire the piece with the cherry). In this chapter we dis...

Thermal decomposition and kinetics behaviour of the de-oiled seed cake of African star apple (Chrosophyllum albidum) has been investigated using thermogravimetry under the nitrogen atmosphere from ambient temperature to 900 °C. The thermogravimetric data for the cake decomposition at six different heating rates (5, 10, 15, 20, 30 and 40 °C/min) were used to evaluate the kinetic decomposition of...

We study truthful mechanisms in the context of cake cutting when agents not only value their own pieces of cake but also care for the pieces assigned to other agents. In particular, agents derive benefits or costs from the pieces of cake assigned to other agents. This phenomenon is often referred to as positive or negative externalities. We propose and study the following model: given an alloca...

In this paper we present a generalization of kernel density estimation called Convex Adaptive Kernel Density Estimation (CAKE) that replaces single bandwidth selection by a convex aggregation of kernels at all scales, where the convex aggregation is allowed to vary from one training point to another, treating the fundamental problem of heterogeneous smoothness in a novel way. Learning the CAKE ...