Two-Sided Random Matching Markets: Ex-Ante Equivalence of the Deferred Acceptance Procedures

Stable matching in a community consisting of N men and N women is a classical combinatorial problem that has been the subject of intense theoretical and empirical study since its introduction in 1962 in a seminal work by Gale and Shapley. When the input preference profile is generated from a distribution, we study the output distribution of two stable matching procedures: women-proposing-deferred-acceptance and men-proposing-deferred-acceptance. We show that the two procedures are ex-ante equivalent—that is, under certain conditions on the input distribution, their output distributions are identical. In terms of technical contributions, we generalize (to the non-uniform case) an integral formula, due to Knuth and Pittel, which gives the probability that a fixed matching is stable. Using an inclusion-exclusion principle on the set of rotations, we give a new formula that gives the probability that a fixed matching is the women/men-optimal stable matching.

Two poisson limit theorems for the coupon collector’s problem with group drawings

In the collector’s problem with group drawings, s out of n different types of coupon are sampled with replacement. In the uniform case, each s-subset of the types has the same probability of being sampled. For this case, we derive a Poisson limit theorem for the number of types that are sampled at most $c-1$ times, where $c \ge 1$ is fixed. In a specified approximate nonuniform setting, we prove a Poisson limit theorem for the special case $c=1$ . As corollaries, we obtain limit distributions for the waiting time for c complete series of types in the uniform case and a single complete series in the approximate nonuniform case.

Algorithmic Information Distortions in Node-Aligned and Node-Unaligned Multidimensional Networks

In this article, we investigate limitations of importing methods based on algorithmic information theory from monoplex networks into multidimensional networks (such as multilayer networks) that have a large number of extra dimensions (i.e., aspects). In the worst-case scenario, it has been previously shown that node-aligned multidimensional networks with non-uniform multidimensional spaces can display exponentially larger algorithmic information (or lossless compressibility) distortions with respect to their isomorphic monoplex networks, so that these distortions grow at least linearly with the number of extra dimensions. In the present article, we demonstrate that node-unaligned multidimensional networks, either with uniform or non-uniform multidimensional spaces, can also display exponentially larger algorithmic information distortions with respect to their isomorphic monoplex networks. However, unlike the node-aligned non-uniform case studied in previous work, these distortions in the node-unaligned case grow at least exponentially with the number of extra dimensions. On the other hand, for node-aligned multidimensional networks with uniform multidimensional spaces, we demonstrate that any distortion can only grow up to a logarithmic order of the number of extra dimensions. Thus, these results establish that isomorphisms between finite multidimensional networks and finite monoplex networks do not preserve algorithmic information in general and highlight that the algorithmic information of the multidimensional space itself needs to be taken into account in multidimensional network complexity analysis.

Algorithmic Information Distortions in Node-Aligned and Node-Unaligned Multidimensional Networks

In this article, we investigate limitations of importing methods based on algorithmic information theory from monoplex networks into multidimensional networks (such as multilayer networks) that have a large number of extra dimensions (i.e., aspects). In the worst-case scenario, it has been previously shown that node-aligned multidimensional networks with non-uniform multidimensional spaces can display exponentially larger algorithmic information (or lossless compressibility) distortions with respect to their isomorphic monoplex networks, so that these distortions grow at least linearly with the number of extra dimensions. In the present article, we demonstrate that node-unaligned multidimensional networks, either with uniform or non-uniform multidimensional spaces, can also display exponentially larger algorithmic information distortions with respect to their isomorphic monoplex networks. However, unlike the node-aligned non-uniform case studied in previous work, these distortions in the node-unaligned case grow at least exponentially with the number of extra dimensions. On the other hand, for node-aligned multidimensional networks with uniform multidimensional spaces, we demonstrate that any distortion can only grow up to a logarithmic order of the number of extra dimensions. Thus, these results establish that isomorphisms between finite multidimensional networks and finite monoplex networks do not preserve algorithmic information in general and highlight that the algorithmic information of the multidimensional space itself needs to be taken into account in multidimensional network complexity analysis.

Algorithmic Information Distortions in Node-Aligned and Node-Unaligned Multidimensional Networks

In this article, we investigate limitations of importing methods based on algorithmic information theory from monoplex networks into multidimensional networks (such as multilayer networks) that have a large number of extra dimensions (i.e., aspects). In the worst-case scenario, it has been previously shown that node-aligned multidimensional networks with non-uniform multidimensional spaces can display exponentially larger algorithmic information (or lossless compressibility) distortions with respect to their isomorphic monoplex networks, so that these distortions grow at least linearly with the number of extra dimensions. We demonstrate that node-unaligned multidimensional networks, either with uniform or non-uniform multidimensional spaces, can also display exponentially larger algorithmic information distortions with respect to their isomorphic monoplex networks. However, unlike the previous node-aligned non-uniform case, these distortions grow at least exponentially with the number of extra dimensions. On the other hand, for node-aligned multidimensional networks with uniform multidimensional spaces, we demonstrate that any distortion can only grow up to a logarithmic order of the number of extra dimensions. Thus, these results establish that isomorphisms between finite multidimensional networks and finite monoplex networks do not preserve algorithmic information in general and highlight that the algorithmic information of the multidimensional space itself needs to be taken into account in multidimensional network complexity analysis.

An impulse-based derivation of the Kutta–Joukowsky equation for wind turbine thrust

Using the concept of impulse in control volume analysis, we derive general expressions for wind turbine thrust in a constant, spatially uniform wind. The absence of pressure in the impulse equations allows for their application in the near wake, where the flow is assumed to be steady in the frame of reference rotating with the blades. The assumption of circumferential uniformity in the near wake – as applies when the number of blades or the tip speed ratio tends to infinity – is needed to reduce these general expressions to the Kutta–Joukowsky (KJ) equation for blade-element thrust. The present derivation improves upon the classical derivation based on the Bernoulli equation by allowing the flow to be rotational in the near wake. The present derivation also yields intermediate expressions for thrust that are valid for a finite number of blades and trailing vortex sheets of finite thickness. For the circumferentially uniform case, our analysis suggests that the magnitudes of the radial velocity and the axial induction factor must be equal somewhere on the plane containing the rotor, and we cite previous studies that show this to occur near the rotor tip across a wide range of thrust coefficients. The derivation reveals one further complication; when deriving the KJ equations using annular control volumes, the existence of vorticity on the lateral control surfaces may cause the local blade loading to differ from the KJ equation, but the magnitude of these deviations is not explored. This complication is not visible to the classical derivation due to its neglect of vorticity.

Examining vehicular speed characteristics through divergences from prior distributions

A variety of approaches, within literature, has been conducted to interpret vehicular speed characteristics. This study turns the attention to the entropy-based approaches, and thus focuses on the maximum entropy method of statistical mechanics and the Kullback–Leibler (KL) divergence approach to examining the vehicular speeds. The vehicle speeds at the selected highway are analyzed in order to find out the disparities among them. However, it is turned out that the speed dynamics could not be distinguished over the speed distributions; hence the maximization of Shannon entropy seems insufficient to compare the speed distributions of each data set. For this reason, the KL divergence approach was performed. This approach displays the comparison, among the speed distributions, based on two prior distribution models, i.e., uniform and Gauss. The examination of the trends of KL divergences obtained from both distributions was made. It was concluded that the KL divergence values for the highway speed data sets ranged between about 0.53 and 0.70 for the uniform case, while for the Gaussian case the obtained values are between 0.16 and 0.33. The KL divergence trends for the real speeds were obtained analogous for both cases, but they differed significantly when the synthetic data sets were employed. As a result, the KL divergence approach proves suitable as an appropriate indicator to compare the speed distributions.

The Concept And Legal Nature Of Amendments To The Contract

The article examines the concept and legal nature of amendments to the contract. It is established that this legal phenomenon due to the rapid development of market relations is becoming widespread. The meaning of the concept of «change» and its legal nature are analyzed. An exhaustive list of possible options for exercising the right to amend the contract has been made. According to the results of scientific research of different points of view of civilians of different epochs on the concept of «change of contract terms», the own definition of the mentioned concept is formulated and it is proposed to make appropriate changes to the current Civil Code of Ukraine. After all, the perfect legislative regulation of certain relations begins with the consolidation of their definition, in order to have a common understanding of the concept. In our opinion, the lack of a legislative definition of the term “change of contract terms” is a negative phenomenon, as it leads to unequal understanding of the concept by parties, substitution of concepts, violation of the rights of parties, different case law, which is based on the results of consideration of the category of cases related to amendments to the contract. Therefore, we believe that the CC of Ukraine should be amended by including Article 650 1 «The concept of changing the terms of the contract», by which we mean the process, active behavior of the party (parties), which aims to create new or edit existing terms of the contract to bring it in accordance with certain life circumstances, legal norms, as well as its optimization, the purpose of which is to obtain by the party (parties) a certain positive result of property and / or non-property nature. The implementation of these changes will make it possible to avoid the above-mentioned negative consequences in the future and will ensure the introduction of uniform case law in resolving disputes related to amendments to the contract.

Legal Thinking in the Interwar Latvian Senate

In this article, the author researches and analyses the legal thinking created by Senators A. Lēbers and M. Čakste, aspects of interdisciplinary interaction of subsidiary sources and branches of law in the interwar Latvian Senate. The findings made by the senators of interwar Latvia (in the capacity of rapporteurs on a case, members of the court’s composition) help to explore the historical events and the legal culture of the respective age by reading the primary sources. In the conditions of contemporary legal system, the described cases and legal institutions mostly have similar regulation and could be useful for the development of the case law on the respective matter. Interaction of subsidiary sources, in particular, the used findings of the doctrine build bridges across ages and promote the continuity of uniform case law and understanding of law. The range of resolved legal matters pertained to a broad area of law – branches of law, interdisciplinary aspects of the legal system. The scientific contribution by Senators A. Lēbers and M. Čakste is universal and significant in the context of European and global thinking. Notwithstanding the circumstances, they remained loyal to democratic Latvia.

Clothing Taskscape as an Approach Toward Assessment of User Needs

This study highlights use of the clothing taskscape (CT) to assess user needs, characterize design problems, and develop design criteria by considering relationships across people, their clothing, environments, activities, and tasks. Two case studies—a liquor store uniform and outdoor winter sporting clothing for seated clients—are used to illustrate how the CT may be operationalized. Data collection included observation and interviews to identify problems and determine design attributes needed in our respective clothing categories. Data were thematically analyzed. Findings in the uniform case study included problems related to uniform styling, fit, fabric, branding, and visual identity. Findings in the winter sporting clothing case study included procurement, garment styling, fit, branding, visual identity, storage of personal effects, storage of large-sized garments, and laundering practices. Use of the CT has the potential to guide designers toward more holistic assessment of the use scenario to assess user needs and develop design criteria.