Group averaging and the Gini deviation

It is known that partitioning a society into groups with subsequent averaging in each group decreases the Gini coefficient. The resulting Lorenz function is piecewise linear. This study deals with a natural question: by how much the Gini coefficient could decrease when passing to a piecewise linear Lorenz function? Obtained results are quite illustrative (since they are expressed in terms of the geometric parameters of the polygon Lorenz curve, such as the lengths of its segments and the angles between successive segments) upper bound estimates for the maximum possible change in the Gini coefficient with a restriction on the group shares, or on the difference between the averaged values of the attribute for consecutive groups. It is shown that there exist Lorenz curves with the Gini coefficient arbitrarily close to one, and at the same time with the Gini coefficient of the averaged society arbitrarily close to zero.

Lempel-Ziv Parsing for Sequences of Blocks

The Lempel-Ziv parsing (LZ77) is a widely popular construction lying at the heart of many compression algorithms. These algorithms usually treat the data as a sequence of bytes, i.e., blocks of fixed length 8. Another common option is to view the data as a sequence of bits. We investigate the following natural question: what is the relationship between the LZ77 parsings of the same data interpreted as a sequence of fixed-length blocks and as a sequence of bits (or other “elementary” letters)? In this paper, we prove that, for any integer b>1, the number z of phrases in the LZ77 parsing of a string of length n and the number zb of phrases in the LZ77 parsing of the same string in which blocks of length b are interpreted as separate letters (e.g., b=8 in case of bytes) are related as zb=O(bzlognz). The bound holds for both “overlapping” and “non-overlapping” versions of LZ77. Further, we establish a tight bound zb=O(bz) for the special case when each phrase in the LZ77 parsing of the string has a “phrase-aligned” earlier occurrence (an occurrence equal to the concatenation of consecutive phrases). The latter is an important particular case of parsing produced, for instance, by grammar-based compression methods.

Essays On Deep Trade Agreements

<div>The number of preferential trade agreements (PTAs) has increased tremendously since 1990. The natural question to ask is why. PTAs are not only about lowering down tariffs further than the most favoured nation (MFN) tariff levels. There are many economic and non-economic policies other than border policies that are addressed in PTAs. Trade agreements dealing with border policies (tariffs) are referred to as “shallow”; and those that are dealing with a broader set of policies are referred to as “deep”. Therefore, PTAs are about something deeper. Parallel to the increase in the PTAs, trade in intermediate inputs has grown exponentially over past decades. Therefore, the first question that arises is whether trade in intermediate inputs generates the need for deep integration. In this dissertation, we show that the nature of trade in intermediate goods requires deep integration. The second question to be addressed is whether the deep trade agreements need to be preferential. With a three-country model, we show that the deep bilateral agreements are rarely chosen over the shallow agreements. Finally, by introducing the deep integration in the multilateral trading system, we conclude that although trade in intermediate inputs calls for deep integration, they do not call for deep PTAs. However, deep integration is better implied under multilateral agreements. Therefore, the deep integration in economic policies does not contribute to the increase in the number of PTAs.</div>

Essays On Deep Trade Agreements

<div>The number of preferential trade agreements (PTAs) has increased tremendously since 1990. The natural question to ask is why. PTAs are not only about lowering down tariffs further than the most favoured nation (MFN) tariff levels. There are many economic and non-economic policies other than border policies that are addressed in PTAs. Trade agreements dealing with border policies (tariffs) are referred to as “shallow”; and those that are dealing with a broader set of policies are referred to as “deep”. Therefore, PTAs are about something deeper. Parallel to the increase in the PTAs, trade in intermediate inputs has grown exponentially over past decades. Therefore, the first question that arises is whether trade in intermediate inputs generates the need for deep integration. In this dissertation, we show that the nature of trade in intermediate goods requires deep integration. The second question to be addressed is whether the deep trade agreements need to be preferential. With a three-country model, we show that the deep bilateral agreements are rarely chosen over the shallow agreements. Finally, by introducing the deep integration in the multilateral trading system, we conclude that although trade in intermediate inputs calls for deep integration, they do not call for deep PTAs. However, deep integration is better implied under multilateral agreements. Therefore, the deep integration in economic policies does not contribute to the increase in the number of PTAs.</div>

Noise-induced barren plateaus in variational quantum algorithms

Variational Quantum Algorithms (VQAs) may be a path to quantum advantage on Noisy Intermediate-Scale Quantum (NISQ) computers. A natural question is whether noise on NISQ devices places fundamental limitations on VQA performance. We rigorously prove a serious limitation for noisy VQAs, in that the noise causes the training landscape to have a barren plateau (i.e., vanishing gradient). Specifically, for the local Pauli noise considered, we prove that the gradient vanishes exponentially in the number of qubits n if the depth of the ansatz grows linearly with n. These noise-induced barren plateaus (NIBPs) are conceptually different from noise-free barren plateaus, which are linked to random parameter initialization. Our result is formulated for a generic ansatz that includes as special cases the Quantum Alternating Operator Ansatz and the Unitary Coupled Cluster Ansatz, among others. For the former, our numerical heuristics demonstrate the NIBP phenomenon for a realistic hardware noise model.

Ploidy Mosaics - Does Endopolyploidy in Explants Affects the Cytogenetic Stability of Orchids Regenerated From PLBs?

The induction and regeneration of protocorm-like bodies (PLBs) is a morphogenetic pathway widely used for orchid micropropagation. As endopolyploidy, i.e., the coexistence of cells with different ploidy levels, is a common feature in orchid tissues, a natural question arises when using somatic tissues as explants for orchid micropropagation: does endopolyploidy in explants affect the cytogenetic stability of regenerated plantlets? To answer this question, Epidendrum fulgens was used as a model plant, and flow cytometry (FC) was used to analyze endopolyploidy in pollinia, petals, labella, leaf bases, leaf tips, root tips, protocorms bases and protocorms apexes, which were subsequently used as explants for PLB induction and plant regeneration. The ploidy screening showed contrasting ploidy patterns in the samples. Endopolyploidy was detected in all tissues, with C-values ranging from 1C to 16C. Protocorm bases and root tips presented the highest proportion of endopolyploidy, while petals and protocorm apexes showed the lowest proportion. Flower parts presented high oxidation for PLB induction and pollinia failed to produce PLB or callus. The highest induction rate was observed at 10 µM TDZ, with 92%, 22%, and 0.92% for protocorm bases, leaves, and root tips, respectively. Plantlets were more easily regenerated from PLBs induced from protocorm bases than from leaves and roots. Doubled ploidy levels were registered in a proportion of 11% and 33% for PLB-regenerated plantlets obtained from protocorm bases and leaf bases, respectively, which was not directly associated with the proportion of endopolyploid cells or cycle value of explants.

New Results on the Aggregation of Norms

It is a natural question if a Cartesian product of objects produces an object of the same type. For example, it is well known that a countable Cartesian product of metrizable topological spaces is metrizable. Related to this question, Borsík and Doboš characterized those functions that allow obtaining a metric in the Cartesian product of metric spaces by means of the aggregation of the metrics of each factor space. This question was also studied for norms by Herburt and Moszyńska. This aggregation procedure can be modified in order to construct a metric or a norm on a certain set by means of a family of metrics or norms, respectively. In this paper, we characterize the functions that allow merging an arbitrary collection of (asymmetric) norms defined over a vector space into a single norm (aggregation on sets). We see that these functions are different from those that allow the construction of a norm in a Cartesian product (aggregation on products). Moreover, we study a related topological problem that was considered in the context of metric spaces by Borsík and Doboš. Concretely, we analyze under which conditions the aggregated norm is compatible with the product topology or the supremum topology in each case.

Is there supersymmetric Lee–Yang fixed point in three dimensions?

The supersymmetric Lee–Yang model is arguably the simplest interacting supersymmetric field theory in two dimensions, albeit nonunitary. A natural question is if there is an analogue of supersymmetric Lee–Yang fixed point in higher dimensions. The absence of any [Formula: see text] symmetry (except for fermion numbers) makes it impossible to approach it by using perturbative [Formula: see text] expansions. We find that the truncated conformal bootstrap suggests that candidate fixed points obtained by the dimensional continuation from two dimensions annihilate below three dimensions, implying that there is no supersymmetric Lee–Yang fixed point in three dimensions. We conjecture that the corresponding phase transition, if any, will be the first-order transition.

Results in Strongly Minihedral Cone and Scalar Weighted Cone Metric Spaces and Applications

The convergence of sequences and non-unique fixed points are established in ℳ-orbitally complete cone metric spaces over the strongly mini-hedral cone, and scalar weighted cone assuming the cone to be strongly mini-hedral. Appropriate examples and applications validate the established theory. Further, we provide one more answer to the question of the existence of the contractive condition in Cone metric spaces so that the fixed point is at the point of discontinuity of a map. Also, we provide a negative answer to a natural question of whether the contractive conditions in the obtained results can be replaced by its metric versions.

Kemeny Consensus Complexity

The computational study of election problems generally focuses on questions related to the winner or set of winners of an election. But social preference functions such as Kemeny rule output a full ranking of the candidates (a consensus). We study the complexity of consensus-related questions, with a particular focus on Kemeny and its qualitative version Slater. The simplest of these questions is the problem of determining whether a ranking is a consensus, and we show that this problem is coNP-complete. We also study the natural question of the complexity of manipulative actions that have a specific consensus as a goal. Though determining whether a ranking is a Kemeny consensus is hard, the optimal action for manipulators is to simply vote their desired consensus. We provide evidence that this simplicity is caused by the combination of election system (Kemeny), manipulative action (manipulation), and manipulative goal (consensus). In the process we provide the first completeness results at the second level of the polynomial hierarchy for electoral manipulation and for optimal solution recognition.