Perfect Set Research Articles

Gruff ultrafilters

We investigate the question of whether Q carries an ultrafilter generated by perfect sets (such ultrafilters were called gruff ultrafilters by van Douwen). We prove that one can (consistently) obtain an affirmative answer to this question in three different ways: by assuming a certain parametrized diamond principle, from the cardinal invariant equality d=c, and in the Random real model.

Perfect independent sets with respect to infinitely many relations

We prove a result on perfect cliques with respect to countably many $$G_{\delta }$$Gź relations on a complete metric space. As an application, we show that a Polish group contains a free subgroup generated by a perfect set as long as it contains any uncountable free subgroup. This answers a recent question of Glaźb and Strobin.

Perfect Difference Number System

A new number system, Perfect Difference Number System (PDNS), based on the mathematical notion of Perfect Difference Sets (PDS) is proposed. It is expected that PDNS will take its place in the family of other number systems and will benefit computer theory and applications.

On cut sets of attractors of iterated function systems

In this paper, we study cut sets of attractors of iterated function systems (IFS) in R d \mathbb {R}^d . Under natural conditions, we show that all irreducible cut sets of these attractors are perfect sets or single points. This leads to a criterion for the existence of cut points of IFS attractors. If the IFS attractors are self-affine tiles, our results become algorithmically checkable and can be used to exhibit cut points with the help of Hata graphs. This enables us to construct cut points of some self-affine tiles studied in the literature.

Hardy–Rellich inequalities in domains of the Euclidean space

For test functions supported in a domain of the Euclidean space we consider the Hardy–Rellich inequality: ∫|Δf|2dx≥C2∫|f|2δ−4(x)dx, where C2=const≥0 and δ(x) is the distance from x to the boundary of the domain. M.P. Owen proved that this inequality is valid in any convex domain with C2=9/16 (M.P. Owen (1999) [21]). We examine the inequality in non-convex domains. It is proved that a positive constant C2 for a plane domain exists if and only if its boundary is a uniformly perfect set. For a domain of dimension d≥2 we prove that the inequality holds with the sharp constant C2=9/16, if the domain satisfies an exterior sphere condition with certain restriction on the radius of the sphere. In addition, we obtain similar results for the inequality ∫δ2(x)|Δf|2dx≥C2⁎∫|f|2δ−2(x)dx.

Code-Based Neighbor Discovery Protocols in Mobile Wireless Networks

In mobile wireless networks, the emerging proximity-based applications have led to the need for highly effective and energy-efficient neighbor discovery protocols. However, existing works cannot realize the optimal worst-case latency in the symmetric case, and their performances with asymmetric duty cycles can still be improved. In this paper, we investigate asynchronous neighbor discovery through a code-based approach, including the symmetric and asymmetric cases. We derive the tight worst-case latency bound in the case of symmetric duty cycle. We design a novel class of symmetric patterns called Diff-Codes, which is optimal when the Diff-Code can be extended from a perfect difference set. We further consider the asymmetric case and design ADiff-Codes. To evaluate (A)Diff-Codes, we conduct both simulations and testbed experiments. Both simulation and experiment results show that (A)Diff-Codes significantly outperform existing neighbor discovery protocols in both the median case and worst case. Specifically, in the symmetric case, the maximum worst-case improvement is up to 50%; in both symmetric and asymmetric cases, the median case gain is as high as 30%.

Models of optimum discrete signals on the ring combinatorial configurations

The innovative techniques for improving the quality indices of radio-signals for communications and radars with non-uniform structure (e.g. code sequences) with respect to error protection, using novel combinatorial configurations such as cyclic difference sets and Ideal Ring Bundles (IRB)s was regarded. Method for construction of optimum discrete signals, based on these models is proposed. IRBs are cyclic sequences of positive integers, which form perfect partitions of a finite interval [1,N]. The sums of connected sub-sequences of an IRB enumerate the set of integers [1,N-1] exactly R-times. This property makes IRBs useful in applications, which need to partition sets with the smallest possible number of intersections. The models of optimum discrete signals, having previously unknown favorable property, which hold for the same set of the IRBs in varieties permutations of its terms, named the "Glory to Ukraine Stars” have been indicated as a cluster of combinatorial configurations. Some algorithms and useful examples for constructing of optimum cyclic error-correcting codes presented. It shows that remarkable properties of IRBs have encoded in fine structure of circular symmetry and asymmetry ensembles. There are great classes of new two- and multidimensional IRBs, which being in excess classic models of optimum discrete signals with respect to number and combinatorial varieties. Indicate that the IRBs to be in exceed of classic perfect difference sets multiply.

Type‐II quasi‐cyclic low‐density parity‐check codes from Sidon sequences

From Sidon sequences, two explicit methods for constructing (2J, 2L)-regular type-II quasi-cyclic (QC) low-density parity-check (LDPC) codes are proposed without four cycles. The advantage of the new methods is two-fold: (i) compared with the existing lower bound of circulant permutation matrix (CPM) sizes above which type-II QC-LDPC codes exist without four cycles, the novel constructions enable much tighter lower bounds; and (ii) the codes constructed by the first method perform almost as well as type-II QC-LDPC codes from perfect cyclic difference set, while possessing a much more flexible CPM sizes.

DISTINGUISHING PERFECT SET PROPERTIES IN SEPARABLE METRIZABLE SPACES

AbstractAll spaces are assumed to be separable and metrizable. Our main result is that the statement “For every space X, every closed subset of X has the perfect set property if and only if every analytic subset of X has the perfect set property” is equivalent to b > ω1 (hence, in particular, it is independent of ZFC). This, together with a theorem of Solecki and an example of Miller, will allow us to determine the status of the statement “For every space X, if every Γ subset of X has the perfect set property then every Γ′ subset of X has the perfect set property” as Γ, Γ′ range over all pointclasses of complexity at most analytic or coanalytic.Along the way, we define and investigate a property of independent interest. We will say that a subset W of 2ω has the Grinzing property if it is uncountable and for every uncountable Y ⊆ W there exists an uncountable collection consisting of uncountable subsets of Y with pairwise disjoint closures in 2ω. The following theorems hold.(1)There exists a subset of 2ω with the Grinzing property.(2)Assume MA + ¬CH. Then 2ω has the Grinzing property.(3)Assume CH. Then 2ω does not have the Grinzing property.The first result was obtained by Miller using a theorem of Todorčević, and is needed in the proof of our main result.

Perfect set of Euler tours of K_{p,p,p}

Perfect set of Euler tours of K_{p,p,p}

Rellich type inequalities in domains of the Euclidean space

For smooth functions supported in a domain of the Euclidean space we investigate two Rellich type inequalities with weights which are powers of the distance function. We prove that for an arbitrary plane domain there exist positive Rellich constants in these inequalities if and only if the boundary of the domain is a uniformly perfect set. Moreover, we obtain explicit estimates of constants in function of geometric domain characteristics. Also, we find sharp constants in these Rellich type inequalities for all non-convex domains of dimension d ≥ 2 provided that the domains satisfy the exterior sphere condition with certain restriction on the radius of spheres.

Automatic brain hemorrhage segmentation and classification algorithm based on weighted grayscale histogram feature in a hierarchical classification structure

Brain hemorrhage is the first cause of death in ages between 15 and 24, and the third after heart diseases and cancers in other ages. Saving the lives of such patients completely depends on detecting the correct location and type of the hemorrhage in an early stage. In this paper, an automatic brain hemorrhage detection and classification algorithm on CT images is proposed. To achieve this purpose, after preprocessing, a modified version of Distance Regularized Level Set Evolution (MDRLSE) is used to detect and separate the hemorrhage regions. Then a perfect set of shape and texture features from each detected hemorrhage region are extracted. Moreover, we define a synthetic feature that is called weighted grayscale histogram feature. In this feature, valuable information from shape, position and area of the hemorrhage are integrated with the grayscale histogram of hemorrhage region. After that a synthetic feature selection algorithm is applied to select the most convenient features. Eventually, the segmented regions are classified into four types of the hemorrhages such as EDH, ICH, SDH and IVH by a hierarchical structure of classification. Our proposed algorithm is evaluated on a perfect set of CT-scan images and obtains the accuracy rate of 96.15%, 95.96% and 94.87% for the segmentation of the EDH, ICH, and SDH types, respectively. Also our proposed classification structure provides the accuracy rate of 92.46% and 94.13% for the first and second classifiers of the hierarchical classification structure for classifying the IVH from normal class and the EDH, ICH and SDH hemorrhage classes, respectively.

Three red herrings around Vaught’s conjecture

We give a model theoretic proof that if there is a counterexample to Vaught’s conjecture there is a counterexample such that every model of cardinality א1 is maximal (strengthening a result of Hjorth’s). We also give a new proof of Harrington’s theorem that any counterexample to Vaught’s conjecture has models in א1 of arbitrarily high Scott rank below א2. The three red herrings1 are false leads towards solving Vaught’s conjecture. Here is one strategy for establishing Vaught’s conjecture that there is no sentence of Lω1,ω that has exactly א1 countable models. Hjorth [9, 8], using descriptive set theoretic results of Mackey and others, has established that if there is a counterexample then there is one that has no model in א2. On the other hand, unpublished results of Harrington show that every counterexample has models with arbitrarily large Scott ranks below א2. This supports the notion that one might construct a model of an arbitrary counterexample that has cardinality א2. The resulting contradiction would yield the conjecture. In the introduction we give some background on the conjecture and describe the three red herrings. Recall that Vaught’s conjecture [19] concerns the number of countable models of a countable first-order theory, or more generally, of a sentence in the infinitary logic Lω1ω , where countable conjunctions and disjunctions but only finite strings of quantifiers are allowed for some countable vocabulary τ . The conjecture states: Vaught Conjecture. If φ is a sentence of Lω1ω then φ either has countably many or continuum-many countable models up to isomorphism. The more “absolute” version replaces “continuum-many” by “a perfect set of ” in the conclusion, where a perfect set of countable models is a perfect set of reals, each of which codes a countable model, such that distinct reals in the perfect set code non-isomorphic models. We say a sentence φ of Lω1ω is scattered if it does not have a perfect set of countable models. Morley [20] defined scattered as: for every countable ∗Research partially supported by Simons travel grant G5402 †Research supported by FWF (Austrian Science Fund) Grant P24654-N25. ‡Research supported by the Austrian Science Fund (FWF) Lise Meitner Grant M1410-N25 §Partially supported by NSF grant DMS-1308546 1See http://en.wikipedia.org/wiki/Red_herring for a fairly comprehensive account of the meaning of the red herring idiom.

Rational behavior under correlated uncertainty

In complete information games, Dekel and Fudenberg (1990) and Börgers (1994) have proposed the solution concept S∞W (one round of elimination of weakly dominated strategies followed by iterated elimination of strongly dominated strategies), motivating it by a characterization in terms of “approximate common certainty” of admissibility. We examine the validity of this characterization of S∞W in an incomplete information setting. We argue that in Bayesian games with a nontrivial state space, the characterization is very sensitive to the way in which uncertainty in the form of approximate common certainty of admissibility is taken to interact with the uncertainty already captured by players' beliefs about the states of nature: We show that S∞W corresponds to approximate common certainty of admissibility when this is not allowed to coincide with any changes to players' beliefs about states. If approximate common certainty of admissibility is accompanied by vanishingly small perturbations to beliefs, then S∞W is a (generally strict) subset of the predicted behavior, which we characterize in terms of a generalization of Hu's (2007) perfect p-rationalizable set.

Dido, Queen of Carthage by Globe Young Players (review)

Reviewed by: Dido, Queen of Carthageby Globe Young Players Miranda Fay Thomas Dido, Queen of CarthagePresented by the Globe Young Playersat the Sam Wanamaker Playhouse, Shakespeare’s Globe, London, England. April 9–18, 2015. Directed by Jacqui Somerville. Design by Mia Fiodquist. Music composed by Olly Fox. Musical Direction by Rob Millett. Choreography by Paul Harris. With Guy Amos (Aeneas), Alexander Clarke (Jupiter/Sergestus), Benjamin Clarke (Cupid), Martha Lily Dean (Ganymede/Ilioneus), Isaac Deayton (Achates), Ed Easton (Ascanius/Carthaginian Lord), Brogan Gilbert (Nurse/Carthaginian Lady), Jasmine Jones (Dido), Ben Lynn (Iarbus), Joseph Marshall (Cloanthus), Sekela Nancy Ngamilo (Anna), Yasmin Prince (Juno/Hermes), and Tamla Tutankhamun (Venus). The Sam Wanamaker Playhouse’s (SWP’s) second season closed, as its first season had done, with an effort from the Globe Young Players. Featuring the same cast of young actors as last year’s The Malcontent, this latest offering showed a pleasing level of maturation within the company while still retaining a sense of fun and play, despite the play’s generic expectation of tragedy. Clearly growing in confidence within the intimate, candlelit space of the SWP, the youngsters’ performance of Marlowe’s tragedy was, overall, far more successful than their first endeavor. This was largely due to the direction of Jacqui Somerville, who enabled the young actors to play on their youthfulness rather than trying to hide it. As we awaited the play’s beginning, the usual architecture and lighting of the SWP were augmented by the sight of grape vines creeping up the pillars to the left and right of the stage, indicating a more classical source than that of last year’s satirical comedy. All three stage doors were left ajar, tantalizing the audience’s gaze with glittering things in the dark beyond. The performance itself began with the young cast emerging out of the darkness, attired in the grays of school uniform complimented by more playful materials of silver and sequins. One boy, who we later discovered to be Cupid (Benjamin Clarke), had safety pins arranged in a pattern of wings on the back of his blazer, and a sling shot rather than a bow. The production consciously evoked the idea of a school play, and chose to acknowledge the youth of the actors and use it to their advantage. Seen in this light, the fake moustaches worn by the children were witty rather than twee. The musical instruments, from oboes to trumpets, played by the cast during the opening revealed that, unlike in many a school production, some of these youthful players had talent and nous beyond their years. [End Page 531] The direction by Jacqui Somerville was clever and sensitive, with the actors encouraged to make imaginative use of the space. Music composed by Olly Fox was deployed subtly and aptly, and helped unify the production, with chimes used to accompany the gods and goddesses as they executed their powers. The costumes, designed by Mia Fiodquist, were some of the most consistent seen at the SWP this season, with the Trojans entering in swish half capes, camouflage pants, and caps with feathers. They also had mud on their shins, and there was a sense of touching ambivalence as to whether the earth stains were redolent of the toil the soldiers had endured or the playground scrapes in which the children might have been embroiled. Jasmine Jones and Guy Amos more than justified their casting as Dido and Aeneas. Jones, in particular, had real stage presence as the Queen of Carthage: her regal orange satin gown embellished with shells, beads, and sequins supported her impressive sense of gravitas (Fig.12). Amos too showed increasing maturity in his portrayal of the Trojan hero. Aeneas’s woeful tale of Troy speech (see Marlowe, The Complete Plays, eds. Romany and Lindsey [Penguin, 2003], 2.1.121) is a perfect set piece for a young actor, and clearly written for a rising star capable of staring powerfully into the middle distance. Yet despite their ability to play beyond their years, the chemistry of the two remained enjoyably teenaged. Dido’s line, “I love thee not. / And yet I hate thee not” (3.1.170–71) came across as delightfully...

PRICE DISCRIMINATION AND PRICING-TO-MARKET BEHAVIOR OF BLACK SEA REGION WHEAT EXPORTERS

AbstractSubstantial changes in the world wheat market have resulted in a shift in the market shares of the main wheat exporting countries. Since 2002, Kazakhstan, Russia, and Ukraine (KRU) have become important wheat exporters on the world market, and their pricing behavior has become a vital issue. By applying the pricing-to-market model to wheat exports, this study analyzes the price-discriminating behavior of KRU wheat exports from 1996 to 2012. The results demonstrate that KRU are able to exercise price discrimination in some importing countries, but in most they either face perfect competition or set common markups in imperfectly competitive markets.

AXIOM I0 AND HIGHER DEGREE THEORY

AbstractIn this paper, we analyze structures of Zermelo degrees via a list of four degree theoretic questions (see §2) in various fine structure extender models, or under large cardinal assumptions. In particular we give a detailed analysis of the structures of Zermelo degrees in the Mitchell model for ω many measurable cardinals. It turns out that there is a profound correlation between the complexity of the degree structures at countable cofinality singular cardinals and the large cardinal strength of the relevant cardinals. The analysis applies to general degree notions, Zermelo degree is merely the author’s choice for illustrating the idea.I0(λ) is the assertion that there is an elementary embedding j : L(Vλ+1) → L(Vλ+1) with critical point < λ. We show that under I0(λ), the structure of Zermelo degrees at λ is very complicated: it has incomparable degrees, is not dense, satisfies Posner–Robinson theorem etc. In addition, we show that I0 together with a mild condition on the critical point of the embedding implies that the degree determinacy for Zermelo degrees at λ is false in L(Vλ+1). The key tool in this paper is a generic absoluteness theorem in the theory of I0, from which we obtain an analogue of Perfect Set Theorem for “projective” subsets of Vλ+1, and the Posner–Robinson follows as a corollary. Perfect Set Theorem and Posner–Robinson provide evidences supporting the analogy between $$AD$$ over L(ℝ) and I0 over L(Vλ+1), while the failure of degree determinacy is one for disanalogy. Furthermore, we conjecture that the failure of degree determinacy for Zermelo degrees at any uncountable cardinal is a theorem of $$ZFC$$.

Mining Sequential Patterns from mFUSP - Tree

Mining sequential patterns from sequence database has consequential responsibility in the data mining region as it can find the association from the ordered list of events. Mining methods that predicated on the pattern growth approach, such as PrefixSpan, are well-organized enough to denude the sequential patterns, but engendering a projection database for each pattern regards as bottleneck of these methods. Lin (2008) first commenced the concept of tree structure to sequential pattern mining, which is acknowledged as Fast updated sequential pattern tree (FUSP tree). However, link information stored in each node of FUSP tree structure increases the complication of this method due to its link updating process. In this paper, at first, we have proposed a modified fast updated sequential pattern tree (called a mFUSP tree) arrangement for storing the complete set of sequences with just frequent items, their frequencies and their relations among items in the given sequence into a compact data structure; excluding this tree structure avoids storing link information along to the next node of the following branch in the tree that carries the same item. Afterward, we have established by a mining method that our mFUSP tree structure is proficient enough to ascertain out the perfect set of frequent sequential patterns from sequence databases without generating any intermediate projected tree and without calling for repeated scanning of the original database during mining. Our experimental result proves that, the performance of our proposed mFUSP tree mining approach is a lot more trustworthy than other existing algorithms like GSP, PrefixSpan and FUSP tree based mining.

Triggering cascades on strongly connected directed graphs

Consider the following process of activation on a strongly connected directed graph G(V,E) with threshold function ϕ:V→N. In round zero, a set of vertices, called the seeds, are active. Thereafter, a vertex v∈V is activated in a round if it has at least ϕ(v) active in-neighbors in the previous round. Once a vertex is activated, it remains active. Sets of seeds activating all vertices after a finite number of rounds are known in the literature as irreversible dynamic monopolies (a.k.a. perfect target sets) corresponding to (G,ϕ). With ϕ(v)=⌈ρdeg−(v)⌉ for all v∈V, where deg−(v) denotes the indegree of v and ρ∈(0,1], this paper proves the existence of an irreversible dynamic monopoly of size no more than max⁡{4.92ρ|V|,1}.

Combination multinitrogen with good oxygen balance: molecule and synthesis design of polynitro-substituted tetrazolotriazine-based energetic compounds.

We investigated 5,8-dinitro-5,6,7,8-tetrahydrotetrazolo[1,5-b][1,2,4]triazine (short for DNTzTr (1)) using various ab initio quantum chemistry methods. We proposed an additional three novel polynitro-substituted tetrazolotriazine-based compounds with exceptional performance, including 5,8-dinitro-5,6-dioxotetrazolo[1,5-b][1,2,4]triazine, DNOTzTr (2), 4,5,9,10-tetranitro[1,2,4,5]tetrazolo[3,4-b][1,2,4,5]tetrazolo[3',4':5,6]triazino[2,3-e]triazine, TNTzTr (3), and 4,5,6,10,11,12-hexanitro-bis[1,2,4,5]tetrazolo[3',4':5,6]triazino[2,3-b:2',3'-e]triazine, HNBTzTr (4). The optimized structure, electronic density, natural bond orbital (NBO) charges and HOMO-LUMO orbitals, electrostatic potential on surface of molecule, IR- and NMR-predicted spectra, as well as thermochemical parameters were calculated with the B3LYP/6-311+G(2d) level of theory. Critical parameters such as density, enthalpy of formation (EOF), and detonation performance have also been predicted. Characters with positive EOF (1386.00 and 1625.31 kJ/mol), high density (over 2.00 g/cm(3)), outstanding detonation properties (D = 9.82 km/s, P = 45.45 GPa; D = 9.94 km/s, P = 47.30 GPa), the perfect oxygen balance set to zero, and acceptable impact sensitivity led novel compounds 3 and 4 to be very promising energetic materials. This work provides the theoretical molecule design and a reasonable synthesis path for further experimental synthesis and testing.