Subsets S1 Research Articles

Can we recover an integral quadratic form by representing all its subforms?

Let o be the ring of integers of a number field. If f is a quadratic form over o and g is another quadratic form over o which represents all proper subforms of f, does g represent f? We show that if g is indefinite, then g indeed represents f. However, when f is positive definite and indecomposable, then there exists a g which represents all proper subforms of f but not f itself. Along the way we give a new characterization of positive definite decomposable quadratic forms over o and a number-field generalization of the finiteness theorem of representations of quadratic forms by quadratic forms over Z[13, Theorem 3.3] which asserts that given any infinite set S of classes of positive definite integral quadratic forms over o of a fixed rank, there exists a finite subset S0 of S with the property that a positive definite quadratic form over o represents all classes in S if and only if it represents all classes in S0.

Minimal universality criterion sets on the representations of binary quadratic forms

For a set S of (positive definite and integral) quadratic forms with bounded rank, a quadratic form f is called S-universal if it represents all quadratic forms in S. A subset S0 of S is called an S-universality criterion set if any S0-universal quadratic form is S-universal. We say S0 is minimal if there does not exist a proper subset of S0 that is an S-universality criterion set. In this article, we study various properties of minimal universality criterion sets. In particular, we show that for ‘most’ binary quadratic forms f, minimal S-universality criterion sets are unique in the case when S is the set of all subforms of the binary form f.

Forcing a □(κ)-like principle to hold at a weakly compact cardinal

Hellsten [17] proved that when κ is Πn1-indescribable, the n-club subsets of κ provide a filter base for the Πn1-indescribability ideal, and hence can also be used to give a characterization of Πn1-indescribable sets which resembles the definition of stationarity: a set S⊆κ is Πn1-indescribable if and only if S∩C≠∅ for every n-club C⊆κ. By replacing clubs with n-clubs in the definition of □(κ), one obtains a □(κ)-like principle □n(κ), a version of which was first considered by Brickhill and Welch [7]. The principle □n(κ) is consistent with the Πn1-indescribability of κ but inconsistent with the Πn+11-indescribability of κ. By generalizing the standard forcing to add a □(κ)-sequence, we show that if κ is κ+-weakly compact and GCH holds then there is a cofinality-preserving forcing extension in which κ remains κ+-weakly compact and □1(κ) holds. If κ is Π21-indescribable and GCH holds then there is a cofinality-preserving forcing extension in which κ is κ+-weakly compact, □1(κ) holds and every weakly compact subset of κ has a weakly compact proper initial segment. As an application, we prove that, relative to a Π21-indescribable cardinal, it is consistent that κ is κ+-weakly compact, every weakly compact subset of κ has a weakly compact proper initial segment, and there exist two weakly compact subsets S0 and S1 of κ such that there is no β<κ for which both S0∩β and S1∩β are weakly compact.

Quality evaluation of signals collected by portable ECG devices using dimensionality reduction and flexible model integration

Objective: Portable devices for collecting electrocardiograms (ECGs) and telemedicine systems for diagnosis are available to residents in deprived areas, but ECGs collected by non-professionals are not necessarily reliable and may impair the accuracy of diagnosis. We propose an algorithm for accurate ECG quality assessment, which can help improve the reliability of ECGs collected by portable devices. Approach: Using challenge data from CinC (2019), signals were classified as ‘acceptable’ and ‘unacceptable’ by annotators. The training set contained 998 12-lead ECGs and the test set contained 500. A 998 × 84 feature matrix, S, was formed by feature extraction and three basic models were obtained through training SVM, DT and NBC on S. The feature subsets S1, S2 and S3 were obtained by dimensionality reduction on S using SVM, DT and NBC, respectively. Three other basic models were obtained through training SVM on S1, DT on S2 and NBC on S3. By combining these six basic models, several integrated models were formed. An iterative method was proposed to select the integrated model with the highest accuracy on the training set. Having compared differences between the output labels and the original data labels, evaluation criteria were calculated. Main results: An accuracy of 98.70% and 98.60% was achieved on the training and test datasets, respectively. High F1 score and Kappa values were also obtained. Significance: The proposed algorithm has advantages over previously reported approaches during automatic assessment of ECG quality and can thus help to reduce reliance on highly trained professionals when assessing the quality of ECGs.

Algorithms for Average Regret Minimization

In this paper, we study a problem from the realm of multicriteria decision making in which the goal is to select from a given set S of d-dimensional objects a minimum sized subset S0 with bounded regret. Thereby, regret measures the unhappiness of users which would like to select their favorite object from set S but now can only select their favorite object from the subset S0. Previous work focused on bounding the maximum regret which is determined by the most unhappy user. We propose to consider the average regret instead which is determined by the sum of (un)happiness of all possible users. We show that this regret measure comes with desirable properties as supermodularity which allows to construct approximation algorithms. Furthermore, we introduce the regret minimizing permutation problem and discuss extensions of our algorithms to the recently proposed k-regret measure. Our theoretical results are accompanied with experiments on a variety of inputs with d up to 7.

Estimates of the mean size of the subset image under composition of random mappings

AbstractLet XNbe a set ofNelements andF1,F2,… be a sequence of random independent equiprobable mappings XN→N. For a subsetS0⊂ XN, |S0|=m, we consider a sequence of its imagesSt=Ft(…F2(F1(S0))…),t=1,2… An approach to the exact recurrent computation of distribution of |St| is described. Two-sided inequalities forM{|St|||S0|=m} such that the difference between the upper and lower bounds iso(m)form,t,N→ ∞,mt=o(N) are derived. The results are of interest for the analysis of time-memory tradeoff algorithms.

Set Partitions with Isolated Singletons

We give results for partitions of the set of the first n natural numbers into k nonempty subsets S1, S2, …, Sk with respect to the number of isolated elements, that is, with respect to integers a such that if a belongs to Si, then neither a − 1 nor a + 1 belongs to Si.

Limit theorem for the size of an image of subset under compositions of random mappings

Abstract Let 𝓧𝓝 be a set consisting of N elements and F 1,F 2, … be a sequence of random independent equiprobable mappings 𝓧𝓝 → 𝓧𝓝. For a subset S 0 ⊂ 𝓧𝓝, |S 0| = n, we consider a sequence of its images St = Ft (… F 2(F 1(S 0))…), t = 1, 2 … The conditions on n, t, N → ∞ under which the distributions of image sizes |St | are asymptotically connected with the standard normal distribution are presented.

Global Increase in Replication Fork Speed during a p57KIP2-Regulated Erythroid Cell Fate Switch

Cell cycle regulators are increasingly implicated in cell fate decisions such as the acquisition or loss of pluripotency and self-renewal potential. The cell cycle mechanisms that regulate these cell fate decisions are largely unknown. Here we studied an S phase- dependent cell fate switch in the erythroid fetal liver, in which murine early erythroid progenitors transition in vivo from a self-renewal state into a phase of active erythroid gene transcription and concurrent maturational cell divisions. In the fetal liver, this transition corresponds to the transition from subset S0 (CD71-low, Ter119-negative) to subset S1 (CD71-high, Ter119-negative). We found that the S0 to S1 transition takes place during an S phase that is abruptly shorter (decreasing from 7 hours to 4 hours). Further, self-renewing S0 cells uniquely express the cyclin-dependent kinase (CDK) inhibitor p57KIP2 during S phase. To investigate its potential role, we studied DNA replication in vitro and in vivo in p57KIP2 -deficient fetal liver progenitors, employing a variety of techniques, including DNA combing. We found that S0 erythroid progenitors are dependent on p57KIP2-mediated slowing of replication forks for self-renewal, either in vivo, or in dexamethasone-dependent expansion cultures in vitro. The switch from self-renewal in S0 to differentiation in wild-type S1 progenitors entails rapid downregulation of p57KIP2 with a consequent global increase in replication fork speed and an abruptly shorter S phase. In the absence of p57KIP2, replication fork processivity increases prematurely in self-renewing S0 cells, prior to the activation of the erythroid transcriptional program (Figure 1), resulting in replicative stress and cell death.It is well established that differentiation leads to reprogramming of DNA replication, reflected by changes to origin usage and to the timing of replication of chromatin domains. Here we find that the replication program is fundamentally altered in additional key respects: the global processivity of replication forks, regulated by CDK activity, increases abruptly with the switch from self-renewal to differentiation, affecting DNA synthesis rates and S phase duration. Our results are also of interest since the regulation of replication kinetics was thought to be primarily via the regulation of origin firing efficiency, rather than via fork processivity. Here we found no difference in the former (there was no significant change in inter-origin distances, Figure 1). While the full significance of faster forks to the activation of the erythroid transcriptional program is yet to be understood, a recent report found that T cell help leads to faster forks and a shorter S phase in B cells (Gitlin et al., Science 349, 643-646 2015). Regulation of global fork speed may therefore be an intrinsic part of physiological developmental programs. [Display omitted] DisclosuresNo relevant conflicts of interest to declare.

On the minimum number of mutually disjoint holes in planar point sets

Let P be a set of n points in general position in the plane. In 1996, Urabe considered a partition of P into subsets S1∪⋯∪Sl such that each Si forms a hole (or an empty convex polygon) of P and these holes are mutually disjoint. Let f(P) be the minimum number of holes over all such partitions of P and F(n)=max{f(P)} over all sets P of n points. Then the current best bounds are given by ⌈n+14⌉≤F(n)≤⌈5n18⌉. In this paper, we prove that F(n)≤⌈n4⌉+1.

On the commutative equivalence of semi-linear sets of [formula omitted

Given two subsets S1,S2 of Nk, we say that S1 is commutatively equivalent to S2 if there exists a bijection f:S1⟶S2 from S1 onto S2 such that, for every v∈S1, |v|=|f(v)|, where |v| denotes the sum of the components of v. We prove that every semi-linear set of Nk is commutatively equivalent to a recognizable subset of Nk.

Probability Bounds for Polynomial Functions in Random Variables

Random sampling is a simple but powerful method in statistics and in the design of randomized algorithms. In a typical application, random sampling can be applied to estimate an extreme value, say maximum, of a function f over a set S ⊆ ℝn. To do so, one may select a simpler (even finite) subset S0 ⊆ S, randomly take some samples over S0 for a number of times, and pick the best sample. The hope is to find a good approximate solution with reasonable chance. This paper sets out to present a number of scenarios for f, S and S0 where certain probability bounds can be established, leading to a quality assurance of the procedure. In our setting, f is a multivariate polynomial function. We prove that if f is a d-th order homogeneous polynomial in n variables and F is its corresponding super-symmetric tensor, and ξi (i = 1, 2, …, n) are i.i.d. Bernoulli random variables taking 1 or −1 with equal probability, then Prob{f(ξ1, ξ2, …, ξn) ≥ τn−d/2 ‖F‖1} ≥ θ, where τ, θ &gt; 0 are two universal constants and ‖·‖1 denotes the summation of the absolute values of all its entries. Several new inequalities concerning probabilities of the above nature are presented in this paper. Moreover, we show that the bounds are tight in most cases. Applications of our results in optimization are discussed as well.

The Erythropoietin Receptor Regulates The Number Of Cell Divisions and The Duration Of Erythroblast Terminal Differentiation By Regulating Erythroblast Iron

Signaling by the erythropoietin receptor (EpoR) is essential for the survival of definitive colony-forming unit-erythroid (CFU-e) progenitors and their erythroblast progeny. Here we used EpoR-null embryos to ask whether EpoR signaling might also exert essential non-survival functions in erythropoiesis.To address this, we rescued EpoR-null fetal liver cells from death by transducing them in vitro with either the anti-apoptotic protein Bcl-xL, or, as control, with the wild-type EpoR. The Bcl-xL-transduced EpoR-null cells survived, expressed hemoglobin and underwent morphological erythroid maturation and enucleation. However, unlike exogenous EpoR, exogenous Bcl-xL was unable to support the formation of EpoR-null CFU-e colonies in methylcellulose. The absence of colonies was explained by the finding that the Bcl-xL-transduced progenitors underwent fewer cell divisions than equivalent EpoR-transduced cells (1.1 vs. 2.9 in 24 hr, respectively) and had a slower rate of intra-S phase DNA synthesis, suggesting longer S phase duration. Multispectral imaging showed that the Bcl-xL-transduced cells matured prematurely, attaining smaller cell and nuclear size and a lower nuclear/cytoplasmic ratio at earlier time points than EpoR-transduced cells. Premature maturation was also evident by flow cytometric analysis. Thus, EpoR-null fetal liver cells in vivo arrest in their differentiation at the transition from subset S0 (Ter119-neg CD71-low) to S1 (Ter119-neg CD71-high) (Pop et al, PLoS Biology 2010). Rescue with EpoR in vitro allows EpoR-null progenitors to resume differentiation, sequentially upregulating CD71 and Ter119. By contrast, rescue of EpoR-null cells with Bcl-xL results in their premature upregulation of Ter119 and failure to upregulate CD71 to high levels. The cell cycle and differentiation deficits in Bcl-xL-supported, EpoR-null erythropoiesis were associated with a slower loss of DNA methylation from the erythroid genome, and with slower erythroid gene transcription.CD71 (the transferrin receptor) is a known target of EpoR and Stat5 signaling. We asked whether the deficits of EpoR-null erythropoiesis might be the result of low cell surface CD71 and the consequent reduced iron transport. In support of this hypothesis, we found that EpoR-null fetal liver cells that are transduced with both CD71 and Bcl-xL resume the normal maturation rate characteristic of EpoR-supported differentiation, as judged by multispectral imaging measurements of cell size and nuclear/cytoplasmic ratio. Further, we were able to restore rapid S phase to Bcl-xL-transduced EpoR-/- erythroblasts by culturing them in the presence of the cell-permeant iron chelator Fe-SIH (salicylaldehyde isonicotinoyl hydrazone), which is able to supply the cell interior with iron even in the absence of CD71 (Figure 1). We suggest that EpoR-mediated upregulation of CD71 at the onset of erythroid terminal differentiation determines the number and duration of erythroblast cell divisions by regulating iron homeostasis. [Display omitted] Disclosures:No relevant conflicts of interest to declare.

Connectivities for k-knitted graphs and for minimal counterexamples to Hadwigerʼs Conjecture

For a given subset S⊆V(G) of a graph G, the pair (G,S) is knitted if for every partition of S into non-empty subsets S1,S2,…,St, there are disjoint connected subgraphs C1,C2,…,Ct in G so that Si⊆Ci. A graph G is ℓ-knitted if (G,S) is knitted for all S⊆V(G) with |S|=ℓ. In this paper, we prove that every 9ℓ-connected graph is ℓ-knitted.Hadwigerʼs Conjecture states that every k-chromatic graph contains a Kk-minor. We use the above result to prove that the connectivity of minimal counterexamples to Hadwigerʼs Conjecture is at least k/9, which was proved to be at least 2k/27 in Kawarabayashi (2007) [4].

Hilbert’s Tenth Problem and Mazur’s Conjectures in Complementary Subrings of Number Fields

We show that Hilbert's Tenth Problem is undecidable for complementary sub- rings of number fields and that the p-adic and archimedean ring versions of Mazur's conjec- tures do not hold in these rings . More specifically, given a number field K, a positive integer t > 1, and t rational nonnegative numbers �1,...,�t whose sum is one, we prove that the nonarchimedean primes of K can be partitioned into t disjoint recursive subsets S1,...,St of densities �1,...,�t, respectively such that Hilbert's Tenth Problem is undecidable for each corresponding ring OK,Si. We also show that we can find a partition as above such that each ring OK,Si possesses an infinite Diophantine set which is discrete in every topology of the field. The only assumption on K we need is that there is an elliptic curve of rank one defined over K. Acknowledgments. Our thanks go to the London Mathematical Society for the very enjoy- able visit by the third author to the University of East Anglia. Also, to whichever barman invented the Marguerita.

On the complexity of reconstructing H-free graphs from their Star Systems

In the Star System problem we are given a set system and asked whether it is realizable by the multi-set of closed neighborhoods of some graph, i.e. given subsets S1, S2, …, Sn of an n-element set V does there exist a graph G = (V, E) with {N[v]: v∈V} = {S1, S2, …, Sn}? For a fixed graph H the H-free Star System problem is a variant of the Star System problem where it is asked whether a given set system is realizable by closed neighborhoods of a graph containing no H as an induced subgraph. We study the computational complexity of the H-free Star System problem. We prove that when H is a path or a cycle on at most four vertices the problem is polynomial time solvable. In complement to this result, we show that if H belongs to a certain large class of graphs the H-free Star System problem is NP-complete. In particular, the problem is NP-complete when H is either a cycle or a path on at least five vertices. This yields a complete dichotomy for paths and cycles. Copyright © 2010 John Wiley & Sons, Ltd. 68:113-124, 2011

Normal triangulations in o-minimal structures

AbstractLet be an o-minimal structure over a real closed field R. Given a simplicial complex K and some definable subsets S1, …, Sl of its realization ∣K∣ in R we prove that there exist a subdivision K′ of K and a definable triangulation φ′: ∣K′∣ → ∣K∣ of ∣K∣ partitioning S1, …, Sl with φ′ definably homotopic to id∣K∣. As an application of this result we obtain the semialgebraic Hauptvermutung.

A finiteness theorem for representability of quadratic forms by forms

In this article, Conway-Schneeberger's and Bhargava's results on representation positive integers by positive denite integral quadratric forms are fully generalized as follows : for any innite set S of positive denite integral quadratic forms of bounded rank, there is a nite subset S0 of S such that any positive denite integral quadratic form that represents every element of S0 represents all elements of S. Some interesting corollaries are obtained and applications to Fourier coecients of Siegel theta-series are introduced.

Revenue Management Under a General Discrete Choice Model of Consumer Behavior

Customer choice behavior, such as buy-up and buy-down, is an important phenomenon in a wide range of revenue management contexts. Yet most revenue management methodologies ignore this phenomenon—or at best approximate it in a heuristic way. In this paper, we provide an exact and quite general analysis of this problem. Specifically, we analyze a single-leg reserve management problem in which the buyers' choice behavior is modeled explicitly. The choice model is very general, simply specifying the probability of purchase for each fare product as a function of the set of fare products offered. The control problem is to decide which subset of fare products to offer at each point in time. We show that the optimal policy for this problem has a quite simple form. Namely, it consists of identifying an ordered family of “efficient“ subsets S1, …, Sm, and at each point in time opening one of these sets Sk, where the optimal index k is increasing in the remaining capacity x and decreasing in the remaining time. That is, the more capacity (or less time) available, the further the optimal set is along this sequence. We also show that the optimal policy is a nested allocation policy if and only if the sequence of efficient sets is nested, that is S1 ⊆ S2 ⊆ … ⊆Sm. Moreover, we give a characterization of when nesting by fare order is optimal. We also develop an estimation procedure for this setting based on the expectation-maximization (EM) method that jointly estimates arrival rates and choice model parameters when no-purchase outcomes are unobservable. Numerical results are given to illustrate both the model and estimation procedure.

Homogeneous and ultrahomogeneous Steiner systems

AbstractA Steiner system (or t — (v, k, 1) design) S is said to be homogeneous if, whenever the substructures induced on two finite subsets S1 and S2 of S are isomorphic, there is at least one automorphism of S mapping S1 onto S2, and is said to be ultrahomogeneous if each isomorphism between the substructures induced on two finite subsets of S can be extended to an automorphism of S. We give a complete classification of all homogeneous and ultrahomogeneous Steiner systems. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 153–161, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jcd.10034