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This paper elaborates on a sieving technique that has first been applied in 2018 for improving bounds on deterministic integer factorization. We will generalize the sieve in order to obtain a polynomial-time reduction from integer factorization to a specific instance of the multiple choice subset-sum problem. As an application, we will improve upon special purpose factorization algorithms for integers composed of divisors with small difference. In particular, we will refine the runtime complexity of Fermat's factorization algorithm by a large subexponential factor. Our first procedure is deterministic, rigorous, easy to implement and has negligible space complexity. Our second procedure is heuristically faster than the first, but has non-negligible space complexity. [ABSTRACT FROM AUTHOR]

In recent years, multi-stable origami structures have garnered increasing attention for their applications in dynamic scenarios such as robotic arm motions, impact energy absorption, and spectrum gap regulation. Understanding the intricate working mechanisms and exploring the rich dynamics of these structures necessitate the development of dynamic models. However, existing dynamic modeling methods for origami structures are often cumbersome, and the resulting dynamic models often lack interpretability. To overcome these limitations, we propose a novel data-driven dynamic modeling approach based on B-spline Galerkin method and subset selection strategy. This approach directly captures the dynamics of multi-stable origami structures using measured data, eliminating the need for reliance on empirical or prior knowledge. To validate the effectiveness of the proposed approach, we first evaluate it on the Duffing system, which has explicit expressions, successfully reconstructing the governing equation. Subsequently, we apply this method to the dynamic modeling of the origami ball structure with tri-stability and the multi-cell stacked Miura-origami (SMO) structure with high dimensionality through simulation, showing favorable results. Finally, using experimental data collected from a bi-stable SMO structure prototype, we employ the proposed method to obtain a global model that can accurately predict different dynamic behaviors over a broad range of excitation frequencies, including intra-well periodic vibrations, inter-well periodic vibrations, and inter-well chaotic vibrations. Overall, our method showcases outstanding efficacy in formulating interpretable, manageable, and comprehensive dynamic models. It plays a pivotal role in delving into the intricate dynamics of multi-stable origami structures. [ABSTRACT FROM AUTHOR]

Building upon the exact methods presented in our earlier work (2022) [5] , we introduce a heuristic approach for the star discrepancy subset selection problem. The heuristic gradually improves the current-best subset by replacing one of its elements at a time. While it does not necessarily return an optimal solution, we obtain promising results for all tested dimensions. For example, for moderate sizes 30 ≤ n ≤ 240 , we obtain point sets in dimension 6 with L ∞ star discrepancy up to 35% better than that of the first n points of the Sobol' sequence. Our heuristic works in all dimensions, the main limitation being the precision of the discrepancy calculation algorithms. We provide a comparison with an energy functional introduced by Steinerberger (2019) [31] , showing that our heuristic performs better on all tested instances. Finally, our results give further empirical information on inverse star discrepancy conjectures. [ABSTRACT FROM AUTHOR]

The spleen represents an important contributor to tumor immune escape, but the relevance of increased splenic metabolic activity remains to be fully elucidated. We retrospectively measured the spleen-to-liver standard uptake value (SLR) on 18F-FDG PET/CT examinations of 92 consecutive patients with FIGO stage IB1 to IVA cervical cancer and integrated the results with survival, response to treatment, tumor immune infiltrate, and baseline characteristics. SLR max > 0.92 (p =.026) and SLR mean > 0.94 (p =.005) were significantly associated with decreased DFS in univariable analysis. Multivariable models were built using best subset selection; ΔSLR max and either SLR max or SLR mean were consistently selected, strongly reinforcing the association between SLR variables and DFS in relation to potential confounders (all models p ≤.002). Independent associations were found for SLR max using multivariable Cox regression models for DFS (all p ≤.003). Further, uni- and multivariable analyses demonstrated the negative impact of higher SLR values on pathological complete response. A statistically significant higher proportion of patients with high SLR max had a dense infiltrate of CD20+ (p =.036) and CD68+ (p =.015) immune cells, as well as PD-L1+ tumor cells (p =.019) as compared to those with low SLR max. Finally, high SLR max status was neither associated with systemic inflammatory markers (except for an increased white blood cell count; p =.038), nor with clinically overt infection. This hypothesis-generating study provides the first evidence that increased splenic metabolic activity is a negative prognostic and predictive biomarker in locally advanced cervical cancer. In addition, it might help to discriminate immunologically 'hot' from 'cold' cervical tumors. • Splenic metabolic activity was a prognostic factor for recurrence in cervical cancer. • Splenic metabolic activity was a predictive factor for pathological complete response in cervical cancer. • Splenic metabolic activity was associated with the cervical tumor immune infiltrate. [ABSTRACT FROM AUTHOR]

This note discusses an interesting matrix factorization called the CUR Decomposition. We illustrate various viewpoints of this method by comparing and contrasting them in different situations. Additionally, we offer a new characterization of CUR decompositions which synergizes these viewpoints and shows that they are indeed the same in the exact decomposition case. [ABSTRACT FROM AUTHOR]

A significant issue that has plagued data-driven Vibration-based Structural Health Monitoring (VSHM) is the mitigation of Environmental and Operational Variations (EOVs). The Damage Sensitive Features (DSFs) that are obtained from the vibration response of the structure are influenced by EOVs. Regression analysis, such as multivariate nonlinear regression, can be used to create relationships between Environmental and Operational Parameters (EOPs) and the DSFs, with EOV-insensitive DSFs created by taking the regression residuals. Inherent issues, originating from nuances in their design, exist within the design of the regression models, following from the overall uncertainty and redundancy in predictors and explained variables, leading to poor performance. To overcome this, a comprehensive nonlinear stepwise regression methodology has been developed to scour the regression models of as much uncertainty as possible. The proposed methodology addresses a number of crucial ideas: removing co-nonlinear variables, identifying the most influential EOPs, facilitating the selection of compact regression bases and determining which DSFs should be regressed. Robust DSFs are created by combining non-regressed DSFs with critically thought-out regressed DSFs. Ultimately, reducing the uncertainty within the models will lead to more confidence in the decision making within a VSHM methodology. • Long-term monitoring of an operational wind turbine blade with artificial damage. • Mitigation of environmental and operational variations for robust damage detection. • Nonlinear forward stepwise method for automatic subset and model order selection. • Informed combination of regressed and non-regressed damage sensitive features. [ABSTRACT FROM AUTHOR]

Each (equigenerated) squarefree monomial ideal in the polynomial ring S = K [ x 1 , ... , x n ] represents a family of subsets of [ n ] , called a (uniform) clutter. In this paper, we introduce a class of uniform clutters, called decomposable clutters, whose associated ideal has linear quotients and hence linear resolution over all fields. We show that chordality of these clutters guarantees the correctness of a conjecture raised by R.S. Simon [23] on extendable shellebility of d -skeletons of a simplex 〈 [ n ] 〉 , for all d. We then prove this conjecture for d ≥ n − 3. [ABSTRACT FROM AUTHOR]

We provide geometric methods and algorithms to verify, construct and enumerate pairs of words (of specified length over a fixed m -letter alphabet) that form identities in the semigroup UT n (T) of n × n upper triangular tropical matrices. In the case n = 2 these identities are precisely those satisfied by the bicyclic monoid, whilst in the case n = 3 they form a subset of the identities which hold in the plactic monoid of rank 3. To each word we associate a signature sequence of lattice polytopes, and show that two words form an identity for UT n (T) if and only if their signatures are equal. Our algorithms are thus based on polyhedral computations and achieve optimal time complexity in some cases. For n = m = 2 we prove a Structural Theorem, which allows us to quickly enumerate the pairs of words of fixed length which form identities for UT 2 (T). This allows us to recover a short proof of Adjan's theorem on minimal length identities for the bicyclic monoid, and to construct minimal length identities for UT 3 (T) , providing counterexamples to a conjecture of Izhakian in this case. We conclude with six conjectures at the intersection of semigroup theory, probability and combinatorics, obtained through analysing the outputs of our algorithms. [ABSTRACT FROM AUTHOR]

Abstract The aim of this paper is two-fold. We first prove several new interpretations of a kind of (q , t) -Catalan numbers along with their corresponding γ -expansions using pattern avoiding permutations. Secondly, we give a complete characterization of certain (− 1) -phenomenon for each subset of permutations avoiding a single pattern of length three, and discuss their q -analogues utilizing the newly obtained q - γ -expansions, as well as the continued fraction of a quint-variate generating function due to Shin and the fourth author. Moreover, we enumerate the alternating permutations avoiding simultaneously two patterns, namely (2413 , 3142) and (1342 , 2431) , of length four, and consider such (− 1) -phenomenon for these two subsets as well. [ABSTRACT FROM AUTHOR]

Abstract We consider random sampling in finitely generated shift-invariant spaces V (Φ) ⊂ L 2 (R n) generated by a vector Φ = (φ 1 , ... , φ r) ∈ (L 2 (R n)) r. Following the approach introduced by Bass and Gröchenig, we consider certain relatively compact subsets V R , δ (Φ) of such a space, defined in terms of a concentration inequality with respect to a cube with side lengths R. Under very mild assumptions on the generators, we show that for R sufficiently large, taking O (R n l o g (R)) many random samples (taken independently uniformly distributed within C R) yields a sampling set for V R , δ (Φ) with high probability. We give explicit estimates of all involved constants in terms of the generators φ 1 , ... , φ r. [ABSTRACT FROM AUTHOR]

Abstract Given a Furstenberg family F of subsets of N , an operator T on a topological vector space X is called F -transitive provided for each non-empty open subsets U , V of X the set { n ∈ Z + : T n (U) ∩ V ≠ ∅ } belongs to F. We classify the topologically transitive operators with a hierarchy of F -transitive subclasses by considering families F that are determined by various notions of largeness and density in Z +. [ABSTRACT FROM AUTHOR]

In this paper, we define the class of ( α , β ) -nonexpansive mappings which is properly larger than the class of α -nonexpansive mappings and prove that every ( α , β ) -nonexpansive mapping T : C → C has an approximate fixed point sequence, where C is a nonempty bounded subset of a Banach space X , α > 0 and β ≥ 0 . This, in particular, gives an affirmative answer to the open question posed by Ariza-Ruiz and et al. concerning the existence of an approximate fixed point sequence for α -nonexpansive mappings, Ariza-Ruiz et al. (2016) [4] . [ABSTRACT FROM AUTHOR]

Highlights • Prototype learning is formulated as a structurally regularized model. • Intrinsic structures of dataset are exploited via l p , 1 -norm grouping. • The prototypes are learnt more representatively and diversely. • An alternative re-weighting solver is presented to solve the proposed model. • The proposed method works well on video summarization and classification tasks. Abstract Prototype learning aims to eliminate redundancy of large-scale data by selecting an informative subset. It is at the center of visual data analysis and processing. However, due to intrinsic structures among sample groups, the learnt prototypes are generally less representative and diversified. To alleviate this issue, we develop in this paper a structurally regularized model via ℓ p , 1 -norm grouping, in which both the intra-group and inter-group structures of source data in object-space are rationally exploited. Thus, while the learnt representative prototypes are prone to distribute in different groups at the inter-group level, the grouping constraint via ℓ p , 1 -norm will enforce the greatest diversity for intra-group prototypes. Considering the convexity in the formulated model, an alternative re-weighting solver is presented to efficiently solve the proposed optimization problem. Experimental results on video summarization, scene categorization and handwriting recognition demonstrate that the proposed method is considerably superior to the state-of-the-art methods in prototype learning. [ABSTRACT FROM AUTHOR]

The cop number of a graph G is the smallest k such that k cops win the game of cops and robber on G . We investigate the maximum cop number of geometric intersection graphs, which are graphs whose vertices are represented by geometric shapes and edges by their intersections. We establish the following dichotomy for previously studied classes of intersection graphs: • The intersection graphs of arc-connected sets in the plane (called string graphs ) have cop number at most 15, and more generally, the intersection graphs of arc-connected subsets of a surface have cop number at most 10 g + 15 in case of orientable surface of genus g , and at most 10 g ′ + 15 in case of non-orientable surface of Euler genus g ′ . For more restricted classes of intersection graphs, we obtain better bounds: the maximum cop number of interval filament graphs is two, and the maximum cop number of outer-string graphs is between 3 and 4. • The intersection graphs of disconnected 2-dimensional sets or of 3-dimensional sets have unbounded cop number even in very restricted settings. For instance, it follows from known results that the cop number is unbounded on intersection graphs of two-element subsets of a line. We further show that it is also unbounded on intersection graphs of 3-dimensional unit balls, of 3-dimensional unit cubes or of 3-dimensional axis-aligned unit segments. [ABSTRACT FROM AUTHOR]

Abbott et al.'s notion of containers characterises a subset of parametric data types which can be described by a set of shapes and set of positions for each shape. This captures common data types such as tuples, lists, trees, arrays, and graphs. Various useful categorical structures can be derived for containers given additional information. For example, directed containers give rise to container comonads (Ahman et al.). Containers provide a useful reasoning principle and asbtraction mechanism for programming. This paper studies the performance characteristics of traversal schemes over containers, modelled by functor and comonad structures. A cost model for container transformations is defined from which complexity bounds for the operations of container functors and comonads are derived. These bounds suggest optimisations for programs structured using these idioms. The abstract interface provided by the syntax of containers and category theory provides complexity bounds and subsequent optimisations that are implementation agnostic (machine free). [ABSTRACT FROM AUTHOR]

Hypersimplices are well-studied objects in combinatorics, optimization, and representation theory. For each hypersimplex, we define a new family of subpolytopes, called r -stable hypersimplices, and show that a well-known regular unimodular triangulation of the hypersimplex restricts to a triangulation of each r -stable hypersimplex. For the case of the second hypersimplex defined by the two-element subsets of an n -set, we provide a shelling of this triangulation that sequentially shells each r -stable sub-hypersimplex. In this case, we utilize the shelling to compute the Ehrhart h ⁎ -polynomials of these polytopes, and the hypersimplex, via independence polynomials of graphs. For one such r -stable hypersimplex, this computation yields a connection to CR mappings of Lens spaces via Ehrhart–MacDonald reciprocity. [ABSTRACT FROM AUTHOR]

We consider the action of the 2-dimensional projective special linear group P S L ( 2 , q ) on the projective line P G ( 1 , q ) over the finite field F q , where q is an odd prime power. A subset S of P S L ( 2 , q ) is said to be an intersecting family if for any g 1 , g 2 ∈ S , there exists an element x ∈ P G ( 1 , q ) such that x g 1 = x g 2 . It is known that the maximum size of an intersecting family in P S L ( 2 , q ) is q ( q − 1 ) / 2 . We prove that all intersecting families of maximum size are cosets of point stabilizers for all odd prime powers q > 3 . [ABSTRACT FROM AUTHOR]

We investigate the regularity of random attractors for the non-autonomous non-local fractional stochastic reaction–diffusion equations in H s ( R n ) with s ∈ ( 0 , 1 ) . We prove the existence and uniqueness of the tempered random attractor that is compact in H s ( R n ) and attracts all tempered random subsets of L 2 ( R n ) with respect to the norm of H s ( R n ) . The main difficulty is to show the pullback asymptotic compactness of solutions in H s ( R n ) due to the noncompactness of Sobolev embeddings on unbounded domains and the almost sure nondifferentiability of the sample paths of the Wiener process. We establish such compactness by the ideas of uniform tail-estimates and the spectral decomposition of solutions in bounded domains. [ABSTRACT FROM AUTHOR]

Wright's conjecture states that the origin is the global attractor for the delay differential equation y ′ ( t ) = − α y ( t − 1 ) [ 1 + y ( t ) ] for all α ∈ ( 0 , π 2 ] when y ( t ) > − 1 . This has been proven to be true for a subset of parameter values α . We extend the result to the full parameter range α ∈ ( 0 , π 2 ] , and thus prove Wright's conjecture to be true. Our approach relies on a careful investigation of the neighborhood of the Hopf bifurcation occurring at α = π 2 . This analysis fills the gap left by complementary work on Wright's conjecture, which covers parameter values further away from the bifurcation point. Furthermore, we show that the branch of (slowly oscillating) periodic orbits originating from this Hopf bifurcation does not have any subsequent bifurcations (and in particular no folds) for α ∈ ( π 2 , π 2 + 6.830 × 10 − 3 ] . When combined with other results, this proves that the branch of slowly oscillating solutions that originates from the Hopf bifurcation at α = π 2 is globally parametrized by α > π 2 . [ABSTRACT FROM AUTHOR]

Background The presence of a trauma surgeon during patient resuscitations is required at most American College of Surgeons–verified trauma centers despite little evidence showing improved patient outcomes in the less-than-critically injured (Tier 2) trauma patients. This study was designed to identify the impact of extending required surgeon response times on outcomes in tier 2 trauma patients. Methods An American College of Surgeons–verified level 2 trauma center extended the maximum allowed surgeon response time for tier 2 activations from 60 min to 120 min on November 1, 2011. Surgeon response time and patient outcomes of the retrospective control group (January 1, 2008-October 31, 2011) were then compared with the prospective test group (November 1, 2011-December 31, 2014). Primary outcomes included mortality and hospital length of stay (HLOS). Secondary outcomes were emergency department length of stay, and time from ED arrival to CT scan. A subset analysis of all patients evaluated by a surgeon within 60 min of arrival versus those evaluated by a surgeon after 60 min was also performed. Results The control and test groups were composed of 757 and 792 patients, and their mean injury severity score was 9.0 and 6.0, respectively. Emergency department length of stay showed a statistically significant increase of 12 min, whereas HLOS was unchanged throughout the study. Mortality was not significantly different between the groups. Subset analysis revealed a median surgeon arrival time of 15 min in the <60-min group and 85 min in the >60-min group, whereas the injury severity score, HLOS, and mortality were not significantly different between these subsets. No correlation existed between these outcomes and surgeon arrival time. Conclusions Doubling required surgeon response time in tier 2 trauma patients does not produce negative outcomes in this patient group. Mandatory surgeon response times in similar patient groups can be re-evaluated to allow for greater flexibility of a limited surgeon workforce while still providing safe care. [ABSTRACT FROM AUTHOR]

To efficiently execute the variance-based global sensitivity analysis, the law of total variance in the successive intervals without overlapping is proved at first, on which an efficient space-partition sampling-based approach is subsequently proposed in this paper. Through partitioning the sample points of output into different subsets according to different inputs, the proposed approach can efficiently evaluate all the main effects concurrently by one group of sample points. In addition, there is no need for optimizing the partition scheme in the proposed approach. The maximum length of subintervals is decreased by increasing the number of sample points of model input variables in the proposed approach, which guarantees the convergence condition of the space-partition approach well. Furthermore, a new interpretation on the thought of partition is illuminated from the perspective of the variance ratio function. Finally, three test examples and one engineering application are employed to demonstrate the accuracy, efficiency and robustness of the proposed approach. [ABSTRACT FROM AUTHOR]

It is shown that certain lower semi-continuous maps from a paracompact space to the family of closed subsets of the bundle space of a Banach bundle admit continuous selections. This generalization of the theorem of Douady, dal Soglio-Herault, and Hofmann on the fullness of Banach bundles has applications to establishing conditions under which the induced maps between the spaces of sections of Banach bundles are onto. Another application is to a generalization of the theorem of Bartle and Graves [3] for Banach bundle maps that are onto their images. Other applications of the selection theorem are to the study of the M-ideals of the space of bounded sections begun in [4] and continued in [9] . A class of Banach bundles that generalizes the class of trivial Banach bundles is introduced and some properties of these Banach bundles are discussed. [ABSTRACT FROM AUTHOR]

Let E be a subset of an integral domain D with quotient field K . A subset S of E is said to be an n -universal subset of E if every integer-valued polynomial f ( X ) ∈ K [ X ] on S (that is, such that f ( S ) ⊆ D ), with degree at most n , is integer-valued on E (that is, f ( E ) ⊆ D ). A sequence a 0 , … , a n of elements of E is said to be a Newton sequence of E of length n if, for each k ≤ n , the subset { a 0 , … , a k } is a k -universal subset of E . Our main results concern the case where D is a Dedekind domain, where both notions are strongly linked to p -orderings, as introduced by Bhargava. We extend and strengthen previous studies by Volkov, Petrov, Byszewski, Fra̧czyk, and Szumowicz that concerned only the case where E = D . In this case, but also if E is an ideal of D , or if E is the set of prime numbers > n + 1 (in D = Z ), we prove the existence of sequences in E of which n + 2 consecutive terms always form an n -universal subset of E . [ABSTRACT FROM AUTHOR]

Let p be a prime number and let q = p r . If C and D are large subsets of F q ⁎ we study the trace of products cd with c ∈ C and d ∈ D and show that it is well distributed in F p . We give an optimal condition (up to an absolute constant factor) on the size of the subsets C and D to ensure that the trace of products cd takes any given value in F p . We also give a condition (optimal up to an absolute constant factor in most cases) on the size of the subsets C and D to ensure that the trace of cd meets the set of k -th powers for k ≥ 1 , respectively the set of generators. Our method will enable us to take sets C and D whose size is substantially below q . Character sums and Gaussian sums over F p and F q will play an important role in the proofs. Some estimates lead to interesting combinatorial questions in finite fields. [ABSTRACT FROM AUTHOR]

The subset sum problem is an important theoretical problem with many applications, such as in coding theory, cryptography, graph theory and other fields. One of the many aspects of this problem is to answer the solvability of the k -subset sum problem. It has been proven to be NP-hard in general. However, if the evaluation set has some special algebraic structure, it is possible to obtain some good conclusions. Zhu, Wan and Keti proposed partial results of this problem over two special kinds of evaluation sets. We generalize their conclusions in this paper, and propose asymptotical results of the solvability of the k -subset sum problem by using estimates of additive character sums over the evaluation set, together with the Brun sieve and the new sieve proposed by Li and Wan. We also apply the former two examples as application of our results. [ABSTRACT FROM AUTHOR]

For integers n ≥ k ≥ 2 let m ( n , k ) denote the maximum size of a family of subsets of an n -set without k pairwise disjoint members. In an important classical paper Kleitman determined m ( n , k ) for n ≡ 0 or −1 (mod k ). We present a simple short proof of this result and prove the uniqueness of the optimal families as well. [ABSTRACT FROM AUTHOR]

In this paper, we study the Heisenberg uniqueness pairs corresponding to a finite number of parallel lines Γ. We give a necessary condition and a sufficient condition for a subset Λ of R 2 so that ( Γ , Λ ) becomes a HUP. [ABSTRACT FROM AUTHOR]

Higher order Bernstein- and Markov-type inequalities are established for trigonometric polynomials on compact subsets of the real line and for algebraic polynomials on compact subsets of the unit circle. In the case of Markov-type inequalities we assume that the compact set satisfies an interval condition. [ABSTRACT FROM AUTHOR]

Let G be an abelian group (written additively), X be a subset of G and S be a minimal zero-sum sequence over X . S is called unsplittable in X if there do not exist an element g in S and two elements x , y in X such that g = x + y and the new sequence S g − 1 x y is still a minimal zero-sum sequence. In this paper, we mainly investigate the case when G = Z and X = 〚 − m , n 〛 with m , n ∈ N . We obtain the structure of unsplittable minimal zero-sum sequences of length at least n + ⌊ m ∕ 2 ⌋ + 2 provided that n ≥ m 2 ∕ 2 − 1 and m ≥ 6 . As a corollary, the Davenport constant D ( 〚 − m , n 〛 ) is determined when n ≥ m 2 ∕ 2 − 1 . The Davenport constant D ( X ) for a general set X ⊂ Z is also discussed. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a complete description of the structure of finite subsets S of torsion-free nilpotent groups of class 2 satisfying | S 2 | = 3 | S | − 2 . In view of results in [12], this gives a complete description of the structure of finite subsets with the above property in any torsion-free nilpotent group. [ABSTRACT FROM AUTHOR]

Consider 2 n points on the unit circle and a reference dissection D ∘ of the convex hull of the odd points. The accordion complex of D ∘ is the simplicial complex of subsets of pairwise noncrossing diagonals with even endpoints that cross a connected set of diagonals of the dissection D ∘ . In particular, this complex is an associahedron when D ∘ is a triangulation, and a Stokes complex when D ∘ is a quadrangulation. We exhibit a bijection between the facets of the accordion complex of D ∘ and some dual objects called the serpent nests of D ∘ . This confirms in particular a prediction of F. Chapoton (2016) in the case of Stokes complexes. [ABSTRACT FROM AUTHOR]

The classic Rosenthal–Lacey theorem asserts that the Banach space C ( K ) of continuous real-valued maps on an infinite compact space K has a quotient isomorphic to c or ℓ 2 . More recently, Ka̧kol and Saxon [20] proved that the space C p ( K ) endowed with the pointwise topology has an infinite-dimensional separable quotient algebra iff K has an infinite countable closed subset. Hence C p ( β N ) lacks infinite-dimensional separable quotient algebras. This motivates the following question: (⁎) Does C p ( K ) admit an infinite-dimensional separable quotient (shortly SQ ) for any infinite compact space K? Particularly, does C p ( β N ) admit SQ ? Our main theorem implies that C p ( K ) has SQ for any compact space K containing a copy of β N . Consequently, this result reduces problem (⁎) to the case when K is an Efimov space (i.e. K is an infinite compact space that contains neither a non-trivial convergent sequence nor a copy of β N ). Although, it is unknown if Efimov spaces exist in ZFC, we show, making use of some result of R. de la Vega (2008) (under ◊), that for some Efimov space K the space C p ( K ) has SQ . Some applications of the main result are provided. [ABSTRACT FROM AUTHOR]

In this paper, for a finite subset A ⊂ { 2 , 3 , ⋯ } , we introduce the notion of longest block function L n ( x , A ) for the Lüroth expansion of x ∈ [ 0 , 1 ) with respect to A and consider the asymptotic behavior of L n ( x , A ) as n tends to ∞. We also obtain the Hausdorff dimensions of the level sets and exceptional set arising from the longest block function. [ABSTRACT FROM AUTHOR]

We show that for a real-analytic connected holomorphically nondegenerate 5-dimensional CR-hypersurface M and its symmetry algebra s one has either: (i) dim ⁡ s = 15 and M is spherical (with Levi form of signature either ( 2 , 0 ) or ( 1 , 1 ) everywhere), or (ii) dim ⁡ s ≤ 11 where dim ⁡ s = 11 can only occur if on a dense open subset M is spherical with Levi form of signature ( 1 , 1 ) . Furthermore, we construct a series of examples of pairwise nonequivalent CR-hypersurfaces with dim ⁡ s = 11 . [ABSTRACT FROM AUTHOR]

In this paper, we obtain the lower dimension formula of compact subsets in the doubling metric space. Using this result, we give the lower dimension formula of the homogeneous sets. We also study the lower dimensions of Moran sets under the suitable condition. [ABSTRACT FROM AUTHOR]

This manuscript is devoted to the study of the concept of a generating subset (a.k.a. Hopf image of a morphism) in the setting of locally compact quantum groups. The aim of this paper is to provide an accurate description of the Hopf image of a given morphism. We extend and unify the previously existing approaches for compact and discrete quantum groups and present some results that can shed light on some local perspective in the theory of quantum groups. In particular, we provide a characterization of fullness of Hopf image in the language of partial actions as well as in representation-theoretic terms in the spirit of representation C ⁎ -categories, extending some known results to the broader setting of non-compact quantum groups. [ABSTRACT FROM AUTHOR]

Let V be a highest weight module over a Kac–Moody algebra g , and let conv V denote the convex hull of its weights. We determine the combinatorial isomorphism type of conv V , i.e. we completely classify the faces and their inclusions. In the special case where g is semisimple, this brings closure to a question studied by Cellini and Marietti (2015) [7] for the adjoint representation, and by Khare (2016, 2017) [17,18] for most modules. The determination of faces of finite-dimensional modules up to the Weyl group action and some of their inclusions also appears in previous works of Satake (1960) [25] , Borel and Tits (1965) [3] , Vinberg (1990) [26] , and Casselman (1997) [6] . For any subset of the simple roots, we introduce a remarkable convex cone which we call the universal Weyl polyhedron, which controls the convex hulls of all modules parabolically induced from the corresponding Levi factor. Namely, the combinatorial isomorphism type of the cone stores the classification of faces for all such highest weight modules, as well as how faces degenerate as the highest weight gets increasingly singular. To our knowledge, this cone is new in finite and infinite type. We further answer a question of Michel Brion, by showing that the localization of conv V along a face is always the convex hull of the weights of a parabolically induced module. Finally, as we determine the inclusion relations between faces representation-theoretically from the set of weights, without recourse to convexity, we answer a similar question for highest weight modules over symmetrizable quantum groups. [ABSTRACT FROM AUTHOR]

Let ( Σ , g ) be a closed connected surface equipped with a riemannian metric. Let ( λ n ) n ∈ N and ( ψ n ) n ∈ N be the increasing sequence of eigenvalues and the sequence of corresponding L 2 -normalized eigenfunctions of the laplacian on Σ. For each L > 0 , we consider ϕ L = ∑ 0 < λ n ≤ L ξ n λ n ψ n where the ξ n are i.i.d centered gaussians with variance 1. As L → ∞ , ϕ L converges a.s. to the Gaussian Free Field on Σ in the sense of distributions. We first compute the asymptotic behavior of the covariance function for this family of fields as L → ∞ . We then use this result to obtain the asymptotics of the probability that ϕ L is positive on a given open proper subset with smooth boundary. In doing so, we also prove the concentration of the supremum of ϕ L around 1 2 π ln ⁡ L . [ABSTRACT FROM AUTHOR]

In this paper, we study the pro-nilpotent group topology on a free group. First we describe the closure of the product of finitely many finitely generated subgroups of a free group in the pro-nilpotent group topology and then present an algorithm to compute it. We deduce that the nil-closure of a rational subset of a free group is an effectively constructible rational subset and hence has decidable membership. We also prove that the G nil -kernel of a finite monoid is computable and hence pseudovarieties of the form V ⓜ G nil have decidable membership problem, for every decidable pseudovariety of monoids V . Finally, we prove that the semidirect product J ⁎ G nil has a decidable membership problem. [ABSTRACT FROM AUTHOR]

Two families A and B of k -subsets of an n -set are called cross-intersecting if A ∩ B ≠ 0̸ for all A ∈ A , B ∈ B . Strengthening the classical Erdős–Ko–Rado theorem, Pyber proved that | A | | B | ≤ n − 1 k − 1 2 holds for n ≥ 2 k . In the present paper we sharpen this inequality. We prove that assuming | B | ≥ n − 1 k − 1 + n − i k − i + 1 for some 3 ≤ i ≤ k + 1 the stronger inequality | A | | B | ≤ ( n − 1 k − 1 + n − i k − i + 1 ) ( n − 1 k − 1 − n − i k − 1 ) holds. These inequalities are best possible. We also present a new short proof of Pyber’s inequality and a short computation-free proof of an inequality due to Frankl and Tokushige (1992). [ABSTRACT FROM AUTHOR]

We show the classical q -Stirling numbers of the second kind can be expressed compactly as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in q and 1 + q . We extend this enumerative result via a decomposition of a new poset Π ( n , k ) which we call the Stirling poset of the second kind. Its rank generating function is the q -Stirling number S q [ n , k ] . The Stirling poset of the second kind supports an algebraic complex and a basis for integer homology is determined. A parallel enumerative, poset theoretic and homological study for the q -Stirling numbers of the first kind is done. Letting t = 1 + q we give a bijective argument showing the ( q , t ) -Stirling numbers of the first and second kinds are orthogonal. [ABSTRACT FROM AUTHOR]

Let G be a finite Abelian group and A a subset of G . The spectrum of A is the set of its large Fourier coefficients. Known combinatorial results on the structure of spectrum, such as Chang's theorem, become trivial in the regime | A | = | G | α whenever α ≤ c , where c ≥ 1 / 2 is some absolute constant. On the other hand, there are statistical results, which apply only to a noticeable fraction of the elements, which give nontrivial bounds even to much smaller sets. One such theorem (due to Bourgain) goes as follows. For a noticeable fraction of pairs γ 1 , γ 2 in the spectrum, γ 1 + γ 2 belongs to the spectrum of the same set with a smaller threshold. Here we show that this result can be made combinatorial by restricting to a large subset. That is, we show that for any set A there exists a large subset A ′ , such that the sumset of the spectrum of A ′ has bounded size. Our results apply to sets of size | A | = | G | α for any constant α > 0 , and even in some sub-constant regime. [ABSTRACT FROM AUTHOR]

In this paper we study the Hausdorff dimension of a elliptic measure μ f in space associated to a positive weak solution to a certain quasilinear elliptic PDE in an open subset and vanishing on a portion of the boundary of that open set. We show that this measure is concentrated on a set of σ -finite n − 1 dimensional Hausdorff measure for p > n and the same result holds for p = n with an assumption on the boundary. We also construct an example of a domain in space for which the corresponding measure has Hausdorff dimension ≤ n − 1 − δ for p ≥ n for some δ which depends on various constants including p . The first result generalizes the authors previous work in [3] when the PDE is the p -Laplacian and the second result generalizes the well known theorem of Wolff in [24] when p = 2 and n = 2 . [ABSTRACT FROM AUTHOR]

Let G be a group. An element g ∈ G is called a test element of G if for every endomorphism φ : G → G , φ ( g ) = g implies that φ is an automorphism. Let F ( X ) be a free group on a finite non-empty set X , and let X = X 1 ∐ X 2 ∐ … ∐ X r be a finite partition of X into r ≥ 2 non-empty subsets. For i = 1 , 2 , … , r , let u i ∈ 〈 X i 〉 ≤ F ( X ) , and let w ( z 1 , … , z r ) be a word in the variables z 1 , … , z r . We give several sufficient conditions on u i ( 1 ≤ i ≤ r ) and w for w ( u 1 , … , u r ) to be a test element of F ( X ) . As an application of these results, we give examples of test elements of a free group of rank greater than two that are not test elements in any pro- p completion of the group. [ABSTRACT FROM AUTHOR]

Several recent papers have considered the Ramsey-theoretic problem of how large a subset of integers can be without containing any 3-term geometric progressions. This problem has also recently been generalized to number fields, determining bounds on the greatest possible density of ideals avoiding geometric progressions. We study the analogous problem over F q [ x ] , first constructing a set greedily which avoids these progressions and calculating its density, and then considering bounds on the upper density of subsets of F q [ x ] which avoid 3-term geometric progressions. This new setting gives us a parameter q to vary and study how our bounds converge to 1 as it changes, and positive characteristic introduces some extra combinatorial structure that increases the tractability of common questions in this area. [ABSTRACT FROM AUTHOR]

If E is a subset of the integers then the n -th characteristic ideal of E is the fractional ideal of Z consisting of 0 and the leading coefficients of the polynomials in Q [ x ] of degree no more than n which are integer valued on E . For p a prime the characteristic sequence of Int ( E , Z ) is the sequence α E ( n ) of negatives of the p -adic valuations of these ideals. The asymptotic limit lim n → ∞ ⁡ α E , p ( n ) n of this sequence, called the valuative capacity of E , gives information about the geometry of E . We compute these valuative capacities for the sets E of sums of ℓ ≥ 2 integers to the power of d , by observing the p -adic closure of these sets. [ABSTRACT FROM AUTHOR]

[Display omitted] • Many current methods of species tree inference assume that there is no selection. • We derive a new approximation of the Wright-Fisher model with selection and mutation. • We use our approximate to study the distribution of gene genealogies with selection. • Selection does not greatly affect the distribution of genealogies in many cases. • Many methods of species tree inference will thus not be greatly affected by selection. The effect of selection acting on regions of the genome on the accuracy of species-level phylogenetic inference using methods that do not explicitly model selection is an open question that is relevant to most, if not all, phylogenomic studies. To address this, we derive a mathematical approximation to the Wright-Fisher model with mutation and selection in the limit as the population size becomes large. In contrast to previous approximations based on diffusion processes, our approximation can be used to study the distribution of coalescent times for an arbitrary number of lineages, allowing calculation of the probability distribution of gene genealogies under the coalescent model. We use these calculations to show that direct selection at strengths typically encountered in practice has only a small effect on the distribution of coalescent times, and hence on the distribution of gene trees. This implies that many coalescent-based methods for estimating the species tree topology will be robust to the presence of selection in a subset of the underlying genes. Selection will, however, bias the estimation of speciation times, causing them to underestimate the true speciation times. Our model captures the effects of selection on the genealogies that generate the observed sequence data, but does not model selective pressures that act only on the subsequent sequences or that negatively impact gene tree estimation. [ABSTRACT FROM AUTHOR]

We prove that a nonempty closed and geodesically convex subset of the l ∞ plane R ∞ 2 is hyperconvex and we characterize the tight spans of arbitrary subsets of R ∞ 2 via this property: Given any nonempty X ⊆ R ∞ 2 , a closed, geodesically convex and minimal subset Y ⊆ R ∞ 2 containing X is isometric to the tight span T ( X ) of X . [ABSTRACT FROM AUTHOR]

We prove transference results for sparse random and pseudo-random subsets of Z N , which are analogous to the quantitative version of the well-known Furstenberg–Sárközy theorem due to Balog, Pintz, Steiger and Szemerédi. In the dense case, Balog et al. showed that there is a constant C > 0 such that for all integer k ≥ 2 any subset of the first N integers of density at least C ( log ⁡ N ) − 1 4 log ⁡ log ⁡ log ⁡ log ⁡ N contains a configuration of the form { x , x + d k } for some integer d > 0 . Let [ Z N ] p denote the random set obtained by choosing each element from Z N with probability p independently. Our first result shows that for p > N − 1 / k + o ( 1 ) asymptotically almost surely any subset A ⊂ [ Z N ] p ( N prime) of density | A | / p N ≥ ( log ⁡ N ) − 1 5 log ⁡ log ⁡ log ⁡ log ⁡ N contains the polynomial configuration { x , x + d k } , 0 < d ≤ N 1 / k . This improves on a result of Nguyen in the setting of Z N . Moreover, let k ≥ 2 be an integer and let γ > β > 0 be real numbers satisfying γ + ( γ − β ) / ( 2 k + 1 − 3 ) > 1 . Let Γ ⊆ Z N ( N prime) be a set of size at least N γ and linear bias at most N β . Then our second result implies that every A ⊆ Γ with positive relative density contains the polynomial configuration { x , x + d k } , 0 < d ≤ N 1 / k . For instance, for squares, i.e., k = 2 , and assuming the best possible pseudo-randomness β = γ / 2 our result applies as soon as γ > 10 / 11 . [ABSTRACT FROM AUTHOR]