Exceptional Set Research Articles

Exceptional set estimates for radial projections in R^n

We prove two conjectures in this paper. The first conjecture is by Lund, Pham and Thu: Given a Borel set \(A\subset \mathbb{R}^n\) such that \(\dim A\in (k,k+1]\) for some \(k\in\{1,\dots,n-1\}\). For \(0<s<k\), we have \(\text{dim}(\{y\in \mathbb{R}^n \setminus A\mid \text{dim} (\pi_y(A)) < s\})\leq \max\{k+s -\dim A,0\}.\) The second conjecture is by Liu: Given a Borel set \(A\subset \mathbb{R}^n\), then \(\text{dim} (\{x\in \mathbb{R}^n \setminus A \mid \text{dim}(\pi_x(A))<\text{dim} A\}) \leq \lceil \text{dim} A\rceil.\)

Deformations of some local Calabi-Yau manifolds

We study deformations of certain crepant resolutions of isolated rational Gorenstein singularities. After a general discussion of the deformation theory, we specialize to dimension $3$ and consider examples which are good (log) resolutions as well as the case of small resolutions. We obtain some partial results on the classification of canonical threefold singularities that admit good crepant resolutions. Finally, we study a noncrepant example, the blowup of a small resolution whose exceptional set is a smooth curve.

The geometric exceptional set in Manin’s conjecture for Batyrev and Tschinkel’s example

The geometric exceptional set in Manin’s conjecture for Batyrev and Tschinkel’s example

On clusters and exceptional sets

In this paper, we first study clusters in type [Formula: see text] by collecting them into a finite number of infinite families given by Dehn twists of their corresponding triangulations, and show that these families are counted by the Catalan numbers. We also highlight the similarities and differences between the annuli diagrams used to study clusters and those used to study exceptional sets in type [Formula: see text]. We then focus on exceptional collections (sets) of modules over path algebras of quivers by first showing that the notion of relative projectivity in exceptional sets is well defined. We finish by counting the number of exceptional sets of representations of type [Formula: see text] quivers with straight orientation and using this to count the number of families of exceptional sets of type [Formula: see text] with straight orientation.

The Paradox of Sanctuary: How Punitive Exceptions Converge to Criminalize and Punish Latinos/as

Abstract Sanctuary cities define themselves as metropoles that refuse to share information, personnel, and facilities with federal immigration authorities to police immigrants. While research suggests that sanctuary cities contest the criminalization of migration, a growing literature depicts how these urban sanctuaries could, in practice, perpetuate hierarchies and exclusionary politics against noncitizens. Yet, most of these studies conceive of urban sanctuary as local policies designed to challenge federal power and, thus, fail to fully capture how sanctuary policies could actually rely on the criminalization of migration to govern cities’ political problems. Drawing upon 1,900 pages of archival materials and 100 newspaper articles, this article takes the case of Chicago to study how and why the urban sanctuary expands immigrants’ rights while reinforcing policing with punitive implications for Latino “undeserving” noncitizens. As a form of racialized governance, I argue that Chicago’s sanctuary policies activate a set of punitive exceptions that—in response to distinct political urgencies—allow law and immigration enforcement to converge and control Latino undocumented workers, “criminals,” and “gangs.” This study not only challenges the premise that sanctuary cities necessarily resist federal power but also illustrates how they could strengthen the legitimacy of the state and racialized police power.

A note on the exceptional set for sums of unlike powers of primes

A note on the exceptional set for sums of unlike powers of primes

On the Liouville function at polynomial arguments

abstract: Let $\lambda$ denote the Liouville function. A problem posed by Chowla and by Cassaigne--Ferenczi--Mauduit--Rivat--S\'ark\"ozy asks to show that if $P(x)\in\mathbb{Z}[x]$, then the sequence $\lambda(P(n))$ changes sign infinitely often, assuming only that $P(x)$ is not the square of another polynomial. We show that the sequence $\lambda(P(n))$ indeed changes sign infinitely often, provided that either (i) $P$ factorizes into linear factors over the rationals; or (ii) $P$ is a reducible cubic polynomial; or (iii) $P$ factorizes into a product of any number of quadratics of a certain type; or (iv) $P$ is any polynomial not belonging to an exceptional set of density zero. Concerning (i), we prove more generally that the partial sums of $g(P(n))$ for $g$ a bounded multiplicative function exhibit nontrivial cancellation under necessary and sufficient conditions on $g$. This establishes a ``99\% version'' of Elliott's conjecture for multiplicative functions taking values in the roots of unity of some order. Part (iv) also generalizes to the setting of $g(P(n))$ and provides a multiplicative function analogue of a recent result of Skorobogatov and Sofos on almost all polynomials attaining a prime value.

Negative photo conductivity triggered with visible light in wide bandgap oxide-based optoelectronic crossbar memristive array for photograph sensing and neuromorphic computing applications

Photoresponsivity studies of wide-bandgap oxide-based devices have emerged as a vibrant and popular research area. Researchers have explored various material systems in their quest to develop devices capable of responding to illumination. In this study, we engineered a mature wide-bandgap oxide-based bilayer heterostructure synaptic memristor to emulate the human brain for applications in neuromorphic computing and photograph sensing. The device exhibits advanced electric and electrophotonic synaptic functions, such as long-term potentiation (LTP), long-term depression (LTD), and paired-pulse facilitation (PPF), by applying successive electric and photonic pulses. Moreover, the device exhibits exceptional electrical SET and photonic RESET endurance, maintaining its stability for a minimum of 1200 cycles without any degradation. Density functional theory calculations of the band structures provide insights into the conduction mechanism of the device. Based on this memristor array, we developed an autoencoder and convolutional neural network for noise reduction and image recognition tasks, which achieves a peak signal-to-noise ratio of 562 and high accuracy of 84.23%, while consuming lower energy by four orders of magnitude compared with the Tesla P40 GPU. This groundbreaking research not only opens doors for the integration of our device into image processing but also represents a significant advancement in the realm of in-memory computing and photograph-sensing features in a single cell.

Non-triviality of an asymptotically flat vacuum spacetime in pure R2 gravity

In [Phys. Rev. D 107, 104008 (2023)], we reported a novel exact closed-form solution which describes asymptotically flat spacetimes in pure [Formula: see text] gravity. The solution is Ricci scalar flat, viz. [Formula: see text] everywhere. Whereas any metric with a null Ricci scalar would trivially satisfy the [Formula: see text] vacuo field equation, [Formula: see text], in this paper, we shall show that our solution satisfies a “stronger” version of the [Formula: see text] vacuo field equation, viz. [Formula: see text], despite the term [Formula: see text] being singular. Even though [Formula: see text] identically vanishes, for our solution, the combinations [Formula: see text] and [Formula: see text] are free of singularity. This exceptional property sets our solution apart from the set of null-Ricci-scalar metrics and makes it a genuinely nontrivial solution. We further demonstrate that, as a member of a larger class of asymptotically de Sitter metrics, our solution is resilient against perturbations in the scalar curvature at largest distances, making it relevant for physical situations where the background deviates from asymptotic flatness.

Morphologies, Compatibilization and Properties of Immiscible PLA-Based Blends with Engineering Polymers: An Overview of Recent Works.

Poly(L-Lactide) (PLA), a fully biobased aliphatic polyester, has attracted significant attention in the last decade due to its exceptional set of properties, such as high tensile modulus/strength, biocompatibility, (bio)degradability in various media, easy recyclability and good melt-state processability by the conventional processes of the plastic/textile industry. Blending PLA with other polymers represents one of the most cost-effective and efficient approaches to develop a next-generation of PLA-based materials with superior properties. In particular, intensive research has been carried out on PLA-based blends with engineering polymers such as polycarbonate (PC), poly(ethylene terephthalate) (PET), poly(butylene terephthalate) (PBT) and various polyamides (PA). This overview, consequently, aims to gather recent works over the last 10 years on these immiscible PLA-based blends processed by melt extrusion, such as twin screw compounding. Furthermore, for a better scientific understanding of various ultimate properties, processing by internal mixers has also been ventured. A specific emphasis on blend morphologies, compatibilization strategies and final (thermo)mechanical properties (tensile/impact strength, ductility and heat deflection temperature) for potential durable and high-performance applications, such as electronic parts (3C parts, electronic cases) to replace PC/ABS blends, has been made.

A self-healing multispectral transparent adhesive peptide glass.

Despite its disordered liquid-like structure, glass exhibits solid-like mechanical properties1. The formation of glassy material occurs by vitrification, preventing crystallization and promoting an amorphous structure2. Glass is fundamental in diverse fields of materials science, owing to its unique optical, chemical and mechanical properties as well as durability, versatility and environmental sustainability3. However, engineering a glassy material without compromising its properties is challenging4-6. Here we report the discovery of a supramolecular amorphous glass formed by the spontaneous self-organization of the short aromatic tripeptide YYY initiated by non-covalent cross-linking with structural water7,8. This system uniquely combines often contradictory sets of properties; it is highly rigid yet can undergo complete self-healing at room temperature. Moreover, the supramolecular glass is an extremely strong adhesive yet it is transparent in a wide spectral range from visible to mid-infrared. This exceptional set of characteristics is observed in a simple bioorganic peptide glass composed of natural amino acids, presenting a multi-functional material that could be highly advantageous for various applications in science and engineering.

Structuration and yield strength characterization of hybrid alkali-activated cement composites (HACC) for ultra-rapid 3D construction printing

This study investigates the critical time-dependent rheological properties, including structuration rate and yield stress, of hybrid alkali-activated cement composites (HACC). HACC, a blend of geopolymer and cement paste, is examined to evaluate its suitability for ultra-rapid 3D construction printing (3DCP), which refers to a material’s ability to attain rapid structuration and buildability within 5–10 min after mixing, essential for additive manufacturing of 3D structures. The investigation extends to the impact of filler materials like silica sand and PVA fibers on HACC’s rapid structuration. Vicat needle, fast- and slow-penetration resistance measurements indicate that setting time increases with decreasing the Silicon/Aluminum (S/A) molar ratio and the addition of silica sand. Conversely, the addition of fibers reduces setting time while enhancing yield strength. Specifically, HACC mixtures with 0.5% and 1% PVA fiber additions, alongside combined sand (30%, 40%) and fiber (0.5%) mixtures, demonstrate ultra-rapid setting behavior within 3–5 min. Notably, mixtures with 40% silica sand and 0.5% PVA fiber exhibit highest yield strength development, i.e., 14 MPa at 12 min. Remarkably, incorporating 1% fibers into the HACC mixture yields the highest 28-day compressive strength (∼80 MPa) and flexural strength (∼11 MPa). The synergistic interplay between geopolymer and cement facilitates rapid setting time and high early-age strength development in HACC mixtures. The varying effects of silica sand and PVA fiber on setting time and yield strength provide opportunities for designing suitable structuration profiles. These findings underscore the exceptional rapid setting, structuration, and mechanical properties of HACC, positioning it as a promising material for ultra-rapid 3D construction printing.

A complex case of Vojta’s general abc conjecture and cases of Campana’s orbifold conjecture

We show a truncated second main theorem of level one with explicit exceptional sets for analytic maps into P 2 \mathbb P^2 intersecting the coordinate lines with sufficiently high multiplicities. The proof is based on a greatest common divisor theorem for an analytic map f : C ↦ P n f:\mathbb C\mapsto \mathbb P^n and two homogeneous polynomials in n + 1 n+1 variables with coefficients which are meromorphic functions of the same growth as the analytic map f f . As applications, we study some cases of Campana’s orbifold conjecture for P 2 \mathbb P^2 and finite ramified covers of P 2 \mathbb P^2 with three components admitting sufficiently large multiplicities. In addition, we explicitly determine the exceptional sets. Consequently, it implies the strong Green-Griffiths-Lang conjecture for finite ramified covers of G m 2 \mathbb G_m^2 .

Composition and distribution of sedimentary organic matter in nearshore Berriasian strata (‘German Wealden’) of the eastern Lower Saxony Basin, NW Germany

Abstract In the Lower Saxony Basin of Northern Germany, the Earliest Cretaceous (Berriasian, ‘German Wealden’) is represented by organic-rich shales assigned to the Isterberg Formation, which have been deposited under brackish-lacustrine conditions in a partially restricted basin. Towards the southern basin margin, these shales interfinger with nearshore sandstones sourced from the Deister-Hils delta. In order to better understand the environmental conditions and depositional dynamics of the Isterberg Fm, the sedimentary organic matter has been studied with a combined approach including bulk rock organic geochemistry and palynofacies analysis. The lower part (W 1 to W 3) is interpreted to reflect very shallow-water conditions and episodic emergence, indicated by multiple occurrences of palaeosol horizons, rhizoliths and intercalated coal beds formed within a mud-dominated and protected domain of the delta plain. The overlying sand-rich interval represents a nearshore deltaic setting. A distinct change is observed in the upper part (W 4), indicating more distal conditions characterized by clay-dominated sedimentation, abundant lumachelle interbeds and a predominance of aquatic-derived organic matter. The newly acquired data fill a gap in the record between the proximal and distal strata of the Berriasian Lower Saxony Basin and provide new insights into the accumulation and preservation of organic matter within this exceptional depositional setting.

FFKM combines high-temperature performance with exceptional low compression set

UK-based Precision Polymer Engineering Ltd (PPE) has developed a perfluoroelastomer (FFKM) sealing material that combines the ability to operate and perform well at a high temperature with exceptional low compression set.

Exceptional sets related to the product of consecutive digits in L\"{u}roth expansions

Exceptional sets related to the product of consecutive digits in L\"{u}roth expansions

On the h-measure of an exceptional set in Fenton-type theorem for Taylor-Dirichlet series

We consider the class $S(\lambda,\beta,\tau)$ of convergent for all $x\ge0$
 Taylor-Dirichlet type series of the form
 $$F(x) =\sum_{n=0}^{+\infty}{b_ne^{x\lambda_n+\tau(x)\beta_n}},\ 
 b_n\geq 0\ (n\geq 0),$$
 where $\tau\colon [0,+\infty)\to
 (0,+\infty)$\ is a continuously differentiable non-decreasing function,
 $\lambda=(\lambda_n)$ and $\beta=(\beta_n)$ are such that $\lambda_n\geq 0, \beta_n\geq 0$ $(n\geq 0)$.
 In the paper we give a partial answer to a question formulated by Salo T.M., Skaskiv O.B., Trusevych O.M. on International conference ``Complex Analysis and Related Topics'' (Lviv, September 23-28, 2013) ([2]). We prove the following statement: For each increasing function $h(x)\colon [0,+\infty)\to (0,+\infty)$, $h'(x)\nearrow +\infty$ $ (x\to +\infty)$, every sequence $\lambda=(\lambda_n)$ such that 
 $\displaystyle\sum_{n=0}^{+\infty}\frac1{\lambda_{n+1}-\lambda_n}<+\infty$
 and for any non-decreasing sequence $\beta=(\beta_n)$ such that
 $\beta_{n+1}-\beta_n\le\lambda_{n+1}-\lambda_n$ $(n\geq 0)$ 
 there exist a function $\tau(x)$ such that $\tau'(x)\ge 1$ $(x\geq x_0)$, a function $F\in S(\alpha, \beta, \tau)$, a set $E$ and a constant $d>0$ such that $h-\mathop{meas} E:=\int_E dh(x)=+\infty$ and $(\forall x\in E)\colon\ F(x)>(1+d)\mu(x,F),$ where $\mu(x,F)=\max\{|a_n|e^{x\lambda_n+\tau(x)\beta_n}\colon n\ge 0\}$ is
 the maximal term of the series.
 
 At the same time, we also pose some open questions and formulate one conjecture.

Hausdorff dimension of certain sets related to random αβ-orbits which are not dense

Let α,β∈[0,1)=R/Z such that at least one of them is irrational. It is known that the random αβ-orbits of the Bernoulli mixture of rotations of α and β by choosing them with equal probability 12 are uniformly distributed modulo 1 with probability one. Chen et al. (2021) [2] showed that the exceptional set in the probability space has full Hausdorff dimension. In this note, we prove that certain sets related to random αβ-orbits which are not dense in R/Z have Hausdorff dimension 1. Our result can be viewed as an improvement of the dimension result obtained by Chen, Wang and Wen.

Hausdorff dimension of certain sets arising by the maximal run-length function over factorial language

Let A be a language with infinite number of words over the alphabet Σ={0,1} and let (xi)i=1+∞ be the digits sequence in the unique terminating dyadic expansion of x∈[0,1). The maximal run-length function ln(x,A) of x over A is defined byln(x,A)=max⁡{k:xi+1xi+2⋯xi+k∈Afor some0≤i≤n−k}. In a general setting, Wu (2024) [18] proved a metrical property on the asymptotic behavior of ln(x,A), which extends the famous Erdős-Rényi limit theorem that represents for the special case A={1k:k≥1}. Under the assumption that A is factorial (i.e. the language A is closed under subword), Wu studied the exceptional setE={x∈[0,1):limn→∞⁡ln(x,A)mn=0andlimn→∞⁡ln(x,A)=∞}, and he obtained a lower bound of the Hausdorff dimension of E with some conditions on mn. In this paper, we determine the Hausdorff dimension of E without any additional assumption on mn and we obtain the Hausdorff dimension of two other kinds of exceptional sets of elements whose maximal run-length functions have different growth rates. Our results significantly extend the existing dimension results in this topic.

Delta‐convex structure of the singular set of distance functions

AbstractFor the distance function from any closed subset of any complete Finsler manifold, we prove that the singular set is equal to a countable union of delta‐convex hypersurfaces up to an exceptional set of codimension two. In addition, in dimension two, the whole singular set is equal to a countable union of delta‐convex Jordan arcs up to isolated points. These results are new even in the standard Euclidean space and shown to be optimal in view of regularity.