The paper offers the first systematic study of ordinary and essential numerical ranges of operators on $\ell_p$, $1<p<\infty$, as an atomic picture within a broader $L^p$ project. The paper begins with Banach-space foundations, including the finite-codimensional description of the essential numerical range and a Banach-space convex-hull inclusion for the essential spectrum. It then turns to finite-dimensional $\ell_p$ geometry, where one finds both positive star-shapedness phenomena and explicit $2\times2$ counterexamples. On $\ell_p$, we prove that the essential numerical range is compact and convex, identify it with the algebraic numerical range of the Calkin image, obtain a compact-perturbation formula, and show that, moreover, the closure of the numerical range is star-shaped, while points in the interior of the essential numerical range are exact star-centres of the numerical range. The paper illustrates the developed theory with sequence-space examples, covering tridiagonal Toeplitz operators and the discrete Hilbert transform, and, after relating our study to a variant of the Crouzeix inequality, closes with a brief discussion of extensions to spaces of class $(P)$ and
A ring $R$ is said to be centrally essential if for every its non-zero element $a$, there exist non-zero central elements $x$ and $y$ with $ax = y$. A ring $R$ is said to be completely centrally essential if all its factor rings are centrally essential rings. It is proved that completely centrally essential semiprimary rings are Lie nilpotent; noetherian completely centrally essential rings are strongly Lie nilpotent (in particular, every such a ring is a $PI$-ring). Every completely centrally essential ring has the classical ring of fractions which is a completely centrally essential ring. If $R$ is a commutative domain and $G$ is an arbitrary group, then any completely centrally essential group ring $RG$ is commutative.
We introduce a variational notion of essential spectrum for the Dirichlet $p-$Laplacian. We then extend the classical Persson Theorem to this nonlinear setting. This result provides a geometric characterization of the bottom of the essential spectrum, in terms of the sharp $L^p$ Poincaré constant ``at infinity''. We also show that in the case $p=2$ our construction of the essential spectrum is perfectly consistent with the classical theory. Finally, as an example, we compute the full spectrum of the Dirichlet $p-$Laplacian on a rectilinear strip: it is purely essential, with no embedded eigenvalues. The arguments of the proofs are elementary and new already for the linear case $p=2$.
We study two notions of largeness for closed submodules of Hilbert C*-modules: essentiality and topological essentiality. While the analogous properties are known to be equivalent for closed two-sided ideals of C*-algebras, the one-sided case is more subtle. We prove that these two notions remain equivalent for closed right ideals of an arbitrary C*-algebra. Next, using the correspondence between submodules and right ideals of the algebra of compact operators, we extend this result to closed submodules of Hilbert C*-modules. In the commutative case, where a Hilbert module can be realized as a continuous field of Hilbert spaces, we give a geometric reformulation of essentiality and derive a fiberwise criterion.
A non-zero unital ring $R$ is said to be centrally essential if for every nonzero element $a$ of $R$, there exist non-zero central elements $x$ and $y$ with $ax = y$. In the paper, almost fully prime centrally essential rings are described in terms of ideal extensions, centrally essential Dorroh extensions, and trivial extensions.
In this paper, a new notion, essential contractibility of Banach algebras, is introduced and some of its properties are examined. The main result is to investigate the essential contractibility of (symmetric abstract) Segal algebras.
Data plays the most prominent role in how language models acquire skills and knowledge. The lack of massive, well-organized pre-training datasets results in costly and inaccessible data pipelines. We present Essential-Web v1.0, a 24-trillion-token dataset in which every document is annotated with a twelve-category taxonomy covering topic, format, content complexity, and quality. Taxonomy labels are produced by EAI-Distill-0.5b, a fine-tuned 0.5b-parameter model that achieves an annotator agreement within 3% of Qwen2.5-32B-Instruct. With nothing more than SQL-style filters, we obtain competitive web-curated datasets in math (-8.0% relative to SOTA), web code (+14.3%), STEM (+24.5%) and medical (+8.6%). Essential-Web v1.0 is available on HuggingFace: https://huggingface.co/datasets/EssentialAI/essential-web-v1.0
We introduce and study $μ$-elements, that generalize a lattice-theoretic abstraction (namely, essential elements) of essential ideals of rings, essential submodules of modules, and dense subsets of topological spaces. Exploring several examples, we show that $μ$-elements are indeed a genuine extension of essential elements. We study preservation of $μ$-elements under contractions and extensions of quantale homomorphisms. We introduce $μ$-complements and $μ$-closedness and study their properties. We determine $μ$-elements for several distinguished quantales, including ideals of $\mathbb{Z}_n$ and open subsets of topological spaces. Finally, we provide a complete characterization of $μ$-elements in modular quantales.
Gene essentiality refers to the degree to which a gene is necessary for the survival and reproductive efficacy of a living organism. Although the essentiality of non-coding genes has been documented, there are still aspects of non-coding genes' essentiality that are unknown to us. For example, We do not know the contribution of sequence features and network spatial features to essentiality. As a consequence, in this work, we propose DeepHEN that could answer the above question. By buidling a new lncRNA-proteion-protein network and utilizing both representation learning and graph neural network, we successfully build our DeepHEN models that could predict the essentiality of lncRNA genes. Compared to other methods for predicting the essentiality of lncRNA genes, our DeepHEN model not only tells whether sequence features or network spatial features have a greater influence on essentiality but also addresses the overfitting issue of those methods caused by the low number of essential lncRNA genes, as evidenced by the results of enrichment analysis.
The notion of essential submodules and essential extensions of modules are extended to groups (typically nonabelian), and several necessary and sufficient conditions for a group to possess a proper essential subgroup are investigated. Further, we have completely characterized groups that do not possess a proper essential extension. These observations are used in concluding several properties of groups having essential subgroups. Finally, a short proof of the well-known theorem of Eilenberg and Moore that the only injective object in the category of groups is the trivial group is given.
Positroids are a family of matroids introduced by Postnikov in the study of non-negative Grassmannians. Postnikov identified several combinatorial objects in bijections with positroids, among which are bounded affine permutations. On the other hand, the notion of essential sets, introduced for permutations by Fulton, was used by Knutson in the study of the special family of interval rank positroids. We generalize Fulton's essential sets to bounded affine permutations. The bijection of the latter with positroids, allows study of the relationship between them. From the point of view of positroids, essential sets are maximally dependent cyclic interval. We define connected essential sets and prove that they give a facet description of the positroid polytope, as well as equations defining the positroid variety. We define a subset of essential sets, called core, which contains minimal rank conditions to uniquely recover a positroid. We provide an algorithm to retrieve the positroid satisfying the rank conditions in the core or any compatible rank condition on cyclic intervals.
The class of nil-essential ideals is a generalisation of the class of essential ideals. Every nil-essential ideal of a reduced ring is essential. Therefore the intersection of all nil-essential ideals over a reduced ring $R$ is the socle of $R$. In this note, we apply this generalisation to give a new criteria of semisimplicity in terms of nil-essentiality of ideals.
A ring $R$ with center $C$ is said to be centrally essential if the module $R_C$ is an essential extension of the module $C_C$. In this paper, we study properties of ideals of centrally essential rings, centrally essential quaternion algebras, and group rings of Hamiltonian groups.
In this paper we study \emph{essential hereditary undecidability}. Theories with this property are a convenient tool to prove undecidability of other theories. The paper develops the basic facts concerning essentially hereditary undecidability and provides salient examples, like a construction of \ehu\ theories due to Hanf and an example of a rather natural essentially hereditarily undecidable theory strictly below {\sf R}. We discuss the (non-)interaction of essential hereditary undecidability with recursive boolean isomorphism. We develop a reduction relation \emph{essential tolerance}, or, in the converse direction, \emph{lax interpretability} that interacts in a good way with essential hereditary undecidability. We introduce the class of $Σ^0_1$-friendly theories and show that $Σ^0_1$-friendliness is sufficient but not necessary for essential hereditary undecidability. Finally, we adapt an argument due to Pakhomov, Murwanashyaka and Visser to show that there is no interpretability minimal essentially hereditarilyundecidable theory.
We establish the essential normality of a large new class of homogeneous submodules of the finite rank d-shift Hilbert module. The main idea is a notion of essential decomposability that determines when an arbitrary submodule can be decomposed into the sum of essentially normal submodules. We prove that every essentially decomposable submodule is essentially normal, and using ideas from convex geometry, we introduce methods for establishing that a submodule is essentially decomposable. It turns out that many homogeneous submodules of the finite rank d-shift Hilbert module have this property. We prove that many of the submodules considered by other authors are essentially decomposable, and in addition establish the essential decomposability of a large new class of homogeneous submodules. Our results support Arveson's conjecture that every homogeneous submodule of the finite rank d-shift Hilbert module is essentially normal.
The Cremona dimension of a group $G$ is the minimal $n$ such that $G$ is isomorphic to a subgroup of the Cremona group of birational transformations of an $n$-dimensional rational variety. In this survey article, we give many examples that gives evidence to the conjecture that the Cremona dimension of a finite group over the field of complex numbers is less than or equal to the essential dimension of the group.
For a cancellative semigroup S and a field F, it is proved that the semigroup algebra FS is centrally essential if and only if the group of fractions $G_S$ of the semigroup $S$ exists and the group algebra $FG_S$ of $G_S$ is centrally essential. The semigroup algebra of a cancellative semigroup is centrally essential if and only if it has the classical right ring of fractions which is a centrally essential ring. There exist non-commutative centrally essential semigroup algebras over fields of zero characteristic (this contrasts with the known fact that centrally essential group algebras over fields of zero characteristic are commutative).
A group action has essential holonomy if the set of points with non-trivial holonomy has positive measure. If such an action is topologically free, then having essential holonomy is equivalent to the action not being essentially free, which means that the set of points with non-trivial stabilizer has positive measure. In this paper, we investigate the relation between the property of having essential holonomy and structure of the acting group for minimal equicontinuous actions on Cantor sets. We show that if such a group action is locally quasi-analytic and has essential holonomy, then every commutator subgroup in the group lower central series has elements with positive measure set of points with non-trivial holonomy. In particular, this gives a new proof that a minimal equicontinuous Cantor action by a nilpotent group has no essential holonomy. We also show that the property of having essential holonomy is preserved under return equivalence and continuous orbit equivalence of minimal equicontinuous Cantor actions. Finally, we give examples to show that the assumption on the action that it is locally quasi-analytic is necessary.
We investigate the relation between essential divisors and F-blowups, in particular, address the problem whether all essential divisors appear on the $e$-th F-blowup for large enough $e$. Focusing on the case of normal affine toric varieties, we establish a simple sufficient condition for a divisor over the given toric variety to appear on the normalized limit F-blowup as a prime divisor. As a corollary, we show that if a normal toric variety has a crepant resolution, then the above problem has a positive answer, provided that we use the notion of essential divisors in the sense of Bouvier and Gonzalez-Sprinberg. We also provide an example of toric threefold singularities for which a non-essential divisor appears on an F-blowup.
We prove new inequalities for the essential generalized and the essential joint spectral radius of Hadamard (Schur) weighted geometric means of bounded sets of infinite nonnegative matrices that define operators on suitable Banach sequence spaces and of bounded sets of positive kernel operators on $L^2$. To our knowledge the obtained inequalities are new even in the case of singelton sets.