arXiv Math @arxiv_math@qoto.org

Let $θ$ be an algebraic integer and $f(x)=x^{n}+ax^{n-1}+bx+c$ be the minimal polynomial of $θ$ over the rationals. Let $K=\mathbb{Q}(θ)$ be a number field and $\mathcal{O}_{K}$ be the ring of integers of $K.$ In this article, we characterize all the prime divisors of the discriminant of $f(x)$ which do not divide the index of $f(x).$ As a fascinating corollary, we deduce necessary and sufficient conditions for the monogenity of the field $K=\mathbb{Q}(θ),$ where $θ$ is associated with certain quadrinomials.

We prove an interesting identity for the sum of determinants, which is a generalization of the sum of a geometric progression. The proof is quite long and a number of other identities are proved along the way. Some of the more elementary ones are deferred to another section at the end.

In 1986, composer Wendy Carlos introduced three unusual musical scales she called alpha, beta and gamma, equal temperament-inspired scales that de-emphasize the octave as the primary interval in favor of the major and minor thirds and the perfect fifth. A derivation of the alpha, beta, and gamma scales due to David Benson is generalized to produce many Carlos-type scales.

A vertex-girth-regular $vgr(v,k,g,λ)$-graph is a $k$-regular graph of girth $g$ and order $v$ in which every vertex belongs to exactly $λ$ cycles of length $g$. While all vertex-transitive graphs are necessarily vertex-girth-regular, the majority of vertex-girth-regular graphs are not vertex-transitive. Similarly, while many of the smallest $k$-regular graphs of girth $g$, the so-called $(k,g)$-cages, are vertex-girth-regular, infinitely many vertex-girth-regular graphs of degree $k$ and girth $g$ exist for many pairs $k,g$. Due to these connections, the study of vertex-girth-regular graphs promises insights into the relations between the classes of extremal, highly symmetric, and locally regular graphs of given degree and girth. This paper lays the foundation to such study by investigating the fundamental properties of $vgr(v,k,g,λ)$-graphs, specifically the relations necessarily satisfied by the parameters $v,k,g$ and $λ$ to admit the existence of a corresponding vertex-girth-regular graph, by presenting constructions of infinite families of $vgr(v,k,g,λ)$-graphs, and by establishing lower bounds on the number $v$ of vertices in a $vgr(v,k,g,λ)$-graph. It also includes computational results determining the orders of smallest cubic and quartic graphs of small girths.

The problem of coding for the uplink and downlink of cloud radio access networks (C-RAN's) with $K$ users and $L$ relays is considered. It is shown that low-complexity coding schemes that achieve any point in the rate-fronthaul region of joint coding and compression can be constructed starting from at most $4(K+L)-2$ point-to-point codes designed for symmetric channels. This reduces the seemingly hard task of constructing good codes for C-RAN's to the much better understood task of finding good codes for single-user channels. To show this result, an equivalence between the achievable rate-fronthaul regions of joint coding and successive coding is established. Then, rate-splitting and quantization-splitting techniques are used to show that the task of achieving a rate-fronthaul point in the joint coding region can be simplified to that of achieving a corner point in a higher-dimensional C-RAN problem. As a by-product, some interesting properties of the rate-fronthaul region of joint decoding for uplink C-RAN's are also derived.

The Thouless theory of quantum pumps establishes the conditions for quantized particle transport per cycle, and determines its value. When describing the pump from a moving reference frame, transported and existing charges transform, though not independently. This transformation is inherent to Galilean space and time, but it is underpinned by a transformation of vector bundles. Different formalisms can be used to describe this transformation, including one based on Bloch theory. Depending on the chosen formalism, the two types of charges will be realized as indices of either the same or different kinds. Finally, we apply the bulk-edge correspondence principle, so as to implement the transformation law within Büttiker's scattering theory of quantum pumps.

Cut-elimination theorems constitute one of the most important classes of theorems of proof theory. Since Gentzen's proof of the cut-elimination theorem for the system $\mathbf{LK}$, several other proofs have been proposed. Even though the techniques of these proofs can be modified to sequent systems other than $\mathbf{LK}$, they are essentially of a very particular nature; each of them describes an algorithm to transform a given proof to a cut-free proof. However, due to its reliance on heavy syntactic arguments and case distinctions, such an algorithm makes the fundamental structure of the argument rather opaque. We, therefore, consider rules abstractly, within the framework of logical structures familiar from universal logic à la Jean-Yves Béziau, and aim to clarify the essence of the so-called ``elimination theorems''. To do this, we first give a non-algorithmic proof of the cut-elimination theorem for the propositional fragment of $\mathbf{LK}$. From this proof, we abstract the essential features of the argument and define something called ``normal sequent structures'' relative to a particular rule. We then prove a version of the rule-elimination theorem for these. Abstracting even more, we define ``abstract sequent structures'' and show that for these structures, the corresponding version of the ``rule''-elimination theorem has a converse as well.

The concepts of spread and spread dimension of a metric space were introduced by Willerton in the context of quantifying biodiversity of ecosystems. In previous work, we developed the theoretical basis for applications of spread dimension as an intrinsic dimension estimator. In this paper we introduce the pseudo spread dimension which is an efficient approximation of spread dimension, and we derive a formula for the standard error associated with this approximation.

To meet the unprecedented mobile traffic demands of future wireless networks, a paradigm shift from conventional cellular networks to distributed communication systems is imperative. Cell-free massive multiple-input multiple-output (CF-mMIMO) represents a practical and scalable embodiment of distributed/network MIMO systems. It inherits not only the key benefits of co-located massive MIMO systems but also the macro-diversity gains from distributed systems. This innovative architecture has demonstrated significant potential in enhancing network performance from various perspectives, outperforming co-located mMIMO and conventional small-cell systems. Moreover, CF-mMIMO offers flexibility in integration with emerging wireless technologies such as full-duplex (FD), non-orthogonal transmission schemes, millimeter-wave (mmWave) communications, ultra-reliable low-latency communication (URLLC), unmanned aerial vehicle (UAV)-aided communication, and reconfigurable intelligent surfaces (RISs). In this paper, we provide an overview of current research efforts on CF-mMIMO systems and their promising future application scenarios. We then elaborate on new requirements for CF-mMIMO networks in the context of these technological breakthroughs. We also present several current open challenges and outline future research directions aimed at fully realizing the potential of CF mMIMO systems in meeting the evolving demands of future wireless networks.

In this paper we cover a few topics on how to treat inverse problems. There are two different flows of ideas. One approach is based on Morse Lemma. The other is based on analyticity which proves that the number of solutions to the inverse problems is generically isolated for some particular class of dynamical systems.

We describe singularities of height functions on singular surfaces in $\mathbb{R}^3$ parameterized by smooth map-germs $\mathcal{A}$-equivalent to one of $S_k$, $B_k$, $C_k$ and $F_4$ singularities in terms of extended geometric language via finite succession of blowing-ups. We investigate singularities of dual surfaces of such singular surfaces.

A new integral representation is derived using a definite integral given by Cauchy and used to evaluate a number of integrals containing the finite series of special functions.

This work is related to the extension of the well-known problem of Roman domination in graph theory to fuzzy graphs. A variety of approaches have been used to explore the concept of domination in fuzzy graphs. This study uses the concept of strong domination, considering the weights of the strong edges. We introduce the strong-neighbors Roman domination number of a fuzzy graph and establish some correlations with the Roman domination in graphs. The strong-neighbors Roman domination number is determined for specific fuzzy graphs, including complete and complete bipartite fuzzy graphs. Besides, several general bounds are given. In addition, we characterize the fuzzy graphs that reach the extreme values with particular attention to fuzzy strong cycles and paths.

In this paper, we use concept of q-calculus and technique of convolution to study the q-Ruscheweyeh derivative by the concept of Janowski function, then we define new Subclass of analytic functions. Coefficients Estimates, radii of starlikeness, close to convexity, extreme points and many interesting properties are investigate, obtained and studied.

We construct explicit non-isotrivial families of polynomials over $\mathbb{Q}$ satisfying uniform boundedness for their rational preperiodic points.

In this paper we introduce and study the notion of I-convergence of sequences in a metric-like space, where I is an ideal of subsets of the set N of all natural numbers. Further introducing the notion of I*-convergence of sequences in a metric-like space we study its relationship with I-convergence.

We answer a question raised in ``Peripheral elements in reduced Alexander modules'' [J. Knot Theory Ramifications 31 (2022), 2250058]. We also correct a minor error in that paper.

We study the fusion 3-categorical symmetries for quantum theories in (3+1)d with self-duality defects. Such defects have been realized physically by half-space gauging in theories with 1-form symmetries $A[1]$ for an abelian group $A$, and have found applications in the continuum and the lattice. These fusion 3-categories will be called (generalized) Tambara-Yamagami fusion 3-categories $(\mathbf{3TY})$. We consider the Brauer-Picard and Picard 4-groupoids to construct these categories using a 3-categorical version of the extension theory introduced by Etingof, Nikshych and Ostrik. These two 4-groupoids correspond to the construction of duality defects either directly in 4d, or from the 5d Symmetry Topological Field Theory (SymTFT). The Witt group of non-degenerate braided fusion 1-categories naturally appears in the aforementioned 4-groupoids and represents enrichments of standard duality defects by (2+1)d TFTs. Our main objective is to study graded extensions of the fusion 3-category $\mathbf{3Vect}(A[1])$. Firstly, we use invertible bimodule 3-categories and the Brauer-Picard 4-groupoid. Secondly, we use that the Brauer-Picard 4-groupoid of $\mathbf{3Vect}(A[1])$ can be identified with the Picard 4-groupoid of its Drinfeld center. Moreover, the Drinfeld center of $\mathbf{3Vect}(A[1])$, which represents topological defects of the SymTFT, is completely described by a sylleptic strongly fusion 2-category formed by topological surface defects of the SymTFT. These are classified by a finite abelian group equipped with an alternating 2-form. We relate the Picard 4-groupoid of the corresponding braided fusion 3-categories with a generalized Witt group constructed from certain graded braided fusion 1-categories using a twisted Deligne tensor product. We perform explicit computations for $\mathbb{Z}/2$ and $\mathbb{Z}/4$ graded $\mathbf{3TY}$ categories.

Solutions of bilevel optimization problems tend to suffer from instability under changes to problem data. In the optimistic setting, we construct a lifted, alternative formulation that exhibits desirable stability properties under mild assumptions that neither invoke convexity nor smoothness. The upper- and lower-level problems might involve integer restrictions and disjunctive constraints. In a range of results, we at most invoke pointwise and local calmness for the lower-level problem in a sense that holds broadly. The alternative formulation is computationally attractive with structural properties being brought out and an outer approximation algorithm becoming available.

We propose a nonlinear filter for noise removal based on the Laplacian for 1D and 2D data. The method utilizes the solution to a fourth-order nonlinear PDE involving the Laplacian for data reconstruction. Evolution equations are introduced to solve this fourth-order nonlinear equation. Numerical experiments show that the new filter preserves discontinuities while filtering out noise. The restored data are piecewise linear and avoid the staircase effect commonly observed with total variation denoising methods.