In this partly expository paper we discuss and describe some of our old and recent results on partial orders on the set (m,n)-graphs (i.e. graphs with n vertices and m edges) and some operations on graphs that are monotone with respect to these partial orders. The partial orders under consideration include those related with some Laplacian characteristics of graphs as well as with some probabilistic characteristics of graphs with randomly deleted edges. Section 2 provides some notions, notation, and simple observations. Section 3 contains some basic facts on the Laplacian polynomial of a graph. Section 4 describes various graph operation and their properties. In Section 5 we introduce some partial orders on the set of (m,n)-graphs related, in particular, with the graph Laplacian and the graph reliability (Laplacian posets and reliability posets}). Section 6 contains some old and recent results on the monotonicity of some graph operations with respect to Laplacian posets. Section 7 and 8 include some old and recent results on the monotonicity of some graph operations with respect to reliability posets and to some other parameters of graphs as well as some open problems. Section 9 co
We constructed some tensor functors that send each exceptional sequence in a module category to another exceptional sequence in another module category by using split extensions and recollements.
This paper is devoted to a general presentation of anti-topological spaces. These structures have been initially proposed by Şahin, Kargın and M. Yücel in 2021. We analyse their basic definition, showing some of its subtleties and implications. The framework thus obtained is used to investigate anti-topological interpretation of some basic topological notions. For example, we discuss the idea of interior and closure and we show some results on door spaces. Moreover, we introduce two non-equivalent types of continuity. Finally, we investigate the idea of density and nowhere density. Finally, we give some preliminary suggestions concerning the modal logic of anti-topological spaces. It is noteworthy that the paper contains some additional remarks on infra-topological and weak spaces. They may be considered as a clarification or correction of some earlier results present in literature.
We point out that the ideas underlying some test procedures recently proposed for testing post-model-selection (and for some other test problems) in the econometrics literature have been around for quite some time in the statistics literature. We also sharpen some of these results in the statistics literature. Furthermore, we show that some intuitively appealing testing procedures, that have found their way into the econometrics literature, lead to tests that do not have desirable size properties, not even asymptotically.
The first section of this modest survey reviews some basic notions and describes some families of examples, and the second section briefly indicates some general aspects of analysis on metric spaces. The remaining three sections are concerned with some particular situations involving sub-Riemannian geometry, hyperbolic groups, and p-adic numbers.
We offer some further applications of some Bailey pairs related to some mock theta functions which were established in a recent study. We discuss and offer some double-sum $q$-series, with new relationships among mock theta functions. We also offer a new relationship between the Bailey pair of Bringmann and Kane with that of Andrews.
In this study, we explore the interrelation between hypergraph symmetries represented by equivalence relations on the vertex set and the spectra of operators associated with the hypergraph. We introduce the idea of equivalence relation compatible operators related to hypergraphs. Some eigenvalues and the corresponding eigenvectors can be computed directly from the equivalence classes of the equivalence relation. The other eigenvalues can be computed from a quotient operator obtained by identifying each equivalence class as an element. We provide an equivalence relation $\mathfrak{R}_s$ on the vertex set of a hypergraph such that the Adjacency, Laplacian, and signless Laplacian operators associated with that hypergraph become $\mathfrak{R}_s$-compatible. The $\mathfrak{R}_s$-equivalence classes are named as units. Using units, we find some more symmetric substructures of hypergraphs called twin units, regular sets, co-regular sets, and symmetric sets. We collectively classify them as building blocks of hypergraphs. We show that the presence of these building blocks leaves certain traces in the spectrum and the corresponding eigenspaces of the $\mathfrak{R}_s$-compatible operators as
In communication field, an important issue is to group users and base stations to as many as possible subnetworks satisfying certain interference constraints. These problems are usually formulated as a graph partition problems which minimize some forms of graph cut. Previous research already gave some results about the cut bounds for unweighted regular graph. In this paper, we prove a result about the lower bound for weighted graphs that have some regular properties and show similar results for more general case.
The aim of these notes is to study some of the structural aspects of the ring of arithmetical functions. We prove that this ring is neither Noetherian nor Artinian. Furthermore, we construct various types of prime ideals. We show arithmetic ring has infinite Krull dimension and its associated prime ideal set is nonempty.
From some works of P. Furtwängler and H.S. Vandiver, we put the basis of a new cyclotomic approach to Fermat's last theorem for p>3 and to a stronger version called SFLT, by introducing governing fields of the form Q(exp(2 i pi/q-1)) for prime numbers q. We prove for instance that if there exist infinitely many primes q, q not congruent to 1 mod p, q^(p-1) not congruent to 1 mod p^2, such that for Q dividing q in Q(exp(2 i pi /q-1)), we have Q^(1-c) = A^p . (alpha), with alpha congruent to 1 mod p^2 (where c is the complex conjugation), then Fermat's last theorem holds for p. More generally, the main purpose of the paper is to show that the existence of nontrivial solutions for SFLT implies some strong constraints on the arithmetic of the fields Q(exp(2 i pi /q-1)). From there, we give sufficient conditions of nonexistence that would require further investigations to lead to a proof of SFLT, and we formulate various conjectures. This text must be considered as a basic tool for future researchs (probably of analytic or geometric nature) - This second version includes some corrections in the English language, an in depth study of the case p=3 (especially Theorem 8), further detail
In 2010, Grigoriev and Shpilrain, introduced some graph-based authentication schemes. We present a cryptanalysis of some of these protocols, and introduce some new schemes to fix the problems.
We explore some aspects of monodromies of D-branes in the Kahler moduli space of Calabi-Yau compactifications. Here a D-brane is viewed as an object of the derived category of coherent sheaves. We compute all the interesting monodromies in some nontrivial examples and link our work to recent results and conjectures concerning helices and mutations. We note some particular properties of the 0-brane.
The middle binomial coefficients can be interpreted as numbers of Motzkin paths which have no horizontal steps at positive heights. Assigning suitable weights gives some nice polynomial extensions. We determine the Hankel determinants and their generating functions for the middle binomial coefficients and derive many conjectures for their polynomial extensions. Finally, we explore experimentally some modifications of the middle binomial coefficients whose Hankel determinants show an interesting modular pattern and obtain some q-analogs.
Let $L=-Δ+V$ with non-negative potential $V$ satisfying some appropriate reverse Hölder inequality. In this paper, we study the boundedness of the commutators of some singular integrals associated to $L$ such as Riesz transforms and fractional integrals with the new BMO functions introduced in \cite{BHS1} on the weighted spaces $L^p(w)$ where $w$ belongs to the new classes of weights introduced by \cite{BHS2}.
Let $r\geq2$ and $r$ be even. An $r$-hypergraph $G$ on $n$ vertices is called odd-colorable if there exists a map $\varphi:[n]\rightarrow\lbrack r]$ such that for any edge $\{j_{1},j_{2},\cdots,j_{r}\}$ of $G$, we have $\varphi(j_{1})+\varphi(j_{2})+\cdot\cdot\cdot+\varphi(j_{r})\equiv r/2(\operatorname{mod}r).$ In this paper, we first determine that, if $r=2^{q}(2t+1)$ and $n\ge 2^{q}(2^{q}-1)r$, then the maximum chromatic number in the class of the odd-colorable $r$-hypergraphs on $n$ vertices is $2^q$, which answers a question raised by V. Nikiforov recently in [V. Nikiforov, Hypergraphs and hypermatrices with symmetric spectrum. Prinprint available in arXiv:1605.00709v2, 10 May, 2016]. We also study some applications of the symmetric spectral property of the odd-colorable $r$-graphs given in that same paper by V. Nikiforov. We show that the Laplacian spectrum and the signless Laplacian spectrum of an $r$-hypergraph $G$ are equal if and only if $G$ is odd-colorable, and then study some further applications of these spectral properties.
We offer two new Mellin transform evaluations for the Riemann zeta function in the region $0<\Re(s)<1.$ Some discussion is offered in the way of evaluating some further Fourier integrals involving the Riemann xi function.
We prove compactness of solutions to some fourth order equations with exponential nonlinearities on four manifolds. The proof is based on a refined bubbling analysis, for which the main estimates are given in integral form. Our result is used in a subsequent paper to find critical points (via minimax arguments) of some geometric functional, which give rise to conformal metrics of constant $Q$-curvature. As a byproduct of our method, we also obtain compactness of such metrics.
The Toba supereruption 74,000 years ago was so massive it may have plunged Earth into years of darkness and cold, leading some scientists to believe humanity nearly went extinct。 Yet archaeological evidence from Africa and Asia suggests early humans were far more resilient than once thought。 Instead of disappearing, some communities adapted with ne
In this letter, we study some evolution networks that grow with linear preferential attachment. Based upon some recent results on the quotient Gamma function, we give a rigorous proof of the asymptotic Mandelbrot law for the degree distribution $p_k \propto (k + c)^{-γ}$ in certain conditions. We also analytically derive the best fitting values for the scaling exponent $γ$ and the shifting coefficient $c$.
Present-day deep neural networks for video semantic segmentation require a large number of fine-grained pixel-level annotations to achieve the best possible results. Obtaining such annotations, however, is very expensive. On the other hand, raw, unannotated video frames are practically free to obtain. Similarly, coarse annotations, which do not require precise boundaries, are also much cheaper. This paper investigates approaches to reduce the annotation cost required for video segmentation datasets by utilising such resources. We show that using state-of-the-art segmentation foundation models, Segment Anything Model (SAM) and Segment Anything Model 2 (SAM 2), we can utilise both unannotated frames as well as coarse annotations to alleviate the effort required for manual annotation of video segmentation datasets by automating mask generation. Our investigation suggests that if used appropriately, we can reduce the need for annotation by a third with similar performance for video semantic segmentation. More significantly, our analysis suggests that the variety of frames in the dataset is more important than the number of frames for obtaining the best performance.