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In this paper, a new sub-family of Hypercubes called the \textit{associated Mersenne graphs} $\mathcal{M}_{n}$ are introduced. The definition of associated Mersenne graphs is motivated from the Fibonacci-run graphs ({Ö}. Eǧecioǧlu, V. Iršič, 2021) by extending run-constrained strings to circularly-run-constrained strings. The name of this new family of graphs is identified with the interesting fact that $|V(\mathcal{M}_{n})|$ is equal to the $n$-th associated Mersenne number. Various interesting structural and enumerative properties of associated Mersenne graphs are investigated, including the analogue of the fundamental recursion, number of vertices and edges, radius, diameter, center, periphery and medianicity. Some future research directions and open problems concerning associated Mersenne graphs are also proposed.

The hexablock is a domain arising from a special case of the $μ$-synthesis problem. We study the commuting operator tuples having the hexablock as a spectral set. Such a tuple is called a hexablock-contraction or simply $\mathbb H$-contraction. We characterize the unitaries and isometries associated with $\mathbb H$-contractions. Two different types of dilation results for $\mathbb H$-contractions are obtained. We find connection of this theory with the operators associated with the symmetrized bidisc and tetrablock, two other domains related to the $μ$-synthesis problem.

In this paper, we study the Euler-Seidel matrices with coefficients and determine the associated Riordan matrix to a given matrix, if it does exist. Computation of the generating fonction of the final sequence is established by the associated Riordan matrix. Applications are given.

We study differential operators associated with families of polynomials orthonormal with respect to certain measures. These operators, when applied to the Fourier transforms of such measures, produce basis functions for expansions of functions analytic on some complex domains. For many classical families of orthogonal polynomials these basis functions are the familiar special functions, such as the Bessel and the spherical Bessel functions. Many familiar identities involving such special functions turn out to be just special cases of such expansions. We also use these differential operators to introduce some new spaces of almost periodic functions. The notions we study here have been successfully applied to signal processing, for example to recovery of band-limited signals from their non-uniform samples as well as from their zero crossings and the locations of their extremal points.

We develop a theory of capacities associated with local Muckenhoupt weights. Fundamental properties of local Muckenhoupt weights will be revisited. Weak type boundedness of nonlinear potential and capacitary strong type inequalities associated with such weights will be addressed. The boundedness of the local maximal function on the spaces of Choquet integrals associated with such weighted capacities will be justified as an application of the theory. We also address on the thinness of sets with the Kellogg property as another application.

Background: Metabolic dysfunction-associated steatotic liver disease (MASLD) affects 30-40% of US adults and is the most common chronic liver disease. Although often asymptomatic, progression can lead to cirrhosis. The objective of the study was to develop and evaluate an electronic health record (EHR) based prediction model to support early detection of MASLD in primary care settings. Methods: We evaluated LASSO logistic regression, random forest, XGBoost, and a neural network model for MASLD prediction using clinical feature subsets from a large EHR database, including the top 10 ranked features. To reduce disparities in true positive rates across racial and ethnic subgroups, we applied an equal opportunity postprocessing method in a prediction model called MASLD EHR Static Risk Prediction (MASER). Results: This retrospective cohort study included 59,492 participants in the training data, 24,198 in the validating data, and 25,188 in the testing data. The LASSO logistic regression model with the top 10 features was selected for its interpretability and comparable performance. Before fairness adjustment, the model achieved AUROC of 0.84, accuracy of 78%, sensitivity of 72%, specifi

This article is devoted to the classification of anti-dendriform algebras that are associated with associativity. They are characterized as algebras with two operations whose sum is associative. In the paper all four-dimensional complex anti-dendriform algebras associated to four-dimensional associative algebras with one-dimensional center are classified

In this paper analysis of the concept of {\it associated homogeneous distributions} (generalized functions) is given, and some problems related to these distributions are solved. It is proved that (in the one-dimensional case) there exist {\it only} {\it associated homogeneous distributions} of order $k=1$. Next, we introduce a definition of {\it quasi associated homogeneous distributions} and provide a mathematical description of all quasi associated homogeneous distributions and their Fourier transform. It is proved that the class of {\it quasi associated homogeneous distributions} coincides with the class of distributions introduced by Gel$'$fand and Shilov \cite[Ch.I,§4.]{G-Sh} as the class of {\it associated homogeneous distributions}. For the multidimensional case it is proved that $f$ is a {\it quasi associated homogeneous distribution} if and only if it satisfies the Euler type system of differential equations. A new type of $Γ$-functions generated by quasi associated homogeneous distributions is defined.

We study the homotopy theory of polyhedral products associated to a combinatorial generalisation of manifolds known as pseudomanifolds. As special cases, we show that loop spaces of moment-angle manifolds associated to triangulations of $S^2$ and $S^3$ decompose as a product of spheres and loops on spheres.

Characterizations of the associated spaces and second associated spaces of the Hardy space on $\mathbb{R}^n$ are given. Some results on the associated spaces of the $\textrm{BMO}(\mathbb{R}^n)$ space are proved also.

In cooperative game theory, associated games allow for providing meaningful characterizations of solution concepts. Moreover, generalized values allow computing an influence or power index of each coalition in a game. In this paper, we view associated games through the lens of game maps and we define the novel Shapley-Hodge Game, briefly ``SHoGa''. We characterize SHoGa via an axiomatic approach as a generalized value, and we thoroughly discuss the consistency properties of its associated game. Furthermore, we describe the Hodge decomposition of an oriented graph representing the transitive closure of the Hasse diagram of coalitions in the game. Finally, we show how SHoGa is linked to the solution of the Poisson equation derived from such a decomposition.

For a Banach space $X$ with a shrinking Schauder frame $(x_i,f_i)$ we provide an explicit method for constructing a shrinking associated basis. In the case that the minimal associated basis is not shrinking, we prove that every shrinking associated basis of $(x_i,f_i)$ dominates an uncountable family of incomparable shrinking associated bases of $(x_i,f_i)$. By adapting a construction of Pełczy{ń}ski, we characterize spaces with shrinking Schauder frames as spaces having the $w^*$-bounded approximation property.

In this paper we continue to study {\it quasi associated homogeneous distributions \rm{(}generalized functions\rm{)}} which were introduced in the paper by V.M. Shelkovich, Associated and quasi associated homogeneous distributions (generalized functions), J. Math. An. Appl., {\bf 338}, (2008), 48-70. [arXiv:math/0608669]. For the multidimensional case we give the characterization of these distributions in the terms of the dilatation operator $U_{a}$ (defined as $U_{a}f(x)=f(ax)$, $x\in \bR^n$, $a >0$) and its generator $\sum_{j=1}^{n}x_j\frac{\partial}{\partial x_j}$. It is proved that $f_k\in {\cD}'(\bR^n)$ is a quasi associated homogeneous distribution of degree $λ$ and of order $k$ if and only if $\bigl(\sum_{j=1}^{n}x_j\frac{\partial}{\partial x_j}-λ\bigr)^{k+1}f_{k}(x)=0$, or if and only if $\bigl(U_a-a^λI\bigr)^{k+1}f_k(x)=0$, $\forall \, a>0$, where $I$ is a unit operator. The structure of a quasi associated homogeneous distribution is described.

The objective of a genome-wide association study (GWAS) is to associate subsequences of individuals' genomes to the observable characteristics called phenotypes (e.g., high blood pressure). Motivated by the GWAS problem, in this paper we introduce the information-theoretic problem of \emph{associated subsequence retrieval}, where a dataset of $N$ (possibly high-dimensional) sequences of length $G$, and their corresponding observable (binary) characteristics is given. The sequences are chosen independently and uniformly at random from $\mathcal{X}^G$, where $\mathcal{X}$ is a finite alphabet. The observable (binary) characteristic is only related to a specific unknown subsequence of length $L$ of the sequences, called \textit{associated subsequence}. For each sequence, if the associated subsequence of it belongs to a universal finite set, then it is more likely to display the observable characteristic (i.e., it is more likely that the observable characteristic is one). The goal is to retrieve the associated subsequence using a dataset of $N$ sequences and their observable characteristics. We demonstrate that as the parameters $N$, $G$, and $L$ grow, a threshold effect appears in the

An orthogonality space is a set equipped with a symmetric, irreflexive relation called orthogonality. Every orthogonality space has an associated complete ortholattice, called the logic of the orthogonality space. To every poset, we associate an orthogonality space consisting of proper quotients (that means, nonsingleton closed intervals), equipped with a certain orthogonality relation. We prove that a finite bounded poset is a lattice if and only if the logic of its orthogonality space is an orthomodular lattice. We prove that that a poset is a chain if and only if the logic of the associated orthogonality space is a Boolean algebra.

Intracellular transport of vesicular cargos, organelles, and other macromolecules is an essential process to move large items through a crowded, and inhomogeneous cellular environment. In an effort to dissect the fundamental effects of crowding and an increasingly complex cellular environment on the transport of individual motor proteins, we have performed in vitro reconstitution experiments with single kinesin-1 motors walking on microtubules in the presence of crowding agents and transient microtubule-associated proteins that more closely emulate the cellular environment. Macromolecular crowding due to inert polymers caused enhanced run lengths of motors, but displayed an increased tendency for non-specific motor association and diffusion, most likely due to depletion interactions. We found that transiently bound associated proteins slowed forward motion, but did not drastically affect the association times, in opposition to previously reported obstacle properties of stably associated microtubule-associated proteins, such as the neuronal protein tau. Such studies of the transport properties of molecular motors in increasingly complex reconstituted environments are important to il

In this paper we use the theory of $ε$-constants associated to tame finite group actions on arithmetic surfaces to define a Brauer group invariant $μ(\X,G,V)$ associated to certain symplectic motives of weight one. We then discuss the relationship between this invariant and $w_2(π)$, the Galois theoretic invariant associated to tame covers of surfaces.

We introduce and study the notion of affine varieties associated to ordered bases and establish Galois connection between the power set of $A^n_K$ and the power set of $K[x_1, . . ., x_n]$, and then induce a Galois correspondence. We generalize the idea by defining affine varieties associated to linear operators. We produce Hilbert's Nullstellensatz version for such varieties and show that there is a 1-1 correspondence between this kind of varieties in $A^n_K$ and the "usual" affine varieties in $A^n_K$. We prove that the \usual" affine varieties forms a skeleton for the category of all affine varieties associated to linear operators, and hence they are equivalent categories.