Multivariate Generalization Research Articles

Advances in the generalized Cesàro polynomials

Cesàro polynomials have been extended in various ways and applied in diverse areas. In this paper, we aim to introduce a multivariable and multiparameter generalization of Cesàro polynomials. Then we explore several generating functions, an addition formula, a differential-recurrence relation, a multiple integral formula for this extended Cesàro polynomial, as well as a multiple integral formula whose kernel is this extended Cesàro polynomial. Also we present several bilinear and bilateral generating functions for this extended Ces`aro polynomial, two of whose examples are demonstrated.

Estimates for Zero Loci of Bernstein–Sato Ideals

We give estimates for the zero loci of Bernstein–Sato ideals. An upper bound is proved as a multivariate generalisation of the upper bound by Lichtin for the roots of Bernstein–Sato polynomials. The lower bounds generalise the fact that log-canonical thresholds, small jumping numbers of multiplier ideals, and their real versions provide roots of Bernstein–Sato polynomials.

Multivariable generalizations of bivariate means via invariance

AbstractFor a given p-variable mean $$M :I^p \rightarrow I$$ M : I p → I (I is a subinterval of $${\mathbb {R}}$$ R ), following (Horwitz in J Math Anal Appl 270(2):499–518, 2002) and (Lawson and Lim in Colloq Math 113(2):191–221, 2008), we can define (under certain assumptions) its $$(p+1)$$ ( p + 1 ) -variable $$\beta $$ β -invariant extension as the unique solution $$K :I^{p+1} \rightarrow I$$ K : I p + 1 → I of the functional equation $$\begin{aligned}&K\big (M(x_2,\dots ,x_{p+1}),M(x_1,x_3,\dots ,x_{p+1}),\dots ,M(x_1,\dots ,x_p)\big )\\&\quad =K(x_1,\dots ,x_{p+1}), \text { for all }x_1,\dots ,x_{p+1} \in I \end{aligned}$$ K ( M ( x 2 , ⋯ , x p + 1 ) , M ( x 1 , x 3 , ⋯ , x p + 1 ) , ⋯ , M ( x 1 , ⋯ , x p ) ) = K ( x 1 , ⋯ , x p + 1 ) , for all x 1 , ⋯ , x p + 1 ∈ I in the family of means. Applying this procedure iteratively we can obtain a mean which is defined for vectors of arbitrary lengths starting from the bivariate one. The aim of this paper is to study the properties of such extensions.

Robust tensor ring-based graph completion for incomplete multi-view clustering

Incomplete multi-view clustering (IMVC) aims to enhance clustering performance by leveraging complementary information from multi-view data, even in the presence of missing instances. This is challenging due to the interference caused by these missing data points. Current IMVC algorithms generally adopt two main strategies: either disregarding the missing instances and focusing on observable data for clustering, or complementing the missing instances within each view to improve the clustering results. However, effectively leveraging the latent information within these missing instances remains a challenge. In response, we propose a novel IMVC framework that complements the tensor consisting of similarity matrices learned from the available instances to enhance the relationships among samples from different views. In addition, we integrate tensor ring completion and M-estimator-based methods into the IMVC approach. This promotes the integration of inter-view information and mitigates errors introduced by missing instances in the model, respectively. Furthermore, we develop two efficient half-quadratic (HQ) based iterative algorithms with soft/hard threshold or multivariate generalization of minimax-concave (GMC) penalty to regularize the low-rank property. Comprehensive evaluations on seven benchmark datasets demonstrate that our method outperforms state-of-the-art IMVC approaches across various metrics.

Log-Concavity of the Alexander Polynomial

Abstract The central question of knot theory is that of distinguishing links up to isotopy. The first polynomial invariant of links devised to help answer this question was the Alexander polynomial (1928). Almost a century after its introduction, it still presents us with tantalizing questions, such as Fox’s conjecture (1962) that the absolute values of the coefficients of the Alexander polynomial $\Delta _{L}(t)$ of an alternating link $L$ are unimodal. Fox’s conjecture remains open in general with special cases settled by Hartley (1979) for two-bridge knots, by Murasugi (1985) for a family of alternating algebraic links, and by Ozsváth and Szabó (2003) for the case of genus $2$ alternating knots, among others. We settle Fox’s conjecture for special alternating links. We do so by proving that a certain multivariate generalization of the Alexander polynomial of special alternating links is Lorentzian. As a consequence, we obtain that the absolute values of the coefficients of $\Delta _{L}(t)$, where $L$ is a special alternating link, form a log-concave sequence with no internal zeros. In particular, they are unimodal.

Lattice points in slices of prisms

Abstract We conduct a systematic study of the Ehrhart theory of certain slices of rectangular prisms. Our polytopes are generalizations of the hypersimplex and are contained in the larger class of polypositroids introduced by Lam and Postnikov; moreover, they coincide with polymatroids satisfying the strong exchange property up to an affinity. We give a combinatorial formula for all the Ehrhart coefficients in terms of the number of weighted permutations satisfying certain compatibility properties. This result proves that all these polytopes are Ehrhart positive. Additionally, via an extension of a result by Early and Kim, we give a combinatorial interpretation for all the coefficients of the $h^*$ -polynomial. All of our results provide a combinatorial understanding of the Hilbert functions and the h-vectors of all algebras of Veronese type, a problem that had remained elusive up to this point. A variety of applications are discussed, including expressions for the volumes of these slices of prisms as weighted combinations of Eulerian numbers; some extensions of Laplace’s result on the combinatorial interpretation of the volume of the hypersimplex; a multivariate generalization of the flag Eulerian numbers and refinements; and a short proof of the Ehrhart positivity of the independence polytope of all uniform matroids.

DIFFERENTIATED APPROACH TO ANTIHYPOXIC, ENDOPORTAL AND PHAGE THERAPY FOR ADVANCED PURULENT PERITONITIS

The purpose of this study was to evaluate the results of differentiated use of enteral antihypoxic therapy (EAT), endoportal therapy (EPT), as well as phage therapy (PT) in patients with diffuse purulent peritonitis complicated by acute intestinal failure (AIF). Materials and Methods: 144 patients with diffuse peritonitis and AIF were treated. The patients were divided into two equal comparable groups: main (n=72) and comparison (n=72). In addition, each group was divided into three subgroups (24 patients each) depending on the degree of AIF. The groups were distributed using adaptive randomization. EAT was carried out by introducing oxygenated water through a nasogastrointestinal (NGI) tube. EPT was carried out by introducing an antibiotic, a hepatoprotector and an antihypoxant through a catheter into the portal system. PT was carried out by introducing a polyvalent bacteriophage through a NGI tube. The following were dynamically studied: EMFC, APACHE II, abdominal cavity index (ACI), acid-base status and blood gas composition, integral intestinal pressure (IIP), intra-abdominal pressure (IAP). Comparison of quantitative indicators in the study groups was carried out using the Wilcoxon-Mann-Whitney U test, as well as its multivariate generalization - the Kruskal-Wallis test, for checking the equality of medians of several samples. Results: The dynamic study of IIP and IAP in the study groups on the day 3 of treatment, showed significantly lower values in the main group (p≤0.05). There was a more rapid decrease in the manifestations of hypoxia in the main group, which was confirmed by an increase in the oxygen partial pressure in the blood. The average number of bed days in the intensive care unit was 4.2±0.3 days in the comparison group, and 3.3±0.2 days in the study group. The overall average number of bed-days was 15.8 in the study group, and 18.5 in the comparison group. Mortality was 16.3% in the comparison group, and 11.3% in the study group (a decrease of 30.7%). Thus, when prescribing NGI, EAT, EPT and PT, one should be guided by the severity of organ dysfunction and the AIF stage. With values of ACI <13, APACHE II <10 and EMFC <5, NGI and EAT are indicated. With ACI 14-22, APACHE II 10-15, EMFC 5-25 - NGI and EAT are supplemented with PT. In the most severe cases, with ACI ≥23, APACHE II >16 and EMFC >25 points, EPT must be added to the above treatment methods. Conclusion: The proposed scheme for the differentiated use of EPT, EAT and PT in patients with diffuse peritonitis complicated by AIF allows to relatively quickly stop inflammation and prevent the development of multiple organ failure and sepsis.

Parameter Estimation of the Dirichlet Distribution Based on Entropy

The Dirichlet distribution as a multivariate generalization of the beta distribution is especially important for modeling categorical distributions. Hence, its applications vary within a wide range from modeling cell probabilities of contingency tables to modeling income inequalities. Thus, it is commonly used as the conjugate prior of the multinomial distribution in Bayesian statistics. In this study, the parameters of a bivariate Dirichlet distribution are estimated by entropy formalism. As an alternative to maximum likelihood and the method of moments, two methods based on the principle of maximum entropy are used, namely the ordinary entropy method and the parameter space expansion method. It is shown that in estimating the parameters of the bivariate Dirichlet distribution, the ordinary entropy method and the parameter space expansion method give the same results as the method of maximum likelihood. Thus, we emphasize that these two methods can be used alternatively in modeling bivariate and multinomial Dirichlet distributions.

Counting derangements with signed right-to-left minima and excedances

Recently Alexandersson and Getachew proved some multivariate generalizations of a formula for enumerating signed excedances in derangements. In this paper we first relate their work to a recent continued fraction for permutations and confirm some of their observations. Our second main result is two refinements of their multivariate identities, which clearly explain the meaning of each term in their main formulas. We also explore some similar formulas for permutations of type B.

Systematic and Durable Analysis of the Rule based Cauchy’s Inequality in solving the n-order Differential Equations

This paper instigates the multivariate generalization of the widely used Cauchy inequality 1 + x ≤ ex where x can be any non-negative real number. The results of this study can be the solution for the Cauchy’s problem for particular Ordinary Differential Equation (ODE). This is also related to study of the complete monotone function and the divided differences theory. The proof is based on the empty product convention notion and the Beppo Levi theorem of Monotone convergence. This study is also extended to multivariate generalization of the simultaneous inequalities. Key Words: ODEs, Inequalities, Population Dynamics, Simultaneous Inequalities, Divided differences.

A model of sustainable development of science and high technology cities taking into account the area's life cycle

Subject. This article analyzes the cyclical nature of urban development. Objectives. The article aims to identify new areas for the development of cities of science and high technology. Methods. For the study, I used the methods of general systems, logical, comparative, and multivariate analyses, grouping, generalization, expert assessments, and optimal control. Results. The article proposes a methodology for constructing an economic and mathematical model of sustainable development of cities of science and high technology. Conclusions. A self-developing innovation ecosystem will help ensure the sustainable development of science cities in the long term.

Multivariate correlation inequalities for P-partitions

Motivated by the Lam--Pylyavskyy inequalities for Schur functions, we give a far reaching multivariate generalization of Fishburn's correlation inequality for the number of linear extensions of posets. We then give a multivariate generalization of the Daykin--Daykin--Paterson inequality proving log-concavity of the order polynomial of a poset. We also prove a multivariate $P$-partition version of the cross-product inequality by Brightwell--Felsner--Trotter. The proofs are based on a multivariate generalization of the Ahlswede--Daykin inequality.

A unified approach to multivariate polynomial sequences with real stability

We give new sufficient conditions for a sequence of multivariate polynomials to be real stable. As applications, we obtain several known results, such as the real stability of multivariate Eulerian polynomials, multivariate Bell polynomials and multivariate polynomials over Stirling permutations, in a unified manner. And we also show some new results, such as the real stability of multivariate Lah polynomials and multivariate polynomials over Jacobi-Stirling permutations, and the proper position property of multivariate orthogonal polynomials and multivariate matching polynomials. Finally, we obtain a multivariate generalization of Fisk's result by presenting a 2×2 matrix, which preserves the proper position property.

Exact Permutation and Bootstrap Distribution of Generalized Pairwise Comparisons Statistics

To analyze multivariate outcomes in clinical trials, several authors have suggested generalizations of the univariate Mann–Whitney test. As the Mann–Whitney statistic compares the subjects’ outcome pairwise, the multivariate generalizations are known as generalized pairwise comparisons (GPC) statistics. For GPC statistics such as the net treatment benefit, the win ratio, and the win odds, asymptotic based or re-sampling tests have been suggested in the literature. However, asymptotic methods require a sufficiently high sample size to be accurate, and re-sampling methods come with a high computational burden. We use graph theory notation to obtain closed-form formulas for the expectation and the variance of the permutation and bootstrap sampling distribution of the GPC statistics, which can be utilized to develop fast and accurate inferential tests for each of the GPC statistics. A simple example and a simulation study demonstrate the accuracy of the exact permutation and bootstrap methods, even in very small samples. As the time complexity is O(N2), where N is the total number of patients, the exact methods are fast. In situations where asymptotic methods have been used to obtain these variance matrices, the new methods will be more accurate and equally fast. In situations where bootstrap has been used, the new methods will be both more accurate and much faster.

Extensions of the Kahn-Saks inequality for posets of width two

The Kahn-Saks inequality is a classical result on the number of linear extensions of finite posets. We give a new proof of this inequality for posets of width two and both elements in the same chain using explicit injections of lattice paths. As a consequence we obtain a \(q\)-analogue, a multivariate generalization and an equality condition in this case. We also discuss the equality conditions of the Kahn-Saks inequality for general posets and prove several implications between conditions conjectured to be equivalent.Mathematics Subject Classifications: 05A15, 05A19, 05A20, 05A30, 06A07Keywords: Poset inequality, Stanley's inequality, Kahn-Saks inequality, log-concavity, q-analogues, equality conditions, lattice paths

Mixed cumulative probit: a multivariate generalization of transition analysis that accommodates variation in the shape, spread and structure of data.

Biological data are frequently nonlinear, heteroscedastic and conditionally dependent, and often researchers deal with missing data. To account for characteristics common in biological data in one algorithm, we developed the mixed cumulative probit (MCP), a novel latent trait model that is a formal generalization of the cumulative probit model usually used in transition analysis. Specifically, the MCP accommodates heteroscedasticity, mixtures of ordinal and continuous variables, missing values, conditional dependence and alternative specifications of the mean response and noise response. Cross-validation selects the best model parameters (mean response and the noise response for simple models, as well as conditional dependence for multivariate models), and the Kullback-Leibler divergence evaluates information gain during posterior inference to quantify mis-specified models (conditionally dependent versus conditionally independent). Two continuous and four ordinal skeletal and dental variables collected from 1296 individuals (aged birth to 22 years) from the Subadult Virtual Anthropology Database are used to introduce and demonstrate the algorithm. In addition to describing the features of the MCP, we provide material to help fit novel datasets using the MCP. The flexible, general formulation with model selection provides a process to robustly identify the modelling assumptions that are best suited for the data at hand.

Multivariate Birnbaum–Saunders distribution based on a skewed distribution and associated EM-estimation

We develop here a multivariate generalization of Birnbaum–Saunders (BS) distribution based on the multivariate skew-normal distribution. Some distributional characteristics and properties are presented, as well as a simple and efficient EM algorithm for the iterative computation of the maximum likelihood (ML) estimates of model parameters, through the hierarchical representation of the proposed model. The standard errors of the maximum likelihood estimates are calculated from the observed Fisher information matrix. Moreover, by using the tools, we present a log-linear regression model, where the the ML estimates are once again obtained using an EM algorithm. Finally, simulation studies and two applications to real data sets are presented for illustrating the model and the inferential results developed here.

The cotype zeta function of [formula omitted

We give an asymptotic formula for the number of sublattices Λ⊆Zd of index at most X for which Zd/Λ has rank at most m, answering a question of Nguyen and Shparlinski. We compare this result to work of Stanley and Wang on Smith normal forms of random integral matrices and discuss connections to the Cohen–Lenstra heuristics. Our arguments are based on Petrogradsky’s formulas for the cotype zeta function of Zd, a multivariable generalization of the subgroup growth zeta function of Zd.

The relation between alternating sign matrices and descending plane partitions: n + 3 pairs of equivalent statistics

There is the same number of n×n alternating sign matrices (ASMs) as there is of descending plane partitions (DPPs) with parts no greater than n, but finding an explicit bijection is an open problem for about 40 years now. So far, quadruples of statistics on ASMs and on DPPs that have the same joint distribution have been identified. We introduce extensions of ASMs and of DPPs along with n+3 statistics on each extension, and show that the two families of statistics have the same joint distribution. The ASM-DPP equinumerosity is obtained as an easy consequence by considering the (−1)-enumerations of these extended objects with respect to one pair of the n+3 pairs of statistics. One may speculate that the fact that these extensions might be necessary to have this significant increase in the number of statistics, as well as the involvement of signs when specializing to ASMs and DPPs may hint at the obstacles in finding an explicit bijection between ASMs and DPPs. One important tool for our proof is a multivariate generalization of the operator formula for the number of monotone triangles with prescribed bottom row that generalizes Schur functions.

Prospects for the Development of Information and Analytical Activities in the System of Public Authority Under the Conditions of Wartime Posture in Ukraine

Ensuring the effective implementation of the basic functions by the country’s public authorities significantly depends on the level of development of its information and analytical activities and the efficiency of the formation and implementation of state policy. The intensification of democratization and the increasing flow of information in the system of public power lead to the activation of the process of formation and development of information and analytical activity, which in the conditions of wartime posture is especially actualized and acquires significant importance. The purpose of the research lies in substantiating the theoretical aspects and practical recommendations for determining the prospects for the development of information and analytical activities in the system of public power in the conditions of wartime posture in Ukraine. General scientific and special methods of economic analysis have been used in the research, in particular as follows: analysis and synthesis, analogies and comparisons, grouping and multivariate cluster analysis, systematization and generalization, graphical and tabular methods. The results of the conducted research of the theoretical and applied principles regarding the determination of prospects for the development of information and analytical activities in the system of public power in the conditions of wartime posture in Ukraine has made it possible to establish that the full-scale invasion of the Russian Federation in Ukraine has caused significant destructive changes in the system of public power, which requires the introduction of effective methods of information and analytical activity and innovative technological means of information protection. It has been proven that Ukraine, like other countries of the Eastern Partnership, are unable to fully ensure the stable development of information and analytical activities in the system of public authorities, for as much as they are characterized by a low level of their digitization, resistance to misinformation and countering cyber threats. It has been revealed that the parameters of Ukraine according to the analysed indicators are moderate from among the countries of the Eastern Partnership, as evidenced by the value of the Multidimensional Index of Digitization 0,42‒0,48;