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When predicting crop yield using both functional and multivariate predictors, the prediction performances benefit from the inclusion of the interactions between the two sets of predictors. We assume the interaction depends on a nonparametric, single-index structure of the multivariate predictor and reduce each functional predictor’s dimension using functional principal component analysis (FPCA). Allowing the number of FPCA scores to diverge to infinity, we consider a sequence of semiparametric working models with a diverging number of predictors, which are FPCA scores with estimation errors. We show that the parametric component of the model is root-n consistent and asymptotically normal, the overall prediction error is dominated by the estimation of the nonparametric interaction function, and justify a CV-based procedure to select the tuning parameters.

Z-complementary pairs (ZCPs) have been widely used in different communication systems. In this paper, we first investigate the odd-periodic correlation property of ZCPs, and propose a new class of ZCPs, called ZOC-ZCPs with zero correlation zone (ZCZ) width \begin{document}$ Z $\end{document} and zero odd-period correlation zone (ZOCZ) width \begin{document}$ Z_{odd} = Z $\end{document} by horizontal concatenation of a certain combination of some known ZCPs. Particularly, based on any known Golay pair, we can generate a class of GCPs of more flexible length whose ZOCZ width is larger than a quarter of the sequence length.

In this paper, we propose a new class of optimal one-coincidence FHS (OC-FHS) sets with respect to the Peng-Fan bounds, including prime sequence sets and HMC sequence sets as special cases. Thereafter, through investigating their properties, we determine all of the FHS distances in the OC-FHS set. Finally, for a given positive integer, we also propose a new class of wide-gap one-coincidence FHS (WG-OC-FHS) sets where the FHS gap is larger than the given positive integer. Moreover, such a WG-OC-FHS set is optimal with respect to the WG-Lempel-Greenberger bound and the WG-Peng-Fan bounds simultaneously.

In this paper, for each of six families of three-valued \begin{document}$ m $\end{document} -sequence correlation, we construct an infinite family of five-weight codes from trace codes over the ring \begin{document}$ R = \mathbb{F}_2+u\mathbb{F}_2 $\end{document} , where \begin{document}$ u^2 = 0. $\end{document} The trace codes have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. Their support structure is determined. An application to secret sharing schemes is given. The parameters of the binary image are \begin{document}$ [2^{m+1}(2^m-1),4m,2^{m}(2^m-2^r)] $\end{document} for some explicit \begin{document}$ r. $\end{document}

A Boolean function in \begin{document}$ n $\end{document} variables is 2-rotation symmetric if it is invariant under even powers of \begin{document}$ \rho(x_1, \ldots, x_n) = (x_2, \ldots, x_n, x_1) $\end{document} , but not under the first power (ordinary rotation symmetry); we call such a function a 2-function. A 2-function is called monomial rotation symmetric (MRS) if it is generated by applying powers of \begin{document}$ \rho^2 $\end{document} to a single monomial. If the quartic MRS 2-function in \begin{document}$ 2n $\end{document} variables has a monomial \begin{document}$ x_1 x_q x_r x_s $\end{document} , then we use the notation \begin{document}$ {2-}(1,q,r,s)_{2n} $\end{document} for the function. A detailed theory of equivalence of quartic MRS 2-functions in \begin{document}$ 2n $\end{document} variables was given in a \begin{document}$ 2020 $\end{document} paper by Cusick, Cheon and Dougan. This theory divides naturally into two classes, called \begin{document}$ mf1 $\end{document} and \begin{document}$ mf2 $\end{document} in the paper. After describing the equivalence classes, the second major problem is giving details of the linear recursions that the Hamming weights for any sequence of functions \begin{document}$ {2-}(1,q,r,s)_{2n} $\end{document} (with \begin{document}$ q say), \begin{document}$ n = s, s+1, \ldots $\end{document} can be shown to satisfy. This problem was solved for the \begin{document}$ mf1 $\end{document} case only in the \begin{document}$ 2020 $\end{document} paper. Using new ideas about "short" functions, Cusick and Cheon found formulas for the \begin{document}$ mf2 $\end{document} weights in a \begin{document}$ 2021 $\end{document} sequel to the \begin{document}$ 2020 $\end{document} paper. In this paper the actual recursions for the weights in the \begin{document}$ mf2 $\end{document} case are determined.

In this paper, we study the complexity of solving overdetermined generic systems of bilinear polynomials over a finite field \begin{document}$ \mathbb{F} $\end{document} . Given a generic bilinear sequence \begin{document}$ B\in \mathbb{F}[ \textbf{x}, \textbf{y}] $\end{document} , with respect to a partition of variables \begin{document}$ \textbf{x} $\end{document} , \begin{document}$ \textbf{y} $\end{document} , we show that, the solutions of the system \begin{document}$ B = \textbf{0} $\end{document} can be efficiently found on the \begin{document}$ \mathbb{F}[ \textbf{y}] $\end{document} -module generated by \begin{document}$ B $\end{document} . Following this observation, we define notions of regularity for overdetermined bilinear systems, and we conjecture that they are generic properties. Also, we propose three variations of Grobner basis algorithms, that only involve multiplication by monomials in the \begin{document}$ \textbf{y} $\end{document} -variables, namely, \begin{document}$ \textbf{y} $\end{document} -XL, based on the XL algorithm, \begin{document}$ \textbf{y} $\end{document} -MXL, based on the mutant XL algorithm, and \begin{document}$ \textbf{y} $\end{document} -HXL, based on a hybrid approach. We develop the necessary theoretical tools to estimate the complexity of the algorithms for such sequences. We present experimental evidence for testing our conjecture, verifying our results, and comparing the complexity of the various methods. Based on the experimental data, we can conclude that \begin{document}$ \textbf{y} $\end{document} -MXL outperforms F4 on bilinear systems.

In this paper, we present a new hybrid spectral-conjugate gradient (SCG) algorithm for finding approximate solutions to nonlinear monotone operator equations. The hybrid conjugate gradient parameter has the Polak–Ribiere–Polyak (PRP), Dai–Yuan (DY), Hestenes–Stiefel (HS) and Fletcher–Reeves (FR) as special cases. Moreover, the spectral parameter is selected such that the search direction has the descent property. Also, the search directions are bounded and the sequence of iterates generated by the new hybrid algorithm converge globally. Furthermore, numerical experiments were conducted on some benchmark nonlinear monotone operator equations to assess the efficiency of the proposed algorithm. Finally, the algorithm is shown to have the ability to recover disturbed signals.

In this article, we study the finite-time stabilization and the asymptotic stabilization with probability one of Markovian jump Boolean control networks (MJBCNs) by sampled-data state feedback controls (SDSFCs). Based on the semi-tensor product (STP), we introduce an augmented variable multiplied by the vector form of the switching signal and the state of MJBCN. We find that under SDSFC, the sequence of the states of the augmented variable at sampling instants satisfies the Markov property. Based on the convergences of the switching signal and the augmented variable, we obtain the sufficient and necessary criteria for the finite-time stabilization and the asymptotic stabilization of MJBCNs by SDSFCs, respectively. Moreover, for the two kinds of stabilization, the feedback matrices of SDSFCs are constructed, respectively. Finally, the obtained results are applied to an apoptosis network and a model of the lactose operon in the Escherichia Coli.

This paper addresses the approximate and exact controllability issues of switched infinite dimensional linear systems. Firstly, a controllability map that is related to switching sequence and a controllability Gramian operator are proposed, based on which two necessary and sufficient conditions are established respectively for approximate and exact controllability; Secondly, it shows that the obtained conditions can degenerate into the existing controllability criteria of infinite dimensional linear systems, switched finite dimensional linear systems and single finite dimensional linear system; Finally, the new results are applied to analyze the approximate controllability of switched linear parabolic systems and the exact controllability of switched linear wave systems. Two examples are given to illustrate the effectiveness of the proposed approaches.

We introduce an object called a tree growing sequence (TGS) in an effort to generalize bijective correspondences between G -parking functions, spanning trees, and the set of monomials in the Tutte polynomial of a graph G . A tree growing sequence determines an algorithm which can be applied to a single function, or to the set P G , q of G -parking functions. When the latter is chosen, the algorithm uses splitting operations – inspired by the recursive definition of the Tutte polynomial – to iteratively break P G , q into disjoint subsets. This results in bijective maps τ and ρ from P G , q to the spanning trees of G and Tutte monomials, respectively. We compare the TGS algorithm to Dhar’s algorithm and the family described by Chebikin and Pylyavskyy in 2005. Finally, we compute a Tutte polynomial of a zonotopal tiling using analogous splitting operations.

Let G be a finite group. For a positive integer d, let s d N ( G ) denote the smallest integer l such that every sequence S over G of length | S | ≥ l has a nonempty product-one subsequence T with | T | ≡ 0 (mod d). In this paper, we mainly study this invariant for dihedral groups D 2 n and metacyclic groups C p ⋉ s C q .

We define and analyze the notion of variational Wardrop equilibrium for nonatomic aggregative games with an infinity of players types. These equilibria are characterized through an infinite-dimensional variational inequality. We show, under monotonicity conditions, a convergence theorem enables to approximate such an equilibrium with arbitrary precision. To this end, we introduce a sequence of nonatomic games with a finite number of players types, which approximates the initial game. We show the existence of a symmetric Wardrop equilibrium in each of these games. We prove that those symmetric equilibria converge to an equilibrium of the infinite game, and that they can be computed as solutions of finite-dimensional variational inequalities. The model is illustrated through an example from smart grids: the description of a large population of electricity consumers by a parametric distribution gives a nonatomic game with an infinity of different players types, with actions subject to coupling constraints., arXiv admin note: substantial text overlap with arXiv:1806.06230

In 1979, Pisier proved remarkably that a sequence of independent and identically distributed standard Gaussian random variables determines, via random Fourier series, a homogeneous Banach algebra P strictly contained in C(T), the class of continuous functions on the unit circle T and strictly containing the classical Wiener algebra A(T), that is, A(T)⫋P⫋C(T). This improved some previous results obtained by Zafran in solving a long-standing problem raised by Katznelson. In this paper, we extend Pisier’s result by showing that any probability measure on the unit circle defines a homogeneous Banach algebra contained in C(T). Thus Pisier algebra is not an isolated object but rather an element in a large class of Pisier-type algebras. We consider the case of spectral measures of stationary sequences of Gaussian random variables and obtain a sufficient condition for the boundedness of the random Fourier series ∑ n∈Zfˆ(n)ξnexp(2πint) in the general setting of dependent random variables (ξn).

Change point analysis aims to detect structural changes in a data sequence. It has always been an active research area since it was introduced in the 1950s. In modern statistical applications, however, high-throughput data with increasing dimensions are ubiquitous in fields ranging from economics, finance to genetics and engineering. For those problems, the earlier works are typically no longer applicable. As a result, the problem of testing a change point for high dimensional data sequences has been an important yet challenging task. In this paper, we first focus on models for at most one change point, and review recent state-of-art techniques for change point testing of high dimensional mean vectors and compare their theoretical properties. Based on that, we provide a survey of some extensions to general high dimensional parameters beyond mean vectors as well as strategies for testing multiple change points in high dimensions. Finally, we discuss some open problems for possible future research directions.

We define a new metric between natural numbers induced by the $\ell_\infty$ norm of their unique prime signatures. In this space, we look at the natural analog of the number line and study the arithmetic function $L_\infty(N)$, which tabulates the cumulative sum of distances between consecutive natural numbers up to $N$ in this new metric. Our main result is to identify the positive and finite limit of the sequence $L_\infty(N)/N$ as the expectation of a certain random variable. The main technical contribution is to show with elementary probability that for $K=1,2$ or $3$ and $\omega_0,\ldots,\omega_K\geq 2$ the following asymptotic density holds $$ \lim_{n\to\infty}\frac{\big|\big\{M\leq n:\; \|M-j\|_\infty, Comment: This article supersedes arXiv:1711.02903. v3: accepted version in J. Number Theory with additional Appendix. v2: Conjecture 1 and 2 of v1 are now proved, major revision of exposition

A portmanteau-type statistic based on sample covariance is proposed to test for the existence of the unit-root of an autoregressive model under a heavy-tailed linear noise, where the noise ɛ t = ∑ j = 0 ∞ d j η t − j and { η t } is a sequence of i.i.d. innovations belonging to the domain of attraction of an α -stable distribution for some α ∈ ( 0 , 2 ) . The proposed procedure is nonparametric and easy to implement. Under certain regular assumptions, the limit distribution of the statistics is shown to be a functional of a standard stable distribution. Further, the good finite sample studies show that the new test outperforms PP, DF and ADF tests in size and power. Applications to the daily world crude oil price and 3-month AA financial commercial paper rate are also given to illustrate the performance of the new test.

Let $s_0,s_1,s_2,\ldots$ be a sequence of rational numbers whose $m$th divided difference is integer-valued. We prove that $s_n$ is a polynomial function in $n$ if $s_n \ll \theta^n$ for some positive number $\theta$ satisfying $\theta < e^{1 + \tfrac{1}{2} + \cdots+ \tfrac{1}{m}} -1$., Comment: 18 pages; revised for clarity

In this paper we consider the fractional parts of a general sequence, for example the sequence $\alpha \sqrt{n}$ or $\alpha n^2$. We give a general method, which allows one to show that long-range correlations (correlations where the support of the test function grows as we consider more points) are Poissonian. We show that these statements about convergence can be reduced to bounds on associated Weyl sums. In particular we apply this methodology to the aforementioned examples. In so doing, we recover a recent result of Technau-Walker (2020) for the triple correlation of $\alpha n^2$ and generalize the result to higher moments. For both of the aforementioned sequences this is one of the only results which indicates the pseudo-random nature of the higher level ($m \ge 3$) correlations., Comment: 132 pages

Let M be a positive integer and q ∈ ( 1 , M + 1 ] . A q-expansion of a real number x is a sequence ( c i ) = c 1 c 2 ⋯ with c i ∈ { 0 , 1 , … , M } such that x = ∑ i = 1 ∞ c i q − i . Let U q j denote the set of numbers having exactly j expansions. Contrary to the unique expansions, we prove for each j ≥ 2 that the set U q j is closed only if it is empty. Then we generalize an important example of Sidorov by exhibiting many non-trivial bases q for which the Hausdorff dimension of U q j is independent of j.

The problems discussed in this paper are motivated by general ratio consensus algorithms, introduced by Kempe, Dobra, and Gehrke (2003) in a simple form as the push-sum algorithm, later extended by B\'en\'ezit et al. (2010) under the name weighted gossip algorithm. We consider a communication protocol described by a strictly stationary, ergodic, sequentially primitive sequence of non-negative matrices, applied iteratively to a pair of fixed initial vectors, the components of which are called values and weights defined at the nodes of a network. The subject of ratio consensus problems is to study the asymptotic properties of ratios of values and weights at each node, expecting convergence to the same limit for all nodes. The main results of the paper provide upper bounds for the rate of the almost sure exponential convergence in terms of the spectral gap associated with the given sequence of random matrices. It will be shown that these upper bounds are sharp. Our results complement previous results of Picci and Taylor (2013) and Iutzeler, Ciblat and Hachem (2013).

The aim of this work is to give some fixed point results based on the technique of measure of noncompactness which extend the classical Darbo’s theorem. With the help of our Darbo-type theorem, we obtain the existence of solution of implicit fractional integral equations in C ( I , l p α ) (collection of all continuous functions from I = [ 0 , a ] ( a > 0 ) to l p α ), where l p α is a tempered sequence space. Finally, we present a numerical example to see the validity and practicability of our existence result.

The problem of constructing the most powerful test for random number generators (RNGs) is considered, where the generators are modelled by stationary ergodic processes. At present, RNGs are widely used in data protection, modelling and simulation systems, computer games, and in many other areas where the generated random numbers should look like binary numbers of a Bernoulli equiprobable sequence. Another problem considered is that of constructing effective statistical tests for random number generators (RNG). Currently, effectiveness of statistical tests for RNGs is mainly estimated based on experiments with various RNGs. We find an asymptotic estimate for the p -value of an optimal test in the case where the alternative hypothesis is a known stationary ergodic source, and then describe a family of tests each of which has the same asymptotic estimate of the p -value for any (unknown) stationary ergodic source. This model appears to be acceptable for binary sequences generated by physical devices that are used in cryptographic data protection systems.

In this paper, to accurately estimate the rare failure rates for large-scale circuits (e.g., SRAM) where process variations are modeled as truncated normal distributions in high-dimensional space, we propose a novel truncated scaled-sigma sampling (T-SSS) method. Similar to scaled-sigma sampling (SSS), T-SSS distorts the truncated normal distributions by a scaling factor, resulting in an analytical model for failure rate estimation. By drawing random samples from the distorted distribution and estimating a sequence of scaled failure rates, we can solve all unknown model coefficients and predict the original failure rate by extrapolation. The accuracy of T-SSS is further assessed by estimating its confidence interval (CI) based on re-sampling. Our numerical results demonstrate that the proposed T-SSS method can achieve superior accuracy over the state-of-the-art method without increasing the computational cost.

Based on the two-point Hermite interpolation technique, the paper proposes a two-point generalized Hermite interpolation and its inversion in terms of weight functions. We prove that upon combining fourth-order optimal iterative scheme to the double Newton’s method (DNM), we can yield a generalized Hermite interpolation formula to relate the first-order derivatives at two points, and the converse is also true. Resorted on the DNM and the derived formula for the generalized inverse Hermite interpolation, some new third-order iterative schemes of quadrature type are constructed. Then, the fourth-order optimal iterative schemes are devised by using a double-weight function. A functional recursion formula is developed which can generate a sequence of two-point generalized Hermite interpolations for any two given weight functions with certain constraints; hence, a more general class of fourth-order optimal iterative schemes is developed from the functional recursion formula. The constructions of fourth-order optimal iterative schemes by using the techniques of double-weight function and the recursion formula obtained from a single weight function are appeared in the literature at the first time. The novelties involve deriving a two-point generalized Hermite interpolation and its inversion in terms of weight functions subjected to two conditions and through the recursion formula, relating the DNM to the third-order iterative schemes by the inverse Hermite interpolation, formulating a functional recursion formula, deriving a broad class fourth-order optimal iterative schemes through double-weight functions rather than the previous technique with a single-weight function, and finding that the new double-weight function and the newly developed fourth-order optimal iterative schemes are inclusive being convergent faster and competitive to other iterative schemes.

A temporal graph is a pair ( G , λ ) where G is a simple graph and λ is a function assigning to each edge time labels telling at which snapshots each edge is active. As recently defined by Mertzios, Molter and Zamaraev (2021), a temporal coloring of ( G , λ ) is a sequence of colorings of the vertices of the snapshots such that each edge is properly colored at least once. We first focus on t-persistent and t-occurrent temporal graphs. The former (resp. the latter) are temporal graphs where each edge of G stays active for at least t consecutive (resp. not-necessarily consecutive) snapshots. We study which values of t make the problem polynomial-time solvable. We also investigate the complexity of the problem when restricted to temporal graphs ( G , λ ) such that G has bounded treewidth.

Assume the initial data of compressible Euler equations has finite energy and total mass. We can construct a sequence of solutions of one-dimensional compressible Navier-Stokes equations (density-dependent viscosity) with stress-free boundary conditions, so that, up to a subsequence, the sequence of solutions of compressible Navier-Stokes equations converges to a finite-energy weak solution of compressible Euler equations. Hence the inviscid limit of the compressible Navier-Stokes is justified. It is worth pointing out that our result covers the interesting case of the Saint-Venant model for shallow water (i.e., α = 1 , γ = 2 ).

In this paper, we propose two inertial projection methods to solve mixed variational inequalities without assuming monotonicity on the cost operator. Consequently, we show that the sequence of iterates generated by our proposed methods converge globally to a solution of the mixed variational inequality under the condition that the set of solutions of the dual mixed variational inequality is nonempty. Our convergence results are obtained without assuming any further stringent condition imposed in other related results in the literature. Moreover, our results improve on many important related results in the literature. Some numerical illustrations are given to show the efficiency of our proposed methods.

An online algorithm is typically unaware of the length of the input request sequence that it is called upon. Consequently, it cannot determine whether it has already processed most of its input or whether the bulk of work is still ahead. In this paper, we are interested in whether some sort of orientation within the request sequence is nevertheless possible. For a real number 0 f 1 , our objective is to preemptively guess the position f n within the request sequence of unknown length n : While processing the input, the online algorithm maintains a guess and is only allowed to update its guess to the position of the current element under investigation. We give a complete characterization of the optimal guessing strategies with respect to f : For f ≤ f 0 ≈ 0 . 1867 , the trivial algorithm that never updates its initial guess is (asymptotically) optimal. For f > f 0 , we give a randomized algorithm that in expectation places the guess at distance β ( f ) n from f n , where β is a non-elementary function, and we prove that this is optimal. Our findings show that the position f max n ≈ 0 . 2847 n is hardest to guess. We also give upper and lower bounds for a natural extension to weighted sequences. Our work can also be seen as the first randomized approach to the online checkpointing problem.

The subject of this paper is the problem of optimal stopping of a sequence of independent and identically distributed random variables with unknown distribution. We propose a stopping rule that is based on relative ranks and study its performance as measured by the maximal relative regret over suitable nonparametric classes of distributions. It is shown that the proposed rule is first-order asymptotically optimal and nearly rate optimal in terms of the rate at which the relative regret converges to zero. We also develop a general method for numerical solution of sequential stopping problems with no distributional information and use it in order to implement the proposed stopping rule. Some numerical experiments illustrating performance of the rule are presented as well.

One of the most interesting properties of Pascal’s triangle is that the sequence of the sums of the elements on its diagonals is the best known recurrence sequence, the Fibonacci sequence. It is also known that other diagonals can be associated with other relevant recurrence sequences, such as the Padovan and k -Padovan sequences. In this paper, we see that similar properties also hold for diagonals of generalized Pascal’s triangles. We show that the diagonal sums in generalized Pascal’s triangles belong to the family of the so-called ‘ k -Padovan-like sequences’ which are linear recurrences of order k with constant coefficients. A recurrence connection between the k -Padovan and k -Padovan-like sequences is derived.

We prove an asymptotic result on the maximum number of k -vertex subtrees in binary trees of given order. This problem turns out to be equivalent to determine the maximum number of k + 2 -cycles in n -vertex outerplanar graphs, thus we settle the generalised outerplanar Turan number for all cycles. We also determine the exponential growth of the generalised outerplanar Turan number of paths P k as a function of k which implies the order of magnitude of the generalised outerplanar Turan number of arbitrary trees. The bounds are strongly related to the sequence of Catalan numbers.

In this paper, we consider a new family of multivariate copulas described by a sequence of functions, named as AMO copula. The set of AMO copulas corresponds to a class of multivariate shock models with the Archimedean type of dependence. Sufficient conditions on the involved sequence of functions to obtain a multivariate copula are given and the probabilistic structure of the AMO copula is provided. We show that the family of AMO copulas is a generalization of Archimedean and Marshall-Olkin copulas family, and it includes some well-known copulas as specific cases. An alternative method for generating random vectors from AMO copulas via distortion functions is provided. In addition, a singular component along the main diagonal of the AMO copula is also verified. Finally, the tail behaviors of AMO copulas are discussed and some numerical illustrations are provided.

Given a real number α ∈ ( 0 , 1 ) , we define the Webster sequence of density α to be W α = ( ⌈ ( n − 1 / 2 ) / α ⌉ ) n ∈ N , where ⌈ x ⌉ is the ceiling function. It is known that if α and β are irrational with α + β = 1 , then W α and W β partition N . On the other hand, an analogous result for three-part partitions does not hold: There does not exist a partition of N into sequences W α , W β , W γ with α , β , γ irrational. In this paper, we consider the question of how close one can come to a three-part partition of N into Webster sequences with prescribed densities α , β , γ . We show that if α , β , γ are irrational with α + β + γ = 1 , there exists a partition of N into sequences of densities α , β , γ , in which one of the sequences is a Webster sequence and the other two are “almost” Webster sequences that are obtained from Webster sequences by perturbing some elements by at most 1. We also provide two efficient algorithms to construct such partitions. The first algorithm outputs the n th element of each sequence in O ( 1 ) time and the second algorithm gives the assignment of the m th positive integer to a sequence in O ( 1 ) time. We show that the results are best-possible in several respects. Moreover, we describe applications of these results to apportionment and optimal scheduling problems.

We study entropy as a measure of randomness in a sequential clinical trial. For any randomization design, we consider the well-known sequence of conditional treatment assignment probabilities, P n , which complements the random allocation proportion. We then use P n to formulate and prove a statement about conditions for the asymptotic mean entropy of a randomization design to achieve its optimum value. We compare both response-adaptive and non response adaptive randomization designs with respect to the asymptotic mean entropy, connecting the variance of P n to the asymptotic mean entropy under normality. Then we explore whether both P n , as well as the allocation proportion, may have zero asymptotic variance.

The problem consists in bounding the number of tetrahedra that share the same volume V, circumradius R, and face areas A 1 , A 2 , A 3 , A 4 . A symbolic computation approach is used to prove that the upper bound of the number of tetrahedra is eight. First, the problem is reduced to a counting root problem using the Cayley-Menger determinant formula for the volume of a simplex and some metric equations of the tetrahedron to construct a system of algebraic equations. Then, an investigation of its real roots is done by analyzing the discriminant sequence and extensive numerical computations.

This letter studies the attainable performance in the discrete-time $\boldsymbol {H_{2}}$ optimization under process-independent irregular information transmission between sensor- and actuator-side parts of the controller. In particular, we are concerned with effects of the variability of the sampling rate on the attainable optimal cost. Tight upper and lower bounds on the cost are derived for fixed maximal and average sampling intervals. These bounds are shown to correspond to the maximal and minimal possible variances of sampling intervals. Further insight into the dependence of the performance on properties of the sampling sequence is provided in several special cases.

In this research study, we first define the strong degree of a vertex in an $ m $-polar fuzzy graph. Then we present various useful properties and prove some results concerning this new concept, in the case of complete $ m $-polar fuzzy graphs. Further, we introduce the concept of $ m $-polar fuzzy strength sequence of vertices, and we also investigate it in the particular instance of complete $ m $-polar fuzzy graphs. We discuss connectivity parameters in $ m $-polar fuzzy graphs with precise examples, and we investigate the $ m $-polar fuzzy analogue of Whitney's theorem. Furthermore, we present a clustering method for vertices in an $ m $-polar fuzzy graph based on the strength of connectedness between pairs of vertices. In order to formulate this method, we introduce terminologies such as $ \epsilon_A $-reachable vertices in $ m $-polar fuzzy graphs, $ \epsilon_A $-connected $ m $-polar fuzzy graphs, or $ \epsilon_A $-connected $ m $-polar fuzzy subgraphs (in case the $ m $-polar fuzzy graph itself is not $ \epsilon_A $-connected). Moreover, we discuss an application for clustering different companies in consideration of their multi-polar uncertain information. We then provide an algorithm to clearly understand the clustering methodology that we use in our application. Finally, we present a comparative analysis of our research work with existing techniques to prove its applicability and effectiveness.

We determine the exact value of the Wiener index, the edge-Wiener index, and the vertex-edge-Wiener index of the Basilica graphs, i.e., the sequence of finite Schreier graphs associated with the action of the Basilica group on the rooted binary tree. Moreover, we give a formula for the total distance of every vertex in the Basilica graphs, and we are able to make it explicit for some special vertices. We finally introduce the notions of asymptotic Wiener index and asymptotic total distance, which are compatible with that of convergence of the sequence of finite Basilica graphs to an infinite orbital limit graph in the Gromov–Hausdorff topology: the asymptotic values are explicitly computed.