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In this paper we propose a novel approach to estimating and testing skewness in a stochastic volatility (SV) model. Our key idea is to replace a normal return error in the standard SV model with a split normal error. We show that this simple variation in the model brings about two large computational advantages. First, the SV can be simulated fast and efficiently using a one-block Gibbs sampling technique. Second, more importantly, this is the first to provide a marginal likelihood calculation method to formally test the skewness and SV in a Bayesian framework. We subsequently demonstrate the efficiency and reliability of our posterior sampling and model comparison methods through a simulation study. The simulation study results also show that neglecting skewness leads to inaccurate SV estimates and conditional expected returns. Our empirical applications to daily stock return data also show strong evidence of negative skewness.

In 2002 Mauduit and Sarkozy started to study finite sequences of \begin{document}$ k $\end{document} symbols \begin{document}$ E_{N} = \left(e_{1},e_{2},\cdots,e_{N}\right)\in \mathcal{A}^{N}, $\end{document} where \begin{document}$ \mathcal{A} = \left\{a_{1},a_{2},\cdots,a_{k}\right\}(k\in \mathbb{N},k\geq 2) $\end{document} is a finite set of \begin{document}$ k $\end{document} symbols. Later many pseudorandom sequences of \begin{document}$ k $\end{document} symbols have been given and studied by using number theoretic methods. In this paper we study the pseudorandom properties of the \begin{document}$ k $\end{document} -ary Sidel'nikov sequences with length \begin{document}$ q-1 $\end{document} by using the estimates for certain character sums with exponential function, where \begin{document}$ q $\end{document} is a prime power. Our results show that Sidel'nikov sequences enjoy good well-distribution measure and correlation measure. Furthermore, we prove that the set of size \begin{document}$ \phi(q-1) $\end{document} of \begin{document}$ k $\end{document} -ary Sidel'nikov sequences is collision free and possesses the strict avalanche effect property provided that \begin{document}$ k = o(q^{\frac{1}{4}}) $\end{document} , where \begin{document}$ \phi $\end{document} denotes Euler's totient function.

We use known characterizations of generalized Paley graphs which are cartesian decomposable to explicitly compute the spectra of the corresponding associated irreducible cyclic codes. As applications, we give reduction formulas for the number of rational points in Artin-Schreier curves defined over extension fields and to the computation of Gaussian periods., 20 pages, 2 tables. Added information on Cartesian products of graphs. A general reviwe was made, with small additions improving readability. Some typos corrected. arXiv admin note: text overlap with arXiv:1908.08097

Agriculture is the back bone of India, one of the major sources for some notable population from past decade, slowly the agriculture is facing downward growth. Due to less rain fall, they are not getting proper yield, and also expected price for their crops. The farmers are getting loss or less profit. If they know the price of particular commodity, accordingly they can grow selected crop depending upon their geographical area. We can use some of the machine leaning regression algorithms to predict the price of the agricultural commodities that helps the formers to do the decision making. This research work focus on detailed study of predicting the price of the cotton using five different ML regression algorithms namely Linear Regression, Bayesian Linear Regression, Boosted Decision Tree Regression, Decision forest Regression, Poisson Regression. Metrics for evaluating the performance of ML model. The Data set contains the cotton prices of different states of India. The cotton price 2019 data set is used for this work and the prediction values are compared with 2020 data set and the predicted prices are nearing to the actual values. Compare to all other algorithms Boosted decision tree giving the good accuracy, after that Bayesian and Linear regression also performing well.

This article mainly explores the local input-to-state stability (LISS) property of a class of nonlinear systems via a saturated control strategy, where both the external disturbance and impulsive disturbance being fully considered. In terms of the Lyapunov method and inequality techniques, some sufficient conditions under which the system can be made LISS are proposed, and the elastic constraint relationship among saturated control gain, rate coefficients, external disturbance, and domain of initial value is revealed. Moreover, the optimization design procedures are provided with the hope of obtaining the estimates of admissible external disturbance and domain of initial value as large as possible, where the corresponding saturated control law can be designed by solving LMI-based conditions. In the absence of an external disturbance, the locally exponential stability (LES) property can also be presented with a set of more relaxed conditions. Finally, two examples are presented to reveal the validity of the obtained results.

The degree distributions of complex networks are usually considered to be power law. However, it is not the case for a large number of them. We thus propose a new model able to build random growing networks with (almost) any wanted degree distribution. The degree distribution can either be theoretical or extracted from a real-world network. The main idea is to invert the recurrence equation commonly used to compute the degree distribution in order to find a convenient attachment function for node connections - commonly chosen as linear. We compute this attachment function for some classical distributions, as the power-law, broken power-law, geometric and Poisson distributions. We also use the model on an undirected version of the Twitter network, for which the degree distribution has an unusual shape. We finally show that the divergence of chosen attachment functions is heavily links to the heavy-tailed property of the obtained degree distributions., Comment: 23 pages, 3 figures

Summary Introduction The purpose of the study was to assess acoustic measures of fundamental frequency (fo), standard deviation of fo (SD of fo), jitter%, shimmer%, noise-to-harmonic ratio (NHR), smoothed cepstral peak prominence (CPPS), and acoustic voice quality index analyzed through multiple Praat versions automatically by VoiceEvalU8 or manually by two raters. In addition, default settings to calculate CPPS in two Praat versions manually analyzed by two raters were compared to Maryn and Weenik 20 procedures for CPPS automatically analyzed by VoiceEvalU8. Methods Nineteen vocally healthy females used VoiceEvalU8 to record three 5-s sustained /a/ trials, the all voiced phrase “we were away a year ago,” and a 15-s speech sample twice a day for five consecutive days. Two raters manually completed acoustic analysis using different versions of Praat and compared that analysis to measures automatically generated through a version of Praat used by VoiceEvalU8. One-way analyses of variance were run for all acoustic measures with post-hoc testing by the Bonferroni method. For acoustic measures that demonstrated significant differences, intraclass correlation coefficients were conducted. Results Results showed no significant differences across automatic and manual analysis for different versions of Praat for all acoustic measures during /a/, for fo, jitter%, shimmer%, and NHR during the phrase, for jitter%, shimmer%, NHR, and CPPS during speech, and for acoustic voice quality index calculated from both sustained /a/ and the phrase. The default Praat settings for CPPS were not significantly different from the Maryn and Weenik 20 procedures for sustained /a/ and speech. Significant differences were present for SD of fo and CPPS during the phrase and fo and SD of fo during speech. SD of fo and CPPS in the phrase were moderately correlated and fo and SD of fo during speech demonstrated good to excellent correlations across the different versions of Praat. Conclusions Acoustic measures analyzed through sustained /a/ and some of the acoustic measures during the phrase and speech were not different across multiple versions of Praat. Automatic analysis by VoiceEvalU8 produced similar mean values as compared to manual analysis by two raters. Even though SD of fo and CPPS in the phrase and fo and SD of fo in speech were different across the versions of Praat, the measures demonstrated moderate to excellent reliability.

Some valuable results over rings have a promising utilization in coding theory and error-correcting code theory. In this paper, we study character sums over a certain non-chain ring and their applications in codebooks. There are two major ingredients in this study. The first ingredient is to investigate Gaussian sums, hyper Eisenstein sums, Jacobi sums over a certain non-chain ring and study the properties of these character sums. For their applications, the second ingredient is to present three classes of asymptotically optimal codebooks with respect to the Welch bound and a family of optimal codebooks with respect to the Levenshtein bound, which are constructed from character sums over a certain non-chain ring.

With view to applications to harmonic and stochastic analysis of infinite network/graph models, we introduce new tools for realizations and transforms of positive definite kernels (p.d.) \begin{document}$ K $\end{document} and their associated reproducing kernel Hilbert spaces. With this we establish two kinds of factorizations: (i) Probabilistic: Starting with a positive definite kernel \begin{document}$ K $\end{document} we analyze associated Gaussian processes \begin{document}$ V $\end{document} . Properties of the Gaussian processes will be derived from certain factorizations of \begin{document}$ K $\end{document} , arising as a covariance kernel of \begin{document}$ V $\end{document} . (ii) Geometric analysis: We discuss families of measure spaces arising as boundaries for \begin{document}$ K $\end{document} . Our results entail an analysis of a partial order on families of p.d. kernels, a duality for operators and frames, optimization, Karhunen–Loeve expansions, and factorizations. Applications include a new boundary analysis for the Drury-Arveson kernel, and for certain fractals arising as iterated function systems; and an identification of optimal feature spaces in machine learning models.

We characterize the connection between \begin{document}$ p $\end{document} -ary linear codes and Ramanujan Cayley graphs. We explicitly determine an equivalence between \begin{document}$ t $\end{document} -weight linear codes over the finite field \begin{document}$ \Bbb F_p $\end{document} and Ramanujan Cayley graphs with \begin{document}$ t+1 $\end{document} eigenvalues. In particular, we get an explicit criterion on the equivalence between two-weight linear codes and Ramanujan strongly regular graphs with explicit parameters. Using this characterization, we construct several families of Ramanujan Cayley graphs with two or three eigenvalues from known linear codes with two or three weights, respectively.

A generalized Stirling numbers of the second kind \begin{document}$ S_{a,b}\left(p,k\right) $\end{document} , involved in the expansion \begin{document}$ \left(an+b\right)^{p} = \sum_{k = 0}^{p}k!S_{a,b}\left(p,k\right) \binom{n}{k} $\end{document} , where \begin{document}$ a \neq 0, b $\end{document} are complex numbers, have studied in [ 16 ]. In this paper, we show that Bernoulli polynomials \begin{document}$ B_{p}(x) $\end{document} can be written in terms of the numbers \begin{document}$ S_{1,x}\left(p,k\right) $\end{document} , and then use the known results for \begin{document}$ S_{1,x}\left(p,k\right) $\end{document} to obtain several new explicit formulas, recurrences and generalized recurrences for Bernoulli numbers and polynomials.

In this work, we propose a new inertial method for solving strongly monotone variational inequality problems over the solution set of a split variational inequality and composed fixed point problem in real Hilbert spaces. Our method uses stepsizes that are generated at each iteration by some simple computations, which allows it to be easily implemented without the prior knowledge of the operator norm as well as the Lipschitz constant of the operator. In addition, we prove that the proposed method converges strongly to a minimum-norm solution of the problem without using the conventional two cases approach. Furthermore, we present some numerical experiments to show the efficiency and applicability of our method in comparison with other methods in the literature. The results obtained in this paper extend, generalize and improve results in this direction.

This paper investigates a class of unified stochastic linear-quadratic-Gaussian (LQG) social optima problems involving a large number of weakly-coupled interactive agents under a generalized setting. For each individual agent, the control and state process enters both diffusion and drift terms in its linear dynamics, and the control weight might be indefinite in cost functional. This setup is innovative and has great theoretical and realistic significance as its applications in mathematical finance (e.g., portfolio selection in mean-variation model). Using some fully-coupled variational analysis under the person-by-person optimality principle, and the mean-field approximation method, the decentralized social control is derived by a class of new type consistency condition (CC) system for typical representative agent. Such CC system is some mean-field forward-backward stochastic differential equation (MF-FBSDE) combined with embedding representation. The well-posedness of such forward-backward stochastic differential equation (FBSDE) system is carefully examined. The related social asymptotic optimality is related to the convergence of the average of a series of weakly-coupled backward stochastic differential equation (BSDE). They are verified through some Lyapunov equations.

The number of proper vertex-3-colorings of every triangle-free planar graph with n vertices and with no separating cycle of length 4 or 5 is at least 2 n / 17700000 . On the other hand, for infinitely many n, there exists a triangle-free planar graph with separating cycles of length 4 and 5 whose number of proper vertex-3-colorings is 2 15 n / log 2 ⁡ ( n ) .

We prove the following theorem. Given a planar graph $G$ and an integer $k$, it is possible in polynomial time to randomly sample a subset $A$ of vertices of $G$ with the following properties: (i) $A$ induces a subgraph of $G$ of treewidth $\mathcal{O}(\sqrt{k}\log k)$, and (ii) for every connected subgraph $H$ of $G$ on at most $k$ vertices, the probability that $A$ covers the whole vertex set of $H$ is at least $(2^{\mathcal{O}(\sqrt{k}\log^2 k)}\cdot n^{\mathcal{O}(1)})^{-1}$, where $n$ is the number of vertices of $G$. Together with standard dynamic programming techniques for graphs of bounded treewidth, this result gives a versatile technique for obtaining (randomized) subexponential parameterized algorithms for problems on planar graphs, usually with running time bound $2^{\mathcal{O}(\sqrt{k} \log^2 k)} n^{\mathcal{O}(1)}$. The technique can be applied to problems expressible as searching for a small, connected pattern with a prescribed property in a large host graph, examples of such problems include Directed $k$-Path, Weighted $k$-Path, Vertex Cover Local Search, and Subgraph Isomorphism, among others. Up to this point, it was open whether these problems can be solved in subexponential parameterized time on planar graphs, because they are not amenable to the classic technique of bidimensionality. Furthermore, all our results hold in fact on any class of graphs that exclude a fixed apex graph as a minor, in particular on graphs embeddable in any fixed surface.

The concept of graph tessellation cover was defined in the context of quantum walk models, and is a current research area in graph theory. In this work, we propose a generalization called total tessellation cover. A tessellation of a graph G is a partition of its vertex set V ( G ) into vertex disjoint cliques. A tessellation cover of G is a set of tessellations that covers its edge set E ( G ) . A total tessellation cover of G consists of a tessellation cover together with a compatible vertex coloring, such that the color of each vertex is different from the tessellation labels of the edges incident to the vertex. The total tessellation cover number T t ( G ) is the size of a minimum total tessellation cover of G . We present lower bounds T t ( G ) ≥ ω ( G ) and T t ( G ) ≥ s ( G ) + 1 , where ω ( G ) is the size of a maximum clique, and s ( G ) is the number of edges of a maximum induced star subgraph. A graph G is called a good total tessellable if T t ( G ) = ω ( G ) or T t ( G ) = s ( G ) + 1 . We study the complexity of the k - total tessellability problem, which aims to decide whether a given graph G has T t ( G ) ≤ k . We prove that k - total tessellability is in P for good total tessellable graphs. We establish the NP -completeness of the k - total tessellability when restricted to the following graph classes: bipartite graphs, line graphs of triangle-free graphs, universal graphs, ( 2 , 1 ) -chordal graphs and planar graphs.

For solving dynamic generalized Lyapunov equation, two robust finite-time zeroing neural network (RFTZNN) models with stationary and nonstationary parameters are generated through the usage of an improved sign-bi-power (SBP) activation function (AF). Taking differential errors and model implementation errors into account, two corresponding perturbed RFTZNN models are derived to facilitate the analyses of robustness on the two RFTZNN models. Theoretical analysis gives the quantitatively estimated upper bounds for the convergence time (UBs-CT) of the two derived models, implying a superiority of the convergence that varying parameter RFTZNN (VP-RFTZNN) possesses over the fixed parameter RFTZNN (FP-RFTZNN). When the coefficient matrices and perturbation matrices are uniformly bounded, residual error of FP-RFTZNN is bounded, whereas that of VP-RFTZNN monotonically decreases at a super-exponential rate after a finite time, and eventually converges to 0. When these matrices are bounded but not uniform, residual error of FP-RFTZNN is no longer bounded, but that of VP-RFTZNN still converges. These superiorities of VP-RFTZNN are illustrated by abundant comparative experiments, and its application value is further proved by an application to robot.

What does it mean today to study a problem from a computational point of view? We focus on parameterized complexity and on Column 16 “Graph Restrictions and Their Effect” of D.S. Johnson’s Ongoing guide, where several puzzles were proposed in a summary table with 30 graph classes as rows and 11 problems as columns. Several of the 330 entries remain unclassified into Polynomial or NP -complete after 35 years. We provide a full dichotomy for the Steiner Tree column by proving that the problem is NP -complete when restricted to Undirected Path graphs. We revise Johnson’s summary table according to the granularity provided by the parameterized complexity for NP -complete problems.

Rainbow coloring problems, of noteworthy applications in Information Security, have been receiving much attention in the last years in Combinatorics. In particular, the rainbow connection number of a connected graph G , denoted r c ( G ) , is the least k for which G admits a (not necessarily proper) k -edge-coloring such that between any pair of vertices there is a path whose edge colors are all distinct. When G is disconnected, we define r c ( G ) = ∞ . A graph G is rainbow critical if the deletion of any edge of G increases r c ( G ) . In this paper we determine the rainbow connection number for the shadow graphs of paths, some circulant graphs, and some Cartesian products involving cycles and paths. We also determine conditions for the rainbow criticality and non-criticality of wheels, fans, and some Cartesian products involving cycles, paths, pans, and complete graphs.

Mahogany (Swietenia macrophylla King) is one of the most valuable wood species found in the American Continent. This study is an evaluation of the production of particleboards using wood residues from the branches and trunk of a mahogany tree. It was designed to test the feasibility of production and the mechanical and physical characteristics of the finished particleboards, as well as to compare the results with similar studies and the requirements of normative documents. The particles were generated from the trunk and branches of a mahogany tree separately and mixed in different proportions to test their influence on the final product. They were bonded using 12% polyurethane resin from renewable resources and pressed under 90º C and 3.5 MPa for 10 minutes. The specimens were prepared and tested according to the guidelines established by the NBR 14810 regarding the static bending, internal bond, stiffness and thickness swelling (24h) tests. The results indicated that there is no significant correlation between the trunk-branches proportion and the Modulus of Elasticity/Rupture values. The production of the particleboards using the residues is feasible and the results of most of the tests fitted the normative requirements, which means that with the refinement of the production process it is possible to have a good product that fully complies with the technical standards. Resumo A madeira de mogno (Swietenia macrophylla King) é uma das mais valiosas encontradas no Continente Americano. Esse trabalho visou avaliar a produção de painéis de partículas de resíduos dos galhos e troncos dessa espécie. Foram avaliados a viabilidade de produção, as propriedades físicas e mecânicas e, além disso, foi realizada a comparação dos resultados com estudos similares e requisitos de normas. Os painéis foram fabricados a partir dos resíduos do tronco e dos galhos, de forma separada e misturados em diferentes proporções para testar sua influência no produto final. Foram aglomerados usando resina de poliuretano e prensados a 90°C e 3,5 MPa por 10 minutos. As amostras foram testadas para flexão estática, aderência interna, rigidez e inchamento (24 h), de acordo com as recomendações da NBR 14810. Os resultados indicaram que não houve correlação significativa entre a proporção tronco-galhos e os valores de módulo de elasticidade/ruptura. A produção dos painéis foi considerada viável e os resultados da maioria dos testes atenderam aos requisitos normativos.

Dunkl operators may be regarded as differential-difference operators parameterized by finite reflection groups and multiplicity functions. In this paper, the Littlewood--Paley square function for Dunkl heat flows in $\mathbb{R}^d$ is introduced by employing the full "gradient" induced by the corresponding carr\'{e} du champ operator and then the $L^p$ boundedness is studied for all $p\in(1,\infty)$. For $p\in(1,2]$, we successfully adapt Stein's heat flows approach to overcome the difficulty caused by the difference part of the Dunkl operator and establish the $L^p$ boundedness, while for $p\in[2,\infty)$, we restrict to a particular case when the corresponding Weyl group is isomorphic to $\mathbb{Z}_2^d$ and apply a probabilistic method to prove the $L^p$ boundedness. In the latter case, the curvature-dimension inequality for Dunkl operators in the sense of Bakry--Emery, which may be of independent interest, plays a crucial role. The results are dimension-free., Comment: Comments welcome!

In this article, we mainly consider the existence of solutions and global Mittag-Leffler stability of delayed fractional-order coupled reaction-diffusion neural networks without strong connectedness. Using the Leary-Schauder's fixed point theorem and the Lyapunov method, some criteria for the existence of solutions and global Mittag-Leffler stability are given. Finally, the correctness of the theory is verified by a numerical example.

A multivariable finite-time control algorithm is proposed for double integrator systems with matched disturbances. The remarkable feature of the developed method is to ensure the finite-time stability of the closed-loop system with uniformly bounded control signal with respect to arbitrary initial conditions. The convergence criteria is derived by using the Lyapunov-based techniques, and the upper bound of the control signal can be calculated based on the obtained criteria. Finally, the effectiveness of the proposed algorithm is confirmed by some numerical simulations.

In this paper, for the mapping of a sphere into a compact orientable manifold S n → M , n ⩾ 1 , we solve the problem of determining whether it represents a nontrivial element in the homotopy group of the manifold π n ( M ) πn(M ). For this purpose, we consistently use the theory of iterated integrals developed by K.-T. Chen. It should be noted that the iterated integrals as repeated integration were previously meaningfully used by Lappo-Danilevsky to represent solutions of systems of linear differential equations and by Whitehead for the analytical description of the Hopf invariant for mappings f : S 2 n - 1 → S n , n ⩾ 2 . We give a brief description of Chen’s theory, representing Whitehead’s and Haefliger’s formulas for the Hopf invariant and generalized Hopf invariant. Examples of calculating these invariants using the technique of iterated integrals are given. Further, it is shown how one can detect any element of the fundamental group of a Riemann surface using iterated integrals of holomorphic forms. This required to prove that the intersection of the terms of the lower central series of the fundamental group of a Riemann surface is a unit group.

We prove the first polynomial bound on the number of monotonic homotopy moves required to tighten a collection of closed curves on any compact orientable surface, where the number of crossings in the curve is not allowed to increase at any time during the process. The best known upper bound before was exponential, which can be obtained by combining the algorithm of De Graaf and Schrijver [ J. Comb. Theory Ser. B , 1997] together with an exponential upper bound on the number of possible surface maps. To obtain the new upper bound, we apply tools from hyperbolic geometry, as well as operations in graph drawing algorithms—the cluster and pipe expansions—to the study of curves on surfaces. As corollaries, we present two efficient algorithms for curves and graphs on surfaces. First, we provide a polynomial-time algorithm to convert any given multicurve on a surface into minimal position. Such an algorithm only existed for single closed curves, and it is known that previous techniques do not generalize to the multicurve case. Second, we provide a polynomial-time algorithm to reduce any k -terminal plane graph (and more generally, surface graph) using degree-1 reductions, series-parallel reductions, and Δ Y -transformations for arbitrary integer k . Previous algorithms only existed in the planar setting when k ≤ 4, and all of them rely on extensive case-by-case analysis based on different values of k . Our algorithm makes use of the connection between electrical transformations and homotopy moves and thus solves the problem in a unified fashion.

We introduce and study a new graph representation where vertices are embedded in three or more dimensions, and in which the edges are drawn on the projections onto the axis-parallel planes. We show that the complete graph on $n$ vertices has a representation in $\lceil \sqrt{n/2}+1 \rceil$ planes. In 3 dimensions, we show that there exist graphs with $6n-15$ edges that can be projected onto two orthogonal planes, and that this is best possible. Finally, we obtain bounds in terms of parameters such as geometric thickness and linear arboricity. Using such a bound, we show that every graph of maximum degree 5 has a plane-projectable representation in 3 dimensions., Accepted at CALDAM 2020

The eternal vertex cover problem is a variant of the classical vertex cover problem defined in terms of an infinite attacker–defender game played on a graph. In each round of the game, the defender reconfigures guards from one vertex cover to another in response to a move by the attacker. The minimum number of guards required in any winning strategy of the defender when this game is played on a graph G is the eternal vertex cover number of G , denoted by evc ( G ) . It is known that given a graph G and an integer k , checking whether evc ( G ) ≤ k is NP-hard. Further, it is known that for any graph G , mvc ( G ) ≤ evc ( G ) ≤ 2 mvc ( G ) , where mvc ( G ) is the vertex cover number of G . Though a characterization is known for graphs for which evc ( G ) = 2 mvc ( G ) , a characterization of graphs for which evc ( G ) = mvc ( G ) remained as an open problem, since 2009. We achieve such a characterization for a class of graphs that includes chordal graphs and internally triangulated planar graphs. For biconnected chordal graphs, our characterization leads to a polynomial time algorithm for precisely determining evc ( G ) and an algorithm for determining a safe strategy for guard movement in each round of the game using only evc ( G ) guards. Though the eternal vertex cover problem is only known to be in PSPACE in general, it follows from our new characterization that the problem is in NP for locally connected graphs, a graph class which includes all biconnected internally triangulated planar graphs. We also provide reductions establishing NP-completeness of the problem for biconnected internally triangulated planar graphs. As far as we know, this is the first NP-completeness result known for the problem for any graph class.

In this paper, we study the Cartesian product of signed graphs as defined by Germina, Hameed and Zaslavsky (2011). Here we focus on its algebraic properties and look at the chromatic number of some Cartesian products. One of our main results is the unicity of the prime factor decomposition of signed graphs. This leads us to present an algorithm to compute this decomposition in linear time based on a decomposition algorithm for oriented graphs by Imrich and Peterin (2018). We also study the chromatic number of a signed graph, that is the minimum order of a signed graph to which the input signed graph admits a homomorphism, of graphs with underlying graph of the form P n □ P m , of Cartesian products of signed paths, of Cartesian products of signed complete graphs and of Cartesian products of signed cycles.

Several combinatorial actions of the affine Weyl group of type C ˜ n on triangulations, trees, words and permutations are compared. Addressing a question of David Vogan, we show that, modulo a natural involution, these permutation representations are multiplicity-free. The proof uses a general construction of Gelfand subgroups in the affine Weyl groups of types C ˜ n and B ˜ n .

In the paper, the exponential stability of impulsive control systems with time delay is studied. By using the average impulsive interval method, some sufficient Lyapunov-based conditions are established for the stability of impulsive time-delay systems, and the impacts of delay on the stability analysis method are further revealed. It is interesting to show that some unstable impulsive time-delay systems may be stabilized by increasing the time delay in continuous dynamics. More interestingly, it is proved that along with the increase of the delay within a certain range, the convergence rate of such impulsive timedelay systems also increases correspondingly. Further, a strict comparison principle for impulsive control systems with delay is established. Then by utilizing this comparison principle, it can be shown that for some stable impulsive systems with delay, under certain conditions, the stability is robust against any large but bounded delay. Compared with the previous results on delay-free impulsive systems, some potential impacts of delay on the stability are investigated. Particularly, the obtained results are extended to the case of impulsive control systems with hybrid im

Restricted star colouring is a variant of star colouring introduced to design heuristic algorithms to estimate sparse Hessian matrices. For k ∈ N , a k -restricted star colouring ( k -rs colouring) of a graph G is a function f : V ( G ) → { 0 , 1 , … , k − 1 } such that (i) f ( x ) ≠ f ( y ) for every edge x y of G , and (ii) there is no bicoloured 3-vertex path ( P 3 ) in G with the higher colour on its middle vertex. We show that for k ≥ 3 , it is NP-complete to test whether a given planar bipartite graph of maximum degree k and arbitrarily large girth admits a k -rs colouring, and thereby answer a problem posed by Shalu and Sandhya (2016). In addition, it is NP-complete to test whether a 3-star colourable graph admits a 3-rs colouring. We also prove that for all ϵ > 0 , the optimization problem of restricted star colouring a 2-degenerate bipartite graph with the minimum number of colours is NP-hard to approximate within n 1 3 − ϵ . On the positive side, we design (i) a linear-time algorithm to test 3-rs colourability of trees, and (ii) an O ( n 3 ) -time algorithm to test 3-rs colourability of chordal graphs.

Given a graph G , we call a partition ( V 1 , V 2 , … , V k ) of its vertex set an induced star partition of G if each set in the partition induces a star. In this paper, we consider the problem of finding an induced star partition of a graph of minimum size and its decision versions. This problem may be viewed as an amalgamation of the well-known dominating set and coloring problems. We obtain the following main results: (1) Deciding whether a graph can be partitioned into k induced stars is NP-complete for each fixed k ≥ 3 and has a polynomial time algorithm for each k ≤ 2 . (2) It is NP-hard to approximate the minimum induced star partition size within n 1 2 − ϵ for all ϵ > 0 . (3) The decision version of the induced star partition problem is NP-complete for subcubic bipartite planar graphs, line graphs (a subclass of K 1 , r -free graphs, r ≥ 3 ), K 1 , 5 -free split graphs and co-tripartite graphs. (4) The minimum induced star partition problem has an r 2 -approximation algorithm for K 1 , r -free graphs ( r ≥ 2 ) and a 2-approximation algorithms for split graphs. We also identify some fixed parameter tractable and exact exponential time algorithms that follow from the literature.

The problem of computing induced subgraphs that satisfy some specified restrictions arises in various applications of graph algorithms and has been well studied. In this paper, we consider the following Balanced Connected Subgraph (shortly, BCS) problem. The input is a graph $G=(V,E)$, with each vertex in the set $V$ having an assigned color, "red" or "blue". We seek a maximum-cardinality subset $V'\subseteq V$ of vertices that is color-balanced (having exactly $|V'|/2$ red nodes and $|V'|/2$ blue nodes), such that the subgraph induced by the vertex set $V'$ in $G$ is connected. We show that the BCS problem is NP-hard, even for bipartite graphs $G$ (with red/blue color assignment not necessarily being a proper 2-coloring). Further, we consider this problem for various classes of the input graph $G$, including, e.g., planar graphs, chordal graphs, trees, split graphs, bipartite graphs with a proper red/blue $2$-coloring, and graphs with diameter $2$. For each of these classes either we prove NP-hardness or design a polynomial time algorithm., 15 pages, 3 figures

In this paper, the finite-time and fixed-time attractiveness problems are investigated for nonlinear impulsive systems. The aim of the proposed problems is to establish some general Lyapunov theorems and settling-time estimates for finite-time and fixed-time attractiveness of nonlinear impulsive systems by analysis technics. Moreover, some comparisons with the existing results are also given. The classical finite-time and fixed-time convergence theorems for nonlinear impulse-free systems are well extended to nonlinear impulsive systems by our results. Furthermore, the existing results of finite-time and fixed-time attractiveness for nonlinear impulsive systems are well improved. Finally, some simulations are utilized to illustrate the usefulness of the theoretical analysis.

In this paper, we consider the following Schrödinger equation (0.1) − Δ u − μ u | x | 2 + V ( x ) u = K ( x ) | u | 2 ∗ − 2 u + f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , where N ⩾ 4, 0 ⩽ μ < μ ‾, μ ‾ = ( N − 2 ) 2 4 , V is periodic in x, K and f are asymptotically periodic in x, we take advantage of the generalized Nehari manifold approach developed by Szulkin and Weth to look for the ground state solution of (0.1).

We establish short-time asymptotics with rates of convergence for the Laplace Dirichlet heat kernel in a ball. So far, such results were only known in simple cases where explicit formulae are available, i.e., for sets as half-line, interval and their products. Presented asymptotics may be considered as a complement or a generalization of the famous “principle of not feeling the boundary” in case of a ball. Following the metaphor, the principle reveals when the process does not feel the boundary, while we describe what happens when it starts feeling the boundary.

We develop a theory of modular forms on the groups SO ( 3 , n + 1 ) , n ≥ 3 . This is very similar to, but simpler, than the notion of modular forms on quaternionic exceptional groups, which was initiated by Gross-Wallach and Gan-Gross-Savin. We prove the results analogous to those of earlier papers of the author on modular forms on exceptional groups, except now in the familiar setting of classical groups. Moreover, in the setting of SO ( 3 , n + 1 ) , there is a family of absolutely convergent Eisenstein series, which are modular forms. We prove that these Eisenstein series have algebraic Fourier coefficients, like the classical holomorphic Eisenstein series on SO ( 2 , n ) . As an application, using a local result of Savin, we prove that the so-called “next-to-minimal” modular form on quaternionic E 8 has rational Fourier expansion.

Nonconvex Stochastic Optimization Nonconvex stochastic optimization problems arise in many machine learning problems, including deep learning. The stochastic gradient Hamiltonian Monte Carlo (SGHMC) is a variant of stochastic gradients with a momentum method in which a controlled and properly scaled Gaussian noise is added to the stochastic gradients to steer the iterates toward a global minimum. SGHMC has shown empirical success in practice for solving nonconvex stochastic optimization problems. In “Global convergence of stochastic gradient Hamiltonian Monte Carlo for nonconvex stochastic optimization: Nonasymptotic performance bounds and momentum-based acceleration,” Gao, Gürbüzbalaban, and Zhu provide, for the first time, the finite-time performance bounds for the global convergence of SGHMC in the context of both population and empirical risk minimization problems and show that acceleration with momentum is possible in the context of global nonconvex stochastic optimization.

Let f be an algebraic valued periodic arithmetical function and L ( s , f ) , defined as L ( s , f ) : = ∑ n = 1 ∞ f ( n ) / n s for ℜ ( s ) > 1 , be the associated Dirichlet series. In this paper, we study the vanishing and arithmetic nature of the special values L ( k , f ) when k > 1 is a positive integer. We prove a generalization of the Baker-Birch-Wirsing theorem conditional on the Polylog conjecture. Adopting a new approach, we define an induction operator on the space of periodic arithmetic functions, which makes precise the notion of an “imprimitive” arithmetic function. This enables us to obtain an analog of Okada's criterion for L ( 1 , f ) = 0 and derive a natural decomposition of the vector space O k ( N ) = { f : Z → Q | f ( n + N ) = f ( n ) for all n ∈ Z , L ( k , f ) = 0 } .

A new Wald-type statistic is proposed for hypothesis testing based on Bayesian posterior distributions under the correct model specification. The new statistic can be explained as a posterior version of the Wald statistic and has several nice properties. First, it is well-defined under improper prior distributions. Second, it avoids Jeffreys–Lindley–Bartlett’s paradox. Third, under the null hypothesis and repeated sampling, it follows a χ 2 distribution asymptotically, offering an asymptotically pivotal test. Fourth, it only requires inverting the posterior covariance for parameters of interest. Fifth and perhaps most importantly, when a random sample from the posterior distribution (such as MCMC output) is available, the proposed statistic can be easily obtained as a by-product of posterior simulation. In addition, the numerical standard error of the estimated proposed statistic can be computed based on random samples. A robust version of the test statistic is developed under model misspecification and inherits many nice properties of the new posterior statistic. The finite sample performance of the statistics is examined in Monte Carlo studies. The method is applied to two latent variable models used in microeconometrics and financial econometrics.

We show that, on the Riemann hypothesis, $\limsup_{X\to\infty}I(X)/X^{2} \leq 0.8603$, where $I(X) = \int_X^{2X} (\psi(x)-x)^2\,dx.$ This proves (and improves on) a claim by Pintz from 1982. We also show unconditionally that $\frac{1}{5\,374}\leq I(X)/X^2 $ for sufficiently large $X$, and that the $I(X)/X^{2}$ has no limit as $X\rightarrow\infty$., Comment: 23 pages

The main purpose of this paper is to investigate the existence of a general evaluatory formula for integral parts of reciprocals of tails of the series defining the Riemann ζ-function for a fixed value of s: a non-unitary positive integer. To that end, we introduce two families of polynomials - one given by a convolution recurrence involving Bernoulli numbers, and the other defined in terms of the previous family. Through an asymptotic argument, we provide the desired formula in terms of said polynomials. Subsequently, using the methodology established, a direct derivation of some particular formulae (corresponding to the cases s = 7 and s = 9 ) is made. We then proceed to study the properties of the polynomials mentioned, in particular obtaining a rather surprising symmetry property about the ‘critical line’.

We generalize Birch's construction of the Heegner points on X 0 ( N ) to construct new points X 1 ( N ) (and therefore construct points on the abelian varieties associated to J 1 ( N ) ). Then, we show that these points form an Euler system, and we improve Kolyvagin's Euler system techniques to show that for our point P τ K / c and any ring class character χ of the extended ring class field of conductor c satisfying χ = χ ‾ , if P τ K / c χ is non-torsion and G K → Aut A f [ π ] is surjective, then the corank of Sel ( A χ / K ) is 1, which implies the rank of A f ( K ) χ is 1.

In this second part of our work, we study the steady state of the population and the social utility for the three dynamics SSD, NBRD and NRPM; which were introduced in the first part. We provide sufficient conditions on the network based on a maximum payoff density parameter of each node under which there exists a unique Nash equilibrium. We then utilize positive correlation properties of the dynamics to reduce the flow graph in order to provide an upper bound on the steady state social utility. Finally we extend the idea behind the sufficient condition for the existence of a unique Nash equilibrium to partition the graph appropriately in order to provide a lower bound on the steady state social utility. We also illustrate interesting cases as well as our results using simulations.

The absolute zeta function for a scheme X of finite type over Z satisfying a certain condition is defined as the limit as p → 1 of the zeta function of X ⊗ F p . In 2016, after calculating absolute zeta functions for a few specific schemes, Kurokawa suggested that an absolute zeta function for a general scheme of finite type over Z should have an infinite product structure which he called the absolute Euler product. In this article, formulating his suggestion using a torsion free Noetherian F 1 -scheme defined by Connes and Consani, we give a proof of his suggestion. Moreover, we show that each factor of the absolute Euler product is derived from the counting function of the F 1 -scheme.

Most left ventricular (LV) Doppler measurements vary significantly with age and gender, making it necessary to correct them for physiological variances. We aimed to verify the hypothesis that different Doppler measurements correlate nonlinearly with different biometric variables raised to different scaling factors and exponents. In this work, a total of 23 LV Doppler parameters were measured in 1224 healthy Chinese adults. An optimized multivariable allometric model (OMAM) and scaling equations were developed in 70% of the subjects (group A), and the reliability of the model and equations was verified using the remaining 30% of the subjects (group B) as well as 183 overweight subjects (group C). The single-variable isometric model (SVIM) with body surface area (BSA) as a scaling variable was used for comparison. Before correction, all 23 LV Doppler parameters correlated significantly with one or more of the biometric variables. In group B, gender differences were found in 47.8% (11/23) of the parameters and were eliminated in 81.8% (9/11) of the parameters after correction with OMAM. The successful correction rate with OMAM was 100% (23/23) in group B and 82.6% (19/23) in group C. New reference values for corrected Doppler measurements independent of biometric variables were established. The SVIM with BSA successfully corrected none of the 23 parameters. In conclusion, different LV Doppler parameters allometrically correlated with one or more of the biometric variables. The novel OMAM developed in this study successfully corrected the effects of the physiological variances of most biometric variables on Doppler measurements in healthy and overweight subjects, and was found to be far superior to the SVIM. However, whether the OMAM equations can be applied to other ethnicities, obese subjects, and pathological conditions requires further investigation.

We show that a A-linear map of Hilbert A-modules is induced by a unitary Hilbert module operator if and only if it extends to an ordinary unitary on appropriately defined enveloping Hilbert spaces. Applications to the theory of multiplicative unitaries let us to compute the equivalence classes of Hilbert modules over a class of C*-algebraic quantum groups. We, thus, develop a theory that, for example, could be used to show non-existence of certain co-actions. In particular, we show that the Cuntz semigroup functor takes a co-action to a multiplicative action.

Given a set $P$ of $n$ points in $\mathbb{R}^2$ and an input line $\gamma$ in $\mathbb{R}^2$, we present an algorithm that runs in optimal $\Theta(n\log n)$ time and $\Theta(n)$ space to solve a restricted version of the $1$-Steiner tree problem. Our algorithm returns a minimum-weight tree interconnecting $P$ using at most one Steiner point $s \in \gamma$, where edges are weighted by the Euclidean distance between their endpoints. We then extend the result to $j$ input lines. Following this, we show how the algorithm of Brazil et al. ("Generalised k-Steiner Tree Problems in Normed Planes", arXiv:1111.1464) that solves the $k$-Steiner tree problem in $\mathbb{R}^2$ in $O(n^{2k})$ time can be adapted to our setting. For $k>1$, restricting the (at most) $k$ Steiner points to lie on an input line, the runtime becomes $O(n^{k})$. Next we show how the results of Brazil et al. ("Generalised k-Steiner Tree Problems in Normed Planes", arXiv:1111.1464) allow us to maintain the same time and space bounds while extending to some non-Euclidean norms and different tree cost functions. Lastly, we extend the result to $j$ input curves., Comment: 31 pages (26 of content), 4 figures (last one as 3 subfigures). Minor corrections since publication: notably our analysis of the Brazil et al. k-steiner tree algorithm in section 4.1 (results unchanged), and a correction to some corollary statements to correct the space usage. Also added short discussion of how the algorithm still works when the MST of the input point set is not unique