Nonuniform biorthogonal wavelets on positive half line via Walsh Fourier transform

In this article, we introduce the notion of nonuniform biorthogonal wavelets on positive half line. We first establish the characterizations for the translates of a single function to form the Riesz bases for their closed linear span. We provide the complete characterization for the biorthogonality of the translates of scaling functions of two nonuniform multiresolution analysis and the associated biorthogonal wavelet families in $$L^2({\mathbb {R}}^+)$$ L 2 ( R + ) . Furthermore, under the mild assumptions on the scaling functions and the corresponding wavelets associated with nonuniform multiresolution analysis, we show that the wavelets can generate Reisz bases.

Do Investment-Based Models Explain Equity Returns? Evidence from Euler Equations

We investigate the empirical implications of the investment-based model of asset pricing for the Hansen-Jagannathan and Kozak-Nagel-Santosh discount factors in the linear span of equity returns. We find that the stochastic discount factors satisfying the Euler equation for equity returns cannot satisfy the Euler equation for investment returns because returns on corporate investment covary inversely with the sources of equity risk relative to returns on equity. As a result, the model fails to replicate the level of the risk premium. Our results suggest that joint restrictions on the optimality of investment and consumption pose stringent conditions for candidate production models.

Influence of Estuarine Water on the Microbial Community Structure of Patagonian Fjords

Fjords are sensitive areas affected by climate change and can act as a natural laboratory to study microbial ecological processes. The Chilean Patagonian fjords (41–56°S), belonging to the Subantarctic ecosystem (46–60°S), make up one of the world’s largest fjord systems. In this region, Estuarine Water (EW) strongly influences oceanographic conditions, generating sharp gradients of oxygen, salinity and nutrients, the effects of which on the microbial community structure are poorly understood. During the spring of 2017 we studied the ecological patterns (dispersal and oceanographic factors) underlying the microbial community distribution in a linear span of 450 km along the estuarine-influenced Chilean Patagonian fjords. Our results show that widespread microbial dispersion existed along the fjords where bacterioplankton exhibited dependence on the eukaryotic phytoplankton community composition. This dependence was particularly observed under the low chlorophyll-a conditions of the Baker Channel area, in which a significant relationship was revealed between SAR11 Clade III and the eukaryotic families Pyrenomonadaceae (Cryptophyte) and Coccomyxaceae (Chlorophyta). Furthermore, dissolved oxygen and salinity were revealed as the main drivers influencing the surface marine microbial communities in these fjords. A strong salinity gradient resulted in the segregation of the Baker Channel prokaryotic communities from the rest of the Patagonian fjords. Likewise, Microbacteriaceae, Burkholderiaceae and SAR11 Clade III, commonly found in freshwater, were strongly associated with EW conditions in these fjords. The direct effect of EW on the microbial community structure and diversity of the fjords exemplifies the significance that climate change and, in particular, deglaciation have on this marine region and its productivity.

When is there a representer theorem?

We consider a general regularised interpolation problem for learning a parameter vector from data. The well-known representer theorem says that under certain conditions on the regulariser there exists a solution in the linear span of the data points. This is at the core of kernel methods in machine learning as it makes the problem computationally tractable. Most literature deals only with sufficient conditions for representer theorems in Hilbert spaces and shows that the regulariser being norm-based is sufficient for the existence of a representer theorem. We prove necessary and sufficient conditions for the existence of representer theorems in reflexive Banach spaces and show that any regulariser has to be essentially norm-based for a representer theorem to exist. Moreover, we illustrate why in a sense reflexivity is the minimal requirement on the function space. We further show that if the learning relies on the linear representer theorem, then the solution is independent of the regulariser and in fact determined by the function space alone. This in particular shows the value of generalising Hilbert space learning theory to Banach spaces.

On the Lattice Properties of Almost L-Weakly and Almost M-Weakly Compact Operators

We establish the domination property and some lattice approximation properties for almost L-weakly and almost M-weakly compact operators. Then, we consider the linear span of positive almost L-weakly (resp., almost M-weakly) compact operators and give results about when they form a Banach lattice and have an order continuous norm.

Independent, Incidence Independent and Weakly Reversible Decompositions of Chemical Reaction Networks

Chemical reaction networks (CRNs) are directed graphs with reactant or product complexes as vertices, and reactions as arcs. A CRN is weakly reversible if each of its connected components is strongly connected. Weakly reversible networks can be considered as the most important class of reaction networks. Now, the stoichiometric subspace of a network is the linear span of the reaction vectors (i.e., difference between the product and the reactant complexes). A decomposition of a CRN is independent (incidence independent) if the direct sum of the stoichiometric subspaces (incidence maps) of the subnetworks equals the stoichiometric subspace (incidence map) of the whole network. Decompositions can be used to study relationships between positive steady states of the whole system (induced from partitioning the reaction set of the underlying network) and those of its subsystems. In this work, we revisit our novel method of finding independent decomposition, and use it to expand applicability on (vector) components of steady states. We also explore CRNs with embedded deficiency zero independent subnetworks. In addition, we establish a method for finding incidence independent decomposition of a CRN. We determine all the forms of independent and incidence independent decompositions of a network, and provide the number of such decompositions. Lastly, for weakly reversible networks, we determine that incidence independence is a sufficient condition for weak reversibility of a decomposition, and we identify subclasses of weakly reversible networks where any independent decomposition is weakly reversible.

ANOVA-HD: Analysis of variance when both input and output layers are high-dimensional

Modern genomic data sets often involve multiple data-layers (e.g., DNA-sequence, gene expression), each of which itself can be high-dimensional. The biological processes underlying these data-layers can lead to intricate multivariate association patterns. We propose and evaluate two methods to determine the proportion of variance of an output data set that can be explained by an input data set when both data panels are high dimensional. Our approach uses random-effects models to estimate the proportion of variance of vectors in the linear span of the output set that can be explained by regression on the input set. We consider a method based on an orthogonal basis (Eigen-ANOVA) and one that uses random vectors (Monte Carlo ANOVA, MC-ANOVA) in the linear span of the output set. Using simulations, we show that the MC-ANOVA method gave nearly unbiased estimates. Estimates produced by Eigen-ANOVA were also nearly unbiased, except when the shared variance was very high (e.g., >0.9). We demonstrate the potential insight that can be obtained from the use of MC-ANOVA and Eigen-ANOVA by applying these two methods to the study of multi-locus linkage disequilibrium in chicken (Gallus gallus) genomes and to the assessment of inter-dependencies between gene expression, methylation, and copy-number-variants in data from breast cancer tumors from humans (Homo sapiens). Our analyses reveal that in chicken breeding populations ~50,000 evenly-spaced SNPs are enough to fully capture the span of whole-genome-sequencing genomes. In the study of multi-omic breast cancer data, we found that the span of copy-number-variants can be fully explained using either methylation or gene expression data and that roughly 74% of the variance in gene expression can be predicted from methylation data.

Reconstruction of a homogeneous polynomial from its additive decompositions when identifiability fails

Let X⊂ℙr be an integral and non-degenerate complex variety. For any q∈ℙr let rX(q) be its X-rank and S(X,q) the set of all finite subsets of X such that |S|=rX(q) and q ∈ 〈S〉, where 〈〉 denotes the linear span. We consider the case |S(X,q)|>1 (i.e. when q is not X -identifiable) and study the set W(X)q:=∩ S∈S(X,q)〈S〉, which we call the non-uniqueness set of q. We study the case dimX=1 and the case X a Veronese embedding of ℙn. We conclude the paper with a few remarks concerning this problem over the reals.

On decay rates of the solutions of parabolic Cauchy problems

We consider the Cauchy problem for a general class of parabolic partial differential equations in the Euclidean space ℝ N . We show that given a weighted L p -space $L_w^p({\mathbb {R}}^N)$ with 1 ⩽ p < ∞ and a fast growing weight w, there is a Schauder basis $(e_n)_{n=1}^\infty$ in $L_w^p({\mathbb {R}}^N)$ with the following property: given an arbitrary positive integer m there exists n m > 0 such that, if the initial data f belongs to the closed linear span of e n with n ⩾ n m , then the decay rate of the solution of the problem is at least t−m for large times t. The result generalizes the recent study of the authors concerning the classical linear heat equation. We present variants of the result having different methods of proofs and also consider finite polynomial decay rates instead of unlimited m.