Arbitrary Weights Research Articles

Correlators of N=4 Supersymmetric Yang-Mills Theory on Real Projective Space at Strong Coupling

We consider type IIB supergravity on a Z2 quotient of AdS5×S5 as the holographic dual of strongly coupled 4D N=4 SYM on RP4 space with the gauging of charge conjugation. Using bootstrap techniques, we determine all two-point functions of 12-BPS operators of arbitrary weights at the leading order in the large central charge expansion. Published by the American Physical Society 2024

The minimal model of Rota-Baxter operad with arbitrary weight

The minimal model of Rota-Baxter operad with arbitrary weight

Simple zeros of $\operatorname{GL}(2)$ $L$-functions

Let f \in S_{k}(\Gamma_{1}(N)) be a primitive holomorphic form of arbitrary weight k and level N . We show that the completed L -function of f has \Omega(T^{\delta}) simple zeros with imaginary part in [-T, T] , for any \delta <{2/27} . This is the first power bound in this problem for f of non-trivial level, where previously the best results were \Omega(\log\log\log{T}) for N odd, due to Booker, Milinovich, and Ng, and infinitely many simple zeros for N even, due to Booker. In addition, for f of trivial level ( N=1 ), we also improve an old result of Conrey and Ghosh on the number of simple zeros.

Multi-Objective Combinatorial Optimization Algorithm Based on Asynchronous Advantage Actor–Critic and Graph Transformer Networks

Multi-objective combinatorial optimization problems (MOCOPs) are designed to identify solution sets that optimally balance multiple competing objectives. Addressing the challenges inherent in applying deep reinforcement learning (DRL) to solve MOCOPs, such as model non-convergence, lengthy training periods, and insufficient diversity of solutions, this study introduces a novel multi-objective combinatorial optimization algorithm based on DRL. The proposed algorithm employs a uniform weight decomposition method to simplify complex multi-objective scenarios into single-objective problems and uses asynchronous advantage actor–critic (A3C) instead of conventional REINFORCE methods for model training. This approach effectively reduces variance and prevents the entrapment in local optima. Furthermore, the algorithm incorporates an architecture based on graph transformer networks (GTNs), which extends to edge feature representations, thus accurately capturing the topological features of graph structures and the latent inter-node relationships. By integrating a weight vector layer at the encoding stage, the algorithm can flexibly manage issues involving arbitrary weights. Experimental evaluations on the bi-objective traveling salesman problem demonstrate that this algorithm significantly outperforms recent similar efforts in terms of training efficiency and solution diversity.

The underlying mechanisms of alignment in error backpropagation through arbitrary weights

Understanding the mechanisms by which plasticity in millions of synapses in the brain is orchestrated to achieve behavioral and cognitive goals is a fundamental question in neuroscience. In this regard, insights from learning methods in artificial neural networks (ANNs) and in particular supervised learning using backpropagation (BP) seem inspiring. However, the implementation of BP requires exact matching between forward and backward weights, which is unrealistic given the known connectivity pattern in the brain (known as the “weight transport problem”). Notably, it has been shown that under certain conditions, error BackPropagation Through Arbitrary Weights (BP-TAW) can lead to a partial alignment between forward and backward weights (weight alignment or WA). This learning algorithm, which is also known as feedback alignment (FA), can result in surprisingly good degrees of accuracy in simple classification tasks. However, the underlying mechanisms and mathematical basis of WA are not thoroughly understood. In this work, we demonstrate the mathematical basis of WA and answer the question of why and in what conditions WA occurs. We show that the occurrence of WA in ANNs is induced by statistical properties of the output and error signals of neurons, such as autocorrelation and cross-correlation, and can happen even in the absence of learning or reduction of the loss function. Moreover, we show that WA can be improved significantly by limiting the norm of input weights to neurons and that such a weight normalization (WN) method can improve the classification accuracy of BP-TAW. The findings presented can be used to further improve the performance of BP-TAW and open new ways for exploring possible learning mechanisms in biological neural networks without exact matching between forward and backward weights.

Parameterized approximation algorithms for weighted vertex cover

A vertex cover of a graph is a set of vertices of the graph such that every edge has at least one endpoint in it. In this work, we study Weighted Vertex Cover with solution size as a parameter. Formally, in the (k,W)-Vertex Cover problem, given a graph G, an integer k, a positive rational W, and a weight function w:V(G)→Q+, the question is whether G has a vertex cover of size at most k of weight at most W, with k being the parameter. An (a,b)-bi-criteria approximation algorithm for (k,W)-Vertex Cover either produces a vertex cover S such that |S|≤ak and w(S)≤bW, or decides that there is no vertex cover of size at most k of weight at most W. We obtain the following results.•A simple (2,2)-bi-criteria approximation algorithm for (k,W)-Vertex Cover in polynomial time by modifying the standard LP-rounding algorithm.•A simple exact parameterized algorithm for (k,W)-Vertex Cover running in O⁎(1.4656k) time1.•A (1+ϵ,2)-approximation algorithm for (k,W)-Vertex Cover running in O⁎(1.4656(1−ϵ)k) time.•A (1.5,1.5)-approximation algorithm for (k,W)-Vertex Cover running in O⁎(1.414k) time.•A (2−δ,2−δ)-approximation algorithm for (k,W)-Vertex Cover running in O⁎(∑i=δk(1−2δ)1+2δδk(1−2δ)2δ(δk+iδk−2iδ1−2δ)) time for any δ<0.5. For example, for (1.75,1.75) and (1.9,1.9)-approximation algorithms, we get running times of O⁎(1.272k) and O⁎(1.151k) respectively.Our algorithms (expectedly) do not improve upon the running times of the existing algorithms for the unweighted version of Vertex Cover. When compared to algorithms for the weighted version, our algorithms are the first ones to the best of our knowledge which work with arbitrary weights, and they perform well when the solution size is much smaller than the total weight of the desired solution.

Bessel functions on GL(n), I

In the context of the Kuznetsov trace formula, we outline the theory of the Bessel functions on GL(n) as a series of conjectures designed as a blueprint for the construction of Kuznetsov-type formulas with given ramification at infinity. We are able to prove one of the conjectures at full generality on GL(n) and most of the conjectures in the particular case of the long Weyl element; as with previous papers, we give some unconditional results on Archimedean Whittaker functions, now on GL(n) with arbitrary weight. We expect the heuristics here to apply at the level of real reductive groups. A forthcoming paper will address the initial conjectures up to Mellin-Barnes integral representations in the case of GL(4) Bessel functions.

Formal deformations, cohomology theory and L∞[1]-structures for differential Lie algebras of arbitrary weight

We introduced a cohomology theory for differential Lie algebras of arbitrary weight which generalised a previous work of Jiang and Sheng. The underlying L∞[1]-structure on the cochain complex is also determined via a generalised version of higher derived brackets. The equivalence between L∞[1]-structures for absolute and relative differential Lie algebras is established. Formal deformations and abelian extensions are interpreted by using lower degree cohomology groups. Also we introduce the homotopy differential Lie algebras.

Kramers-Wannier Duality and Random-Bond Ising Model.

We present a new combinatorial approach to the Ising model incorporating arbitrary bond weights on planar graphs. In contrast to existing methodologies, the exact free energy is expressed as the determinant of a set of ordered and disordered operators defined on a planar graph and the corresponding dual graph, respectively, thereby explicitly demonstrating the Kramers-Wannier duality. The implications of our derived formula for the Random-Bond Ising Model are further elucidated.

Generalized NS-algebras

We generalize the notion of an NS-algebra, which was previously only considered for associative, Lie and Leibniz algebras, to arbitrary categories of binary algebras with one operation. We do this by defining these algebras using a bimodule property, as we did in our earlier work for defining the notions of a dendriform and tridendriform algebra for such categories of algebras. We show that several types of operators lead to NS-algebras: Nijenhuis operators, twisted Rota-Baxter operators and relative Rota-Baxter operators of arbitrary weight. We thus provide a general framework in which several known results and constructions for associative, Lie and Leibniz-NS-algebras are unified, along with some new examples and constructions that we also present.

Glass Box and Black Box Machine Learning Approaches to Exploit Compositional Descriptors of Molecules in Drug Discovery and Aid the Medicinal Chemist.

The synthetic medicinal chemist plays a vital role in drug discovery. Today there are AI tools to guide next syntheses, but many are "Black Boxes" (BB). One learns little more than the prediction made. There are now also AI methods emphasizing visibility and "explainability" (thus explainable AI or XAI) that could help when "compositional data" are used, but they often still start from seemingly arbitrary learned weights and lack familiar probabilistic measures based on observation and counting from the outset. If probabilistic methods were used in a complementary way with BB methods and demonstrated comparable predictive power, they would provide guidelines about what groups to include and avoid in next syntheses and quantify the relationships in probabilistic terms. These points are demonstrated by blind test comparison of two main types of BB methods and a probabilistic "Glass Box" (GB) method new outside of medicine, but which appears well suited to the above. Because many probabilities can be involved, emphasis is on the predictive power of its simplest explanatory models. There are usually more inactive compounds by orders of magnitude, often a problem for machine learning methods. However, the approaches used here appear to work well for such "real world data".

Korenblum’s Principle for Bergman Spaces with Radial Weights

AbstractWe show that the Korenblum maximum (domination) principle is valid for weighted Bergman spaces $$A^p_w$$ A w p with arbitrary (non-negative and integrable) radial weights w in the case $$1\le p&lt;\infty $$ 1 ≤ p &lt; ∞ . We also notice that in every weighted Bergman space the supremum of all radii for which the principle holds is strictly smaller than one. Under the mild additional assumption $$\liminf _{r\rightarrow 0^+} w(r)&gt;0$$ lim inf r → 0 + w ( r ) &gt; 0 , we show that the principle fails whenever $$0&lt;p&lt;1$$ 0 &lt; p &lt; 1 .

Deformations and cohomology theory of Ω-Rota-Baxter algebras of arbitrary weight

In this paper, we introduce the concepts of relative and absolute Ω-Rota-Baxter algebras of weight λ, which can be considered as a family algebraic generalization of relative and absolute Rota-Baxter algebras of weight λ. We study the deformations of relative and absolute Ω-Rota-Baxter algebras of arbitrary weight. Explicitly, we construct an L∞[1]-algebra via the method of higher derived brackets, whose Maurer-Cartan elements correspond to relative Ω-Rota-Baxter algebra structures of weight λ. For a relative Ω-Rota-Baxter algebra of weight λ, the corresponding twisted L∞[1]-algebra controls its deformations, which leads to the cohomology theory of it, and this cohomology theory can interpret the formal deformations of the relative Ω-Rota-Baxter algebra. Moreover, we also obtain the corresponding results for absolute Ω-Rota-Baxter algebras of weight λ from the relative version.

Bootstrapping AdS2× S2 hypermultiplets: hidden four-dimensional conformal symmetry

We bootstrap the 4-point amplitude of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 hypermultiplets in AdS2 × S2 at tree-level and for arbitrary external weights. We hereby explicitly demonstrate the existence of a hidden four-dimensional conformal symmetry that was used as an assumption in previous studies to derive this result.

Shifted convolution sums for GL(3)×GL(2)$GL(3)\times GL(2)$ averaged over weighted sets

AbstractLet be the Fourier coefficients of an Hecke–Maass cusp form and be those of an Hecke holomorphic or Hecke–Maass cusp form . Let and be a sequence. We show that if for some , for any , and a similar bound holds when . This improves Sun's bound and generalizes it to an average with arbitrary weights . Moreover, we demonstrate how one can recover the factorizable moduli structure given by Jutila's variant of the circle method via studying a shifted sum with weighted average. This allows us to recover Munshi's bound on the shifted sum with a fixed shift without using Jutila's circle method.

Embeddings of a Multi-Weighted Anisotropic Sobolev Type Space

Parameters such as various integral and differential characteristics of functions, smoothness properties of regions and their boundaries, as well as many classes of weight functions cause complex relationships and embedding conditions for multi-weighted anisotropic Sobolev type spaces. The desire not to restrict these parameters leads to the development of new approaches based on the introduction of alternative definitions of spaces and norms in them or on special localization methods. This article examines the embeddings of multi-weighted anisotropic Sobolev type spaces with anisotropy in all the defining characteristics of the norm of space, including differential indices, summability indices, as well as weight coefficients. The applied localization method made it possible to obtain an embedding for the case of an arbitrary domain and weights of a general type, which is important in applications in differential operators’ theory, numerical analysis.

Simplicity of AdS super Yang-Mills at one loop

We perform a systematic bootstrap analysis of four-point one-loop Mellin amplitudes for super gluons in AdS5 × S3 with arbitrary Kaluza-Klein weights. The analysis produces the general expressions for these amplitudes at extremalities two and three, as well as analytic results for many other special cases. From these results we observe remarkable simplicity. We find that the Mellin amplitudes always contain only simultaneous poles in two Mellin-Mandelstam variables, extending a previous observation in the simplest case with the lowest Kaluza-Klein weights. Moreover, we discover a substantial extension of the implication of the eight-dimensional hidden conformal symmetry, which goes far beyond the Mellin poles associated with the leading logarithmic singularities. This leaves only a small finite set of poles which can be determined on a case-by-case basis from the contributions of protected operators in the OPE.

Modeling Structured Dependency Tree with Graph Convolutional Networks for Aspect-Level Sentiment Classification.

Aspect-based sentiment analysis is a fine-grained task where the key goal is to predict sentiment polarities of one or more aspects in a given sentence. Currently, graph neural network models built upon dependency trees are widely employed for aspect-based sentiment analysis tasks. However, most existing models still contain a large amount of noisy nodes that cannot precisely capture the contextual relationships between specific aspects. Meanwhile, most studies do not consider the connections between nodes without direct dependency edges but play critical roles in determining the sentiment polarity of an aspect. To address the aforementioned limitations, we propose a Structured Dependency Tree-based Graph Convolutional Network (SDTGCN) model. Specifically, we explore construction of a structured syntactic dependency graph by incorporating positional information, sentiment commonsense knowledge, part-of-speech tags, syntactic dependency distances, etc., to assign arbitrary edge weights between nodes. This enhances the connections between aspect nodes and pivotal words while weakening irrelevant node links, enabling the model to sufficiently express sentiment dependencies between specific aspects and contextual information. We utilize part-of-speech tags and dependency distances to discover relationships between pivotal nodes without direct dependencies. Finally, we aggregate node information by fully considering their importance to obtain precise aspect representations. Experimental results on five publicly available datasets demonstrate the superiority of our proposed model over state-of-the-art approaches; furthermore, the accuracy and F1-score show a significant improvement on the majority of datasets, with increases of 0.74, 0.37, 0.65, and 0.79, 0.75, 1.17, respectively. This series of enhancements highlights the effective progress made by the STDGCN model in enhancing sentiment classification performance.

Інфляція як мірило незалежності центральних банків

The aim of the article is to study the process of formation of the modern paradigm of independence of central banks, determine the role of inflation in it, outline the criteria for independence and determine the dynamics of legal indicators of autonomy of monetary institutions. It is noted that the impetus for changes in relations between governments and central banks of many industrialized countries was the episodes of high inflation in the 1970s. Awareness of the problem of the time discrepancy between the short-term perspective of political decisions and their long-term impact on the economy provoked institutional changes in the status of monetary institutions. They envisaged the prioritization of central banks’ activities to ensure a consistently low level of inflation and provide independence to achieve it. It is also noted that in countries with economies in transition, the tendency to separate central banks into an independent institution was implemented during the 1990s. The criteria for the independence of central banks are determined by legislation and their charters. These usually include the following: isolation of central bank leadership from political pressure; prohibition to give instructions to central banks in the conduct of monetary policy; a clearly defined purpose of monetary policy; restrictions on lending to government programs and budget deficits. In addition to the formally defined independence, there is also the actual independence of central banks, which is usually outlined through the index of management turnover of monetary institutions and its correlation with the inflation rate. But this does not take into account other (exogenous) factors that affect the inflation rate, which makes the studies less relevant. In addition, the construction of any indices includes an arbitrary weight value of each of them, which reflects the sensitivity of the results obtained depending on their composition. As noted in the study, instead of a static analysis of the indicators of independence of central banks, it is more expedient to determine their modification from the beginning of the formation of the modern paradigm of independence to the present day. The analysis showed that since the beginning of the 1970s, most of the countries studied have changed the rules prohibiting central banks from financing (in any form) the budget deficit. At the same time, obstacles to interference in monetary policy were established. This suggests that the change in these criteria had the greatest impact on the downward trend in the level and volatility of inflation in the period under study.

Approximating Minimum k-Tree Cover of a Connected Graph Inspired by the Multi-Ferry Routing in Delay Tolerant Networks

In this paper, we consider the problem of covering the vertex set of a given graph by [Formula: see text] trees so as to minimize the maximum weight of the trees, as a subproblem of the multi-ferry scheduling problem proposed by Zhao and Ammar. After pointing out that the approximation ratio of a greedy scheme based on the Kruskal’s algorithm is provably bad, we show that the approximation ratio cannot be better than 3/2 for [Formula: see text] even when the edge selection criterion is modified so as to minimize the increase in the maximum weight in the collection of trees. We then propose two polynomial-time algorithms with a guaranteed approximation ratio. The first algorithm achieves 3-approximation for the class of graphs in which the edge weights satisfy the triangle inequality. The second algorithm achieves 4-approximation for any connected graph with arbitrary edge weights.