Product Space Research Articles

Operators in the internal space and locality

Realizations of the holographic correspondence in String/M theory typically involve spacetimes of the form AdS × Y where Y is some internal space which geometrizes an internal symmetry of the dual field theory, hereafter referred to as an “R symmetry”. It has been speculated that areas of Ryu-Takayanagi surfaces anchored on the boundary of a subregion of Y, and smeared over the base space of the dual field theory, quantify entanglement of internal degrees of freedom. A natural candidate for the corresponding operators are linear combinations of operators with definite R charge with coefficients given by the “spherical harmonics” of the internal space: this is natural when the product spaces appear as IR geometries of higher dimensional AdS spaces. We study clustering properties of such operators both for pure AdS × Y and for flow geometries, where AdS × Y arises in the IR from a different spacetime in the UV, for example higher dimensional AdS or asymptotically flat spacetime. We show, in complete generality, that the two point functions of such operators separated along the internal space obey clustering properties at scales sufficiently larger than the AdS scale. For non-compact Y, this provides a notion of approximate locality. When Y is compact, clustering happens only when the size of Y is parametrically larger than the AdS scale. This latter situation is realized in flow geometries where the product spaces arise in the IR from an asymptotically AdS geometry at UV, but not typically when they arise near black hole horizons in asymptotically flat spacetimes. We discuss the significance of this result for entanglement and comment on the role of color degrees of freedom.

The class of grim reapers in [formula omitted]

We study translators of the mean curvature flow in the product space H2×R. In H2×R there are three types of translations: vertical translations due to the factor R and parabolic and hyperbolic translations from H2. A grim reaper in H2×R is a translator invariant by a one-parameter group of translations. The variety of translators and translations in H2×R makes the family of grim reapers particularly rich. In this paper we give a full classification of the grim reapers of H2×R with a description of their geometric properties. In some cases, we obtain explicit parametrizations of the surfaces.

On Inner Products Derived from the Standard n-Inner Product on an Inner Product Space

In this paper, we study relations between inner products derived from the standard n-inner product defined on an inner product space. In particular, we are interested in knowing when orthogonality with respect to the original inner product is preserved by the derived inner product.

Characterization of Entanglement via Non‐Existence of a Subquantum Random Field

AbstractAny pure state of a compound system with the state space determines a kind of covariance operator acting in the Cartesian product . If this operator is positively defined, then it determines a random field valued in . In this case compound quantum system can be treated as a classical random field system whose configuration space is not tensor, but Cartesian product space. It happens that and a subquantum process exists if and only if quantum state is not entangled. The technical framework used in this note is already presented by von Neumann.

Stability and Instability of an Apollonius-Type Functional Equation

For the inner product space, we have Appolonius’ identity. From this identity, Park and Th. M. Rassias induced and investigated the quadratic functional equation of the Apollonius type. And Park and Th. M. Rassias first introduced an Apollonius-type additive functional equation. In this work, we investigate an Apollonius-type additive functional equation in 2-normed spaces. We first investigate the stability of an Apollonius-type additive functional equation in 2-Banach spaces by using Hyers’ direct method. Then, we consider the instability of an Apollonius-type additive functional equation in 2-Banach spaces.

Global well-posedness for a system of quasilinear wave equations on a product space

Global well-posedness for a system of quasilinear wave equations on a product space

Geometric Inequalities of Slant Submanifolds in Locally Metallic Product Space Forms

In this particular article, our focus revolves around the establishment of a geometric inequality, commonly referred to as Chen’s inequality. We specifically apply this inequality to assess the square norm of the mean curvature vector and the warping function of warped product slant submanifolds. Our investigation takes place within the context of locally metallic product space forms with quarter-symmetric metric connections. Additionally, we delve into the condition that determines when equality is achieved within the inequality. Furthermore, we explore a number of implications of our findings.

Hybrid Virtual Screening Approach to Predict Novel Natural Compounds against HIV-1 CCR5.

Human immunodeficiency virus (HIV) infection continues to pose a major global health challenge. HIV entry into host cells via membrane fusion mediated by the viral envelope glycoprotein gp120/gp41 is a key step in the HIV life cycle. CCR5, expressed on CD4+ T cells and macrophages, acts as a coreceptor facilitating HIV-1 entry. The CCR5 antagonist maraviroc is used to treat HIV infection. However, it can cause adverse effects and has limitations such as only inhibiting CCR5-tropic viruses. There remains a need to develop alternative CCR5 inhibitors with improved safety profiles. Natural products may offer advantages over synthetic inhibitors including higher bioavailability, binding affinity, effectiveness, lower toxicity, and molecular diversity. However, screening the vast chemical space of natural compounds to identify novel CCR5 inhibitors presents challenges. This study aimed to address this gap through a hybrid ligand-based pharmacophore modeling and molecular docking approach to virtually screen large natural product databases. A reliable pharmacophore model was developed based on 311 known CCR5 antagonists and validated against an external data set. Five natural product databases containing over 306,000 compounds were filtered based on drug-likeness rules. The validated pharmacophore model screened the databases to identify 611 hits. Key residues of the CCR5 receptor crystal structure were identified for docking. The top hits were docked, and interactions were analyzed. Molecular dynamics simulations were conducted to examine complex stability. Computational prediction evaluated pharmacokinetic properties. Three compounds exhibited similar interactions and binding energies to maraviroc. MD simulations demonstrated complex stability comparable to maraviroc. One compound showed optimal predicted absorption, minimal metabolism, and a lower likelihood of interactions than maraviroc. This computational screening workflow identified three natural compounds with promising CCR5 inhibition and favorable pharmacokinetic profiles. One compound emerged as a lead based on bioavailability potential and minimal interaction risk. These findings present opportunities for developing alternative CCR5 antagonists and warrant further experimental investigation. Overall, the hybrid virtual screening approach proved effective for mining large natural product spaces to discover novel molecular entities with drug-like properties.

Analyzing the Ricci Tensor for Slant Submanifolds in Locally Metallic Product Space Forms with a Semi-Symmetric Metric Connection

This article explores the Ricci tensor of slant submanifolds within locally metallic product space forms equipped with a semi-symmetric metric connection (SSMC). Our investigation includes the derivation of the Chen–Ricci inequality and an in-depth analysis of its equality case. More precisely, if the mean curvature vector at a point vanishes, then the equality case of this inequality is achieved by a unit tangent vector at the point if and only if the vector belongs to the normal space. Finally, we have shown that when a point is a totally geodesic point or is totally umbilical with n=2, the equality case of this inequality holds true for all unit tangent vectors at the point, and conversely.

Perturbations of non-autonomous second-order abstract Cauchy problems

In this paper we present time-dependent perturbations of second-order non-autonomous abstract Cauchy problems associated to a family of operators with constant domain. We make use of the equivalence to a first-order non-autonomous abstract Cauchy problem in a product space, which we elaborate in full detail. As an application we provide a perturbed non-autonomous wave equation.

Vector-valued Gaussian processes on non-Euclidean product spaces: constructive methods and fast simulations based on partial spectral inversion

Vector-valued Gaussian processes on non-Euclidean product spaces: constructive methods and fast simulations based on partial spectral inversion

Topology Meets Number Theory

Liouville proved the existence of a set L of transcendental real numbers now known as Liouville numbers. Erdős proved that while L is a small set in that its Lebesgue measure is zero, and even its s-dimensional Hausdorff measure, for each s > 0, equals zero, it has the Erdős property, that is, every real number is the sum of two numbers in L . He proved L is a dense G δ -subset of R and every dense G δ -subset of R has the Erdős property. While being a dense G δ -subset of R is a purely topological property, all such sets contain c Liouville numbers. Each dense G δ -subset of R , including L , is homeomorphic to the product N ℵ 0 of copies of the discrete space N of all natural numbers. Also this product space is homeomorphic to the space P of all irrational real numbers and the space T of all transcendental real numbers. Hence every dense G δ -subset of R has cardinality c . Indeed, any dense G δ -subset of R has a chain Xm , m ∈ ( 0 , ∞ ) of homeomorphic dense G δ -subsets such that X m ⊂ X n , for n < m, and X n ∖ X m has cardinality c . Finally, every real number r ≠ 1 is equal to ab , for some a , b ∈ L .

Metal-Hydride C-C Cross-Coupling of Alkenes Through a Double Outer-Sphere Mechanism.

This Synopsis covers recent reports of metal-catalyzed alkene functionalizations that likely involve iterative outer-sphere reactions in which the substrate reacts directly with a metal ligand instead of with the metal center itself. Traditional metal hydride-catalyzed alkene functionalizations involve this latter pathway whereby the alkene forms part of the metal ligand sphere (i.e. an inner-sphere reaction). In contrast, alkenes do not ligate the metal in so-called outer-sphere reactions and instead react with a metal ligand. These transformations have proved crucial for the synthesis of high fraction sp3 (Fsp3) targets, especially in hindered fragment couplings of relevance to natural product space.

Seeking efficiency or price gouging? Evidence from pharmaceutical mergers

We examine the impact of mergers on drug prices and document significant differences between the post-merger pricing strategies of highly innovative pharmaceutical firms and other firms. While the former raise prices of overlapping drugs, especially brand name drugs that tend to be first-in-class and patented, we find pervasive evidence of price reductions by generic manufacturers. Our evidence suggests that the price reductions are due to cost cuts realized by less innovative firms in overlapping product spaces. We also show that less innovative acquirers cut R&D and shift product development from high-novelty products to cheaper, less-risky products.

Sparse kernel k-means clustering

Clustering is an essential technique that groups similar data points to uncover the underlying structure and features of the data. Although traditional clustering methods such as k-means are widely utilized, they have limitations in identifying nonlinear clusters. Thus, alternative techniques, such as kernel k-means and spectral clustering, have been developed to address this issue. However, another challenge arises when irrelevant variables are present in the data; this can be mitigated by employing variable selection methods such as the filter, wrapper, and embedded approaches. In this study, with a particular focus on kernel k-means clustering, we propose an embedded variable selection method using a tensor product space along with a general analysis of variance kernel for nonlinear clustering. Comprehensive experiments involving simulations and real data analysis demonstrated that the proposed method achieves competitive performance compared to existing approaches. Thus, the proposed method may serve as a reliable tool for accurate cluster identification and variable selection to gain insights into complex datasets.

Shopping behavior and the effect of monetary policy on inflation heterogeneity along the income distribution

This paper studies the effect of monetary policy on inflation along the income distribution in the euro area and shows that monetary policy has differential effects. In response to monetary tightenings, high-income households engage in more intense product substitution and trade down in the product and store space relative to low-income households, such that their inflation responds relatively more. This channel stands in contrast to the effect arising from different consumption bundles, which imply that the inflation experienced by high-income households responds less to monetary policy. The paper also reveals substantial cross-country heterogeneity, with the product substitution channel being more pronounced in countries where price differences between possible substitutes are larger.

Nonlinear two-component system of time-fractional PDEs in [formula omitted]-dimensions: Invariant subspace method combined with variable transformation

In this article, we develop a systematic approach of the invariant subspace method combined with variable transformation to find the generalized separable exact solutions of the nonlinear two-component system of time-fractional PDEs (TFPDEs) in (2+1)-dimensions for the first time. Also, we explicitly explain how to construct various kinds of finite-dimensional invariant linear product spaces for the given system using the invariant subspace method combined with variable transformation. Additionally, we present how to use the obtained invariant linear product spaces to derive the generalized separable exact solutions of the discussed system. We also note that the discussed method will help to reduce the nonlinear two-component system of TFPDEs in (2+1)-dimensions into the nonlinear two-component system of TFPDEs in (1+1)-dimensions, which again reduces to a system of time-fractional ODEs through the obtained invariant linear product spaces. More specifically, the significance and efficacy of the systematic investigation of the discussed method have been investigated through the initial and boundary value problems of the generalized nonlinear two-component system of time-fractional reaction–diffusion equations (TFRDEs) in (2+1)-dimensions for finding the generalized separable exact solutions, which can be expressed in terms of the exponential, trigonometric, polynomial, Euler-Gamma and Mittag-Leffler functions. Also, 2D and 3D graphical representations of some of the obtained solutions are presented for different values of fractional orders.

Spaces determined by countably many locally compact subspaces

The theory of compactly generated spaces, alternatively k-spaces, plays an important role in general and algebraic topology. In this paper, we develop a theory of compact generation for non-Hausdorff spaces using locally compact spaces. We are particularly interested in the case that countably many locally compact spaces suffice and restrict our attention to that case. The notion of ℓcω-spaces (which can be considered as a type of kω-spaces) is introduced, which are determined by countably many locally compact spaces. It is proved that the product space of two ℓcω-spaces is still an ℓcω-space. We slightly modify the notion of an ℓcω-space to apply it to posets equipped with the Scott topology. The notions of ℓcω-posets, ℓfω-posets and c-posets are introduced. We develop a corresponding theory for these three kinds of posets and investigate the conditions under which the Scott topology on the product of two posets is equal to the product of the individual Scott topologies and under which the Scott topology on a dcpo is sober. Several such conditions are given.

Information economies with taste diversity and bounded attention spans

We consider an economy with a countably infinite number of consumers and pure public information goods. Each of these differentiated products is produced by a single monopoly firm that enters the market if it can cover costs. Thus, the product space is endogenous. We assume that the population of agents has diverse tastes, but bounded attention spans for content. We show that this implies that at all Pareto efficient allocations, all agents consume a finite number of public goods, and that each public good is consumed by a finite number of agents. In effect, these two taste assumptions turn pure public goods into what amount to club goods, despite the lack of rivalry in consumption or crowding of any type. Unfortunately, the equilibrium outcomes of Tiebout-like competition between public good providers do not satisfy the First Welfare Theorem. Even non-anonymous Lindahlian price systems are not sufficient to signal all profit opportunities to firms. We conclude that information markets are likely to be inefficient, and there will always remain opportunities for economic profits in an information economy.

Related or unrelated diversification: What is smart specialization?

In this paper, we investigate the nature of path dependency in specialization of countries in terms of international trade. Our focus is on the product space density metric, which is employed in the literature on smart specialization and the product space. We find that although density is supposed to capture relatedness between a country's current specialization pattern and potential products that it may diversify into, density also contains a large component of what we call unrelated diversity, i.e., country- and product fixed effects. Together, these fixed effects capture 97 % of the variance of density. We isolate the related variety component in density and then estimate logit regressions that predict gains or losses of specialization. The relative influence of unrelated variety on changes of specialization increases with the level of diversification of the country: only countries that are not already diversified are strongly influences of related variety. We also show that the impact of the country effect in the impact of unrelated diversity in gains or keeps of specializations increases with diversity. This suggests that existing capabilities breed new capabilities, i.e., diversification is a virtuous circle that is largely independent of relatedness, and that the source of path dependency in international specialization patterns lies much more in generic production capabilities rather than relatedness in product space.