A scalar product for quasinormal mode solutions to Teukolsky’s homogeneous radial equation is presented. Evaluation of this scalar product can be performed either by direct integration, or by evaluation of a confluent hypergeometric functions. The related scalar product will be useful for better understanding analytic solutions to Teukolsky’s radial equation, particularly the quasinormal modes, their potential spatial completeness, and whether the quasinormal mode overtone excitations may be estimated by spectral decomposition rather than fitting. With that motivation, the scalar product is applied to confluent Heun polynomials where it is used to derive their peculiar orthogonality and eigenvalue properties. A potentially new relationship is derived between the confluent Heun polynomials’ scalar products and eigenvalues. Using these results, it is shown for the first time that Teukolsky’s radial equation (and perhaps similar confluent Heun equations) are, in principle, exactly tridiagonalizable. To this end, “canonical” confluent Heun polynomials are conjectured.

The Ehrenfest theorem is a well-known theoretical result of quantum mechanics. It relates the dynamical evolution of the expectation value of a quantum operator to the expectation value of its corresponding commutator with the Hermitian Hamiltonian operator. However, the proof of validity of the Ehrenfest theorem for quantum gravity field theory has remained elusive, while its validation poses challenging conceptual questions. In fact, this presupposes a number of minimum requirements, which include the prescription of quantum Hamiltonian operator, the definition of scalar product, and the identification of dynamical evolution parameter. In this paper, it is proven that the target can be established in the framework of the manifestly covariant quantum gravity theory (CQG theory). This follows as a consequence of its peculiar canonical Hamiltonian structure and the commutator-bracket algebra that characterizes its representation and probabilistic interpretation. The theoretical proof of the theorem for CQG theory permits to elucidate the connection existing between quantum operator variables of gravitational field and the corresponding expectation values to be interpreted as dynamical physical observables set in the background metric space-time.

This paper explores the Monge–Ampere equation in the context of isotropic geometry. The study begins with an overview of the fundamental properties of isotropic space, including its scalar product, distance formula, and the nature of surfaces and curvatures within this geometric framework. A special focus is placed on dual transformations with respect to the isotropic sphere, and the self-inverse property of the dual surface is established. The article formulates the Monge–Ampere equation for isotropic space and studies its invariant solutions under isotropic motions. Several lemmas are proved to demonstrate how solutions transform under linear modifications and isotropic motions. A specific class of Monge–Ampere-type nonlinear partial differential equations is solved analytically using dual transformations and separation of variables. Additionally, translation surfaces and their curvature properties are studied in detail, particularly through the lens of dual curvature. The results demonstrate the deep relationship between curvature invariants and Monge–Ampere-type equations and show how duality simplifies the solution of nonlinear PDEs. These methods can be used for surface reconstruction and modeling in isotropic spaces.

The work further develops the results of studies [1]–[5] based on a systematic analysis using an interpretative-formal approach in a new scientific and technical direction – the technological interpretation of provisions of vector and tensor analysis that are similar in meaning. This allows expanding the formal field of representation of technological processes within the concept of their “technological meaning” and increasing the formal capacity of TP description. The results of research are revealed in the practical and production aspects related to TP, providing for the application of vector and tensor analysis in the coordinate approach, technological space, scalar product, matrix tensor, as well as examples of tensor and vector analysis in composite AKZ technology. It is determined that vectors (tensors) can be specified in different ways, depending on the technological context (polymer composite materials technology – PCM), and the set of components is only its representation in a certain (in terms of detail) basis. A coordinate approach is used, as well as the possibility of other methods of specifying and working with vectors (tensors) using the example of ordinary vectors and simple second-rank tensors, characterized by the powerful idea of orthogonality. Since the second vector and tensor represent real technological objects, including: autonomous dynamic systems (ADS), structural and technological solutions (STS), technological processes (TP), in the form of contravariant and covariant vectors, etc.The interpretative correspondence of the technological interpretation of contravariant and covariant coordinates of a vector is shown, and the nature of the relationships between technological contravariant and technological covariant coordinates is established.The example demonstrates the invariance of the enlarged stages of a complex technological process in different coordinate systems, which confirms the invariance of the technological vector under the condition of transformation of its coordinates.

We introduce a multi-scale generalization of the notion of scalar product on real and complex vector spaces and study the corresponding Voronoi tilings. This framework allows to analyze limiting metric structures involving several levels of convergence. In particular, we describe metric degenerations of polarized tori and Hausdorff limits of Voronoi tilings.

A search for pseudoscalar or scalar bosons decaying to a top quark pair (tt¯) in final states with one or two charged leptons is presented. The analyzed proton-proton collision data was recorded ats=13TeVby the CMS experiment at the CERN LHC and corresponds to an integrated luminosity of 138 fb-1. The invariant massmtt¯of the reconstructedtt¯system and variables sensitive to its spin and parity are used to discriminate against the standard modeltt¯background. Interference between pseudoscalar or scalar boson production and the standard modeltt¯continuum is included, leading to peak-dip structures in themtt¯distribution. An excess of the data above the background prediction, based on perturbative quantum chromodynamics (QCD) calculations, is observed near the kinematictt¯production threshold, while good agreement is found for highmtt¯. The data are consistent with the background prediction if the contribution from a simplified model of a color-singlet1S0[1]tt¯quasi-bound stateηt, inspired by nonrelativistic QCD, is added. Upper limits at 95% confidence level are set on the coupling between the pseudoscalar or scalar bosons and the top quark for boson masses in the range 365-1000 GeV, relative widths between 0.5% and 25%, and two background scenarios with or withoutηtcontribution.

Establishing a methodological framework for evaluating gait symmetry through muscle synergies: A pilot study.

The article presents the development of an intelligent system for classifying user-generated texts in social networks under conditions of linguistic uncertainty. It addresses the growing need for automated identification of professional interests among social media users, which is especially relevant for recruitment, education, and professional orientation. Traditional methods for identifying target audiences are time-consuming, costly, and inefficient when dealing with unstructured data. The proposed system utilizes and compares modern text vectorization methods — TF-IDF, FastText, and BERT — to improve classification accuracy. The system architecture includes modules for collecting user data from VKontakte via API, preprocessing text (normalization, lemmatization, noise reduction), and thematic classification using a predefined set of IT-related terms. The classification decision is based on the scalar product between text vectors and reference vectors of key terms. The system also supports an ensemble decision mechanism combining multiple models to increase reliability. The study provides a comparative analysis of the effectiveness of the selected vectorization methods in binary classification tasks (IT-related or not) using real user data. Experiments demonstrate the superiority of BERT in terms of accuracy, followed by FastText. TF-IDF showed lower sensitivity to thematic content in short and informal mes­sages. A web-based interface was developed to automate user classification. It allows the input of a VK community ID, retrieves and analyzes user data, and generates reports on IT interest distribution. The system can be applied in career guidance, educational analytics, and HR processes to identify suitable candidates based on their digital footprint.

In this work, chiral molecular descriptors were defined using 2 distinct approaches: (1) scalar triple products of vectorial molecular properties, and (2) descriptors that attempt to quantify the amount of twist in the overall molecular shape. Because both approaches give rise to conformation dependence, descriptor values were averaged over a conformational ensemble obtained by Molecular Dynamics. In addition, a method is introduced that attempts to quantify the asymmetry of the distribution of the descriptor values over the conformational ensemble. The totality of the resulting descriptors were named “Ensemble Steric and Electrostatic Chirality (ESEC) descriptors”. A pilot validation study was performed by building Quantitative Structure-Enantioselectivity Relationships (QSER), i.e. mathematical models to predict the chromatographic separation of enantiomers, using a test set of 43 structurally diverse pharmaceuticals analyzed on a polysaccharide-based chiral stationary phase. The best linear regression model (7 descriptors) for the chiral separation (expressed as selectivity factor) featured a low leave-one-out cross validation error (0.0814), a well-predicted elution sequence of the separated enantiomers (21 out of 23 molecules) and a well-predicted αRS for 27 out of 42 molecules. To the best of our knowledge, this is the first time that acceptable linear QSER models were obtained for chiral chromatographic separations of such a chemically diverse set of pharmaceuticals.

We consider the open XYZ spin chain with boundary fields. We solve the model by the new Separation of Variables approach introduced in [J. M. Maillet and G. Niccoli, J. Stat. Mech.: Theory Exp. 094020 (2019)]. In this framework, the transfer matrix eigenstates are obtained as a particular sub-class of the class of so-called separate states. We consider the problem of computing scalar products of such separate states. As usual, they can be represented as determinants with rows labelled by the inhomogeneity parameters of the model. We notably focus on the special case in which the boundary parameters parametrising the two boundary fields satisfy one constraint, hence enabling for the description of part of the transfer matrix spectrum and eigenstates in terms of some elliptic polynomial QQ-solution of a usual TQTQ-equation. In this case, we show how to transform the aforementioned determinant for the scalar product into some more convenient form for the consideration of the homogeneous and thermodynamic limits: as in the open XXX or XXZ cases, our result can be expressed as some generalisation of the so-called Slavnov determinant.

This paper presents a study of hardware implementation of basic multi-operand neural ope- rations for artificial neural networks. The operational basis of neural networks is identified, which includes groups of preprocessing operations, processor operations, and transfer function computations. The selection of basic multi-operand neural operations is justified: finding extreme values in one- dimensional arrays, calculating the sum of squared differences, scalar product computation, and group summation. Vertical computation methods for the specified operations have been improved through simultaneous processing of bit slices of all operands and adaptive complexity changes of operations in pipeline stages. This ensures synchronization of data arrival time with computation time and high equipment utilization efficiency. An integrated approach to developing parallel-pipeline devices is proposed, which is based on the capabilities of modern element base and considers the requirements of specific applications. The principles of developing parallel-pipeline devices for vertical-group compu- tation have been defined: using the basis of elementary arithmetic operations, modularity, pipelining, spatial parallelism, structural uniformity, timing parameter coordination, and specialization for specific tasks. A serial-parallel data format converter has been developed for transforming the stream of word- sequential input data into a parallel one-dimensional data array. Basic parallel-pipeline structures have been created that implement vertical computation algorithms in hardware. The synthesis method for parallel-pipeline devices has been improved using mechanisms for coordinating pipeline cycle duration with data arrival time. It is shown that the application of the developed methods and structures ensures real-time data processing with high equipment utilization efficiency.

Abstract Searches for pair-produced long-lived particles (LLPs) at the LHC commonly operate under the assumption that the two LLPs are identical. In this paper we entertain the possibility that the targeted final states are, instead, induced by a LLP pair with different masses and/or lifetimes. We propose a simple and intuitively-parametrised toy model in order to study such asymmetric production of LLPs. Using the recasting material of a recent search for displaced jets by the ATLAS collaboration, we demonstrate that we can set constraints on the production cross-section times branching fraction into jets for a variety of asymmetric LLP and mediator mass combinations.

Scalar products and Left LCD codes

We compute scalar products of off-shell Bethe vectors in models with \mathfrak{o}_{2n+1}𝔬2n+1 symmetry. The scalar products are expressed as a sum over partitions of the Bethe parameter sets, the building blocks being the so-called highest coefficients. We prove some recurrence relations and a residue theorem for these highest coefficients, and prove that they are consistent with the reduction to \mathfrak{g}\mathfrak{l}_n𝔤𝔩n invariant models. We also express the norm of on-shell Bethe vectors as a Gaudin determinant.

We introduce the conformally invariant scalar product, originally devised for radiation fields, to the study of the modes of optical resonators. This scalar product allows one to normalize and compare resonant modes using their corresponding radiation fields. Such fields are polychromatic fields free of divergences, which are determined from the complex frequencies and the modal fields on the surface of the resonator. The scalar product is expressed as surface integrals involving the modal fields, multiplied by closed-form factors incorporating the complex frequencies. In a practical application, we study the modes of disk-shaped whispering gallery resonators and show that the proposed scalar product accurately predicts the geometry-dependent anticrossings between modes. Published by the American Physical Society 2025

Abstract Dark matter (DM) genesis via Ultraviolet (UV) freeze-in embeds the seed of reheating temperature and dynamics in its relic density. Thus, discovery of such a DM candidate can possibly open the window for post-inflationary dynamics. However, there are several challenges in this exercise, as freezing-in DM possesses feeble interaction with the visible sector and therefore very low production cross-section at the collider. We show that mono-photon (and dilepton) signal at the ILC, arising from DM effective operators connected to the SM field strength tensors, can still warrant a signal discovery. We study both the scalar and fermionic DM production during reheating via UV freeze-in, when the inflaton oscillates at the bottom of a general monomial potential. Interestingly, we see, right DM abundance can be achieved only in the case of bosonic reheating scenario, satisfying bounds from big bang nucleosynthesis (BBN). This provides a unique correlation between collider signal and the post-inflationary dynamics of the Universe within single-field inflationary models.

Gravitational particle production in cosmology is a mechanism through which particles of different natures are produced during the very early universe. It is a general mechanism that explains how the universe became populated with the particles of the Standard Model after cosmic inflation and may also account for the origin of dark matter. In this work, we study the non-perturbative production of massive scalar particles in the Higgs-R2 inflation model, a two-field scalar inflation model within the Einstein frame. We consider spectator scalar fields that are conformally or minimally coupled to the gravitational field through the curvature scalar R which in turn is a time-dependent function determined by the fields driving inflationary dynamics. We numerically compute the production of these particles using the Bogolyubov transformation method for each scenario, aiming to assess the spectrum of the produced particles. For both scenarios, we consider light particles with masses mχ≪Hend and large masses that exceed the Hubble scale at the end of inflation mχ≳Hend. We use these numerical results to calculate the relic abundance Ωχh2 to find out if the model is viable as a dark matter candidate.

Abstract We study quasinormal mode expansions by adopting a Keldysh scheme for the spectral construction of asymptotic resonant expansions. Quasinormal modes are first cast in terms of a non-selfadjoint problem by adopting, in a black hole perturbation setting, a spacetime hyperboloidal approach. Then the Keldysh expansion of the resolvent, built on bi-orthogonal systems, provides a spectral version of Lax-Phillips expansions on scattering resonances. We clarify the role of scalar product structures in the Keldysh setting [1], that prove non-necessary to construct the resonant expansions (in particular the quasinormal mode time-series at null infinity), but are required to define the (constant) excitation coefficients in the bulk resonant expansion. We demonstrate the efficiency and accuracy of the Keldysh spectral approach to (non-selfadjoint) dynamics, even beyond its limits of validity, in particular recovering Schwarzschild black hole late power-law tails. We also study early dynamics by exploring i) the existence of an earliest time of validity of the resonant expansion and ii) the interplay between overtones extracted with the Keldysh scheme and regularity. Specifically, we address convergence aspects of the series and, on the other hand, we implement non-modal analysis tools, namely assessing $$H^p$$ H p -Sobolev dynamical transient growths and constructing $$H^p$$ H p -pseudospectra. Finally, we apply the Keldysh scheme to calculate “second-order” quasinormal modes and complement the qualitative study of overtone distribution by presenting the Weyl law for the counting of quasinormal modes in black holes with different (flat, De Sitter, anti-De Sitter) spacetime asymptotics.

The object of this study is the processes of parallel vertical-group data processing and minimization of equipment costs, which enable the synthesis of real-time recursive neural elements with high efficiency of equipment use. A model of a recursive-type neural element has been built, which, through the use of a parallel vertical-group method for calculating the scalar product and the ability to choose the number of bits in the group for the formation of partial products, coordinates the time of receipt of weights and input data with the time of calculating the result at the output of the neural element. This approach provides a hardware implementation of the neural element with minimal use of equipment. The basic structure of the neural element has been designed, which, through the use of hardware mapping of the constructed graph model, regularity, and modularity of the structure, provides the synthesis of hardware for a specific application. The application of pipelines and spatial parallelism of data processing, as well as the organization of the process of calculating the scalar product, as the performance of a single operation, enables the implementation of a neural element for real-time operation. Analytical expressions have been built to estimate the parameters of a neural element depending on the bit depth of operands, the number of data inputs, and the number of bits in the group. A method for synthesizing a recursive-type neural element has been devised, which, due to the use of the basic structure, enables mechanisms for matching the time of receipt of weight coefficients and input data with the time of calculating the output, thus ensuring its implementation for specific applications. Considering ways to minimize equipment costs ensures the construction of a neural element with minimal hardware costs. The synthesized neural element for a data depth of 16 bits with an increase in the number of bits that are simultaneously processed in a group, from 2 to 8, provides a decrease in the processing time by 2.8 times with a reduction in the efficiency of using the equipment of the neural element by no more than 1.6 times

Charged scalars appear in many motivated extensions beyond the Standard Model. We analyze the constraints on charged scalar pair production via the Drell-Yan process at the Large Hadron Collider and interpret them in terms of weak isospin quantum numbers. Leveraging the experimental limits from existing LHC data and phenomenological recast analyses, we place bounds on the branching ratio of the charged scalar as a function of its mass, electric charge, and isospin. This approach enables us to determine limits on the branching ratios directly from experimental data, without appealing to a specific model. We provide a detailed analysis for singly and doubly charged scalars across various weak isospin scenarios, focusing on decays into leptonic and bosonic final states, and we validate this approach in extended Higgs sectors such as the Higgs triplet model and the Georgi-Machacek model. Published by the American Physical Society 2025