Irreducible Representations Research Articles

Investigation of the Fork–Join System with Markovian Arrival Process Arrivals and Phase-Type Service Time Distribution Using Machine Learning Methods

This paper presents a study of fork–join systems. The fork–join system breaks down each customer into numerous tasks and processes them on separate servers. Once all tasks are finished, the customer is considered completed. This design enables the efficient handling of customers. The customers enter the system in a MAP flow. This helps create a more realistic and flexible representation of how customers arrive. It is important for modeling various real-life scenarios. Customers are divided into K≥2 tasks and assigned to different subsystems. The number of tasks matches the number of subsystems. Each subsystem has a server that processes tasks, and a buffer that temporarily stores tasks waiting to be processed. The service time of a task by the k-th server follows a PH (phase-type) distribution with an irreducible representation (βk, Sk), 1≤k≤K. An analytical solution was derived for the case of K=2 when the input MAP flow and service time follow a PH distribution. We have efficient algorithms to calculate the stationary distribution and performance characteristics of the fork–join system for this case. In general cases, this paper suggests using a combination of Monte Carlo and machine learning methods to study the performance of fork–join systems. In this paper, we present the results of our numerical experiments.

Matrix factorizations of the discriminant of Sn

Consider the symmetric group Sn acting as a reflection group on the polynomial ring k[x1,…,xn] where k is a field, such that Char(k) does not divide n!. We use Higher Specht polynomials to construct matrix factorizations of the discriminant of this group action: these matrix factorizations are indexed by partitions of n and respect the decomposition of the coinvariant algebra into isotypical components. The maximal Cohen–Macaulay modules associated to these matrix factorizations give rise to a noncommutative resolution of the discriminant and they correspond to the nontrivial irreducible representations of Sn. All our constructions are implemented in Macaulay2 and we provide several examples. We also discuss extensions of these results to Young subgroups of Sn and indicate how to generalize them to the reflection groups G(m,1,n).

Solving Particle–Antiparticle and Cosmological Constant Problems

We solve the particle-antiparticle and cosmological constant problems proceeding from quantum theory, which postulates that: various states of the system under consideration are elements of a Hilbert space H with a positive definite metric; each physical quantity is defined by a self-adjoint operator in H; symmetry at the quantum level is defined by a representation of a real Lie algebra A in H such that the representation operator of any basis element of A is self-adjoint. These conditions guarantee the probabilistic interpretation of quantum theory. We explain that in the approaches to solving these problems that are described in the literature, not all of these conditions have been met. We argue that fundamental objects in particle theory are not elementary particles and antiparticles but objects described by irreducible representations (IRs) of the de Sitter (dS) algebra. One might ask why, then, experimental data give the impression that particles and antiparticles are fundamental and there are conserved additive quantum numbers (electric charge, baryon quantum number and others). The reason is that, at the present stage of the universe, the contraction parameter R from the dS to the Poincare algebra is very large and, in the formal limit R→∞, one IR of the dS algebra splits into two IRs of the Poincare algebra corresponding to a particle and its antiparticle with the same masses. The problem of why the quantities (c,ℏ,R) are as are does not arise because they are contraction parameters for transitions from more general Lie algebras to less general ones. Then the baryon asymmetry of the universe problem does not arise. At the present stage of the universe, the phenomenon of cosmological acceleration (PCA) is described without uncertainties as an inevitable kinematical consequence of quantum theory in semiclassical approximation. In particular, it is not necessary to involve dark energy the physical meaning of which is a mystery. In our approach, background space and its geometry are not used and R has nothing to do with the radius of dS space. In semiclassical approximation, the results for the PCA are the same as in General Relativity if Λ=3/R2, i.e., Λ>0 and there is no freedom for choosing the value of Λ.

Nodal superconductivity in miassite Rh17S15

Solid state chemistry has produced a plethora of materials with properties not found in nature. For example, high-temperature superconductivity in cuprates is drastically different from the superconductivity of naturally occurring metals and alloys and is frequently referred to as unconventional. Unconventional superconductivity is also found in other synthetic compounds, such as iron-based and heavy-fermion superconductors. Here, we report compelling evidence of unconventional nodal superconductivity in synthetic samples of Rh17S15 (Tc = 5.4 K), which is also found in nature as the mineral miassite. We investigated the temperature-dependent variation of the London penetration depth Δλ(T) and the disorder evolution of the critical superconducting temperature Tc and the upper critical field Hc2(T) in single crystalline Rh17S15. We found a T − linear temperature variation of Δλ(T) below 0.3Tc, which is consistent with the presence of nodal lines in the superconducting gap of Rh17S15. The nodal character of the superconducting state is supported by the observed suppression of Tc and Hc2(T) in samples with a controlled level of non-magnetic disorder introduced by 2.5 MeV electron irradiation. We propose a nodal sign-changing superconducting gap in the A1g irreducible representation, which preserves the cubic symmetry of the crystal and is in excellent agreement with the derived superfluid density. To the best of our knowledge, this establishes miassite as the only mineral known so far that reveals unconventional superconductivity in its clean synthetic form, though it is unlikely that it is present in natural crystals because of unavoidable impurities that quickly destroy nodal superconductivity.

Magnetic structure and pressure-induced spin-crossover in Me3B2O6 (Me = Mn, Fe, Co, Ni) kotoites: Representation analysis and DFT calculations

The magnetic and electronic properties of transition metal borates Me3B2O6 (Me = Mn, Fe, Co, Ni) with kotoite structure have been investigated at ambient and high pressures via a combination of representation analysis and density functional theory (DFT + U) calculations. Several magnetic configurations corresponding to the different irreducible representations have been considered. The total-energy calculations reveal that the magnetic ground state of Me3B2O6 kotoites is composition-dependent. The lowest energy magnetic phase of manganese and nickel kotoites is characterized by the antiferromagnetic ordering of the transition metal magnetic moments along the c- axis and along the b-axis for cobalt and iron kotoites. The magnetic cell of Ni3B2O6 kotoite corresponds to k = (1/2, 0, 1/2) vector and four time larger than the unit cell. The calculated exchange constants indicate the competition between ferromagnetic and antiferromagnetic interactions. At a critical pressure, Me ions undergo a high-spin to low-spin state crossover. This magnetic moments collapse is analyzed in terms of change in electronic structure under pressure.

Understanding Self-Supervised Learning of Speech Representation via Invariance and Redundancy Reduction

Self-supervised learning (SSL) has emerged as a promising paradigm for learning flexible speech representations from unlabeled data. By designing pretext tasks that exploit statistical regularities, SSL models can capture useful representations that are transferable to downstream tasks. Barlow Twins (BTs) is an SSL technique inspired by theories of redundancy reduction in human perception. In downstream tasks, BTs representations accelerate learning and transfer this learning across applications. This study applies BTs to speech data and evaluates the obtained representations on several downstream tasks, showing the applicability of the approach. However, limitations exist in disentangling key explanatory factors, with redundancy reduction and invariance alone being insufficient for factorization of learned latents into modular, compact, and informative codes. Our ablation study isolated gains from invariance constraints, but the gains were context-dependent. Overall, this work substantiates the potential of Barlow Twins for sample-efficient speech encoding. However, challenges remain in achieving fully hierarchical representations. The analysis methodology and insights presented in this paper pave a path for extensions incorporating further inductive priors and perceptual principles to further enhance the BTs self-supervision framework.

Scalar two-point functions at the late-time boundary of de Sitter

We calculate two-point functions of scalar fields of mass m and their conjugate momenta at the late-time boundary of de Sitter with Bunch-Davies boundary conditions, in general d + 1 spacetime dimensions. We perform the calculation using the wavefunction picture and using canonical quantization. With the latter one clearly sees how the late-time field and conjugate momentum operators are linear combinations of the normalized late-time operators αN and βN that correspond to unitary irreducible representations of the de Sitter group with well-defined inner products. The two-point functions resulting from these two different methods are equal and we find that both the autocorrelations of αN and βN and their cross correlations contribute to the late-time field and conjugate momentum two-point functions. This happens both for light scalars m<d2H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left(m<\\frac{d}{2}H\\right) $$\\end{document}, corresponding to complementary series representations, and heavy scalars m>d2H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left(m>\\frac{d}{2}H\\right) $$\\end{document}, corresponding to principal series representations of the de Sitter group, where H is the Hubble scale of de Sitter. In the special case m = 0, only the βN autocorrelation contributes to the conjugate momentum two-point function in any dimensions and we gather hints that suggest αN to correspond to discrete series representations for this case at d = 3.

Urban Green Space Usage and Life Satisfaction During the Covid-19 Pandemic

Whereas research in Germany has focused mainly on how environmental burdens such as noise or air pollution affect health or well-being, little is known about the effects of environmental goods. Our study examines urban green spaces as one such environmental good, and asks how their usage affected life satisfaction during the coronavirus pandemic. Drawing on attention restoration and stress reduction theory as well as on theories of social integration, we further ask how different kinds of activities such as meeting friends or doing sports mediate the effects of green spaces on life satisfaction. Our empirical analysis builds on unique individual panel data for residents of Hannover and Braunschweig taken from interviews before and during the coronavirus pandemic. Using causal difference-in-difference models, our results reveal no robust effect of increased or decreased green space usage on life satisfaction. However, results depend on the operationalization: Analyses point to a negative effect of reduced green space visits only when green space use is assessed retrospectively. Hence, our study highlights the need for further causality-oriented research on the relationship between green space usage and life satisfaction. Moreover, it stresses the value of performing robustness checks by applying alternative operationalizations in causal analyses.

Twisted conjugacy in residually finite groups of finite Prüfer rank

Abstract Suppose 𝐺 is a residually finite group of finite upper rank admitting an automorphism 𝜑 with finite Reidemeister number R ⁢ ( φ ) R(\varphi) (the number of 𝜑-twisted conjugacy classes). We prove that such a 𝐺 is soluble-by-finite (in other words, any residually finite group of finite upper rank that is not soluble-by-finite has the R ∞ R_{\infty} property). This reduction is the first step in the proof of the second main theorem of the paper: suppose 𝐺 is a residually finite group of finite Prüfer rank and 𝜑 is its automorphism. Then R ⁢ ( φ ) R(\varphi) (if it is finite) is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations of 𝐺, which are fixed points of the dual map φ ̂ : [ ρ ] ↦ [ ρ ∘ φ ] \hat{\varphi}\colon[\rho]\mapsto[\rho\circ\varphi] (i.e. we prove the TBFT𝑓, the finite version of the conjecture about the twisted Burnside–Frobenius theorem, for such groups).

An Analogue of Ladder Representations for Classical Groups

Abstract In this paper, we introduce a notion of ladder representations for split odd special orthogonal groups and symplectic groups over a non-archimedean local field of characteristic zero. This is a natural class in the admissible dual, which contains both strongly positive discrete series representations and irreducible representations with irreducible $A$-parameters. We compute Jacquet modules and the Aubert duals of ladder representations, and we establish a formula for describing ladder representations in terms of linear combinations of standard modules.

An Integrated Model Based Prediction of Machining Accuracy for Milling Machine

The accuracy of machining can be predicted during the design process of CNC machine tools, while the prediction accuracy depends on the prediction model. However, when establishing the prediction model, the coupling effect among the machine tool’s different components is always ignored for modeling efficiency, so this will lead to its reliability cannot be guaranteed. To address this problem, the model reduction theory and the lumped parameter method are used in this study to set the state-space equations of the essential structural parts. Moreover, the electro-mechanical coupling parameter equation of the servo system is transformed into a state-space equation. Then the state-space equations of the components and the servo system are coupled to establish the machine tool dynamic model. Furthermore, a theoretical model of instantaneous undeformed chip thickness is found, and according to the linear relationship between the micro-cutting force and the instantaneous undeformed chip thickness, the instantaneous cutting force model of the cylindrical milling cutter is established. Based on these, the integrated model, which considers the mechanical structure, control system and cutting force, is obtained and further used in Simulink to build the machining accuracy prediction platform, as well as the accuracy of the integrated prediction model is verified. Moreover, the prediction model established by the integrated modeling method can effectively simulate the actual machining conditions of the machine tool, and the superiority of prediction is confirmed by comparing the simulated and measured results in peripheral milling applications. The results show the error in the system response efficiency is 14.8%, while the error in position accuracy and delay characteristics is less than 10%, the error between the predicted size of the prediction model and the actual size is within 28 μm, a minimum prediction error is 7 μm.

How are newcomer proactive behaviors received by leaders and peers? A relational perspective.

Although most studies have shown that newcomers benefit from proactive behaviors, these behaviors are not always viewed positively by colleagues, resulting in negative consequences for newcomers. Drawing on uncertainty reduction and social cognitive theories, we contend that newcomer proactive behaviors are viewed positively by competent leaders and peers but negatively by those with low competence. Further, we argue that newcomer proactive behaviors impact leader and peer threat perceptions, affecting subsequent workplace relationships, which in turn influence newcomer voluntary turnover. We empirically test our hypotheses in a three-time multisource study, utilizing a sample of 377 newcomers, 132 leaders, and 721 peers. Supporting our hypotheses, when leaders and peers are less competent, newcomer proactive behaviors, through impacting leader and peer threat perceptions, result in lower quality relationships with leaders and peers. These cascading effects positively correlate with newcomer voluntary turnover. Conversely, opposite effects arise for more competent leaders and peers. Implications of how newcomer proactive behaviors impact workplace relationships and turnover are discussed. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Finite dimensional irreducible representations of Lie superalgebra D (2, 1; α)

This paper focuses on the finite dimensional irreducible representations of Lie superalgebra D(2, 1; α). We explicitly construct the finite dimensional representations of the superalgebra D(2, 1; α) by using the shift operator and differential operator representations. Unlike ordinary Lie algebra representation, there are typical and atypical representations for most superalgebras. Therefore, its typical and atypical representation conditions are also given. Our results are expected to be useful for the construction of primary fields of the corresponding current superalgebra of D(2, 1; α).

Row–column duality and combinatorial topological strings

Integrality properties of partial sums over irreducible representations, along columns of character tables of finite groups, were recently derived using combinatorial topological string theories (CTST). These CTST were based on Dijkgraaf-Witten theories of flat G-bundles for finite groups G in two dimensions, denoted G-TQFTs. We define analogous combinatorial topological strings related to two dimensional topological field theories (TQFTs) based on fusion coefficients of finite groups. These TQFTs are denoted as R(G)-TQFTs and allow analogous integrality results to be derived for partial row sums of characters over conjugacy classes along fixed rows. This relation between the G-TQFTs and R(G)-TQFTs defines a row-column duality for character tables, which provides a physical framework for exploring the mathematical analogies between rows and columns of character tables. These constructive proofs of integrality are complemented with the proof of similar and complementary results using the more traditional Galois theoretic framework for integrality properties of character tables. The partial row and column sums are used to define generalised partitions of the integer row and column sums, which are of interest in combinatorial representation theory.

Collaboration as a Village Self-Reliance Ecosystem

This study aims to analyze the process of collaboration between the village government, community, and interest groups in achieving village self-reliance programs in Ketapang Village, Indonesia. Using descriptive analysis qualitative research, the study employed purposive sampling to select the village head and village secretary as informants. Data collection techniques included observation and interviews, and data analysis followed Miles and Hubermen's theory of data collection, reduction, presentation, and conclusion. The study used Collaborative Governance theory from Ansell and Gash to measure the questions in the indicators. The results showed that the collaboration between actors was successful, resulting in the construction of the Kalifor tourism village icon as a UMKM development area. The program had a positive impact on economic and social resilience in Ketapang Village. The study highlights the importance of collaborative governance in achieving village self-reliance programs and provides insights for policymakers and practitioners in Indonesia and beyond Highlight : Collaborative Governance: The study used Collaborative Governance theory from Ansell and Gash to measure the success of the collaboration between actors in Ketapang Village. Village Self-Reliance: The study aimed to analyze the process of collaboration between the village government, community, and interest groups in achieving village self-reliance programs in Ketapang Village. Economic and Social Resilience: The program had a positive impact on economic and social resilience in Ketapang Village, highlighting the importance of collaborative governance in achieving village self-reliance programs. Keywords: Collaborative Governance, Village Self-Reliance, Ketapang Village, Indonesia, Economic and Social Resilience

Non-Strict Plurisubharmonicity of Energy on Teichmüller Space

Abstract For an irreducible representation $\rho :\pi _{1}(\Sigma _{g})\to \textrm{GL}(n,\mathbb{C})$, there is an energy functional $\textrm{E}_{\rho }: {{\mathcal{T}}}_{g}\to \mathbb{R}$, defined on Teichmüller space by taking the energy of the associated equivariant harmonic map into the symmetric space $\textrm{GL}(n,\mathbb{C})/\textrm{U}(n)$. It follows from a result of Toledo that $\textrm{E}_{\rho }$ is plurisubharmonic, that is, its Levi form is positive semi-definite. We describe the kernel of this Levi form, and relate it to the $\mathbb{C}^{*}$ action on the moduli space of Higgs bundles. We also show that the points in $ {{\mathcal{T}}}_{g}$ where strict plurisubharmonicity fails (i.e., this kernel is non-zero) are critical points of the Hitchin fibration. When $n\geq 2$ and $g\geq 3$, we show that for a generic choice $(S,\rho )$, the map $\textrm{E}_{\rho }$ is strictly plurisubharmonic. We also describe the kernel of the Levi form for $n=1$.

Functional equations and gamma factors of local zeta functions for the metaplectic cover of SL2

We introduce a local zeta-function for an irreducible admissible supercuspidal representation π of the metaplectic double cover of SL2 over a non-archimedean local field of characteristic zero. We prove a functional equation of the local zeta-functions showing that the gamma factor is given by a Mellin type transform of the Bessel function of π. We obtain an expression of the gamma factor, which shows its entireness on C. Moreover, we show that, through the local theta-correspondence, the local zeta-function on the covering group is essentially identified with the local zeta-integral for spherical functions on PGL2≅SO3 associated with the prehomogenous vector space of symmetric matrices of degree 2.

Minimum codimension of eigenspaces in irreducible representations of simple classical linear algebraic groups

Let k be an algebraically closed field of characteristic p ≥ 0 , let G be a simple simply connected classical linear algebraic group of rank l and let T be a maximal torus in G with rational character group X ( T ) . For a nonzero p-restricted dominant weight λ ∈ X ( T ) , let V be the associated irreducible kG-module. We define ν G ( V ) as the minimum codimension of any eigenspace on V for any non-central element of G. In this paper, we determine lower-bounds for ν G ( V ) for G of type A l and dim ( V ) ≤ l 3 2 , and for G of type B l , C l , or D l and dim ( V ) ≤ 4 l 3 . Moreover, we give the exact value of ν G ( V ) for G of type A l with l ≥ 15 ; for G of type B l or C l with l ≥ 14 ; and for G of type D l with l ≥ 16 .

A certain twisted Jacquet module of GL(6) over a finite field: The rank 2 case

Let F be a finite field and G=GL(6,F). In this paper, we explicitly describe the structure of the twisted Jacquet module πN,ψA where A is a rank 2 matrix and π is an irreducible cuspidal representation of G.

Robust Sparse Sufficient Dimension Reduction via Adaptive Lasso Penalty

In some multi-index models applications, there is an important role for dimension reduction and variable selection (VS) methods. The ALMAVE is a method for variables selection under sufficient dimension reduction theory settings. It combines the adaptive Lasso and MAVE (minimum average variance estimation) to produce sparse and accurate solutions. The ALMAVE is a very sensitive method to outliers in the response y due to use the least-squares criterion. In this article, we proposed robust ALMAVE. Also, an efficient estimation algorithm was proposed. The simulation studies and Logo design data analysis were employed to check the effectiveness of ALMAVE.