Dimension Formula Research Articles

Anomalous dimensions via on-shell methods: Operator mixing and leading mass effects

We elaborate on the application of on shell and unitarity-based methods for evaluating renormalization group coefficients, and generalize this framework to account for the mixing of operators with different dimensions and leading mass effects. We derive a master formula for anomalous dimensions stemming from the general structure of operator mixings, up to two-loop order, and show how the Higgs low-energy theorem can be exploited to include leading mass effects. A few applications on the renormalization properties of popular effective field theories showcase the strength of the proposed approach, which drastically reduces the complexity of standard loop calculations. Our results provide a powerful tool to interpret experimental measurements of low-energy observables, such as flavor violating processes or electric and magnetic dipole moments, as induced by new physics emerging above the electroweak scale. Published by the American Physical Society 2024

Young walls and graded dimension formulas for finite quiver Hecke algebras of types A2n−1(2), Dn(1) and Bn(1)

In this paper, we establish the graded dimension formulas of finite quiver Hecke algebras using level-[Formula: see text] Fock space representations of quantum affine algebras of types [Formula: see text], [Formula: see text] and [Formula: see text].

On the Dimension of Limit Sets on ℙ(ℝ3) via Stationary Measures: Variational Principles and Applications

Abstract This paper investigates the (semi)group action of $\textrm{SL}_{n}({\mathbb R})$ on ${\mathbb P}({\mathbb R}^{n})$, a primary example of non-conformal, non-linear, and non-strictly contracting action. We establish variational principles of the affinity exponent for two main examples: the Borel Anosov representations and the Rauzy gasket. In [ 32], they obtain a dimension formula for the stationary measures on ${\mathbb P}({\mathbb R}^{3})$. Combined with our result, it allows us to study the Hausdorff dimension of limit sets of Anosov representations in $\textrm{SL}_{3}({\mathbb R})$ and the Rauzy gasket. It yields the equality between the Hausdorff dimensions and the affinity exponents in both settings, generalizing the classical Patterson–Sullivan formula. In the appendix, we improve the numerical lower bound of the Hausdorff dimension of Rauzy gasket to $1.5$.

Are there fractals in sign language?

Sign language is analyzed from the perspective of Menzerath-Altmann law (MAL). In reply to a question posed in the title, this study involves a comprehensive fractal analysis of a concrete Czech sign language utterance at multiple linguistic levels. Through this examination, a novel linguistic level, the so-called “pseudosyllables”, is unearthed and subjected to thorough investigation. While the results of this experimental exploration do not conclusively confirm fractal structure in sign language overall via an isomorphism between the fractal dimension formula and MAL, we undeniably detect the existence of pseudofractal objects of sign language in the sense of the modified (whence the prefix “pseudo”) conjecture by Hřebíček on the basis of our arguments, particularly linked to this newly identified linguistic level.

Dimensions of Level-1 Group-Based Phylogenetic Networks

Phylogenetic networks represent evolutionary histories of sets of taxa where horizontal evolution or hybridization has occurred. Placing a Markov model of evolution on a phylogenetic network gives a model that is particularly amenable to algebraic study by representing it as an algebraic variety. In this paper, we give a formula for the dimension of the variety corresponding to a triangle-free level-1 phylogenetic network under a group-based evolutionary model. On our way to this, we give a dimension formula for codimension zero toric fiber products. We conclude by illustrating applications to identifiability.

Machine Learning Assisted Exploration for Affine Deligne–Lusztig Varieties

AbstractThis paper presents a novel, interdisciplinary study that leverages a Machine Learning (ML) assisted framework to explore the geometry of affine Deligne–Lusztig varieties (ADLV). The primary objective is to investigate the non-emptiness pattern, dimension, and enumeration of irreducible components of ADLV. Our proposed framework demonstrates a recursive pipeline of data generation, model training, pattern analysis, and human examination, presenting an intricate interplay between ML and pure mathematical research. Notably, our data-generation process is nuanced, emphasizing the selection of meaningful subsets and appropriate feature sets. We demonstrate that this framework has a potential to accelerate pure mathematical research, leading to the discovery of new conjectures and promising research directions that could otherwise take significant time to uncover. We rediscover the virtual dimension formula and provide a full mathematical proof of a newly identified problem concerning a certain lower bound of dimension. Furthermore, we extend an open invitation to the readers by providing the source code for computing ADLV and the ML models, promoting further explorations. This paper concludes by sharing valuable experiences and highlighting lessons learned from this collaboration.

Research on Energy Loss of Optimization of Inducer–Impeller Axial Fit Dimensions Based on Wave-Piercing Theory

With the development of modern fluid machinery, the energy density of pumps is gradually being improved, and at the same time, higher demands are being placed on the cavitation performance, hence the introduction of the inducer and centrifugal impeller to form a dynamic–dynamic series structure. However, there are strict constraints on the axial size of pumps in fields such as firefighting and aerospace. The traditional empirical formula no longer satisfies the need to fit the axial dimensions between the induced wheel and the impeller at high velocities. Therefore, based on the wave-piercing theory, the drag reduction coefficient is introduced to explore the optimal axial fit size from the perspective of energy characteristics. This paper focuses on the influence of the inducer’s wake on the energy characteristics of downstream impellers, and conducts the following research: by adjusting the axial matching dimensions between the upstream inducer and the centrifugal impeller in the initial model, ten sets of axial distance models with matching dimensions of KD are designed, and the drag reduction coefficient is embedded to determine the optimal axial distance. The results show that the optimal axial distance is 0.2D, which is far lower than the axial distance value of 0.42D obtained from the traditional empirical formula for axial matching dimensions. Meanwhile, this paper uses tangential velocity, the inlet flow angle of the impeller, entropy production theory, and other indicators to analyze the internal energy loss of the high-speed vehicular fire pumps one by one. All of them confirm that the impeller in the high-speed vehicular fire pump has the lowest energy loss and optimal performance at an axial distance of 0.2D. Specifically, at this axial distance, the head can reach 259 m, and the hydraulic efficiency is as high as 83.62%. Thus, the feasibility of determining the axial placement of the impeller using the drag coefficient is validated. This research provides new insights into determining the axial coordination dimensions between the inducer and the impeller.

IMPLEMENTATION IN SСAS KIDYM OF SOFTWARE DETERMINATION OF DIMENSIONS OF UNKNOWNS IN PROBLEMS OF MECHANICS BASED ON ENERGY RELATIONS

The paper presents the results of the study of an algorithm for processing measurement units of physical and geometric quantities in the problems of mechanics implemented in the specialized computer algebra system (SCAS) KiDyM. The KiDyM software complex is used for solving problems of mechanics of engineering and scientific orientation of any complexity. For the completeness of the initial data preparation process for such tasks, the use of measurement units in the initial and resulting data is essential. The output data of KiDyM tasks have a rather slow appearance due to their ana- lytical form. To provide assistance to the user, KiDyM programs have a special diagnostic block of output data. Therefore, the implementation of the dimensions used in KiDyM data pursues the additional goal of increasing the diagnostic capabilities of the system. Dimensional formulas are imple- mented here as ordinary analytical expressions, which enables the built-in computer algebra system to form expressions of data measurement units during their computer analytical transformations in the process of solving problems of dynamics, statics and kinematics. Thus, working out the meas- urement units in the SCAS KiDyM includes reading the values of the variables with the dimensions specified in formula form the initial data, diagnos- ing the correctness of dimensionising and compliance of the dimensions with the data, storing them in instances of the "variable" and "element" classes, calculating the dimensions according to data conversion formulas to obtain calculation results. The paper demonstrates how to efficiently es- tablish dimensions of geometrical and physical quantities using energy relations between coordinates and characteristics of elements of the mechanical model. For the demonstration of the logic of the implemented algorithm, simple compact examples from educational problems of kinematics, statics, and dynamics of flat and spatial systems are provided. The article shows how KiDyM constructs the necessary equations for solving tasks, and how units of measurement for equation components and their solutions can be obtained. Since dimensions are represented as formulas here, their simplifica- tion – reduction and substitution with derivative units – can be easily implemented. Moreover, since the variables involved in dimension formulas and the variables of the task itself, within KiDyM, are situated in different data spaces, they can have the same names without any confusion.

Scale recurrence lemma and dimension formula for Cantor sets in the complex plane

Abstract We prove a multidimensional conformal version of the scale recurrence lemma of Moreira and Yoccoz [Stable intersections of regular Cantor sets with large Hausdorff dimensions. Ann. of Math. (2)154(1) (2001), 45–96] for Cantor sets in the complex plane. We then use this new recurrence lemma, together with Moreira’s ideas in [Geometric properties of images of Cartesian products of regular Cantor sets by differentiable real maps. Math. Z.303 (2023), 3], to prove that under the right hypothesis for the Cantor sets $K_1,\ldots ,K_n$ and the function $h:\mathbb {C}^{n}\to \mathbb {R}^{l}$ , the following formula holds: $$ \begin{align*}HD(h(K_1\times K_2 \times \cdots\times K_n))=\min \{l,HD(K_1)+\cdots+HD(K_n)\}.\end{align*} $$

Assouad type dimensions of infinitely generated self-conformal sets

We study the dimension theory of limit sets of iterated function systems consisting of a countably infinite number of conformal contractions. Our focus is on the Assouad type dimensions, which give information about the local structure of sets. Under natural separation conditions, we prove a formula for the Assouad dimension and prove sharp bounds for the Assouad spectrum in terms of the Hausdorff dimension of the limit set and dimensions of the set of fixed points of the contractions. The Assouad spectra of the family of examples which we use to show that the bounds are sharp display interesting behaviour, such as having two phase transitions. Our results apply in particular to sets of real or complex numbers which have continued fraction expansions with restricted entries, and to certain parabolic attractors.

Heron’s Formula in Higher Dimensions

This paper shows how geometric algebra can be used to derive a novel generalization of Heron’s classical formula for the area of a triangle in the plane to higher dimensions. It begins by illustrating some of the many ways in which the conformal model of three-dimensional Euclidean space yields provocative insights into some of our most basic intuitive notions of solid geometry. It then uses this conceptual framework to elucidate the geometric meaning of Heron’s formula in the plane, and explains in detail how it extends naturally to the volumes of tetrahedra in space. The paper closes by outlining a proof of a previously conjectured extension of the formula to the hyper-volumes of simplices in all dimensions.

The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula

For a Hermitian Lie group G of tube type we find the contribution of the holomorphic discrete series to the Plancherel decomposition of the Whittaker space L2(G/N,ψ), where N is the unipotent radical of the Siegel parabolic subgroup and ψ is a certain non-degenerate unitary character on N. The holomorphic discrete series embeddings are constructed in terms of generalized Whittaker vectors for which we find explicit formulas in the bounded domain realization, the tube domain realization and the L2-model of the holomorphic discrete series. Although L2(G/N,ψ) does not have finite multiplicities in general, the holomorphic discrete series contribution does.Moreover, we obtain an explicit formula for the formal dimensions of the holomorphic discrete series embeddings, and we interpret the holomorphic discrete series contribution to L2(G/N,ψ) as boundary values of holomorphic functions on a domain Ξ in a complexification GC of G forming a Hardy type space H2(Ξ,ψ).

Modular representations in type A with a two-row nilpotent central character

We study the category of representations of slm+2n over a field of characteristic p with p≫0, whose p-character is a nilpotent whose Jordan type is the two-row partition (m+n,n). In a previous paper with Anno, we used Bezrukavnikov-Mirkovic-Rumynin's theory of positive characteristic localization and exotic t-structures to give a geometric parametrization of the simples using annular crossingless matchings. Building on this, here we give combinatorial dimension formulae for the simple objects, and compute the Jordan-Hölder multiplicities of the simples inside the baby Vermas. We use Cautis-Kamnitzer's geometric categorification of the tangle calculus to study the images of the simple objects under the BMR equivalence. Our results generalize Jantzen's formulae in the subregular nilpotent case (i.e. when n=1), and may be viewed as a positive characteristic analogue of the combinatorial description for Kazhdan-Lusztig polynomials of Grassmannian permutations due to Lascoux and Schutzenberger.

Multiphysics analytical and numerical studies of biomolecule preconcentration utilizing ion concentration polarization: a case study of convergent microchannels.

A convergent sector in microfluidic devices utilizing ion concentration polarization (ICP) can help increase the preconcentration rate and the concentration enhancement factor (CEF) of biomolecules. In this work, we present a detailed study of the nozzle-like-squeeze effect of a convergent channel on the preconcentration of biomolecules. By numerically solving coupled Nernst-Planck-Poisson-Navier-Stokes governing equations for the 2D channel model, we report the first study on the critical width of a convergent region in the channel to retain the advantage of the nozzle-like-squeeze effect in speeding up preconcentration and augmenting CEF. In addition, we investigated the impact of the location and the dimensions of a convergent sector on the mechanism of biomolecule preconcentration. The location of an ion-selective membrane was also determined to ensure that biomolecules are focused on the convergent region of the channel. Moreover, we introduce analytical studies to compare and verify simulation findings. Specifically, the formulas for the critical dimensions of a convergent channel, location of a preconcentrated biomolecule plug, and position of an ion-selective membrane are presented. Furthermore, important working parameters, including electric potentials and hydrostatic pressures, were examined to scrutinize their effect on convergent concentrators. These detailed analytical solutions and numerical simulation results are consistent with experimental observations, providing deep insights into the ICP phenomenon and the preconcentration mechanism of biomolecules in convergent microfluidic concentration devices.

Mixed metric dimension over (edge) corona products

A subset S of V(G) is called a mixed resolving set for G if, for every two distinct elements x and y of V ( G )∪ E ( G ) , there exists v ∈ S such that d ( v , x )≠ d ( v , y ) . The mixed metric dimension of G, denoted by dim m ( G ) , is the minimum cardinality of a mixed resolving set in G. In this paper, a closed formula for the mixed metric dimension of corona product of graphs is proved. Also, a sharp upper bound and a closed formula for the mixed metric dimension of edge corona product of graphs is presented.

Microfabrication of Li4SiO4 pebbles with complex pore structure via digital light processing: Accuracy control and properties optimizing

Li4SiO4 ceramic pebble is deemed as the most attractive tritium breeder for fusion reactor blanket. In this work, we fabricated a novel Li4SiO4 pebble with micrometer-scale pore structure by digital light processing to verify the feasibility of complex pore structure manufacturing for future breeders. The effects of processing parameters on printing accuracy, microstructure and mechanical properties of Li4SiO4 pebbles were investigated. For synthesizing the pure Li4SiO4 phase, the optimal stoichiometric ratio of Li2SiO3 to Li2CO3 was 1:1.09 in the premixed powder. The photosensitivity of dual-phase ceramic suspension was investigated according to an attenuation model, and the overgrowth was reduced to ∼50 μm by using the dimension formula. After optimizing the post-processing, the radial pore structure Li4SiO4 pebbles with excellent crush load (50.3 N, standard deviation = 4.4 N) and suitable open porosity (18.3 %) can be yielded. This work provides a fresh solution to control the pore structure of tritium breeding pebbles in an unprecedented micrometer scale.

A variational principle in the parametric geometry of numbers

We extend the parametric geometry of numbers (initiated by Schmidt and Summerer, and deepened by Roy) to Diophantine approximation for systems of m linear forms in n variables, and establish a new connection to the metric theory via a variational principle that computes fractal dimensions of a variety of sets of number-theoretic interest. The proof of our variational principle relies on two novel ingredients: a variant of Schmidt's game capable of computing the Hausdorff and packing dimensions of any set, and the notion of templates, which generalize Roy's rigid systems. We use our variational principle to compute the Hausdorff and packing dimensions of the set of singular systems of linear forms and show they are equal, resolving a conjecture of Kadyrov, Kleinbock, Lindenstrauss and Margulis, as well as a question of Bugeaud, Cheung and Chevallier. As a corollary of Dani's correspondence principle, the divergent trajectories of a one-parameter diagonal action on the space of unimodular lattices with exactly two Lyapunov exponents with opposite signs have equal Hausdorff and packing dimensions. Other applications include quantitative strengthenings of theorems due to Cheung and Moshchevitin, which originally resolved conjectures due to Starkov and Schmidt respectively; as well as dimension formulas with respect to the uniform exponent of irrationality for simultaneous and dual approximation in two dimensions, completing partial results due to Baker, Bugeaud, Cheung, Chevallier, Dodson, Laurent and Rynne.

A convex analysis approach to the metric mean dimension: Limits of scaled pressures and variational principles

We introduce the concept of upper metric mean dimension of a one-parameter family of scaled pressure functions, which extends the corresponding notion for a single potential and satisfies a variational principle. This approach, supported by Convex Analysis, conveys a definition of measure-theoretic upper metric mean dimension, which is concave and upper semi-continuous, and therewith equilibrium states. In the context of dynamical systems, we establish a variational principle for the metric mean dimension with potential in terms of Katok entropy. As an application, we provide a simple formula for the upper metric mean dimension with potential for the shift on the space ([0,1]D)N, for every D∈N, which links mean dimension theory with ergodic optimization.

Influence of collar's shape on scour hole geometry at circular pier

Bridge collapses, which are frequently caused by scouring or other hydraulic complexities, have disastrous socioeconomically consequences. Accordingly, this paper offers clear water scour experimental investigation that assesses the impacts of circular and novel hexagonal collar shapes on scouring around circular piers which embedded in a uniform sand bed. Scour geometry was analyzed for both protected and unprotected piers, along with evaluating the percentage reductions of the scour hole dimensions and volume. The results demonstrate that the suggested collars have a considerable effect on separation, stagnation, and dissipation of the impinging downflow and vortices, resulting in a substantial impact on the scour geometry and significantly mitigates local scour. In conjunction with the laboratory data, empirical predicting formulas for scour dimensions and novel coefficients for the effect of collar shape are proposed. To incorporate a novel consideration of estimating scour depth at collared piers, the common prediction equations were updated by adding the newly developed collar shape coefficient. The proposed and upgraded formulas are compared and validated, and the findings demonstrate robust consistency.

Dimensions of projected sets and measures on typical self-affine sets

Let T1,…,Tm be a family of d×d invertible real matrices with ‖Ti‖<1/2 for 1≤i≤m. For a=(a1,…,am)∈Rmd, let πa:Σ={1,…,m}N→Rd denote the coding map associated with the affine IFS {Tix+ai}i=1m. We show that for every Borel probability measure μ on Σ, each of the following dimensions (lower and upper Hausdorff dimensions, lower and upper packing dimensions) of π⁎aμ is constant for Lmd-a.e. a∈Rmd, where π⁎aμ stands for the push-forward of μ by πa. In particular, we give a necessary and sufficient condition on μ so that π⁎aμ is exact dimensional for Lmd-a.e. a∈Rmd. Moreover, for every analytic set E⊂Σ, each of the Hausdorff, packing, lower and upper box-counting dimensions of πa(E) is constant for Lmd-a.e. a∈Rmd. Formal dimension formulas of these projected measures and sets are given. The Hausdorff dimensions of exceptional sets are estimated.