Arc Space Research Articles

Standard monomials and invariant theory for arc spaces III: Special linear group

This is the third in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field K, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the special linear group. This is more subtle than the results for the general linear and symplectic groups obtained in the first two papers because the arc space of the corresponding affine quotients can be nonreduced.

Geotechnical performance of X-section cast-in-place concrete piles under torsion loads in clay soil – numerical study

ABSTRACT Deep foundations are inexorably subjected to torsional load due to eccentric lateral loads. The conventional piles face shortcoming under torsional loads due to its fair face. So, this study employed an innovative X-section cast in place concrete pile to improvement the torsional capacity of piles. PLAXIS 3D V20 Software was used to study the performance of XCC pile under torsional load, and combined torsional, and vertical loads in clay soil. The numerical results reflect that at the ratio of pile length to pile diameter (Lp/Dp) = 15, and the ratio of the open arc spacing to pile diameter (ap/Dp) = 0.14, open arc degrees (θ°) = 90, the torsional capacity can be improved by 3.96, 5.06, and 5.71 times of torsional capacity of conventional pile with the same cross-sectional area at clay cohesions of 70 KN/m2, 40 KN/m2, and 30 KN/m2, respectively. Also, The XCC pile recorded an increase ratio of 35% in the vertical capacity of piles compared to conventional pile with the same cross-sectional area. It can be concluded that XCC piles exhibit significant resistance to torsional forces due to their cross-sectional geometry which enhances soil-pile interaction.

Transient characteristics of ultra-high frequency adjustable multi-pulse gas tungsten welding arc

In response to the shortcomings of traditional Gas Tungsten Arc Welding (GTAW), such as low penetration, slow welding speed, and low production efficiency, an Ultra-High-Frequency Adjustable Multi-Pulse Gas Tungsten Arc Welding (UFMP-GTAW) process is proposed. This process more effectively concentrates the temperature distribution of the welding arc. To characterize the arc morphology and its dynamic changes, high-speed photography captured ten images of the UFMP-GTAW welding arc within one cycle. The arc image processing algorithm proposed in the paper was used to study the arc's variation characteristics over one cycle. This algorithm includes binarization, arc segmentation, and edge detection to obtain arc morphology parameters. The arc symmetry algorithm is used to symmetrize the arc images for Abel inversion, resulting in the emission coefficient distribution. Further calculations provide the temperature distribution, electrical conductivity distribution, and current density distribution of the arc. Analysis reveals that changes in arc temperature, electrical conductivity, and current density lag behind changes in current. The temperature of the arc spreads from the central axis to the surrounding arc space. When high-frequency current variations occur, the heat input generated by the current increase takes time to transfer to the entire arc space. Conversely, if the current suddenly decreases, the heat input rapidly diminishes. However, because heat transfer takes time, the arc space remains relatively stable for a short period, making the ultra-high-frequency arc form more stable.

Standard monomials and invariant theory of arc spaces II: Symplectic group

This is the second in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field K K , we construct a standard monomial basis for the arc space of the Pfaffian variety over K K . As an application, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the symplectic group.

Multiplicity structure of the arc space of a fat point

Multiplicity structure of the arc space of a fat point

The Formation Mechanism and Suppression Strategies of Spatter in Pulsed Gas Metal Arc Welding for Titanium Alloy

The GMAW process for titanium alloy is not commonly applied within the industry due to the occurrence of severe spatter. This research endeavors to elucidate the mechanism underlying spatter formation and explore efficacious strategies to suppress spatter. The experimental results demonstrated the existence of two distinct spatter types: large and small spatter particles. The high- speed images and synchronous electrical signals were utilized for determining the spatter formation mechanism, with force analysis serving to mutually validate the inferences. The large spatter particles originated from the whole transitional molten droplet as it descended within the arc space, while the small spatter particles were formed by the partial transitional molten droplet as it contacted the weld pool. The cathode jet force accounted for the formation of large spatter particles, whereas the electromagnetic force was responsible for the small spatter particles. To suppress spatter, increasing detachment current and decreasing pulsing frequency were employed. Consequently, the spatter rate witnessed a remarkable decrease from 14.00% to 3.33% with a progressive increment in detachment current from 100 A to 300 A, and a corresponding decline from 12.67% to 1.33% upon decrementing the pulsing frequency from 90 Hz to 50 Hz. This research suggests that a judicious increase in the detachment current can effectively decrease the spatter rate while concurrently preserving welding efficiency.

Electric Arc Furnace Electrode Movement Control System Based on a Fuzzy Arc Length Identifier

From the point of view of the synthesis of control influences, arc steelmaking furnaces are complex nonlinear objects with strongly expressed mutual influences. It has been demonstrated that at a given supply voltage, the distribution of the current values of the phase currents in the quasi-steady-state mode makes it possible to estimate the situation in the arc space of an arc steelmaking furnace and identify the value of arc lengths. This dependence is preserved in transient modes. In order to identify arc lengths from the phase currents, it is proposed to use an approach based on the theory of fuzzy sets. The construction of the fuzzy arc length identifier rule base was carried out in two stages: the first stage used data from quasi-steady-state modes and the Wang–Mendel algorithm; the second stage involved adding a new rule to the database if the activity level of the formed rules was lower than the established level α for the data obtained in the dynamic mode. Further optimization of the parameters of the fuzzy identifier for operation in dynamic modes was carried out using the “back-propagation” algorithm. Based on the identified values of arc lengths, a control system for the movement of electrodes in an arc steelmaking furnace was synthesised. The proposed control system makes it possible to eliminate unproductive electrode movements due to changes in the situation in other phases of the arc steelmaking furnace and simplifies the application of modern methods of synthesising a control system for such complex objects. The results obtained in the mathematical model have confirmed the effectiveness of the proposed control system for the movement of electrodes in an arc steelmaking furnace.

The mechanism study of TIG-MIG hybrid welding process based on simulation

Tungsten inert gas-metal inert gas (TIG-MIG) hybrid welding combines the advantages of tungsten inert gas (TIG) welding and metal inert gas (MIG) welding. Due to the benefits of the two welding techniques, it is an effective strategy to enhance welding quality and productivity. To investigate the mechanism of arcs interaction in TIG-MIG hybrid welding, a three-dimensional transient model for arc-droplet-weld pool behavior was established. The temperature field, metal vapor content, flow fluid, current density and electromagnetic force in the arc space and weld pool were simulated and analyzed. The calculation results showed that the introduction of the TIG arc decreased the content of metal vapor and increased the arc temperature. Compared with single MIG welding, the amplitudes of maximum arc temperature and plasma velocity in TIG-MIG hybrid welding were lower. Experimental results demonstrated that the droplet deflection angle, droplet transfer frequency, weld bead cross-section, and arc temperature distribution all agreed well with calculated values.

Arc spaces and wedge spaces for toric varieties

Arc spaces and wedge spaces for toric varieties

Standard monomials and invariant theory for arc spaces I: General linear group

This is the first in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field [Formula: see text], we construct a standard monomial basis for the arc space of the determinantal variety over [Formula: see text]. As an application, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the general linear group.

Cosets of Free Field Algebras via Arc Spaces

AbstractUsing the invariant theory of arc spaces, we find minimal strong generating sets for certain cosets of affine vertex algebras inside free field algebras that are related to classical Howe duality. These results have several applications. First, for any vertex algebra ${{\mathcal {V}}}$, we have a surjective homomorphism of differential algebras $\mathbb {C}[J_{\infty }(X_{{{\mathcal {V}}}})] \rightarrow \text {gr}^{F}({{\mathcal {V}}})$; equivalently, the singular support of ${{\mathcal {V}}}$ is a closed subscheme of the arc space of the associated scheme $X_{{{\mathcal {V}}}}$. We give many new examples of classically free vertex algebras (i.e., this map is an isomorphism), including $L_{k}({{\mathfrak {s}}}{{\mathfrak {p}}}_{2n})$ for all positive integers $n$ and $k$. We also give new examples where the kernel of this map is nontrivial but is finitely generated as a differential ideal. Next, we prove a coset realization of the subregular ${{\mathcal {W}}}$-algebra of ${{\mathfrak {s}}}{{\mathfrak {l}}}_{n}$ at a critical level that was previously conjectured by Creutzig, Gao, and the 1st author. Finally, we give some new level-rank dualities involving affine vertex superalgebras.

Unsupervised anomalous sound detection for industrial monitoring based on ArcFace classifier and gaussian mixture model

The operation state of machine can be monitored by performing anomalous sound detection (ASD). Unsupervised-ASD is a detection task in which the model detects unknown anomalous sounds without the use of anomalous sounds to train it. However, when detecting completely unknown anomalous samples, it is challenging to classify samples with high similarities and determine decision boundaries, leading to the poor detection performance. In light of the above deficiencies, we propose the ArcFace classifier and Gaussian mixture model (GMM) based unsupervised-ASD method. The Arcface loss-based classifier is proposed to aggregate the hidden features of different classes into the corresponding arc space, which increases the separability of samples. The GMM-based anomaly score calculation is presented to determine the more complex decision boundary. Experiments are carried out on the datasets provided by DCASE 2020 Task 2 and CRWU. As demonstrated by the areas under the receiver operating characteristic curve (AUC) and the partial AUC (pAUC), the proposed method has better performance compared with other methods in comparison.

Deformations of arcs and comparison of formal neighborhoods for a curve singularity

Abstract Let $ {\mathcal C} $ be an algebraic curve and c be an analytically irreducible singular point of ${\mathcal C}$ . The set ${\mathscr {L}_{\infty }}({\mathcal C})^c$ of arcs with origin c is an irreducible closed subset of the space of arcs on ${\mathcal C}$ . We obtain a presentation of the formal neighborhood of the generic point of this set which can be interpreted in terms of deformations of the generic arc defined by this point. This allows us to deduce a strong connection between the aforementioned formal neighborhood and the formal neighborhood in the arc space of any primitive parametrization of the singularity c. This may be interpreted as the fact that analytically along ${\mathscr {L}_{\infty }}({\mathcal C})^c$ the arc space is a product of a finite dimensional singularity and an infinite dimensional affine space.

Formal Neighbourhoods and Formal Models of Non-Degenerate Arcs

An important result about the geometry of the arc space of an algebraic variety is the theorem of Drinfeld–Grinberg–Kazhdan, representing the formal neighbourhood of a non-degenerate arc. We start with a brief review of some important results and notions. Then the complete proof of the theorem with examples is given. A generalization to the relative case is discussed for smooth and ´etale morphisms.

Small irreducible components of arc spaces in positive characteristic

We study the Irreducibility Problem for the arc space X ∞ of an irreducible singular algebraic variety X defined over a perfect field k of characteristic p > 0 . The existence of reducible such X ∞ is related to the fact that Kolchin's Irreducibility Theorem fails to hold in positive characteristic. We obtain a complete description of the irreducible components of X ∞ when X is a surface. Section 5 introduces two main new problems in arbitrary dimension: (1) in the line of O. Zariski and H. Hironaka, blowing up any X to get Y → X with Y ∞ irreducible; (2) in the line of J. Nash's work on arcs, characterizing irreducibility in terms of Resolution of Singularities.

Reduced arc schemes for Veronese embeddings and global Demazure modules

We consider arc spaces for the compositions of Plücker and Veronese embeddings of the flag varieties for simple Lie groups of types ADE. The arc spaces are not reduced and we consider the homogeneous coordinate rings of the corresponding reduced schemes. We show that each graded component of a homogeneous coordinate ring is a cocyclic module over the current algebra and is acted upon by the algebra of symmetric polynomials. We show that the action of the polynomial algebra is free and that the fiber at the special point of a graded component is isomorphic to an affine Demazure module whose level is the degree of the Veronese embedding. In type A1 (which corresponds to the Veronese curve), we give the precise list of generators of the reduced arc space. In general type, we introduce the notion of global higher level Demazure modules, which generalizes the standard notion of the global Weyl modules, and identify the graded components of the homogeneous coordinate rings with these modules.

The Singular Support of the Ising Model

Abstract We prove a new quasiparticle sum expression for the character of the Ising model vertex algebra, related to the Jackson–Slater $q$-series identity of Rogers–Ramanujan type and to Nahm sums for the matrix $\left (\begin {smallmatrix}8&3\\3&2 \end {smallmatrix}\right ) $. We find, as consequences, an explicit monomial basis for the Ising model and a description of its singular support. We find that the ideal sheaf of the latter, defining it as a subscheme of the arc space of its associated scheme, is finitely generated as a differential ideal. We prove three new $q$-series identities of the Rogers–Ramanujan–Slater type associated with the three irreducible modules of the Virasoro Lie algebra of central charge $1/2$. We give a combinatorial interpretation to the identity associated with the vacuum module.

GROTHENDIECK RINGS OF DEFINABLE SUBASSIGNMENTS AND EQUIVARIANT MOTIVIC MEASURES

This paper studies categories of definable subassignments with some category equivalences to semi-algebraic and constructible subsets of arc spaces of algebraic varieties. These equivalences lead to the identity of certain Grothendieck rings, which allows us to compare the motivic measure of Cluckers-Loeser with that of Denef-Loeser for certain classes of definable subassignments.

Embedding codimension of the space of arcs

Abstract We introduce a notion of embedding codimension of an arbitrary local ring, establish some general properties and study in detail the case of arc spaces of schemes of finite type over a field. Viewing the embedding codimension as a measure of singularities, our main result can be interpreted as saying that the singularities of the arc space are maximal at the arcs that are fully embedded in the singular locus of the underlying scheme, and progressively improve as we move away from said locus. As an application, we complement a theorem of Drinfeld, Grinberg and Kazhdan on formal neighbourhoods in arc spaces by providing a converse to their theorem, an optimal bound for the embedding codimension of the formal model appearing in the statement, a precise formula for the embedding dimension of the model constructed in Drinfeld’s proof and a geometric meaningful way of realising the decomposition stated in the theorem.

Higher Derivations of Modules and the Hasse–Schmidt Module

In this paper we revisit Ribenboim’s notion of higher derivations of modules and relate it to the recent work of de Fernex and Docampo on the sheaf of differentials of the arc space. In particular, we derive their formula for the Kähler differentials of the Hasse–Schmidt algebra as a consequence of the fact that the Hasse–Schmidt algebra functors commute.