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Within the last decade, thin ultra-low velocity zone (ULVZ) layering, sitting directly on top of the core–mantle boundary (CMB), has begun to be investigated using the flip-reverse-stack (FRS) method. In this method, pre- and post-cursor arrivals that are symmetrical in time about the ScS arrival, but with opposite polarities, are stacked. This same methodology has also been applied to high velocity layering, with indications that ultra-high velocity zones (UHVZs) may also exist. Thus far, studies using the FRS technique have relied on 1-D synthetic predictions to infer material properties of ULVZs. 1-D ULVZ models predominantly show a SdS precursor that reflects off the top of the ULVZ and an ScscS reverberation within the ULVZ that arrives as a post-cursor. 1-D UHVZ models are more complex and have a different number of arrivals depending on a variety of factors including UHVZ thickness, velocity contrast, and lateral extent. 1-D modelling approaches assume that lower mantle heterogeneity is constant and continuous everywhere across the lower mantle. However, lower mantle features display lateral heterogeneity and are either finite in extent or display local thickness variations. We examine the interaction of the ScS wavefield with ULVZs and UHVZs in 2.5-D geometries of finite extent. We show that multiple additional arrivals exist that are not present in 1-D predictions. In particular, multipath ScS arrivals as well as additional post-cursor arrivals are generated. Subsequent processing by the FRS method generates complicated FRS traces with multiple peaks. Furthermore, post-cursor arrivals can be generated even when the ScS ray path does not directly strike the heterogeneity from above. Analysing these predictions for 2.5-D models using 1-D modelling techniques demonstrates that a cautious approach must be adopted in utilization and interpretation of FRS traces to determine if the ScS wavefield is interacting with a ULVZ or UHVZ through a direct strike on the top of the feature. In particular, traveltime delays or advances of the ScS arrival should be documented and symmetrical opposite polarity arrivals should be demonstrated to exist around ScS. The latter can be quantified by calculation of a time domain multiplication trace. Because multiple post-cursor arrivals are generated by finite length heterogeneities, interpretation should be confined to single layer models rather than to interpret the additional peaks as internal layering. Furthermore, strong trade-offs exist between S -wave velocity perturbation and thickness making estimations of ULVZ or UHVZ elastic parameters highly uncertain. We test our analysis methods using data from an event occurring in the Fiji-Tonga region recorded in North America. The ScS bounce points for this event sample the CMB region to the southeast of Hawaii, in a region where ULVZs have been identified in several recent studies. We see additional evidence for a ULVZ in this region centred at 14°N and 153°W with a lateral scale of at least 250 km × 360 km. Assuming a constant S -wave velocity decrease of −10 or −20 per cent with respect to the PREM model implies a ULVZ thickness of up to 16 or 9 km, respectively. [ABSTRACT FROM AUTHOR]

An affine spread is a set of subspaces of AG (n , q) of the same dimension that partitions the points of AG (n , q) . Equivalently, an affine spread is a set of projective subspaces of PG (n , q) of the same dimension which partitions the points of PG (n , q) \ H ∞ ; here H ∞ denotes the hyperplane at infinity of the projective closure of AG (n , q) . Let Q be a non-degenerate quadric of H ∞ and let Π be a generator of Q , where Π is a t-dimensional projective subspace. An affine spread P consisting of (t + 1) -dimensional projective subspaces of PG (n , q) is called hyperbolic, parabolic or elliptic (according as Q is hyperbolic, parabolic or elliptic) if the following hold: Each member of P meets H ∞ in a distinct generator of Q disjoint from Π ; Elements of P have at most one point in common; If S , T ∈ P , | S ∩ T | = 1 , then ⟨ S , T ⟩ ∩ Q is a hyperbolic quadric of Q . In this note it is shown that a hyperbolic, parabolic or elliptic affine spread of PG (n , q) is equivalent to a spread of Q + (n + 1 , q) , Q (n + 1 , q) or Q - (n + 1 , q) , respectively. [ABSTRACT FROM AUTHOR]

We develop a theory of quasi post-critically finite degenerations of Blaschke products. This gives us tools to study the boundaries of hyperbolic components of rational maps in higher-dimensional moduli spaces. We use it to obtain a combinatorial classification of geometrically finite polynomials on the boundary of the main hyperbolic component ℋd, i.e., the hyperbolic component in the space of monic and centered polynomials that contains zd. We also show that the closure ℋd is not a topological manifold with boundary for d ≥ 4 by constructing self-bumps on its boundary. [ABSTRACT FROM AUTHOR]

In several applications, ranging from computational geometry and finite element analysis to computer graphics, there is a need to approximate functions defined on triangular domains rather than rectangular ones. For this purpose, frequently used interpolation methods include barycentric interpolation, piecewise linear interpolation, and polynomial interpolation. However, the use of polynomial interpolation methods may suffer from the Runge phenomenon, affecting the accuracy of the approximation in the presence of equidistributed data. In these situations, the constrained mock-Chebyshev least squares approximation on rectangular domains was shown to be a successful approximation tool. In this paper, we extend it to triangular domains, by using both Waldron and discrete Leja points. This paper is dedicated to Len Bos on the occasion of his retirement. Len, for us, is a master of mathematics and also a big friend. He introduced us to the fascinating world of "finding good interpolation nodes and effective interpolation strategies", mostly in the multivariate setting. The set of points we are using in this paper, Waldron and Leja, have been introduced to us by him and we hope that this note can be of some interest for him and all people working on approximation theory. [ABSTRACT FROM AUTHOR]

This study focuses on the optimization dynamics of racing go-karts, which is heavily influenced by the frame's stiffness. Lacking suspensions and differentials, go-karts rely on the frame stiffness for wheel balancing and skid prevention by lifting the inner rear wheel during turns. Utilizing a rigid-flexible model in MSC Software ADAMS View, validated by frame deformation measurements, this research integrates finite element analysis with multibody techniques. The model, leverages computer aided design files for frame geometry and employs finite element analysis for frame validation. It facilitates evaluating go-kart dynamics through simulations, aiding in maneuver testing and design optimization. This approach provides a comprehensive framework for advancing go-kart designs. [ABSTRACT FROM AUTHOR]

This paper is concerned about a class of finite time collapsing along continuity method over a compact Kähler manifold. It is shown that along the continuity method the diameter is uniformly bounded and its Gromov–Hausdorff limit is homeomorphic and locally isometric to the regular part of base space. Furthermore, we also prove that the Gromov–Hausdorff limit is unique under the algebraic assumption of the Kähler class by applying the recent work of Song–Tian–Zhang [2019, arXiv:1904.08345]. [ABSTRACT FROM AUTHOR]

In this paper we investigate a model for quantum gravity on finite noncommutative spaces using the theory of blobbed topological recursion. The model is based on a particular class of random finite real spectral triples (A,H,D,γ,J), called random matrix geometries of type (1,0), with a fixed fermion space (A,H,γ,J), and a distribution of the form e-S(D)dD over the moduli space of Dirac operators. The action functional S(D) is considered to be a sum of terms of the form ∏si=1Tr(Dni) for arbitrary s⩾1. The Schwinger-Dyson equations satisfied by the connected correlators Wn of the corresponding multi-trace formal 1-Hermitian matrix model are derived by a differential geometric approach. It is shown that the coefficients Wg,n of the large N expansion of Wn's enumerate discrete surfaces, called stuffed maps, whose building blocks are of particular topologies. The spectral curve (Σ,ω0,1,ω0,2) of the model is investigated in detail. In particular, we derive an explicit expression for the fundamental symmetric bidifferential ω0,2 in terms of the formal parameters of the model. [ABSTRACT FROM AUTHOR]

In 1975, Erd̋s asked the following question: What is the maximum number of edges that an |$n$| -vertex graph can have without containing a cycle with all diagonals? Erd̋s observed that the upper bound |$O(n^{5/3})$| holds since the complete bipartite graph |$K_{3,3}$| can be viewed as a cycle of length six with all diagonals. In this paper, we resolve this old problem. We prove that there exists a constant |$C$| such that every |$n$| -vertex graph with at least |$Cn^{3/2}$| edges contains a cycle with all diagonals. Since any cycle with all diagonals contains cycles of length four, this bound is best possible using well-known constructions of graphs without four-cycles based on finite geometry. Among other ideas, our proof involves a novel lemma about finding an almost-spanning robust expander, which might be of independent interest. [ABSTRACT FROM AUTHOR]

Determining the maximal length of MDS codes with certain dimension is one of the central topics in coding theory and finite geometry. The MDS Main Conjecture states that the maximal length of a non-trivial |$q$| -ary MDS code of dimension |$k$| is |$q+1$| except when |$q$| is even and |$k=3$| or |$k=q-1$|⁠. We prove that the maximal length of non-trivial |$q$| -ary MDS elliptic codes is close to |$\frac{q}{2}$|⁠ , which gives an affirmative answer to a conjecture of Li, Wan, and Zhang. Moreover, we apply our result to derive an answer to a question on subset sums in finite abelian groups from elliptic curves. [ABSTRACT FROM AUTHOR]

Mixed-level designs are widely used in various experiments. This paper considers how to construct minimum aberration 4 1 2 n mixed-level designs. Based on finite projective geometry, secondary complementary set is defined as part of the design which plays a crucial role in constructing the optimal designs. Then, algebraic connection between the wordlength pattern of a 4 1 2 n design and that of its secondary complementary set is established. According to the connection, some general rules for identifying type 0 minimum aberration 4 1 2 n mixed-level designs are proposed via their secondary complementary sets. Those rules can help construct minimum aberration 4 1 2 n designs conveniently when the secondary complementary set contains fewer elements than the complementary set. The type 0 minimum aberration 4 1 2 n designs with large n are tabulated via secondary complementary sets. [ABSTRACT FROM AUTHOR]

In this work, we present an analytical model to describe atomic force microscopy force curves of standard linear solid and power-law viscoelastic materials by taking indenter geometry and finite thickness effects into account. We show that conical/pyramidal cantilevers are less susceptible to finite thickness effects than other shapes, such as spherical and flat punch indenters. This is confirmed by finite element modeling of the stress field distribution within the sample. We also perform a systematic error analysis for the cases where finite thickness samples are analyzed with an infinite thickness force model. In particular, we show that for spherical indenters and indentation depth up to 20% of sample thickness, the mechanical response of viscoelastic materials will wrongly mimic a purely elastic behavior even within timescales where viscoelastic relaxation should appear. [ABSTRACT FROM AUTHOR]

It is known that Mermin–Peres-like proofs of quantum contextuality can furnish non-local games with a guaranteed quantum strategy, when classically no such guarantee can exist. This phenomenon, also called quantum pseudo-telepathy, has been studied in the case of the so-called Mermin magic square game. In this paper, we review in detail two different ways of implementing on a quantum computer such a game and propose a new Doily game based on the geometry of 2-qubit Pauli group. We show that the quantumness of these games is almost revealed when we play them on the IBM Quantum Experience; however, the inherent noise in the available quantum machines prevents a full demonstration of the non-classical aspects. [ABSTRACT FROM AUTHOR]

A function f : Fq → Fq is perfect nonlinear or planar if, for every nonzero b ∈ Fq, the discrete derivative f(x + b) − f(x) − f(b) of f(x) is a bijection of Fq. Perfect nonlinear functions have a wide range of applications in Combinatorics, Coding Theory, Finite Geometry, Cryptography, and Design Theory. A polynomial of the form Σi,j aij xpi+pj is called a Dembowski-Ostrom polynomial (DO polynomial). In this article, we construct four classes of perfect nonlinear functions of DO-type polynomials over finite fields Fptn, for t = 2 and 3. [ABSTRACT FROM AUTHOR]

A full classification (up to equivalence) of all minimal blocking sets in PG (2 , 9) was obtained by computer. The resulting numbers of minimal blocking sets are tabulated according to size of the set and order of the automorphism group. For the minimal blocking sets with the larger automorphism groups explicit (geometric) descriptions are given. Some of these results can also be generalised to Desarguesian projective planes of higher order. We also give a complete list of all blocking semiovals in PG (2 , 9) (up to equivalence). [ABSTRACT FROM AUTHOR]

Research and development in the field of space debris (SD) removal technologies is one of the most important and urgent tasks facing all spacefaring nations. By now, a large number of different concepts of removal have already been proposed. One of them is the SD removal by ion beam generated by an ion source installed on board a service spacecraft (SSC). A detailed understanding of the process of ion beam interaction with the SD surface is required to develop an optimal control algorithm for the virtual SSC-SD mated configuration in order to improve this method of SD removal. In addition, when designing the SSC, it is necessary to take into account back flows of the material sputtered from the surface of removed SD by the ion beam, which can contaminate the SSC operating surfaces during the removal maneuver. Within the framework of the presented work, experimental studies were performed to determine the coefficient of accommodation of the ion impulse during the ion beam interaction with the surface of the target simulating the external surface of the removed object. Besides, the flow density of the sputtered target material was experimentally evaluated during its exposure to the ion beam. To do this, we used the radio-frequency ion source (injector) with a beam diameter of 160 mm, which is supposed to be used as an actuating unit on board the SSC designed for transporting the SD objects to the disposal orbit or to the orbits with limited life. When measuring the ion beam accommodation coefficient, the presence of additional force arising on the target due to the flow of sputtered material leaving its surface, and the ion-electron emission process on the target, which could distort the experimental data, were taken into account. To estimate the magnitude of the flows of sputtered target material under the influence of the ion beam, a special experimental design was developed for simulating the presence of a point source of sputtered material. The results obtained in this work will be used for modeling the impact of the ion beam on the surface of real SD objects with complex geometry using the finite element method, for refining the control algorithm of the virtual SSC-SDO mated configuration and for developing proposals on the SSC layout, which should reduce or completely eliminate the negative impact associated with the material sputtering from the SDO surface onto the SSC elements. Successful implementation of the developed method of SDO removal will contribute to improving the safety of space activities for all countries and to the development of the rocket-and-space industry in the Russian Federation, in particular. • The accommodation coefficient of the ion beam momentum was determined. • The back flow of sputtered target particles was estimated. • In a first approximation, the form of the target material sputtering indicatrix was obtained. [ABSTRACT FROM AUTHOR]

It has been argued by Iliesiu, Kologlu and Turiaci in arXiv:2107.09062 that one can compute the supersymmetric index of black holes using black hole geometry carrying finite temperature but a specific complex angular velocity. We follow their prescription to compute the logarithmic correction to the entropy of BPS states in four dimensions, defined as the log of the index of supersymmetric black holes, and find perfect agreement with the previous results for the same quantity computed using the near horizon AdS2 × S2 geometry of zero temperature black holes. Besides giving an independent computation of supersymmetric black hole entropy, this analysis also provides a test of the procedure used previously for computing logarithmic corrections to Schwarzschild and other non-extremal black hole entropy. [ABSTRACT FROM AUTHOR]

In this paper we consider a family F of 2n-dimensional F q -linear rank metric codes in F q n × n arising from polynomials of the form x q s + δ x q n 2 + s ∈ F q n [ x ] . The family F was introduced by Csajbók et al. (JAMA 548:203–220) as a potential source for maximum rank distance (MRD) codes. Indeed, they showed that F contains MRD codes for n = 8 , and other subsequent partial results have been provided in the literature towards the classification of MRD codes in F for any n. In particular, the classification has been reached when n is smaller than 8, and also for n greater than 8 provided that s is small enough with respect to n. In this paper we deal with the open case n = 8 , providing a classification for any large enough odd prime power q. The techniques are from algebraic geometry over finite fields, since our strategy requires the analysis of certain 3-dimensional F q -rational algebraic varieties in a 7-dimensional projective space. We also show that the MRD codes in F are not equivalent to any other MRD codes known so far. [ABSTRACT FROM AUTHOR]

Total ankle arthroplasty is the gold standard surgical treatment for severe ankle arthritis and fracture. However, revision surgeries due to the in vivo failure of the ankle implant are a serious concern. Extreme bone density loss due to bone remodelling is one of the main reasons for in situ implant loosening, with aseptic loosening of the talar component being one of the primary reasons for total ankle arthroplasty revisions. This study is aimed at determining the performance and potential causes of failure of the talar component. Herein, we investigated the stress, strain, and bone density changes that take place in the talus bone during the first 6 months of bone remodelling due to the total ankle arthroplasty procedure. Computed tomography scans were used to generate the 3D geometry used in the finite element (FE) model of the Intact and implanted ankle. The Scandinavian Total Ankle Replacement (STAR™) CAD files were generated, and virtual placement within bone models was done following surgical guidelines. The dorsiflexion physiological loading condition was investigated. The cortical region of the talus bone was found to demonstrate the highest values of stress (5.02 MPa). Next, the adaptive bone remodelling theory was used to predict bone density changes over the initial 6-month post-surgery. A significant change in bone density was observed in the talus bone due to bone remodelling. The observed quantitative changes in talus bone density over 6-month period underscore potential implications for implant stability and fracture susceptibility. These findings emphasise the importance of considering such biomechanical factors in ankle implant design and clinical management. [ABSTRACT FROM AUTHOR]

For fixed s≥3, we prove that if optimal Ks-free pseudorandom graphs exist, then the Ramsey number r(s,t) is t s−1+o(1) as t→∞. Our method also improves the best lower bounds for r(Cℓ ​,t) obtained by Bohman and Keevash from the random Cℓ-free process by polylogarithmic factors for all odd ℓ≥5 and ℓ∈{6,10}. For ℓ=4 it matches their lower bound from the C4-free process. We also prove, via a different approach, that r(C5,t)>(1+o(1))t 11/8 and r(C7 ​,t)>(1+o(1))t 11/9 . These improve the exponent of t in the previous best results and appear to be the first examples of graphs F with cycles for which such an improvement of the exponent for r(F,t) is shown over the bounds given by the random F-free process and random graphs. [ABSTRACT FROM AUTHOR]

The flipped classroom approach has been used as a variation of digital learning even before the covid-19 pandemic. Using this approach and the First Principles of Effective Instruction, this study determined and compared the proving skills of higher education students in finite geometries using true experimental research involving two equivalent classes in Modern Geometry. Data on proving competencies were gathered from the 27 pairs of randomly selected respondents and were subjected to data exploratory analysis to ensure the appropriateness of statistical tools for data analysis. Results reveal that both classes performed equivalently in the pretest. However, the flipped class exhibited a statistically significant improvement in the posttest and in-class activities than the non-flipped class. Additionally, the flipped class was found to have equivalent performances in both individual and group in-class activities. The results showed that necessary competencies in proving theorems can be attained using the flipped classroom approach following the activation, demonstration, application, and integration phases of instruction. The study recommended using the approach to support students learning achievement, performance enhancement, and active learning environment. It can also be incorporated into crafting an adaptive learning continuity plan for the post-pandemic recovery period and beyond. [ABSTRACT FROM AUTHOR]

Inspired by the natural bifurcating structures, tree-like microchannels have been applied for microelectronics cooling. In order to understand the thermal-hydraulic performance of a flat-plat tree-like microchannel, successive branching ratios of tree-like structure are optimized based on minimization of flow resistance. It is shown that the optimal successive diameter ratio of symmetrical and dichotomous structures under volume constraint follows Murray's law, while the optimal successive length ratio under the constraint of fixed channel area follows the power law 2-2/3. A mathematical model of convection in disc-shaped heat sink composed of a tree-like microchannel with selfaffine rough surface is developed by the fractal geometry and finite element method. The flat-plate tree-like micro-channel with optimal successive diameter and length ratio shows enhanced thermalhydraulic performance. The Nusselt number of the flat-plat tree-like micro-channel increases with the inlet Reynolds number and the self-affine fractal dimension of the rough wall. The present optimization method and mathematical model for the flat-plate tree-like microchannel shed light on the design of flat-plate micro-channel heat sinks and flow channel in fuel cell among other potential cooling applications. [ABSTRACT FROM AUTHOR]

Three‐level fractional factorial split‐plot (FFSP) designs with the whole plot (WP) factors being more important than the subplot factors are considered in the article. An aliased component‐number pattern of type WP (WP‐ACNP) is introduced for ranking such designs. The criterion of general minimum lower‐order confounding of type WP (WP‐GMC) is proposed based on WP‐ACNP. The expressions of the key components in WP‐ACNP are given via complementary sets. Some necessary conditions for FFSP designs to be WP‐GMC FFSP designs are given and some three‐level WP‐GMC FFSP designs are constructed and tabulated. [ABSTRACT FROM AUTHOR]

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Plant diversity includes over 300,000 species, and leaf structure is one of the main targets of selection, being highly variable in shape and size. On the other hand, the optimization of antenna design has no unique solution to satisfy the current range of applications. We analyzed the foliar geometries of 100 plant species and applied them as a biomimetic design template for microstrip patch antenna systems. From this set, a subset of seven species were further analyzed, including species from tropical and temperate forests across the phylogeny of the Angiosperms. Foliar geometry per species was processed by image processing analyses, and the resultant geometries were used in simulations of the reflection coefficients and the radiation patterns via finite differences methods. A value below −10 dB is set for the reflection coefficient to determine the operation frequencies of all antenna elements. All species showed between 3 and 15 operational frequencies, and four species had operational frequencies that included the 2.4 and 5 GHz bands. The reflection coefficients and the radiation patterns in most of the designs were equal or superior to those of conventional antennas, with several species showing multiband effects and omnidirectional radiation. We demonstrate that plant structures can be used as a biomimetic tool in designing microstrip antenna for a wide range of applications. [ABSTRACT FROM AUTHOR]

We study the gravity solution dual to the D0 brane quantum mechanics, or BFSS matrix model, in the 't Hooft limit. The classical physics described by this gravity solution is invariant under a scaling transformation, which changes the action with a specific critical exponent, sometimes called the hyperscaling violating exponent. We present an argument for this critical exponent from the matrix model side, which leads to an explanation for the peculiar temperature dependence of the entropy in this theory, S ∝ T9/5. We also present a similar argument for all other Dp-brane geometries. We then compute the black hole quasinormal modes. This involves perturbing the finite temperature geometry. These perturbations can be easily obtained by a mathematical trick where we view the solution as the dimensional reduction of an AdS2+9/5× S8 geometry. [ABSTRACT FROM AUTHOR]

In recent years, many useful applications of the polynomial method have emerged in finite geometry. Indeed, algebraic curves, especially those defined by Rédei-type polynomials, are powerful in studying blocking sets. In this paper, we reverse the engine and study when blocking sets can arise from rational points on plane curves over finite fields. We show that irreducible curves of low degree cannot provide blocking sets and prove more refined results for cubic and quartic curves. On the other hand, using tools from number theory, we construct smooth plane curves defined over F p of degree at most 4 p 3 / 4 + 1 whose points form blocking sets. [ABSTRACT FROM AUTHOR]

Proto-exact categories , introduced by Dyckerhoff and Kapranov, are a generalization of Quillen exact categories which provide a framework for defining algebraic K-theory and Hall algebras in a non-additive setting. This formalism is well-suited to the study of categories whose objects have strong combinatorial flavor. In this paper, we show that the categories of modules over semirings and hyperrings - algebraic structures which have gained prominence in tropical geometry - carry proto-exact structures. In the first part, we prove that the category of modules over a semiring is equipped with a proto-exact structure; modules over an idempotent semiring have a strong connection to matroids. We also prove that the category of algebraic lattices L has a proto-exact structure, and furthermore that the subcategory of L consisting of finite lattices is equivalent to the category of finite B -modules as proto-exact categories, where B is the Boolean semifield. We also discuss some relations between L and geometric lattices (simple matroids) from this perspective. In the second part, we prove that the category of modules over a hyperring has a proto-exact structure. In the case of finite modules over the Krasner hyperfield K , a well-known relation between finite K -modules and finite incidence geometries yields a combinatorial interpretation of exact sequences. [ABSTRACT FROM AUTHOR]

The implementation of discontinuous functions occurs in many of today's state-of-the-art partial differential equation solvers. However, in finite element methods, this poses an inherent difficulty: efficient quadrature rules available when integrating functions whose discontinuity falls in the element's interior are for low order degree polynomials, not easily extended to higher order degree polynomials, and cover a restricted set of geometries. Many approaches to this issue have been developed in recent years. Among them, one of the most elegant and versatile is the equivalent polynomial technique. This method replaces the discontinuous function with a polynomial, allowing integration to occur over the entire domain rather than integrating over complex subdomains. Although eliminating the issues involved with discontinuous function integration, the equivalent polynomial tactic introduces its problems. The exact subdomain integration requires a machinery that quickly grows in complexity when increasing the polynomial degree and the geometry dimension, restricting its applicability to lower order degree finite element families. The current work eliminates this issue. We provide algebraic expressions to exactly evaluate the subdomain integral of any degree polynomial on parent finite element shapes cut by a planar interface. These formulas also apply to the exact evaluation of the embedded interface integral. We provide recursive algorithms that avoid overflow in computer arithmetic for standard finite element geometries: triangle, square, cube, tetrahedron, and prism, along with a hypercube of arbitrary dimensions. [ABSTRACT FROM AUTHOR]

Fracture prognosis and characterization efforts require knowledge of crack tip position and the Stress Intensity Factors (SIFs) acting in the vicinity of the crack. Here, we present an efficient numerical approach to infer both of these characteristics under a consistent theoretical framework from noisy, unstructured displacement data. The novel approach utilizes the separability of the asymptotic linear elastic fracture mechanics fields to expedite the search for crack tip position and is particularly useful for noisy displacement data. The manuscript begins with an assessment of the importance of accurately locating crack tip position when quantifying the SIFs from displacement data. Next, the proposed separability approach for quickly inferring crack tip position is introduced. Comparing to the widely used displacement correlation approach, the performance of the separability approach is assessed. Cases involving both noisy data and systematic deviation from the asymptotic linear elastic fracture mechanics model are considered, e.g. inelastic material behavior and finite geometries. An open source python implementation of the proposed approach is available for use by those doing field and laboratory work involving digital image correlation and simulations, e.g. finite element, discrete element, molecular dynamics and peridynamics, where the crack tip position is not explicitly defined. [ABSTRACT FROM AUTHOR]

Factorial designs are often used in various industrial and sociological experiments to identify significant factors and factor combinations that may affect the process response. In the statistics literature, several studies have investigated the analysis, construction, and isomorphism of factorial and fractional factorial designs. When there are multiple choices for a design, it is helpful to have an easy-to-use tool for identifying which designs are distinct, and which of those can be efficiently analyzed/has good theoretical properties. For this task, we present an R library called IsoCheck that checks the isomorphism of multi-stage 2n factorial experiments with randomization restrictions. Through representing the factors and their combinations as a finite projective geometry, IsoCheck recasts the problem of searching over all possible relabeling as a search over collineations, then exploits projective geometric properties of the space to make the search much more efficient. Furthermore, a bitstring representation of the factorial effects is used to characterize all possible rearrangements of designs, thus facilitating quick comparisons after relabeling. This paper presents several detailed examples with R codes that illustrate the usage of the main functions in IsoCheck. Besides checking equivalence and isomorphism of 2n multi-stage factorial designs, we demonstrate how the functions of the package can be used to create a catalog of all non-isomorphic designs, and find good designs as per a suitably defined ranking criterion. [ABSTRACT FROM AUTHOR]

We demonstrate the influence of edge effects on the photothermal-induced phase shift measured by a homodyne quadrature laser interferometer and compare the experiments with rigorous theoretical descriptions of thermoelastic surface displacement of metals. The finite geometry of the samples is crucial in determining how the temperature is distributed across the material and how this affects the interferometer phase shift measurements. The optical path change due to the surface thermoelastic deformation and thermal lens in the surrounding air is decoded from the interferometric signal using analytical and numerical tools. The boundary/edge effects are found to be relevant to properly describe the interferometric signals. The tools developed in this study provide a framework for the study of finite size effects in heat transport in opaque materials and are applicable to describe not only the phase shift sensed by the interferometer but also to contribute to the photothermal-based technologies employing similar detection mechanisms. [ABSTRACT FROM AUTHOR]

Analytic gradients of electronic eigenvalues require one calculation per nuclear geometry, compared to at least 3n + 1 calculations for finite difference methods, where n is the number of nuclei. Analytic nonadiabatic derivative coupling terms (DCTs), which are calculated in a similar fashion, are used to remove nondiagonal contributions to the kinetic energy operator, leading to more accurate nuclear dynamics calculations than those that employ the Born-Oppenheimer approximation, i.e., that assume off-diagonal contributions are zero. The current methods and underpinnings for calculating both of these quantities, gradients and DCTs, for the State-Averaged MultiReference Configuration Interaction with Singles and Doubles (MRCI-SD) wavefunctions in COLUMBUS are reviewed. Before this work, these methods were not available for wavefunctions of a relativistic MRCI-SD Hamiltonian. Calculation of these terms is critical in successfully modeling the dynamics of systems that depend on transitions between potential energy surfaces split by the spin-orbit operator, such as diode-pumped alkali lasers. A formalism for calculating the transition density matrices and analytic derivative coupling terms for such systems is presented. [ABSTRACT FROM AUTHOR]

In this article, we establish an ℓ2 decoupling inequality for the surface F 4 2 : = { (ξ 1 , ξ 2 , ξ 1 4 + ξ 2 4) : (ξ 1 , ξ 2) ∈ [ 0 , 1 ] 2 } associated with the decomposition adapted to finite type geometry from our previous work [Li, Z., Miao, C., Zheng, J.: A restriction estimate for a certain surface of finite type in ℝ3. J. Fourier Anal. Appl., 27(4), Paper No. 63, 24 pp. (2021)]. The key ingredients of the proof include the so-called generalized rescaling technique, an ℓ2 decoupling inequality for the surfaces { (ξ 1 , ξ 2 , ϕ 1 (ξ 1) + ξ 2 4) : (ξ 1 , ξ 2) ∈ [ 0 , 1 ] 2 } with ϕ1 being non-degenerate, reduction of dimension arguments and induction on scales. [ABSTRACT FROM AUTHOR]

In this article, we establish an ℓ2 decoupling inequality for the surface F 4 2 : = { (ξ 1 , ξ 2 , ξ 1 4 + ξ 2 4) : (ξ 1 , ξ 2) ∈ [ 0 , 1 ] 2 } associated with the decomposition adapted to finite type geometry from our previous work [Li, Z., Miao, C., Zheng, J.: A restriction estimate for a certain surface of finite type in ℝ3. J. Fourier Anal. Appl., 27(4), Paper No. 63, 24 pp. (2021)]. The key ingredients of the proof include the so-called generalized rescaling technique, an ℓ2 decoupling inequality for the surfaces { (ξ 1 , ξ 2 , ϕ 1 (ξ 1) + ξ 2 4) : (ξ 1 , ξ 2) ∈ [ 0 , 1 ] 2 } with ϕ1 being non-degenerate, reduction of dimension arguments and induction on scales. [ABSTRACT FROM AUTHOR]

In 1980, U. Faigle introduced a sort of finite geometries on posets that are in bijective correspondence with finite semimodular lattices. His result has almost been forgotten in lattice theory. Here we simplify the axiomatization of these geometries, which we call Faigle geometries. To exemplify their usefulness, we give a short proof of a theorem of Grätzer and E. Knapp (2009) asserting that each slim semimodular lattice L has a congruencepreserving extension to a slim rectangular lattice of the same length as L. As another application of Faigle geometries, we give a short proof of G. Grätzer and E.W. Kiss' result from 1986 (also proved by M. Wild in 1993, the present author and E.T. Schmidt in 2010, and B. Skublics in 2013) that each finite semimodular lattice L has an extension to a geometric lattice of the same length as L. [ABSTRACT FROM AUTHOR]

An alternative proof of the necessary conditions on A , B ∈ q 2 * for f (X) = X + A X 1 + q 2 (q − 1) / 4 + B X 1 + 3 q 2 (q − 1) / 4 to be a permutation polynomial in q 2 , q even, is given. This proof involves standard arguments from algebraic geometry over finite fields and fast symbolic computations. [ABSTRACT FROM AUTHOR]

The notion of cross-intersecting set pair system of size $m$ , $ (\{A_i\}_{i=1}^m, \{B_i\}_{i=1}^m)$ with $A_i\cap B_i=\emptyset$ and $A_i\cap B_j\ne \emptyset$ , was introduced by Bollobás and it became an important tool of extremal combinatorics. His classical result states that $m\le\binom{a+b}{a}$ if $|A_i|\le a$ and $|B_i|\le b$ for each $i$. Our central problem is to see how this bound changes with the additional condition $|A_i\cap B_j|=1$ for $i\ne j$. Such a system is called $1$ -cross-intersecting. We show that these systems are related to perfect graphs, clique partitions of graphs, and finite geometries. We prove that their maximum size is at least $5^{n/2}$ for $n$ even, $a=b=n$ , equal to $\bigl (\lfloor \frac{n}{2}\rfloor +1\bigr)\bigl (\lceil \frac{n}{2}\rceil +1\bigr)$ if $a=2$ and $b=n\ge 4$ , at most $|\cup _{i=1}^m A_i|$ , asymptotically $n^2$ if $\{A_i\}$ is a linear hypergraph ( $|A_i\cap A_j|\le 1$ for $i\ne j$), asymptotically ${1\over 2}n^2$ if $\{A_i\}$ and $\{B_i\}$ are both linear hypergraphs. [ABSTRACT FROM AUTHOR]

Horizontal geostatic stress estimated from self-bored pressuremeter (SBP) data using the 'lift off' method is uncertain because of even small deficiencies in self-boring, but that uncertainty can be minimised by modelling the complete test. Iterative forward modelling based on large-strain cavity expansion in frictional dilating (non-associated Mohr–Coulomb) soil, with correction for finite SBP geometry, is both easily implemented in a spreadsheet and computes quickly. Such modelling of a campaign of SBP tests in oil-sand tailings shows a baseline geostatic stress ratio K0 = 0.6 for those tailings that are truly normally consolidated. Other geological history factors, including compaction by tracking and wetting–drying cycles, adds about a Δσh ≈ 60 kPa 'locked-in' stress to this normally consolidated trend; an alternative view is that these factors produce K0 ≈ 1. The modelling spreadsheet is provided as a downloadable Excel application in the online supplementary material. [ABSTRACT FROM AUTHOR]

Combinatorial t-designs have nice applications in coding theory, finite geometries and engineering areas. t-designs can be constructed from image sets of a fixed size of some special polynomials. This paper constructs t-designs from the quadratic polynomial x 4 + x 3 over GF (2 n) and determine their parameters. We yield 2- 2 n , 3 · 2 n - 2 , 3 · 2 n - 2 3 · 2 n - 2 - 1 designs for n even and 3- 2 n , 2 n - 1 , 2 n - 1 2 n - 2 - 1 designs for n odd. [ABSTRACT FROM AUTHOR]

We consider projective Hjelmslev geometries over finite chain rings of length 2 with residue field of order q. In these geometries we introduce and investigate the so-called homogeneous arcs defined as multisets of points with respect to a special weight function. These arcs are associated with linearly representable q-ary codes. We establish a relation between the parameters of a linearly representable code and the parameters of the associated arc. We prove an inequality for the largest homogeneous weight of an arc of given size N. We consider the τ -dual of a homogeneous arc and give an explicit formula for the homogeneous weights of the hyperplanes with respect to the dual arc. We prove that if a homogeneous arc is of constant weight, this weight is forced to be zero and the arc is a sum of neighbour classes of points. We prove various numerical conditions on the parameters of homogeneous two-weight arcs, as well as an interesting identity involving the parameters of a projective homogeneous two-weight arc and its dual, which isturns out to be also a homogeneous two-weight arc. Finally, we present examples of two-weight homogeneous arcs some of which are new. [ABSTRACT FROM AUTHOR]

A simple finite element implementation of the Mott model for fragmentation of a thin walled ring has been implemented and used to explore the effect of local variations in fracture strain around the ring. The model has successfully reproduced the fragment size distributions previously reported, which follow a characteristic "Mott distribution" form, providing sufficient (1000) simulations are run. It has been shown that this form is retained even when there are large differences in the random distribution of fracture strains or a different choice of function used to describe the fracture strain scatter. In these cases, the strain rate has a much stronger effect than fracture strain distribution the on the average fragment size and fragment distribution. However, for cases where there are a small number of local defects that strongly reduce the fracture strain at certain locations around the ring, the predicted fragment size distribution develops a bimodal character. This is also the case for large but gradual variations in fracture strain with position around the ring. The results have implications for cases where a small number of large pre-existing defects exist, or processing has led to macrozones in the microstructure. The utility of a simple fast running model to study these cases is discussed. [ABSTRACT FROM AUTHOR]

A linear code is called a locally repairable code (LRC) with locality r if one can recover an erased code symbol by accessing at most r other code symbols. Constructions of LRCs have been widely investigated in recent years. In this paper, we give a step forward in this direction. Firstly, we propose a novel concept of perfect LRCs whose size exactly achieves the Hamming-type bound, similar to the perfect codes that achieving the Hamming bound in classical coding theory. By the parity-check matrix approach, we establish some important connections between the existence of LRCs and the existence of some subsets of finite geometry and finite fields with certain properties, respectively. By employing q-Steiner systems and sunflowers in projective geometry and difference sets in finite fields, we obtain two new constructions of perfect LRCs with flexible parameters and present several new constructions of k-optimal LRCs achieving another Hamming-type bound under the integers restriction. Moreover, for fixed q and r, the code lengths of all the q-ary r-LRCs constructed in this paper can be arbitrarily large and the code rates can asymptotically achieve the upper bound r r + 1 . [ABSTRACT FROM AUTHOR]

In this study, we introduce a Graph network‐enhanced Finite Element approach to accelerate Finite Element simulations. We utilize the discretized geometry from a Finite Element pre‐processor to establish the graph and use the Graph Neural Network to solve the boundary value problem of the discretized domain. The advantage of graph neural networks is that they have a similar structure as compared to a discretized domain with nodes and elements. The underlying dynamics of the system are computed via a learned message‐passing. The goal here is to enhance and accelerate the FEM simulations using the proposed GNN network by incorporating the underlying mechanics knowledge into the network to enhance the generalizing ability of the network on various loading and boundary conditions. All the proposed studies in the literature where graph networks are applied to Finite Element Methods use images as input and output. The advantage of the proposed model is that it takes inputs such as the nodal information, their corresponding edges, nodal coordinates and the boundary conditions for each particular node from a Finite Element pre‐processor and computes the von‐Mises stress at each node along with their edge connections as output that can be read by a Finite Element post‐processor. [ABSTRACT FROM AUTHOR]

A hybrid computing method for predicting the magnetic field of line-start permanent magnet synchronous motor is presented in this article. The vector magnetic potential (VMP) is selected to be the primary objective of the calculation. For the convenience of expressing the VMP in the polar coordinate system, the topology of each motor component is replaced by the assembly of radial sectors. Along with the topology approximation, the VMPs of the stator and air gap are determined analytically, whereas the finite difference method is used to calculate the VMP of the rotor with the eddy current. Combining the numerical technique with the analytical method not only enables solution efficiency higher than using the former one merely but also keeps away from the inefficacy of using the latter one solely to predict magnetic fields with complex geometry. The presented approach is validated by the finite-element analysis and prototyping tests. [ABSTRACT FROM AUTHOR]

We present a novel analogue for finite exchangeable sequences of the de Finetti, Hewitt and Savage theorem and investigate its implications for multi-marginal optimal transport (MMOT) and Bayesian statistics. If (Z 1 , ... , Z N) is a finitely exchangeable sequence of N random variables taking values in some Polish space X, we show that the law μ k of the first k components has a representation of the form μ k = ∫ P 1 N (X) F N , k (λ) d α (λ) for some probability measure α on the set of 1 N -quantized probability measures on X and certain universal polynomials F N , k . The latter consist of a leading term N k - 1 / ∏ j = 1 k - 1 (N - j) λ ⊗ k and a finite, exponentially decaying series of correlated corrections of order N - j ( j = 1 , ... , k ). The F N , k (λ) are precisely the extremal such laws, expressed via an explicit polynomial formula in terms of their one-point marginals λ . Applications include novel approximations of MMOT via polynomial convexification and the identification of the remainder which is estimated in the celebrated error bound of Diaconis and Freedman (Ann Probab 8(4):745–764, 1980) between finite and infinite exchangeable laws. [ABSTRACT FROM AUTHOR]

A blocking set in a graph G is a subset of vertices that intersects every maximum independent set of G. Let mmbs (G) be the size of a maximum (inclusion-wise) minimal blocking set of G. This parameter has recently played an important role in the kernelization of Vertex Cover with structural parameterizations. We provide a panorama of the complexity of computing mmbs parameterized by the natural parameter and the independence number of the input graph. We also consider the closely related parameter mmhs , which is the size of a maximum minimal hitting set of a hypergraph. Finally, we consider the problem of computing mmbs parameterized by treewidth, especially relevant in the context of kernelization. Since a blocking set intersects every maximum-sized independent set of a given graph and properties involving counting the sizes of arbitrarily large sets are typically non-expressible in monadic second-order logic, its tractability does not seem to follow from Courcelle's theorem. Our main technical contribution is a fixed-parameter tractable algorithm for this problem. [ABSTRACT FROM AUTHOR]

The long-time behaviour of spin-spin correlators in the slow relaxation of systems undergoing phase-ordering kinetics is studied in geometries of finite size. A phenomenological finite-size scaling ansatz is formulated and tested through the exact solution of the kinetic spherical model, quenched to below the critical temperature, in 2 Ÿ 𝑑 Ÿ 4 dimensions. [ABSTRACT FROM AUTHOR]

Constructing hemisystems of the Hermitian surface is a well known, apparently difficult, problem in Finite geometry. So far, a few infinite families and some sporadic examples have been constructed. One of the different approaches relies on the Fuhrmann-Torres maximal curve and provides a hemisystem in P G (3 , p 2) for every prime p of the form p = 1 + 4 a 2 , a even. Here we show that this approach also works in P G (3 , p 2) for every prime p = 1 + 4 a 2 , a odd. The resulting hemisystem gives rise to two weight linear codes and strongly regular graphs whose properties are also investigated. [ABSTRACT FROM AUTHOR]

Given a Henselian and Japanese discrete valuation ring A and a flat and projective A-scheme X, we follow the approach of Biswas and dos Santos [J. Inst. Math. Jussieu 10 (2011), no. 2, 225-234] to introduce a full subcategory of coherent modules on X which is then shown to be Tannakian. We then prove that, under normality of the generic fibre, the associated affine and flat group is pro-finite in a strong sense (so that its ring of functions is a Mittag-Leffler A-module) and that it classifies finite torsors Q - X. This establishes an analogy to Nori's theory of the essentially finite fundamental group. In addition, we compare our theory with the ones recently developed by Mehta-Subramanian and Antei-Emsalem-Gasbarri. Using the comparison with the former, we show that any quasi-finite torsor Q - X has a reduction of the structure group to a finite one. [ABSTRACT FROM AUTHOR]