An important first step in understanding or solving a problem can be the selection of coordinates. Insight can be gained from finding invariants within a class of coordinate systems. For example, an important feature of rectangular coordinates is that the Euclidean distance between two points is an invariant of a change to another rectangular system by a rigid motion, which consists of translations, rotations and reflections. Indeed, the form of the distance function is an invariant. In calculus courses, students learn about polar coordinates, so that useful curves can be simply expressed and more easily studied.

This article begins with a new look at earlier work on this topic and continues with the presentation of a graphical technique for the determination of the months of a year for which the 13th is a Friday.

It might be thought that the likelihood of the 13 th day of a month being a Friday is the same as that of the 13th being any other day, but this is not so. The full day/date repeat cycle is 400 years, this being the interval between century years which are leap years. The outcome of an extraordinary counting exercise some 50 years ago by a 13-year-old Eton schoolboy, S. R. Baxter [1, 2], was to show that, over this period, the 13th of a month will be a Friday at least once more than any other day. Of necessity, Baxter’s calculation was intricate and an independent confirmation was undertaken as a matter of interest. The 400-year period beginning 1 March 2014 was considered. The work is described in four stages.

The direct derivation (DD) method is a technique for quantitative phase analysis (QPA). It can be characterized by the use of the total sums of scattered/diffracted intensities from individual components as the observed data. The crystal structure parameters are required when we calculate the intensities of reflections or diffraction patterns. Intensity can, however, be calculated only with the chemical composition data if it is not of individual reflections but of a total sum of diffracted/scattered intensities for that material. Furthermore, it can be given in a form of the scattered intensity per unit weight. Therefore, we can calculate the weight proportion of a component material by dividing the total sum of observed scattered/diffracted intensities by the scattered intensity per unit weight. The chemical composition data of samples under investigation are known in almost all cases at the stage of QPA. Thus, a technical problem is how to separate the observed diffraction pattern of a mixture into individual component patterns. Various pattern decomposition techniques currently available can be used for separating the pattern of a mixture. In this report, the theoretical background of the DD method and various techniques for pattern decompositions are reviewed along with the examples of applications.

In this paper we study a large system of N servers, each with capacity to process at most C simultaneous jobs; an incoming job is routed to a server if it has the lowest occupancy amongst d (out of N) randomly selected servers. A job that is routed to a server with no vacancy is assumed to be blocked and lost. Such randomized policies are referred to JSQ(d) (Join the Shortest Queue out of d) policies. Under the assumption that jobs arrive according to a Poisson process with rate $N\lambda^{(N)}$ where $\lambda^{(N)}=\sigma-\frac{\beta}{\sqrt{N}\,}$, $\sigma\in\mathbb{R}_+$ and $\beta\in\mathbb{R}$, we establish functional central limit theorems for the fluctuation process in both the transient and stationary regimes when service time distributions are exponential. In particular, we show that the limit is an Ornstein–Uhlenbeck process whose mean and variance depend on the mean field of the considered model. Using this, we obtain approximations to the blocking probabilities for large N, where we can precisely estimate the accuracy of first-order approximations.

In this work, we study a new model for continuum line-of-sight percolation in a random environment driven by the Poisson–Voronoi tessellation in the d-dimensional Euclidean space. The edges (one-dimensional facets, or simply 1-facets) of this tessellation are the support of a Cox point process, while the vertices (zero-dimensional facets or simply 0-facets) are the support of a Bernoulli point process. Taking the superposition Z of these two processes, two points of Z are linked by an edge if and only if they are sufficiently close and located on the same edge (1-facet) of the supporting tessellation. We study the percolation of the random graph arising from this construction and prove that a 0–1 law, a subcritical phase, and a supercritical phase exist under general assumptions. Our proofs are based on a coarse-graining argument with some notion of stabilization and asymptotic essential connectedness to investigate continuum percolation for Cox point processes. We also give numerical estimates of the critical parameters of the model in the planar case, where our model is intended to represent telecommunications networks in a random environment with obstructive conditions for signal propagation.

The paper discusses the risk of ruin in insurance coverage of an epidemic in a closed population. The model studied is an extended susceptible–infective–removed (SIR) epidemic model built by Lefèvre and Simon (Methodology Comput. Appl. Prob. 22, 2020) as a block-structured Markov process. A fluid component is then introduced to describe the premium amounts received and the care costs reimbursed by the insurance. Our interest is in the risk of collapse of the corresponding reserves of the company. The use of matrix-analytic methods allows us to determine the distribution of ruin time, the probability of ruin, and the final amount of reserves. The case where the reserves are subjected to a Brownian noise is also studied. Finally, some of the results obtained are illustrated for two particular standard SIR epidemic models.

We provide upper and lower bounds for the mean $\mathscr{M}(H)$ of $\sup_{t\geq 0} \{B_H(t) - t\}$, with $B_H(\!\cdot\!)$ a zero-mean, variance-normalized version of fractional Brownian motion with Hurst parameter $H\in(0,1)$. We find bounds in (semi-) closed form, distinguishing between $H\in(0,\frac{1}{2}]$ and $H\in[\frac{1}{2},1)$, where in the former regime a numerical procedure is presented that drastically reduces the upper bound. For $H\in(0,\frac{1}{2}]$, the ratio between the upper and lower bound is bounded, whereas for $H\in[\frac{1}{2},1)$ the derived upper and lower bound have a strongly similar shape. We also derive a new upper bound for the mean of $\sup_{t\in[0,1]} B_H(t)$, $H\in(0,\frac{1}{2}]$, which is tight around $H=\frac{1}{2}$.

We prove a higher genus version of the genus $0$ local-relative correspondence of van Garrel-Graber-Ruddat: for $(X,D)$ a pair with X a smooth projective variety and D a nef smooth divisor, maximal contact Gromov-Witten theory of $(X,D)$ with $\lambda _g$-insertion is related to Gromov-Witten theory of the total space of ${\mathcal O}_X(-D)$ and local Gromov-Witten theory of D.

Specializing to $(X,D)=(S,E)$ for S a del Pezzo surface or a rational elliptic surface and E a smooth anticanonical divisor, we show that maximal contact Gromov-Witten theory of $(S,E)$ is determined by the Gromov-Witten theory of the Calabi-Yau 3-fold ${\mathcal O}_S(-E)$ and the stationary Gromov-Witten theory of the elliptic curve E.

Specializing further to $S={\mathbb P}^2$, we prove that higher genus generating series of maximal contact Gromov-Witten invariants of $({\mathbb P}^2,E)$ are quasimodular and satisfy a holomorphic anomaly equation. The proof combines the quasimodularity results and the holomorphic anomaly equations previously known for local ${\mathbb P}^2$ and the elliptic curve.

Furthermore, using the connection between maximal contact Gromov-Witten invariants of $({\mathbb P}^2,E)$ and Betti numbers of moduli spaces of semistable one-dimensional sheaves on ${\mathbb P}^2$, we obtain a proof of the quasimodularity and holomorphic anomaly equation predicted in the physics literature for the refined topological string free energy of local ${\mathbb P}^2$ in the Nekrasov-Shatashvili limit.

Let $\mathbb {S}^{d-1}$ denote the unit sphere in Euclidean space $\mathbb {R}^d$, $d\geq 2$, equipped with surface measure $\sigma _{d-1}$. An instance of our main result concerns the regularity of solutions of the convolution equation

$$\begin{align*}a\cdot(f\sigma_{d-1})^{\ast {(q-1)}}\big\vert_{\mathbb{S}^{d-1}}=f,\text{ a.e. on }\mathbb{S}^{d-1}, \end{align*}$$

$a\in C^\infty (\mathbb {S}^{d-1})$

$q\geq 2(d+1)/(d-1)$

$f\in L^2(\mathbb {S}^{d-1})$

$C^\infty (\mathbb {S}^{d-1})$

$\mathbb {S}^{d-1}$

$C^\infty $

This article deals with the problem of when, given a collection $\mathcal {C}$ of weakly compact operators between separable Banach spaces, there exists a separable reflexive Banach space Z with a Schauder basis so that every element in $\mathcal {C}$ factors through Z (or through a subspace of Z). In particular, we show that there exists a reflexive space Z with a Schauder basis so that for each separable Banach space X, each weakly compact operator from X to $L_1[0,1]$ factors through Z.

We also prove the following descriptive set theoretical result: Let $\mathcal {L}$ be the standard Borel space of bounded operators between separable Banach spaces. We show that if $\mathcal {B}$ is a Borel subset of weakly compact operators between Banach spaces with separable duals, then for $A \in \mathcal {B}$, the assignment $A \to A^*$ can be realised by a Borel map $\mathcal {B}\to \mathcal {L}$.

There are growing efforts to mine public and common-sense semantic network databases for engineering design ideation stimuli. However, there is still a lack of design ideation aids based on semantic network databases that are specialized in engineering or technology-based knowledge. In this study, we present a new methodology of using the Technology Semantic Network (TechNet) to stimulate idea generation in engineering design. The core of the methodology is to guide the inference of new technical concepts in the white space surrounding a focal design domain according to their semantic distance in the large TechNet, for potential syntheses into new design ideas. We demonstrate the effectiveness in general, and use strategies and ideation outcome implications of the methodology via a case study of flying car design idea generation.

Early childhood education has long-lasting influences on people, and an appropriate companion toy can play an essential role in children's brain development. This paper establishes a complete framework to guide the design of intelligent companion toys for preschool children from 2 to 6 years old, which is child-centered and environment-oriented. The design process is divided into three steps: requirement confirmation, the smart design before the sale, and the iterative update after the sale. This framework considers the characteristics of children and highlights the integration of human and artificial intelligence in design. A case study is provided to prove the superiority of the new framework. In addition to enriching the research on intelligent toy design, this paper also guides for practitioners to design smart toys and helps in children's cognitive development.

Learning from human demonstration (LfD), among many speedup techniques for reinforcement learning (RL), has seen many successful applications. We consider one LfD technique called human–agent transfer (HAT), where a model of the human demonstrator’s decision function is induced via supervised learning and used as an initial bias for RL. Some recent work in LfD has investigated learning from observations only, that is, when only the demonstrator’s states (and not its actions) are available to the learner. Since the demonstrator’s actions are treated as labels for HAT, supervised learning becomes untenable in their absence. We adapt the idea of learning an inverse dynamics model from the data acquired by the learner’s interactions with the environment and deploy it to fill in the missing actions of the demonstrator. The resulting version of HAT—called state-only HAT (SoHAT)—is experimentally shown to preserve some advantages of HAT in benchmark domains with both discrete and continuous actions. This paper also establishes principled modifications of an existing baseline algorithm—called A3C—to create its HAT and SoHAT variants that are used in our experiments.

Statistical significance analysis, based on hypothesis tests, is a common approach for comparing classifiers. However, many studies oversimplify this analysis by simply checking the condition p-value < 0.05, ignoring important concepts such as the effect size and the statistical power of the test. This problem is so worrying that the American Statistical Association has taken a strong stand on the subject, noting that although the p-value is a useful statistical measure, it has been abusively used and misinterpreted. This work highlights problems caused by the misuse of hypothesis tests and shows how the effect size and the power of the test can provide important information for better decision-making. To investigate these issues, we perform empirical studies with different classifiers and 50 datasets, using the Student’s t-test and the Wilcoxon test to compare classifiers. The results show that an isolated p-value analysis can lead to wrong conclusions and that the evaluation of the effect size and the power of the test contributes to a more principled decision-making.

In this paper, we have investigated the relativistic electron acceleration by plasma wave in an axially magnetized plasma by considering the self-magnetic field effects. We show that the optimum value of an external axial magnetic field could increase the electron energy gain more than 40% than that obtained in the absence of the magnetic field. Moreover, results demonstrate that the self-magnetic field produced by the electric current of the energetic electrons plays a significant role in the plasma wakefield acceleration of electron. In this regard, it will be shown that taking into account the self-magnetic field can increase the electron energy gain up to 36% for the case with self-magnetic field amplitude Ωs = 0.3 and even up to higher energies for the systems containing stronger self-magnetic field. The effects of plasma wave amplitude and phase, the ion channel field magnitude, and the electron initial kinetic energy on the acceleration of relativistic electron have also been investigated. A scaling law for the optimization of the electron energy is eventually proposed.

In this paper, we report on the acceleration of protons and oxygen ions from tens of micrometer large water droplets by a high-intensity laser in the range of 1020 W/cm2. Proton energies of up to 6 MeV were obtained from a hybrid acceleration regime between classical Coulomb explosion and shocks. Besides the known thermal energy spectrum, a collective acceleration of oxygen ions of different charge states is observed. 3D PIC simulations and analytical models are employed to support the experiential findings and reveal the potential for further applications and studies.

An elastic-plastic B spline finite strip method is proposed to investigate the continuous plate straightening process in this paper. First, the B spline displacement function that satisfies the boundary conditions of simply supported end and free end of the strip element is established, and then the stress-strain matrix is established. Second, the set method of total stiffness matrix based on B spline finite strip method for plate straightening problem is proposed, and the influence of nodal line number and strip elements number on the sparsity of total stiffness matrix is analyzed. Third, the loads on the strip elements are taken as linear uniform distribution, and the transformation matrix between the equivalent linear load and the actual load of the strip element is established. At last, the plate straightening simulation of 11 rolls straightening machine is made based on the elastic-plastic B spline finite strip method, the calculated results agree with the measured results, which approves that the elastic-plastic B spline finite strip method established can be applied to the plate straightening process.

The article deals with the modeling of stiffness properties of the rotors flange joints, which largely determine overall dynamics. Research is conducted on the example of the standard compressor shaft flange connection and the disk of the high-pressure turbine in the gas generator of the gas turbine engine (GTE). It is noted that the bending stiffness of the flange connection is a nonlinear function of the bending moment, whose both experimental and analysis magnitude is related to the rotor deflection from the unbalanced forces. It is shown that the value of the bending stiffness essentially depends not upon the flange connection geometry but on the bolts tightening force, the axial force, the tensile joint, the contact strain of the flange surfaces. Analysis of the effect obtained in different models of the flange connection of the bending stiffness values on the overall dynamics of the rotor showed the necessity of taking into account the entire set of factors acting in the joint.