Papers

Fractions require learners to revise properties associated with whole numbers. This paper presents a design rationale and a critical analysis of an eight-level source progression for fractions (Grades 1-8, nominal ages 6-14), from which the French-curriculum and International Baccalaureate chapters published at https://commeunjeu.org are assembled. Its central hypothesis is a bridge from sharing to numerical measure: a share of one unit is represented by a bar, read as a length relative to a specified unit, and located on the number line. The reference unit is kept stable in the initial tasks to focus attention on fraction magnitude, equivalence and operations; changing the unit is deliberately postponed, without treating this as a mathematical requirement or an established reduction in cognitive load. Three principles organise the meanings: sharing one unit, coordinating bars and number lines, and establishing fractions as numbers before explicitly treating them as scalar operators. Two supporting principles concern valid local justifications before formal generalisation and chains of equalities, introduced in Grades 2-3, with written reasons in lessons from Grade 4. The analysis relates these choices to the research literature and selected curricula, examines task strategies and feedback, and distinguishes realised features from discrepancies and proposed revisions. The source documents and their curriculum projections do not establish the order actually experienced by pupils. No empirical evaluation is reported. Primary hypotheses address transfer between representations, coordination of units and the numerical status of fractions before operators; the audit also identifies limits to the reproducibility of its findings.

In the classical occupancy problem, $B$ balls are placed independently and uniformly into $d$ boxes. We study a graph analogue of this model. Let $G$ be a directed $r$-regular graph on $d$ vertices and fix a vertex $i$. Consider all paths of length $L$ starting from $i$, and let $N_L(i,j)$ denote the number of such paths ending at a vertex $j$. Thus the vertices play the role of boxes and the $r^L$ paths play the role of balls. In general, these occupancies depend strongly on the geometry of $G$. We show that inserting independent uniform permutations between successive layers restores the classical Poisson behavior at the empirical level. Although the path endpoints remain strongly dependent, their empirical occupancy distribution has the same limiting Poisson law as independent balls thrown into boxes. More precisely, at the critical scale $r^L/d\to\lambda\in(0,\infty)$, provided $rL/d \to 0$, the empirical path multiplicity distribution converges in Wasserstein distance to $\operatorname{Pois}(\lambda)$, independently of the geometry of the underlying graphs. We also prove a local Poisson limit theorem and a Berry-Esseen bound for the empirical distribution functions. These estimates determine the full total-variation transition of the normalized path multiplicities toward uniformity. For arbitrary doubly stochastic kernels, we further establish exact covariance and Frobenius norm identities, together with an exact contraction formula for the chi-square divergence. As applications, we study synchronous opinion averaging, obtaining an explicit cutoff profile for equal-size groups and quantitative convergence to consensus for heterogeneous group sizes.

Geometrically exact beams naturally arise in physical phenomena and engineering practice for modeling flexible structures, such as soft robotic manipulators and DNA structures. To the best of our knowledge, there appear to be no geometric integrators for stochastic geometrically exact beams in a variational formulation. In this paper, we introduce variational principles for the dynamics of stochastic geometrically exact beams, in both continuous- and discrete-time settings. By introducing perturbations via stochastic potentials and stochastic kinematics, we derive the equations of motion from a variational viewpoint. The resulting stochastic geometric integrators simultaneously discretize time and space in a parallel manner, which provides a distributed computational framework for solving the stochastic dynamics of complex beam systems. In particular, the proposed integrators are shown to be a stochastic extension of the well-known deterministic staggered leapfrog scheme for the wave equation, and also a stochastic extension of Yee's scheme for spatially one-dimensional computational electromagnetism. Numerical results are shown to demonstrate the effectiveness of the proposed approaches.

Let $u_k^{(r)}(H,p)$ be the maximum, over all $k$-vertex $r$-uniform hypergraphs $F$, of the absolute difference between the number of induced copies of $F$ in $H$ and its expectation in the binomial random $r$-graph with the same number of vertices as $H$ and edge probability $p$. Let $u_k^{(r)}(n,p)$ be the minimum of $u_k^{(r)}(H,p)$ over all $n$-vertex $r$-uniform hypergraphs, and let $D_r(n,p)$ be the distance from $p\binom nr$ to the nearest integer. For fixed $r\ge3$, $k\ge2r$ and $\varepsilon>0$, the quantity $u_k^{(r)}(n,p)$ has order $\max\!\left\{D_r(n,p),p(1-p)\right\}n^{k-r}$ whenever $p(1-p)n^{\lfloor r/2\rfloor/(\binom{2r-2}{r}-1)}\ge\varepsilon$. Using local moment estimates and switchings in a sparse reservoir, we construct one hypergraph with the nearest-integer edge count that controls all induced orders simultaneously.

We introduce the notion of essential tracial rank at most one for simple $C^*$-algebras. We show that separable simple $C^*$-algebras with essential tracial rank at most one have several regularity properties, including stable rank one, strict comparison, and tracial approximate divisibility. A finite amenable simple $C^*$-algebras with this property is $\mathcal Z$-stable. We also prove the converse. Thus this property gives a new characterization of separable simple finite amenable $\mathcal Z$-stable $C^*$-algebras and shows that these algebras admit a one-dimensional tracial approximation. In the unital real-rank-zero case, this approximation can be strengthened to show that the algebra has tracial rank zero. This completes a long-standing project. As a byproduct, we prove that every tracial state on a unital separable simple finite amenable $C^*$-algebra is quasidiagonal and that the algebra itself is quasidiagonal. Consequently, every separable simple finite amenable $C^*$-algebra of finite nuclear dimension has finite decomposition rank. None of these results requires the UCT.

When an outcome is missing not at random, identification is commonly restored by a shadow variable: a covariate that predicts the outcome but is excluded from the response mechanism. The literature treats the first requirement, relevance, as a yes-or-no condition. We make it continuous, parametrizing it by a scalar that scales the shadow variable's entire contribution to the moment conditions, and ask what inference costs as that scalar vanishes. One functional of the inverse weight governs the answer. We characterize it exactly, show that it fixes the effective sample size, and use it to sharpen the rates of the estimator that avoids the weight. The characterization is concrete. Whether the functional saturates or diverges turns not on boundedness of the outcome but on whether the tilting variable has an atom at its essential supremum: a binary outcome saturates, a uniform one does not. It also bounds how far a scanned set reaches, and that reach grows only as the square root of the logarithm of the sample size, so a thousandfold increase buys one unit of tilt while the set to be scanned widens without limit.

In this article, we present the diagonalization of skew-Hermitian dual quaternion matrices in the sense of unitary equivalence. For a unit quaternion $\phi$, we also obtain the diagonalization result of $\phi$-Hermitian dual quaternion matrices in the sense of unitary $\phi$-congruence, which generalizes the Autonne–Takagi factorization of symmetric complex matrices and $\phi$-Hermitian quaternion matrices.

We solve Brauer's Problem 33 by determining the finite $2$-groups that occur as Sylow $2$-subgroups of finite groups with a single conjugacy class of involutions. Besides the Sylow $2$-subgroups arising from the relevant non-abelian simple groups in Table 1, these are precisely the homocyclic, generalized quaternion, and Suzuki $2$-groups. The classification of finite simple groups is used in this paper.

In this paper we study pairwise compatibility graphs whose witness tree is required to be an unweighted caterpillar. This is a natural restriction of PCGs, in the same line of research as the study of star witnesses, which led to the class of double-threshold graphs. In the caterpillar case, we show that this restriction is equivalent to a one-dimensional difference-labelling problem: a graph admits such a representation if and only if its vertices can be assigned integer labels so that adjacency is determined exactly by whether the absolute difference of two labels belongs to a given interval. Using this characterization, we derive structural properties of the class, including bounded twin-width, and study its relation with familiar graph families. We show that the class contains cycles and two-dimensional grids, while it excludes natural obstructions such as the subdivided star $S_{2,2,2,2,2}$, the cube graph $Q_3$, the wheel $W_6$, and the complement of the $3$-sunlet. We also consider restricted representations, including centipede witnesses and representations with a given interval and we give a polynomial-time recognition algorithm when the interval is fixed.

We study upper tail large deviations for the first passage time in the frog model on $\mathbb Z^d$, $d\ge 2$. Let $\mathrm{T}(0,x)$ denote the first time at which the infection, started from the origin, reaches $x$, and let $\mu$ be the associated time constant. We prove that, for any Euclidean unit vector $v\in\mathbb R^d$ and $\xi>0$, \[ \lim_{n\to\infty} -\frac{1}{\mathfrak{s}_d(n)} \log \mathbb{P}\bigl(\mathrm{T}(0,\lfloor nv\rfloor) \ge (\mu(v)+\xi)n\bigr) =\mathfrak{r}_d\xi, \] where $\mathfrak s_2(n)=n/\log n$ and $\mathfrak r_2=\pi$ for $d=2$; $\mathfrak s_d(n)=n$ and $\mathfrak r_d$ is the escape probability of simple random walk for $d\ge3$. Moreover, on the upper tail large deviation event, we show that a certain slowdown mechanism occurs near the endpoint. To rule out other mechanisms, we analyze the growth of the infection near the starting point using a modified exploration process, and investigate its propagation through the intermediate region using the renormalization and covering arguments.

We developed a staff rostering system for complex employment and staffing requirements at a municipal water utility and deployed it in routine operations at the Water Purification Office of Suita City Waterworks Bureau, Japan, in May 2026. We estimated that preparing a monthly roster took approximately 2-3 h after deployment, compared with 12-18 h previously. The system represents roster states, effective staffing and office assignments using binary and auxiliary integer variables and optimises the entire planning horizon. When the normal model yields no feasible solution, a second stage attempts to generate a reference roster by penalising weekly day-off shortages while retaining the other hard constraints. Outputs identify shortages and meeting-date changes applied before solving. Computational experiments used operational inputs for 14 staff over 245 days. The baseline was solved to optimality in approximately 177 s; a modified meeting-day scenario produced a reference roster with five staff-days of shortage. Independent monthly solving without history produced 16 staff-week cases of excess days off across two month-boundary weeks. This case demonstrates the practical value of optimisation-based decision support for staff rostering in municipal water utilities by showing how mathematical optimisation can be integrated into roster preparation and checking workflows with reduced preparation time.

We consider stable and effective restarts of the SS–RR and SS–RRR methods for computing the eigenpairs of a large Hermitian matrix $A\in\mathbb{C}^{n\times n}$ with the $n_{ev}$ eigenvalues in a given interval. Taking the CJ–SS–RR and CJ–SS–RRR methods for instance, we prove that the commonly used restart scheme and its orthonormalized variant, named the natural restart schemes, are not numerically stable, thereby lowering attainable accuracy of approximate eigenpairs and missing some of the $n_{ev}$ desired eigenpairs. To cure these deficiencies, we propose a new restart scheme, called the combination restart, that takes the $n\times \ell$ starting matrix to be the product of the matrix, which consists of the selected $n_{ev}$ approximate eigenvectors, and an $n_{ev}\times \ell$ Gaussian random matrix. We prove the remarkable mathematical and numerical superiority of the combination restart over the natural restarts, then present a hybrid restart scheme that fully exploits the advantages of the natural restart and combination restart, and finally develop CJ–SS–RR and CJ–SS–RRR algorithms with the hybrid restart. Numerical experiments confirm that the new algorithms are numerically backward stable to the working precision and more efficient than the CJ–SS–RRR algorithm with the natural restart. The combination restart and hybrid restart schemes can be straightforwardly extended to the SS–RR and SS–RRR methods for general standard and generalized eigenvalue problems.

We reduce the problem of establishing pointwise upper decay estimates in thin domains for viscosity subsolutions of fully nonlinear partial differential equations to solving a one-dimensional differential inequality. The resulting estimates make explicit the interaction between the transverse geometry of the domain and the ellipticity of the equation, and are shown to be sharp. Our framework allows the ellipticity to degenerate at infinity and accommodates nonhomogeneous equations with lower-order terms. As an application, we obtain decay estimates for $p$-subharmonic functions when $1<p\leq 2$

Conventional control approaches to optimization problems under uncertainty entail a practical compromise: deterministic controls offer robustness at the expense of pathwise accuracy while stochastic controls can adapt to individual realizations but suffer from the curse of dimensionality. To achieve both robustness and adaptivity, a neural operator-based approach is proposed in this paper for distributed elliptic optimal control problems with high-dimensional random inputs. A data-driven training algorithm is first established to learn the mapping from coefficient realizations directly to their optimal pathwise controls. To bypass the burden of generating training data pairs, a physics-informed training algorithm is then proposed and recast as weakly supervised learning using pseudo label targets produced by standard adjoint-based control updates. Numerical experiments with rough coefficients are conducted to validate the effectiveness and efficiency of our robust-pathwise controls, which outperforms the benchmark robust deterministic control on unseen coefficient realizations owing to the generalization capability and fast inference of trained models.

We maximise expected reward per unit of marginal tail risk, a reward-to-CVaR ratio against a baseline. The standard Dinkelbach route re-solves a Rockafellar–Uryasev LP or QP with one auxiliary variable per tail scenario at each multiplier update. Our method eliminates the multiplier iteration and keeps scenario auxiliaries out of the master, while still evaluating tail risk to generate cuts. A Charnes–Cooper/Schaible homogenisation removes the reward/risk multiplier, and the remaining convex constraint is separated on demand: each cut coefficient is an Euler allocation vector, read off a ranking and an average. The result is a single convex program, finitely terminating at the global optimum, with size independent of S — an LP in $M{+}1$ variables, or a second-order cone program in $M{+}2$ for a concave quadratic numerator. The method runs 20–100 times faster than the exact Dinkelbach formulation and only needs single-digit cut counts.

We study stable solutions of the mean curvature equation $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) =|u|^{p-1}u $$ in domains of $\mathbb R^n$. We establish Liouville-type results in three settings. First, for globally stable solutions in $\mathbb R^n$, we prove that $u\equiv0$ if $1<p<p_{JL}$, where $p_{JL}$ is the Joseph–Lundgren exponent. Second, for solutions in $\mathbb R^n$ that are stable outside a compact set, we show that if $n>2$ and $p=p_S$, where $p_S$ is the Sobolev critical exponent, then $u\equiv0$. Third, for solutions of the Dirichlet problem on connected unbounded domains with smooth boundary satisfying an epigraph-type geometric condition, we prove that if $n>2$, $1<p\le p_S$, and the solution is stable outside a compact set, then $u\equiv0$. The proofs are based on a gradient estimate for stable solutions, an integrability result, conservation laws and Pohozaev-type identities, and a unique continuation argument. These results extend Liouville-type theorems for stable solutions from the semilinear Lane–Emden equation to the mean curvature equation.

We study a forward-backward McKean-Vlasov system with jumps whose backward equation contains a source obtained by evaluating the minimal-norm selection of a maximal monotone operator along the forward state. This selection changes branch across an interface that moves with the state law, the control, and an imprecisely known parameter described by fuzzy cuts. Assuming normal non-degeneracy and finite-statistics dependence of the interface on the law, we prove that the first and second moments of the time spent within a given distance of the interface are bounded, respectively, by a constant times that distance and by a constant times its square. The second-moment estimate makes it possible to integrate the source error before squaring it, which yields a Yosida error of first order in the regularization parameter in the natural solution norm; a first-moment argument gives only order one half. An explicit example shows that the first-order rate is sharp. For compact classes of admissible selections, the corresponding solutions form the cuts of a fuzzy solution set. These cuts are compact, stable with respect to the fuzzy level and the data, and approximated at the same first-order rate in Hausdorff distance. All estimates are uniform over a non-dominated family of probability measures representing volatility uncertainty. The quantitative results are obtained for separable quadratic potentials with finitely many affine branches.

A receiver can change its observation state while the physical signal remains the same. In rate-splitting multiple access (RSMA) downlinks, this distinction separates receive-beam design and phase-reference transport from the motion of the satellite–UAV line of sight. We formulate receive optimization within the physical–observation dual-axis (PODA) structure and obtain three complementary results. For a known line-of-sight response and positive-definite receiver-noise covariance, a noise-weighted matched combiner simultaneously maximizes every decoding SINR and the minimum user rate. Under directional uncertainty, the principal eigenvector of the conditional response covariance globally maximizes mean array gain; for a Gaussian directional model, an equal-amplitude receiver is optimized by steering towards the conditional mean. Mean rate requires a different objective. We prove that the RSMA common-rate water level is jointly concave in the users' receive gains and derive a semidefinite relaxation with a computable global upper bound. A feasible rank-one beam can therefore be compared directly with the optimum of the finite-scenario problem. We also derive a precision-weighted linear predictor that suppresses uncertain physical-velocity estimates after observation-frame transport. Paired dynamic simulations reduce direction-cosine RMSE by 51.29% relative to unrestricted nine-point extrapolation. In a separate uncertainty experiment, direct rate optimization increases mean minimum spectral efficiency from 0.32425 to 0.41712 bit/s/Hz, an improvement of 28.64%; the corresponding design objective is within $2.28\times10^{-4}$ bit/s/Hz of a global upper bound. The phase-transport equation fixes a coherent representative of each optimal beam without changing its gain.

Let $k$ be a perfect field and let $A$ be a finite-dimensional representation-finite $k$-algebra. We prove that finite global dimension implies that the integral Cartan determinant of $A$ is one, extending Enomoto's result from algebraically closed fields to perfect fields.

The reflection complexity of a sequence, introduced recently by Allouche, Campbell, Li, Shallit, and Stipulanti, counts the factors of a given length up to reversal. We generalize this notion considering a group $G$ generated by antimorphisms over a given alphabet $\mathcal A$. The $G$-complexity of a sequence over $\mathcal A$ counts the factors of a given length up to the action of $G$. For $G$ generated by the reflection alone, the $G$-complexity coincides with the reflection complexity. We characterize eventual periodicity in terms of the $G$-complexity. Then we focus on the binary alphabet and the group $H$ generated by reflection and the antimorphism exchanging letters. We provide a simple formula for the $H$-complexity in the case of $H$-rich sequences and determine the precise values for complementary symmetric Rote sequences and generalized binary Thue-Morse sequences. Finally, we describe the aperiodic recurrent sequences having the minimum $H$-complexity – they are exactly the complementary symmetric Rote sequences, while the binary sequences with the minimum reflection complexity correspond to the Sturmian sequences.

In 2010, M. A. Chaudhry et al. introduced a class of functions Sigma satisfying an L^1-condition. They proved that every function phi in Sigma gives rise to the functional equation for the Riemann zeta function. They also provided an example of a function outside Sigma for which the functional equation still holds. In this paper, we present several examples of Bessel functions of order zero that do not belong to Sigma but nevertheless give rise to the functional equation for the Riemann zeta function. In particular, we show that J_0(a sqrt(x)), J_0(ax), and Y_0(x) exhibit this phenomenon.

Many applications in science and engineering require solving sparse Hermitian eigenvalue problems that are so large that even algorithms whose runtime scales linearly with the matrix dimension are unacceptably slow. Responding to this challenge, there is a growing body of work on sparsified iterative methods and submatrix selection methods that often perform well in practice. However, theoretical understanding of these methods is limited. In this paper, we take a step toward better understanding sublinear-time eigenvalue computation by analyzing the sparsified power method. We prove that the method approximates a leading eigenpair to a prescribed accuracy in a number of iterations comparable to the standard power method. The required number of retained nonzeros depends explicitly on the target accuracy, the initial overlap with the leading eigenvector, the compressibility of the leading eigenvector, and the relative spectral gap. When these quantities and the column sparsity are independent of dimension, the runtime is independent of dimension. In contrast, we show that a comparable worst-case guarantee cannot hold for a prototypical sparsified submatrix selection method. Submatrix selection is the approach of choice in many quantum chemistry applications. However, we show that, even with substantial initial eigenvector overlap, a single iteration of the method can produce a vector that is orthogonal to the leading eigenvector, after which the method cannot recover. Nevertheless, experiments on matrices from the SuiteSparse matrix collection and quantum chemistry problems illustrate the practical effectiveness of submatrix selection methods in a variety of domains despite these worst-case limitations.

We study random contingency tables obtained by conditioning a matrix of independent Poisson entries on its row and column sums. We identify such a table with the block counts of a random permutation drawn from a Gibbs measure of Mallows type, an exact coupling whose normalizing constants recover Barvinok's permanent identity. We use this correspondence to study the table at three scales. For finite dimensions, a transference principle compares the table with a matrix of independent Poisson entries whose means form the static Schrödinger bridge, the solution of a discrete entropic optimal transport problem. As the dimensions grow and the margins and prior means converge, the normalized histogram of the table converges to a limiting Schrödinger bridge at a quantitative rate. The fluctuations of the table around the bridge are characterized in two complementary ways. Linear statistics satisfy a central limit theorem for bounded test functions, with limiting variance equal to the squared $L^2$-distance, under the bridge, of the test function to the subspace of additively separable functions; as a process they converge to white noise with intensity given by the bridge, conditioned to have vanishing marginals. Spectrally, the limiting spectral distribution of the centered table is governed by a Dyson equation whose variance profile is the density of the bridge. The key ingredient is an asymptotic formula for the partition function of the Gibbs measure in terms of a Fredholm determinant given by the canonical correlations of the bridge. For Mallows-type random permutations, our results prove Pal's conjecture on the limiting partition function and Mukherjee's conjecture on the central limit theorem for linear statistics, both in greater generality.

We study autoregressive token generation through the notion of sequential samplers: A circuit that samples the next symbol only given the previous ones and possibly a small amount of state. We show that (1) a sequential sampler with $k$ bits of state can be simulated by a stateless one at a roughly $O(2^{k})$ size overhead; (2) efficient insertional sampling (Stern, Chan, Kiros, Uszkoreit, ICML 2019) is more powerful than efficient sequential sampling in any fixed order; (3) sequential samplers with small state cannot establish certain forms of covert cryptographic channels. Result (1), together with the pseudoentropy vs. sequential sampleability duality of Vadhan and Zheng (STOC 2012), characterizes efficient sequential sampleability with small state. Result (3) provides a potential safety mechanism against subliminal coordination among malicious LLM agents.

The focus of this article is to establish the existence portion of the Strauss conjecture for wave equations with spatially dependent scattering damping. In the current setting, the damping is not sufficiently strong to alter the critical Strauss exponent. In a subsequent study, the authors will consider the case of damping coefficients with critical decay where a shift of the Strauss exponent is expected. While several recent studies of the Strauss conjecture have relied upon weighted Strichartz estimates, here, instead, an approach that is based only on integrated local energy estimates, in particular those with both $r^p$ and ghost weights, is developed. This technique also applies to the problem for the free equation. Space-time Klainerman-Sobolev estimates provide the necessary decay to close the argument.

In this paper, by normalizing the trace-zero norm forms according to the parity of orders of definite rational quaternion algebras, we give an explicit formulation to the even Clifford correspondence followed by the reciprocity of ternary forms. We obtain an exact identity between ternary representation numbers and the number of elements of an order with prescribed reduced norm and trace, which applies to all orders without squarefreeness constraint on the reduced discriminant. Applications include a criterion for realizing ternary forms as trace-zero norm forms, a formula for traces of Brandt matrices in terms of representation numbers, and explicit theta series relations associated with local reciprocity. We also describe reciprocal orders and order chains, and interpret the ternary forms occurring in the congruent number problem through their associated quaternion orders.

The Strauss conjecture concerns wave equations with power-type nonlinearities and seeks the optimal range of powers that admit global solutions for all sufficiently nice and sufficiently small initial data. In three spatial dimensions, it is well known that small data global existence is available provided that the power $p>p_c(3)=1+\sqrt{2}$, where $p_c(n)$ denotes the critical power in $n$-spatial dimensions. In the current article, the analogous problem is considered for damped wave equations where the damping coefficient is at the critical scaling. When $a$ denotes the constant multiple applied to the damping term, it is shown that the corresponding critical power is $p_c(3+a)$ provided that $a$ is sufficiently small.

In these lecture notes we review some constructions and results on the relation between topology of the moduli space of stable curves and integrable systems of evolutionary PDEs.

We consider an elliptic Cauchy problem for the Poisson equation in which both Dirichlet and Neumann boundary data are prescribed only on a portion of the boundary, while no boundary conditions are available on the remaining part of the boundary. Such problems are severely ill-posed. In this work, we study the convergence of an iterative method based on the alternating imposition of Dirichlet and Neumann boundary conditions. We formulate the scheme within a Hilbert space framework, where a yet another alternation becomes apparent: each iteration can now be interpreted as an orthogonal projection onto an appropriate closed subspace. This perspective provides a transparent geometric interpretation of the algorithm and yields a concise convergence proof based on von Neumann's classical result on alternating (orthogonal) projections. In addition to elucidating the convergence of the method, this alternating projections perspective also offers a general rate of convergence in terms of the Friedrichs angle (between two closed subspaces). We point out by means of an example that such convergence rate estimate degenerates for the elliptic Cauchy problem under consideration.

We study products $B_{\alpha,m}^{(d)}B_{\beta,m}^{(d)}$ of Berezin transforms on generalized Fock spaces over $\mathbb C^d$. We establish a result for $m\neq2$: if $B_{\alpha,m}^{(d)}B_{\beta,m}^{(d)}$ coincides with another such product, then the weight parameters of the second product must be $(\alpha,\beta)$. As consequences, we characterize when two Berezin transforms commute and when the product of two Berezin transforms is again a Berezin transform.