In this paper, we define θ -orthotomicoids of spherical curves in spherical space. The relationship between θ -orthotomicoids and θ -evolutoids of spherical curves is established. We generalize the notion to the category of spherical frontals and find that θ -orthotomicoids of spherical frontals may have singularities, while θ -orthotomicoids of spherical unit speed curves are regular. Then, we investigate properties of θ -orthotomicoids of spherical frontals by using the singularity theory. From the viewpoint of the contact geometry, we prove the singularities of θ -orthotomicoids are deeply related to the order of contact between θ -orthotomicoids and specific circles on the unit sphere. Finally, we provide some examples to demonstrate main results.
Graph generation has attracted considerable attention in recent years. Most existing work focuses on generating graphs from collections, such as proteins, while the problem of generating a proxy from a single graph with comparable properties remains underexplored. Prior approaches often emphasize modifying neural architectures or training on edges with fixed node sets, which tends to produce high edge overlap and limited diversity. In addition, many neural graph generation methods overlook the mathematical structure of the input graph. For instance, key metrics such as wedge counts, which are determined by degree distributions, are often neglected. To address these limitations, we introduce a graph generation method that incorporates graph-theoretic principles into the learning process. Our approach preserves both global and local characteristics of the input graph while correcting the degree distribution to avoid duplicating the original topology. Unlike other single-graph methods that fix the node set, we generate new node embeddings and sample fresh node sets, yielding new connection topologies with substantially lower edge overlap.
Using recently developed algorithms, we compute and compare best L^2 and L^∞ rational approximations of analytic functions on the unit disk. Although there is some theory for these problems going back decades, this may be the first computational study. To compute the L^2 best approximations, we employ a new formulation of TF-IRKA in barycentric form.
We introduce a parameterized computational framework for the evolution of strategic behavior in continuous games. We consider the collective dynamics of players through a time-dependent probability density over the strategy space, representing the likelihood of each strategy being chosen at any given time. Instead of directly solving the high-dimensional Fokker–Planck equation that governs this evolution, we represent the probability density as the pushforward of a reference distribution with parameterized pushforward maps and consider the evolution of the parameterized equations. The motivation for this work comes from a limitation of the parameterized Wasserstein gradient flow (PWGF) framework [14] when it is applied to game dynamics. PWGF provides a parameterized approach for evolution equations of probability density that can be formulated as Wasserstein gradient flows. However, not all game dynamics admit such a gradient flow formulation. We generalize the parameterized pushforward map framework to the non-gradient flows and apply it to the game dynamics with provable error bound in Wasserstein metric. Numerical experiments with several non-gradient systems in economics are provided to demonstrate the effectiveness of this new framework.
Abstract We determine the action of the Hecke operators $$T_{\mathfrak {p},i}$$ T p , i on the coefficient forms $$g_{1}, \ldots , g_{r-1}, g_{r} = \Delta $$ g 1 , … , g r - 1 , g r = Δ , and h , which together generate the ring of modular forms for $${{\,\textrm{GL}\,}}(r, \mathbb {F}_{q}[T])$$ GL ( r , F q [ T ] ) . All these are eigenforms with powers of $$\pi $$ π as eigenvalues, where $$\pi $$ π is the monic generator of the prime ideal $$\mathfrak {p}$$ p of $$\mathbb {F}_{q}[T]$$ F q [ T ] . We further describe the growth of the t -expansion coefficients of the discriminant function $$\Delta $$ Δ . It is such that the product expansion of $$\Delta $$ Δ as well as the t -expansion of each modular form converges on the natural fundamental domain for $${{\,\textrm{GL}\,}}(r, \mathbb {F}_{q}[T])$$ GL ( r , F q [ T ] ) .
Identifying the precise boundary between quantum and classical computational power is a central challenge in quantum computing. We introduce polyhedral classical simulators, a framework for classical simulation grounded in polyhedral geometry. This framework encompasses well-known methods such as the Gottesman–Knill algorithm, while extending naturally to more recent models including quantum computation with magic states and measurement-based quantum computation. The framework is compositional: The correctness of a simulation reduces to verifying a preservation property for the basic instruments of the model, from which the full adaptive simulation is assembled. Beyond unifying existing simulation methods, this provides a geometric roadmap for pushing the boundary of efficient classical simulation further.
The theory of descent, originating from the work of Évariste Galois and further developed by mathematicians such as Weil, Grothendieck, and Janelidze, has become a fundamental tool in algebraic geometry and category theory. This paper presents a new categorical framework for Galois descent and establishes a Galois descent theorem.
We consider generalized quantifiers which are either closed under embeddings or closed under monomorphisms or consist of structures that contain a substructure in a fixed Σ ^1_2 -definable set (which we call a Σ ^1_2 -quantifier). We study the relativization of these quantifiers with respect to these closure properties. We also consider when formulas built from such quantifiers are absolute with respect to these closure properties. We show that formulas built from embedding-closed or monomorphism-closed quantifiers need not be absolute (with respect to these definitions), whereas formulas built from Σ ^1_2 -quantifiers are absolute (with respect to the Σ ^1_2 -definitions). The latter result makes use of an absoluteness result, which may be of independent interest.
W. Kohnen introduced kernel functions to study the nonvanishing of L-functions attached to Hecke eigenforms. Y. Martin defined L-functions for Jacobi forms of arbitrary index and studied the analytic properties of these L-functions. In this paper, we study the nonvanishing of L-functions and Poincaré series for Jacobi forms defined on ℋ×ℂ^g,1 using kernel functions.
We introduce a quadratic form Q on the space of functions on the gap poset G of the numerical semigroup ⟨ a,b⟩ . We prove combinatorially that when evaluated on the indicator function of an upward closed subset D, this quadratic form precisely recovers the Gorsky–Mazin statistic of D, viewed as a Young subdiagram of G. Furthermore, we prove Theorem 1.2 that when evaluated on a pair of subdiagrams of G, the symmetric bilinear form associated with Q is equal to a novel cross- statistic, which is non-negative. Combining these, we prove the inequality Q(𝐧)≥1|G| ‖𝐧‖ _∞ ^2 if 𝐧 is a real-valued decreasing function on G, showing an effective positive definiteness of Q on the corresponding cone. Theorem 1.2, the main engine of the paper, was autoformalized in Lean/Mathlib by AxiomProver.
For a fixed tracial unital Banach * -probability space ( A,τ) , we constructed the corresponding definite or indefinite inner product space ( A_0,[ ,] _τ) , where A_0=A/ker( τ) is the quotient Banach space and [ ,] _τ is a definite or indefinite inner product on A_0 induced by the trace τ on A. The A_0 -Hardy space H_A_0:2( D_1) was constructed and adjointable Block Toeplitz Banach space operators over A_0 acting on H_A_0:2( D_1) were studied in Cho (Block-Toeplitz operators on the hardy space induced by a Tracial Unital Banach * -probability space, 2024, to submitted). In this paper, we are interested in the cases where ( A,τ) is a free product Banach * -probability space, k∈Λ⋆( A_k,τ _k) of Banach * -probability spaces {( A_k,τ _k) } _k∈Λ of A, where Λ is a countable (finite or infinite) index set. As applications, we consider a case where such A is a unital Banach * -algebra generated by mutually free multi-semicircular elements.
This paper presents an analysis of lower bounds on local criticality within the family of piecewise quadratic reversible centers. Through a study of perturbations within this class, we determine that a minimum of five (in the linear case) or four (in the nonlinear case) local critical periods can bifurcate from the isochronous center. Notably, five represents the highest level of weakness observed in this context.
For a given finite extension K over ℚ , let L/K be a finite Galois extension with Galois group G. Then, by the normal basis theorem, there exists α∈ L such that L = K[G] ·α , where K[G] is the group ring. Such an element α is called a normal basis generator. We say α∈ L is a completely normal basis generator, if α is a normal basis generator for L/F for every intermediate field F such that K⊂ F⊂ L . In this article, we prove the following result. Let L/K be a finite Galois extension of number fields with Galois group G. If L⊂ℝ and G is an abelian group or dihedral group, then there exists a Pisot–Vijayaraghavan number α∈ L such that for any natural number m, α ^m is a primitive element as well as a completely normal basis generator of L/K. As an application of our result, we prove the following upper bound for the index, when L = K and K = ℚ in the above result, to get [𝒪_K: ℤ[G]·α ] ≤( τ ^n-1|d_K|^1/2 - 1/2n+1) ^n where n = [K:ℚ] , τ =ϕ (n)n >1 when G is abelian and τ = n+1 when G is dihedral group of order n. We also classify the extension of prime degree related to this. We use resolvents and technique from geometry of numbers.
Morse functions on a closed manifold M need not realize the minimal number of critical points of smooth functions on M since several critical points often may be fused into a single degenerate one. We study conditions under which critical points of smooth functions on a manifold M of dimension 3 can be fused. We say that a function f:M→ [a, c] on a compact manifold is regular if the boundary of M is the union of f^-1(a) and f^-1(c) , and f has no critical points on ∂ M . We prove that if f^-1(a) is a sphere and f^-1(c) is a non-empty surface, then the Conley index is a complete obstruction to fusing critical points of a regular function. On the other hand, we prove that the Conley index is not a complete obstruction to fusing critical points if, for example, M is F_0,3× S^1 , where F_0,3 is a 2-sphere with the interior of three disjoint disks removed.
For every nuclear Zl-algebra Lambda and every small v-stack X on perfectoid spaces, we construct an infinity-category Dnuc(X,Lambda) of nuclear (i.e., "ind-Banach") Lambda-modules on X. We then construct a full 6-functor formalism for these sheaves, generalizing the & eacute;tale 6-functor formalism for Lambda=Fl. Prominent choices for Lambda l are Zl, Ql and Ql. We also provide and study an abstract notion of ULA sheaves in this setting, whose definition and basic properties can be carried over to any 6-functor formalism. Applied to classifying stacks, we obtain a robust theory of nuclear representations, i.e., continuous representations on filtered colimits of Banach spaces.
