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The question of when a vertex algebra is a quantization of the arc space of its associated scheme has recently received a lot of attention in both the mathematics and physics literature. This property was first studied by Tomoyuki Arakawa and Anne Moreau (see their paper in the references), and was given the name \lq\lq classical freeness" by Jethro van Ekeren and Reimundo Heluani [Comm. Math. Phys. 386 (2021), no. 1, pp. 495-550] in their work on chiral homology. Later, it was extended to vertex superalgebras by Hao Li [Eur. J. Math. 7 (2021), pp. 1689–1728]. In this note, we prove the classical freeness of the simple affine vertex superalgebra L_n(\mathfrak {o}\mathfrak {s}\mathfrak {p}_{m|2r}) for all positive integers m,n,r satisfying -\frac {m}{2} + r +n+1 > 0. In particular, it holds for the rational vertex superalgebras L_n(\mathfrak {o}\mathfrak {s}\mathfrak {p}_{1|2r}) for all positive integers r,n. [ABSTRACT FROM AUTHOR]

In this paper we resolve a conjecture of Kresch and Tamvakis [Duke Math. J. 110 (2001), pp. 359–376]. Our result is the following. Theorem : For any positive integer D and any integers i,j (0\leq i,j \leq D), \; the absolute value of the following hypergeometric series is at most 1: \begin{equation*} {_4F_3} \left [ \begin {array}{c} -i, \; i+1, \; -j, \; j+1 \\ 1, \; D+2, \; -D \end{array} ; 1 \right ]. \end{equation*} To prove this theorem, we use the Biedenharn-Elliott identity, the theory of Leonard pairs, and the Perron-Frobenius theorem. [ABSTRACT FROM AUTHOR]

Let (T_n)_{n\in {\mathbb Z}} be the Tribonacci sequence and for a prime p and an integer m let \nu _p(m) be the exponent of p in the factorization of m. For p=2 Marques and Lengyel found some formulas relating \nu _p(T_n) with \nu _p(f(n)) where f(n) is some linear function of n (which might be constant) according to the residue class of n modulo 32 and asked if similar formulas exist for other primes p. In this paper, we give an algorithm which tests whether for a given prime p such formulas exist or not. When they exist, our algorithm computes these formulas. Some numerical results are presented. [ABSTRACT FROM AUTHOR]

A chain complex can be viewed as a representation of a certain self-injective quiver with relations, Q. To define Q, include a vertex q_n and an arrow q_n \xrightarrow {\partial } q_{n-1} for each integer n. The relations are \partial ^2 = 0. Replacing Q by a general self-injective quiver with relations, it turns out that some of the key properties of chain complexes generalise. Indeed, consider the representations of such a Q with values in {}_{A}\mspace {-1mu}\operatorname {Mod} where A is a ring. We showed in earlier work that these representations form the objects of the Q-shaped derived category , \mathcal {D}_{Q}(A), which is triangulated and generalises the classic derived category \mathcal {D}_{}(A). This follows ideas of Iyama and Minamoto. While \mathcal {D}_{Q}(A) has many good properties, it can also diverge dramatically from \mathcal {D}_{}(A). For instance, let Q be the quiver with one vertex q, one loop \partial, and the relation \partial ^2 = 0. By analogy with perfect complexes in the classic derived category, one may expect that a representation with a finitely generated free module placed at q is a compact object of \mathcal {D}_{Q}(A), but we will show that this is, in general, false. The purpose of this paper, then, is to compare and contrast \mathcal {D}_{Q}(A) and \mathcal {D}_{}(A) by investigating several key classes of objects: Perfect and strictly perfect, compact, fibrant, and cofibrant. [ABSTRACT FROM AUTHOR]

The object of this paper is to provide a new and systematic tauberian approach to quantitative long time behaviour of C_{0}-semigroups \left (\mathcal {V}(t)\right)_{t \geqslant 0} in L^{1}(\mathbb {T}^{d}\times \mathbb {R}^{d}) governing conservative linear kinetic equations on the torus with general scattering kernel \boldsymbol {k}(v,v') and degenerate (i.e. not bounded away from zero) collision frequency \sigma (v)=\int _{\mathbb {R}^{d}}\boldsymbol {k}(v',v)\boldsymbol {m}(\mathrm {d}v'), (with \boldsymbol {m}(\mathrm {d}v) being absolutely continuous with respect to the Lebesgue measure). We show in particular that if N_{0} is the maximal integer s \geqslant 0 such that \begin{equation*} \frac {1}{\sigma (\cdot)}\int _{\mathbb {R}^{d}}\boldsymbol {k}(\cdot,v)\sigma ^{-s}(v)\boldsymbol {m}(\mathrm {d}v) \in L^{\infty }(\mathbb {R}^{d}), \end{equation*} then, for initial datum f such that \displaystyle \int _{\mathbb {T}^{d}\times \mathbb {R}^{d}}|f(x,v)|\sigma ^{-N_{0}}(v)\mathrm {d}x\boldsymbol {m}(\mathrm {d}v) <\infty it holds \begin{equation*} \left \|\mathcal {V}(t)f-\varrho _{f}\Psi \right \|_{L^{1}(\mathbb {T}^{d}\times \mathbb {R}^{d})}=\dfrac {{\varepsilon }_{f}(t)}{(1+t)^{N_{0}-1}}, \qquad \varrho _{f}≔\int _{\mathbb {R}^{d}}f(x,v)\mathrm {d}x\boldsymbol {m}(\mathrm {d}v), \end{equation*} where \Psi is the unique invariant density of \left (\mathcal {V}(t)\right)_{t \geqslant 0} and \lim _{t\to \infty }{\varepsilon }_{f}(t)=0. We in particular provide a new criteria of the existence of invariant density. The proof relies on the explicit computation of the time decay of each term of the Dyson-Phillips expansion of \left (\mathcal {V}(t)\right)_{t \geqslant 0} and on suitable smoothness and integrability properties of the trace on the imaginary axis of Laplace transform of remainders of large order of this Dyson-Phillips expansion. Our construction resorts also on collective compactness arguments and provides various technical results of independent interest. Finally, as a by-product of our analysis, we derive essentially sharp "subgeometric" convergence rate for Markov semigroups associated to general transition kernels. [ABSTRACT FROM AUTHOR]

In 2000, Héthelyi and Külshammer [Bull. London Math. Soc. 32 (2000), pp. 668–672] proposed that if G is a finite group, p is a prime dividing the group order, and k(G) is the number of conjugacy classes of G, then k(G)\geq 2\sqrt {p-1}, and they proved this conjecture for solvable G and showed that it is sharp for those primes p for which \sqrt {p-1} is an integer. This initiated a flurry of activity, leading to many generalizations and variations of the result; in particular, today the conjecture is known to be true for all finite groups. In this note, we put forward a natural new and stronger conjecture, which is sharp for all primes p, and we prove it for solvable groups, and when p is large, also for arbitrary groups. [ABSTRACT FROM AUTHOR]

The Wallis power function is defined on \left (-\min \left \{ p,q\right \},\infty \right) by \begin{equation*} W_{p,q}\left (x\right) =\left (\dfrac {\Gamma \left (x+p\right) }{\Gamma \left (x+q\right) }\right) ^{1/\left (p-q\right) }\text { if }p\neq q\text { and }W_{p,p}\left (x\right) =e^{\psi \left (x+p\right) }. \end{equation*} Let p_{i},q_{i}\in \mathbb {R} with 0<\delta _{i}=p_{i}-q_{i}\leq 1, \theta _{i}=\left (1-\delta _{i}\right) /2 for i=1,2 and p_{1}+q_{1}=p_{2}+q_{2}=2\sigma +1. We prove that, if q_{1}>q_{2} then for any integer m\in \mathbb {N}, the function \begin{equation*} x\mapsto \left (-1\right) ^{m}\left [ \ln \frac {W_{p_{1},q_{1}}\left (x\right) }{W_{p_{2},q_{2}}\left (x\right) }-\sum _{k=1}^{m}\frac {a_{2k}\left (\theta _{1},\theta _{2}\right) }{\left (2k+1\right) \left (2k\right) \left (x+\sigma \right) ^{2k}}\right ] \end{equation*} is completely monotonic on \left (-\sigma,\infty \right), where \begin{equation*} a_{2k}\left (\theta _{1},\theta _{2}\right) =\frac {B_{2k+1}\left (\theta _{2}\right) }{\theta _{2}-1/2}-\frac {B_{2k+1}\left (\theta _{1}\right) }{\theta _{1}-1/2}. \end{equation*} This extends and generalizes some known results. [ABSTRACT FROM AUTHOR]

Starting from billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in d-dimensional Euclidean space, we introduce a new type of separable integer partition classes, called type B. We study the numbers of basis partitions with d parts and relate them to the Fibonacci sequence and its natural generalizations. Remarkably, the generating series of basis partitions can be related to the quiver generating series of symmetric quivers corresponding to the framed unknot via knots-quivers correspondence, and to the count of Schröder paths. [ABSTRACT FROM AUTHOR]

Motivated by the many roles that hook lengths play in mathematics, we study the distribution of the number of t-hooks in the partitions of n. We prove that the limiting distribution is normal with mean \[ \mu _t(n)\sim \frac {\sqrt {6n}}{\pi }-\frac {t}{2} \] and variance \[ \sigma _t^2(n)\sim \frac {(\pi ^2-6)\sqrt {6n}}{2\pi ^3}. \] Furthermore, we prove that the distribution of the number of hook lengths that are multiples of a fixed t\geq 4 in partitions of n converge to a shifted Gamma distribution with parameter k=(t-1)/2 and scale \theta =\sqrt {2/(t-1)}. [ABSTRACT FROM AUTHOR]

A new highly symmetrical model of the compact Lie algebra \mathfrak {g}^c_2 is provided as a twisted ring group for the group \mathbb {Z}_2^3 and the ring \mathbb {R}\oplus \mathbb {R}. The model is self-contained and can be used without previous knowledge on roots, derivations on octonions or cross products. In particular, it provides an orthogonal basis with integer structure constants, consisting entirely of semisimple elements, which is a generalization of the Pauli matrices in \mathfrak {su}(2) and of the Gell-Mann matrices in \mathfrak {su}(3). As a bonus, the split Lie algebra \mathfrak {g}_2 is also seen as a twisted ring group. [ABSTRACT FROM AUTHOR]

In this note we present a construction of an infinite family of diagonal quintic threefolds defined over \mathbb {Q} each containing infinitely many rational points. As an application, we prove that there are infinitely many quadruples B=(B_{0}, B_{1}, B_{2}, B_{3}) of co-prime integers such that for a suitable chosen integer b (depending on B), the equation B_{0}X_{0}^5+B_{1}X_{1}^5+B_{2}X_{2}^5+B_{3}X_{3}^{5}=b has infinitely many positive integer solutions. [ABSTRACT FROM AUTHOR]

We provide an algorithm to compute generators of the orthogonal group of the discriminant group associated to an integral quadratic lattice over the integers. We give a closed formula for its order. [ABSTRACT FROM AUTHOR]

We prove a general result concerning the paucity of integer points on a certain family of 4-dimensional affine hypersurfaces. As a consequence, we deduce that integer-valued polynomials have small asymmetric additive energy. [ABSTRACT FROM AUTHOR]

Say that (x, y, z) is a positive primitive integral Pythagorean triple if x, y, z are positive integers without common factors satisfying x^2 + y^2 = z^2. An old theorem of Berggren gives three integral invertible linear transformations whose semi-group actions on (3, 4, 5) and (4, 3, 5) generate all positive primitive Pythagorean triples in a unique manner. We establish an analogue of Berggren's theorem in the context of a one-variable polynomial ring over a field of characteristic \neq 2. As its corollaries, we obtain some structure theorems regarding the orthogonal group with respect to the Pythagorean quadratic form over the polynomial ring. [ABSTRACT FROM AUTHOR]

Let f^{(r)}(n;s,k) be the maximum number of edges of an r-uniform hypergraph on n vertices not containing a subgraph with k edges and at most s vertices. In 1973, Brown, Erdős and Sós conjectured that the limit \begin{equation*} \lim _{n\to \infty } n^{-2} f^{(3)}(n;k+2,k) \end{equation*} exists for all positive integers k\ge 2. They proved this for k=2. In 2019, Glock proved this for k=3 and determined the limit. Quite recently, Glock, Joos, Kim, Kühn, Lichev and Pikhurko proved this for k=4 and determined the limit; we combine their work with a new reduction to fully resolve the conjecture by proving that indeed the limit exists for all positive integers k\ge 2. [ABSTRACT FROM AUTHOR]

We present new algorithms for computing the low n bits or the high n bits of the product of two n-bit integers. We show that these problems may be solved in asymptotically 75% of the time required to compute the full 2n-bit product, assuming that the underlying integer multiplication algorithm relies on computing cyclic convolutions of sequences of real numbers. [ABSTRACT FROM AUTHOR]

The set of degrees of maps D(M,N), where M,N are closed oriented n-manifolds, always contains 0 and the set of degrees of self-maps D(M) always contains 0 and 1. Also, if a,b\in D(M), then ab\in D(M); a set A\subseteq \mathbb {Z} so that ab\in A for each a,b\in A is called multiplicative. On the one hand, not every infinite set of integers (containing 0) is a mapping degree set (Neofytidis, Wang, and Wang [Bull. Lond. Math. Soc. 55 (2023), pp. 1700–1717]) and, on the other hand, every finite set of integers (containing 0) is the mapping degree set of some 3-manifolds (Costoya, Muñoz and Viruel [ Finite sets containing zero are mapping degree sets , arXiv: 2301.13719 ]). We show the following: Not every multiplicative set A containing 0,1 is a self-mapping degree set. For each n\in \mathbb {N} and k\geq 3, every D(M,N) for n-manifolds M and N is D(P,Q) for some (n+k)-manifolds P and Q. As a consequence of (ii) and Costoya, Muñoz and Viruel, every finite set of integers (containing 0) is the mapping degree set of some n-manifolds for all n\neq 1,2,4,5. [ABSTRACT FROM AUTHOR]

We compute the Chow rings with integral coefficients of moduli stacks of minimal Weierstrass fibrations over the projective line. For each integer N\geq 1, there is a moduli stack \mathcal {W}^{\mathrm {min}}_N parametrizing minimal Weierstrass fibrations with fundamental invariant N. Following work of Miranda and Park–Schmitt, we give a quotient stack presentation for each \mathcal {W}^{\mathrm {min}}_N. Using these presentations and equivariant intersection theory, we determine a complete set of generators and relations for each of the Chow rings. For the cases N=1 (respectively, N=2), parametrizing rational (respectively, K3) elliptic surfaces, we give a more explicit computation of the relations. [ABSTRACT FROM AUTHOR]

We derive Itô–type change of variable formulas for smooth functionals of irregular paths with nonzero pth variation along a sequence of partitions, where p \geq 1 is arbitrary, in terms of fractional derivative operators. Our results extend the results of the Föllmer–Itô calculus to the general case of paths with 'fractional' regularity. In the case where p is not an integer, we show that the change of variable formula may sometimes contain a nonzero 'fractional' Itô remainder term and provide a representation for this remainder term. These results are then extended to functionals of paths with nonzero \phi-variation and multidimensional paths. Using these results, we derive an isometry property for the pathwise Föllmer integral in terms of \phi-variation. [ABSTRACT FROM AUTHOR]

We prove that for even integer k\geq k_0, almost all of zeros of the even period polynomial associated to a newform of weight k on \Gamma _0(N) are on the circle |z|=1/\sqrt {N}. [ABSTRACT FROM AUTHOR]

We provide a simple criterion on a family of functions that implies a square function estimate on L^p for every even integer p \geq 2. This defines a new type of superorthogonality that is verified by checking a less restrictive criterion than any other type of superorthogonality that is currently known. [ABSTRACT FROM AUTHOR]

This note proves two theorems regarding Fermat-type equation x^r + y^r = dz^p where r \geq 5 is a prime. Our main result shows that, for infinitely many integers d, the previous equation has no non-trivial primitive solutions such that 2 \mid x+y or r \mid x+y, for a set of exponents p of positive density. We use the modular method with a symplectic argument to prove this result. [ABSTRACT FROM AUTHOR]