Generalized q,t-Catalan numbers

Let C be a smooth projective curve of genus $$g\ge 2$$ . Fix an integer $$r\ge 0$$ , and let $$\underline{k}=(k_1,\ldots ,k_n)$$ be a sequence of positive integers with $$\sum _{i=1}^n k_i =g-1$$ . In this paper, we study n-pointed curves $$(C,p_1,\ldots ,p_n)$$ such that the line bundle $$L:=O_C\left( \sum _{i=1}^n k_i p_i\right) $$ is a theta-characteristic with $$h^0\left( C,L\right) \ge r+1$$ and $$h^0\left( C,L\right) \equiv r+1 \,\mathrm {(mod\,2)}$$ . We prove that they describe a sublocus $${\mathcal {G}}^r_g(\underline{k})$$ of $${\mathcal {M}}_{g,n}$$ having codimension at most $$g-1+\frac{r(r-1)}{2}$$ . Moreover, for any $$r\ge 0$$ , $$\underline{k}$$ as above, and g greater than an explicit integer g(r) depending on r, we present irreducible components of $${\mathcal {G}}^r_g(\underline{k})$$ attaining the maximal codimension in $${\mathcal {M}}_{g,n}$$ , so that the bound turns out to be sharp.

Let {an} be an increasing sequence of positive integers and let be the family of all functions from the positive integers into the positive integers, which satisfy Assume that are random functions with probabilities and for all n > 1 and 0 elsewhere, i.e., all permissible values of f, given the past, are equally likely.

Exact formulas for moments of sums of classical parking functions

Consider the difference equation oo (E) (-l)"+,A"4, + ;[>4,-4-0.k=0 where m is a positive integer, (Pk)k>o is a sequence of positive real numbers and (/(t)/t>o 's a sequence of integers with 0 < o < l\ < 12 < .The characteristic equation of (E) is CO (*) -(i-Ar + x>_/* = o.*=0We prove the following theorem.Theorem, (i) For m even, (E) has a positive solution (An)nez w'th limsup"_00 A" < oo if and only if (*) has a root in (0, 1).(ii) For m odd, (E) has a positive solution (A")nez if and only if (*) has a root in (0,1).

This is the first part of a two-part paper. Our paper was motivated by two classical papers: A paper of Sir Charles Wheatstone published in 1844 on representing certain powers of an integer as sums of arithmetic progressions and a paper of J. J. Sylvester published in 1882 for determining the number of ways a positive integer can be represented as the sum of a sequence of consecutive integers. There have been many attempts to extend Sylvester Theorem to the number of representations for an integer as the sums of different types of sequences, including sums of certain arithmetic progressions. In this part of the paper, we will make yet one more extension: We will describe a procedure for computing the number of ways a positive integer can be represented as the sums of all possible arithmetic progressions, together with an example to illustrate how this procedure can be carried out. In the process of doing this, we will also give an extension of Wheatstone’s work. In the second part of the paper, we will continue on the problems initiated by Wheatstone by studying certain relationships among the representations for different powers of an integer as sums of arithmetic progressions.

A new construction for the complementary sets of polyphase sequences is presented. The new construction is based on the application of Zadoff—Chu, as well as Ipatov polyphase sequences with ideal periodic autocorrelation function. It is shown that cyclic time shifted versions of either polyphase sequences form the complementary set, if the length of sequences is N = 5. m2 and the corresponding time shifts are different multiples of m, where s and m are any positive integers.

Let L be a line bundle on a projective variety X, and let s ≥ 0 be a non-negative integer. L is said to be s-ample if there exists an m > 0 such that Lm is generated by its global sections and the fibers of the associated morphism \(\varphi \left| {{L^m}} \right|:X \to \mathbb{P}({H^0}{(X,{L^m})^ * })\) are all of dimension ≤ s. For example L is 0-ample if and only if L is ample in the usual sense.KeywordsVector BundleLine BundleAbelian VarietyGlobal SectionAmple Line BundleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let $X$ be a compact Kähler manifold, endowed with an effective reduced divisor $B=\sum Y_{k}$ having simple normal crossing support. We consider a closed form of $(1,1)$-type $\unicode[STIX]{x1D6FC}$ on $X$ whose corresponding class $\{\unicode[STIX]{x1D6FC}\}$ is nef, such that the class $c_{1}(K_{X}+B)+\{\unicode[STIX]{x1D6FC}\}\in H^{1,1}(X,\mathbb{R})$ is pseudo-effective. A particular case of the first result we establish in this short note states the following. Let $m$ be a positive integer, and let $L$ be a line bundle on $X$, such that there exists a generically injective morphism $L\rightarrow \bigotimes ^{m}T_{X}^{\star }\langle B\rangle$, where we denote by $T_{X}^{\star }\langle B\rangle$ the logarithmic cotangent bundle associated to the pair $(X,B)$. Then for any Kähler class $\{\unicode[STIX]{x1D714}\}$ on $X$, we have the inequality $$\begin{eqnarray}\displaystyle \int _{X}c_{1}(L)\wedge \{\unicode[STIX]{x1D714}\}^{n-1}\leqslant m\int _{X}(c_{1}(K_{X}+B)+\{\unicode[STIX]{x1D6FC}\})\wedge \{\unicode[STIX]{x1D714}\}^{n-1}.\end{eqnarray}$$ If $X$ is projective, then this result gives a generalization of a criterion due to Y. Miyaoka, concerning the generic semi-positivity: under the hypothesis above, let $Q$ be the quotient of $\bigotimes ^{m}T_{X}^{\star }\langle B\rangle$ by $L$. Then its degree on a generic complete intersection curve $C\subset X$ is bounded from below by $$\begin{eqnarray}\displaystyle \biggl(\frac{n^{m}-1}{n-1}-m\biggr)\int _{C}(c_{1}(K_{X}+B)+\{\unicode[STIX]{x1D6FC}\})-\frac{n^{m}-1}{n-1}\int _{C}\unicode[STIX]{x1D6FC}.\end{eqnarray}$$ As a consequence, we obtain a new proof of one of the main results of our previous work [F. Campana and M. Păun, Orbifold generic semi-positivity: an application to families of canonically polarized manifolds, Ann. Inst. Fourier (Grenoble) 65 (2015), 835–861].

On strings containing all subsets as substrings

In this paper, we examine the distribution of the last digits in integer sequences generated by functions of positive integers. Specifically, we investigate the distribution of last digits in squares of integers and the last digits of expressions such as n2 + n and ax2 + bx + c, for positive integer values. We derive the theoretical probabilities of these last digits and provide proofs for the distribution properties, including symmetry characteristics.

Given an increasing sequence of positive integers { m n } \left \{ {{m_n}} \right \} , a non-decreasing sequence of positive integers { b n } \left \{ {{b_n}} \right \} , and a measurable, measure-preserving ergodic transformation τ \tau on a probability space ( Ω , F , μ ) \left ( {\Omega ,\mathcal {F},\mu } \right ) , the a.s. convergence of the moving averages T n ( f ) = b n − 1 ∑ k = m n + 1 m n + b n f ( τ k ) {T_n}\left ( f \right ) = b_n^{ - 1}\sum \nolimits _{k = {m_n} + 1}^{{m_n} + {b_n}} {f\left ( {{\tau ^k}} \right )} is considered, for f ∈ L p ( Ω ) f \in {L_p}\left ( \Omega \right ) . A counterexample is constructed in the case of polynomial-like { m n } \left \{ {{m_n}} \right \} .

A pair of sequences is called a Golay complementary&#13;\npair (GCP) if their aperiodic auto-correlation sums are zero for&#13;\nall out-of-phase time shifts. Existing known binary GCPs only&#13;\nhave even-lengths in the form of 2 10 26&#13;\n (where ; ; &#13;\n are&#13;\nnon-negative integers). To fill the gap left by the odd-lengths,&#13;\nwe investigate the optimal odd-length binary pairs which display&#13;\nthe closest correlation property to that of GCPs. Our criteria&#13;\nof “closeness” is that each pair has the maximum possible zerocorrelation&#13;\nzone (ZCZ) width and minimum possible out-of-zone&#13;\naperiodic auto-correlation sums. Such optimal pairs are called&#13;\noptimal odd-length binary Z-complementary pairs (OB-ZCP) in&#13;\nthis paper. We show that each optimal OB-ZCP has maximum&#13;\nZCZ width of (N + 1)=2, and minimum out-of-zone aperiodic&#13;\nsum magnitude of 2, where N denotes the sequence length (odd).&#13;\nSystematic constructions of such optimal OP-ZCPs are proposed&#13;\nby insertion and deletion of certain binary GCPs, which settle&#13;\nthe 2011 Li-Fan-Tang-Tu open problem positively. The proposed&#13;\noptimal OB-ZCPs may serve as a replacement for GCPs in&#13;\nmany engineering applications where odd sequence lengths are&#13;\npreferred. In addition, they give rise to a new family of base-two&#13;\nalmost difference families (ADF) which are useful in studying&#13;\npartially balanced incomplete block design (BIBD).

This letter is focused on increasing the zero correlation zone (ZCZ) of even-length binary Z-complementary pairs (EB-ZCPs). Till date, the maximum ZCZ ratio (i.e., ZCZ width over the sequence length) for systematically constructed EB-ZCPs is 2/3. In this letter, we give a construction of EB-ZCPs with lengths 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α+2</sup> 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">β</sup> 26 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">γ</sup> + 2 (where α, β, and γ are nonnegative integers) and ZCZ widths 3 × 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sup> 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">β</sup> 26 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">γ</sup> + 1, thus achieving asymptotic ZCZ ratio of 3/4. The proposed EB-ZCPs are constructed via proper insertion of concatenated odd-length binary ZCPs. The ZCZ width is proved by exploiting several newly identified intrinsic structure properties of binary Golay complementary pairs, obtained from Turyn's method. The proposed EB-ZCPs have aperiodic autocorrelation sums (AACS) magnitude of 4 outside the ZCZ region (except for the last time-shift taking AACS value of zero).

We propose a variation of the classical Hilbert scheme of points, thedouble nested Hilbert scheme of points, which parametrizes flags of zero-dimensional subschemes whose nesting is dictated by a Young diagram. Over a smooth quasi-projective curve, we compute the generating series of topological Euler characteristic of these spaces, by exploiting the combinatorics of reversed plane partitions. Moreover, we realize this moduli space as the zero locus of a section of a vector bundle over a smooth ambient space, which therefore admits a virtual fundamental class. We apply this construction to the stable pair theory of a local curve, that is the total space of the direct sum of two line bundles over a curve. We show that the invariants localize to virtual intersection numbers on double nested Hilbert scheme of points on the curve, and that the localized contributions to the invariants are controlled by three universal series for every Young diagram, which can be explicitly determined after the anti-diagonal restriction of the equivariant parameters. Under the anti-diagonal restriction, the invariants are matched with the Gromov–Witten invariants of local curves of Bryan–Pandharipande, as predicted by the Maulik–Nekrasov–Okounkov–Pandharipande (MNOP) correspondence. Finally, we discuss$K$-theoretic refinements à la Nekrasov–Okounkov.

On the existence of symmetric chain decompositions in a quotient of the Boolean lattice