Multiplication Theorem Research Articles

Prime forms and higher genus deformed Eisenstein series

Using the theory of Szegő kernel on a genus g Riemann surfaces obtained as a result of the multiple ρ-parameter formalism of sewing of g handles to the complex sphere, we derive new formulas related prime forms, theta functions, and deformed Eisenstein series. We establish recurrent formulas for genus g prime forms and Szegő kernel as well as further identities. Using the above results, we introduce finally another definition of genus g counterpart of genus one deformed Eisenstein series. The results obtained are then useful in computation of vertex algebra related cohomologies.

Joint discrete universality for periodic zeta-functions. II

In the paper, for certain classes of operators F in the space of analytic functions, we prove the discrete universality for compositions F (ζ(s, α ; 𝔞, 𝔟 )), where ζ(s, α ; 𝔞 ; 𝔟 ) is a collection consisting from periodic and periodic Hurwitz zeta-functions, i. e., the approximation of analytic functions by discrete shifts F (ζ(s + ikh, α ; 𝔞 ; 𝔟 )) with h > 0 and k = 0, 1, . . For this, a theorem of [12] is applied.

The regular representation of 𝑈_{𝜐}(𝔤𝔩_{𝔪|𝔫})

Using quantum differential operators, we construct a super representation of U υ ( g l m | n ) U_{\boldsymbol {\upsilon }}(\mathfrak {gl}_{m|n}) on a certain polynomial superalgebra. We then extend the representation to its formal power series algebra which contains a U υ ( g l m | n ) U_{\boldsymbol {\upsilon }}(\mathfrak {gl}_{m|n}) -submodule isomorphic to the regular representation of U υ ( g l m | n ) U_{\boldsymbol {\upsilon }}(\mathfrak {gl}_{m|n}) . In this way, we obtain a presentation of U υ ( g l m | n ) U_{\boldsymbol {\upsilon }}(\mathfrak {gl}_{m|n}) by a basis together with explicit multiplication formulas of the basis elements by generators.

Discrete Moments of Additive Twists. I: The Mean-Square

In this note, we prove an asymptotic formula for the mean-square of the so-called periodic zeta-function with respect to the parameter. This may be compared with similar formulae for Dirichlet L-functions to (multiplicative) residue class characters due to Paley and others. The periodic zeta-function is the twist of the Riemann zeta-function with an additive character.

Connection and duplication formulas for the Boas-Buck-Appell polynomials

The present paper conduct to introduce the connection and duplication formulas associated with the Boas-Buck-Appell polynomials. Examples providing the analogues results for certain members related to the Boas-Buck-Appell polynomials are considered.

Cluster multiplication theorem in the quantum cluster algebra of type [formula omitted] and the triangular basis

The objective of the present paper is to prove cluster multiplication theorem in the quantum cluster algebra of type A2(2). As corollaries, we obtain bar-invariant Z[q±12]-bases established in [6], and naturally deduce the positivity of the elements in these bases. One bar-invariant basis as the triangular basis of this quantum cluster algebra is also explicitly described.

An OAM Patch Antenna Design and Its Array for Higher Order OAM Mode Generation

In this letter, a systematic design procedure is proposed on how to implement the characteristic mode analysis to design and optimize an orbital angular momentum (OAM) antenna with circular polarization. The main idea of this work is by combining proper high-order modes to fit the current distribution of the analytical model. It is shown that by using proper high-order modes, a single antenna element could act like an array making the antenna system more compact. As a prototype, a patch antenna working at the 2.4 GHz Wi-Fi bands to generate the right-hand circularly polarized OAM beam with topology charge ${l_{{{\bf OAM}}}} = -1$ is designed, fabricated, and measured to demonstrate the design method. In addition, an array composed by the proposed antenna element is designed and experimented. New physical phenomenon is found that the OAM modes of the antenna element and the array can be combined considering the multiplication theorem. The combination relaxes the restriction of $|{l_{\bf OAM}}| to some extent in comparison with the traditional uniform circular arrays for OAM generation.

Algebraic curves with many automorphisms

Let X be a (projective, geometrically irreducible, non-singular) algebraic curve of genus g≥2 defined over an algebraically closed field K of odd characteristic p. Let Aut(X) be the group of all automorphisms of X which fix K element-wise. It is known that if |Aut(X)|≥8g3 then the p-rank (equivalently, the Hasse–Witt invariant) of X is zero. This raises the problem of determining the (minimum-value) function f(g) such that whenever |Aut(X)|≥f(g) then X has zero p-rank. For eveng we prove that f(g)≤900g2. The odd genus case appears to be much more difficult although, for any genus g≥2, if Aut(X) has a solvable subgroup G such that |G|>252g2 then X has zero p-rank and G fixes a point of X. Our proofs use the Hurwitz genus formula and the Deuring Shafarevich formula together with a few deep results from Group theory characterizing finite simple groups whose Sylow 2-subgroups have a cyclic subgroup of index 2. We also point out some connections with the Abhyankar conjecture and the Katz–Gabber covers.

Remarks on PBW bases of Ringel–Hall algebras of cyclic quivers

In this paper, we give a recursive formula for the interesting PBW basis [Formula: see text] of composition subalgebras [Formula: see text] of Ringel–Hall algebras [Formula: see text] of cyclic quivers after [Generic extensions and canonical bases for cyclic quivers, Canad. J. Math. 59(6) (2007) 1260–1283], and another construction of canonical bases of [Formula: see text] from the monomial bases [Formula: see text] following [Multiplication formulas and canonical basis for quantum affine, [Formula: see text], Canad. J. Math. 70(4) (2018) 773–803]. As an application, we will determine all the canonical basis elements of [Formula: see text] associated with modules of Loewy length [Formula: see text]. Finally, we will discuss the canonical bases between Ringel–Hall algebras and affine quantum Schur algebras.

Dicuil (9th century) on triangular and square numbers

Dicuil was a ninth-century Irish monk who taught at the Carolingian school of Louis the Pious. He wrote a Computus or astronomical treatise in Latin in about 814–16, which contains a chapter on triangular and square numbers. Dicuil describes two methods for calculating triangular numbers: the simple method of summing the natural numbers, and the more complex method of multiplication, equivalent to the formula n(n + 1)/2. He also states that a square number is equal to twice a triangular number minus the generating number, equivalent to n2 = 2[n(n + 1)/2] – n. The multiplication formula for triangular numbers was first explicitly described in about the third century AD by the Greek authors Diophantus and Iamblichus. It was also known as a solution to other mathematical problems as early as 300 BC. It reappeared in the West in the sixteenth century. Dicuil thus fills a gap in our medieval knowledge.

Gromov–Witten Invariants of Calabi–Yau Manifolds With Two Kähler Parameters

Abstract We study the Gromov–Witten theory of $K_{{{\mathbb{P}}}^1\times{{\mathbb{P}}}^1}$ and some Calabi–Yau hypersurfaces in toric varieties. We give a direct geometric proof of the holomorphic anomaly equation for $K_{{{\mathbb{P}}}^1\times{{\mathbb{P}}}^1}$ in the form predicted by B-model physics. We also calculate the closed formula of genus one quasimap invariants of Calabi–Yau hypersurfaces in ${{\mathbb{P}}}^{m-1}\times{{\mathbb{P}}}^{n-1}$ after restricting the 2nd Kähler parameter to zero. By the wall-crossing theorem between Gromov–Witten and quasimap invariants, we thus obtain their genus one Gromov–Witten invariants.

Multiplicative properties of certain elements in Bridgeland Hall algebras

Let A be a finite dimensional algebra of finite global dimension over a finite field. In the present paper, we introduce certain elements in the Bridgeland Hall algebra of A, and give a multiplication formula for these elements. In particular, this generalizes the main result in Geng and Peng (2015, J. Algebra 437:116–132).

Formal Affine Demazure and Hecke Algebras of Kac-Moody Root Systems

We define the formal affine Demazure algebra and a formal affine Hecke algebra associated to a Kac-Moody root system. We prove structure theorems for these algebras, and use them to extend several results and constructions (presentation in terms of generators and relations, coproduct and product structures, filtration by codimension of Bott-Samelson classes, root polynomials and multiplication formulas) that were previously known for finite root systems.

Reproducing kernel orthogonal polynomials on the multinomial distribution

Diaconis and Griffiths (2014) study the multivariate Krawtchouk polynomials orthogonal on the multinomial distribution. In this paper we derive the reproducing kernel orthogonal polynomials Qn(x,y;N,p) on the multinomial distribution which are sums of products of orthonormal polynomials in x and y of fixed total degree n=0,1,…,N. The Poisson kernel ∑n=0NρnQn(x,y;N,p) arises naturally from a probabilistic argument. An application to a multinomial goodness of fit test is developed, where the chi-squared test statistic is decomposed into orthogonal components which test the order of fit. A new duplication formula for the reproducing kernel polynomials in terms of the 1-dimensional Krawtchouk polynomials is derived. The duplication formula allows a Lancaster characterization of all reversible Markov chains with a multinomial stationary distribution whose eigenvectors are multivariate Krawtchouk polynomials and where eigenvalues are repeated within the same total degree. The χ2 cutoff time, and total variation cutoff time is investigated in such chains. Emphasis throughout the paper is on a probabilistic understanding of the polynomials and their applications, particularly to Markov chains.

Affine flag varieties and quantum symmetric pairs, II. Multiplication formula

We establish a multiplication formula for a tridiagonal standard basis element in the idempotent version, i.e., the Lusztig form, of the coideal subalgebras of quantum affine gln arising from the geometry of affine partial flag varieties of type C. We apply this formula to obtain the stabilization algebras K˙nc, K˙nȷı, K˙nıȷ and K˙ηıı, which are idempotented coideal subalgebras of quantum affine gln. The symmetry in the formula leads to an isomorphism of the idempotented coideal subalgebras K˙nȷı and K˙nıȷ with compatible monomial, standard and canonical bases.

Presenting affine Schur algebras

The universal enveloping algebra U ( g l ^ n ) \mathcal {U}(\widehat {\frak {gl}}_n) of g l ^ n \widehat {\frak {gl}}_n was realized in [A double Hall algebra approach to affine quantum Schur–Weyl theory, Cambridge University Press, Cambridge, 2012, Ch. 6] using affine Schur algebras. In particular some explicit multiplication formulas in affine Schur algebras were derived. We use these formulas to study the structure of affine Schur algebras. In particular, we give a presentation of the affine Schur algebra S △ ( n , r ) Q {\mathcal S}_{{\!\vartriangle }}(n,r)_{\mathbb Q} over Q \mathbb {Q} .

Experimental study of neutron counting in a zero-power reactor driven by a neutron source inherent in highly enriched uranium fuels

ABSTRACT Even a zero-power reactor core containing highly enriched uranium has a weak neutron source inherent in uranium 235, and consequently, a neutron counter placed closely to the core without external neutron source registers a certain counting rate. The study of the counting is very important for zero-power reactor physics experiments with a high precision. In this experimental study, first, at a shutdown state of the UTR-Kinki reactor without start-up neutron source, a pulse height distribution of output signals from a neutron proportional counter was measured to confirm that these signals resulted from neutron detections. At several subcritical states of the UTR, then, the Feynman-α analysis was carried out to confirm that the neutrons detected by the counter must be fission neutrons multiplied by fission chain reactions. The correlation amplitude measured in the Feynman-α analysis was much higher than that measured in a previous drive by start-up source. Further, it was also confirmed that the subcriticality dependence of neutron counting rate followed the source multiplication formula. This feature indicated that the one-point model was very successful in the subcritical range including the shutdown state.

Genus formulas for abelian $p$-extensions

We apply a result of Kani relating genera and Hasse-Witt invariants of Galois extensions to a family of abelian $p$-extensions. Our formulas generalize the case of elementary abelian $p$-extensions found by Garcia and Stichtenoth.

On a theta product of Jacobi and its applications to q-gamma products

We give a new proof for a product formula of Jacobi which turns out to be equivalent to a q -trigonometric product which was stated without proof by Gosper. We apply this formula to derive a q -analogue for the Gauss multiplication formula for the gamma function. Furthermore, we give explicit formulas for short products of q -gamma functions.

${\boldsymbol~q}$-Schur superalgebras

This paper provides a survey on the latest developments of $q$-Schur superalgebras, quantum linear supergroups and their canonical bases and irreducible (polynomial) representations. We first introduce the definitions of $q$-Schur superalgebras in three different contexts and their associated bases,and describe the relationships between the three bases. We then move on to display certain multiplication formulas in $q$-Schur superalgebras and discuss their applications to the new realizationof quantum linear supergroups and to the regular representation of a $q$-Schur superalgebra. We provide another application for the construction of canonical bases for the positive part of thequantum linear supergroup. As a by-product, a semi-simplicity criterion is given for $q$-Schur superalgebras. Moreover, by generalising the idea of Alperins weight conjecture and Scottspermutation representation theory, we provide a classification of irreducible modules for a $q$-Schur superalgebra. We also mention a new approach to introduce infinitesimal and little $q$-Schur superalgebras without using quantum coordinate superalgebras.