product formula
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Author(s):  
Louis H. Kauffman ◽  
Eiji Ogasa

We use the terms, knot product and local-move, as defined in the text of this paper. Let [Formula: see text] be an integer [Formula: see text]. Let [Formula: see text] be the set of simple spherical [Formula: see text]-knots in [Formula: see text]. Let [Formula: see text] be an integer [Formula: see text]. We prove that the map [Formula: see text] is bijective, where [Formula: see text]Hopf, and Hopf denotes the Hopf link. Let [Formula: see text] and [Formula: see text] be 1-links in [Formula: see text]. Suppose that [Formula: see text] is obtained from [Formula: see text] by a single pass-move, which is a local-move on 1-links. Let [Formula: see text] be a positive integer. Let [Formula: see text] denote the knot product [Formula: see text]. We prove the following: The [Formula: see text]-dimensional submanifold [Formula: see text] [Formula: see text] is obtained from [Formula: see text] by a single [Formula: see text]-pass-move, which is a local-move on [Formula: see text]-submanifolds contained in [Formula: see text]. See the body of this paper for the definitions of all local-moves in this abstract. We prove the following: Let [Formula: see text], and [Formula: see text] be positive integers. If the [Formula: see text] torus link is pass-move-equivalent to the [Formula: see text] torus link, then the Brieskorn manifolds, [Formula: see text] and [Formula: see text], are diffeomorphic as abstract manifolds. Let [Formula: see text] and [Formula: see text] be (not necessarily connected or spherical) 2-dimensional closed oriented submanifolds in [Formula: see text]. Suppose that [Formula: see text] is obtained from [Formula: see text] by a single ribbon-move, which is a local-move on 2-dimensional submanifolds contained in [Formula: see text]. Let [Formula: see text] be an integer [Formula: see text]. We prove the following: The [Formula: see text]-submanifold [Formula: see text] [Formula: see text] is obtained from [Formula: see text] by a single [Formula: see text]-pass-move, which is a local-move on [Formula: see text]-dimensional submanifolds contained in [Formula: see text].


Author(s):  
Valeriano Aiello ◽  
Daniele Guido ◽  
Tommaso Isola

Given a spectral triple on a [Formula: see text]-algebra [Formula: see text] together with a unital injective endomorphism [Formula: see text], the problem of defining a suitable crossed product [Formula: see text]-algebra endowed with a spectral triple is addressed. The proposed construction is mainly based on the works of Cuntz and [A. Hawkins, A. Skalski, S. White and J. Zacharias, On spectral triples on crossed products arising from equicontinuous actions, Math. Scand. 113(2) (2013) 262–291], and on our previous papers [V. Aiello, D. Guido and T. Isola, Spectral triples for noncommutative solenoidal spaces from self-coverings, J. Math. Anal. Appl. 448(2) (2017) 1378–1412; V. Aiello, D. Guido and T. Isola, A spectral triple for a solenoid based on the Sierpinski gasket, SIGMA Symmetry Integrability Geom. Methods Appl. 17(20) (2021) 21]. The embedding of [Formula: see text] in [Formula: see text] can be considered as the dual form of a covering projection between noncommutative spaces. A main assumption is the expansiveness of the endomorphism, which takes the form of the local isometricity of the covering projection, and is expressed via the compatibility of the Lip-norms on [Formula: see text] and [Formula: see text].


Author(s):  
Kengo Fukunaga ◽  
Kohta Gejima

Let [Formula: see text] be a normalized cuspidal Hecke eigenform. We give explicit formulas for weighted averages of the rightmost critical values of triple product [Formula: see text]-functions [Formula: see text], where [Formula: see text] and [Formula: see text] run over an orthogonal basis of [Formula: see text] consisting of normalized cuspidal Hecke eigenforms. Those explicit formulas provide us an arithmetic expression of the rightmost critical value of the individual triple product [Formula: see text]-functions.


2021 ◽  
Vol 127 (3) ◽  
Author(s):  
Nguyen Quang Dieu ◽  
Tang Van Long

In this note, we establish a product property for $P$-extremal functions in the same spirit as the original product formula due to J. Siciak in Ann. Polon. Math., 39 (1981), 175–211. As a consequence, we obtain convexity for the sublevel sets of such extremal functions. Moreover, we also generalize the product property of $P$-extremal functions established by L. Bos and N. Levenberg in Comput. Methods Funct. Theory 18 (2018), 361–388, and later by N. Levenberg and M. Perera, in Contemporary Mathematics 743 (2020), 11–19, in which no restriction on $P$ is needed.


2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Sam Hopkins ◽  
Martin Rubey

AbstractKreweras words are words consisting of n$$\mathrm {A}$$ A ’s, n$$\mathrm {B}$$ B ’s, and n$$\mathrm {C}$$ C ’s in which every prefix has at least as many $$\mathrm {A}$$ A ’s as $$\mathrm {B}$$ B ’s and at least as many $$\mathrm {A}$$ A ’s as $$\mathrm {C}$$ C ’s. Equivalently, a Kreweras word is a linear extension of the poset $$\mathsf{V}\times [n]$$ V × [ n ] . Kreweras words were introduced in 1965 by Kreweras, who gave a remarkable product formula for their enumeration. Subsequently they became a fundamental example in the theory of lattice walks in the quarter plane. We study Schützenberger’s promotion operator on the set of Kreweras words. In particular, we show that 3n applications of promotion on a Kreweras word merely swaps the $$\mathrm {B}$$ B ’s and $$\mathrm {C}$$ C ’s. Doing so, we provide the first answer to a question of Stanley from 2009, asking for posets with ‘good’ behavior under promotion, other than the four families of shapes classified by Haiman in 1992. We also uncover a strikingly simple description of Kreweras words in terms of Kuperberg’s $$\mathfrak {sl}_3$$ sl 3 -webs, and Postnikov’s trip permutation associated with any plabic graph. In this description, Schützenberger’s promotion corresponds to rotation of the web.


Information ◽  
2021 ◽  
Vol 12 (11) ◽  
pp. 483
Author(s):  
Michel Riguidel

From the functional equation of Riemann’s zeta function, this article gives new insight into Hadamard’s product formula. The function and its family of associated functions, expressed as a sum of rational fractions, are interpreted as meromorphic functions whose poles are the poles and zeros of the function. This family is a mathematical and numerical tool which makes it possible to estimate the value of the function at a point in the critical strip from a point on the critical line .Generating estimates of at a given point requires a large number of adjacent zeros, due to the slow convergence of the series. The process allows a numerical approach of the Riemann hypothesis (RH). The method can be extended to other meromorphic functions, in the neighborhood of isolated zeros, inspired by the Weierstraß canonical form. A final and brief comparison is made with the and functions over finite fields.


2021 ◽  
pp. 1-11
Author(s):  
Tyrone Crisp

By computing the completely bounded norm of the flip map on the Haagerup tensor product [Formula: see text] associated to a pair of continuous mappings of locally compact Hausdorff spaces [Formula: see text], we establish a simple characterization of the Beck-Chevalley condition for base change of operator modules over commutative [Formula: see text]-algebras, and a descent theorem for continuous fields of Hilbert spaces.


Author(s):  
Anatoly Korybut

Abstract An analogue of the Moyal star product is presented for the deformed oscillator algebra. It contains several homotopy-like additional integration parameters in the multiplication kernel generalizing the differential Moyal star-product formula exp[iεαβ ∂α∂β]. Using Pochhammer formula [1], integration over these parameters is carried over a Riemann surface associated with the expression of the type zx(1 − z)y where x and y are arbitrary real numbers.


Author(s):  
Stephen L. Adler

We show that the recipe for computing the expansions [Formula: see text] and [Formula: see text] of outgoing and ingoing null geodesics normal to a surface admits a covariance group with nonconstant scalar [Formula: see text], corresponding to the mapping [Formula: see text], [Formula: see text]. Under this mapping, the product [Formula: see text] is invariant, and thus the marginal surface computed from the vanishing of [Formula: see text], which is used to define the apparent horizon, is invariant. This covariance group naturally appears in comparing the expansions computed with different choices of coordinate system.


2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Surya Mahadevan ◽  
Jayanthi Thanigan ◽  
Srinivasa Reddy

Research methodology The case is written based on general experience. Case overview/synopsis Zealvita is a challenger brand to NutriMalt in the white malted food drink (MFD) category. It has a product formula that compares favorably on taste and equally on nutrition. However, Zealvita is not able to translate the power product formula to a winning market formula. Drawing on its legacy and strong adoption route, NutriMalt built a dominant 88% market share in the White MFD category. The market rule of “disproportionate market share for the leading brand” applies with inexorable force in MFD. Smarting at the low market share, Zealvita is in search of a marketing strategy to create churn. Rajiv Product Manager of Zealvita believes that consumer sales promotion of a higher order and at a higher frequency than what is normal can tilt the scales. From Zealvita’s perspective is there a strategic advantage in operating consumer promotion? Is it safe to assume that NutriMalt will not retaliate with consumer promotion? Can consumer sales promotion be sustained at planned frequency? What is the logic in a continuous consumer promotion program? Complexity academic level This case can be used at the post-graduate level in the Marketing Strategy course or in a course that has a sales promotion management or competition management segment. This case is also appropriate for use in executive education programs. Supplementary materials Teaching notes are available for educators only. Please contact your library to gain login details or email [email protected] to request teaching notes.


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