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Journal articles on the topic 'K)-Symmetric functions'

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Published: 3 June 2025

Last updated: 6 July 2025

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1

Anh, V. V. "K-Fold symmetric starlike univalent functions." Bulletin of the Australian Mathematical Society 32, no. 3 (1985): 419–36. http://dx.doi.org/10.1017/s0004972700002537.

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This paper establishes the radius of convexity, distortion and covering theorems for the classwhere−1 ≤ B < A ≤ 1, w(0) = 0, |w (z)| < 1 in the unit disc. Coefficient bounds for functions in are also derived.

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2

Calderón-Gómez, José E., Luis A. Medina, and Carlos A. Molina-Salazar. "Short k-rotation symmetric Boolean functions." Discrete Applied Mathematics 343 (January 2024): 49–64. http://dx.doi.org/10.1016/j.dam.2023.10.003.

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3

Asakly, Walaa. "Enumerating symmetric and non-symmetric peaks in words." Online Journal of Analytic Combinatorics, no. 13 (December 31, 2018): 1–7. https://doi.org/10.61091/ojac-1302.

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Let \([k] = \{1, 2, \ldots, k\}\) be an alphabet over \(k\) letters. A word \(\omega\) of length \(n\) over alphabet \([k]\) is an element of \([k]^n\) and is also called \(k\)-ary word of length \(n\). We say that \(\omega\) contains a peak, if exists \(2 \leq i \leq n-1\) such that \(\omega_{i-1} < \omega_i, \omega_i > \omega_{i+1}\). We say that \(\omega\) contains a symmetric peak, if exists \(2 \leq i \leq n-1\) such that \(\omega_{i-1} = \omega_{i+1} < \omega_i\), and contains a non-symmetric peak, otherwise. In this paper, we find an explicit formula for the generating functions for the number of \(k\)-ary words of length \(n\) according to the number of symmetric peaks and non-symmetric peaks in terms of Chebyshev polynomials of the second kind. Moreover, we find the number of symmetric and non-symmetric peaks in \(k\)-ary word of length \(n\) in two ways by using generating functions techniques, and by applying probabilistic methods.

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4

Rupp, R., та A. Sasane. "Reducibility in Aℝ(K), Cℝ(K), and A(K)". Canadian Journal of Mathematics 62, № 3 (2010): 646–67. http://dx.doi.org/10.4153/cjm-2010-025-9.

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AbstractLet K denote a compact real symmetric subset of ℂ and let Aℝ(K) denote the real Banach algebra of all real symmetric continuous functions on K that are analytic in the interior K◦ of K, endowed with the supremum norm. We characterize all unimodular pairs ( f , g) in Aℝ(K)2 which are reducible. In addition, for an arbitrary compact K in ℂ, we give a new proof (not relying on Banach algebra theory or elementary stable rank techniques) of the fact that the Bass stable rank of A(K) is 1. Finally, we also characterize all compact real symmetric sets K such that Aℝ(K), respectively Cℝ(K), has Bass stable rank 1.

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5

SERGEEV, A. N., and A. P. VESELOV. "JACK–LAURENT SYMMETRIC FUNCTIONS FOR SPECIAL VALUES OF PARAMETERS." Glasgow Mathematical Journal 58, no. 3 (2015): 599–616. http://dx.doi.org/10.1017/s0017089515000361.

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AbstractWe consider the Jack–Laurent symmetric functions for special values of parametersp0=n+k−1m, wherekis not rational andmandnare natural numbers. In general, the coefficients of such functions may have poles at these values ofp0. The action of the corresponding algebra of quantum Calogero–Moser integrals$\mathcal{D}$(k,p0) on the space of Laurent symmetric functions defines the decomposition into generalised eigenspaces. We construct a basis in each generalised eigenspace as certain linear combinations of the Jack–Laurent symmetric functions, which are regular atp0=n+k−1m, and describe the action of$\mathcal{D}$(k,p0) in these eigenspaces.

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6

Liu, Hongli. "The Weight and Nonlinearity of 2-rotation Symmetric Cubic Boolean Function." Journal of Mathematics Research 7, no. 2 (2015): 187. http://dx.doi.org/10.5539/jmr.v7n2p187.

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The conceptions of $\chi$-value and K-rotation symmetric Boolean functions are introduced by Cusick. K-rotation symmetric Boolean functions are a special rotation symmetric functions, which are invariant under the $k-th$ power of $\rho$.In this paper, we discuss cubic 2-value 2-rotation symmetric Boolean function with $2n$ variables, which denoted by $F^{2n}(x^{2n})$. We give the recursive formula of weight of $F^{2n}(x^{2n})$, and prove that the weight of $F^{2n}(x^{2n})$ is the same as its nonlinearity.

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7

Zhang, Z. Q., and Y. Y. Shi. "Communication complexities of symmetric XOR functions." Quantum Information and Computation 9, no. 3&4 (2009): 255–63. http://dx.doi.org/10.26421/qic9.3-4-5.

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We call $F:\{0, 1\}^n\times \{0, 1\}^n\to\{0, 1\}$ a symmetric XOR function if for a function $S:\{0, 1, ..., n\}\to\{0, 1\}$, $F(x, y)=S(|x\oplus y|)$, for any $x, y\in\{0, 1\}^n$, where $|x\oplus y|$ is the Hamming weight of the bit-wise XOR of $x$ and $y$. We show that for any such function, (a) the deterministic communication complexity is always $\Theta(n)$ except for four simple functions that have a constant complexity, and (b) up to a polylog factor, both the error-bounded randomized complexity and quantum communication with entanglement complexity are $\Theta(r_0+r_1)$, where $r_0$ and $r_1$ are the minimum integers such that $r_0, r_1\leq n/2$ and $S(k)=S(k+2)$ for all $k\in[r_0, n-r_1)$.

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8

DELENCLOS, JONATHAN, and ANDRÉ LEROY. "NONCOMMUTATIVE SYMMETRIC FUNCTIONS AND W-POLYNOMIALS." Journal of Algebra and Its Applications 06, no. 05 (2007): 815–37. http://dx.doi.org/10.1142/s021949880700251x.

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Let K, S, D be a division ring, an endomorphism and a S-derivation of K, respectively. In this setting we introduce generalized noncommutative symmetric functions and obtain Viète formula and decompositions of differential operators. W-polynomials show up naturally, their connections with P-independency, Vandermonde and Wronskian matrices are briefly studied. The different linear factorizations of W-polynomials are analyzed. Connections between the existence of LLCM of monic linear polynomials with coefficients in a ring and the left duo property are established at the end of the paper.

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9

Zainab, Saira, Mohsan Raza, Qin Xin, Mehwish Jabeen, Sarfraz Nawaz Malik, and Sadia Riaz. "On q-Starlike Functions Defined by q-Ruscheweyh Differential Operator in Symmetric Conic Domain." Symmetry 13, no. 10 (2021): 1947. http://dx.doi.org/10.3390/sym13101947.

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Motivated by q-analogue theory and symmetric conic domain, we study here the q-version of the Ruscheweyh differential operator by applying it to the starlike functions which are related with the symmetric conic domain. The primary aim of this work is to first define and then study a new class of holomorphic functions using the q-Ruscheweyh differential operator. A new class k−STqτC,D of k-Janowski starlike functions associated with the symmetric conic domain, which are defined by the generalized Ruscheweyh derivative operator in the open unit disk, is introduced. The necessary and sufficient condition for a function to be in the class k−STqτC,D is established. In addition, the coefficient bound, partial sums and radii of starlikeness for the functions from the class of k-Janowski starlike functions related with symmetric conic domain are included.

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10

Dib, Hacen. "K-Bessel functions in two variables." International Journal of Mathematics and Mathematical Sciences 2003, no. 14 (2003): 909–16. http://dx.doi.org/10.1155/s0161171203112057.

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The Bessel-Muirhead hypergeometric system (or0F1-system) in two variables (and three variables) is solved using symmetric series, with an explicit formula for coefficients, in order to express theK-Bessel function as a linear combination of the J-solutions. Limits of this method and suggestions for generalizations to a higher rank are discussed.

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11

ZHAO, WENHUA. "NONCOMMUTATIVE SYMMETRIC FUNCTIONS AND THE INVERSION PROBLEM." International Journal of Algebra and Computation 18, no. 05 (2008): 869–99. http://dx.doi.org/10.1142/s0218196708004615.

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Let K be any unital commutative ℚ-algebra and z = (z1, z2, …, zn) commutative or noncommutative variables. Let t be a formal central parameter and K[[t]]〈〈z〉〉 the formal power series algebra of z over K[[t]]. In [29], for each automorphism Ft(z) = z - Ht(z) of K[[t]]〈〈z〉〉 with Ht=0(z) = 0 and o(H(z)) ≥ 1, a [Formula: see text] (noncommutative symmetric) system [28] ΩFt has been constructed. Consequently, we get a Hopf algebra homomorphism [Formula: see text] from the Hopf algebra [Formula: see text] [9] of NCSFs (noncommutative symmetric functions). In this paper, we first give a list for the identities between any two sequences of differential operators in the [Formula: see text] system ΩFt by using some identities of NCSFs derived in [9] and the homomorphism [Formula: see text]. Secondly, we apply these identities to derive some formulas in terms of differential operator in the system ΩFt for the Taylor series expansions of u(Ft) and [Formula: see text]; the D-Log and the formal flow of Ft and inversion formulas for the inverse map of Ft. Finally, we discuss a connection of the well-known Jacobian conjecture with NCSFs.

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12

Alsarari, Fuad, Latha Satyanarayana, and Maslina Darus. "The Second Hankel Determinant for k-symmetrical Functions." Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2(102) (November 2023): 3–10. http://dx.doi.org/10.56415/basm.y2023.i2.p3.

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13

C., Selvaraj, Varadharajan S., and Lakshmi S. "FEKETE-SZEGO INEQUALITIES AND (j, k) SYMMETRIC FUNCTIONS USING q DERIVATIVE." ISSN: 2455 – 5428 International Journal of Current Research and Modern Education, Special Issue (August 20, 2017): 136–47. https://doi.org/10.5281/zenodo.845919.

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In this paper sharp upper bounds of for functions belonging to new subclasses defined using the concept of symmetric functions using derivative are derived. Furthermore, the application of the results are also illustrated.

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14

Liu, Jin-Lin, and Rekha Srivastava. "Convolution properties for certain meromorphically multivalent functions." Filomat 31, no. 1 (2017): 113–23. http://dx.doi.org/10.2298/fil1701113l.

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Let ?p denote the class of functions of the form f(z)=z-p + ??,n=p anzn (p ? N). Two new subclasses Hp,k(?,A,B) and Qp,k(?,A,B) of meromorphically multivalent functions starlike with respect to k-symmetric points in ?p are investigated. Certain convolution properties for these subclasses are obtained.

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15

Devi K, Renuka, Hamid Shamsan, and S. Latha. "Some Results on (j,k) Symmetric Starlike Harmonic Functions." Journal of Mathematics and Informatics 13 (April 1, 2018): 29–39. http://dx.doi.org/10.22457/jmi.v13a4.

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16

Fischer, P., and J. D. Stegeman. "Functions operating on positive-definite symmetric matrices." Journal of Mathematical Analysis and Applications 171, no. 2 (1992): 461–70. http://dx.doi.org/10.1016/0022-247x(92)90358-k.

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17

Lam, Thomas, Anne Schilling, and Mark Shimozono. "K-theory Schubert calculus of the affine Grassmannian." Compositio Mathematica 146, no. 4 (2010): 811–52. http://dx.doi.org/10.1112/s0010437x09004539.

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AbstractWe construct the Schubert basis of the torus-equivariant K-homology of the affine Grassmannian of a simple algebraic group G, using the K-theoretic NilHecke ring of Kostant and Kumar. This is the K-theoretic analogue of a construction of Peterson in equivariant homology. For the case where G=SLn, the K-homology of the affine Grassmannian is identified with a sub-Hopf algebra of the ring of symmetric functions. The Schubert basis is represented by inhomogeneous symmetric functions, calledK-k-Schur functions, whose highest-degree term is a k-Schur function. The dual basis in K-cohomology is given by the affine stable Grothendieck polynomials, verifying a conjecture of Lam. In addition, we give a Pieri rule in K-homology. Many of our constructions have geometric interpretations by means of Kashiwara’s thick affine flag manifold.

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18

Colzani, Leonardo. "Regularity of spherical means and localization of spherical harmonic expansions." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 41, no. 3 (1986): 287–97. http://dx.doi.org/10.1017/s1446788700033723.

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AbstractLet G/K be a compact symmetric space, and let G = KAK be a Cartan decomposition of G. For f in L1(G) we define the spherical means f(g, t) = ∫k∫k ∫(gktk′) dk dk′, g ∈ G, t ∈ A. We prove that if f is in Lp(G), 1 ≤ p ≤ 2, then for almost every g ∈ G the functions t → f(g, t) belong to certain Soblev spaces on A. From these regularity results for the spherical means we deduce, if G/K is a compact rank one symmetric space, a theorem on the almost everywhere localization of spherical harmonic expansions of functions in L2 (G/K).

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19

Mortini, Raymond, and Rudolf Rupp. "Approximation by invertible elements and the generalized $E$-stable rank for $A({\boldsymbol D})_{\mathsf R}$ and $C({\boldsymbol D})_{\mathrm{sym}}$." MATHEMATICA SCANDINAVICA 109, no. 1 (2011): 114. http://dx.doi.org/10.7146/math.scand.a-15180.

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We determine the generalized $E$-stable ranks for the real algebra, $C(\boldsymbol{D})_{\mathrm{sym}}$, of all complex valued continuous functions on the closed unit disk, symmetric to the real axis, and its subalgebra $A(\boldsymbol{D})_{\mathsf R}$ of holomorphic functions. A characterization of those invertible functions in $C(E)$ is given that can be uniformly approximated on $E$ by invertibles in $A(\boldsymbol {D})_{\mathsf R}$. Finally, we compute the Bass and topological stable rank of $C(K)_{\mathrm{sym}}$ for real symmetric compact planar sets $K$.

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20

PARK, DAESHIK. "APPROXIMATING GREEN'S FUNCTIONS ON ℙ1 POSITIVE CHARACTERISTIC". Communications in Contemporary Mathematics 12, № 04 (2010): 537–67. http://dx.doi.org/10.1142/s0219199710003919.

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Fix a finite K-symmetric set [Formula: see text] and a K-symmetric probability vector [Formula: see text]. Let 𝔇v be a finite union of balls [Formula: see text] for some ah ∈ Kv and some [Formula: see text], where the balls 𝔅(ah, rh) are disjoint from 𝔛. Put 𝔈v := 𝔇v ∩ ℙ1(Kv). Then there exists a positive integer Nv such that for each sufficiently large integer N divisible by Nv, there are a number Rv, with [Formula: see text], and an [Formula: see text]-function fv(z) ∈ Kv(z) of degree N whose zeros form a "well-distributed" sequence in 𝔈v such that [Formula: see text] is a disjoint union of balls centered at the zeros of fv(z) and for all z ∉ 𝔇v, [Formula: see text]

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21

DING, HONGMING, and WEI HE. "SERIES EXPANSIONS OF THE GENERALIZED K-BESSEL FUNCTIONS ON SYMMETRIC CONES." Analysis and Applications 06, no. 01 (2008): 1–10. http://dx.doi.org/10.1142/s021953050800102x.

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22

GAZZINI, MARITA, and ROBERTA MUSINA. "HARDY–SOBOLEV–MAZ'YA INEQUALITIES: SYMMETRY AND BREAKING SYMMETRY OF EXTREMAL FUNCTIONS." Communications in Contemporary Mathematics 11, no. 06 (2009): 993–1007. http://dx.doi.org/10.1142/s0219199709003636.

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Denote points in ℝk × ℝN - k as pairs ξ = (x,y), and assume 2 ≤ k < N. In this paper, we study the problem [Formula: see text] where [Formula: see text] and [Formula: see text], the Hardy constant. Our results are the following: (i) Let [Formula: see text]. Then there exists at least an entire cylindrically symmetric solution. (ii) Let [Formula: see text] and λ ≥ 0. Then any solution v ∈ Lp(ℝN;|x|-bdξ) is cylindrically symmetric. (iii) Let [Formula: see text] and [Formula: see text]. Then ground state solutions are not cylindrically symmetric, and therefore there exist at least two distinct entire solutions. We prove also similar results for the degenerate problem [Formula: see text] namely, for the Euler–Lagrange equations of the Maz'ya inequality with cylindrical weights.

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23

Kailey, N., and S. Sonali. "Higher-order symmetric duality in nondifferentiable multiobjective optimization over cones." Filomat 33, no. 3 (2019): 711–24. http://dx.doi.org/10.2298/fil1903711k.

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In this paper, a new pair of higher-order nondifferentiable multiobjective symmetric dual programs over arbitrary cones is formulated, where each of the objective functions contains a support function of a compact convex set. We identify a function lying exclusively in the class of higher-order K-?-convex and not in the class of K-?-bonvex function already existing in literature. Weak, strong and converse duality theorems are then established under higher-order K-?-convexity assumptions. Self duality is obtained by assuming the functions involved to be skew-symmetric. Several known results are also discussed as special cases.

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24

Shtrakov, Slavcho, and Jörg Koppitz. "Finite Symmetric Functions with Non-Trivial Arity Gap." Serdica Journal of Computing 6, no. 4 (2013): 419–36. http://dx.doi.org/10.55630/sjc.2012.6.419-436.

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Given an n-ary k-valued function f, gap(f) denotes the essential arity gap of f which is the minimal number of essential variables in f which become fictive when identifying any two distinct essential variables in f. In the present paper we study the properties of the symmetric function with non-trivial arity gap (2 ≤ gap(f)). We prove several results concerning decomposition of the symmetric functions with non-trivial arity gap with its minors or subfunctions. We show that all non-empty sets of essential variables in symmetric functions with non-trivial arity gap are separable. ACM Computing Classification System (1998): G.2.0.

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25

Akgül, Arzu, and Luminita-Ioana Cotîrlă. "Coefficient Estimates for a Family of Starlike Functions Endowed with Quasi Subordination on Conic Domain." Symmetry 14, no. 3 (2022): 582. http://dx.doi.org/10.3390/sym14030582.

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In 1999, for (0≤k<∞), the concept of conic domain by defining k-uniform convex functions were introduced by Kanas and Wisniowska and then in 2000, they defined related k-starlike functions denoted by k−UCV and k−ST respectively. Motivated by their studies, in our work, we define the class of k-parabolic starlike functions, denoted k−SHm,q, by using quasi-subordination for m-fold symmetric analytic functions, making use of conic domain Ωk. We determine the coefficient bounds and estimate Fekete–Szegö functional by the help of m-th root transform and quasi subordination for functions belonging the class k−SHm,q.

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26

Elumalai, M., and C. Selvaraj. "Some Differential Subordination and Superordination Properties Of K-Symmetric Functions." International Journal of Computational and Applied Mathematics 11, no. 2 (2016): 167. http://dx.doi.org/10.37622/ijcam/11.2.2016.167-181.

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27

Dastouri, A., та A. Ranjbari. "Функции с однородными подуровнями на конусах". Владикавказский математический журнал, № 2 (19 червня 2023): 56–64. http://dx.doi.org/10.46698/c4468-3841-3187-l.

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Extended real-valued functions on a real vector space with uniform sublevel sets are important in optimization theory. Weidner studied these functions in [1]. In the present paper, we study the class of these functions, which coincides with the class of Gerstewitz functionals, on cones. These cone are not necessarily embeddable in vector spaces. Almost any Weidner's results are not true on cones without extra conditions. We show that the mentioned conditions are necessary, by nontrivial examples. Specially for element k from the cone $\mathcal{P}$, we define $k$-directional closed subsets of the cone and prove some properties of them. For a subset $A$ of the cone $\mathcal{P}$, we characterize domain of the $\varphi_{A,k}$ (function with uniform sublevel set) and show that this function is $k$-transitive. One of the important conditions for satisfying the results, is that $k$ has the symmetric element in the cone. Also, we prove that, under some conditions, the class of Gerstewitz functionals coincides with the class of $k$-translative functions on $\mathcal{P}$.

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28

Ólafsson, Gestur, and Henrik Schlichtkrull. "Fourier transforms of spherical distributions on compact symmetric spaces." MATHEMATICA SCANDINAVICA 109, no. 1 (2011): 93. http://dx.doi.org/10.7146/math.scand.a-15179.

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In our previous articles [27] and [28] we studied Fourier series on a symmetric space $M=U/K$ of the compact type. In particular, we proved a Paley-Wiener type theorem for the smooth functions on $M$, which have sufficiently small support and are $K$-invariant, respectively $K$-finite. In this article we extend these results to $K$-invariant distributions on $M$. We show that the Fourier transform of a distribution, which is supported in a sufficiently small ball around the base point, extends to a holomorphic function of exponential type. We describe the image of the Fourier transform in the space of holomorphic functions. Finally, we characterize the singular support of a distribution in terms of its Fourier transform, and we use the Paley-Wiener theorem to characterize the distributions of small support, which are in the range of a given invariant differential operator. The extension from symmetric spaces of compact type to all compact symmetric spaces is sketched in an appendix.

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29

Rychtář, Jan. "On Gâteaux Differentiability of Convex Functions in WCG Spaces." Canadian Mathematical Bulletin 48, no. 3 (2005): 455–59. http://dx.doi.org/10.4153/cmb-2005-042-7.

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AbstractIt is shown, using the Borwein–Preiss variational principle that for every continuous convex function f on a weakly compactly generated space X, every x0 ∈ X and every weakly compact convex symmetric set K such that , there is a point of Gâteaux differentiability of f in x0 +K. This extends a Klee's result for separable spaces.

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30

YAKOUBI, FATMA, ALI BOUSSAYOUD, BAGHDADI ALOUI, and HIND MERZOUK. "k-BALANCING NUMBERS AND NEW GENERATING FUNCTIONS WITH SOME SPECIAL NUMBERS AND POLYNOMIALS." Journal of Science and Arts 22, no. 4 (2022): 929–40. http://dx.doi.org/10.46939/j.sci.arts-22.4-a14.

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In this paper, we consider some operators including symmetric functions. From those operators, we obtain the generating functions of -balancing numbers with some special numbers and Chebyshev polynomials.

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31

Saito, Hiroshi. "On L-functions associated with the vector space of binary quadratic forms." Nagoya Mathematical Journal 130 (June 1993): 149–76. http://dx.doi.org/10.1017/s0027763000004475.

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The purpose of this paper is to prove functional equations of L-functions associated with the vector space of binary quadratic forms and determine their poles and residues. For a commutative ring K, let V(K) be the set of all symmetric matrices of degree 2 with coefficients in K. In V(C), we consider the inner productwhere for .

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32

Sun, Lei, Jian Liu, and Fang-Wei Fu. "Balanced $$2^k$$ 2 k -variable rotation symmetric Boolean functions with optimal algebraic immunity." Journal of Applied Mathematics and Computing 61, no. 1-2 (2019): 185–203. http://dx.doi.org/10.1007/s12190-019-01245-2.

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33

Koike, Kazuhiko. "On a conjecture of Stanley on Jack symmetric functions." Discrete Mathematics 115, no. 1-3 (1993): 211–16. http://dx.doi.org/10.1016/0012-365x(93)90490-k.

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34

Huang, Jing, Huafeng Liu, and Fuxia Xu. "Two-Dimensional Divisor Problems Related to Symmetric L-Functions." Symmetry 13, no. 2 (2021): 359. http://dx.doi.org/10.3390/sym13020359.

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In this paper, we study two-dimensional divisor problems of the Fourier coefficients of some automorphic product L-functions attached to the primitive holomorphic cusp form f(z) with weight k for the full modular group SL2(Z). Additionally, we establish the upper bound and the asymptotic formula for these divisor problems on average, respectively.

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35

Sergeev, Igor S. "On the complexity of monotone circuits for threshold symmetric Boolean functions." Discrete Mathematics and Applications 31, no. 5 (2021): 345–66. http://dx.doi.org/10.1515/dma-2021-0031.

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Abstract The complexity of implementation of a threshold symmetric n-place Boolean function with threshold k = O(1) via circuits over the basis {∨, ∧} is shown not to exceed 2 log2 k ⋅ n + o(n). Moreover, the complexity of a threshold-2 function is proved to be 2n + Θ( $\begin{array}{} \sqrt n \end{array} $ ), and the complexity of a threshold-3 function is shown to be 3n + O(log n), the corresponding lower bounds are put forward.

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36

Boubellouta, Khadidja, Ali Boussayoud, and Mohamed Kerada. "Symmetric Functions for k-Fibonacci Numbers and Orthogonal Polynomials." Turkish Journal of Analysis and Number Theory 6, no. 3 (2018): 98–102. http://dx.doi.org/10.12691/tjant-6-3-6.

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37

Wang, Hui, Jie Peng, Yuan Li, and Haibin Kan. "On $2k$-Variable Symmetric Boolean Functions With Maximum Algebraic Immunity $k$." IEEE Transactions on Information Theory 58, no. 8 (2012): 5612–24. http://dx.doi.org/10.1109/tit.2012.2201350.

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38

Cusick, Thomas W., and Younhwan Cheon. "Affine equivalence for rotation symmetric Boolean functions with 2 k variables." Designs, Codes and Cryptography 63, no. 2 (2011): 273–94. http://dx.doi.org/10.1007/s10623-011-9553-6.

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39

Cusick, T. W., and Yuan Li. "k-th order symmetric SAC boolean functions and bisecting binomial coefficients." Discrete Applied Mathematics 149, no. 1-3 (2005): 73–86. http://dx.doi.org/10.1016/j.dam.2005.02.006.

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40

Dubey, Ramu, Lakshmi Narayan Mishra, Luis Manuel Sánchez Ruiz, and Deepak Umrao Sarwe. "Nondifferentiable Multiobjective Programming Problem under Strongly K-Gf-Pseudoinvexity Assumptions." Mathematics 8, no. 5 (2020): 738. http://dx.doi.org/10.3390/math8050738.

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In this paper we consider the introduction of the concept of (strongly) K- G f -pseudoinvex functions which enable to study a pair of nondifferentiable K-G- Mond-Weir type symmetric multiobjective programming model under such assumptions.

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41

Boughaba, Souhila, Ali Boussayoud, Serkan Araci, Mohamed Kerada, and Mehmet Acikgoz. "Construction of a new class of generating functions of binary products of some special numbers and polynomials." Filomat 35, no. 3 (2021): 1001–13. http://dx.doi.org/10.2298/fil2103001b.

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In this paper, we derive some new symmetric properties of k-Fibonacci numbers by making use of symmetrizing operator. We also give some new generating functions for the products of some special numbers such as k-Fibonacci numbers, k-Pell numbers, Jacobsthal numbers, Fibonacci polynomials and Chebyshev polynomials.

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42

Khan, Mohammad Faisal, Shahid Khan, Nazar Khan, Jihad Younis, and Bilal Khan. "Applications of q-Symmetric Derivative Operator to the Subclass of Analytic and Bi-Univalent Functions Involving the Faber Polynomial Coefficients." Mathematical Problems in Engineering 2022 (July 7, 2022): 1–9. http://dx.doi.org/10.1155/2022/4250878.

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In this paper, using the basic concepts of symmetric q -calculus operator theory, we define a symmetric q -difference operator for m -fold symmetric functions. By considering this operator, we define a new subclass ℛ b φ , m , q of m -fold symmetric bi-univalent functions in open unit disk U . As in applications of Faber polynomial expansions for f m ∈ ℛ b φ , m , q , we find general coefficient a m k + 1 for n ≥ 4 , Fekete–Szegő problems, and initial coefficients a m + 1 and a 2 m + 1 . Also, we construct q -Bernardi integral operator for m -fold symmetric functions, and with the help of this newly defined operator, we discuss some applications of our main results. For validity of our result, we have chosen to give some known special cases of our main results in the form of corollaries and remarks.

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43

Mortini, Raymond, and Rudolf Rupp. "The Symmetric Versions of Rouché’s Theorem via ∂--Calculus." Journal of Complex Analysis 2014 (February 4, 2014): 1–9. http://dx.doi.org/10.1155/2014/260953.

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Let (f,g) be a pair of holomorphic functions. In this expositional paper we apply the ∂--calculus to prove the symmetric version “|f+g|<|f|+|g| on ∂K” as well as the homotopic version of Rouché's theorem for arbitrary planar compacta K. Using Eilenberg's representation theorem we also give a converse to the homotopic version. Then we derive two analogs of Rouché's theorem for continuous-holomorphic pairs (a symmetric and a nonsymmetric one). One of the rarely presented properties of the non-symmetric version is that in the fundamental boundary hypothesis, |f+g|≤|g|, equality is allowed.

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SABA, NABIHA, and ALI BOUSSAYOUD. "ORDINARY GENERATING FUNCTIONS OF BINARY PRODUCTS OF (p,q)-MODIFIED PELL NUMBERS AND k-NUMBERS AT POSITIVE AND NEGATIVE INDICES." Journal of Science and Arts 20, no. 3 (2020): 627–48. http://dx.doi.org/10.46939/j.sci.arts-20.3-a11.

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In this paper, we introduce a operator in order to derive some new symmetric properties of (p,q)-modified Pell numbers and we give some new generating functions of the products of (p,q)-modified Pell numbers with k-Fibonacci and k-Lucas numbers, k-Pell and k-Pell Lucas numbers, k-Jacobsthal and k-Jacobsthal Lucas numbers at positive and negative indices. By making use of the operator defined in this paper, we give some new generating functions of the products of (p,q)-modified Pell numbers with k-balancing and k-Lucas-balancing numbers.

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45

Swarup, Chetan, Ramesh Kumar, Ramu Dubey, and Dowlath Fathima. "New Class of K-G-Type Symmetric Second Order Vector Optimization Problem." Axioms 12, no. 6 (2023): 571. http://dx.doi.org/10.3390/axioms12060571.

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In this paper, we present meanings of K-Gf-bonvexity/K-Gf-pseudobonvexity and their generalization between the above-notice functions. We also construct various concrete non-trivial examples for existing these types of functions. We formulate K-Gf-Wolfe type multiobjective second-order symmetric duality model with cone objective as well as cone constraints and duality theorems have been established under these aforesaid conditions. Further, we have validates the weak duality theorem under those assumptions. Our results are more generalized than previous known results in the literature.

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Weth, Tobias, and Tolga Yeşil. "Fourier extension estimates for symmetric functions and applications to nonlinear Helmholtz equations." Annali di Matematica Pura ed Applicata (1923 -) 200, no. 6 (2021): 2423–54. http://dx.doi.org/10.1007/s10231-021-01086-6.

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AbstractWe establish weighted $$L^p$$ L p -Fourier extension estimates for $$O(N-k) \times O(k)$$ O ( N - k ) × O ( k ) -invariant functions defined on the unit sphere $${\mathbb {S}}^{N-1}$$ S N - 1 , allowing for exponents p below the Stein–Tomas critical exponent $$\frac{2(N+1)}{N-1}$$ 2 ( N + 1 ) N - 1 . Moreover, in the more general setting of an arbitrary closed subgroup $$G \subset O(N)$$ G ⊂ O ( N ) and G-invariant functions, we study the implications of weighted Fourier extension estimates with regard to boundedness and nonvanishing properties of the corresponding weighted Helmholtz resolvent operator. Finally, we use these properties to derive new existence results for G-invariant solutions to the nonlinear Helmholtz equation $$\begin{aligned} -\Delta u - u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}({\mathbb {R}}^{N}), \end{aligned}$$ - Δ u - u = Q ( x ) | u | p - 2 u , u ∈ W 2 , p ( R N ) , where Q is a nonnegative bounded and G-invariant weight function.

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47

Tariq, Muhammad, Soubhagya Kumar Sahoo, Hijaz Ahmad, Thanin Sitthiwirattham, and Jarunee Soontharanon. "Several Integral Inequalities of Hermite–Hadamard Type Related to k-Fractional Conformable Integral Operators." Symmetry 13, no. 10 (2021): 1880. http://dx.doi.org/10.3390/sym13101880.

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In this paper, we present some ideas and concepts related to the k-fractional conformable integral operator for convex functions. First, we present a new integral identity correlated with the k-fractional conformable operator for the first-order derivative of a given function. Employing this new identity, the authors have proved some generalized inequalities of Hermite–Hadamard type via Hölder’s inequality and the power mean inequality. Inequalities have a strong correlation with convex and symmetric convex functions. There exist expansive properties and strong correlations between the symmetric function and various areas of convexity, including convex functions, probability theory, and convex geometry on convex sets because of their fascinating properties in the mathematical sciences. The results of this paper show that the methodology can be directly applied and is computationally easy to use and exact.

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48

Koelink, Erik, and Jie Liu. "$\mathrm{BC}_{2}$-Type Multivariable Matrix Functions and Matrix Spherical Functions." Publications of the Research Institute for Mathematical Sciences 60, no. 2 (2024): 305–49. http://dx.doi.org/10.4171/prims/60-2-2.

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Matrix spherical functions associated to the compact symmetric pair (\mathrm{SU}(m+2), \mathrm{S}(\mathrm{U}(2)\times \mathrm{U}(m))) , m\geq 2 , having a reduced root system of type \mathrm{BC}_{2} , are studied. We consider an irreducible K -representation (\pi,V) arising from the \mathrm{U}(2) -part of K , and the induced representation \mathrm{Ind}_{K}^{G} \pi splits multiplicity-free. The corresponding spherical functions, i.e. \Phi \colon G \to \mathrm{End}(V) satisfying \Phi(k_{1}gk_{2})=\pi(k_{1})\Phi(g)\pi(k_{2}) for all g\in G , k_{1},k_{2}\in K , are studied by examining certain leading terms which involve hypergeometric functions. This is done explicitly using the action of the radial part of the Casimir operator on these functions and their leading terms. To suitably grouped matrix spherical functions we associate two-variable matrix orthogonal polynomials giving a matrix analogue of Koornwinder’s 1970s two-variable orthogonal polynomials, which are Heckman–Opdam polynomials for \mathrm{BC}_{2} . In particular, we find explicit orthogonality relations with the matrix polynomials being eigenfunctions to an explicit second-order matrix partial differential operator. The scalar part of the matrix weight is less general than Koornwinder’s weight.

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49

Stadler, Peter F. "Random walks and orthogonal functions associated with highly symmetric graphs." Discrete Mathematics 145, no. 1-3 (1995): 229–37. http://dx.doi.org/10.1016/0012-365x(94)00038-k.

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50

Borogovac, Muhamed. "Characterization of Weyl Functions in the Class of Operator-Valued Generalized Nevanlinna Functions." Sarajevo Journal of Mathematics 20, no. 1 (2024): 149–71. http://dx.doi.org/10.5644/sjm.20.01.13.

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We provide the necessary and sufficient conditions for a generalized Nevanlinna function $Q$ ($Q\in N_{\kappa }\left( \mathcal{H} \right)$) to be a Weyl function (also known as a Weyl-Titchmarch function). We also investigate an important subclass of $N_{\kappa }(\mathcal{H})$, the functions that have a boundedly invertible derivative at infinity $Q'\left( \infty \right):=\lim \limits_{z \to \infty}{zQ(z)}$. These functions are regular and have the operator representation $Q\left( z \right)=\tilde{\Gamma}^{+}\left( A-z \right)^{-1}\tilde{\Gamma},z\in \rho \left( A \right)$, where $A$ is a bounded self-adjoint operator in a Pontryagin space $\mathcal{K}$. We prove that every such strict function $Q$ is a Weyl function associated with the symmetric operator $S:=A_{\vert (I-P)\mathcal{K}}$, where $P$ is the orthogonal projection, $P:=\tilde{\Gamma} \left( \tilde{\Gamma}^{+} \tilde{\Gamma} \right)^{-1} \tilde{\Gamma}^{+} $. Additionally, we provide the relation matrices of the adjoint relation $S^{+}$ of $S$, and of $\hat{A}$, where $\hat{A}$ is the representing relation of $\hat{Q}:=-Q^{-1}$. We illustrate our results through examples, wherein we begin with a given function $Q\in N_{\kappa }\left( \mathcal{H} \right)$ and proceed to determine the closed symmetric linear relation $S$ and the boundary triple $\Pi$ so that $Q$ becomes the Weyl function associated with $\Pi$. 2020 Mathematics Subject Classification. 34B20, 47B50, 47A06, 47A56

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