scholarly journals Topology on the spectrum of the algebra of entire symmetric functions of bounded type on the complex $L_\infty$

2017 ◽  
Vol 9 (1) ◽  
pp. 22-27 ◽  
Author(s):  
T.V. Vasylyshyn
Keyword(s):  
Symmetric Functions ◽  
Complex Banach Space ◽  
Continuous Homomorphism ◽  
The Other ◽  
Quotient Topology ◽  
Abelian Topological Group ◽  
Point Evaluation ◽  
Other Hand ◽  
Continuous Complex ◽  
Complex Valued

It is known that the so-called elementary symmetric polynomials $R_n(x) = \int_{[0,1]}(x(t))^n\,dt$ form an algebraic basis in the algebra of all symmetric continuous polynomials on the complex Banach space $L_\infty,$ which is dense in the Fr\'{e}chet algebra $H_{bs}(L_\infty)$ of all entire symmetric functions of bounded  type on $L_\infty.$ Consequently, every continuous homomorphism $\varphi: H_{bs}(L_\infty) \to \mathbb{C}$ is uniquely determined by the sequence $\{\varphi(R_n)\}_{n=1}^\infty.$ By the continuity of the homomorphism $\varphi,$ the sequence $\{\sqrt[n]{|\varphi(R_n)|}\}_{n=1}^\infty$ is bounded. On the other hand, for every sequence $\{\xi_n\}_{n=1}^\infty \subset \mathbb{C},$ such that the sequence $\{\sqrt[n]{|\xi_n|}\}_{n=1}^\infty$ is bounded,  there exists  $x_\xi \in L_\infty$ such that $R_n(x_\xi) = \xi_n$ for every $n \in \mathbb{N}.$ Therefore, for the point-evaluation functional $\delta_{x_\xi}$ we have $\delta_{x_\xi}(R_n) = \xi_n$ for every $n \in \mathbb{N}.$ Thus, every continuous complex-valued homomorphism of $H_{bs}(L_\infty)$ is a point-evaluation functional at some point of $L_\infty.$ Note that such a point is not unique. We can consider an equivalence relation on $L_\infty,$ defined by $x\sim y \Leftrightarrow \delta_x = \delta_y.$ The spectrum (the set of all continuous complex-valued homomorphisms) $M_{bs}$ of the algebra $H_{bs}(L_\infty)$ is one-to-one with the quotient set $L_\infty/_\sim.$ Consequently, $M_{bs}$ can be endowed with the quotient topology. On the other hand, it is naturally to identify $M_{bs}$ with the set of all sequences $\{\xi_n\}_{n=1}^\infty \subset \mathbb{C}$ such that the sequence $\{\sqrt[n]{|\xi_n|}\}_{n=1}^\infty$ is bounded.We show that the quotient topology is Hausdorffand that $M_{bs}$ with the operation of coordinate-wise addition of sequences forms an abelian topological group.

scholarly journals Metric on the spectrum of the algebra of entire symmetric functions of bounded type on the complex $L_\infty$

2018 ◽  
Vol 9 (2) ◽  
pp. 198-201 ◽  
Author(s):  
T.V. Vasylyshyn
Keyword(s):  
Symmetric Functions ◽  
Complex Banach Space ◽  
Quotient Topology ◽  
Natural Representation ◽  
Abelian Topological Group ◽  
Point Evaluation ◽  
Continuous Complex ◽  
Fréchet Algebra ◽  
Quotient Set ◽  
Complex Valued

It is known that every complex-valued homomorphism of the Fréchet algebra $H_{bs}(L_\infty)$ of all entire symmetric functions of bounded type on the complex Banach space $L_\infty$ is a point-evaluation functional $\delta_x$ (defined by $\delta_x(f) = f(x)$ for $f \in H_{bs}(L_\infty)$) at some point $x \in L_\infty.$ Therefore, the spectrum (the set of all continuous complex-valued homomorphisms) $M_{bs}$ of the algebra $H_{bs}(L_\infty)$ is one-to-one with the quotient set $L_\infty/_\sim,$ where an equivalence relation "$\sim$'' on $L_\infty$ is defined by $x\sim y \Leftrightarrow \delta_x = \delta_y.$ Consequently, $M_{bs}$ can be endowed with the quotient topology. On the other hand, $M_{bs}$ has a natural representation as a set of sequences which endowed with the coordinate-wise addition and the quotient topology forms an Abelian topological group. We show that the topology on $M_{bs}$ is metrizable and it is induced by the metric $d(\xi, \eta) = \sup_{n\in\mathbb{N}}\sqrt[n]{|\xi_n-\eta_n|},$ where $\xi = \{\xi_n\}_{n=1}^\infty,\eta = \{\eta_n\}_{n=1}^\infty \in M_{bs}.$

scholarly journals Point-evaluation functionals on algebras of symmetric functions on $(L_\infty)^2$

2019 ◽  
Vol 11 (2) ◽  
pp. 493-501
Author(s):  
T.V. Vasylyshyn
Keyword(s):  
Symmetric Functions ◽  
Topological Algebra ◽  
Complex Banach Space ◽  
Symmetric Polynomials ◽  
Point Evaluation ◽  
Cartesian Power ◽  
Bounded Complex ◽  
Continuous Complex ◽  
Elementary Symmetric Polynomials ◽  
Complex Valued

It is known that every continuous symmetric (invariant under the composition of its argument with each Lebesgue measurable bijection of $[0,1]$ that preserve the Lebesgue measure of measurable sets) polynomial on the Cartesian power of the complex Banach space $L_\infty$ of all Lebesgue measurable essentially bounded complex-valued functions on $[0,1]$ can be uniquely represented as an algebraic combination, i.e., a linear combination of products, of the so-called elementary symmetric polynomials. Consequently, every continuous complex-valued linear multiplicative functional (character) of an arbitrary topological algebra of the functions on the Cartesian power of $L_\infty,$ which contains the algebra of continuous symmetric polynomials on the Cartesian power of $L_\infty$ as a dense subalgebra, is uniquely determined by its values on elementary symmetric polynomials. Therefore, the problem of the description of the spectrum (the set of all characters) of such an algebra is equivalent to the problem of the description of sets of the above-mentioned values of characters on elementary symmetric polynomials. In this work, the problem of the description of sets of values of characters, which are point-evaluation functionals, on elementary symmetric polynomials on the Cartesian square of $L_\infty$ is completely solved. We show that sets of values of point-evaluation functionals on elementary symmetric polynomials satisfy some natural condition. Also, we show that for any set $c$ of complex numbers, which satisfies the above-mentioned condition, there exists an element $x$ of the Cartesian square of $L_\infty$ such that values of the point-evaluation functional at $x$ on elementary symmetric polynomials coincide with the respective elements of the set $c.$

scholarly journals Peterson Isomorphism in K-theory and Relativistic Toda Lattice

2018 ◽  
Vol 2020 (19) ◽  
pp. 6421-6462 ◽  
Author(s):  
Takeshi Ikeda ◽  
Shinsuke Iwao ◽  
Toshiaki Maeno
Keyword(s):  
Symmetric Functions ◽  
Toda Lattice ◽  
Flag Variety ◽  
The Other ◽  
Initial Condition ◽  
Birational Morphism ◽  
Affine Grassmannian ◽  
Other Hand ◽  
K Theory ◽  
Relativistic Toda Lattice

Abstract The K-homology ring of the affine Grassmannian of $SL_{n}(\mathbb{C})$ was studied by Lam, Schilling, and Shimozono. It is realized as a certain concrete Hopf subring of the ring of symmetric functions. On the other hand, for the quantum K-theory of the flag variety $F\,\! l_{n}$, Kirillov and Maeno provided a conjectural presentation based on the results obtained by Givental and Lee. We construct an explicit birational morphism between the spectrums of these two rings. Our method relies on Ruijsenaars’s relativistic Toda lattice with unipotent initial condition. From this result, we obtain a K-theory analogue of the so-called Peterson isomorphism for (co)homology. We provide a conjecture on the detailed relationship between the Schubert bases, and, in particular, we determine the image of Lenart–Maeno’s quantum Grothendieck polynomial associated with a Grassmannian permutation.

Montel Algebras on the Plane

1970 ◽  
Vol 22 (1) ◽  
pp. 116-122 ◽  
Author(s):  
W. E. Meyers
Keyword(s):  
Topological Algebra ◽  
Maximum Modulus ◽  
Continuous Homomorphism ◽  
Continuous Functions ◽  
Maximum Modulus Principle ◽  
Compact Open Topology ◽  
Bounded Functions ◽  
Continuous Complex ◽  
Complex Valued

The results of Rudin in [7] show that under certain conditions, the maximum modulus principle characterizes the algebra A (G) of functions analytic on an open subset G of the plane C (see below). In [2], Birtel obtained a characterization of A(C) in terms of the Liouville theorem; he proved that every singly generated F-algebra of continuous functions on C which contains no non-constant bounded functions is isomorphic to A(C) in the compact-open topology. In this paper we show that the Montel property of the topological algebra A (G) also characterizes it. In particular, any Montel algebra A of continuous complex-valued functions on G which contains the polynomials and has continuous homomorphism space M (A) homeomorphic to G is precisely A(G).

scholarly journals A Bijection between Atomic Partitions and Unsplitable Partitions

10.37236/494 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
William Y. C. Chen ◽  
Teresa X. S. Li ◽  
David G. L. Wang
Keyword(s):  
Symmetric Functions ◽  
The Other ◽  
Generating Set ◽  
Power Sum ◽  
Other Hand

In the study of the algebra $\mathrm{NCSym}$ of symmetric functions in noncommutative variables, Bergeron and Zabrocki found a free generating set consisting of power sum symmetric functions indexed by atomic partitions. On the other hand, Bergeron, Reutenauer, Rosas, and Zabrocki studied another free generating set of $\mathrm{NCSym}$ consisting of monomial symmetric functions indexed by unsplitable partitions. Can and Sagan raised the question of finding a bijection between atomic partitions and unsplitable partitions. In this paper, we provide such a bijection.

scholarly journals Algebras of symmetric analytic functions on Cartesian powers of Lebesgue integrable in a power $p\in [1,+\infty)$ functions

2021 ◽  
Vol 13 (2) ◽  
pp. 340-351
Author(s):  
T.V. Vasylyshyn
Keyword(s):  
Banach Space ◽  
Analytic Functions ◽  
Entire Functions ◽  
Complex Banach Space ◽  
Cartesian Power ◽  
Entire Complex ◽  
Continuous Complex ◽  
Fréchet Algebra ◽  
Algebraic Basis ◽  
Complex Valued

The work is devoted to the study of Fréchet algebras of symmetric (invariant under the composition of every of components of its argument with any measure preserving bijection of the domain of components of the argument) analytic functions on Cartesian powers of complex Banach spaces of Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued functions on the segment $[0,1]$ and on the semi-axis. We show that the Fréchet algebra of all symmetric analytic entire complex-valued functions of bounded type on the $n$th Cartesian power of the complex Banach space $L_p[0,1]$ of all Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued functions on the segment $[0,1]$ is isomorphic to the Fréchet algebra of all analytic entire functions on $\mathbb C^m,$ where $m$ is the cardinality of the algebraic basis of the algebra of all symmetric continuous complex-valued polynomials on this Cartesian power. The analogical result for the Fréchet algebra of all symmetric analytic entire complex-valued functions of bounded type on the $n$th Cartesian power of the complex Banach space $L_p[0,+\infty)$ of all Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued functions on the semi-axis $[0,+\infty)$ is proved.

A study of cometary eruptions through OH radio-lines

1999 ◽  
Vol 173 ◽  
pp. 249-254
Author(s):  
A.M. Silva ◽  
R.D. Miró
Keyword(s):  
Short Duration ◽  
Growth Curves ◽  
The Other ◽  
Initial Mass ◽  
Radio Lines ◽  
Other Hand ◽  
Sudden Rise

AbstractWe have developed a model for theH2OandOHevolution in a comet outburst, assuming that together with the gas, a distribution of icy grains is ejected. With an initial mass of icy grains of 108kg released, theH2OandOHproductions are increased up to a factor two, and the growth curves change drastically in the first two days. The model is applied to eruptions detected in theOHradio monitorings and fits well with the slow variations in the flux. On the other hand, several events of short duration appear, consisting of a sudden rise ofOHflux, followed by a sudden decay on the second day. These apparent short bursts are frequently found as precursors of a more durable eruption. We suggest that both of them are part of a unique eruption, and that the sudden decay is due to collisions that de-excite theOHmaser, when it reaches the Cometopause region located at 1.35 × 105kmfrom the nucleus.

Closing the GAP—A 5Å Scanning Microscope

1969 ◽  
Vol 27 ◽  
pp. 6-7
Author(s):  
A. V. Crewe
Keyword(s):  
Normal Form ◽  
The Other ◽  
Resolving Power ◽  
Scanning Microscope ◽  
Conventional Microscope ◽  
Other Hand ◽  
Closing The Gap ◽  
Thermal Sources ◽  
Better Than ◽  
Transmission Microscope

We have become accustomed to differentiating between the scanning microscope and the conventional transmission microscope according to the resolving power which the two instruments offer. The conventional microscope is capable of a point resolution of a few angstroms and line resolutions of periodic objects of about 1Å. On the other hand, the scanning microscope, in its normal form, is not ordinarily capable of a point resolution better than 100Å. Upon examining reasons for the 100Å limitation, it becomes clear that this is based more on tradition than reason, and in particular, it is a condition imposed upon the microscope by adherence to thermal sources of electrons.

Life Beyond 1MeV

1980 ◽  
Vol 38 ◽  
pp. 30-33
Author(s):  
K.H. Westmacott
Keyword(s):  
Electron Microscopy ◽  
Past Experience ◽  
The Other ◽  
Good Case ◽  
Other Hand ◽  
The U.S

Life beyond 1MeV – like life after 40 – is not too different unless one takes advantage of past experience and is receptive to new opportunities. At first glance, the returns on performing electron microscopy at voltages greater than 1MeV diminish rather rapidly as the curves which describe the well-known advantages of HVEM often tend towards saturation. However, in a country with a significant HVEM capability, a good case can be made for investing in instruments with a range of maximum accelerating voltages. In this regard, the 1.5MeV KRATOS HVEM being installed in Berkeley will complement the other 650KeV, 1MeV, and 1.2MeV instruments currently operating in the U.S. One other consideration suggests that 1.5MeV is an optimum voltage machine – Its additional advantages may be purchased for not much more than a 1MeV instrument. On the other hand, the 3MeV HVEM's which seem to be operated at 2MeV maximum, are much more expensive.

Can the Academic Self-Concept be Positive and Realistic?

2005 ◽  
Vol 19 (3) ◽  
pp. 129-132 ◽  
Author(s):  
Reimer Kornmann
Keyword(s):  
Academic Performance ◽  
Opposite Effect ◽  
Self Esteem ◽  
Point Of View ◽  
The Other ◽  
Self Concept ◽  
High Ability ◽  
Self Image ◽  
Other Hand

Summary: My comment is basically restricted to the situation in which less-able students find themselves and refers only to literature in German. From this point of view I am basically able to confirm Marsh's results. It must, however, be said that with less-able pupils the opposite effect can be found: Levels of self-esteem in these pupils are raised, at least temporarily, by separate instruction, academic performance however drops; combined instruction, on the other hand, leads to improved academic performance, while levels of self-esteem drop. Apparently, the positive self-image of less-able pupils who receive separate instruction does not bring about the potential enhancement of academic performance one might expect from high-ability pupils receiving separate instruction. To resolve the dilemma, it is proposed that individual progress in learning be accentuated, and that comparisons with others be dispensed with. This fosters a self-image that can in equal measure be realistic and optimistic.

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