Surjective Map Research Articles

Nonlinear maps preserving similarity on B(X)

Let X be a real or complex Banach space with dimension at least 3. Denote by the Banach algebra of all bounded linear operators on X. In this paper, we characterize surjective maps satisfying if and only if for all .

Combination of optical emission spectroscopy and multivariate data analysis techniques as a versatile non-invasive tool for characterizing xenon/krypton mixed gas plasma inside operating ion thrusters

Non-invasive assessment of the plasma parameters is a useful tool for a reliable characterization of many electric thrusters for space applications. Due to high costs, limited availability, and growing use of electric propulsion in spaceflight, alternatives to Xe as a propellant are becoming increasingly important. One option is to use the lighter noble gas krypton or xenon/krypton gas mixtures as a propellant. We propose a versatile analytical approach for establishing empirical correlations between plasma parameters and optical emission (OE) spectroscopy utilizing principal component analysis (PCA). Our approach allows us to establish a surjective mapping of individual OE spectra via their PCA scores onto the corresponding plasma parameters. We prove the feasibility of this approach for Xe, Kr, and Xe/Kr mixed plasmas demonstrating that it is applicable for a wide range of propellant candidates. A major advantage is that the approach does not rely on any microscopic modeling of the OE spectra of the plasma. After having established corresponding reference mappings, the approach can be explored for determining non-invasively and spatially resolved plasma parameters of the propellant plasma of various kinds of operating ion thrusters, which operate in the same plasma regime as the reference plasma. Thus, this method may contribute to shorter qualification and testing times of ion thrusters.

Quotients of L-domains

It is well known that the image of a bounded complete domain, respectively a continuous lattice T under a surjective map preserving infs of nonempty subsets and directed sups is a bounded complete domain, respectively a continuous lattice and a congruence relation on T is an equivalence relation on T and a subalgebra of T × T . In this paper, we propose some counterexamples to explain that these results do not persist in L -domains. The concepts of strong homomorphisms among L -domains and congruence relation on L -domains are introduced. We prove that the image of an L -domain under a surjective map which preserves infs of nonempty subsets bounded above and directed sups and satisfies an equation is an L -domain. Meanwhile, we obtain the quotients of L -domains. Moreover, we give the Isomorphism Theorem for L -domains.

Nonlinear Birkhoff–James orthogonality preservers in smooth normed spaces

Let X and Y be smooth normed spaces which are either real and dim⁡X≥3, or infinite dimensional complex, and one of them is reflexive. Then a surjective mapping from X to Y preserves Birkhoff–James orthogonality in both directions if and only if it has the form x↦τ(x)Ux for some surjective linear or conjugate linear isometry U:X→Y and some scalar–valued mapping τ on X. In particular, there exists a surjective mapping from X to Y preserving Birkhoff–James orthogonality in both directions if and only if X and Y are isometrically isomorphic or conjugate isometrically isomorphic. Several illustrative examples and relations with Wigner's theorem are also given.

Modular parametrization as Polyakov path integral: cases with CM elliptic curves as target spaces

For an elliptic curve E over an abelian extension k/K with CM by K of Shimura type, the L-functions of its [k:K] Galois representations are Mellin transforms of Hecke theta functions; a modular parametrization (surjective map) from a modular curve to E pulls back the 1-forms on E to give the Hecke theta functions. This article refines the study of our earlier work and shows that certain class of chiral correlation functions in Type II string theory with [E]_C (E as real analytic manifold) as a target space yield the same Hecke theta functions as objects on the modular curve. The Kahler parameter of the target space [E]_C in string theory plays the role of the index (partially ordered) set in defining the projective/direct limit of modular curves.

A note on the Jacobian Conjecture

Let $F:\mathbb C^n\to \mathbb C^n$ be a polynomial mapping with non-vanishing Jacobian. If the set $S_F$ of non-properness of $F$ is smooth, then $F$ is a surjective mapping. Moreover, if $S_F$ is connected, then $\chi (S_F) \gt 0.$ Additionally, if $n=2$

Algebraic intermediate hyperbolicities

We extend Lang's conjectures to the setting of intermediate hyperbolicity and prove two new results motivated by these conjectures. More precisely, we first extend the notion of algebraic hyperbolicity (originally introduced by Demailly) to the setting of intermediate hyperbolicity and show that this property holds if the appropriate exterior power of the cotangent bundle is ample. Then, we prove that this intermediate algebraic hyperbolicity implies the finiteness of the group of birational automorphisms and of the set of surjective maps from a given projective variety. Our work answers the algebraic analogue of a question of Kobayashi on analytic hyperbolicity.

Spectrally additive group homomorphisms on Banach algebras

Let A and B be unital complex Banach algebras. A surjective map ϕ:A→B is called a spectrally additive group homomorphism if the spectrum of x±y is equal to the spectrum of ϕ(x)±ϕ(y) for each x,y∈A. If A is semisimple and either A or B has an essential socle, then we prove that a spectrally additive group homomorphism ϕ:A→B is a continuous Jordan-isomorphism. If, in addition, either A or B is prime, then we conclude that ϕ is either a continuous algebra isomorphism or anti-isomorphism. It is noteworthy that the continuity and linearity (or even additivity) of the map ϕ does not form part of the hypothesis, but is rather obtained in the conclusion. The techniques employed in the proof of these results utilizes the spectral rank, trace and determinant, and yields a new additive characterization of finite rank elements in a Banach algebra which is of independent interest.

A cliquey subgraph approach to sparsified UAV visual inertial odometry by surjective Bayes-tree-to-factor-graph mapping

A new method is proposed to improve trajectory estimation accuracy by visual inertial odometry (VIO) subject to information sparsification. Current practices assume that the result of sparsification, i.e. a subgraph of a factor graph, takes the form of a tree and impose mutually independencies upon its nodes. However, this oversimplification may undermine a close approximation to the complete information matrix and eventually results in excessive estimation errors. Therefore, we propose to use a cliquey subgraph with preserved mutual correlation between selected pairs of the nodes. Sparsification is further accelerated by the new discovery that connectivity between arbitrary nodes can be directly read from Bayes tree, by transforming its surjective mapping to the underlying factor graph to a bijective mapping through a downstream-upstream traversing operation. Results from public dataset EuRoC indicate that the proposed method, while maintaining low computational profile, achieves lower absolute trajectory error (ATE) than that of the existing ones in the easy- and hard-level sequences and performs equally well in medium-level sequences. This result renders the proposed method a potent candidate for power saving in visual navigation-purposed application specific integrated circuit (ASIC) chip designs.

Open and surjective mapping theorems for differentiable maps with critical points

Let $\Omega$ be an open subset of $\mathbb R^n (n \ge 2)$, and let $F : \Omega \rightarrow \mathbb R^n$ be a continuously differentiable map with countably many critical points. We show that $F$ is an open map. Let $G :\mathbb R^n \rightarrow \mathbb R^n (n \ge 1)$ be a continuously differentiable map such that $G(x) \rightarrow \infty$ as $x \rightarrow \infty$. Then it is proved that $G$ is surjective if and only if each connected component of the complement of the set of critical values of $G$ contains at least one image of $G$. Several applications of both theorems especially to complex analysis are presented.

On linear continuous operators between distinguished spaces $$C_p(X)$$

As proved in Ka̧kol and Leiderman (Proc AMS Ser B 8:86–99, 2021), for a Tychonoff space X, a locally convex space $$C_{p}(X)$$ is distinguished if and only if X is a $$\Delta $$ -space. If there exists a linear continuous surjective mapping $$T:C_p(X) \rightarrow C_p(Y)$$ and $$C_p(X)$$ is distinguished, then $$C_p(Y)$$ also is distinguished (Ka̧kol and Leiderman Proc AMS Ser B, 2021). Firstly, in this paper we explore the following question: Under which conditions the operator $$T:C_p(X) \rightarrow C_p(Y)$$ above is open? Secondly, we devote a special attention to concrete distinguished spaces $$C_p([1,\alpha ])$$ , where $$\alpha $$ is a countable ordinal number. A complete characterization of all Y which admit a linear continuous surjective mapping $$T:C_p([1,\alpha ]) \rightarrow C_p(Y)$$ is given. We also observe that for every countable ordinal $$\alpha $$ all closed linear subspaces of $$C_p([1,\alpha ])$$ are distinguished, thereby answering an open question posed in Ka̧kol and Leiderman (Proc AMS Ser B, 2021). Using some properties of $$\Delta $$ -spaces we prove that a linear continuous surjection $$T:C_p(X) \rightarrow C_k(X)_w$$ , where $$C_k(X)_w$$ denotes the Banach space C(X) endowed with its weak topology, does not exist for every infinite metrizable compact C-space X (in particular, for every infinite compact $$X \subset {\mathbb {R}}^n$$ ).

Jordan triple homomorphisms on {mathcal {T}}_infty (F)

Let {mathcal {T}}_infty (F) be the algebra of all {mathbb {N}}times {mathbb {N}} upper triangular matrices defined over a field F of characteristic different from 2. We consider the Jordan triple homomorphisms of {mathcal {T}}_infty (F), i.e. the additive maps that satisfy the condition phi (xyx)=phi (x)phi (y)phi (x) for all x,yin {mathcal {T}}_infty (F). For the case when F is a prime field we find the form of all such maps phi . For the general case we present the form of the surjective maps phi .

Shannon Entropy Loss in Mixed-Radix Conversions.

This paper models a translation for base-2 pseudorandom number generators (PRNGs) to mixed-radix uses such as card shuffling. In particular, we explore a shuffler algorithm that relies on a sequence of uniformly distributed random inputs from a mixed-radix domain to implement a Fisher–Yates shuffle that calls for inputs from a base-2 PRNG. Entropy is lost through this mixed-radix conversion, which is assumed to be surjective mapping from a relatively large domain of size to a set of arbitrary size n. Previous research evaluated the Shannon entropy loss of a similar mapping process, but this previous bound ignored the mixed-radix component of the original formulation, focusing only on a fixed n value. In this paper, we calculate a more precise formula that takes into account a variable target domain radix, n, and further derives a tighter bound on the Shannon entropy loss of the surjective map, while demonstrating monotonicity in a decrease in entropy loss based on increased size J of the source domain . Lastly, this formulation is used to specify the optimal parameters to simulate a card-shuffling algorithm with different test PRNGs, validating a concrete use case with quantifiable deviations from maximal entropy, making it suitable to low-power implementation in a casino.

A non-Borel special alpha-limit set in the square

AbstractWe consider the complexity of special $\alpha $ -limit sets, a kind of backward limit set for non-invertible dynamical systems. We show that these sets are always analytic, but not necessarily Borel, even in the case of a surjective map on the unit square. This answers a question posed by Kolyada, Misiurewicz, and Snoha.

Thermodynamic Formalism for Random Weighted Covering Systems

We develop for the first time a quenched thermodynamic formalism for random dynamical systems generated by countably branched, piecewise-monotone mappings of the interval that satisfy a random covering condition. Given a random contracting potential $$\varphi $$ (in the sense of Liverani–Saussol–Vaienti), we prove there exists a unique random conformal measure $$\nu _\varphi $$ and unique random equilibrium state $$\mu _\varphi $$ . Further, we prove quasi-compactness of the associated transfer operator cocycle and exponential decay of correlations for $$\mu _\varphi $$ . Our random driving is generated by an invertible, ergodic, measure-preserving transformation $$\sigma $$ on a probability space $$(\Omega ,{\mathscr {F}},m)$$ ; for each $$\omega \in \Omega $$ we associate a piecewise-monotone, surjective map $$T_\omega :I\rightarrow I$$ . We consider general potentials $$\varphi _\omega :I\rightarrow {\mathbb {R}}\cup \{-\infty \}$$ such that the weight function $$g_\omega =e^{\varphi _\omega }$$ is of bounded variation. We provide several examples of our general theory. In particular, our results apply to new examples of linear and non-linear systems including random $$\beta $$ -transformations, randomly translated random $$\beta $$ -transformations, countably branched random Gauss–Renyi maps, random non-uniformly expanding maps (such as intermittent maps and maps with contracting branches) composed with expanding maps, and a large class of random Lasota–Yorke maps.

Irreducible cone spherical metrics and stable extensions of two line bundles

A cone spherical metric is called irreducible if any developing map of the metric does not have monodromy in U(1). By using the theory of indigenous bundles, we construct on a compact Riemann surface X of genus gX≥1 a canonical surjective map from the moduli space of stable extensions of two line bundles to that of irreducible metrics with cone angles in 2πZ>1, which is generically injective in the algebro-geometric sense as gX≥2. As an application, we prove the following two results about irreducible metrics:• as gX≥2 and d is even and greater than 12gX−7, the effective divisors of degree d which could be represented by irreducible metrics form an arcwise connected Borel subset of Hausdorff dimension ≥2(d+3−3gX) in Symd(X);• as gX≥1, for almost every effective divisor D of degree odd and greater than 2gX−2 on X, there exist finitely many cone spherical metrics representing D.

Nilpotent structures and collapsing Ricci-flat metrics on the K3 surface

We exhibit families of Ricci-flat Kahler metrics on K3 surfaces which collapse to an interval, with Tian-Yau and Taub-NUT metrics occurring as bubbles. There is a corresponding continuous surjective map from the K3 surface to the interval, with regular fibers diffeomorphic to either 3-tori or Heisenberg nilmanifolds.

Hyperspaces through regular and meager subcontinua

Given a metric continuum X, let C(X) and F1(X) be the hyperspaces of subcontinua and the one-point sets of X, respectively. Let D(X) be the collection of all regular subsets in X belonging to C(X) and let M(X) be all the subcontinua of X with empty interior. In the first part of this paper we are interested in the problem to classify continua that satisfies one of the following equalities: F1(X)=M(X), D(X)=D(X)−F1(X), D(X)=C(X)−M(X) and C(X)=D(X)∪M(X). We show that for hereditarily locally connected continua these equalities and the property of not contain a continuum called dendrite D1 are equivalent each other. In the second part, we consider a continuum X for which there exists a continuous surjective and monotone function f:X→[0,1]. We show that the conditions that X is a λ-type continuum and every t in [0,1] is a cohesion point are equivalent to the equalities D(X)={A∈C(X):f(A)∈D([0,1])} and M(X)={A∈C(X):f(A)∈M([0,1])}. Throughout this paper we pose open problems.

Section problems for graph configuration spaces

We consider, for a finite graph G, when the surjective map Confn+1(G)→Confn(G) of configuration spaces admits a section. We study when the answer depends only on the homotopy type of G, and give a complete answer. We also provide basic techniques for construction of sections.

Operators preserving mutual strong Birkhoff–James orthogonality on [formula omitted

We classify surjective linear maps on B(H) that preserve mutual strong Birkhoff–James orthogonality.