Nonseparable Case Research Articles

Hulls of double cyclic codes

In this paper, we characterize hulls of double cyclic codes over Z2. Firstly, we give the generators of hulls of double cyclic codes under the separable case and the non-separable case with coprime odd block lengths. Then, using the concept of a generating function in combinatorics, the solution of dimensions of hulls of double cyclic codes over Z2 is given. Furthermore, the enumeration of double cyclic codes over Z2 with hulls of a fixed dimension is determined. As an application, some good entanglement-assisted quantum error-correcting codes (EAQECCs) are obtained by the hulls of non-separable double cyclic codes.

A Scalable and Robust Vertex-Star Relaxation for High-Order FEM

Pavarino proved that the additive Schwarz method with vertex patches and a low-order coarse space gives a $p$-robust solver for symmetric and coercive problems. However, for very high polynomial degree it is not feasible to assemble or factorize the matrices for each patch. In this work we introduce a direct solver for separable patch problems that scales to very high polynomial degree on tensor product cells. The solver constructs a tensor product basis that diagonalizes the blocks in the stiffness matrix for the internal degrees of freedom of each individual cell. As a result, the non-zero structure of the cell matrices is that of the graph connecting internal degrees of freedom to their projection onto the facets. In the new basis, the patch problem is as sparse as a low-order finite difference discretization, while having a sparser Cholesky factorization. We can thus afford to assemble and factorize the matrices for the vertex-patch problems, even for very high polynomial degree. In the non-separable case, the method can be applied as a preconditioner by approximating the problem with a separable surrogate.

On solvability in the small and Schauder-type estimates for higher order elliptic equations in grand Sobolev spaces (nonseparable case)

In this work, it is considered an elliptic operator L of mth order with nonsmooth coefficients in a non-standard grand Sobolev space on a bounded domain generated by the norm of the grand Lebesgue space . Under weaker restrictions on the coefficients of the operator, we prove the solvability (in the strong sense) in the small in and also establish interior Schauder-type estimates for these spaces. These estimates play the main role in establishing the Fredholmness of the Dirichlet problem for the equation Lu = f. The considered spaces are not separable, infinitely differentiable functions are not dense in them, and therefore many classical methods concerning Sobolev spaces are not applicable in this case. Nevertheless, it is possible to obtain the corresponding results under the assumption that the coefficients of the principal terms of the operator L are continuous, and the rest are essentially bounded in Ω.

Choquet order and hyperrigidity for function systems

We establish a dilation-theoretic characterization of the Choquet order on the space of measures on a compact convex set using ideas from the theory of operator algebras. This yields an extension of Cartier's dilation theorem to the non-separable case, as well as a non-separable version of Šaškin's theorem from approximation theory. We show that a slight variant of this order characterizes the representations of a commutative C*-algebras that have the unique extension property relative to a set of generators. This reduces the commutative case of Arveson's hyperrigidity conjecture to the question of whether measures that are maximal with respect to the classical Choquet order are also maximal with respect to this new order. An example shows that these orders are not the same in general.

Global convergence and geometric characterization of slow to fast weight evolution in neural network training for classifying linearly non-separable data

In this paper, we study the dynamics of gradient descent in learning neural networks for classification problems. Unlike in existing works, we consider the linearly non-separable case where the training data of different classes lie in orthogonal subspaces. We show that when the network has sufficient (but not exceedingly large) number of neurons, (1) the corresponding minimization problem has a desirable landscape where all critical points are global minima with perfect classification; (2) gradient descent is guaranteed to converge to the global minima. Moreover, we discovered a geometric condition on the network weights so that when it is satisfied, the weight evolution transitions from a slow phase of weight direction spreading to a fast phase of weight convergence. The geometric condition says that the convex hull of the weights projected on the unit sphere contains the origin.

C*-algebras with and without ≪-increasing approximate units

For elements a,b of a C*-algebra we denote a=ab by a≪b. We show that all ω1-unital C*-algebras have ≪-increasing approximate units, extending a classical result for σ-unital C*-algebras. We also construct (in ZFC) the first examples of C*-algebras with no ≪-increasing approximate unit. One of these examples is a C*-subalgebra of B(ℓ2(ω1)). These examples are, by necessity, not approximately finite dimensional (AF), but some of them are still scattered and so locally finite dimensional (LF) in the sense of Farah and Katsura. It follows that there are scattered C*-algebras which are not AF. We further show that the existence of a C*-subalgebra of B(ℓ2) with no ≪-increasing approximate unit or the existence of an LF but not AF C*-subalgebra of B(ℓ2) are independent of ZFC. Our examples also show that an extension of an AF algebra by an AF algebra does not have to be an AF algebra in the nonseparable case.

Yield degradation in inertial-confinement-fusion implosions due to shock-driven kinetic fuel-species stratification and viscous heating

Anomalous thermonuclear yield degradation (i.e., that not describable by single-fluid radiation hydrodynamics) in Inertial Confinement Fusion (ICF) implosions is ubiquitously observed in both Omega and National Ignition experiments. Multiple experimental and theoretical studies have been carried out to investigate the origin of such a degradation. Relative concentration changes of fuel-ion species, as well as kinetically enhanced viscous heating, have been among possible explanations proposed for certain classes of ICF experiments. In this study, we investigate the role of such kinetic plasma effects in detail. To this end, we use the iFP code to perform multi-species ion Vlasov-Fokker-Planck simulations of ICF capsule implosions with the fuel comprising various hydrodynamically equivalent mixtures of deuterium (D) and helium-3 (3He), as in the original Rygg experiments [J. R. Rygg et al., Phys. Plasmas 13, 052702 (2006)]. We employ the same computational setup as in O. Larroche [Phys. Plasmas 19, 122706 (2012)], which was the first to simulate the experiments kinetically. However, unlike the Larroche study, and in partial agreement with experimental data, we find a systematic yield degradation in multi-species simulations versus averaged-ion simulations when the D-fuel fraction is decreased. This yield degradation originates in the fuel-ion species stratification induced by plasma shocks, which imprints the imploding system and results in the relocation of the D ions from the core of the capsule to its periphery, thereby reducing the yield relative to a non-separable averaged-ion case. By comparing yields from the averaged-ion kinetic simulations and from the hydrodynamic scaling, we also observe yield variations associated with ion kinetic effects other than fuel-ion stratification, such as ion viscous heating, which is typically neglected in hydrodynamic implosions' simulations. Since our kinetic simulations are driven by hydrodynamic boundary conditions at the fuel-ablator interface, they cannot capture the effects of ion viscosity on the capsule compression, or effects associated with the interface, which are expected to be important. Studies of such effects are left for future work.

Noncommutative Cantor–Bendixson derivatives and scattered C⁎-algebras

We analyze the sequence obtained by consecutive applications of the Cantor–Bendixson derivative for a noncommutative scattered C⁎-algebra A, using the ideal IAt(A) generated by the minimal projections of A. With its help, we present some fundamental results concerning scattered C⁎-algebras, in a manner parallel to the commutative case of scattered compact or locally compact Hausdorff spaces and superatomic Boolean algebras. It also allows us to formulate problems which have motivated the “cardinal sequences” programme in the classical topology, in the noncommutative context. This leads to some new constructions of noncommutative scattered C⁎-algebras and new open problems. In particular, we construct a type IC⁎-algebra which is the inductive limit of stable ideals Aα, along an uncountable limit ordinal λ, such that Aα+1/Aα is ⁎-isomorphic to the algebra of all compact operators on a separable Hilbert space and Aα+1 is σ-unital and stable for each α<λ, but A is not stable and where all ideals of A are of the form Aα. In particular, A is a nonseparable C⁎-algebra with no ideal which is maximal among the stable ideals. This answers a question of M. Rørdam in the nonseparable case. All the above C⁎-algebras Aαs and A satisfy the following version of the definition of an AF algebra: any finite subset can be approximated from a finite-dimensional subalgebra. Two more complex constructions based on the language developed in this paper are presented in separate papers [21,22].

Revisiting Kneser’s Theorem for Field Extensions

A Theorem of Hou, Leung and Xiang generalised Kneser’s addition Theorem to field extensions. This theorem was known to be valid only in separable extensions, and it was a conjecture of Hou that it should be valid for all extensions. We give an alternative proof of the theorem that also holds in the non-separable case, thus solving Hou’s conjecture. This result is a consequence of a strengthening of Hou et al.’s theorem that is inspired by an addition theorem of Balandraud and is obtained by combinatorial methods transposed and adapted to the extension field setting.

The tetrahexahedric Calogero model

We consider the spherical reduction of the rational Calogero model (of type $A_{n-1}$, without the center of mass) as a maximally superintegrable quantum system. It describes a particle on the $(n{-}2)$-sphere in a very special potential. A detailed analysis is provided of the simplest non-separable case, $n{=}4$, whose potential blows up at the edges of a spherical tetrahexahedron, tesselating the two-sphere into 24 identical right isosceles spherical triangles in which the particle is trapped. We construct a complete set of independent conserved charges and of Hamiltonian intertwiners and elucidate their algebra. The key structure is the ring of polynomials in Dunkl-deformed angular momenta, in particular the subspaces invariant and antiinvariant under all Weyl reflections, respectively.

Borel structures coming from various topologies on $\mathbf{B}(\mathcal{H})$

Although there exist different types of (well-known) locally convex topologies on $\mathbf{B}(\mathcal{H})$, the notion of measurability on the set of operator valued functions $f:\Omega\to \mathbf{B}(\mathcal{H})$ is unique when $\mathcal{H}$ is separable (see [1]). In this current discussion we observe that unlike the separable case, in the non-separable case we have to face different types of measurability. Moreover the algebraic operations “addition and product” are not compatible with the set of operator valued measurable functions.

Nonseparability and von Neumann's theorem for domains of unbounded operators

A classical theorem of von Neumann asserts that every unbounded self-adjoint operator $A$ in a separable Hilbert space $H$ is unitarily equivalent to an operator $B$ in $H$ such that $D(A)\cap D(B)=\{0\}$. Equivalently this can be formulated as a property for nonclosed operator ranges. We will show that von Neumann's theorem does not directly extend to the nonseparable case. In this paper we prove a characterisation of the property that an operator range $\mathcal{R}$ in a general Hilbert space $H$ admits a unitary operator $U$ such that $U\mathcal{R}\cap\mathcal{R}=\{0\}$. This allows us to study stability properties of operator ranges with the aforementioned property.

The tetrahexahedric angular Calogero model

The spherical reduction of the rational Calogero model (of type A n−1 and after removing the center of mass) is considered as a maximally superintegrable quantum system, which describes a particle on the (n−2)-sphere subject to a very particular potential. We present a detailed analysis of the simplest non-separable case, n=4, whose potential is singular at the edges of a spherical tetrahexahedron. A complete set of independent conserved charges and of Hamiltonian intertwiners is constructed, and their algebra is elucidated. They arise from the ring of polynomials in Dunkl-deformed angular momenta, by classifying the subspaces invariant and antiinvariant under all Weyl reflections, respectively.

Metrizable DH-spaces of the first category

We show that if a separable space X contains an open subset which is of the first category in itself and is not a λ-space, then X has c many types of countable dense subsets. We introduce Λ-spaces as a generalization of the λ-spaces for non-separable case and consider properties of these spaces. In particular, we prove that if X is a non-σ-discrete h-homogeneous Λ-space, then X is densely homogeneous and X∖A is homeomorphic to X for every σ-discrete subset A⊂X.

Compact covers and function spaces

For a Tychonoff space X, we denote by Cp(X) and Cc(X) the space of continuous real-valued functions on X equipped with the topology of pointwise convergence and the compact-open topology respectively. Providing a characterization of the Lindelöf Σ-property of X in terms of Cp(X), we extend Okunevʼs results by showing that if there exists a surjection from Cp(X) onto Cp(Y) (resp. from Lp(X) onto Lp(Y)) that takes bounded sequences to bounded sequences, then υY is a Lindelöf Σ-space (respectively K-analytic) if υX has this property. In the second part, applying Christensenʼs theorem, we extend Pelantʼs result by proving that if X is a separable completely metrizable space and Y is first countable, and there is a quotient linear map from Cc(X) onto Cc(Y), then Y is a separable completely metrizable space. We study also a non-separable case, and consider a different approach to the result of J. Baars, J. de Groot, J. Pelant and V. Valov, which is based on the combination of two facts: Complete metrizability is preserved by ℓp-equivalence in the class of metric spaces (J. Baars, J. de Groot, J. Pelant). If X is completely metrizable and ℓp-equivalent to a first-countable Y, then Y is metrizable (V. Valov). Some additional results are presented.

Testing blind separability of complex Gaussian mixtures

The separation of a complex mixture based solely on second-order statistics can be achieved using the Strong Uncorrelating Transform (SUT) if and only if all sources have distinct circularity coefficients. However, in most problems we do not know the circularity coefficients, and they must be estimated from observed data. In this work, we propose a detector, based on the generalized likelihood ratio test (GLRT), to test the separability of a complex Gaussian mixture using the SUT. For the separable case (distinct circularity coefficients), the maximum likelihood (ML) estimates are straightforward. On the other hand, for the non-separable case (at least one circularity coefficient has multiplicity greater than one), the ML estimates are much more difficult to obtain. To set the threshold, we exploit Wilks' theorem, which gives the asymptotic distribution of the GLRT under the null hypothesis. Finally, numerical simulations show the good performance of the proposed detector and the accuracy of Wilks' approximation.

An ant colony system for transportation user equilibrium analysis in congested networks

In this paper we present Ant Colony System for Traffic Assignment (ACS-TA) for the solution of deterministic and stochastic user equilibria (DUE and SUE, respectively) problems. DUE and SUE are two well known transportation problems where the transportation demand has to be assigned to an underlying network (supply in transportation terminology) according to single user satisfaction rather than aiming at some global optimum. ACS-TA turns the classic ACS meta-heuristic for discrete optimization into a technique for equilibrium computation. ACS-TA can be easily adapted to take into account all aspects characterizing the traffic assignment problem: multiple origin-destination pairs, link congestion, non-separable cost link functions, elasticity of demand, multiple classes of demand and different user cost models including stochastic cost perception. Applications to different networks, including a non-separable costs case study and the standard Sioux Falls benchmark, are reported. Results show good performance and wider applicability with respect to conventional approaches especially for stochastic user equilibrium computation.

Normal systems over ANR’s, rigid embeddings and nonseparable absorbing sets

Most of results of Bestvina and Mogilski [\textit{Characterizing certain incomplete infinite-dimensional absolute retracts}, Michigan Math. J. \textbf{33} (1986), 291--313] on strong $Z$-sets in ANR's and absorbing sets is generalized to nonseparable case. It is shown that if an ANR $X$ is locally homotopy dense embeddable in infinite-dimensional Hilbert manifolds and $w(U) = w(X)$ (where `$w$' is the topological weight) for each open nonempty subset $U$ of $X$,then $X$ itself is homotopy dense embeddable in a Hilbert manifold. It is also demonstrated that whenever $X$ is an AR, its weak product $W(X,*) = \{(x_n)_{n=1}^{\infty} \in X^{\omega}:\ x_n = * \textup{for almost all} n\}$ is homeomorphic to a pre-Hilbert space $E$ with $E \cong \Sigma E$. An intrinsic characterization of manifolds modelled on such pre-Hilbert spaces is given.

Analysis of a class of nonlinear and non-separable multiscale representations

In this paper, we introduce a particular class of nonlinear and non-separable multiscale representations which embeds most of these representations. After motivating the introduction of such a class on one-dimensional examples, we investigate the multi-dimensional and non-separable case where the scaling factor is given by a non-diagonal dilation matrix M. We also propose new convergence and stability results in L p and Besov spaces for that class of nonlinear and non-separable multiscale representations. We end the paper with an application of the proposed study to the convergence and the stability of some nonlinear multiscale representations.

On properties of h-homogeneous spaces with the Baire property

We describe how to assign an h-homogeneous space b ( X , k ) with a dense complete subspace and of weight k to any strongly zero-dimensional metric space X of weight ⩽ k. We investigate the properties of such spaces and obtain the conditions when b ( X 1 , k ) is homeomorphic to b ( X 2 , k ) . The h-homogeneous separable space T which is a union of B ( ω ) and a countable subspace was constructed by E. van Douwen. Similarly, the h-homogeneous separable space S which is a union of B ( ω ) and a σ-compact subspace was described by J. van Mill. These spaces are generalized for the non-separable case. We prove that if Ind X = 0 and X = G ∪ L , where G is an absolute G δ and L is of first category, then X ω is an h-homogeneous space. We consider certain cases when A × X ω is homeomorphic to X ω providing A ⊂ X ω .