Equivariant compactifications of reductive groups

We study conformal conserved currents in arbitrary irreducible representations of the Lorentz group using the embedding space formalism. With the help of the operator product expansion, we first show that conservation conditions can be fully investigated by considering only two- and three-point correlation functions. We then find an explicitly conformally-covariant differential operator in embedding space that implements conservation based on the standard position space operator product expansion differential operator ∂μ, although the latter does not uplift to embedding space covariantly. The differential operator in embedding space that imposes conservation is the same differential operator mathcal{D} ijA used in the operator product expansion in embedding space. We provide several examples including conserved currents in irreducible representations that are not symmetric and traceless. With an eye on four-point conformal bootstrap equations for four conserved vector currents 〈JJJJ〉 and four energy-momentum tensors 〈TTTT〉, we mostly focus on conservation conditions for leftlangle JJmathcal{O}rightrangle and leftlangle TTmathcal{O}rightrangle . Finally, we reproduce and extend the consequences of conformal Ward identities at coincident points by determining three-point coefficients in terms of charges.

We develop techniques useful for obtaining conformal blocks in embedding space. We construct a unique differential operator in embedding space and use it to construct a function that will be an important ingredient in assembling conformal blocks. We show a number of relations that the components of conformal blocks satisfy and find invariance of our expressions under the dihedral group.

Problems of solving different linear difference equation is given to study the properties of the matrix operators in various functional spaces. One of the important problems of functional analysis is to establish criteria of boundedness of the linear operators in functional spaces. Question of the boundedness of matrix operators in sequence spaces is a classic problem of functional analysis and there are many unsolved problems in it. For example, in the general case it is impossible to establish the boundedness of the matrix operator in the spaces of sequences by the given matrix. Therefore, various classes of matrix operators are considered for which the criteria of their boundedness are known. Due to the variety of encountered problems in practice, it is necessary to have various alternative criteria for the boundedness of matrix operators. In this paper, we establish a new alternative criterion for the boundedness of one class of matrix operators.

Major neuropsychiatric disorders such as major depressive disorder (MDD) and schizophrenia (SCZ), as well as the neurodevelopmental disorder autism spectrum disorder (ASD), are traditionally treated as distinct clinical entities. However, genome-wide association studies indicate shared genetic risks, motivating a transdiagnostic view. Resting-state functional connectivity (rsFC) is a promising biomarker for these disorders, but its high dimensionality complicates inference of inter-disorder relationships in the native feature space. Here, we develop an rsFC-based embedding-relation workflow that quantifies disorder relationships in a connectivity-informed, low-dimensional embedding space. Central to the workflow is a mutual information-based embedding framework that evaluates candidate embedding approaches and selects an optimal strategy. Using synthetic connectivity data, the framework indicates that rsFC embeddings are best represented in a spherical space under a moderate level of supervision. Building on this insight, we applied the workflow to curated, multi-disorder rsFC datasets to derive shared embedding spaces encompassing the connectivity features of ASD, MDD, and SCZ. In these spaces, we consistently observed a robust three-way relationship: a pronounced neurobiological dissimilarity between ASD and MDD, contrasted with greater similarity between SCZ and both disorders. These findings support a dimensional, transdiagnostic perspective on neuropsychiatric disorders and offer new insights into their shared and distinct neural underpinnings.

A bounded linear operator is said to be nice if its adjoint preserves extreme points of the dual unit ball. Motivated by a description due to Labuschagne and Mascioni [9] of such maps for the space of compact operators on a Hilbert space, in this article we consider a description of nice surjections onK(X, Y) for Banach spacesX, Y. We give necessary and sufficient conditions when nice surjections are given by composition operators. Our results imply automatic continuity of these maps with respect to other topologies on spaces of operators. We also formulate the corresponding result forL(X, Y) thereby proving an analogue of the result from [9] forL p (1 <p ≠ 2 < ∞) spaces. We also formulate results when nice operators are not of the canonical form, extending and correcting the results from [8].

Image reconstruction algorithms seek to reconstruct a sought-after object from a collection of measurements. However, complete measurements such that an object can be uniquely reconstructed are seldom available. Analysis of the null components of the imaging system can guide both physical design of the imaging system and algorithmic design of reconstruction algorithms to more closely reconstruct the true object. Characterizing the null space of an imaging operator is a computationally demanding task. While computationally efficient methods have been proposed to iteratively estimate the null space components of a single or a small number of images, full characterization of the null space remains intractable for large images using existing methods. This work proposes a novel learning-based framework for constructing a null space projection operator of linear imaging operators utilizing an artificial neural network autoencoder. To illustrate the approach, a stylized 2D accelerated MRI reconstruction problem (for which an analytical representation of the null space is known) was considered. The proposed method was compared to state-of-the-art randomized linear algebra techniques in terms of accuracy, computational cost, and memory requirements. Numerical results show that the proposed framework achieves comparable or better accuracy than randomized singular value decomposition. It also has lower computational cost and memory requirements in many practical scenarios, such as when the dimension of the null space is small compared to the dimension of the object.

Geophysical Inverse Theory and Regularization Problems

In this chapter, we develop a basic theory of quasisymmetric embeddings in metric spaces, following for the most part the paper by Tukia and Väisälä [176]. We use the notation f : X → Y for an embedding f of a metric space X in a metric space Y. Thus, note in particular that in this notation f is not supposed to be onto. Recall that an embedding is a map that is a homeomorphism onto its image.

Generalizations of Self-Adjointness to Banach Spaces

We investigate properties of order continuous operators on pre-Riesz spaces with respect to the embedding of the range space into a vector lattice cover or, in particular, into its Dedekind completion. We show that order continuity is preserved under this embedding for positive operators, but not in general. For the vector lattice \ell_0^\infty of eventually constant sequences, we consider the pre-Riesz space of regular operators on \ell_0^\infty and show that making the range space Dedekind complete does not provide a vector lattice cover of the pre-Riesz space. A similar counterexample is obtained for the directed part of the space of order continuous operators on \ell_0^\infty .

In the first part of this paper we present a representation theorem for the directional derivative of the metric projection operator in an arbitrary Hilbert space. As a consequence of the representation theorem, we present in the second part the development of the theory of projected dynamical systems in infinite dimensional Hilbert space. We show that this development is possible if we use the viable solutions of differential inclusions. We use also pseudomonotone operators.

Unconditional optimal [formula omitted]-norm error estimate and superconvergence analysis of a linearized nonconforming finite element variable-time-step BDF2 method for the nonlinear complex Ginzburg–Landau equation

We describe how to implement the conformal bootstrap program in the context of the embedding space OPE formalism introduced in previous work. To take maximal advantage of the known properties of the scalar conformal blocks for symmetric-traceless exchange, we construct tensorial generalizations of the three-point and four-point scalar conformal blocks that have many nice properties. Further, we present a special basis of tensor structures for three-point correlation functions endowed with the remarkable simplifying property that it does not mix under permutations of the external quasi-primary operators. We find that in this approach, we can write the M-point conformal bootstrap equations explicitly in terms of the standard position space cross-ratios without the need to project back to position space, thus effectively deriving all conformal bootstrap equations directly from the embedding space. Finally, we lay out an algorithm for generating the conformal bootstrap equations in this formalism. Collectively, the tensorial generalizations, the new basis of tensor structures, as well as the procedure for deriving the conformal bootstrap equations lead to four-point bootstrap equations for quasi-primary operators in arbitrary Lorentz representations expressed as linear combinations of the standard scalar conformal blocks for spin-ℓ exchange, with finite ℓ-independent terms. Moreover, the OPE coefficients in these equations conveniently feature trivial symmetry properties. The only inputs necessary are the relevant projection operators and tensor structures, which are all fixed by group theory. To illustrate the procedure, we present one nontrivial example involving scalars S and vectors V, namely ⟨SSSV⟩.

The Cox ring of a spherical embedding

We propose a simple method for constructing positive operator-valued measures (POVMs) using any set of matrices which form an orthonormal basis for the space of complex matrices. Considering the orthonormal set of irreducible spherical tensors, we examine the properties of the construction on the [Formula: see text]-dimensional subspace of the [Formula: see text]-dimensional Hilbert space of [Formula: see text] qubits comprising the permutationally symmetric states. Using the notion of vectorization, the constructed POVMs are interpretable as projection operators in a higher-dimensional space. We then describe a method to physically realize the constructed POVMs for symmetric states using the Clebsch–Gordan decomposition of the tensor product of irreducible representations of the rotation group. We illustrate the proposed construction on a spin-1 system, and show that it is possible to generate entangled states from separable ones.