This paper studies the existence and construction of bases consisting of tangential and anisotropic vectors in finite-dimensional quadratic spaces over fields of characteristic different from two. While classical theory guarantees the existence of orthogonal bases in regular quadratic spaces, the existence of bases governed by alternative geometric constraints such as tangency or isotropy has remained largely unexplored. We introduce determinant-based constructive methods extending the Gram–Schmidt process to arbitrary quadratic spaces, yielding systematic criteria for generating orthogonal, tangential, and isotropic families of vectors. Our main results establish necessary and sufficient conditions for the existence of tangential bases, including a characterization of regular spaces of positive index and strong algebraic obstructions in the hyperbolic case. In addition, we prove a general constructive existence theorem for isotropic bases in real regular quadratic spaces.

In this paper, we discuss the geometric structure, i.e., Dirac structure, underlying port-Hamiltonian systems. The paper has a tutorial character, and thus it contains questions/exercises. We start with the general definition of a Dirac structure and show that on finite-dimensional spaces, there is a simple matrix characterization. By simple examples, we show that, even in the finite-dimensional case, a Dirac structure does not guarantee the existence of solutions for an associated ordinary differential or difference equation. For associated partial differential equations, i.e., on an infinite-dimensional Dirac structure, the existence problem becomes even more challenging. We show that the spaces have to be chosen with care, but when we have shown the existence of solutions, then the Dirac structure will give us the desired properties, such as conservation of energy. The Dirac structure also implies that the associated transfer function has nice properties.

Objective.We propose a new formulation for ideal observers (IOs) that incorporate stochastic object models (SOMs) for data acquisition optimization.Approach. A data acquisition system is considered as a (possibly nonlinear) discrete-to-discrete mapping from a finite-dimensional object space,x∈Rnd, to a finite-dimensional measurement space,y∈Rm. For binary tasks, the two underlying SOMs,H0andH1, are specified by two probability density functions (PDFs)p0(x),p1(x). This leads to the notion of intrinsic likelihood ratio (LR)ΛI(x)=p1(x)/p0(x)and intrinsic class separability (ICS), the latter quantifies the population separability that is independent of data acquisition. With respect to ICS, the IO employs the 'extrinsic' LRΛ(y)=pr(y|H1)/pr(y|H0)of the data and quantifies the extrinsic class separability (ECS). The difference between ICS and ECS measures the efficiency of data acquisition. We show that the extrinsic LRΛ(y)is the expectation of the intrinsic LRΛI(x), where the expectation is with respect to the posterior PDFpr(x|y,H0)underH0.Main results. We use two examples, one to clarify the new IO and the second to demonstrate its potential for real world applications. Specifically, we apply the new IO to spectral optimization in dual-energy CT projection domain material decomposition (pMD), for which SOMs are used to describe variability of basis material line integrals. The performance rank orders obtained by IO agree with physics predictions.Significance.The main computation in the new IO involves sampling from the posterior PDFpr(x|y,H0), which are similar to (fully) Bayesian reconstruction. Thus our IO computation is amenable to standard techniques already familiar to CT researchers. The example of dual-energy pMD serves as a prototype for other spectral optimization problems, e.g., for photon counting CT or multi-energy CT with multi-layer detectors.

While the expected time of first visit to a subspace under some quantum evolution is a relevant statistic, it is also important to confront this information with the associated variance, so that one may have a better understanding of how representative such means are. In this work, we study these matters in the context of quantum channels acting on finite-dimensional Hilbert spaces, and we provide practical methods for calculating the above data with respect to goal subspaces of interest. By considering an assumption which is strictly weaker than irreducibility, we are also able to consider unitary quantum walks. Our approach is based on the monitored evolution of positive, trace-preserving operators given by quantum Markov chains as described by Gudder. Then, given any channel and some goal subspace of interest, one may induce an absorbing chain in a natural way so that the monitored evolution of the original map can be examined.

This paper investigates a class of implicit neutral fractional integro-differential equations of Volterra–Fredholm type. The equations incorporate a tempered fractional derivative in the Caputo sense, along with both retarded (delay) and advanced arguments. The problem is formulated on a time domain segmented into past, present, and future intervals and includes nonlinear mixed integral operators. Using Banach’s contraction mapping principle and Schauder’s fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions within the space of continuous functions. The study is then extended to general Banach spaces by employing Darbo’s fixed point theorem combined with the Kuratowski measure of noncompactness. Ulam–Hyers–Rassias stability is also analyzed under appropriate conditions. To demonstrate the practical applicability of the theoretical framework, explicit examples with specific nonlinear functions and integral kernels are provided. Furthermore, detailed numerical simulations are conducted using MATLAB-based specialized algorithms, illustrating solution convergence and behavior in both finite-dimensional and Banach space contexts.

Abstract We present a systematic quantization scheme for bounded symplectic domains of the form D × G ⊂ T ∗ G , where D ⊂ g ∗ is a bounded region defined by algebraic inequalities and G is a compact Lie group with Lie algebra g . The finiteness of the symplectic volume implies that quantization yields a finite-dimensional Hilbert space, with observables represented by Hermitian matrices, for which we provide an explicit realization. Boundary effects necessitate modifications of the standard von Neumann and Dirac conditions, which usually underlie the correspondence principle. Physically, the compact group G plays the role of momentum space, while g ∗ corresponds to the (noncommutative) position space of a particle. The assumption of compact momentum space has profound physical consequences, including the supertunneling phenomenon and the emergence of a maximal fermion density.

This article discusses the question of the exact formula for K–functional for a pair(λn1 (ω), ln∞(h)) and the equivalent formula for Kp–functional for a pair (λnp(ω), ln∞(h)).

In this paper, we propose two projected dynamical systems for solving inverse quasi-variational inequality problems in finite-dimensional Hilbert spaces–one ensuring finite-time stability and the other guaranteeing fixed-time stability. We first establish the connection between these dynamical systems and the solutions of inverse quasi-variational problems. Then, under mild conditions on the operators and parameters, we analyze the finite-time and fixed-time stability of the proposed systems. Both approaches offer accelerated convergence; however, while the settling time of a finite-time stable dynamical system depends on initial conditions, the fixed-time stable system achieves convergence within a predefined time, independent of initial conditions. Furthermore, we consider an explicit forward Euler discretization of the dynamical system, ensuring a consistent discretization of the fixed-time stable dynamics. This leads to a novel forward-backward algorithm, for which we present a detailed convergence analysis. To demonstrate their effectiveness, we provide numerical experiments, including an application to the traffic assignment problem.

The functions under approximation here have as a domain a finite dimensional Banach space with dimension N∈N and are with values in RN. Exploiting some topological properties of the above we are able to perform multi-composite general Neural Network multivariate approximation to the above functions. The treatment is quantitative. We produce multivariate multi-composite general Jackson type inequalities involving the modulus of continuity of the function under approximation. The established convergences are pointwise and uniform. Our technique is expected to lead to accelerated speeds of convergence.

Abstract Let $\mathbb {k}$ be a field, and let $\mathcal {C}$ be a Cauchy complete $\mathbb {k}$ -linear braided category with finite-dimensional morphism spaces and . We call an indecomposable object X of $\mathcal C$ non-negligible if there exists $Y\in \mathcal {C}$ such that is a direct summand of $Y\otimes X$ . We prove that every non-negligible object $X\in \mathcal {C}$ such that $\dim \operatorname {End}(X^{\otimes n})<n!$ for some n is automatically rigid. In particular, if $\mathcal {C}$ is semisimple of moderate growth and weakly rigid, then $\mathcal {C}$ is rigid. As applications, we simplify Huang’s proof of rigidity of representation categories of certain vertex operator algebras, and we get that for a finite semisimple monoidal category $\mathcal {C}$ , the data of a $\mathcal {C}$ -modular functor is equivalent to a modular fusion category structure on $\mathcal {C}$ , answering a question of Bakalov and Kirillov. Furthermore, we show that if $\mathcal {C}$ is rigid and has moderate growth, then the quantum trace of any nilpotent endomorphism in $\mathcal {C}$ is zero. Hence $\mathcal {C}$ admits a semisimplification, which is a semisimple braided tensor category of moderate growth. Finally, we discuss rigidity in braided r-categories which are not semisimple, which arise in logarithmic conformal field theory. These results allow us to simplify a number of arguments of Kazhdan and Lusztig.

Abstract Traced monoidal categories model processes that can feed their outputs back to their own inputs, abstracting iteration. The category of finite-dimensional Hilbert spaces with the direct sum tensor is not traced. But surprisingly, in 2014, Bartha showed that the monoidal subcategory of isometries is traced. The same holds for coisometries, unitary maps, and contractions. This suggests the possibility of feeding outputs of quantum processes back to their own inputs, analogous to iteration. In this paper, we show that Bartha’s result is not specifically tied to Hilbert spaces, but works in any dagger additive category with Moore–Penrose pseudoinverses (a natural dagger-categorical generalization of inverses).

While quantum statistical mechanics triumphs in explaining many equilibrium phenomena, there is an increasing focus on going beyond conventional scenarios of thermalization. Traditionally examples of nonthermalizing systems are either integrable or disordered. Recently, examples of translationally invariant physical systems have been discovered whose excited energies avoid thermalization either due to local constraints (whether exact or emergent) or due to higher-form symmetries. In this article, we extend these investigations for the case of 3D U ( 1 ) quantum dimer models, which are lattice gauge theories with finite-dimensional local Hilbert spaces (also generically called quantum link models) with staggered charged static matter. Using a combination of analytical and numerical methods, we uncover a class of athermal states that arise in large winding sectors, when the system is subjected to external electric fields. The polarization of the dynamical fluxes in the direction of applied field traps excitations in 2D planes, while an interplay with the Gauss law constraint in the perpendicular direction causes exotic athermal behavior due to the emergence of new conserved quantities. This causes a geometric fragmentation of the system. We provide analytical arguments showing that the scaling of the number of fragments is exponential in the linear system size, leading to weak fragmentation. Further, we identify sectors which host fractonic excitations with severe mobility restrictions. The unitary evolution of fragments dominated by fractons is qualitatively different from the one dominated by nonfractonic excitations.

We reduce the polynomial cluster value problem for the algebra of bounded analytic functions, H ∞ , on the ball of Banach spaces X to the same polynomial cluster value problem for H ∞ but on the ball of those spaces which are `1 -sums of finite dimensional spaces.

Given a quantum state in the finite-dimensional Hilbert space Cn, the range of possible values of a quantum observable is usually identified with the discrete spectrum of eigenvalues of a corresponding Hermitian matrix. Here any such observable is identified with (i) an 'ortho-measurable' function defined on the Boolean 'ortho-algebra' generated by the eigenspaces that form an orthogonal decomposition of Cn, and (ii) a 'numerically identified' orthogonal decomposition of Cn. The latter means that each subspace of the orthogonal decomposition can be uniquely identified by its own attached real number, just as each eigenspace of a Hermitian matrix can be uniquely identified by the corresponding eigenvalue. Furthermore, any density matrix on Cn is identified with a Bayesian prior 'ortho-probability' measure defined on the linear subspaces that make up the Boolean ortho-algebra induced by its eigenspaces. Then any pure quantum state is identified with a degenerate density matrix, and any mixed state with a probability measure on a set of orthogonal pure states. Finally, given any quantum observable, the relevant Bayesian posterior probabilities of measured outcomes can be found by the usual trace formula that extends Born's rule. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.

We establish a solvability criterion for nonautonomous time-evolution inclusions governed by the right-hand side without the convexity assumption. In this study, we examine the problem x˙(t)∈F(t,x(t))a.e.onI0:=[0,T0], for some T0>0, where F has a closed graph, constrained within the subdifferential operator of a convex, time-dependent potential g(t,·). This work extends the existing literature, which has primarily focused on the autonomous case or time-dependent mappings with a time-independent convex potential in finite-dimensional spaces. Under novel assumptions that the potential g is convex in the state variable and Lipschitz in time, we establish a solvability criterion. An example of the applicability of our result for nonconvex nonautonomous differential inclusions is stated. As a significant application of our main theorem, we demonstrate that a certain class of implicit nonconvex sweeping processes with an unbounded perturbative term admits solutions.

We prove that the norm of X⊗ˆπY is SSD if either X=ℓp(I) for p>2 and Y is a finite-dimensional Banach space such that the modulus of convexity is of power type q<p (e.g. if Y∗ is a subspace of Lq), or if X=c0(I) and Y∗ is any uniformly convex finite-dimensional Banach space. We also provide a characterisation of SSD elements of a projective tensor product which attain its projective norm in terms of a strengthening of the local Bollobás property for bilinear mappings.

In this paper, we design some generalized split problems which can be seen as an extended form of the split variational inequality problems. We present several iterative algorithms for solving generalized split problems and demonstrate the weak convergence results under some appropriate assumptions within the context of real Hilbert spaces. Finally, we support these results with the help of numerical examples in both the finite and infinite dimensional spaces. As a result of this work, a new direction will be opened in studying split problems.

Abstract Let G and H be finite-dimensional vector spaces over $\mathbb{F}_p$ . A subset $A \subseteq G \times H$ is said to be transverse if all of its rows $\{x \in G \colon (x,y) \in A\}$ , $y \in H$ , are subspaces of G and all of its columns $\{y \in H \colon (x,y) \in A\}$ , $x \in G$ , are subspaces of H . As a corollary of a bilinear version of the Bogolyubov argument, Gowers and the author proved that dense transverse sets contain bilinear varieties of bounded codimension. In this paper, we provide a direct combinatorial proof of this fact. In particular, we improve the bounds and evade the use of Fourier analysis and Freiman’s theorem and its variants.

Abstract We study universal minimal flows of connected and locally contractible topological groups by means of tools from homotopy theory and dynamical cohomology. We provide methods for computing their first cohomotopy groups and illustrate them by computations related to connected Lie groups and identity components of homeomorphism groups of compact connected surfaces with or without boundary. Since metrizable spaces have countable first cohomotopy groups, these methods serve as a useful new instrument for proving nonmetrizability of various universal minimal flows. In this way we reprove several results in this direction known before and also prove some new ones. Finally, we indicate how our methods can be used to construct finite-dimensional minimal spaces for infinite-dimensional topological groups, having all orbits of first category.

We refine the well-known Blanco-Koldobsky-Turnšek theorem, which states that a norm one linear operator defined on a Banach space is an isometry if and only if it preserves orthogonality at every element of the space. We improve the result for Banach spaces in which the set of all smooth points forms a dense $G_{\delta}$ set. We further demonstrate that if a norm one operator preserves orthogonality on a hyperplane not passing through the origin, then it is an isometry. In the context of finite-dimensional Banach spaces, we prove that preserving orthogonality on the set of all extreme points of the unit ball forces the norm one operator to be an isometry, which substantially refines the Blanco-Koldobsky-Turnšek theorem. Finally, for finite-dimensional polyhedral spaces, we establish the significance of the set of all $k$-smooth points for any possible $k$ in the study of isometric theory.