Usual Operations Research Articles

Semiclassical Szegö Limit of Eigenvalue Clusters for the Hydrogen Atom Zeeman Hamiltonian

We prove a limiting eigenvalue distribution theorem (LEDT) for suitably scaled eigenvalue clusters around the discrete negative eigenvalues of the hydrogen atom Hamiltonian formed by the perturbation by a weak constant magnetic field. We study the hydrogen atom Zeeman Hamiltonian $H_V(h,B) = (1/2)( - i h {\mathbf \nabla} - {\mathbf A}(h))^2 - |x|^{-1}$, defined on $L^2 (R^3)$, in a constant magnetic field ${\mathbf B}(h) = {\mathbf \nabla} \times {\mathbf A}(h)=(0,0,\epsilon(h)B)$ in the weak field limit $\epsilon(h) \rightarrow 0$ as $h\rightarrow{0}$. We consider the Planck's parameter $h$ taking values along the sequence $h=1/(N+1)$, with $N=0,1,2,\ldots$, and $N\rightarrow\infty$. We prove a semiclassical $N \rightarrow \infty$ LEDT of the Szeg\"o-type for the scaled eigenvalue shifts and obtain both ({\bf i}) an expression involving the regularized classical Kepler orbits with energy $E=-1/2$ and ({\bf ii}) a weak limit measure that involves the component $\ell_3$ of the angular momentum vector in the direction of the magnetic field. This LEDT extends results of Szeg\"o-type for eigenvalue clusters for bounded perturbations of the hydrogen atom to the Zeeman effect. The new aspect of this work is that the perturbation involves the unbounded, first-order, partial differential operator $w(h, B) = \frac{(\epsilon(h)B)^2}{8} (x_1^2 + x_2^2) - \frac{ \epsilon(h)B}{2} hL_3 ,$ where the operator $hL_3$ is the third component of the usual angular momentum operator and is the quantization of $\ell_3$. The unbounded Zeeman perturbation is controlled using localization properties of both the hydrogen atom coherent states $\Psi_{\alpha,N}$, and their derivatives $L_3(h)\Psi_{\alpha,N}$, in the large quantum number regime $N\rightarrow\infty$.

The Degree of Irreversibility in Deterministic Finite Automata

Recently, a method to decide the NL-complete problem of whether the language accepted by a given deterministic finite automaton (DFA) can also be accepted by some reversible deterministic finite automaton (REV-DFA) has been derived. Here, we show that the corresponding problem for nondeterministic finite automata (NFA) is PSPACE-complete. The recent DFA method essentially works by minimizing the DFA and inspecting it for a forbidden pattern. We here study the degree of irreversibility for a regular language, the minimal number of such forbidden patterns necessary in any DFA accepting the language, and show that the degree induces a strict infinite hierarchy of language families. We examine how the degree of irreversibility behaves under the usual language operations union, intersection, complement, concatenation, and Kleene star, showing tight bounds (some asymptotically) on the degree.

Regularity and solvability of linear differential operators in Gevrey spaces

AbstractWe are interested in the following question: when regularity properties of a linear differential operator imply solvability of its transpose in the sense of Gevrey ultradistributions? This question is studied for a class of abstract operators that contains the usual differential operators with real‐analytic coefficients. We obtain a new proof of a global result on compact manifolds (global Gevrey hypoellipticity implying global solvability of the transpose), as well as some results in the non‐compact case by means of the so‐called property of non‐confinement of singularities. We provide applications to Hörmander operators, to operators of constant strength and to locally integrable systems of vector fields. We also analyze a conjecture stated in a recent paper of Malaspina and Nicola, which asserts that, in differential complexes naturally arising from locally integrable structures, local solvability in the sense of ultradistributions implies local solvability in the sense of distributions. We establish the validity of the conjecture when the cotangent structure bundle is spanned by the differential of a single first integral.

Hyperbolic geometry on noncommutative polyballs

This paper is an introduction to the hyperbolic geometry of noncommutative polyballs Bn(H) in B(H)n1+⋯+nk, where n=(n1,…,nk)∈Nk and B(H) is the algebra of all bounded linear operators on a Hilbert space H. We use the theory of free pluriharmonic functions on polyballs and noncommutative Poisson kernels on tensor products of full Fock spaces to define hyperbolic type metrics on Bn(H), study their properties, and obtain hyperbolic versions of Schwarz–Pick lemma for free holomorphic functions on polyballs. As a consequence, the polyballs can be viewed as noncommutative hyperbolic spaces. When specialized to the regular polydisk Dk(H) (which corresponds to the case n1=⋯=nk=1), our hyperbolic metric δH is complete and invariant under the group Aut(Dk) of all free holomorphic automorphisms of Dk(H), and the δH-topology induced on Dk(H) is the usual operator norm topology. The restriction of δH to the scalar polydisk Dk is equivalent to the Kobayashi distance on Dk. Most of the results of this paper are presented in the more general setting of Harnack (resp. Poisson) parts of the closed polyball Bn(H)−.

Simplified D = 11 pure spinor b ghost

A b-ghost was constructed for the D = 11 non-minimal pure spinor superparticle by requiring that {Q, b} = T where Q={Lambda}^{alpha }{D}_{alpha }+{R}^{alpha }{overline{W}}_{alpha } is the usual non-minimal pure spinor BRST operator. As was done for the D = 10 b-ghost, we will show that the D = 11 b-ghost can be simplified by introducing an SO(10, 1) fermionic vector {overline{Sigma}}^i constructed out of the fermionic spinor Dα and pure spinor variables. This simplified version will be shown to satisfy {Q, b} = T and {b, b} = BRST - trivial.

Non-local order parameters as a probe for phase transitions in the extended Fermi-Hubbard model

The Extended Fermi-Hubbard model is a rather studied Hamiltonian due to both its many applications and a rich phase diagram. Here we prove that all the phase transitions encoded in its one dimensional version are detectable via non-local operators related to charge and spin fluctuations. The main advantage in using them is that, in contrast to usual local operators, their asymptotic average value is finite only in the appropriate gapped phases. This makes them powerful and accurate probes to detect quantum phase transitions. Our results indeed confirm that they are able to properly capture both the nature and the location of the transitions. Relevantly, this happens also for conducting phases with a spin gap, thus providing an order parameter for the identification of superconducting and paired superfluid phases

Modeling electron fractionalization with unconventional Fock spaces

It is shown that certain fractionally-charged quasiparticles can be modeled on D−dimensional lattices in terms of unconventional yet simple Fock algebras of creation and annihilation operators. These unconventional Fock algebras are derived from the usual fermionic algebra by taking roots (the square root, cubic root, etc) of the usual fermionic creation and annihilation operators. If the fermions carry non-Abelian charges, then this approach fractionalizes the Abelian charges only. In particular, the mth-root of a spinful fermion carries charge e/m and spin 1/2. Just like taking a root of a complex number, taking a root of a fermion yields a mildly non-unique result. As a consequence, there are several possible choices of quantum exchange statistics for fermion-root quasiparticles. These choices are tied to the dimensionality of the lattice by basic physical considerations. One particular family of fermion-root quasiparticles is directly connected to the parafermion zero-energy modes expected to emerge in certain mesoscopic devices involving fractional quantum Hall states. Hence, as an application of potential mesoscopic interest, I investigate numerically the hybridization of Majorana and parafermion zero-energy edge modes caused by fractionalizing but charge-conserving tunneling.

Asymmetric Composition of Possibilistic Operators in Formal Concept Analysis: Application to the Extraction of Attribute Implications from Incomplete Contexts

Formal concept analysis theory (FCA) classically relies on the use of the Galois powerset operator. Formal similarities between possibility theory and formal concept analysis have led to the use of possibilistic operators in FCA, which were ignored before. In this paper, an approach based on the use of asymmetric composition of the two most usual possibilistic operators is proposed. It enables us to complement the stem base, by deriving attribute implications with disjunctions on both sides of the implications. Besides, the approach is also generalized to incomplete contexts involving explicit positive and negative information. We outline the potential application of these results to the completion of TBoxes in description logic.

Towards tests of quark-hadron duality with functional analysis and spectral function data

The presence of terms that violate quark-hadron duality in the expansion of QCD Green's functions is a generally accepted fact. Recently, a new approach was proposed for the study of duality violations (DVs), which exploits the existence of a rigorous lower bound on the functional distance, measured in a certain norm, between a "true" correlator and its approximant calculated theoretically along a contour in the complex energy plane. In the present paper we pursue the investigation of functional-analysis based tests towards their application to real spectral function data. We derive a closed analytic expression for the minimal functional distance based on the general weighted $L^2$ norm and discuss its relation with the distance measured in $L^\infty$ norm. Using fake data sets obtained from a realistic toy model in which we allow for covariances inspired from the publicly available ALEPH spectral functions, we obtain by Monte Carlo simulations the statistical distribution of the strength parameter that measures the magnitude of the DV term added to the usual operator product expansion (OPE). The results show that, if the region with large errors near the end-point of the spectrum in $\tau$ decays is excluded, the functional-analysis based tests using either $L^2$ or $L^\infty$ norms are able to detect, in a statistically significant way, the presence of DVs in realistic spectral function pseudodata.

History of residual stresses during the production phases of AlSi10Mg parts processed by powder bed additive manufacturing technology

ABSTRACTResidual stresses can be detrimental to the proper functioning and the structural integrity of built parts. During the selective laser melting (SLM) process, the scanned layers are subjected to rapid thermal cycles. Steep temperature gradients generate residual stresses. In this paper, the semi-destructive hole-drilling method has been used to measure the residual stresses on AISi10Mg parts fabricated by means of an SLM process. The evolution of the profile of the residual stresses of the samples is illustrated in function of the process phase (as-built, after the thermal treatment, after shot-peening) for samples with and without supports. The outcomes have shown the presence, on the as-built components, of high tensile stresses that the usual post-processing operations are not able to minimise. This paper provides a new knowledge on the history of residual stresses during the production phases of metal parts through powder-bed additive manufacturing technology.

Quaternionic Aharonov–Bohm Effect

A quaternionic analog of the Aharonov-Bohm effect is developed without the usual anti-hermitian operators in quaternionic quantum mechanics (QQM). A quaternionic phase links the solutions obtained to ordinary complex wave functions, and new theoretical studies and experimental tests are possible for them.

Numerical radius inequalities for certain 2 × 2 operator matrices II

We give several sharp numerical radius inequalities for certain 2×2 operator matrices. Among other inequalities, it is shown that if A and B be operators in B(H). Thenw([AB00])≤12(‖A‖+‖AA⁎+BB⁎‖12) and2w([0AB0])≤max⁡(‖A‖,‖B‖)+12(‖|A|t|B⁎|1−t‖+‖|B|t|A⁎|1−t‖) for all t∈[0,1], where w(⋅) and ‖⋅‖ denote the numerical radius and the usual operator norm, respectively. The second inequality refines and generalizes earlier inequalities.

Metamathematical Properties of a Constructive Multi-typed Theory

This paper describes an axiomatic theory BT, which is a suitable formal theory for developing constructive mathematics, due to its expressive language with countable number of set types and its constructive properties such as the existence and disjunction properties, and consistency with the formal Church thesis. BT has a predicative comprehension axiom and usual combinatorial operations. BT has intuitionistic logic and is consistent with classical logic. BT is mutually interpretable with a so called theory of arithmetical truth PATr and with a second-order arithmetic SA that contains infinitely many sorts of sets of natural numbers. We compare BT with some standard second-order arithmetics and investigate the proof-theoretical strengths of fragments of BT, PATr and SA.

The first eigenvalue for a quasilinear Schrödinger operator and its application

In this paper, we study the first eigenvalue for a quasilinear Schrödinger operator, which is greater than the first eigenvalue of the usual laplacian operator. As an application we treat a quasilinear resonance problem involving a subcritical growth perturbation.

Jeffrey Offutt and Jita Printing: Getting to Yes

Jeffrey Offutt had overcome many hardships and personal difficulties to better himself and build his own graphic-design and corporate-printing business: Jita Printing. Just as it seemed his efforts had begun to pay off in the form of steady work and a loyal clientele, Offutt was hit with two problems. First, designing and printing marketing materials for a small local theater had started as a quick, small undertaking for deeply discounted prices, but had turned into a drawn-out job with multiple design stages. Second, a hardcover book design for a flower shop, involved an entirely new business venture for Jita, and Offutt was having a difficult time coming up with a quote that was both competitive and profitable. How should Offutt deal with these two problems? This case is used in entrepreneurship electives at Darden, but would also fit well in a small business strategy course for non-MBA audiences. Excerpt UVA-ENT-0195 Rev. Feb. 27, 2020 Jeffrey Offutt and Jita Printing: Getting to Yes Introduction The sun had long set on the small Houston office that housed Jita Printing (Jita), a graphic design and corporate printing business, by the time Jeffrey Offutt sat down to his computer. A familiar pang of anxiety gripped him as he pulled up his email. Offutt had put off responding to two of his newer clients all day, and he knew they expected answers by the morning. It was July 2011, a mere two years after Offutt had started Jita with his wife out of their home, and the company now had its own office, an industrial printer, and a loyal client base. Offutt had built Jita from the ground up; he delivered every printing job personally, sometimes adding a handwritten thank-you note. Building and maintaining strong, personal client relationships were, in part, what set Jita above its competitors. Offutt had spent much of the week contemplating his responses to both clients and had yet to come up with the right decisions for his company. Both jobs were proving to be more complex than Offutt had anticipated. The first, designing and printing marketing materials for the Performing Arts Theater, was supposed to be a quick, small undertaking, but had turned into a drawn-out job with multiple design stages. The second, a hardcover book design for Bloom, a local flower shop, was a step outside the usual operations at Jita, and Offutt was having a difficult time coming up with a suitable quote. Offutt stared at the blinking cursor in his blank email document, knowing that his responses would affect the future of Jita's revenues and daily operations. He began to think back on how he ended up at this juncture. . . .

The coupled cluster method and entanglement in three fermion systems

The Coupled Cluster (CC) and full CI expansions are studied for three fermions with six and seven modes. Surprisingly the CC expansion is tailor made to characterize the usual stochastic local operations and classical communication (SLOCC) entanglement classes. It means that the notion of a SLOCC transformation shows up quite naturally as a one relating the CC and CI expansions, and going from the CI expansion to the CC one is equivalent to obtaining a form for the state where the structure of the entanglement classes is transparent. In this picture, entanglement is characterized by the parameters of the cluster operators describing transitions from occupied states to singles, doubles, and triples of non-occupied ones. Using the CC parametrization of states in the seven-mode case, we give a simple formula for the unique SLOCC invariant J. Then we consider a perturbation problem featuring a state from the unique SLOCC class characterized by J≠0. For this state with entanglement generated by doubles, we investigate the phenomenon of changing the entanglement type due to the perturbing effect of triples. We show that there are states with real amplitudes such that their entanglement encoded into configurations of clusters of doubles is protected from errors generated by triples. Finally we put forward a proposal to use the parameters of the cluster operator describing transitions to doubles for entanglement characterization. Compared to the usual SLOCC classes, this provides a coarse grained approach to fermionic entanglement.

Experimental Analysis of Residual Stresses on AlSi10Mg Parts Produced by Means of Selective Laser Melting (SLM)

During the Selective Laser Melting (SLM) process, the scanned layers are subjected to rapid thermal cycles. Steep temperature gradients generate residual stresses. Residual stresses can be detrimental to the proper functioning and the structural integrity of built parts. In this paper, the semi-destructive hole-drilling method has been used to measure the residual stresses on AISi10Mg parts after building, stress relieving and shot-peening, respectively. The outcomes have shown the presence, on the as-built components, of high tensile stresses that the usual post-processing operations are not able to minimize. The adopted method has proved to be a suitable tool to identify optimal process parameters for each step of the production cycle.

The topology of θω-open sets

We define the ??-closure operator as a new topological operator. We show that ??-closure of a subset of a topological space is strictly between its usual closure and its ?-closure. Moreover, we give several sufficient conditions for the equivalence between ??-closure and usual closure operators, and between ??-closure and ?-closure operators. Also, we use the ??-closure operator to introduce ??-open sets as a new class of sets and we prove that this class of sets lies strictly between the class of open sets and the class of ?-open sets. We investigate ??-open sets, in particular, we obtain a product theorem and several mapping theorems. Moreover, we introduce ?-T2 as a new separation axiom by utilizing ?-open sets, we prove that the class of !-T2 is strictly between the class of T2 topological spaces and the class of T1 topological spaces. We study relationship between ?-T2 and ?-regularity. As main results of this paper, we give a characterization of ?-T2 via ??-closure and we give characterizations of ?-regularity via ??-closure and via ??-open sets.

A spectral characterization of absolutely norming operators on S. N. ideals

The class of absolutely norming operators on complex Hilbert spaces of arbitrary dimensions was introduced in [6] and a spectral characterization theorem for these operators was established in [11]. In this paper we extend the concept of absolutely norming operators to various symmetric norms. We establish a few spectral characterization theorems for operators on complex Hilbert spaces that are absolutely norming with respect to various symmetric norms. It is also shown that for many symmetric norms the absolutely norming operators have the same spectral characterization as proven earlier for the class of operators that are absolutely norming with respect to the usual operator norm. Finally, we prove the existence of a symmetric norm on the algebra $\mathcal B (\mathcal H)$ with respect to which even the identity operator does not attain its norm.

A spectral characterization of absolutely norming operators on S. N. ideals

The class of absolutely norming operators on complex Hilbert spaces of arbitrary dimensions was introduced in [6] and a spectral characterization theorem for these operators was established in [11]. In this paper we extend the concept of absolutely norming operators to various symmetric norms. We establish a few spectral characterization theorems for operators on complex Hilbert spaces that are absolutely norming with respect to various symmetric norms. It is also shown that for many symmetric norms the absolutely norming operators have the same spectral characterization as proven earlier for the class of operators that are absolutely norming with respect to the usual operator norm. Finally, we prove the existence of a symmetric norm on the algebra $\mathcal B (\mathcal H)$ with respect to which even the identity operator does not attain its norm.