Tensor product semigroups

The concept of Arithmetic Odd Decomposition [AOD] was introduced by E. Ebin Raja Merly and N. Gnanadhas. A decomposition {G 1 , G 2 ,. .. , G n } G is said to be Arithmetic Decomposition if each G i is connected and | E(G i)| = a+ (i-1) d , for 1  i  n and a, d  ℤ. When a =1 and d = 2, we call the Arithmetic Decomposition as Arithmetic Odd Decomposition. A decomposition { G 1 , G 3 ,. .. , G 2n-1 } of G is said to be AOD if | E

The Latin Tableau Conjecture, a longstanding open problem in algebraic combinatorics, characterizes the existence of fillings of Young diagrams with prescribed symbol multiplicities under row and column non-repetition constraints. Proposed over two decades ago in connection with Rota's basis conjecture, it has remained unresolved. This paper presents an attempt at a constructive proof using an inductive approach that removes shape-preserving matchings from the outer antidiagonal of the Young diagram, ensuring the chromatic difference sequence (CDS) majorizes the residual type at each step. The proof includes a key lemma guaranteeing the preservation of the majorization condition. We discuss applications to Rota's basis conjecture, highlighting how the resolution advances matroid theory and multiflow problems. To illustrate, we provide detailed numerical examples with figures. Both Grok's and Gemini's lemma proofs are featured. This method offers a transparent, algorithmic path to generating such tableaux, with implications for broader tableau theory.

The local multiplier algebra Mloc(A) of a C∗-algebra A is the C∗-algebraic direct limit of multiplier algebras M(K) along the downward-directed system E(A) of all (closed) essential ideals K of A. Such algebras first arose in the study of derivations and were formally introduced by ...

Batanin and Leinster's work on globular operads has provided one of many potential definitions of a weak ω-category. Through the language of globular operads they construct a monad whose algebras encode weak ω-categories. The purpose of this work is to show how to construct a similar monad which will allow us to formulate weak ω-categorifications of any equational algebraic theory. We first review the classical theory of operads and PROs. We then present how Leinster's globular operads can be extended to a theory of globular PROs via categorical enrichment over the category of collections. It is then shown how a process called globularization allows us to construct from a classical PRO P a globular PRO whose algebras are those algebras for P which are internal to the category of strict ω-categories and strict ω-functors. Leinster's notion of a contraction structure on a globular operad is then extended to this setting of globular PROs in order to build a monad whose algebras are globular PROs with contraction over the globularization of the classical PRO P . Among these PROs with contraction over P is the globular PRO whose algebras are by construction the fully weakened ω-categorifications of the algebraic theory encoded by P . Globular PROs and the weak ω-categorification of algebraic theories by

Christensen and Evans showed that, in the language of Hilbert modules, a bounded derivation on a von Neumann algebra with values in a two-sided von Neumann module (i.e. a sufficiently closed two-sided Hilbert module) are inner. Then they use this result to show that the generator of a normal uniformly continuous completely positive (CP-) semigroup on a von Neumann algebra decomposes into a (suitably normalized) CP-part and a derivation like part. The backwards implication is left open. In these notes we show that both statements are equivalent among themselves and equivalent to a third one, namely, that any type I tensor product system of von Neumann modules has a unital central unit. From existence of a central unit we deduce that each such product system is isomorphic to a product system of time ordered Fock modules. We, thus, find the analogue of Arveson's result that type I product systems of Hilbert spaces are symmetric Fock spaces. On the way to our results we have to develop a number of tools interesting on their own right. Inspired by a very similar notion due to Accardi and Kozyrev, we introduce the notion of semigroups of completely positive definite kernels (CPD-semigroups), being a generalization of both CP-semigroups and Schur semigroups of positive definite C-valued kernels. The structure of a type I system is determined completely by its associated CPD-semigroup

This letter presents an improved Toom's algorithm that allows hardware savings without slowing down the processing speed. We derive formulae for the number of multiplications and additions required to compute the linear convolution of size . We demonstrate the computational advantage of the proposed improved algorithm when compared to previous algorithms, such as the original matrix-vector multiplication and the FFT algorithms.

In this paper we introduce a new flatness property of acts over monoids which is an extension of Conditions (P ) and (E), called Condition (EP ) and will give a classification of monoids over which all (finitely generated, cyclic, monocyclic) right acts satisfying Condition (EP ) have other flatness properties and also monoids over which all (cyclic) right acts satisfy Condition (EP ).

Near-openly generated groups are introduced. It is a topological and multiplicative subclass of R-factorizable groups. Dense and open subgroups, quotients and Raikov completion of a near-openly generated group are near-openly generated. Almost connected pro-Lie groups, lindelöff almost metrizable groups and the spaces C p (X) of all continuous real-valued functions on a Tychonoff space X with pointwise convergence topology are nearopenly generated. We provide characterizations of near-openly generated groups using methods of inverse spectra and topological game theory.

Let A,B,C be C -algebras. Given A-B and B-C normed bi- modules V and W respectively, whose unit ball is convex with respect to the actions of the C -algebras, we study the reasonable seminorms on the relative tensor product V B W, having the same convexity property. This kind of bimodule is often encountered and retains many features of the usual normed space. We show that the classical Grothendieck program extends nicely in this setting. Fixing B, we then establish that there exists an unique such seminorm on V B W for any V,W if and only if B is infinite in a weaker sense than proper infiniteness and stronger than the non existence of tracial states (the equivalence of these two latter notions still remaining open). Applying this result when B is a stable C -algebra, we show that the relative Haagerup tensor product of operator bimodules is both injective and projective.

Problems related to symmetries and dimensional reduction are common in the mathematical and physical literature, and are intensively studied presently. As a rule, the symmetry group ("reducing group") and its orbits ("external dimensions") are compact, and this is essential in models where the volume of the orbits is related to physical quantities. However, this case is only a part of the natural problems related to dimensional reduction. In the present paper, we consider an action of a (generally non-compact) Lie group on a vector bundle, construct a formalism of reduced bundles for description of all invariant sections of the original bundle, and study the algebraic structures that occur in the reduced bundle. We show that in the case of a non-compact reducing group it is possible that the reduction is non-standard ("non-canonical"), and construct an explicit obstruction for canonical reduction in terms of cohomology of groups. We consider in detail the reduction of tangent and cotangent bundles, and show that, in general, the duality between the two is violated in the process of reduction. The reduction of the tensor product of tangent and cotangent bundles is also discussed. We construct examples of non-canonical dimensional reduction and of violation of duality between the tangent and cotangent bundles in the reduction.

In this paper we consider a polynomial collocation method for the numerical solution of a singular integral equation over the interval. More precisely, the operator of our integral equation is supposed to be of the form aI + µ −1 bSµI + K with S the Cauchy integral operator, with piecewise continuous coefficients a and b , with a regular integral operator K , and with a Jacobi weight µ. To the equation [aI + µ −1 bSµI + K]u = f we apply a collocation method, where the collocation points are the Chebyshev nodes of the second kind and where the trial space is the space of polynomials multiplied by another Jacobi weight. For the stability and convergence of this collocation in weighted L 2 spaces, we derive necessary and sufficient conditions.

We analyze how tensor product constructions and direct sum decompositions interact with automorphisms of the unitary group U(H) and with the property of norm-attainability. New structural theorems identify classes of operators whose norm-attainability is preserved (or destroyed) by tensoring with identity or by forming infinite direct sums, and we characterize automorphisms that decompose along tensor factors. Applications to block-operator matrices and weighted shifts are discussed.

In this paper, we build a multidimensional wavelet decomposition based on polyharmonic B-splines. The prewavelets are polyharmonic splines and so not tensor products of univariate wavelets. Explicit construction of the filters specified by the classical dyadic scaling relations is given and the decay of the functions and the filters is shown. We then design the decomposition/recomposition algorithm by means of downsampling/upsampling and convolution products.

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The following paper is a variation on a theme of Gianfranco Cimmino on some integral representation formulas for the solution of a linear equations system. Cimmino was probably motivated for giving a representation formula suitable not only for theoretical investigations but also for applied computation. In this paper we will prove that the Cimmino integrals are strictly related to the residues of some zeta-like functions associated to the linear system.

This article is concerned with an extensive study of a infinite-dimensional Lie algebra sv, introduced in [14] in the context of non-equilibrium statistical physics, containing as subalgebras both the Lie algebra of invariance of the free Schrödinger equation and the central charge-free Virasoro algebra Vect(S 1 ). We call sv the Schrödinger-Virasoro Lie algebra. We choose to present sv from a Newtonian geometry point of view first, and then in connection with conformal and Poisson geometry. We turn afterwards to its representation theory: realizations as Lie symmetries of field equations, coadjoint representation, coinduced representations in connection with Cartan's prolongation method (yielding analogues of the tensor density modules for Vect(S 1 )), and finally Verma modules with a Kac determinant formula. We also present a detailed cohomogical study, providing in particular a classification of deformations and central extensions; there appears a non-local cocycle.

We present an approximate implicitization method for planar curves. The computed implicit representation is a piecewise rational approximation of the distance function to the given parametric curve. The proposed method consists of four main steps: quadratic B-spline approximation of the given parametric curve, data reduction, segments-wise implicitization, multiplying with suitable polynomial factors. These segments are joined such that the collection generate a global C r spline function which approximates the distance function, for r = 0, 1.

The Holevo quantity provides an upper bound for the mutual information between the sender of a classical message encoded in quantum carriers and the receiver. Applying the strong subadditivity of entropy we prove that the Holevo quantity associated with an initial state and a given quantum operation represented in its Kraus form is not larger than the exchange entropy. This implies upper bounds for the coherent information and for the quantum Jensen-Shannon divergence. Restricting our attention to classical information we bound the transmission distance between any two probability distributions by the entropic distance, which is a concave function of the Hellinger distance.

Adem and Reichstein introduced the ideal of truncated symmetric polynomials to present the permutation invariant subring in the cohomology of a finite product of projective spaces. Building upon their work, I describe a generating set of the ideal of truncated symmetric polynomials in arbitrary positive characteristic, and offer a conjecture for minimal generators.

We obtain a complete asymptotic expansion of the integrated density of states of operators of the form H = (-Δ) w + B in R d . Here w > 0 and B belong to a wide class of almost-periodic self-adjoint pseudodifferential operators of order less than 2w. In particular, we obtain such an expansion for magnetic Schrödinger operators with either smooth periodic or generic almost-periodic coefficients. ln λ) when d > 3. In the multidimensional almost-periodic case, formula (1.6) was known only with K = 0 and R(λ 2 ), see . In , formula (1.5) was obtained for operators (1.1) assuming that the real-valued potential V is either smooth periodic, or generic quasi-periodic, or Vol. 15 (2014) Asymptotic Expansion of IDS 265 belongs to a reasonably wide class of almost-periodic functions (see for a complete set of conditions on V ). In the case of magnetic Schrödinger operator one expects the asymptotic formula (1.5) to be still valid (assuming similar restrictions on the magnetic potential A = A(x) and the electric potential V = V (x)). However, until now only the partial asymptotic formula (1.6) with K = 0 and R(λ) = O(λ d-2+ε 2

Connections between set-theoretic Yang-Baxter and reflection equations and quantum integrable systems are investigated. We show that set-theoretic R-matrices are expressed as twists of known solutions. We then focus on reflection and twisted algebras and we derive the associated defining algebra relations for R-matrices being Baxterized solutions of the A-type Hecke algebra H N (q = 1). We show in the case of the reflection algebra that there exists a "boundary" finite sub-algebra for some special choice of "boundary" elements of the B-type Hecke algebra B N (q = 1, Q). We also show the key proposition that the associated double row transfer matrix is essentially expressed in terms of the elements of the B-type Hecke algebra. This is one of the fundamental results of this investigation together with the proof of the duality between the boundary finite subalgebra and the B-type Hecke algebra. These are universal statements that largely generalize previous relevant findings and also allow the investigation of the symmetries of the double row transfer matrix.

Let k be an algebraically closed field of characteristic 0 and let A be a finitely generated k-algebra that is a domain whose Gelfand-Kirillov dimension is in [2, 3). We show that if A has a nonzero locally nilpotent derivation then A has quadratic growth. In addition to this, we show that A either satisfies a polynomial identity or A is isomorphic to a subalgebra of D(X), the ring of differential operators on an irreducible smooth affine curve X, and A is birationally isomorphic to D(X).

In this paper we study initial topological properties of the (non-)finitely-generated locus of Frobenius Algebra coming from Stanley-Reisner rings defined through face ideals. More specifically, we will give a partial answer to a conjecture made by M. Katzman about the openness of the finitely generated locus of such Frobenius algebras. This conjecture can be formulated in a precise manner: Let us define where F denotes the Frobenius functor and E Rp the injective hull of the residual field of the local ring (Rp, pRp). Is the locus U an open set in the Zariski Topology? In the case where R is a ring of the form R = K[[x 1 , . . . , xn]]/I, where I ⊂ R is a face ideal, i.e., square-free monomial ideal, we show that the corresponding U has non-empty interior. Even more, we prove that in general U contains two different types of opens sets and that in specific situation its complement contains intersections of opens and closes sets in the Zariski topology. Furthermore, we explicitly verify in some non-trivial examples that U is an non-trivial open set.

We systematically and comprehensively study the statistics of the spectrum of gauge groups of the weakly coupled free fermionic heterotic region of the string landscape. Specifically, we are seeking to generate all possible gauge group sectors for consistent models ...

In this paper we have showed that the qubit can be expressed through the coherent states. Consequently, a message, i.e. a sequence of qubits, is expressed as a tensor product of coherent states. In the quantum information theory and practice, only the code and key message are expressed as a sequence of qubits, i.e. through a quantum channel, the properly information will be transmitted by using a classical channel. Even if the most used coherent states in the quantum information theory are the coherent states of the harmonic oscillator (particularly, expressing by them the Schrödinger "cat states" and the Bell states), several authors have been demonstrated that other kind of coherent states may be used in quantum information theory. For the ensembles of qubits, we must use the density operator, in order to describe the informational content of the ensemble. The diagonal representation of the density operator, in the coherent state representation, is also useful to examine the entanglement of the states.

The shallow water equations (SWE), which describe the flow of a thin layer of fluid in two dimensions have been used by the atmospheric modelling community as a vehicle for testing promising numerical methods for solving atmospheric and oceanic problems. The SWE are important for the study of the dynamics of large-scale flows, as well for the development of new numerical schemes that are applied to more complex models. In this paper we present a finite difference p-adaptive method based on high order finite differences that is applied using an error indicator for solving the SWE on the sphere. A standard test set is used to evaluate the accuracy of the new method. The results obtained are compared with the pseudo-spectral method.

In a preceding article the authors and Tran Ngoc Nam constructed a minimal injective resolution of the mod 2 cohomology of a Thom spectrum. A Segal conjecture type theorem for this spectrum was proved. In this paper one shows that the above mentioned resolutions can be realized topologically. In fact there exists a family of cofibrations inducing short exact sequences in mod 2 cohomology. The resolutions above are obtained by splicing together these short exact sequences. Thus the injective resolutions are realizable in the best possible sense. In fact our construction appears to be in some sense an injective closure of one of Takayasu. It strongly suggests that one can construct geometrically (not only homotopically) certain dual Brown-Gitler spectra.

─ A novel, efficient, and simple modification to standard marching-on-in-time (MOT)–based time-domain integral equation (TDIE) solvers is presented. It allows for the use of high-order temporal interpolators without the need to extrapolate and predict future unknowns. The order of these temporal interpolators is increased as the distance of source and testing quadrature points increases. The proposed TDIE solver significantly increases the accuracy of solutions by exploiting high-order temporal interpolation at no significant extra computational cost. Numerical examples are presented to validate the proposed method. Index Terms Marching-on-in-time (MOT), temporal interpolator, and time-domain integral equation (TDIE).

In this article, we elucidate the structure and properties of a class of anomalous high-energy states of matter-free U (1) quantum link gauge theory Hamiltonians using numerical and analytical methods. Such anomalous states, known as quantum many-body scars in the literature, have generated a lot of interest due to their athermal nature. Our starting Hamiltonian is H = O kin + λOpot, where λ is a real-valued coupling, and O kin (Opot) are summed local diagonal (off-diagonal) operators in the electric flux basis acting on the elementary plaquette □. The spectrum of the model in its spin-1 2 representation on Lx × Ly lattices reveal the existence of sublattice scars, |ψs⟩, which satisfy O pot,□ |ψs⟩ = |ψs⟩ for all elementary plaquettes on one sublattice and O pot,□ |ψs⟩ = 0 on the other, while being simultaneous zero modes or nonzero integer-valued eigenstates of O kin . We demonstrate a "triangle relation" connecting the sublattice scars with nonzero integer eigenvalues of O kin to particular sublattice scars with O kin = 0 eigenvalues. A fraction of the sublattice scars have a simple description in terms of emergent short singlets, on which we place analytic bounds. We further construct a long-ranged parent Hamiltonian for which all sublattice scars in the null space of O kin become unique ground states and elucidate some of the properties of its spectrum. In particular, zero energy states of this parent Hamiltonian turn out to be exact scars of another U (1) quantum link model with a staggered short-ranged diagonal term. CONTENTS I. Introduction 1 II. The models 3 A. Two U (1) quantum link lattice gauge theories 4 B. An index theorem and mid-spectrum zero modes 4 III. Sublattice scars 5 A. Short singlet sublattice scars 6 B. Non-singlet sublattice scars 7 C. Sublattice scars with O kin = ±2 9 IV. Efficient algorithm to generate sublattice scars 10 V. Parent Hamiltonian for sublattice scars 11 VI. More quantum scars from zero modes 12 VII. Conclusion and outlook 13 Acknowledgments 15 References 15

We show that the tensor product of two cyclic A∞-algebras is, in general, not a cyclic A∞-algebra, but an A∞-algebra with homotopy inner product. More precisely, following Markl and Shnider in [MS], we construct an explicit combinatorial diagonal on the pairahedra, which are contractible polytopes controlling the combinatorial structure of an A∞-algebra with homotopy inner products, and use it to define a categorically closed tensor product. A cyclic A∞-algebra can be thought of as an A∞-algebra with homotopy inner products whose higher inner products are trivial. However, the higher inner products on the tensor product of cyclic A∞-algebras are not necessarily trivial.

An A_∞-bialgebra is a DGM H equipped with structurally compatible operations ω^j,i : H^⊗ i --> H^⊗ j such that (H,ω^1,i) is an A_∞-algebra and (H,ω^j,1) is an A_∞-coalgebra. Structural compatibility is controlled by the biderivative operator Bd, defined in terms of two kinds of cup products on certain cochain algebras of pemutahedra over the universal PROP U = End(TH).

An A∞-bialgebra of type (m, n) is a Hopf algebra H equipped with a "compatible" operation ω n m : H ⊗m → H ⊗n of positive degree. We determine the structure relations for A∞-bialgebras of type (m, n) and construct a purely algebraic example for each m ≥ 2 and m + n ≥ 4.

An A∞-bialgebra is a DGM H equipped with structurally compatible operations { ω : H → H } such that ( H, ω ) is an A∞-algebra and ( H, ω ) is an A∞-coalgebra. Structural compatibility is controlled by the biderivative operator Bd, defined in terms of two kinds of cup products on certain cochain algebras of pemutahedra over the universal PROP U = End (TH). To Jim Stasheff on the occasion of his 68th birthday.

Two new quasi-linear elliptic systems of partial differential equations to automatically generate two-dimensional boundary conforming structured grids are formulated. One of the new systems generates grids with near-uniform cell areas. The other produces meshes with near-uniform coordinate line spacings. In both cases, the resulting grids conform to complex boundaries with severe singularities without self-overlapping. In contrast with other elliptic generators, the control functions are held as dependent variables. They obey Poisson-type equations with appropriate forcing. Grid quality analysis reveals the advantage in terms of smoothness and cell area uniformity of the new grids compared with other structured grids. An efficient procedure to combine the novel elliptic grids with algebraic grids for large domains is devised.

In this work, we study the Codeword Stabilized Quantum Codes (CWS codes) a generalization of the stabilizers quantum codes using a new approach, the algebraic structure of modules, a generalization of linear spaces. We show then a new result that relates CWS codes with stabilizer codes generalizing results in the literature.

We introduce and study a notion of analytic loop group with a Riemann-Hilbert factorization relevant for the representation theory of quantum affine algebras at roots of unity Uǫ(ĝ) with non trivial central charge. We introduce a Poisson structure and study properties of its Poisson dual group. We prove that the Hopf-Poisson structure is isomorphic to the semi-classical limit of the center of Uǫ(ĝ) (it is a geometric realization of the center). Then the symplectic leaves, and corresponding equivalence classes of central characters of Uǫ(ĝ), are parameterized by certain G-bundles on an elliptic curve.

Based on the monoid classifier ∆, we give an alternative axiomatization of Freyd's paracategories, which can be interpreted in any bicategory of partial maps. Assuming furthermore a free-monoid monad T in our ambient category, and coequalisers satisfying some exactness conditions, we give an abstract envelope construction, putting paramonoids (and paracategories) in the more general context of partial algebras. We introduce for the latter the crucial notion of saturation, which characterises those partial algebras which are isomorphic to the ones obtained from their enveloping algebras. We also set up a factorisation system for partial algebras, via epimorphisms and (monic) Kleene morphisms and relate the latter to saturation.

We propose a uniform, category-theoretic account of structural induction for inductively defined data types. The account is based on the understanding of inductively defined data types as initial• algebras for certain kind of endofunctors T: llll-.JB on a bicartesian/distributive category B. Regarding a predicate logic as a fibration p : !Pl-+llll over llll, we consider a logical predicate lifting of T to the total category lF. Then, a predicate is inductive precisely when it carries an algebra structure for such lifted endofunctor. The validity of the induction principle is formulated by requiring that the 'truth' predicate functor T: JBl-.JP' preserve initial algebras. We then show that when the fibration admits a comprehension principle, analogous to the one in set theory, it satisfies the induction principle. We also consider the appropriate extensions of the above formulation to deal with initiality (and induction) in arbitrary contexts, i.e. the 'stability' property of the induction principle.

We consider pseudo-descent in the context of 2-fibrations. A 2-category of descent data is associated to a 3-truncated simplicial object in the base 2-category. A morphism q in the base induces (via comma-objects and pullbacks) an internal category whose truncated nerve allows the definition of the 2-category of descent data for q. When the 2-fibration admits direct images, we provide the analogous of the Beck-Bénabou-Roubaud theorem, identifying the 2-category of descent data with that of pseudo-algebras for the pseudo-monad q * Σq. We introduce a notion of 2-regularity for a 2-category R, so that its basic 2-fibration of internal fibrations cod : Fib(R) → R admits direct images. In this context, we show that essentiallysurjective-on-objects morphisms, defined by a certain lax colimit, are of effective descent by means of a Beck-style pseudo-monadicity theorem.

We propose a new representation of quantum circuits that eliminates the projection steps traditionally associated with measurement, resulting in a fundamentally static depiction of the circuit without intrinsic time ordering. In this framework, the evolution of the quantum system is fully unitary and deterministic, with no irreversible processes. By decoupling physical evolution from subjective measurement outcomes, this approach offers a novel, time-symmetric view of quantum mechanics, free from traditional notions of causality and the directional flow of time.