Product AB Research Articles

RATIONAL TIME-FREQUENCY GABOR FRAMES ASSOCIATED WITH PERIODIC SUBSETS OF THE REAL LINE

For a, b > 0 and g ∈ L2(ℝ), write 𝒢(g, a, b) for the Gabor system: [Formula: see text] Let S be an aℤ-periodic measurable subset of ℝ with positive measure. It is well-known that the projection 𝒢(gχS, a, b) of a frame 𝒢(g, a, b) in L2(ℝ) onto L2(S) is a frame for L2(S). However, when ab > 1 and S ≠ ℝ, 𝒢(g, a, b) cannot be a frame in L2(ℝ) for any g ∈ L2(ℝ), while it is possible that there exists some g such that 𝒢(g, a, b) is a frame for L2(S). So the projections of Gabor frames in L2(ℝ) onto L2(S) cannot cover all Gabor frames in L2(S). This paper considers Gabor systems in L2(S). In order to use the Zak transform, we only consider the case where the product ab is a rational number. With the help of a suitable Zak transform matrix, we characterize Gabor frames for L2(S) of the form 𝒢(g, a, b), and obtain an expression for the canonical dual of a Gabor frame. We also characterize the uniqueness of Gabor duals of type I (respectively, type II).

Plünnecke and Kneser type theorems for dimension estimates

Given a division ring K containing the field k in its center and two finite subsets A and B of K*, we give some analogues of Plünnecke and Kneser Theorems for the dimension of the k-linear span of the Minkowski product AB in terms of the dimensions of the k-linear spans of A and B. We also explain how they imply the corresponding more classical theorems for abelian groups. These Plünnecke type estimates are then generalized to the case of associative algebras. We also obtain an analogue in the context of division rings of a theorem by Tao describing the sets of small doubling in a group.

Non-intersecting squared Bessel paths with one positive starting and ending point

We consider a model of n non-intersecting squared Bessel processes with one starting point a > 0 at time t = 0 and one ending point b > 0 at time t = T. After proper scaling, the paths fill out a region in the tx-plane. The region may come to the hard edge at 0 or may not, depending on the value of the product ab. We formulate a vector equilibrium problem for this model, which is defined for three measures, with upper constraints on the first and third measures and an external field on the second measure. It is shown that the limiting mean distribution of the paths at time t is given by the second component of the vector that minimizes this vector equilibrium problem. The proof is based on a steepest descent analysis for a 4 × 4 matrix-valued Riemann-Hilbert problem which characterizes the correlation kernel of the paths at time t. We also discuss the precise locations of the phase transitions.

Some results on the reverse order law in rings with involution

We investigate some necessary and sufficient conditions for the hybrid reverse order law \({(ab)^\# = b^{\dag} a^{\dag}}\) in rings with involution. Assuming that a and b are Moore–Penrose invertible, we present an equivalent condition for the product ab to be an EP element.

On the closedness, the self-adjointness and the normality of the product of two unbounded operators

Abstract Let A and B be two unbounded densely defined operators on a Hilbert space ℋ. The purpose of this work is to give simple conditions that make the product AB closed, self-adjoint and normal provided the two operators are so.

Around Operator Monotone Functions

We show that the symmetrized product AB + BA of two positive operators A and B is positive if and only if \({f(A+B)\leq f(A)+f(B)}\) for all non-negative operator monotone functions f on [0,∞) and deduce an operator inequality. We also give a necessary and sufficient condition for that the composition \({f \circ g}\) of an operator convex function f on [0,∞) and a non-negative operator monotone function g on an interval (a, b) is operator monotone and present some applications.

Special Identities for Quasi-Jordan Algebras

Semispecial quasi-Jordan algebras (also called Jordan dialgebras) are defined by the polynomial identities These identities are satisfied by the product ab = a ⊣ b + b ⊢ a in an associative dialgebra. We use computer algebra to show that every identity for this product in degree ≤7 is a consequence of the three identities in degree ≤4, but that six new identities exist in degree 8. Some but not all of these new identities are noncommutative preimages of the Glennie identity.

Mixed-Type Reverse-Order Laws for the Generalized Inverses of an Operator Product

In this paper, we study mixed-type reverse-order laws for the Moore–Penrose inverse of an operator product AB, and obtain necessary and sufficient conditions for these mixed-type reverse-order laws. Results related to other generalized inverses are also proved.

Fitting cores and supersolvable groups

Let A be a group. What can be said about the group B to ensure that A and the normal product AB belong to the same prescribed class of groups? Results in this direction are given for the classes of supersolvable groups, absolutely solvable groups and Lagrange groups.

A characterization of *-automorphism on ℬ(H)

Let H be a Hilbert space and \( A \) be a standard *-subalgebra of ℬ(H). We show that a bijective map Φ: \( A \uparrow A \) preserves the Lie-skew product AB − BA* if and only if there is a unitary or conjugate unitary operator U ∈ ℬ(H) such that Φ(A) = UAU* for all A ∈ \( A \), that is, Φ is a linear *-isomorphism or a conjugate linear *-isomorphism.

A Few Remarks on Milburn Equation

After presenting the infinite operator-sum form solution to the Milburn equation dρ/dt = γ(UρU† – ρ) = γU[ρ,U†], where U = e-iH/ℏγ, and verifying that this equation preserves the three necessary conditions of density operators during time evolution, we prove that the von Neumann entropy increases with time. We also point out that if A and B both obey the Milburn equation, then the product AB obeys (d/dt) (AB) = γU[AB, U†] – (1/γ) (dA/dt) (dB/dt), which violates the Milburn equation, this reflects that a pure state will evolve to a mixture in general.

Common Lyapunov solutions for two matrices whose difference has rank one

Real stable matrices A and B with rank of A - B equal to one have a common Lyapunov solution if and only if their product AB has no real negative eigenvalue. This was proved by Shorten and Narendra [R.N. Shorten, K.S. Narendra, On common quadratic Lyapunov functions for pairs of stable LTI systems whose system matrices are in companion form, IEEE Trans. Automat. Control 48 (4) (2003) 618–621], whose proof is based on the fundamental results of Kalman on Luré’s problem. In this paper we give an alternative proof of this result and its generalization to the general regular inertia case, and to the case when the matrices A and B are complex.

Continuity of the generalized spectral radius in max algebra

Let ‖ · ‖ be an induced matrix norm associated with a monotone norm on R n and β be the collection of all nonempty closed and bounded subsets of n × n nonnegative matrices under this matrix norm. For Ψ , Φ ∈ β , the Hausdorff metric H between Ψ and Φ is given by H ( Ψ , Φ ) = max { sup A ∈ Ψ inf B ∈ Φ ‖ A - B ‖ , sup B ∈ Φ inf A ∈ Ψ ‖ A - B ‖ } . The max algebra system consists of the set of nonnegative numbers with sum a ⊗ b = max { a , b } and the standard product ab for a , b ⩾ 0 . For n × n nonnegative matrices A , B their product is denoted by A ⊗ B , where [ A ⊗ B ] ij = max 1 ⩽ k ⩽ n a ik b kj . For each Ψ ∈ β , the max algebra version of the generalized spectral radius of Ψ is μ ( Ψ ) = limsup m → ∞ [ sup A ∈ Ψ ⊗ m μ ( A ) ] 1 m , where Ψ ⊗ m = { A 1 ⊗ A 2 ⊗ ⋯ ⊗ A m : A i ∈ Ψ } . Here μ ( A ) is the maximum circuit geometric mean. In this paper, we prove that the max algebra version of the generalized spectral radius is continuous on the Hausdorff metric space ( β , H ) . The notion of the max algebra version of simultaneous nilpotence of matrices is also proposed. Necessary and sufficient conditions for the max algebra version of simultaneous nilpotence of matrices are presented as well.

Multiplicative decomposability of shifted sets

The following two problems are open: 1) Do two sets of positive integers A and B exist, with at least two elements each, such that A+ B coincides with the set of primes P, for sufficiently large elements? 2) Let A = {6, 12, 18}. Is there an infinite set B of positive integers such that AB + 1 ⊂ P ? A positive answer would imply that there are infinitely many Carmichael numbers with 3 prime factors. In this paper we prove the multiplicative analogue of the first problem, namely that there are no two sets A and B, with at least two elements each, such that the product AB coincides with any additively shifted copy P + c of the set of primes for sufficiently large elements. We also prove that shifted copies of sets of integers which are generated by certain subsets of the primes cannot be multiplicatively decomposed.

Hamiltonicity of cubic Cayley graphs

Following a problem posed by Lovász in 1969, it is believed that every finite connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arising from finite groups having a (2,s,3) -presentation, that is, for groups G=\langle a,b\mid a^2=1, b^s=1, (ab)^3=1, \dots \rangle generated by an involution a and an element b of order s\geq3 such that their product ab has order 3 . More precisely, it is shown that the Cayley graph X=\operatorname{Cay}(G,\{a,b,b^{-1}\}) has a Hamilton cycle when |G| (and thus s ) is congruent to 2 modulo 4 , and has a long cycle missing only two adjacent vertices (and thus necessarily a Hamilton path) when |G| is congruent to 0 modulo 4 .

Basic Results of Gabor Frame on Local Fields*

Basic facts for Gabor frame {E u(m)b T u(n)a g} m, n ∈ρ on local field are investigated. Accurately, that the canonical dual of frame {E u(m)b T u(n)a g} m, n ∈ρ also has the Gabor structure is showed; that the product ab decides whether it is possible for {E u(m)b T u(n)a g} m, n ∈ρ to be a frame for L 2(K) is discussed; some necessary conditions and two sufficient conditions of Gabor frame for L 2(K) are established. An example is finally given.

Solvent and Temperature Effects on the Reduction and Amination Reactions of Electrophiles by Lithium Dialkylaminoborohydrides

The influence of temperature and solvent effects on the reduction and amination mechanisms of iodomethane by lithium N,N-diisopropylaminoborohydride (iPr-LAB) was examined in varying concentrations of THF and dioxane. The reactions of benzyl chloride and trimethylsilyl chloride with iPr-LAB in THF were also studied. The amination of iodomethane is favored over reduction at low and room temperatures in pure THF and with increasing the amount of dioxane in THF. At higher temperatures, the reduction reaction appears to compete with the amination. In dioxane solvent, however, iodomethane yields exclusively the amination product regardless of temperature. On the other hand, reduction by iPr-LAB to the aminoborane is the only product observed in THF when benzyl chloride and trimethylsilyl chloride are used. To understand the solvent effects on the product distribution, ab initio and density functional theory (DFT) calculations were used to examine the mechanisms of reduction and amination of chloromethane and bromomethane by lithium dimethylaminoborohydride (LAB) in THF and dioxane. The results of these calculations show that the relative reaction barrier heights are significantly affected by the nature of the coordinated solvent molecule and thus lend support to the experimental observations.

On the existence of a common quadratic Lyapunov function for a rank one difference

Suppose that A and B are real Hurwitz matrices, and that their difference A − B is rank one. Then A and B have a common quadratic Lyapunov function if and only if the product AB has no real negative eigenvalue. This result is due to Shorten and Narendra, who showed that it follows as a consequence of the Kalman–Yacubovich–Popov lemma and the solution of the Lur’e problem. Here we present a new and independent proof based on results from convex analysis and the theory of moments.

A novel approach for bit-serial AB2 multiplication in finite fields GF(2 m)

This paper presents a new inner product AB 2 multiplication algorithm and effective hardware architecture for exponentiation in finite fields GF(2 m ). Exponentiation is more efficiently implemented by applying AB 2 multiplication repeatedly rather than AB multiplication. Thus, efficient AB 2 multiplication algorithms and simple architectures are the key to implementing exponentiation. Accordingly, this paper proposes an efficient inner product multiplication algorithm based on an irreducible all one polynomial (AOP) and simple architecture, which has the same hardware equipment as Fenn's AB multiplier. The proposed bit-serial multiplication algorithm and architecture are highly regular and simpler than those of previous works.

Modeling of color values for nondestructive evaluation of maturity of mango

Mango, an important tropical fruit, is marketed throughout the world. Quality of ripened mango depends on its maturity at harvest, which is hitherto determined subjectively by experience. In the present investigation, the potential of a nondestructive method for predicting maturity using color values taken by a handheld colorimeter was explored. Maturity of mango was defined in term of maturity index ( I m), a ratio of total soluble solids (TSS) of mango under experiments and minimum level of TSS (8 °Brix) of matured mango, expressed in percentage. Hunter L, a, and b values of 315 freshly harvested mangoes ranging from immature to over matured were measured using HunterLab colorimeter. TSS of whole mango juice was measured by a handheld refractometer and maturity index was computed. The maturity index and L, a, and b values of 160 samples were fitted in different forms of models using multiple linear regression (MLR), partial least square and principal component regression. The model using MLR on variables a, b and the product ab was selected. The prediction performance of the model was tested with another set of 100 unripe samples. Precision of prediction was also verified by sensory evaluation of 55 ripe mangoes and was found that the fruits predicted to be mature could ripe with high-satisfied taste while the ones predicted to be immature or over mature were mostly rejected by the panels.