Betti numbers of mixed product ideals

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Better bound on the exponent of the radius of the multipartite separable ball

m/2−1
,
(1)
in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semideﬁnite matrices (actually we do slightly better, but with the same asymptotic exponent). For qubits (d0 = 2) this radius is is (2/3)m/2−1 , to be compared to (1/2)m/2−1 from
ln 3 − 1) ≈ .29248125, compared [6]. If we express it as as κ2−γm , the exponent is γ = 0.5( ln 2
2
to [6]’s exponent of γ = 1/2. The non-qubit exponent is better, too, but approaches our earlier one as d0 → ∞. From this, we easily obtain a lower bound on the radius of the largest Frobenius-norm ball of separable normalized density matrices: for example, for m qubits it is (3/2) × 2−(1+γ )m ≡ (3/2) × 6−m/2 (versus our earlier 2 × 2−3m/2 ). A slightly better, but more complicated, version of our new bound lets us improve the factor 3/2 to √ 3m+1 /(3m + 3), which rapidly approaches 3. This gives a number of qubits below which NMR with standard pseudopure-state preparation techniques can access only unentangled states; with parameters realistic for current experiments, this is 36 qubits (compared to 23 qubits via our earlier results). We also address several points not strictly necessary for obtaining these results, but which relate to the power and nature of our methods, and the possibilities for strengthening the results. Szarek [7] found the ﬁrst upper bound below unity on ball size, and recently Aubrun and Szarek [8] found an upper bound on ball size which matches (up to a logarithmic factor) the lower bound we obtain here for qubits, though for qudits with d > 2 there is still an exponential gap. One of the most natural mathematical methods for tackling this problem is to use a general result of F. John [9] relating the inner and outer ellipsoids of a convex set. We show that straightforward application of this natural method gives results weaker than we obtain here; weaker, in fact, than our earlier ones [6]. Our methods may appear technical; nevertheless, many of the intermediate results are mathematically interesting in their own right and have applications to quantum information problems other than the one at hand. Along the way we explain some of these, notably a variant proof of the result that the eigenvalues of a separable bipartite quantum state are majorized by those of its marginal density operators [10], and an example of the use of John’s theorem to bound the radii of other inner balls of quantum information-theoretic interest, in this case the inner ball of the convex hull of all maximally entangled states (related to an application-oriented entanglement measure, the fully entangled fraction of [11]). Many of our results use bounds on induced norms of various classes of maps on matrices, which we expect to be useful in other contexts. An appendix includes an additional bound, closely related to one used in the main argument, on the 2-to-∞ induced norm of stochastic linear maps that are positive on a radius-a ball of matrices around the identity.

The International Takeoff of New Products The Role of Economics, Culture, and Country Innov

The International Takeoff of New Products: The Role of Economics,Culture,andCountry InnovativenessGerard J.Tellis•Stefan Stremersch•Eden YinMarshall School of Business,The University of Southern California,P.O.Box90089-1421,Los Angeles,California90089-1421Erasmus University Rotterdam,P.O.Box1738,Burg.Oudlaan50,3000DR Rotterdam,The NetherlandsJudge Institute,Cambridge University,Trumpington Street,Cambridge CB21AG,UKtellis@•stremersch@few.eur.nl•e.yin@S ales takeoff is vitally important for the management of new products.Limited prior research on this phenomenon covers only the United States.This study addresses the following questions about takeoff in Europe:1)Does takeoff occur as distinctly in other countries,as it does in the United States?2)Do different categories and countries have consistently different times-to-takeoff?3)What economic and cultural factors explain the intercountry differences?4)Should managers use a sprinkler or waterfall strategy for the introduction of new prod-ucts across countries?We gathered data on137new products across10categories and16European countries.We adapted the threshold rule for identifying takeoff(Golder and Tellis1997)to this multinational context.We specify a parametric hazard model to answer the questions above.The major results are as follows:1)Sales of most new products display a distinct takeoff in various European countries,atan average of six years after introduction.2)The time-to-takeoff varies substantially across countries and categories.It is four timesshorter for entertainment products than for kitchen and laundry appliances.It is almost halfas long in Scandinavian countries as in Mediterranean countries.3)While culture partially explains intercountry differences in time-to-takeoff,economicfactors are neither strong nor robust explanatory factors.4)These results suggest distinct advantages to a waterfall strategy for introducing prod-ucts in international markets.(International New Product Growth;New Product Takeoff;New Product Growth;InternationalDiffusion;Diffusion of Innovations)IntroductionNew products do not grow into maturity at a steady rate.Rather,their sales pattern is marked by a long introduction period when sales linger at low levels. At a certain point in time,the new product breaks into rapid growth,associated with a huge jump in sales. Academic literature and the business press refer to this point as the takeoff in sales.It is the point of tran-sition between the introduction and growth stage of a new product.The time-to-takeoff is the duration of theTHE INTERNATIONAL TAKEOFF OF NEW PRODUCTSintroductory stage or the period from the introduction to the takeoff.The takeoff of new products is a vitally important phenomenon in the management of new products for several reasons.First,growth rate in sales at takeoff may exceed400%(Golder and Tellis1997).Such rapid growth requires extensive resources in terms of adver-tising,sales staff,manufacturing,distribution,and inventory support.Second,many new product man-agers are under extensive pressure to pull the plug on new products that show sluggish sales.However, research shows that the introduction period may be quite long and variable.Knowing the likelihood and timing of the takeoff of a new product helps to with-stand the pressure to pull the plug and preempt pre-mature cancellation of promising new products.Third, takeoff is often a signal of the mass adoption of a prod-uct and its ultimate commercial success.Knowing the impact of company decisions on the likelihood and timing of takeoff is important for effectively manag-ing such success.Fourth,the introduction and takeoff of new products across various countries are criti-cal events in international marketing strategies.This issue has gained importance with increasing global-ization and uniﬁcation among countries within trade zones(EU,NAFTA,GATT,etc.).Knowing how take-off varies across countries is important for designing effective international strategies.Recently,Agarwal and Bayus(2002)and Golder and Tellis(1997)modeled the takeoff of consumer durables in the United States.These studies raise a number of questions about takeoff in the international realm,which we attempt to address in the context of Europe:•Does the phenomenon of takeoff occur as dis-tinctly in other countries,especially in European countries,as it does in the United States?•Do different categories and countries have con-sistently different times-of-takeoff?•What economic and cultural factors explain the intercountry differences?•Should managers use a sprinkler or waterfall strategy for the introduction of new products across countries?The only relevant literature on this topic occurs in the relatedﬁeld of diffusion modeling.Although this literature is extensive,with more than a hundred studies(Mahajan et al.1990),only a few of these stud-ies address the international diffusion of new prod-ucts.Most of them examine how the parameters of the Bass diffusion model vary by country(Ganesh et al.1997,Gatignon et al.1989,Heeler and Hustad 1980,Helsen et al.1993,Putsis and Sen2001,Takada and Jain1991,and Talukadar et al.2001).Putsis et al. (1997)model the inﬂuence of“mixing”between coun-tries on cross-country diffusion patterns.Dekimpe et al.(2000)examine the sequential adoption of a tech-nological innovation by various countries.None of these studies addresses the takeoff of new products across multiple countries.The present study focuses on the variation in time-to-takeoff of10new consumer durables across16 European countries(see Figure1for examples).As such,this is the most extensive study ever conducted on the growth of new products in Europe.Besides country-speciﬁc differences,the study also examines how underlying economic and cultural characteris-tics explain the variation in time-to-takeoff across countries.We chose Europe as a domain of analysis for two reasons.First,European countries have a relatively long history of developed capitalist markets.As such, data on new product introductions are available in these countries,though neither complete nor easy to obtain.Second,Europe has a long post-WW II his-tory of uniﬁcation toward a single market,through institutes such as the European Council,the Benelux, the West-European Union,the North American Treaty Organization,the Organization for Economic Coop-eration and Development,the European Free Trade Association,The European Union,and the European Monetary Union.1With the introduction of a single European currency(the EURO),the culmination of the uniﬁcation movement,there is a growing percep-tion among some economists and non-Europeans that the western half of the continent constitutes a single European market.Yet at the same time,these coun-tries differ substantially in economic strength and 1These institutions have changed names over the course of their existence;we use the most recent ones.Figure1Illustration of Takeoff in Selected Countriescultural identity.Thus there is an equally strong per-ception among nationalists and Europeans that thesecountries have distinct market identities.The takeoffof new products is an important context in which totest some of these rival perceptions.This paper is organized as follows.The second sec-tion presents the model and research hypotheses.Thethird section explains the data.The fourth sectionpresents the results.Theﬁnal section discusses theﬁndings,implications,and limitations.Modeling International TakeoffAs Golder and Tellis(1997)explain,takeoff is a time-dependent binary event that is best modeled by thehazard model.However,unlike Golder and Tellis(1997),we use a parametric hazard model ratherthan the proportional hazard model that they use.We modify the threshold rule they use and alsouse raters to identify the takeoff.We explain ourapproach below in two parts:model development,and conceptual framework and hypotheses.Model DevelopmentThe hazard model is often used in medicine and biol-ogy,where the focal event,failure,typically corre-sponds to accident,death,or recurrence of disease.For us the focal event,takeoff,although a success,corresponds to the“failure”in the hazard literature.Similarly,being in the introductory stage corresponds to “survival”or “success”in the hazard literature.Because the model for international takeoff we pro-pose includes both time-varying (such as market pen-etration and GDP)and time-invariant covariates (such as cultural characteristics),a fully parametric hazard model is most appropriate (Helsen and Schmittlein 1993,Kalbﬂeisch and Prentice 1980).In our treatment of the parametric hazard model and the derivation of the likelihood,we draw heavily upon Petersen (1986a,1986b).Let T be a nonnegative continuous random variable that denotes the duration in a state.In the presence of time-varying covariates—which we denote as X t —the hazard function or the rate of transition from one state to another (e.g.,introduction to growth)ish t X t =lim t →0P t ≤T <t + t T ≥t X t / t(1)Let t k be the duration in a state at the time when either the state is left or censoring occurs.We divide t k into k nonoverlapping but adjacent segments of time,which need not be of the same length.Let t 0=0and t 0<t 1<···<t j −1<t j <···<t k .We obtain the follow-ing probability of surviving beyond duration t j given survival at duration t j −1,conditional upon the path taken by X t up to duration t j :P T ≥t j T ≥t j −1 X t j=exp −tjt j −1h s X t j ds (2)The survival function for duration of t k can then bewritten asS t k X t k =k j =1P T ≥t j T ≥t j −1 X t j=exp −t 1h s X 0 ds∗···∗exp −t k t k −1h s X t k −1 ds=exp −kj =1 t jt j −1h s X t j −1 ds(3)The probability density function becomesf t k X t k =h t k X t k S t k X t k(4)We propose to test a log-logistic function because it allows for nonmonotonic baseline hazards.The hazard function (h t )and survival function (S t )for a logistic distribution can,in a general form,be written ash t =p t p −11+ t p (5)S t =11+ t p(6)where:=e −x i p =1(7)The log-likelihood isln L i = i ln h t k x k −kj =1 t jt j −1h s x j ds(8)An interpretation of this model is that the haz-ard function is adjusted by the independent vari-ables of each individual category in each country ateach time period.This adjustment occurs by the haz-ard ratio,which is deﬁned as e − .Positive coefﬁ-cients decrease the probability of takeoff and negative coefﬁcients increase the hazard of takeoff.The parameter characterizes the distribution of the hazard rate. i is a right-censoring indicator.Similar to Helsen and Schmittlein (1993),but unlike Jain and Vilcassim (1991),we do not include a term for unobserved heterogeneity.We capture hetero-geneity across categories and countries by including theory-based category and country variables in our model.Conceptual Framework and HypothesesWe develop a theory to explain takeoff around two broad groups of factors,country characteristics and category characteristics.We can further classify country characteristics into economic,cultural,and information access variables.The subsequent discus-sion develops speciﬁc hypotheses for each of these variables.Economic Variables.Four economic constructs are likely to play an important role in the takeoff of newproducts:a country’s wealth,economic progressive-ness,economic roles in the household,and openness of the economic system.Wealth:Prior research posits that wealth strongly inﬂuences the speed with which inhabitants of a spe-ciﬁc country adopt a new product(Helsen et al.1993). So we expect that wealth will have a strong effect on time-to-takeoff.A well-known conclusion of the diffusion literature is that innovators generally are wealthier than later adopters(Rogers1995).Wealthier people attach a lower utility to money,which is what economists call the“wealth”effect.The lower util-ity of money has two consequences.First,wealthier people can better afford the risks of adopting a new product early(Dickerson and Gentry1983).Risk is an important determinant of timing of adoption(Sheth 1968).Second,because prices of new products tend to start high and drop steadily(Golder and Tellis1998), wealthier people will be able to better afford new products early when prices are high.In sum,wealth-ier people are expected to adopt a new product earlier than less-wealthy people.So,we hypothesize: Hypothesis1(H1).New products take off faster in countries with higher average wealth than in those with less average wealth.Economic Progressiveness:One meaning of economic progressiveness is the extent to which wealth is dis-tributed within countries.Even when a population at large in a country has high average wealth,it may be concentrated in a few homes.In this case,the vast majority of people may still be poor and may be unable to afford the new product.Thus,after control-ling for average wealth per capita,high disparity in wealth may mean that many people cannot afford a new product,causing it to take off later than when income disparity is low.Economic progressiveness also refers to the extent to which countries participate in economic unions. Country unions facilitate the movement of capi-tal,labor,suppliers,and goods between countries. Unions reduce economic disparity among countries and encourage the formation of a common market. This economic atmosphere is more likely to promote the dispersion and growth of new products in coun-tries that belong to the union.Thus,sales of new products may grow more rapidly within countries that take part in such unions.This argument is also in line with theﬁnding of Mahajan and Muller(1994), that a borderless Europe leads to faster diffusion.We hypothesize:Hypothesis2(H2).New products take off faster in countries with greater economic progressiveness than in countries with lower economic progressiveness. Economic Roles in the Household:Over the second half of the century there has been a steady change in work roles in the family.Families have moved from having predominantly one income earner to two income earners(with the woman,in addition to the man,working outside the home).Such households experience intense pressure on time,and the family has less time for housework,relaxation,and entertain-ment.Such families put a high value on any appli-ances that help them save time.All new products have at least some time-saving features.For example, dryers and washers free a great deal of time com-pared to manual washing and hanging clothes to dry. VCRs enable convenient watching of movies at home instead of time-consuming trips to movie theaters. Home computers save time in word processing and home accounts.Thus,we expect that countries in which more women work outside the home are likely to adopt durables faster than those in which fewer women do so.This position is consistent with that of Gatignon et al.(1989).We hypothesize:Hypothesis3(H3).New products take off faster in countries with a high activity rate of women than in coun-tries with a low activity rate of women.Openness of the Economic System:The openness of the economic system of a country refers to the extent to which the country is involved in international trade.Because of increasing international free trade between countries,this may be an important and increasingly relevant factor in the international take-off of new products.Open economic systems may speed the takeoff of new products for two reasons. First,economic openness encourages the develop-ment or opening of uniﬁed infrastructure between countries,such as freeways,phone lines,railways,TV broadcasting,etc.Such infrastructure may facili-tate the faster spread of new products through obser-vation or word of mouth.Economic openness also fosters greater competition,which increases produc-tion and distribution efﬁciency(Talukdar et al.2001). Savings from these efﬁciencies should make new durables more affordable to consumers,with a faster takeoff as a consequence.So,we hypothesize: Hypothesis4(H4).New products take off faster in countries that have a more open economic system than in countries that have a less open economic system. Cultural Variables.Prior research suggests that a country’s culture strongly affects the speed at which its citizens adopt a new product(Dekimpe et al.2000,Gatignon et al.1989,Takada and Jain 1991).We identify four cultural variables that can affect time-to-takeoff across countries:uncertainty avoidance,masculinity,2need for achievement,and industriousness.Uncertainty Avoidance:Uncertainty avoidance refers to the level of anxiety about the future(Hofstede1980, 2001).Societies that are high in uncertainty avoidance continuously feel the inherent uncertainty in life as a threat that must be fought,while societies low in uncertainty avoidance more easily accept uncertainty and take“each day as it comes”(Hofstede1980). We expect countries that are high in uncertainty avoidance to show later takeoffs than those that are low in uncertainty avoidance,for two reasons.First, societies that are low in uncertainty avoidance are more willing to take risks.Therefore,they will more readily accept new products(Rogers1995).Second, societies high in uncertainty avoidance consider novel ideas as dangerous and are more intolerant toward change than societies low in uncertainty avoidance (Hofstede1980).Thus,countries that are low in uncer-tainty avoidance will embrace a new product more easily than countries that are higher in uncertainty avoidance.Our expectation is also consistent with prior research.For example,Lynn and Gelb(1996)ﬁnd2We found neither theoretical support nor empirical evidence in our data to include the two other Hofstede dimensions,power dis-parity and individualism,as explanatory variables.a negative correlation between a country’s uncer-tainty avoidance and the penetration of six con-sumer durables.Steenkamp et al.(1999)ﬁnd that consumers in countries high in uncertainty avoidance are less innovative than consumers in countries low in uncertainty avoidance.Hypothesis5(H5).New products take off faster in countries low in uncertainty avoidance than in countries high in uncertainty avoidance.Masculinity:In most cultures men tend to be more assertive,while women tend to be more nurturing. Male behavior is associated with autonomy,aggres-sion,exhibition,and dominance,while female behav-ior is associated with nurturance,afﬁliation,and humility(Hofstede1980,2001).Masculinity and fem-ininity refer to the sex role pattern in society at large, to the extent it is characterized by male or female characteristics.We expect masculinity to affect the speed of takeoff for two reasons.First,masculine societies attach more value to recognition and wealth,while feminine societies attach more value to human contacts and living envi-ronment(Hofstede1980).The adoption of new prod-ucts allows consumers to exhibit their wealth and success,which may be more compatible with mascu-line societies.Consumers in masculine societies may thus show higher innovativeness,as compared to con-sumers in more feminine societies(Steenkamp et al. 1999).Second,in masculine societies people tend to make decisions independently and admire the strong and the independent(Hofstede1980).When a new prod-uctﬁrst emerges,adoptions are few and require inde-pendent decisions by innovators.This trait of mas-culine societies may also lead to better acceptance of new products.In contrast,in feminine societies the tendency to make group decisions may lead to less acceptance of a product.For all these reasons,we expect masculine countries to show faster takeoff than feminine countries.So,we hypothesize the following.Hypothesis6(H6).New products take off faster in countries high in masculinity than in countries low in masculinity.Need for Achievement:In one of the earliest discus-sions of need for achievement,Murray(1938)describes it as the tendency or desire to do things as rapidly and/or as well as possible.Need for achievement includes successfully and independently overcom-ing obstacles,competing with and surpassing others, and high self-regard.Veroff et al.(1962)associate a high need for achievement particularly with working harder,being less satisﬁed with current success,and being more oriented to the future fruits of work.All these traits may lend themselves to greater eagerness for adopting new products and greater willingness to experiment with new products as soon as they are available.Thus,we expect:Hypothesis7(H7).New products take off faster in countries in which the inhabitants have a high need for achievement than in countries in which the inhabitants have a low need for achievement. Industriousness:Industrious people are inclined to work and tend to value the fruits of work more than less industrious people.The industriousness of a pop-ulation can affect the speed of takeoff for supply and demand reasons.Such people realize that innovations can make work more productive as well as make their rest from work more productive,so industrious peo-ple tend to be more receptive to innovations as well as to work harder to develop innovations.Thus,when a new product is available,industrious entrepreneurs, retailers,and distributors are likely to work harder to make this product available to the general popu-lation.At the same time,the people themselves are more likely to search for,try out,and adopt the new product.Thus,the new product is likely to take off faster in an industrious culture than in one that is not industrious.So we expect the following. Hypothesis8(H8).New products take off faster in countries with a more industrious culture than in one that is less industrious.Information Access.Prior research suggests that people’s access to information strongly affects the speed at which they adopt a new product(Rogers 1995).Therefore,we may expect that a new product takes off faster in countries in which inhabitants have easy access to information than in other countries.We identify three factors that capture different dimen-sions of information access:media intensity,mobility, and education.Media Intensity:Mass media such as newspapers, radio,and television play an important role in cre-ating awareness of a new product among poten-tial adopters(Beal and Rogers1960)and inﬂuencing acceptance of a new product(Katz and Lazarsfeld 1955).Mass media may also lead to greater ability of consumers to detect superior new products,and thus increase the rate at which,and the likelihood that,consumers will adopt them.Mass media also contribute to the cosmopolitanism of consumers of a country,especially if it concerns“cosmopolite chan-nels”(Rogers1995).Through cosmopolite channels, consumers in a target country can access information about innovations that have been introduced in other countries,even before the innovation is introduced in the target country(Gatignon et al.1989).Such infor-mation can hasten the takeoff of the innovation in the target country.For all these reasons,we hypothesize: Hypothesis9(H9).New products take off faster in countries high in media intensity than in countries low in media intensity.Mobility:Interpersonal communication affects the rate at which consumers learn about new products. An important facilitator of such communication is mobility.Gatignon et al.(1989)have shown that the higher the mobility of a country’s inhabitants, the more rapidly new products penetrate the social system.So,we hypothesize the following. Hypothesis10(H10).New products take off faster in countries in which the inhabitants have high mobility than in countries in which the inhabitants have low mobility. Education:Education involves the exposure of peo-ple to a constant stream of new ideas,which makes them more receptive to cation also sensitizes people to the importance of technology in human progress.That again makes them more receptive to innovations.Indeed,a generalﬁnding in diffusion research is that educated people tend to adopt new products earlier than noneducated people (Rogers1995).So,we hypothesize the following.Hypothesis11(H11).New products take off faster in countries in which the inhabitants have enjoyed higher edu-cation than in countries in which the inhabitants have not enjoyed higher education.Category CharacteristicsWe identify four category characteristics that may affect the probability of takeoff:product class,mar-ket penetration,number of prior takeoffs,and year of introduction.(Although these variables are intrinsi-cally category characteristics,some,like market pen-etration,number of prior takeoffs,and introduction year,could also vary by countries.)Product Class:Product class may affect the probabil-ity of a new product’s takeoff(Gatignon et al.1989, Golder and Tellis1997).In particular,we distinguish between white goods,such as kitchen and laundry appliances,and brown goods,such as entertainment and information products.We expect brown goods to take off earlier than white goods because they appeal to all members of a household,provide more instant gratiﬁcation,and are more visible to guests.Thus,we hypothesize:Hypothesis12(H12).Brown goods take off faster than white goods.Market Penetration:We deﬁne market penetration as the percentage of households that have purchased the new product.Prior research posits that product cate-gories reach takeoff at an average market penetration of2.5–3%(Golder and Tellis1997).The diffusion liter-ature also suggests that market penetration may be an important correlate of the event that we call takeoff (Sultan et al.1990).Thus,an increase in market pene-tration increases the likelihood of a takeoff.To avoid problems of simultaneity,we include the lagged value of market penetration as an independent variable in the model.Thus,we hypothesize:Hypothesis13(H13).The higher the market penetra-tion,the higher the probability of takeoff.Prior Takeoffs:A new product’s prior takeoff in other countries can stimulate takeoff in a target country for at least four reasons.First,as the product takes off in other countries,the media are more likely to report its use or popularity,increasing its attractiveness in the target country.Second,on seeing a takeoff of a new product in other countries,manufacturers or retailers are more likely to promote sales in the target coun-try,triggering a takeoff.Third,takeoff in other coun-tries implies more adopters,so that a consumer in a target country has a higher probability of contacting an adopter from the other countries.Such a contact can increase acceptance of the new product and thus takeoff in the target country.Fourth,when a prod-uct takes off in other countries,potential adopters in a target country are more likely to perceive the new product to be a success.This perception is likely to reduce the perceived risk associated with adopting the new product,increasing acceptance of the new product and takeoff in the target country.Thus,we hypothesize:Hypothesis14(H14).The higher the prior takeoffs in other countries,the higher the probability of takeoff in a target country.Year of Introduction:The literature is ambiguous about the effect of the year of introduction.Golder and Tellis(1997)argue that due to the faster speed of technological innovation in more recent years, new products improve faster.Thus they are likely to appeal to consumers and takeoff off sooner than prod-ucts introduced in prior decades.On the other hand, Bayus(1992,1994)argues that technological change is not occurring any faster in more recent decades than it did in earlier decades.By this logic,takeoff should not occur any faster in more recent decades than in earlier decades.Recently,Van den Bulte(2000)showed that although on average there has been an increase in dif-fusion speed,this effect disappears when one controls for economic and demographic evolutions as well as the nature of products studied.Therefore,we will not posit a hypothesis for this effect,but merely include it as a control variable.DataThis section describes our data collection and measures.Data CollectionThe data collection was difﬁcult and slow.It took the periodic efforts of several research assistants and。

英文版产业经济学教程第10章串谋

第二章垄断协议第十三条禁止具有竞争关系的经营者达成下列垄断协议：（一）固定或者变更商品价格；（二）限制商品的生产数量或者销售数量；（三）分割销售市场或者原材料采购市场；（四）限制购买新技术、新设备或者限制开发新技术、新产品；（五）联合抵制交易；（六）国务院反垄断执法机构认定的其他垄断协议。本法所称垄断协议，是指排除、限制竞争的协议、决定或者其他协同行为。第十四条禁止经营者与交易相对人达成下列垄断协议：（一）固定向第三人转售商品的价格；（二）限定向第三人转售商品的最低价格；《中华人民共和国反垄断法》
M p p 1
• • •
•
那么其未来报酬为零。企业1的最优偏离值就是使短期利润最大化。 M p （是企业1短期利润最大化的价格是一个很小的数值）。所以，最优偏离后的报酬是 V' M ，即短期利润加上未来的零利润。 1 M 1 M 从均衡角度，应满足或简化为
• 行业衰落的反面是行业正在发展。假定需求以速度g在增长，这表明在其他条件不变的情况下， t+1时期的利润将较t时期增长一个因子（1+g）。 • 可以把这种现象描述为假定利润函数不变，而考虑成折现因子的变化。由于行业增长率为（1+g），所以折现因子为 1 h ( 1g)
1r/ f
• 结论：企业间相互作用越频繁，行业存在和增长的可能性越大，合谋价格就越趋于均衡。
n n n n i 1 i i i 1 i i 1 i i i 1
( a b q )( q ) C ( q )
均衡条件
• 一阶条件为 • 对称的合作古诺均衡为
MR a 2 b q MC 2 cq i i i j 1j

CGMP--Current-Good-Manufacturing-Practice(中英文对照版)

Subpart A—General Provisions§211。

1 Scopea)The regulations in this part contain theminimum current good manufacturing practice for preparation of drug products for administration to humans or animals.b)The current good manufacturing practiceregulations in this chapter, as they pertain to drug products, and in parts 600 through 680 of this chapter，as they pertain to biological products for human use，shall be considered to supplement，not supersede, the regulations in this part unless the regulations explicitly provide otherwise。

In the event it is impossible to comply with applicable regulations both in this part and in other parts of this chapter or in parts 600 through 680 of this chapter, the regulation specifically applicable to the drug product in question shall supersede the regulation in this part。

概率统计中英术语对照表

概率统计中英术语对照表Probability Theory概率论Trial 试验intersection交union 并frequency 频率difference 差additivity 可加性complementation 对立contain 包含equivalent 等价mean 均值convolution [，kɔnvə'lu：ʃən］卷积variance 方差covariance 协方差correlated 相关standard deviation 标准差Random experiment 随机试验random event 随机事件certain event 必然事件impossible event 不可能事件elementary/fundamental event 基本事件the probability of event A 事件的概率sample point 样本点sample space 样本空间Classical probability 古典概型geometric probability 几何概型conditional probability 条件概型total probability 全概率formula of multiplication 乘法公式pair wise independence 两两相互独立Distribution function 分布函数discrete random variable 离散型随机变量two—point distribution （0-1)分布binomial distribution 二次分布Poisson distribution 泊松分布hyper geometric distribution 超几何分布Continuous random variable 连续型随机变量probability density function 概率密度函数uniform distribution 均匀分布Exponential distribution 指数分布standard normal distribution 标准正态分布Cauchy distribution 柯西分布n—dimensional random vector n维随机变量bivariate random variable [bai’vεəriət] 二维随机变量joint distribution function 联合分布函数bivariate discrete random variable 二维离散型随机变量joint distribution law 联合分布律bivariate continuous random variable 二维连续型随机变量joint probability density function 联合概率密度函数bivariate normal distribution 二维正态分布marginal distribution function 边缘分布函数marginal distribution law 边缘分布律marginal probability density function 边缘概率密度函数conditional distribution function 条件分布函数conditional probability density function 条件概率密度函数mathematical expectation 数学期望standard random variable 标准随机变量moment generating function 矩母函数characteristic function 特征函数positive correlated 正相关mixed moment 混合矩negative correlated 负相关mixed central moment 混合中心矩moment of order k about the origin 阶原点矩central moment of order k 阶中心矩covariance matrix 协方差矩阵convergence in probability 依概率收敛Bernouli large numbers law 伯努力大数定律Mathematical statistics数理统计individuality 个体population 总体sample size 样本大小simple random sample 随机样本efficiency有效statistic 统计量sample mean 样本均值sample variance样本方差sample standard deviation 样本标准差sample central moment of order k样本的阶中心矩skewness ［'skju：nis］偏度coefficient of variation 变异系数order statistics 次序统计量degrees or freedom 自由度sampling distribution 抽样分布parameter estimation 参数估计point estimation 点估计estimator 估计量estimate 估计值likelihood function 似然函数method of moment 矩估计法unbiased estimator 无偏估计量maximum likelihood estimate 最大似然估计system of likelihood equations似然方程组consistent estimator 一致估计量confidence level 置信水平confidence interval 置信区间upper confidence limit 置信上限parametric hypothesis 参数估计non-parametric hypothesis 非参数估计alternative hypothesis 备择假设null hypothesis 零假设Significance level 显著性水平rejection region 拒绝域acceptance region 接受域test for goodness of fit 拟和优度检验contingency table 列连表regression function 回归函数regression equation 回归方程linear regression model 线形回归模型regression coefficient 回归系数normal linear model 正态线形模型least squares estimate 最小二乘估计method of least squares 最小二乘法sum of squares of residual 残差平方和sum of squares of regression 回归平方和sum of residual 剩余平方和total sum of squares of deviations 总变差平方和coefficient of determination 判定系数point interval 点预测prediction interval 预测区间one—way analysis variance 单因素方差分析two—way analysis of variance 双因素方差分析interaction effect 交互效应。

Productivity and Undesirable Outputs

230
A directional distance function approach
or Fa ¨ re et al. (1993) one could estimate a shadow price. One possible solution is to use a productivity index that does not require information on prices of eﬄuents, for example, the Malmquist index; see Fa ¨ re and Grosskopf (1996). However, in the presence of undesirable outputs, this index may not be computable. Here we propose a new index, which we call the Malmquist–Luenberger productivity index which overcomes the shortcomings of the original Malmquist index. This index readily allows for inclusion of undesirable outputs without requiring information on shadow prices. It also explicitly credits ﬁrms or industries for reductions in undesirable outputs, providing a measure of productivity which will tell managers whether their “true” productivity has improved over time. This index also tells managers if there has been technical progress (a shift in the best practice frontier) and whether they are catching up to the frontier. Since the index is computed using a data envelopment analysis type approach, information concerning benchmark ﬁrms and technical eﬃciency is also generated for individual ﬁrms. In order to illustrate the applicability of this index, we compute productivity for data from the Swedish paper and pulp industry. We begin with a discussion of the way in which we model technology, and then turn to our measure of productivity based on this model. Section 4 includes a discussion of our data and results. Section 5 provides a brief conclusion. 2. Modelling technology with good and bad outputs The basic pollution problem is that production of “good” outputs, such as paper or electricity, is typically accompanied by the joint production of undesirable by-products, such as suspended solids or SO2. The fact that goods and bads are jointly produced means that reduction of bads will be “costly”: either resources must be diverted to “clean-up” (e.g. scrubbers), production must be cut back, or ﬁnes must be paid. M I More formally, if we denote good outputs by yvR+ , bad outputs by bvR+ , and inputs N by xvR+, then we can describe technology in a very general way via the output sets P(x)={( y, b): x can produce ( y, b)}. (2.1)

lecture23(博弈论讲义(Carnegie Mellon University))

June 20, 2003
73-347 Game Theory--Lecture 23
6
Cournot duopoly model of incomplete information (version one) cont'd
Firm 2's marginal cost depends on some factor (e.g. technology) that only firm 2 knows. Its marginal cost can be
* * * Firm 1 chooses q1 which is its best response to firm 2's ( q2 (cH ) , q2 (cL ) ) (and the probability). * If firm 2's marginal cost is HIGH then firm 2 chooses q2 (cH ) which is its * best response to firm 1's q1 . * If firm 2's marginal cost is LOW then firm 2 chooses q2 (cL ) which is its * best response to firm 1's q1 .
June 20, 2003
73-347 Game Theory--Lecture 23
3
Static (or simultaneous-move) games of complete information
A set of players (at least two players) For each player, a set of strategies/actions Payoffs received by each player for the combinations of the strategies, or for each player, preferences over the combinations of the strategies All these are common knowledge among all the players.

企业全要素生产率估计方法比较_尹恒

2
2015 年第 4 期
2007 ） ①。因此， TFP 估计对宏观经济学家和评估企业的相对绩效（ Van Biesebroeck， ②。微观经济学家来说一直都是充满魅力的话题 TFP 估计又是一个令人多少有些沮丧的话题。即使在顶级期刊上发表的然而， Young（ 1995 ）估计 1966 至 1990 年间中国香港地区和中研究结果也大相径庭。例如，国台湾地区的 TFP 增长率在 2% 到 3% 之间，韩国的 TFP 增长率为 1． 7% ，而 Hsieh （ 1999 ）估计的 TFP 增长率数值比 Young 多出 3% 。最明显的是新加坡， Young 的估计为－ 0． 68% ，而 Hsieh 的估计达 2． 39% 。这不是由于估计方法的缺乏，相反，学界 TFP ，。对估计方法的探讨已有近百年的历史发展出了大量的工具姑且不说投入和产出度量方面的差异，即便针对相同的样本和变量，不同的方法也可能得到完全不同的结果。这说明传统 TFP 估计方法缺乏可靠性。例如，根据本文用传统方法对中国服装皮革行业同一时期（ 2003 —2007 ）、相同样本和相同变量定义（总产值）的估计， DEA 估计的 TFP 年增长率为 6． 87% 、 OLS 为 2． 72% 、指数方法为 3． 22% 、固定效应动态面板为0． 17% （见表 2 ）。由此，呈现出大量相互矛盾的结论就不足为为 4． 24% 、奇了。这甚至严重到动摇人们对 TFP 估计的信心③。本文以中国竞争性行业为例，全面比较企业层面 TFP 估计方法的结果。以下第二部分介绍数据和变量构建；第三部分分析估计 TFP 的传统方法的缺陷；第四部分总结 TFP 结构估计的基本思路，提出针对数据产生过程特征的估计方法；第五部分讨论使用增加值估计企业 TFP 可能面临的问题；第六部分处理进入退出，并就函数变量定义进行敏感性分析；第七部分总结全文。形式、

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a r X i v :0804.0756v 1 [m a t h .A C ] 4 A p r 2008Betti numbers of mixed product ideals Giancarlo Rinaldo Abstract.We compute the Betti numbers of the resolution of a special class of square-free monomial ideals,the ideals of mixed products.Moreover when these ideals are Cohen-Macaulay we calculate their type.Mathematics Subject Classiﬁcation (2000).Primary 13P10;Secondary 13B25.Keywords.Mixed product ideals,Betti numbers,Cohen-Macaulay rings.1.Introduction The class of ideals of mixed products is a special class of square-free monomial ideals.They were ﬁrst introduced by G.Restuccia and R.Villarreal (see [6]and [7]),who studied the normality of such ideals.In [4]C.Ionescu and G.Rinaldo studied the Castelnuovo-Mumford regularity,the depth and dimension of mixed product ideals and characterize when they are Cohen-Macaulay.Let S =K [x ,y ]be a polynomial ring over a ﬁeld K in two disjoint sets of variables x =(x 1,...,x n ),y =(y 1,...,y m ).The ideals of mixed products are the ideals I q J r +I s J t ,q,r,s,t ∈N ,q +r =s +t,where I l (resp.J p )is the ideal of S generated by all the square-free monomials of degree l (resp.p )in the variables x (resp.y ).We set I 0=J 0=S .By symmetry,essentially there are 2cases:i)L =I k J r +I s J t ,0≤k <s ;ii)L =I k J r ,k ≥1or r ≥1.Let F be the minimal free resolution of an ideal I :F :0→F p →···→F p −1→···→F 0→I →0and let F i =⊕jS (−j )βij .We want to calculate the Betti numbers,βij ,when I is a mixed product ideals.2Giancarlo RinaldoTo reach this goal we consider S/I as a Stanley-Reisner ring(see[1],Chapter 5),we calculate the Alexander duality of I and we use the Hochster’s formula(see [1],Chapter5.5or[5],Chapter1.5).In section2we calculate the dual of the mixed product ideal I q and I q J r.Hochster’s formula give us a powerful tool to calculate Betti numbers of the resolution of a squarefree monomial ideal I.For this reason we have to calculate I∗and the homology of the chain complex of the simplicial complex∆∗related to I∗(section3).In the last section as an application we compute the type of Cohen-Macaulay mixed product ideals.2.Alexander duality of mixed product idealsWe recall the following(see[5],Deﬁnition1.35):Deﬁnition2.1.Let I=(xα1,...,xαq)⊂K[x]=K[x1,...,x n]be a square-freemonomial ideal,withαi=(αi1,...,αin)∈{0,1}n.The Alexander dual of I is theidealI∗= qi=1mαi,where mαi =(x j:αij=1).Proposition2.2.Let S=K[x1,...,x n],I q⊂S with1≤q≤n.ThenI∗q=I n−q+1.Proof.Let I q=(x i1···x iq:1≤i1<···<i q≤n),we consider I q as a Stanley-Reisner ideal of the simplicial complex∆,that isI q=I∆=(xτ:τ/∈∆),whereτ={i1,...,i q}and xτ=x i1···x iq.This implies that the simplicial complex∆is pure and its facets are all the facets of dimension q−2,that areσ={i1,...,i q−1}with1≤i1<i2<...< i q−1≤n.We want to calculate∆∗={τ={i′1,...,i′n−q},with1≤i′1<···<i′n−q≤n,are the facets of∆∗.The minimal non faces of∆∗are{i′1,...,i′n−q+1}and its Stanley-Reisner ideal isI∆∗=(x i′1···x i′n−q+1:1≤i′1<···<i′n−q+1≤n)=I n−q+1.Betti numbers of mixed product ideals 3Proposition 2.3.Let S =K [x 1,...,x n ,y 1,...,y m ],(I q +J r )⊂S with 1≤q ≤n ,1≤r ≤m .Then(I q +J r )∗=I n −q +1J m −r +1.Proof.LetI q +J r =(x i 1···x i q ,y j 1···y j r :1≤i 1<···<i q ≤n,1≤j 1<···<j r ≤m ),by Alexander duality we have(I q +J r )∗=( αm α)∩( βn β),where m α=(x i 1,...,x i q ),n β=(y j 1,...,y j r ).By the coprimality of m and n weobtain ( αm α)∩( βn β)=( αm α)( βn β)=(I q )∗(J r )∗,and by Proposition 2.2we have the assertion.Corollary 2.4.Let S =K [x 1,...,x n ,y 1,...,y m ],I q J r ⊂S with 1≤q ≤n ,1≤r ≤m .Then(I q J r )∗=I n −q +1+J m −r +1.Proof.By Alexander duality (I ∗)∗=I ,therefore the assertion follows by Propo-sition 2.3. Remark 2.5.We observe that in general the class of mixed product ideals is not closed under Alexander duality.In fact(I q J r +I s J t )∗=I n −q +1+I n −s +1J m −r +1+J m −t +1,with 1≤q <s ≤n,1≤t <r ≤m .To calculate this duality we observe that(I q J r +I s J t )∗=( α,β(m α+n β))∩( γ,δ(m γ+n δ)),where m α=(x i 1,...,x i q ),n β=(y j 1,...,y j r ),m γ=(x i ′1,...,x i ′s )and n δ=(y j ′1,...,y j ′t ).By Corollary 2.4we obtain( α,βm α+n β)∩( γ,δm γ+n δ)=(I n −q +1+J m −r +1)∩(I n −s +1+J m −t +1),and since I n −s +1⊃I n −q +1,J m −r +1⊃J m −t +1we have the assertion.4Giancarlo Rinaldo3.Betti numbers of mixed product idealsLet K be aﬁeld,S=K[x1,...,x n]be a polynomial ring,I⊂S,and let F be the minimal free resolution of the ideal I.ThenF:0→F p→···→F p−1→···→F0→I→0S(−j)βij.We want to calculate the Betti numbers,βij,when I is a with F i=⊕jmixed product ideals.Lemma3.1.Let S=K[x1,...,x n],I q⊂S with1≤q≤n.βi(I q)=βi,q+i(I q)= n q+i q+i−1i ,i≥0.Proof.We observe that I q has a q-linear resolution(see Example2.2,[3])therefore the Betti number of each free-module in the FFR of I q isβi(I q)=βi,q+i(I q).Let I∆=I q,where∆is the simplicial complex deﬁned in the proof of Propo-sition2.2.By Hochster’s formula(see Corollary1.40,[5])we haveβi,q+i(I∆)= |σ|=q+iβi,σ(I∆)= σ;K).By the symmetry between simplices in∆of the same dimension,we obtainσ;K)= n q+i dim K H i−1(link∆∗(σ0={q+i+1,...,n}∈∆∗and n q+i is the number of simplices of dimension q+i−1in∆.Since I∆∗=I n−q+1(see Proposition2.2)the facets of∆∗are{i1,...,i n−q}, 1≤i1<...<i n−q≤n,thereforelinkσ0is→···∂1→C0∂0→C−1→0,0→C i−1∂i−1where dim K C j= q+i j+1 ,j=−1,0,...,i−1.It is easy to observe that H j(link∆∗(σ0;K))∼=ker∂i−1).We want to show that dim K ker∂j−1= q+i−1j and we make induction on the length of the exact sequence of vector spaces,j,→···∂1→C0∂0→C−1→0.0→ker∂j−1→C j−1∂j−1For j=1we obtainBetti numbers of mixed product ideals50→ker∂0→C0∂0→C−1→0anddim K ker∂0=dim K C0−dim K C−1=q+i−1as expected.Let dim K ker∂j−1= q+i−1j .We consider the short exact sequence0→ker∂j→C j∂j→ker∂j−1→0,and since q+i−1j+1 + q+i−1j = q+i j+1we obtain the assertion. Theorem3.2.Let S=K[x1,...,x n,y1,...,y m],I q J r⊂S with1≤q≤n,1≤r≤m.Then for i≥0,we haveβi(I q J r)=βi,q+r+i(I q J r)= j+k=i n q+j m r+k q+j−1j r+k−1k .Proof.Since I q J r has a(q+r)-linear resolution(see Lemma2.5,[4])the Betti number of each free-module in the FFR of I q J r isβi(I q J r)=βi,q+r+i(I q J r).Let I∆=I q J r,by Hochster’s formula we have thatβi,q+r+i(I∆)= |σ|=q+r+iβi,σ(I∆)= σ;K).Let deg a bidegree on the ring S=K[x1,...,x n,y1,...,y m],where deg(x i)= (1,0)and deg(y j)=(0,1).It is easy to observe that the number of squarefree monomials in S of the same bidegree(q+j,r+k)isn q+j m r+k .By the symmetry between simplices with the“same bidegree”in∆,we ﬁxσ0={x1,...,x q+j,y1,...,y r+k}and its corresponding Alexander dualσ0;K),with i=j+k.By Corollary2.4,I∆∗=I n−q+1+J m−r+1,and the facets of∆∗are{x i1,...,x in−q,y j1,...,y jm−r},6Giancarlo Rinaldo1≤i1<...<i n−q≤n,1≤j1<...<j m−r≤m,therefore linkσ0. Therefore H i−1(link∆∗(Betti numbers of mixed product ideals7 We may assume q+r≥s+t.1)Let j=s+r−2.Then Tor i(K,S/I s J r)i+j=0and Tor i−1(K,S/I q J r⊕S/I s J t)i+j=0.Therefore we have the exact sequence0→Tor i(K,S/I q J r⊕S/I s J t)i+j→→Tor i(K,S/(I q J r+I s J t))i+j→Tor i−1(K,S/I s J r)i+j→0.If s+r−2>q+r−1we obtainTor i(K,S/(I q J r+I s J t))i+j∼=Tor i−1(K,S/I s J r)i+j,and toﬁnish we have to study degree2)and3).If s+r−2=q+r−1>s+t+1,that is degree1)and2)coincide,we obtain dim K Tor i(K,S/(I q J r+I s J t))i+j=dim K Tor i(K,S/I q J r)i+j+dim K Tor i−1(K,S/I s J r)i+j, and we have to study degree3)toﬁnish.If s+r−2=q+r−1=s+t+1,that is cases1),2)and3)coincide, dim K Tor i(K,S/(I q J r+I s J t))i+j=dim K Tor i(K,S/I q J r)i+j++dim K Tor i(K,S/I s J t)i+j+dim K Tor i−1(K,S/I s J r)i+j, and this case is complete.2)The case j=q+r−1=s+r−2has been already studied in case1), therefore we assume j=q+r−1<s+r−2.We obtain the exact sequence 0→Tor i(K,S/I q J r⊕S/I s J t)i+j→Tor i(K,S/(I q J r+I s J t))i+j→0and we continue as in case1).3)This case is similar to2).The assertion follows. Remark3.4.For completness we compute the Betti numbers of(I s+I q J t+J r) (see Remark2.5).That isβi(I s+I q J t+J r)=βi(I s)+βi(I q J t)+βi(J r)+βi−1(I q J r)+βi−1(I s J t),i≥1. To prove this is enough to observe that(I s)∩(I q J t+J r)=I s J t+I s J r=I s J t and we have the exact sequence0→S/I s J t→S/I s⊕S/(I q J t+J r)→S/(I s+I q J t+J r)→0.(3.2) To obtain the assertion we continue as in the proof of Theorem3.3.4.Cohen-Macaulay Type of mixed product idealsLet K be aﬁeld,S=K[x1,...,x n]be a polynomial ring,I⊂S a graded ideal. We consider S/I as a standard graded K-algebra.We have the following(see[7],Proposition4.3.4):Proposition4.1.Let S/I be a Cohen-Macaulay ring then the type of S/I is equal to the last Betti number in the minimal free resolution of S/I as an S-module.8Giancarlo RinaldoIn[4]we classiﬁed the Cohen-Macaulay algebra S/I when I is a mixed prod-uct ideal.From now on we say CM for short.Proposition4.2.Let S=K[x1,...,x n,y1,...,y m].1.Let0<q≤n then S/I q is CM with dim S/I q=m+q−1;2.Let0<q≤n and0<r≤m then S/(I q J r)is CM if and only if r=m andq=n with dim S/I n J m=m+n−1;3.Let0<q<s≤n and0<r≤m then S/(I q J r+I s)is CM if and only ifs=q+1and r=m with dim S/(I q J m+I q+1)=m+q−1;4.Let0<q<s≤n and0<t<r≤m then S/(I q J r+I s J t)is CM if and only ifr=m,s=n,t=m−1,q=n−1with dim S/(I n−1J m+I n J m−1)=m+n−2. Proof.See Proposition3.1,Theorem3.2and Corollary3.8of[4],and observe that a CM ideal is unmixed.By the results of Section3we want to calculate the type of the CM algebras classiﬁed in Proposition4.2.Proposition4.3.Let S=K[x1,...,x n],I q⊂S and1≤q≤n thentype S/I q= n−1n−qProof.By Auslander-Buchsbaum Theorem(Theorem1.3.3,[1])and Proposition 4.2.(1),we havepd S/I q=dim S−depth S/I q=dim S−dim S/I q=n−q+1.By Lemma3.1and observing thatβi(I q)=βi+1(S/I q)we havetype(S/I q)=βn−q+1(S/I q)= n−1n−q .From now on let S=K[x1,...,x n,y1,...,y m].Proposition4.4.Let S/I q J r be a Cohen-Macaulay algebra thentype S/I q J r=1with q=n,r=m.Proof.By Corollary4.2.(2),we have that S/I q J r is CM if and only if I q J r is a principal ideal and the assertion follows. Proposition4.5.Let S/(I q J r+I s)be a Cohen-Macaulay algebra thentype S/(I q J r+I s)= n−1n−q + n−1n−q−1 .with s=q+1and r=m.Betti numbers of mixed product ideals9 Proof.By Proposition4.2.(3),we have that S/(I q J r+I s)is CM if and only if s=q+1and r=m therefore by Theorem3.2.(2)of[4]dim S/(I q J m+I q+1)=m+q−1.By Auslander-Buchsbaum Theorem we obtainpd S/(I q J m+I q+1)=dim S−dim S/(I q J m+I q+1)=n−q+1. Applying Theorem3.3,with i=n−q,s=q+1,r=m,we have βn−q+1(S/(I q J m+I q+1))=βn−q(I q J m)+βn−q(I q+1)+βn−q−1(I q+1J m). Since the pd S/I q+1=n−q,we have that pd I q+1=n−q−1,that isβn−q(I q+1)= 0.Thereforeβn−q+1(S/(I q J m+I q+1))=βn−q(I q J m)+βn−q−1(I q+1J m),and by Theorem3.2βn−q(I q J m)= j+k=n−q n q+j m m+k q+j−1j m+k−1k ,βn−q−1(I q+1J m)= j+k=n−q−1 n q+1+j m m+k q+j j m+k−1k . It is easy to see that for k>0the summands inβn−q(I q J m)and inβn−q−1(I q+1J m) vanish.If k=0we obtain the assertion. Proposition4.6.Let S/(I q J r+I s J t)be a Cohen-Macaulay algebra thentype S/(I q J r+I s J t)=m+n−1,with q=n−1,r=m,s=n,t=m−1.Proof.By Proposition4.2.(4),we have that S/(I q J r+I s J t)is CM if and only if q=n−1,r=m,s=n,t=m−1,therefore by Theorem3.2.(2)[4]dim S/(I n−1J m+I n J m−1)=m+n−2.By Auslander-Buchsbaum Theorem we obtainpd S/(I n−1J m+I n J m−1)=dim S−dim S/(I n−1J m+I n J m−1)=2.Applying Theorem3.3,with i=1,q=n−1,r=m,s=n,t=m−1,we obtainβ2(S/(I n−1J m+I n J m−1))=β1(I n−1J m)+β1(I n J m−1)+β0(I n J m).We observe that I n J m is a principal ideal thereforeβ0(I n J m)=1.By Theorem3.2β1(I n−1J m)= j+k=1 n n−1+j m m+k n−1+j−1j m+k−1k ,10Giancarlo Rinaldoβ1(I n J m−1)= j+k=1 n n+j m m−1+k n+j−1j m−1+k−1k . It is easy to see that for k>0the summands inβ1(I n−1J m)vanish.For j>0the same happens toβ1(I n J m−1).Therefore we obtainβ2(S/(I n−1J m+I n J m−1))=n−1+m−1+1.Remark4.7.We observe that the mixed product ideal that are Gorenstein(e.g. CM ring of type1),are:1.K[x1,...,x n]/I n;2.K[x1,...,x n]/(x1,...,x n);3.K[x1,...,x n,y1,...,y m]/I n J m.References[1]W.Bruns,J.Herzog,Cohen-Macaulay rings,Cambridge Univ.Press,Cambridge,1997.[2]D.Eisenbud,Commutative Algebra with a View Toward Algebraic Geometry,Springer Verlag,New-York,1994.[3]J.Herzog,T.Hibi,Cohen-Macaulay polymatroidal ideals,b.,27(2006),513-517.[4]C.Ionescu,G.Rinaldo,Some algebraic invariants related to mixed product ideals,Accepted for publication in Archiv der Mathematik.[5]ler, B.Sturmfels,Combinatorial Commutative Algebra,Springer-Verlag,Berlin,2005.[6]G.Restuccia,R.Villarreal,On the normality of monomial ideals of mixed products,Commun.Algebra,29(2001),3571-3580.[7]R.Villarreal,Monomial algebras,Marcel Dekker,New-York,2001.Giancarlo RinaldoDipartimento di MatematicaUniversit`a di MessinaContrada Papardo,salita Sperone,3198166Messina,Italye-mail:rinaldo@dipmat.unime.it。